The importance of reference frame for pressure at the liquid–vapour interface

Edward R Smith

doi:10.1080/08927022.2021.1953697

Abstract

ABSTRACT The local pressure tensor is non-unique, a fact which has generated confusion and debate in the 70 years since the seminal work by Irving Kirkwood. This non-uniqueness is normally attributed to the interaction path between molecules, especially in the interfacial-science community. In this work, we reframe this discussion of non-uniqueness in terms of the location, or reference frame, used to measure the pressure. By using a general mathematical description of the liquid–vapour interface, we obtain a reference frame that moves with the interface through time, providing new insight into the pressure. We compare this instantaneous moving reference frame with the fixed Eulerian one. Through this process, we show the requirement that normal pressure balance at the moving surface is satisfied by surface fluxes; however, an additional corrective term based on surface curvature is required for the average pressure in a volume. We make the case that a focus on the path of integration is the cause of confusion in the literature. Using an explicit reference frame with a more general derivation of pressure clarifies some of the issues of uniqueness, providing a pressure tensor which is defined at any instant in time and valid away from thermodynamic equilibrium.

Highlights

Since the pioneering work of Irving and Kirkwood [1], we have had a firm theoretical foundation for the pressure tensor in statistical mechanics
The appropriate definition of pressure in molecular dynamics (MD) simulation has been the subject of some debate
By considering a reference frame described by an arbitrary function j = j(x, y, t) fitted to the surface every time, we get a general form of interface pressure

Summary

Introduction

Since the pioneering work of Irving and Kirkwood [1], we have had a firm theoretical foundation for the pressure tensor in statistical mechanics. The simplest approach is to set the delta functions to one and use the tensor version of the Virial [2] at a local point in space to get the pressure This is the IK1 approximation, so called because it is a first-orderterm in the full Irving Kirkwood expressions [3]. The difference between the Harasima and Irving Kirkwood forms is attributed to the different interaction paths, or interaction contours, assumed between the molecules [17] This is generalised in the work of Schofield and Henderson [18] who show that the path of interaction between two particles, which defines their interaction force, can occur in an infinite number of ways.

Methods

Results

Conclusion