Physical Chemistry of Foods

vNcat>5,ymax&1. The results shown in Figure 14.6a have been obtained in
this way; by using emulsions of variousvvalues, the validity of Eq. (14.14)
can be checked.
From results onyas a function of timetatTc, the nucleation rateJ
can be derived (assuming the time needed for crystallization in a droplet
once a nucleus has been formed to be relatively short). However, it has been
observed thatJdecreases with time, approximately according to

J¼J 0 ð 1 

y

ymax

Þð 14 : 15 Þ

A decrease inJis to be expected, since the catalytic impurities will show
variation in efficiency and the most efficient ones tend to act first. Actually,
this makes the concept of a fixed Ncat value at a given temperature
questionable, since the value will depend on the duration of the experiment.
Equation (14.15) is an empirical one; it is well obeyed in one kind of
system (triglyceride mixtures), but whether it applies in the same form to
other systems is uncertain. Assuming it to be correct and combining it with
Eq. (14.14), integration leads to

y¼

1 
expð
vNcatÞ

1 þ 1 =J 0 vt

¼

ymax

1 þ 1 =J 0 vt

ð 14 : 16 Þ

Note that the number of catalytic impurities per dropletvNcatwill determine
what proportion of the droplets will eventually contain crystals; the initial
rate of formation of nuclei per dropletJ 0 vwill determine the value ofy/ymax
as a function of time. Naturally, the values ofJ 0 and ofNcatare correlated in
a given system; see e.g., Figure 14.6. The time needed to obtain a certain
ratio ofyoverymaxgreatly varies, e.g., between a minute and some hours for
a ratio of 0.8. According to Eq. (14.16), the time will increase with
decreasing value ofvfor a given value ofT; it also tends to increase with
decreasing supersaturation (increasing temperature) for a given value ofv.
Figure 14.9b. gives some examples ofymax(T) for various values ofv,
calculated for heterogeneous nucleation. Note that near 5 8 C, homogeneous
nucleation will take over. It is seen that droplet size has a very large effect on
the extent of crystallization at a given temperature.

Surface nucleation. Thus far, we have only considered volume
nucleation, implying that the number of catalytic impurities in a droplet is
proportional to its volume. Another possibility is nucleation at the inside of
the droplet boundary, if that surface is catalytic for nucleus formation. This
is called heterogeneous surface nucleation. An example of the resulting
relation betweenymaxandTcis given in Figure 14.9b, curve S. Over a very