Chemistry Reference
In-Depth Information
For self-diffusivity,
is the Cartesian atom position, and the time correlation
function in Eq. [19] is of the molecular velocities. For the shear viscosity, the
integral in Eq. [19] is of the time correlation of the off-diagonal elements of
the stress tensor. For the thermal conductivity the integral is over the energy cur-
rent, and for the electrical conductivity the integral is over the electric current.
99
An important implicit assumption in Eqs. [19] and [20] is that
the time
over which these expressions are evaluated is much larger than the correlation
time of the variable
x
. This assumption is often satisfied easily for simple
liquids, where relaxation times are fast. For ionic liquids, however, we have
seen that correlation times can be very long—on the order of 10 ns or more.
Moreover, a very large separation of time scales exists for different motions of
the ions. Application of Eqs. [19] and [20] to systems with such long correla-
tion times can be problematic. Equation [19] is particularly suspect. If it is
assumed that
x
obeys Gaussian statistics, the standard error in the time
correlation function is approximated as
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x
ð
Þ
t
s
2
t
t
run
2
Error
h
x
i
½
21
where
is the characteristic correlation time and
t
run
is the length of the simu-
lation. The problem with ionic liquids is that, even though the relevant
correlation functions decay rather quickly, it is not clear that all the relevant
dynamical processes contributing to a particular transport coefficient are being
probed on this time scale. An example of this was demonstrated clearly by
Urahata and Ribeiro
100
who computed various single-particle time correlation
functions for [C
1
mim][Cl] (see Figure 15) and showed that a vast separation of
time scales exist in this system. The correlation functions associated with the
ring center of mass and the alkyl chain dihedral angles decorrelate quickly.
However, reorientational motion of the ring takes place on time scales that
are orders of magnitude slower. Not surprisingly, the mean-square displace-
ment over these time scales also shows distinct regions.
Computing a transport coefficient to within 1% accuracy requires a
simulation that is several orders of magnitude longer than the relaxation
time, assuming Eq. [21] is valid. With ionic liquids, this is challenging for
properties such as the viscosity because the high viscosity is the result of a
low modulus and a long relaxation time. Thus the stress correlation function
is low amplitude but long ranged in time and easily overwhelmed by the noise
associated with rapid intramolecular modes that have nothing to do with the
long-time relaxation processes.
The first attempt to compute the self-diffusivity of an ionic liquid was by
Hanke and co-workers,
30
who determined the mean-square displacement of
[C
1
mim][Cl] over 15 ps and extracted a self-diffusivity using Eq. [15]. As we
now know, this time frame is almost certainly too short a simulation to
t
