Chemistry Reference
In-Depth Information
Derivation of the step speed dependence on concentration under many different
assumptions leads to a similar relationship, the difference lying in the physical
parameters of which
is comprised. One important exception is in the case of smooth
steps. Then the density of kink sites depends on concentration increasing as (
C
/
C
B
e
B
)
P
1/2
P
for
sufficiently small
β
and one can no longer replace the expression for the detachment rate
with the equilibrium expression. Then Equation (18) becomes:
σ
C
B
e
B
f
(
n
B
1
B
)] (19)
where
n
B
1
B
is the density of nucleated kinks on a step edge and the function
f
(
n
B
1
B
) rises
linearly at low
v
=
Ωβ
[(
C
−
C
B
e
B
)
−
and approaches a constant at high concentrations. In other words, the
step speed is non-linear at low concentrations, but becomes linear at high concentrations
and then just looks like the rough limit with a lateral offset, as shown in Figure 17. Note
the similarity of
v
vs.
C
for the smooth limit to the data on calcite shown in Figure 16.
One of the important consequences of Equations (18) and (19) is that the step speed
scales with the
U
absolute
U
supersaturation, not the
U
actual
U
supersaturation. This means that if
two types of crystals are placed in solutions of the same supersaturation, the crystal that is
more soluble will grow faster than the other, simply because there is a larger flux of
molecules to the surface. In other words,
step speed scales with solubility
and one cannot
assume that faster growth rates imply faster kinetics at the kink sites.
Step generation: 2D nucleation vs. growth at dislocations
So far we have assumed that steps are pre-existing on a crystal surface. But without a
new source of steps, any pre-existing steps would rapidly grow out to the edge of the
crystal leaving a featureless terrace and no source for growth (For a demonstration of this
phenomenon see Rashkovich 1991). One way to generate new steps is to generate two-
dimensional islands of molecules that then spread outward (see Fig. 13a). But as in the
case of three-dimensional nucleation, there is a critical size and a free energy barrier
associated with this process, although in this case they are due to the excess free energy
of creation of the new step edge rather than creation of a new surface (see Fig. 13b). (We
will derive the equivalent 2D nucleation expressions later.) As Figure 18 shows, at
sufficiently high supersaturations, crystal surfaces do grow from solutions by this
mechanism (Teng et al. 2000; Land et al. 1997; Land and De Yoreo 1999). (Table 2
σ
Figure 17.
Dependence of step speed on
concentration in the rough limit (Eqn. 18)
and when steps are smooth due to low kink
density (Eqn. 19).
