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 $D_e\propto {\rho }^{-3/5}$ and $G\propto {\rho }^{7/5}$ 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 $D_e\propto {\rho }^{-1/3}$ and $G\propto {\rho }^{4/3},$ which is found to be the correct scaling in the limit $\rho L_p^2\gg 1.$ 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.