Physical Chemistry of Foods

energy involved per pair of particles aggregating is given by

DGmix&kBTjðlnj
 1 Þ

This gives forj¼0.001 and 0.1, values of 0.008 and 0.33 timeskBT,

respectively. Since the attractive free energy generally amounts to

several timeskBT, this means that the loss in mixing entropy is in

most cases too small to affect the aggregation significantly.

However, for very small particles, say,d<0.1mm, the attractive

free energy tends to be much smaller, e.g., of the order ofkBT, and

the change in mixing entropy may then play a part, especially ifjis

not very small.

13.2.1 Perikinetic Aggregation

The termperikineticsignifies that the particles encounter each other because
of their Brownian motion or diffusion.

‘‘Fast’’ Aggregation. Smoluchowski has worked out a theory for
this case, assuming that particles will stick and remain aggregated when
encountering each other. He considered particles diffusing to a ‘‘central’’
particle. Because any particle colliding with the central one is, as it were,
annihilated, a concentration gradient of particles is formed. By solving
Fick’s equation (5.17) for spherical coordinates, he obtained for the fluxJperi
(in s
^1 ) of particles (2) toward a central particle (1)

Jperi¼ 4 pDpairRcollN ð 13 : 1 Þ

whereNis the number of particles per unit volume.Dpairis the mutual
diffusion coefficient of two particles, and it is assumed to equalD 1 þD 2.
Rcollis thecollision radius, which equals, for spheres,a 1 þa 2 .* For equal
spheres we can just use a single diffusion coefficientDand a single diameter
d. Since, moreover,D¼kBT= 3 pZd[Eq. (5.16)], the particle diameter is
eliminated from the result.
Assuming that each pairwise collision reduces the particle number by
unity, the change in particle concentration with time is obtained by
multiplyingJbyNand dividing by 2 to prevent counting every collision

• In this chapter we will use the symbolafor the radius of a spherical particle.