Contact sphere

The composites with the conducting fibers may also be considered as the structurized systems in their way. The fiber with diameter d and length 1 may be imagined as a chain of contacting spheres with diameter d and chain length 1. Thus, comparing the composites with dispersed and fiber fillers, we may say that N = 1/d particles of the dispersed filler are as if combined in a chain. From this qualitative analysis it follows that the lower the percolation threshold for the fiber composites the larger must be the value of 1/d. This conclusion is confirmed both by the calculations for model systems [27] and by the experimental data [8, 15]. So, for 1/d 103 the value of the threshold concentration can be reduced to between 0.1 and 0.3 per cent of the volume. 

Jefri, M.A., Nichols, K.L., and Jayaraman, K. "Sedimentation of Two Contacting Spheres in Dilute Polymer Solutions," Proc. Svmp. Recent Dev. Struct. Continua.. 1985, 21-5. 

Lund et al. (1999) reported finite element simulations of conduction between two contacting spheres without fluid flow. They modeled a small separation between the spheres to allow for interparticle micro-asperity gaps, and then the... 

Note that the elastic displacement of a single sphere equals half of the total displacement of two contacting spheres. Therefore, from Eq. (2.74), we have... 

Figure 5.7 Neighbor-joining trees of the sequence homology of each binding site for monoamine-related GPCRs. The underlying sequence blocks correspond to transmembrane residues identified within the 6-A contact spheres of 5-HT (panel A), propranolol (panel B) and 8-OH-DPAT (panel C), respectively, in the rho-dopsin-based models of the 5-HT1A receptor-ligand complexes. Details as in the legend for Figure 5.2.

In addition to the needs following from the stoichiometry the VEP also takes those requirements of structural geometry into consideration which coincide with the principles of symmetry and connection. The space principle is also discussed here, however, with the difference that we do not start from a model of rigid mutually contacting spheres but also plastical or soft contacts between atoms and ions are admitted. Let us try and delimit the physical meaning of the VEP by the following consideration ... 

Below the percolation line, there is predicted to be a sample-spanning cluster of contacting spheres. Woutersen et al. (1994) found that the gel point for 47-nm octadecyl-grafted silica spheres in benzene is in reasonable agreement with the predicted percolation transition. However, Grant and Russel (1993) found that the gelation line is below the percolation ... 

Near the contact, the vertical arrows at the dashed contour schematically represent the surface forces which cause an additional deformation of the elastic sphere thus increasing the contact radius from aH (Hertz) to aJKR (JKR). The contact radius for the JKR model is a function of the external load, the work of adhesion, the radius of the contacting sphere (or the reduced radii of the contacting spheres, if two spheres are in contact) and the elastic constant K (a combination of the Young s moduli and the Poisson s ratios of the contacting materials), defined as... 

When the contacting sphere is about to jump off, the force that is necessary to separate the sphere from the flat plane, Lpf is written as... 

For the Hertzian contact, no force is needed to pull away the contacting sphere from the flat plane in excess of the weight of the sphere. However, for the JKR contact, due to adhesion forces, this does not hold. The value of the nonzero pull-off force represents the adhesion of the contacting sphere with the flat plane. Strictly speaking, this force corresponds to adherence of the surfaces as energy dissipation, surface relaxation, etc. also influence its value. It should be stressed that the value of the JKR pull-off force only depends on the sphere (lens) radius and the work of adhesion in the medium in which the JKR experiment is conducted. Thus, the contact area and mechanical properties for true JKR contacts do not play a role for its value. All the above considerations for contact mechanics were based on pairwise additivity of molecular forces. 

F12 radiation view factor between two contacting spheres... 

M. Barquins and D. Maugis, "Adhesive Contact of Axisymmetric Punches on an Elastic Half-Space The Modified Hertz-Huber s Stress Tensor For Contacting Spheres," J. Mech. Theor. Appl., 1, 331 (1982). 

FIG. 5 (a) Derivative of the reduced conductivity dGla° )/d(, and (b) coupling coefficient 377 in a cubic array of contacting spheres (solid lines) or oblate ellipsoids (dashed lines) along the horizontal (O) and vertical (+) directions as functions of the inverse dimensionless double layer thickness kR. The dash-dotted line in (a) is the result for spheres of a matched asymptotic expansions technique [13,27],... 

Infinitely stiff spheres do not deform when brought in contact and pressed together. Hence the contact between them is a point contact. If the Young s modulus of the contacting spheres is finite, then the contact point becomes a contact circle with a radius a . The value of the contact radius a depends in such cases on the elastic properties of the spheres, on the Young s moduli El and 2. and on the Poisson s ratios v and V2, of the two contacting materials, respectively. The value of the contact radius a can be calculated from the following formula ... 

However, when surface (adhesive) forces are present, the shape of the contacting spheres in the vicinity of the rim of the contact area will further be deformed. Due to van der Waals attraction, this additional deformation of the elastic body will pull the two contacting objects closer together and hence further increases the contact radius with regard to the Hertzian value. 

The value of can only become zero (at rupture of contact) if the normal force is negative, i.e. if the contacting sphere is pulled up (away from the flat). If this is the case, a physical solution can only exist if... 

Jefri, M. A., K. L. Nichols, and K. Jayaraman, Sedimentation of two contacting spheres in dilute polymer solutions. Recent Developments in Structured Continua (D. DeKee and P. N. Kaloni, eds.), Longman, London, 1985, p. 21. 

Our treatment of adhesive interactions is based on an energy balance, and is very similar to the derivation presented by Johnson et al. in their classic paper [7]. These authors assumed a geometry of contacting spheres with small contact areas a/h = 0). Here we use a more generalized version in order to readily account for geometrical effects [8]. 

Classical pattern of binary reactions in liquids is the following reagents, randomly walking in solution, occasionaly find each other and having entered in a direct contact, react. If the process of walks may be considered as continual encounter diffusion, then the frequency of reagents encounter is determined by "diffusive" rate constant k = 47tRD, where R is a contact sphere... 

Figure 14.20. Contact sphere according to Suzuki and Oshima...

The area of the contact sphere divided by the area of the surface cut (on Sc) by a sphere of co-ordination ntrmber j defines z°. 

Yij the radius of the lens of fluid between two contacting or near contacting spheres, m the radius of the isothermal core used in Model B, m Re, local relative Reynolds number for particle i, dimensionless Rj radius of particle i, m Rj radius of particle j, m Rjj mean radius of particles i andj, m tc collision contact duration between particles i and j, s tj static contact duration between particles i and j, s Te the environmental temperature, K Tj, bed temperature, K... 

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