Statistical Thermodynamics of Polymer Solutions. I. Theory of the Second Virial Coefficient for a Homogeneous Solute

An approximate theory of the second virial coefficient A₂ for a system of identical random flight polymer chains of n steps, of average length b, is developed on the basis of a treatment by Zimm in power series form which, though formally exact, is not rapidly enough convergent to be practically useful. By introducing, as an approximation, a spherically symmetrical distribution of the chain segments of two interacting molecules averaged relative to the locus of an initial intersegmental contact, it becomes possible to sum the series, the terms of which relate to molecular configurations with progressively greater numbers of simultaneous contacts. The result, a double integral, is evaluated numerically; but the simple relation $$A_2=\mathrm{const}\text{[1−}\exp \text{(1.42ψ)]/1.42ψ,}$$ in which ${\text{ψ=4βn}}^{\text{1/2}}{\text{(3/2πb}}^2{)}^{\text{3/2}}$ and β is the mutually excluded volume for a pair of segments, differs from the integral by less than one percent for all positive values of the parameter ψ. The analytical form of this equation is justified in terms of the theory. Virial coefficients calculated from the theory with parameters derived from intrinsic viscosity measurements agree fairly well with the observed dependence on solute molecular weight for two systems in good solvents, but the magnitudes obtained are too small. If, however, the parameters of the theory are considered arbitrary, correspondence with experiment, to within errors of measurement, can be achieved.