Statistical Mechanics of Dilute Polymer Solutions. II

Abstract

New thermodynamic relationships applicable to dilute solutions of heterogeneous high polymers are derived. The function previously given to express the interaction between two polymer molecules of equal size is modified to include chemically similar polymer molecules differing in size. With the aid of this more general pair partition function the mutually excluded volume for a pair of polymer molecules is easily computed. The osmotic pressure for a dilute solution of a heterogeneous polymer may be expressed in virial form through the application of conventional methods for treating imperfect gases. The second virial coefficient in the series expansion for the osmotic pressure is readily obtained but it is necessary to resort to the hard sphere approximation for estimation of the third coefficient. The third coefficient is much larger than was indicated by earlier theories now known to be inapplicable to dilute solutions, and plots of π/c vs. c must be appreciably curved if the slope is great. Similar expressions are derived for Hc/τ obtained from light scattering measurements. For a dilute solution of a homogeneous polymer the value of the second coefficient in either the osmotic pressure or light scattering expression is not entirely independent of molecular weight, as predicted by earlier theories. The effect of heterogeneity on the second coefficient in these expressions is investigated for the special case of a dilute solution of two polymers differing only in molecular weight, from which it follows that the ratio of the second coefficients in the light scattering and osmotic pressure relations may be either equal to two, or greater or less than that value, depending on the heterogeneity ratio. At the critical miscibility temperature Θ, the excluded volume vanishes and the present treatment becomes identical with previous theories. The latter should be valid for all concentrations only at the unique temperature T=Θ. The primary parameters κ1 and ψ1 governing the heat and entropy of interaction between polymer segments and solvent molecules are difficult to evaluate solely from thermodynamic measurements on dilute solutions except at T=Θ.