On the nature and propagation of errors in roughness parameters obtained from spectral analysis of atomic force microscopy topographic images

Abstract

Scale-dependent surface roughness strongly affects critical surface properties of materials, including adhesion, wettability, and optical/thermal properties. As a particular example, tuning the ratio of the true to nominal area—a parameter that depends on the root mean square (RMS) local slope of the finest scales of topography—is an effective approach to tailor the wetting characteristics of solid surfaces. While power spectral density (PSD) analysis of atomic force microscopy (AFM) topographic images allows for directly assessing the scale-dependence of surface roughness, this approach to analyze AFM height maps requires power-law modeling and extrapolation of a PSD with inherently non-normal error distributions. Here, we use a Monte Carlo approach based on synthetic AFM images of known input power-law parameters to (1) evaluate the accuracy of fitting techniques based on the expected distribution of the PSD; (2) evaluate the error propagation from the standard errors of the fitted power-law parameters to the computed RMS slope and area ratio; and (3) evaluate the statistical power of various PSD regression techniques when differentiating surfaces. The results indicated that standard error for ordinary least squares on a log-log PSD (log OLS) underpredicts the observed variance by ∼50%. This underprediction can be eliminated by implementing a log-link gamma regression. Moreover, when propagating the standard error to derived parameters (e.g., the RMS slope), the propagated error is generally conservative relative to the observed variance and closely predicts the observed variance when extrapolating to the finest scale. This result demonstrates the possibility of accurately estimating roughness parameters that are critical for evaluating surface phenomena on the basis of fitting and extrapolating AFM data using self-affine models. Finally, our results provided evidence for the possibility of statistically differentiating surfaces with similar power-law parameters when using weighted gamma regression with a mean of 10 images, as opposed to unweighted log-OLS that requires as many as 10 000 images to differentiate images.