Polydispersity of particles

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. 

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. 

First, there is the problem of polydispersity of particles and aggregates. It is one thing to attempt to deduce the structure of a set of objects that are all similar in density... 

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 ... 

Fig. 15. Average size and polydispersity of particles as a function of the center of mass of the cluster in units of the average diameter of particles in the bulk liquid. Here the average size is scaled to the diameter of the particles in the liquid a) / < Hq)- The polydispersity is defined...

Rowell and co-workers [62-64] have developed an electrophoretic fingerprint to uniquely characterize the properties of charged colloidal particles. They present contour diagrams of the electrophoretic mobility as a function of the suspension pH and specific conductance, pX. These fingerprints illustrate anomalies and specific characteristics of the charged colloidal surface. A more sophisticated electroacoustic measurement provides the particle size distribution and potential in a polydisperse suspension. Not limited to dilute suspensions, in this experiment, one characterizes the sonic waves generated by the motion of particles in an alternating electric field. O Brien and co-workers have an excellent review of this technique [65]. 

Figure C2.17.4. Transmission electron micrograph of a field of Zr02 (tetragonal) nanocrystals. Lower-resolution electron microscopy is useful for characterizing tire size distribution of a collection of nanocrystals. This image is an example of a typical particle field used for sizing puriDoses. Here, tire nanocrystalline zirconia has an average diameter of 3.6 nm witli a polydispersity of only 5% 1801.

Although most cakes consist of polydisperse, nonspherical particle systems theoretically capable of producing more closely packed deposits, the practical cakes usually have large voids and are more loosely packed due to the lack of sufficient particle relaxation time available at the time of cake deposition hence the above-derived value of 17.6 pm becomes nearer the 10 pm limit when air pressure dewatering becomes necessary. 

Issues to be considered in selecting the best stabilizing system are polymeric chain branching which increases with high temperature and the presence of some stabilizers, polydispersity of the particles produced, and grafting copolymerization, which may occur because of the reaction of vinyl acetate with emulsifiers such as poly(vinyl alcohol) (43,44). 

Size Recovery and Yield Centiifuges have been apphed to classify polydispersed fine particles. The size distribution of the paiticles is quantified by the cumulative weight fraction F less than a given particle size d for both the feed and the centrate streams. It is measured by a particle size counter which operates based on piinciples such as sedimentation or optical scatteiing. 

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. 

In order to develop integral equations for the correlation functions, we consider the system composed of N polydisperse spheres. The average density of particles with diameter <7, is given by... 

FIG. 2 A polydisperse suspension consists of particles of different sizes. 

The essence of this idea is that there is a limit to which particles of like-size can occupy a given space, even when arranged in closely packed arrays (e.g., cubic or tetrahedral arrays). The voids that are left are usually smaller than the parent particles and may be filled by particles of smaller size to increase the concentrations of particles in space. Thus, polydispersity can give a lower viscosity at the same volume fraction or permit higher volume loading at the equivalent monodisperse viscosity. 

However, in subsequent studies [23-25,88-90] it was demonstrated that in reality the particle deposition is not a purely geometric effect, and the maximum surface coverage depends on several parameters, such as transport of particles to the surface, external forces, particle-surface and particle-particle interactions such as repulsive electrostatic forces [25], polydispersity of the particles [89], and ionic strength of the colloidal solution [23,88,90]. Using different kinds of particles and substrates, values of the maximum surface coverage varied by as much as a factor of 10 between the different studies. 

For instance, nanoparticles of silver chloride have been synthesized by mixing two mi-croemnlsions, one containing silver ions and the other containing chloride ions. It was shown that the average particle size, the polydispersity and the number of particles formed depend on the intermicellar exchange rate and/or the rigidity of the surfactant shell [228],... 

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. 

Templates made of surfactants are very effective in order to control the size, shape, and polydispersity of nanosized metal particles. Surfactant micelles may enclose metal ions to form amphiphilic microreactors (Figure 11a). Water-in-oil reverse micelles (Figure 11b) or larger vesicles may function in similar ways. On the addition of reducing agents such as hydrazine nanosized metal particles are formed. The size and the shape of the products are pre-imprinted by the constrained environment in which they are grown. 

Silver nanoparticles can be deposited on Ti02 by UV-irradiation. Deposition of polydisperse silver particles is a key to multicolor photochromism. The nanoparticles with different size have different resonant wavelength. Upon irradiation with a monochromatic visible light, only the resonant particle is excited and photoelectrochemically dissolved, giving rise to a decrease in the extinction at around the excitation wavelength. This spectral change is the essence of the multicolor photochromism. The present photoelectrochemical deposition/dissolution processes can be applied to reversible control of the 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,... 

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