Electroviscous Coefficient

Following the theory of Watterson and White [9], we consider the volume averaged stress tensor (c) defined by 

We first consider the case of a suspension of uncharged soft particles, that is, ZeN = 0. In this case D3 = 5b L2lLi and Eq. (27.40) becomes 

The coefficient Q(Aa, a/b) in Eq. (27.45) expresses the effects of the presence of the uncharged polymer layer upon As A — 00 or a b, soft particles tend to hard particles and Q(Aa, alb) 1 so that Eq. (27.45) tends to Eq. (27.1). For 1 0, on the other hand, the polymer layer vanishes so that a suspension of soft particles of outer radius b becomes a suspension of hard particles of radius a and the particle volume fraction changes from cj) to 4 , which is given by 

Indeed, in this case Q(Aa, a/b) tends to a /b so that rjs becomes 

FIGURE 27.1 Function Q Xa, alb) for a suspension of uncharged spherical soft particles as a function of Xa for several values of alb. The line for a b= 1 corresponds to a suspension of spherical hard particles. From Ref. 26. 

---

Finally, pi is the primary electroviscous coefficient which is a function of the charge on the particle or, more conventionally, the electrostatic potential,, on the "slip-ping plane" which defines the hydrodynamic radius of the particle, and properties (charge, bulk density number, and limiting conductance) of the electrolyte ions (Rubio-Hernandez et al. 2000). 

The slopes of the different curves correspond to the fuU electrohydrodynamic effect, ( ) + ( ) pj, where the first term expresses the hydrodynamic effect, and the second is the consequence of the distortion of the electrical double layer that surrounds the particles. To determine this second term and, more exactly, the primary electroviscous coefficient, pi. 

Garda-Salinas M. J., Romero-Cano M. S., de las Nieves F. J.. Zeta potential study of a polystyrene latex with variable surface charge influence on the electroviscous coefficient. Progr Colloid Polym Sci (2000) 115 112-116. 

In this chapter, we first present a theory of the primary electroviscous effect in a dilute suspension of soft particles, that is, particles covered with an ion-penetrable surface layer of charged or uncharged polymers. We derive expressions for the effective viscosity and the primary electroviscous coefficient of a dilute suspension of soft particles [26]. We then derive an expression for the effective viscosity of uncharged porous spheres (i.e., spherical soft particles with no particle core) [27]. 

One can calculate the primary electroviscous coefficient p for a suspension of soft particles with low ZeN via Eq. (27.53). 

For a suspension of charged spherical soft particles carrying low ZeN, the electroviscous coefficient p can be calculated via Eq. (27.53) as combined with Eq. (27.54) for L ku, la, alb). Figure 27.2 show some examples of the calculation of L Ka, la, alb) as a function of Ka obtained via Eq. (27.54) at la = 50. This figure shows that as Ka increases, L(Ka, la, alb) first decreases, reaching a minimum around ca. no =10, and then increases. Note that the primary electroviscous coefficient p for a suspension of hard particles (shown as a curve with alb = 1 in Eig. 27.2) does not exhibit a minimum. 

Approximate results calculated via Eq. (27.57) are also shown as dotted lines in Fig. 27.2. It is seen that Ka > 100, the agreement with the exact result is excellent. The presence of a minimum of L Ka, la, alb) as a function of Ka can be explained qualitatively with the help of Eq. (27.57) as follows. That is, L Ka, la, alb) is proportional to 1/k at small Ka and to k at large Ka, leading to the presence of a minimum of L Ka, la, alb). As is seen in Fig. 27.3, for the case of a suspension of hard particles, the function L ko) decreases as Ka increases, exhibiting no minimum. This is the most remarkable difference between the effective viscosity of a suspension of soft particles and that for hard particles. It is to be noted that although L Ka, la, alb) increases with Ka at large Ka, the primary electroviscous coefficient p itself decreases with increasing electrolyte concentration. The reason is that the... 

Equation (27.64) shows that the electroviscous coefficient p for large Ka decreases with increasing Ka. 

The intrinsic viscosity [17] in the above expression includes the primary electroviscous effect. The experimental data of Stone-Masui and Watillon (1968) for polymer latices seem to be consistent with the above equation (Hunter 1981). Corrections for a for large values of kRs are possible, and the above equation can be extended to larger Peclet numbers. However, because of the sensitivity of the coefficients to kRs and the complications introduced by multiparticle and cooperative effects, the theoretical formulations are difficult and the experimental measurements are uncertain. For our purpose here, the above outline is sufficient to illustrate how secondary electroviscous effects affect the viscosity of charged dispersions. 

A charged particle in suspension with its inner immobile Stern layer and outer diffuse Gouy (or Debye-ffiickel) layer presents a different problem from that arising with a smooth and small nonpolar sphere. In movement such particles experience electroviscous effects which have two sources (a) the resistance of the ion cloud to deformation, and (b) the repulsion between particles in close contact. When particles interact, for example to form pairs in the system, the new particle will have a different shape from the original and will have different flow properties. The coefficient 2.5 in Einstein s equation (7.30)... 

The second electroviscous effect is due to fire electrostatic repulsion between particles approaching each other and is directly proportional to the square of the particle concentration. The essential feature about this effect is that it occurs at high concentrations of the suspensions (unlike the first electrostatic effects) and when there is an overlap of the double layer. The additional dissipative effects that appear as a result of the repulsion bring about an increase in the viscosity. Chan et al. [164] showed that an expression of the form (4.17) can be used to account for the second electroviscous effect but the coefficient a would then strongly depend on the distance between the centers of the particles and consequently, on the particle concentration. The second electroviscous effect is, at times, known to give rise to non-Newtonian behavior of a suspension as observed by Harmsen et al. [165]. 

---