Thermodynamics of Heterogeneous Polymers and Their Solutions

The statistical mechanical treatment previously applied to homogeneous polymer solutions has been extended to heterogeneous polymers composed of numerous molecular species differing in polymer chain length. Entropies of mixing with solvent are derived with reference to three standard states: the oriented pure components, the disoriented pure components, and the disoriented mixture of species. The entropy of mixing with respect to the last-mentioned state is identical with that previously derived neglecting polymer heterogeneity. Upon introducing a van Laar heat of mixing term, expressions for the partial molal free energies are derived. These should be reasonably correct in view of the satisfactory agreement between observed solvent activities in polymer-solvent solutions and theory, as shown by Huggins. The relationships should be useful not only in dealing with solutions, but also in equilibria involving heterogeneous polymers in the absence of solvent. The thermodynamic equations are applied to the problem of equilibrium in the range of partial miscibility for solvent-heterogeneous polymer systems. The requisites for efficient fractionation of high polymers are discussed. The efficiency of separation depends on the ratio of the volumes of the supernatant and the precipitated phases. In order to attain a high value for this ratio, very dilute solutions must be employed. The higher the molecular weight at which separation occurs the greater the dilution required for the same sharpness of separation. The free energy expressions are also applied to the formulation of equilibrium constants to be employed in reversible polymerization-degradation processes. It is shown that concentrations should be expressed in moles per unit volume in the equilibrium constant. The equilibrium state for a polyfunctional condensation, considered as a reaction between functional groups, is not affected by the polymeric nature of the reacting species; the position of equilibrium should be the same as is found for analogous monofunctional reactants under the same conditions.