Numerical calculations of the viscoelastic properties of solutions of branched polymers based on the Zimm-Kilb theory

Numerical calculations were performed for the viscoelastic properties of dilute solutions of branched star polymers with equal branch lengths as formulated in terms of a bead-spring model by Zimm and Kilb without using the integrodifferential equation approximation method to calculate the eigenvalues. The complex modulus and complex viscosity were calculated as functions of frequency for various combinations of the number of branches f (4, 8, and 13), the number of beads in one branch N_b (= N/f; 20 to 100, where N + 1 is the total number of beads, N the number of springs in the molecule) and the reduced hydrodynamic interaction parameter h^* (= h/N^1/2 0.05 to 0.3, where h is the hydrodynamic interaction parameter of Zimm and Kilb). The frequency dependence of the complex modulus in the low-frequency range depends mainly on h^* and not on N_b if N_b is large enough, and it is very close to that calculated from the eigenvalues for h→∞ obtained by Zimm and Kilb, if h^* is about 0.25. As h^* decreases from 0.25, the frequency dependence gradually approaches that of the free-draining cash (h→0). Calculations may be carried out for h^* values somewhat larger than 0.25 and result in a frequency dependence that is not intermediate to the h → 0 and h → ∞ cases as evaluated by Zimm and Kilb. The physical meaning of such “super-non-free-draining” values of h^* is uncertain, however. The intrinsic viscosity ratio g′ = [η]_f/[η]_lin is an increasing function of h^* and changes very slowly with N. For h^* = 0.25, g′ is close to the non-free-draining limit for any value of N.