Depth-sensing indentation of spherical particles on corrugated substrates — An asymptotic model

Abstract

An AFM-based nanoindentation of a spherical particle absorbed on a corrugated surface of rigid substrate is considered in the framework of continuum mechanics. The corresponding problem of unilateral frictionless contact with multiple contact zones is formulated using isotropic infinitesimal elasticity. Both the AFM probe surface and the surfaces of rigid supports, which are exposed to contact with the elastic sphere, are assumed to be spherical. The method of matched asymptotic expansions is utilized for constructing the leading order asymptotic model of multiple contact. Explicit formulas are derived for the force-displacement relationship and the incremental indentation stiffness. It is shown that the difference between the one-point contact and the multi-point contact (e.g., with two and three local supports) is found to be significant. The case of tight hexagonal packing of the support spheres is considered in detail and illustrated by a comparison with the Hertzian theory of local contact.