Physical Chemistry of Foods

Note This does not imply that the density decreases when going

from the center to the periphery of a cluster: that would be

incompatible with the scale-invariance.

The decrease in particle density of the clusters as they grow in size has
another important consequence, viz:

3. Gel formation. The volume fraction of particles in an aggregatejA
   [Eq. (13.14)] decreases until it equals the volume fraction of particles in the
   whole systemj. At that point, the system can be considered to be fully
   packed with aggregates, which thus form a space-filling network, spanning
   the vessel containing the system, implying that a gel results. The average
   radius of the aggregates at the moment of gelationRgthen is given by

Rg¼aj^1 =ðD
^3 Þ ð 13 : 15 Þ

The average number of particles in an aggregate of radiusRgis given by

Np;g¼jD=ðD
^3 Þ ð 13 : 16 Þ

Np,gandRgare smaller for largerjand smallerD. For example, the values
j¼0.1 andD¼2 yieldNp,g¼100, andRg¼ 10 a. This implies that a gel
would soon be formed after aggregation starts.
The reader should be cautioned that Eqs. (13.12)–(13.16), as well as
any derived equations, are in factscaling relations. They should contain
numerical constants in order to be precise. These constants generally do not
differ greatly from unity.

Another point is that not all aggregation leads to fractal structures.
Conditionsfor the formation of perfect fractal aggregates and of a space-
filling network are following:

During aggregation the liquid should be quiescent (perikinetic

aggregation), since velocity gradients can deform and possibly

break up aggregates. In practice, one often adds a reagent to a sol to

induce aggregation, e.g., acid, salt, or ethanol, while stirring to

disperse it throughout the liquid. In the meantime, aggregation

occurs, and fractal clusters formed are maltreated and tend to

become compacted. The resulting dense aggregates will generally

sediment; in other words, precipitation occurs. To obtain a gel, the

transition of the particles from stable to reactive (i.e., prone to

aggregation) has to occur in the absence of significant velocity

gradients.