Chemistry Reference
In-Depth Information
Recall that Del Popolo and Voth
62
computed the dynamics of [C
2
mim]
[NO
3
] and saw evidence of what they termed dynamic heterogeneity. This can
be quantified using a so-called non-Gaussian parameter,
94
a
ð
t
Þ
, given by
4
3
5
hj
r
ð
t
Þj
i
a
ð
t
Þ¼
2
1
½
13
2
hj
r
ð
t
Þj
i
where
is the displacement of an ion center of mass over some time
t
, and
the angle brackets refer to an ensemble average. For a Gaussian distribution of
displacements, characteristic of normal liquid diffusive motion,
r
ð
t
Þ
0. The
extent to which this parameter is nonzero is indicative of dynamic heterogene-
ity characteristic of subdiffusive motion. They also characterized the dynamics
by computing the van Hove correlation function, defined as
a
ð
t
Þ¼
*
+
N
X
N
1
r
i
ð
r
i
ð
G
s
ð
r
;
t
Þ¼
1
d
½
t
Þ
0
Þ
r
½
14
i
¼
This correlation function measures the probability that the center of mass of a
molecule is at position
r
c
at time
t
given that it was at the origin,
r
c
ð
0
Þ
, at time
0.
G
s
exhibits Gaussian behavior at very short times due to free particle
behavior, and in the long time, hydrodynamic limit it will also be Gaussian.
Most liquids have Gaussian behavior at intermediate time scales, but super-
cooled liquids display non-Gaussian behavior at intermediate time scales
due to dynamic heterogeneity. Figure 11 shows the van Hove correlation
function and the non-Gaussian parameter for [C
2
mim][NO
3
]. At intermediate
times the non-Gaussian parameter (inset) for both the cation and anion are
nonzero, and the van Hove correlation function deviates significantly from
Gaussian behavior.
Hu and Margulis
95,96
studied the dynamics of [C
4
mim][PF
6
] by carrying
out MD simulations on the neat liquid for 3 ns at 400 and 500 K, and for 9 ns
at 300 K. At 300 K, the non-Gaussian parameter
ð
r
;
t
Þ
reached a maximum at
about 2.5 ns, while the maximum occurred at just over 100 ps at 400 K, in
agreement with Del Popolo and Voth's findings.
62
From their computed van
Hove correlation function, they found that most ions diffuse more slowly
than what would be expected for Gaussian dynamics. However, a small group
of ions diffuse much faster than what is expected. Interestingly, a small num-
ber of highly mobile ions move further in 200 ps than the less mobile ions do
over 2000-3000 ps. Hu and Margulis also found that translational mobility is
totally decoupled from rotational mobility for the [PF
6
] anion, but the two are
coupled tightly for the [C
4
mim] cation. They then went on to simulate the
absorption and emission spectra of the organic probe molecule 2-amino-
7-nitrofluorene (ANF) by immersing a single ANF molecule,
a
ð
t
Þ
in both the
