Relationship of the Second Virial Coefficient to Polymer Chain Dimensions and Interaction Parameters

Abstract

A useful approximation has been found for the excluded volume integral for the interaction of a pair of polymer molecules, represented by Gaussian distributions of chain segments about their respective centers of gravity. By means of this approximation, the theoretical expression for the second virial coefficient in the expansion of the osmotic pressure is represented over the entire range of polymer‐solvent interaction by A 2 = const ([η]/M) ln [1+(π 1 2 /4)X 1 +(π 1 2 3 3 2 /32)X 2 ]. X 1 and X 2 are related to the thermodynamic interaction parameters χ1 and χ2, respectively, in the semiempirical expression for the solventchemical potential μ1—μ1 0=RT[ln(1—v 2)+(1–1/x)v 2+χ1 v 2 2+ χ2 v 2 3+···] where v 2 is the volume fraction of polymer and x the ratio of molar volumes of polymer and solvent. Inclusion of the higher term χ2 v 2 3 (and X 2) constitutes a refinement over the treatment previously published. It is shown that the influence of X 2 may be appreciable for low molecular weights and in poor solvents; its effect vanishes as the molecular weight becomes large. However, if X 2≠0, the temperature at which A 2 for a given polymer‐solvent pair becomes zero will, in general, depend upon the molecular weight. The similar influence of this term on the expression for the intramolecular expansion factor is smaller, although not necessarily negligible. Interaction parameters (χ1) are calculated from second virial coefficients for a number of polymer‐solvent systems, and these are compared with the χ1 values obtained from intrinsic viscosities (intramolecular theory). The good agreement obtained offers strong evidence for the general validity of the intermolecular (A 2) and intramolecular theories. A 2 increases slightly more rapidly with decrease in M than theory predicts. On the whole, however, the consistency of results is gratifyingly good. The intrinsic viscosity increases linearly with A 2 M in poor solvents. This relation must be replaced by one of approximate direct proportionality in good solvents.