Reptation as a dynamic mean-field theory: Self and tracer diffusion in a simple model of rodlike polymers

The simplest model of entangling polymers, a semidilute solution of hard infinitely thin rods that perform only translational Brownian motion, is studied. An approximate microscopic theory of self and tracer diffusion is presented. Within this theory independent binary collisions are modified to account for the influence of the surrounding rods on the two-particle dynamics. The interaction with the other rods is taken into account in an average self-consistent way. In the semidilute regime the theory leads to the same scaling law for the transversal self-diffusion constant as that derived from a reptation-tube theory. The relaxation time and the localization length asymptotically follow the same scaling law as the disentanglement time and the tube radius, respectively. For the tracer diffusion problem, if the length of the matrix rods is larger or equal to the length of the test rod the reptation prediction for the transversal diffusion constant is asymptotically recovered. For matrix rods much shorter than the test rod the transversal diffusion constant follows a different scaling law.