*5.3. Adhesion Map*

To clarify the effect of *A*, *λ* and *µ* on the pull-off force *W pull*−*o f f* we fixed the sphere radius *R* † = 100 and change *µ* = - 10−<sup>1</sup> , 5 and *A*/*λ* = [10−<sup>3</sup> , 10<sup>0</sup> ]. Figures 14 and 15 shows the contour plot of the pull-off force respectively for *λ* † = 20 and *λ* † = 5. One immediately notices that larger adhesive forces are reached with longer wavelengths. Figure 14 shows that adhesion enhancement happens in a limited parameter region. For very low ratio *A*/*λ* the contact problem reduces to that of the smooth sphere on a smooth halfspace, hence by changing the Tabor parameter from *µ* = 0.01 to *µ* = 5 one moves from the Bradley *W pull*−*o f f* <sup>=</sup> 2 to the JKR solution *W pull*−*o f f* <sup>=</sup> 1.5. Increasing *<sup>A</sup>*/*<sup>λ</sup>* for small Tabor parameter (*<sup>µ</sup>* <sup>&</sup>lt; <sup>10</sup>−0.6 <sup>≈</sup> 0.25) leads to a strong reduction of the pull-off force, as indeed we are in the range where the rigid solution of the Guduru problem holds (cfr. Section 2, Figure 7). Notice that keeping *λ* constant and increasing *A*/*λ* leads to both increasing of *A* † and *α* in Figure 7 heading to very strong reduction of the macroscopic pull-off force. Instead, if *A*/*λ* is increased at large Tabor parameter (*µ* & 0.25 for *λ* = 20), adhesion enhancement takes place and high pull-off forces can be reached (in Figure 14 up to *W pull*−*o f f* <sup>≃</sup> 4 for *<sup>µ</sup>* <sup>≃</sup> 5). Contrary to JKR theory predictions, further increasing of the amplitude to wavelength ratio *A*/*λ* does not lead to stronger adhesive forces, but adhesion is destroyed by roughness.

**Figure 14.** Contour plot of the dimensionless pull-off force (absolute value) as a function of *µ* = - 10−<sup>1</sup> , 5 and *A*/*λ* = [10−<sup>3</sup> , 10<sup>0</sup> ] for *λ* † = 20.

**Figure 15.** Contour plot of the dimensionless pull-off force (absolute value) as a function of *µ* = - 10−<sup>1</sup> , 5 and *A*/*λ* = [10−<sup>2</sup> , 10<sup>0</sup> ] for *λ* † = 5.
