Random close packing

Samples can be concentrated beyond tire glass transition. If tliis is done quickly enough to prevent crystallization, tliis ultimately leads to a random close-packed stmcture, witli a volume fraction (j) 0.64. Close-packed stmctures, such as fee, have a maximum packing density of (]) p = 0.74. The crystallization kinetics are strongly concentration dependent. The nucleation rate is fastest near tire melting concentration. On increasing concentration, tire nucleation process is arrested. This has been found to occur at tire glass transition [82]. 

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 ... 

Random close packing (applicable to metallic glasses)... 

Equation (48) has been derived under the assumption that the volume fraction can reach unity as more and more particles are added to the dispersion. This is clearly physically impossible, and in practice one has an upper limit for , which we denote by max. This limit is approximately 0.64 for random close packing and roughly 0.71 for the closest possible arrangement of spheres (face-centered cubic packing or hexagonal close packing). In this case, d in Equation (46) is replaced by d(j>/[ 1 — (/m[Pg.169]

If there are sufficiently strong repulsive interactions, such as from Ihe electric double-layer lorce. then the gas bubbles at the lop of u froth collect together without bursting. Furthermore, their interfaces approach as closely as these repulsive forces allow typically on the order of 100 nm. Thus bubbles on top of a froth can pack together very closely and still allow most uf the liquid to escape downward under the influence of gravity while maintaining their spherical shape. Given sufficient liquid, such a foam can resemble the random close-packed structure formed by hard spheres. 

A possible approach that we have presented in detail elsewhere C6, 7) is to consider the centroids of these globular segments to be distributed as the centres of randomly close-packed spheres. 

The assumption of a symmetric potential-well may not be too far off from the physical reality in that in a random-close-packed liquid state, atoms would tend to slid-by one another tangentially, instead of colliding head-on. 

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). 

In principle, it is possible to calculate Xg from the Gmin values given in Fig. 6, using equation (5). If the floes are assumed to be close-packed structures, then n must be between 8 (random-close packing) and 12 (hexagonal close-packing). However, equation (5) is based on a monodisperse suspension. 

So, let us imagine that at very low temperatures in a material that for whatever reason has not or cannot crystallize, the molecules are more or less randomly close packed. The total volume of the system is then that of the hard core of the molecules plus the unoccupied volume between them. As the tem-... 

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

An aqueous colloidal suspension also has an osmotic pressure associated with both the double layer of the particles in solution and the structure of the particles. The osmotic pressure term for the structure is given in Section 11.6 for both ordered and random close packing. The osmotic pressure associated with the double layer surrounding the ceramic particles in aqueous solution is discussed here. 

This divergence occurs at random close packing of 0.63 — 0.64, consistent with the idea that higher volume fractions require some degree of order to be thermodynamically stable. 

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

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