Third Virial Coefficient of Polymer Solutions

The osmotic third virial coefficient A3 of solutions of linear chain polymers is calculated using the random flight model. Some preliminary comments are given somewhat in detail on the approximations introduced, and the theory is developed in the two-parameter scheme. First, the perturbation theory is presented with a revision of the result obtained by Stockmayer. An approximate closed expression for A3 is then derived applying a variational method as used in the previous paper on the second virial coefficient A2, the procedure being equivalent to the coupling-parameter method. Evaluation is performed by introduction of approximations similar to the Casassa—Markovitz approximation in their theory of A2. It is shown that the dimensionless quantity A3/A22M with the polymer molecular weight M becomes zero at the theta point and has the asymptotic value of 0.753 for high-molecular-weight polymers in good solvents if the third-power law equation of Fixman is assumed for the expansion factor of the polymer. The values of A3/A22M are somewhat larger than the corresponding values predicted by Stockmayer and Casassa. A comparison of the theory with experiment is made for polystyrene fractions in toluene and polyisobutylene fractions in cyclohexane, and an agreement between the theoretical and observed values of A3/A22M is discussed.