Dilute suspension model, spherical

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) ... 

In this approach, the unknown parameters, AjS and/or BjS, are calculated directly from differential equations in partial derivatives for an appropriate model. In the simpl t case of a dilute suspension of spherical inclusions of component 1 in a continuous matrix of component 2, the temperatum field around any particle is presumed to be unperturbed by its neighbors (i.e., the external field gradient is equal to the mean gradient in MHM hence, one may write... 

Electrokinetic equations describing the electrical conductivity of a suspension of colloidal particles are the same as those for the electrophoretic mobility of colloidal particles and thus conductivity measurements can provide us with essentially the same information as that from electrophoretic mobihty measurements. Several theoretical studies have been made on dilute suspensions of hard particles [1-3], mercury drops [4], and spherical polyelectrolytes (charged porous spheres) [5], and on concentrated suspensions of hard spherical particles [6] and mercury drops [7] on the basis of Kuwabara s cell model [8], which was originally applied to electrophoresis problem [9,10]. In this chapter, we develop a theory of conductivity of a concentrated suspension of soft particles [11]. The results cover those for the dilute case in the limit of very low particle volume fractions. We confine ourselves to the case where the overlapping of the electrical double layers of adjacent particles is negligible. 

So far we have discussed electrophoresis of particles and drops in electrolyte solutions, i.e., salt-containing media. For those in salt-free media containing only counterions (e.g., nonaqueous media [63]), special consideration is needed. To treat a suspension of spherical particles in a salt-free medium, one usually employs a free volume model, in which each sphere of radius a is surrounded by a spherical free volume of radius R within which counterions are distributed so that electrical neutrality as a whole is satisfied. The particle volume fraction is given by 4>= (fl// ). We treat the case of dilute suspensions, viz., < C lara/R< 1. Let the concentration (number density) and valence of the counterions be n and — z, respectively. For a spherical particle carrying surface charge density a or total surface charge Q = 4ira a, it follows from the electroneutrality condition that... 

A comparable cell model is that of Kuwabara (1959), the only difference being that the spherical cell surface is in this case assumed to be at zero vorticity rather than at zero shear stress. The coefficient of c / in Eq. (28a) for dilute suspensions becomes 1.8 in Kuwabara s solution, instead of 1.5. 

FIGURE41.3 Stiffness vs. volume fraction for Voigt and Reuss models, as well as for dilute isotropic suspensions of platelets, fibers, and spherical particles embedded in a matrix. Phase moduli are 200 and 3 GPa. 

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