Chemistry Reference
In-Depth Information
Equation (23) was originally derived for spirals with isotropic step edge energies and
kinetics, and assumes a high kink site density such that the rate of attachment is not
limited by the availability of kinks. Voronkov (1973) proposed that for highly
polygonized spirals, the distance between kink sites along a step is large so that the
attachment rate is limited by their availability. He pointed out that the equilibrium shape
of a step should be curved towards the corners and that the straight, central portion—
which determines the speed—comprises only a small fraction of the step. (This prediction
of step shape is verified in Fig. 20.) When the neighboring step advances, there is a rapid
increase in the relative size of the straight portion and hence in the number of available
kink sites. As a result, when
L
exceeds
L
B
c
B
, the step speed rises rapidly to its limiting value
as its length increases from
L
B
c
B
to a length
L
' whose order of magnitude is given by
L
B
c
B
+
2
,
h
and
c
specific to calcite (see Teng et al.
1998), we obtain
L
ƍ/
L
B
c
B
~ 1.1, a prediction which is consistent with the results in Figure 22.
This result calls into question the use of data like those of Figure 21 to derive step edge
energies by applying the Gibbs-Thomson formalism. Unfortunately, this is currently an
unresolved area of crystal growth science.
Both the existence of a critical length and the length dependence of the step speed
impact the terrace width and hence the growth rate. Consider an isotropic square spiral. If
the step speed is zero for
L
<
L
B
c
B
and jumps discontinuously to its maximum step speed,
v
B
B
at
L
=
L
B
c,
B
then the terrace width would be given by 4
L
B
c
B
(Burton et al. 1951; Rashkovich
1991). However, because the step speed rises gradually from zero to
v
B
B
, the terrace width is
larger by a factor that we will refer to as the Gibbs factor,
G
. Equation (23) leads to
G
= 2.4
for small
Ω
/
ch
σ
. Substituting in the constants of
Ω
.
For the rhombohedral spiral of calcite, the terrace widths for the two step directions
become:
σ,
giving a terrace width of 9.6
L
B
c
B
, which decreases to
G
= 1 in the limit of large
σ
W
B
+
B
= 2
G
(1+
B
)<
L
B
c
B
>/sin
θ
(24a)
W
B
−
B
= 2
G
(1+1/
B
)<
L
B
c
B
>/sin
θ
(24b)
where
B
is
v
P
+
P
/
v
P
−
P
, <
L
B
c
B
> is the average value of
L
B
c
B
for the two step directions and
is the
angle between adjacent turns of the spiral. Figure 23 shows the measured dependence of
W
B
±
B
on
θ
σ
as well as
W
= 9.6<
L
B
c
B
> obtained using Equation (24) at small
σ
. As predicted,
the terrace widths scale inversely with
, but due to the anomalously rapid rise in
v
(
L
),
the measured
G
factor is close to one. As a consequence, the growth rate of the calcite
surface is about 2.5 times that predicted from classical growth theory (Burton et al. 1951;
Chernov 1961).
σ
MODIFYING THE SHAPES OF GROWTH
HILLOCKS AND CRYSTALS
The shapes of crystals are controlled by a combination of energetic and kinetic
factors (Burton et al. 1951; Chernov 1961). The equilibrium shape is that which
minimizes the total surface free energy of the crystal, which is in turn the sum of the
individual products of surface area times interfacial energy (see Fig. 24a). Thus low
energy faces are preferentially expressed. On the other hand, crystal shape is rarely
achieved by equilibration in a solution at equilibrium conditions, but rather during
exposure to growth conditions. The slowest growing faces become the largest, and
rapidly growing faces either become small or disappear altogether. Not surprisingly, the
faces which are the low energy faces tend to also be those that grow slowly. Thus faces
expressed during growth tend to be those expressed at equilibrium as well, although the
sizes of the faces depend more critically on kinetic factors as well as the details of the
