COMPLEX VISCOSITY FOR THE RIGOROUS FORMULATION OF THE MULTIBEAD INTERNAL VISCOSITY MODEL WITH HYDRODYNAMIC INTERACTION

The internal viscosity (IV) model for a polymer chain resembles the classical Rouse-Zimm model consisting of a linear sequence of N "springs" (submolecules) connecting N + 1 "beads" (discrete frictional centers) but adds an additional viscous component within the submolecule in parallel with the spring. While many qualitative improvements in rheological predictions result from introducing the IV concept, previous IV formulations have been generally unsatisfactory; either they were severely restricted in generality (e.g., to N = 1 or to very small values of the IV parameter-phi) or were suspect due to use of intuitive but unproven dynamical assertions (e.g., the "linearized rotational velocity" approximation, LRV). Some model predictions also were unrealistic. Here, we achieve a rigorous and complete solution for eta'(omega) and eta"(omega) by expressing the IV force in terms of the vectorially proper deformational velocity (Booij and van Wiechen) and expanding the bead distribution function psi in terms of strain. Only zero-order and first-order terms in strain are subsequently needed when computing configurational moments of psi for representing shear stress. Closed-form analytical expressions are obtained for [eta'] and [eta"] in terms of parameters phi, N, and h* (hydrodynamic interaction); because a conventional normal-mode analysis succeeds, traditional eigenvalues a(k)(N,h*) can be used. Graphical displays are given of behavior for N = 1 to 100, h* = 0 and 0.262, and phi = 0 to infinity. None of the unrealistic features of earlier IV models, based on the LRV approximation, are seen. Overall, the influence of phi on predictions is quite weak, a surprising but satisfactory result. A realistic prediction for the high-omega limit [eta' infinity] appears, showing the necessity for phi but also the importance of N and h*. Implications of results and chain modeling practices are discussed.