Biomedical Engineering Reference
In-Depth Information
Theoretical treatment of crystal growth mechanisms in dilute environments at
the atomic scale was given in the form of the Burton-Cabrera-Frank (BCF) model
in 1951 [
26
]. The growth rate of crystal face
R
in which the growth unit goes
through the surface diffusion before incorporating in the growth step is expressed as
a function of the driving force of growth (supersaturation,
):
R
D
C
2
1
tanh
1
;
(3.1)
where
C
and
1
are constants determined by the growth conditions. A characteristic
of this formula is that it is possible to approximate
R
2
at low supersaturation
and
at high supersaturation, which was frequently used for discussing the
growth mechanisms.
The research conducted on NaClO
3
is an example in which detailed analysis
of growth mechanism using the BCF model was performed on crystal grown
from aqueous solution [
27
]. The relationship between the face growth rates and
supersaturation was investigated with precise temperature control (
R
1/100
ı
Cof
fluctuation) at low supersaturation (
1%), and the data were analyzed using (
3.1
).
It was concluded that spiral growth via surface diffusion occurred. This was the
first time that spiral growth via surface diffusion in aqueous solution was quantita-
tively shown. Denk and Botsaris performed the measurements of potassium alum
(K-Alum) under high-supersaturation conditions and concluded that the data could
be interpreted by considering the combination of spiral growth and two-dimensional
nucleation [
28
]. Subsequently, growth mechanisms approximated using face growth
rate data and growth theory equations became widely used, and the growth
mechanisms of numerous aqueous inorganic and organic crystals were elucidated
(for example, [
29
]).
These studies focused on two important topics. The first was the growth mode
in an aqueous solution system (spiral or two-dimensional nucleation growth), and
the second was the investigation of the rate-limiting process of growth, in which
emphasis was placed on the assertion that the importance of surface diffusion that
had originally been applied to crystals grown in the gas system also holds for
aqueous solution systems. Chernov proposed a model in which the growth unit is
directly incorporated from the environmental phase into growth steps [
30
]. This
model showed that the surface diffusion of the growth unit is not important in a
solution system, particularly an aqueous solution system.
Following the tests of the spiral growth model by measuring the crystal growth
rate, attempts to obtain direct evidence for spiral growth from surface observations
were actively conducted from the 1970s onward, even for aqueous solution crystals.
Examples include the observations of NaCl and KCl [
31
], K-Alum [
32
], potassium
hydrogen phthalate (KAP) [
33
], and NiSO
4
6H
2
O[
34
].
In a study of crystal growth mechanisms, van Erk et al. vividly showed the
importance of surface observations although not in regard to crystals grown in
aqueous solutions. They measured the growth rate of garnet crystals grown by
liquid phase epitaxy and concluded that spiral growth through surface diffusion had
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