DMT theory

To account for some of the shortcomings of the JKR theory, Derjaguin and coworkers [19] developed an alternative theory, known as the DMT theory. According to the DMT theory, the attractive force between the surfaces has a finite range and acts outside the contact zone, where the surface shape is assumed to be Hertzian and not deformed by the effect of the interfacial forces. The predictions of the DMT theory are significantly different compared to the JKR theory. 

There was some argument in the literature over the relative merits and demerits of the JKR and the DMT theories [23-26], but the controversy has now been satisfactorily resolved. A critical comparison of the JKR and DMT theories can be obtained from the literature [23-30]. According to Tabor [23], JKR theory is valid when the dimensionless parameter given by Eq. 25 exceeds a value of about five. 

In an attempt to determine the applicability of JKR and DMT theories, Lee [91] measured the no-load contact radius of crosslinked silicone rubber spheres in contact with a glass slide as a function of their radii of curvature (R) and elastic moduli (K). In these experiments, Lee found that a thin layer of silicone gel transferred onto the glass slide. From a plot of versus R, using Eq. 13 of the JKR theory, Lee determined that the work of adhesion was about 70 7 mJ/m". a value in clo.se agreement with that determined by Johnson and coworkers 6 using Eqs. 11 and 16. 

As indicated, an implicit assumption of the JKR theory is that there are no interactions outside the contact radius. More specifically, the energy arguments used in the development of the JKR theory do not allow specific locations of the adhesion forces to be determined except that they must be associated with the contact line where the two surfaces of the particle and substrate become joined. Adhesion-induced stresses act at the surface and not a result of action-at-a-distance interatomic forces. This results in a stress singularity at the circumference of the contact radius [41]. The validity of this assumption was first questioned by Derjaguin et al. [42], who proposed an alternative model of adhesion (commonly referred to as the DMT theory ). Needless to say, the predictions of the JKR and DMT models are vastly different, as discussed by Tabor [41]. 

Whereas the JKR model approached the topic of particle adhesion from a contact mechanics viewpoint, the DMT theory simply assumes that the adhesion-induced contact has the same shape as a Hertzian indentor. The normal pressure distribution Ph(p) for the Hertzian indentor is related to the repulsive force and the distance from the center of the contact circle to the point represented by r according to the relationship [49]... 

Upon comparison of Eqs. 29 and 36, it is readily apparent that both theories predict the same power law dependence of the contact radius on particle radius and elastic moduli. However, the actual value of the contact radius predicted by the JKR theory is that predicted by the DMT model. This implies that, for a given contact radius, the work of adhesion would have to be six times as great in the DMT theory than in the JKR model. It should be apparent that it is both necessary and important to establish which theory correctly describes a system. 

The discrepancies between the predictions of the JKR and DMT theories were first discussed by Tabor [41]. Tabor s discussion resulted in a rather heated exchange in the literature [42,43]. Subsequently, Muller et al. [44,45] attempted to... 

However, if one attempted to determine iua from the DMT theory, one would get an unrealistically large value. In the same paper, the authors also presented micrographs of particles in contact with the substrate under a negative applied load that was not quite sufficient to effect detachment. It was reported that the observed contact radius under those circumstances was approximately 70% of the expected contact in the absence of the applied load. This observation is in apparent agreement with the JKR prediction that detachment occurs under negative loads that reduce the contact to about 63% of the equilibrium contact radius. 

Hertzian mechanics alone cannot be used to evaluate the force-distance curves, since adhesive contributions to the contact are not considered. Several theories, namely the JKR [4] model and the Derjaguin, Muller and Torporov (DMT) model [20], can be used to describe adhesion between a sphere and a flat. Briefly, the JKR model balances the elastic Hertzian pressure with attractive forces acting only within the contact area in the DMT theory attractive interactions are assumed to act outside the contact area. In both theories, the adhesive force is predicted to be a linear function of probe radius, R, and the work of adhesion, VFa, and is given by Eqs. 1 and 2 below. 

This is the same as Eq. [2], the value for nondeformable solids in vacuum. However, in the case of deformable solids, DMT theory gives a finite contact radius at zero applied load ... 

Although the DMT theory attempts to incorporate distance-dependent surface interactions into the adhesion problem, it does not take into account the effect surface forces have on the elastic deformation. In other words, it does not predict the neck formation predicted by JKR. 

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. 

Unfortunately, important results of the DMT theory cannot be expressed as convenient analytic expressions. There is, however, one simple result. For the adhesive force they obtained ... 

The interaction between fine particles is often dominated by their mechanical properties such as Young s modulus. This was first considered by Hertz theory. Adhesion between spherical particles increases with the radius of the particles and is described by JKR and DMT theories. 

As with DMT, theories have again been advanced that schizophrenia is associated with increased production of harmala alkaloids. As Shulgin has remarked, consensus among researchers now is that this approach is "a red-herring. ... 

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. 

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. 

From the DMT theory [7], which establishes a relationship between the adhesion force F and the thermodynamic work of adhesion Wo P=2tiRWq, R=tip... 

Q Fig. 3.4 Thermodynamic work of adhesion deduced from the DMT theory, versus surface energy of SAMs grafted on wafers. 

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