Random close-packing of spheres

Value expected for ordered close packing of spheres = 0.74 value expected for random close packing of spheres = 0.67... 

Random Close Packing of Spheres Well Defined ... 

FIGURE 10-55 Schematic diagram showing the unoccupied volume and oscillations around a mean position in ordered and random close packing of spheres. 

Torquato, S., Truskett, T. M., and Debenedetti, P. G., Is random close packing of spheres well defined Phys. Rev. Lett. 84,2064 (2000). 

Other phenomena may also be observed in concentrated dispersions. Thus, as the concentration increases, direct particle—particle contacts increase in importance, and as the limiting packing fraction is approached (j) = 0.625 for the random close-packing of spheres) the suspension ceases to flow and the viscosity becomes infinite. Shortly before this limit is reached the suspension (especially if the particles are monodisperse spheres) often exhibits dilatency. When the particles are nearly close-packed, flow can occur only by the particles rolling past one another (Figure 8.8), and this results in an increase in volume the shear thickening is accompanied by dilatence of the system. If the amount of liquid present is insufficient to fill the extra void volume produced, the surface may become dry. This is the familiar phenomenon observed when one steps on wet sand. 

The density of random close packing of spheres. J. Phys. D Appl. Phys., % 863-866. 

A. Z. Zinchenko. Algorithm for random close packing of spheres with periodic boundary conditions./. Comp. Phys., 114 298, 1994. 

The universal packing fraction, t] = 0.625, for the mers of rubber-like polymer systems corresponds to the random close packing of hard spheres. The existence of this universal value may be motivated as follows Assume first the absence of nonbonded interactions and consider a network of Gaussian chains y with chain vectors R(y) occupying a volume v. The force f(y) required to maintain the chain vector fixed at R(y) is... 

Scott, G. D. Radial Distribution of the Random Close Packing of Equal Spheres. Nature 194, 956 (1962). 

FIGURE 12.11 Packing structures of cubic close packing, hexagonal close packing, and random close packing of 0.31 Mm diameter Ti02 spheres. 

The total energy U for a random close-packing of N spheres is obtained as a sum of all the contact energies ... 

The atomic packing in disordered solids was investigated first by Bernal (1964), who considered the problem in the context of a model of a simple liquid that consisted of randomly close-packed hard spheres of uniform size and described the structure as a distribution of five different canonical polyhedra with well-defined volume fractions. 

Scott GD, Knight KR, Bernal ID, Mason J (1962) Radial distribution of random close packing of equal spheres. Nature 194 956-957... 

Scott GD, Mader DL (1964) Angular distribution of random close-packed equal spheres. Nature 201 382-383... 

The threshold volume fraction of percolation (( >,) is guided by the amphiphile shell length and the overall volume fraction of the dispersed phase. For zero shell length and no interparticle attractive interaction, according to the randomly close-packed hard sphere model, 4, = 0.65 systems with strong attractive interactions end up with (f), being lowered from 0.65 to 0.10. 

Berryman JG (1983) Random close packing of hard spheres and disks. Phys Rev A27 1053-1061... 

Schematic representations of two mesoporous xerogels. (A) Randonutlose packing of low-density spheres. (B) Hierarchical random-close packing of dense spheres. (C) Schematic representation of microporous xerogei illustrating uniform distribution of terminal sites. (D) Schematic representation of the particulate xerogei illustrating the dense oxide core and nonuniform distribution of terminal sites.

At concentrations lower than random close packing of hard spheres (63%), simple emulsions are viscous liquids. Multiple emulsions behave in the same way. Nevertheless, it has been observed that attractive interactions between... 

Both of the above equations reduce to the Einstein limit [eqn. (5.1)] at low concentrations ( —> 0) and to oo as > K K The crowding factor K can be treated either as an adjustable parameter for fitting experimental data or as a theoretical parameter equal to the reciprocal of the volume fraction at which diverges to infinity. For random close packing of monodisperse hard spheres, we have 0niax = 0.64 and K = 1.56. An alternative theoretical route to K is to expand... 

Here, we employ (4.5) and variants of it that are more appropriate for dense and very dissipative systems to describe the dependence of the transport coefficients on the volume fraction. The product G = vgo will be shown to arise prominently in the expression for the mean free path or, equivalently, the distance between the edges of the spheres, quantities that rapidly go to zero as the random close packed limit is approached (volume fraction v 0.64). We note that as the limit is approached, G increases rapidly (e.g., at v = 0.5, G = 3). In what follows, we make use of this to simplify the dependence of the theory on the volume fraction. Also, for volume fractions above 0.49, a more accurate form of 5 0 introduced by Torquato (1985) replaces the singularity of (4.5) at unity with a singularity at the value of 0.64 appropriate to random close packing of identical, frictionless spheres ... 

D.V. Xellor, Random Close Packing of Equal Spheres Structure and Implications for Use as a Model Porous Medium, PhD Thesis, Open University, 1989. 

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