By Jaiswal, Ravi; Mittal, K. L

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This Monte Carlo process is repeated hundreds of thousands of times, which allows the system to reach ‘equilibrium’ [52]. 19) [54]. Additionally, the effects of RH on the pull-off force between AFM cantilevers and substrates of varying hydrophilicity can be predicted qualitatively with this method. Specifically, a maximum in the adhesion force is predicted around 30% RH for a strongly hydrophilic tip; a plateau above 34% is predicted for a hydrophobic tip, due to the interaction of two confined layers of water; and a monotonic increase is predicted for a slightly hydrophilic tip [46, 55].

To avoid this difficulty, the variable transformation x → p has been introduced, and the range of integration has changed correspondingly from [rn, ), for which rn can be arbitrarily small, to [1, ). Note also that the functions εi and μi are complex. 61, the integrand must be complex-differentiable. Complex-valued functions often contain points, zeroes, poles, asymptotes, and other features at which they are not complex-differentiable in Cartesian coordinates. 66. 61 to be mapped onto a space over which it is complex-differentiable over the range of integration.

21 Field-emission SEM micrograph of an alumina particle mounted on the tip of an AFM cantilever. Reproduced with permission [132]. 21. In an attempt to more accurately characterize the Hamaker coefficient, the geometries of the tips are often characterized by either reverse imaging [130] or scanning electron microscopy [11, 130, 131], while the surface roughness is often estimated based on AFM topographical images [132]. 53, to model this force [132–136]. 22 demonstrates another experimental approach to measure the Hamaker coefficients for the silicon nitride/ruthenium system in air [128].