Physical Chemistry of Foods

spherical crystals and takingJ(lnb) from independent measurements. It is
seen that the results almost fit a single curve, as is to be expected.

Crystal Size Distribution. Crystal size was briefly mentioned in
Section 14.2.3; Figure 14.7 gives a rough indication of the effects of
supersaturation on growth rate, nucleation rate, and crystal size.
If crystallization occurs in a steady state so that supersaturation is kept
constant—which can, e.g., be realized in a continuously operating crystal-
lizer—the following equation for the number frequency of crystals of radius
rwill hold:

fðrÞ¼

J

LC

exp

r

LCt



ð 15 : 6 Þ

See Section 9.3.1 for the mathematics of size distributions. t is a
characteristic time scale; it would be the mean residence time in the
crystallizer mentioned. A plot of log f(r) versusris a straight line of slope

1/LCt. This generally implies a wide size distribution. The number average
radiusr 10 is simply given byLCt. In a steady-state crystallizer, the average
size is thus independent of nucleation rate.
However, in many situationstwould be a kind of induction time for
nucleation and hence be proportional to 1/J. In such a case, the averager
then is proportional to LC/J, as may be intuitively expected. In most
situations, the relations will be more complicated, although the trends are as
given. If copious secondary nucleation occurs, the crystals formed will be
quite small, and it is not simple to achieve formation of large crystals.

Agitation. Often, crystallization from solution occurs while the
mixture is stirred or otherwise agitated. This generally increases the overall
crystallization rate, and it may also affect crystal size. The following effects
can be involved:

Sedimentation and aggregation of crystals is generally prevented.

For diffusion-controlled crystal growth, crystallization rate is

enhanced.

More effective removal of the heat of fusion results in a higher

supersaturation near the crystal surface.

Nucleation rate is generally enhanced because (a) the effective value of

lnb is larger; (b) any heterogeneous nucleation at macroscopic

surfaces will be enhanced; and (c) secondary nucleation can be

enhanced, be it due to contact nucleation, breaking off protrusions

from crystals, or ‘‘sweeping off’’ clusters of oriented molecules from

growing crystal faces.