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 are 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(2)(p)>>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.