Tube diameter in tightly entangled solutions of semiflexible polymers

A statistical mechanical treatment is given of the confinement of a wormlike polymer in an entangled solution to a tube, yielding quantitative predictions for the average tube diameter D-e and macroscopic plateau modulus G, in the tightly entangled regime in which D-e is much less than the persistence length L-p. Three approaches an pursued. A self-consistent binary collision approximation, which explicitly describes the topological constraints imposed by neighboring chains, yields predictions consistent with the scaling laws D-e proportional to rho (-3/5) and G proportional to rho (7/5) proposed previously, where rho is the contour length per unit volume. An effective medium approximation, which treats the network as a continuum with a modulus G, instead yields D-e proportional to rho (-1/3) and G proportional to rho (4/3), which is found to be the correct scaling in the limit rhoL(p)(2) much greater than 1. An elastic network approximation treats the displacement of a test chain as the sum of a collective displacement of the network, which is treated as a continuum, plus a local displacement, which is treated in a binary collision approximation. Predictions are compared to measurements of both D-e and G in actin protein filament (F-actin) solutions.