Adhesion of nanoscale asperities with power-law profiles

The behavior of single-asperity micro- and nanoscale contacts in which adhesion is present is important for the performance of many small-scale mechanical systems and processes, such as atomic force microscopy (AFM). When analyzing such problems, the bodies in contact are often assumed to have paraboloidal shapes, thus allowing the application of the familiar Johnson–Kendall–Roberts (JKR), Derjaguin–Müller–Toporov (DMT), or Maugis–Dugdale (M–D) adhesive contact models. However, in many situations the asperities do not have paraboloidal shapes and, instead, have geometries that may be better described by a power-law function. An M–D-n analytical model has recently been developed to extend the M–D model to asperities with power-law profiles. We use a combination of M–D-n analytical modeling, finite element (FE) analysis, and experimental measurements to investigate the behavior of nanoscale adhesive contacts with non-paraboloidal geometries. Specifically, we examine the relationship between pull-off force, work of adhesion, and range of adhesion for asperities with power-law-shaped geometries. FE analysis is used to validate the M–D-n model and examine the effect of the shape of the adhesive interaction potential on the pull-off force. In the experiments, the extended M–D model is applied to analyze pull-off force measurements made on nanoscale tips that are engineered via gradual wear to have power-law shapes. The experimental and modeling results demonstrate that the range of the adhesive interaction is a crucial parameter when quantifying the adhesion of non-paraboloidal tips, quite different than the familiar paraboloidal case. The application of the M–D-n model to the experimental results yields an unusually large adhesion range of 4–5 nm, a finding we attribute to either the presence of long-range van der Waals forces or deviations from continuum theory due to atomic-scale roughness of the tips. Finally, an adhesion map to aid in analysis of pull-off force measurements of non-paraboloidal tips is presented. The map delineates the cases in which a simplified rigid analysis can be used to analyze experimental data.

A map is constructed that depicts the region in R/h and σ0/E⁎ parameter space where a simple rigid approximation for calculating the pull-off force of a power-law-shaped tip is valid. Black lines showing four example cases demonstrate how the location on the map changes as h decreases when R, WA, and E⁎ are assumed to be constant. Figure optionsDownload as PowerPoint slideHighlights
► Finite element analysis of adhesion between elastic, power-law shaped asperities.
► Validate a previously published M–D-n model of adhesive contact.
► Measure the work of adhesion and the adhesion range for power-law-shaped AFM tips.
► Present a map that identifies a valid regime for simplified adhesion analysis.