How Thermal Fluctuations Affect Hard-Wall Repulsion and Thereby Hertzian Contact Mechanics

Contact problems as they occur in tribology and colloid science are often solved with the assumption of hard-wall and hard-disk repulsion between locally smooth surfaces. This approximation is certainly meaningful at sufficiently coarse scales. However, at small scales, thermal fluctuations can become relevant. In this study, we address the question how they render non-overlap constraints into finite-range repulsion. To this end, we derive a closed-form analytical expression for the potential of mean force between a hard wall and a thermally fluctuating, linearly elastic counterface. Theoretical results are validated with numerical simulations based on the Green's function molecular dynamics technique, which is generalized to include thermal noise while allowing for hard-wall interactions. Applications consist of the validation of our method for flat surfaces and the generalization of the Hertzian contact to finite temperature. In both cases, similar force-distance relationships are produced with effective potentials as with fully thermostatted simulations. Analytical expressions are identified that allow the thermal corrections to the Hertzian load-displacement relation to be accurately estimated. While these corrections are not necessarily small, they turn out surprisingly insensitive to the applied load.