Correlations between adhesion hysteresis and friction at molecular scales

A self-affine fractal surface satisfies a transformation $x→ζx$ and $z→zζH$⁠, where $x$ is a 2D position vector in a surface plane and $H=3−D$ is a Hurst exponent and $D$ is a fractal dimension.

The critical shear stress $τ$ depends on sample elasticity and does not change with roughness. The critical tip-sample separation distance $δc$ (where the contact is ruptured) may alter, but then also an initial contact point shifts by the same amount so that both effects cancel out.

The contact length $λ$ between smooth surfaces within used here JKRS approximation is $λ=(R∕Y)(FSP+2Fadh+2FSPFadh+Fadh2)3$ with $R=100nm$⁠, $FSP=20nN$⁠, and $Fadh$ tip-sample adherence force from Fig. 4.

The digital lateral AFM images resolution was about $(3000nm)∕(256points)≈12nm$⁠. Our tip, however, has higher radii of curvature limiting our experimental resolution at higher values depending on the asperities heights. Thus, some bigger length of $600nm$ was arbitrarily chosen to count asperities and obtain their mean separation distance (see Fig. 5 for an AFM topography line cross section with asperities separations).