Supplemental Materials for “Are pressure fluctuation-based
equilibrium methods really worse than nonequilibrium methods
for calculating viscosities?”
Ting Chen, Berend Smit*, and Alexis T. Bell*
Energy Biosciences Institute
and
Department of Chemical Engineering
University of California, Berkeley, CA 94720-1462
Viscosity of Lennard-Jones Fluid
Fig.1 shows a typical pressure-pressure time correlation function (TCF) of LJ fluid at
T*= 2.0 and * = 0.452. As can be seen, after about a correlation time of 2, the TCF starts
to oscillate between -9e-5 and 9e-5 (the oscillation on the negative side can’t be seen
because of logarithmic scale), where the integration of TCFs basically contribute nothing
to the GK integral in viscosity calculation. This time (t = 2) thus can be considered as the
earliest correlation time we can make viscosity estimate.
Fig. 2 compares our simulation results with literature results1, 2 for * = 0.4.
Viscosity of SPC and SPC/E Water Models
The calculations suggest that for a simple LJ fluid both the equilibrium and the
nonequilibrium methods give very similar results. To assess whether a similar conclusion
holds for a more complex fluid, we also computed the viscosity of water.
Our system consisted of 256 water molecules. The time step was 2 fs and SHAKE
algorithm was employed to constrain bond lengths and bond angles in the rigid versions of
*
To whom correspondence
alexbell@berkeley.edu
should
be
addressed:
berend-smit@berkeley.edu,
1
the SPC and SPC/E models of water. The cut-off distance was taken as 10 Å and longrange tail corrections for both energy and pressure were applied. A particle-particle particle
mesh solver with a precision of 104 was employed to treat long-range electrostatic
interactions. The systems were first run for 1 ns in a NPT ensemble at 1 atm and T = 300.2

K to make sure the correct density is reproduced, followed by a 2ns of NVT simulation to
equilibrate the system before the actual 20 ns production run. The densities obtained from
NPT simulations of water were 976 kg/m3 and 998 kg/m3 for the for SPC and SPC/E water
models, respectively, and are consistent with reported densities obtained by literature
simulation results.3
Fig. 3 shows viscosity calculation for the SPC model of water using Eq. 1 of the article.
The inset shows the short correlation time behavior. Results for the SPC/E model (not
shown here) were similar. Here too, the viscosity calculations converged rapidly (within 3
ps of correlation time). The averaged viscosities over a correlation time between 3 to 15 ps
(or 6000 samples) are 0.399±0.003 cP and 0.670±0.003 cP for the SPC and SPC/E water
models, respectively. Simulations on a larger 1728 water molecule system as the one used
in Hess’ work4 give nearly identical viscosity estimates, as shown in Fig. 4. An average
between a correlation time of 3 and 15 ps gives a viscosity of 0.400 ±0.004 cP, which is
almost identical to the viscosity estimate on our smaller 256-water system. Because in an
equilibrium simulation one does not need to perturb the system, the viscosity computation
is much less sensitive to the size of the system.
These results are in good agreement with the results obtained by Hess,4 who obtained
water viscosities of 0.38±0.02, 0.397, 0.407±0.007, and 0.405±0.005 cP for the SPC model
at 300 K using the Einstein relation, the transverse method, the SLLOD method, and the
2
periodic perturbation method, respectively. Hess also reported a viscosity of 0.642±0.008
cP for the SPC/E model using periodic perturbation. Wu et al.3 recently obtained a
viscosity of 0.40 cP for SPC model and 0.72 cP for the SPC/E model using periodic
perturbation at 298.5 K.
References:
1.
2.
3.
4.
R. L. Rowley and M. M. Painter, Int. J. Thermophys. 18, 1109 (1997).
D. M. Heyes, Physical Review B 37, 5677 (1988).
Y. J. Wu, H. L. Tepper, and G. A. Voth, J. Chem. Phys. 124, 024503 (2006).
B. Hess, J. Chem. Phys. 116, 209 (2002).
3
Fig.1 Pressure-pressure time correlation function of LJ fluid. T*= 2.0, * =
0.452, Total simulation time t *  110 5 . Data is the same as the one used in Fig.1 in main
text.

4
Fig. 2 (Color online) Viscosities of LJ fluids as a function of temperature at * = 0.4.
Triangles and stars are viscosity data from Ref. 1 and 2. Circles and crosses are viscosity
estimates obtained using EMD and RNEMD methods from this work.
5
Fig. 3 The viscosity of SPC water as a function of maximum time taken in the
integration of Eq. 1 for the LJ fluid. Inset shows the short time behavior.
6
Fig. 4 Viscosity calculation for SPC water model. N=1728 water molecules were used.
7