Effects of finite probe size on self-affine roughness measurements

TL;DR

This paper investigates how finite probe size affects the measurement of self-affine roughness on fracture surfaces, revealing scale-dependent biases and proposing a correction method based on theoretical and experimental analysis.

Contribution

It introduces a scaling-based analysis of probe size effects on roughness measurements, validated through numerical simulations and AFM experiments, providing a new metrological procedure.

Findings

Probe size limits accurate roughness measurement to a scale dependent on probe size and surface properties.

Theoretical predictions of the correlation length scale match numerical simulations.

Experimental AFM data confirm the relevance of the proposed correction method.

Abstract

The roughness of fracture surfaces has been shown to exhibit self-affne scale invariance for a wide variety of materials and loading conditions. The range of scales over which this regime extends remains a matter of debate, together with the universality of the associated exponents. The topography of these surfaces is however often investigated with a contact probe that is larger than the micro-structure. In this case, we show that the correlation function of the roughness and the corresponding Hurst exponent $\zeta$ can only be measured down to a length scale $\Delta xc$ which depends on the probe size $R$, on $\zeta$ and on the surface topothesy $l$, and exhibit spurious behavior at smaller scales. First, we derive the dependence of $\Delta xc$ on these parameters from a simple scaling argument. Then we study this dependence numerically and verify our theoretical prediction. 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.