Chemistry Reference
In-Depth Information
that the molecules in question undergo a random walk. One empirical
criterion for determining if a random walk takes place is that a molecule
must move over some distance longer than a nominal length scale associated
with that particular molecule. One measure of this length scale is the maxi-
mum distance between two atoms in the molecule, given as
q
ð
2
R
max
¼
r
i
r
j
Þ
½
18
where
r
i
and
r
j
are the positions of atoms furthest apart in the molecule. For
most ionic liquids, this value is on the order of 5-10
˚
. To observe diffusive
motion, a simulation should be run long enough for the MSD to reach at least
25-100
˚
.
2
Figure 12 shows that for [C
8
mpy[Tf
2
N] at 298 K the MSD has not
reached 6
˚
2
even after 5 ns. This means that, on average, most of the ions
have not even moved the length of two carbon-carbon bonds, making it highly
unlikely that diffusive motion is operative.
COMPUTING SELF-DIFFUSIVITIES, VISCOSITIES,
ELECTRICAL CONDUCTIVITIES, AND THERMAL
CONDUCTIVITIES FOR IONIC LIQUIDS
There have been several studies undertaken to compute macroscopic
transport properties of ionic liquids, despite the difficulties mentioned above.
These properties include the self-diffusivity, viscosity, electrical conductivity,
and thermal conductivity. In this section we review some of these
works, but first some background is given on how these transport properties
are computed.
The most common method for computing a transport property is to car-
ry out an equilibrium molecular dynamics simulation and compute the integral
of the appropriate time correlation function. The general formula is given as
ð
1
h
x
ð
Þ
x
ð
g
¼
dt
t
0
Þi
½
19
0
where
is the perturbation in the Hamilto-
nian associated with the particular transport property under consideration and
x
g
is the transport coefficient and
x
signifies a time derivative. Integrals of the form given by Eq. [19] are known
as
Green
-
Kubo integrals
.
99
Detailed discussions of the theory behind this
approach may be found in standard references.
94,99
It is easy to show that an integrated form of Eq. [19] results in an
''Einstein'' formula similar to Eq. [15]. Thus an equivalent expression for
g
is
2
2
t
g
¼hð
x
ð
t
Þ
x
ð
0
ÞÞ
i
½
20
