Particle Packing Theory

The packing behaviour of particulate materials depends largely on their particle size, shape and surface characteristics. The behaviour of model systems with closely defined size and shape distributions is now well understood. Real particulate materials are much harder to treat, largely due to the difficulty in determining and describing their size and shape distributions accurately. Nevertheless, the principles derived for the model systems can be applied in a semi-quantitative way and appear to work reasonably well. 

The simplest case to consider first is the packing of smooth, regular, mono-sized particles. This has been well studied for smooth spherical particles, which can readily move past one another. With these particles a maximum ordered packing fraction of 0.74 has been established, although other, less dense packings are feasible. Random loose packing fraction is difficult to predict accurately but is about 0.60. Random dense packing is more readily predicted and is about 0.64. 

The case of spherical particles provides the simplest example. As we have already seen, the maximum ordered packing fraction is 0.74. From geometric principles it can be shown that this structure has two sizes of pores present, which can be filled by spheres with 22.5 % and 41.4% of the diameter of the largest spheres. This results in an increase 

Unfortunately, such simple geometric calculations are not possible for the more relevant random packing situation, even when spherical particles are involved. This is because there will be a random distribution of pore sizes and the small particles will not spread out evenly among them. In particular, the small particles show a tendency to cluster around the 

Small particles fitting into the gaps between large particles 

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Although binder levels increase as particle size is reduced, and they are greatest in aH-flour mixes where surface area is very high, the principle of minimum binder level stiU appHes. The appHcation of particle packing theory to achieve minimum binder level in all-flour mixes is somewhat more complex because of the continuous gradation in sizes encountered (4). 

German R. Fundamentals of particle packing theory. Proceedings of the 30th Annual St. Louis Section of the American Ceramic Society Symposium on Refractories, St. Louis, MO, 1994. 

CEC is a miniaturized separation technique that combines capabilities of both interactive chromatography and CE. In Chapter 17, the theory of CEC and the factors affecting separation, such as the stationary phase and mobile phase, are discussed. The chapter focuses on the preparation of various types of columns used in CEC and describes the progress made in the development of open-tubular, particle-packed, and monolithic columns. The detection techniques in CEC, such as traditional UV detection and improvements made by coupling with more sensitive detectors like mass spectrometry (MS), are also described. Furthermore, some of the applications of CEC in the analysis of pharmaceuticals and biotechnology products are provided. 

Given the diversity of relevant applications, it is not surprising that the characterization of voids in disordered systems has an appreciable history, which can be traced back to primitive hole theories of the liquid state (Frenkel, 1955 Ono and Kondo, 1960). While the early theories offer an admittedly rudimentary lattice description of voids, recent computational advances permit an exact (and highly efficient) characterization of the continuum void geometry present in particle packings in two (Rintoul and Torquato, 1995) and three dimensions (Sastry et al., 1997a). 

Pore structure for unconsolidated media is inferred from a particle size distribution, the geometry of the particles and flie packii arrangement of particles. The theory of packing is well established for symmetrical geometries such as spheres. Information on particle size, geometry and the theory of packii allows relationships between pore size distributions and particle size distributions to be established. 

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... 

Current theories do not consider the effect of particle packing structure (i.e. bulk density) on sintering behaviour. Particle packing is a major process variable. One example concerning the inconsistency of theory and reality concerns the theoretical result that densification kinetics should increase with decreasing particle size. Common experience indicates that powders with a very small crystallite size (e.g. <0.1 pm) can be very difficult to pack and densify. It is now commonly accepted that strong. 

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