Kuwabara’s cell model

In this chapter, we extend the electrokinetic theory of soft particles (Chapter 21), which is applicable for dilute suspensions, to cover the case of concentrated suspensions [1-3] on the basis of Kuwabara s cell model [4], which has been applied to theoretical studies of various electrokinetic phenomena in concentrated suspensions of hard colloidal particles [5-23]. 

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 bb). 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,... 

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. 

According to Kuwabara s cell model [7], we assume that at the outer surface of the unit cell (r = c), the liquid velocity u is parallel to the electrophoretic velocity U of the particle. 

Figure 6, Illustration of Kuwabara s cell model used for calculating IPS in a particulate system.

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. 

---