Area constrained SOS models of interfaces

TL;DR

This paper extends the 2D SOS model by introducing a constant total area constraint, deriving the free energy and entropy, and analyzing how this affects interface properties compared to canonical models.

Contribution

It presents the first analytical and numerical study of the SOS model with a fixed total area constraint, expanding understanding of interface models under new physical conditions.

Findings

The area constraint reduces the ratio of total area to projected area at high temperatures.

The entropy per column collapses onto a common curve in reduced variables.

Analytical results are supported by numerical calculations.

Abstract

The solid-on solid (SOS) model in two dimensions ($d=2$) is now solved under the constraint of constant energy and then under the new constraint of constant total area. From the combinatorial factors $g(E;L,M)$, the new ensemble is constructed with its free energy $F(A_{tot},T)$ of a membrane of constant (onedimensional) area $A_{tot}$. The entropy per column $Y=(1/L)\log g(E;L,M)$ of rectangular $L\times M$ strips reduces to a common curve in reduced variables. Definitions of the "area" and of the interfacial tension, are compared or discussed. Analytical calculations are supported with numerical ones and vice versa. Overall the constraint reduces the ratio of $A_{tot}$ to the projected area $L$, as compared with canonical calculation, strongly at high temperatures.