Random close packing theory

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 ... 

Finally, numerical simulations (Mitarai and Nakanishi 2007, Reddy and Kumaran 2007) indicate that for dense shearing flows of very dissipative spheres, the singularity depends on the amount of dissipation in a collision and that it is less than the value at random close packing. We employ the various forms of G associated with these collisional radial distribution functions as we consider theories that apply for increasing volume fractions and increasing amounts of collisional dissipation. 

Particles in the liquid state are randomly packed and relatively close to each other (see Figure 6.3B). They are in constant, random motion, sliding freely over one another, but without sufficient kinetic energy to separate completely from each other. The liquid state is a situation in which cohesive forces dominate slightly. The characteristic properties of liquids are also explained by the kinetic theory ... 

Such short-chain dispersants cannot adopt a random coil configuration they will be more or less linear molecules no matter the ability of the polymer to solvate the chains. In fact, molecules like stearic acid on calcium carbonate pack together so tightly that there is no possibility for the polymer to penetrate between e chains and interact with them anyway. Thus, it is found that in reality, short-chain dispersants are often rather efficacious and need not be tuned to exactly match the polymer they are used in. The length of the chain can be tuned but in reality, stearate groups are so inexpensive that other types cannot compete in any but the most specialized applications. Similarly, in theory, the best dispersant tail is one with a polarity matched closely to that of the matrix polymer. However, in practice, hydrocarbon tail types predominate as they are available and at low cost. 

These data indicate that when the concentration of a stable sol is increased, the particles pack more closely together, but remain randomly distributed. Similar behavior is observed in sols that are stabilized by a steric barrier. Van Helden and Vrij [45] stabilized monodisperse silica particles (radius -17 nm) in cyclohexane or chloroform by chemisorbing a layer of stearyl alcohol on the surface. For volume fractions up to 0.4, the compressibility, light scattering [45], and SANS [46] from the sols were in close accord with that expected from a suspension of hard spheres, and the particle size obtained by fitting the theory to the data was in agreement with that seen... 

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