Contact radius

The JKR theory, much like the Hertz theory, assumes a parabolic approximation for the profile of sphere, which is valid for small ratios of contact radii to the sphere s radius. Maugis [34] has shown that for small particles on a soft substrate, this ratio could be so large that such parabolic approximation is no longer valid. Under such conditions, the use of exact expression for the sphere profile is necessary for the applicability of the JKR theory, which is expressed as... 

Another manifestation of a time dependence to particle adhesion involves the phenomenon of total engulfment of the particle by the substrate. It is recognized that both the JKR and MP theories of adhesion assume that the contact radius a is small compared to the particle radius R. Realistically, however, that may not be the case. Rather, the contact radius depends on the work of adhesion between the two materials, as well as their mechanical properties such as the Young s modulus E or yield strength Y. Accordingly, there is no fundamental reason why the contact radius cannot be the same size as the particle radius. For the sake of the present discussion, let us ignore some mathematical complexities and simply assume that both the JKR and MP theories can be simply expanded to include large contact radii. Let us further assume that, under conditions of no externally applied load, the contact and particle radii are equal, that is a(0) = R. Under these conditions, Eq. 29 reduces to... 

For purpose of interchangeability contact radii shall comply with Table 4-5. 

Figures 4-16, 4-17, 4-18, and Table 4-5 show recommended radii of hoisting tool contact surfaces. These recommendations cover hoisting tools used in drilling, and tubing hooks, but all other workover tools. Contact radii are intended to cover only points of contact between two elements and are not intended to define other physical dimensions of the connecting parts.

A chemical bond produces proximity of the two involved atomic nuclei, whose separation must be less than the sum of commonly accepted random-contact radii (if there is arguing about the exact value of the contact radii then the coimection is not a chemical bond). This is a necessary but not sufficient condition. Especially in the intermolecular case, proximity between two nuclei does not always imply a chemical bond. 

Covalent bond distances and angles tell us how the atomic nuclei are arranged in space but they do not tell us anything about the outside surfaces of molecules. The distance from the center of an atom to the point at which it contacts an adjacent atom in a packed structure such as a crystal (Fig. 2-1) is known as the van der Waals radius. The ways in which biological molecules fit together are determined largely by the van der Waals contact radii. These, too, are listed in Table 2-1. In every case they are approximately equal to the covalent radius plus 0.08 nm. Van der Waals radii... 

The JKR theory predicts correct contact radii for relative soft surfaces with effective radii larger than 100 /an. This was shown in direct force measurements by the surface forces apparatus [217, 218] or specifically designed systems. For smaller spheres it was verified using the colloidal probe technique [219],... 

The ability of this one parameter model to represent our experimental data is shown in Figure 3. This is a plot of the contact radii (p = D/2) of the gold clusters as measured by STM as a function of their free space volumes. The solid line in this figure is the prediction of the theoretical model, which is... 

Figure 3. Plot of the contact radii of Au clusters supported on a Au substrate as a function of the volumes of the clusters.

Eq. (13) makes the distinction between bonded (interactions between atoms that are connected by no more than three bonds) and nonbonded (interactions between atoms that are not connected to each other at all) interactions. The nonbonded energy accounts for repulsion, van der Waals attraction, and electrostatic interactions, van der Waals attraction occurs at short range and rapidly dies off as the interacting atoms move apart by a few angstroms. Repulsion occurs when the distance between interacting atoms becomes even slightly less than the sum of their contact radii. 

Here a, Oy, and are the Lennard-Jones contact radii (the radii at which the interaction potential has a value of zero) for atoms pairs i-i, j-j, and i-j, respectively and e, and e y are the Lennard-Jones well depths of the respective component interactions. The Lennard-Jones potential then has the following form ... 

There are many other nonplanar conformations of cyclohexane, one of which is the boat conformation. You can visualize the interconversion of a chair conformation to a boat conformation by twisting the ring as illustrated in Figure 3.9. A boat conformation is considerably less stable than a chair conformation. In a boat conformation, torsional strain is created by four sets of eclipsed hydrogen interactions, and steric strain is created by the one set of flagpole interactions. Steric strain (also called nonbonded interaction strain) results when nonbonded atoms separated by four or more bonds are forced abnormally close to each other—that is, when they are forced closer than their atomic (contact) radii allow. The difference in potential energy between chair and boat conformations is approximately 27 kj/mol (6.5 kcal/mol), which means that, at room temperature, approximately 99.99% of all cyclohexane molecules are in the chair conformation. 

Batsanov SS (2000) Intramolecular contact radii similar to van der Waals ones. Rus J Inorg Chem 45 892-896... 

The intermolecular contact radii Ric in crystals of MX with tetrahedral coordination coincide with the vdW radii of the elements of the fifth period and their compounds. The Rjc of metals determined from the M- -C(CH3) distances in M(C5Mes)n molecules, are close to the vdW radii determined by independent methods. The likely reason is that the CH3 group can tilt away from the plane of cy-clopentadienyl ring without significant energy loss and hence the repulsion between M and C(CH3) is similar to the intermolecular interaction, see Table S4.13 [103]. 

For the BX3 and MX4, more complicated procedures were devised, taking into account the deformation of atomic volumes of the central atoms in these molecules and the intra-molecular contact radii (see [129, 130] for details). The results of the calculations of anisotropic vdW radii for the abovementioned molecular structures are listed in Table S4.18 as one can see, the anisotropy of the X radii in molecules depends on the molecular environment. A summary of the averaged anisotropic vdW radii in the X2 and AX2 molecules is given in Table 4.7. 

Steric strain (Section 2.6A) The strain that arises when nonbonded atoms separated by four or more bonds are forced closer to each other than their atomic (contact) radii would allow. Steric strain is also called non-bonded interaction strain, or van der Waals strain. 

Fig. 10 Experimental solid line), simulated hollow squares), and difference hollow triangles) inclination of silicon cantilevers versus time upon evaporation of water microdrops. T 25 °C, RH 30%. Drop data various initial volumes, contact radii, and contact angles Pl = 0.072 N/m. Cantilever data Zq = 750 pm, w = 90 pm, d = l.S pm, u b = 1-5 pm, and dc = 1 7 pm gold layer thickness = 30nm Young s moduli Esi = 180 GPa, au = 78 GPa Poisson s ratios vsi - 0.26, uau = 0.44...

Fig. 10. Schematic representation of the contact radius as a function of time for the case where the contact radius increases to a maximum value and then decreases. The contact radii at times I (from the receding portion of the experiment) and /,(/) (from the advancing portion of the experiment) are equal to one another.

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