Physical Chemistry of Foods

These relations imply that foru!0 alsoD!0, or in other words the
particles would never touch each other. Although the theory breaks down
for very small distances, the aggregation rate will be significantly decreased.
How then is it possible that it has been observed that particles can aggregate
at a rate predicted by Eq. (13.2) for fast aggregation? The answer is that
attractive forces acting at small interparticle distances will enhance the
mutual diffusion rate, just as repulsive forces decrease the rate. Sample
calculations have shown that for particles that attract each other by van der
Waals attraction (other interaction forces being negligible), the attraction
induced increase and the hydrodynamically induced decrease of the
aggregation rate may compensate each other.
If repulsive interaction is significant, Eq. (13.5) should be modified to
include the retardation due to hydrodynamic interactions. The result is

W& 2

Z?

2

D?

DðzÞ

z
^2 exp

VðzÞ

kBT

dz ð 13 : 7 Þ

whereD?/D(z) can be obtained from Eq. (13.6) withu¼z
2. The result
for the case of Figure 13.3, i.e., the integrandy*, is given in the figure (note
the difference in scale). From Eq. (13.7), we now calculateW¼24,000, i.e.,
larger than the result according to Eq. (13.4) by a factor 65, but still smaller
by a factor 15 than exp (Vmax/kBT).

Reaction-Limited Aggregation. The validity of Eq. (13.7) can be
questioned. It is unlikely to be correct if the particles are not very small (say,
a>0.1mm), and moreover the Fuchs stability ratioWis of considerable
magnitude. The reason is that for a small value ofDpair, combined with a
significant repulsive barrier, not only the close approach but also the drifting
apart of two particles is greatly slowed down. This means that a significant
concentration gradient of particles near a ‘‘central’’ particle cannot form. In
other words, the system is almost ideally mixed, implying that the
aggregation would proceed as in the classical theory for bimolecular
reactions, discussed in Section 4.3.3. In principle, a relation comparable to
Eq. (4.11) would apply, but it has not yet been worked out.
We now can distinguish three situations (regimes):

1. Diffusion-controlled aggregation according to Smoluchowski, i.e.,
   Eq. (13.1).


2. The Fuchs treatment, if the colloidal interaction curve is known,
   and as modified by hydrodynamic retardation, i.e., use of Eq.
   (13.7). This equation can be considered as a solution of the
   problem discussed in Section 4.3.5.