Chemistry Reference
In-Depth Information
What are the implications of these studies on the calculation of
macroscopic dynamical properties of ionic liquids? At the very least,
they suggest that one should be careful when applying standard computa-
tional techniques used for simple liquids to ionic liquids. Most of these
techniques assume ergodic behavior, but the work described above shows
this may not always be the case. Due to the sluggish dynamics of ionic
liquid systems, one should carry out very long simulations to ensure ade-
quate sampling.
Consider the simplest dynamic property one can compute, the self-
diffusivity,
D
s
. The standard approach for computing
D
s
is to conduct an
equilibrium MD simulation and accumulate the mean-square displacement
as a function of time. The self-diffusivity is then computed using the Einstein
equation:
1
6
lim
d
dt
hj
2
D
s
¼
r
ð
t
Þ
r
ð
0
Þj
i
½
15
t
!1
where the term in angle brackets is the mean-square displacement (MSD). By
plotting the MSD as a function of time and taking the slope, one gets an esti-
mate of the self-diffusivity. The problem with Eq. [15] is that it is valid only in
the limit of ''infinite'' time, where ''infinite'' implies times much longer than
the longest relevant relaxation times. How long is this for an ionic liquid?
As the results above suggest, even at temperatures as high as 400 K, these times
can be longer than 10 ns, which makes for an expensive and time-consuming
simulation. There are at least three tests one can apply to check if a system
exhibits diffusive behavior. First, one can compute the non-Gaussian para-
meter for the system using Eq. [13], making sure that the MSD is tracked
for times long enough for this parameter to reach zero. The other approaches
involve observing the MSD itself.
Figure 12 is a plot of MSD of the [C
8
mpy] cation at 298K, obtained
from a simulation of the neat ionic liquid [C
8
mpy][Tf
2
N].
64
The dashed lines
are the individual
x
,
y
,and
z
components of the MSD, while the solid line is
the overall displacement. For a homogeneous system, the
x
,
y
,and
z
compo-
nents should all be equal, and they appear to be so for this system. A slope of
this plot can be calculated, and a self-diffusivity estimated using Eq. [15], but
the question is, does applying Eq. [15] to the data in Figure 12 give a reliable
self-diffusivity? This can be determined simply by testing whether the mean-
square displacement versus time has a slope of unity, as it must for a diffusive
system. In general the mean-square displacement will have a power law
dependence on time, according to
2
t
b
hj
r
ð
t
Þ
r
ð
0
Þj
i/
½
16
