Polydisperse size distribution

An approximate mathematical correction of a measured polydisperse size distribution has been carried out using a deconvolution technique [170]. This was verified with suspension droplets the advantage being that it can measure such distributions the disadvantages being that the correlation between particle size and velocity is lost and at least 5,000 data points are required for deconvolution [171]... 

With a polydisperse size distribution, the total fraction remaining, F, can be determined by summing over the differential size distribution 0 where 0 is the fraction of total particle volume with initial diameters from dj... 

The total fraction remaining, F, calculated from the volume of HC1 added is given as a function of dimensionless time, t/t50, where t50 is the time required to dissolve 50% of the CaC03. Calculated curves are also given using the simplified mass transfer model with a single particle size (monodisperse) and with the actual polydisperse size distribution (Table 2). The polydisperse model fits the shape of the curve very well at all times. The monodisperse model is only satisfactory for t/t50 less than 1. 

Obviously, while Equation 15.88 strictly holds for monodis-perse nanocrystal size distribution, it holds on average for polydisperse size distribution. The combination of Equations 15.86a and 15.88 leads to our working equation... 

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]. 

Even when carefully prepared, model colloids are almost never perfectly monodisperse. The spread in particle sizes, or polydispersity, is usually expressed as the relative widtli of tire size distribution,... 

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.

The diametei of average mass and surface area are quantities that involve the size raised to a power, sometimes referred to as the moment, which is descriptive of the fact that the surface area is proportional to the square of the diameter, and the mass or volume of a particle is proportional to the cube of its diameter. These averages represent means as calculated from the different powers of the diameter and mathematically converted back to units of diameter by taking the root of the moment. It is not unusual for a polydispersed particle population to exhibit a diameter of average mass as being one or two orders of magnitude larger than the arithmetic mean of the diameters. In any size distribution, the relation ia equation 4 always holds. 

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. 

There are many complications with interpreting MWCO data. First, UF membranes have a distribution of pore sizes. In spite of decades of effort to narrow the distribution, most commercial membranes are not notably sharp. What little is known about pore-size distribution in commercial UF membranes fits the Poisson distribution or log-normal distribution. Some pore-size distributions may be polydisperse. 

A further feature of anionic polymerisation is that, under very carefully controlled eonditions, it may be possible to produee a polymer sample which is virtually monodisperse, i.e. the molecules are all of the same size. This is in contrast to free-radical polymerisations which, because of the randomness of both chain initiation and termination, yield polymers with a wide molecular size distribution, i.e. they are said to be polydisperse. In order to produce monodisperse polymers it is necessary that the following requirements be met ... 

Aerosol, polydisperse An aerosol with a geometric standard of deviation of size-distribution greater than 1.5. 

When applied to the SEC column, the calibrated polydisperse polymer solution provides a large number of data points in a single run. Use of a standard with a molecular size distribution that encompasses the full separation range for the column allows the entire separation range to be calibrated in a single run (Fig. 2.4). 

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) ... 

An illustration of the potential for improved resolution in the determination of size distributions with particles in the small size range is indicated in Figures 12 to ik. These results are for a polydisperse polystyrene (labelled 2D2) which has been doped by the addition (29% by mamber) of the Dow 380 X polystyrene standard. 

V. Mishra, S. M. Kresta, J. H. Masliyah 1998, (Self-preservation of the drop size distribution function and variation in the stability ratio for rapid coalescence of a polydisperse emulsion in a simple shear field), J. Colloid Interface Sci. 197, 57. 

Illustration Short-time behavior in well mixed systems. Consider the initial evolution of the size distribution of an aggregation process for small deviations from monodisperse initial conditions. Assume, as well, that the system is well-mixed so that spatial inhomogeneities may be ignored. Of particular interest is the growth rate of the average cluster size and how the polydispersity scales with the average cluster 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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