Continuum Analysis of Atomistic Contacts

We have recently applied a multilevel multiintegration technique to numerically solve the normal and tangential loading of rough surfaces in Hertz-type contact problems [1]. After the qualitative analysis presented in [2], a more refined attempt to generalise the results obtained using continuum mechanics-based approaches and to extend their validity to nanoscale contacts is performed here. The results of the continuum simulations are compared to those reported in the literature at atomic level [3, 4]. It is shown that the use of the rough contact idealisation described here is capable of partially reconciling continuum mechanics and atomistic simulations by capturing some of the features that cannot be captured by the means of conventional Hertzian theory. The potential insight gained using atomistic simulations and the limitations of continuum mechanics in describing some of the physics governing the interactions between rubbing bodies are also discussed. INTRODUCTION The use of continuum mechanics for the analysis of contacts is often a most acceptable and convenient approach. With the increasingly common and important cases of atomistic scale contacts, however, the limitations (and potential errors) of its fundamental assumptions have become highlighted. As the scale of analysis approaches atomic level, the characterisation of the substrate as a continuum becomes less and less credible. Molecular dynamics simulations may then appear to be a more appropriate method for modelling contacts on this scale, but is not without downsides, not least of which is the computational time required. A large proportion of the errors associated with continuum mechanics can come from the specification of the continuum surface. Often the surfaces are assumed to be smooth on a macro scale; this is an unnecessary simplification and a more accurate representation of the surfaces will extend the validity of the continuum approach. With fast solution techniques, continuum mechanics may then still be the more suitable method for analysis of nanoscale contacts. Here, we assess the use of continuum mechanics at the lowest feasible limit of atomic scale contacts. Recently, Luan and Robbins have modelled the contact of a spherical tip, of the size found in atomic force microscopes, on a flat substrate using atomistic simulations based upon Lennard-Jones potentials [4]. They demonstrated the significant errors that are produced by the usual smooth continuum analysis. Different crystal lattice arrangements for the tips were analysed, for which varying contact responses were observed. They suggested that each tip form was identical on the continuum scale. In particular, although recognising that the local surface roughness at atomic scale could play a very important role in reconciling continuum and atomistic simulations, they asserted that one of the main difficulties in applying continuum theory to atomistic contacts is that it not capable of representing nanoscale asperities. This is due to the fact that it seems not possible to uniquely define the root mean squared (rms) roughness of such surfaces. Luan and Robbins, however, did not consider the possibility of analysing the contacts under investigation as “real” rough surfaces instead of relying on classical representation of rough surfaces in contact based on the use of such parameter. This is indeed possible and here we show how to differentiate each such tip on a continuum scale by representing the continuum boundary as a rough surface formed by the contours of the atomic lattice. Each atom is considered as a sphere of diameter σ, where σ corresponds to the atomic diameter of the LennardsJones solid. Although the notion of varying pressures over an atomic diameter is clearly inaccurate, the representation of the surface features at the subnanometer scale may be sufficient to allow analysis of AFM tips and similarly sized contacts. The contact is solved using the Multilevel Multiintegration technique of Venner and Lubrecht [5] which we recently applied to the analysis of multiple asperity and rough macroscale contacts [1]. RESULTS & DISCUSSION Results were obtained for the contact between a flat substrate and various forms of tip, each with mean diameter 100σ. The flat substrate was representative of an FCC crystal with [001] surface. Five different tip forms were analysed, similar to those investigated in [4]. A bent commensurate tip was formed by maintaining the lattice form of the substrate but displacing the atoms to generate the required radius. A bent incommensurate tip was formed identically, but with σ and thus atomic spacing of 0.99437 times that of the substrate (leading to non-aligned atoms). A stepped cut tip was formed by removing atoms that fell outside of the required tip radius; an amorphous tip was formed likewise, but from an amorphous arrangement of atoms. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 P / E *