Particle geometries

To offer better value, fillers are usually coated and surface modified, and special manufacturing processes are used to control the size and geometry of the particles. Researchers are now going down to the microscopic - and even to nanoscopic - scales to modify the surface of the material and improve the interface bond. The latest work on nano-sized particles show that these sub-microscopic particles can give, at 5% addition, the sort of mechanical reinforcement that needs around 40% of a conventional filler such as talc. 

Small particle geometry generally does not improve mechanical characteristics, except for providing more rigidity. Lamina or fibre structures with larger particle geometry normally give stabilization with improved mechanical characteristics. 

But smaller particles may well also give increased stabilization by the increased cohesion between filler surface and polymer chain. 

Fillers can be surface treated to improve adhesion and improve mechanical properties. The process can also improve moisture resistance, reduce surface energy and melt viscosity, improve dispersion and processing characteristics, reduce the need for stabilizers and lubricants, and improve the end-product surface. 

Special silane coupling agents that produce a chemical reaction with the polymer may improve stiffness and/or toughness considerably, but they tend to be expensive, and other routes are worth investigating. 

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Regardless of the particle geometry, the excluded volume exceeds the actual volume of the molecules by a factor which depends on the shape of the particles. 

The most important stmctural variables are again polymer composition, density, and ceU size and shape. Stmctural foams have relatively high densities (typically >300 kg/m ) and ceU stmctures similar to those in Figure 2d which are primarily comprised of holes in contrast to a pentagonal dodecahedron type of ceU stmcture in low density plastic foams. Since stmctural foams are generally not uniform in ceU stmcture, they exhibit considerable variation in properties with particle geometry (103). 

Particle geometry largely Particle geometry strongly... 

The analysis of fluid-solid reactions is easier when the particle geometry is independent of the extent of reaction. Table 11.6 lists some situations where this assumption is reasonable. However, even when the reaction geometry is fixed, moving boundary problems and sharp reaction fronts are the general rule for fluid-solid reactions. The next few examples explore this point. 

It is of interest to determine the extent to which there is flow through the interior holes of the particles, as the reaction activity is proportional to geometric surface area under these conditions. So, it is important to know whether the extra surface area provided by the holes is accessible to the flow. It is not easy to see this internal flow from the path lines in Fig. 25, although there appears to be flow through the center particle. To determine this more clearly we constructed a surface that passed through the midpoint of the center particle, perpendicular to its axis, for each of the particle geometries. This is shown as the dark square in Fig. 26, which illustrates the results for the 4-hole particle. 

The conclusions about asymptotic values of tj summarized in Tables 8.2 and 8.3, and the behavior of tj in relation to Figure 8.11, require a generalization of the definition of the Thiele modulus. The result for " in equation 8.520 is generalized with respect to particle geometry through Le, but is restricted to power-law kinetics. However, since... 

Two approaches are common in modelling the SSP process. For the first approach, an overall reaction rate is used which describes the polycondensation rate in terms of the increase of intrinsic viscosity with time. Depending on the size and shape of the granules, the reaction temperature, the pressure, and the amount and type of co-monomers, the overall polycondensation rate lies between 0.01 and 0.03 dL/g/h. The reaction rate has to be determined experimentally and can be used for reactor scale-up, but cannot be extrapolated to differing particle geometry and reaction conditions. 

First-order kinetics is easier to apply to transport and degradation models because they do not require knowledge about particle geometry. 

For within-particle AT the simple analysis by Prater (1958) for any particle geometry and kinetics gives the desired expression. Since the temperature and concentration within the particle are represented by the same form of differential equation (Laplace equation) Prater showed that the T and distributions must have the same shape thus at any point in the pellet x... 

The factors that control separation and dispersion are quite different. The relative separation of two solutes is solely dependent on the nature and magnitude of the Interactions between each solute and the two phases. Thus, the relative movement of each solute band would appear to be Independent of column dimensions or particle geometry and be determined only by the choice of the stationary phase and the mobile phase. However, there is a caveat to this statement. It assumes that any exclusion properties of the stationary phase are not included in the term particle geometry. The pore size of the packing material can control retention directly and exclusively, as in exclusion chromatography or, indirectly, by controlling the access of the solute to the stationary phase in normal and reverse phase chromatography. As all stationary phases based on silica gel exhibit some exclusion properties, the ideal situation where the selective retention of two solutes is solely controlled by phase interactions is rarely met in practice. If the molecular size of the solutes differ, then the exclusion properties of the silica gel will always play some part in solute retention. 

Table 8 Summary of the different models for the gas permeation Model Ref Filler Particle geometry Formula...

It is important to remember that the radius of gyration provides an unambiguous measure of a particle s extension in space. This quantity is evaluated from experimental data—as illustrated in Example 5.3 —with no assumptions as to particle geometry. If we happen to know the shape of a particle as well, Rg can be translated into a geometrical dimension of the particle. We examine this further in the next section. 

Throughout most of this chapter the emphasis has been on the evaluation of zeta potentials from electrokinetic measurements. This emphasis is entirely fitting in view of the important role played by the potential in the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory of colloidal stability. From a theoretical point of view, a fairly complete picture of the stability of dilute dispersions can be built up from a knowledge of potential, electrolyte content, Hamaker constants, and particle geometry, as we discuss in Chapter 13. From this perspective the fundamental importance of the f potential is evident. Below we present a brief list of some of the applications of electrokinetic measurements. 

To evaluate fission product release in a reactor, it is necessary to supply the appropriate particle geometry, diffusion coefficients, and distribution coefficients. This is a formidable task. To approach this problem, postirradiation fission product release has been studied as a function of temperature. The results of these studies are complex and require considerable interpretation. The SLIDER code without a source term has proved to be of considerable value in this interpretation. Parametric studies have been made of the integrated release of fission products, initially wholly in the fueled region, as a function of the diffusion coefficients and the distribution coefficients. These studies have led to observations of critical features in describing integrated fission product releases. From experimental values associated with these critical features, it is possible to evaluate at least partially diffusion coefficients and distribution coefficients. These experimental values may then be put back into SLIDER with appropriate birth and decay rates to evaluate inreactor particle fission product releases. Figure 11 is a representation of SLIDER simulation of a simplified postirradiation fission product release experiment. Calculations have been made with the following pertinent input data ... 

This correlation gives, for perfectly wettable solids, fairly good estimates of the static holdup for different particle-geometries and sizes. Saez and Carbonnel [26] used the hydraulic diameter, instead of the nominal particle diameter, as the characteristic length in the Eotvos number, to include the influence of the particle geometry on the static hold-up. However, no improvement could be obtained in correlating the data with this new representation. 

Other factors which may also be responsible to some extent for the small coefficient obtained for the coconut carbon include the approximation of spherical particle geometry, the assumption of an isotropic medium, and the assumption of a radial diffusion path (13). 

For spherical particle geometry, as in the case of a microbial floe, a pellet of mould or a bead of gel-entrapped enzyme, the expression for the effectiveness factor can again be derived by a procedure similar to that used in Chapter 3 for a spherical pellet of conventional catalyst. A material balance for the substrate across an elementary shell of radius r and thickness dr within the pellet will yield ... 

Several investigators have used radioactive tracer methods to determine diffusion rates. Bangham et al. (32) and Papahadjopoulos and Watkins (33) studied transport rates of radioactive Na+, K+, and Cl" from small particles or vesicles of lamellar liquid crystal to an aqueous solution in which the particles were dispersed. Liquid crystalline phases of several different phospholipids and phospholipid mixtures were used. Because of uncertainties regarding particle geometry and size distribution, diffusion coefficients could not be calculated. Information was obtained, however, showing that the transport rates of K+ and Cl" in a given liquid crystal could differ by as much as a factor of 100. Moreover, relative transport rates of K+ and Cl" were quite different for different phospholipids. The authors considered that ions had to diffuse across platelike micelles to reach the aqueous phase. 

Structure Ca10(PO4)e(OH)2 crystalline, nonstoichiometric mineral rich in surface ions (primarily carbonate) Analytical Properties Separation of proteins overcomes some difficulties associated with ion exchange selectivity and efficiency depend to some extent on particle geometry (i.e., sphere, plate, etc.)... 

Modern electron microscopes are very well capable of imaging individual particles, but of course it is impossible to do so, even for a representative fraction of particles in a supported catalyst. Carlsson et al. [18] described an interesting method to obtain the particle geometry distribution, such that the fraction of edge and corner sites in a supported catalyst can be estimated. An assumption must be made on the shape of the particles, for which these authors used the truncated octahedron, and were able to demonstrate the procedure for gold particles on three different supports. 

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