Bidisperse systems

Several authors have sought to provide equations for particles varying in shape and density (see Patwardhan and ChiTien, 1985 Davis and Gecol, 1994). Whilst these have been successfully tested on bidisperse systems, more work... 

FIG. 6 A bidisperse system consisting of large and small particles. 

Wakao and Smith [20] originally developed the random pore model to account for the behaviour of bidisperse systems which contain both micro- and macro-pores. Many industrial catalysts, for example, when prepared in pellet form, contain not only the smaller intraparticle pores, but also larger pores consisting of the voids between compressed particles. Transport within the pellet is assumed to occur through void regions... 

The parameters D and Dk > whether for macro (denoted by subscript m) or for micro (denoted by subscript ju) regions, are normal bulk and Knudsen diffusion coefficients, respectively, and can be estimated from kinetic theory, provided the mean radii of the diffusion channels are known. Mean radii, of course, are obtainable from pore volume and surface area measurements, as pointed out in Sect. 3.1. For a bidisperse system, two peaks (corresponding to macro and micro) would be expected in a differential pore size distribution curve and this therefore provides the necessary information. Macro and micro voidages can also be determined experimentally. 

Two major entry models - the diffusion-controlled and propagation-controlled models - are widely used at present. However, Liotta et al. [28] claim that the collision entry is more probable. They developed a dynamic competitive growth model to understand the particle growth process and used it to simulate the growth of two monodisperse polystyrene populations (bidisperse system) at 50 °C. Validation of the model with on-line density and on-line particle diameter measurements demonstrated that radical entry into polymer particles is more likely to occur by a collision mechanism than by either a propagation or diffusion mechanism. 

We will now look at the integer moment-transport equations for two systems. The first corresponds to a monodisperse case with kinetic equation given by Eq. (6.1). The second is a bidisperse system with kinetic equations... 

For a bidisperse system the moment-transport equations derived from Eq. (6.129) are... 

In the past, similar bidispersed systems have been investigated and modeled in the catalyst deactivation area (5-7). However, modeling of immobilized affinity adsorbent beads is more complex due to the non-linearity introduced by the rapid ligand binding reaction which is dependent on the product concentration. 

Experimental data (C4, S2, S6) show that t varies from about 1.5 to over 10. A reasonable range for many commercial porous solids is about 2-6 (S2). If the porous solid consists of a bidispersed system of micropores and macropores instead of a monodispersed pore system, the. approach above should be modified (C4, S6). 

What has gone wrong in the analysis Why doesn t Eq. 4.21 work for the hypothetical mixture of polystyrene spheres in water The answer is that particle motions in a bidisperse system are not a Markoff process, because each particle has a memory. Each particle remembers its own size and diffusion coefficient. In fact, (, t)... 

Videomicroscopy offers possibilities that other techniques do not, such as the ability to visualize three-dimensional clusters in which cluster membership is determined by the particle dynamic properties, e.g., whether the particles are fast- or slow-moving(33). Videomicroscopy also allows measurement of quantities not directly accessible from other techniques, such as the van Hove self- and distinct distribution functions (x, t) and Gj(x, t) of the particle displacements. The Gs(x, t) gives the likelihood that a particle will have a displacements during a time t Gd(x, t) gives the likelihood that if one particle is at the origin at time 0 then a different particle will be at s at time t. A dilute bidisperse system in which particles of two different sizes all perform independent Brownian motion would have d.Gs(x, t) that was a sum of two Gaussians, a field correlation function t)... 

Figure 3.5a shows the collision frequency / (in units of T/ Tnaj)) versus packing fraction ( ) for the LS method for a 2D bidisperse system with N = 6 disks. The collision frequency diverges as / (( )y -( )) near a MS packing with 4>j [30] the compressions are stopped when the collision frequency exceeds a large value, f -10 ... 

Figure 3.6 Two-dimensional projection of the paths in configuration space for the LS (circles) and MD (squares) packing-generation methods with two different initial conditions for 2D bidisperse systems with N = 6. For these initial conditions, both methods generate the same MS packing. For the calculation of Vp = i) + iVi -yi)y]>...

Figure 3.7 Number of times out of = 10 trials that each of the 20 distinct MS packings at dpj is obtained using the LS (squares) and MD (circles) packing-generation methods for 2D bidisperse systems with N = 6.

Figure 6.5 The characteristic deviation in packing fraction (Acj)), = < )c - ())j at which the adjacency matrix deviates from that at (pj for each of the 20 MS packings at jamming onset for N = 6 bidisperse systems using small successive compressions 10 (circles) and 10 (crosses). The dashed line indicates the average ((A( ))c) = 0.028 over the 20 MS packings, (b) The average ((Acf)) ) over 100 2D bidisperse MS packings as a function of N. The solid line has slope -1.9.

Because of the nonlinear (with respect to the applied field) nature of the electro-orientation effects, with the added difficulty of the nonspherical shape of the LPs, a quantitative explanation of the phenomenon is still lacking. However, these results have stimulated the investigation of electro-kinetic phenomena in bidisperse systems for cases where the quantitative evaluation and comparison with models are easier because of the sphericity and monodispersity of both the LPs and the SPs. 

Figure 6 Impact of the relative amounts of long and short rods on the percolation threshold of a bidisperse system of rods of equal diameter but different lengths /.short and /. ng. The normalized percolation threshold is plotted as a function of the fraction of long rods, x, for length ratios (/-long/Z-short) = 2, 4, 6, and 8, respectively, from top to bottom. Reprinted with permission from Kyrylyuk, A. V. van der Schoot, P. Proc. of the Natl. Acad. Sci. USA 2008, 105, 8221-8226. Copyright 2008 National Academy of Sciences, U.S.A.

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