Physical Chemistry of Foods

twice. The result is

dN

dt

¼

1

2

JperiN¼ 2 pDpairRcollN^2 ¼ 4 pDdN^2 ¼

4 kBT

3 Z

N^2 ð 13 : 2 Þ

where the last two equalities are valid for equal-sized spheres only. A useful
quantity is thehalving time t0.5, i.e., the time needed for halving the number
of particles, which is given by

t 0 : 5 ¼

2

J 0

¼

3 Z

4 kBTN 0

¼

pd^30 Z

8 kBTj

ð 13 : 3 Þ

whereJ 0 ,N 0 , andd 0 are the initial values ofJ,N, andd, respectively. The
second part of the equation is obtained by using the relationj¼pd^3 N=6,j
being the volume fraction of particles.
Of course, the aggregation rate decreases as aggregation proceeds,
sinceNkeeps decreasing (and the aggregate size increasing). Nevertheless,
the number concentration as a function of time follows a simple relation,
viz.,

Nt¼

N 0

1 þt=t 0 : 5

ð 13 : 4 Þ

Incidentally, this relation is valid in every case where
dN=dt!N^2. Figure
13.2 gives a calculated example of the decrease inNwith time, as well as the
concentration of doublets, triplets, etc. Notice that doublets do not only
form but also disappear again by aggregation.
For spherical particles in water at room temperature, Eq. (13.3)
reduces tot 0 : 5 & 0 : 1 d^3 =jseconds, ifdis expressed inmm. Assumingd¼ 1 mm
andj¼0.01, the halving time would be only 10 seconds, implying that
aggregation proceeds very fast. Note that the initial aggregation rate
strongly increases with decreasing particle diameter for constant volume
fraction. The observed results for unhindered perikinetic aggregation
generally agree well with the Smoluchowski predictions for the first few
aggregation steps.
The relations are different if the collision radius is larger than the
hydrodynamic radius determining the diffusion coefficient [although Eqs.
(13.1) and (13.4) would still be valid]. Such a difference between radii occurs
if the particles are strongly anisometric or irregularly shaped, and the
aggregation rate will then be faster than for spherical particles of the same
volume. For the same reason, polymer chains that extend from the particles’
surface and that can cause interparticle bridging, may markedly increase
perikinetic aggregation rate over the Smoluchowski prediction.Polydisper-