Physical Chemistry of Foods

then often diffuse back again after coming so close that it senses significant
repulsion. This implies that the particle concentration gradient near the
central particle will be smaller than in the absence of repulsion. Fuchs found
a general solution for the resulting retardation for the case of equal-sized
spheres. The result is

W¼

1

a

¼ 2

Z?

2

z
^2 exp

VðzÞ

kBT

dz ð 13 : 5 Þ

z¼ 2 þ

h

a

whereais particle radius andhinterparticle distance.
A calculated example is in Figure 13.3, which gives the magnitude of
V/kBTand of the integrandyof Eq. (13.5) (i.e., the expression between the
integral sign and dz) as a function of the relative distancez. Twice the area
under the y-curve corresponds toW&365, implying that the assumed
repulsion would lead to a slowing down of the aggregation by this factor. It
may further be noticed that a significant contribution of the repulsion toW
is restricted, in the present case, to the range whereVis larger than about
10 kBT.
It is tempting to take the maximum value ofVas an activation free
energy for aggregation, which would imply thatW¼expðVmax=kBTÞ, but
this is not correct. In the example of Figure 13.3,Vmax& 12 : 8 kBT, which
would lead toW& 360 ;000, almost 1000 times the value according to Fuchs.
If Vmaxis only 7 times kBT, Eq. (13.5) predicts W¼1.2, i.e., almost
negligible, whereas e^7 ¼1100, which would imply strong retardation.
The reason thatWcan be small despiteVmaxbeing large is that the
range ofzover which the value ofyis significant is so small. In other words,
the distance between the particlesh over which repulsion effectively affects
the value ofWis very small relative to particle radiusa. A larger value ofh/
aleads to a higherW-value. Broadly speaking,h*/atends to be larger for a
smaller value ofa, and for very small particles, say one nm, the use ofVmax
as an activation free energy may be reasonable.
The Fuchs theory has been applied to colloids stabilized by
electrostatic repulsion. It can then be derived that the dependence of the
aggregation rate on salt concentration is roughly as given in Figure 13.4.
The critical concentration mcris defined as the value above which the
aggregation rate does not further increase; presumably, the stability factor
then equals about unity. As shown, the magnitude ofmcrgreatly depends on
the valence of the ions, even stronger than expected on the basis of the
Debye expression for the ionic strengthðI¼ð 1 = 2 ÞSmiz^2 iÞ, i.e., with the