Physical Chemistry of Foods

concentration would be limited, and the larger a droplet is, the larger the
probability that it contains at least one catalytic impurity, which would be
sufficient to induce crystallization. Droplets devoid of catalytic impurities
would then freeze at the homogeneous nucleation temperature.
To understand how catalytic impurities may work, it will be useful to
consult Section 10.6.1 first. In Figure 14.5a an embryo is shown that may
lead to homogeneous nucleation, if it is small enough. If now a surface of a
materialkis present, an embryo may be formed on that surface, as depicted
in Figure 14.5b, provided that cosyis finite. Its value depends on the three
interfacial tensions (specific interfacial free energies) according to theYoung
equation(10.10), which can be written as

cosy¼

gak
gbk

gab

ð 14 : 11 Þ

At the same supersaturation (which generally means at the same
temperature), the curvature will be the same according to Eq. (14.9), and
this implies that the volume of the embryo is smaller. The volume of the
embryo divided by that of a sphere of the same radius is given by

fcat¼

1

4

ð 2 þcosyÞð 1 
cosyÞ^2 ð 14 : 12 Þ

which value is smaller for smallery. The value ofDGmaxas given in Eq.
(14.7) should now be multiplied byfcat.

Heterogeneous Nucleation Rate. Insertingfcatinto Eq. (14.10)
then would give the rate of heterogeneous nucleationJhet. This would apply
for a high concentration of identical catalytic impurities. In practice, this is
rarely the case. The concentration of impurities will generally determine the
magnitude of the preexponential factor. The particles present may be of
various materials, leading to a range ofyvalues, as illustrated in Figure
14.5b, frames 1 and 2. The particles may also have different shapes, and
frame 3 shows how that can affect the size of the embryo. Moreover, as
depicted in frame 4, the boundarya–bmay even be flat (depending on the
shape of the indentation), which would mean that nucleation can occur at
zero undercooling. The effectiveness of particles in catalyzing nucleation will
thus show considerable statistical variation, which implies that the lower the
temperature (or the higher the supersaturation), the greater the number of
impurities that are catalytic.