Polymers with nearest- and next nearest-neighbor interactions on the Husimi lattice

TL;DR

This paper provides an exact solution for a generalized self-avoiding walk model with nearest- and next-nearest-neighbor interactions on a Husimi lattice, revealing complex phase behavior including tricritical points and coil-globule transitions.

Contribution

It introduces a novel exact analysis of a self-avoiding walk model with extended interactions, exploring various NNN definitions and their impact on phase diagrams.

Findings

Existence of a tricritical ($ heta$-) line in phase diagrams.

A coil-globule transition occurs even with infinite repulsion between NN monomers.

Discontinuous NP-P transition observed on the square lattice with infinite NNN repulsion.

Abstract

The exact grand-canonical solution of a generalized interacting self-avoid walk (ISAW) model, placed on a Husimi lattice built with squares, is presented. In this model, beyond the traditional interaction $\omega_1=e^{\epsilon_1/k_B T}$ between (nonconsecutive) monomers on nearest-neighbor (NN) sites, an additional energy $\epsilon_2$ is associated to next-NN (NNN) monomers. Three definitions of NNN sites/interactions are considered, where each monomer can have, effectively, at most 2, 4 or 6 NNN monomers on the Husimi lattice. The phase diagrams found in all cases have (qualitatively) the same thermodynamic properties: a non-polymerized (NP) and a polymerized (P) phase separated by a critical and a coexistence surface that meet at a tricritical ($\theta$-) line. This $\theta$-line is found even when one of the interactions is repulsive, existing for $\omega_1$ in the range $[0,\infty)$, i. e., for $\epsilon_1/k_B T$ in the range $[-\infty,\infty)$. Counterintuitively, a $\theta$-point exists even for an infinite repulsion between NN monomers ($\omega_1=0$), being associated to a coil-"soft globule" transition. In the limit of an infinite repulsive force between NNN monomers, however, the coil-globule transition disappears and only a NP-P continuous transition is observed. This particular case, with $\omega_2=0$, is also solved exactly on the square lattice, using a transfer matrix calculation, where a discontinuous NP-P transition is found. For attractive and repulsive forces between NN and NNN monomers, respectively, the model becomes quite similar to the semiflexible-ISAW one, whose crystalline phase is not observed here, as a consequence of the frustration due to competing NN and NNN forces. The mapping of the phase diagrams in canonical ones is discussed and compared with recent results from Monte Carlo simulations.