Entropy of polydisperse chains: solution on the Husimi lattice

TL;DR

This paper derives an exact analytical expression for the entropy of polydisperse chains on a Husimi lattice, revealing how polydispersivity influences entropy and molecular weight distribution, with implications for understanding equilibrium polymerization.

Contribution

It provides an exact solution for the entropy of polydisperse chains on a Husimi lattice, extending previous one-dimensional results to more complex lattice structures.

Findings

Entropy excess due to polydispersivity matches one-dimensional case.

Molecular weight distribution becomes exponential at large mean molecular weight.

Derived entropy expression as a function of monomer density and mean molecular weight.

Abstract

We consider the entropy of polydisperse chains placed on a lattice. In particular, we study a model for equilibrium polymerization, where the polydispersivity is determined by two activities, for internal and endpoint monomers of a chain. We solve the problem exactly on a Husimi lattice built with squares and with arbitrary coordination number, obtaining an expression for the entropy as a function of the density of monomers and mean molecular weight of the chains. We compare this entropy with the one for the monodisperse case, and find that the excess of entropy due to polydispersivity is identical to the one obtained for the one-dimensional case. Finally, we obtain a distribution of molecular weights with a rather complex behavior, but which becomes exponential for very large mean molecular weight of the chains, as required by scaling properties which should apply in this limit.