Chemistry Reference
In-Depth Information
Fig. 1.3
Density distribution
ρ(
plotted vs.
x
for a
system of 900 particles in a
D
x
)
L
geometry, with
periodic boundary conditions
in the
y
-dire
c
tion,
L
×
n
y
a
0
,
D
=
n
x
a
0
√
3
/
2, with lattice
spacing
a
0
=
(
=
/
√
3
1
/
2
ρ)
=
.
2
1
049.
Top part
refers to
T
=
1
.
2
and
bottom part
refers to
T
6. Only the
left part
of
the strip is shown in each
case (the center of the strip,
x
=
1
.
2, is marked by a
vertical line
). A strip with
“planar wall” and with
“structured wall” boundary
condition is included, cf. text,
as well as the density
distribution of a
corresponding bulk system
=
D
/
this correlation length in the liquid phase, one finds a pronounced increase as the
temperature is lowered toward the melting transition temperature [
5
], as expected
due to the KTHNY [
6
-
8
] character of the transition.
Orientational order, appropriate for a triangular lattice, is characterized by local
order parameter
ψ
6
(
r
k
)
and associated correlation function
g
6
(
y
)
ψ
6
(
r
k
)
=
(
1
/
6
)
exp
(
6i
φ
jk
),
g
6
(
y
)
=|
ψ
6
(
r
k
)ψ
6
(
r
)
|
(1.3)
j
(
n
.
n
.
of
k
)
where
. The angle
between a bond connecting particles
k
and
j
and a reference direction (e.g., the
y
-direction) is denoted as
r
k
=
(
x
k
,
y
k
)
is the position of the
x
th particle and
y
=|
y
k
−
y
|
φ
jk
.Onesumsexp
(
6i
φ
jk
)
over all six nearest neighbors
ψ
6
(
r
k
)
=
of
k
,so
1 for a perfect triangular lattice. Also this orientational order is
strongly enhanced by the walls (Fig.
1.4
), and the orientational correlation length
that one can extract from this decay also increases as one approaches melting [
5
]
(the correlation length for orientational order is found to be somewhat larger than
its counterpart for positional order (as expected, since it is only the former length
