Derjaguin, Muller, and 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. 

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. 

In the derivation of Eq. 37, no assumption is made about the relative size of the cohesive zone. The underlying assumption of the fracture mechanics treatment employed throughout this chapter is that the cohesive zone is small in comparison to the overall dimensions of the contact zone, i.e. d < a. The opposite case, where d a, corresponds to the DMT theoiy of Derjaguin, Muller and Toporov [15]. The DMT condition of large cohesive zone size is almost never met in practice, with the exception of very stiff, nano-scale contacts, or in cases where the cohesive zone corresponds to a liquid meniscus... 

The analysis of the adhesion data thus resolves to that of two curved elastic bodies in contact. The strength of the adhesive junction will be determined By a balance between the surface attractive forces and the bulk elastic forces opposing deformation. Two theories have been proposed to describe such a contact. The first is due to Derjaguin, Muller and Toporov (DMT theory) which was developed for hard materials (E > 10 Nm 2) On the assumption that the deformation is Hertzian and that separation occurs when the contact area is reduced to zero, they derived the following expression for the force of detachment... 

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]. 

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 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... 

Modem theories of adhesion mechanics of two contacting solid surfaces are based on the Johnson, Kendall, and Roberts (JKR theory), or on the Derjaguin-Muller-Toporov (DMT) theory. The JKR theory is applicable to easily deformable, large bodies with high surface energy, whereas the DMT theory better describes very small and hard bodies with low surface energy. The JKR theory gives an important result about the adhesion or pull-off force. That is, the adhesion force is related to the work of adhesion, and the reduced radius, R, of the AFM tip-surface contact as... 

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... 

From the JKR theory, the pull-oflf force or adhesion force of a sphere (radius R) from a flat is >nRy, where y is the interfacial tension between the solid and the indenting sphere. The Derjaguin-Muller-Toporov (DMT) theory predicts that the pull-off force is AnRy, i.e., greater by a factor of 4/3 than the adhesion force predicted by the JKR theory. The two theories thus disagree in their predictions of interfadal tension from force measurements by this constant. However, there are two major differences between the theories that can be tested experimentally. One is the shape of the interface in contact under no load conditions and the second is the contact radius at which the surfaces separate. The JKR theory predicts that when the pull-ofif adhesion force equals 3 tRy, the surfaces separate fi-om a finite area of contact and the radius at pull-off is 0.63 times the radius under no load conditions. On the other hand, the DMT theory predicts that at the instant of separation, the contact radius is 0, i.e., the surfaces separate only at the point where they have achieved their original undistorted shape. 

Derjaguin, B.V. Muller, V.M. Toporov, Y.P. (1975) Effect of contact deformations on the adhesion of particles. Journal of Colloid and Interface Science 53, 314—326. 

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