Scale-Area Analysis of Scanning Tunneling Microscopy/Atomic Force Microscopy Data by the Patchwork Method

Abstract

A new method of analyzing large, topographic data sets, such as those generated by scanning probe microscopy (e.g., STM, AFM) is described. The method uses triangular patches to calculate areas of the topographic data set as a function of patch areas, i.e., as a function of scale. This patchwork method is analogous to the coastline method of fractal analysis for profiles, however, because the calculations are based on areas rather than profiles or coastlines, the patchwork method makes use of the information inherent in the proximity of the individual scans. This method develops characterizations for the data sets that have clear applications in engineering design and research. It has the potential for elucidating mechanisms of interactions with surfaces, analyzing surface creation and wear mechanisms, and for designing surface topographies and the manufacturing processes to produce them.

Using the patchwork method, one can identify the smooth-rough crossover in scale which separates the relatively large scales where the surface is smooth and best described by Euclidean geometry, from those smaller scales where the surface is rough and best described by fractal geometry. One also obtains parameters that characterize the intricacies, or complexities, of the topography below the smooth-rough crossover, and the scales over which different levels of geometric complexity exist. These complexity parameters can be related to fractal dimensions; more than one fractal dimension for a topographic data set indicates that the geometry has a multifractal character.

The surface areas, calculated as a function of scale from the topographic data sets, can also be used directly to determine the strengths of interactions with the surface. To determine the strength of interaction (e.g., adhesion) the interaction is modeled as a collection of discrete interactions (e.g., adhesive bonds) of determinable size and strength. The strength of the interaction is then determined from the surface area calculated at the size, or scale, of the interaction.