Particles crowding effect

Particle Crowding" Effect. The growth of a crystal in a suspension can be influenced by the presence of other crystals when the crystal number concentration (number/volume) exceeds a certain level so that the interparticle distance becomes smaller than 20 particle diameters. For most precipitation systems, the crystal number concentration is typically higher than lO" per cm, and the interparticle distances are normally less than 15 times the crystal size. In such a crowded system, the diffusion fields around the individual crystals begin to influence one another and one may expect some interparticle effects on crystal growth. 

We have already indicated that the coefficient kx in Equation (42) has been calculated for spheres by various theoretical models. While this coefficient is a measure of concentration effects, we do not pursue its derivation. Instead, we qualitatively examine the effect of particle crowding as the origin of the positive deviations from the Einstein theory that inevitably set in at higher concentrations, as seen in Figures 4.10 and 4.11. 

FIG. 4.11 The effect of particle crowding on viscosity. The solid line is drawn according to Equation (48) the points are experimental results. (Data from R. Roscoe, Br. J. Appl. Phys., 3, 267(1952).)... 

Figure 15.3 indicates that for the silica microspheres, the potential and viscosity both follow the expected behavior predicted by the classical DLVO theory. On the other hand, the nanosize fnmed silica exhibits a discrepancy between the expectation of DLVO theory and the experimental results that is, as the of the nanosize fumed silica increases, viscosity sharply increases. Hence factors such as particle crowding, particle ordering, and electroviscous effects will also impact viscosity, in addition to aggregate or network formation. 

Precoat slurry concentration will depend primarily on the ratio of filter area to filter and piping volume. If it is much below 0.3%, precoating may be difficult because the formation of the bridge depends partly on the crowding effect of the particles of Celite trying to get through the septum openings. 

In 1974, Tunstall and Hird (87) published a paper entitled "Effect of Particle Crowding on Scattering Power of Titanium Pigments" in which they agreed with Stieg s 1959 interpretation of... 

This theory was first suggested by Fahlstrom who proposed that the cut size is primarily a function of the capacity of the underflow orifice and of the particle size analysis of the feed. He argued that the crowding effect, or hindered discharge through the apex, can swamp the primary interaction to the extent that the cut size can be estimated from the mass recovery to the underflow. 

For a typical ACA with a volume fraction of particles ranging from 3 to 15 vol%, the open circuit probabihty on a 100 pm pad varies from 10 33 to 10 3, which is extremely small. However, in reahty, there is always a crowding effect that must be taken into account. In this case, the particle distribution can be described using a binominal distribution model ... 

Small particle size extenders are the other class of opacifying aids acting as spacers, they increase the average distance between Ti02 particles in a fihn, thereby reducing crowding effects. The primary particle size of these specialized extenders is quite small (typically <0.5 pm), and the colloidal components of a coating need to be properly formulated and stabihzed in order for these materials to work effectively. 

Notice that in Table 5.2 all the deviations are negative the observed molar volume is less than that predicted by the ideal gas law. This effect can be attributed to attractive forces between gas particles. These forces tend to pull the particles toward one another reducing the space between them. As a result, the particles are crowded into a smaller volume, just as if an additional external pressure were applied. The observed molar volume, Vm, becomes less than V , and the deviation from ideality is negative ... 

We focus on the effects of crowding on small molecule reactive dynamics and consider again the irreversible catalytic reaction A + C B + C asin the previous subsection, except now a volume fraction < )0 of the total volume is occupied by obstacles (see Fig. 20). The A and B particles diffuse in this crowded environment before encountering the catalytic sphere where reaction takes place. Crowding influences both the diffusion and reaction dynamics, leading to nontrivial volume fraction dependence of the rate coefficient fy (4>) for a single catalytic sphere. This dependence is shown in Fig. 21a. The rate constant has the form discussed earlier,... 

Barnea and Mizrahi (1973) considered the effects of the modified density and viscosity of the suspending fluid, as represented by Eq. (14-21), as well as a crowding or hindrance effect that decreases the effective space around the particles and increases the drag. This additional crowding factor is 1 + k2(pl/3, which, when included in Eq. (14-21), gives... 

This effective medium or mean field assumption is easy to understand if there is a very large size difference between the newly added particles and any there previously, for example if we think of adding particles to a molecular liquid, we merely treat the liquid as a structureless continuum. However when the dimensions become comparable, the finite volume of the particles present prior to each addition must be considered, i.e. new particles can only replace medium and not particles. The consequence of this crowding is that the concentration change is greater than expressed in Equation (3.52) and it must be corrected to the volume available ... 

The mean-field theory has a number of shortcomings, including the approximations of a mean concentration around all particles and the establishment of spherically symmetric diffusion fields around every particle, similar to those that would exist around a single particle in a large medium. The larger the particles total volume fraction and the more closely they are crowded, the less realistic these approximations are. No account is taken in the classical model of such volume-fraction effects. Ratke and Voorhees provide a review of this topic and discuss extensions to the classical coarsening theory [8]. 

The remedy is not to attempt the reduction of chemistry to the one-particle solutions of quantum physics, without taking the emergent properties of chemical systems into account. Chemical reactions occur in crowded environments where the presence of matter in molar quantities is not without effect on the behaviour of the quantum objects that mediate the interactions. It is only against this background that quantum theory can begin to make a useful contribution to the understanding of chemical systems. 

One physical consequence of the excluded volume effect is that solutions crowded with macromolecules tend to be highly nonideal. In an ideal solution, a particle has access to any region of the solution. This is clearly not true for the intracellular milieu (figure 6.22). As a result of excluded volume effects, the chemical activities of both large and small solutes are considerably higher than their actual concentrations (this is another aspect of the nonideality of the intracellular milieu). The nonideal characteristics of the intracellular solution favor a minimization of excluded volume, that is, they... 

Similar results to those obtained here by the stability measurements have been reported by Roe and Brass (7.8) They studied polystyrene latex stabilized by potassium palmitate. The analysis supplied by these authors shows that the order of magnitude of the slope of the stability curves can be accounted for as an entropic effect of crowding of adsorbed molecules during an encounter between two particles. They pointed this out as a possible explanation as the amount of emulsifier adsorbed strongly affects the stability without altering the electrophoreti-cally derived double-layer potential. 

Diffusion within cells is even more complex than within simple gels first the cytoplasm is a molecularly crowded zone, a complex gel with structural obstacles such as actin and myosin fibers and strands. There is the additional tortuosity that occurs in gels as the moving particle avoids the regions of the macromolecular chains and the obstruction effects from the impenetrable regions of the cytoplasm. If we designate as the effective... 

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