Non-equilibrium molecular dynamics calculation of the shear viscosity of liquid rubidium

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1 J. Phys. F: Met. Phys. 17 (1987) Printed in the UK Non-equilibrium molecular dynamics calculation of the shear viscosity of liquid rubidium P T Cummingsf and G P Morrisst + Department of Chemical Engineering, Thornton Hall, University of Virginia. Charlottesville, VA 22901, USA Research School of Chemistry. Australian National University, GPO Box 4, Canberra, ACT 260 I, Australia Received 18 April in final form 25 June 1986 Abstract. We report non-equilibrium molecular dynamics simulations of liquid rubidium using the isokinetic sllod algorithm for the dynamics and the Price potential for the interactions. The three state points simulated are the same three chosen by Bansal and Bruns who used a quite different non-equilibrium method. However. our results differ substantially from theirs. Specifically, we find that the Price potential consistently predicts better results for the shear viscosity. A more fundamental disagreement between our results and theirs is that our results underestimate the experimental data where theirs overestimate and vice versa. This suggests that the method used by Bansal and Bruns has systematic errors. 1. Introduction Over the past decade, the non-equilibrium molecular dynamics (NEMD) technique has developed into a powerful tool for the study of transport properties of both simple and molecular fluids (Hoover and Ashurst 1975, Evans and Morriss 1984a, Evans 1985, Evans and Hoover 1986). The most promising recent developments include the derivation of synthetic techniques for calculating such properties as the shear viscosity, thermal conductivity and self-diffusion and the use of Gauss's principle of least constraint as a tool for constraining simulations in physically meaningful ways (yielding, for example, isokinetic and/or isobaric ensemble simulations (Hoover et a1 1982, Evans and Morriss 1983)). Finally, the formal status of these and other NEMD techniques has been clarified tremendously in recent years through linear response theory, so that previously raised objections to the orthodoxy of the NEMD methods have now been effectively answered (Evans and Morriss 1984a, Morriss and Evans 1985). Of all the NEMD techniques, methods for calculating the shear viscosity have the longest history and therefore have come under the greatest scrutiny. The methodsreviewed in $ 3 of Evans and Morriss (1984a)-amount to setting up a steady state planar Couette flow in a homogeneous system. The streaming velocity has a non-zero component in the x direction, U,, which satisfies du,/d>x= y where y is the constant strain rate. Equation (1) implies that there is a linear velocity profile in the x direction, the characteristic of planar Couette flow (Bird et a1 1960). The J'S /87/ $ IOP Publishing Ltd 593 (1)

2 594 P T Cummings and G P Morriss component, Pyx, of the pressure tensor Pis related to the velocity profile by the constitutive relation PYx=-~(d~,/d.v)=-~~ (2) where for Newtonian fluids q, the shear viscosity, is dependent on the state variables (pressure, temperature T and composition, for example). Non-Newtonian fluids are fluids for which q depends on y in addition to the dependence on the thermodynamic state. The pressure tensor P, which is of second rank, is calculated from the expression Y. Y Y I = I where m is the mass of each molecule, U, is the velocity of molecule i, Vis the volume of the system, r!, = r, - r, with rk being the position of the centre of mass of molecule k and F, is the force exerted by molecule i on moleculej and so is given by the gradient of the pair potential q(r) between the molecules. Notice that the velocities in equation (3) have the streaming part U subtracted out and are thus peculiar velocities (Evans and Morriss 1984a). The shear viscosity is therefore calculated from equation (2) using the time average of the jix-component of P and the hydrostatic pressure is the time average of the trace of P. In order to perform a molecular dynamics simulation of planar Couette flow, one necessary modification of the usual formalism is that the images of the central simulation cell located above and below it must move in time in opposite directions at a constant rate determined by the strain rate 7, and the x-component of the velocity of a molecule leaving either the top or the bottom of the central cell in the y direction must be modified to be consistent with the overall velocity profile (the so-called sliding brick boundary conditions). These boundary conditions were first introduced by Lees and Edwards (1972) and in their original NEMD algorithm the velocity profile was set up and maintained solely by these moving boundaries, the dynamics of the particles in the simulation cell being given by Newton s equations of motion, dri pi - dt m _- dpi dt - Fi where pi is the laboratory momentum of molecule i and Fi is the force on molecule i due to all the other molecules in the system. A convenient reformulation in terms of first-order equations of motion (Evans and Morriss 1984b) is the sllod algorithm given by dri Pi - +ri.vu dt m -- dpi - Fi -pi * VU dt combined with the Lees-Edwards sliding brick boundary conditions. As far as the trajectories r,(t) of the individual molecules are concerned, the sllod algorithm can be shown to be equivalent to the Lees-Edwards algorithm except at time f=o since elimination of pi from equations (5a) and (5b) leads to Newton s equations of motion. However, from equations (5) it is clear that the sllod equations embody the velocity profile

3 Shear riscositv of liquid rubidium 5 95 Table I. A comparison of predicted and experimental shear viscosities for liquid rubidium. The theoretical predictions of this work and Bansal and Bruns (1984) are obtained using two different non-equilibrium molecular dynamics techniques. 1 S O I a This work. Bansal and Bruns (1984). Landolt-Bornstein (1 969). (since the gradient of the streaming velocity is added to the peculiar velocity pi/m). The sllod algorithm is thus homogeneous and consistent with the boundary conditions. Evans and Morriss (1984a) review the development of the sllod algorithm (as well as the closely related Doll s tensor method) and the reasons for their utility in non-equilibrium simulations of planar Couette flow; we refer the reader to their paper for details. Other methods for calculating the shear viscosity include the Green-Kubo relation (which requires long equilibrium molecular dynamics simulations of large systems for accuracy equivalent to the synthetic sllod algorithm (Holian and Evans 1983)), the differential trajectory method (Ciccotti and Jacucci 1975, Ciccotti et a1 1976, 1979) and the method of Singer and co-workers (Singer et a1 1980). It has recently been shown (Paolini et a1 1986) that the differential trajectory method is more reliable when the perturbation is a delta function rather than a step function (as used by Singer et a1 1980). This last method was used by Bansal and Bruns (1984) to determine the shear viscosity of liquid Rb using the Price potential (Price et a1 1970, Price 1971) at three state points: T= K, p= g ~ m-~, T= K, p= g ~ m - and ~ T= K, p= g cm-3 where T is the absolute temperature and p is the density. Their results (shown in table 1) suggest that the Price potential is only moderately accurate in predicting the transport properties of liquid Rb. In fact, according to Bansal and Bruns. the error in the shear viscosity predicted by the Price potential ranges from 32 to 45%. Given the success of the Price potential in predicting thermodynamic and static and dynamic structural properties of liquid Rb (Rahman 1974a, b, Mountain 1978, Arlinghaus and Cummings 1987), this discrepancy seems somewhat large and prompted the reexamination of the predictions of the Price potential given in this paper. Using the best available technique-that is, Gaussian thermostatted sllod dynamics-we find the results given in table 1, which indicate that the Price potential results range in error from 3 to 11%, a significant improvement over that of Bansal and Bruns (1984). In 8 2 of this paper, we describe the method of calculation, the results obtained and their analysis in more detail. Section 3 contains our conclusions. 2. NEMD simulation method The potential used in the simulations reported in this paper is the Price potential calculated according to the prescription given by Price and co-workers (Price et a1 1970, Price 1971). As with all effective interionic potentials derived using pseudopotential theory (Harrison 1966), the Price potential is density dependent; thus the potential differs at each of the

4 596 P T Cutnmings and G P Morriss three state points simulated due to the different densities involved. The NEMD equations solved are the isokinetic sllod equations given by where,i is given by where 0 indicates the full contraction of two second-order tensors and N is the number of particles (pseudoatoms in the current context) in the simulation cell. The formula for 1 is dictated by requiring that the temperature T is held constant, which is equivalent to demanding that EL I p: remain a constant of the motion. All the simulations were performed using 108 rubidium ions. The calculations were carried out on a Computer Signal Processing Inc. (CSPI) bit word array processor in the Center for Computer Aided Engineering at the University of Virginia. Since the potential and the force are not given analytically, they were discretised onto a grid of 900 points equally spaced in r2 corresponding to the range O<r<3a where Y is the interparticle separation. Both the force and potential were obtained by linear interpolation of this grid. The potential and force were assumed to be zero for r > 3a and the potential shifted to make it continuous at the cut-off 3a. Those simulation runs which start from a crystalline configuration begin with a face-centred cubic lattice which is equilibrated over time steps. At each state point, planar Couette flow was simulated at five values of the shear rate y given in dimensionless form by y* = ya(m/~) /~ where a is the diameter of an ion which can be defined to be the interionic separation at which the Price potential is zero so that the potential is positive for all r < U (= 4.4 8, for the densities simulated), m is the mass of the ions (= x g) and E is the depth of the Price potential (which is taken to be c/kb =400 K, kb being Boltzmann s constant, at the three state points simulated). In each simulation, the dimensionless time step used in the fourth-order predictor-corrector integration routine (Evans and Morriss 1984a) is Ar* = Ar(c/ma2)1/2 = which corresponds to a real time step of x s. The values of )I* along with the number of time steps per value of y* and the calculated values of the shear viscosity are given in tables 2 to 4. Notice that at each state point the shear viscosity reduces with Table 2. NEMD simulation results for liquid rubidium at a temperature of K and a density of g cm-j. Least-squares fitting of the data to equation (9) gives 7= y* I2 cp. The zero strain rate value of CP compares with the experimental value of CP and the Bansal and Bruns (1984) value of o 0.3 IO * i i i i * k * i k * 0.0 I 4.54 i 0.0 I

5 3.086 Shear riscositj' oj'liquid rubidium 597 Table 3. NEMD simulation results for liquid rubidium at a temperature of K and density g ~ m - ~ Least-squares, fitting of the data to equation (9) gives ~= cp. The zero strain rate value of CP compares with the experimental value of CP and the Bansal and Bruns (1984) value of o 0. I85 i ~ i I4 k 0.01 I z i i i I72 i i 0. I i T i i increasing shear rate-that is, rubidium exhibits shear thinning, and so is non-newtonian (Bird et a1 1960). in the range of strain rates used in the NEMD simulations. This is typical of the NEMD technique. Because of the small system sizes involved in these simulations, the strain rates must be quite large in order for there to be a discernible velocity profile across the simulation sample (so that a value of P,, can be determined with small fluctuations). For liquid rubidium, a reduced strain rate of y* = 1.0 corresponds to a strain rate in real units of y=4.482 x loii s-' which exceeds considerably strain rates currently accessible to experiment. (Note, however, that for many systems-particularly colloidal suspensions where the particles have a very high molecular mass-a reduced strain rate of y* = 1.O corresponds to strain rates in real units well within present laboratory capabilities.) Clearly then, the viscosity of interest which must be compared with experiment is the Newtonian viscosity obtained as the zero strain rate extrapolation of the simulation results for 7. Theoretical considerations (Kawasaki and Gunton 1973, Yamada and Kawasaki 1975, Ernst et a/ 1978) and extensive "ID simulation results (Evans 1979, 1981, Evans and Hanley 1980) indicate that to leading order v is related to the strain rate by where vo and vi are independent of 7. Consequently, we illustrate the simulation results for From these figures, it is clear the shear viscosity in figures 1 to 3 by plotting v against e. that the simulation results correlate very well with equation (9). To calculate vo and vl, the simulation results are least-squares fitted to equation (9) with the results given in tables 2 to 4. The zero-strain rate results are then reported in table 1. As an aside, the error estimates given in tables 2 to 4 and illustrated as error bars in figures 1 to 3 deserve some explanation. In the appendix, the individual simulation runs which were used to obtain the means and error estimates given in tables 2 to 4 are Table 4. hemd simulation results for liquid rubidium at a temperature of K and density g cm-3. Least-squares fitting of the data to equation (9) gives q= y*'" cp. The zero strain rate value of 38 CP compares with the experimental value of 55 CP and the Bansal and Bruns (1984) value of I.o 28 f ko i i l i i i f I 0. I34 i i i i i i 0.01

6 598 P T Cummings and G P Morriss Figure 1. The shear viscosity q of liquid rubidium at K and g cm-3 as a function of the square root of the reduced strain rate y*. The circles and the error bars represent the simulation results and the estimated error respectively (see text for explanation). The straight line is the least-squares fit of the experimental data to equation (9) and is given in table 2. described in detail. At most state points (defined by the density, temperature and strain rate), several simulation runs were performed, each of which yielded a value (or measurement ) for q, the pressure P and the configurational energy Uconf. The values reported in tables 2 to 4 are then obtained as the statistical mean of the measurements (weighted according to run length) and the errors represent one standard deviation. For Figure 2. The shear viscosity q of liquid rubidium at K and g cm-3 as a function of the square root of the reduced strain rate y*. The circles and the error bars represent the simulation results and the estimated error respectively (see text for explanation). The straight line is the least-squares fit of the experimental data to equation (9) and is given in table 3.

7 Shear viscosity of liquid rubidium o 112 Figure 3. The shear viscosity q of liquid rubidium at K and g cm-3 as a function of the square root of the reduced strain rate y*. The circles and the error bars represent the simulation results and the estimated error respectively (see text for explanation). The straight line is the least-squares fit of the experimental data to equation (9) and is given in table 4. those state points where only one long simulation run was performed, the errors have been conjectured, the conjecture being based in part on the block averages of 2000 timesteps performed throughout every simulation run. Thus, the reported values and error estimates in tables 2 to 4 represent block size weighted block averages and standard deviations. Recently, Straatsma et a1 (1986) have examined the formal correctness of estimating errors based on standard deviations obtained from block averaging and they conclude that the quality of the error estimate depends on the choice of the block size, which must be neither too small (as this increases correlation between measurements ) nor too large (as this reduces the number of measurements ). On the basis of their conclusions, it is possible that the errors reported in this paper may be artificially low. In view of this, we have included details of the individual simulation runs in the appendix for the reader to draw his or her own conclusions about the quality of our results. From the results in table 1, we conclude that the Price potential is quite accurate in predicting the shear viscosity of liquid rubidium at the three state points shown. It is interesting to note that at the high-density state point (which is very near the triple point), the Price potential prediction overestimates the experimental result while it underestimates it at the medium and lower density. Since the temperature is also varying at each state point, it is difficult to draw a firm conclusion about where the Price potential might yield exact viscosities, but it is clear that this region of exactitude of the Price potential is bounded by the temperatures and densities of the medium- and high-density state points simulated in this study. That the Price potential is most accurate near the triple point and less accurate at expanded (lower density) states is consistent with results obtained by Rahman (1974b) and Mountain (1978) for the structure factor of liquid rubidium. Using equilibrium molecular dynamics, Rahman found that the Price potential predicted the structure factor and inelastic neutron scattering data for rubidium at the triple point very well while Mountain found using Monte Carlo simulation that at expanded states the Price potential yielded less accurate predictions for the structure factor. The source of the inaccuracy of the Price

8 600 P T Crrtntiiitigs and G P Morriss potential at expanded states is not clear and will undoubtedly by the subject of further investigation. It is known (Kawasaki and Gunton 1973, Yamada and Kawasaki 1975, Ernst et al 1978; Evans 1979, 1981, Evans and Hanley 1980) that the pressure P depends on the strain rate according to the equation P=Po +Ply? Figure 4 shows the results obtained by plotting P against y*3'2 and clearly equation (10) is satisfied by the simulation results. (Strictly speaking, the results presented in figure 4 correspond to the trace of the pressure tensor P given in equation (3). The full expression for the pressure for liquid metals involves terms which take into account the electronic contribution and the density dependence of the Price potential.) According to Evans (1 983), the configurational part of the internal energy depends on the strain rate according to uconf = L'O + U] y3 2. The results obtained in this simulation are consistent with this dependence and can be fitted to equation (I 1) as indicated in figure 5. The NEMD results at the highest density state point, given in table 2, warrant further comment. Of the three state points simulated, the results at this state point show the most (10) (1 1) 'f T* 312 Figure 4. The dimensionless trace of the pressure tensor-see equation (3)-as a function of the cube of the square root of the strain rate Y* for liquid rubidium at the three state points T= K, p= g cm-3 (labelled I), T= K, p= g CITI-~ (labelled 11) and T= K.p= g ~ m ((labelled - ~ 111).

9 Shear riscositj- of liquid rubidium 60 1 deviation from the strain rate dependence for viscosity given in equations (9). In fact, plotting v against y* on a log-log plot suggests that these results fit a strain rate dependence of the form V=Vo + V I f (12) where a is slightly less than i. It has been observed that a failure to satisfy the strain rate dependence (9) for the shear viscosity over the whole range of simulated strain rates may indicate that there are several distinct regions where (9) is valid with different values of v0 and ql for each region corresponding to Bingham plastic behaviour (Evans 1982). This behaviour would indicate that at the density and temperature in question the simulated fluid would, at zero strain rate, be a crystalline solid rather than a liquid and that the imposition of a non-zero strain rate induces shear melting. However, there is no evidence in the present simulations nor in that of Rahman (1947b) that at zero strain rate the Price potential is predicting a solid rather than a liquid state. On the basis of the available information, we therefore conclude that the deviation from equation (9) is not due to shear melting effects. In view of the discussion above, we suspect that the errors in the results tabulated in table 2 represent underestimates and that equation (9) is satisfied to within the actual error of the simulation. To confirm this, we are in the process of performing further simulations at this state point using more values of y and evaluating more carefully the dependence of the results on system size. (To date, our investigation of system size dependence has been limited to relatively short calculations at this state point using 256 molecules for y* equal to 1.0 and which differ from the results reported in table 1 by less than 3%, suggesting that system size dependence of the results in table 2 is minimal.) 3. Conclusions We have demonstrated the accuracy of the Price potential in calculating the shear viscosity of liquid rubidium. The NEMD results presented in this paper represent errors of only 3-11% when compared with the corresponding experimental data. These results contrast

10 602 P T Cummings and G P Morriss with those obtained by Bansal and Bruns (1984) who found errors of 32-45% using a somewhat different NEMD simulation technique. Moreover, as is clear from table 1, Bansal and Bruns underestimate the experimental results where the results presented in this paper overestimate and tiice iwsa. Thus there is a strong discrepancy between the results presented here and those of Bansal and Bruns (1984) which must be attributed to the simulation methodology since the same pair potentials are used in both studies. (Since Bansal and Bruns also used the Price potential and the same size system (108 molecules), apart from the difference in simulation methodology the only other possible source of the discrepancy is the cut-off (the point beyond which the potential is assumed to be zero). In this paper, as noted in 0 2 a cut-off of 3a was used while Bansal and Bruns utilised a cut-off equal to half the box size which for the three densities g, g and g cm-3 correspond to cut-offs of 2.47, 2.57 and 2.67~ respectively. To probe the importance of the cut-off. we performed several short simulation runs with a cut-off of 2.5~ at the intermediate density state point. The results obtained for the shear viscosity lie within the error estimates given in table 3. This eliminates the difference in cut-off as a source of the discrepancy between the present results and those of Bansal and Bruns.) Given the success of the NEMD technique used in this paper on many other systems (both simple and molecular fluids), we are confident of the accuracy of the results presented in this paper. Accepting our results and considering the agreement of the simulation results presented in this paper with experimental data suggests that the Price potential is capable of predicting shear viscosities of liquid rubidium with quantitative accuracy. In future research, the accuracy of the Price potential for other transport and thermodynamic properties of liquid rubidium, as well as other liquid metals, will be examined. Acknowledgments PTC acknowledges the Camille and Henry Dreyfus Foundation for their support of this research through the award of a Grant for Newly Appointed Faculty in the Chemical Sciences. The simulations reported in this paper were performed on a CSPI bit word array processor in the Center for Computer Aided engineering at the University of Virginia. The authors are indebted to the Center for the provision of computational facilities in support of the array processor. The support of the National Science Foundation in providing matching funds (CPE ) for the purchase of the array processor to support research in statistical mechanics and chemical process dynamics is gratefully acknowledged. Appendix Details of the individual simulation runs are provided in table Al. At each density and temperature, the simulations began with strain rate y* = 1.O and the molecules initially in a face-centred cubic crystalline configuration. For smaller values of the strain rate, a melted configuration obtained from an earlier simulation run was used as the initial configuration. At each value of the strain rate, a run of between and timesteps was performed and discarded in order to reach a steady state. The discarded runs are not recorded in table A 1.

11 Shear viscosity of liquid rubidium 603 Table A 1. Individual non-equilibrium molecular dynamics simulation runs performed to obtain the results given in tables 1 to 4. Timesteps T(K) p(gcm- ) y II (CP) Uconf/N& Pa3/& l o I.o I.O 1.o I.o 1.o o 1.o 1.o I I References Arlinghaus R T and Cummings P T I987 J. Phys. F: Met. Phys Bansal R and Bruns W 1984 J. Chern. Phys Bird R 8, Stewart W E and Lightfoot E N 1960 Transport Phenomena (New York: Wiley)

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