Electrophoretic Mobility of Concentrated Soft Particles

Consider a concentrated suspension of charged spherical soft particles moving with a velocity 17 in a liquid containing a general electrolyte in an applied electric field E. We assume that the particle core of radius a is coated with an ion-penetrable layer of polyelectrolytes with a thickness d. The polyelectrolyte-coated particle has thus an inner radius a and an outer radius b = a + d. We employ a cell model [4] in which each particle is surrounded by a concentric spherical shell of an electrolyte solution, having an outer radius c such that the particle/cell volume ratio in the unit cell is equal to the particle volume fraction 4 throughout the entire dispersion (Fig. 22.1), namely. 

The origin of the spherical polar coordinate system (r, 9, cp) is held fixed at the center of one particle and the polar axis (9 = 0) is set parallel to E. Let the electrolyte be composed of M ionic mobile species of valence zt and drag coefficient A,-(/ = 1, 2,. . . , M), and let nf be the concentration (number density) of the ith ionic species in the electroneutral solution. We also assume that fixed charges are distributed with a density of pflx. We adopt the model of Debye-Bueche where the polymer segments are regarded as resistance centers distributed in the polyelectrolyte 

Biophysical Chemistry of Biointerfaces By Hiroyuki Ohshima Copyright 2010 by John Wiley Sons, Inc. 

FIGURE 22.1 Spherical particles in concentrated suspensions in a ceU model. Each sphere is surrounded by a virtual shell of outer radius c. The particle volume fraction p is given hy (f — (b/c). The volume fraction of the particle core of radius a is given by (a/c).  

The fundamental equations for the flow velocity of the liquid ii(r) at position r and that of the /th ionic species v,(r) are the same as those for the dilute case (Chapter 5) except that Eq. (5.10) applies to the region b r c (not to the region r b). The boundary conditions for u(r) and v,(r) are also the same as those for the dilute case, but we need additional boundary conditions to be satisfied at r = c. According to Kuwabara s cell model [4], we assume that at the outer surface of the unit cell (r = c) the liquid velocity is parallel to the electrophoretic velocity U of the particle, 

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The electrophoretic mobility of spherical soft particles in a concentrated suspension is defined by = UIE. It must be mentioned here that the electrophoretic mobility fx in this chapter is defined with respect to the externally applied electric field E so that the boundary condition (22.8) has been employed following Levine and Neale [5]. There is another way of defining the electrophoretic mobility in the concentrated case, where the mobility /i is defined as /i = U/ E), (E) being the magnitude of the average electric field (E) within the suspension [8, 19-21]. It follows from the continuity condition of electric current that K E) = K°°E, where K and K°° are, respectively, the electric conductivity of the suspension and that of the electrolyte solution in the absence of the particles. Thus, jx and ix are related to each other by = K /K°°. For the dilute case, there is no difference between jx and jx. ... 

For low potentials, it can be shown that sed is related to the electrophoretic mobility p of concentrated soft particles (22.10) by... 

Equation (22.23) is the required approximate expression for the electrophoretic mobility of soft particles in concentrated suspensions when the condition [22.22] (which holds for most cases) is satisfied. In the limit 0 0, Eq. (22.23) tends to Eq. (21.51) for the dilute case. For low potentials, Eq. (22.23) reduces to... 

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. 

By evaluating h(r) at r = c, we obtain the following general expression for the electrophoretic mobility fx of soft particles in a concentrated suspension [1,3] ... 

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