A simple generalization of Prandtl–Tomlinson model to study nanoscale rolling friction

Abstract

Prandtl–Tomlinson (PT) model can be used to explain nanoscale friction in a variety of situations, except when a nanoscale object undergoes rolling. To alleviate this problem, we generalize the PT model as a collection of interacting point particles arranged on a ring of radius R. The center of mass of the ring is connected to a spring of stiffness k, whose other end is attached to a fictitious mass that moves with a constant velocity v. The entire assembly is driven in a composite force field, which is a product of (i) the familiar sinusoidal function used in the PT model and (ii) a parametrically controlled (λ) exponentially varying function that is dependent on the vertical coordinates of the particles. Our generalized model degenerates to the standard PT model if R≪1 and λ→0. With increasing k, for R≪1 and λ≠0, the ring undergoes a transition from sticky to smooth dynamics for both x and y directions. The dynamics, investigated numerically for the general case of R∼1 and λ≠0, reveals several interesting aspects of nanoscale tribology including the regimes where energy dissipation due to friction is minimum. Furthermore, the results from our proposed model are in agreement with those from molecular dynamics simulations as well. We believe that the simplicity of our model along with its similarity to the PT model may make it a popular tool for analyzing complicated nanotribological regimes.