Dependence of friction on roughness, velocity, and temperature.

We study the dependence of friction on surface roughness, sliding velocity, and temperature. Expanding on the classic treatment of Greenwood and Williamson, we show that the fractal nature of a surface has little influence on the real area of contact and the static friction coefficient. A simple scaling argument shows that the static friction exhibits a weak anomaly mu ~ A(0)(-chi/4), where A0 is the apparent area and chi is the roughness exponent of the surface. We then develop a method to calculate atomic-scale friction between a microscopic asperity, such as the tip of a friction force microscope (FFM) and a solid substrate. This method, based on the thermal activation of the FFM tip, allows a quantitative extraction of all the relevant microscopic parameters and reveals a universal scaling behavior of atomic friction on velocity and temperature. This method is extended to include a soft atomic substrate in order to simulate FFM scans more realistically. The tip is connected with the support of the cantilever by an ideal spring and the substrate is simulated with a ball-spring model. The tip and substrate are coupled with repulsive potentials. Simulations are done at different temperatures and scanning velocities on substrates with different elastic moduli. Stick-slip motion of the tip is observed, and the numerical results of the friction force and distribution of force maxima match the theoretical framework.