92168.pdf

230 Colloid stability

spherical particles, and considering only the aggregation of single

particles to form doublets

where a is the effective radius of the particles and D is the diffusion

coefficient. Substituting D = kTI6rrrja (equation (2.6)) and combining

equations (8.16) and (8.17) gives

(8.18)

where fcj> is the rate constant for diffusion-controlled coagulation.
For a hydrosol at room temperature, the time f1/2 in which the
number of particles is halved by diffusion-controlled coagulation is
calculated from the above equations to be of the order of 10 H/n 0
seconds, if « 0 is expressed in the unit, particles cm"^3. In a typical
dilute hydrosol, the number of particles per cm^3 may be about 1010 -
10
11
, and so, on this basis, *1/2 should be of the order of a few seconds.
Rapid coagulation is, in fact, not quite as simple as this, because
the last part of the approach of two particles is (a) slowed down
because it is difficult for liquid to flow away from the narrow gap
between the particles, and (b) accelerated by the van de Waals
attraction between the particles. Lichtenbelt and co-workers^205 have
measured rapid coagulation rates by a stopped-flow method and
found them, typically, to be about half the rate predicted according to
equation (8.18).
When there is a repulsive energy barrier, only a fraction l/Wof the
encounters between particles lead to permanent contact. W is known
as the stability ratio - i.e.
k°
W = -^~ (8.19)
k 2
A theoretical expression relating the stability ratio to the potential
energy of interaction has been derived by Fuchs^110 :

a (8.20)

Theoretical relationships between the stability ratio and electrolyte
concentration can be obtained by numerical solution of this integral
for given values of A and tf/d. Figure 8.7 shows the results of