Polydispersed particles particle-size distributions

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]. 

Particle size distribution Polydisperse, requires refinement to give narrower fraction before use in column packing Monodisperse as produced in the reactor... 

In order to cadculate a particle size distribution directly from the output chromatogram for a polydisperse system, the integral, dispersion equation for the chromatogram signal, F(V), as a function of elution volume, V, needs to be evaluated (27) ... 

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,... 

Leblanc and Fogler developed a population balance model for the dissolution of polydisperse solids that included both reaction controlled and diffusion-controlled dissolution. This model allows for the handling of continuous particle size distributions. The following population balance was used to develop this model. 

Equation (28) is valid only for monodisperse drugs. For polydisperse drugs, the overall fraction of dose absorbed may be estimated on the basis of the particle size distribution [44],... 

The distribution of molecular weights of each generation was determined from measurements on about 50 molecules, with results shown in Figure 12.19 (the weight fraction is the percent dendrimer in each interval of molecular weight under consideration). Based on these distributions, the polydispersity index (.MJMa) of G5 to G10 can be calculated, with results shown in Table 12.1 [39], They are all less than 1.08, which means that the particle size distribution is very uniform for each generation. 

In the case of a polydisperse system the calculation of the particle size distribution is possible by using special transformation algorithms. For this purpose certain requirements need to be fulfilled, such as a spherical particle shape, sufficient dilution, and a large difference between the refractive indices of the inner and the outer phases. Since usually not all requirements can be fulfilled, the z-average is preferred as a directly accessible parameter rather than the distribution fimction depending on models. 

In practice, we have a number of solid fuels, for example biofuels (forest or agricultural derived biofuels), coal, municipal solid waste (MSW) and many others [23]. A fuel bed is composed of varying sizes of solid-fuel particles, also called polydispersed solid-.fuels [15]. The fuel chemistry is different depending on whether it is coal, biofuel or MSW. The fuel bed can be dry or consist of moisture. The fuel physics are for example, particle size distribution, particle shape, particle density and bed permeability. 

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

Solid particles are not likely to be uniform spheres, even if the sample is carefully fractionated rather, they will be irregularly shaped and polydisperse, although the particle size distribution may be narrow. The smallest particles will have the largest effect on the solubility, but they may be the hardest to measure. 

Colloidal systems are generally of a polydispersed nature - i.e. the molecules or particles in a particular sample vary in size. By virtue of their stepwise build-up, colloidal particle and polymer molecular sizes tend to have skew distributions, as illustrated in Figure 1.2, for which the Poisson distribution often offers a good approximation. Very often, detailed determination of relative molecular mass or particle size distribution is impracticable and less perfect experimental methods, which yield average values, must be accepted. The significance of the word average depends on the relative contributions of the various molecules or particles to the property of the system which is being measured. 

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... 

The minimum fluidisation velocity represents the lower limit of the range of operative conditions at which a fluid-bed process can be operated, while the particle terminal fall velocity represents the upper limit beyond which the particles will start leaving or elutriating from the bed. To avoid or reduce carryover of particles from a fluidised bed, gas velocity has to be kept between um( and ut. For polydisperse systems, in calculating wmr, the mean diameter dvs is used for the particle size distribution present in the bed. In calculating nt, the smallest size of solids present in appreciable quantity in the bed is used. 

Thus, given gparticle size distribution. For narrow size distributions, the autocorrelation function is satisfactorily analyzed by the method of cumulants to give the moments of the particle size distribution.(7) However, the analysis of QELS data for samples with polydisperse or multimodal distributions remains an area of active research.(8)... 

For polydisperse systems, replacement of equation 13 into equation 2 for every particle diameter, yields an approximation to the turbidity in terms of ratios of moments of the particle size distribution without having to make assumptions regarding the shape of the distribution ... 

If all the particles in a sample are of the same size, then the powder is monodis-perse. Truly, the particles have more than one size in polydisperse samples. The monodisperse particle size distribution is more desirable than the polydisperse one. Therefore, the shape and surface area of the individual particles, the size ranges based on number or weight of particles, as well as the total surface area are variable. The commonly illustrated particle forms are sphere, rod, fiber, granular, cubical, flake, condensation Hoc, and aggregate. 

The velocity can also be calculated for polydispersed systems with a particle size distribution function, x(<0 (Aldushin et al., 1976a). For a unimodal particle size distribution, an effective particle size, d s, defined as... 

SFFF can measure gold particles down to 15 nm. Particle size distribution was determined from peak broadening caused by polydispersity of the sample. Peak broadening due to instrument imperfections was also detected. The results were compared with data from SEM and PCS. SEM gave a mean diameter of 20 nm sizing by PC was not possible due to aggregation [86]. 

Weiner et. al. [291] determined the particle size distribution as a function of concentration for a number of colloid suspensions. The results showed the advantage of using single mode fiber optics as a practical tool. A comparison made between this and a previous design was presented and several limiting features summarized. A recent review with 54 references covers basic physics, and experimental methods [292]. Applications of the technique to the determination of mean particle diameter, polydispersity and higher order moments are discussed. 

The size of droplets in a polydisperse emulsion may be expressed by one or two numbers, rather than stipulating the full particle size distribution (9). The most useful numbers are the mean diameter d, which is a measure of the central tendency of the distribution, and the standard deviation, a, which is a measure of the width of the distribution ... 

In a polydisperse system, the calculation of particle size distribution is possible, in addition, by using special transformation algorithms. For this, certain... 

Macromolecular or particulate samples fractionated by the FFF are usually not uniform but exhibit a distribution of the concerned extensive or intensive parameter [8] or, in other words, a polydispersity. Molar mass distribution (MMD), sometimes called molecular weight distribution (MWD), or particle size distribution (PSD) describes the relative proportion of each molar mass (molecular weight), M, or particle size (diameter), d, species composing the sample. This proportion can be expressed as a number of the macromolecules or particles of a given molar mass or diameter, respectively, relative to the number of aU macromolecules or particles in the sample ... 

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