Shear viscosity
I. Argon-krypton
of model mixtures by nonequilibrium
mixtures
molecular
dynamics.
Song Hi Leea) and Peter T. Cummingsb)
Department of ChemicaI Engineering, Thornton Hall, University of Virginia, Charlottesville,
22903-2442
Virginia
(Received 22 March 1993; accepted 11 May 1993)
We present nonequilibrium molecular dynamics simulations of argon-krypton liquid mixtures at
constant temperature and pressure ( T= 135 K and P=40 bar) as the basecasefor a consistent
study of the role of intermolecular potentials on the transport properties of molecular liquids
and their mixtures. Using Lennard-Jonesmodels for the two species,very good agreementwith
experiment is obtained for the thermodynamic and transport properties of the two pure fluids.
Simulations at constant temperature and density using the Barker-Fisher-Watts potentials for
pure fluids yields predictions for thermodynamic properties and viscosity in excellent agreement
with experiment.
1.INTRODUCTION
The development of accurate, widely applicable, predictive methods for thermophysical properties estimation
baaedon an understanding of the molecular processesdetermining the properties of interest continues to be an enduring goal for chemical engineering thermodynamicists.
Microscopic level simulation plays a key role in understanding the relationship between microscopic interactions
and macroscopic thermophysical properties. This is because microscopic level simulation permits the researcher
to selectively switch on and off key intermolecular interactions (such as dispersion attraction, repulsion, shape,electrostatics, and hydrogen bonding) and evaluate their effect
on the property of interest.
In this paper, we begin a series of nonequilibrium molecular dynamics (NEMD) studies of model mixtures that
examines the role of intermolecular forces in determining
the shear viscosity of liquids. NEMD has become a standard tool for the study of transport properties by molecular
simulation. The theory of NEMD has been described in
great detail by Evans and Morriss in their monograph’and
applications have beenreviewed by Cummings and Evans.’
NEMD methods rely on measuring the responseof a system to a perturbing field and relating the linear responseto
a transport coefficient. The microscopic system is simulated at steady state away from equilibrium and the transport coefficient calculated from the ratio of the current to
the applied field. In the caseof viscosity, the applied field is
a boundary driven planar Couette flow field characterized
by strain rate y=du,/dy
(where u, is the velocity in the
x-direction) and the responseis the off-diagonal element of
the pressure tensor, Px,, , so that the viscosity 7 is given by
P
T=-7*
The alternative to obtaining viscosity by NEMD is to
“)On sabbatical leave from Department of Chemistry, Kyungsung University, Pusan 608-736, Korea.
“Author to whom correspondence should be addressed,
use an equilibrium method such as the Green-Kubo (GK)
method3” or one of the Einstein methods.’However, such
methods have strong system size dependencies.For example, Holian and Evans6 found that the GK method gave
results for viscosity consistent with 108-moleculeNEMD
only in the limit of very large numbers of molecules (2000)
while Chialvo et a2.’demonstrated an anomalous number
dependencein the Einstein approachesthat could only be
overcome by prohibitively long simulations. For these reasons, we have used NEMD in this work.
The goal of this series of papers is to elucidate the
effect of dipolar and quadrupolar interactions and linear
shapeon the viscosity of liquids and their mixtures. To this
end, we begin by considering a mixture of argon and krypton at a constant temperature and pressure of T= 135 K
and P=40 bar. We model the two fluids as Lennard-Jones
fluids so that the intermolecular potential Z.+(Y)between a
species i and a speciesj molecule whose intermolecular
separation is r is given by
z+j(r> =4qj
[(y-(%)6]
f
(2)
where Eij and aij are the energy minimum and zero point of
the Lennard-Jones (LJ) interaction. For the argon and
krypton models used in this paper, these parameters are
defined in Sec. II, along with the NEMD algorithms.
II. SIMULATION
DETAILS
In this section, we describe the intermolecular potentials and the details of the NEMD simulation algorithm.
A. Intermolecular
potentials
Two sets of intermolecular potentials are used in this
work. For the study of mixture properties, the LJ potential
is used for the Ar/Ar, Ar/Kr, and Kr/Kr potentials. The
LJ parametersfor these potentials are given in Table I and
are derived on the basis of liquid state thermodynamic
properties.8-10The cross interaction parameters between
Ar and Kr are calculated from the simple Lorentz-
Berthelotrules
0021-9606/93/99(5)/391
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(5), 1 September
1993
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3919
S. H. Lee and P. T. Cummings: Shear viscosity of mixtures. I
3920
TABLE I. Lennard-Jones parameters for Ar/Ar,
interactions.
Interaction
U(A)
Ar/Ar
Kr/Kr
Armr
3.405
3.633
3.519
eij= (E+j) "'3
Kr/Kr,
and Ar/Kr
t-Y
bar. Experimental
E/~B W
Property
119.8
161.0
141.4
(b%)
(g/Em3)
40.0
40.0
26.2
1.034
1.062
1.034
Expt.
NpT MD for LJ
NVT MD for BPW
(kJ?zi)
(lo-’
z s/m’)
2.432
2.637
2.485
740
885
780
Oij
Oi+aj
1
a:
=2'
The use of the LJ potentials is motivated by the extensive
existing published results on the viscosity of the LJ fluid, as
well as the ease with which it can be extended to more
complex systems by the addition of dipolar and quadrupolar interactions, and by the addition of sites to introduce
shape. However, the LJ potential is clearly limited in its
accuracy for the noble gases,so we have also performed
NEMD simulations of pure Ar and Kr using the highly
accurate Barker-Fisher-Watts (BFW) potentials.““2 The
functional form of the BFW potentials is given by
q(x) = [P(x-
f
(5)
-K.
I
The effect of the thermostatting term involving api in Eq.
(4b) is to hold the translational kinetic energy constant
(hence the term isokinetic) . The functional form of this
term is derived by Gauss’principle of least constraint. The
momenta in Eqs. (4a) and (4b) are measuredwith respect
to the streaming velocity of the fluid and are known as
peculiar momenta. In similar fashion to the thermostatting
constant, the dilation rate k=drc/dt controls the volume of
the system in order to constrain the pressurep. The pressure is one third of the trace of the pressure tensor P,
which is expressedin terms of molecular quantities by
N PiPi
1)4_tQ<X- 1)5]6?a’(1--x), x> 1
X<l,
;g+
pv= ;gl G+
lci~.GNrijFij~
I
- i. r~yg+m)]~
=o,
TABLE II. Pure argon system at T= 135 K andp=40
results are from Ref. 24.
(3b)
where E is the well depth of the potential and x=r/l$,
with R, being the position of the potential minimum. The
values of the parameters in Eq. (3) are given by Barker
and co-workers.“f’2 For AI-, the BFW potential is designed
to be most accurate when supplementedwith a three-body
potential, which we have neglected in the present study.
(6)
where V is the volume of the system, rij=ri-rj
is the
vector joining the centers of molecules i andj, and Fii is the
force between them. The equation of motion for the dilation rate for a pure fluid is given by Hood et al. I3 The
extension to mixtures is straightforward and given by
k=
C
@ij[‘ij*
I<i< j$N
(E.+)+yx,Y,]
(7)
’
B. NpT NEMD algorithm
For the LJ fluid mixtures, the NEMD algorithm used
in this paper is a mixture version of the isobaric isokinetic
sllod algorithm described by Hood et al. I3 The equations of
motion for this algorithm are given by
dri
Pi
x=-g+ri’
VU+riri,
I
hi
dt=Fi-pi’
VU--lipi--pi,
and Uij~Uij( rij> is the intermolecular potential between
speciesi and speciesj molecules.These equationsof motion
are combined with the Lees-Edwards “sliding brick”
boundary conditions.i4 In the absence of the thermostat
and the isobaric constraint, the terms in Eqs. (4a) and
TABLE III. Pure krypton system at T= 135 K and p=40
mental results are from Ref. 24.
where rj, mi, and pi are the position, mass, and momentum, respectively, of molecule i, Fi is the force exerted by
the other molecules on molecule i, u= (u,,O,O) with U,
=yy is the velocity field corresponding to planar Couette
flow, and a, is the thermostatting constant, given by
Property
U-&
Expt.
NpT MD for LJ
NVT MD for BFW
J. Chem. Phys., Vol. 99, No. 5, 1 September
40.0
40.0
-141.3
--%td
(g/Cm3)
(k.J/mol)
2.315
2.327
2.315
5.563
6.261
6.017
bar. Experi-
(lo-’
g s/m*)
3016
3425
3146
1993
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S. H. Lee and P. T. Cummings:
Shear viscosity of mixtures. I
3921
TABLE IV. Densities, configurational energies, and viscosities of pure argon and pure krypton modeled by
the LJ potential as function of strain rate at T=135 K and p=40 bar.
Pure liquid argon (x=0)
Pure liquid krypton (x= 1)
Y(Ps-‘)
Run
length
Wcm3)
(k.J/mol)
1) (lo-’
N s/m2)
(g/Em”)
--Ewnr
(kI/mol)
r] (lo-’
N s/m*)
1.0
0.81
0.64
0.49
0.36
0.25
0.16
0.09
0.04
60 K
60K
60K
1OOK
1OOK
IOOK
160 K
16OK
2OQK
0.8832
0.9496
0.9709
1.017
1.009
1.028
1.058
1.054
1.063
3.476
3.714
3.893
4.104
4.080
4.172
4.295
4.282
4.326
449.0
556.1
615.8
663.9
673.7
702.2
744.7
741.7
796.9
1.819
1.934
2.053
2.129
2.193
2.249
2.298
2.318
2.322
5.823
6.217
6.164
7.092
1.366
7.606
7.818
1.902
1.927
865
1104
1375
1649
1944
2124
2335
2678
2926
P
- Econf
(4b) involving the strain field, Vu, cancel to yield Newton’s equations of motion relating ri and Fj. This implies
that the sllod algorithm truly generatesboundary driven
planar Couette flow, leading to the conclusion that it is
correct to arbitrary order in the strain rate.” In order to
obtain a good signal to noise ratio, with the NEMD algorithms it is necessaryto use high strain rates y which turn
out to be high enough to cause the shear viscosity to be
strain rate dependent. In order to compute the shear viscosity of a Newtonian fluid using the shod algorithm, after
the simulation reachessteady state at a given strain rate y
one computes and averagesthe pressure tensor defined in
Eq. (6). The strain rate dependent shear viscosity is then
obtained from Newton’s law of viscosity
rl=-
PXJJ-PYX
.Q
9
. _
where Pxy and Pyx are the averagedxy and yx components
of P. From kinetic and mode coupling theories, it is known
that the strain rate dependenceof the shear viscosity is
linear in y 1’2.16-‘8Hence, to apply the shod algorithm to a
Newtonian fluid, one performs several simulations at differing strain rates y and fits the resulting strain rate dependent viscosities to the equation
(10)
rl=%+w’“.
The zero strain rate extrapolation of q, 70, is thus the
Newtonian viscosity.
C. NVT NEMD algorithm
For pure Ar and Kr described by the BFW potentials,
the algorithm used is the isochoric isokinetic (NVT) sllod
algorithm which is obtained simply by putting I&O in Eqs.
(3) and (4).
III. RESULTS
All simulations reported in this section were performed on 108 molecules initially located in a fee lattice
and equilibrated for at least 200 000 time steps. The intermolecular potentials were subject to spherical cutoffs as
follows: For the LJ fluids, the cutoff distance was 2.25~~~
for pure Ar, and 2.250,~ for pure Kr and the Ar/Kr mixtures. For the BFW potentials, the cutoff was 0.5L, where
L is the length of the side of the simulation cell and so is
determined by the number density. Long range corrections
to the energy and pressure were included in these properties by assuming that the pair distribution function was
uniform beyond the cutoff distance.
We begin by reporting pure fluid results using the LJ
potentials and the BFW potentials. The results are summarized in Table II for Ar and in Table III for Kr. Note
TABLE V. Pressures, configurational energies, and viscosities of pure argon and pure krypton as function
of strain rate at T= 135 K and p= 1.034 and 2.315 g/cm3, respectively for the BFW potential.
Pure liquid argon (x=0)
Y(Ps-‘)
1.0
0.81
0.64
0.49
0.36
0.25
0.16
0.09
0.04
Run
length
60
60
60
100
100
100
160
160
200
K
K
K
K
K
K
K
K
K
7] (lo-’
N s/m2)
(‘&
144.5
104.5
70.0
49.5
23.7
25.9
11.0
22.5
22.2
3.988
4.042
4.084
4.122
4.141
4.151
4.162
4.167
4.177
648.1
684.5
669.2
691.7
716.9
727.8
742.5
153.5
732.4
Pure liquid krypton
(b:r)
978.0
691.7
441.9
286.5
100.8
8.0
-88.1
-96.5
-127.4
(x= 1)
- -%,nf
(kI/mol)
r] (lo-’
N s/m*)
6.916
7.186
7.348
7.414
7.597
7.662
7.129
7.746
1.754
1640
1801
1938
2149
2273
2401
2470
2715
2866
J. Chem. Phys., Vol. 99, No. 5, 1 September 1993
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S. H. Lee and P. T. Cummings: Shear viscosity of mixtures. I
2500
2000
77
1500
1000
t
0.2
0
0.4
lnO.6
0.8
-I
It--c--c---l-l
500
1
0.2
0
Y
0.4
Y
FIG. 1. Shear viscosities (in units of lo-’ N s/m*) for argon/krypton
mixtures as a function of the square root of the strain rate (in units of
ps-‘“) for various mole fractions of krypton. The straight lines are least
squared linear fits to the simulation results.
that for the LJ fluid, since the simulations are performed in
the NpT ensemblethe density, energy, and viscosity are the
predicted quantities, while for the BFW potential, the pressure, energy,and viscosity are the predicted quantities. For
both Ar and Kr, it is clear that the BFW potentials predict
the energy and viscosity highly accurately (within 2.2% of
the Ar energy and 9.2% of the Kr energy, and within 5.5%
of the Ar viscosity and 4.3% of the Kr viscosity). The
pressuresare less accurately predicted. However, pressure
as measuredin molecular simulations exhibits considerable
sensitivity to the details of the cutoff procedure, so that the
error in the pressure for Ar can be regarded as quite small.
For Kr, however, the pressure is negative. The T= 135 K
and p=40 bar is quite close to the experimental vapor-
0
0.2
0.4
0.6
0.8
1
l/2
Y
FIG. 2. Strain rate dependent shear viscosities (in units of 10e7 N s/m2)
for pure argon and pure krypton obtained using the BFW intermolecular
potentials. The straight lines are least squared linear fits to the simulation
results.
3/z
0.6
0.8
1
FIG. 3. Strain rate dependent internal energies of pure argon and krypton
obtained using the BFW intermolecular potentials. The straight lines are
least squared linear fits to the simulation results.
liquid coexistence boundary for Kr, and it may be that
small inaccuracies introduced by neglecting higher order
interactions in our simulations are sufficient to push the
model system into the two phase region.
The results for the LJ fluid are not as good as for the
BFW potential (within 2.7% of the Ar density and 0.5%
of the Kr density, within 8.4% of the Ar energyand 11.3%
of the Kr energy, and within 19.6% of the Ar viscosity and
13.6% of the Kr viscosity). All quantities are overestimates in magnitude, both in comparison with experiment
and with the BFW potential. The likely causeof this overestimation is the difference in the long range attractive
terms in the LJ and BFW potentials. Both LJ and BFW
potentials have l/r6 terms at long range; however, the coefficient of this term in the LJ potential is nearly twice that
of the corresponding term in the BFW potential. Thus, the
LJ potentials can be regarded as more attractive; greater
attraction would result in lower energies, higher
viscosities-since greater attraction is equivalent to lower
temperature if the energy is held constant-and lower
pressures-and thus higher density to achieve a particular
pressure.
The strain rate dependentdensity, energy and viscosity
of pure liquid argon and pure liquid krypton are reported
in Table IV for the LJ potential and the strain rate dependent density, energy and viscosity in Table V for the BFW
potential. Some general trends can be noted. First, in both
cases the shear viscosity satisfies Eq. (IO) as can be seen
from Fig. 1 (the x=0 results for Ar and x= 1 results for
Kr) for the LJ potential and Fig. 2 for the BFW potential.
The extrapolations to zero strain rate of the least squares
fits to Eq. (10) are used to determine the Newtonian viscosities reported in Tables II and III. Second,for the NVT
simulations it is known that the configurational energy,
E
and the pressure, p, satisfy the asymptotic relati~~~i6-ZZ
J. Chem. Phys., Vol. 99, No. 5, 1 September
1993
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S. H. Lee and P. T. Cummings: Shear viscosity of mixtures. I
TABLE VII. Density, internal energy, and viscosity of mixtures of argon
and krypton at T= 135 K and p=40 bar as a function of krypton mole
fraction, x.
800
x
600
5
E2
P Wcm3)
0
l/6
l/3
l/2
400
&
2/3
6/6
1
0
0.2
0.4
0.6
0.8
1.062
1.261
1.444
1.664
1.842
2.048
2.327
FIG. 4. Strain rate dependent pressure of pure argon and krypton obtained using the BFW intermolecular potentials. The straight lines are
least squared linear fits to the simulation results.
TABLE VI. Densities, configurational energies, and viscosities of argon/krypton
of strain rate at T= 135 K and p=40 bar.
n= l/6
1.0
0.81
0.64
0.49
0.36
0.25
0.16
0.09
0.04
Wcm3)
60K
60 K
60 K
100 K
100 K
100 K
160 K
160 K
200 K
0.9815
1.085
1.131
1.191
1.226
1.244
1.256
1.270
1.267
60
60
60
100
100
100
160
160
200
P
1.301
1.401
1.485
1.548
1.578
1.606
1.627
1.642
1.662
N s/m2)
885
1271
1502
1775
2019
2470
3425
3.555
3.976
4.181
4.445
4.594
4.675
4.734
4.790
4.780
4.256
4.627
4.958
5.236
5.365
5.486
5.568
5.634
5.720
It is evident from Figs. 3 and 4 that the NVT simulation
results for the BFW potentials are consistent with these
equations.
The strain rate dependent density, configurational energy and viscosity of the Ar/Kr mixtures, increasing in
steps of l/6 in the Kr mole fraction X, are given in Table
VI. The strain rate dependentviscosities are shown in Fig.
liquid mixtures as function
x= l/3
r] (lo-’
N s/m2)
Wcm3)
(kI/mol)
7 (lo-’
N s/m’)
453.9
582.7
657.2
772.1
857.4
901.1
962.9
1049
1067
1.167
1.238
1.292
1.403
1.408
1.419
1.423
1.425
1.441
3.978
4.289
4.508
4.992
5.014
5.063
5.08 1
5.088
5.155
533.8
632.2
752.8
892.9
985.4
1055
1103
1191
1306
P
x= l/2
1.0
0.81
0.64
0.49
0.36
0.25
0.16
0.09
0.04
4.321
4.776
5.158
5.725
6.170
6.761
7.945
r] (lo-’
(11)
Y
Run
length
- Econf ( Wmol)
1
3/2
Y (Ps-‘1
3923
--Ecmf
x=2/3
558.3
701.8
853.7
969.8
1072
1176
1224
1353
1623
1.412
1.526
1.621
1.712
1.768
1.793
1.823
1.831
1.839
4.461
4.898
5.269
5.634
5.859
5.973
6.090
6.133
6.157
610.0
701.6
972.1
1123
1265
1385
1492
1587
1668
x=5/6
1.0
0.81
0.64
0.49
0.36
0.25
0.16
0.09
0.04
60
60
60
100
100
100
160
160
200
K
K
K
K
K
K
K
K
K
1.532
1.650
1.785
1.857
1.930
1.955
2.004
2.024
2.028
4.732
5.176
5.686
5.980
6.275
6.385
6.582
6.667
6.687
648.4
854.7
1069
1233
1441
1586
1687
1885
2162
J. Chem. Phys., Vol. 99, No. 5, 1 September
1993
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3924
S. H. Lee and P. T. Cummings: Shear viscosity of mixtures. I
The composition dependenciesof the viscosity, energy,
and density are shown in Figs. 5 and 6. For ideal liquid
mixtures, one would expect that each of the properties
would depend linearly on composition, i.e.,
p=xp,,+ (1 -X)PAr,
(12)
where P is q,, Econf, and p. The linear model for each
property is shown in the figures as a solid line. Clearly, the
viscosity and energy differ noticeably from the linear
model. In Fig. 5, an exponential model, given by
r]=exp[x
I,
0
0.2
I
I
I,,
.,
0.4
1
0.6
,
,
,
0.8
.,
,
1
x
FIG. 5. Composition dependence of the shear viscosity in argon/krypton
mixtures obtained using LJ intermolecular potentials. The solid line is the
linear model and the dashed line is the exponential model described in
the text.
In r]Kr+
(l-x%
%+I
(13)
is also shown. This model is an engineering correlation
recommendedfor predicting liquid mixtures in the absence
of mixture viscosity data.23It is clear that this correlation
underpredicts at low Kr mole fraction and overpredicts at
high Kr mole fraction. However, it is superior to the simple linear model and correctly predicts that the mixture
viscosity is lower than the linear model.
IV. CONCLUSIONS
1 along with the pure LJ fluid results. In general, agreement with Eq. ( 10) is very good, and the extrapolation to
zero strain rate is used to obtain the viscosities reported in
Table VII.
ACKNOWLEDGMENTS
-6
-8
0
0.2
0.4
0.6
0.8
1
x
2.4
2.2
2
P
In this paper, we have presented NEMD simulations
of pure Ar and Kr described by both LJ and BFW potentials, and of mixtures of Ar and Kr described by LJ potentials. We found that the mixtures are somewhat nonideal in
that properties calculated from the simulations were nonlinear in composition, with the energy and viscosity showing the greatest nonideality. In future publications, we will
consider these same models but with the addition of multipolar interactions and nonspherical shape, for which the
simulations presentedhere will represent the base case.
1.8
1.6
1.4
1.2
1
0
0.2
0.4
0.6
0.8
1
X
FIG. 6. Composition dependence of the energy (top) and density (bottom) in argon/krypton mixtures obtained using LJ intermolecular potentials. In each case, the straight line is the linear dependence expected in
the case of ideal liquid mixtures.
The authors gratefully acknowledge support of this research by the National ScienceFoundation through Grant
No. CTS-9101326.One of us (S.H.L.) acknowledgesthe
sabbatical leave support of Kyungsung University and the
hospitality of the Department of Chemical Engineering at
the University of Virginia during the course of this work.
’D. J. Evans and G. P. Morriss, Statistical Mechanics of Nonequilibrium
Liquids (Academic, New York, 1990).
‘P. T. Cummings and D. J. Evans, Ind. Eng. Chem. Res. 31, 1237
(1992).
‘R. Zwanzig, Annu. Rev. Phys. Chem. 16, 67 ( 1965).
4D. J. Evans, in Molecular Dynamics Simulation of StatisticalMechanicalSystems (Italian Physical Society, North-Holland, Amsterdam, 1986), pp. 221-240.
5D. A. McQuarrie, Statistical Mechanics (Harper and Row, New York,
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