Calculation of phase diagrams not requiring the derivatives of the Gibbs energy for multinary mixtures

A method for the calculation of phase diagrams (tie lines and binodal, spinodal, critical points and their stability) based exclusively on the Gibbs energy of mixing with no need of its derivatives with respect to the composition variables is extended to multinary mixtures for any number of components. The mathematical description of the (K-1)-dimensional phase diagram of a K-component mixture is presented. The method is demonstrated for a quinternary blend of five polymers exhibiting a closed miscibility gap; all binary, ternary and quaternary subsystems are completely miscible. The phase separation in the quinternary system is caused by very favorable interactions in the ternary subsystem…

Complex miscibility behaviour for polymer blends in flow

Abstract Experimental observations of the effect of shear flow on the miscibility of binary polymer blends are compared to calculations based on a generalized Gibbs energy of mixing Gγ˙. This mixing free energy characterizes the steady state established at shear rateγ˙, as the sum of G z , the equilibrium Gibbs energy and E s , the energy the system stores while flowing.

Phase diagrams of the system tetrahydrofuran/γ-butyrolactone/poly(ether imide) and determination of interaction parameters

Abstract The thermodynamic interactions in the ternary mixture tetrahydrofuran/γ-butyrolactone/poly(ether imide) (THF/γ-BL/PEI) are investigated from 30 to 50°C. This is a membrane-forming system with a high He-selectivity. Cloud point measurements show that the two binary polymer solutions THF/PEI and γ-BL/PEI exhibit miscibility gaps which close towards the centre of the ternary phase diagram where the mixtures become homogeneous (cosolvency). Vapour pressures were determined for the subsystem THF/γ-BL. For the theoretical calculations the Gibbs energy of mixing is formulated according to the Flory-Huggins theory and the interaction parameter χ THF γ-BL calculated from the vapour pressure…

Phase diagrams calculated for flowing polymer solutions: spinodal and three phase conditions

Spinodal lines and critical points (CPs) are calculated for flowing solutions of polystyrene in trans-decalin. Three types of CPs can be distinguished: The first consists of stable CPs (ordinary critical line) and originates from the CP of the quiescent system. The other two CPs are bound to shear. Additional stable CPs (extraordinary critical line) result for higher polymer concentrations and unstable CPs for intermediate concentrations. Ordinary and unstable critical line merge in a heterogeneous double CP. The coexistence of three phases in the flowing system (eulytic points) comes to an end as two of them merge upon an increase in shear rate at a critical end point.

Influence of Molar Mass Distribution on the Compatibility of Polymers

Abstract Phase equilibria were calculated by means of a new method (direct minimization of the Gibbs energy of mixing) for polymer blends consisting of monodisperse polymer A and polydisperse polymer B. The results obtained for a Schulz-Flory distribution of B (molecular nonuniformity U = (M w/M n) −1 = 1 and 100 components of model B) agree quantitatively with that of computations on the basis of continuous thermodynamics. The influence of U B on the miscibility of A and B in 1:1 mixtures was studied for constant M w of B, quantifying the incompatibility of the polymers by the length of the tie lines. The outcome of these calculations demonstrates that the typical effect of an augmentation…

Shear Effects on the Phase Diagrams of Solutions of Highly Incompatible Polymers in a Common Solvent. 2. Experiment and Theory

Cloud point temperatures (Tcp) were measured at different constant shear rates for three representatives of the ternary system cyclohexanone/polystyrene/poly(n-butyl methacrylate) (CHO/PS/PBMA) by means of a newly constructed rheo-optical apparatus that can be operated in the temperature range from 0 to 100 °C up to maximum shear rates of 1440 s-1 and maximum stresses of 384 Pa. In all cases one observes an extension of the homogeneous region as the shear rate γ is raised. With the system CHO/PS 196w/PBMA 2050 (the figures denote the molar masses of the polymers in kilograms/mole) the effects become maximum for high concentrations of PBMA, where the demixing temperatures increase by more th…

Shear influences on the solubility of LDPE in ethene

Abstract The influence of shear on the demixing pressures of the binary system ethene/polyethene was determined viscometrically in the temperature range 147–195°C up to pressures of 1400 bar at polymer concentrations of 18.4, 22.5 and 27.4 wt%, respectively. Autoclaves were used to prepare the solutions and to measure cloud points for the stagnant system. A high-pressure viscometer was used to perform experiments at shear rates γ up to 1500 s −1 . Under all conditions investigated here the two-phase area grows as γ becomes larger and the demixing pressures increase up to 23 bar. This observation is in contrast to theoretical expectation according to which it should be the one-phase region w…

Calculation of phase diagrams not requiring the derivatives of the Gibbs energy demonstrated for a mixture of two homopolymers with the corresponding copolymer

A method is presented which allows the calculation of phase diagrams (spinodal, binodal and tie lines) on the basis of the Gibbs energy of mixing ΔG. No derivatives of ΔG with respect to the composition variables are required. This method is particularly useful in cases where the composition dependence of ΔG is very complex and no analytical representation of the derivatives can be given. The method is applied to a ternary mixture of two homopolymers with a copolymer consisting of the same monomers. The sequence distribution of the copolymer is kept constant between random and purely alternating, and phase diagrams are calculated for different chemical compositions of the copolymer. The com…

Calculation of shear influences on the phase separation of polymer blends exhibiting upper critical solution temperatures

Calculations were performed on the basis of a generalized Gibbs energy of mixing G γ, which is the sum of the Gibbs energy of mixing of the stagnant system and E s, the energy stored in the system during stationary flow. With increasing shear rate γ, the demixing temperatures shift to lower values (shear-induced mixing; diminution of the heterogeneous area), then to higher values (shear-induced demixing), and finally to lower values again before the effects fade out. The details of the rather complex phase diagrams resulting for a given shear rate are primarily determined by a band in the T/χ plane (χ = mole fraction) within which (∂2 E s/∂χ2) T UCST where they submerge. Bound to a suitable…

Phase diagrams calculated for sheared ternary polymer blends

Abstract On the basis of the generalized Gibbs energy of mixing G γ (which is the sum of the Gibbs energy for zero shear and the energy the system stores in steady flow) phase diagrams were calculated as a function of shear rate γ for ternary model blends. This modelling uses simple equations for the description of the stagnant systems (Flory-Huggins) and for the contributions resulting from flow. Surface and alignment effects are neglected. A new procedure, which does not require the derivatives of G γ with respect to composition, was used to that end. Choosing typical values for the binary interaction parameters and molar masses, four classes of ternary systems were studied in greater det…

Calculation of miscibility behavior of multinary polymer blends

A method for the calculation of phase diagrams (tie lines and binodal, spinodal, critical points and their stability) based exclusively on the Gibbs energy of mixing, δG, is presented which does not require the calculation of the derivatives with respect to the composition. The method is demonstrated for ternary mixtures of two homopolymers and the corresponding copolymer, and for quaternary and quinternary blends of five polymers exhibiting a closed miscibility gap. The advantages of the presented method become most obvious in the mathematical description of measured phase diagrams, where complex composition dependencies of the interaction parameter are observed.

Shear induced mixing/demixing: blends of homopolymers, of homopolymers plus copolymers, and blends in solution

Shear may shift the phase boundary towards the homogeneous state (shear induced mixing, SIM), or in the opposite direction (shear induced demixing, SID). SIM is the typical behavior of mixtures of components of low molar mass and polymer solutions, SID can be observed with solutions of high molar mass polymers and polymer blends at higher shear rates. The typical sequence with increasing shear rate is SIM, then occurrence of an isolated additional immiscible area (SLD), melting of this island into the main miscibility gap, and finally SIM again. A three phase line originates and ends in two critical end points. Raising pressure increases the shear effects. For copolymer containing systems S…

The viscosities of moderately concentrated solutions of poly(butyl methacrylate)—Mw = 1820 kg/mol–in the thermodynamically good solvent toluene and in the theta solvent 2-propanol were measured up to pressures of 2000 bar as a function of shear rate in the temperature interval from 40 to 70°C. Both systems obey Graessley's equation so that it is possible to obtain the following rheological parameters in addition to the zero-shear viscosity η0 from the dependences . η can be split into an entanglement part ηent plus a frictional contribution ηfric, and the characteristic viscometric relaxation time τ0 becomes accessible. The present results demonstrate that the application of pressure raises…