Derjaguin-Muller-Toporov model

The JKR approximation works well for high adhesion, large radii of curvature and compliant materials but may underestimate surface forces. An alternative theory have been developed by Derjaguin, Muller, Toporov (DMT) to include noncontact adhesion forces acting in a ring-shaped zone around the contact area [81]. On the other hand, the DMT approximation constrains the tip-sample geometry to remain Hertzian, as if adhesion forces could not deform the surfaces. The DMT model applies to rigid systems with small adhesion and radius of curvature, but may underestimate the contact area. For many SFM s, the actual situation is likely to lie somewhere between these two models [116]. The transition between the models their applicability for SFM problems were analysed elsewhere [120,143]. 

In Section 3.2.2 we treated Hertzian contact. We must consider that the case, where adhesive forces are negligible, is rather special. If adhesive effects are no longer negligible, one must switch to the adhesive contact model developed by Johnson, Kendall, and Roberts (the JKR model) [Johnson et al., 1971]. In the limit case of weak adhesive force, Fadh, one can also use the Derjaguin-Muller-Toporov... 

Derjaguin-Muller-Toporov assumed that there is Hertz deformation and developed another model that included the effect of adhesion force. According to the DMT model, the pull-off force is given as... 

In the limits of established contact mechanics models, including those developed by Johnson-Kendall-Roberts (JKR) [5] or by Derjaguin, Muller, and Toporov (DMT) [6], the measured forces are a function of the chemical identity of the contacting surfaces (via the work of adhesion W12 that depends on the surface and interfacial free energies involved). In addition, we need to consider the nature of the medium, the radius of the AFM tip, and also temperature and loading rate. 

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