Characteristic Length and Time Scales Associated with the Impact of an Elasto-plastic Particle with a Large Hard Plane Surface

In this paper we analyze plastic collisions between spherical nano-scaled particles and a large hard plane. We assume for most of the paper that surface tension between the particle and plane accelerates the particle to a speed which is large compared with its initial speed. The subsequent maximum width and depth of plastic indentation of the particle by the plane is independent of the initial speed of the particle, being a function of particle size, and material properties. This depth of indentation is independent of particle size, and if it is assumed equal to one interatomic spacing in the colliding particle, then a relationship is derived linking nano-scale yield strength to the elastic modulus of the particle. Similarly, should surface tension overcome the elastic recoil process so that the particle is captured on the surface of the plane, then the period of the surface oscillations of the particle depends on the particle size and on material properties. This special regime for nano-scale collisions allows a dimensional argument to approximate the relative importance of fluid viscosity between the particle and plane, relative to plastic processes, even though the corresponding continuum mathematical problem is ill-defined. Finally, we identify a condition which determines when a collision process can be assumed to be a localized phenomenon, and argue that when this condition is not satisfied, then complex dynamical motions are likely.