Monte Carlo simulation of many-arm star polymers in two-dimensional good solvents in the bulk and at a surface

A Monte Carlo technique is proposed for the simulation of statistical properties of many-arm star polymers on lattices. In this vectorizing algorithm, the length of each arml is increased by one, step by step, from a starting configuration withl=1 orl=2 which is generated directly. This procedure is carried out for a large sample (e.g., 100,000 configurations). As an application, we have studied self-avoiding stars on the square lattice with arm lengths up tol max=125 and up tof=20 arms, both in the bulk and in the geometry where the center of the star is adsorbed on a repulsive surface. The total number of configurations, which behaves asN∼l γ G–1μ fl , whereμ=2.6386 is the usual effective…

Continuum Monte Carlo simulation of phase transitions in rod-like molecules at surfaces

Stiff rod-like chain molecules with harmonic bond length potentials and trigonometric bond angle potentials are used to model Langmuir monolayers at high densities. One end of the rod-like molecules is strongly bound to a flat two-dimensional substrate which represents the air-water interface. A ground-state analysis is performed which suggests phase transitions between phases with and without collective uniform tilt. Large-scale off-lattice Monte Carlo simulations over a wide temperature range show in addition to the tilting transition the presence of a strongly constrained melting transition at high temperatures. The latter transition appears to be related to two-dimensional melting of th…

Elastic properties, structures and phase transitions in model colloids

The nature of the melting transition for a system of hard discs with translational degrees of freedom in two spatial dimensions has been analysed by a combination of computer simulation methods and a finite size scaling technique. The behaviour of the system is consistent with the predictions of the Kosterlitz–Thouless–Halperin–Nelson–Young (KTHNY) theory. The structural and elastic properties of binary colloidal mixtures in two and three spatial dimensions are discussed as well as those of colloidal systems with quenched point impurities. Hard and soft discs in external periodic (light-) fields show rich phase diagrams including freezing and melting transitions when the density of the syst…

Conformations and orientational ordering of semiflexible polymers in spherical confinement.

Semiflexible polymers in lyotropic solution confined inside spherical nanoscopic “containers” with repulsive walls are studied by molecular dynamics simulations and density functional theory, as a first step to model confinement effects on stiff polymers inside of miniemulsions, vesicles, and cells. It is shown that the depletion effects caused by the monomer-wall repulsion depend distinctly on the radius R of the sphere. Further, nontrivial orientational effects occur when R, the persistence length ℓp, and the contour length L of the polymers are of similar magnitude. At intermediate densities, a “shell” of wall-attached chains is forming, such that the monomers belonging to those chains a…

Scaling theory of star polymers and general polymer networks in bulk and semi-infinite good solvents

Theorie d'echelle utilisant l'equivalence entre la fonction generatrice du nombre total de configuration et la fonction de correlation a plusieurs spins du modele de Heisenberg classique a n composantes dans la limite n→0

Pearl-necklace structures of molecular brushes with rigid backbone under poor solvent conditions: A simulation study

Bottle-brush polymers, where flexible side chains containing N=20 to 50 effective monomers are grafted to a rigid backbone, are studied by molecular dynamics simulations, varying the grafting density σ and the solvent quality. Whereas for poor solvents and large enough σ the molecular brush is a cylindrical object, homogeneous in axial direction, for intermediate values of σ an axially inhomogeneous structure of "pearl-necklace" type is formed. The "pearls," however, have a strongly nonspherical ellipsoidal shape, due to the fact that several side chains cluster together in one pearl, qualitatively consistent with predictions of Sheiko et al. [Eur. Phys. J. E 13, 125 (2004)] We analyze the …

Bending or buckling: Compression-induced phase transition in a semi-flexible polymer brush

Molecular-dynamics simulations are presented for systems of densely grafted semiflexible macromolecules grafted to a planar non-adsorbing substrate, studying the case where the persistence length of the polymers is of the same order as their contour length so that the polymer brush may exhibit nematic order. We focus our attention on the case where the first bond must orient perpendicularly to the substrate (so the structure resembles a "Fakir's bed" for short chains and a "polymer bristle" for longer chains). When such layers are exposed to uniform compression, the pressure vs. distance relationship exhibits two stages: i) for very small compression the chains exhibit "buckling" yet mainta…

Inter-Chain Structure Factors of Flexible Polymers in Solutions: A Monte Carlo Investigation

Off-lattice Monte Carlo simulations of both the single chain structure factor h(q) and the inter-chain structure factor HD(q) of flexible polymers in solutions are presented over a wide range of both wavenumber q and concentration c from the dilute to the concentrated regime, for chain lengths up to N = 256. The single chain properties $\{$gyration radius 〈Rg2〉, $h(q)\}$ are in reasonable agreement with the expected theoretical behavior, showing a crossover from swollen chains $\{\langle R_{\rm g}^2\rangle \propto N^{2\nu} ,~ h(q) \propto q^{-1/\nu}\}$ to Gaussian chains, and the data comply with a scaling description, with a correlation length ξ∝c-ν/(3ν-1). However, the inter-chain structu…

Spinodal Decomposition Kinetics of Colloid-Polymer Mixtures Including Hydrodynamic Interactions

The phase separation dynamics of a model colloid-polymer mixture is studied by taking explicitly the hydrodynamic interactions caused by the solvent into account. Based on the studies on equilibrium phase behavior we perform a volume quench from the homogeneous region of the phase diagram deep into the region where colloid-rich and polymer-rich phases coexist. We demonstrate that the Multiparticle Collision Dynamics (MPCD) algorithm is well suited to study spinodal decomposition and present first results on the domain growth behavior of colloid-polymer mixtures in quasi two-dimensional confinement. On the one hand side we find that the boundary condition of the solvent with respect to the r…

Do the contact angle and line tension of surface-attached droplets depend on the radius of curvature?

Results from Monte Carlo simulations of wall-attached droplets in the three-dimensional Ising lattice gas model and in a symmetric binary Lennard-Jones fluid, confined by antisymmetric walls, are analyzed, with the aim to estimate the dependence of the contact angle $(\Theta)$ on the droplet radius $(R)$ of curvature. Sphere-cap shape of the wall-attached droplets is assumed throughout. An approach, based purely on "thermodynamic" observables, e.g., chemical potential, excess density due to the droplet, etc., is used, to avoid ambiguities in the decision which particles belong (or do not belong, respectively) to the droplet. It is found that the results are compatible with a variation $[\Th…

Monte Carlo Study of Critical Point Shifts in Thin Films

We report preliminary results of Monte Carlo simulations of critical point shifts in thin slit-like capillaries. By making use of the isomorphism with an Ising model subject to bulk and surface fields and employing a multi-cluster update algorithm with ghost-spin term we obtain the coexistence curve and the behavior at the critical point for various film thicknesses D.

Structural and dynamical properties of sodium silicate melts: An investigation by molecular dynamics computer simulation

We present the results of large scale computer simulations in which we investigate the static and dynamic properties of sodium disilicate and sodium trisilicate melts. We study in detail the static properties of these systems, namely the coordination numbers, the temperature dependence of the Q^(n) species and the static structure factor, and compare them with experiments. We show that the structure is described by a partially destroyed tetrahedral SiO_4 network and the homogeneously distributed sodium atoms which are surrounded on average by 16 silicon and other sodium atoms as nearest neighbors. We compare the diffusion of the ions in the sodium silicate systems with that in pure silica a…

Properties of the Ising magnet confined in a corner geometry

Abstract The properties of Ising square lattices with nearest neighbor ferromagnetic exchange confined in a corner geometry, are studied by means of Monte Carlo simulations. Free boundary conditions at which boundary magnetic fields ± h are applied, i.e., at the two boundary rows ending at the lower left corner a field + h acts, while at the two boundary rows ending at the upper right corner a field − h acts. For temperatures T less than the critical temperature T c of the bulk, this boundary condition leads to the formation of two domains with opposite orientation of the magnetization direction, separated by an interface which for T larger than the filling transition temperature T f ( h ) …

Theory of first-order phase transitions

An introductory review of various concepts about first-order phase transitions is given. Rules for classification of phase transitions as second or first order are discussed, as well as exceptions to these rules. Attention is drawn to the rounding of first-order transitions due to finite-size or quenched impurities. Computational methods to calculate phase diagrams for simple model Hamiltonians are also described. Particular emphasis is laid on metastable states near first-order phase transitions, on the 'stability limits' of such states (e.g. the 'spinodal curve' of the gas-liquid transition) and on the dynamic mechanisms by which metastable states decay (nucleation and growth of droplets …

Wetting and phase separation at surfaces

We study the problem ofsurfacedirected spinodal decomposition, viz., the dynamical interplay of wetting and phase separation at surfaces. In particular, we focus on the kinetics of wetting-layer growth in a semi-infinite geometry for arbitrary surface potentials and mixture compositions. We also present representative results for phase separation in confined geometries, e.g., cylindrical pores, thin films, etc.

Critical Phenomena at the Surface of Systems Undergoing a Bulk First Order Transition: Are They Understood?

Systems that exhibit a first-order phase transition in the bulk, such as binary alloys where the order parameter vanishes discontinuously at some critical value of a control parameter, may show a continuous vanishing of the local order parameter at the surface. This “surface-induced disordering” is described theoretically as a variant of critical wetting, where an interface between the locally disordered surface and the ordered bulk gradually moves towards the bulk. We test this description by Monte Carlo simulations for a body centered cubic model alloy, with interactions between nearest and next nearest neighbors, for which the phase diagram in the bulk has been calculated very accurately…

Understanding the properties of liquid-crystalline polymers by computational modeling

Abstract A topical review of recent theoretical work on the properties of lyotropic solutions and melts containing semiflexible polymers in thermal equilibrium is given, with a focus on the liquid-crystalline and smectic order of these systems in the bulk and under confinement. Starting with a discussion of single chain properties in terms of the Kratky-Porod worm-like chain model and its limitations, extensions along the lines of Onsager’s theory for the isotropic-nematic transition of solutions of hard rods are briefly reviewed. This discussion is followed by a review of recent Molecular Dynamics simulations and classical Density Functional Theory calculations. It is argued that, even in …

Monte Carlo simulations are presented for binary (AB) symmetric polymer mixtures (chain lengths NANBN) for the case that an attractive interaction ϵ exists between monomers of the same kind, and the limiting case that one species (B) is very diluted. It is shown that with increasing interaction strength ϵ/(kB · T), T being the absolute temperature, the minority chains collapse to a very dense configuration, while the majority chains stay nearly Gaussian. Both chain radii, structure factors and numbers of nearest neighbour contacts are discussed.

Orientational ordering transitions of semiflexible polymers in thin films: A Monte Carlo simulation

Athermal solutions (from dilute to concentrated) of semiflexible macromolecules confined in a film of thickness D between two hard walls are studied by means of grand-canonical lattice Monte Carlo simulation using the bond fluctuation model. This system exhibits two phase transitions as a function of the thickness of the film and polymer volume fraction. One of them is the bulk isotropic-nematic first-order transition, which ends in a critical point on decreasing the film thickness. The chemical potential at this transition decreases with decreasing film thickness ("capillary nematization"). The other transition is a continuous (or very weakly first-order) transition in the layers adjacent …

Simulations of Glassforming Network Fluids: Classical Molecular Dynamics versus Car-Parrinello Molecular Dynamics

Abstract Static and dynamic Properties of molten germanium dioxide are studied by two simulation methods, classical Molecular Dynamics (MD) using the Oeffner-Elliott (OE) potential, and “ab initio” Car-Parrinello Molecular Dynamics (CPMD). While CPMD provides a (presumably) more accurate description of the local structure and the forces, it severely suffers from finite size effects when the structure beyond the first neighbor shells is considered. For glassforming fluids, the demanding equilibrium needs are a further reason, why simply MD is still preferable, when a “good” effective potential is available.

Critical phenomena without “hyper scaling”: How is the finite-size scaling analysis of Monte Carlo data affected?

Abstract The finite size scaling analysis of Monte Carlo data is discussed for two models for which hyperscaling is violated: (i) the random field Ising model (using a model for a colloid-polymer mixture in a random matrix as a representative) (ii) The Ising bi-pyramid in computing surface fields.

Diluted Heisenberg Ferromagnets with Competing Ferro- and Antiferromagnetic Interactions: Evidence for a New Universality Class?

The site-diluted classical face-centered cubic (fee) Heisenberg model with exchange between nearest and (J nn > 0) next nearest (J nnn =-J nn /2) neighbors is studied by Monte Carlo simulations using the heatbath algorithm in conjunction with histogram reweighting techniques. Finite size scaling analysis suggests that the diluted system crosses over to a new type of critical behavior, different from that of the pure system, in contrast to the prediction of the Harris criterion. But this model possibly can explain related experimental findings in Eu x Sr 1-x S.

Study of the dynamical approach to the interface localization–delocalization transition of the confined Ising model

Confined magnetic Ising films in a L ? D geometry (), with short-range competing magnetic fields?(h) acting at opposite walls along the D-direction, exhibit a slightly rounded localization?delocalization transition of the interface between domains of different orientations that runs parallel to the walls. This transition is the precursor of a wetting transition that occurs in the limit of infinite film thickness () at the critical curve Tw(h). For T Tw(h)) such an interface is bounded (unbounded) to the walls, while right at Tw(h) the interface is freely fluctuating around the centre of the film. Starting from disordered configurations, corresponding to , we quench to the wetting critical t…

Dynamics of confined polymer melts: Recent Monte Carlo simulation results

The dynamic behavior of thin polymer films is studied by Monte Carlo simulations of a simplified lattice model. The film geometry is realized by two opposite hard walls whose distance is varied in the simulations. In the films the dynamics is accelerated with respect to the bulk, leading to a decrease of the extrapolated glass transition temperature with decreasing film thickness.

Rejection-Free Monte Carlo

So far, we have been using the rejection Monte Carlo algorithms. To remind us, the algorithms proceed from state x to possible state \(x'\) as outlined in Algorithm 1.

Monte Carlo Simulation of Crystal-Liquid Phase Coexistence

When a crystal nucleus is surrounded by coexisting fluid in a finite volume in thermal equilibrium, the thermodynamic properties of the fluid (density, pressure, chemical potential) are uniquely related to the surface excess free energy of the nucleus. Using a model for weakly attractive soft colloidal particles, it is shown that this surface excess free energy can be determined accurately from Monte Carlo simulations over a wide range of nucleus volumes, and the resulting nucleation barriers are completely independent from the size of the total volume of the system. A necessary ingredient of the analysis, the pressure at phase coexistence in the thermodynamic limit, is obtained from the in…

Brownian dynamics of grafted polymer chains: time dependent properties

Results of computer simulations of polymer layers consisting of chains grafted by one end on an unpenetrable plane are presented. Characteristics of translational and rotational motion of different chain segments and correlation functions of chain radii were calculated both for single layers at different grafting densities s and for two interacting layers at different distances D between parallel grafting planes. Two values of grafting density were used in the latter case. The behavior of different correlation times as function of s and D and the interplay between the interpenetration of the brushes and rotational and translational motion are discussed. Both relaxation functions and mean sq…

Avoiding Boundary Effects in Wang-Landau Sampling

A simple modification of the ``Wang-Landau sampling'' algorithm removes the systematic error that occurs at the boundary of the range of energy over which the random walk takes place in the original algorithm.

The Observation of Formation and Annihilation of Solitons and Standing Strain Wave Superstructures in a Two-Dimensional Colloidal Crystal

Confining a colloidal crystal within a long narrow channel produced by two parallel walls can be used to impose a meso-scale superstructure of a predominantly mechanical elastic character [Chui et al., EPL 2008, 83, 58004]. When the crystal is compressed in the direction perpendicular to the walls, we obtain a structural transition when the number of rows of particles parallel to the walls decreases by one. All the particles of this vanishing row are distributed throughout the crystal. If the confining walls are structured (say with a corrugation along the length of the walls), then these extra particles are distributed neither uniformly nor randomly; rather, defect structures are created a…

Finite-size scaling at the dynamical transition of the mean-field 10-state Potts glass

We use Monte Carlo simulations to study the static and dynamical properties of a Potts glass with infinite range Gaussian distributed exchange interactions for a broad range of temperature and system size up to N=2560 spins. The results are compatible with a critical divergence of the relaxation time tau at the theoretically predicted dynamical transition temperature T_D, tau \propto (T-T_D)^{-\Delta} with Delta \approx 2. For finite N a further power law at T=T_D is found, tau(T=T_D) \propto N^{z^\star} with z^\star \approx 1.5 and for T>T_D dynamical finite-size scaling seems to hold. The order parameter distribution P(q) is qualitatively compatible with the scenario of a first order glas…

Path-integral Monte Carlo study of crystalline Lennard-Jones systems.

The capability of the path-integral Monte Carlo (PIMC) method to describe thermodynamic and structural properties of solids at low temperatures is studied in detail, considering the noble-gas crystals as examples. In order to reduce the systematic limitations due to finite Trotter number and finite particle number we propose a combined Trotter and finite-size scaling. As a special application of the PIMC method we investigate $^{40}\mathrm{Ar}$ at constant volume and in the harmonic approximation. Furthermore, isotope effects in the lattice constant of $^{20}\mathrm{Ne}$ and $^{22}\mathrm{Ne}$ are computed at zero pressure. The obtained results are compared with classical Monte Carlo result…

Phase Separation in a Binary Mixture of Semiflexible Polymers Confined in a Repulsive Sphere

A polymer chain trapped between two parallel repulsive walls: A Monte-Carlo test of scaling behavior

An off-lattice bead-spring model of a polymer chain trapped between two parallel walls a distance D apart is studied by Monte-Carlo methods, using chain lengths N in the range $$32 \le N \le 512$$ and distances D from 4 to 32 (in units of the maximum spring extension). The scaling behavior of the coil linear dimensions parallel to the plates and of the force on the walls is studied and discussed with the help of current theoretical predictions. Also the density profiles of the monomers across the slit are obtained and it is shown that the predicted variation with the distance z from a wall, $$\rho (z) \propto {z^{1/\nu }}$$ , is obtained only when one introduces an extrapolation length λ in…

The liquid-solid transition of hard discs: first-order transition or Kosterlitz-Thouless-Halperin-Nelson-Young scenario?

We consider the question of whether a two-dimensional hard-disc fluid has a first-order transition from the liquid state to the solid state as in the three-dimensional melting-crystallization transition or whether one has two subsequent continuous transitions, from the liquid to the hexatic phase and then to the solid phase, as proposed by Kosterlitz, Thouless, Halperin, Nelson and Young (KTHNY). Monte Carlo (MC) simulations of the fluid that study the growth of the bond orientational correlation length, and of the crystal are discussed. The emphasis is on a recent consistency test of the KTHNY renormalization group (RG) scenario, where MC simulations are used to estimate the bare elastic c…

Critical wetting in the square Ising model with a boundary field

The Ising square lattice with nearest-neighbor exchangeJ>0 and a free surface at which a boundary magnetic fieldH1 acts has a second-order wetting transition. We study the surface excess magnetization and the susceptibility ofL×M lattices by Monte Carlo simulation and probe the critical behavior of this wetting transition, applying finite-size scaling methods. For the cases studied, the results are not consistent with the presumably exactly known values of the critical exponents, because the asymptotic critical region has not yet been reached. Implication of our results for critical wetting in three dimensions and for the application of the present model to adsorbed wetting layers at surfac…

Phase separation in thin films: Effect of temperature gradients

We study the phase-separation kinetics of a binary (AB) mixture confined in a thin film of thickness D with a temperature gradient. Starting from a Kawasaki-exchange kinetic Ising model, we use a master-equation approach to systematically derive an extension of the Cahn-Hilliard model for this system. We study the effect of temperature gradients perpendicular to the film with "neutral" (no preference for either A or B) surfaces. We highlight the rich phenomenology and pattern dynamics which arises from the interplay of phase separation and the temperature gradient.

New Results on the Collapse Transition(s) of Flexible Homopolymers

We analyze the collapse transition of flexible homopolymer chains in the bond-fluctuation model employing the Wang-Landau Monte Carlo algorithm. The coil-globule transition is followed by a first order transition into a solid state occurring in the collapsed globule. In the thermodynamic limit (chain length to infinity) the topology of the phase diagram depends on the range of the attractive interaction between the monomers. For sufficiently large interaction range a normal behaviour of a continuous coil-globule transition at the Θ-temperature followed by a crystallization transition at lower temperature is observed. For short interaction range the first-order transition asymptotically can …

Star polymers confined in a nanoslit: a simulation test of scaling and self-consistent field theories

The free energy cost of confining a star polymer where f flexible polymer chains containing N monomeric units are tethered to a central unit in a slit with two parallel repulsive walls a distance D apart is considered, for good solvent conditions. Also the parallel and perpendicular components of the gyration radius of the star polymer, and the monomer density profile across the slit are obtained. Theoretical descriptions via Flory theory and scaling treatments are outlined, and compared to numerical self-consistent field calculations (applying the Scheutjens–Fleer lattice theory) and to Molecular Dynamics results for a bead-spring model. It is shown that Flory theory and self-consistent fi…

Phase transitions in adsorbed layers formed on crystals of square and rectangular surface lattice

Abstract This article gives a survey of phase transitions in adsorbed films on well defined surfaces of square and rectangular symmetry of the lattice. The discussion concentrates on the effects of periodic changes of the adsorbate–substrate potential on the structure and thermodynamic properties of adsorbed films. Different theoretical approaches are briefly reviewed, with an emphasis on those which explicitly take into account final corrugation of the surface potential. Several aspects of statistical mechanical description of phase transitions in surface layers, such as order–disorder, melting, commensurate–incommensurate transitions in monolayer films as well as transitions connected wit…

Linear Dimensions of Adsorbed Semiflexible Polymers: What can be learned about their persistence length?

Conformations of partially or fully adsorbed semiflexible polymer chains are studied varying both contour length $L$, chain stiffness, $\ensuremath{\kappa}$, and the strength of the adsorption potential over a wide range. Molecular dynamics simulations show that partially adsorbed chains (with ``tails,'' surface attached ``trains,'' and ``loops'') are not described by the Kratky-Porod wormlike chain model. The crossover of the persistence length from its three-dimensional value (${\ensuremath{\ell}}_{p}$) to the enhanced value in two dimensions ($2{\ensuremath{\ell}}_{p}$) is analyzed, and excluded volume effects are identified for $L\ensuremath{\gg}{\ensuremath{\ell}}_{p}$. Consequences fo…

What is the order of the two-dimensional polymer escape transition?

An end-grafted flexible polymer chain in three-dimensional space between two pistons undergoes an abrupt transition from a confined coil to a flowerlike conformation when the number of monomers in the chain, $N$, reaches a critical value. In two-dimensional (2D) geometry, excluded-volume interactions between monomers of a chain confined inside a strip of finite length $2L$ transform the coil conformation into a linear string of blobs. However, the blob picture raises questions about the nature of this escape transition. To check theoretical predictions based on the blob picture we study 2D single-polymer chains with excluded-volume interactions and with one end grafted in the middle of a st…

Phase Separation of Colloid Polymer Mixtures Under Confinement

Colloid polymer mixtures exhibit vapor-liquid like and liquid-solid like phase transitions in bulk suspensions, and are well-suited model systems to explore confinement effects on these phase transitions. Static aspects of these phenomena are studied by large-scale Monte Carlo simulations, including novel “ensemble switch” methods to estimate excess free energies due to confining walls. The kinetics of phase separation is investigated by a Molecular Dynamics method, where hydrodynamic effects due to the solvent are included via the multiparticle collision dynamics method.

Monte Carlo simulation of polymers at interfaces

Abstract Polymers at interfaces pose challenging problems to statistical physics because their configurations often differ greatly from the bulk. Computer simulation of coarse-grained models then gives valuable insight and allows stringent tests of various theoretical predictions. Three examples are briefly treated: chain configurations of B-chains in the surface-enriched B-rich layer of an (AB) binary polymer mixture; “frustrated” lamellar ordering in ultra-thin block-copolymer films; and the collapse of polymer brushes in bad solvents.

When does Wenzel's extension of Young's equation for the contact angle of droplets apply? A density functional study.

he contact angle of a liquid droplet on a surface under partial wetting conditions differs for a nanoscopically rough or periodically corrugated surface from its value for a perfectly flat surface. Wenzel's relation attributes this difference simply to the geometric magnification of the surface area (by a factor $r_{\rm w}$), but the validity of this idea is controversial. We elucidate this problem by model calculations for a sinusoidal corrugation of the form $z_{\rm wall}(y) = \Delta\cos(2\pi y/\lambda)$ , for a potential of short range $\sigma_{\rm w}$ acting from the wall on the fluid particles. When the vapor phase is an ideal gas, the change of the wall-vapor surface tension can be co…

Towards the Quantitative Prediction of the Phase Behavior of Polymer Solutions by Computer Simulation

The phase diagram of polymer solutions (cf. e.g. alkanes dissolved in supercritical carbon dioxide) is complicated, since there are four control parameters (temperature, pressure, monomer volume fraction, chain length of the polymer) and due to the interplay of liquid-vapor transitions and fluid-fluid unmixing. As a result I very intricate phase diagram topologies can result. An attempt to develop coarse-1 grained models that can deal with this task will be described. As usual, the polymers I will be modelled as off-lattice bead-spring chains, where several chemical monomers I are integrated into one effective bond, torsional degrees of freedom being dis-I regarded. But also a coarse-graine…

Monte Carlo simulations of the polymer glass transition: From the test of theories to material modeling

We present results on the glass transition in polymer melts using Monte Carlo simulations of the bond fluctuation lattice model. There are two questions we address in this work. What is the temperature dependence of the entropy density in such a model polymer melt and how well is it described by theories like the Gibbs-DiMarzio theory of the glass transition? And to what degree is one able to map the Hamiltonian of such an abstract lattice model onto a specific polymer material and use it to model the large scale and long time properties of a realistic polymer melt?

Monte Carlo simulation of a lyotropic first-order isotropic-nematic phase transition in a lattice polymer model

We present a Monte Carlo simulation of the bond-fluctuation lattice model, using a Hamiltonian which introduces a change in the conformational statistics of the polymer chains from Gaussian behavior at high temperatures to rigid rod behavior at low temperatures. We do not introduce any attractive interaction between the chains. Upon cooling, the aspect ratio of the chains increases above the critical value for the density employed in the simulation, and we observe an entropically driven phase transition into a nematic phase. We examine this transition quantitatively by a careful finite size scaling study using an optimized cumulant intersection method, and show that the transition is of fir…

Growth of Domains and Scaling in the Late Stages of Phase Separation and Diffusion-Controlled Ordering Phenomena

These lectures consider the kinetics of phase changes, induced by a sudden change of external thermodynamic parameters. E.g., we treat a system with a second-order transition at a critical temperature Tc (Fig. 1, left part). For T0 > Tc the system is disordered, while for T < Tc there is an order parameter ± ψ (implying one-component orderings, e.g., an Ising model; later we discuss generalizations). We consider a “quenching experiment”: The system is brought from an initially disordered state at T0 to a state at T where in equilibrium the system should be orderedl. Since no sign of ψ is preferred, the system starts forming locally ordered regions of either sign, separated by domain walls. …

Can one detach a fully adsorbed flexible polymer chain by an ultra-small external force?

Full adsorption of flexible chains onto typical solid substrates occurs at a surface interaction energy of (5–10) kBT. The corresponding detachment force is in the range 10–50 pN. In contrast to “bare” solid substrates common to non-living materials, surfaces coated with brush-like polymer layers are very common in biological soft matter. We employ a simple mean-field approach to describe the effects of weak attraction between a floating long macromolecule and the brush. We show that even for a moderately thick brush a very small effective attraction is enough to produce complete binding of the long chain. The detachment force scales as , where W is the brush thickness. Hence the force coul…

Comparison of Dissipative Particle Dynamics and Langevin thermostats for out-of-equilibrium simulations of polymeric systems

In this work we compare and characterize the behavior of Langevin and Dissipative Particle Dynamics (DPD) thermostats in a broad range of non-equilibrium simulations of polymeric systems. Polymer brushes in relative sliding motion, polymeric liquids in Poiseuille and Couette flows, and brush-melt interfaces are used as model systems to analyze the efficiency and limitations of different Langevin and DPD thermostat implementations. Widely used coarse-grained bead-spring models under good and poor solvent conditions are employed to assess the effects of the thermostats. We considered equilibrium, transient, and steady state examples for testing the ability of the thermostats to maintain const…

Reply to “Comment on ‘Elastic constants from microscopic strain fluctuations’ ”

We agree with Coupier et al. [Phys. Rev. E 81, 013101 (2010)] that their technique for extracting elastic constants from microscopic strain fluctuations improves upon ours because of a more accurate computation of the integral of the elastic correlation function over sub-blocks. However, we believe that their interpretation of the physical relevance of the elastic correlation length extracted from the fits is misleading.

Stretching of Free Chains Confined in Concave Brush-Coated Nanocylinders

The structure of a free flexible macromolecule confined in a cylindrical nanopore whose wall is coated by a polymer brush is studied by Monte Carlo simulation, varying the grafting density as well as the radius of the cylindrical pore. Because of this confinement, the free chain is stretched in axial direction; while for small grafting densities of the brush the end-to-end distance increases monotonously with decreasing pore radius, a nonmonotonic variation occurs for larger grafting densities. We show that this effect is due to strong interpenetration of the free chain and the brush chains; for very narrow pores a strong layering of cylindrical shells is found, and comparison with self-con…

Universal monomer dynamics of a two dimensional semi-flexible chain

We present a unified scaling theory for the dynamics of monomers for dilute solutions of semiflexible polymers under good solvent conditions in the free draining limit. Our theory encompasses the well-known regimes of mean square displacements (MSDs) of stiff chains growing like t^{3/4} with time due to bending motions, and the Rouse-like regime t^{2 \nu / (1+ 2\nu)} where \nu is the Flory exponent describing the radius R of a swollen flexible coil. We identify how the prefactors of these laws scale with the persistence length l_p, and show that a crossover from stiff to flexible behavior occurs at a MSD of order l^2_p (at a time proportional to l^3_p). A second crossover (to diffusive moti…

Interfaces between coexisting phases of polymer mixtures: Comparison between Monte Carlo simulations and theoretical predictions

Large scale Monte Carlo investigations of the interface between A-rich and B-rich phases of symmetric binary (AB) polymer mixtures are presented, using the bond fluctuation model of flexible chains with NA=NB=N=32 effective monomers. The temperature range studied, 0.144<T/Tc0.759, includes both the strong and the weak segregation limit. Interfacial free energy and interfacial structure are studied, and compared to predictions based on the selfconsistent field theory. Also the broadening of the interfacial width due to capillary waves is considered, and finite size effects due to the confinement of interfaces in thin films of polymer blends are discussed.

Osmotic pressure, atomic pressure and the virial equation of state of polymer solutions: Monte Carlo simulations of a bead-spring model

A recently introduced coarse-grained model of polymer chains is studied analyzing various contributions to the pressure as obtained from the virial theorem as a function of chain length N, temperature T and density ϕ. The off-lattice model of the polymer chains has anharmonic springs between the beads, but of finite extensibility, and the Morse-type interaction between beads is repulsive at very short distances and attractive at intermediate distances. Solvent molecules are not explicitly included. It is found that the covalent forces along the chain (modelled by the spring potentials) contribute a negative term to the pressure, irrespective of temperature, which vanishes linearly in ϕ as ϕ…

Investigation of Finite-Size Effects in the Determination of Interfacial Tensions

The interfacial tension between coexisting phases of a material is an important parameter in the description of many phenomena such as crystallization, and even today its accurate measurement remains difficult. We have studied logarithmic finite-size corrections in the determination of the interfacial tension with large scale Monte Carlo simulations, and have identified several novel contributions which not only depend on the ensemble, but also on the type of the applied boundary conditions. We present results for the Lennard-Jones system and the Ising model, as well as for hard spheres, which are particularly challenging. In the future, these findings will contribute to the understanding a…

Polymer Brushes on Flat and Curved Substrates: What Can be Learned from Molecular Dynamics Simulations

Ising systems with pairwise competing surface fields

The magnetization distribution and phase behaviour of large but finite Ising simple cubic L × L × L lattices in d = 3 dimensions and square L × L lattices in d = 2 dimensions are studied for the case where four free boundaries are present, at which surface fields +Hs act on one pair of opposite boundaries while surface fields −Hs act on the other pair (in d = 3, periodic boundary conditions are used for the remaining pair). Both the distribution PL(m) of the global magnetization and also the distribution of the local magnetization m(x,z) are obtained by Monte Carlo simulations, where x and z denote the coordinates when the boundaries are oriented along the x-axis and z-axis (in d = 2); or a…

Efficient prediction of thermodynamic properties of quadrupolar fluids from simulation of a coarse-grained model: the case of carbon dioxide.

Monte Carlo simulations are presented for a coarse-grained model of real quadrupolar fluids. Molecules are represented by particles interacting with Lennard-Jones forces plus the thermally averaged quadrupole-quadrupole interaction. The properties discussed include the vapor-liquid coexistence curve, the vapor pressure along coexistence, and the surface tension. The full isotherms are also accessible over a wide range of temperatures and densities. It is shown that the critical parameters (critical temperature, density, and pressure) depend almost linearly on a quadrupolar parameter q=Q(*4)T*, where Q* is the reduced quadrupole moment of the molecule and T* the reduced temperature. The mode…

The mean field to Ising crossover in the critical behavior of polymer mixtures : a finite size scaling analysis of Monte Carlo simulations

Monte Carlo simulations of the bond fluctuation model of symmetrical polymer mixtures (chain lengths N A =N B =N) are analyzed near the critical temperature T c (N) of their unmixing transition. Two choices of interaction range are studied, using a square-well potential with effective coordination number z eff ≃ 14 or z eff ≃ 5, respectively, at a volume fraction O= 0.5 of occupied lattice sites, and chain lengths in the range 8≤ N≤ 512. A linear relation between N and T c (N) is established, T c (N)= AN+B, where the correction term B is positive for z eff = 14 but negative for z eff = 5. The critical behavior of the models is analyzed via finite size scaling techniques, paying attention to…

Finite size effects at thermally-driven first order phase transitions: A phenomenological theory of the order parameter distribution

We consider the rounding and shifting of a firstorder transition in a finited-dimensional hypercubicL d geometry,L being the linear dimension of the system, and surface effects are avoided by periodic boundary conditions. We assume that upon lowering the temperature the system discontinuously goes to one ofq ordered states, such as it e.g. happens for the Potts model ind=3 forq≧3, with the correlation length ξ of order parameter fluctuation staying finite at the transition. We then describe each of theseq ordered phases and the disordered phase forL≫ξ by a properly weighted Gaussian. From this phenomenological ansatz for the total distribution of the order parameter, all moments of interest…

LARGE-SCALE SIMULATIONS IN CONDENSED MATTER PHYSICS —THE NEED FOR A TERAFLOP COMPUTER

The introduction of vector processors {“supercomputers” with a performance in the range of 109 floating point operations (1 GFLOP) per second} has had an enormous impact on computational condensed matter physics. The possibility of a substantially enhanced performance by massively parallel processors (“teraflop” machines with 1012 floating point operations per second) will allow satisfactory treatment of a large range of important scientific problems which have to a great extent thus far escaped numerical resolution. The present paper describes only a few examples (out of a long list of interesting research problems!) for which the availability of “teraflops” will allow spectacular progres…

Theoretical Foundations of the Monte Carlo Method and Its Applications in Statistical Physics

In this chapter we first introduce the basic concepts of Monte Carlo sampling, give some details on how Monte Carlo programs need to be organized, and then proceed to the interpretation and analysis of Monte Carlo results.

Semiflexible Polymers in the Bulk and Confined by Planar Walls

Semiflexible polymers in solution under good solvent conditions can undergo an isotropic-nematic transition. This transition is reminiscent of the well-known entropically-driven transition of hard rods described by Onsager’s theory, but the flexibility of the macromolecules causes specific differences in behavior, such as anomalous long wavelength fluctuations in the ordered phase, which can be understood by the concept of the deflection length. A brief review of the recent progress in the understanding of these problems is given, summarizing results obtained by large-scale molecular dynamics simulations and density functional theory. These results include also the interaction of semiflexib…

On the polymer physics origins of protein folding thermodynamics

A remarkable feature of the spontaneous folding of many small proteins is the striking similarity in the thermodynamics of the folding process. This process is characterized by simple two-state thermodynamics with large and compensating changes in entropy and enthalpy and a funnel-like free energy landscape with a free-energy barrier that varies linearly with temperature. One might attribute the commonality of this two-state folding behavior to features particular to these proteins (e.g., chain length, hydrophobic/hydrophilic balance, attributes of the native state) or one might suspect that this similarity in behavior has a more general polymer-physics origin. Here we show that this behavi…

Kinetics of the Formation of Ordered Domains on Surfaces: Theoretical Considerations and Monte-Carlo Simulation

When an adsorbed monolayer which initially is in a disordered state is suddenly brought to a temperature in the regime of the ordered phase, domains of the ordered phase are predicted to form and grow with time t after the quench according to a power law, i.e. linear dimension L(t) ∞ tx. At the same time, the structure function S(k,t) is predicted to satisfy a scaling law, S(k,t) = S(k,tx), k being the difference between the wave vector observed in the scattering and the Bragg wave vector describing the long range order. The theoretical ideas which lead to this behaviour are briefly reviewed, and evidence from simulations of simple lattice gas models and Potts models is presented. Particula…

Monte Carlo renormalization group methods

Theory of glass transition in spin glasses, orientational glasses and structural glasses

Theoretical concepts about the glass transition are briefly reviewed, and the test of these ideas by Monte Carlo simulations of simple lattice models is described, with an emphasis on isotropic and anisotropic orientational glasses, and the bond fluctuation model of polymer melts. It is suggested that orientational glasses do have an equilibrium phase transition at zero temperature (in d = 3 dimensions!) only, in contrast to the Ising spin glass which orders at nonzero temperature. A diverging glass correlation length is identified that is responsible for the anomalous slowing down. For the Potts glass, the divergence seems to be exponential, implying that the model is at its lower critical…

Monte Carlo Simulations of Surfaces and Interfaces in Materials

Many applications of materials are controlled by their surface and interface properties. In particular, metallic alloys (but also mixed dielectric materials and amorphous polymer blends) are not homogeneously mixed on a microscopic length scale, although they are macroscopically homogeneous. Depending on the preparation of the sample, there exists a heterophase microstructure, with typical domain sizes, e.g. in the 1 to 102 µm range, separated by interfaces between them. The physical properties of such intrinsic interfaces (grain boundaries between small crystallites, antiphase domain boundaries in ordered alloys, Bloch walls in magnetic materials, etc.) are not only an important controllin…

Finite-size effects on liquid-solid phase coexistence and the estimation of crystal nucleation barriers.

A fluid in equilibrium in a finite volume $V$ with particle number $N$ at a density $\rho = N/V$ exceeding the onset density $\rho_f $ of freezing may exhibit phase coexistence between a crystalline nucleus and surrounding fluid. Using a method suitable for the estimation of the chemical potential of dense fluids we obtain the excess free energy due to the surface of the crystalline nucleus. There is neither a need to precisely locate the interface nor to compute the (anisotropic) interfacial tension. As a test case, a soft version of the Asakura-Oosawa model for colloid polymer-mixtures is treated. While our analysis is appropriate for crystal nuclei of arbitrary shape, we find the nucleat…

The smectic phase in semiflexible polymer materials: A large scale Molecular Dynamics study

Abstract Semiflexible polymers in concentrated lyotropic solution are studied within a bead-spring model by molecular dynamics simulations, focusing on the emergence of a smectic A phase and its properties. We systematically vary the density of the monomeric units for several contour lengths that are taken smaller than the chain persistence length. The difficulties concerning the equilibration of such systems and the choice of appropriate ensemble (constant volume versus constant pressure, where all three linear dimensions of the simulation box can fluctuate independently) are carefully discussed. Using HOOMD-blue on graphics processing units, systems containing more than a million monomeri…

Unmixing of Polymer Blends Confined in Ultrathin Films:  Crossover between Two-Dimensional and Three-Dimensional Behavior

The interplay between chain conformations and phase separation in binary symmetric polymer mixtures confined into thin films by "neutral" hard walls (i.e., walls that do not preferentially attract or repel one of the two components of the mixture) is studied by Monte Carlo simulations. Using the bond fluctuation model on a simple cubic lattice in the semi grand canonical ensemble, we locate the critical temperature of demixing via finite size scaling methods for a wide range of chain lengths (16/= N/= 256 effective monomers per chain) and film thicknesses (2/= D/= 19 lattice spacings). Simultaneously, we investigate the geometrical structure of the chains, showing that despite using melt de…

Spinodal decomposition in a binary polymer mixture: Dynamic self-consistent-field theory and Monte Carlo simulations

We investigate how the dynamics of a single chain influences the kinetics of early stage phase separation in a symmetric binary polymer mixture. We consider quenches from the disordered phase into the region of spinodal instability. On a mean field level we approach this problem with two methods: a dynamical extension of the self consistent field theory for Gaussian chains, with the density variables evolving in time, and the method of the external potential dynamics where the effective external fields are propagated in time. Different wave vector dependencies of the kinetic coefficient are taken into account. These early stages of spinodal decomposition are also studied through Monte Carlo…

Slowing down in the three-dimensional three-state Potts glass with nearest neighbor exchange : A Monte Carlo study

,Static and dynamic properties of the Potts model on the simple cubic lattice with nearest neighbor ±Ĵ-interaction are obtained from Monte Carlo simulations in a temperature range where full thermal equilibrium still can be achieved (T/Ĵ ≥ 0.6). For a lattice size L = 16, in this range finite size effects are still negligible, but the data for the spin glass susceptibility agree with previous extrapolations based on finite size scaling of very small lattices. While the static properties are compatible with a zero temperature transition, they certainly do not prove it. Unlike the Ising spin glass, the decay of the time-dependent order parameter is compatible with a simple Kohlrausch function…

What can be learned from the rotational motion of single molecules in a polymer melt near the glass transition?

We develop a framework for the interpretation of single-molecule (SM) spectroscopy experiments of probe dynamics in a complex glass-forming system. Specifically, from molecular dynamics simulations of a single probe molecule in a coarse-grained model of a polymer melt, we show the emergence of sudden large angular reorientations (SLARs) of the SM as the mode coupling critical temperature is closely approached. The large angular jumps are intimately related to meta-basin transitions in the potential energy landscape of the investigated system and cause the appearance of stretched exponential relaxations of various rotational observables, reported in the SM literature as dynamic heterogeneity…

Phase Behavior of Active Swimmers in Depletants: Molecular Dynamics and Integral Equation Theory

We study the structure and phase behavior of a binary mixture where one of the components is self-propelling in nature. The inter-particle interactions in the system were taken from the Asakura-Oosawa model, for colloid-polymer mixtures, for which the phase diagram is known. In the current model version the colloid particles were made active using the Vicsek model for self-propelling particles. The resultant active system was studied by molecular dynamics methods and integral equation theory. Both methods produce results consistent with each other and demonstrate that the Vicsek model based activity facilitates phase separation, thus broadening the coexistence region.

Growing length scales in a supercooled liquid close to an interface

We present the results of molecular dynamics computer simulations of a simple glass former close to an interface between the liquid and the frozen amorphous phase of the same material. By investigating F_s(q,z,t), the incoherent intermediate scattering function for particles that have a distance z from the wall, we show that the relaxation dynamics of the particles close to the wall is much slower than the one for particles far away from the wall. For small z the typical relaxation time for F_s(q,z,t) increases like exp(Delta/(z-z_p)), where Delta and z_p are constants. We use the location of the crossover from this law to the bulk behavior to define a first length scale tilde{z}. A differe…

Hydrokinetic simulations of nanoscopic precursor films in rough channels

We report on simulations of capillary filling of high-wetting fluids in nano-channels with and without obstacles. We use atomistic (molecular dynamics) and hydrokinetic (lattice-Boltzmann) approaches which point out clear evidence of the formation of thin precursor films, moving ahead of the main capillary front. The dynamics of the precursor films is found to obey a square-root law as the main capillary front, z^2(t) ~ t, although with a larger prefactor, which we find to take the same value for the different geometries (2D-3D) under inspection. The two methods show a quantitative agreement which indicates that the formation and propagation of thin precursors can be handled at a mesoscopic…

Surface effects on spinodal decomposition in binary mixtures: The case with long-ranged surface fields

We present detailed numerical results for phase-separation kinetics of critical binary mixtures in the vicinity of a surface that exerts a long-ranged attractive force on one of the components of the mixture. We consider surface potentials of the form $V(Z)\ensuremath{\sim}{Z}^{\ensuremath{-}n}$, where $Z$ is the distance from the surface and $n=1,2,3$. In particular, we investigate the interplay of the surface wetting layer with the dynamics of domain growth. We find that the wetting layer at the surface exhibits power-law growth with an exponent that depends on $n$, in contrast to the case with a short-ranged surface potential, where the growth is presumably logarithmic. From correlation …

Depletion induced isotropic-isotropic phase separation in suspensions of rod-like colloids

When non-adsorbing polymers are added to an isotropic suspension of rod-like colloids, the colloids effectively attract each other via depletion forces. We performed Monte Carlo simulations to study the phase diagram of such rod-polymer mixture. The colloidal rods were modeled as hard spherocylinders; the polymers were described as spheres of the same diameter as the rods. The polymers may overlap with no energy cost, while the overlap of polymers and rods is forbidden. Large amounts of depletant cause phase separation of the mixture. We estimated the phase boundaries of isotropic-isotropic coexistence both in the bulk and in confinement. To determine the phase boundaries we applied the gra…

How does stiffness of polymer chains affect their adsorption transition?

The adsorption transition and the structure of semiflexible adsorbed macromolecules are studied by a molecular dynamics simulation of a coarse-grained, bead-spring type model. Varying chain length N and stiffness κ (which is proportional to the persistence length lp in d = 3 dimensions) as well as the strength ϵwall of the adsorption potential, the adsorbed monomer fraction, orientational bond order parameter, and chain linear dimensions are studied. In the simulations, excluded volume interactions normally are included but can be “switched off,” and thus, the influence of excluded volume (leading to deviations from predictions of the wormlike chain model) can be identified. It is shown tha…

Coil-bridge transition in a single polymer chain as an unconventional phase transition: theory and simulation.

The coil-bridge transition in a self-avoiding lattice chain with one end fixed at height H above the attractive planar surface is investigated by theory and Monte Carlo simulation. We focus on the details of the first-order phase transition between the coil state at large height H ⩾ Htr and a bridge state at H ⩽ Htr, where Htr corresponds to the coil-bridge transition point. The equilibrium properties of the chain were calculated using the Monte Carlo pruned-enriched Rosenbluth method in the moderate adsorption regime at (H/Na)tr ⩽ 0.27 where N is the number of monomer units of linear size a. An analytical theory of the coil-bridge transition for lattice chains with excluded volume interact…

Anomalous Fluctuations of Nematic Order in Solutions of Semiflexible Polymers

The nematic ordering in semiflexible polymers with contour length $L$ exceeding their persistence length $\ell_p$ is described by a confinement of the polymers in a cylinder of radius $r_{eff}$ much larger than the radius $r_\rho$, expected from the respective concentration of the solution. Large scale Molecular Dynamics simulations combined with Density Functional Theory are used to locate the Isotropic-Nematic ($I-N$)-transition and to validate this cylindrical confinement. Anomalous fluctuations, due to chain deflections from neighboring chains in the nematic phase are proposed. Considering deflections as collective excitations in the nematically ordered phase of semiflexible polymers el…

On melting of two-dimensional monolayer films

The melting of two-dimensional films formed on the (100) fcc crystal is studied by Monte Carlo simulation. The results obtained suggest that in systems with only weakly corrugated surface potential, exhibiting the hexagonal close packed solid structure, the melting transition is followed by the lsing-like transition as predicted by the theory of Nelson and Halperin. In the case of highly corrugated surface potential, the film forms registered structure which disorders gradually as the temperature is raised.

Adsorption of Oligomers and Polymers into a Polymer Brush Formed from Grafted Ring Polymers

The interaction of a ring polymer brush with a solution containing oligomers or free linear flexible macromolecules is studied by Monte Carlo simulation, varying the chain length of the free chains, and in selected cases also the lengths of the rings. Two grafting densities are studied, corresponding to semidilute and very concentrated conditions, and a comparison with the corresponding case of brushes formed from grafted linear chains is made. Although the ring polymer linear dimensions in the brushes show an anomalous scaling with ring length, similar to (noncatenated) ring polymer melts, the concentration profiles of oligomers and long macromolecules in ring polymer brushes differ only v…

Classical Heisenberg antiferromagnets with nearest and next-nearest neighbor interactions on the face-centered cubic lattice: a model for EuTe?

Magnetic properties of the Heisenberg antiferromagnet with spin quantum numberS→∞ on the face-centered cubic lattice are studied as function of temperature and magnetic field, using molecular field approximation and Monte Carlo methods. In order to model Europiumtelluride, we use isotropic exchange interactions between nearest- and nextnearest neighbors; the values of these exchange constants are taken from experiments. In addition, a pseudo-dipolar anisotropy (truncated after the next-nearest neighbor distance) is included; the molecular field calculations also are performed with the full dipolar of real EuTe in two respects: the structure in zero magnetic field involves 8 sublattices in t…

Monte Carlo investigations of phase transitions: status and perspectives

Using the concept of finite-size scaling, Monte Carlo calculations of various models have become a very useful tool for the study of critical phenomena, with the system linear dimension as a variable. As an example, several recent studies of Ising models are discussed, as well as the extension to models of polymer mixtures and solutions. It is shown that using appropriate cluster algorithms, even the scaling functions describing the crossover from the Ising universality class to the mean-field behavior with increasing interaction range can be described. Additionally, the issue of finite-size scaling in Ising models above the marginal dimension (d*=4) is discussed.

Monte Carlo and molecular dynamics simulation of the glass transition of polymers

Two coarse-grained models for polymer chains in dense glass-forming polymer melts are studied by computer simulation: the bond-fluctuation model on a simple cubic lattice, where a bond-length potential favors long bonds, is treated by dynamic Monte Carlo methods, and a bead-spring model in the continuum with a Lennard-Jones potential between the beads is treated by Molecular Dynamics. While the dynamics of both models differ for short length scales and associated time scales, on mesoscopic spatial and temporal scales both models behave similarly. In particular, the mode coupling theory of the glass transition can be used to interpret the slowing down of the undercooled polymer melt. For the…

Stochastic dynamics of polymer brushes under shear deformation

The dynamical properties of polymer brushes under shear deformation have been studied by computer simulation. Both the local and the global dynamic properties at various shear rates have been calculated. The distribution of orientational and translational mobilities of the different monomers along the chain have been obtained. It was shown that the local mobility of the brushes changes very slowly with increasing of shear rates up to the largest rates. Increase in grafting density leads to an increasingly step like dependence of the correlation times as a function of shear rate.

Monte Carlo calculation of free energy for a fcc lattice-gas model

A face-centered-cubic Ising lattice-gas model with nearest- and next-nearest-neighbor interactions is studied, and an accurate determination of the transition temperature for the discontinuous order-disorder transition is obtained. This model is of interest in the studies of phase diagrams for metallic alloys. The location of the transition was previously not known accurately, and its estimation has a number of applications. Very accurate absolute free-energy densities for the two coexisting phases have been obtained from a combination of the standard thermodynamic integration method and the method of sampling finite-size dependence. The latent-heat also is calculated with good precision.

Crossover scaling in semidilute polymer solutions: a Monte Carlo test

Formation of metastable structures by phase separation triggered by initial composition gradients in thin films.

Phase separation kinetics of a binary (A,B) mixture contained in a thin film of thickness D induced by a quench from the one-phase region into the miscibility gap is studied by simulations using a Cahn-Hilliard-Cook model. The initial randomly mixed state (50% A, 50% B) contains a concentration gradient perpendicular to the film, while the surfaces of the film are "neutral" (no preference for either A or B). In thermal equilibrium, a pattern of large A-rich and B-rich domains must result, separated by domain walls oriented perpendicularly to the external surfaces of the thin film. However, it is shown that for many choices of D and the strength of the initial gradient Ψ(g), instead a very l…

Unconventional ordering behavior of semi-flexible polymers in dense brushes under compression

Using a coarse-grained bead-spring model for semi-flexible macromolecules which form a polymer brush, the structure and dynamics of the polymers were investigated, varying the chain stiffness and the grafting density. The anchoring conditions for the grafted chains were chosen such that their first bonds were oriented along the normal to the substrate plane. The compression of such a semi-flexible brush by a planar piston was observed to be a two-stage process: for a small compression the chains were shown to contract by "buckling" deformation whereas for a larger compression the chains exhibited a collective (almost uniform) bending deformation. Thus, the stiff polymer brush underwent a 2n…

Dynamics of a supercooled polymer melt above the mode-coupling critical temperature: cage versus polymer-specific effects

This paper reports results of molecular dynamics simulations for a glassy polymer melt consisting of short, non-entangled chains. The temperature region studied covers the supercooled state of the melt above the mode-coupling critical temperature. The analysis focuses on the interplay of simple-liquid and polymer-specific effects. One can clearly distinguish two regimes: a regime of small and one of large monomer displacements. The first regime corresponds to motion of a monomer in its local environment. It is dominated by the cage effect and well described by the idealized mode-coupling theory. The second regime is governed by the late-β/early-α process. In this regime the connectivity of …

Simulations of critical phenomena: from Ising models to fluids

A brief retrospective is given, how simulations of critical phenomena started about 45 years ago, and how finite size scaling concepts helped to make such studies quantitative.

Structure of bottle-brush polymers in solution: A Monte Carlo test of models for the scattering function

Extensive Monte Carlo results are presented for a lattice model of a bottle-brush polymer under good solvent or Theta solvent conditions. Varying the side chain length, backbone length, and the grafting density for a rigid straight backbone, both radial density profiles of monomers and side chain ends are obtained, as well as structure factors describing the scattering from a single side chain and from the total bottle-brush polymer. To describe the structure in the interior of a very long bottle-brush, a periodic boundary condition in the direction along the backbone is used, and to describe effects due to the finiteness of the backbone length, a second set of simulations with free ends of…

Conformational studies of bottle-brush polymers absorbed on a flat solid surface.

The adsorption of a bottle-brush polymer end-grafted with one chain end of its backbone to a flat substrate surface is studied by Monte Carlo simulation of a coarse-grained model, that previously has been characterized in the bulk, assuming a dilute solution under good solvent conditions. Applying the bond fluctuation model on the simple cubic lattice, we vary the backbone chain length $N_b$ from $N_b=67$ to $N_b = 259$ effective monomeric units, the side chain length $N$ from N=6 to N=48, and the grafting density $\sigma=1$, i.e., parameters that correspond well to the experimentally accessible range. When the adsorption energy strength $\epsilon$ is varied, we find that the adsorption tra…

Surface Effects on Block Copolymer Melts Above the Order-Disorder Transition: Linear Theory of Equilibrium Properties and the Kinetics of Surface-Induced Ordering

A phenomenological theory is developed for static and dynamic aspects ofsurface- induced ordering of symmetrical block copolymers taking fluctuation corrections into account, but, considering conditions where the bulk block copolymer melt is still disordered, and a lin- earized version of the resulting Ginzburg-Landau type equation suffices. Both the semi-infinite geometry and symmetrical films of thickness 2L are treated, applying the same boundary con- ditions as used previously for a treatment of wetting in polymer blends, assuming short range surface forces and a long Wavelength approximation. For the static order parameter profile in thin film geometry, We derive an oscillatory converg…

Evidence for the time-temperature superposition principle from Monte-Carlo simulations of the glass transition in two-dimensional polymer melts

The bond fluctuation model on a square lattice with a bond-length dependent potential exhibits in simulations of slow cooling a kinetic glass transition where the system falls out of equilibrium. Extending previous work, the relaxation functions of gyration radius and end-to-end distance, and the bond autocorrelation function of the polymers are presented and related to the time-dependent displacements of inner monomeric units and center of gravity of the whole chains, respectively. Over a wide temperature range the data can be collapsed on master curves satisfying the time-temperature superposition principle for Rouse dynamics.

Structure and dynamics of thin polymer films: a case study with the bond-fluctuation model

Abstract This paper reports Monte Carlo simulation results of a polymer melt of short, non-entangled chains which are embedded between two impenetrable walls. The melt is simulated by the bond-fluctuation lattice model under athermal conditions, i.e. only excluded volume interactions between the monomers and between the monomers and the walls are taken into account. In the simulations, the wall separation is varied from about one to about 15 times the bulk radius of gyration R g . The confinement influences both static and dynamic properties of the films: Chains close to the walls preferentially orient parallel to it. This parallel orientation decays with increasing distances from the wall …

Entropic Unmixing in Nematic Blends of Semiflexible Polymers

Binary mixtures of semiflexible polymers with the same chain length but different persistence lengths separate into two coexisting different nematic phases when the osmotic pressure of the lyotropic solution is varied. Molecular Dynamics simulations and Density Functional Theory predict phase diagrams either with a triple point, where the isotropic phase coexists with two nematic phases, or a critical point of unmixing within the nematic mixture. The difference in locally preferred bond angles between the constituents drives this unmixing without any attractive interactions between monomers.

Interfaces in immiscible polymer blends: A Monte Carlo simulation approach on the CRAY T3E

Polymeric materials pose a challenge for Monte Carlo simulations because of the widely spread length and time scales involved. Using large scale computer simulations we investigate the interfacial structure in a partially compatible polymer mixture. The problem is studied in the framework of a coarse grained lattice model - the bond fluctuation model on the simple cubic lattice, choosing N = 32 and lattice linear dimensions L × L × D up to 512 × 512 × 64. We employ a two dimensional geometric decomposition scheme to implement this algorithm on the CRAY T3E. The algorithm scales very well with the number of processors. The structure of polymer coils near interfaces between coexisting phases …

Computer simulation of models for orientational glasses

Abstract Monte Carlo studies of two- and three-dimensional lattice models where quadrupoles interact with a nearest-neighbor Gaussian coupling are reviewed. None of these models has a thermodynamic glass phase transition at non-zero temperature like the Ising spin glass: rather, phase transitions at zero temperature occur that exhibit a dynamical freeze-in spread out over a wide temperature range and are characterized by a strongly non-exponential relaxation. The time-dependent glass order parameter, q(t), decays with time, t, compatible with a stretched exponential decay q(t) ∼ exp [− (t/τ)y] with a strongly temperature-dependent exponent. While the static glass ‘susceptibility’ for isotro…

Unmixing of binary alloys by a vacancy mechanism of diffusion: a computer simulation

The initial stages of phase separation are studied for a model binary alloy (AB) with pairwise interactions e AA , e AB , e BB between nearest neighbors, assuming that there is no direct interchange of neighboring atoms possible, but only an indirect one mediated by vacancies (V) occurring in the system at a concentrationc v and which are strictly conserved, as are the concentrationsc A andc B of the two species.A-atoms may jump to vacant sites with jump rateГ A , B-atoms with jump rateГ B (in the absence of interactions). Particular attention is paid to the question to what extent nonuniform distribution of vacancies affects the unmixing kinetics. Our study focuses on the special caseГ A =…

The relaxation dynamics of a supercooled liquid confined by rough walls

We present the results of molecular dynamics computer simulations of a binary Lennard-Jones liquid confined between two parallel rough walls. These walls are realized by frozen amorphous configurations of the same liquid and therefore the structural properties of the confined fluid are identical to the ones of the bulk system. Hence this setup allows us to study how the relaxation dynamics is affected by the pure effect of confinement, i.e. if structural changes are completely avoided. We find that the local relaxation dynamics is a strong function of z, the distance of the particles from the wall, and that close to the surface the typical relaxation times are orders of magnitude larger tha…

Interfaces in the confined Ising system with competing surface fields

Abstract When a magnetic Ising film is confined in a L × M geometry ( L ⪡ M ) short-range competing magnetic fields ( h 1 ) are applied at opposite walls along the M -direction, a (weakly rounded) localization–delocalization transition of the interface between domains of different orientation that runs parallel to walls can be observed. This transition is the precursor of a wetting phase transition that occurs in the limit of infinite film thickness ( L → ∞ ) at the critical curve T w ( h 1 ) . For T T w ( h 1 ) ( T > T w ( h 1 ) ) such an interface is bound to (unbound from) the walls, while right at T w ( h 1 ) the interface is freely fluctuating around the center of the film. We present …

Understanding the Multiple Length Scales Describing the Structure of Bottle-brush Polymers by Monte Carlo Simulation Methods

Bottle-brush polymers contain a long flexible macromolecule as a backbone to which flexible side chains are grafted. Through the choice of the grafting density and the length of the side chains the local stiffness of this cylindrical molecular brush can be controlled, but a quantitative understanding of these phenomena is lacking. Monte Carlo simulation results are presented and discussed which address this issue, extractingmesoscopic length scales (such as the cross-sectional radius, persistence length, and contour length of these objects). Large-scale simulations of the bond fluctuation model are combined with simulations of the simple selfavoiding walk (SAW) model with flexibility contro…

Dynamics of Glassy Polymer Melts in Confined Geometry: A Monte Carlo Simulation

Dynamic properties of a dense polymer melt confined between two hard walls are investigated over a wide range of temperatures by dynamic Monte Carlo simulation. The temperature interval ranges from the ordinary liquid to the strongly supercooled melt. The influence of temperature, density and confinement on the polymer dynamics is studied by various mean-square displacements, structural relaxation functions and quantities derived from them (relaxation times, apparent diffusion coefficients, monomer relaxation rates), yielding the following results: The motion of the monomers and polymers close to the walls is enhanced in parallel, but reduced in perpendicular direction. This dynamic anisotr…

Finite-size scaling in a microcanonical ensemble

The finite-size scaling technique is extended to a microcanonical ensemble. As an application, equilibrium magnetic properties of anL×L square lattice Ising model are computed using the microcanonical ensemble simulation technique of Creutz, and the results are analyzed using the microcanonical ensemble finite-size scaling. The computations were done on the multitransputer system of the Condensed Matter Theory Group at the University of Mainz.

Monte Carlo Tests of Nucleation Concepts in the Lattice Gas Model

The conventional theory of homogeneous and heterogeneous nucleation in a supersaturated vapor is tested by Monte Carlo simulations of the lattice gas (Ising) model with nearest-neighbor attractive interactions on the simple cubic lattice. The theory considers the nucleation process as a slow (quasi-static) cluster (droplet) growth over a free energy barrier $\Delta F^*$, constructed in terms of a balance of surface and bulk term of a "critical droplet" of radius $R^*$, implying that the rates of droplet growth and shrinking essentially balance each other for droplet radius $R=R^*$. For heterogeneous nucleation at surfaces, the barrier is reduced by a factor depending on the contact angle. U…

A new boundary-controlled phase transition: Phase separation in an Ising bi-pyramid with competing surface fields

We study phase coexistence of an Ising ferromagnet in a bi-pyramid geometry with a square basal plane of linear extension 2L + 1. Antisymmetric surface fields act on the pyramid surfaces above and below the basal plane. In the limit L → ∞, the magnetisation stays zero at the bulk critical temperature, but becomes discontinuously non-zero at the cone filling critical temperature associated with a single pyramid. Monte Carlo simulations and scaling considerations show that this transition is described by a Landau theory with size-dependent coefficients that give rise to singular critical amplitudes.

Characteristic Length Scales and Radial Monomer Density Profiles of Molecular Bottle-Brushes: Simulation and Experiment

Extensive Monte Carlo simulations are presented for bottle-brush polymers under good solvent conditions, using the bond fluctuation model on the simple cubic lattice. Varying the backbone length (from Nb = 67 to Nb = 259 effective monomers) as well as the side chain length (from N = 6 to N = 48), for a physically reasonable grafting density of one chain per backbone monomer, we find that the structure factor describing the total scattering from the bottle-brush provides an almost perfect match for some combinations of (Nb, N) to experimental data of Rathgeber et al. [J. Chem. Phys. 2005, 122, 124904], when we adjust the length scale of the simulation to reproduce the experimental gyration r…

Neutrons detect order in glasses

The first glassy material was probably made in ancient Egypt some 4500 years ago, so the fact that the structure of glass is still one of the biggest puzzles in physics may come as a surprise. When a liquid is cooled very quickly, the atoms do not have time to arrange themselves into an ordered crystalline solid. Instead, the super cooled liquid falls out of equilibrium and into a disordered amorphous network, more commonly known as a glass.

Classical and ab-initio molecular dynamic simulation of an amorphous silica surface

We present the results of a classical molecular dynamic simulation as well as of an ab initio molecular dynamic simulation of an amorphous silica surface. In the case of the classical simulation we use the potential proposed by van Beest et al. (BKS) whereas the ab initio simulation is done with a Car-Parrinello method (CPMD). We find that the surfaces generated by BKS have a higher concentration of defects (e.g. concentration of two-membered rings) than those generated with CPMD. In addition also the distribution functions of the angles and of the distances are different for the short rings. Hence we conclude that whereas the BKS potential is able to reproduce correctly the surface on the …

Surface-directed spinodal decomposition in a thin-film geometry: A computer simulation

The phase separation kinetics of a two-dimensional binary mixture at critical composition confined between (one-dimensional) straight walls which preferentially attract one component of the mixture is studied for a wide range of distancesD between the walls. Following earlier related work on semiinfinite systems, two choices of surface forces at the walls are considered, one corresponding to an incompletely wet state of the walls, the other to a completely wet state (forD→∞). The nonlinear Cahn-Hilliard-type equation, supplemented with appropriate boundary conditions which account for the presence of surfaces, is replaced by a discrete equivalent and integrated numerically. Starting from a …

Conformational Properties of Polymer Mushrooms Under Spherical and Cylindrical Confinement

A coarse grained model of a flexible macromolecule end-grafted on the inside of a sphere or a cylinder under good solvent conditions is studied by Monte Carlo simulations. For cylindrical confinement, two regimes are found: when the cylinder radius R exceeds the gyration radius R 90 of the polymer mushroom grafted to a planar surface, a simple scaling description holds. In the opposite case, a non-monotonic crossover to a cigar-like quasi-one-dimensional structure occurs, and the distribution P e (x) of the free chain end in the x-direction along the cylinder axis becomes bimodal. Spherical confinement, on the other hand, causes a crossover from dilute to semidilute behavior of the structur…

Determination of the origin and magnitude of logarithmic finite-size effects on interfacial tension: Role of interfacial fluctuations and domain breathing

The ensemble-switch method for computing wall excess free energies of condensed matter is extended to estimate the interface free energies between coexisting phases very accurately. By this method, system geometries with linear dimensions $L$ parallel and $L_z$ perpendicular to the interface with various boundary conditions in the canonical or grandcanonical ensemble can be studied. Using two- and three-dimensional Ising models, the nature of the occurring logarithmic finite size corrections is studied. It is found crucial to include interfacial fluctuations due to "domain breathing".

An Ising ferromagnet with an antiferromagnetic surface layer: A simple model for magnetic surface reconstruction

Simple cubic Ising lattices are studied by Monte Carlo simulation, using a thin film geometry (usually 40 atomic layers thick), with nearest neighbour ferromagnetic exchange J in the bulk and nearest neighbour antiferromagnetic interaction Js between surface spins. Applying a technique of preferential sampling in the surface layers, we investigate the ordering for a variety of values of JsJ and for various temperatures. For JsAF < Js < − 0.25J (where JsAF ≈ − 2.01J) ferromagnetic ordering occurs at a higher temperature than the antiferromagnetic surface ordering, while for − 0.25J < Js no antiferromagnetic long range order is possible. For Js < JsAF the surface transition occurs at a higher…

Monte Carlo simulation of the glass transition in polymeric systems: Recent developments

Abstract The bond fluctuation model on square and s.c. lattices is used as a coarse-grained model for flexible polymers in dense melts. Using an energy that favours long bonds, a conflict is created between the tendency of the bonds to stretch at low temperatures and packing constraints. This simple concept of ‘geometric frustration’ leads to glass transition. Both static and dynamic properties of this model are investigated by Monte Carlo simulations, paying attention to effects found by varying the cooling rate and the chain length N of the polymers. In two and three spatial dimensions an effective (cooling-rate dependent) glass transition temperature T g can be defined, where the system …

Finite-size scaling analysis of the ?4 field theory on the square lattice

Monte-Carlo calculations are performed for the model Hamiltonian ℋ = ∑i[(r/2)Φ 2(i)+(u/4)/gF4(i)]+∑ (C/2)[Φ (i)−Φ(j)]2 for various values of the parametersr, u, C in the crossover region from the Ising limit (r→-∞,u+∞) to the displacive limit (r=0). The variableφ(i) is a scalar continuous spin variable which can lie in the range-∞<φ(i)<+∞, for each lattice site (i).φ(i) is a priori selected proportional to the single-site probability in our Monte Carlo algorithm. The critical line is obtained in very good agreement with other previous approaches. A decrease of apparent critical exponents, deduced from a finite-size scaling analysis, is attributed to a crossover toward mean-field values at t…

Low-temperature anharmonic lattice deformations near rotator impurities: A quantum Monte Carlo approach.

At zero temperature the equilibrium structures of a system consisting of a quantum rotator (${\mathrm{N}}_{2}$) embedded in a relaxing lattice (Ar) surrounding are studied with a variational approach. With symmetric wave functions (para-${\mathrm{N}}_{2}$), we obtain a cubic lattice deformation near the rotator, while with antisymmetric wave functions (ortho-${\mathrm{N}}_{2}$), we obtain a tetragonal lattice deformation forming a stable oriented ground state. At low temperatures, we investigate the properties of this system with a quantum Monte Carlo simulation. On top of the tetragonal deformation the width of the nearest-neighbor oscillations follows classical ``scaling'' laws according …

Perspective: The Asakura Oosawa model: A colloid prototype for bulk and interfacial phase behavior

In many colloidal suspensions, the micrometer-sized particles behave like hard spheres, but when non-adsorbing polymers are added to the solution a depletion attraction (of entropic origin) is created. Since 60 years the Asakura-Oosawa model, which simply describes the polymers as ideal soft spheres, is an archetypical description for the statistical thermodynamics of such systems, accounting for many features of real colloid-polymer mixtures very well. While the fugacity of the polymers (which controls their concentration in the solution) plays a role like inverse temperature, the size ratio of polymer versus colloid radii acts as a control parameter to modify the phase diagram: when this …

A new insight into the isotropic–nematic phase transition in lyotropic solutions of semiflexible polymers: density-functional theory tested by molecular dynamics

Semiflexible polymers in solution are studied for a wide range of both contour length L and persistence length lp as a function of monomer concentration under good solvent conditions. Both density-functional theory (DFT) and molecular dynamics (MD) simulation methods are used, and a very good agreement between both techniques is observed for rather stiff polymers. Evidence for a new mechanism of order parameter fluctuations in the nematic phase is presented, namely collective deformations of bundles of wormlike chains twisted around each other, and the typical wavelengths and amplitudes of these modes are estimated. These long wavelength fluctuations cause a reduction of the order parameter…

Finite-Size Scaling Study of the Simple Cubic Three-State Potts Glass

During the last few years the Potts glass model has attracted more and more attention. It is considered as a first step towards modelling the phase transition of structural and orientational glasses. A mean-field approach /1/ predicts a low temperature behavior completely different from what is known from Ising spin glasses /2/. But short range models differ markedly from mean-field-predictions. So it is natural to ask, how the short range Potts glass behaves. Especially the question of the lower critical dimension d l is important, below which a finite temperature transition ceases to occur. We tried to answer this by combining Monte-Carlo simulations with a finite-size scaling analysis. T…

Phase transitions of single polymer chains and of polymer solutions: insights from Monte Carlo simulations

The statistical mechanics of flexible and semiflexible macromolecules is distinct from that of small molecule systems, since the thermodynamic limit can also be approached when the number of (effective) monomers of a single chain (realizable by a polymer solution in the dilute limit) is approaching infinity. One can introduce effective attractive interactions into a simulation model for a single chain such that a swollen coil contracts when the temperature is reduced, until excluded volume interactions are effectively canceled by attractive forces, and the chain conformation becomes almost Gaussian at the theta point. This state corresponds to a tricritical point, as the renormalization gro…

Pulling Single Adsorbed Bottle-Brush Polymers off a Flat Surface: A Monte Carlo Simulation

Force versus extension behavior of flexible chains and semiflexible bottle-brush polymers adsorbed from a good solvent on a planar substrate is studied by Monte Carlo simulation of the bond fluctua...

Scattering function of semiflexible polymer chains under good solvent conditions

Using the pruned-enriched Rosenbluth Monte Carlo algorithm, the scattering functions of semiflexible macromolecules in dilute solution under good solvent conditions are estimated both in $d=2$ and $d=3$ dimensions, considering also the effect of stretching forces. Using self-avoiding walks of up to $N = 25600$ steps on the square and simple cubic lattices, variable chain stiffness is modeled by introducing an energy penalty $\epsilon_b$ for chain bending; varying $q_b=\exp (- \epsilon_b/k_BT)$ from $q_b=1$ (completely flexible chains) to $q_b = 0.005$, the persistence length can be varied over two orders of magnitude. For unstretched semiflexible chains we test the applicability of the Krat…

Molecular Dynamics Simulations

A tutorial introduction to the technique of Molecular Dynamics (MD) is given, and some characteristic examples of applications are described. The purpose and scope of these simulations and the relation to other simulation methods is discussed, and the basic MD algorithms are described. The sampling of intensive variables (temperature T, pressure p) in runs carried out in the microcanonical (NVE) ensemble (N= particle number, V = volume, E = energy) is discussed, as well as the realization of other ensembles (e.g. the NVT ensemble). For a typical application example, molten SiO2, the estimation of various transport coefficients (self-diffusion constants, viscosity, thermal conductivity) is d…

Structure and dynamics of grafted polymer layers: A Monte Carlo simulation

The bond fluctuation model of polymer chains on lattices is used to study layers of polymers anchored with one end at a hard wall, assuming good solvent conditions and repulsive interactions between the monomers and the wall. Chain lengths from N=10 to N=80 and grafting densities σ from 0.025 to 0.20 are considered, both for the ‘‘quenched’’ case, where the anchor points are kept fixed at randomly chosen surface sites, and the ‘‘annealed’’ case, where lateral diffusion of the anchored ends at the wall is considered. Profiles of monomer density and free end density, chain linear dimensions parallel and perpendicular to the wall, as well as corresponding mean square displacements of inner and…

Shape of cross-over between mean-field and asymptotic critical behavior three-dimensional Ising lattice

Abstract Recent numerical studies of the susceptibility of the three-dimensional Ising model with various interaction ranges have been analyzed with a cross-over model based on renormalization-group matching theory. It is shown that the model yields an accurate description of the cross-over function for the susceptibility.

Static and dynamical properties of a supercooled liquid confined in a pore

We present the results of a Molecular Dynamics computer simulation of a binary Lennard-Jones liquid confined in a narrow pore. The surface of the pore has an amorphous structure similar to that of the confined liquid. We find that the static properties of the liquid are not affected by the confinement, while the dynamics changes dramatically. By investigating the time and temperature dependence of the intermediate scattering function we show that the dynamics of the particles close to the center of the tube is similar to the one in the bulk, whereas the characteristic relaxation time tau_q(T,rho) of the intermediate scattering function at wavevector q and distance rho from the axis of the p…

Phase separation versus wetting: A mean field theory for symmetrical polymer mixtures confined between selectively attractive walls

Partially compatible symmetrical (N A # N B = N) binary mixtures of linear flexible polymers (A, B) are considered in the presence of two equivalent walls a distance D apart, assuming that both walls preferentially adsorb the same component. Using a Flory-Huggins type mean field approach analogous to previous work studying wetting phenomena in the semi-infinite version of this model, where D → ∞, it is shown that a single phase transition occurs in this thin film geometry, namely a phase separation between an A-rich and a B-rich phase (both phases include the bulk of the film). The coexistence curve is shifted to smaller values of the inverse Flory-Huggins parameter x -1 with decreasing D, …

Chain linear dimensions in the surface-enriched layer of polymer mixtures

We calculate the mean-square end-to-end distances and mean-square gyration radii using the bond fluctuation model for a binary polymer blend in the presence of a wall by Monte Carlo simulation. In the bulk, the size of the minority, low-concentration polymer species is compressed compared to the majority one. In the vicinity of the wall, where the minority polymer concentration is enriched due to attraction from the wall, the dimensions of the two types of polymers are approximately equal and are essentially the same as in an athermal polymer melt. Thus, the geometric constraint is more important to the structure of the polymers than the polymer-polymer and polymer-wall interactions.

Nucleation phenomena in polymeric systems

Materials formed from long flexible macromolecules differ from their small-molecule analogs, because corresponding collective length scales are distinctly larger and many dynamical phenomena are very much slower; in addition, the variation of chain length N yields a control parameter that leaves intermolecular forces invariant, but allows a stringent test of theories. These concepts are exemplified in a discussion of nucleation barriers for symmetrical polymer (A, B)-mixtures (chain lengths NA = NB = N) near the critical temperature Tc, and for symmetrical block copolymers near the (fluctuation-induced) first order transition between the disordered melt and the lamellar mesophase. While in …

Second inflection point of water surface tension in the deeply supercooled regime revealed by entropy anomaly and surface structure using molecular dynamics simulations

The surface tension of supercooled water is of fundamental importance in physical chemistry and materials and atmospheric sciences. Controversy, however, exists over its temperature dependence in the supercooled regime, especially on the existence of the second inflection point (SIP). Here, we use molecular dynamics simulations of the SPC/E water model to study the surface tension of water (sigma(w)) as a function of temperature down to 198.15 K, and find a minimum point of surface excess entropy per unit area around approximate to 240-250 K. Additional simulations with the TIP4P/2005 water model also show consistent results. Hence, we predict an SIP of sigma(w) roughly in this region, at t…

Capillary Waves in a Colloid-Polymer Interface

The structure and the statistical fluctuations of interfaces between coexisting phases in the Asakura-Oosawa (AO) model for a colloid--polymer mixture are analyzed by extensive Monte Carlo simulations. We make use of a recently developed grand canonical cluster move with an additional constraint stabilizing the existence of two interfaces in the (rectangular) box that is simulated. Choosing very large systems, of size LxLxD with L=60 and D=120, measured in units of the colloid radius, the spectrum of capillary wave-type interfacial excitations is analyzed in detail. The local position of the interface is defined in terms of a (local) Gibbs surface concept. For small wavevectors capillary wa…

Monte Carlo Simulations in Polymer Science

Monte Carlo methods are useful for computing the statistical properties of both single macromolecules of various chemical architectures and systems containing many polymers (solutions, melts, blends, etc.). Starting with simple models (lattice models such as the self-avoiding walk or the bond fluctuation model, as well as coarse-grained or chemically realistic models in the continuum) various algorithms exist to generate conformations typical for thermal equilibrium, but dynamic Monte Carlo methods can also model diffusion and relaxation processes (as described by the Rouse and the reptation models for polymer melt dynamics). Limitations of the method are explained, and also the measures to…

Specific Heat of Amorphous Silica within the Harmonic Approximation

We investigate to what extent the specific heat of amorphous silica can be calculated within the harmonic approximation. For this we use molecular dynamics computer simulations to calculate, for a simple silica model (the BKS potential), the velocity autocorrelation function and hence an effective density of states g(ν). We find that the harmonic approximation is valid for temperatures below 300 K but starts to break down at higher temperatures. We show that, to obtain a reliable description of the low-frequency part of g(ν), i.e., where the boson peak is observed, it is essential to use large systems for the simulations and small cooling rates to quench the samples. We find that the calcul…

On the commensurate–incommensurate transition in adsorbed monolayers

Abstract A Monte Carlo simulation method is used to study the commensurate–incommensurate phase transition in monolayers and the formation of bilayer films on the (100) face of an fcc crystal. The phase diagram for the system which forms the registered (1×1) and high density incommensurate phases in the monolayer has been determined. It is shown that the registered phase undergoes the transition to a denser incommensurate solid phase when the film density increases. The mechanism of melting of the monolayer film is found to depend on the film density. In particular, the melting of dense incommensurate solid monolayer film is found to be accompanied by the transfer of adsorbed molecules into…

Computer Simulations of the Dynamics of Amorphous Silica

We present the results of a large scale computer simulation we performed to investigate the dynamical properties of supercooled silica. We show that parallel supercomputers such as the CRAY-T3E are very well suited to solve these type of problems. We find that at low temperatures the transport properties such as the diffusion constants and the viscosity agree well with the experimental data. At high temperatures this simulation predicts that in the transport quantities significant deviations from the Arrhenius law should be observed. Finally we show that such types of simulations can be used to investigate also complex dynamical quantities, such as the dynamical structure factor, and that t…

Phase separation of symmetrical polymer mixtures in thin-film geometry

Monte Carlo simulations of the bond fluctuation model of symmetrical polymer blends confined between two “neutral” repulsive walls are presented for chain lengthNA=NB=32 and a wide range of film thicknessD (fromD=8 toD=48 in units of the lattice spacing). The critical temperaturesTc(D) of unmixing are located by finite-size scaling methods, and it is shown that\(T_c (\infty ) - T_c (D) \propto D^{ - {1 \mathord{\left/ {\vphantom {1 {v_3 }}} \right. \kern-\nulldelimiterspace} {v_3 }}} \), wherev3≈0.63 is the correlation length exponent of the three-dimensional Ising model universality class. Contrary to this result, it is argued that the critical behavior of the films is ruled by two-dimensi…

Phase Separation and Nematic Order in Lyotropic Solutions: Two Types of Polymers with Different Stiffnesses in a Common Solvent

The interplay of the isotropic-nematic transition and phase separation in lyotropic solutions of two types of semiflexible macromolecules with pronounced difference in chain stiffness is studied by Density Functional Theory and Molecular Dynamics simulations. While the width of the isotropic-nematic two-phase coexistence region is narrow for solutions with a single type of semiflexible chain, the two-phase coexistence region widens for solutions containing two types of chains with rather disparate stiffness. In the nematic phase, both types of chains contribute to the nematic order, with intermediate values of the order parameter compared to the corresponding single component solutions. As …

Water adsorption on amorphous silica surfaces: A Car-Parrinello simulation study

A combination of classical molecular dynamics (MD) and ab initio Car-Parrinello molecular dynamics (CPMD) simulations is used to investigate the adsorption of water on a free amorphous silica surface. From the classical MD SiO_2 configurations with a free surface are generated which are then used as starting configurations for the CPMD.We study the reaction of a water molecule with a two-membered ring at the temperature T=300K. We show that the result of this reaction is the formation of two silanol groups on the surface. The activation energy of the reaction is estimated and it is shown that the reaction is exothermic.

Monte Carlo simulation of crystalline polyethylene

Abstract We consider here the problem of constructing an efficient algorithm for a classical Monte Carlo simulation of crystalline polyethylene with unconstrained bond lengths and angles. This macromolecular crystal presents a particular example of a system with many different energy scales, ranging from soft ones represented by nonbonded van der Waals interactions, to stiff ones, represented in particular by bond stretching. A proper sampling of all the energy scales poses a problem and it is shown that a standard Metropolis algorithm employing just local moves is not very efficient at low temperatures. As a solution it is proposed to employ also global moves consisting of displacements of…

Small-Angle Excess Scattering: Glassy Freezing or Local Orientational Ordering?

We present Monte Carlo simulations of a dense polymer melt which shows glass-transition-like slowing-down upon cooling, as well as a build up of nematic order. At small wave vectors q this model system shows excess scattering similar to that recently reported for light-scattering experiments on some polymeric and molecular glass-forming liquids. For our model system we can provide clear evidence that this excess scattering is due to the onset of short-range nematic order and not directly related to the glass transition.

Two-state protein-like folding of a homopolymer chain

Many small proteins fold via a first-order "all-or-none" transition directly from an expanded coil to a compact native state. Here we study an analogous direct freezing transition from an expanded coil to a compact crystallite for a simple flexible homopolymer. Wang-Landau sampling is used to construct the 1D density of states for square-well chains of length 128. Analysis within both the micro-canonical and canonical ensembles shows that, for a chain with sufficiently short-range interactions, the usual polymer collapse transition is preempted by a direct freezing or "folding" transition. A 2D free-energy landscape, built via subsequent multi-canonical sampling, reveals a dominant folding …

Statics and Dynamics of Bidisperse Polymer Melts:  A Monte Carlo Study of the Bond-Fluctuation Model

As a first step toward the computer simulation of polydisperse polymeric melts, a lattice model containing two types of chains with lengths N1 = 20 − x and N2 = 20 + 4x (0 ≤ x ≤ 10 ) is studied. This variation of x, together with the fixed composition of 80% of short and 20% of long chains, leads to a polydispersity of 1 ≤ Nw/Nn ≤ 2 (Nw, Nn:  weight-, number-average chain lengths). To represent dense melts, the bond-fluctuation model at a volume fraction, φ = 1/2, of occupied lattice sites is used. The simulation treats both the athermal case (chain connectivity and excluded volume interaction only) and a thermal case, where additionally a choice for the bond length and bond angle potential…

Polymer Films in the Normal-Liquid and Supercooled State: A Review of Recent Monte Carlo Simulation Results

This paper reviews recent Monte Carlo simulation studies of the glassy behavior in thin polymer films. The simulations employ a version of the bond-fluctuation lattice model, in which the glass transition is driven by the competition between a stiffening of the polymers and their dense packing in the melt. The melt is geometrically confined between two impenetrable walls separated by distances ranging from once to about fifteen times the bulk radius of gyration. The confinement influences static and dynamic properties of the films: Chains close to the wall preferentially orient parallel to it. This orientation tendency propagates through the film and leads to a layer structure at low temper…

ChemInform Abstract: Specific Heat of Amorphous Silica within the Harmonic Approximation.

We investigate to what extent the specific heat of amorphous silica can be calculated within the harmonic approximation. For this we use molecular dynamics computer simulations to calculate, for a simple silica model (the BKS potential), the velocity autocorrelation function and hence an effective density of states g(ν). We find that the harmonic approximation is valid for temperatures below 300 K but starts to break down at higher temperatures. We show that, to obtain a reliable description of the low-frequency part of g(ν), i.e., where the boson peak is observed, it is essential to use large systems for the simulations and small cooling rates to quench the samples. We find that the calcul…

Simulation of Dense Polymer Systems in Two and Three Dimensions

Dense polymer systems are modeled by self- and mutually avoiding walks on lattices. Both simple models where the step length is one lattice spacing and more complicated models where the step length is distinctly longer and may fluctuate (“bond fluctuation model”) are discussed, and it is shown that the computer simulation of such models gives useful insight to understand the thermodynamic phase behavior and the relaxational dynamics of dense polymer solutions and polymer melts. The huge demands in computing power needed for a successful simulation of such systems can be covered by parallel computers such as the multitransputer facility of the University of Mainz.

3D Conformations of Thick Synthetic Polymer Chains Observed by Cryogenic Electron Microscopy.

The backbone conformations of individual, unperturbed synthetic macromolecules have so far not been observed directly in spite of their fundamental importance to polymer physics. Here we report the dilute solution conformations of two types of linear dendronized polymers, obtained by cryogenic transmission electron stereography and tomography. The three-dimensional trajectories show that the wormlike chain model fails to adequately describe the scaling of these thick macromolecules already beyond a few nanometers in chain length, in spite of large apparent persistence lengths and long before a signature of self-avoidance appears. This insight is essential for understanding the limitations o…

Interfaces in polymer blends

We investigate the structure and thermodynamics of interfaces in dense polymer blends using Monte Carlo (MC) simulations and self-consistent field (SCF) calculations. For structurally symmetric blends we find quantitative agreement between the MC simulations and the SCF calculations for excess quantities of the interface (e.g., interfacial tension or enrichment of copolymers at the interface). However, a quantitative comparison between profiles across the interface in the MC simulations and the SCF calculations has to take due account of capillary waves. While the profiles in the SCF calculations correspond to intrinsic profiles of a perfectly flat interface the local interfacial position f…

Three-step decay of time correlations at polymer-solid interfaces

Two-step decay of relaxation functions, i.e., time scale separation between microscopic dynamics and structural relaxation, is the defining signature of the structural glass transition. We show that for glass-forming polymer melts at an attractive surface slow desorption kinetics introduces an additional time scale separation among the relaxational degrees of freedom leading to a three-step decay. The inherent length scale of this process is the radius of gyration in contrast to the segmental scale governing the glass transition. We show how the three-step decay can be observed in incoherent scattering experiments and discuss its relevance for the glass transition of confined polymers by an…

Phase transitions in a single polymer chain: A micro-canonical analysis of Wang–Landau simulations

Abstract We present simulation results for the phase behavior of a single chain for a flexible lattice polymer model using the Wang–Landau sampling idea. Using the micro-canonical density of states obtained with this method we will discuss the ability of an analysis in the micro-canonical ensemble to locate the coil-globule (continuous) and liquid–solid (first-order) transitions found for this problem using a canonical analysis.

Continuous Phase Transitions at Surfaces of CuAu Alloy Models — A Monte Carlo Study of Surface Induced Order and Disorder

The influence of surface on phase transitions has found significant attention in recent years, and a number of excellent reviews exists. [1, 2, 3] A variety of complex phenomena occur which are also related to the physics of adsorption and wetting. The scenario of wetting requires three distinct phases, for instance the vacuum, the bulk phase and a third phase intervening in between at equilibrium. In case of surface induced disorder (SID, a film of disordered layers at the surface “wets” the bulk phase as the temperature approaches the bulk transition temperature T c,b. The transition at the surface may be continuous (standard critical wetting phenomena), and, as theoretically investigated…

Smectic C and Nematic Phases in Strongly Adsorbed Layers of Semiflexible Polymers

Molecular dynamics simulations of semiflexible polymers in a good solvent reveal a dense adsorbed layer when the solution is exposed to an attractive planar wall. This layer exhibits both a nematic and a smectic phase (smA for short and smC for longer chains) with bond vectors aligned strictly parallel to the wall. The tilt angle of the smC phase increases strongly with the contour length of the polymers. The isotropic-nematic transition is a Kosterlitz-Thouless transition and also the nematic-smectic transition is continuous. Our finding demonstrates thus a two-dimensional realization of different liquid crystalline phases, ubiquitous in three dimensions, that occurs in a single monomolecu…

How do droplets on a surface depend on the system size?

Abstract We investigate the thermodynamics of inhomogeneous polymer melts in the framework of a coarse grained off-lattice model. Properties of the liquid–vapour interface and the packing of the melt in contact with an attractive wall are considered. We employ Monte Carlo simulations in the grand canonical ensemble to determine excess free energies, the wetting temperature and the pre-wetting line, as well as the pre-wetting critical point. Having determined the wetting properties and the phase diagram of the model polymer, we perform canonical Monte Carlo simulations of small droplets on a surface. This allows us to study the dependence of droplet size on the wetting properties. It is foun…

Surface-induced ordering and disordering in face-centered-cubic alloys: A Monte Carlo study

Using extensive Monte Carlo simulations we have studied phase transitions in a fcc model with antiferromagnetic nearest-neighbor couplings $J$ in the presence of different free surfaces which lead either to surface-induced order or to surface-induced disorder. Our model is a prototype for CuAu-type ordering alloys and shows a strong first-order bulk transition at a temperature $\frac{k{T}_{\mathrm{cb}}}{|J|}=1.738005(50)$. For free (100) surfaces, we find a continuous surface transition at a temperature ${T}_{\mathrm{cs}}g{T}_{\mathrm{cb}}$ exhibiting critical exponents of the two-dimensional Ising model. Surface-induced ordering occurs as the temperature approaches ${T}_{\mathrm{cb}}$ and …

Gas transport through polymer membranes and free volume percolation

We consider the influence of structural and dynamical properties of a polymer membrane on the gas transport through this matrix. The diffusant and the polymer only interact through repulsive interactions. In the case of a glassy polymer, when one can consider the matrix as frozen, the gas particle diffusion is determined by the free volume structure of the system. We show how the percolation properties of the free volume show up in a subdiffusive behavior of the diffusant. When one takes matrix mobility into account the ideal percolation transition vanishes but its trace can still be found in a subdiffusive regime in the gas particle mean square displacement. In the statically non-percolati…

Dynamics of a Spreading Nanodroplet: A Molecular Dynamic Simulation

The spreading of polymer nanodroplets upon a sudden change from partial to complete wetting on an ideally flat and structureless solid substrate has been studied by molecular dynamic simulations using a coarse-grained bead-spring model of flexible macromolecules. Tanner's law for the growth of the lateral droplet radius {R(f) t 0.1 } is found to hold as long as the droplet does not disintegrate into individually moving chains. The data for the contact angle θ following from Tanner's law correspond to a dependence on time {θ(t) t -0.3 }. Our analysis of the mean square displacements of the polymer centers of mass reveals several dynamic regimes during the process of spreading. PACS numbers: …

Confinement-induced screening of hydrodynamic interactions and spinodal decomposition: Multiscale simulations of colloid-polymer mixtures

Phase separation kinetics of a colloid-polymer mixture confined between two planar repulsive walls is studied by a multiscale simulation approach. Colloids and polymers are described by particles interacting with continuous potentials suitable for molecular-dynamics simulation, while hydrodynamic interactions mediated by solvent particles are accounted for by the multiparticle collision dynamics method. Varying the distance D between the walls and the character of the boundary conditions, the interplay of structure formation parallel and perpendicular to the walls is studied, and the effect of hydrodynamics on the growth of domain size ld(t) with time t is elucidated. Only for slip boundary…

From capillary condensation to interface localization transitions in colloid-polymer mixtures confined in thin-film geometry.

Monte Carlo simulations of the Asakura-Oosawa (AO) model for colloid-polymer mixtures confined between two parallel repulsive structureless walls are presented and analyzed in the light of current theories on capillary condensation and interface localization transitions. Choosing a polymer to colloid size ratio of q=0.8 and studying ultrathin films in the range of D=3 to D=10 colloid diameters thickness, grand canonical Monte Carlo methods are used; phase transitions are analyzed via finite size scaling, as in previous work on bulk systems and under confinement between identical types of walls. Unlike the latter work, inequivalent walls are used here: while the left wall has a hard-core rep…

Finite-size tests of hyperscaling.

The possible form of hyperscaling violations in finite-size scaling theory is discussed. The implications for recent tests in Monte Carlo simulations of the d = 3 Ising model are examined, and new results for the d = 5 Ising model are presented.

Simulation studies of gas-liquid transitions in two dimensions via a subsystem-block-density distribution analysis

The finite-size scaling analysis of the density distribution function of subsystems of a system studied at constant total density is studied by a comparative investigation of two models: (i) the nearest-neighbor lattice gas model on the square lattice, choosing a total lattice size of 64×64 sites. (ii) The two-dimensional off-lattice Lennard-Jones system (truncated at a distance of 2.5 σ, σ being the range parameter of the interaction) withN=4096 particles, applying the NVT ensemble. In both models, the density distribution functionPL(ρ) is obtained forL×L subsystems for a wide range of temperaturesT, subblock linear dimensionsL and average densities . Particular attention is paid to the qu…

Monte Carlo Simulations of Body Centered Cubic Alloys

We illustrate the use of Monte Carlo simulations in the study of order-disorder phenomena in metallic alloys by presenting detailed work on a fairly realistic lattice model for iron aluminum. The model has been constructed based on recent measurements of effective interaction parameters and includes a description of the magnetism of iron within a Heisenberg Hamiltonian. We show that it reproduces the bulk phase diagram in a qualitatively correct way. Then internal antiphase boundaries and free surfaces in the (100)-direction are studied. An interfacial roughening transition is predicted as well as critical broadening of the profiles as the bulk approaches a second order transition. Nonstoec…

Monte Carlo Study of a Lattice Gas Model with Nonadditive Lateral Interactions

Interactions between polymer brush-coated spherical nanoparticles: the good solvent case.

The interaction between two spherical polymer brushes is studied by molecular dynamics simulation varying both the radius of the spherical particles and their distance, as well as the grafting density and the chain length of the end-grafted flexible polymer chains. A coarse-grained bead-spring model is used to describe the macromolecules, and purely repulsive monomer-monomer interactions are taken throughout, restricting the study to the good solvent limit. Both the potential of mean force between the particles as a function of their distance is computed, for various choices of the parameters mentioned above, and the structural characteristics are discussed (density profiles, average end-to…

Model calculations of phase diagrams of magnetic alloys on the body-centered-cubic lattice.

We treat a model for a binary (AB) alloy, where species A is magnetic (Ising spin σi = ± 1) while species B is not, and repulsive interactions are assumed between first and second neighbors of the same kind, in addition to a nearest-neighbor ferromagnetic exchange interaction. Both the mean-field approximation, the cluster variation (CV) method in the tetrahedron approximation and the Monte Carlo (MC) method are applied; comparing the phase diagrams obtained by the various approximations their accuracy is tested. It is shown that the CV method is in rather close agreement with the MC method for the present problem.

Curvature dependence of surface free energy of liquid drops and bubbles: A simulation study.

We study the excess free energy due to phase coexistence of fluids by Monte Carlo simulations using successive umbrella sampling in finite LxLxL boxes with periodic boundary conditions. Both the vapor-liquid phase coexistence of a simple Lennard-Jones fluid and the coexistence between A-rich and B-rich phases of a symmetric binary (AB) Lennard-Jones mixture are studied, varying the density rho in the simple fluid or the relative concentration x_A of A in the binary mixture, respectively. The character of phase coexistence changes from a spherical droplet (or bubble) of the minority phase (near the coexistence curve) to a cylindrical droplet (or bubble) and finally (in the center of the misc…

The phase diagram of a single polymer chain: New insights from a new simulation method

We present simulation results for the phase behavior of a single chain for a flexible lattice polymer model using the Wang-Landau sampling idea. Applying this new algorithm to the problem of the homopolymer collapse allows us to investigate not only the high temperature coil–globule transition but also an ensuing crystallization at lower temperature. Performing a finite size scaling analysis on the two transitions, we show that they coincide for our model in the thermodynamic limit corresponding to a direct collapse of the random coil into the crystal without intermediate coil–globule transition. As a consequence, also the many chain phase diagram of this model can be predicted to consist o…

Semiflexible Polymers in Spherical Confinement: Bipolar Orientational Order Versus Tennis Ball States

Densely packed semiflexible polymers with contour length L confined in spheres with radius R of the same order as L cannot exhibit uniform nematic order. Depending on the chain stiffness (which we vary over a wide range), highly distorted structures form with topological defects on the sphere surface. These structures are completely different from previously observed ones of very long chains winding around the inner surface of spheres and from nematic droplets. At high densities, a thin shell of polymers close to the sphere surface exhibits a tennis ball texture due to the confinement-induced gradual bending of polymer bonds. In contrast, when the contour length of the chains is significant…

Polymer droplets on substrates with striped surface domains: molecular dynamics simulations of equilibrium structure and liquid bridge rupture

The structure of a polymer nanodroplet adsorbed on a flat lyophobic substrate chemically decorated with a lyophilic stripe of width 2RD is studied by molecular dynamics simulation of a coarse-grained bead–spring model of short macromolecules (containing N = 20 effective monomers). Varying the stripe width, the strength of the monomer–wall attraction and the temperature, the equilibrium morphology of the resulting droplets is studied and discussed in terms of current phenomenological theories. In the second part, the behaviour of a liquid bridge connecting two such lyophilic stripes a distance L apart is analysed. It is shown that for large enough L such free-standing films are unstable and …

Estimation of Nucleation Barriers from Simulations of Crystal Nuclei Surrounded by Fluid in Equilibrium

Nucleation rates for homogeneous nucleation are commonly estimated in terms of an Arrhenius law involving the nucleation barrier, written in terms of a competition of the contribution in surface free energy of the nucleus and the free energy gain proportional to the nucleus volume. For crystal nuclei this “classical nucleation theory” is hampered by the problem that the nucleus in general is non spherical, since the interfacial excess free energy depends on the orientation of the interface relative to the crystal axes. This problem can be avoided by analyzing the equilibrium of a crystal nucleus surrounded by fluid in a small simulation box in thermal equilibrium. Estimating the fluid press…

Phase Behavior and Microscopic Transport Processes in Binary Metallic Alloys: Computer Simulation Studies

In a binary liquid mixture, different kinds of phase transitions can occur that are associated with various mass transport phenomena in the liquid. First, there is the possibility that the liquid undergoes a liquid-liquid demixing transition [1]. Near the critical point of this transition, a slowing down of dynamic properties is observed which is characterized, e.g., by a vanishing interdiffusion coefficient at the critical point [2, 3]. Another possible phase transition is a first-order transition of the liquid into a crystalline structure. In this case, crystal nucleation and growth are limited by the diffusive transport in the liquid [1, 4]. In a binary liquid, crystal nucleation process…

Monte Carlo Simulations of Semi-Flexible Polymers

We present Monte Carlo simulations on the phase behavior of semiflexible macromolecules. For a single chain this question is of biophysical interest given the fact that long and stiff DNA chains are typically folded up into very tight compartments. So one can ask the question how the state diagram of a semiflexible chain differs from the coilglobule behavior of a flexible macromolecule. Another effect connected with rigidity of the chains is their tendency to aggregate and form nematically ordered structures. As a consequence one has two competing phase transitions: a gas-liquid and an isotropic-nematic transition potentially giving rise to a complicated phase diagram.

Surface-induced disorder in body-centered-cubic alloys

We present Monte Carlo simulations of surface induced disordering in a model of a binary alloy on a bcc lattice which undergoes a first order bulk transition from the ordered DO3 phase to the disordered A2 phase. The data are analyzed in terms of an effective interface Hamiltonian for a system with several order parameters in the framework of the linear renormalization approach due to Brezin, Halperin and Leibler. We show that the model provides a good description of the system in the vicinity of the interface. In particular, we recover the logarithmic divergence of the thickness of the disordered layer as the bulk transition is approached, we calculate the critical behavior of the maxima o…

Structure of Polymer Brushes in Cylindrical Tubes: A Molecular Dynamics Simulation

Molecular Dynamics simulations of a coarse-grained bead-spring model of flexible macromolecules tethered with one end to the surface of a cylindrical pore are presented. Chain length $N$ and grafting density $\sigma$ are varied over a wide range and the crossover from ``mushroom'' to ``brush'' behavior is studied for three pore diameters. The monomer density profile and the distribution of the free chain ends are computed and compared to the corresponding model of polymer brushes at flat substrates. It is found that there exists a regime of $N$ and $\sigma$ for large enough pore diameter where the brush height in the pore exceeds the brush height on the flat substrate, while for large enoug…

Glass physics: still not transparent

Glass is a commonplace word. One immediately thinks of windows or bottles and of properties like brittleness or transparency. However, for a glass blower another feature is more important: glass does not melt abruptly, as a crystal does, but gradually over a range of temperatures. This means that he or she can alter the temperature at which glass solidifies or becomes a liquid by changing the rate at which it is cooled or heated. This is in stark contrast to the behaviour observed when the crystalline form of a material is heated: it will always melt at the same temperature.

Surface critical behaviour near the uniaxial Lifshitz point of the axial next-nearest-neighbour Ising model

The semi-infinite axial next-nearest-neighbour Ising (ANNNI) model in the disordered phase is treated within a molecular-field approximation, and the singularities of various response functions characterizing the critical behaviour at the surface are obtained. In previous work (Binder K and Frisch H L 1999 Eur. Phys. J. B 10 71) the axis where a nearest-neighbour ferromagnetic (J 1 ) and next-nearest-neighbour antiferromagnetic (J 2 ) exchange compete was chosen perpendicular to the surface plane. In the present work we consider an orientation of this axis parallel to the surface, allowing also for different values of these exchange interactions (j 1 ,j 2 ) in the surface plane. We derive t…

The dielectric α -relaxation in polymer films: A comparison between experiments and atomistic simulations

The question of whether the glass transition temperature in thin polymer films depends on the film thickness or not has given rise to heated debate for almost two decades now. One of the most puzzling findings is the seemingly universal thickness independence of the dielectric α-relaxation observed for supported films. It is puzzling not only in view of the fact that other techniques or other geometries sometimes showed a significant shift of as a function of film thickness, but more so, because computer simulations for all types of polymer film models revealed changes in the structure and dynamics close to a hard surface or a free surface. Our results suggest to explain this apparent contr…

Confined Crystals on Substrates: Order and Fluctuations in Between One and Two Dimensions

The effect of lateral confinement on a crystal of point particles in d = 2 dimensions in a strip geometry is studied by Monte Carlo simulations and using phe- nomenological theoretical concepts. Physically, such systems confined in long strips of width D can be realized via colloidal particles at the air-water interface, or by adsorbed monolayers at suitably nanopatterned substrates, etc. As a generic model, we choose a repulsive interparticle potential decaying with the twelfth inverse power of distance. This system has been well studied in the bulk as a model for two- dimensional melting. The state of the system is found to depend very sensitively on the boundary conditions providing the …

Unexpectedly normal phase behavior of single homopolymer chains

Employing Monte Carlo simulations, we show that the topology of the phase diagram of a single flexible homopolymer chain changes in dependence on the range of an attractive square well interaction between the monomers. For a range of attraction larger than a critical value, the equilibrium phase diagram of the single polymer chain and the corresponding polymer solution phase diagram exhibit vapor (swollen coil, dilute solution), liquid (collapsed globule, dense solution), and solid phases. Otherwise, the liquid-vapor transition vanishes from the equilibrium phase diagram for both the single chain and the polymer solution. This change in topology of the phase diagram resembles the behavior k…

Finite-size scaling in Ising-like systems with quenched random fields: Evidence of hyperscaling violation

In systems belonging to the universality class of the random field Ising model, the standard hyperscaling relation between critical exponents does not hold, but is replaced by a modified hyperscaling relation. As a result, standard formulations of finite size scaling near critical points break down. In this work, the consequences of modified hyperscaling are analyzed in detail. The most striking outcome is that the free energy cost \Delta F of interface formation at the critical point is no longer a universal constant, but instead increases as a power law with system size, \Delta F proportional to $L^\theta$, with $\theta$ the violation of hyperscaling critical exponent, and L the linear ex…

Monte Carlo simulation of phase separation and clustering in the ABV model

As a model for a binary alloy undergoing an unmixing phase transition, we consider a square lattice where each site can be either taken by an A atom, a B atom, or a vacancy (V), and there exists a repulsive interaction between AB nearest neighbor pairs. Starting from a random initial configuration, unmixing proceeds via random jumps of A atoms or B atoms to nearest neighbor vacant sites. In the absence of any interaction, these jumps occur at jump ratesΓ A andΓ B, respectively. For a small concentration of vacancies (c v=0.04) the dynamics of the structure factorS(k,t) and its first two momentsk 1(t),k 2 2 (t) is studied during the early stages of phase separation, for several choices of co…

Heterogeneous nucleation of a droplet pinned at a chemically inhomogeneous substrate: A simulation study of the two-dimensional Ising case

Heterogeneous nucleation is studied by Monte Carlo simulations and phenomenological theory, using the two-dimensional lattice gas model with suitable boundary fields. A chemical inhomogeneity of length b at one boundary favors the liquid phase, while elsewhere the vapor is favored. Switching on the bulk field Hb favoring the liquid, nucleation and growth of the liquid phase starting from the region of the chemical inhomogeneity are analyzed. Three regimes occur: for small fields, Hb bcrit, the critical droplet radius is so large that a critical droplet having the contact angle θc required by Young's equation in the region of the chemical inhomogeneity does not yet "fit" there since the base…

Monte Carlo studies ofd= 2 Ising strips with long-range boundary fields

A two-dimensional Ising model with nearest-neighbour ferromagnetic exchange confined in a strip of width L between two parallel boundaries is studied by Monte Carlo simulations. `Free' boundaries are considered with unchanged exchange interactions at the boundary but long-range boundary fields of the form H (n ) = ? h [n -3 - (L - n + 1) -3 ], where n = 1, 2, ... ,L labels the rows across the strip. In the case of competing fields and L , the system exhibits a critical wetting transition of a similar type as in the well studied case of short-range boundary fields. At finite L , this wetting transition is replaced by a (rounded) interface localization-delocalization transition at Tc (h , L )…

Monte carlo studies of phase transitions in polymer blends and block copolymer melts

The unmixing transition of both symmetrical polymer blends AB (i.e. chain lengthsNA=NB=N) and asymmetrical ones (NB/NA=2,3) is studied by large-scale Monte Carlo simulations of the bond fluctuation model. Combination of semi-grand-canonical simulation techniques, «histogram reweighting» and finitesize scaling allows an accurate location of the coexistence curve in the critical region. The variation of the critical temperature with chain length (N) is studied and compared to theoretical predictions. For the symmetrical case, use of chain lengths up toN=512 allows a rough estimation of crossover scaling functions for the crossover from Ising to mean-field exponents. The order-disorder transit…

On the Ground State Structure of Adsorbed Monolayers: Can One Find them by Monte Carlo Simulation?

While the classical ground state structure of an atomic monolayer adsorbed at a noncorrugated perfectly flat substrate trivially is a triangular lattice, the spacing being the minimum of the interatomic potential, nontrivial structures occur on corrugated substrates. This problem is exemplified for the (100) face of a face-centered cubic crystal, varying both the density of the adsorbed monolayer and the strength of the potential due to the surface. Increasing the density beyond that of the commensurate c(2 x 2) structure, incommensurate patterns become stable with “heavy” walls (HW) oriented along the face diagonals [including the “crossing heavy walls” (CRHW) phase]. It is shown that slow…

Simulation of vapor-liquid coexistence in finite volumes: a method to compute the surface free energy of droplets.

When a fluid at a constant density $\ensuremath{\rho}$ in between the densities of coexisting vapor $({\ensuremath{\rho}}_{\text{v}})$ and liquid $({\ensuremath{\rho}}_{\ensuremath{\ell}})$ at temperatures below criticality is studied in a (cubic) box of finite linear dimension $L$, phase separation occurs in this finite volume, provided $L$ is large enough. For a range of densities, one can observe a liquid droplet (at density ${\ensuremath{\rho}}_{\ensuremath{\ell}}^{\ensuremath{'}}$ slightly exceeding ${\ensuremath{\rho}}_{\ensuremath{\ell}}$) coexisting in stable thermal equilibrium with surrounding vapor (with density ${\ensuremath{\rho}}_{\text{v}}^{\ensuremath{'}}g{\ensuremath{\rho}}…

Interface Localization-Delocalization in a Double Wedge: A New Universality Class with Strong Fluctuations and Anisotropic Scaling

Using Monte Carlo simulations and finite-size scaling methods we study ``wetting'' in Ising systems in a $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}{L}_{y}$ pore with quadratic cross section. Antisymmetric surface fields ${H}_{s}$ act on the free $L\ifmmode\times\else\texttimes\fi{}{L}_{y}$ surfaces of the opposing wedges, and periodic boundary conditions are applied along the $y$ direction. In the limit $L\ensuremath{\rightarrow}\ensuremath{\infty}$, ${L}_{y}/{L}^{3}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$, the system exhibits a new type of phase transition, which is the analog of the ``filling transition'' that occurs in a single wedge. It is charac…

Structure and dynamics of polymer brushes near the Θ point: A Monte Carlo simulation

Grafted polymer layers under variable solvent conditions are studied by Monte Carlo simulations using the bond fluctuation model. Structural information such as monomer density profiles, brush thickness, mean‐square displacement of monomers, and positions of the monomers along the chain are obtained for temperatures above, at, and below the Θ point. In particular, the scaling of the brush thickness is formulated and verified by the simulation data. At the Θ point, more extensive simulations are performed to investigate the structural and dynamical properties. While the brush thickness at the Θ point agrees very well with the scaling and self‐consistent field predictions, the latter deviate …

Numerical investigations of complex nano-systems

The nature of the melting transition for a system of hard disks with translational degrees of freedom in two spatial dimensions has been analysed by a combination of computer simulation methods and a finite size scaling technique. The behaviour of the system is consistent with the predictions of the Kosterlitz–Thouless–Halperin–Nelson–Young (KTHNY) theory. The structural and elastic properties of binary colloidal mixtures in two and three spatial dimensions are discussed as well as those of colloidal systems with quenched point impurities. Hard and soft disks in external periodic (light) fields show rich phase diagrams, including freezing and melting transitions when the density of the syst…

Conformational Changes of a Single Semiflexible Macromolecule Near an Adsorbing Surface: A Monte Carlo Simulation

The properties of a single semiflexible chain tethered to a planar surface with a long-ranged attractive potential are studied by means of Monte Carlo simulations. We employ the bond fluctuation lattice model and the Wang-Landau sampling technique. We present the diagram of states for semiflexible chains consisting of N = 64 and 128 monomer units as a function of temperature T and strength of the adsorption potential, epsilon(w), and also compare this with the diagram of states for flexible chains of these two lengths. The diagram of states consists of the regions of a coil, liquid globule, solid isotropic globule, adsorbed coil, and quasi-two-dimensional solid globule with nematic bond ord…

Critical properties and finite-size effects of the five-dimensional Ising model

Monte Carlo calculations of the thermodynamic properties (energy, specific heat, magnetization suceptibility, renormalized coupling) of the nearest-neighbour Ising ferromagnet on a five-dimensional hypercubic lattice are presented and analyzed. Lattices of linear dimensionsL=3, 4, 5, 6, 7 with periodic boundary conditions are studied, and a finite size scaling analysis is performed, further confirming the recent suggestion thatL does not scale with the correlation length ξ (the temperature variation of which near the critical temperatureT c is ξ∝|1-T/T c |−1/2), but rather with a “thermodynamic length”l (withl∝|1-T/T c |−2/d ,d=5 here). The susceptibility (extrapolated to the thermodynamic …

Interfaces between coexisting phases in polymer mixtures: What can we learn from Monte Carlo simulations?

Symmetric binary polymer mixtures are studied by Monte Carlo simulation of the bond fluctuation model, considering both interfaces between coexisting bulk phases and interfaces confined in thin films. It is found that the critical behavior of interfacial tension and width is compatible with that of the Ising model, as expected from the universality principle. In the strong segregation limit, only qualitative but not quantitative agreement with the self-consistent field (SCF) theory is found. It is argued that the SCF theory requires √ 6 X √D for short-range forces, in agreement with experiment.

Monte Carlo simulation in polymer physics: Some recent developments

The computer simulation of macromolecular materials has to deal with phenomena on length scales from 1A to 100A, as well as with time scales ranging over many orders of magnitude, and thus still presents a challenge. With suitably coarse-grained models which disregard detailed information on chemical structure nevertheless collective phenomena can be described, such as unmixing of polymer blends, mesophase ordering of block-copolymer melts, “blob formation” in semidilute solutions, etc. Simulations of such models provide a sensitive test of approximate theories and give valuable hints for experiments.

Competition between liquid-crystalline ordering and glassy freezing in melts of semiflexible polymers: A monte carlo simulation

We present results of a Monte Carlo simulation of dense melts of semiflexible polymers using the bond-fluctuation model. The chosen Hamiltonian increases the chain stiffness upon cooling which in turn leads to glass-transition like freezing of the polymer mobility. Employing an efficient simulation algorithm, which is able to equilibrate the simulated systems to lower temperature than the Rouse-type algorithm showing the glassy freezing, we are able to observe an isotropic-nematic phase transition. This transition lies above the glass transition temperature one would extrapolate from the observed freezing behavior.

Polymer translocation through a nanopore induced by adsorption: Monte Carlo simulation of a coarse-grained model

Dynamic Monte Carlo simulation of a bead-spring model of flexible macromolecules threading through a very narrow pore in a very thin rigid membrane are presented, assuming at the cis side of the membrane a purely repulsive monomer-wall interaction, while the trans side is attractive. Two choices of monomer-wall attraction epsilon are considered, one choice is slightly below and the other slightly above the "mushroom to pancake" adsorption threshold epsilon(c) for an infinitely long chain. Studying chain lengths N=32, 64, 128, and 256 and varying the number of monomers N(trans) (time t=0) that have already passed the pore when the simulation started, over a wide range, we find for epsiloneps…

Monte Carlo Methods: a powerful tool of statistical physics

Statistical mechanics of condensed matter systems (solids, fluids) tries to express macroscopic equilibrium properties of matter as averages computed from a Hamiltonian that expresses interactions of an atomistic many body system. While analytic methods for most problems involve crude and uncontrolled approximations, the Monte Carlo computer simulation method allows a numerically exact treatment of this problem, apart from “statistical errors” which can be made as small as desired, and the systematic problem that a system of finite size is treated rather than the thermodynamic limit. However, the simulations of phase transitions then elucidate how a symmetry breaking arises via breaking of …

Computer Simulation Studies of Chain Dynamics in Polymer Brushes

Center-of-mass and single monomer motion in grafted chains comprising a strongly stretched polymer brush in thermal equilibrium are studied by large scale molecular dynamics and Monte Carlo simulations of a coarse-grained model. Good solvent conditions are assumed. Our findings seriously question earlier theoretical predictions about the relaxation described by Rouse dynamics of brush coatings. Thus, the correlation functions of parallel and perpendicular components of the mean distance of the center-of-mass from the grafting site, the squared gyration radius and end-to-end distance, are found to deviate strongly from a simple exponential decay. While the relaxation times extracted from the…

Kinetics of domain growth in finite Ising strips

Abstract Monte Carlo simulations are presented for the kinetics of ordering of the two-dimensional nearest-neighbor Ising models in an L x M geometry with two free boundaries of length M ⪢ L . This geometry models a “terrace” of width L on regularly stepped surfaces, adatoms adsorbed on neighboring terraces being assumed to be noninteracting. Starting out with an initially random configuration of the atoms in the lattice gas at coverage θ = 1 2 in the square lattice, quenching experiments to temperatures in the range 0.85⩽ T / T c ⩽1 are considered, assuming a dynamics of the Glauber model type (no conservation laws being operative). At T c the ordering behavior can be described in terms of…

Interface localisation-delocalisation transition in a symmetric polymer blend: a finite-size scaling Monte Carlo study

Using extensive Monte Carlo simulations we study the phase diagram of a symmetric binary (AB) polymer blend confined into a thin film as a function of the film thickness D. The monomer-wall interactions are short ranged and antisymmetric, i.e, the left wall attracts the A-component of the mixture with the same strength as the right wall the B-component, and give rise to a first order wetting transition in a semi-infinite geometry. The phase diagram and the crossover between different critical behaviors is explored. For large film thicknesses we find a first order interface localisation/delocalisation transition and the phase diagram comprises two critical points, which are the finite film w…

Nanodroplets on a solid plane: wetting and spreading in a Monte Carlo simulation

Abstract The wetting behavior and spreading dynamics of small polymer melt droplets in the course of transition from partial to complete wetting conditions on a flat structureless solid substrate have been studied by dynamic Monte Carlo simulation. From the density profiles of the drops we determine the contact angles at varying strength of the van der Waals surface forces in the whole interval of partial wetting. The validity of Young's equation is then tested whereby the surface tension of the melt/vapor interface is derived independently from interfacial fluctuation analysis, and the surface free energy of the melt at the substrate—from the anisotropy of the local pressure at the wall. T…

The Ising model as a playground for the study of wetting and interface behavior

Computer simulations have played an important role in the elucidation of wetting and interface unbinding phenomena. In particular, use of the Ising-lattice-gas model in a film geometry and subject to diverse surface and bulk magnetic fields has permitted extensive Monte Carlo simulations to reveal new features of the phase diagrams associated with these phenomena and to provoke new theoretical studies. The status of our knowledge about the nature of wetting and interface-delocalization transitions which has resulted from these Ising model simulations will be summarized.

Molecular dynamics simulations of capillary rise experiments in nanotubes coated with polymer brushes.

The capillary filling of a nanotube coated with a polymer brush is studied by molecular dynamics simulations of a coarse-grained model, assuming various conditions for the fluid-wall and fluid-brush interactions. Whereas the fluid is modeled by simple point particles interacting with Lennard-Jones forces, the (end-grafted, fully flexible) polymers that form the brush coating are described by a standard bead-spring model. Our experiments reveal that capillary filling is observed even for walls that would not be wetted by the fluid, provided the polymer brush coating itself wets. Generally, it is found that the capillary rise always proceeds through a t1/2 law with time t while the underlying…

Scaling concepts for polymer brushes and their test with computer simulation

After a brief review of the scaling concepts for static and dynamic properties of polymer brushes in good solvents and Theta solvents, the Monte Carlo evidence is discussed. It is shown that under typical conditions the diameter of the last blob is of the order of 10-20% of the brush height, and therefore pronounced deviations from the self-consistent field predictions occur. In bad solvents, lateral microphase separation occurs leading to an irregular pattern of "dimples". Particularly interesting is the response of brushes to shear deformation, and the interaction between two interpenetrating brushes. Recent attempts to understand the resulting shear forces via molecular-dynamics simulati…

Critical behavior of active Brownian particles

We study active Brownian particles as a paradigm for a genuine nonequilibrium phase transition requiring steady driving. Access to the critical point in computer simulations is obstructed by the fact that the density is conserved. We propose a method based on arguments from finite-size scaling to determine critical points and successfully test it for the two-dimensional (2D) Ising model. Using this method allows us to accurately determine the critical point of two-dimensional active Brownian particles at ${\mathrm{Pe}}_{\text{cr}}=40(2), {\ensuremath{\phi}}_{\text{cr}}=0.597(3)$. Based on this estimate, we study the corresponding critical exponents $\ensuremath{\beta}, \ensuremath{\gamma}/\…

Monte Carlo tests of theoretical predictions for critical phenomena: still a problem?

Two Monte Carlo studies of critical behavior in ferromagnetic Ising models are described: the first one deals with the crossover from the Ising class to the mean field class, when the interaction range increases. The second study deals with the finite size behavior at dimensionalities above the marginal dimension where Landau theory applies. The numerical results are compared to pertinent theoretical predictions, and unsolved problems are briefly described.

Surface-directed spinodal decomposition: Phenomenology and numerical results.

We present a phenomenological theory for surface effects on spinodal decomposition in mixtures and related phenomena such as the dynamics of surface segregation. Numerical solutions of our equations show striking similarity to recent results from experiments on polymer mixtures with one component preferentially attracted to a wall.

Anomalous diffusion: Summary

Monte Carlo studies of finite-size effects at first-order transitions

Abstract First-order phase transitions are ubiquitous in nature but their presence is often uncertain because of the effects which finite size has on all transitions. In this article we consider a general treatment of size effects on lattice systems with discrete degrees of freedom and which undergo a first-order transition in the thermodynamic limit. We review recent work involving studies of the distribution functions of the magnetization and energy at a first-order transition in a finite sample of size N connected to a bath of size N′. Two cases: N′ = ∞ and N′ = finite are considered. In the former (canonical ensemble) case, the distributions are approximated by a superposition of Gaussi…

On the first-order collapse transition of a three-dimensional, flexible homopolymer chain model

We present simulation results for the phase behavior of flexible lattice polymer chains using the Wang-Landau sampling idea. These chains display a two-stage collapse through a coil-globule transition followed by a crystallization at lower temperatures. Performing a finite-size scaling analysis on the two transitions, we show that they coincide in the thermodynamic limit corresponding to a direct collapse of the random coil into the crystal without intermediate coil-globule transition.

Ordering of two-dimensional crystals confined in strips of finite width.

Monte Carlo simulations are used to study the effect of confinement on a crystal of point particles interacting with an inverse power law potential $\ensuremath{\propto}{r}^{\ensuremath{-}12}$ in $d=2$ dimensions. This system can describe colloidal particles at the air-water interface, a model system for experimental study of two-dimensional melting. It is shown that the state of the system (a strip of width $D$) depends very sensitively on the precise boundary conditions at the two ``walls'' providing the confinement. If one uses a corrugated boundary commensurate with the order of the bulk triangular crystalline structure, both orientational order and positional order is enhanced, and suc…

Elastic moduli, dislocation core energy and melting of hard disks in two dimensions

Elastic moduli and dislocation core energy of the triangular solid of hard disks of diameter $\sigma$ are obtained in the limit of vanishing dislocation- antidislocation pair density, from Monte Carlo simulations which incorporates a constraint, namely that all moves altering the local connectivity away from that of the ideal triangular lattice are rejected. In this limit, we show that the solid is stable against all other fluctuations at least upto densities as low as $\rho \sigma^2 = 0.88$. Our system does not show any phase transition so diverging correlation lengths leading to finite size effects and slow relaxations do not exist. The dislocation pair formation probability is estimated …

Spherical polymer brushes under good solvent conditions: molecular dynamics results compared to density functional theory.

A coarse grained model for flexible polymers end-grafted to repulsive spherical nanoparticles is studied for various chain lengths and grafting densities under good solvent conditions, by Molecular Dynamics methods and density functional theory. With increasing chain length the monomer density profile exhibits a crossover to the star polymer limit. The distribution of polymer ends and the linear dimensions of individual polymer chains are obtained, while the inhomogeneous stretching of the chains is characterized by the local persistence lengths. The results on the structure factor of both single chain and full spherical brush as well as the range of applicability of the different theoretic…

Structure and transport properties of amorphous aluminium silicates: computer simulation studies

The structure and transport properties of SiO2-Al2O3 melts containing 13 mol% and 47 mol% Al2O3 are investigated by means of large scale molecular dynamics computer simulations. The interactions between the atoms are modelled by a pair potential which is a modified version of the one proposed by Kramer et al. [J. Am. Chem. Soc. 64, 6435 (1991)]. Fully equilibrated melts in the temperature range 6000 K &gt;= T &gt; 2000 K are considered as well as glass configurations, that were obtained by a rapid quench from the lowest melt temperatures. Each system is simulated at two different densities in order to study the effect of pressure on structural and dynamic properties. We find that the Al ato…

The relaxation dynamics of a simple glass former confined in a pore

We use molecular dynamics computer simulations to investigate the relaxation dynamics of a binary Lennard-Jones liquid confined in a narrow pore. We find that the average dynamics is strongly influenced by the confinement in that time correlation functions are much more stretched than in the bulk. By investigating the dynamics of the particles as a function of their distance from the wall, we can show that this stretching is due to a strong dependence of the relaxation time on this distance, i.e. that the dynamics is spatially very heterogeneous. In particular we find that the typical relaxation time of the particles close to the wall is orders of magnitude larger than the one of particles …

Finite-size scaling above the upper critical dimension revisited: The case of the five-dimensional Ising model

Monte Carlo results for the moments of the magnetization distribution of the nearest-neighbor Ising ferromagnet in a L^d geometry, where L (4 \leq L \leq 22) is the linear dimension of a hypercubic lattice with periodic boundary conditions in d=5 dimensions, are analyzed in the critical region and compared to a recent theory of Chen and Dohm (CD) [X.S. Chen and V. Dohm, Int. J. Mod. Phys. C (1998)]. We show that this finite-size scaling theory (formulated in terms of two scaling variables) can account for the longstanding discrepancies between Monte Carlo results and the so-called ``lowest-mode'' theory, which uses a single scaling variable tL^{d/2} where t=T/T_c-1 is the temperature distan…

Estimation of the critical behavior in an active colloidal system with Vicsek-like interactions

We study numerically the critical behavior of a modified, active Asakura-Oosawa model for colloid-polymer mixtures. The colloids are modeled as self-propelled particles with Vicsek-like interactions. This system undergoes phase separation between a colloid-rich and a polymer-rich phase, whereby the phase diagram depends on the strength of the Vicsek-like interactions. Employing a subsystem-block-density distribution analysis, we determine the critical point and make an attempt to estimate the critical exponents. In contrast to the passive model, we find that the critical point is not located on the rectilinear diameter. A first estimate of the critical exponents $\beta$ and $\nu$ is consist…

Simulation of Transport in Partially Miscible Binary Fluids: Combination of Semigrandcanonical Monte Carlo and Molecular Dynamics Methods

Binary Fluids that exhibit a miscibility gap are ubiquitous in nature (glass melts, polymer solutions and blends, mixtures of molten metals, etc.) and exhibit a delicate interplay between static and dynamic properties. This is exemplified for a simple model system, the symmetrical AB Lennard-Jones mixture. It is shown how semigrandcanonical Monte Carlo methods, that include A→B (B→A) identity switches as Monte Carlo moves, can yield the phase diagram, the interfacial tension between coexisting phases, and various pair correlation functions and structure factors. In addition to the build-up of long-ranged concentration correlations near the critical point, unmixing is also accompanied by the…

Recent Developments in Monte Carlo Simulations of Lattice Models for Polymer Systems

A brief review is given of methodological advances made during the past decade with the Monte Carlo sampling of equilibrium properties of simple lattice models of polymer systems, and representative applications of these new algorithms are summarized. These algorithms include Wang−Landau (WL) sampling, the pruned-enriched Rosenbluth method (PERM), and topology violating dynamic Monte Carlo algorithms such as combinations of local moves, slithering snake moves, and “double bridging” moves for the bond fluctuation model. The applications mentioned concern phase-transition-like phenomena of single chains (collapse and crystallization in bad solvents; interplay of collapse and adsorption; escap…

Stability of thin polymer films: influence of solvents.

The interface and surface properties and the wetting behavior of polymer-solvent mixtures are investigated using Monte Carlo simulations and self-consistent field calculations. We carry out Monte Carlo simulations in the framework of a coarse-grained bead-spring model using short chains (oligomers) of N(P)=5 beads and a monomeric solvent, N(S)=1. The self-consistent field calculations are based on a simple phenomenological equation of state for compressible binary mixtures and we employ Gaussian chain model. The bulk behavior of the polymer-solvent mixture belongs to type III in the classification of van Konynenburg and Scott [Phil. Trans. R. Soc. London, Ser. A 298, 495 (1980)]. It is char…

Simulation of Order-Disorder Phenomena and Diffusion in Metallic Alloys

The application of the Monte Carlo method to lattice-statistics problems in metallurgy is reviewed. Examples are given for the prediction of phase diagrams from simple model assumptions for effective interatomic potentials and for the calculation of parameters describing long- and short-range order, ordering energy, etc., both for face-centered cubic (fcc) and body-centered cubic (bcc) lattices. Applications to real systems such as Cu—Au and Fe—Al alloys are discussed.

Polymeric alloys: Model materials for the understanding of the statistical thermodynamics of mixtures

Polymeric materials find industrial applications that are comparable to those of metals and ceramics.1 In addition to the great variability via the synthesis of various monomers and the choice of the degree of polymerization (N), alloying of polymers finds increasing attention for combining favorable materials properties.1,2 But polymeric (binary) alloys (A,B) of flexible polymers with chain lengths NA, NB are also most interesting for testing theoretical concepts: changing NA, NB one controls the entropy of mixing, keeping intermolecular forces invariant. Variation of these control parameters thus allows stringent tests of the theories on miscibility, unmixing etc. Furthermore, the large s…

Friction between Polymer Brushes in Good Solvent Conditions:  Steady-State Sliding versus Transient Behavior

Previous molecular dynamics simulations of friction between polymer brushes in relative sliding motion [Kreer, T.; Muser, M. H.; Binder, K.; Klein, J. Langmuir 2001, 17, 7804] are extended beyond steady-state conditions. We study two different protocols:  (i) stop and return and (ii) stop and go. In protocol (i), the relative, lateral motion between the two surfaces is stopped abruptly and reimposed opposite to the initial direction after the system could relax for some time. Protocol (ii) is similar except that the sliding direction is maintained. In the constant-velocity steady state, the average lateral extension lc of the polymers is found to be a power law of the sliding velocity v, na…

Phase behaviour of heteronuclear dimers in three-dimensional systems—a Monte Carlo study

Monte Carlo simulation in the grand canonical ensemble, the histogram reweighting technique and finite size scaling are used to study the phase behaviour of dimers in three-dimensional systems. A single molecule is composed of two segments A and B, and the bond between them cannot be broken. The phase diagrams have been estimated for a set of model systems. Different structures formed by heteronuclear dimers have been found. The results show a great variety of vapour–liquid coexistence behaviour depending on the strength of the interactions between segments.

Two-dimensional isotropic orientational glasses: a computer-simulation study

The analogue of the Edwards-Anderson model for isotropic vector spin glasses, but taking three-component quadrupoles instead of spins at each lattice site, is studied on the square lattice with extensive Monte Carlo calculations, using a nearest-neighbor symmetric gaussian interaction. It is shown that at low temperaturesT the model develops a short range order both with respect to glass like correlations and with respect to “ferromagnetic” correlations among the quadrupoles. The associated correlation lengths and susceptibilities diverge asT→0, and the critical exponents for this zero-temperature phase transition are estimated. Dynamic correlation functions are analyzed as well and it is s…

Blends of Semiflexible Polymers: Interplay of Nematic Order and Phase Separation

Mixtures of semiflexible polymers with a mismatch in either their persistence lengths or their contour lengths are studied by Density Functional Theory and Molecular Dynamics simulation. Considering lyotropic solutions under good solvent conditions, the mole fraction and pressure is systematically varied for several cases of bending stiffness κ (the normalized persistence length) and chain length N. For binary mixtures with different chain length (i.e., NA=16, NB=32 or 64) but the same stiffness, isotropic-nematic phase coexistence is studied. For mixtures with the same chain length (N=32) and large stiffness disparity (κB/κA=4.9 to 8), both isotropic-nematic and nematic-nematic unmixing oc…

Breakdown of the Kratky-Porod Wormlike Chain Model for Semiflexible Polymers in Two Dimensions

By large-scale Monte Carlo simulations of semiflexible polymers in $d=2$ dimensions the applicability of the Kratky-Porod model is tested. This model is widely used as "standard model" for describing conformations and force versus extension curves of stiff polymers. It is shown that semiflexible polymers in $d=2$ show a crossover from hard rods to self-avoiding walks, the intermediate Gaussian regime (implied by the Kratky-Porod model) is completely absent. Hence the latter can also describe force versus extension curves only if the contour length is only a few times larger than the persistence length. Consequences for experiments on biopolymers at interfaces are briefly discussed.

The phase coexistence method to obtain surface free energies and nucleation barriers: a brief review

A recently developed method where one analyses the finite size effects associated with liquid–solid phase equilibria including vapour–crystal coexistence is briefly reviewed. It is shown that the e...

Finite-size scaling for a first-order transition where a continuous symmetry is broken: The spin-flop transition in the three-dimensional XXZ Heisenberg antiferromagnet

Finite-size scaling for a first-order phase transition where a continuous symmetry is broken is developed using an approximation of Gaussian probability distributions with a phenomenological ``degeneracy'' factor included. Predictions are compared with data from Monte Carlo simulations of the three-dimensional, $XXZ$ Heisenberg antiferromagnet in a field in order to study the finite-size behavior on a $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ simple cubic lattice for the first-order ``spin-flop'' transition between the Ising-like antiferromagnetic state and the canted, $XY$-like state. Our theory predicts that for large linear dimension $L$ the field dependen…

A finite size scaling study of the five-dimensional Ising model

For systems above the marginal dimension d*, where mean field theory starts to become valid, such as Ising models in d = 5 for which d* = 4, hyperscaling is invalid and hence it was suggested that finite size scaling is not ruled by the correlation length ξ (∝ |t| −1/2 in Landau theory, t being the distance from the critical point) but by a “thermodynamic length” l (∝ |t| −2/d). Early simulation work by Binder et al. using nearest neighbor hypercubic L5 lattices with L ⩽ 7 yielded some evidence for this prediction, but the renormalized coupling constant gL = −3 + 〈M4〉/〈M2〉2 at Tc was gL ≈ −1.0 instead of the prediction of Brezin and Zinn-Justin, gL(Tc) = −3 + Γ4(1/4)/(8 π2) ≈ −0.812. In the…

Capillary condensation in cylindrical pores: Monte Carlo study of the interplay of surface and finite size effects.

When a fluid that undergoes a vapor to liquid transition in the bulk is confined to a long cylindrical pore, the phase transition is shifted (mostly due to surface effects at the walls of the pore) and rounded (due to finite size effects). The nature of the phase coexistence at the transition depends on the length of the pore: For very long pores the system is axially homogeneous at low temperatures. At the chemical potential where the transition takes place fluctuations occur between vapor-like and liquid-like states of the cylinder as a whole. At somewhat higher temperatures (but still far below bulk criticality) the system at phase coexistence is in an axially inhomogeneous multi-domain …

Monte Carlo simulations of the solid-liquid transition in hard spheres and colloid-polymer mixtures

Monte Carlo simulations at constant pressure are performed to study coexistence and interfacial properties of the liquid-solid transition in hard spheres and in colloid-polymer mixtures. The latter system is described as a one-component Asakura-Oosawa (AO) model where the polymer's degrees of freedom are incorporated via an attractive part in the effective potential for the colloid-colloid interactions. For the considered AO model, the polymer reservoir packing fraction is eta_p^r=0.1 and the colloid-polymer size ratio is q=sigma_p/\sigma=0.15 (with sigma_p and sigma the diameter of polymers and colloids, respectively). Inhomogeneous solid-liquid systems are prepared by placing the solid fc…

Capillary Rise in Nanopores: Molecular Dynamics Evidence for the Lucas-Washburn Equation

When a capillary is inserted into a liquid, the liquid will rapidly flow into it. This phenomenon, well studied and understood on the macroscale, is investigated by Molecular Dynamics simulations for coarse-grained models of nanotubes. Both a simple Lennard-Jones fluid and a model for a polymer melt are considered. In both cases after a transient period (of a few nanoseconds) the meniscus rises according to a $\sqrt{\textrm{time}}$-law. For the polymer melt, however, we find that the capillary flow exhibits a slip length $\delta$, comparable in size with the nanotube radius $R$. We show that a consistent description of the imbibition process in nanotubes is only possible upon modification o…

Multiscale Computer Simulations in Physics, Chemistry, and Biology: The Example Of Silica

We show to what extent molecular dynamics simulations (MD) can explore struc-tural and dynamic properties of atomic systems whereby the system under consideration is amorphous silica (SiO2). Two studies are presented: (i) a large scale simulation of the dynam-ics of a SiO2 melt and (ii) the investigation of free silica surfaces where a mixture of a classical MD and a Car-Parrinello molecular dynamics is used.

Universal critical behavior of curvature-dependent interfacial tension.

From the analysis of Monte Carlo simulations of a binary Lennard-Jones mixture in the coexistence region, we provide evidence that the curvature dependence of the interfacial tension can be described by a simple theoretical function σ(R)ξ(2)=C(1)/[1+C(2)(ξ/R)(2)], where ξ is the correlation length and R is the droplet radius. The universal constants C(1) and C(2) are estimated. In the model, a Tolman length is strictly absent, but, since its critical behavior is believed to be much weaker than ξ, we argue that it only provides a correction to scaling and does not affect the leading critical behavior, which should be described by the above function for any system in the Ising universality cl…

Surface-directed phase separation with off-critical composition: Analytical and numerical results

We study the interplay of wetting and phase separation in an unstable binary mixture $(\mathrm{AB})$ with off-critical composition, placed in contact with a surface which prefers the component $A.$ We consider surface potentials $V(z)\ensuremath{\sim}{z}^{\ensuremath{-}n},$ where z is the distance from the surface, and present analytical arguments and detailed numerical results to elucidate wetting-layer kinetics for arbitrary mixture compositions. If the preferred component is the minority phase, the wetting-layer thickness exhibits a potential-specific behavior at early times $\ensuremath{\tau},$ ${R}_{1}\ensuremath{\sim}{\ensuremath{\tau}}^{1/(n+2)},$ before crossing over to the universa…

Computer simulations of SiO2 and GeO2

Classical Molecular Dynamics (MD) simulations are used to study structural and dynamic properties of amorphous germania (GeO2) in comparison to those of silica (SiO2). The total structure factor, as obtained from these simulations, is in very good agreement with that of neutron scattering experiments, both for germania and silica. The tetrahedral network structure in silica and germania leads to a prepeak in the structure factor that appears at slightly smaller wavenumbers in GeO2 than in SiO2. At high temperatures the diffusion constants are very similar in both systems whereas at low temperatures diffusion is significantly faster in germania than in silica. We also outline the strategy fo…

Evidence against a glass transition in the 10-state short range Potts glass

We present the results of Monte Carlo simulations of two different 10-state Potts glasses with random nearest neighbor interactions on a simple cubic lattice. In the first model the interactions come from a \pm J distribution and in the second model from a Gaussian one, and in both cases the first two moments of the distribution are chosen to be equal to J_0=-1 and Delta J=1. At low temperatures the spin autocorrelation function for the \pm J model relaxes in several steps whereas the one for the Gaussian model shows only one. In both systems the relaxation time increases like an Arrhenius law. Unlike the infinite range model, there are only very weak finite size effects and there is no evi…

Phase Transitions and Relaxation Processes in Macromolecular Systems: The Case of Bottle-brush Polymers

As an example for the interplay of structure, dynamics, and phase behavior of macromolecular systems, this article focuses on the problem of bottle-brush polymers with either rigid or flexible backbones. On a polymer with chain length $N_b$, side-chains with chain length $N$ are endgrafted with grafting density $\sigma$. Due to the multitude of characteristic length scales and the size of these polymers (typically these cylindrical macromolecules contain of the order of 10000 effective monomeric units) understanding of the structure is a challenge for experiment. But due to excessively large relaxation times (particularly under poor solvent conditions) such macromolecules also are a challen…

Brownian dynamics simulation of grafted polymer brushes

We present results of computer simulations by the method of Brownian dynamics of polymeric brushes attached to impenetrable planes. For testing both model and method we have used one polymer brush attached to a repulsive plane and compare some results with Monte Carlo results of Lai and Binder on the bond fluctuation model. We have also studied two polymeric brushes attached to two parallel planes at different distances between planes, and investigate the interplay between the interpenetration of the brushes and the configurational properties of the grafted chains.

Phase Transitions of Polymer Blends and Block Copolymer Melts in Thin Films

Static and dynamic glass transitions in the 10-state Potts glass: What can Monte Carlo simulations contribute?

The p-state Potts glass with infinite range Gaussian interactions can be solved exactly in the thermodynamic limit and exhibits an unconventional phase behavior if p >4: A dynamical transition from ergodic to non-ergodic behavior at a temperature T D is followed by a first order transition at T 0 < T D, where a glass order parameter appears discontinuously, although the latent heat is zero. If one assumes that a similar scenario occurs for the structural glass transition as well (though with the singular behavior at T D rounded off), the p-state Potts glass should be a good test case to develop methods to deal with finite size effects for the static as well as the dynamic transition, and to…

Free-energy barriers for crystal nucleation from fluid phases.

Monte Carlo simulations of crystal nuclei coexisting with the fluid phase in thermal equilibrium in finite volumes are presented and analyzed, for fluid densities from dense melts to the vapor. Generalizing the lever-rule for two-phase coexistence in the canonical ensemble to finite volume, "measurements" of the nucleus volume together with the pressure and chemical potential of the surrounding fluid allows to extract the surface free energy of the nucleus. Neither the knowledge of the (in general non-spherical) nucleus shape nor of the angle-dependent interface tension is required for this task. The feasibility of the approach is demonstrated for a variant of the Asakura-Oosawa model for c…

Cooperative motion and growing length scales in supercooled confined liquids

Using molecular dynamics simulations we investigate the relaxation dynamics of a supercooled liquid close to a rough as well as close to a smooth wall. For the former situation the relaxation times increase strongly with decreasing distance from the wall whereas in the second case they strongly decrease. We use this dependence to extract various dynamical length scales and show that they grow with decreasing temperature. By calculating the frequency dependent average susceptibility of such confined systems we show that the experimental interpretation of such data is very difficult.

Simulation of Phase Transitions of Single Polymer Chains: Recent Advances

The behaviour of a flexible polymer chain in solvents of variable quality in dilute solution is discussed both in the bulk and in the presence of an adsorbing wall. Monte Carlo simulations of coarse-grained bead-spring models and of the bond fluctuation model are presented and interpreted in terms of phenomenological theories and scaling concepts. Particular attention is paid to the behaviour of the polymer chain when the temperature of the polymer solution gets lower than the Theta temperature. It is argued that the adsorption transition line at the Theta temperature splits into lines of wetting and drying transitions of polymer globules attached to the wall. In addition, it is shown that …

Quantum effects on the herringbone ordering ofN2on graphite

The effects of quantum fluctuations on the ``2-in'' herringbone ordering in a realistic model of 900 ${\mathrm{N}}_{2}$ molecules adsorbed in the (\ensuremath{\surd}3 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}3 )R30\ifmmode^\circ\else\textdegree\fi{} structure on graphite are studied via path-integral Monte Carlo (PIMC) simulations. Quasiclassical and quasiharmonic calculations agree for high and low temperatures, respectively, but only PIMC gives satisfactory results over the entire temperature range. We can quantify the lowering of the transition temperature and the depression of the ground state order to 10% as compared to classical modeling.

Molecular-dynamics simulation of a glassy polymer melt: Incoherent scattering function

We report results of molecular-dynamics simulations for a glassy polymer melt consisting of short, linear bead-spring chains. It was shown in previous work that this onset of the glassy slowing down is compatible with the predictions of the mode coupling theory. The physical process of `caging' of a monomer by its spatial neighbors leads to a distinct two step behavior in the particle mean square displacements. In this work we analyze the effects of this caging process on the Rouse description of the melt's dynamics. We show that the Rouse theory is applicable for length and time scales above the typical scales for the caging process. Futhermore, the monomer displacement is compared with si…

Structure of bottle brush polymers on surfaces: weak versus strong adsorption.

Large-scale Monte Carlo simulations are presented for a coarse-grained model of cylindrical molecular brushes adsorbed on a flat structureless substrate, varying both the chain length N of the side chains and the backbone chain length N(b). For the case of good solvent conditions, both the cases of weak adsorption (only 10 to 15% of the monomers being bound to the surface) and strong adsorption (~40% of the monomers being bound to the surface, forcing the bottle brush into an almost 2D conformation) are studied. We focus on the scaling of the total linear dimensions of the cylindrical brush with both chain lengths N and N(b), demonstrating a crossover from rod-like behavior (for not very la…

Nonmonotonical crossover of the effective susceptibility exponent

We have numerically determined the behavior of the magnetic susceptibility upon approach of the critical point in two-dimensional spin systems with an interaction range that was varied over nearly two orders of magnitude. The full crossover from classical to Ising-like critical behavior, spanning several decades in the reduced temperature, could be observed. Our results convincingly show that the effective susceptibility exponent gamma_eff changes nonmonotonically from its classical to its Ising value when approaching the critical point in the ordered phase. In the disordered phase the behavior is monotonic. Furthermore the hypothesis that the crossover function is universal is supported.

The high-temperature dynamics of a mean-field Potts glass

Abstract We use Monte Carlo simulations to investigate the dynamic properties of the ten-state infinite-range Potts glass. By analyzing the spin autocorrelation function for system sizes up to N = 2560, we show that strong finite size effects are present around the predicted dynamic transition temperature. The autocorrelation function shows strong self-averaging at high temperatures, whereas close to the dynamic transition shows lack of self-averaging.

Computer Simulation of Molten and Glassy Silica and its Mixtures with Sodium Oxide and Aluminium Oxide

Dynamics of wetting transitions: A time-dependent Ginzburg-Landau treatment

The dynamic behavior at wetting transitions is studied for systems with short-range forces and nonconserved order parameter. From a continuum limit of a purely relaxational lattice model in mean-field approximation, a time-dependent Ginzburg-Landau equation with a time-dependent boundary condition at the surface is derived in the long wavelength approximation. The dynamics of relaxation close to stable and metastable states is treated in linear response. A divergence of the relaxation time occurs both for critical wetting and along the surface spinodal lines (in the case of first-order wetting), although the static surface layer susceptibilities χ1, χ11 stay finite at the surface spinodal i…

Capillary Rise in Nanotubes Coated with Polymer Brushes

The spontaneous rise of a fluid in a brush-coated nanocapillary is studied by molecular dynamics simulation of a coarse-grained model. The cases of changing wettability of both the capillary walls and the brush were examined. We also investigated the impact of polymer chain length on the transport of fluid along the nanotube. We found that capillary filling takes place in both lyophilic and lyophobic tubes, provided that the polymer brush coating is wetted by the fluid. In all the cases studied, capillary rise proceeds by a time-square law, but the mechanisms behind them (Lucas-Washburn or diffusive propagation) differ, depending on the chain length N. For a wettable wall, the speed of flui…

Anomalous diffusion of polymers in supercooled melts near the glass transition

Two coarse-grained models for polymer chains in dense melts near the glass transition are investigated: the bond fluctuation lattice model, where long bonds are energetically favored, is studied by dynamic Monte Carlo simulation, and an off-lattice bead-spring model with Lennard-Jones forces between the beads is treated by Molecular Dynamics. We compare the time-dependence of the mean square displacements of both models, and show that they become very similar on mesoscopic scales (i.e., displacements larger than a bond length). The slowing down of motions near the glass transition is discussed in terms of the mode coupling theory and other concepts.

Dynamics of Polymer Melts above the Glass Transition:  Monte Carlo Studies of the Bond Fluctuation Model

The bond fluctuation model on the simple cubic lattice with a bond-length dependent potential energy favoring long bonds exhibits a glassy freezing in as the temperature is lowered, many properties being qualitatively similar to experiment. The present paper studies the dynamical properties of the model (as they result from the random hopping algorithm), using configurations of undercooled polymer melts that have been carefully equilibrated by the slithering snake algorithm. In this way quantitatively reliable data can be obtained for distinctly lower temperatures than in the previous work on the dynamics of this model that used the random hopping algorithm for equilibration as well. If var…

Temperature dependence of single chain properties in a binary polymer blend

The temperature dependence of the correlation length of composition fluctuations and single chain statics and dynamics is studied in a symmetric, binary polymer blend. Our Monte Carlo simulation reveals a pronounced shrinking of the chain in the minority phase at low temperatures. However, only a weak temperature dependence of the single chain properties is found above criticality. Especially there is only a weak coupling between the correlation length of composition fluctuations and the relaxation of the internal chain structure. The coherent dynamic structure factor does not show any signs of a spatial restricted motion for our chain length N = 16, which is far below the entanglement leng…

Competition between submonolayer ordering and multilayer adsorption: Studies of simple lattice gas models

Abstract We model condensation of adatoms at a substrate surface by a semi-infinite simple cubic lattice gas system. While in the bulk there is just a nearest-neighbour attractive interaction, in the first layer adjacent to the surface we allow for a periodic potential due to the substrate with a period of two lattice spacings, or for a next-nearest-neighbour repulsive interaction mediated by the substrate. Hence order-disorder phenomena may occur in the first layer, while only gas-liquid condensation transitions can occur in layers further away from the substrate surface. The ground-state phase diagrams of this model are obtained exactly, while the behaviour at nonzero temperatures is obta…

Macromol. Theory Simul. 7/2007

Calculation of phase diagrams for models of metallic alloys

We briefly review a longstanding problem of metallurgy and statistical physics, namely, the prediction of phase diagrams of binary alloys from simple model assumptions on the atomic interactions, such as Ising-type models. Various methods of statistical mechanics which have been applied to this problem are introduced and compared to each other, such as the cluster-variation method and Monte-Carlo simulation. The merits as well as the limitations of these methods are discussed, emphasizing examples of fcc and bcc lattices which are potentially relevant for the problem of short-range order and long-range order in metallic alloys such as Cu−Au, Ni−Cr, and Fe−Al.A brief comparison with correspo…

Computer simulations of a Lennard-Jones model for Ar1—x(N2)x: A prototype system for quadrupolar glasses

Abstract Recent theoretical studies of orientational ordering in pure and diluted nitrogen crystals are summarized. While pure N2 has a first order phase transition from a plastic crystal to a phase with long-range orientational order, dilution with argon atoms leads to a quadrupolar glass phase. Monte Carlo simulations are used to study these phases, considering also the behavior of isolated N2 impurities in Ar crystals. It is shown that a simple model that neglects electrostatic interactions and takes only Lennard-Jones interactions into account can describe already many properties in qualitative agreement with experiment. Even the slow dynamics of the quadrupole moments can be modeled by…

Hard sphere fluids at a soft repulsive wall: A comparative study using Monte Carlo and density functional methods

Hard-sphere fluids confined between parallel plates at a distance D apart are studied for a wide range of packing fractions including also the onset of crystallization, applying Monte Carlo simulation techniques and density functional theory. The walls repel the hard spheres (of diameter σ) with a Weeks-Chandler-Andersen (WCA) potential V(WCA)(z) = 4ε[(σ(w)/z)(12) - (σ(w)/z)(6) + 1/4], with range σ(w) = σ/2. We vary the strength ε over a wide range and the case of simple hard walls is also treated for comparison. By the variation of ε one can change both the surface excess packing fraction and the wall-fluid (γ(wf)) and wall-crystal (γ(wc)) surface free energies. Several different methods t…

How Well Can Coarse-Grained Models of Real Polymers Describe Their Structure? The Case of Polybutadiene

Coarse-graining of chemical structure of macromolecules in the melt is investigated using extensive molecular dynamics simulation data which are based on a united atom force-field model of polybutadiene. Systematically increasing the number, n, of the united atoms approximated by an effective coarse-grained monomer, we study the influence of degree of coarse-graining on the structure functions such as the segment-segment intermolecular and intramolecular correlation functions. These results are compared to Monte Carlo simulations of the corresponding coarse-grained bead-spring model and Chen-Kreglewski potential for chain molecules. In contrast to the atomistic chemically realistic model of…

Capillary Nematization of Semiflexible Polymers

emiflexible polymers under good solvent conditions confined by two planar parallel repulsive walls are investigated for a wide range of monomer concentrations and distances between the walls, for a case where persistence length and contour length of the macromolecules are almost equal. Chain conformations and local nematic ordering near the walls are studied by both molecular dynamics methods and density functional theory, putting it in perspective with the recent work where the isotropic phase of semiflexible polymer solutions in the vicinity of a single repulsive wall in semi-infinite geometry is considered. Profiles of the total density of monomers as well as densities of end- and middle…

Understanding the glass transition and the amorphous state of matter: can computer simulation solve the challenge?

The glass transition of supercooled fluids is one of the big puzzles of condensed matter physics, because there occurs a dramatic slowing down (the viscosity η can increase from about η = 1 Poise at the melting transition to η 10 13 Poise at the glass transition temperature T g ), but one hardly sees any accompanying change in the static structure. Theoretical concepts are very controversial - e.g., the Gibbs-di Marzio theory attributes glassy freezing to an underlying entropy catastrophe (the entropy of the supercooled fluid would fall below the crystal entropy at the Kauzmann temperature T 0 T g . Computer simulations offer the advantage that atomistically detailed information on structur…

Equation of State for Macromolecules of Variable Flexibility in Good Solvents: A Comparison of Techniques for Monte Carlo Simulations of Lattice Models

The osmotic equation of state for the athermal bond fluctuation model on the simple cubic lattice is obtained from extensive Monte Carlo simulations. For short macromolecules (chain length N=20) we study the influence of various choices for the chain stiffness on the equation of state. Three techniques are applied and compared in order to critically assess their efficiency and accuracy: the repulsive wall method, the thermodynamic integration method (which rests on the feasibility of simulations in the grand canonical ensemble), and the recently advocated sedimentation equilibrium method, which records the density profile in an external (e.g. gravitation-like) field and infers, via a local …

Guide to Practical Work with the Monte Carlo Method

The guide is structured such that we proceed from the “easy” simulation methods and algorithms to the more sophisticated. For each method the algorithms are presented by the technique of stepwise refinement. We first present the idea and the basic outline. From then on we proceed by breaking up the larger logical and algorithmic structures into smaller ones, until we have reached the level of single basic statements. Sometimes we may elect not to go to such a depth and the reader is asked to fill in the gaps.

Quantum Monte Carlo methods

Introduction In most of the discussion presented so far in this book, the quantum character of atoms and electrons has been ignored. The Ising spin models have been an exception, but since the Ising Hamiltonian is diagonal (in the absence of a transverse magnetic field), all energy eigenvalues are known and the Monte Carlo sampling can be carried out just as in the case of classical statistical mechanics. Furthermore, the physical properties are in accord with the third law of thermodynamics for Ising-type Hamiltonians (e.g. entropy S and specific heat vanish for temperature T → 0, etc.) in contrast to the other truly classical models dealt with in previous chapters (e.g. classical Heisenbe…

Monte Carlo Calculations on Phase Transitions in Adsorbed Layers

How to define variation of physical properties normal to an undulating one-dimensional object.

One-dimensional flexible objects are abundant in physics, from polymers to vortex lines to defect lines and many more. These objects structure their environment and it is natural to assume that the influence these objects exert on their environment depends on the distance from the line-object. But how should this be defined? We argue here that there is an intrinsic length scale along the undulating line that is a measure of its "stiffness" (i.e., orientational persistence), which yields a natural way of defining the variation of physical properties normal to the undulating line. We exemplify how this normal variation can be determined from a computer simulation for the case of a so-called b…

Spinodal decomposition in thin films: Molecular-dynamics simulations of a binary Lennard-Jones fluid mixture

We use molecular dynamics (MD) to simulate an unstable homogeneous mixture of binary fluids (AB), confined in a slit pore of width $D$. The pore walls are assumed to be flat and structureless, and attract one component of the mixture (A) with the same strength. The pair-wise interactions between the particles is modeled by the Lennard-Jones potential, with symmetric parameters that lead to a miscibility gap in the bulk. In the thin-film geometry, an interesting interplay occurs between surface enrichment and phase separation. We study the evolution of a mixture with equal amounts of A and B, which is rendered unstable by a temperature quench. We find that A-rich surface enrichment layers fo…

How does the pattern of grafting points influence the structure of one-component and mixed polymer brushes?

Using Monte Carlo simulations of a coarse-grained bead-spring model we study the lateral structure formation of one-component polymer brushes in a bad solvent and of a mixed polymer brush upon increasing the incompatibility of the two species. We compare the morphology of the brush with a regular distribution of grafting points and with a random arrangement. Density or composition fluctuations of the grafting points enhance the formation of irregular structures but randomness prevents the formation of long-range order. Even small fluctuations of the grafting points are sufficient to pin the lateral structures of the brush.

Corner wetting in the two-dimensional Ising model: Monte Carlo results

Square L ? L (L = 24?128) Ising lattices with nearest neighbour ferromagnetic exchange are considered using free boundary conditions at which boundary magnetic fields ? h are applied, i.e., at the two boundary rows ending at the lower left corner a field +h acts, while at the two boundary rows ending at the upper right corner a field ?h acts. For temperatures T less than the critical temperature Tc of the bulk, this boundary condition leads to the formation of two domains with opposite orientations of the magnetization direction, separated by an interface which for T larger than the filling transition temperature Tf (h) runs from the upper left corner to the lower right corner, while for T …

Finite-size-scaling study of the simple cubic three-state Potts glass: Possible lower critical dimension d=3.

For small lattices with linear dimension L ranging from L=3 to L=8 we obtain the distribution function P(q) of the overlap q between two real replicas of the three-state Potts-glass model with symmetric nearest-neighbor interaction with a Gaussian distribution. A finite-size-scaling analysis suggests a zero-temperature transition to occur with an exponentially diverging correlation length ${\ensuremath{\xi}}_{\mathrm{SG}}$\ensuremath{\sim}exp(C/${\mathit{T}}^{\mathrm{\ensuremath{\sigma}}}$). This implies that d=3 is the lower critical dimension.

Elastic constants from microscopic strain fluctuations

Fluctuations of the instantaneous local Lagrangian strain $\epsilon_{ij}(\bf{r},t)$, measured with respect to a static ``reference'' lattice, are used to obtain accurate estimates of the elastic constants of model solids from atomistic computer simulations. The measured strains are systematically coarse- grained by averaging them within subsystems (of size $L_b$) of a system (of total size $L$) in the canonical ensemble. Using a simple finite size scaling theory we predict the behaviour of the fluctuations $$ as a function of $L_b/L$ and extract elastic constants of the system {\em in the thermodynamic limit} at nonzero temperature. Our method is simple to implement, efficient and general e…

Atomistic Simulation of Transport Phenomena in Simple and Complex Fluids and Fluid Mixtures

Computer simulations of fluids in thermal equilibrium can yield information on transport coefficients such as self—diffusion and interdiffusion coefficients, viscosity, and thermal conductivity. While the estimation of self—diffusion coefficients from the mean square displacements of the respective particles is rather straightforward, the estimation of other transport coefficients is less straightforward, and can be based on either an analysis of time correlation functions of the appropriate collective variables, or on nonequilibrium techniques where the linear response to appropriate perturbations is measured.

Finite-size scaling approach for critical wetting: rationalization in terms of a bulk transition with an order parameter exponent equal to zero.

Clarification of critical wetting with short-range forces by simulations has been hampered by the lack of accurate methods to locate where the transition occurs. We solve this problem by developing an anisotropic finite-size scaling approach and show that then the wetting transition is a ``bulk'' critical phenomenon with order parameter exponent equal to zero. For the Ising model in two dimensions, known exact results are straightforwardly reproduced. In three dimensions, it is shown that previous estimates for the location of the transition need revision, but the conclusions about a slow crossover away from mean-field behavior remain unaltered.

MONTE CARLO METHODS FOR FIRST ORDER PHASE TRANSITIONS: SOME RECENT PROGRESS

This brief review discusses methods to locate and characterize first order phase transitions, paying particular attention to finite size effects. In the first part, the order parameter probability distribution and its fourth-order cumulant is discussed for thermally driven first-order transitions (the 3-state Potts model in d=3 dimensions is treated as an example). First-order transitions are characterized by a minimum of the cumulant, which gets very deep for large enough systems. In the second part, we discuss how to locate first order phase boundaries ending in a critical point in a large parameter space. As an example, the study of the unmixing transition of asymmetric polymer mixtures…

Double-well thermodynamic potentials and spinodal curves: how real are they?

The concept of double-well thermodynamic potentials, ubiquitous since the van der Waals description of the vapour-to-liquid transition and the Landau theory of phase transitions, is critically re-examined. Particular emphasis is put on the extent to which spinodal curves (separating ‘metastable’ from ‘unstable’ states) are meaningful. It is argued that in full thermodynamic equilibrium spinodals are well-defined when one either considers finite subsystems of an infinitely large system, or systems with all linear dimensions finite. Evidence is given that in a finite (cubic) d-dimensional box the spinodals correspond (in a fluid) to the rounded ‘droplet evaporation’ or ‘bubble condensation’ t…

Critical phenomena in colloid-polymer mixtures: interfacial tension, order parameter, susceptibility, and coexistence diameter.

The critical behavior of a model colloid-polymer mixture, the so-called AO model, is studied using computer simulations and finite size scaling techniques. Investigated are the interfacial tension, the order parameter, the susceptibility and the coexistence diameter. Our results clearly show that the interfacial tension vanishes at the critical point with exponent 2\nu ~ 1.26. This is in good agreement with the 3D Ising exponent. Also calculated are critical amplitude ratios, which are shown to be compatible with the corresponding 3D Ising values. We additionally identify a number of subtleties that are encountered when finite size scaling is applied to the AO model. In particular, we find …

Phase Transitions in Polymeric Systems

The study of collective phenomena in polymeric systems is a particular challenge,because there occurs structure simultaneously on many length scales. Restricting attention to neutral flexible linear macromolecules, we disregard here interesting problems like polyelectrolytes, stiff chains or chains with stiff parts, branched chains and polymer networks, and all combinations thereof including e.g. liquid crystalline polymers, and treat only the simplest case. But already then there occurs structure from the scale of a covalent bond along the backbone of a chain (~ 1 A) over the scale of the persistence length (~ 10 A) to the coil gyration radius R g (~ 102 A), in the description of a single …

Statistical Mechanics of Polymers: New Developments - International Workshop

Cylindrical confinement of solutions containing semiflexible macromolecules: surface-induced nematic order versus phase separation

Solutions of semiflexible polymers confined in cylindrical pores with repulsive walls are studied by Molecular Dynamics simulations for a wide range of polymer concentrations. Both the case where both lengths are of the same order and the case when the persistence length by far exceeds the contour length are considered, and the enhancement of nematic order along the cylinder axis is characterized. With increasing density the character of the surface effect changes from depletion to the formation of a layered structure. For binary 50 : 50 mixtures of the two types of polymers an interplay between surface enrichment of the stiffer component and the isotropic-nematic transition is found, and a…

Orthorhombic Phase of Crystalline Polyethylene: A Constant Pressure Path Integral Monte Carlo Study

In this paper we present a Path Integral Monte Carlo (PIMC) simulation of the orthorhombic phase of crystalline polyethylene, using an explicit atom force field with unconstrained bond lengths and angles. This work represents a quantum extension of our recent classical simulation (J. Chem. Phys. 106, 8918 (1997)). It is aimed both at exploring the applicability of the PIMC method on such polymer crystal systems, as well as on a detailed assessment of the importance of quantum effects on different quantities. We used the $NpT$ ensemble and simulated the system at zero pressure in the temperature range 25 - 300 K, using Trotter numbers between 12 and 144. In order to investigate finite-size e…

Enrichment of the chain ends in polymer melts at interfaces

How Does the Relaxation of a Supercooled Liquid Depend on Its Microscopic Dynamics?

Using molecular dynamics computer simulations we investigate how the relaxation dynamics of a simple supercooled liquid with Newtonian dynamics differs from the one with a stochastic dynamics. We find that, apart from the early beta-relaxation regime, the two dynamics give rise to the same relaxation behavior. The increase of the relaxation times of the system upon cooling, the details of the alpha-relaxation, as well as the wave vector dependence of the Edwards-Anderson-parameters are independent of the microscopic dynamics.

Theory of the evaporation/condensation transition of equilibrium droplets in finite volumes

Abstract A phenomenological theory of phase coexistence of finite systems near the coexistence curve that occurs in the thermodynamic limit is formulated for the generic case of d-dimensional ferromagnetic Ising lattices of linear dimension L with magnetization m slightly less than mcoex. It is argued that in the limit L→∞ an unconventional first-order transition occurs at a characteristic value mt

Phase diagram of a model anticlustering binary mixture in two dimensions: A semi-grand-canonical Monte Carlo study

The temperature-density phase diagram of a model binary mixture in two dimensions is investigated using a semi-grand-canonical Monte Carlo simulation scheme which allows for exchange between the two species while keeping the total number of atoms fixed. The gas-liquid and the gas-solid regions of the phase diagram are mapped out using the efficient block analysis method incorporating finite-size scaling of the various coexisting densities. An ordered square lattice structure is seen to be stable at low temperatures. Interesting short-range ordering phenomena resulting in a ``disorder line'' in the fluid phase are also analyzed and compared with results from liquid-state integral equation th…

Dynamics of multilayer adsorption: a Monte Carlo simulation

Abstract The growth of an adsorbed film at an initially empty surface which is exposed at time t = 0 to a gas is studied within the framework of a kinetic lattice gas model by Monte Carlo simulation. The model includes an attractive potential V ( z ) between adsorbed particles at distance z from the surface, V(z) = −A z 3 and a nearest-neighbor attractive interaction between the gas atoms. Several choices of the surface potential depth A , corresponding to different sequence of layering transitions, are considered. The Monte Carlo process assumes random evaporation/condensation events of gas atoms in adsorbed layers close to the surface, while surface diffusion is disregarded. For temperatu…

Critical Dynamics in a Binary Fluid: Simulations and Finite-Size Scaling

We report comprehensive simulations of the critical dynamics of a symmetric binary Lennard-Jones mixture near its consolute point. The self-diffusion coefficient exhibits no detectable anomaly. The data for the shear viscosity and the mutual-diffusion coefficient are fully consistent with the asymptotic power laws and amplitudes predicted by renormalization-group and mode-coupling theories {\it provided} finite-size effects and the background contribution to the relevant Onsager coefficient are suitably accounted for. This resolves a controversy raised by recent molecular simulations.

Effect of the solvent quality on the structural rearrangement of spherical brushes: coarse-grained models

A coarse-grained model for flexible polymers end-grafted to repulsive spherical nanoparticles is studied for various polymer lengths, grafting densities, and nanoparticle sizes by molecular dynamics simulations, considering variable solvent quality in the framework of an implicit solvent treatment. Below the theta point, the tuning of the temperature strongly influences the coverage of the nanoparticle surface by collapsed single chains or clusters of several chains. The shape and size of the aggregates depend on the number of monomers and surface density of the polymers. Specifically we analyzed the effect of the solvent quality on the density profiles and radius of gyration of the single …

Computer simulations of undercooled fluids and the glass transition

Abstract Two model studies are presented that attempt to describe the static and dynamic properties of glass-forming fluids via molecular dynamics simulations: The first model is an atomistically realistic model of SiO 2 , the second model provides a coarse-grained description of polymer liquids, i.e., typical `fragile' glassformers, while SiO 2 is the prototype of a `strong glassformer'. For both models, attention is given to the questions as to which range of temperatures are properties in equilibrium, and whether such simulations can help to interpret experiments and/or check theoretical predictions. While in the simulation of SiO 2 using the potential of van Beest, Kramer and van Santen…

Molecular Simulation of Polymer Melts and Blends: Methods, Phase Behavior, Interfaces, and Surfaces

Kinetics of phase separation in thin films: simulations for the diffusive case.

We study the diffusion-driven kinetics of phase separation of a symmetric binary mixture (AB), confined in a thin-film geometry between two parallel walls. We consider cases where (a) both walls preferentially attract the same component (A), and (b) one wall attracts A and the other wall attracts B (with the same strength). We focus on the interplay of phase separation and wetting at the walls, which is referred to as {\it surface-directed spinodal decomposition} (SDSD). The formation of SDSD waves at the two surfaces, with wave-vectors oriented perpendicular to them, often results in a metastable layered state (also referred to as ``stratified morphology''). This state is reminiscent of th…

Computer Simulations of Undercooled Fluids and Glasses

An introduction to the Molecular Dynamics (MD) simulation of chemically realistic models for undercooled fluids and glasses is given, emphasizing silicatic materials such as molten silicon dioxide and its mixtures with sodium oxide and aluminium oxide, and comparing the simulation results to experimental data whenever possible.

Polymer solutions confined in slit-like pores with attractive walls: An off-lattice Monte Carlo study of static properties and chain dynamics

Using a bead spring model of flexible polymer chains, the density profiles and chain configurational properties of polymer solutions confined between parallel plates were studied. A wide range of density ϕ, chain length N, and strength e of a short-range attractive wall potential was investigated. Both a temperature T in the good solvent regime (T > θ, θ being the Theta temperature where a chain in unconfined bulk three-dimensional solution would behave ideally) and a temperature in the bad solvent regime (T θ) show a crossover from two-dimensional excluded volume behavior (Rg ∝ N2ν with ν = 3/4) to ideal random walk behavior (ν = 1/2), the relaxation times show effective exponents Zeff (τ …

Calculation of local pressure tensors in systems with many-body interactions

Local pressures are important in the calculation of interface tensions and in analyzing micromechanical behavior. The calculation of local pressures in computer simulations has been limited to systems with pairwise interactions between the particles, which is not sufficient for chemically detailed systems with many-body potentials such as angles and torsions. We introduce a method to calculate local pressures in systems with n-body interactions (n=2,3,4,) based on a micromechanical definition of the pressure tensor. The local pressure consists of a kinetic contribution from the linear momentum of the particles and an internal contribution from dissected many-body interactions by infinitesim…

Reduction of the glass transition temperature in polymer films: A molecular-dynamics study

We present results of molecular dynamics (MD) simulations for a non-entangled polymer melt confined between two completely smooth and repulsive walls, interacting with inner particles via the potential $U_{\rm wall}\myeq (\sigma/z)^9$, where $z \myeq |z_{\rm particle}-z_{\rm wall}|$ and $\sigma$ is (roughly) the monomer diameter. The influence of this confinement on the dynamic behavior of the melt is studied for various film thicknesses (wall-to-wall separations) $D$, ranging from about 3 to about 14 times the bulk radius of gyration. A comparison of the mean-square displacements in the film and in the bulk shows an acceleration of the dynamics due to the presence of the walls. %Consistent…

Effects of inhomogeneities of cross-links on a microphase separation of polymer mixtures

We generalize de Gennes' theory of the microphase separation of cross-linked polymer mixtures to take into account the spatial fluctuations of the elasticity constant c, preventing the mixture from complete segregation. Within a mean-field analysis we found that the spatial fluctuations of c(r), which are assumed to obey the Poisson distribution, enlarge the size of the domains. The latter is obtained to be temperature dependent.


Computer simulation of bottle-brush polymers with flexible backbone: good solvent versus theta solvent conditions.

By Molecular Dynamics simulation of a coarse-grained bead-spring type model for a cylindrical molecular brush with a backbone chain of $N_b$ effective monomers to which with grafting density $\sigma$ side chains with $N$ effective monomers are tethered, several characteristic length scales are studied for variable solvent quality. Side chain lengths are in the range $5 \le N \le 40$, backbone chain lengths are in the range $50 \le N_b \le 200$, and we perform a comparison to results for the bond fluctuation model on the simple cubic lattice (for which much longer chains are accessible, $N_b \le 1027$, and which corresponds to an athermal, very good, solvent). We obtain linear dimensions of …

Monte Carlo simulations of phase transitions of systems in nanoscopic confinement

Abstract When simple or complex fluids are confined to ultrathin films or channels or other cavities of nanoscopic linear dimensions, the interplay of finite size and surface controls the phase behavior, and may lead to phase transitions rather different from the corresponding phenomena in the bulk. Monte Carlo simulation is a very suitable tool to clarify the complex behavior of such systems, since the boundary conditions providing the confinement can be controlled and arbitrarily varied, and detailed structural information on the inhomogeneous states of the considered systems is available. Examples used to illustrate these concepts include simple Ising models in pores and double-pyramid-s…

Monte Carlo methods for polymer chains in two - dimensional geometries (polymers at surfaces and interfaces)

Coarse-grained models of polymers at interfaces can be defined such that their treatment by Monte Carlo simulation is most convenient and efficient for the problem at hand. This simulation strategy is briefly illustrated with three examples: (1) The orientational ordering of rigid rod-like polymers grafted to a surface, where “table methods” can be used, applying a fine discretization of the angles describing rod orientation. (2) Surface enrichment of one species in a polymer blend is treated by a semi-grand-canonical technique. (3) The number of configurations and structure of a star polymer attached with its center to a wall is studied by a “growth technique” generalizing simple sampling …

High-temperature series analysis of the p-state Potts glass model on d-dimensional hypercubic lattices

We analyze recently extended high-temperature series expansions for the “Edwards-Anderson” spin-glass susceptibility of the p-state Potts glass model on d-dimensional hypercubic lattices for the case of a symmetric bimodal distribution of ferro- and antiferromagnetic nearest-neighbor couplings \(\). In these star-graph expansions up to order 22 in the inverse temperature \(\), the number of Potts states p and the dimension d are kept as free parameters which can take any value. By applying several series analysis techniques to the new series expansions, this enabled us to determine the critical coupling Kc and the critical exponent \(\) of the spin-glass susceptibility in a large region of …

Monte Carlo simulation studies of the interfaces between polymeric and other solids as models for fiber-matrix interactions in advanced composite materials

As a coarse-grained model for dense amorphous polymer systems interacting with solid walls (i.e., the fiber surface in a composite), the bond fluctuation model of flexible polymer chains confined between two repulsive surfaces is studied by extensive Monte Carlo simulations. Choosing a potential for the length of an effective bond that favors rather long bonds, the full temperature region from ordinary polymer melts down to the glass transition is accessible. It is shown that in the supercooled state near the glass transition an “interphase” forms near the walls, where the structure of the melt is influenced by the surface. This “interphase” already shows up in static properties, but also h…

Calculation of local pressure tensors in systems with many-body interactions

Local pressures are important in the calculation of interface tensions and in analyzing micromechanical behavior. The calculation of local pressures in computer simulations has been limited to systems with pairwise interactions between the particles, which is not sufficient for chemically detailed systems with many-body potentials such as angles and torsions. We introduce a method to calculate local pressures in systems with n-body interactions (n=2,3,4, . . .) based on a micromechanical definition of the pressure tensor. The local pressure consists of a kinetic contribution from the linear momentum of the particles and an internal contribution from dissected many-body interactions by infin…

Scattering function and the dynamics of phase separation in polymer mixtures under shear flow

The phenomenological mean-field theory describing concentration fluctuations and spinodal decomposition of binary mixtures of long flexible macromolecules is generalized to mixtures under steady shear flow. This shear flow leads to a partial orientation and stretching of the coils, as well as to an anisotropic deformation of concentration fluctuations. Generalizing the approach of Onuki and Kawasaki, we obtain the collective scattering function describing these concentration fluctuations in the mixture under shear flow. Both the steady-state situation in the one-phase region and the initial stages of spinodal decomposition for concentrations inside of the spinodal curve are considered.

Intra- and Interchain Correlations in Semidilute Polymer Solutions:  Monte Carlo Simulations and Renormalization Group Results

We investigate the intra- and intermolecular correlations in semidilute polymer solutions by large-scale computer simulations and renormalization group calculations. In the framework of the bond fluctuation model we study polymers with chain lengths up to N = 2048 monomers and determine the intermolecular pair correlation function, the coherent scattering intensity, and its distinct part at all length scales. The simulations are compared quantitatively to renormalization group calculations of the universal crossover scaling function. Special attention is paid to length scales smaller than the density screening length ξ, where the distinct part of the scattering function in the simulations i…

One- and two-component bottle-brush polymers: simulations compared to theoretical predictions

Scaling predictions and results from self-consistent field calculations for bottle-brush polymers with a rigid backbone and flexible side chains under good solvent conditions are summarized and their validity and applicability is assessed by a comparison with Monte Carlo simulations of a simple lattice model. It is shown that under typical conditions, as they are also present in experiments, only a rather weak stretching of the side chains is realized, and then the scaling predictions based on the extension of the Daoud-Cotton blob picture are not applicable. Also two-component bottle brush polymers are considered, where two types (A,B) of side chains are grafted, assuming that monomers of …

GLASS TRANSITION IN THIN POLYMER FILMS: A MOLECULAR DYNAMICS STUDY

A melt of nonentangled polymer chains confined between two smooth and purely repulsive walls is studied for various film thicknesses D and temperatures. The dynamics of the supercooled films is qualitatively identical to that of the bulk, but the walls lead to faster relaxation. To quantify this observation we analyze the data by the mode-coupling theory (MCT) of the glass transition. We find that the critical temperature of MCT, Tc(D), decreases with D and that T - Tc(D) is a relevant temperature scale. The static structure factor and dynamic correlation functions at intermediate times coincide with bulk behavior when compared to the same T - Tc(D).

Selfdiffusion of polymer chains in solutions and melts

Anomalous diffusion of monomers of polymer chains, as well as motion of these chains as a whole, is discussed with an emphasis on Monte Carlo simulations and simple scaling concepts. While the behavior of isolated chains in good solvents or Theta-solvents without excluded volume interactions is fully accounted for by the Rouse model, the behavior is less clear both for isolated chains in bad solvents and for chains in dense melts. Collapsed chains are shown to diffuse as g3(t) = <([rCM (t) -rCM(0)]2〉 ∝ tξ3 where the (effective?) exponent ξ3 simply seems to be linearly temperature-dependent for temperatures T lower than the Σ-temperature, ξ3 T/Θ. A relaxation time τ oc N3 is found, and scali…

Anomalous Diffusion and Relaxation of Collapsed Polymer Chains

Time-dependent displacement of monomers and the centre-of-gravity motion of a polymer chain at various temperatures below the theta-temperature are studied by Monte Carlo simulation of an off-lattice model. While inner monomers diffuse Rouse-like, [ri(t) − ri(0)]2  t1/2, the centre of mass exhibits pronounced anomalous diffusion, [rc.m.(t) − rc.m.(0)]2  ta, where the exponent a seems to depend on temperature. The resulting anomalous dependence of the relaxation times on chain length is discussed in terms of scaling ideas. A possible relation to a glasslike freezing in of the collapsed globules is pointed out.

The droplet evaporation/condensation transition in a finite volume

A fluid in the NVT ensemble at T less than the critical temperature T_c and rho = N/V somewhat in excess of rho_coex (density of the saturated gas in the gas-liquid transition) is considered. For V-&gt;infinity, a macroscopic liquid droplet coexists with surrounding saturated gas according to the lever rule. For finite V, droplets can only exist if they exceed a minimum size. A (rounded) first order transition of the system occurs when the droplet evaporates into the supersaturated gas.Simulation evidence for this transition is given for a Lennard-Jones model and interpreted by a phenomenological theory. At the transition, the chemical potential difference mu_t-mu_coex scales like L^(-d/(d+…

Phase diagram of polymer blends in confined geometry

Within self-consistent field theory we study the phase behavior of a symmetrical binary AB polymer blend confined into a thin film. The film surfaces interact with the monomers via short range potentials. One surface attracts the A component and the corresponding smei-infinite system exhibits a first order wetting transition. The surface interaction of the opposite surface is varied as to study the crossover from capillary condensation for symmetric surfaces fields to the interface localization/delocalization transition for antisymmetric surface fields. In the former case the phase diagram has a single critical point close to the bulk critical point. In the latter case the phase diagram exh…

Mechanical desorption of a single chain: unusual aspects of phase coexistence at a first-order transition.

The phase transition occurring when a single polymer chain adsorbed at a planar solid surface is mechanically desorbed is analyzed in two statistical ensembles. In the force ensemble, a constant force applied to the nongrafted end of the chain (that is grafted at its other end) is used as a given external control variable. In the $z$-ensemble, the displacement $z$ of this nongrafted end from the surface is taken as the externally controlled variable. Basic thermodynamic parameters, such as the adsorption energy, exhibit a very different behavior as a function of these control parameters. In the thermodynamic limit of infinite chain length the desorption transition with the force as a contro…

Monte Carlo Simulation of a Homopolymer−Copolymer Mixture Interacting with a Surface: Bulk versus Surface Micelles and Brush Formation

Using Monte Carlo simulations of the bond fluctuation model, we study the formation of micelles in a confined mixture of asymmetric AB-diblock copolymers and homopolymers. The composition of the sphere-forming AB-diblock copolymers is fA = 1/8. The mixture is confined into a thin film. The film surfaces attract the minority component of the diblock with strength, eW. To efficiently sample the micelle size distribution and establish equilibrium between the surface and the bulk, we work in the semigrandcanonical ensemble, i.e. at fixed density and fixed chemical potential difference between the two types of chains, choosing a large incompatibility χN ≃ 100 (strong segregation regime). The com…

Spherically averaged versus angle-dependent interactions in quadrupolar fluids

Employing simplified models in computer simulation is on the one hand often enforced by computer time limitations but on the other hand it offers insights into the molecular properties determining a given physical phenomenon. We employ this strategy to the determination of the phase behaviour of quadrupolar fluids, where we study the influence of omitting angular degrees of freedom of molecules via an effective spherically symmetric potential obtained from a perturbative expansion. Comparing the liquid-vapor coexistence curve, vapor pressure at coexistence, interfacial tension between the coexisting phases, etc., as obtained from both the models with the full quadrupolar interactions and th…

Polymer brushes on flat and curved surfaces: How computer simulations can help to test theories and to interpret experiments

Theoretical descriptions of static properties of polymer brushes are reviewed, with an emphasis on monodisperse macromolecules grafted to planar, cylindrical, or spherical substrates. Blob concepts and resulting scaling relations are outlined, and various versions of the self-consistent field theory are summarized: the classical approximation and the strong stretching limit, as well as the lattice formulation. The physical justification of various inherent assumptions is discussed, and computer simulation results addressing the test of the validity of these approximations are reviewed. Also, alternative theories, such as the single chain mean field theory and the density functional theory, …

Interplay of order-disorder phenomena and diffusion in rigid binary alloys in the presence of vacancies: Monte Carlo simulations

Transport phenomena are studied for a binary $(AB)$ alloy on a rigid square lattice with nearest-neighbor attraction between unlike particles, assuming a small concentration ${c}_{v}$ of vacancies $V$ being present, to which $A$ $(B)$ particles can jump with rates ${\ensuremath{\Gamma}}_{A}$ $({\ensuremath{\Gamma}}_{B})$ in the case where the nearest-neighbor attractive energy ${ϵ}_{AB}$ is negligible in comparison with the thermal energy ${k}_{B}T$ in the system. This model exhibits a continuous order-disorder transition for concentrations ${c}_{A},{c}_{B}=1\ensuremath{-}{c}_{A}\ensuremath{-}{c}_{V}$ in the range ${c}_{A,1}^{\mathit{crit}}\ensuremath{\leqslant}{c}_{A}\ensuremath{\leqslant}…

Activity mediated phase separation: Can we understand phase behavior of the nonequilibrium problem from an equilibrium approach?

We present results for structure and dynamics of mixtures of active and passive particles, from molecular dynamics (MD) simulations and integral equation theory (IET) calculations, for a physically motivated model. The perfectly passive limit of the model corresponds to the phase-separating Asakura-Oosawa model for colloid-polymer mixtures in which, for the present study, the colloids are made self-propelling by introducing activity in accordance with the well known Vicsek model. Such activity facilitates phase separation further, as confirmed by our MD simulations and IET calculations. Depending upon the composition of active and passive particles, the diffusive motion of the active specie…

Monte Carlo study of surface phase transitions in the three-dimensional Ising model.

We present the results of extensive Monte Carlo simulations of phase transitions and critical behavior at the surface of a simple cubic Ising model. Profiles of the magnetization and internal energy are determined as a function of the distance from the surface, and we extract surface and bulk properties as a function of temperature and surface coupling ${\mathit{J}}_{\mathit{s}}$. The surface-bulk multicritical point is located with improved precision, ${\mathit{J}}_{\mathit{s}}$/J=1.52\ifmmode\pm\else\textpm\fi{}0.02, and crossover behavior is studied. New estimates for critical exponents are extracted, ${\ensuremath{\gamma}}_{1}$=0.78\ifmmode\pm\else\textpm\fi{}0.06, ${\ensuremath{\gamma}…

Intramolecular phase separation of copolymer "bottle brushes": No sharp phase transition but a tunable length scale

A lattice model for a symmetrical copolymer "bottle brush" molecule, where two types (A,B) of flexible side chains are grafted with one chain end to a rigid backbone, is studied by a variant of the pruned-enriched Rosenbluth method (PERM), allowing for simultaneous growth of all side chains in the Monte Carlo sampling. Choosing repulsive binary interactions between unlike monomers and varying the solvent quality, it is found that phase separation into an $A$-rich part of the cylindrical molecule and a $B$-rich part can occur only locally. Long range order (in the direction of the backbone) does not occur, and hence the transition from the randomly mixed state of the bottle brush to the phas…

Modelling of Orientational Ordering in Lipid Monolayers

Lipid monolayers at high densities are modelled as rigid rods grafted to an interface at the sites of a regular lattice. The transition between the state where the rods are uniformly tilted to a disordered state with no (average) tilt is studied by computer simulation methods. For the one-dimensional model, the molecular dynamics approach is found much less suitable to equilibrate the system rather than Monte Carlo methods. Both in d=2 discretized versions of Monte Carlo codes are much more efficient than continuum Monte Carlo methods, in spite of huge storage requirements. While in d=l the transition occurs at temperature T=0 via the spontaneous creation of solitons, at d=2 a finite temper…

Structural properties of concave cylindrical brushes interacting with free chains

We present a self-consistent field theoretical study of the microstructure of concave cylindrical brushes as a function of the cylinder radius, grafting density, grafted chain length, and the solvent quality. We show that the results for the radial monomer density profile and the distribution of the free ends are in good agreement with the corresponding molecular dynamics results. Part of the investigation is focused on the conformational behavior of a free macromolecule in a cylindrical brush. A central result is the observed non-monotonous variation of the size of a free chain in a brush-coated tube when the tube radius is systematically changed. An interpretation of this behavior which d…

Simple sampling Monte Carlo methods

Monte Carlo Methods for the Sampling of Free Energy Landscapes

In this chapter, we return to classical statistical mechanics, wherein the canonical ensemble averages of an observable \(A(\overrightarrow{x})\), where \(\overrightarrow{x} \) stands symbolically for the “microstate” coordinate in the configurational part of the phase space of the system, are given by (cf. Sect. 2.1.1)

Wetting in fluid systems. Wetting and capillary condensation of lattice gases in thin film geometry

Monte Carlo studies of lattice gas models with attractive interactions between nearest neighbors on a simple cubic lattice are carried out for a L×L×D geometry with two hard walls of size L×L and periodic boundary conditions parallel to the wall. Two types of short-range forces at the walls are considered: (i) Both walls are of the same type and exert an attractive force of the same strength (in Ising model terminology, surface fields HD = H1 occur). (ii) The walls differ, one attracts and the other repels particles, again with the same strength (HD = −H1). In the first case, capillary condensation occurs at a chemical potential differing from its value for phase coexistence in the bulk, an…

The Ising square lattice in aL�M geometry: A model for the effect of surface steps on phase transitions in adsorbed monolayers

Critical phenomena in adsorbed monolayers on surfaces are influenced by limited substrate homogeneity, such as surface steps. We consider the resulting finite-size and boundary effects in the framework of a lattice gas system with nearest neighbor attraction in aL×M geometry, with two free boundaries of lengthM≫L, and periodic boundary conditions in the other direction (along the direction of the steps). This geometry thus models a “terrace” of the stepped surface, and adatoms adsorbed on neighboring terraces are assumed to be non-interacting. Also the effect of boundary “fields” is considered (describing the effects of missing neighbors and changed binding energy to the substrate near the …

Phase transition shifts in films

Abstract We present a Monte Carlo computer simulation study of phase transitions in a three-dimensional Ising/lattice gas model with nearest neighbor attractive coupling and confined to a slit-like capillary with absorbing walls. Data are generated for thicknesses D ⩽ 40 and are used to study the shift of the phase boundaries due to finite wall separation.

On the order of the herringbone transition of N2 on graphite: a Monte Carlo study

Using the anisotropic planar-rotor model we investigate the herringbone phase transition of N2 in the (√3 × √3)R30° commensurate phase on graphite by large scale Monte Carlo simulations. The effective correlation length ξ is measured near the transition temperature T0. The data, extrapolated to T0, yield a large but finite ξ at T0 demonstrating that the herringb ordering is a weak first order transition.

First Versus Second Order Phase Transitions in the Three-Dimensional Three-State Potts Model in Random Fields

The ordering of three-states Potts ferromagnets on the simple cubic lattice exposed to random fields is investigated by extensive Monte Carlo simulations. Evidence is presented that the transition is second order for intermediate strength of the fields, while it presumably is first order for large field strength. The implications for various theoretical predictions are briefly discussed.

Comparison of model potentials for molecular-dynamics simulations of silica.

Structural, thermomechanical, and dynamic properties of pure silica SiO2 are calculated with three different model potentials, namely, the potential suggested by van Beest, Kramer, and van Santen (BKS) [Phys. Rev. Lett. 64, 1955 (1990)], the fluctuating-charge potential with a Morse stretch term for the short-range interactions proposed by Demiralp, Cagin, and Goddard (DCG)[Phys. Rev. Lett. 82, 1708 (1999)], and a polarizable force field proposed by Tangney and Scandolo (TS) [J. Chem. Phys. 117, 8898 (2002)]. The DCG potential had to be modified due to flaws in the original treatment. While BKS reproduces many thermomechanical properties of different polymorphs rather accurately, it also sh…

Critical behavior of the surface-layer magnetization at the extraordinary transition in the three-dimensional Ising model.

We have used a vectorized multispin-coding Monte Carlo method to determine the behavior of the surface-layer magnetization ${\mathit{m}}_{1}$ at the bulk transition in a simple-cubic Ising film with strongly enhanced surface coupling, i.e., at the extraordinary transition. In contrast to recent renormalization-group calculations we find no evidence for a discontinuous slope in the temperature dependence of ${\mathit{m}}_{1}$; the data are consistent with a free-energy-like (T-${\mathit{T}}_{\mathit{c}}$${)}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$ behavior plus background terms.

Simulation of binary fluids exposed to selectively adsorbing walls: a method to estimate contact angles and line tensions

For an understanding of interfacial phenomena of fluids on the nanoscale a detailed knowledge of the excess free energies of fluids due to walls is required, as well as of the interfacial tension between coexisting fluid phases. A description of simulation approaches to solve this task is given for a suitable model binary (A + B) fluid. Sampling the order parameter distribution of the system without walls, the curvature dependent and flat interfacial tensions of coexisting ‘bulk’ phases is extracted. In a thin film geometry, the difference in wall free energies is found via a new thermodynamic integration method. Thus the contact angle θ of macroscopic droplets is estimated from Young's equ…

Semi-flexible polymer chains in quasi-one-dimensional confinement: a Monte Carlo study on the square lattice

Single semi-flexible polymer chains are modeled as self-avoiding walks (SAWs) on the square lattice with every 90° kink requiring an energy eb. While for eb = 0 this is the ordinary SAW, varying the parameter qb = exp(−eb/kBT) allows the variation of the effective persistence length p over about two decades. Using the pruned-enriched Rosenbluth method (PERM), chain lengths up to about N = 105 steps can be studied. In previous work it has already been shown that for contour lengths L = Nb (the bond length b is the lattice spacing) of order p a smooth crossover from rods to two-dimensional self-avoiding walks occurs, with radii R ∝ p1/4L3/4, the Gaussian regime predicted by the Kratky–Porod m…

Interplay between Chain Collapse and Microphase Separation in Bottle-Brush Polymers with Two Types of Side Chains

Conformations of a bottle-brush polymer with two types (A,B) of grafted side chains are studied by molecular dynamics simulations, using a coarse-grained bead−spring model with side chains of up to...

Heterogeneous nucleation at a wall near a wetting transition: a Monte Carlo test of the classical theory

While for a slightly supersaturated vapor the free energy barrier ΔF(hom)(*), which needs to be overcome in a homogeneous nucleation event, may be extremely large, nucleation is typically much easier at the walls of the container in which the vapor is located. While no nucleation barrier exists if the walls are wet, for incomplete wetting of the walls, described via a nonzero contact angle Θ, classical theory predicts that nucleation happens through sphere-cap-shaped droplets attracted to the wall, and their formation energy is ΔF(het)(*) = ΔF(hom)(*)f(Θ), with f(Θ) = (1-cosΘ)(2)(2+cosΘ)/4. This prediction is tested through simulations for the simple cubic lattice gas model with nearest-nei…

Dynamics of surface enrichment: A theory based on the Kawasaki spin-exchange model in the presence of a wall

A mean-field theory is developed for the description of the dynamics of surface enrichment in binary mixtures, where one component is favored by an impenetrable wall. Assuming a direct exchange (Kawasaki-type) model of interdiffusion, a layerwise molecular-field approximation is formulated in the framework of a lattice model. Also the corresponding continuum theory is considered, paying particular attention to the proper derivation of boundary conditions for the differential equation at the hard wall. As an application, we consider the explicit solutions of the derived equations in the case where nonlinear effects can be neglected, studying the approach of an initially flat (homogeneous) co…

Excess free energy of nanoparticles in a polymer brush

Abstract We present an efficient method for direct determination of the excess free energy Δ F of a nanoparticle inserted into a polymer brush. In contrast to Widom's insertion method, the present approach can be efficiently implemented by Monte Carlo or Molecular Dynamics methods also in a dense environment. In the present investigation the method is used to determine the free energy penalty Δ F ( R , D ) for placing a spherical particle with an arbitrary radius R at different positions D between the grafting plane and the brush surface. Deep inside the brush, or for dense brushes, one finds Δ F  ∝  R 3 whereas for shallow nanoclusions Δ F  ∝  R 2 , regardless of the particle interaction (…

Polymer chains confined into tubes with attractive walls: A Monte Carlo simulation

A bead-spring off-lattice model of a polymer chain with repulsive interactions among repeating units confined into straight tubes of various cross sections, DT2, is studied by Monte Carlo simulation. We are also varying the chain length from N = 16 to 128 and the strength of a short-range attractive interaction between the repeating units and the walls of the tube. Longitudinal and perpendicular static linear dimensions of the chains are analyzed, as well as the density profile of repeating units across the tube. These data are interpreted in terms of scaling concepts describing the crossover between three-dimensional and quasi-one-dimensional chain conformations and the adsorption transiti…

A combined molecular dynamics and Monte Carlo study of the approach towards phase separation in colloid-polymer mixtures.

A coarse-grained model for colloid-polymer mixtures is investigated where both colloids and polymer coils are represented as point-like particles interacting with spherically symmetric effective potentials. Colloid-colloid and colloid-polymer interactions are described by Weeks-Chandler-Andersen potentials, while the polymer-polymer interaction is very soft, of strength k(B)T/2 for maximum polymer-polymer overlap. This model can be efficiently simulated both by Monte Carlo and molecular dynamics methods, and its phase diagram closely resembles that of the well-known Asakura-Oosawa model. The static and dynamic properties of the model are presented for systems at critical colloid density, va…

Phase Transitions in Dense Lipid Monolayers Grafted to a Surface:  Monte Carlo Investigation of a Coarse-Grained Off-Lattice Model

Semiflexible amphiphilic molecules end-grafted at a flat surface are modeled by a bead-spring chain with stiff bond angle potentials. Constant density Monte Carlo simulations are performed varying temperature, density, and chain length of the molecules, whose effective monomers interact with Lennard-Jones potentials. For not too large densities and low temperatures the monolayer is in a quasi-two-dimensional crystalline state, characterized by uniform tilt of the (stretched) chains. Raising the temperature causes a second-order transition into a (still solid) phase with no tilt. For the first time, finite size scaling concepts are applied to a model of a surfactant monolayer, and it is foun…

The ensemble switch method and related approaches to obtain interfacial free energies between coexisting phases from simulations: a brief review

The accurate estimation of the excess free energy due to an interface between coexisting phases of a model system by computer simulation often is a challenging task. We review here two methods, whi...

Copolymer Melts in Disordered Media

The symmetric AB block copolymer melt in a gel matrix with preferential adsorption of A monomers on the gel gives an example of a random-field system, which is described near the point of the microphase separation transition by the random field Landau-Brazovskii Hamiltonian. By using the technique of the 2-nd Legendre transform, the phase diagram of the system is calculated. We found that the preferential adsorption of the copolymer on the gel results in two effects: a) It decreases the temperature of the first order phase transition between disordered and ordered phase. b) There exists a region on the phase diagram at some small but finite value of the adsorption energy in which the replic…

Surface-induced disordering at first-order transitions in body-centered cubic binary alloys: A Monte-Carlo simulation

Surface effects on the phase transition from theDO3 phase to the disordered phase are studied for a bcc Ising antiferromagnet with nearest and next-nearest neighbor exchange interactions in a magnetic field. This model can also be considered to represent binary alloys such as the FeAl-system; missing interactions near the surface translate then into surface magnetic fields. The change of the local magnetization near the surface then corresponds to “surface enrichment” of one component. For a plausible choice of parameters surface-induced disordering is found and the associated critical behavior is studied. Varying the bulk fieldH near the transition fieldHc, we find that the thickness of th…

Computer simulation of macromolecular materials

Computer simulation of model systems with Monte Carlo methods enables the detailed study of structure and thermodynamic properties of these systems and thus constitutes a link between analytic theory and experiment. Typical applications that are discussed include polymer blends, dynamics of local motions in polymer melts, and the adsorption of polymers on walls.

Nearest-neighbor Ising antiferromagnet on the fcc lattice: Evidence for multicritical behavior.

The phase behavior of the Ising model with nearest-neighbor antiferromagnetic interactions on the fcc lattice in a homogeneous magnetic field is studied by means of large-scale Monte Carlo simulations. In accordance with the most recent of the previous investigations, but with significantly higher accuracy, it is found that the ``triple'' point at which the disordered phase coexists with both the AB phase as well as with the ${\mathit{A}}_{3}$B phase (corresponding to the model's lattice gas interpretation as a binary alloy ${\mathit{A}}_{\mathit{xB}1\mathrm{\ensuremath{-}}\mathit{x}}$ such as ${\mathrm{Cu}}_{\mathit{x}}$${\mathrm{Au}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$) occurs at a nonz…

The Importance of Intermediate Range Order in Silicates: Molecular Dynamics Simulation Studies

We present the results of large scale computer simulations in which we investigate the structural and dynamic properties of silicate melts with the compositions (Na2O)2(SiO2) and (Al2O3)2(Si02). In order to treat such systems on a time scale of several nanoseconds and for system sizes of several thousand atoms it is necessary to use parallel supercomputers like the CRAY T3E. We show that the silicates under consideration exhibit additional intermediate range order as compared to silica (SiO2) where the characteristic intermediate length scales stem from the tetrahedral network structure. For the sodium silicate system it is demonstrated that the latter structural features are intimately con…

Monte Carlo simulations of the 2d-Ising model in the geometry of a long stripe

Abstract The two-dimensional Ising model in the geometry of a long stripe can be regarded as a model system for the study of nanopores. As a quasi-one-dimensional system, it also exhibits a rather interesting “phase behavior”: At low temperatures the stripe is either filled with “liquid” or “gas” and “densities” are similar to those in the bulk. When we approach a “pseudo-critical point” (below the critical point of the bulk) at which the correlation length becomes comparable to the length of the stripe, several interfaces emerge and the systems contains multiple “liquid” and “gas” domains. The transition depends on the size of the stripe and occurs at lower temperatures for larger stripes.…

Influence of a continuous quenching procedure on the initial stages of spinodal decomposition

Instead of the standard assumption in the theory of phase separation where an instantaneous quench from an initial equilibrium state to the final state in the two-phase region is assumed, we consider the more realistic situation that the change of the external control parameter (e.g. temperature) can only be performed with finite rates. During the initial stages of spinodal decomposition the system then has some “memory” of the states intermediate between the initial and the final one. This influence of the finite quench rate in continuous quenching procedures is studied within the linearized theory of spinodal decomposition, with the Langer-Baron-Miller decoupling, and with Monte Carlo sim…

Do crossover functions depend on the shape of the interaction profile?

We examine the crossover from classical to non-classical critical behaviour in two-dimensional systems with a one-component order parameter. Since the degree of universality of the corresponding crossover functions is still subject to debate, we try to induce non-universal effects by adding interactions with a second length scale. Although the crossover functions clearly depend on the range of the interactions, they turn out to be remarkably robust against further variation of the interaction profile. In particular, we find that the earlier observed non-monotonic crossover of the effective susceptibility exponent occurs for several qualitatively different shapes of this profile.

All-or-none proteinlike folding transition of a flexible homopolymer chain.

Here we report a first-order all-or-none transition from an expanded coil to a compact crystallite for a flexible polymer chain. Wang-Landau sampling is used to construct the complete density of states for square-well chains up to length 256. Analysis within both the microcanonical and canonical ensembles shows a direct freezing transition for finite length chains with sufficiently short-range interactions. This type of transition is a distinctive feature of "one-step" protein folding and our findings demonstrate that a simple homopolymer model can exhibit protein-folding thermodynamics.

Escape transition of a polymer chain from a nanotube: How to avoid spurious results by use of the force-biased pruned-enriched Rosenbluth algorithm

A polymer chain containing $N$ monomers confined in a finite cylindrical tube of diameter $D$ grafted at a distance $L$ from the open end of the tube may undergo a rather abrupt transition, where part of the chain escapes from the tube to form a "crown-like" coil outside of the tube. When this problem is studied by Monte Carlo simulation of self-avoiding walks on the simple cubic lattice applying a cylindrical confinement and using the standard pruned-enriched Rosenbluth method (PERM), one obtains spurious results, however: with increasing chain length the transition gets weaker and weaker, due to insufficient sampling of the "escaped" states, as a detailed analysis shows. In order to solve…

Polymer mixtures in confined geometries: Model systems to explore phase transitions

While binary (A,B) symmetric polymer mixtures ind = 3 dimensions have an unmixing critical point that belongs to the 3d Ising universality class and crosses over to mean field behavior for very long chains, the critical behavior of mixtures confined into thin film geometry falls in the 2d Ising class irrespective of chain length. The critical temperature always scales linearly with chain length, except for strictly two-dimensional chains confined to a plane, for whichT; c ∝N; 5/8 (this unusual exponent describes the fractal contact line between segregated chains in dense melts in two spatial dimensions, d = 2). When the walls of the thin film are not neutral, but preferentially attract one …

Phase separation of binary mixtures in thin films: Effects of an initial concentration gradient across the film.

We study the kinetics of phase separation of a binary (A,B) mixture confined in a thin film of thickness $D$ by numerical simulations of the corresponding Cahn-Hilliard-Cook (CHC) model. The initial state consisted of 50$%$ A:50$%$ B with a concentration gradient across the film, i.e., the average order parameter profile is ${\ensuremath{\Psi}}_{\mathrm{av}}(z,t=0)=(2z/D\ensuremath{-}1){\ensuremath{\Psi}}_{g},\phantom{\rule{0.28em}{0ex}}0\ensuremath{\leqslant}z\ensuremath{\leqslant}D$, for various choices of ${\ensuremath{\Psi}}_{g}$ and $D$. The equilibrium state (for time $t\ensuremath{\rightarrow}\ensuremath{\infty}$) consists of coexisting A-rich and B-rich domains separated by interfac…

Ergodicity breaking in a mean field Potts glass: A Monte Carlo investigation

We use Monte Carlo simulations, single spin-flip as well as parallel tempering techniques to investigate the 10-state fully connected Potts glass for system sizes of up to N = 2560. We find that the α-relaxation shows a strong dependence on N and that for the system sizes considered the system remains ergodic even at temperatures below T D , the dynamical critical temperature for this model. However, if one uses the data for the finite size systems, such as the relaxation times or the time dependence of the spin autocorrelation function, and extrapolates them to the thermodynamic limit, one finds that they are indeed compatible with the results for N = ∞ (which are known from analytical cal…

Statistical and systematic errors in Monte Carlo sampling

We have studied the statistical and systematic errors which arise in Monte Carlo simulations and how the magnitude of these errors depends on the size of the system being examined when a fixed amount of computer time is used. We find that, depending on the degree of self-averaging exhibited by the quantities measured, the statistical errors can increase, decrease, or stay the same as the system size is increased. The systematic underestimation of response functions due to the finite number of measurements made is also studied. We develop a scaling formalism to describe the size dependence of these errors, as well as their dependence on the “bin length” (size of the statistical sample), both…

Monte Carlo Simulations of Polymer Systems

The impact of Monte Carlo “computer experiments” in polymer physics is described, emphasizing three examples taken from the author’s research group. The first example is a test of the classical Flory—Huggins theory for polymer mixtures, including a discussion of cricital phenomena. Also “technical aspects” of such simulations (“grand-canonical” ensemble, finite—size scaling, etc.) are explained briefly. The second example refers to configurational statistics and dynamics of chains confined to cylindrical tubes; the third example deals with the adsorption of polymers at walls. These simulations check scaling concepts developed along the lines of de Gennes.

Monte Carlo investigation of a model for a three-dimensional orientational glass with short-range gaussian interaction

The analogue of the Edwards-Anderson model for isotropic vector spin glasses, but taking quadrupoles instead of unit vectors at each lattice site of the considered simple cubic lattice, is studied as a model for an orientational glass. We study both the case where the quadrupole moment can orient in a three-dimensional space (m=3) and the case where the orientation is restricted to a plane (m=2), but otherwise the Hamiltonian is fully isotropic. ℋ= $$ - \sum\limits_{\left\langle {i,j} \right\rangle } {J_{ij} } \left[ {\left( {\sum\limits_{\mu = 1}^m {S_i^\mu S_j^\mu } } \right)^2 - \frac{1}{m}} \right]$$ , whereJ ij is a random gaussian interaction between nearest neighbors, andS i μ the μ'…

Monte-Carlo Methods

The article conbtains sections titled: 1 Introduction and Overview 2 Random-Number Generation 2.1 General Introduction 2.2 Properties That a Random-Number Generator (RNG) Should Have 2.3 Comments about a Few Frequently Used Generators 3 Simple Sampling of Probability Distributions Using Random Numbers 3.1 Numerical Estimation of Known Probability Distributions 3.2 “Importance Sampling” versus “Simple Sampling” 3.3 Monte-Carlo as a Method of Integration 3.4 Infinite Integration Space 3.5 Random Selection of Lattice Sites 3.6 The Self-Avoiding Walk Problem 3.7 Simple Sampling versus Biased Sampling: the Example of SAWs Continued 4 Survey of Applications to Simulation of Transport Processes 4.…

Orthorhombic Phase of Crystalline Polyethylene: A Monte Carlo Study

In this paper we present a classical Monte Carlo simulation of the orthorhombic phase of crystalline polyethylene, using an explicit atom force field with unconstrained bond lengths and angles and periodic boundary conditions. We used a recently developed algorithm which apart from standard Metropolis local moves employs also global moves consisting of displacements of the center of mass of the whole chains in all three spatial directions as well as rotations of the chains around an axis parallel to the crystallographic c-direction. Our simulations are performed in the NpT ensemble, at zero pressure, and extend over the whole range of temperatures in which the orthorhombic phase is experime…

Single chain structure in thin polymer films: Corrections to Flory's and Silberberg's hypotheses

Conformational properties of polymer melts confined between two hard structureless walls are investigated by Monte Carlo simulation of the bond-fluctuation model. Parallel and perpendicular components of chain extension, bond-bond correlation function and structure factor are computed and compared with recent theoretical approaches attempting to go beyond Flory's and Silberberg's hypotheses. We demonstrate that for ultrathin films where the thickness, $H$, is smaller than the excluded volume screening length (blob size), $\xi$, the chain size parallel to the walls diverges logarithmically, $R^2/2N \approx b^2 + c \log(N)$ with $c \sim 1/H$. The corresponding bond-bond correlation function d…

Simulation of surface-controlled phase separation in slit pores: Diffusive Ginzburg-Landau kinetics versus Molecular Dynamics

The phase separation kinetics of binary fluids in constrained geometry is a challenge for computer simulation, since nontrivial structure formation occurs extending from the atomic scale up to mesoscopic scales, and a very large range of time needs to be considered. One line of attack to this problem is to try nevertheless standard Molecular Dynamics (MD), another approach is to coarse-grain the model to apply a time-dependent nonlinear Ginzburg–Landau equation that is numerically integrated. For a symmetric binary mixture confined between two parallel walls that prefer one species, both approaches are applied and compared to each other. There occurs a nontrivial interplay between the forma…

Dynamics of star polymers in a good solvent: A Kramers potential treatment

The ‘‘effective’’ relaxation time τ of isolated star polymers with excluded volume interactions in the Rouse model limit (i.e., disregarding hydrodynamic interactions present in real solvents) is studied varying both the number of arms f and the number of monomers per arm l. Here τ is defined from the response of the gyration radius of the star polymer to a Kramers potential that describes the effect of shear flow in lowest order in the shear rate. Monte Carlo simulations are performed with two different techniques (simple sampling with enrichment or dynamic Monte Carlo, respectively) for two different models (simple self‐avoiding walks with an extended core or the bond fluctuation model, r…

The escape transition of a compressed star polymer: Self-consistent field predictions tested by simulation

The escape transition of a polymer "mushroom" (a flexible chain grafted to a flat non-adsorbing substrate surface in a good solvent) occurs when the polymer is compressed by a cylindrical piston of radius $R$, that by far exceeds the chain gyration radius. At this transition, the chain conformation abruptly changes from a two-dimensional self-avoiding walk of blobs (of diameter $H$, the height of the piston above the substrate) to a "flower conformation", i.e. stretched almost one-dimensional string of blobs (with end-to-end distance $\approx R$) and an "escaped" part of the chain, the "crown", outside the piston. The extension of this problem to the case of star polymers with $f$ arms is c…

Adsorption Transition of a Polymer Chain at a Weakly Attractive Surface: Monte Carlo Simulation of Off-Lattice Models

A bead-spring model of a polymer chain with one end attached to a wall is studied by Monte Carlo simulations for chain lengths 16 ≤ N ≤ 256. Two types of adsorption potentials, 9-3 and 10-4 Lennard-Jones (LJ) potentials, between the effective monomers and the wall are assumed. For both cases the adsorption transition where the chain changes its asymptotic statistical properties from a three-dimensional to a two-dimensional configuration is located using a scaling analysis. It is shown that the crossover exponent φ = 0.50 ± 0.02 is the same for both LJ potentials. This value is compatible with recent theoretical predictions and simulation results for lattice models with short-range wall pote…

Phase separation of symmetric polymer mixtures in a common good solvent in the semidilute concentration regime

Monte Carlo simulations of lattice models of binary (AB) symmetric polymer mixtures (chain lengthsN A=N B=N) in a common good solvent are carried out and the phase diagrams and critical properties of the unmixing transitions are estimated and interpreted in terms of recent theories. Polymers are modeled by self-avoiding walks of lengthN=16, 32 and 64 on the simple cubic lattice. Data for vacancy concentrations of φV=0.6, 0.8 and 0.85 are analyzed. It is shown that forN=16, φV=0.85 no phase separation occurs, down to the lowest temperature, while forN=32, φV=0.85 still phase separation occurs but no longer is complete. Our results are compatible with a scaling theory based on a “renormalizat…

Theory of orientational glasses models, concepts, simulations

Abstract This review describes the various attempts to develop a theoretical understanding for ordering and dynamics of randomly diluted molecular crystals, where quadrupole moments freeze in random orientations upon lowering the temperature, as a result of randomness and competing interactions. While some theories attempt to model this freezing into a phase with randomly oriented quadrupole moments in terms of a bond-disorder concept analogous to the Edwards-Anderson model of spin glasses, other theories attribute the freezing to random field-like terms in the Hamiltonian. While models of the latter type have been studied primarily by microscopic molecular field-type treatments, the former…

Order and Disorder Phenomena at Surfaces of Binary Alloys

We present recent Monte Carlo results on surfaces of bcc-structured binary alloys which undergo an order-disorder phase transformation in the bulk. In particular, we discuss surface order and surface induced disorder at the bulk transition between the ordered (DO3) phase and the disordered (A2) phase. An intricate interplay between different ordering and segregation phenomena leads to a complex surface behavior, which depends on the orientation of the surface under consideration.

Computer simulations of critical phenomena and phase behaviour of fluids

Computer simulation techniques such as Monte Carlo (MC) and Molecular Dynamics (MD) methods yield numerically exact information (apart from statistical errors) on model systems of classical statistical mechanics. However, a systematic limitation is the restriction to a finite (and often rather small) particle number N (or box linear dimension L, respectively). This limitation is particularly restrictive near critical points (due to the divergence of the correlation length of the order parameter) and for the study of phase equilibria (possibly involving interfaces, droplets, etc.). Starting out with simple lattice gas (Ising) models, finite size scaling analyses have been developed to overco…

Effect of Chain Stiffness on the Adsorption Transition of Polymers

Polymers grafted with one chain end to an impenetrable flat hard wall which attracts the monomers with a short-range adsorption potential (of strength e) are studied by large scale Monte Carlo simulations, using the pruned–enriched Rosenbluth method (PERM). Chain lengths up to N = 25600 steps are considered, and the intrinsic flexibility of the chain is varied via an energy penalty for nonzero bond angles, eb. Choosing qb = exp(−eb/kBT) in the range from qb = 1 (fully flexible chains) to qb = 0.005 (rather stiff chains with a persistence length of about lp=52 lattice spacings), the adsorption transition is found to vary from about e/kBTc ≈ 0.286 to e/kBTc ≈ 0.011, confirming the theoretical…

From orientational glasses to structural glasses: What computer simulations have contributed to understand experiments

Abstract Orientational glasses, produced by random dilution of molecular crystals, exhibit a freezing transition of the quadrupole moments. Monte Carlo simulations of lattice models (generalization of the Edwards–Anderson spin glass model) have been used to elucidate this behavior. While short range models exhibit a static glass transition at zero temperature only, the infinite range Potts glass exhibits a transition where a glass order parameter appears discontinuously. At higher temperature, a dynamical transition occurs, described by mode-coupling theory (MCT). MCT has also been tested by Monte Carlo and molecular dynamics simulations of coarse-grained models of glass-forming polymers. W…

Amorphous silica between confining walls and under shear: a computer simulation study

Molecular dynamics computer simulations are used to investigate a silica melt confined between walls at equilibrium and in a steady-state Poisseuille flow. The walls consist of point particles forming a rigid face-centered cubic lattice and the interaction of the walls with the melt atoms is modelled such that the wall particles have only a weak bonding to those in the melt, i.e. much weaker than the covalent bonding of a Si-O unit. We observe a pronounced layering of the melt near the walls. This layering, as seen in the total density profile, has a very irregular character which can be attributed to a preferred orientational ordering of SiO4 tetrahedra near the wall. On intermediate lengt…

Comparative classical and ab initio Molecular Dynamics study of molten and glassy germanium dioxide

A Molecular Dynamics (MD) study of static and dynamic properties of molten and glassy germanium dioxide is presented. The interactions between the atoms are modelled by the classical pair potential proposed by Oeffner and Elliott (OE) [Oeffner R D and Elliott S R 1998, Phys. Rev. B, 58, 14791]. We compare our results to experiments and previous simulations. In addition, an ab initio method, the so-called Car-Parrinello Molecular Dynamics (CPMD), is applied to check the accuracy of the structural properties, as obtained by the classical MD simulations with the OE potential. As in a similar study for SiO2, the structure predicted by CPMD is only slightly softer than that resulting from the cl…

Critical wetting with short-range forces: Is mean-field theory valid?

Single Molecules Probing the Freezing of Polymer Melts: A Molecular Dynamics Study for Various Molecule-Chain Linkages

8 pages; International audience; We present molecular dynamics simulations of coarse-grained model systems of a glassforming polymer matrix containing fluorescent probe molecules. These probe molecules are either dispersed in the matrix or covalently attached to the center or the end of a dilute fraction of the polymer chains. We show that in all cases the translational and rotational relaxation of the probe molecules is a faithful sensor for the glass transition of the matrix as determined from a mode-coupling analysis or Vogel-Fulcher analysis of their R-relaxation behavior. Matrix and dumbbell related relaxation processes show a clear violation of the Stokes-Einstein-Debye laws. In accor…

Atomistic modeling of materials properties by Monte Carlo Simulation

In order to optimize materials properties, in many cases a deeper understanding of the relationship between the chemical-atomistic structure and the physical properties of the solid and fluid phases of the material is necessary. Monte Carlo simulation is a tool that allows the reliable calculation of thermodynamic properties of strongly interacting many-body condensed matter systems. Given a model of effective interatomic or intermolecular interactions (drawn either from quantum-chemical-type interactions or from analysis of suitable experimental data), macroscopic bulk properties of a material can be simulated, as well as interfacial phenomena and certain kinds of slow dynamic processes (o…

Grafted polymer layers under variable solvent conditions: A Monte Carlo simulation

Polymer chains anchored with one end at a hard wall under variable solvent conditions are investigated by Monte Carlo simulations using the bond- fluctuation model. Detail information on the structural properties are obtained above, at, and below the Θ-point and discussed in terms of the appropriate theories. In particular, the scaling of the brush thickness is formulated and verified by the simulation data. For the dynamics at the Θ-point, both the relaxation time of the chain configuration and the mean-square time displacement are studied. At temperatures distinctly below the Θ-point, we find that the layer develops considerable lateral inhomogeneity in its density, which has not been pre…

Monte Carlo simulation of polymeric materials: Recent progress

Monte Carlo simulations are presented, dealing with phase diagrams of block copolymer melts and polymer blends, including the unmixing kinetics of the latter systems. The theoretical background is briefly reviewed: Ginzburg-type criteria reveal that in mixtures of long flexible polymers a “crossover” occurs from mean-field behavior (as described by Flory-Huggins theory) to nonclassical Ising-type behavior, and spinodal curves can be unusually sharp. This crossover is demonstrated by large scale simulations of the bond fluctuation model, and it is also shown that for symmetric mixtures the critical temperature scales with chain length as Tc α N. The prefactor in this relation is distinctly s…

Does Young's equation hold on the nanoscale? A Monte Carlo test for the binary Lennard-Jones fluid

When a phase-separated binary ($A+B$) mixture is exposed to a wall, that preferentially attracts one of the components, interfaces between A-rich and B-rich domains in general meet the wall making a contact angle $\theta$. Young's equation describes this angle in terms of a balance between the $A-B$ interfacial tension $\gamma_{AB}$ and the surface tensions $\gamma_{wA}$, $\gamma_{wB}$ between, respectively, the $A$- and $B$-rich phases and the wall, $\gamma _{AB} \cos \theta =\gamma_{wA}-\gamma_{wB}$. By Monte Carlo simulations of bridges, formed by one of the components in a binary Lennard-Jones liquid, connecting the two walls of a nanoscopic slit pore, $\theta$ is estimated from the inc…

Molecular Dynamics simulation of evaporation processes of fluid bridges confined in slit-like pore

A simple fluid, described by point-like particles interacting via the Lennard-Jones potential, is considered under confinement in a slit geometry between two walls at distance Lz apart for densities inside the vapor-liquid coexistence curve. Equilibrium then requires the coexistence of a liquid "bridge" between the two walls, and vapor in the remaining pore volume. We study this equilibrium for several choices of the wall-fluid interaction (corresponding to the full range from complete wetting to complete drying, for a macroscopically thick film), and consider also the kinetics of state changes in such a system. In particular, we study how this equilibrium is established by diffusion proces…

Dynamical scaling of surface growth in simple lattice models

We present extensive simulations of the atomistic Edwards-Wilkinson (EW) and Restricted Edwards-Wilkinson (REW) models in 2+1 dimensions. Dynamic finite-size scaling analyses of the interfacial width and structure factor provide the estimates for the dynamic exponent z=1.65+/-0.05 for the EW model and z=2.0+/-0.1 for the REW model. The stochastic contribution to the interface velocity U due to the deposition and diffusion of particles is characterized for both the models using a blocking procedure. For the EW model the time-displaced temporal correlations in U show nonexponential decay, while the temporal correlations decay exponentially for the REW model. Dynamical scaling of the temporal …

Spin glasses: Experimental facts, theoretical concepts, and open questions

This review summarizes recent developments in the theory of spin glasses, as well as pertinent experimental data. The most characteristic properties of spin glass systems are described, and related phenomena in other glassy systems (dielectric and orientational glasses) are mentioned. The Edwards-Anderson model of spin glasses and its treatment within the replica method and mean-field theory are outlined, and concepts such as "frustration," "broken replica symmetry," "broken ergodicity," etc., are discussed. The dynamic approach to describing the spin glass transition is emphasized. Monte Carlo simulations of spin glasses and the insight gained by them are described. Other topics discussed …

Amorphous Silica at Surfaces and Interfaces: Simulation Studies

The structure of surfaces and interfaces of silica (SiO2) is investigated by large scale molecular dynamics computer simulations. In the case of a free silica surface, the results of a classical molecular dynamics simulation are compared to those of an ab initio method, the Car—Parrinello molecular dynamics. This comparative study allows to check the accuracy of the model potential that underlies the classical simulation. By means of a pure classical MD, the interface between amorphous and crystalline SiO2 is investigated, and as a third example the structure of a silica melt between walls is studied in equilibrium and under shear. We show that in the latter three examples important structu…

Finite size effects at phase transitions

For many models of statistical thermodynamics and of lattice gauge theory computer simulation methods have become a valuable tool for the study of critical phenomena, to locate phase transitions, distinguish whether they are of first or second order, and so on. Since simulations always deal with finite systems, analysis of finite size effects by suitable finite size scaling concepts is a key ingredient of such applications. The phenomenological theory of finite size scaling is reviewed with emphasis on the concept of probability distributions of order parameter and/or energy. Attention is also drawn to recent developments concerning anisotropic geometries and anisotropic critical behavior, …

Formation of Ordered Structures in Quenching Experiments: Scaling Theory and Simulations

In this note we want to address the particular problem of the formation of ordered structures resulting from “quenching experiments”. The generic experimental situation is depicted in Figure 1. Initially the system is in an unordered random state in the one-phase region. Then the temperature is lowered (for some systems like polymers the coexistence curve is inverted so that the temperature must be raised) until the system is in the two phase region. The system is now in a non-equilibrium situation and evolves toward equilibrium. It is during the evolution toward equilibrium that the system develops ordered structures /1,2/.

Unusual finite size effects in the Monte Carlo simulation of microphase formation of block copolymer melts

Extensive Monte Carlo simulations are presented for the Fried-Binder model of block copolymer melts, where polymer chains are represented as self and mutually avoiding walks on a simple cubic lattice, and monomer units of different kind (A, B) repel each other if they are nearest neighbors (e AB > O). Choosing a chain length N = 20, vacancy concentration Φ v = 0,2, composition f = 3/4, and a L × L × L geometry with periodic boundary conditions and 8 ≤ L ≤ 32, finite size effects on the collective structure factor S(q) and the gyration radii are investigated. It is shown that already above the microphase separation transition, namely when the correlation length ζ(T) of concentration fluctuat…

Static and dynamic properties of supercooled thin polymer films

The dynamic and static properties of a supercooled (non-entangled) polymer melt are investigated via molecular-dynamics (MD) simulations. The system is confined between two completely smooth and purely repulsive walls. The wall-to-wall separation (film thickness), D, is varied from about 3 to about 14 times the bulk radius of gyration. Despite the geometric confinement, the supercooled films exhibit many qualitative features which were also observed in the bulk and could be analyzed in terms of mode-coupling theory (MCT). Examples are the two-step relaxation of the incoherent intermediate scattering function, the time-temperature superposition property of the late time alpha-process and the…

Interfacial properties of glassy polymer melts: A Monte Carlo study

The properties of the interface between a polymer melt and a solid wall are studied over a wide range of temperatures by dynamic Monte Carlo simulations. It is shown that in the supercooled state near the glass transition of the melt an “interphase” forms, the structure of which is influenced by the wall. The thickness of this interphase is determined from the monomer density profile near the surface and is strongly temperature dependent. At low glass-like temperatures it is larger than the bulk radius of gyration of the chains.

Overview: Understanding nucleation phenomena from simulations of lattice gas models

Monte Carlo simulations of homogeneous and heterogeneous nucleation in Ising/lattice gas models are reviewed with an emphasis on the general insight gained on the mechanisms by which metastable states decay. Attention is paid to the proper distinction of particles that belong to a cluster (droplet), that may trigger a nucleation event, from particles in its environment, a problem crucial near the critical point. Well below the critical point, the lattice structure causes an anisotropy of the interface tension, and hence nonspherical droplet shapes result, making the treatment nontrivial even within the conventional classical theory of homogeneous nucleation. For temperatures below the rough…

Recipes for successful simulation

A Collection of recipes could be called a cookbook, but Daan Frenkel and Berend Smit's book achieves more than that. By explaining the physics behind the algorithms, the authors let you learn how molecular dynamics and Monte Carlo simulation methods work, how to apply these methods in a sensible way and what information can be extracted from them. Although computer simulation, lying between analytical theory and experiment, is now regarded as the third branch science, it is still viewed with skepticism some researchers precisely because of this interdisciplinary character.

Structures of stiff macromolecules of finite chain length near the coil-globule transition: A Monte Carlo simulation

Using a coarse-grained model of a semiflexible macromolecule, the equilibrium shapes of the chain have been studied varying both the temperature and the chain stiffness. We have applied Monte Carlo techniques using the bond fluctuation model for a chain length of N = 80 effective monomers, and two different types of interactions: a potential depending on the angle between successive bonds along the chain to control the chain stiffness, and an attractive interaction between non-bonded effective monomers to model variable solvent quality. In a diagram of states where chain stiffness and inverse temperature and used as variables, we find regions where the chain exists as coil, as spherical glo…

Conformations, Transverse Fluctuations and Crossover Dynamics of a Semi-Flexible Chain in Two Dimensions

We present a unified scaling description for the dynamics of monomers of a semiflexible chain under good solvent condition in the free draining limit. We consider both the cases where the contour length $L$ is comparable to the persistence length $\ell_p$ and the case $L\gg \ell_p$. Our theory captures the early time monomer dynamics of a stiff chain characterized by $t^{3/4}$ dependence for the mean square displacement(MSD) of the monomers, but predicts a first crossover to the Rouse regime of $t^{2\nu/{1+2\nu}}$ for $\tau_1 \sim \ell_p^3$, and a second crossover to the purely diffusive dynamics for the entire chain at $\tau_2 \sim L^{5/2}$. We confirm the predictions of this scaling descr…

Phase transitions in polymer blends and block copolymer melts: Some recent developments

The classical concepts about unmixing of polymer blends (Flory-Huggins theory) and about mesophase ordering in block copolymers (Leibler's theory) are briefly reviewed and their validity is discussed in the light of recent experiments, computer simulations and other theoretical concepts. It is emphasized that close to the critical point of unmixing non-classical critical exponents of the Ising universality class are observed, in contrast to the classical mean-field exponents implied by the Flory-Huggins theory. The temperature range of this non-mean-field behavior can be understood by Ginzburg criteria. The latter are also useful to discuss the conditions under which the linearized (Cahn-li…

Dynamics of Ising spin glasses far below the lower critical dimension: The one-dimensional case and small clusters

The Glauber model is studied for symmetric distributions (±J and gaussian) of the nearest-neighbour interactionJ, including a magnetic field. For small clusters of spins (closed rings ofN bonds, withN≦7) the complex magnetic susceptibility χ(ω) and the time-dependent remanent magnetizationm(t) are found exactly for given bond configurations {Jij} by diagonalization of the Liouville operator; apart from the ±J model, the average over {Jij} must be done numerically by simple random sampling Monte Carlo. Nevertheless our accuracy is much better than corresponding dynamic Monte Carlo simulations, even if one considers the extrapolation toN→∞.

Chain length dependence of the state diagram of a single stiff-chain macromolecule: Theory and Monte Carlo simulation

We present a Monte Carlo computer simulation and theoretical results for the dependence of the state diagram of a single semiflexible chain on the chain length. The calculated transition lines between different structures in the state diagrams for both studied chain lengths N=40 and N=80 can be described by theoretical predictions which include chain length dependence explicitly. The stability criteria of different structures are discussed. The theoretically predicted exponent in the dependence of the toroid size on the chain length is compatible with computer simulation results.

Adsorption of Semiflexible Polymers in Cylindrical Tubes

Conformations of wormlike chains in cylindrical pores with attractive walls are explored for varying pore radius and strength of the attractive wall potential by molecular dynamics simulations of a coarse-grained model. Local quantities such as the fraction of monomeric units bound to the surface and the bond-orientational order parameter as well as the radial density distribution are studied, as well as the global chain extensions parallel to the cylinder axis and perpendicular to the cylinder surface. A nonmonotonic convergence of these properties to their counterparts for adsorption on a planar substrate is observed due to the conflict between pore surface curvature and chain stiffness. …

Surface-directed spinodal decomposition: Lattice model versus Ginzburg-Landau theory

When a binary mixture is quenched into the unstable region of the phase diagram, phase separation starts by spontaneous growth of long-wavelength concentration fluctuations ("spinodal decomposition"). In the presence of surfaces, the latter provide nontrivial boundary conditions for this growth. These boundary conditions can be derived from lattice models by suitable continuum approximations. But the lattice models can also be simulated directly, and thus used to clarify the conditions under which the Ginzburg–Landau type theory is valid. This comparison shows that the latter is accurate only in the immediate vicinity of the bulk critical point, if thermal fluctuations can also be neglecte…

Escape Transition of a Grafted Polymer Chain

The escape transition of a flexible polymer chain of chain length N, end-grafted at a hard wall and compressed by a piston of radius R in good solvent conditions, is studied by Monte Carlo simulation and by phenomenological arguments. In contrast to previous theories which have predicted a jump in the force f at a critical value H t of the height H of the piston above the wall, we find that the transition (which is sharp only for N → ∞) is characterized by a flat region of f in the f — H isotherm, i. e. a jump in the height occurs at the transition from H esc , t to H imptt , with (H imp , t — H esc , t )/H esc , t ≈ 0.26. At the transition the constant force f t is predicted and observed t…

Statics and dynamics of colloid-polymer mixtures near their critical point of phase separation: A computer simulation study of a continuous Asakura–Oosawa model

We propose a new coarse-grained model for the description of liquid-vapor phase separation of colloid-polymer mixtures. The hard-sphere repulsion between colloids and between colloids and polymers, which is used in the well-known Asakura-Oosawa (AO) model, is replaced by Weeks-Chandler-Anderson potentials. Similarly, a soft potential of height comparable to thermal energy is used for the polymer-polymer interaction, rather than treating polymers as ideal gas particles. It is shown by grand-canonical Monte Carlo simulations that this model leads to a coexistence curve that almost coincides with that of the AO model and the Ising critical behavior of static quantities is reproduced. Then the …

Mobility, interdiffusion, and tracer diffusion in lattice-gas models of two-component alloys

The transport properties of lattice-gas models of alloys with two particle species are studied. The numbers of the particles and vacancies are conserved, and the two particle species have different exchange rates with the vacancies. The mobility and interdiffusion is described by the linear Onsager theory of transport. The Onsager coefficients are estimated from numerical simulations of the mobilities. A recently proposed relation between the Onsager coefficients of the random-alloy model is verified. The interdiffusion of the two species is directly monitored in the simulations; it is well described by the estimated Onsager coefficients. The results on interdiffusion are compared with simu…

Dielectric Relaxation of a Polybutadiene Melt at a Crystalline Graphite Surface: Atomistic Molecular Dynamics Simulations

Dielectric experiments are an indispensable tool to further our understanding of the relaxation behavior of polymers, not only in bulk samples but also in confined situations. A chemically realistic Molecular Dynamics simulation, in which all information about molecular motions is available, can shed light onto the questions of heterogeneity and anisotropy of the underlying molecular relaxation processes which lead to the ensemble averaged experimental dielectric signal. In this contribution, we present a careful analysis of the dielectric response of a weakly polar and confined polymer, 1,4-polybutadiene between graphite walls. The relaxation of the segmental dipole moments was obtained in…

Simulations clarify when supercooled water freezes into glassy structures

Although liquid water is a ubiquitous substance and its properties are crucial for all living species, the precise understanding of these properties is still a matter of active scientific research. One rather mysterious aspect concerns the conditions when undercooled water freezes not into ice crystals but into glass-like structures. Based on a rather novel type of computer simulation approach, in PNAS, Limmer and Chandler (1) propose a nonequilibrium phase diagram that attempts to clarify the conditions (temperature, pressure, cooling protocol) under which one should observe transitions from undercooled water to different forms of amorphous ice.

How Do Droplets Depend on the System Size? Droplet Condensation and Nucleation in Small Simulation Cells

Using large scale grandcanonical Monte Carlo simulations in junction with a multicanonical reweighting scheme we investigate the liquid-vapor transition of a Lennard—Jones fluid. Particular attention is focused on the free energy of droplets and the transition between different system configurations as the system tunnels between the vapor and the liquid state as a function of system size. The results highlight the finite size dependence of droplet properties in the canonical ensemble and free energy barriers along the path from the vapor to the liquid in the grandcanonical ensemble.

Shape of crossover between mean-field and asymptotic critical behavior in a three-dimensional Ising lattice

Recent numerical studies of the susceptibility of the three-dimensional Ising model with various interaction ranges have been analyzed with a crossover model based on renormalization-group matching theory. It is shown that the model yields an accurate description of the crossover function for the susceptibility.

Soliton staircases and standing strain waves in confined colloidal crystals

We show by computer simulation of a two-dimensional crystal confined by corrugated walls that confinement can be used to impose a controllable mesoscopic superstructure of predominantly mechanical elastic character. Due to an interplay of the particle density of the system and the width D of the confining channel, "soliton staircases" can be created along both parallel confining boundaries, that give rise to standing strain waves in the entire crystal. The periodicity of these waves is of the same order as D. This mechanism should be useful for structure formation in the self-assembly of various nanoscopic materials.

Molecular dynamics study of phase separation kinetics in thin films.

We use molecular dynamics to simulate experiments where a symmetric binary fluid mixture (AB), confined between walls that preferentially attract one component (A), is quenched from the one-phase region into the miscibility gap. Surface enrichment occurs during the early stages, yielding a B-rich mixture in the film center with well-defined A-rich droplets. The droplet size grows with time as l(t) proportional t(2/3) after a transient regime. The present atomistic model is also compared to mesoscopic coarse-grained models for this problem.

Spinodal Decomposition in Binary Polymer Blends: Monte Carlo Simulations and Dynamic Mean Field Theory

Using large scale computer simulations we have investigated the interplay between single chain dynamics and the kinetics of phase separation in a symmetric binary polymer blend. In the framework of a coarse grained lattice model — the bond fluctuation model on a three dimensional lattice — we monitor the growth of concentration fluctuations after a quench from the one phase region into the miscibility gap. Chains of 64 effective segments are simulated in a cell of linear dimension L = 160, i.e., each simulation box contains 256 000 particles. The growth rate of composition fluctuations is averaged over 64 realizations of the temperature quench.

Kinetics of phase separation in thin films: Lattice versus continuum models for solid binary mixtures

A description of phase separation kinetics for solid binary (A,B) mixtures in thin film geometry based on the Kawasaki spin-exchange kinetic Ising model is presented in a discrete lattice molecular field formulation. It is shown that the model describes the interplay of wetting layer formation and lateral phase separation, which leads to a characteristic domain size $\ell(t)$ in the directions parallel to the confining walls that grows according to the Lifshitz-Slyozov $t^{1/3}$ law with time $t$ after the quench. Near the critical point of the model, the description is shown to be equivalent to the standard treatments based on Ginzburg-Landau models. Unlike the latter, the present treatmen…

Simulation of Copolymer Bottle-Brushes

The structure of bottle-brush polymers with a rigid backbone and flexible side chains is studied in three dimensions, varying the grafting density, the side chain length, and the solvent quality. Some preliminary results of theoretical scaling considerations for one-component bottle-brush polymers in a good solvent are compared with Monte Carlo simulations of a simple lattice model. For the simulations a variant of the pruned-enriched Rosenbluth method (PERM) allowing for simultaneous growth of all side chains in the Monte Carlo sampling is employed. For a symmetrical binary (A,B) bottle-brush polymer, where two types (A,B) of flexible side chains are grafted with one chain end to the backb…

Compressed Polymer Brushes under the Shear Flow

The structure and dynamics of two compressed polymer brushes under the shear deformation were investigated by the methods of stochastic dynamics. The shear was created by the move-ment of grafting planes with the constant velo-cities in opposite directions along the planes.

Adsorption-induced polymer translocation through a nanopore: a Monte Carlo investigation

Abstract We study the translocation of a coarse-grained flexible polymer through a nanopore in a membrane induced by its adsorption on the trans side of the membrane. Dramatic differences in the threading behavior are observed if the adhesion to the membrane wall, e w , is below or above the adsorption threshold e c r . For e w e c r (weak adsorption) the activation barrier for translocation is at c cis 0 ≈ N / 2 (in terms of the fraction of chain c cis = N cis / N before the pore), independent of chain length N. For e w > e c r this barrier is at a constant (vanishing) number of passed trans monomers for all N. The mean time of chain passage τ trans ∝ c cis 1.3 when c cis c cis 0 . It scal…

Coarse-graining dipolar interactions in simple fluids and polymer solutions: Monte Carlo studies of the phase behavior

In this paper we investigate the phase diagram of pure dipolar substances and their mixtures with short alkanes, using grand canonical Monte Carlo simulations of simplified coarse-grained models. Recently, an efficient coarse-grained model for simple quadrupolar molecules, based on a Lennard-Jones (LJ) interaction plus a spherically averaged quadrupolar potential, has been shown to be successful in predicting single-component and mixture phase diagrams. Motivated by these results, we investigate the phase diagrams of simple dipolar molecules (and their mixtures with alkanes) using a spherically averaged potential. First, we test the model on pure components. A generalized (state-dependent) …

Scattering from concentration fluctuations in polymer blends: A monte carlo investigation

The collective scattering function Scoll( $$\vec q$$ ), which describes light (neutron-, x-ray) scattering under wavevector $$\vec q$$ , is obtained from Monte Carlo simulations for a symmetrical polymer mixture. The polymers are modelled by self-avoiding walks ofN A=NB=N steps on a simple cubic lattice, where a fractionφ V of sites is left vacant, and an attractive energye occurs if two neighboring sites are taken by the same kind of monomer. Spinodal curves are estimated from linear extrapolation of S coll −1 (0) vs.e/k B T, whereT is the temperature. Also the single chain structure factor is obtained and the de Gennes random phase approximation (RPA) can thus be tested. Unexpectedly, str…

N2monolayers physisorbed on graphite: the herringbone transition revisited

Monte Carlo simulations were undertaken of the orientational herringbone phase transition of N2 adsorbed on graphite in the complete monolayer (✓3 × ✓3) R30° structure. The non-universal aspects (c...

Monte Carlo modelling of the polymer glass transition

We are proposing a lattice model with chemical input for the computer modelling of the polymer glass transition. The chemical input information is obtained by a coarse graining procedure applied to a microscopic model with full chemical detail. We use this information on Bisphenol-A-Polycarbonate to predict it's Vogel-Fulcher temperature out of a dynamic Monte Carlo Simulation. The microscopic structure of the lattice model is that of a genuine amorphous material, and the structural relaxation obeys the time temperature superposition.

Some necessary background

Field-induced ordering phenomena and non-local elastic compliance in two-dimensional colloidal crystals

Ordering phenomena in colloidal dispersions exposed to external one-dimensional, periodic fields or under confinement are studied systematically by Monte Carlo computer simulations. Such systems are useful models for the study of monolayers on a substrate. We find that the interaction with a substrate potential completely changes the miscibility of a binary, hard disc mixture at low external field amplitudes. The underlying ordering mechanisms leading to this laser-induced de-mixing differ, depending on which components interact with the substrate potential. Generic effects of confinement on crystalline order in two dimensions are studied in a model system of point particles interacting via…

Static and Dynamic Properties of Adsorbed Chains at Surfaces:  Monte Carlo Simulation of a Bead-Spring Model

The adsorption of flexible polymers from dilute solution in good solvents at attractive walls is studied by Monte Carlo simulation of a coarse-grained off-lattice model, varying chain length N and ...

HOW MONTE CARLO SIMULATIONS CAN CLARIFY COMPLEX PROBLEMS IN STATISTICAL PHYSICS

Statistical mechanics of condensed matter systems in physics (fluids and solids) derives macroscopic equilibrium properties of these systems as averages computed from a Hamiltonian that describes the atomistic interactions in the system. While analytic methods for most problems involve uncontrolled approximations, Monte Carlo simulations allow numerically exact treatments, apart from statistical errors and from the systematic problem that finite systems are treated rather than the thermodynamic limit. However, this problem can be overcome by finite size scaling methods, and thus Monte Carlo methods have become a very powerful tool to study even complex phase transitions. Examples given wil…

Anisotropic interfacial tension, contact angles, and line tensions: A graphics-processing-unit-based Monte Carlo study of the Ising model

As a generic example for crystals where the crystal-fluid interface tension depends on the orientation of the interface relative to the crystal lattice axes, the nearest neighbor Ising model on the simple cubic lattice is studied over a wide temperature range, both above and below the roughening transition temperature. Using a thin film geometry $L_x \times L_y \times L_z$ with periodic boundary conditions along the z-axis and two free $L_x \times L_y$ surfaces at which opposing surface fields $\pm H_{1}$ act, under conditions of partial wetting, a single planar interface inclined under a contact angle $\theta < \pi/2$ relative to the yz-plane is stabilized. In the y-direction, a generaliza…

Mechanical Properties of Single Molecules and Polymer Aggregates

This chapter deals with the mechanical properties of single polymer chains, aggregates, and supramolecular complexes. The topics discussed cover a broad range from fundamental statistical mechanics of the equilibrium elastic properties of single polymer chains to details of the behavior of binding pockets in biomolecular assemblies as observed by force spectroscopy. The first section treats the equilibrium mechanical properties of single polymer chains in various environments, investigated via extensive simulations employing coarse-grained models that have proven extremely successful in many branches of polymer physics, namely the bond-fluctuation model and the self-avoiding walk model. Apa…

Monte Carlo Simulation of Alloy Phase Diagrams and Short-Range Order

As a prototype model for order-disorder phenomena in binary alloys, a face-centered cubic lattice is considered,the sites of which can be taken by either A-atoms or B-atoms, assuming pair-wise interactions between nearest (J) and next nearest neighbours (J). The phase diagram is constructed from Monte Carlo calculations. Some technical aspects essential for the success of such calculations are briefly mentioned (use of grand-canonical rather than canonical ensemble, how to obtain the free energy needed to locate first-order phase transitions, etc.). It is shown that the topology of the phase diagram changes when the ratio R = Jnnn/Jnn is varied, and this behaviour is discussed in the contex…

The dynamics of sodium in sodium disilicate: Channel relaxation and sodium diffusion

We use molecular dynamics computer simulations to study the dynamics of amorphous (Na_2O)2(SiO_2). We find that the Na ions move in channels embedded in a SiO_2 matrix. The characteristic distance between these channels gives rise to a prepeak in the structure factor at around q=0.95 A^-1. The dynamics of sodium is given by a fast process which can be seen in the incoherent scattering function and a slow process which is seen in the coherent function. The relaxation time of the latter coincides with the alpha-relaxation time of the matrix. The Kohlrausch exponent of the fast process for q&gt;1.6 A^1 is the same as the von Schweidler exponent for the slow one, demonstrating that the two proc…

Out of Equilibrium Characteristics of a Forced Translocating Chain through a Nanopore

Polymer translocation through a nano-pore in a thin membrane is studied using a coarse-grained bead-spring model and Langevin dynamics simulation with a particular emphasis to explore out of equilibrium characteristics of the translocating chain. We analyze the out of equilibrium chain conformations both at the $cis$ and the $trans$ side separately either as a function of the time during the translocation process or as as function of the monomer index $m$ inside the pore. A detailed picture of translocation emerges by monitoring the center of mass of the translocating chain, longitudinal and transverse components of the gyration radii and the end to end vector. We observe that polymer confi…

Self-diffusion in polymer solutions using the bond-fluctuation MC-algorithm

Abstract A lattice Monte Carlo study of the self-diffusion of polymer chains in an athermal solution of equal chains is presented. The examined chain lengths, N (= 20–200), and volume fractions, φ (= 0.025-0.5), cover the range from dilute solution to concentrated solution, respectively. The dynamics show a gradual crossover from Rouse to reptation-like behaviour. Analysing the data according to a scaling theory and taking into account the density dependence of the microscopic length and time-scales, an almost perfect scaling of the self-diffusion coefficient is achieved. The high statistical accuracy of the data (103–104 chains per parameter combination) was obtainable by using a transpute…

Evidence of thin-film precursors formation in hydrokinetic and atomistic simulations of nano-channel capillary filling

We present hydrokinetic Lattice Boltzmann and Molecular Dynamics simulations of capillary filling of high-wetting fluids in nano-channels, which provide clear evidence of the formation of thin precursor films, moving ahead of the main capillary front. The dynamics of the precursor films is found to obey the Lucas-Washburn law as the main capillary front, z2(t) proportional to t, although with a larger prefactor, which we find to take the same value for both geometries under inspection. Both hydrokinetic and Molecular Dynamics approaches indicate a precursor film thickness of the order of one tenth of the capillary diameter. The quantitative agreement between the hydrokinetic and atomistic m…

Phase transitions in nanosystems caused by interface motion: the Ising bipyramid with competing surface fields.

The phase behavior of a large but finite Ising ferromagnet in the presence of competing surface magnetic fields +/- H_s is studied by Monte Carlo simulations and by phenomenological theory. Specifically, the geometry of a double pyramid of height 2L is considered, such that the surface field is positive on the four upper triangular surfaces of the bi-pyramid and negative on the lower ones. It is shown that the total spontaneous magnetization vanishes (for L -&gt; infinity) at the temperature T_f(H), related to the "filling transition" of a semi-infinite pyramid, which can be well below the critical temperature of the bulk. The discontinuous vanishing of the magnetization is accompanied by a…

Frictional drag between polymer bearing surfaces

Some fundamental features of friction between two polymer bearing surfaces in relative sliding motion are investigated by molecular dynamics simulations. End-tethered and adsorbed polymers are considered under good and poor solvent conditions. The shear stress is measured while varying the solvent's viscosity, surface separation, degree of polymerization and grafting density. For all systems we observe shear thinning that is attributed to the orientation of the chains along the shear direction. This effect is particularly strong for brushes, for which the shear stress during the steady sliding state is mainly determined by the degree of overlap between the brushes.

Statics and dynamics of dense polymer systems studied by monte carlo simulation

Monte Carlo simulations of coarse–grained models of macromolecules offer a unique tool to study the interplay between coil conformations, thermodynamic properties, and chain configurational relaxation and diffusion. Two examples are discussed where the chain conformation strongly differs from a gaussian coil: (i) collapsed chains in a bad solvent, where anomalous diffusion occurs in the Rouse limit and the relaxation time increases at least with the third power of chain length. (ii) Expulsion of a chain from a semidilute polymer brush. The initially stretched chain contracts to a gaussian coil and the center of mass moves outward with constant velocity until it reaches the region of the “la…

Monte Carlo study of the ising model phase transition in terms of the percolation transition of “physical clusters”

Finite squareL×L Ising lattices with ferromagnetic nearest neighbor interaction are simulated using the Swendsen-Wang cluster algorithm. Both thermal properties (internal energyU, specific heatC, magnetization 〈|M|〉, susceptibilityχ) and percolation cluster properties relating to the “physical clusters,” namely the Fortuin-Kasteleyn clusters (percolation probability 〈P∞〉, percolation susceptibilityχp, cluster size distributionnl) are evaluated, paying particular attention to finite-size effects. It is shown that thermal properties can be expressed entirely in terms of cluster properties, 〈P∞〉 being identical to 〈|M|〉 in the thermodynamic limit, while finite-size corrections differ. In contr…

Richteret al.Respond

On the equation of state for thermal polymer solutions and melts with attractive interaction

We perform Monte Carlo simulations of a lattice model for polymer melts, i. e., the bond fluctuation model in three dimensions. By using an energy parameter that prefers relatively long bonds, the model exhibits a glass transition at low temperatures, in close qualitative similarity to experiment. We modify this model by adding an attractive interaction of variable strength. We demonstrate that a small interaction strength has only a very small effect on the static properties of the melt. For a fixed strength of the potential, the chemical potential is measured by a modified particle-insertion method over a large range of temperatures and densities. The osmotic pressure is obtained by therm…

Symmetric diblock copolymers confined into thin films: A Monte Carlo investigation on the CRAY T3E

We present the results of large scale computer simulations targeted at investigating the phase stability and the structure of symmetric AB diblock copolymers in thin films. The connectivity of the two different monomer species A and B in the diblock copolymer prevents macrophage separation and the molecules assemble into A-rich and B-rich domains on the scale of the molecule’s extension. This large length scale of the ordering phenomena makes these polymeric systems a promising candidate for revealing the universal features of self-assembling in amphiphilic molecules. However, the widely spread length and time scales impart protracted long relaxation times to the systems and pose a challeng…

Appendix: listing of programs mentioned in the text

Bridging the Gap Between Atomistic and Coarse-Grained Models of Polymers: Status and Perspectives

Recent developments that increase the time and distance scales accessible in the simulations of specific polymers are reviewed. Several different techniques are similar in that they replace a model expressed in fully atomistic detail with a coarse-grained model of the same polymer, atomistic → coarse-grained (and beyond!), thereby increasing the time and distance scales accessible within the expenditure of reasonable computational resources. The bridge represented by the right-pointing arrow can be constructed via different procedures, which are reviewed here. The review also considers the status of methods which reverse this arrow, atomistic ← coarse-grained. This “reverse-mapping” recover…

Monte Carlo simulation of micelle formation in block copolymer solutions

Short block copolymers in selective solvents (bad for A-block, good for B-block) are modeled by flexible bead-spring chains, where beads interact with short range Morse potentials of variable strength. It is shown that already very short chains (N A = N B = 2) exhibit a rather well-defined critical micelle concentration (cmc). The mass distribution of the micelles and their gyration tensor components as well as their internal structure are studied. It is shown that the relaxation time increases exponentially with the strength E AA of the attractive energy between the A-monomers, and thus frozen-in micelles of medium size are obtained when E AA is chosen too large. Our results are compared t…

Grafted polymer layers under shear: A Monte Carlo simulation

Endgrafted polymers at surfaces exposed to a shear flow are modeled by a nonequilibrium Monte Carlo method where the jump rate of effective monomers to neighboring lattice sites against the flow direction is smaller than in the flow direction, assuming that this difference in jump rates is proportional to the local velocity of the flowing fluid. In the dilute case of isolated chains, the velocity profile is assumed linearly increasing with the distance from the surface, while for the case of polymer brushes the screening of the velocity field is calculated using a parabolic density profile for the brush whose height is determined self‐consistently. Linear dimensions of isolated chains are o…

Polymer-brush lubricated surfaces with colloidal inclusions under shear inversion.

We characterize the response of compressed, sheared polymer-brush bilayers with colloidal inclusions to highly nonstationary inversion processes by means of molecular dynamics simulations and scaling theory. Bilayers with a simple (dimeric) solvent reveal an overshoot for the shear stress, while simulations of dry brushes without explicit solvent molecules fail to display this effect. We demonstrate that mechanical instabilities can be controlled by the inclusion of macromolecular structures, such as colloids of varying softness. Based on a recently developed theory, we suggest a scaling approach to determine a characteristic time for conformational and collective responses.

Droplets pinned at chemically inhomogenous substrates: A simulation study of the two-dimensional Ising case

As a simplified model of a liquid nanostripe adsorbed on a chemically structured substrate surface, a two-dimensional Ising system with two boundaries at which surface fields act is studied. At the upper boundary, the surface field is uniformly negative, while at the lower boundary (a distance L apart), the surface field is negative only outside a range of extension b, where a positive surface stabilizes a droplet of the phase with positive magnetization for temperatures T exceeding the critical temperature Tw of the wetting transition of this model. We investigate the local order parameter profiles across the droplet, both in the directions parallel and perpendicular to the substrate, vary…

Phase diagram and structure of colloid-polymer mixtures confined between walls

The influence of confinement, due to flat parallel structureless walls, on phase separation in colloid-polymer mixtures, is investigated by means of grand-canonical Monte Carlo simulations. Ultra-thin films, with thicknesses between $D=3-10$ colloid diameters, are studied. The Asakura-Oosawa model [J. Chem. Phys. 22, 1255 (1954)] is used to describe the particle interactions. To simulate efficiently, a ``cluster move'' [J. Chem. Phys. 121, 3253 (2004)] is used in conjunction with successive umbrella sampling [J. Chem. Phys. 120, 10925 (2004)]. These techniques, when combined with finite size scaling, enable an accurate determination of the unmixing binodal. Our results show that the critica…

Transitions of tethered chain molecules under tension

An applied tension force changes the equilibrium conformations of a polymer chain tethered to a planar substrate and thus affects the adsorption transition as well as the coil-globule and crystallization transitions. Conversely, solvent quality and surface attraction are reflected in equilibrium force-extension curves that can be measured in experiments. To investigate these effects theoretically, we study tethered chains under tension with Wang-Landau simulations of a bond-fluctuation lattice model. Applying our model to pulling experiments on biological molecules we obtain a good description of experimental data in the intermediate force range, where universal features dominate and finite…

Polymer brushes under flow and in other out-of-equilibrium conditions

Polymer brushes are formed from flexible linear macromolecules tethered at one chain end to a solid substrate, forming a dense polymeric layer of polymer chains which are more or less stretched in the direction perpendicular to the substrate surface. These systems find interest also due to numerous applications (colloid stabilization, improvement of lubrication properties when the surfaces are exposed to shear, protection of the surface against adsorption of nanoparticles or proteins, etc.), for which often the dynamic non-equilibrium response of these brushes to external perturbation is important. The present review summarizes recent computer simulation studies pertinent to these questions…

Dynamical Heterogeneities Below the Glass Transition

We present molecular dynamics simulations of a binary Lennard-Jones mixture at temperatures below the kinetic glass transition. The ``mobility'' of a particle is characterized by the amplitude of its fluctuation around its average position. The 5% particles with the largest/smallest mean amplitude are thus defined as the relatively most mobile/immobile particles. We investigate for these 5% particles their spatial distribution and find them to be distributed very heterogeneously in that mobile as well as immobile particles form clusters. The reason for this dynamic heterogeneity is traced back to the fact that mobile/immobile particles are surrounded by fewer/more neighbors which form an ef…

Spinodal decomposition of polymer solutions: A parallelized molecular dynamics simulation

In simulations of phase separation kinetics, large length and time scales are involved due to the mesoscopic size of the polymer coils, and the structure formation on still larger scales of length and time. We apply a coarse-grained model of hexadecane dissolved in supercritical carbon dioxide, for which in previous work the equilibrium phase behavior has been established by Monte Carlo methods. Using parallelized simulations on a multiprocessor supercomputer, large scale molecular dynamics simulations of phase separation following pressure jumps are presented for systems containing $N=435\phantom{\rule{0.2em}{0ex}}136$ coarse-grained particles, which correspond to several millions of atoms…

Anomalous size-dependence of interfacial profiles between coexisting phases of polymer mixtures in thin-film geometry: A Monte Carlo simulation

The interfacial profile between coexisting phases of a binary mixture (A,B) in a thin film of thickness D and lateral linear dimensions L depends sensitively on both linear dimensions and on the nature of boundary conditions and statistical ensembles applied. These phenomena generic for systems in confined geometry are demonstrated by Monte-Carlo simulations of the bond fluctuation model of symmetric polymer mixtures. Both the canonical and semi-grand-canonical ensemble are studied. In the canonical ensemble, the interfacial width w increases (from small values which are of the same order as the intrinsic profile) like sqrt{D}, before a crossover to a saturation value w_max (w_max^2 proport…

Disordered and Frustrated Spin Systems

A brief review on the effects of quenched disorder on magnetic ordering is given. This disorder can be due to dilution of a ferro- or antiferromagnetic crystal with nonmagnetic atoms, or due to noncrystallinity (amorphous magnetic systems). This disorder in the positions of the magnetic atoms leads to disorder in the exchange interactions between spins. If the disorder is sufficiently weak, the critical temperature of magnetic ordering is somewhat decreased, and the critical behavior may change, but the nature of ordering is maintained. However, if the disorder is sufficiently strong, magnetic long-range order may disappear altogether at a percolation threshold, or a new type of order may a…

Conformational Properties of Semiflexible Chains at Nematic Ordering Transitions in Thin Films: A Monte Carlo Simulation

Athermal solutions of semiflexible macromolecules with excluded volume interactions and with varying concentration (dilute, semidilute, and concentrated solutions) in a film of thickness D between ...

Critical phenomena in polymer mixtures: Monte Carlo simulation of a lattice model

A lattice model of a symmetrical binary (AB) polymer mixture is studied, modelling the polymer chains by self-avoiding walks withN A =N B =N steps on a simple cubic lattice. If a pair of nearest neighbour sites is taken by different monomersAB orBA, an energye ab is won; if the pair of sites is taken by anAA or aBB pair, an energye is won, while the energy is reduced to zero if at least one of the sites of the pair is vacant. To allow enough chain mobility, 20% of the lattice sites are vacancies. In addition to local motions of the chain segments we use a novel “grand-canonical” simulation technique:A chains are transformed intoB chains and vice versa, keeping the chemical potential differe…

On the Adsorption Process in Polymer Brushes:  A Monte Carlo Study

The adsorption process of the single polymer chain in a polymer brush of varying surface coverages is studied by means of Monte Carlo simulations of the bond-fluctuation lattice model. Only the end monomers can adsorb at the grafting surface, whereas inner monomers interact repulsively with it. The brush builds up a steric hindrance which forces the penetrating polymer to stretch strongly and which is responsible for small adsorption probabilities at surface coverages close to the overlap density. The final step of the adsorption process is determined by a fluctuation of the end monomer around its average position, which is comparable to the initial step of the desorption process.

The intermediate coherent scattering function of entangled polymer melts: a Monte Carlo test of des Cloizeaux' theory

Using the bond fluctuation model for flexible polymer chains in a dense melt the intermediate coherent scattering function for chains containing N=200 monomers is calculated and interpreted in terms of a recent theory of des Cloizeaux. The theory yields an explicit description for the crossover from the Rouse model to the regime where reptation prevails, for the limit N→∞. While the Monte Carlo data are qualitatively compatible with this description, an accurate estimation of the tube diameter is prevented due to the onset of a diffusive decay of the scattering function, not included in the theory. For a full quantitative analysis of the Monte Carlo data (as well as of experiments on chains…

Melting transition in two dimensions: A finite-size scaling analysis of bond-orientational order in hard disks

We describe a general and efficient method, based on computer simulations and applicable to a general class of fluids, that allows us to determine (i) bounds on the transition densities of the melting transition that are valid in the thermodynamic limit and (ii) the order of the phase transition. The bond-orientational order parameter, its susceptibility, and the compressibility are measured simulataneously on many length scales, and the latter two quantities are extrapolated to the thermodynamic limit by application of the subblock analysis method of finite-size scaling. We include a detailed analysis, related to the subblock method, of the cross correlations of the fluctuations of the den…

Equilibrium between a Droplet and Surrounding Vapor: A Discussion of Finite Size Effects

In a theoretical description of homogeneous nucleation one frequently assumes an "equilibrium" coexistence of a liquid droplet with surrounding vapor of a density exceeding that of a saturated vapor at bulk vapor-liquid two-phase coexistence. Thereby one ignores the caveat that in the thermodynamic limit, for which the vapor would be called supersaturated, such states will at best be metastable with finite lifetime, and thus not be well-defined within equilibrium statistical mechanics. In contrast, in a system of finite volume stable equilibrium coexistence of droplet and supersaturated vapor at constant total density is perfectly possible, and numerical analysis of equilibrium free energie…

Semiflexible Polymers Interacting with Planar Surfaces: Weak versus Strong Adsorption

Semiflexible polymers bound to planar substrates by a short-range surface potential are studied by Molecular Dynamics simulations to clarify the extent to which these chain molecules can be considered as strictly two-dimensional. Applying a coarse-grained bead-spring model, the chain length N and stiffness &kappa

Monte Carlo simulation of block copolymers

Monte Carlo simulations deal with crudely simplified but well-defined models and have the advantage that they treat the statistical thermodynamics of the considered model exactly (apart from statistical errors and problems due to finite size effects). Therefore, these simulations are well suited to test various approximate theories of block copolymer ordering, e.g. the self-consistent field theory. Recent examples of this approach include the study of block copolymer ordering at melt surfaces and confinement effects in thin films, adsorption of block copolymers at interfaces of unmixed homopolymer blends, the phase behavior of ternary mixtures of two homopolymers and their block copolymer, …

Controlling the wetting properties of the Asakura-Oosawa model and applications to spherical confinement.

We demonstrate for the Asakura-Oosawa model and an extension of this model that uses continuous rather than hard potentials, how wetting properties at walls can be easily controlled. By increasing the interaction range of the repulsive wall potential acting on the colloids (while keeping the polymer-wall interactions constant) polymers begin to substitute colloids at walls and the system can be driven from complete wetting of colloids via partial wetting to complete wetting of polymers. As an application, we discuss the morphology and wetting behavior of colloid-polymer mixtures in spherical confinement. We apply the recently developed 'ensemble switch method' where the Hamiltonian is exten…

Multiscale modeling of polymers at interfaces

A brief review of modeling and simulation methods for a study of polymers at interfaces is provided. When studying truly multiscale problems as provided by realistic polymer systems, coarse graining is practically unavoidable. In this process, degrees of freedom on smaller scales are eliminated to the favor of a model suitable for efficient study of the system behavior on larger length and time scales. We emphasize the need to distinguish between dynamic and static properties regarding the model validation. A model which accurately reproduces static properties may fail completely, when it comes to the dynamic behavior of the system. Furthermore, we comment on the use of Monte Carlo method i…

Mechanisms for the Dynamics of Phase Transformations

An introductory review of the dynamics of (first- order) phase transitions is given. Concepts describing the initial stages of the transition, such as nucleation, spinodal decomposition (in the case of unmixing) are introduced, and their validity is critically discussed. The theoretical results are compared to recent computer simulations and pertinent experiments. Then the scaling concepts describing the late stages of domain growth are discussed, and open problems are outlined.

Structure Formation of Polymeric Building Blocks: Complex Polymer Architectures

This chapter describes macromolecules with a complex structure, their defined aggregation in solution, their adsorption to surfaces, and their possible aggregation on surfaces. The term “complex structure” implies that the macromolecules show different, distinct structural elements or building blocks on a supra-atomic length scale. Key to understanding the complex structure of macromolecules, their aggregation, and adsorption to surfaces are intra- and intermolecular interactions such as van der Waals, electrostatic, π–π interactions, and hydrogen bonds.

Molecular-dynamics simulation of a glassy polymer melt: Rouse model and cage effect

We report results of molecular-dynamics simulations for a glassy polymer melt consisting of short, linear bead-spring chains. It was shown in previous work that this onset of the glassy slowing down is compatible with the predictions of the mode coupling theory. The physical process of `caging' of a monomer by its spatial neighbors leads to a distinct two step behavior in the particle mean square displacements. In this work we analyze the effects of this caging process on the Rouse description of the melt's dynamics. We show that the Rouse theory is applicable for length and time scales above the typical scales for the caging process. Futhermore, the monomer displacement is compared with si…

Glass transition of polymer melts: Test of theoretical concepts by computer simulation.

Abstract Polymers are good glass formers and allow for the study of melts near the glass transition in (meta-)stable equilibrium. Theories of the glass transition imply such an equilibrium and can, hence, be tested by the study of polymer melts. After a brief summary of the basic experimental facts about the glass transition in polymers, the main theoretical concepts are reviewed: mode coupling theory (MCT), entropy theory, free-volume theory, the idea of a growing length describing the size of cooperative regions, etc. Then, two basic coarse-grained models of polymers are described, which have been developed aiming at a test of these concepts. The first model is the bond-fluctuation model …

Critical Wetting and Interface Localization—Delocalization Transition in a Double Wedge

Using Monte Carlo simulations and finite-size scaling methods we study “wetting” in Ising systems in a L x L x L y pore with quadratic cross section. Antisymmetric surface fields H s act on the free L x L y surfaces of the opposing wedges, and periodic boundary conditions are applied along the y-direction. Our results represent the first simulational observation of fluctuation effects in three dimensional wetting phenomena and corroborate recent predictions on wedge filling. In the limit L → ∞ L y /L 3 = const the system exhibits a new type of phase transition, which is the analog of the “filling transition” that occurs in a single wedge. It is characterized by critical exponents α = 3/4, β…

Finite-size scaling analysis of the anisotropic critical behavior of the two-dimensional Ising model under shear

The critical behavior of the two-dimensional Ising Model with non-conserved order parameter in steady-state shear is studied by large-scale Monte Carlo simulations. Studying the structure factor S(qx,qy) in the disordered phase, the ratio of correlation length exponents νx/νy in the two lattice directions (x,y) is estimated, and the critical temperature is determined as a function of the shear rate as Tc() − Tc(0) ∝ with ≈0.45. Critical exponents β≈0.37, γ≈1.1, ; ν⊥≈0.46, ν∥≈1.38 are roughly compatible with anisotropic hyperscaling.

Transitions between imperfectly ordered crystalline structures: A phase switch Monte Carlo study

A model for two-dimensional colloids confined laterally by ``structured boundaries'' (i.e., ones that impose a periodicity along the slit) is studied by Monte Carlo simulations. When the distance $D$ between the confining walls is reduced at constant particle number from an initial value ${D}_{0}$, for which a crystalline structure commensurate with the imposed periodicity fits, to smaller values, a succession of phase transitions to imperfectly ordered structures occur. These structures have a reduced number of rows parallel to the boundaries (from $n$ to $n\ensuremath{-}1$ to $n\ensuremath{-}2$, etc.) and are accompanied by an almost periodic strain pattern, due to ``soliton staircases'' …

Glassy dynamics in thin polymer films: recent MD results

The influence of a film geometry on the glass transition is investigated via molecular dynamics (MD) simulations of a (non-entangled) polymer melt. The confinement is realized by two identical potential barriers of the form U wall = z -9 , where z denotes the distance of a particle from the wall. Despite the geometric confinement, basic qualitative features of the system dynamics can be well described in the framework of the mode-coupling theory (MCT). Examples are the two-step relaxation of the incoherent intermediate scattering function, the time-temperature superposition property of the late time α-process and the space-time factorization of the scattering function on the intermediate ti…

Monte Carlo Simulations of Alloy Phase Transformations

The use of Monte Carlo simulation methods for study of order-disorder phase transitions in lattice models of alloys is reviewed, with an emphasis on interfacial phenomena and the kinetics of ordering and/or phase separation. Topics discussed include the attempt to predict the phase diagram of Fe-Al alloys from recent measurements of effective interaction parameters, competition between magnetic and crystallographic ordering in such alloys, and the structure of their antiphase domain boundaries. Both an interfacial roughening transition of this domain wall and interfacial enrichment phenomena are predicted. Then simulations of alloy-vacuum surfaces are discussed, and it is shown that both ca…

Confinement effects on phase behavior of soft matter systems.

When systems that can undergo phase separation between two coexisting phases in the bulk are confined in thin film geometry between parallel walls, the phase behavior can be profoundly modified. These phenomena shall be described and exemplified by computer simulations of the Asakura-Oosawa model for colloid-polymer mixtures, but applications to other soft matter systems (e.g. confined polymer blends) will also be mentioned. Typically a wall will prefer one of the phases, and hence the composition of the system in the direction perpendicular to the walls will not be homogeneous. If both walls are of the same kind, this effect leads to a distortion of the phase diagram of the system in thin …

Suspensions of rod-like colloids and a depleting agent under confinement

We present a computer simulation study of suspensions of rod-like colloids and a depletant in confinement to a slit-pore. Mixtures of hard spherocylinders and ideal spheres were studied by means of Monte Carlo simulations in the grand canonical ensemble. By use of finite size scaling analysis we determined the critical behaviour. In order to overcome large barriers in the free energy we applied the successive umbrella sampling method (Virnau and Muller 2004 J. Chem. Phys. 120 10925). We find that, under confinement, the critical point of gas–liquid demixing shifts to higher concentrations of rods and smaller concentrations of spheres due to the formation of an orientationally ordered surfac…

Critical adsorption of a single macromolecule in polymer brushes.

The adsorption of long flexible macromolecules by polymer brush-coated surfaces is studied by molecular dynamics simulations and by calculations using density functional and self-consistent field theories. The case of repulsive interactions between the substrate surface and the monomers of both the brush polymers and the extra chains that can get absorbed into the brush is considered. Under good solvent conditions, critical absorption can occur, if the interaction between the monomers of the brush polymers and the extra chain is (weakly) attractive. It is shown that it is possible to map out the details of the critical absorption transition, if the chain length and/or the grafting density o…

Thin Ising films with competing walls: A Monte Carlo study.

Ising magnets with a nearest neighbor ferromagnetic exchange interaction J on a simple cubic lattice are studied in a thin film geometry using extensive Monte Carlo simulations. The system has two large L\ifmmode\times\else\texttimes\fi{}L parallel free surfaces, a distance D apart from each other, at which competing surface fields act, i.e., ${\mathit{H}}_{\mathit{D}}$=-${\mathit{H}}_{1}$. In this geometry, the phase transition occurring in the bulk at a temperature ${\mathit{T}}_{\mathit{c}\mathit{b}}$ is suppressed, and instead one observes the gradual formation of an interface between coexisting phases stabilized by the surface fields. While this interface is located in the center of th…

Scaling theory for radial distributions of star polymers in dilute solution in the bulk and at a surface, and scaling of polymer networks near the adsorption transition

Monomer density profiles ρ(r) and center–end distribution functions g(rCE) of star polymers are analyzed by using a scaling theory in arbitrary dimensions d, considering dilute solutions and the good solvent limit. Both the case of a free star in the bulk and of a center‐adsorbed star at a free surface are considered. In the latter case of a semi‐infinite problem, a distinction is made between repulsive walls, attractive walls—where for large arm length l the configuration of the star is quasi‐(d−1) dimensional—, and ‘‘marginal walls’’ where for l→∞ the transition from d‐dimensional structure occurs. For free stars, ρ(r) behaves as r−d+1/ν for small r, where ν is the exponent describing the…

More on importance sampling Monte Carlo methods for lattice systems

Motion, relaxation dynamics, and diffusion processes in two-dimensional colloidal crystals confined between walls

The dynamical behavior of single-component two-dimensional colloidal crystals confined in a slit geometry is studied by Langevin dynamics simulation of a simple model. The colloids are modeled as pointlike particles, interacting with the repulsive part of the Lennard-Jones potential, and the fluid molecules in the colloidal suspension are not explicitly considered. Considering a crystalline strip of triangular lattice structure with n=30 rows, the (one-dimensional) walls confining the strip are chosen as two rigidly fixed crystalline rows at each side, commensurate with the lattice structure and, thus, stabilizing long-range order. The case when the spacing between the walls is incommensura…

Energetic analysis of succinic acid in water droplets: insight into the size-dependent solubility of atmospheric nanoparticles

&amp;lt;p&amp;gt;Size-dependent solubility is prevalent in atmospheric nanoparticles, but a molecular level understanding is still insufficient, especially for organic compounds. Here, we performed molecular dynamics simulations to investigate the size dependence of succinic acid solvation on the scale of ~1-4 nm with the potential of mean forces method. Our analyses reveal that the surface preference of succinic acid is stronger for a droplet than the slab of the same size, and the surface propensity is enhanced due to the curvature effect as the droplet becomes smaller. Energetic analyses show that such surface preference is primarily an enthalpic effect in both systems, while the entropi…

Quantum Monte Carlo Simulations: An Introduction

To be specific, let us consider for the moment the problem of N atoms in a volume V at temperature T, and we wish to calculate the average of some observable A which in quantum mechanics is described by an operator Â.

Molecular-dynamics computer simulation of crystal growth and melting in Al 50 Ni 50

The melting and crystallization of Al50Ni50} are studied by means of molecular dynamics computer simulations, using a potential of the embedded atom type to model the interactions between the particles. Systems in a slab geometry are simulated where the B2 phase of AlNi in the middle of an elongated simulation box is separated by two planar interfaces from the liquid phase, thereby considering the (100) crystal orientation. By determining the temperature dependence of the interface velocity, an accurate estimate of the melting temperature is provided. The value k=0.0025 m/s/K for the kinetic growth coefficient is found. This value is about two orders of magnitude smaller than that found in …

Incommensurate phases in adsorbed monolayers: structure and energy of domain walls

Abstract The properties of incommensurate films of domain-wall structure formed on the (1 0 0) plane of face centered cubic crystals are studied by Monte Carlo simulation. The wall energies, wall structure and the wall–wall interaction are determined for different types of domain walls occurring in films which form the c(2×2) registered structure. The systems characterized by different strength and corrugation of the surface potential and of different misfit between adsorbate and adsorbent are discussed. It is demonstrated that heavy as well as light walls are rather strongly localized. Moreover, it is shown that the incommensurate structure with crossing heavy walls has higher stability th…

Computer Simulations and Coarse-Grained Molecular Models Predicting the Equation of State of Polymer Solutions

Monte Carlo and molecular dynamics simulations are, in principle, powerful tools for carrying out the basic task of statistical thermodynamics, namely the prediction of macroscopic properties of matter from suitable models of effective interactions between atoms and molecules. The state of the art of this approach is reviewed, with an emphasis on solutions of rather short polymer chains (such as alkanes) in various solvents. Several methods of constructing coarse-grained models of the simple bead–spring type will be mentioned, using input either from atomistic models (considering polybutadiene as an example) or from experiment. Also, the need to have corresponding coarse-grained models of t…

Entropy of glassy polymer melts: Comparison between Gibbs-DiMarzio theory and simulation.

We calculate the free energy of a model for a polymer melt in a computer simulation of the bond-fluctuation model and determine the entropy of the melt over a wide range of temperatures, including the region close to the glass transition. The results are compared with the Gibbs-DiMarzio theory, a theory by Flory for semiflexible polymers, and a modification of their theories due to Milchev. We can describe the data within the framework of the Flory theory with Milchev's correction and discuss the consequences for the understanding of the glass transition. \textcopyright{} 1996 The American Physical Society.

Densely packed semiflexible macromolecules in a rigid spherical capsule

The ordering of semiflexible polymers with persistence length lp and contour length L confined in a sphere of radius R is studied by molecular dynamics simulations of a coarse-grained model. Monomer densities are chosen where the corresponding bulk lyotropic solution or melt is a well-ordered nematic, and purely repulsive walls of the rigid confining sphere are considered. It is found that polymers close to the walls are bent according to the curvature of the confining spheres with all their monomers in a few layers parallel to the sphere surface, whereas the remaining macromolecules closer to the sphere center have one chain end and their center of mass far from the surface. The latter cha…

Properties of the interface in the confined Ising magnet with competing surface fields

Abstract A two-dimensional magnetic Ising system confined in an L × D geometry ( L ⪡ D ) in the presence of competing magnetic fields (h) acting at opposite walls along the D -direction, exhibits an interface between domains of different orientation that run parallel to the walls. In the limit L → ∞ , this interface undergoes a wetting transition that occurs at the critical curve T w ( h ) , so that for T T w ( h ) such an interface is bound to the walls, while for T w ( h ) ⩽ T T cb the interface is freely fluctuating around the center of the film, where T cb is the bulk critical temperature. By considering both short- and long-range magnetic fields acting at the walls, we study the diverg…

Chain Conformations and Phase Behavior in Confined Polymer Blends

We investigate the chain conformations and phase separation in binary polymer blends. Using large scale semi-grandcanonical Monte Carlo simulations and finite size scaling, we investigate the molecular extension and the intermolecular paircorrelation function in thin films with hard, non-preferentially adsorbing surfaces. The interplay between chain conformations, demixing and the validity of mean field theory is investigated for a large variation of chain lengths 16 ≤ N ≤ 512. Three regimes of film thickness D can be distinguished: (i) For film thicknesses much larger than the unperturbed chain extension R e, bulk behavior is observed, i.e., the critical temperature of demixing T c increas…

Dynamics of single semiflexible polymers in dilute solution

We study the dynamics of a single semiflexible chain in solution using computer simulations, where we systematically investigate the effect of excluded volume, chain stiffness, and hydrodynamic interactions. We achieve excellent agreement with previous theoretical considerations, but find that the crossover from the time τb, up to which free ballistic motion of the monomers describes the chain dynamics, to the times W−1 or τ0, where anomalous monomer diffusion described by Rouse-type and Zimm-type models sets in, requires two decades of time. While in the limit of fully flexible chains the visibility of the anomalous diffusion behavior is thus rather restricted, the t3/4 power law predicted…

Modeling glass materials

Abstract Structural and dynamic properties of silicate melts and glasses (SiO 2 and its mixtures with Na 2 O and Al 2 O 3 ) are derived from Molecular Dynamics simulations and compared to pertinent experimental data. It is shown that these mixtures exhibit additional intermediate order as compared to pure silica, where the characteristic length scales stem from the tetrahedral network structure. While sodium ions show much faster diffusion through percolating channels than the silicon and oxygen ions forming the surrounding network, aluminium ions are incorporated into the network (leading to tricluster formation) and do not show such an enhanced mobility.

Colloid-polymer mixtures between asymmetric walls: Evidence for an interface localization transition

We demonstrate via computer simulation that mixtures of colloids and polymers confined to thin films have the ability to undergo an interface localization transition. While one wall of the film is assumed to be hard for both particles, at the other wall, an additional repulsive potential acts, but on the colloids only. By varying the strength of this repulsion, a crossover from capillary condensation to interface localization is found. The latter occurs under conditions where in the bulk almost complete phase separation has occurred.

Dynamics of Polymer Chains Confined in Slit-Like Pores

Monte Carlo simulations of an off-lattice bead spring model of polymer chains are presented, confining the chains between two repulsive parallel planes a distance D apart. Varying the chain length N from N = 16 to N = 128, we show that under good solvent conditions the chains behave like two-dimensional self-avoiding walks, their mean square gyration radius scales as (R g 2 ) N 2v with v = 3/4. The density profile across the slit is independent of N and maximal in the center of the slit. The dynamical properties of the chains are found to be in full agreement with the Rouse model with excluded volume in d = 2 dimensions, the relaxation times vary like τ N Z with z = 2v +1 = 5/2, the diffusi…

Probing predictions due to the nonlocal interface Hamiltonian: Monte Carlo simulations of interfacial fluctuations in Ising films

Extensive Monte Carlo simulations have been performed on an Ising ferromagnet under conditions that would lead to complete wetting in a semi-infinite system. We studied an L×L×D slab geometry with oppositely directed surface fields so that a single interface is formed and can undergo a localization-delocalization transition. Under the chosen conditions the interface position is, on average, in the middle of the slab, and its fluctuations allow a sensitive test of predictions that the effective interactions between the interface and the confining surfaces are nonlocal. The decay of distance dependent correlation functions are measured within the surface, in the middle of the slab, and betwee…

Logarithmic finite-size effects on interfacial free energies: Phenomenological theory and Monte Carlo studies

The computation of interfacial free energies between coexisting phases (e.g.~saturated vapor and liquid) by computer simulation methods is still a challenging problem due to the difficulty of an atomistic identification of an interface, and due to interfacial fluctuations on all length scales. The approach to estimate the interfacial tension from the free energy excess of a system with interfaces relative to corresponding single-phase systems does not suffer from the first problem but still suffers from the latter. Considering $d$-dimensional systems with interfacial area $L^{d-1}$ and linear dimension $L_z$ in the direction perpendicular to the interface, it is argued that the interfacial …

Surface effects on phase transitions of modulated phases and at Lifshitz points: A mean field theory of the ANNNI model

The semi-infinite axial next nearest neighbor Ising (ANNNI) model in the disordered phase is treated within the molecular field approximation, as a prototype case for surface effects in systems undergoing transitions to both ferromagnetic and modulated phases. As a first step, a discrete set of layerwise mean field equations for the local order parameter mn in the nth layer parallel to the free surface is derived and solved, allowing for a surface field H1 and for interactions JS in the surface plane which differ from the interactions J0 in the bulk, while only in the z-direction perpendicular to the surface competing nearest neighbor ferromagnetic exchange (J1) and next nearest neighbor an…

Numerical evidence of hyperscaling violation in wetting transitions of the random-bond Ising model in d = 2 dimensions

We performed extensive simulations of the random-bond Ising model confined between walls where competitive surface fields act. By properly taking the thermodynamic limit we unambiguously determined wetting transition points of the system, as extrapolation of localization-delocalization transitions of the interface between domains of different orientation driven by the respective fields. The finite-size scaling theory for wetting with short-range fields establishes that the average magnetization of the sample, with critical exponent β, is the proper order parameter for the study of wetting. While the hyperscaling relationship given by γ+2β=ν +ν requires β=1/2 (γ=4, ν =3, and ν =2), the therm…

Finite Size Effects in Thin Film Simulations

Phase transitions in thin films are discussed, with an emphasis on Ising-type systems (liquid-gas transition in slit-like pores, unmixing transition in thin films, orderdisorder transitions on thin magnetic films, etc.) The typical simulation geometry then is a L xL x D system, where at the low confining L x L surfaces appropriate boundary “fields” are applied, while in the lateral directions periodic boundary conditions are used. In the z-direction normal to the film, the order parameter always is inhomogeneous, due to the boundary “fields” at the confining surfaces. When one varies the temperature T from the region of the bulk disordered phase to a temperature below the critical temperatu…

Adsorption and structure formation of semiflexible polymers on spherical surfaces

Abstract Rigid spheres with a short-range attractive potential are taken as a coarse-grained model of vesicles, which contain a solution of semiflexible polymers in their interior. Assuming good solvent conditions with an implicit description of the solvent, effective monomers experience bond-length and bond-angle potentials as well as excluded-volume interaction. Due to the attractive vesicle surface, phase separation occurs between a thin shell of adsorbed monomers at the surface and a rather dilute, and therefore, disordered polymer solution in the sphere interior. While at a planar attractive surface the wormlike chains would exhibit liquid crystalline (nematic and smectic) order, the c…

Polymer absorption in dense polymer brushes vs. polymer adsorption on the brush-solvent interface

Molecular-dynamics simulations of a coarse-grained model of a dense brush of flexible polymers (of type A) interacting with a long flexible macromolecule (of type B) are presented, considering the case of an attractive AB interaction, while effective interactions between AA and BB pairs of monomers are repulsive. Varying the strength of the attraction between unlike monomers, an adsorption transition at some critical value is found, where the B-chain is bound to the brush-solvent interface, similar to the adsorption on a planar solid substrate. However, when is much higher than , the long macromolecule is gradually “sucked in” the brush, developing many pieces that are locally stretched in …

Anomalous scaling of the critical temperature of unmixing with chain length for two-dimensional polymer blends

The thermodynamics, structure and the chain configurations of symmetrical polymer mixtures confined into ultrathin films are studied by Monte Carlo simulations of the bond fluctuation model. It is shown that the Flory-Huggins–type scaling of the critical temperature (Tc ~ N) with chain length N in the bulk is replaced by a weaker increase, Tc ~ N1/2, in an ultrathin film, and this is interpreted in terms of geometric arguments. The pair-correlation function g(r) of monomers from different chains exhibits a pronounced correlation hole, and the density of intermolecular contacts zc decreases with N as zc ~ N−1/2.

Microcanonical Determination of the Interface Tension of Flat and Curved Interfaces from Monte Carlo Simulations

The investigation of phase coexistence in systems with multi-component order parameters in finite systems is discussed, and as a generic example, Monte Carlo simulations of the two-dimensional q-state Potts model (q=30) on LxL square lattices (40&lt;=L&lt;=100) are presented. It is shown that the microcanonical ensemble is well-suited both to find the precise location of the first order phase transition and to obtain an accurate estimate for the interfacial free energy between coexisting ordered and disordered phases. For this purpose, a microcanonical version of the heatbath algorithm is implemented. The finite size behaviour of the loop in the curve describing the inverse temperature vers…

PHASE EQUILIBRIA IN THIN POLYMER FILMS

Within self-consistent field theory and Monte Carlo simulations the phase behavior of a symmetrical binary AB polymer blend confined into a thin film is studied. The film surfaces interact with the monomers via short ranged potentials. One surface attracts the A component and the corresponding semi-infinite system exhibits a first order wetting transition. The surface interaction of the opposite surface is varied as to study the crossover from capillary condensation for symmetric surface fields to interface localization/delocalization transition for antisymmetric surface fields. In the former case the phase diagram has a single critical point close to the bulk critical point. In the latter…

Effects of confinement and external fields on structure and transport in colloidal dispersions in reduced dimensionality

In this work, we focus on low-dimensional colloidal model systems, via simulation studies and also some complementary experiments, in order to elucidate the interplay between phase behavior, geometric structures and transport properties. In particular, we try to investigate the (nonlinear!) response of these very soft colloidal systems to various perturbations: uniform and uniaxial pressure, laser fields, shear due to moving boundaries and randomly quenched disorder.We study ordering phenomena on surfaces or in monolayers by Monte Carlo computer simulations of binary hard-disk mixtures, the influence of a substrate being modeled by an external potential. Weak external fields allow a control…

Semidilute and Concentrated Polymer Solutions near Attractive Walls:  Dynamic Monte Carlo Simulation of Density and Pressure Profiles of a Coarse-Grained Model

Using a bead−spring model of flexible polymer chains, we study polymer adsorption from solutions onto attractive planar walls, varying both the strength of the adsorption potential e and the concentration of the solution over a wide range. Treating the case of good solvents, the profiles of density and pressure are computed and it is shown that thermal equilibrium between the adsorbed layer and the bulk solution is obtained. The case of a wall with purely repulsive potential under otherwise identical conditions is treated for comparison. It is shown that for the strongly adsorbing wall there is a pronounced layering, while a layered structure at the repulsive wall occurs only for high conce…

Lattice gas models for multilayer adsorption: variation of phase diagrams with the strength of the substrate potential

Abstract The simple cubic lattice gas model with nearest-neighbor attractive interaction is considered for the case where the potential V ( z ), that an adatom at a distance z from the surface experiences due to the substrate, is V ( z ) = − A / z 3 . Exact ground state phase diagrams are obtained for different A , while the behavior at nonzero temperatures is studied both by Monte Carlo simulations and the molecular field approximation. We show that the detailed sequence of the layering transitions in the first few layers depends very strongly on the strength of the substrate potential: for strong potentials individual first-order layering transitions in layers 1, 2, 3, …, while for interm…

COMPUTER SIMULATION OF PROFILES OF INTERFACES BETWEEN COEXISTING PHASES: DO WE UNDERSTAND THEIR FINITE SIZE EFFECTS?

Interfaces between coexisting phases are very common in condensed matter physics, and thus many simulations attempt to characterize their properties, in particular, the interfacial tension and the interfacial profile. However, while theory usually deals with the "intrinsic profile", the latter is not a straightforward output of a simulation: The actual profile (observed in simulations and/or experiments!) is broadened by lateral fluctuations. Therefore, in the usual simulation geometry of L × L × L (in three dimensions), where one chooses suitable boundary conditions to stabilize one or two interfaces of (minimal) area L × L, the profile (and in particular the interfacial width) depends on…

Artificial multiple criticality and phase equilibria: an investigation of the PC-SAFT approach

The perturbed-chain statistical associating fluid theory (PC-SAFT) is studied for a wide range of temperature, T, pressure, p, and (effective) chain length, m, to establish the generic phase diagram of polymers according to this theory. In addition to the expected gas-liquid coexistence, two additional phase separations are found, termed "gas-gas" equilibrium (at very low densities) and "liquid-liquid" equilibrium (at densities where the system is expected to be solid already). These phase separations imply that in one-component polymer systems three critical points occur, as well as equilibria of three fluid phases at triple points. However, Monte Carlo simulations of the corresponding sys…

Some Finite Size Effects in Simulations of Glass Dynamics

We present the results of a molecular dynamics computer simulation in which we investigate the dynamics of silica. By considering different system sizes, we show that in simulations of the dynamics of this strong glass former surprisingly large finite size effects are present. In particular we demonstrate that the relaxation times of the incoherent intermediate scattering function and the time dependence of the mean squared displacement are affected by such finite size effects. By compressing the system to high densities, we transform it to a fragile glass former and find that for that system these types of finite size effects are much weaker.

Growth Kinetics of Wetting Layers at Surfaces

Monte Carlo simulation of lattice gas models for the wetting transitions in systems with short range forces are described. A nearest-neighbor simple cubic lattice with nonconserved “Glauber dynamics” is used, applying a slab geometry (LxL cross section). It is shown that the growth proceeds in two stages: for short times t, the thickness of the wetting layer at an initially nonwet wall increases proportional to the logarithm of the time; for t » L2(lnL)2 the thickness increases proportional to t1/2/L. Generalizations to other systems are briefly discussed. Also two-dimensional growth of a wetting film at surface steps is considered, considering “terraces” of an LxM geometry with M»L as subs…

Surface effects on spinodal decomposition in binary mixtures and the interplay with wetting phenomena.

The phase separation of binary mixtures in a semi-infinite geometry is investigated both by a phenomenological theory and by numerical calculations using a discrete equivalent of the descriptive equations. In the framework of ``model B'' (which describes solid binary mixtures), attention is paid to a proper treatment of the boundary conditions at the free surfaces. We confine ourselves to short-range surface forces and consider parameter values that correspond to both nonwet and wet surfaces in thermal equilibrium. During the initial stages of spinodal decomposition, after a quench from considering an initial condition that corresponds to a completely random concentration distribution, one …

Crossover scaling in two dimensions

We determine the scaling functions describing the crossover from Ising-like critical behavior to classical critical behavior in two-dimensional systems with a variable interaction range. Since this crossover spans several decades in the reduced temperature as well as in the finite-size crossover variable, it has up to now largely evaded a satisfactory numerical determination. Using a new Monte Carlo method, we could obtain accurate results for sufficiently large interactions ranges. Our data cover the full crossover region both above and below the critical temperature and support the hypothesis that the crossover functions are universal. Also the so-called effective exponents are discussed …

Entropic Unmixing in Nematic Blends of Semiflexible Polymers.

Binary mixtures of semiflexible polymers with the same chain length, but different persistence lengths, separate into two coexisting different nematic phases when the osmotic pressure of the lyotropic solution is varied. Molecular Dynamics simulations and Density Functional Theory predict phase diagrams either with a triple point, where the isotropic phase coexists with two nematic phases or a critical point of unmixing within the nematic mixture. The difference in locally preferred bond angles between the constituents drives this unmixing without any attractive interactions between monomers.

Dragging a Polymer Chain into a Nanotube and Subsequent Release

We present a scaling theory and Monte Carlo (MC) simulation results for a flexible polymer chain slowly dragged by one end into a nanotube. We also describe the situation when the completely confined chain is released and gradually leaves the tube. MC simulations were performed for a self-avoiding lattice model with a biased chain growth algorithm, the pruned-enriched Rosenbluth method. The nanotube is a long channel opened at one end and its diameter $D$ is much smaller than the size of the polymer coil in solution. We analyze the following characteristics as functions of the chain end position $x$ inside the tube: the free energy of confinement, the average end-to-end distance, the averag…

ON THE CALCULATION OF THE HEAT CAPACITY IN PATH INTEGRAL MONTE CARLO SIMULATIONS

In Path Integral Monte Carlo simulations the systems partition function is mapped to an equivalent classical one at the expense of a temperature-dependent Hamiltonian with an additional imaginary time dimension. As a consequence the standard relation linking the heat capacity Cv to the energy fluctuations, &lt;E2&gt;−&lt;E&gt;2, which is useful in standard classical problems with temperature-independent Hamiltonian, becomes invalid. Instead, it gets replaced by the general relation [Formula: see text] for the intensive heat capacity estimator; β being the inverse temperature and the subscript P indicates the P-fold discretization in the imaginary time direction. This heatcapacity estimator…

Monte Carlo Study of Dense Monolayer and Bilayer Films on the (100) Plane of Face-Centered Cubic Crystals

A Monte Carlo simulation method in the canonical and in the grand canonical ensembles is used to study the behavior and properties of dense monolayer and bilayer films formed on the (100) plane of model face-centered cubic crystals. Systems with different effects due to the periodicity of the gas−solid potential are considered, and the mechanism of melting in the first and the second adsorbed layer is discussed. It is demonstrated that the film structure is very sensitive to the gas−solid potential corrugation, as well as to the temperature and the surface coverage. In particular, it is shown that monolayer films formed on weakly corrugated surfaces exhibit the incommensurate (dense) phase …

Modeling polyethylene with the bond fluctuation model

This work presents an application of recently developed ideas about how to map real polymer systems onto abstract models. In our case the abstract model is the bond fluctuation model with a Monte Carlo dynamics. We study the temperature dependence of chain dimensions and of the self-diffusion behavior in the melt from high temperatures down to 200 K. The chain conformations are equilibrated over the whole temperature range, which is possible for the abstract type of model we use. The size of the chains as measured by the characteristic ratio is within 25% of experimental data. The simulated values of the chain self-diffusion coefficient have to be matched to experimental information at one …

Off-lattice models

Critical behavior of a colloid-polymer mixture confined between walls

We investigate the influence of confinement on phase separation in colloid-polymer mixtures. To describe the particle interactions, the colloid-polymer model of Asakura and Oosawa [J. Chem. Phys. 22, 1255 (1954)] is used. Grand canonical Monte Carlo simulations are then applied to this model confined between two parallel hard walls, separated by a distance D=5 colloid diameters. We focus on the critical regime of the phase separation and look for signs of crossover from three-dimensional (3D) Ising to two-dimensional (2D) Ising universality. To extract the critical behavior, finite size scaling techniques are used, including the recently proposed algorithm of Kim et al. [Phys. Rev. Lett. 91…

First-order and tricritical wetting transitions in the two-dimensional Ising model caused by interfacial pinning at a defect line

We present a study of the critical behavior of the Blume-Capel model with three spin states (S=±1,0) confined between parallel walls separated by a distance L where competitive surface magnetic fields act. By properly choosing the crystal field (D), which regulates the density of nonmagnetic species (S=0), such that those impurities are excluded from the bulk (where D=) except in the middle of the sample [where DM(L/2)≠], we are able to control the presence of a defect line in the middle of the sample and study its influence on the interface between domains of different spin orientations. So essentially we study an Ising model with a defect line but, unlike previous work where defect lines …

Chapter III Phase transitions at surfaces

Abstract The statistical mechanics of phase transitions is briefly reviewed, with an emphasis on surfaces. Flat surfaces of crystals may act as a substrate for adsorption of two-dimensional (d=2) monolayers and multilayers, offering thus the possibility to study phase transitions in restricted dimensionality. Critical phenomena for special universality classes can thus be investigated which have no counterpart in d=3. Also phase transitions can occur that are in a sense “in between” different dimensionalities (e.g., multilayer adsorption and wetting phenomena are transitions in between two and three dimensions, while adsorption of monolayers on stepped surfaces allows phenomena in between o…

Semiflexible polymers confined in a slit pore with attractive walls: two-dimensional liquid crystalline order versus capillary nematization

Semiflexible polymers under good solvent conditions interacting with attractive planar surfaces are investigated by Molecular Dynamics (MD) simulations and classical Density Functional Theory (DFT). A bead-spring type potential complemented by a bending potential is used, allowing variation of chain stiffness from completely flexible coils to rod-like polymers whose persistence length by far exceeds their contour length. Solvent is only implicitly included, monomer-monomer interactions being purely repulsive, while two types of attractive wall-monomer interactions are considered: (i) a strongly attractive Mie-type potential, appropriate for a strictly structureless wall, and (ii) a corrugat…

Wetting transitions near the bulk critical point: Monte Carlo simulations for the Ising model

Critical, tricritical, and first-order wetting transitions are studied near the bulk critical point of a simple cubic nearest-neighbor Ising model by extensive Monte Carlo simulations. The model applies an exchange J in the bulk and exchange ${J}_{s}$ in the surface planes, where surface fields ${H}_{1}$ also act in addition to a possible bulk field H. Lattices in a thin-film geometry L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}D are used, with two free L\ifmmode\times\else\texttimes\fi{}L surfaces (with L up to 256) and film thickness D up to 160, applying a very fast fully vectorizing multispin coding program. Our results present the first quantitative evidence f…

Computer simulation studies of finite-size broadening of solid–liquid interfaces: from hard spheres to nickel

Using Molecular Dynamics (MD) and Monte Carlo (MC) simulations interfacial properties of crystal-fluid interfaces are investigated for the hard sphere system and the one-component metallic system Ni (the latter modeled by a potential of the embedded atom type). Different local order parameters are considered to obtain order parameter profiles for systems where the crystal phase is in coexistence with the fluid phase, separated by interfaces with (100) orientation of the crystal. From these profiles, the mean-squared interfacial width w^2 is extracted as a function of system size. We rationalize the prediction of capillary wave theory that w^2 diverges logarithmically with the lateral size o…

KINETICS OF POLYMER EJECTION FROM CAPSID CONFINEMENT: SCALING CONSIDERATIONS AND COMPUTER EXPERIMENT

We investigate the ejection dynamics of a flexible polymer chain out of confined environment by means of scaling considerations and Monte Carlo simulations. Situations of this kind arise in different physical contexts, including a flexible synthetic polymer partially confined in a nanopore and a viral genome partially ejected from its capsid. In the case of cylindric confinement the entropic driving force which pulls the chain out of the pore is argued to be constant once a few persistent lengths are out of the pore. We demonstrate that in this case the ejection dynamics follows a [Formula: see text]-law with elapsed time t. The mean ejection time τ depends nonmonotonically on chain length…

Monte Carlo simulations of polymer dynamics: Recent advances

A brief review is given of applications of Monte Carlo simulations to study the dynamical properties of coarse-grained models of polymer melts, emphasizing the crossover from the Rouse model toward reptation, and the glass transition. The extent to which Monte Carlo algorithms can mimic the actual chain dynamics is critically examined, and the need for the use of coarse-grained rather than fully atomistic models for such simulations is explained. It is shown that various lattice and continuum models yield qualitatively similar results, and the behavior agrees with the findings of corresponding molecular dynamics simulations and experiments, where available. It is argued that these simulatio…

Lack of long-range order in confined two-dimensional model colloidal crystals.

We investigate the nature of the ordered phase for a model of colloidal particles confined within a quasi-one-dimensional (Q1D) strip between two parallel boundaries, or walls, separated a distance $D$ in two dimensions (2D). Using Monte Carlo simulations we find that at densities typical of the bulk 2D triangular solid the order in the D1D strip is determined by the nature of the boundaries. While the order is enhanced for a suitably corrugated boundary potential, for a uniformly repulsive smooth boundary potential ordering normal to the walls is enhanced (``layering''), but destroyed parallel to the wall.

Positive Tolman Length in a Lattice Gas with Three-Body Interactions

We present a new method to determine the curvature dependence of the interface tension between coexisting phases in a finite volume from free energies obtained by Monte Carlo simulations. For the example of a lattice gas on a 3D fcc lattice with nearest neighbor three-body interactions, we demonstrate how to calculate the equimolar radius ${R}_{e}$ as well as the radius ${R}_{s}$ of the surface of tension and thus the Tolman length $\ensuremath{\delta}({R}_{s})={R}_{e}\ensuremath{-}{R}_{s}$. Within the physically relevant range of radii, $\ensuremath{\delta}({R}_{s})$ shows a pronounced ${R}_{s}$ dependence, such that the simple Tolman parametrization for the interface tension is refutable.…

Monte Carlo simulations of chain dynamics in polymer brushes

The bond-fluctuation model of polymer chains has been used to study layers of end-grafted polymers anchoring at repulsive walls for a broad range of chain length, grafting densities and solvent quality. The dynamics of monomers and associated relaxation times are investigated and interpreted by phenomenological theories and scaling arguments. The case is also considered where a chain is cut off from its anchor point and the chain is subsequently expelled from the brush. Both the velocity with which the chain leaves the brush and the associated conformational changes (chain contraction etc.) are analysed and interpreted in terms of recent theoretical concepts.

Surface effects on kinetics of ordering

We study the effects of surfaces on the kinetics of phase changes in Ising-type systems. If the surface effects can be modelled by a field which couples linearly to the local order parameter, the growth of wetting or drying layers occurs. The numerical solution of the corresponding time-dependent Ginzburg-Landau equation yields a temporally logarithmic growth for the thickness of a wetting (drying) layer growing from an unstable dry (wet) state. On the other hand, if one starts off with a metastable state, the radius of a supercritical plug (wet or dry) grows linearly in time, in accordance with recent experimental results.

Simulation of first- and second-order transitions in asymmetric polymer mixtures

The critical properties of dense asymmetric binary polymer mixtures are studied by grand canonical simulations within the framework of the 3-dimensional bond fluctuation lattice model. The monomers interact with each other via a potential ranging over the entire first peak of the pair distribution. An asymmetry is realized by giving the ratio of interactions λ = ∈AA/∈BB between monomers of the A-species and of the B-species a value different from 1. Using multiple histogram extrapolation techniques for the data analysis, the two phase region, which is a line of first-order transitions driven by the chemical potential difference, and the critical point are determined for a mixture of chains …

Simulations of phase transitions in macromolecular systems

Abstract The study of phase transitions in concentrated solutions and melts of flexible or stiff polymers is a computational challenge for computer simulations, since already a single polymer coil exhibits nontrivial structure from the scale of a chemical bond (1 A) to the coil radius (100 A), and for the simulation of collective phenomena huge simulation boxes containing many polymers are required. A strategy to deal with this problem is the use of highly coarse-grained models on a lattice, such as the bond fluctuation model. Several studies employing such models will be briefly reviewed, e.g.: temperature-driven isotropic-nematic phase transition in concentrated solutions of semiflexible …

Medium-range interactions and crossover to classical critical behavior

We study the crossover from Ising-like to classical critical behavior as a function of the range R of interactions. The power-law dependence on R of several critical amplitudes is calculated from renormalization theory. The results confirm the predictions of Mon and Binder, which were obtained from phenomenological scaling arguments. In addition, we calculate the range dependence of several corrections to scaling. We have tested the results in Monte Carlo simulations of two-dimensional systems with an extended range of interaction. An efficient Monte Carlo algorithm enabled us to carry out simulations for sufficiently large values of R, so that the theoretical predictions could actually be …

Fluctuations and lack of self-averaging in the kinetics of domain growth

The fluctuations occurring when an initially disordered system is quenched at timet=0 to a state, where in equilibrium it is ordered, are studied with a scaling theory. Both the mean-sizel(t)d of thed-dimensional ordered domains and their fluctuations in size are found to increase with the same power of the time; their relative size fluctuations are independent of the total volumeLd of the system. This lack of self-averaging is tested for both the Ising model and the φ4 model on the square lattice. Both models exhibit the same lawl(t)=(Rt)x withx=1/2, although the φ4 model has “soft walls”. However, spurious results withx≷1/2 are obtained if “bad” pseudorandom numbers are used, and if the n…

Effects of finite thickness on interfacial widths in confined thin films of coexisting phases

The capillary broadening of a 2-phase interface is investigated both experimentally and theoretically. When a binary mixture in a thin film with thickness D segregates into two coexisting phases the interface between the two phases may form parallel to the substrate due to preferential surface attraction of one of the components. We show that the interfacial profile (of intrinsic width w0) is broadened due to capillary waves, which lead to fluctuations, of correlation length of the local interface positions in the directions parallel to the confining walls. We postulate that acts as an upper cutoff for the spectrum of capillary waves on the interface, so that the effective mean square inter…

Monte carlo simulation of the glass transition of polymer melts

The bond fluctuation model of polymer melts is presented as a reasonable compromise between simulation efficiency and realistic chemical detail. It is shown that inclusion of a potential energy that depends on the length of the effective bonds connecting the effective monomers easily creates a conflict between configurational entropy of dense packing and the energetic tendency of the bonds to stretch. This competition leads to a glass transition of the model, which very well describes many features of real systems.

Semiflexible polymers under good solvent conditions interacting with repulsive walls.

Solutions of semiflexible polymers confined by repulsive planar walls are studied by density functional theory and Molecular Dynamics simulations, to clarify the competition between the chain alignment favored by the wall and the depletion caused by the monomer-wall repulsion. A coarse-grained bead-spring model with bond bending potential is studied, varying both the contour length and the persistence length of the polymers, as well as the monomer concentration in the solution (good solvent conditions are assumed throughout, and solvent molecules are not included explicitly). The profiles of monomer density and pressure tensor components near the wall are studied, and the surface tension of…

Anomalous Structure and Scaling of Ring Polymer Brushes

A comparative simulation study of polymer brushes formed by grafting at a planar surface either flexible linear polymers (chain length $N_L$) or (non-catenated) ring polymers (chain length $N_R=2 N_L$) is presented. Two distinct off-lattice models are studied, one by Monte Carlo methods, the other by Molecular Dynamics, using a fast implementation on graphics processing units (GPUs). It is shown that the monomer density profiles $\rho(z)$ in the $z$-direction perpendicular to the surface for rings and linear chains are practically identical, $\rho_R(2 N_L, z)=\rho_L(N_L, z)$. The same applies to the pressure, exerted on a piston at hight z, as well. While the gyration radii components of ri…

Polymer Dynamics in a Polymer-Solid Interphase: Molecular Dynamics Simulations of 1,4-Polybutadiene At a Graphite Surface

A chemically realistic model of 1,4-polybutadiene confined by graphite walls in a thin film geometry was studied by molecular dynamics simulations. The chemically realistic approach allows for a quantitative determination of a variety of experimentally accessible relaxation functions (e.g., dielectric, NMR, or neutron scattering responses). The simulations yield these experimental observables. Additionally, the simulations can be resolved as a function of distance to the solid interface on a much finer scale than experimentally possible, providing a detailed mechanistic picture of the segmental and large scale motions of polymers in the interfacial region between bulk polymer melts and soli…

Formation of Micelles in Homopolymer-Copolymer Mixtures:  Quantitative Comparison between Simulations of Long Chains and Self-Consistent Field Calculations

Using Monte Carlo simulations of the bond fluctuation model and self-consistent field calculations, we study the formation of micelles in a mixture of homopolymers and asymmetric AB-diblock copolymers with composition, fA = 1/8. Both types of molecules are fully flexible and have identical length. We work in the semi-grand-canonical ensemble, i.e., we fix the monomer density and incompatibility, χN ≃ 100 (strong segregation regime), and control the composition of the mixture via the exchange chemical potential, δμ ≡ μAB − μB between the copolymers and homopolymers. The Monte Carlo simulation comprises moves that allow homopolymers to mutate into AB-diblock copolymers and vice versa. These m…

Glass Formation in Polymers: Theory of Glass Transition

Phase separation of an asymmetric binary fluid mixture confined in a nanoscopic slit pore: Molecular-dynamics simulations

As a generic model system of an asymmetric binary fluid mixture, hexadecane dissolved in carbon dioxide is considered, using a coarse-grained bead-spring model for the short polymer, and a simple spherical particle with Lennard-Jones interactions for the carbon dioxide molecules. In previous work, it has been shown that this model reproduces the real phase diagram reasonable well, and also the initial stages of spinodal decomposition in the bulk following a sudden expansion of the system could be studied. Using the parallelized simulation package ESPResSo on a multiprocessor supercomputer, phase separation of thin fluid films confined between parallel walls that are repulsive for both types…

Fluorescence Lifetime of a Single Molecule as an Observable of Meta-Basin Dynamics in Fluids Near the Glass Transition

Using single molecule spectroscopy, we show that the fluorescence lifetime trajectories of single probe molecules embedded in a glass-forming polymer melt exhibit strong fluctuations of a hopping character. Using molecular dynamics simulations targeted to explain these experimental observations, we show that the lifetime fluctuations correlate strongly with the average square displacement function of the matrix particles. The latter observable is a direct probe of the meta-basin transitions in the potential energy landscape of glass-forming liquids. We thus show here that single molecule experiments can provide detailed microscopic information on system properties that hitherto have been ac…

Structure and pair correlations of a simple coarse grained model for supercritical carbon dioxide

A recently introduced coarse-grained pair potential for carbon dioxide molecules is used to compute structural properties in the supercritical region near the critical point, applying Monte Carlo simulations. In this model, molecules are described as point particles, interacting with Lennard-Jones (LJ) forces and a (isotropically averaged) quadrupole–quadrupole potential, the LJ parameters being chosen such that gratifying agreement with the experimental phase diagram near the critical point is obtained. It is shown that the model gives also a reasonable account of the pair correlation function, although in the nearest neighbour shell some systematic discrepancies between the model predicti…

Mechanisms for the Decay of Unstable and Metastable Phases: Spinodal Decomposition, Nucleation and Late-Stage Coarsening

The basic concepts on the kinetics of phase separation in alloys are introduced, and the current status of the theory is briefly reviewed. Particular emphasis is given to questions such as the conditions under which the linearized theory of spinodal decomposition is valid, the significance of spinodal curves, the possible description of coarsening in terms of power laws and structure-factor scaling, and non-equilibrium percolation phenomena.

Some Important Recent Developments of the Monte Carlo Methodology

Roughly at the time (1987) when the manuscript for the first three chapters of the present book was completed, several breakthroughs occurred. They had a profound influence on the scope of Monte Carlo simulations in statistical physics, particularly for the study of phase transitions in lattice models.

Hydrodynamic mechanisms of spinodal decomposition in confined colloid-polymer mixtures: A multiparticle collision dynamics study

A multiscale model for a colloid-polymer mixture is developed. The colloids are described as point particles interacting with each other and with the polymers with strongly repulsive potentials, while polymers interact with each other with a softer potential. The fluid in the suspension is taken into account by the multiparticle collision dynamics method (MPC). Considering a slit geometry where the suspension is confined between parallel repulsive walls, different possibilities for the hydrodynamic boundary conditions (b.c.) at the walls (slip versus stick) are treated. Quenching experiments are considered, where the system volume is suddenly reduced (keeping the density of the solvent flui…

Simulation of fluid-solid coexistence in finite volumes: A method to study the properties of wall-attached crystalline nuclei

The Asakura-Oosawa model for colloid-polymer mixtures is studied by Monte Carlo simulations at densities inside the two-phase coexistence region of fluid and solid. Choosing a geometry where the system is confined between two flat walls, and a wall-colloid potential that leads to incomplete wetting of the crystal at the wall, conditions can be created where a single nanoscopic wall-attached crystalline cluster coexists with fluid in the remainder of the simulation box. Following related ideas that have been useful to study heterogeneous nucleation of liquid droplets at the vapor-liquid coexistence, we estimate the contact angles from observations of the crystalline clusters in thermal equil…

Monte Carlo simulation of polymer mixtures: recent progress

‘Intrinsic’ profiles and capillary waves at interfaces between coexisting phases in polymer blends

Abstract Lateral fluctuations in the local position of the center of the interface between coexisting phases in unmixed polymer blends lead to a broadening of interfacial widths; comparing self-consistent field predictions for the ‘intrinsic’ profile to simulations (or experiments), this ‘capillary wave’ broadening needs consideration. This problem has been studied by extensive Monte Carlo simulations of the bond fluctuation model for symmetrical polymer mixtures, both for free interfaces (between bulk phases) and for confined interfaces (in thin films between parallel walls). While the capillary wave predictions at large length scales are confirmed, the extraction of the ‘intrinsic’ profil…

Simulations of Polymers in Confined Geometries

The properties of flexible polymers moving inside porous structures are believed to be relevant to practical problems such as filtration, gel permeation chromatography, heterogeneous catalysis, oil recuperation, etc.1. Similarly the adsorption of macromolecules on interfaces plays an important role for problems such as adhesion, flocculation and stabilisation of colloid particles, biological membrane function, artificial organs in medicine, etc. 2. Aside from this eventual practical application, the configurational statistics of polymers in such confined geometries is a challenging problem of theoretical physics. The present brief review will be concerned with the study of a single long fle…

Capillary condensation in the two-dimensional lattice gas: A Monte Carlo test of fluctuation corrections to the Kelvin equation

A two-dimensional lattice gas model with nearest-neighbour attractive interaction confined in a strip of width L between two parallel boundaries at which an attractive short-range force acts is studied by Monte Carlo simulations, for cases where the system is in the wet phase near the critical wetting transition line for . We study the shift of the chemical potential of the transition in the strip as a function of L by thermodynamic integration methods, , and also obtain the thickness of the wetting film at the chemical potential at which capillary condensation occurs. In the range the data are consistent with a variation according to the Kelvin equation, , as well as with a shifted Kelvin …

Suppression of capillary wave broadening of interfaces in binary alloys due to elastic interactions.

By Monte Carlo simulations in the constant-temperature--constant-pressure ensemble a planar interface between unmixed A-rich and B-rich phases of a binary (A, B) alloy on a compressible diamond lattice is studied. No significant capillary wave broadening of the concentration profile across the interface is observed, unlike lattice models of incompressible mixtures and fluids. The distortion of the lattice structure across the interface is studied.

Computer simulation of models for the structural glass transition

In order to test theoretical concepts on the glass transition, we investigate several models of glassy materials by means of Monte Carlo (MC) and Molecular Dynamics (MD) computer simulations. It is shown that also simplified models exhibit a glass transition which is in qualitative agreement with experiment and that thus such models are useful to study this phenomenon. However, the glass transition temperture as well as the structural properties of the frozen-in glassy phase depend strongly on the cooling history, and the extrapolation to the limit of infinitely slow cooling velocity is nontrivial, which makes the identification of the (possible) underlying equilibrium transition very diffi…

Computer simulation of the glass transition of polymer melts

Bond fluctuation models on square and simple cubic lattices at melt densities are simulated, using potentials depending on the length of the (effective) bond (and also on the bond angle, in d=3 dimensions). Various relaxation functions have the Kohlrausch-Williams-Watts (KWW) form; the associated relaxation time diverges as exp (const/T 2) in d=2 and as exp [const/T−T 0)] in d=3. For d=3 the self-diffusion constant also follows the Vogel-Fulcher law, with T 0=250 K for chain lengths N=20 and potentials adapted to bisphenol-A-polycarbonate [BPA-PC].

On the Glass Transition in Polymer Films: Recent Monte Carlo Results

AbstractThis paper reports results of a Monte Carlo simulation for a simplified lattice modelof a supercooled polymer film. The film geometry is realized by two opposite hard walls.The distance between the walls is varied. The chains exhibit a strong tendency to orientparallel to the walls and are flattened when being very close to them. This deviation of thepolymer structure with respect to the bulk is accompanied by an acceleration of local densityfluctuations. On the other hand, the diffusion coefficient of a chain remains unaffected.

Critical phenomena at surfaces

Abstract The presence of free surfaces adds a rich and interesting complexity to critical phenomena associated with phase transitions occurring in bulk materials. We shall review Monte Carlo computer simulation studies of surface critical behavior in simple cubic Ising- and XY-models with nearest-neighbor interactions J in the bulk and Js at the surface. These studies allow the identification of various critical exponents and critical amplitude ratios involving both the critical behavior of local quantities and of surface excess corrections to the bulk. We consider both the “ordinary” transition (surface criticality controlled by the bulk) and the “special transition” (a multicritical point…

Escape transition of a compressed polymer mushroom under good solvent conditions

The escape transition of a flexible polymer chain of chain length N, endgrafted at a hard wall and compressed by a piston of radius R, is studied by Monte Carlo simulation and by phenomenological arguments. In contrast to previous theories which have considered the transition as a function of a (fixed) height H of the piston above the wall, we consider the transition as a function of the conjugate variable, the force f acting on the piston. We find that the transition (which is sharp only for N → ∞) is characterized by a flat region of f in the f vs. H isotherm, i.e. a jump in the height occurs at the transition from Hesc,t to Himp,t, with (Himp,t − Hesc,t)/Hesc,t ≈ 0.26.

Relaxation processes and glass transition of confined polymer melts: A molecular dynamics simulation of 1,4-polybutadiene between graphite walls.

Molecular dynamics simulations of a chemically realistic model for 1,4-polybutadiene in a thin film geometry confined by two graphite walls are presented. Previous work on melts in the bulk has shown that the model faithfully reproduces static and dynamic properties of the real material over a wide temperature range. The present work studies how these properties change due to nano-confinement. The focus is on orientational correlations observable in nuclear magnetic resonance experiments and on the local intermediate incoherent neutron scattering function, Fs(qz, z, t), for distances z from the graphite walls in the range of a few nanometers. Temperatures from about 2Tg down to about 1.15Tg…

Dynamics of phase separation and critical phenomena in polymer mixtures

The phenomenological mean-field theory for statics and dynamics of polymer mixtures is described, generalizing the approaches of Flory-Huggins, Cahn-Hilliard and de Gennes. Predictions are made for critical behavior, spinodal decomposition and homogeneous nucleation. The validity of the mean-field approximations is discussed with Ginzburg criteria. The results of the theory are compared to computer simulations and recent experiments.

Monte Carlo simulations of Ising models and polymer blends in double wedge geometry: Evidence for novel types of critical phenomena

Abstract Two-phase coexistence in systems with free surfaces is enforced by boundary fields requiring the presence of an interface. Varying the temperature or the surface field, one can observe new types of phase transitions where the interface essentially disappears (it becomes bound to a wall or a wedge or a corner of the system). These transitions are simulated with Monte Carlo for Ising ferromagnets and polymer blends, applying finite size scaling analysis. Anisotropic critical fluctuations may occur, and in the limit where the system becomes macroscopically large in all three directions the order parameter vanishes discontinuously (either because its exponent β = 0 , or its critical am…

Study of the confined Ising magnet with long-range competing boundary fields

We present extensive Monte Carlo simulations of the Ising film confined in an L × M geometry () in the presence of long-range competing magnetic fields h(n) = h1/n3(n = 1,2,...,L) which are applied at opposite walls along the M-direction. Due to the fields, an interface between domains of different orientations that runs parallel to the walls forms and can be located close to one of the two surfaces or fluctuate in the centre of the film (localization–delocalization transition). This transition is the precursor of the wetting phase transition that occurs in the limit of infinite film thickness () at the critical curve Tw(h1). For T<Tw(h1) (T≥Tw(h1)) such an interface is bound to (unbound fr…

Isotropic–isotropic phase separation in mixtures of rods and spheres: Some aspects of Monte Carlo simulation in the grand canonical ensemble

Abstract In this article we consider mixtures of non-adsorbing polymers and rod-like colloids in the isotropic phase, which upon the addition of polymers show an effective attraction via depletion forces. Above a certain concentration, the depletant causes phase separation of the mixture. We performed Monte Carlo simulations to estimate the phase boundaries of isotropic–isotropic coexistence. To determine the phase boundaries we simulated in the grand canonical ensemble using successive umbrella sampling [J. Chem. Phys. 120 (2004) 10925]. The location of the critical point was estimated by a finite size scaling analysis. In order to equilibrate the system efficiently, we used a cluster move…

Phase transitions in thin films with competing surface fields and gradients.

As a generic model for phase equilibria under confinement in a thin-film geometry in the presence of a gradient in the field conjugate to the order parameter, an Ising-lattice gas system is studied by both Monte Carlo simulations and a phenomenological theory. Choosing an $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}D$ geometry with $L\ensuremath{\gg}D$ and periodic boundary conditions in the $x,y$ directions, we place competing surface fields on the two $L\ifmmode\times\else\texttimes\fi{}L$ surfaces. In addition, a field gradient $g$ is present in the $z$ direction across the film, in competition with the surface fields. At temperatures $T$ exceeding the critical…

Monte Carlo studies of anisotropic surface tension and interfacial roughening in the three-dimensional Ising model.

Extensive Monte Carlo simulations of the simple cubic Ising model with nearest-neighbor ferromagnetic interactions with a tilted interface are presented for a wide range of lattice size L, temperature T, and tilt angles \ensuremath{\theta}. The anisotropic interfacial tension is studied in detail. From the small-angle data, we obtain the step free energy density ${f}_{S}$(T,L). Finite-size scaling of the step free energy density is discussed and used to probe the predicted temperature dependence of the correlation length near and above the roughening transition. The square-root temperature dependence predicted by solid-on-solid model calculations is exhibited. Finite-size scaling implies th…

A slow process in confined polymer melts: layer exchange dynamics at a polymer solid interface

Employing Molecular Dynamics simulations of a chemically realistic model of 1,4-polybutadiene between graphite walls we show that the mass exchange between layers close to the walls is a slow process already in the melt state. For the glass transition of confined polymers this process competes with the slowing down due to packing effects and intramolecular rotation barriers.

Stretching semiflexible polymer chains: Evidence for the importance of excluded volume effects from Monte Carlo simulation

Semiflexible macromolecules in dilute solution under very good solvent conditions are modeled by self-avoiding walks on the simple cubic lattice ($d=3$ dimensions) and square lattice ($d=2$ dimensions), varying chain stiffness by an energy penalty $\epsilon_b$ for chain bending. In the absence of excluded volume interactions, the persistence length $\ell_p$ of the polymers would then simply be $\ell_p=\ell_b(2d-2)^{-1}q_b^{-1}$ with $q_b= \exp(-\epsilon_b/k_BT)$, the bond length $\ell_b$ being the lattice spacing, and $k_BT$ is the thermal energy. Using Monte Carlo simulations applying the pruned-enriched Rosenbluth method (PERM), both $q_b$ and the chain length $N$ are varied over a wide r…

Monte Carlo tests of renormalization-group predictions for critical phenomena in Ising models

Abstract A critical review is given of status and perspectives of Monte Carlo simulations that address bulk and interfacial phase transitions of ferromagnetic Ising models. First, some basic methodological aspects of these simulations are briefly summarized (single-spin flip vs. cluster algorithms, finite-size scaling concepts), and then the application of these techniques to the nearest-neighbor Ising model in d=3 and 5 dimensions is described, and a detailed comparison to theoretical predictions is made. In addition, the case of Ising models with a large but finite range of interaction and the crossover scaling from mean-field behavior to the Ising universality class are treated. If one c…

Computer Simulation Techniques in Condensed Matter Physics

Stick-slip motion and plastic flow of a two-dimensional colloidal crystal confined to moving corrugated rigid boundaries

Computer simulations are presented where a model for a two-dimensional colloidal crystal confined to corrugated walls is exposed to a steady-state shear deformation. Following up on an earlier study, where average velocity profiles of the particles in the crystal across the slit have been obtained, we now analyse the time dependence of the particle motions and of the resulting shear forces. We discuss the extent to which the resulting irregular and damped oscillatory motions can be associated with stick-slip motions familiar from friction phenomena.

Semiflexible macromolecules in quasi-one-dimensional confinement: Discrete versus continuous bond angles

The conformations of semiflexible polymers in two dimensions confined in a strip of width D are studied by computer simulations, investigating two different models for the mechanism by which chain stiffness is realized. One model (studied by molecular dynamics) is a bead-spring model in the continuum, where stiffness is controlled by a bond angle potential allowing for arbitrary bond angles. The other model (studied by Monte Carlo) is a self-avoiding walk chain on the square lattice, where only discrete bond angles (0° and ±90°) are possible, and the bond angle potential then controls the density of kinks along the chain contour. The first model is a crude description of DNA-like biopolymer…

Phase behavior of flexible and semiflexible polymers in solvents of varying quality.

The interplay of nematic order and phase separation in solutions of semiflexible polymers in solvents of variable quality is investigated by density functional theory (DFT) and molecular dynamics (MD) simulations. We studied coarse-grained models, with a bond-angle potential to control chain stiffness, for chain lengths comparable to the persistence length of the chains. We varied both the density of the monomeric units and the effective temperature that controls the quality of the implicit solvent. For very stiff chains, only a single transition from an isotropic fluid to a nematic is found, with a phase diagram of "swan-neck" topology. For less stiff chains, however, also unmixing between…

Surfaces have a profound effect on the structure and related properties of multiphase polymeric materials, such as polymer mixtures and block copolymer mesophases. In particular, phase transitions in the bulk (unmixing, microphase separation, etc.) may be complemented by surface-induced transitions (formation of wetting layers, surface-directed spinodal decomposition, surface-induced ordering). This review gives a brief introduction to the phenomenological theories of such phenomena, emphasizing the simplest approach based on Flory—Huggins—de Gennes free energy functionals and associated Monte Carlo simulations. More sophisticated theories and recent experiments are mentioned briefly.

The interplay between wetting and phase behaviour in binary polymer films and wedges: Monte Carlo simulations and mean field calculations

By confining a binary mixture, one can profoundly alter its miscibility behaviour. The qualitative features of miscibility in confined geometry are rather universal and are shared by polymer mixtures as well as small molecules, but the unmixing transition in the bulk and the wetting transition are typically well separated in polymer blends. We study the interplay between wetting and miscibility of a symmetric polymer mixture via large scale Monte Carlo simulations in the framework of the bond fluctuation model and via numerical self-consistent field calculations. The film surfaces interact with the monomers via short-ranged potentials, and the wetting transition of the semi-infinite system …

Character of the Phase Transition in Thin Ising Films with Competing Walls

By extensive Monte Carlo simulations of a lattice gas model we have studied the controversial nature of the gas-liquid transition of a fluid confined between two parallel plates that exert competing surface fields. We find that the transition is shifted to a temperature just below the wetting transition of a semi-infinite fluid but belongs to the two-dimensional Ising universality class. In between this new type of critical point and bulk criticality, a response function ${x}_{\mathrm{nn}}^{max}$ varying exponentially with $D$ is observed, $\frac{2 \mathrm{ln}{\ensuremath{\chi}}_{\mathrm{nn}}^{max}}{D}={\ensuremath{\ell}}^{\ensuremath{-}1}$, where $\ensuremath{\ell}$ is a new length charact…

Introduction: Purpose and Scope of this Volume, and Some General Comments

In recent years the method of “computer simulation” has started something like a revolution of science: the old division of physics (as well as chemistry, biology, etc.) into “experimental” and “theoretical” branches is no longer really complete. Rather, “computer simulation” has become a third branch complementary to the first two traditional approaches.

Spin-one-Ising model for (CO)1?x (N2) x mixtures: A finite size scaling study of random-field-type critical phenomena

A qualitative model for solid mixtures of diatomic molecules, where one species (called CO, to be specific) carries both a dipole moment and a quadrupole moment, while the other species (calledN 2) has only a quadrupole moment, is studied by Monte Carlo methods. We use spinsS i =±1 to represent the orientations of the CO electric dipole moment, if the lattice sitei is taken by a CO molecule, whileS i =0 if the site is taken by anN 2 molecule. Assuming nearest-neighbor antiferroelectric interactions between CO molecules, and a bilinear dipole-quadrupole coupling between CO andN 2, the randomly quenchedN 2 molecules act like random fields do in the random field Ising model. In previous work i…

Polymer Brushes on Flat and Curved Substrates: Scaling Concepts and Computer Simulations

The scaling concepts for isolated flexible macromolecules in good solvent grafted with one chain end to a flat surface (polymer mushrooms) as well as for layers of many overlapping end-grafted chain molecules (polymer brushes) are introduced. Monte Carlo attempts to test these concepts are briefly reviewed. Then the extension of these concepts to polymer brushes grafted to the interior of a cylinder surface is discussed. Molecular Dynamics results on chain average linear dimensions in the direction normal to the grafting surface and in axial direction are described, as well as distribution functions for the density of end monomers and of all monomers of the chains. It is argued that under t…

Rounding of Phase Transitions in Cylindrical Pores

Phase transitions of systems confined in long cylindrical pores (capillary condensation, wetting, crystallization, etc.) are intrinsically not sharply defined but rounded. The finite size of the cross section causes destruction of long range order along the pore axis by spontaneous nucleation of domain walls. This rounding is analyzed for two models (Ising/lattice gas and Asakura-Oosawa model for colloid-polymer mixtures) by Monte Carlo simulations and interpreted by a phenomenological theory. We show that characteristic differences between the behavior of pores of finite length and infinitely long pores occur. In pores of finite length a rounded transition occurs first, from phase coexiste…

MC Study of the p-state Mean-Field Potts Glass

The p-state mean-field Potts glass with ±J-couplings is studied by Monte Carlo (MC) simulations, both for p = 3 and p = 6 states. At the exactly known glass transition temperature Tc, the moments q( k ) of the spin glass order parameter satisfy for p = 3 a simple scaling behavior, q( k ) \({q^{\left( k \right)}}\alpha {N^{ - k/3}}{\tilde f_k}\left\{ {{N^{1/3}}\left( {1 - T/{T_c}} \right)} \right\},k = 1,2,3,...\). The specific-heat maxima exhibit a similar behavior, c V max α const — N -l/3, while the approach of the maxima positions T max to T c as N → ∞ is non-monotonic. For p = 6 the results are compatible with the expected result of a quite peculiar first-order phase transition. The spe…

Phase transitions and phase equilibria in spherical confinement

Phase transitions in finite systems are rounded and shifted and affected by boundary effects due to the surface of the system. This interplay of finite size and surface effects for fluids confined inside of a sphere of radius $R$ is studied by a phenomenological theory and Monte Carlo simulations of a model for colloid-polymer mixtures. For this system the phase separation in a colloid-rich phase and a polymer-rich phase has been previously studied extensively in the bulk. It is shown that spherical confinement can strongly enhance the miscibility of the mixture. Depending on the wall potentials at the confining surface, the wetting properties of the wall can be controlled, and this interpl…

Molecular dynamics simulation of the surface tension of aqueous sodium chloride: from dilute to highly supersaturated solutions and molten salt

Sodium chloride (NaCl) is one of the key components of atmospheric aerosols. The surface tension of aqueous NaCl solution (σNaCl,sol) and its concentration dependence are essential to determine the equilibrium water vapor pressure of aqueous NaCl droplets. Supersaturated NaCl solution droplets are observed in laboratory experiments and under atmospheric conditions, but the experimental data for σNaCl,sol are mostly limited up to subsaturated solutions. In this study, the surface tension of aqueous NaCl is investigated by molecular dynamics (MD) simulations and the pressure tensor method from dilute to highly supersaturated solutions. We show that the linear approximation of concentration de…

First-order interface localization-delocalization transition in thin Ising films using Wang-Landau sampling

Using extensive Monte Carlo simulations, we study the interface localization- delocalization transition of a thin Ising film with antisymmetric competing walls for a set of parameters where the transition is strongly first-order. This is achieved by estimating the density of states (DOS) of the model by means of Wang-Landau sampling (WLS) in the space of energy, using both, single-spin-flip as well as N-fold way updates. From the DOS we calculate canonical averages related to the configurational energy, like the internal energy, the specific heat, as well as the free energy and the entropy. By sampling microcanonical averages during simulations we also compute thermodynamic quantities relat…

Dynamic percolation transition induced by phase separation: A Monte Carlo analysis

The percolation transition of geometric clusters in the three-dimensional, simple cubic, nearest neighbor Ising lattice gas model is investigated in the temperature and concentration region inside the coexistence curve. We consider “quenching experiments,” where the system starts from an initially completely random configuration (corresponding to equilibrium at infinite temperature), letting the system evolve at the considered temperature according to the Kawasaki “spinexchange” dynamics. Analyzing the distributionnl(t) of clusters of sizel at timet, we find that after a time of the order of about 100 Monte Carlo steps per site a percolation transition occurs at a concentration distinctly l…

Spinodal decomposition of a two-dimensional model alloy with mobile vacancies

Abstract Monte Carlo simulations are performed for the initial stages of phase separation in a model binary alloy (AB), where unmixing is caused by a repulsive energy between atoms of different kind ( e AA = e BB = e ), and a small fraction c v of mobile vacancies is present (typically c v = 0.04.) Unlike previous work, where interdiffusion was modelled in an unrealistic way by direct interchange of A and B atoms for c v = 0, were use the vacancy mechanism of diffusion: A-atoms may jump to vacant sites with a rate Γ A and B-atoms may jump to vacant sites with a rate Γ B , no direct A–B interchange being permitted. It is shown that the overall time-scale on which phase separation proceeds ty…

Finite Size Scaling Tools for the Study of Interfacial Phenomena and Wetting

In this chapter, we use the word “interface” in the sense of a boundary between coexisting bulk phases (in thermal equilibrium). An example is the interface between liquid (e.g. water) and gas phases (water vapor) but also interfaces between fluid and solid phases (e.g. water and ice) can be considered, as well as interfaces between coexisting solid phases. The generic example are “domain walls” in magnets, separating domains with opposite orientation of the magnetization, a case that can already be studied in the framework of the simple Ising model (Chaps. 2 and 3) where one has spins on the sites of a rigid perfect lattice pointing up or down.

Phase transitions in polymeric systems: A challenge for Monte Carlo simulation

Polymers are more difficult to simulate than small molecule systems, due to the large size of random polymer coils (and their slow relaxation, that is observed when dynamic simulation algorithms are used). However, variation of the chain length N of a flexible polymer chain provides a very useful additional control parameter, allowing stringent tests of theories, and new physical phenomena may emerge. As an example of these concepts, critical phenomena in polymer mixtures are described. It is shown that unmixing of symmetrical mixtures ( N A = N B = N ) is described by an equation for the critical temperature T c ( N ) = aN + b rather than T c ∝ N as claimed by some theories. While for fini…

Dynamics of macromolecules grafted in spherical brushes under good solvent conditions

Spherical polymer brushes have a structure intermediate between star polymers and polymer brushes on flat substrates, and are important building blocks of polymer nanoparticles. Molecular dynamics simulations are presented for isolated spherical polymer brushes under good solvent conditions, varying the grafting density as well as the chain length, using a coarse-grained bead-spring model of flexible chains. We complement previous work on the static properties of the same model by analyzing the chain dynamics, studying the motions of monomers in relation to their position along the grafted chains, and extract suitable relaxation times. A qualitative discussion in terms of the Rouse model is…

Confined binary two-dimensional colloidal crystals: Monte Carlo simulation of crack formation.

Binary mixtures (A, B) of colloidal particles of different sizes in two dimensions may form crystals with square lattice structure (the A-particles occupying the white sites and the B-particles the black sites of a checkerboard). Confining such a system by two parallel 'walls' a distance D apart, long-range order in the direction parallel to the walls is stabilized by 'corrugated walls' that are commensurate with the lattice structure but destabilized by structureless 'hard walls', even if there is no misfit between the strip width D and the crystal lattice spacing. The crossover to quasi-one-dimensional behavior is studied by Monte Carlo simulations, analyzing Lindemann parameters and disp…

Phase transitions and phase coexistence: equilibrium systems versus externally driven or active systems - Some perspectives

A tutorial introduction to the statistical mechanics of phase transitions and phase coexistence is presented, starting out from equilibrium systems and nonequilibrium steady-state situations in ext...

Cluster Algorithms and Reweighting Methods

Roughly at the time (1987) when the manuscript for the first three chapters of the present book was completed, several breakthroughs occurred. They had a profound influence on the scope of Monte Carlo simulations in statistical physics, particularly for the study of phase transitions in lattice models.

Active nonlinear microrheology in a glass-forming Yukawa fluid.

A molecular dynamics computer simulation of a glass-forming Yukawa mixture is used to study the anisotropic dynamics of a single particle pulled by a constant force. Beyond linear response, a scaling regime is found where a force-temperature superposition principle of a Peclet number holds. In the latter regime, the diffusion dynamics perpendicular to the force can be mapped on the equilibrium dynamics in terms of an effective temperature, whereas parallel to the force a superdiffusive behavior is seen in the long-time limit. This behavior is associated with a hopping motion from cage to cage and can be qualitatively understood by a simple trap model.

Simulation of Models for the Glass Transition: Is There Progress?

The glass transition of supercooled fluids is a particular challenge for computer simulation, because the (longest) relaxation times increase by about 15 decades upon approaching the transition temperature T g. Brute-force molecular dynamics simulations, as presented here for molten SiO2 and coarse-grained bead-spring models of polymer chains, can yield very useful insight about the first few decades of this slowing down. Hence this allows to access the temperature range around T c of the so-called mode coupling theory, whereas the dynamics around the experimental glass transition is completely out of reach. While methods such as “parallel tempering” improve the situation somewhat, a method…

Cover Picture: Macromol. Theory Simul. 7/2006

The Dynamics of Supercooled Silica: Acoustic modes and Boson peak

Using molecular dynamics computer simulations we investigate the dynamics of supercooled silica in the frequency range 0.5-20~THz and the wave-vector range 0.13-1.1\AA^{-1}. We find that for small wave-vectors the dispersion relations are in very good agreement with the ones found in experiments and that the frequency at which the boson-peak is observed shows a maximum at around 0.39\AA^{-1}.

Methods to Compute Pressure and Wall Tension in Fluids containing Hard Particles

Colloidal systems are often modelled as fluids of hard particles (possibly with an additional soft attraction, e.g. caused by polymers also contained in the suspension). in simulations of such systems, the virial theorem cannot be straightforwardly applied to obtain the components of the pressure tensor. In systems confined by walls, it is hence also not straightforward to extract the excess energy due to the wall (the "wall tension") from the pressure tensor anisotropy. A comparative evaluation of several methods to circumvent this problem is presented, using as examples fluids of hard spheres and the Asakura-Oosawa model of colloid-polymer mixtures with a size ratio $q=0.15$ (for which th…

Nematic order in solutions of semiflexible polymers: Hairpins, elastic constants, and the nematic-smectic transition

Coarse-grained models of lyotropic solutions of semiflexible polymers are studied by both molecular dynamics simulations and density functional theory calculations, using an implicit solvent bead-spring model with a bond-angle potential. We systematically vary the monomer density, persistence length, and contour length over a wide range and explore the full range from the isotropic-nematic transition to the nematic-smectic transition. In the nematic regime, we span the entire regime from rigid-rod like polymers to thin wormlike chains, confined in effective straight tubes caused by the collective nematic effective ordering field. We show that the distribution of bond angles relative to the …

COOLING RATE DEPENDENCE AND DYNAMIC HETEROGENEITY BELOW THE GLASS TRANSITION IN A LENNARD–JONES GLASS

We investigate a binary Lennard-Jones mixture with molecular dynamics simulations. We consider first a system cooled linearly in time with the cooling rate gamma. By varying gamma over almost four decades we study the influence of the cooling rate on the glass transition and on the resulting glass. We find for all investigated quantities a cooling rate dependence; with decreasing cooling rate the system falls out of equilibrium at decreasing temperatures, reaches lower enthalpies and obtains increasing local order. Next we study the dynamics of the melting process by investigating the most immobile and most mobile particles in the glass. We find that their spatial distribution is heterogene…

On the interpretation of the experimental Raman spectrum of β-eucryptite LiAlSiO4 from atomistic computer modeling

Abstract The vibrational spectrum of β-eucryptite LiAlSiO4 with stuffed high quartz structure – commercially relevant for zero-expansion glass ceramics – was calculated by lattice energy minimization and diagonalization of the dynamical matrix using an ab initio based ion-pair shell model potential. A full symmetry analysis of the vibrational modes was carried out. Raman activity of vibrations was calculated under parameterization of individual polarizability factors for each type of interatomic bonds in β-eucryptite LiAlSiO4. Calculated vibrational energies agree with the experimental energies within ±2.3%. Agreement of calculated spectroscopic Raman intensities with experimental intensiti…

Phase Behavior of Polymer-Containing Systems: Recent Advances Through Computer Simulation

The dynamics of melts containing mobile ions: computer simulations of sodium silicates

We present the results of large-scale computer simulations in order to discuss the structural and dynamic properties of sodium silicate melts with the compositions (Na2O)2(SiO2) (NS2) and (Na2O)20(SiO2) (NS20). We show that, compared to silica (SiO2), these systems exhibit additional intermediate range order on intermediate length scales that stem from the tetrahedral network structure. By means of intermediate-scattering functions, we characterize the dynamics of sodium in the system under consideration. Whereas in NS2 the incoherent scattering functions for Na decay much faster to zero than the coherent ones for Na–Na, in NS20 this different behaviour of the incoherent and coherent functi…

Model calculations for wetting transitions in polymer mixtures

Partially compatible binary mixtures of linear flexible polymers are considered in the presence of a wall which preferentially adsorbs one component. Using a Flory-Huggins type mean field approach, it is shown that in typical cases at two-phase coexistence the wall is always « wet », i.e. coated with a macroscopically thick layer of the preferred phase, and the transition to the non wet state occurs at volume fractions of the order of 1/~N (where N is the chain length) at the coexistence curve. Both first and second order wetting transitions are found, and the variation of the surface layer thickness, surface excess energy and related quantities through the transition is studied. We discuss…

Crystal nuclei in melts: A Monte Carlo simulation of a model for attractive colloids

As a model for a suspension of hard-sphere like colloidal particles where small nonadsorbing dissolved polymers create a depletion attraction, we introduce an effective colloid-colloid potential closely related to the Asakura-Oosawa model but that does not have any discontinuities. In simulations, this model straightforwardly allows the calculation of the pressure from the Virial formula, and the phase transition in the bulk from the liquid to crystalline solid can be accurately located from a study where a stable coexistence of a crystalline slab with a surrounding liquid phase occurs. For this model, crystalline nuclei surrounded by fluid are studied both by identifying the crystal-fluid …

Microphase separation in bottlebrush polymers under poor-solvent conditions

Molecular-dynamics simulations are used to study the structure of bottlebrush polymers with rigid backbones, for various grafting densities, side chain lengths, and varying solvent quality. While we confirm different states of the bottlebrush proposed by Sheiko et al. (Eur. Phys. J. E, 13 (2004) 125) we find that the transition between stretched and collapsed brushes occurs in a rather gradual manner. The pearl-necklace structure occurring at intermediate grafting densities and rather low temperatures has a pronounced medium-range order along the backbone.

Power laws and crossovers in off-critical surface-directed spinodal decomposition.

We study the dynamics of phase separation in binary mixtures near a surface with a preferential attraction for one of the components of the mixture. We obtain detailed numerical results for a range of mixture compositions. In the case where the minority component is attracted to the surface, wetting layer growth is characterized by a crossover from a surface-potential-dependent growth law to a universal law. We formulate a simple phenomenological model to explain our numerical results.

Monte Carlo investigation of head-tail ordering of CO monolayers on graphite

Abstract Heat capacity measurements recently showed that CO physisorbed on graphite undergoes a head-tail ordering transition at roughly 5 K. The present paper is a detailed Monte Carlo study of this phase transition and the ordered state. The simulations are based on an ab initio pair potential and rely crucially on a thorough finite-size scaling study of various quantities. In agreement with experiments we find that the transition belongs to the universality class of the Ising model in two dimensions. We go beyond experimental knowledge by revealing the particular ferrielectric structure of the ground state, and show that the transition is due to the molecule's shape asymmetry rather than…

The interplay between structure and ionic motions in glasses

We present research examples that demonstrate how molecular dynamics simulations of real materials have reached a high level of sophistication. For simplicity, we focus on examples taken from our own research-although many other groups have done similarly valuable work on other systems and problems.

Interaction Between Polymer Brush-Coated Spherical Nanoparticles: Effect of Solvent Quality

The interaction between two spherical polymer brushes in solvents of variable quality is studied by molecular dynamics simulation and by self-consistent field theory, varying both the radius of the spherical particles and their distance, as well as the grafting density and the chain length of the end-grafted flexible polymer chains. Both the potential of mean force between the particles as a function of their distance is computed, for various choices of the parameters mentioned above, and the structural characteristics are discussed (density profiles, average end-to-end distance of grafted chains, etc.) It is found that for rather short chain lengths and not too large grafting densities, is…

Wedge filling and interface delocalization in finite Ising lattices with antisymmetric surface fields

Theoretical predictions by Parry et al. for wetting phenomena in a wedge geometry are tested by Monte Carlo simulations. Simple cubic $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}{L}_{y}$ Ising lattices with nearest neighbor ferromagnetic exchange and four free $L\ifmmode\times\else\texttimes\fi{}{L}_{y}$ surfaces, at which antisymmetric surface fields $\ifmmode\pm\else\textpm\fi{}{H}_{s}$ act, are studied for a wide range of linear dimensions $(4l~Ll~320,30l~{L}_{y}l~1000),$ in an attempt to clarify finite size effects on the wedge filling transition in this ``double-wedge'' geometry. Interpreting the Ising model as a lattice gas, the problem is equivalent to a li…

Study of the dynamic growth of wetting layers in the confined Ising model with competing surface fields

A two-dimensional magnetic Ising system confined in an L × D geometry () in the presence of competing magnetic fields (h) acting at opposite walls along the D-direction exhibits an interface between domains of different orientation that runs parallel to the walls. In the limit of infinite film thickness () this interface undergoes a wetting transition that occurs at the critical curve Tw(h), so that for T<Tw(h) such an interface is bound to the walls, while for Tw(h)≤T≤Tcb the interface is freely fluctuating around the centre of the film, where Tcb is the bulk critical temperature. Starting from a monodomain structure with the interface bound to one wall, we study the onset of the interface…

Static properties of end-tethered polymers in good solution: A comparison between different models

We present a comparison between results, obtained from different simulation models, for the static properties of end-tethered polymer layers in good solvent. Our analysis includes data from two previous studies--the bond fluctuation model of Wittmer et al. [J. Chem. Phys. 101, 4379 (1994)] and the off-lattice bead-spring model of Grest and Murat [Macromolecules 26, 3108 (1993)]. Additionally, we explore the properties of a similar off-lattice model simulated close to the Theta temperature. We show that the data for the bond fluctuation and the Grest-Murat model can be analyzed in terms of scaling theory because chains are swollen inside the Pincus blob. In the vicinity of the Theta point th…

Standard Definitions of Persistence Length Do Not Describe the Local “Intrinsic” Stiffness of Real Polymer Chains

On the basis of extensive Monte Carlo simulations of lattice models for linear chains under good and Θ solvents conditions, and for bottle-brush polymers under good solvent conditions, different me...

Spinodal decomposition of polymer solutions: molecular dynamics simulations of the two-dimensional case.

As a generic model system for phase separation in polymer solutions, a coarse-grained model for hexadecane/carbon dioxide mixtures has been studied in two-dimensional geometry. Both the phase diagram in equilibrium (obtained from a finite size scaling analysis of Monte Carlo data) and the kinetics of state changes caused by pressure jumps (studied by large scale molecular dynamics simulations) are presented. The results are compared to previous work where the same model was studied in three-dimensional geometry and under confinement in slit geometry. For deep quenches the characteristic length scale ℓ(t) of the formed domains grows with time t according to a power law close to [Formula: see…

Semiflexible polymer brushes and the brush-mushroom crossover.

Semiflexible polymers end-grafted to a repulsive planar substrate under good solvent conditions are studied by scaling arguments, computer simulations, and self-consistent field theory. Varying the chain length N, persistence length lp, and grafting density σg, the chain linear dimensions and distribution functions of all monomers and of the free chain ends are studied. Particular attention is paid to the limit of very small σg, where the grafted chains behave as "mushrooms" no longer interacting with each other. Unlike a flexible mushroom, which has a self-similar structure from the size (a) of an effective monomer up to the mushroom height (h/a ∝ N(v), ν ≈ 3/5), a semiflexible mushroom (l…

Molecular dynamics of supercooled polymer films

We present results of molecular dynamics simulations for a supercooled polymer melt confined between two smooth and purely repulsive walls. The thickness D of the film is about 7 times the bulk radius of gyration. For all temperatures studied, a significant increase of the monomer and chain mobilities with respect to the bulk is observed. Preliminary results suggest that structural relaxation times exhibit a power-law behavior in the vicinity of a critical temperature T c (D) 0.39 (in Lennard-Jones units). This estimate of T c (D) is about 14% smaller than the corresponding bulk value. Despite this significant decrease the time dependence of various mean-square displacements seems to be una…

Monte-Carlo Simulation of 3-Dimensional Glassy Polymer Melts: Reptation Versus Single Monomer Dynamics

A polymer melt is simulated at finite temperature by the Monte-Carlo method. We use a coarse-grained model for the polymer system, the bond-fluctuation model. Static properties of the melt can be obtained by generating configurations not with single-monomer- dynamics which moves individual monomers locally, but reptation-dynamics which allows collec- tive motion of the chains. This algorithm can produce equilibrated configurations much faster. It is demonstrated that static properties do not differ from those obtained by single-monomer- dynamics. Values of the radius of gyration, the mean square bond length and similar quantities for different temperatures and densities are presented.

Local Viscosity in the Vicinity of a Wall Coated by Polymer Brush from Green-Kubo Relations

When fluids are confined in slit pores between parallel walls, their static structures and their dynamical properties exhibit inhomogeneity in the z-direction perpendicular to the wall. Of particular interest are local bulk viscosity η b (z) and shear viscosity η s (z). Here, we discuss an algorithm to estimate these quantities from Green-Kubo relations using equilibrium molecular dynamics. As an application example, a polymer brush (macromolecules end-grafted to a substrate at z= 0) interacting with a solvent formed from point-like particles is given.

Stochastic dynamics simulation of grafted polymer brushes under shear deformation

We present results of computer simulations of polymer brushes (layers of polymer chains attached at one end onto an impermeable planar surface) under shear deformation at constant shear rate. As the first stage of calculations the behavior of a single brush was studied. The monomer density profile, the distribution of the chain ends, the positions and orientations of different monomers along the chain were calculated. Dimensions of the polymer chains as functions of the shear rate were obtained for different grafting densities. An increase in the brush thickness over the grafting plane with an increase in the shear rate as predicted by the theory of Barrat was observed. However, the magnitu…

Monte Carlo Test of the Classical Theory for Heterogeneous Nucleation Barriers

Flat walls facilitate the condensation of a supersaturated vapor: Classical theory of heterogeneous nucleation predicts that the free energy barrier $\Delta F_{\rm het}^*$ which needs to be overcome for the formation of sphere-cap shaped nucleation seeds is smaller than the barrier $\Delta F^*_{\rm hom}$ for spherical droplets in the bulk by a factor $0<f(\theta)<1$, which only depends on the contact angle $\theta$. In this letter we compute both $\Delta F^*_{\rm hom}$ and $\Delta F^*_{\rm het}$ from Monte Carlo simulations and test the theory for the lattice gas model (for which $\theta$ can be readily controlled). Even though the theory is only based on macroscopic arguments, it is shown …

Monte Carlo studies of polymer interdiffusion and spinodal decomposition: A review

Abstract Putting a layer of polymer A on top of a layer of polymer B, the broadening of the interfacial profile is observed in the framework of a lattice model (‘bond fluctuation method’). The interdiffusion constant is studied as a function of chain length, vacancy concentration, and interaction energy between unlike monomers, and a comparison with pertinent theoretical predictions is made. A lattice model where polymers are represented as self-avoiding walks on a simple cubic lattice is used to model ‘spinodal decomposition’, i.e. phase separation by ‘uphill diffusion’ in the unstable part of the phase diagram of a polymer mixture. For chain lengths N ≤ 32, the linearized Cahn-like theory…

Monte Carlo Simulations of Growth Kinetics and Phase Transitions at Interfaces: Some Recent Results

ABSTRACTIn the first part Monte Carlo studies of the kinetics of multilayer adsorption (without screening) are described. The approach to the jamming coverage in each layer is asymptotically exponential. The jamming coverages approach the infinite-layer limit value according to a power law. In the second part, studies of phase transitions in two dimensional fluids are reviewed. With a combination of Monte Carlo and finite size scaling block analysis techniques, accurate values are obtained for the critical temperatures, coexistence densities and the compressibilities of an adsorbed fluid layer in an NVT ensemble.

Monte Carlo simulation of a lattice model for ternary polymer mixtures

Monte Carlo studies of symmetrical polymer mixturesAB, modelled by selfavoiding walks withNA=NB=N steps on a simple cubic lattice, are presented for arbitrary concentrations of vacanciesφv in the range fromφv=0.2 toφv=0.8 and chain lengthsN≤64. We obtained the phase diagrams and the equation of state for three choices of the ratio ∈ / ∈AB (∈ being the energy between monomers of the same kind, ∈AB being the energy between different monomers). Flory-Huggins theory provides only a qualitative understanding of these results. If the equation of state is “fitted” with an effective Flory-Huggins parameterχeff, the latter turns out to be strongly dependent on both concentration and temperature.

Interface localization transition in Ising films with competing walls: Ginzburg criterion and crossover scaling.

Monte Carlo study of surface critical behavior in the XY model.

We have used Monte Carlo simulations to study the behavior of $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}D$ slabs containing classical spins which interact via nearest-neighbor $\mathrm{XY}$ coupling. The coupling constant ${J}_{S}$ for spins in the surface layer is fixed at $0.5J$. Finite-size scaling is used to analyze data for $D=59$ and to extract estimates for the surface critical exponents. We find that ${\ensuremath{\beta}}_{1}$ is in good agreement with theoretical predictions.

Momentum-dependent interfacial tension in polymer solutions

A model for the interface between a concentrated and a very dilute polymer solution is studied by Monte Carlo simulations at temperatures below the Theta temperature (in bad solvent conditions). The wave-number–dependent interfacial tension γ(q) is extracted from an analysis of the capillary wave amplitudes. It is shown that γ(q) decreases monotonically with q2, while no evidence is found for the predicted increase γ(q) ∝ κq2 with a positive bending rigidity κ of the interface at large q. Consequences for the interpretations of simulations and experiments on interfacial widths are briefly discussed.

Influence of Confining Walls on the Dynamics of Supercooled Simple Liquids

The relaxation dynamics of supercooled liquids in the bulk shows many features that are not seen in the dynamics of liquids at elevated temperatures, such as a very slow decay of the time-correlation functions, stretching, etc. The dynamics of liquids that are close to a surface (free space, confining wall, etc.) is even more complex in that the stretching is increased and in that there is evidence for the presence of multiple time scales. In this paper we review some results of recent molecular dynamics computer simulations in which we investigated the dynamics of a simple glass forming liquid in the vicinity of a wall. Two types of walls axe studied: A rough one and a smooth one. We find …

Intermediate Range Order in Silicate Melts and Glasses: Computer Simulation Studies

ABSTRACTWe present the results of large scale computer simulations to discuss the structural and dynamic properties of silicate melts with the compositions (Na2O)(2·SiO2), (Na2O)(20·SiO2) and (Al2O3)(2·SiO2). We show that these systems exhibit additional intermediate range order as compared to silica (SiO2) where the characteristic intermediate length scales stem from the tetrahedral network structure. Furthermore we show that the sodium dynamics in the sodium silicate systems exhibits a very peculiar feature: the long–time decay of the incoherent intermediate scattering function can be described by a Kohlrausch law with a constant exponent β for q &gt; qth whereby qth is smaller than the l…

Method for wettability characterization based on contact line pinning.

We demonstrate an efficient and reliable method for wettability characterization by determining the contact angle theta which a liquid-vapor interface makes with a solid wall. The purpose is to overcome the difficulties, related to the curvature of the liquid-vapor interface, which make measurements of theta rather uncertain, especially on the micro- and nanoscale. The method employs a specially designed slitlike channel in contact with a reservoir whereby the wettability of one of the slit walls is to be examined whereas the other (auxiliary) wall is separated by half into a lyophilic and a lyophobic part so as to pin the incoming fluid and fix the one end of the liquid-vapor interface. In…

Linear chain surfactants at a planar interface: a comparative Monte Carlo study of several lattice models

Linear chain surfactants in a densely packed arrangement (such as alkane chains in lipid monolayers in the “uniform tilt” structures) are described by a crude coarse-grained model where the endgroups grafted on the interface form a regular lattice and the chains are described by the bond fluctuation model with chains containing N = 4 effective monomers only. Square-well interactions between the monomers are studied for both the attractive and repulsive case for three choices of the interaction range. None of these models exhibits a structure with uniform tilt. For attractive interactions the last bond has a strong tendency to fold back thus leading to a very high density close to the interf…

Semiflexible Macromolecules with Discrete Bond Angles Confined in Nanoslits: A Monte Carlo Test of Scaling Concepts

Single semiflexible polymer chains confined in a planar slit geometry between parallel nonadsorbing repulsive walls a distance D apart are studied by Monte Carlo simulations of a lattice model, for the case of good solvent conditions. The polymers are modeled as self-avoiding walks on the simple cubic lattice, where every 90° kink requires a bending energy eb. For small qb = exp(−eb/kBT) the model has a large persistence length lp (given by lp ≈ 1/(4qb) in the bulk three-dimensional dilute solution, in units of the lattice spacing). Unlike the popular Kratky–Porod model of worm-like chains, this model takes both excluded volume into account and approximates the fact that bond angles between…

Langevin dynamics simulations of a two-dimensional colloidal crystal under confinement and shear

Langevin dynamics simulations are used to study the effect of shear on a two-dimensional colloidal crystal (with implicit solvent) confined by structured parallel walls. When walls are sheared very slowly, only two or three crystalline layers next to the walls move along with them, while the inner layers of the crystal are only slightly tilted. At higher shear velocities, this inner part of the crystal breaks into several pieces with different orientations. The velocity profile across the slit is reminiscent of shear banding in flowing soft materials, where liquid and solid regions coexist; the difference, however, is that in the latter case the solid regions are glassy while here they are …

Nature of crossover from classical to Ising-like critical behavior

We present an accurate numerical determination of the crossover from classical to Ising-like critical behavior upon approach of the critical point in three-dimensional systems. The possibility to vary the Ginzburg number in our simulations allows us to cover the entire crossover region. We employ these results to scrutinize several semi-phenomenological crossover scaling functions that are widely used for the analysis of experimental results. In addition we present strong evidence that the exponent relations do not hold between effective exponents.

Wetting and layering in the nearest-neighbor simple-cubic Ising lattice: A Monte Carlo investigation.

Critical, tricritical, and first-order wetting transitions are studied in a simple-cubic nearest-neighbor Ising model, with exchange J in the bulk and exchange ${J}_{s}$ in the surface planes, by applying suitable bulk and surface fields H and ${H}_{1}$. Monte Carlo calculations are presented for systems of size L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}D, in a thin film geometry with D=40 layers and two free L\ifmmode\times\else\texttimes\fi{}L surfaces, with L ranging from L=10 to L=50. In addition, evidence for prewetting transitions and for layering transitions (the latter occur for temperatures T less than the roughening temperature ${T}_{R}$) is presented. …

Cooling-rate effects in amorphous silica: A computer-simulation study

Using molecular dynamics computer simulations we investigate how in silica the glass transition and the properties of the resulting glass depend on the cooling rate with which the sample is cooled. By coupling the system to a heat bath with temperature $T_b(t)$, we cool the system linearly in time, $T(t)=T_i-\gamma t$, where $\gamma$ is the cooling rate. We find that the glass transition temperature $T_g$ is in accordance with a logarithmic dependence on the cooling rate. In qualitative accordance with experiments, the density shows a local maximum, which becomes more pronounced with decreasing cooling rate. The enthalpy, density and the thermal expansion coefficient for the glass at zero t…

Finite-size scaling and the crossover to mean-field critical behavior in the two-dimensional Ising model with medium-ranged interactions.

Critical amplitudes in finite-size scaling relations show a singular dependence on the range of the interactions, R. The respective power laws are predicted from phenomenological crossover scaling considerations. These predictions are tested by Monte Carlo simulations for medium-ranged Ising square lattices. It is speculated that some deviations between the simulation results and corresponding predictions may be due to logarithmic corrections.

Molecular Dynamics Computer Simulation of Cooling Rate Effects in a Lennard-Jones Glass

We present the results of a molecular dynamics computer simulation of a binary Lennard-Jones mixture. We simulate a quench of the system from a liquid state at high temperatures to a glass state at zero temperature by coupling the system to a heat bath that has a temperature that decreases linearly (with slope -γ) with time. We investigate how the residual density of the system varies as a function of the cooling rate γ and rationalize our results by means of the dependence of the coordination number of the particles on the cooling rate.

Simulation of the glass transition in polymeric systems: Evidence for an underlying phase transition?

Abstract The bond fluctuation model of polymer chains on sc lattices with an energy that favours long bonds can describe the slowing down of supercooled melts that approach the glass transition in qualitative similarity with various experiments. In this paper we focus on the question of whether there exists a correlation length that increases to large values when the temperature is lowered towards the glass transition. Two types of analysis are presented: firstly density oscillations near hard walls become long range, and the resulting correlation length becomes larger than the gyration radius, secondly oscillations in the pair correlation function in real space also become long range, and …

Ejection of a Polymer Chain from a Nanopore: Theory and Computer Experiment

We consider the ejection dynamics of a flexible polymer chain out of confined environment. This situation arises in different physical contexts, including a flexible synthetic polymer partially confined in a nanopore and a viral genome partially ejected from its capsid. We describe the chain release from confinement both analytically and by means of dynamic Monte Carlo simulation. We find two distinct regimes of ejection dynamics depending on whether the chain is fully or partially confined. Partially confined chains are ejected from a pore of length L and diameter D after a typical time τ ∝ L2D5/3, regardless of their contour length N. The process is driven by a constant force f ≈ 5kBT/D a…

Statistical Theories of Phase Transitions

The sections in this article are Introduction Phenomenological Concepts Order Parameters and the Landau Symmetry Classification Second-Order Transitions and Concepts about Critical Phenomena (Critical Exponents, Scaling Laws, etc.) Second-Order Versus First-Order Transitions; Tricritical and other Multicritical Phenomena Dynamics of Fluctuations at Phase Transitions Effects of Surfaces and of Quenched Disorder on Phase Transitions: A Brief Overview Computational Methods Dealing with the Statistical Mechanics of Phase Transitions and Phase Diagrams Models for Order–Disorder Phenomena in Alloys Molecular Field Theory and its Generalization (Cluster Variation Method, etc) Computer Simulation T…

Entropy theory and glass transition: A test by Monte Carlo simulation

This article reviews the results of a test of the Gibbs-DiMarzio theory by Monte Carlo Simulation. The simulation employed the bond-fluctuation model on a simple cubic lattice. This model incorporates two kinds of interactions: the excluded volume interaction among all monomers of the melt and an internal energy of the chains, which favors large bonds and makes the chains stiffen with decreasing temperature. The stiffening of the chains leads to an increase of their volume requirements, which competes with the packing constraints at low temperatures. This competition strongly slows down the structural relaxation of the melt and induces the glassy behavior. The model therefore takes into acc…