15.2.3 Overall Crystallization Rate
In practice, one is generally not so much interested in the linear growth rate
LCas in the amount (mass or volume) of crystals formed per unit volume
and unit time. In principle, the latter can be given asLC 6 AC, whereACis
the specific surface area of the crystals. However, both factors tend to vary
with time. Generally,LCdecreases because crystallization implies depletion
of solute and hence a decrease of supersaturation. Moreover, release of the
heat of fusion may cause the temperature to increase significantly, hence the
solubility to increase, hence lnbto decrease.ACincreases because (a) each
crystal increases in size, and (b) more crystals are formed if nucleation goes
on. Several, often complicated, growth rate theories have been worked out
for various conditions. We will only touch on a few aspects.
Nucleation and Growth. A common situation is when a
homogeneous solution (or melt) is rapidly cooled to a constant
temperature where lnbis large enough to induce nucleation. Then the
semiempiricalAvrami equationoften applies:
jðtÞ¼ 1 expðKtnÞ&Ktn ð 15 : 4 Þwherej¼volume fraction of crystals,t¼time, andKandnare about
constant for a given system. The part after the&sign applies for smallj,
say<0.1. The exponentnranges between 0.5 and 4. It is larger for reaction-
limited than for diffusion-limited crystallization; larger for constant-rate
than for depletion-dependent nucleation; and smaller for a more anisometric
crystal shape.
TABLE15.1 Rate of Growth of Some Faces ofa-Lactose Monohydrate Crystals
for Various Solvent Compositions
Supersaturation Growth (nm?s^1 ) of facea
lnb Remarks on composition 010 110 100 110 0 11
0.44 1.05 0.92 0.36 0.08 0.00
0.44 þ10 ppm gelatin 0.33 0.28 0.28 0.11
0.44 þ100 ppm riboflavin 0.75 0.00 0.00 0.00 0.00
0.79 12 9.4 5.8 3.3 1.9
aIndicated by Miller indices.
Source: After results by A. S. Michaels and A. van Kreveld. Neth. Milk Dairy J. 20 (1966) 163.