Diffusion of a polymer in a random medium

2014 The effect of a random potential on the diffusion coefficient of a polymer is shown to be a relevant perturbation in the renormalization group sense below four dimensions. Attempts to « improve » the perturbation expansion by means of the renormalization group failed as it was not possible to locate a stable accessible fixed point. Speculations on the physical implications of these results are given. J. Physique LETTRES 42 (1981) L-413 L-416 15 SEPTEMBRE 1981, Classification Physics Abstracts 61.40K In this letter we examine the statics and dynamics of a flexible linear polymer moving in a fixed (quenched) random potential. This potential might be envisaged as being produced by hard spheres placed at random throughout space whose diameters are small compared to the length of the polymer molecule. Such a model might serve as a crude approximation for a polymer diffusing through a medium of closely packed small particles, e.g. oil through sand. However, our chief motivation in studying this problem was to try to understand what de Gennes refers to as « one of the major unsolved problems of polymer physics » [1]. This is the discrepancy between the reptation model prediction for the molecular weight dependence of the viscosity ~ of a melt and experiment. Experimentally it is found that" ’" Nn where mn is of order 3.3 to 3.4 while the reptation model prediction for mn is 3. The reptation model is usually introduced by first considering a system which is much simpler than a polymer melt but which still shows nontrivial entanglement effects. This is a single chain trapped in a fixed network [1]. The chain has to move through the network in a snake-like fashion called «reptation ». The wriggling of a polymer in two dimensions amongst fixed randomly placed discs is also of this character. In the case we want to understand, viz, polymer motion in a random potential in three dimensions, the randomly placed spheres, do not provide topological constraints on the motion of the chain. We hope, nevertheless, that the motion falls into the same « universality class » as that through a fixed network. The presence of topological constraints makes the network problem intractable. However, the random potential problem is more amenable to standard techniques, such as those provided by the renormalization group. The bead-spring Rouse model [2] will be adopted to describe the dynamics of the polymer. The equations of motion are where Rj(t) is the position vector of the jth bead ( j = 1, 2, ..., N) and a = 1, ..., d labels its Cartesian components. The dimensionality of the system is d. JL is the friction coefficient and we have supposed that the presence of the random potential screens out the long-range hydrodynamic interactions. The effects of random collisions with solvent molecules is represented by the random force fj(t) whose distribution is taken as Gaussian. The potential energy (kB T ) -1