Page 204 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

P1: GAE/LSK P2: FLV Final Pages
Encyclopedia of Physical Science and Technology EN004D-ID159 June 8, 2001 15:47

Crystallization Processes 101

1. Mechanisms An empirical approach also can be used to relate growth
kinetics to supersaturation by simply fitting growth-rate
Among the many models that have been proposed to de-
data with a power-law function of the form:
scribe surface-reaction kinetics are those that assume crys-
tals grow by layers and others that consider growth to oc- G = k G σ g (25)
cur by the movement of a continuous step. Each physical
model results in a specific relationship between growth where k G and g are system-dependent constants. Such an
rate and supersaturation and, although none can predict approach is valid over modest ranges of supersaturation,
growth kinetics a priori, insights regarding the effects of and the power-law function approximates the fundamental
process variables on growth can be obtained. Because of expressions derived from the above models.
the extensive literature on the subject, only the key aspects
of the physical models and (in one case) the resulting re-
2. Impurities
lationship between growth and supersaturation predicted
by each theory will be discussed here. The presence of impurities can alter growth rates substan-
The model used to describe the growth of crystals by tially, usually by decreasing them. Furthermore, as de-
layers is based on a two-step, birth-and-spread mecha- scribed in Section IV.B, impurities can alter crystal mor-
nism. In one of the steps (birth) a two-dimensional nu- phology through their effects on the growth rates of crystal
cleus is formed on the crystal surface, and in the sec- faces. Mechanisms include: (1) adsorption of an impurity
ond step (spread) the two-dimensional nucleus grows to on the crystal surface or at specific growth sites such as
cover the crystal surface. When one or the other of the kinks, thereby blocking access to the site by a growth
steps is controlling growth rates, simplifications of the unit; (2) formation of complexes between an impurity and
more complicated dependence of growth rate on super- a growth unit; and (3) incorporation of an impurity into a
saturation can be developed to give what are known as growing crystal and creating defects or repelling the addi-
the mononuclear two-dimensional nucleation theory and tion of a growth unit to the subsequent crystal layer. Few
the polynuclear two-dimensional nucleation theory. of these mechanistic views result in predictive capabili-
In the mononuclear two-dimensional nucleation theory, ties, and it is usual to rely on experimental data that are
surface nucleation occurs at a finite rate while the spread- often correlated empirically.
ing across the surface occurs at an infinite rate. The re- Because impurities most often result in reduced crys-
verse is true for the polynuclear two-dimensional nucle- tal growth rate, feedstocks to laboratory and bench-scale
ation theory. Theoretical relationships have been derived units should be as similar as possible to that expected in the
between growth rate and supersaturation for each of these full-scale unit. The generation of impurities in upstream
conditions but are considered beyond the scope of this process units can depend on the way those units are oper-
discussion. ated, and protocols of such units should follow a consistent
The screw-dislocation theory (sometimes referred to as practice. It is equally important to monitor the composi-
the BCF theory because of its development by Burton, tion of recycle streams so as to detect any accumulation of
Cabrera, and Frank) is based on a mechanism of continu- impurities that might lead to a reduction in growth rates.
ous movement in a spiral or screw of a step or ledge on the The solvent from which a material is crystallized influ-
crystal surface. The theory shows that the dependence of ences crystal morphology and growth rate. These effects
growth rate on supersaturation can vary from a parabolic have been attributed to two sets of factors. One has to do
relationship at low supersaturations to a linear relationship with the effects of solvent on viscosity, density, and dif-
at high supersaturations. In the BCF theory, growth rate is fusivity and, therefore, mass transfer. The second factor
given by: is concerned with the structure of the interface between
crystal and solvent; a solute–solvent system that has a
2
σ b high solubility is likely to produce a rough interface and,
G = k G tanh (24)
b σ concomitantly, large crystal growth rates.
where is screw dislocation activity and b is a system-
dependent quantity that is inversely proportional to tem- E. Crystal Growth in Mixed Crystallizers
perature. It can be shown that the dependence of growth
Population balances on crystals in a crystallizer require a
rate on supersaturation is linear if the ratio b/ σ is large,
definition of growth rates in terms of the rate of change of
but the dependence becomes parabolic as the ratio be-
a characteristic dimension:
comes small. It is possible, then, to observe variations in
the dependence of growth rate on supersaturation for a dL
G = (26)
given crystal-solvent system. dt