Entropy of polydisperse chains: Solution on the Husimi lattice

Abstract

We consider the entropy of polydisperse chains placed on a lattice. In particular, we study a model for equilibrium polymerization, where the polydispersity 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 polydispersity 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.

Polymer models with competing collapse interactions on Husimi and Bethe lattices

“…Even though the solution of Husimi and Bethe lattice models is a very standard technique [36], we would like to emphasize that our variational-free-energy approach considerably simplifies the derivation of the recursion equations, which may be of use in particular when dealing with models of considerable complexity like the current one. This approach is not fully original, because it has been demonstrated for instance in certain specialized literature dealing with spin glasses (where a quenched variational free energy is considered) [40], but, to the best of our knowledge, it has never been practically exploited, at least in the context of polymer models [20][21][22][23][24][25][26][27][28][29].…”

“…In this article, we study a generalized polymer model that incorporates all the aforementioned ones, in the framework of Husimi [20][21][22][23][24] and Bethe lattices [25][26][27][28][29]. The free parameters characterizing such lattices (building blocks, coordination numbers) are chosen, according to experience, in order to obtain the best possible approximation to a regular 2d square lattice model.…”

Entropy of dimers chains placed on a one-dimensional lattice with q-states

“…This model was studied on Bethe [28] and Husimi [29] lattices with coordination z > 2. We treat the one-dimensional version of a dimer chain with q-states in an exact form as well as by MC algorithm.…”

The first order phase transitions in the multisite spin-1/2 model on a pure Husimi lattice

“…Refs. [22][23][24][25][26][27][28][29][30][31][32][33][34] as well as references cited therein). First of all, it is wellknown that all such classical spin models on pure Husimi trees and lattices can be usually solved exactly at least in the closed form of recursion relations and, at the same time, these models simulate properties of real systems more appropriately than, e.g., the mean field approach [35][36][37].…”