Monodisperse suspension

A similar technique has been applied to the generation of monodisperse suspensions in water. This type of method was first used in medical field and then widely used to spray monodisperse solid particles such as polystyrene latex particles. Aerosols of solutes have also been produced by atomizing solutions of salt, sugar or methylene blue dye dissolved in water. In practical operations, a low concentration of solid particles in a solvent is recommended in order to avoid possible agglomeration of suspensions in the solvent. 

The time-dependent decrease in the concentration of particles (N = number of particles per cubic centimeter) in a monodisperse suspension due to collisions by Brownian motion can be represented by a second-order rate law... 

Fig. 7 Film energy profile for a monodisperse suspension as a fimction of particle volume fraction...

What is the relation between the diffusion coefficient of a monodispersed suspension of spherical particles and the decay of the intensity correlation function ... 

Perikinetic Coagulation. If colloidal particles are of such dimensions that they are subject to thermal motion, the transport of these particles is accomplished by this Brownian motion. Collisions occur when one particle enters the sphere of influence of another particle. The coagulation rate measuring the decrease in the concentration of particles with time, N (in numbers/ml.), of a nearly monodisperse suspension corresponds under these conditions to the rate law for a second order reaction (15) ... 

Monodisperse Suspensions. The absolute accuracy of the instruments were tested with a set of monodisperse solutions. Table I shows the results of this comparative study. 

Polydisperse Suspensions. Polydisperse suspensions were approximated by mixtures of monodisperse suspensions. In order not to introduce any bias in the analysis, all the data from three different instruments were analyzed using the same program. The distribution functions were assumed to be a sum of equally spaced histograms for PCS, delta functions for TS and logarithmically spaced histograms for FDPA. The height and position of each parameter was optimized using a nonlinear least squares minimization process (15). 

For a monodisperse suspension with N non-absorbing spherical particles per cm3, in absence of multiple scattering, the turbidity, t, is given by ... 

Direct particle counting of an initially monodisperse suspension was used to measure the time dependence of the q-moment M0, as given in the following table. Examine these data for conformity to either transport- or reaction-controlled flocculation kinetics and estimate the characteristic time scale, 2/kn p0, wherekn = kmn for m n 1. (Answer k n= 3.05 x 10 22 m3 s"1 = 2KD/Wmn, corresponding to Wmn = 4.07 X 104 for all m, n.)... 

A number of analytical solutions have been developed since that of von Smoluchowski, all of which contain some assumptions and constraints. Friedlander [33] and Swift and Friedlander [34] developed an approach relaxing the above constraint of an initially monodisperse suspension. Using a continuous particle size distribution function, a nonlinear partial integro-differential equation (with no known solution) results from Eq. (5). Friedlander [35] demonstrated the utility of a similarity transformation for representation of experimental particle size distributions. Swift and Friedlander [34] employed this transformation to reduce the partial integro-differential equation to a total integro-differential equation, and dem-... 

For a monodisperse suspension, the average electrophoretic velocity of the particles can be expressed as... 

Perikinetic agglomeration applies to a monodisperse suspension and can be represented by a second-order rate law ... 

Comparisons with experimental data have been effected by Batchelor. In careful experiments, Kops-Werkhoven and Fijnaut (1981) obtained —6 + 1 for the coefficient of the first-order volume fraction term in the sedimentation velocity [cf. Eq. (4.9)] and 1.3 0.2 for the corresponding term in the diffusion coefficient [cf. Eq. (4.12)]. The latter term is derived from light-scattering experiments. For monodisperse suspensions, these results agree reasonably well with Batchelor s predictions. 

In the absence of interactions, particles of differing sizes and shapes are statistically independent. For this reason, we can treat the statistical properties of light scattered from a dilute polydisperse suspension as the sum of contributions of many dilute monodisperse suspensions of particles with characteristic shape and size. Suppose that each characteristic shape/size combination is labeled with the index v. Let (V represent the number of particles having a particular shape and size. Clearly, we require N = Ns. All sums over the (V particles in a suspension can be expressed in terms of sums over the shape/size distribution. Thus we have... 

In principle, it is possible to calculate Xg from the Gmin values given in Fig. 6, using equation (5). If the floes are assumed to be close-packed structures, then n must be between 8 (random-close packing) and 12 (hexagonal close-packing). However, equation (5) is based on a monodisperse suspension. 

For the Cross equation, which is valid for monodisperse suspensions of hard spheres, the shear stress, t,, is given by... 

For a monodisperse suspension the decay rate can be described by a first order rate equation. For a polydisperse suspension the decay rate is a sum of exponentials. Measurement of the decay rate permits computation of particle size [338]. 

If the energy barrier to aggregation is removed (e.g., by adding excess electrolyte) then aggregation is diffusion controlled only Brownian motion of independent droplets or particles is present. For a monodisperse suspension of spheres, Smoluchowski developed an equation for this rapid coagulation ... 

Other problems in deriving a priori equations result from the polydisperse namre of pharmaceutical suspensions. The particle size distribution will determine rj. A polydisperse suspension of spheres has a lower viscosity than a similar monodisperse suspension. 

Hydrodynamic interactions in aggregation by fluid shear are relatively small for monodisperse suspensions and become increasingly important as the... 

These three examples show that Smoluchowski s rectilinear model can provide a useful representation of the initial contact rates in laboratory studies using monodisperse suspensions (case A in Figure 9), that this model can overpredict contacts in lakes and oceans by 1-2 orders of magnitude (case B in Figure 9), and that it overestimates initial contact rates in water- and wastewater-treatment systems by 3 orders of magnitude or more (case C, Figure 9). 

In order to achieve a monodisperse suspension or polydisperse system with particular particle size contribution, it is necessary to control the process of nucleation and particle growth. With most disperse systems, where the particles have some finite solubility, the smaller particles will have higher solubilities than their larger counterparts. With time, molecular diffusion occurs from the smaller to the larger particles, and this results in a shift in the particle size distribution to larger values this process is referred to as Ostwald ripening. 

The field autocorrelation function gi(r) for a monodisperse suspension decays exponentially with r,... 

For a narrow particle size distribution the cumulant analysis is usually satisfactory. The cumulant method is based on the assumption that, for monodisperse suspensions gi(r) is monoexponential. Hence, the log of gi(r) versus t yields a straight line with a slope equal to F,... 

We have chosen hematite oxalate as a model system, since the photochemical properties of colloidal hematite (Stramel and Thomas, 1986) and the photochemistry of iron(III) oxalato complexes in solution (Parker and Hatchard, 1959) have been studied extensively. The experiments presented in this section were carried out as batch experiments with monodispersed suspensions of hematite (diameter of the particles 50 and 100 nm), synthesized according to Penners and Koopal (1986) and checked by electron microscopy and X-ray diffraction. An experimental technique developed for the study of photoredox reactions with colloidal systems (Sulzberger, 1983) has been used. A pH of 3 was chosen to maximize the adsorption of oxalate at the hematite surface. This case study is described in detail by Siffert (1989) and Siffert et al. (manuscript in preparation). 

Experimental sophistication has advanced significantly with the recent availability of high-purity materials of precise configuration such as single crystals, atomically tailored surfaces, monodispersed suspensions, and precisely patterned microporous membranes. The ability to characterize these surfaces with a variety of ex situ techniques, and then to transfer them in a controlled manner into an electrochemical environment for further in situ study, presents a dramatic advance in the sophistication of electrochemical science. The preceding section of this chapter summarizes these advances. 

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