Detailed Balance Condition and Effective Free Energy in the Primitive Chain Network Model

TL;DR

This paper investigates the primitive chain network model for entangled polymers, identifies issues with its dynamic equations not satisfying detailed balance, proposes modifications, and analyzes the resulting equilibrium properties and distribution functions.

Contribution

It introduces heuristic modifications to the PCN model's dynamic equations to satisfy detailed balance and derives the effective free energy including interactions.

Findings

Modified PCN model satisfies detailed balance.

Effective free energy includes Gaussian and interaction terms.

Distribution functions differ from simple slip-link models.

Abstract

We consider statistical mechanical properties of the primitive chain network (PCN) model for entangled polymers from its dynamic equations. We show that the dynamic equation for the segment number of the PCN model does not reduce to the standard Langevin equation which satisfies the detailed balance condition. We propose heuristic modifications for the PCN dynamic equation for the segment number, to make it reduce to the standard Langevin equation. We analyse some equilibrium statistical properties of the modified PCN model, by using the effective free energy obtained from the modified PCN dynamic equations. The PCN effective free energy can be interpreted as the sum of the ideal Gaussian chain free energy and the repulsive interaction energy between slip-links. By using the single chain approximation, we calculate several distribution functions of the PCN model. The obtained distribution functions are qualitatively different from ones for the simple slip-link model without any direct interactions between slip-links.