Dynamical theory of concentration fluctuations in polymer solutions under shear

Abstract

Shear flow has been shown experimentally to lead to large concentration fluctuations in entangled polymer solutions near to their coexistence curves. I develop a dynamical theory of concentration fluctuations coupled to polymer elastic stress, in terms of a ‘‘two-fluid’’ model of polymer plus solvent. At first order in shear rate γ̇, only one component of the stress is coupled to the concentration. The structure factor S(q;γ̇) is given as a ‘‘history integral’’ over thermal-noise amplitudes at wave numbers convected onto q. The characteristic wave number ${\mathit{q}}^{\mathrm{*}}$, at which concentration fluctuations relax in a stress-relaxation time, determines (1) the length scale for the mixing of modes as observed in the dynamic light scattering of quiescent solutions, and (2) the length scale of peaks observed in static light scattering under shear. Both features depend on treating stress dynamics beyond an adiabatic approximation. At O(γ${\mathrm{̇}}^2$), the spinodal is shifted to higher (lower) temperatures for fluctuations in the gradient (vorticity) direction.