Analytic Expressions for the Statistics of the Primitive-Path Length in Entangled Polymers

An analytic expression is proposed for the primitive-path length of entangled polymer chains. The expression is derived from statistical mechanics of a chain that is a random walk with randomly scattered entanglements. The only parameters are the number of Kuhn steps in the chain and a dimensionless parameter beta that contains information about the entanglement density and Kuhn step size. The expression is found to compare very favorably with numerical results recently found from examining topological constraints in microscopic simulations. The comparison also predicts well the plateau modulus of polyethylene, suggesting that the slip-link model is a viable intermediate in the search for true ab initio rheology predictions. Since the expression is analytic, it can be used to make predictions where the simulations cannot reach, and hence is applicable for coarse graining.