Chemistry Reference
In-Depth Information
Fig. 1.4
Plot of the local
orientational order parameter
square
2
vs.
x
for
thesamesystemasin
Fig.
1.3
,for
T
|
ψ
(
x
)
|
6
=
1
.
2(
top
part
)and
T
6(
bottom
part
).
Circles
refer to the
planar wall and
square
to the
structured wall boundary
condition
=
1
.
that diverges at the fluid-hexatic transition [
7
]). From Fig.
1.4
it is clear that for
strips of any finite width
D
the melting transition is strongly affected by finite size
rounding: the walls induce a significant orientational order also in the fluid phase,
and so the average mean square order parameter
d
x
ψ
2
2
D
decays
gradually with increasing temperature and stays nonzero also in the fluid phase for
any value
D
6
=
|
ψ
6
(
x
)
|
/
.
In contrast to this enhancement of order due to the walls, seen in Figs.
1.3
and
1.4
,
a different behavior results when we study the positional order in the
y
-direction par-
allel to the walls. Defining for a crystal row in the
y
-direction the static structure fac-
tor
S
<
∞
(
q
)
, assuming also an orientation of the wavevector
q
along the
y
-direction, as
,
y
)
]
S
(
q
)
=
(
1
/
n
y
)
exp
[
i
q
(
y
−
(1.4)
=
one obtains at
T
1 (i.e., far below the melting temperature) the behavior shown in
Fig.
1.5
: for structured walls, the expected picture for a crystal indeed results: There
are sharp Bragg peaks at the positions
qd
a
0
now, dis-
tinct from the background (“thermal diffusive scattering”). As expected, the height
of these peaks decreases with increasing
q
, due to the effect of the Debye-Waller
/
2
π
=
1
,
2
,
3
,...
, with
d
=
