Methods and numerical aspects of nanoscopic contact area estimation in atomistic tribological simulations

Abstract

We show how data obtained from molecular dynamics (MD) simulations of nanoscale friction should be treated for producing constitutive system parameters with a proper error estimation. A visualisation scheme for discrete atomistic geometries based on the smooth particle method (SPM) was parametrised and validated to yield an accurate and computationally robust estimation of the contact area between two touching nanoscopic asperities. We present some thoughts on the error estimation of the contact forces occurring due to the load and the shearing motion. The variance in the friction force constitutes the main source of error for the fitting of the constitutive system parameters. The dependence of the constitutive system parameters on the number of available data points was also studied. It was shown that an equal spacing (by load) of the data points can result in better values for the system parameters than the convergence trend suggests.

Introduction

By post-processing molecular dynamics (MD) data based on the smooth particle method (SPM)  [1], the authors have previously shown that a three-term kinetic friction law  [2]

$$F\left(L\right)=F_0+\tau A_{\mathrm{asp}}\left(L\right)+\mu L$$

holds for both mixed- and boundary-lubricated nanotribological systems. Here $F\left(L\right)$ is the load-dependent friction force, whereas the load independent Derjaguin-offset $F_0$   [3] together with the shear strength $\tau$ and the kinetic coefficient of friction (CoF) $\mu$ form the unique set of constitutive system parameters which characterises the nanoscopic systems of interest from a tribological point of view. The Bowden–Tabor term $\tau A_{\mathrm{asp}}\left(L\right)$ in Eq. (1) is ascribed to the adhesion-controlled friction  [4] and, in addition to the constant Derjaguin-offset $F_0$, accounts for the irregular departure of the friction-versus-load behaviour $F\left(L\right)$ from the well-known Amontons–Coulomb law $\mu L$   [5], [6] via the load-dependent asperity contact area $A_{\mathrm{asp}}\left(L\right)$.

There are several methods known in the literature to estimate $A_{\mathrm{asp}}\left(L\right)$ at the atomic length scale, especially when dealing with single-asperity contact situations  [7]. In a wide class of approaches, the contact pressure distribution, e.g., the radial one, is used to estimate the contact radius and hence $A_{\mathrm{asp}}\left(L\right)$   [8], [9]. In another group of approximations, the number of atoms within the contact zone is determined, and by multiplying this by an estimate for the atomic contact area, $A_{\mathrm{asp}}\left(L\right)$ is obtained  [10], [11]. In our MD + SPM scheme, a material density is constructed around the atomic positions that lie within the solid regions of the investigated nanotribological system, such that when solid–solid contact occurs during sliding, the corresponding asperity contact area is considered as the minimal cross-section of the solid bridge  [2]. The authors have previously shown that the thus-defined quantity $A_{\mathrm{asp}}\left(L\right)$ does not vary linearly with the load $L$, but is proportional to the number of contact atoms  [1].

In this work we will discuss methods which are applied to analyse the output of MD shear simulations. First we will briefly review the post-processing tool by the authors  [1], [2], which can be used for mapping discrete MD data to the continuum as well as for defining and determining the asperity–asperity contact area in a mixed-lubrication simulation. Furthermore, the theoretical concepts and numerical methods required for calculating contact forces in MD shear simulations will be discussed. Finally, we will show that these forces and the contact area provide a proper basis for obtaining all relevant tribological system parameters, namely the Derjaguin-offset $F_0$, the shear strength $\tau$, and the kinetic CoF $\mu$.