Chemistry Reference
In-Depth Information
within 1 ps. The resulting time integral of the correlation function is affected
by the large statistical noise (see inset of the figure), and, therefore, the accu-
racy in the estimated values of the electrical conductivity is poor; at the two
lowest temperatures the uncertainties are of the same order of magnitude
as the conductivity.
107
Estimated conductivities for this system are about
0.1 S/cm, which is within an order of magnitude or so of the expected value.
The point we make here is that it is unclear whether a transport property
can be computed from a time integral over less than 100 ps in a liquid having
widely varying dynamical time scales, some of which are in the nanosecond
time scale. In general, it has been found that viscosities and conductivities
computed using Green-Kubo integrals tend to result in slower dynamics (higher
viscosities, lower conductivities) than what is observed experimentally, an
effect that is often attributed to the neglect of polarizability. Another
overlooked factor, however, could be with the methods themselves; because
short-time integrals are incapable of capturing the contribution that long-
time relaxation processes have on a transport property, it may be that these
methods are only probing a ''local'' transport characteristic of the system.
Such behavior is well known in the polymer melt literature, where relaxation
times are quite long.
108
It is not that the methods themselves are ''incorrect,''
it is just that numerically evaluating the integrals accurately is difficult and
can lead to incorrect results.
An alternative to using Green-Kubo integrals is to use either the (for-
mally equivalent) integrated Einstein formula (Eq. [20]) or to implement a
nonequilibrium method. In both cases, the appropriate response function
can be averaged over an arbitrarily long time, thereby avoiding the numerical
problems associated with Green-Kubo integrals. Use of these methods will not
necessarily reduce computation time, but they may overcome the numerical
problems associated with ionic liquid systems.
Borodin and Smith
109
used equilibriumMD to compute the self-diffusivity,
viscosity, and electrical conductivity of
N
-methyl-
N
-propylpyrrolidinium bis
(trifluoromethylsulfonyl)imide ([C
3
mpyro][Tf
2
N]) at temperatures between
303 and 393 K. Importantly, in all cases they used an Einstein-type equation
to compute the transport coefficients. They were extremely careful in the way
they equilibrated the system; production runs varied from 8 ns at the highest
temperature to 16 ns at the lowest temperature. They also developed and uti-
lized a many-body polarizable force field for this system. Figure 19 shows the
computed self-diffusivites compared against NMR values from Nicotera and
co-workers.
110
The agreement is outstanding. Figure 20 shows that the electri-
cal conductivity also matches unpublished experimental data collected by
Henderson of the U.S. Naval Academy quite well. Borodin and Smith were
not able to make direct comparisons between their computed viscosities and
experimental results, however, due to a lack of experimental data at which
temperatures the simulations were run, though extrapolated values appeared
reasonable. The results of this study demonstrate that transport coefficients
