Excluded Volume Effect of Linear Polymer Molecules

Abstract

A new closed expression for the excluded volume effect of linear polymer molecules is developed with the aid of an equivalent ellipsoid model, in which the polymer chain with fixed ends is replaced by a uniform distribution of unconnected segments within an ellipsoid whose dimensions are chosen to give the correct principal radii of gyration of the chain. The linear expansion factor α for a chain of N segments is given by (α3−α)/N12={(1+13α−2)−32}−1(4/3)5/2(3/2π)32(β/a3)where a is the effective bond length and β is the binary cluster integral of a segment pair. In contrast to Flory's familiar equation, the ratio (α5—α3)/N½ is an increasing function of α instead of a constant. The new equation is in excellent agreement with the Monte Carlo calculations of Wall and Erpenbeck for a diamond lattice chain and also with existing viscosity data for various polymer solutions, provided that the expansion factor is evaluated from the equation[η]/[η]θ=α2.43in accord with the theories of Kurata-Yamakawa and Ptitsyn-Eisner. It is also shown that the present equation for α is in rather good agreement with the numerical solution of Fixman's differential equation.