Polydispersity, of particle size

Equations (16) and (18) discriminate between intraparticle and interparticle interference effects embodied in bj(q. t) and exp rq- ry(/)—r/(/) ), respectively. The amplitude function bj(q.t) contains information on the internal structure, shape, orientation, and composition of individual particles. Variations of bj(q.t) across the particle population reflect the polydispersity of particle size, shape, orientation, and composition. The phase function expjrq (ry (r) — r/(/)]( carries information on the random motion of individual particles, the collective motion of many particles, and the equilibrium arrangement of particles in the suspension medium. 

The polydispersity of particle size increases 0m. for example, of randomly packed uniform spheres from 0.62 to about 0.9. Thus, while for small and large monodispersed spheres 0m = 0.62, for the mixtures, depending on composition and size ratio, 0m 1-0. For polydispersed spheres of diameter di and average diameter d, Pishvaei et al. [49] used the model of Ouchiyama and Tanaka [50-55] to calculate the maximum packing volume fraction, 0m, and the average number of spherical particles, n ... 

Particulate systems composed of identical particles are extremely rare. It is therefore usefiil to represent a polydispersion of particles as sets of successive size intervals, containing information on the number of particle, length, surface area, or mass. The entire size range, which can span up to several orders of magnitude, can be covered with a relatively small number of intervals. This data set is usually tabulated and transformed into a graphical representation. 

Compai ison with literature experimental and calculation data showed that the model proposed ensures the accurate behavior of the functional dependence of x-ray fluorescence intensity on the particle size. Its main advantage is the possibility to estimate the effect of particle size for polydispersive multicomponent substances. 

As a rule, the dispersed catalysts are polydisperse (i.e., contain crystallites and/or crystalline aggregates of different sizes and shapes). For particles of irregular shape, the concept of (linear) size is indehnite. For such a particle, the diameter d of a sphere of the same volume or number of metal atoms may serve as a measure of particle size. 

As previously discussed, we expect the scaling to hold if the polydisper-sity, P, remains constant with respect to time. For the well-mixed system the polydispersity reaches about 2 when the average cluster size is approximately 10 particles, and statistically fluctuates about 2 until the mean field approximation and the scaling break down, when the number of clusters remaining in the system is about 100 or so. The polydispersity of the size distribution in the poorly mixed system never reaches a steady value. The ratio which is constant if the scaling holds and mass is conserved,... 

Several mathematical models are available for predicting the dissolution of particles of mixed size. Some are more complex than others and require lengthy calculations. The size of polydisperse drug particles can be represented with a distribution function. During the milling of solids, the distribution of particle sizes most often results in a log-normal distribution. A log-normal distribution is positively skewed such that there can exist a significant tail on the distribution, hence a number of large particles. The basic equation commonly used to describe the particle distribution is the log-normal function,... 

Order and polydispersity are key parameters that characterize many self-assembled systems. However, accurate measurement of particle sizes in concentrated solution-phase systems, and determination of crystallinity for thin-film systems, remain problematic. While inverse methods such as scattering and diffraction provide measures of these properties, often the physical information derived from such data is ambiguous and model dependent. Hence development of improved theory and data analysis methods for extracting real-space information from inverse methods is a priority. 

Heterodisperse Suspensions. The rate laws given above apply to monodisperse colloids. In polydisperse systems the particle size and the distribution of particle sizes have pronounced effects on the kinetics of agglomeration (O Melia, 1978). For the various transport mechanisms (Brownian diffusion, fluid shear, and differential settling), the rates at which particles come into contact are given in Table 7.2. 

The results were obtained for the polydispersed mixtures possessing the following characteristic properties of particle size distribution function (Figs. 14.1-14.2) ... 

A monodisperse aerosol is one with a narrow size distribution, which, for log-normal-distributed particles, usually means a geometric standard deviation of about 1.2 or smaller. Monodisperse particles are expected to have simple shapes and uniform composition with respect to size. A polydisperse aerosol, on the other hand, is one containing a wide range of particle sizes, but which may otherwise be homogeneous in terms of the basic physical and chemical properties that are not related to size. The term heterodisperse is also used occasionally this describes aerosols varying widely in physical and chemical characteristics, as well as size. 

Figure 1.8 shows an electron micrograph of latex particles made from polystyrene cross-linked with divinylbenzene. Note that these latex particles are not the same as simple polystyrene molecules in a true solution. The particles shown in the figure display a remarkable degree of homogeneity with respect to particle size. Such a sample is said to be monodisperse (in size), in contrast to polydisperse systems, which contain a variety of particle sizes. We have a good... 

This discussion of aggregates leads us to another important characteristic of dispersions we have not yet considered in sufficient detail polydispersity. Monodisperse systems are the exception rather than the rule. Even in those rare cases in which a monodisperse system exists, any aggregation that occurs will result in a distribution of particle sizes because of the random nature of the aggregation process. 

Effect of Particle Size on Rate of Dissolution Dissolution of Polydisperse Systems 474... 

Figure 16 may also be used to illustrate the problem of polydispersity. For simplicity, consider a small polydispersity in particle size at constant refractive index. The theoretical surface shown in Figure 16 must be modified with a running integration over the size distribution which tends to fill in the minima and round off the maxima so that some of the structure is averaged out. Graphical illustrations of the effect have been reported (9) and the effect has been exploited to obtain both particle size and distribution width (10). It should be realized that both very narrow or very broad distributions tend... 

Only a minority of systems of industrial interest contain powders with a uniform particle size, i.e. monodisperse. Most systems generally show a distribution of sizes (polydisperse) and it is then necessary to define the average dimension. There are many different definitions of particle size,1 2 the most commonly used, particularly in fluidisation, is the so called volume-surface mean or the Sauter mean diameter. This is the... 

Averages of particle size, polydisperse particulate 3y3tems, 161-178... 

The mathematical models of the reacting polydispersed particles usually have stiff ordinary differential equations. Stiffness arises from the effect of particle sizes on the thermal transients of the particles and from the strong temperature dependence of the reactions like combustion and devolatilization. The computation time for the numerical solution using commercially available stiff ODE solvers may take excessive time for some systems. A model that uses K discrete size cuts and N gas-solid reactions will have K(N + 1) differential equations. As an alternative to the numerical solution of these equations an iterative finite difference method was developed and tested on the pyrolysis model of polydispersed coal particles in a transport reactor. The resulting 160 differential equations were solved in less than 30 seconds on a CDC Cyber 73. This is compared to more than 10 hours on the same machine using a commercially available stiff solver which is based on Gear s method. 

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