Chemistry Reference
In-Depth Information
Figure 20.
AFM images showing the birth of a new step segment on calcite. In (a), the smallest step is
below the critical length and is not moving. In (b) it has exceed the critical length and begun to move. A
new sub-critical step segment now appears. (Modified after Teng et al. 1998)
where
L
is the length of the step,
b
is the 6.4 Å intermolecular distance along the step,
c
is
the 3.1 Å distance between rows,
γ
B
and
γ
B
are the step edge free energies along the + and
+
−
P
are contributions to the step edge free energy from the corner
sites as illustrated in Figure 1c. (When step curvature is included, the terms on the right
hand side of Eqns. 20b and 20c become integrals of
P
and
−
steps,
and
γ
B
, γ
γ
++
−−
+−
γ
as a function of orientation.)
±
Setting
shows that free energy only decreases if
the length of the step exceeds a critical value,
L
B
c
B
, given by:
L
B
c±
B
= 2
bc
<
∆
g
to zero and substituting k
T
σ
for
∆µ
γ>
B
/
k
B
b
B
T
σ
(21)
±
Alternatively, setting
g
= 0 for a step of arbitrary length gives the length dependent
equilibrium activity,
a
B
e
B
(
L
):
∆
a
B
e
B
(
L
) =
a
B
e,
B
exp(2
bc
<
γ>
/
Lk
B
b
B
T
)
(22a)
L
B
c
B
/
L
) (22b)
where the subscript refers to the infinitely long step and
a
and
a
B
e,
B
are related by
a
=
a
B
e,
B
exp(
=
a
B
e,
B
exp(
σ
). Equations (22a) and (22b) are statements of the Gibbs-Thomson effect. It
predicts that, even for the growth spiral, the critical length should scale inversely with the
supersaturation as in Equation (21). Figure 21 shows that the results for calcite agree with
the predicted dependence. The slopes of the lines give the values of <
σ
γ>
.
±
Figure 21 also shows that
L
B
c±
B
vs. 1/
σ
exhibits a non-zero intercept. It marks the
supersaturation,
1D nucleation along the step edge
lowers the free energy of the steps. This supersaturation gives the approximate free energy
barrier,
g
B
1D±
B
, to formation of a stable dimer on the step edge through
g
B
1D±
B
=
kT
σ
c±
B
where
L
B
c±
B
goes to zero. For
σ
>
σ
,
c±
B
σ
c±
B
(Chernov
1998)
.
The Gibbs-Thomson effect also affects step speeds for short step segments.
Combining Equations (6), (18), and (22) leads to a length dependent step speed:
v
B
±
B
(
L
) =
v
B
±
B
{1
[e
P
(σ
L
c±
/
L
)
P
1]/[e
P
σ
P
−
−
−
1]}
(23)
where
v
B
±
B
is given by Equation (18) for
a
B
e
B
(
L
→
). When
σ
<< 1, Equation (23) reduces
L
B
c
B
/
L
) (Rashkovich, 1991). Unfortunately,
in the few cases that have been investigated, the prediction of Equation (23) has turned
out to be incorrect (See for example, Teng et al. 1998). Figure 22 shows the predicted
dependence of
v
P
+
P
/
v
B
B
on
L
/
L
B
c
B
for calcite along with the measured dependence. The
measured speed is independent of supersaturation or step direction and rises much more
rapidly than that predicted by Equation (23).
to a commonly used approximation:
v
=
v
B
B
(1
−
