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Encyclopedia of Physical Science and Technology EN004D-ID159 June 8, 2001 15:47

98 Crystallization Processes

The simplicity of Eq. (13) results in the use of relative liquor transformed to a slurry of very fine crystals with
supersaturation in most empirical expressions for nucle- only a slight increase in supersaturation, for example by
ation and growth kinetics. While beguilingly simple, and decreasing the solution temperature.
correct in limiting cases, great care should be taken in ex- The effect of exogenous solid matter (as in heteroge-
tending such expressions beyond conditions for which the neous nucleation) in the supersaturated solution is equiv-
correlations were developed. alent to that of a catalyst in a reactive mixture. Namely,
ν
For ionic solutes, a i = a , which leads to S a i = it is to reduce the energy barrier to the formation of a
±
∗ ν
(a ± /a ) and new phase. In effect, the solid matter reduces the interfa-
±
cial energy surf by what may amount to several orders of
µ i γ i± C i
= ν ln (14) magnitude.
= ν ln S a i
∗
RT γ C ∗
i± i The classical nucleation theory embodied in Eq. (16)
has a number of assumptions and physical properties
∗
Again, for γ i± ≈ γ ,
i±
that cannot be estimated accurately. Accordingly, empir-
µ i C i ical power-law relationships involving the concept of a
≈ ν ln = ν ln S i (15)
RT C ∗ metastable limit have been used to model primary nucle-
i
ation kinetics:
n
◦
B. Primary Nucleation B = k N σ max (17)
The term primary nucleation is used to describe both ho- where k N and n are parameters fit to data and σ max is
mogeneous and heterogeneous nucleation mechanisms in the supersaturation at which nuclei are observed when
which solute crystals play no role in the formation of new the system is subjected to a specific protocol. Although
crystals. Primary nucleation mechanisms involve the for- Eq. (17) is based on empiricism, it is consistent with the
mation of crystals through a process in which constituent more fundamental Eq. (16).
crystal units are stochastically combined. Both homo-
geneous and heterogeneous nucleation require relatively C. Secondary Nucleation
high supersaturations, and they exhibit a high-order de-
pendence on supersaturation. As will be shown shortly, Secondary nucleation is the formation of new crystals
the high-order dependence has a profound influence on through mechanisms involving existing solute crystals; in
the character of crystallization processes in which primary other words, crystals of the solute must be present for
nucleation is the dominant means of crystal formation. secondary nucleation to occur. Several features of sec-
The classical theoretical treatment of primary nucle- ondary nucleation make it important in the operation of
ation that produces a spherical nucleus results in the ex- industrial crystallizers: First, continuous crystallizers and
pression: seeded batch crystallizers have crystals in the magma that
can participate in secondary nucleation mechanisms. Sec-
3 2
16π surf v ond, the requirements for the mechanisms of secondary
B ◦ = A exp −
3
3
3k T [ln(σ + 1)] 2 nucleation to be operative are fulfilled easily in most in-
dustrial crystallizers. Finally, many crystallizers are oper-

σ<0.1 16π 3 surf v 2 ated in a low supersaturation regime so as to maximize
≈ A exp − (16)
3
3
3k T σ 2 yield, and at such supersaturations the growth of crystals
is more likely to produce desired morphologies and high
where k is the Boltzmann constant, surf is the interfacial purity; these low supersaturations can support secondary
surface energy per unit area, v is molar volume of the nucleation but not primary nucleation.
crystallized solute, and A is a constant.
The theory shows that the most important variables af-
1. Mechanisms
fecting the rates at which primary nucleation occur are
interfacial energy surf , temperature T , and supersatura- Secondary nucleation can occur through several mecha-
tion σ. The high-order dependence of nucleation rate on nisms, including initial breeding, contact nucleation (also
these three variables, especially supersaturation, is impor- known as collision breeding), and shear breeding. Al-
tant because, as shown by an examination of Eq. (16), a though a universal expression for the kinetics of secondary
small change in any of the three variables could produce an nucleation does not exist, a working relationship often can
enormous change in nucleation rate. Such behavior gives be obtained by correlating operating data from a crystal-
rise to the often observed phenomenon of having a clear lizer with a semi-empirical expression. Guidance as to the