Shear viscosity of model mixtures by nonequilibrium molecular dynamics. I. Argon-krypton mixtures

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1 Shear viscosity of model mixtures by nonequilibrium molecular dynamics. I. Argon-krypton mixtures Song Hi Leea) and Peter T. Cummingsb) Department of ChemicaI Engineering, Thornton Hall, University of Virginia, Charlottesville, 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 base case for a consistent study of the role of intermolecular potentials on the transport properties of molecular liquids and their mixtures. Using Lennard-Jones models for the two species, very good agreement with 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 baaed on an understanding of the molecular processes determining 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 been reviewed by Cummings and Evans. NEMD methods rely on measuring the response of a system to a perturbing field and relating the linear response to 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 case of 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 response is the off-diagonal element of the pressure tensor, Px,,, so that the viscosity 7 is given by T=-7* P The alternative to obtaining viscosity by NEMD is to )On sabbatical leave from Department of Chemistry, Kyungsung University, Pusan , 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-molecule NEMD only in the limit of very large numbers of molecules (2000) while Chialvo et a2. demonstrated an anomalous number dependence in the Einstein approaches that 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 shape on 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.+() between a species i and a species j 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 parameters for these potentials are given in Table I and are derived on the basis of liquid state thermodynamic properties.8-10 The cross interaction parameters between Ar and Kr are calculated from the simple Lorentz- Berthelot rules J. Chem. Phys. 99 (5), 1 September /93/99(5)/ American Institute of Physics 3919

2 3920 S. H. Lee and P. T. Cummings: Shear viscosity of mixtures. I TABLE I. Lennard-Jones parameters for Ar/Ar, Kr/Kr, and Ar/Kr interactions. t- Interaction Ar/Ar Kr/Kr Armr U(A) E/~B W TABLE II. Pure argon system at T= 135 K andp=40 bar. Experimental results are from Ref. 24. Property (b%) (g/em3) (kj?zi) (lo- z s/m ) Expt NpT MD for LJ NVT MD for BPW eij= (E+j) "'3 Oij =2' Oi+aj 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 - i. r~yg+m)]~ q(x) = [P(x- 1)4_tQ<X- 1)5]6?a (1--x), x> 1 =o, X<l, (3b) where E is the well depth of the potential and x=r/l$, with R, being the position of the potential minimum. The values of the parameters in Eq. (3) are given by Barker and co-workers. f 2 For AI-, the BFW potential is designed to be most accurate when supplemented with a three-body potential, which we have neglected in the present study. a: ;g+ 1 f -K. I The effect of the thermostatting term involving api in Eq. (4b) is to hold the translational kinetic energy constant (hence the term isokinetic). The functional form of this term is derived by Gauss principle of least constraint. The momenta in Eqs. (4a) and (4b) are measured with respect to the streaming velocity of the fluid and are known as peculiar momenta. In similar fashion to the thermostatting constant, the dilation rate k=drc/dt controls the volume of the system in order to constrain the pressure p. The pressure is one third of the trace of the pressure tensor P, which is expressed in terms of molecular quantities by N PiPi pv= ;gl G+ lci~.gnrijfij~ I where V is the volume of the system, rij=ri-rj is the vector joining the centers of molecules i andj, and Fii is the force between them. The equation of motion for the dilation rate for a pure fluid is given by Hood et al. I3 The extension to mixtures is straightforward and given by k= - ij* (E.+)+yx,,] (7) I<i< j$n (5) (6) 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 I hi dt=fi-pi VU+riri, VU--lipi--pi, 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 and Uij~Uij( rij> is the intermolecular potential between species i and species j molecules. These equations of 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 bar. Experimental results are from Ref. 24. Property --%td U-& (g/cm3) (k.j/mol) (lo- g s/m*) Expt NpT MD for LJ NVT MD for BFW

3 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. (Ps- ) Run length Pure liquid argon (x=0) Pure liquid krypton (x= 1) P - Econf 1) (lo- --Ewnr r] (lo- Wcm3) (k.j/mol) N s/m2) (g/em ) (ki/mol) N s/m*) K K K OOK OOK IOOK K OK OQK (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 generates boundary 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 necessary to 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 reaches steady state at a given strain rate y one computes and averages the pressure tensor defined in Eq. (6). The strain rate dependent shear viscosity is then obtained from Newton s law of viscosity PXJJ-PX rl=-.q 9. _ where Pxy and Pyx are the averaged xy and yx components of P. From kinetic and mode coupling theories, it is known that the strain rate dependence of the shear viscosity is linear in y Hence, 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 rl=%+w. (10) 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 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= and g/cm3, respectively for the BFW potential. Pure liquid argon (x=0) Pure liquid krypton (x= 1) (Ps- ) Run length ( & 7] (lo- N s/m2) (b:r) - -%,nf r] (lo- (ki/mol) N s/m*) K K K K K K K K K

4 S. H. Lee and P. T. Cummings: Shear viscosity of mixtures. I lno t -I It--c--c---l-l 3/z 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. 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. that for the LJ fluid, since the simulations are performed in the NpT ensemble the 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 pressures are less accurately predicted. However, pressure as measured in 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- l/2 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. 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 energy and 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 cause of 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 dependent density, 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

5 S. H. Lee and P. T. Cummings: Shear viscosity of mixtures. I E2 400 & 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. x P Wcm3) - Econf ( Wmol) r] (lo- N s/m2) l/ l/ l/ / / /2 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. (11) 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 dependent viscosities are shown in Fig. TABLE VI. Densities, configurational energies, and viscosities of argon/krypton liquid mixtures as function of strain rate at T= 135 K and p=40 bar. (Ps- 1 n= l/6 x= l/3 Run P r] (lo- P --Ecmf 7 (lo- length Wcm3) N s/m2) Wcm3) (ki/mol) N s/m ) K K K K K K K K K x= l/2 x=2/ x=5/ K K K K K K K K K

6 3924 S. H. Lee and P. T. Cummings: Shear viscosity of mixtures. I The composition dependencies of 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 I, I I I,,., 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. 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. r]=exp[x In r]kr+ (l-x% %+I (13) is also shown. This model is an engineering correlation recommended for predicting liquid mixtures in the absence of mixture viscosity data.23 It 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 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 presented here will represent the base case x ACKNOWLEDGMENTS The authors gratefully acknowledge support of this research by the National Science Foundation through Grant No. CTS One of us (S.H.L.) acknowledges the 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. P 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. D. J. Evans and G. P. Morriss, Statistical Mechanics of Nonequilibrium Liquids (Academic, New ork, 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 Statistical- MechanicalSystems (Italian Physical Society, North-Holland, Amsterdam, 1986), pp D. A. McQuarrie, Statistical Mechanics (Harper and Row, New ork, 1976). 6B. L. Hohan and D. J. Evans, J. Chem. Phys. 78, 5147 (1983). A. A. Chialvo, P. T. Cummings, and D. J. Evans, Phys. Rev. E. 47, 1702 (1993). *W. B. Streett and L. A. K. Staveley, J. Chem. Phys. 47, 2449 (1967). J,. LL. Singer and K. Singer, Mol. Phys. 24, 357 (1972). I. R. McDonald, Mol. Phys. 23, 41 (1972). J. A. Barker, R. A. Fisher, and R. 0. Watts, Mol. Phys. 21,657 (1971). J. A. Barker, R. 0. Watts, J. K. Lee, T. P. Schafer, and. T. Lee, J. Chem. Phys. 61, 3081 (1974). 13L. M. Hood, D. J. Evans, and G. P. Morriss, J. Stat. Phys. 57, 729 (1989). 14A. W. Lees and S. F. Edwards, J. Phys. C 5, 1921 (1972). D. J. Evans and G. P. Morriss, Phys. Rev. A 30, 1528 (1984).

7 S. H. Lee and P. T. Cummings: Shear viscosity of mixtures. I T. amada and K. Kawasaki, Prog. Theor. Phys. 53, 111 (1975). K. Kawasaki and J. D. Gunton, Phys. Rev. A 8, 2048 (1973). *M. H. Ernst, B. Cichocki, J. R. Dorfman, J. Sharma, and H. J. van Beijeren, Stat. Phys. 18, 237 (1978). 19D. J. Evans, Phys. Lett. A 74, 229 (1979). D. J. Evans, Phys Rev. A 23, 1988 (1981). *ID. J. Evans and H. J. M. Hanley, Physica A 108, 567 (198 1). 22D. J. Evans, Physica A 118, 51 (1983). 23R. C. Reid, J. M. Prausnitz, and T. K. Sherwood, The Properties of Liquids and Gases, 3rd ed. (McGraw-Hill, New ork, 1977). 24V. A. Rabinovich, A. A. Vasserman, V. I. Nedostup, and L. S. Veksler, Thermodynamic Properties of Neon, Argon, Krypton and Xenon (Hemisphere, Washington, D.C., 1988).

Shear viscosity of liquid rubidium at the triple point

J. Phys. F: Met. Phys. 18 (1988) 1439-1447. Printed in the UK Shear viscosity of liquid rubidium at the triple point P T Cummingsi- and G P Morriss-l: i Department of Chemical Engineering, Thornton Hall,