Monodisperse Systems of Spherical Particles

For the sedimentation of rarefied monodisperse systems of spherical particles, drops, or bubbles, the mean Sherwood number can be calculated by using formulas (4.6.8) and (4.6.17), where the Peclet number must be determined on the basis of the constrained flow velocity. 

To find mass and heat transfer coefficients at high Peclet numbers, just as for isolated particles, it suffices to know the vortex distribution over the surfaces of solid spheres. Therefore, one can use the results of Section 4.6 in the calculations. 

Let us consider mass and heat transfer for a monodisperse system of spherical particles of radius a with volume density f of the solid phase. We use the fluid velocity field obtained at low Reynolds numbers from the Happel cell model (see Section 2.9) to find the mean Sherwood number [74,76  

Here Pe = aU /D is the Peclet number calculated from the constrained flow velocity, which can be calculated by the formula 

Experimental data on mass and heat transfer in a constrained flow are often treated as the dependence of the Kollborn factor Ko = Sh/fScRe, ) on the Reynolds number. A comparison of experimental data for the Kollborn factor for solid spheres [76] with 0.5 4 0.7 with the theoretical values for Re 1 showed that the results of calculations for low Re remain valid for Re 50. 

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The most well known acoustic theory for heterogeneous systems was developed by Epstein and Carhart (3), and Al-legra and Hawley (10). This theory takes into account the foiu most important mechanisms (viscous, thermal, scattering, and intrinsic) and is termed the ECAH theory. This theory describes attenuation for a monodisperse system of spherical particles and isvalid only for dilute systems. 

In order to close these expressions for particulate pressures, we also need equations for the variance of total particle volume concentration in an assemblage of particles belonging to the two different types. For an arbitrary polydisperse particulate pseudo-gas, variances of partial volume concentrations for different particles can be evaluated on the basis of the thermodynamical theory of fluctuations. According to this theory, these variances are expressible in terms of the minors of a matrix that consists of the cross derivatives of the chemical potentials for particles of different species over the partial number concentrations of such particles [39]. For a binary pseudo-gas, these chemical potentials can be expressed as functions of number concentrations using the statistical theory of binary hard sphere mixtures developed in reference [77]. However, such a procedure leads to a very cumbersome and inconvenient final equation for the desired variance. To simplify the matter, it has been suggested in reference [76] to ignore a slight difference between this variance and the similar quantity for a monodisperse system of spherical particles of the same volume concentration. This means that the variance under question may be approximately described by Equation 7.4 even in the case of binary mixtures. 

We note that in [421], the cell flow model was used for the investigation of mass and heat transfer in monodisperse systems of spherical drops, bubbles, or solid particles for Re < 250 and 0 < < 0.5. 

The present chapter is aimed at reviewing the development and the specific applications of the SCGLE theory of colloid dynamics and of dynamic arrest. Thus, it is not aimed at reviewing the state-of-the-art in either of these research areas, for which excellent reviews are available [2,3,43-45]. We must also say that notable topics in both fields are barely or never mentioned here. This includes, for example, the structural, mechanical, and rheological properties, and the effects of hydrodynamic interactions. Instead, we focus on the treatment of the effects of direct conservative interactions in simple colloidal systems. Thus, here we shall primarily deal with monodisperse suspensions of spherical particles in the absence of hydrodynamic interactions, although the extension to multi-component systems will also be an important aspect of this review. 

This equation again demonstrates that particle size and solubility are the main parameters affecting dissolution kinetics of drug powders, which, in turn, could affect the release profile of dosage forms if dissolution is the rate-limiting step of in vivo absorption. Table 5.1 demonstrates several examples of dissolution times of spherical particles (assuming monodispersed systems) as a function of solubility and particle size. 

It is difficult to define precisely the term aqueous silica sols and thereby contrast them with other forms of silica (colloidal silica, colloidal quartz, pyrogenic silica, and so forth). Bulk chemical distinctions are not very useful. The definition chosen here follows Iler s terminology (I). Aqueous silica sols are characteristically composed of spherical particles nucleated and grown by alkaline hydrolysis of sodium silicate solutions. They are often monodisperse systems and have particle diameters in the range 1-100 nm (density, —2.2 g/cm3) that lead to sols that vary from optically transparent to opalescent. 

The relationships between 17 and ( ) have been derived for suspensions of monodispersed hard spheres in Newtonian liquids. However, most real systems are polydispersed in size, and do not necessarily consist of spherical particles. It has been found that here also Simha s Eq 7.24, Mooney s Eq 7.28, or Krieger-Dougherty s Eq 7.8 are useful, provided that the intrinsic viscosity and the maximum packing volume fraction are defined as functions of particle shape and size polydispersity. For example, by allowing ( ) to vary with composition, it was possible to describe the vs. ( ) variation for bimodal suspensions [Chang and Powell, 1994]. Similarly, after values... 

The value of x is determined by the geometry of the system, primarily by the particle size (radius, r, for spherical particles) and by the packing density of particles described by porosity, H. The porosity is a dimensionless characteristic defined as the ratio of the volume of pores, Vp to the total volume of the porous structure, V, that is, n = Vp/V. The x = Xir, H) dependence can be estimated from data on the degree of dispersion of the particles and the porosity of samples by employing the specific models for disperse structures. Eor example, in the case of loose monodisperse structures with spherical particles connected into crossing chains with n particles per chain between the nodes (Figure 3.17), the X function for the case when the porosity H does not exceed 48% can be described as... 

A certain amount of controversy has been caused by the question as to what specific characteristics of the AF function represent the universal conditions for spontaneous dispersion To address this subject, we will analyze changes in the free energy, AF, associated with the dispersion, as a function of the size and number of particles, their concentration, and the value of the free interfacial energy at the interface between the disperse phase and the dispersion medium. This analysis is performed for three characteristic conditions (1) varying the particle size at a constant volume of the disperse phase, (2) varying the number of particles at a constant particle size, and (3) varying the particle size at a constant number of particles. The analysis will be restricted to systems that are monodisperse at every stage of the dispersion process and consist of spherical particles with radius r. The volume of the dispersion medium is assumed to be constant, for example, V = 1000 cm. At 300 K, the kF value is 4.14 x 10" J. 

Generally, alkoxide-derived monodisperse oxide particles have been produced by batch processes on a beaker scale. However, on an industrial scale, the batch process is not suitable. Therefore, a continuous process is required for mass production. The stirred tank reactors (46) used in industrial process usually lead to the formation of spherical, oxide powders with a broad particle size distribution, because the residence time distribution in reactor is broad. It is necessary to design a novel apparatus for a continuous production system of monodispersed, spherical oxide particles. So far, the continuous production system of monodisperse particles by the forced hydrolysis... 

In 1968, Stober et al. (18) reported that, under basic conditions, the hydrolytic reaction of tetraethoxysilane (TEOS) in alcoholic solutions can be controlled to produce monodisperse spherical particles of amorphous silica. Details of this silicon alkoxide sol-gel process, based on homogeneous alcoholic solutions, are presented in Chapter 2.1. The first attempt to extend the alkoxide sol-gel process to microemul-sion systems was reported by Yanagi et al. in 1986 (19). Since then, additional contributions have appeared (20-53), as summarized in Table 2.2.1. In the microe-mulsion-mediated sol-gel process, the microheterogeneous nature (i.e., the polar-nonpolar character) of the microemulsion fluid phase permits the simultaneous solubilization of the relatively hydrophobic alkoxide precursor and the reactant water molecules. The alkoxide molecules encounter water molecules in the polar domains of the microemulsions, and, as illustrated schematically in Figure 2.2.1, the resulting hydrolysis and condensation reactions can lead to the formation of nanosize silica particles. 

Monodisperse spheres are not only uniquely easy to characterize, but also very rarely encountered. Polymerization under carefully controlled conditions allows the preparation of the polystyrene latex shown in Figure 1.8. Latexes of this sort are used as standards for the size calibration of optical and electron micrographs (also see Section 1.5a.3). However, in the majority of colloidal systems, the particles are neither spherical nor monodisperse, but it is often useful to define convenient effective linear dimensions that are representative of the sizes and shapes of the particles. There are many ways of doing this, and whether they are appropriate or not depends on the use of such dimensions in practice. There are excellent books devoted to this topic (see, for example, Allen 1990) and, therefore, we consider only a few examples here for the purpose of illustration. 

Monodispersed sols containing spherical polymer particles (e.g. polystyrene latexes22"24, 135) can be prepared by emulsion polymerisation, and are particularly useful as model systems for studying various aspects of colloidal behaviour. The seed sol is prepared with the emulsifier concentration well above the critical micelle concentration then, with the emulsifier concentration below the critical micelle concentration, subsequent growth of the seed particles is achieved without the formation of further new particles. 

The spherical nature of the particles and the high degree of monodispersity in polystyrene latices makes them ideal systems for testing such a hypothesis and exploring its practical implications. In some preliminary experiments ( 3C) we have determined the c.c.c. values of a series of latices of different particle sizes and then over a range of salt concentrations at and above the c.c.c. examined the effect on the associated state of using dialysis to remove the salt. The results obtained are summarized in Table III. 

In this presentation I have given a brief review of the factors controlling the stability and instability of polymer latices. There is little doubt that as a consequence of the ready availability of dispersions of spherical, monodisperse particles, polymer latices, our knowledge of the behaviour of colloidal dispersions has progressed rapidly over the past fifteen years. However, many phenomena remain to be investigated in quantitative detail and we must remember that the small energy changes involved in these systems, by comparison with molecular reactions, make many of the phenomena very subtle. 

Lee and Lightfoot [229] developed the theoretical basis of Fl-FFF. This theory has been confirmed by numerous works on the fractionation of model systems, including monodisperse spherical polystyrene latexes and a number of proteins [41,228,229,240], some polydextrans [229], viruses [241], and other spherical particles and macromolecules [242,243]. 

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