Toporov

An issue, at present unresolved, is that Derjaguin, Muller and Toporov [24,25] have put forward a different analysis of the contact mechanics from JKR. Maugis has described a theory which comprehends both the theories as special cases [26]. 

In the JKR theory it is assumed that surface forces are active only in the contact area. In reality, surface forces are active also outside of direct contact. This is, for instance, the case for van der Waals forces. Derjaguin, Muller, and Toporov took this effect into account and developed the so-called DMT theory [206], A consequence is that a kind of neck or meniscus forms at the contact line. As one example, the case of a hard sphere on a soft planar surface, is shown in Fig. 6.19. 

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]. 

N.N.Loznetsova, K.A.Pavlov, J.P.Toporov, G.G.Shchegolev. A role of micelle formation in displaying antiwear properties in lubricant oils. // Tezisy dokl. II Mezhd.Confer. Colloid-2003 - 2003. -p.184. 

Finally, we note that several other contact mechanics theories have been put forward, which are not described in detail in this contribution. The most important ones of these theories for AFM applications include the Derjaguin-Muller-Toporov (DMT), the Bumham-Colton-Pollock (BCP), and the Maguis mechanics [11, 12 ]. These theories differ in the assumptions (and limitations) and yield different expressions for the pull-off force. For example, the DMT theory, which assumes that long-range surface forces act only outside the contact area (as opposed to JKR, where adhesion forces only inside the contact area are assumed), predicts a pull-off force of —2 tRW. 

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. 

The DMT (Deijaguin-Muller-Toporov) approach [9.11,13,14], which gives the same magnitude of contact force but is opposite in direction to the JI pull-off force, has been considered for inteipreting data from AFM force-distance contact mechanics measurements. The DMT theory describes the interaction force, F(D), acting between a flat surface and a spherical surface of radius. R, which is related to the interaction energy per unit area, W(D). at some distance of separation, D. 

B. H. BAIRAMOV, V. V. TOPOROV, F. B. BAYRAMOV A.F. Ioffe Pkysico-Technical Institute, RAS 194021 St. Petersburg, Russia... 

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