Colloid stability 213For the case of two spherical particles of radii a\ and a 2 , Sternpotentials, (^0) dl and (^0) d2, and a shortest distance, H, between their
Stern layers, Healy and co-workers^195 have derived the following
expressions for constant-potential, VR, and constant-charge, VR,
double-layer interactions. The low-potential form of the Poisson-
Boltzmann distribution (equation 7.12) is assumed to hold and KU\
and K.ai are assumed to be large compared with unity:
(a,+a 2 )
r
-exp[-K//]
l in(1-exp[-2K//])l (8,2)
J j
where € is the permittivity of the dispersion medium and K is as
defined in equation (7.6)
Table 8.2 shows the signs of VR that accord with equations (8.1)
and (8.2) for different homocoagulation and heterocoagulation
situations. (N.B. Attraction is negative and repulsion positive.)
For equal spheres, with at = a 2 = a and (^0) dl = (^0) d2 = </rd, equations
(8.1) and (8.2) reduce to
(8.3)
and
For small electric double layer overlap, such that exp [-K//] < 1,
these expressions both reduce to
VR = 2ireai^ exp[-K//] (8.5)