Long-time dynamics of Rouse–Zimm polymers in dilute solutions with hydrodynamic memory

Abstract

The dynamics of flexible polymers in dilute solutions is studied taking into account the hydrodynamic memory, as a consequence of fluid inertia. As distinct from the Rouse-Zimm (RZ) theory, the Boussinesq friction force acts on the monomers (beads) instead of the Stokes force, and the motion of the solvent is governed by the nonstationary Navier-Stokes equations. The obtained generalized RZ equation is solved approximately using the preaveraging of the Oseen tensor. It is shown that the time correlation functions describing the polymer motion essentially differ from those in the RZ model. The mean-square displacement (MSD) of the polymer coil is at short times approximately t(2) (instead of approximately t). At long times the MSD contains additional (to the Einstein term) contributions, the leading of which is approximately t. The relaxation of the internal normal modes of the polymer differs from the traditional exponential decay. It is displayed in the long-time tails of their correlation functions, the longest lived being approximately t(-3/2) in the Rouse limit and t(-5/2) in the Zimm case, when the hydrodynamic interaction is strong. It is discussed that the found peculiarities, in particular, an effectively slower diffusion of the polymer coil, should be observable in dynamic scattering experiments.

The joint Rouse-Zimm theory of the dynamics of polymers in dilute solutions

“…The above equations then describe the motion of one bead in the solvent with the effective action of other coils on the motion of the solvent flow. This problem can be transformed to that solved already in [17] (see also [9,15]). The velocity field can be in the Fourier representation in the time written as follows:…”

“…This expression holds in the case of steady flow and takes into account the hydrodynamic interaction. In a more general case, with the hydrodynamic memory [9,12,15] the force (2) should be replaced by the Boussinesq force and equation (1) has to be solved together with the nonstationary hydrodynamic equations for the macroscopic velocity of the solvent. To take into account the presence of other polymers in solution, we use the Brinkman's work [13] (see also [14]) in which a polymer is considered as a porous medium.…”

“…The considered time scales are t τ R = R 2 ρ/η, where R is the hydrodynamic radius of the polymer, ρ is the density and η is the viscosity of the solvent. This means that the effects of hydrodynamic memory (or the viscous aftereffect) are neglected [9,12,15]. The distribution of the coils in solution is considered to be stationary (this is justified at least for the times t τ D ; the choice of this time scale is possible since always τ p τ D , where τ D = R 2 /D is the characteristic time of the coil diffusion with the diffusion coefficient D).…”

“…Our approach differs from the traditional ones in several main points. Firstly, the joint Rouse-Zimm theory is exploited [9]. In the literature, following de Gennes [10], it is assumed that at θ conditions, the Zimm modes with the dispersion of relaxation times τ pZ ∼ p −3/2 , where p is the mode number, at low frequencies, should always dominate the Rouse modes with τ pR ∼ p −2 ; we show that these both contributions to the polymer characteristics should be taken into account.…”

“…Next, the internal modes are distributed discretely (the assumption of their continuous distribution with respect to p is true only in a restricted time domain and often leads to incorrect interpretation of experimental data [11]. The formalism from our theory with the time-dependent hydrodynamics of polymers has been adopted [9,12] and, finally, the Brinkman's theory [13,14] for the flow in porous media is used in order to take into account the effect of other coils on the test polymer. The presented theory has the following limitations.…”