[Paper Review] Statistical mechanics of fluids at an impermeable wall

V. M. Zaskulnikov|arXiv (Cornell University)|May 6, 2010

Phase Equilibria and Thermodynamics26 references3 citations

TL;DR

This paper establishes the equivalence between two long-standing approaches in statistical mechanics of fluids at an impermeable wall: the 'adsorption' (cluster expansion) and 'surface tension' (pressure tensor) methods. By rigorously linking the derivative of the tangential pressure tensor with respect to chemical potential to the average near-surface number density, it proves that both approaches yield identical results, enabling cross-utilization of techniques and deriving an exact virial expansion for the near-surface two-dimensional fluid equation of state.

ABSTRACT

The problem of surface effects at a fluid/force field boundary is investigated. A classical simple fluid with a locally introduced field simulating a solid is considered. For the case of a hard-core field, rigid, exponential, realistic, and macroscopically smooth boundaries are examined. Two approaches to this problem are analyzed. With some degree of arbitrariness, they can be referred to as "adsorption" vs "surface tension" or "cluster expansion" vs "pressure tensor". The "adsorption" approach is used to obtain a series in powers of the activity for gamma. For Mayer-type expansion the integrals of the Ursell functions contain factors which depend on the particle/wall interaction potential. In the case of a hard wall, the coefficients of the series reduce to the first moments of the Ursell functions taken over certain regions. The "surface tension" approach is used to expand the Kirkwood-Buff formula for gamma to the arbitrary localization of the dividing surface. "The surface tension coefficient" breaks up into the term proportional to the Henry constant, depending on the dividing surface position, and universal nonlinear surface coefficient. It is shown that the derivative of the tangential component of the pressure tensor with respect to the chemical potential coincides with the near-surface number density on average over the transition region, that has two consequences. Firstly, it proves complete identity between "tension" and "adsorption" approaches in the domain of their existence. Secondly, it gives the near-surface virial expansion, which determines the exact equation of state of near boundary "two-dimensional" fluid. The tangential component of the pressure tensor averaged over the transition region plays the role of pressure, and the average number density - the role of number density.

Motivation & Objective

• To resolve long-standing inconsistencies in the statistical mechanics of fluid interfaces at rigid walls.
• To clarify the role of the dividing surface localization in surface free energy and surface tension formulations.
• To establish a rigorous mathematical equivalence between the 'adsorption' and 'surface tension' approaches.
• To derive a virial-type expansion for the near-surface two-dimensional fluid using the pressure tensor and chemical potential derivative.
• To clarify the physical meaning of the Henry constant and surface terms in the context of arbitrary wall-fluid potentials.

Proposed method

• Uses the grand canonical ensemble and cluster expansion techniques to derive a series expansion for the surface excess grand potential (γ) in powers of activity.
• Introduces Ursell function integrals modified by particle-wall interaction potentials, with general expressions for each term in the series.
• Applies the Kirkwood-Buff formula for surface tension, generalized to arbitrary dividing surface positions and arbitrary wall potentials.
• Derives the equivalence between the pressure tensor's tangential component derivative and the average near-surface number density across the transition region.
• Uses the equivalence to derive a virial expansion for the two-dimensional fluid near the wall, treating the tangential pressure and average density as thermodynamic variables.
• Validates the method through asymptotic analysis and boundary contribution estimation, showing higher-order terms vanish in the thermodynamic limit.

Experimental results

Research questions

• RQ1 Can the 'adsorption' and 'surface tension' approaches in fluid-wall systems be rigorously shown to be equivalent?
• RQ2 How does the choice of dividing surface position affect the surface terms and the Henry constant in the grand potential expansion?
• RQ3 What is the exact form of the virial expansion for the near-surface two-dimensional fluid, and how is it related to the pressure tensor?
• RQ4 Why do earlier studies yield inconsistent or unphysical results for the surface number density and Henry constant?
• RQ5 How do particle-wall interaction potentials influence the structure of the surface free energy series?

Key findings

• The derivative of the tangential pressure tensor with respect to chemical potential is exactly equal to the average near-surface number density over the transition region.
• This identity proves the complete equivalence between the 'adsorption' and 'surface tension' approaches within their domain of validity.
• The surface free energy expansion in powers of activity is derived in quadrature form via the 'adsorption' method, while the 'surface tension' method yields an analog of the Mayer expansion.
• The surface tension coefficient splits into a position-dependent term proportional to the Henry constant and a universal nonlinear surface term.
• For a hard-wall potential, the coefficients in the activity series reduce to the first moments of Ursell functions over specific spatial regions.
• The near-surface system behaves as a two-dimensional fluid with an exact equation of state derived from the virial expansion, with tangential pressure and average density as state variables.