Evaluation of self-affine surfaces and their implication for frictional dynamics as illustrated with a Rouse material

We present a theory of hysteresis friction of sliding bulk rubber networks using the dynamic Rouse model for the rubber–glass transition region. The hard substrate is described by a self-affine rough surface that is a good representative for real surfaces having asperities within different length scales of different orders of magnitude. We find a general solution of the friction coefficient as a function of sliding velocity and typical surface parameters (e.g. surface fractal dimension, correlation lengths of surface profile). Further, we show the correlation with the viscoelastic loss modulus of the bulk rubber and the applicability of the Williams–Landel–Ferry transform to the velocity and temperature dependence of the frictional force as found experimentally. We demonstrate how the succesive inclusion of relaxation Rouse modes p=1,2,3,… into the final expression for the frictional force leads to a superposition of the contributions of the different modes and, as a consequence, to a broad, bell-shaped frictional curve as observed in the pioneering experiments of Grosch. We show how the theory simplifies for the special case of a Rouse slider interacting with a Brownian surface.