Suspension of Spherical Particles

This is a model, for example, for blood or cell suspensions, and the electrolytic solution of interest may then have a considerable ionic conductivity. An analytical solution (Maxwell, 1873) is relatively simple for a dilute suspension of spherical particles, and with DC real conductivity as parameters (a is for the total suspension, Oa for the external medium, and a, for the particles) the relation is Maxwell s spherical particles mixture equation) (Foster and Schwan, 1989)  

If the particles have a DC conductivity. Maxwell—Wagner effects cause e as a function of frequency to have an additional slope downwards. 

As for the relaxation time, Debye derived a simple expression for a viscosity determined relaxation time of a sphere of radius a in liquid, Eq. (3.39) x = 4Tca T /kT. The relaxation time is therefore proportional to the volume of the sphere and the viscosity of the liquid. 

Maxwell s Eq. (3.45) is rigorous only for dilute concentrations, and Hanai (1960) extended the theory for high-volume fractions  

Hanai s equation gives rise to dispersion curves that are broader than the Maxwell-Wagner equation (Takashima, 1989). Similar work had also earlier been done by Bmggeman (1935). 

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A somewhat similar problem arises in describing the viscosity of a suspension of spherical particles. This problem was analyzed by Einstein in 1906, with some corrections appearing in 1911. As we did with Stokes law, we shall only present qualitative arguments which give plausibility to the final form. The fact that it took Einstein 5 years to work out the bugs in this theory is an indication of the complexity of the formal analysis. Derivations of both the Stokes and Einstein equations which do not require vector calculus have been presented by Lauffer [Ref. 3]. The latter derivations are at about the same level of difficulty as most of the mathematics in this book. We shall only hint at the direction of Lauffer s derivation, however, since our interest in rigid spheres is marginal, at best. 

Figure 9.2 (a) Schematic representation of a unit cube containing a suspension of spherical particles at volume fraction [Pg.589]

How does yield stress depend on the size of particles We have mentioned above that increasing the specific surface, i.e. decreasing an average size of particles of one type, causes an increase in yield stress. This fact was observed in many works (for example [14-16]). Clear model experiments the purpose of which was to reveal the role of a particle s size were carried out in work [8], By an example of suspensions of spherical particles in polystyrene melt it was shown that yield stress of equiconcentrated dispersions may change by a hundred of times according to the diameter d of non-... 

In a study of the viscosity of a solution of suspension of spherical particles (colloids), suggested that the specific viscosity rjsp is related to a shape factor Ua-b in the following way ... 

Jeffrey [181] estimated the effective conductivity in a dilute suspension of spherical particles, and in terms of diffusion coefficient his solution was... 

Carrique F, Arroyo FJ, Jimenez ML, Delgado Av. Influence of double-layer overlap on the electrophoretic mobility and DC conductivity of a concentrated suspension of spherical particles. J. Phys. Chem. B 2003 107 3199-3206. 

Thus, dendrimers exhibit a unique combination of (a) high molecular weights, typical for classical macromolecular substances, (b) molecular shapes, similar to idealized spherical particles and (c) nanoscopic sizes that are larger than those of low molecular weight compounds but smaller than those of typical macromolecules. As such, they provide unique rheological systems that are between typical chain-type polymers and suspensions of spherical particles. Notably, such systems have not been available for rheological study before, nor are there yet analytical theories of dense fluids of spherical particles that are successful in predicting useful numerical results. 

The main result was that regardless of dendrimer generation (i.e. molecular weight) and concentration, all of the examined solutions exhibited characteristic Newtonian flow behavior, as shown in Figure 14.6. This was in striking contrast to the typical behavior of either chain-type polymers of comparable molecular weights [33], or suspensions of spherical particles [34-37], both of which exhibit... 

Khan, A. R. and Richardson, J. F. Chem. Eng. Comm. 78 (1989) 111. Fluid-particle interactions and flow characteristics of fluidized beds and settling suspensions of spherical particles. 

Equality holds for a single sphere or a collection of identical spheres inequality holds if they are distributed in size or composition. This inequality was used by Hunt and Huffman (1973), for example, as an indicator of dispersion in suspensions of spherical particles. It was pointed out by Fry and Kattawar (1981) that the inequalities they derived are useful consistency checks on measurements of all 16 scattering matrix elements. 

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

Figure 11.19(b) plots the steady state ratio of the viscosities of suspensions of spherical particles in Newtonian liquids, /iv, to the viscosity of the Newtonian fluid, /y. It was constructed by Thomas (79) using the data of a number of investigators. A variety of uniform-sized particles having diameters of 1 —4(X) pm were used. They included PS and... 

Fig. 11.19 Viscosity of suspensions of spherical particles in Newtonian fluids, (a) Curve constructed by Bigg. [Reprinted by permission from D. M. Bigg, Rheological Behavior of Highly Filled Polymer Melts, Polym. Eng. Sci, 23, 206 (1983).] (b) Curves presented by Thomas (79).

Brownian motion must be taken into account for suspensions of small (submicron-sized) particles. By their very nature, such stochastic Brownian forces favor the ergodicity of any configurational state. Although no completely general framework for the inclusion of Brownian motion will be presented here, its effects will be incorporated within specific contexts. Especially relevant, in the present rheological context, is the recent review by Felderhof (1988) of the contribution of Brownian motion to the viscosity of suspensions of spherical particles. 

Consider a dilute suspension of spherical particles A in a stationary liquid B. If the spheres are sufficiently small, yet large with respect to the molecules of stationary liquid, the collisions between the spheres and the liquid molecules B lead to a random motion of the spheres. This motion is called the Brownian motion. Dilute diffusion of suspended spherical colloid particles is related to the temperature and the friction coefficient by... 

The first study of the rheology of suspensions was made by Einstein65 who analyzed the flow of dilute suspensions of spherical particles. The relative... 

The hydrodynamic behavior of complex particles in solution is similar to that of suspensions of solid spheres. Applying to the solutions of the PMAA-poly(ethylene glycol) complex, Einstein s equation for the viscosity of suspension of spherical particles, t]Sp/c = 2.54 q> (where tp is the volume fraction of dissolved substance) the solvent content in complex coils has been estimated33. It is about 75 vol%, i.e. the complex particles contain comparatively small quantities of the solvent in comparison with a usual random coil in solution which contains about 97-99 vol% of solvent34. ... 

The surface volume mean diameter for a suspension of spherical particles is given by ... 

If now a magnetic field H is applied to a suspension of spherical particles of diameter d, the orientation-dependent potential energy Wh of a particle in this field is expressed as... 

In deriving an equation for the viscosity of a suspension of spherical particles, Einstein considered particles which were far enough apart to be treated independently. The particle volume fraction (p is defined by... 

In these relations, A is the conductivity of the suspension, and the subscripts m, o, and w refer to the microemulsion, oil and emulsifier combined, and water. The Hanai expression can be considered to be an extension of the Maxwell theory that more consistently accounts for the presence of neighboring particles (8) for the 0/W microemulsions considered here, the predictions of the Maxwell and Hanai formulas (as well as various other mixture theories) are not greatly different. Moreover, while these theories were developed for suspensions of spherical particles, the predictions of the mixture theories are not expected to vary greatly with the geometry of the dispersed particles, provided that the droplets are prolate or oblate ellipsoids whose axial ratios are not greatly... 

The particle-particle interactions lead to a dependence of the viscosity, q, of a colloidal dispersion on the particle volume fraction, ((). Einstein showed that for a suspension of spherical particles in the dilnte limit ... 

In the case of microemulsions and suspensions of spherical particles, it is usually more convenient to use the volume fraction, ((), of the dispersed particles as a measure of their concentration. By using the fact that ([) = pV, we can obtain the virial expansion ... 

Figure 7-14. A comparison of experimental and simulation data for the viscosity of a suspension of spherical particles versus the Einstein prediction (7-197). The maximum volume fraction for random cubic packing of equal-size spheres is 0.64 (figure courtesy of J. E Brady).

The Smoluchowski formula, which is used in the AcoustoSizer from the measured dynamic mobility, is valid for a disperse suspension of spherical particles according to Eq. 1. 

Figure 9.14 shows a map of independent/dependent scattering for packed beds and suspensions of spherical particles [59]. The map is developed based on available experimental results. The experiments are from several investigators, and some of the experiments are reviewed later. The results show that for relatively high temperatures in most packed beds, the scattering of thermal radiation can be considered independent. 

It should be noted that the relative viscosity dependence on the volume fraction for suspensions of spherical particles using the same treatment as above gives... 

The rheological behavior of multiphase systems within the linear, dilute region ((() < 0.05) is relatively well described. For example, for dilute suspensions of spherical particles in Newtonian liquids, Eq 7.2 reduces to Einstein s formula for the relative viscosity, T ... 

BATCHELOR, G.K. 1977. The effect of Brownian motion on the bulk stress in a suspension of spherical particles. ]. Fluid Mech. 83, 97—117. 

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