Physical Chemistry of Foods

dimensionality.* Actually, the equation should also contain a proportion-
ality constant; whenRis taken as the radius of the smallest sphere that can
contain the aggregate (as in Figure 13.7a), this constant is close to unity. The
fractal dimensionality has a noninteger value smaller than three, in the
example shown about 1.8. The equation is not precisely obeyed, but that is
because of statistical variation. The aggregate is therefore called astochastic
fractal.
Figure 13.7 has been produced by computer simulation of the
aggregation process. The simulations can mimic the experimental results
very well, provided (a) that the particles and clusters{move by Brownian
motion, implying that the trajectories have a given randomness; and (b) that
clusters form aggregates with other clusters, not merely with single particles
(‘‘cluster-cluster aggregation’’).

Consequences. The fractal nature of the aggregates implies two
important properties:

1. Scale invariance.The structure of the aggregates is scale-invariant
   or self-similar. The same type of structure is observed (albeit with statistical
   variation) at any length scale that is clearly larger thanaand smaller thanR.
   In other words, observations at various magnifications show roughly the
   same picture. This applies to different clusters as well as to different regions
   in one (large) cluster.


2. Larger clusters are more open(more tenuous or rarefied). This is
   shown as follows. Equation (13.12) gives the number of particles in a cluster.
   We can also calculate the number of sitesNsin the cluster, i.e., the number
   of particles in a sphere of radiusRif closely packed:

Ns¼

R

a

 3

ð 13 : 13 Þ

The volume fraction of particles in a fractal aggregate then is given by

jA:

Np

Ns

¼

R

a

D
3

ð 13 : 14 Þ

Since invariablyD<3 (in three-dimensional space), this means that the
particle density of the clusterjAwill decrease with increasing values ofRor
Np.

• Note that the same symbol is used for diffusion coefficient and fractal dimensionality.
  {The terms cluster and aggregate are used here as synonyms.