Coarsening of Particle Distributions

Figure 15.6 Particle distributions observed in coarsening experiments on semisolid Pb-Sn alloys. The volume fraction of particles is 0.64. The upper row shows a steady increase in mean particle size with aging time. The lower row is scaled so that the apparent mean particle size is invariant—demonstrating that the particle distribution remains essentially constant during coarsening. From Hardy and Voorhees [7).

To model coarsening, consider a powder compact consisting of a distribution of particles, with an average particle radius r. y. Assuming all particles to be spheres, the average partial pressure over the ensemble is given by Eq. (10.6), or... 

Coarsening via Ostwald ripening is analyzed via a population balance. We define the particle size distribution function as f(R,t), where f(R,t) dR is the number of particles per unit volume at a given time in the particle range R to R+dR in a mean field approach. The differential equation that describes f(R,t) is... 

Fang, Z. and Patterson, B. R., Experimental investigation of particle size distribution influence on diffusion controlled coarsening, Acta MetalL Mater., 41, 2017-24, 1993. 

If the coarsening process is limited by the rate at which atoms diffuse through the matrix from the source particles to the sink particles, it can be shown [13] that the mean particle size of the distribution will increase with time according to... 

Question The following mean particle size-time data were obtained for a distribution of particles embedded in a matrix during a coarsening experiment ... 

In a two-phase system consisting of a distribution of fine particles embedded in a matrix, capillary forces can promote the gradual dissolution of the smallest (most soluble) particles compensated by growth of the largest particles. This phenomenon is known as coarsening and leads to a gradual decrease in the number of particles and an increase in the average particle size over time. 

Aggregation means coarsening of the size distribution. The rate of size changes can, therefore, be considered as a measure of the microscopic stability. In principle, one can use any particle sizing technique for the purpose of stability characterisation— provided that the sample preparation does not affect aggregation and that both preparation and measurement time are well below the characteristic time scale(s) of the aggregation process. For dilute suspensions, one usually employs optical sizing techniques. 

A molecular simulation of a macroscopic biosensor is still beyond our computational means. We have therefore used only the field theory to model its dynamical behavior. We have found that the eflfect of the presence of adsorbed particles at the confining walls is to slow down the coarsening of domains in the middle of the cell. When the concentration of adsorbed particles reaches a critical value, the correlation length seems to be frozen and the system never orders into a single cell-encompassing domain, as it does in the absence of adsorbed particles. Hence, the time dependent response of the sensor encodes additional information regarding the amount (and perhaps also the distribution) of adsorbed particles. 

This asymptotic distribution is independent of the initial distribution at the start of coarsening. In this distribution, the average radius a, taken as the arithmetic mean radius, is given by a = (8/9)a, and the maximum particle radius is 2a. The distribution fimction/is shown in Fig. 9.7. The critical radius in this steady-state coarsening regime increases parabolically according to... 

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