Towards the Irving-Kirkwood limit of the mechanical stress tensor.

E R Smith,D M Heyes,D Dini

doi:10.1063/1.4984834

Abstract

The probability density functions (PDFs) of the local measure of pressure as a function of the sampling volume are computed for a model Lennard-Jones (LJ) fluid using the Method of Planes (MOP) and Volume Averaging (VA) techniques. This builds on the study of Heyes, Dini, and Smith [J. Chem. Phys. 145, 104504 (2016)] which only considered the VA method for larger subvolumes. The focus here is typically on much smaller subvolumes than considered previously, which tend to the Irving-Kirkwood limit where the pressure tensor is defined at a point. The PDFs from the MOP and VA routes are compared for cubic subvolumes, . Using very high grid-resolution and box-counting analysis, we also show that any measurement of pressure in a molecular system will fail to exactly capture the molecular configuration. This suggests that it is impossible to obtain the pressure in the Irving-Kirkwood limit using the commonly employed grid based averaging techniques. More importantly, below in LJ reduced units, the PDFs depart from Gaussian statistics, and for , a double peaked PDF is observed in the MOP but not VA pressure distributions. This departure from a Gaussian shape means that the average pressure is not the most representative or common value to arise. In addition to contributing to our understanding of local pressure formulas, this work shows a clear lower limit on the validity of simply taking the average value when coarse graining pressure from molecular (and colloidal) systems.

Highlights

The stress, or a pressure tensor (PT), is a central property in continuum mechanics, defining the load in a structure or the evolution of a fluid
Equilibrium molecular dynamics (MD) simulations have been carried out to explore the impact of the grid-averaging resolution on the pressure probability density functions (PDFs)
Two measures of the local pressure are considered, Volume Averaging (VA) and the Method of Planes (MOP), as the averaging volume sizes are decreased towards the Irving and Kirkwood12 limit

Summary

Introduction

The stress, or a pressure tensor (PT), is a central property in continuum mechanics, defining the load in a structure or the evolution of a fluid. With the increasing interest in microfluidic devices and nano engineering, there is a need to develop computational tools for small scale systems. This requires the motions of individual molecules to be averaged so that they can be understood in terms of flow fields which can be measured by experiments and compared to continuum fluid theory. The purpose of the average quantities is both to understand the flow behavior in terms of macroscopic fields, such as velocity and stress, and to link these to continuum grid based methods. The pressure tensor (PT) remains the subject of a great deal of confusion and debate in the molecular dynamics literature. A detailed understanding of the time and spatial dependence of the PT fluctuations is essential in the context of nanofluidic research and molecular-to-continuum coupling simulations.

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Discussion

Conclusion