Spheres of equal size

Figure B3.2.3. The muffin-tin spheres in the (110) plane of a zineblende erystal. The niielei are surrounded by spheres of equal size, eovering about 34% of the erystal volume. Uuoeeupied tetrahedral positions are iudieated by erosses. The eouveutioual unit eell is shown at the bottom the erystal direetions are noted.

In this category, among the molecular, electrostatic and magnetic interparticle bonds, interest is primarily centered on the van der Waals-type attractive forces that may predominate in the absence of liquid and solid bonds. The force of the van der Waals attraction between two spheres of equal size is (R4)... 

For conducting spheres of equal size and equal and like charge, the force of repulsion is given by... 

The development given for determining pore-size in terms of particle-diameter applies only to an array of spheres of equal size. When the spheres are of different sizes the solution is possible only by involved statistical procedures. However, the average pore-diameter may be determined experimentally by means of the Poiseuille equation (see Chapter 15). Traxler and Baum (1936) used this method in the following manner The Poiseuille equation is... 

The latest models propose to represent electrolyte solutions as a collections of hard spheres of equal size, ions, immersed in a dielectric continuum, the solvent. For such a system, what is called the Mean Spherical Approximation, MSA, has been successful in estimating osmotic and mean activity coefficients for aqueous 1 1 electrolyte solutions, and has provided a reasonable fit to experimental data for dilute solutions of concentrations up to -0.3 mol dm". The advantage in this approach is that only one... 

A micro-density method depending on a comparison of the times of falling drops in an immiscible second liquid of low viscosity and volatility can be used with 0 001-0 01 ml. Liquid spheres of equal size of depsities qi and Q2 falling through a liquid of density q require, according to Stokes s law ( 4. VIII L), the times hence for a given sizeof drop,... 

Among the structural models of cellular or porous materials those characterized by differently ordered packing of balls or spheres of the same diameters have most widely been used. In this approach either the spheres have been considered as real cells or the cell (pore) models have been derived from an analysis of assumed spacings between the contacting solid spheres. However, no system of packed spheres would adequately describe the properties of any real cellular system which never exhibit a regular packing. It is also impossible to describe the structure of most cellular systems via models assuming spheres of equal size. 

The silo is divided into cells containing fuel particles, which are handled as spheres of equal size, and their effective diameter is calculated according to [7]. The evaporative mass flow from the wet surface of the particle is calculated [4] ... 

Figure 5 Total (per unit cell) and local (per space-filling atomic spheres of equal size at both Pt and Sn sites) densities of states of PtsSn, calculated by the tight-binding linear muffin-tin orbitals method. At positive (sample-) bias voltages the unoccupied states above E are imaged in the STM. From Ref. [27].

Sphere Spheres of equal size may be packed in three dimensions to give hexagonal close-packed... 

In the dumbbell model, a polymer chain in a solvent is pictured as two massless spheres of equal size connected by a frictionless spring. The spheres experience a hydrodynamic drag proportional to their size, characterized by the Flory radius. Assume that the displacement of the spring generated by the thermal energy is also characterized by the Flory radius. Write the equation of motion for the dumbbell and show that the characteristic relaxation time for the chain deformation is that given by Eq. (9.2.1). 

These expressions for the collision function assume purely geometrical collisions and do not include electrostatic, van der Waals, or viscous forces. Corrections for these surface and fluid forces are available for Brownian coagulation and have been verifled experimentally by Lichten-belt et al. (12) in the absense of electrostatic forces for particles uniform in size. For shear coagulation, corrections have been computed for collisions between spheres of equal size, and experimental agreement with theory has been obtained only when electrostatic forces are absent (van de Ven and Mason (13) Zeichner and Schowalter (14)). Differential-sedimentation coagulation of hydrosols has not been examined theoretically or experimentally. 

If spheres of equal size are packed together in a plane, they will arrange themselves in one of the patterns shown in Fig. 2. The sphere centers will lie at the comers of equilateral triangles, and each sphere will be in contract with six others. The three-dimensional stmcture of closely packed spheres of equal sizes will consist of such. The conditions that must be satisfied are that the space occupied per sphere is a minimum and that the spheres all have the same environment. The relative orientation of the spheres is disregarded if it is assumed that they possess... 

Gas bubbles dispersed in the bulk electrolyte, common in industrial electrolysis, are essentially randomly distributed spheres having zero conductivity. There are a number of different approaches to describing the effect of such dispersions on the overall conductivity. Simplification of the problem is possible when the dispersed phase is dilute or when a limited range of void fraction is considered. Some writers discuss media in which the dispersed phase occupies well-defined lattice positions while others treat random dispersions. One may also consider spheres of equal size or dispersions containing a distribution of sizes. I classify the approaches by the type of dispersion they aim to describe and compare the theory for these classes to appropriate experimental data. 

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