Contact theory

R. Holm, Electric Contacts Handbook, 4th ed., Sptinget-Vedag, New York, 1967 comprehensive treatment of electric contact theory. 

By combining Hertz s contact theory (Eq. 1) and with Hamaker s functional form for the attractive force (Eq. 17), the Derjaguin model takes the form... 

Contact theories suggest that complete strength may be obtained when the interface has wetted at 0 = 1. Wetting is a prerequisite for strength development but does not imply that the maximum strength has developed when molecular... 

T. Ioannides, and X.E. Verykios, Charge transfer in metal catalysts supported on Doped Ti02 A Theoretical approach based on metal-semiconductor contact theory, J. Catal. 161,560-569 (1996). 

The number of cycles of disk rotation required to initiate the wear track correlated positively with the weight percent of the siloxane modifier in the epoxy. However, the initiation times for the ATBN- and CTBN-modified epoxies showed no significant correlation with the percentage of the incorporated modifier. The initiation of the wear track is assumed to result from the fatigue of the epoxy hence initiation time is related to the surface stresses. Because the surface stresses are inversely related to the elastic modulus as predicted by the Hertzian elastic contact theory 52), the initiation time data at ION load were compared to the elastic moduli of the materials in Fig. 16. The initiation times for the siloxane-modified epoxies were negatively correlated with their elastic moduli while samples modified with ATBN and CTBN showed positive correlations with their moduli. At lower loads the initiation times for the siloxane-modified epoxies increased. The effect of load on the CTBN- and ATBN-modified epoxies was too erratic to show any significant trends. 

In this chapter, two simple cases of stereomechanical collision of spheres are analyzed. The fundamentals of contact mechanics of solids are introduced to illustrate the interrelationship between the collisional forces and deformations of solids. Specifically, the general theories of stresses and strains inside a solid medium under the application of an external force are described. The intrinsic relations between the contact force and the corresponding elastic deformations of contacting bodies are discussed. In this connection, it is assumed that the deformations are processed at an infinitely small impact velocity and for an infinitely long period of contact. The normal impact of elastic bodies is modeled by the Hertzian theory [Hertz, 1881], and the oblique impact is delineated by Mindlin s theory [Mindlin, 1949]. In order to link the contact theories to collisional mechanics, it is assumed that the process of a dynamic impact of two solids can be regarded as quasi-static. This quasi-static approach is valid when the impact velocity is small compared to the speed of the elastic... 

During a collision, the colliding solids undergo both elastic and inelastic (or plastic) deformations. These deformations are caused by the changes of stresses and strains, which depend on the material properties of the solids and the applied external forces. Theories on the elastic deformations of two elastic bodies in contact are introduced in the literature utilizing Hertzian theory for frictionless contact and Mindlin s approach for frictional contact. As for inelastic deformations, few theories have been developed and the available ones are usually based on elastic contact theories. Hence, an introduction to the theories on elastic contact of solids is essential. 

Besides the oblique contact, tangential displacements may also be produced in the contact of two elastic spheres under the actions of a compressional twist, as shown in Fig. 2.13. Since the torsional couple does not give rise to a displacement in the z-direction, the pressure distribution is not influenced by the twist and is thus given by the Hertzian contact theory. 

Collisions between particles with smooth surfaces may be reasonably approximated as elastic impact of frictionless spheres. Assume that the deformation process during a collision is quasi-static so that the Hertzian contact theory can be applied to establish the relations among impact velocities, material properties, impact duration, elastic deformation, and impact force. 

As mentioned, the erosion of a solid surface depends on the collisional force, angle of incidence, and material properties of both surface and particles. Although abrasive erosion rates cannot be precisely predicted at this stage, some quantitative account of erosion modes which relates various impact parameters and properties is useful. In the following, a simple model for the ductile and brittle modes of erosion by dust or granular materials suspended in a gas medium moving at a moderate speed is discussed in light of the Hertzian contact theory [Soo, 1977]. 

The first non-Markovian approach to chemical reactions in solutions, developed by Smoluchowski [1], was designed for contact irreversible reactions controlled by diffusion. Contrary to conventional (Markovian) chemical kinetics in the Smoluchowskii theory, the reaction constant of the bimolecular reaction, k(t), becomes a time-dependent quantity instead of being tmly constant. This feature was preserved in the Collins-Kimball extension of the contact theory, valid not only for diffusional but for kinetic reactions as well [2]. 

However, MET is not a unique theory accounting for multiparticle effects. There are some others competing between themselves, but they all can be reduced to the integral equations of IET distinctive only by their kernels. Depending in a different way on the concentration of quenchers c, the kernels of all contact theories of irreversible quenching coincide with that of IET in the low concentration limit (c —> 0) [46], IET of the reversible dissociation of exciplexes is also the common limit for all multiparticle theories of this reaction, approached at c = 0 [47], This universality and relative simplicity of IET makes it an irreplaceable tool for kinetic analysis in dilute solutions. 

The authors were satisfied correctly that the resulting reaction radius 3.4 A agrees with the literature where it is usually assumed that I is a contact quencher and must approach the fluorophore very closely [62]. The identification of the contact and reaction radii justifies a posteriori the application of contact theory to this kind of the reaction. As will be shown below, for the remote (energy and electron) transfer, these are two different quantities. 

Using this expression in Eq. (3.4) for R(t) and integrating the latter in Eq. (3.10), one can get the general contact q and the corresponding Stern-Volmer constant, which is an increasing function of quencher concentration c. There are also a number of competing contact theories that do the same but with slightly different results. They were compared in Ref. 46, reviewed in Section XII. 

Only at the fastest diffusion, when Rq < a, the contact theory is applicable to electron transfer. Under this condition the Collins-Kimball expression (3.21) integrated in Eq. (3.4) constitutes a reasonable approximation to R(t) and for both long and short times. However, it was recognized long ago that for slower diffusion the Collins-Kimball model works better if a is considered as a fitting... 

In differential contact theory k(t) monotonously decreases with time, starting from the maximum kinetic value (3.145) and approaching the minimal stationary value ... 

As has been proved in Eq. (3.700), the lowest limit of the Stern-Volmer constant Ko = k(0) is the same for all contact theories. However, there is also the upper limit of k(c) reached at largest c. At the very beginning the quenching kinetics is always exponential... 

Figure 3.88. The concentration dependence of the Stern-Volmer constant K in units of ko for a number of contact theories, provided ko = 3.7 x 1010M 1s 1 is the same for all of them. (From Ref. 46.)...

From the comparison of this result with others, it is clear that LESA is just poor interpolation between border limits. This is partially true also for MET/ MPK3, although it is rather good for small concentrations as was actually expected [39,40,44,45]. At least it is better than IET and LSA, which reproduce the ideal Stern-Volmer law and are suitable for calculation of only concentration-independent constants (horizontal dashed line). Two remaining theories, MPK2 and SCRTA, give results that are very close to each other and to the exact one. This comparison is instructive to those who like to employ one of the existing contact theories to the particular problem of interest. 

The results of Schwab s experiments indicate that there is a carrier effect whose direction can be correlated with predictions based on the metal-semiconductor contact theory. However in catalysis research there are many examples of cases where attempts to reproduce complicated experiments of this type have resulted in conflicting results. A question of this basic importance will need confirming evidence before the concept of the carrier effect is widely accepted. 

In the case of insects, decisive proof in favour of the material contact theory was furnished by the findings of Butenandt el al. (1959). As a result of long research work, they isolated the sex attractant substance produced by the female of the silk worm, Bombyx mori, and identified it as ( )-10-(2T)-12-hexadecadienol (bom bykol, 1). Determination of the active structure was achieved by the synthesis and biological testing of the four possible isomers (Butenandt et al., 1959). 

The frictional behavior we are concerned with is a consequence of rubbing contact and involves the interaction of surfaces. Real surfaces are neither geometrically nor molecularly smooth. Discussion of the complexities of contact theory and their influence on rubbing behavior is the subject of Chapter 12. But even a simple treatment of the basic theory of friction involves the role of surface structure. 

N. J. Fisher and M. M. Yovanovich, Thermal Constriction Resistance of Sphere/Layered Flat Contacts Theory and Experiments, ASME Journal of Heat Transfer, Vol. Ill, pp. 249-256,1989. 

Pettigrew, T. E (2008). Future directions for intergroup contact theory and research. International Journal of Intercultural Relations, 32 S), 182-199. 

Equivalent stress distribution which is shown in Figure 13 (loading below yield stress) is in good correlation with Hertzian contact theory. 

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