Fractal Network Model for Contact Conductance

The topography of rough surfaces strongly influences the conduction of heat and electricity between two surfaces in contact. Roughness measurements on a variety of surfaces have shown that their structure follows a fractal geometry whereby similar images of the surface appear under repeated magnification. Such a structure is characterized by the fractal dimension D, which lies between 2 and 3 for a surface and between 1 and 2 for a surface profile. This paper uses the fractal characterization of surface roughness to develop a new network model for analyzing heat conduction between two contacting rough surfaces. The analysis yields the simple result that the contact conductance h and the real area of contact A_t are related as h ~ A_t^D/2 where D is the fractal dimension of the surface profile. Contact mechanics of fractal surfaces has shown that A_t varies with the load F as A_t ~ F^η where η ranges from 1 to 1.33 depending on the value of D. This proves that the conductance and load are related as h ~ F^ηD/2 and resolves the anomaly in previous investigations, which theoretically and experimentally obtained different values for the load exponent. The analytical results agreed well with previous experiments although there is a tendency for overprediction.