New statistical mechanical treatment of systems near surfaces. I. Theory and principles

Abstract

We present a new theoretical framework for a statistical mechanical and thermodynamic description of any general inhomogeneous system (not necessarily polymeric) in the presence of surfaces. The framework is an extension of a lattice theory recently developed for a homogeneous system and requires approximating the original lattice by a recursive lattice which, for simplicity, we take to be a modified tree structure (see Fig. 4), TM as described in the text. The tree is formed recursively by two basic elements, the main tree 𝒯 and the surface tree T¯. The model is solved exactly using a recursion technique. The technique allows us to account for connectivity, architecture, excluded-volume effects, interactions, etc. exactly. The resulting description goes beyond the random-mixing approximation used in most mean-field theories. We consider a general model of a multicomponent system and its exact solution on the modified tree TM provides us with an approximate theory of the inhomogeneous system on the original lattice. We provide a general discussion of the theory and principles involved. Our method produces results similar to those of Monte Carlo simulations but can even be applied to cases where Monte Carlo simulations are not possible. We also obtain surface free energy and the surface entropy that is not easily obtained in a Monte Carlo simulation. Our method is more reliable than the mean-field method of Scheutjens and Fleer, whose predictions are, in many cases, in direct contradiction with the Monte Carlo simulations. Our method is fast by at least three orders of magnitude compared to rival methods.