Effects of finite probe size on self-affine roughness measurements

The roughness of fracture surfaces exhibits self-affinity for a wide variety of materials and loading conditions. The universality and the range of scales over which this regime extends are still debated. The topography of these surfaces is however often investigated with a finite contact probe. In this case, we show that the correlation function of the roughness can only be measured down to a length scale Deltax{c} which depends on the probe size R, the Hurst exponent zeta of the surface and its topothesy l, and exhibits spurious behavior at smaller scales. First, we derive the dependence of Deltax{c} on these parameters from a simple scaling argument. Then, we verify this dependence numerically. Finally, we establish the relevance of this analysis from AFM measurements on an experimental glass fracture surface and provide a metrological procedure for roughness measurements.