Second Virial Coefficient of Linear Polymer Molecules

A variation theory is developed for obtaining a closed expression for the second virial coefficient A2 of linear polymer molecules, which corresponds to the Fixman theory of the excluded volume effect. It is shown that the second virial coefficient of the polymer in good solvents is effectively identical with that of a nonpenetrating sphere whose radius is proportional to the root-mean-square statistical radius 〈S2〉½. In extremely good solvents, A2 becomes a constant independent of molecular weight M. A graphical method is proposed for separate determination of the effective bond length and the polymer—solvent interactions; the method consists of plotting A2M½ against M½. Applications of this method are illustrated by four examples, polystyrene in toluene, isotactic polypropylene in tetralin and in α-chloronaphthalene, and nitrocellulose in acetone. The effective bond lengths obtained are in good agreement with those previously evaluated from the molecular weight dependence of intrinsic viscosity [η], but the polymer—solvent interactions are not. The theoretical value of the ratio A2M/[η] is about 60 in good solvents, which is approximately a half of the ordinary experimental values. A modification of the variation theory is proposed, which corresponds to Ptitsyn's modification of the Fixman theory and leads to a more plausible value 110 for A2M/[η]. The triple contact term in the perturbation expansion series of A2 is also evaluated. It is found that the expansion of a molecular coil due to the volume effect is somewhat suppressed by intermolecular interactions in the proximity of the second molecule.