Physical Chemistry of Foods

For the case of (hypothetical) spherical crystals and reaction-limited
crystallization, the Avrami equation takes the form

jðtÞ¼

1

3

pL^3 CJt^4 ð 15 : 5 Þ

where the growth rateLCand the nucleation rateJare assumed to be
constant; the equation will thus only apply in the very beginning of the
process. When the dependence ofLCandJon lnbis known, a more realistic
numerical relation betweenjandtcan be derived.
An example of experimentally observed growth curves, for various
values of theinitialsupersaturation lnb 0 , is given in Figure 15.12a; under
the conditions of the experiment, the hardened palm oil will behave almost
like a single solute (approximately like tripalmitate). It is seen that there is a
kind ofinduction time tindfor crystallization to take off. In principle,
tind! 1 =J, whether homogeneous, heterogeneous, or secondary nucleation
occurs. The growth rate then rapidly increases with time, for the most part
owing to an increase ofAC. At higher values ofjthe rate slows down
because the decrease of lnbbecomes overriding. A recalculation of the
results to the dependence ofLCon lnbis given in Figure 15.12b, assuming

FIGURE15.12 Rate of crystallization (in theb^0 -polymorph) of 12%hardened
palm oil in sunflower oil. (a) Dependence of the volume fraction of crystalsj(t)on
timetfor various values of the initial supersaturation lnb 0 (indicated near the
curves), varied by varying temperature. (b) Recalculated linear growth ratesLC
during the process as a function of supersaturation lnb; the hatched region contains
all the curves for lnb 0 ¼2.25–3.50. (After results by W. Kloek. Ph.D. thesis,
Wageningen University, 1998.)