Viscoelastic Effect on Hydrodynamic Relaxation in Poly- mer Solutions

– The viscoelastic effect on the hydrodynamic relaxation in semi-dilute polymer solutions is investigated. From the linearized two-fluid model equations we predict that the dynamical asymmetry coupling between the velocity fluctuations and the viscoelastic stress influences on the hydrodynamic relaxation process, resulting in a wave number dependent shear viscosity. Previously, Doi and Onuki [1] showed that the diffusion process is drastically influenced by the dynamical asymmetry coupling between the concentration fluctuations and the longitudinal mode of the viscoelastic force (stress-diffusion coupling). Their analysis predicted an additional wave number dependence of the diffusion coefficient due to the viscoelasticity, which was confirmed by the recent experimental studies (see for example Ref. [2]). However, in their analysis the very viscous systems without the velocity fluctuations were assumed, so the effect of the viscoelasticity on the relaxation of the velocity fluctuations (hydrodynamic relaxation) was not investigated. Here we take the transverse parts of the viscoelastic force and the velocity fluctuations into consideration. We shall then show that the dynamical asymmetry coupling between these transverse modes has a strong influence on the hydrodynamic relaxation, resulting in a wave number dependent shear viscosity. In the present analysis we use the two-fluid model [3, 1] as the basic equations describing the dynamics of polymer solutions. The two-fluid model was put forward by Brochard and de Gennes [3]. Later, the current form of the two-fluid model was derived by Doi and Onuki [1]. It has been successfully used to explain various phenomena such as shear induced concentration fluctuations [1, 5, 6, 7], non-exponential decay in dynamic scattering near equilibrium [3, 1, 8], phase separation dynamics [4,9,10,11], and so on. Here we briefly survey the two-fluid model equations. The readers who want to know the details of the two-fluid model, see the original paper [1] or excellent reviews [4, 11]. Let vp(r, t) and vs(r, t) be the average velocities of polymer and solvent, respectively, and ψ(r, t) is the volume fraction of polymer at point r and time t. ψ satisfies a conservation law