Density, monodisperse particle systems

We consider a system of monodisperse particles of number density N which are oriented at random. The scattering intensity I(q) as function of q, the magnitude of the scattering vector is given by [1-5]... 

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... 

As particle counter a special automatic system devised by Technicon Instr. Corp., USA, has been applied. The method described above allows the preparation of 0.5-5 pm relatively low density monodisperse... 

Several colloidal systems, that are of practical importance, contain spherically symmetric particles the size of which changes continuously. Polydisperse fluid mixtures can be described by a continuous probability density of one or more particle attributes, such as particle size. Thus, they may be viewed as containing an infinite number of components. It has been several decades since the introduction of polydispersity as a model for molecular mixtures [73], but only recently has it received widespread attention [74-82]. Initially, work was concentrated on nearly monodisperse mixtures and the polydispersity was accounted for by the construction of perturbation expansions with a pure, monodispersive, component as the reference fluid [77,80]. Subsequently, Kofke and Glandt [79] have obtained the equation of state using a theory based on the distinction of particular species in a polydispersive mixture, not by their intermolecular potentials but by a specific form of the distribution of their chemical potentials. Quite recently, Lado [81,82] has generalized the usual OZ equation to the case of a polydispersive mixture. Recently, the latter theory has been also extended to the case of polydisperse quenched-annealed mixtures [83,84]. As this approach has not been reviewed previously, we shall consider it in some detail. 

Data analysis methods depend upon the level of order in the sample. The degree of order, in turn, depends upon the scale of distance on which the sample is viewed. For example, casein micelles show great variation in size (20 to 300 nm diameter) and so must be treated as a polydisperse system. However, the density variations ( submicelles ) within the whole micelle are much more uniform in size. They can be treated as a quasi-monodisperse system (Stothart and Cebula, 1982) and analyzed in terms of inter-particle interference (Stothart, 1989). 

Let us consider a system of monodispersed aerosol particles. Because of the collisions between the particles, they will coagulate to form doublets. The rate of formation of a doublet depends, of course, on the particle radius, the particle density, the viscosity and temperature of the suspending medium, the Hamaker constant, and the particle number concentration. If the rate of formation of a doublet is itnuch less than its rate of dissociation, the doublets are unstable. These unstable doublets attain... 

Two major entry models - the diffusion-controlled and propagation-controlled models - are widely used at present. However, Liotta et al. [28] claim that the collision entry is more probable. They developed a dynamic competitive growth model to understand the particle growth process and used it to simulate the growth of two monodisperse polystyrene populations (bidisperse system) at 50 °C. Validation of the model with on-line density and on-line particle diameter measurements demonstrated that radical entry into polymer particles is more likely to occur by a collision mechanism than by either a propagation or diffusion mechanism. 

In this relation, N is the number density of the scattering microemulsion droplets and S(q) is the static structure factor. Equation (2.12) is only strictly valid for the case of monodisperse spheres. However, for the case of low polydispersities the occurring error is small [63, 64]. S(q) describes the interactions between and the spatial correlations of the droplets. These are in general well approximated by hard sphere interactions in microemulsion systems [65], The influence of inter-particle interactions as described by S(q) canbe estimated at least for S(0) using the Carnahan-Starling expression [52,64,66]... 

Let us consider mass and heat transfer for a monodisperse system of spherical particles of radius a with volume density 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 cpmax is the maximum volume fraction (packing density) that the dispersed particles can have. At that value the viscosity becomes infinite (no flow possible). For random packing of monodisperse spheres, polydisperse systems, its value can be appreciably higher. Note that now particle size becomes a variable, though its spread (e.g., relative standard deviation) rather than its average is determinant. 

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 above equations can be used to deduce the properties of the suspension from observations of the front speeds, typically the one separating the clarified layer from the suspension. For example, knowing the fall speed (Eq. 5.4.6), we can determine the effective particle size if the particle density has been found independently. The extension of the results to infinitely dilute systems containing particles of two or more sizes (polydisperse systems) is straightforward and will not be discussed further here. It may only be mentioned that with different fall speeds there will be as many distinct downward-moving fronts as there are particle sizes. From measurements of these front speeds the particle sizes can be determined as for the monodisperse system. 

As an example, the system consisting of metal oxide particles in solutions of Co ions is described here [7]. Figure 2 gives the adsorption densities of cobalt(II) ions on monodispersed magnetite (Fe304> particles as a function of the pH, while Figure 3 shows the temperature effect on the same system. The latter clearly indicates the chemisorption nature of the processes involved. 

At the same time, Masliyah (1979) addressed the density difference problem. For a monodisperse system, i.e., a single particle species, both Eq. (123) and Eq. (127) are equivalent to... 

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