Stephen M. Lambert

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 in Binary Systems:Monte-Carlo Simulations for Calculating the Effect of Nonrandom Mixing

Monte-Carlo simulations of a lattice model for incompressible monomer/r-mer mixtures are used to obtain accurate results for the configurational energy of mixing. Based on simulation results, the energy of mixing is correlated as a function of temperature and composition using an empirical expression. The configurational Helmholtz energy is obtained upon using the Gibbs-Helmholtz equation with Guggenheims athermal entropy of mixing as boundary condition. Since Monte-Carlo simulations give essentially exact results for the lattice model, the effects of nonrandom mixing on the configurational thermodynamic properties of a binary mixture can be determined. The expression generated here produces coexistence curves that are more accurate than those from other models, especially near the critical region.

Miscibilities in binary copolymer systems

Abstract A lattice theory is presented for liquid-liquid equilibria in binary systems containing random copolymers. This theory takes into account deviations from random mixing through a non-randomness factor which follows from a generalization of Monte-Carlo calculations for the three-dimensional Ising model. While the lattice remains incompressible, the effect of specific interactions (hydrogen bonding) is introduced by superimposing on the non-random (Ising model) expression for the Helmholtz energy of mixing a correction based on the lattice-gas model by ten Brinke and Karasz. The resulting theory can predict immiscibility caused by lower critical solution temperatures. Several theoretical miscibility maps at fixed temperature were computed; these are compared with those predicted by the random-mixing Flory-Huggins theory. Theoretical miscibility maps are also compared with experiment for a few systems with strong specific interactions.

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.