Interacting spheres

The interaction of two infinite blocks separated at the surface by a distance d is one of the easiest possible situations to consider and therefore was chosen as the example to develop in detail. Bodies of different geometries have also been analyzed in much the same way that we have the blocks. The results of several such derivations are shown in Table 10.4. The expressions for 4 become more complicated for more complex geometrical situations, but considerable simplification results for spheres in which the separation is much less than the radius. The feature described by Equation (50) for interacting spheres is also evident in the results given in Table 10.4. It is most readily seen by examining the case of equal spheres separated by small distances. Note that both Rs and d can be multiplied by any common factor without changing... 

Of the various quantities that affect the shape of the net interaction potential curve, none is as accessible to empirical adjustment as k. This quantity depends on both the concentration and valence of the indifferent electrolyte, as shown by Equation (11.41). For the present we examine only the consequences of concentration changes on the total potential energy curve. We consider the valence of electrolytes in the following section. To consider the effect of electrolyte concentration on the potential energy of interaction, it is best to use the more elaborate expressions for interacting spheres. Figure 13.8 is a plot of ne, for this situation as a function of separation of surfaces with k as the parameter that varies from one curve to another. 

Batchelor, G. K. and Wen, C. S. (1982) Sedimentation in a dilute polydisperse system of interacting spheres. Part 2. Numerical results. /. Fluid. Mech. 124, 495-528. 

Figure E3.1. Schematic flow pattern of two interactive spheres due to wake attraction.

Collision Cross-Section The model of gaseous molecules as hard, non-interacting spheres of diameter o can satisfactorily account for various gaseous properties such as the transport properties (viscosity, diffusion and thermal conductivity), the mean free path and the number of collisions the molecules undergo. It can be easily visualised that when two molecules collide, the effective area of the target is no1. The quantity no1 is called the collision cross-section of the molecule because it is the cross-sectional area of an imaginary sphere surrounding the molecule into which the centre of another molecule cannot penetrate. 

Consider the interaction energy Vsp(H) between two spheres 1 and 2 of radii Uj and fl2 separated by a distance H between their surfaces (Fig. 12.1). The spherical Poisson-Boltzmann equation for the two interacting spheres has been not solved. If, however, the following conditions are satisfied. 

FIGURE 12.1 Deijaguin s approximation for the two interacting spheres 1 and 2 at separation H, having radii and U2, respectively. 

By applying the above method to two interacting spheres on the basis of the linearized Poisson-Boltzmann equations, we can derive series expansion representations for the double-layer interaction between two spheres 1 and 2 (Fig. 14.3). 

In case (c), the image interaction energy carries both characters of cases (a) and (b). When the unperturbed surface potentials and the radii of the two spheres become similar, these two contributions from cases (a) and (b) tend to cancel each other so that the total image interaction for case (c) becomes small, as shown in Fig. 14.6, in which the interacting spheres are identical Ka = ku2 = 5 and i/ oi = 4 02)- In the opposite case where the difference in the two unperturbed potentials is large, the image interaction for case (c) is determined almost only by the larger unperturbed surface potential. 

The chapter starts with an overview of the nature of major sites of radioactive environmental contamination. A brief summary of the health effects associated with exposure to ionizing radiation and radioactive materials follows. The remainder of the chapter summarizes current knowledge of the properties of radionuclides as obtained and applied in three interacting spheres... 

Figure 7 The simulation box with side length L, and the sub-cells width side length 1. A particle s interaction sphere of radias Vc and its neighborhood sphere of radius < Z.

Figure 8 The group neighborhood of the five particles that happen to be in a particular cell. The arrows extending from the comers of this cell are all of length Vc. They show that the group neighborhood completely covers the interaction sphere of any particle located in the cell. The sub-cells have side length I = rc-...

The correlation length in semidilute solution can be experimentally determined by measuring the diffusion coefficient of very dilute colloidal spheres of various sizes, provided that the spheres do not interact with the polymers. Consider diffusion of a non-interacting sphere in a semidilute unentangled solution. 

L. G. Leal, The slow motion of slender rod-like particles in a second-order fluid, J. Fluid Mech. 69, 305-37 (1975) B. P. Ho and L. G. Leal, Migration of rigid spheres in a two-dimensional unidirectional shear flow of a second-order fluid, J. Fluid Mech. 76, 783-99 (1976) P. C. H. Chan and L. G. Leal, The motion of a deformable drop in a second-order fluid, J. Fluid Mech. 92, 131-70 (1979) L. G. Leal, The motion of small particles in non-Newtonian fluids, J. Non-Newtonian Fluid Mech. 5, 33-78 (1979) R. J. Phillips, Dynamic simulation of hydro-dynamically interacting spheres in a quiescent second-order fluid, J. Fluid Mech. 315, 345-65 (1996). 

The viscosities of the halogens are 2 to times those of the paraffins. They shov. no definite trend y ith increasing molecular weight For rigid rion-interacting spheres,- the viscosity is Predicted... 

Table 12.20. Partial (5, 5, 8 ) and total (5) solubility parameters and radius of interaction sphere (C ) of polymers/blends [Pendyala and Xavier, 1992]...

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