Chemistry Reference
In-Depth Information
free enthalpy of the system has a minimum. In this case:
R
hc
=
R
R
c
+
R u
(
R
)
γ
d
G
d
R
2
πγ
1
−
=
0
,
(5.17a)
d
R
2
R
hc
=
1
R
c
−
d
2
G
u
(
R
)
γ
R
γ
d
u
d
R
2
πγ
−
>
0
,
(5.17b)
where
u
(
R
)
γ
R
F
R
H
1
=
R
H
.
(5.18)
1
+
R
2
/
Using Eqs. (5.17a) and (5.17b) and assuming that
R
hc
,
R
c
and
R
F
are all positive,
the following expression can be derived:
2
R
hc
1
2
>
R
F
>
R
hc
1
R
hc
R
c
R
hc
R
c
+
+
.
(5.19)
From this expression we observe that
R
F
>
R
hc
. The terrace connecting two spirals,
one left-handed and the other right-handed, will grow indefinitely if the diameter of
the critical 2D nucleus, 2
R
c
, is less than the distance between the emerging points of
the two spirals (Frank, 1949). The minimum measured separation between coupled
spirals is about 50 nm, so that
R
c
25 nm (see Section 5.1). Asymmetric hollow
cores can be clearly seen in Fig. 5.13 and are in fact composed of two single hollow
cores
of the same sign
separated by about 2
R
hc
, as can be inferred from the figure.
The step height of each spiral is the modulus of
Λ
, and the superposition of the
spirals, even if they originate from two different hollow cores, is probably facilitated
by the layered structure of the
<
-phase. Positive/negative sign of curvature means
that, after the emergence of the spiral, it maintains/changes the sign of rotation.
Figure 5.12 is an example of spirals with positive sign of curvature: one right-
handed and one left-handed. In fact, the Frank-Read mechanism of growth, which
governs the evolution of the spirals, forces the change of sign of the curvatures of
the spirals upon the advance of the step. The positive sign of curvature illustrated in
Fig. 5.12 will turn to negative after the two hollow cores are connected by a straight
terrace. With the advance of the step, the spirals will have negative sign (see e.g.,
centre of Fig. 5.11) which will turn back to positive when the situation shown in
Fig. 5.12 is restored. In Fig. 5.13 we find examples of two spirals emerging from
hollow cores, each with a different curvature.
Changes in curvature are related to the high elasticity of the material as well
as to its layered structure. The high elasticity is suggested by several examples of
screw dislocations reaching step heights of 1.2 nm very rapidly, that is very close
to the hollow cores (about 30-40 nm away from the centre of them). At larger
distances the steps adopt different shapes. This is due to the strong decay of the
α
