Supplemental Materials for “Are pressure fluctuation-based equilibrium methods really worse than nonequilibrium methods for calculating viscosities”

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 pressur

Supplemental Materials for “Are pressure fluctuation-based equilibrium methods really worse than nonequilibrium methods

for calculating viscosities?”

Ting Chen, Berend Smit1, 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 ρ* = 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 ρ* = 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

the SPC and SPC/E models of water The cut-off distance was taken as 10 Å and long-range 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

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

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

water viscosities of 0.38±0.02, 0.397, 0.407±0.007, and 0.405±0.005 cP for the SPC model

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at 300 K using the Einstein relation, the transverse method, the SLLOD method, and the 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 R L Rowley and M M Painter, Int J Thermophys 18, 1109 (1997).

2 D M Heyes, Physical Review B 37, 5677 (1988).

3 Y J Wu, H L Tepper, and G A Voth, J Chem Phys 124, 024503 (2006).

4 B Hess, J Chem Phys 116, 209 (2002).

Fig.1 Pressure-pressure time correlation function of LJ fluid T*= 2.0, ρ* = 0.452, Total simulation time t*

text

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Fig 2 (Color online) Viscosities of LJ fluids as a function of temperature at ρ* = 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

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

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Fig 4 Viscosity calculation for SPC water model N=1728 water molecules were used.