ABSTRACT
Consider a spherical colloidal particle of radius a. When zeta potential is low, the electrical double layer around a spherical particle maintains its spherical symmetry during electrophoresis (δρel(r)=0 or δni(r)=0) so that
(29)
and
(30)
Here the first term on the right-hand side, −Ercosθ, is the potential of the applied electric field in the absence of the particle. The second term corresponds to the distortion of the applied electric field due to the presence of the particle, which occurs in such a way that the applied field becomes parallel to the particle surface. In this situation, Eq. 26 becomes
(31)
Substituting Eq. 30 into Eq. 25 gives
(32)
with
f(κa)=1−eκa{5E7(Ka)−2E5(κa)} (33)
where En (κa) is the exponential integral of order n. Eq. 32 was first derived by Henry[11] and f(κa) is called Henry’s function. As κa→∞, f(κa)→1 and Henry’s Eq. 32 tends to Smoluchowski’s Eq. 3, whereas if κa→0, then f(κa)→2/3 and Eq. 32 becomes Hückel’s Eq. 4. That is, Henry’s formula (Eq. 32) is applicable for all values of κa provided that ζ is low. Ohshima[20] has derived the following simple approximate formula for Henry’s function ƒ(κa) with relative errors of less than 1%:
(34)
The difference between Smoluchowski’s Eq. 3 and Hüc-kel’s Eq. 4 by a factor of 2/3 can be explained as follows. It is seen from Eq. 30 that the potential of the applied field near the particle surface r≈a is larger than the original undistorted field by a factor of 3/2. The electrophoretic mobility is determined mainly by electrolyte ions in the double layer (of thickness 1/κ). As is seen in Fig. 2, for thick double layers ( ) most electrolyte ions in the double layer experience undistorted original field. For thin double layers ( ) on the other hand, most electrolyte ions in the double layer experience distorted field. This is the reason why Smoluchowski’s mobility Eq. 3 is larger than Hückel mobility Eq. 4 by 3/2.
