DMT model

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]. 

Equilibrium is established when the attractive surface forces are balanced by elastic repulsion forces between the materials. The DMT model states that the elastic repulsion force is related to the attractive force within the contact region Fs by... 

This implies that, according to the DMT model, exactly half of the attractive interactions occur outside the zone of contact and half of the attractive interactions occur inside the zone of contact. 

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. 

As is evident, there are several distinctive characteristics of adhesion-induced plastic deformations, compared to elastic ones. Perhaps the most obvious distinction is the power-law dependence of the contact radius on particle radius. Specifically, the MP model predicts an exponent of 1/2, compared to the 2/3 predicted by either the JKR or DMT models. 

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. 

The above measurements all rely on force and displacement data to evaluate adhesion and mechanical properties. As mentioned in the introduction, a very useful piece of information to have about a nanoscale contact would be its area (or radius). Since the scale of the contacts is below the optical limit, the techniques available are somewhat limited. Electrical resistance has been used in early contact studies on clean metal surfaces [62], but is limited to conducting interfaces. Recently, Enachescu et al. [63] used conductance measurements to examine adhesion in an ideally hard contact (diamond vs. tungsten carbide). In the limit of contact size below the electronic mean free path, but above that of quantized conductance, the contact area scales linearly with contact conductance. They used these measurements to demonstrate that friction was proportional to contact area, and the area vs. load data were best-fit to a DMT model. 

Thomas et al. [72] used a modified force microscope in which a compensatory force was applied to the probe to keep its displacement at zero. With this system they studied interactions between organomercaptan molecules with CH3, NH2, or COOH end groups. All measurements were performed in dry nitrogen. From SEM-measured tip radii and pull-off force they calculated the work of adhesion using the DMT model. They found that the work of adhesion values qualitatively scaled as expected for van der Waals, hydrogen bonding,... 

Exact analysis shows that the two models represent two extremes of the real situation [207-209], For large, soft solids the JKR model describes the situation realistically. For small, hard solids it is appropriate to use the DMT model. A criterion, which model is to be used, results from the height of the neck (Fig. 6.19)... 

If the neck height is larger than some atomic distances, the JKR model is more favorable. With shorter neck heights the DMT model is more suitable. 

The neck height is about as large as an atomic diameter. Therefore the DMT model is suitable and the adhesion, therefore, is... 

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]. 

Similar to Muller et al. [103], several authors performed self-consistent numerical computations of the non-linear equations based on the Lennard-Jones potential instead of dealing with analytical approaches. Greenwood [108, 109] and Feng [112] calculated the dependence of the pull-off force on the transition parameter A. For fixed y and R, softer spheres require less force to separate, though they deform more significantly. At the extreme limits of small and large values of A, however, the pull-off force becomes virtually independent of A. On a local scale, this is consistent with the prediction of the JKR and the DMT model that the pull-off force is independent of elastic moduli of spheres [112]. 

Heekeren K, Neukirch A, Daumann J, Stoll M, Obradovic M, et al. 2007. Prepulse inhibition of the startle reflex and its attentional modulation in the human S-ketamine and N,N-dimethyltryptamine (DMT) models of psychosis. J Psychopharmacol 21 312-320. 

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M.O., and Huber, M.L., NIST Standard Reference Database 23 Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Dillon, H. E., and Penoncello, S. G., A Fundamental Equation for Calculation of the Thermodynamic Properties of Ethanol, Int. J. Thermophys., 25(2) 321-335,2004. The source for viscosity is Kiselev, S. B., Ely, J. E, Abdulagatov, I. M., and Huber, M. L., Generalized SAFT-DFT/DMT Model for the Thermodynamic, Interfacial, and Transport Properties of Associating Fluids Application for n-Alkmols, Ind. Eng. Chem. Res., 44 6916-6927, 2005. The source for thermal conductivity is unpublished, 2004 however, the fit uses functional form found in Marsh, K., Perkins, R., and Ramires, M.L.V, Measurement and Correlation of the Thermal Conductivity of Propane from 86 to 600 K at Pressures to 70 MPa, J. Chem. Eng. Data, 47(4) 932-940, 2002. 

For nanoaerosol, e 1 because of the great rigidity. Several theoretical models of adhesion energy can be found in the literature. Two of the most well-known ones are JKR model and DMT model. These two models contradict each other because they represent two extremes in the Tabor parameter spectrum. JKR model is applied for soft material, large radius, compliant spheres, and large adhesion energy, and DMT model is for hard material, small radius, and low adhesion energy. Therefore, DMT model should be considered first for nanoaerosol thermal rebound analysis. 

DMT Elastic-Adhesive Normal Contact Model The theory of Derjaguin et al. [27], referred to as DMT model, assumes that the attractive forces all lie outside the contact zone. The attraction forces are also assumed not... 

Pashley, M.D. (1984) Further consideration of the DMT model for elastic contact. Colloids and Surfaces 12, 69-11. 

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... 

The DMT model predicts that the contact radius at the separation is zero. That is,... 

While the JKR and the DMT models assume elastic deformation, there are experimental data that suggests that, in many cases, plastic deformation occurs. Maugis... 

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