Yuhua Song

Liquid-liquid equilibria for polymer solutions and blends, including copolymers

Abstract A simplified perturbed hard-sphere-chain (PHSC) theory is applied to interpret, correlate, and (in part) predict liquid-liquid equilibria (LLE) for polymer solutions and blends, including copolymers. The PHSC equation of state uses a hard-sphere-chain reference system plus a van der Waals attractive perturbation. Three pure-component parameters are obtained from readily available thermodynamic properties. Mixture parameters are obtained using pure-component parameters, conventional combining rules, and one or two binary constants. Theoretical and experimental coexistence curves and miscibility maps show good agreement for selected blends containing polymers and copolymers. For LLE of dilute or semi-dilute solvent/polymer solutions, it is necessary to decrease the pure-component polymer chain length in the perturbation term, probably because the mean-field approximation is not suitable for such solutions.

Liquid—liquid phase diagrams for binary polymer solutions from a perturbed hard-sphere-chain equation of state

Abstract The perturbed hard-sphere-chain (PHSC) equation of state for multicomponent mixtures is presented as a generalization from the equation of state for pure fluids. The reference term, based on Chiews equation of state for hard-sphere chains, requires no mixing rules. Only the attractive perturbation requires van der Waals one-fluid mixing rules. Cross parameters needed in the perturbation are obtained using pure-fluid parameters and simple combining rules. The simplifying physical assumptions required to reduce the perturbation term to the Flory χ parameter are given. Specific interactions are included by adapting the model of ten Brinke and Karasz. Model calculations for binary mixtures demonstrate that the PHSC equation can predict lower critical solution temperatures, upper critical solution temperatures and closed partial-miscibility loops. Special attention is given to the effects of polymer molecular weight, pressure, differences in segment size and differences in segment interaction energy.

Liquid-liquid equilibria and theta temperatures in homopolymer-solvent solutions from a perturbed hard-sphere-chain equation of state

The perturbed hard-sphere-chain (PHSC) equation of state is used to calculate liquid-liquid equilibria of binary nonpolar solvent/homopolymer systems exhibiting both an upper critical solution temperature (UCST) and a lower critical solution temperature (LCST). Systems studied include polyisobutylene, polyethylene, and polystyrene solutions. Equation-of-state parameters of homopolymers are obtained by regressing the pressure-volume-temperature data of polymer melts. In polymer solutions, however, theory overestimates the equation-of-state effect which causes the LCST at elevated temperature. To correct the overestimated equation-of-state effect, an empirical adjustable parameter is introduced into the perturbation term of the PHSC equation of state. An entropy parameter is also introduced into the Helmholtz energy of the mixture to correlate quantitatively the dependence of critical temperatures on polymer molecular weight. For systems exhibiting a LCST, two adjustable parameters are required to obtain quantitative agreement of theoretical critical temperatures with experiment as a function of polymer molecular weight. For systems exhibiting both an UCST and a LCST, three adjustable parameters may be necessary. The need for so many empirical binary parameters is probably due to the oversimplified perturbation term which is based on the mean-field assumption.

Liquid‐liquid equilibria and theta temperatures in binary solvent‐copolymer solutions from a perturbed hard‐sphere‐chain equation of state

The perturbed hard-sphere-chain (PHSC) equation of state for copolymer systems is used to calculate liquid-liquid equilibria for binary solvent/copolymer systems that exhibit simultaneously an upper critical solution temperature (UCST) and a lower critical solution temperature. The PHSC equation of state for copolymer solutions uses those binary parameters that represent liquid-liquid equilibria for the two parent homopolymer solutions. An additional intersegmental parameter is also required to define the interaction energy between a pair of chemically dissimilar segments comprising the copolymer molecule. Theory is compared with experiment for binary copolymer solutions containing poly(styrene-co-α-methylstyrene) random copolymers. Using the same K BC , the intersegmental parameter between styrene and α-methylstyrene segments, theory and experiment show fair agreement for the copolymer-composition dependence of theta temperatures associated with UCST in cyclohexane, methyl cyclohexane, and decalin. A comparison is also made between K BC obtained from the copolymer-solution data and that obtained from the coexistence curve of homopolymer blends containing polystyrene and poly(α-methylstyrene). Data for copolymer solutions require a K BC . that represents interactions more unfavorable than those in homopolymer blends.

14 Equations of state for polymer systems

Publisher Summary This chapter summarizes equations of state (EOS) for molten polymers and for mixtures of polymers with solvents or other molten polymers. This chapter compares their theoretical foundations and indicates their usefulness for calculating thermodynamic properties, especially for phase equilibria. Attention is restricted to polymer liquids. No significant attention is given in the chapter to glassy polymers or to crystallinity. To describe polymers and their mixtures with solvents and other polymers, equations of state can be divided into four categories: cell models, lattice-fluid models, hole models, and tangent-sphere models. The cell and lattice-fluid models provide different adaptations of the incompressible-lattice model of polymer mixtures; however, each incorporates compressibility in a different manner. Hole models combine both methods of incorporating compressibility introduced by cell and lattice-fluid models. Finally, recent advances in statistical thermodynamics have brought to the forefront tangent-sphere models of chain-like fluids. These models abandon lattice origins; they model polymers as freely-jointed tangent-spheres where unbonded spheres interact through a specified intermolecular potential.