Shear viscosity of liquid rubidium at the triple point

Transcription

1 J. Phys. F: Met. Phys. 18 (1988) 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, University of Virginia, Charlottesville. VA 22901, USA $ Research School of Chemistry, Australian National University, GPO Box 4, Canberra, ACT Australia Received 11 August in final form 5 January 1988 Abstract. In a recent paper we reported non-equilibrium molecular dynamics simulations of liquid rubidium using the isokinetic sllod algorithm for the dynamics and the Price potential for the interactions. New and more extensive results are presented which confirm that the results exhibit normal square-root behaviour of the shear viscosity as a function of strain rate at the triple point. I. Introduction 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 1984, Evans 1986, Evans and Hoover 1986). Recent developments, such as the sllod algorithm for shear viscosity and the derivation of synthetic techniques for calculating such properties as the thermal conductivity and self-diffusion, 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 er af 1982, Evans and Morriss 1983) and the extension of linear response theory to test the validity of NEMD techniques (Morriss and Evans 1985), are reviewed by Evans (1986). In a recent paper (Cummings and Morriss 1987) we reported non-equilibrium molecular dynamics simulations of liquid rubidium using the isokinetic sllod algorithm for the dynamics and the Price potential for the interactions. In that paper, some concern was expressed about the results obtained at the triple point since the simulation results for shear viscosity available at the time appeared to deviate a little from the expected square-root dependence on the strain rate (see equation (6) below). This raised the possibility that at the triple point density and temperature in the absence of shear the equilibrium state of the simulated system (rubidium pseudoatoms interacting via the Price potential) is a solid rather than a triple point liquid. In the latter case, one expects to observe separate domains of square-root dependence in the shear viscosity associated with Bingham plastic-type behaviour (Evans 1982). It was conjectured that this is not the case and that, within the true error bars for the simulations, the results did satisfy the square-root dependence. Evidence confirming this conjecture is reported in this paper. The simulations reported in the earlier paper were performed on 108 molecules at five different strain /88/ IOP Publishing Ltd 1439

2 1440 P T Cummings and G P Morriss rates; in this paper, we present results for five additional strain rates with 108 molecules and results for six strain rates obtained using 256 molecules. Collectively, these results confirm that the square-root dependence of the shear viscosity is evident in the simulations and that the simulated system is in the liquid state at the experimental triple point. In Q 2, we briefly review the NEMD simulation method and in 0 3 we present results obtained at the triple point. 2. NEMD simulation method The sllod equations of motion induce steady-state planar Couette flow in a homogeneous system such that the streaming velocity has a non-zero component in the x direction, U,, which satisfies du,ldy = y where y is the constant strain rate. This corresponds to the linear velocity profile that distinguishes planar Couette flow (Bird et a1 1960). In order to do this, the positions r, and peculiar momentap, of the molecules in the simulation are obtained by solving the first-order isokinetic sllod equations of motion given by (Evans and Morriss 1984, Cummings and Morriss 1987) dr,ldt = p/m + r, Vu (20) dp,ldt = F, -pi - VU - Ap, (1) (2b) combined with the Lees-Edwards sliding brick boundary conditions (Lees and Edwards 1972). In equation (2), u=(yy, 0, 0) is the streaming velocity of the molecules, F, is the force on molecule i and A is given by A = (PI Ff -pip, Vu) p, * p,. I= I I, =I In this equation, 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 A is dictated by requiring that the temperature T be held constant, which is equivalent to demanding that E:; =,p: remain a constant of the motion. The yx component, Pyx, of the pressure tensor P is related to the velocity profile by the constitutive relation (3) where rj is the shear viscosity. The pressure tensor P, which is of second rank, is calculated from the expression N N N I= I r=l ]=I where m is the mass of each molecule, U, is the velocity of molecule i, V is the volume of the system, rf, = r, - r, and F,, is the force exerted on molecule i by molecule j and so

3 Liquid rubidium ut the triple point 1441 is given by the gradient of the pair potential q(r) between the molecules. Notice that the velocities in equation (5) have the streaming part U subtracted out and are thus peculiar velocities (Evans and Morriss 1984). The shear viscosity is therefore calculated from equation (4) using the time average of the yx component of P and the hydrostatic pressure P is the time average of the trace of P. The strain-rate-dependent q, P and configurational internal energy Uconf are then fitted to q = Til + VI Y I P= Pll+ Ply :? (7) win! = U,, + U,?3i? (8) where the subscript 0 indicates the zero strain rate limit of each quantity. Equations (6)-(8) are the leading-order strain rate dependencies derived theoretically (Kawasaki and Gunton 1973, Yamada and Kawasaki 1975, Ernst er ul 1978) and confirmed by extensive simulations (Evans 1979, 1981, 1983, Evans and Hanley 1980). The Newtonian viscosity of rubidium is given by q(,. The square-root dependence of the shear viscosity on strain rate, equation (6), is a consequence of the long-ranged t-? behaviour of the stress autocorrelation function at large time r. There has been some recent discussion as to whether the rrd? dependence (where d is the dimensionality of the system) is the true asymptotic longtime behaviour of the shear stress autocorrelation function (Dorfman and Kirkpatrick 1986, Kirkpatrick 1984, Kirkpatrick and Nieuwondt 1986, van Beijeren 1984). If this enhanced long-time tail behaviour is not the final asymptotic form then the small-y behaviour of the viscosity may deviate from its usual square-root dependence. Very recently Evans and Morriss (1987) have shown that at sufficiently small values of the shear rate, the viscosity becomes an analytic function, varying as y. This analytic behaviour is observed at shear rates that are much lower than those typically accessible in non-equilibrium molecular dynamics simulations. In practice, this means that the extrapolation to zero strain rate (required to obtain Newtonian viscosity) based on equation (6) may, in some cases, lead to an overestimate of the exact result. (6) 3. NEMD simulations at the triple point As in our previous paper (Cummings and Morriss 1987), all the simulations were performed on a Computer Signal Processing Inc (CSPI) bit word array processor using 108 or 256 rubidium ions using the Price potential (Price et a1 1970, Price 1971) for the interaction between rubidium pseudoatoms. 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. Other mechanical details of the simulation method used are described in P 2 of Cummings and Morriss (1987). In the earlier paper (Cummings and Morriss 1987) simulation results at the triple point (density p= g cm-, temperature T=318.9 K) were obtained for 108 rubidium ions at five reduced strain rates y* = yo(m/e) (equal to 0.1,O. 16, 0.36,0.64 and 1.0). In this paper, we report additional results for 108 molecules at strain rates y* of 0.01, 0.04, 0.25, 0.49 and 0.81 as well as results for 256 pseudoatom simulations at strain rates of 0.04, 0.09, 0.16, 0.36, 0.64 and 1.0. The 108 pseudoatom simulation results are tabulated in table 1 along with those from the earlier paper; the 256 pseudoatom results are given in table 2. All the shear viscosity results are plotted

4 ~ 1442 P T Cummings and G P Morriss Table 1. NEMD simulation results for liquid rubidium at the triple point (T= K and p= g ~m-~) using 108 rubidium pseudoatoms. Time steps y' ~(cp) LF"'/NE Pdls 80 OOO 80 OOO 160 OOO 100 OOO 240 OOO 100 OOO 400 OOO 360 OOO 360 OOO 240 OOO 1.o against y*"' in figure 1 and the pressure and internal energy results against Y"~'' in figure's 2 and 3 respectively. In tables 1 and 2, no numerical estimates of the errors in the shear viscosity are provided. This is in contrast to the previous paper (Cummings and Morriss 1987) in which the errors in the simulation quantities were estimated based on the standard deviation of block averages. As was pointed out in the previous paper, such error estimation is strongly dependent on the block size chosen. Rather than give numerical values of the error in the simulated quantities, we provide the following estimates as guides to the accuracy of the simulation results: the error in the shear viscosity varies from less than 1% for the high strain rate results (f' = 1) to 2-3% at the lowest strain rate; the error in the configurational internal energy and the pressure is 1% or less at all strain rates. The combined (108 and 256 pseudoatom) simulation results are fitted in a leastsquares sense to the strain rate equations (6)-(8). (In computing the least-squares fit, the results for 108 and 256 pseudoatom systems at strain rates 1.O, 0.64, 0.36, 0.16, 0.09 and 0.04 are averaged together to give a single value for 7, P and U at each of these strain rates.) The resulting fitted equations are ~= ~:'"' Pdl~ = ~"~" V'"'INE= t 0.418~*.~". (9) Table 2. NEMD simulation results for liquid rubidium at the triple point (T= K and p= g cm-') using 256 rubidium pseudoatoms. Time steps 20 OOO 40 OOO 4oOOO 40 OOO 40 OOO 80 OOO Y* 1.o tl (CP) LFOnfINe Pdle

5 Liquid rubidium at the triple point L Figure I. The shear viscosity q of liquid rubidium at the triple point as a function of the square root of the reduced strain rate y*. In all the figures, the circles (0) and triangles (A) represent the 108 and 256 pseudoatom simulation results respectively. The straight line is the least-squares fit of the simulation data to equation (6). Thus, the combined simulation results predict that the Newtonian viscosity of liquid rubidium at the triple point is cp. This is within 3.5% of the result of CP obtained using only the limited data from the previous paper (Cummings and Morriss 1987) and within 7% of the experimental result CP (Landolt-Bornstein 1969). It is our contention that the combined simulation results plotted in figure 1 exhibit the expected square-root dependence of the shear viscosity on strain rate, equation (6). There is greater regularity in the deviation of the 108 pseudoatom results from the square-root dependence (except at the lowest strain rate) than in the deviation of the 256 pseudoatom results. Nevertheless, all the simulation results cluster around the Figure 2. The dimensionless trace of the pressure tensor-see equation (5)-as a function of the cube of the square root of the strain rate y" for liquid rubidium at the triple point. The straight line is the least-squares fit of the simulation data to equation (7).

6 1444 P T Cummings and G P Morriss s a Figure3. The dimensionless configurational internal energy as a function of the cube of the square root of the strain rate y" for liquid rubidium at the triple point. The straight line is the least-squares fit of the simulation data to equation (8). square-root behaviour rather well, considering the large change in the viscosity over the range of simulated strain rates. One might be tempted to view the regularity in the deviation of the 108 pseudoatom results as being indicative of a dependence on strain rate that differs from equation (6). In particular, one might question two aspects of the asymptotic behaviour given in equation (6): first, is the large-y'l behaviour asymptotic in y":"'or is the power law different from 1/2? Second, is the small-y" result (q-y'") at all evident in the simulation? To answer these questions, the simulation results were leastsquares fitted to the following equations: q = qi, + q p l + q2y*2 (10) Equation (12) attempts to determine whether equation (6) is valid. If it is valid, one can expect that a will be near 1/2. Equation (1 1) is intended to determine whether the deviation from square-root behaviour in the simulation results is evidence of the transition to the classical small-y" behaviour. If equation (6) is valid, one can expect to see a small q? coefficient compared with the 7, coefficient (i.e., q2/ql<<l). Equation (10) is the most general fit of the simulation data that is considered here: again, validity of equation (6) would correspond to an a in this equation of approximately 112 and a small qrjq, ratio. The results of the least-squares fits are given in table 3 along with the squared error E' defined by I1 In this equation, the q,"""' are the simulation results for the shear viscosity (of which there are eleven when the 108 and 256 pseudoatom results at the same strain rate are combined), the y,* are the reduced strain rates at which the simulations were

7 Liquid rubidium at the triple point 1445 Table 3. Results obtained by least-squares fitting of simulation results to equations (10)- (12) and (6). Equation vi, 'I1 'I2 U E' (10) ( I.2 x 10. (11) X 10.' (12) W x 10 -' (6) I X 10 ' performed and fhci"! is the expression for the shear viscosity obtained from equations ( IO)-( 12) and (6) as appropriate. Simply on the grounds of flexibility, one can expect that equation (12) will exhibit the lowest squared error and equation (6) the highest. Only large differences between the squared errors can be taken as evidence that the asymptotic form of equation (6) is invalid. The resulting curves are plotted along with the simulation results in figure 4. It is evident that none of the curves fits the simulation results dramatically better than equation (6). In fact, equation (12) yields an exponent ( ) that is very close to 1/2. indicating that if the simulation data are dominated by a single power-law decay l a 0.2 Y - E I I I I , Figure4. The shear viscosity 7 of liquid rubidium at the triple point as a function of the reduced strain rate f. The full and dotted curves represent the least-squares fits of equations (IO) and (I 1) respectively in the upper figure. and of equations (12) and (6) respectively in the lower figure.

8 1446 P T Cummings and G P Morriss then the power is very close to 1/2. The equation that might be regarded as having the most theoretical legitimacy, equation (ll), fits the data well and moreover indicates that the correction to equation (6) is small. Perhaps more significant than any single fitted equation is that each equation extrapolates to a Newtonian viscosity in the range cp. This suggests strongly that the value obtained by extrapolating equation (6), 0.690cP, represents a good estimate of the zero strain rate viscosity implied by the simulation results. 4. Conclusions Additional simulation results at the triple point of liquid rubidium have been presented and analysed. It is clear from these results that the behaviour of the strainrate-dependent shear viscosity at the triple point is not anomalous (in this context, exhibiting Bingham plastic behaviour) and the square-root behaviour is that typically seen in non-equilibrium simulations at this range of values of strain rate. Based on the four functional forms used to extrapolate the simulation results to zero strain rate, the Price potential predicts a viscosity of liquid rubidium at the triple point of 0.69k0.04 cp. Acknowledgments PTC acknowledges the support of this research by the National Science Foundation (grant CBT ). The authors thank the National Science Foundation and the Australian Department and Science and Technology which, through the US/Australia Collaborative Research Program, provided travel funds which facilitated this research. 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. The authors are indebted to one of the referees who suggested fitting the shear viscosity results to functional forms other than equation (6), thus resulting in the discussion in the last three paragraphs of $3. References van Beijeren H 1984 Phys. Lett. 105A 191 Bird R B, Stewart W E and Lightfoot E N 1960 Transport Phenomena (New York: Wiley) Cummings P T and Morriss G P 1987 J. Phys. F: Mer. Phys Dorfman J R and Kirkpatrick T R 1986 Moleculur Dynumics Simuluiio,7 of SIuris/icul Mechunicul Sysrems ed. G Ciccotti and W G Hoover (Amsterdam: North-Holland) Ernst M H, Cichocki B, Dorfman J R, Sharma J and van Beijeren H 1978 J. Stat. Phys Evans D J 1979 Phys. Lett. 74A Phys. Rev. A Phys. Rev. A Molecular Dynamics Simulation of Statistical Mechanical Systems ed. G Ciccotti and W G Hoover (Amsterdam: North-Holland)

9 Liquid rubidium ut the triple point 1447 Evans D J and Hanley H J M 1980 Physica A Evans D J and Hoover W G 1986 Ann. Rev. Fluid Mech Evans D J and Morriss G P 1983 Chem. Phys Comput. Phys. Rep Phys. Rev. A Hoover W G and Ashurst W T 1975 Theor. Chem. Adu. Perspect. 1 1 Hoover W G, Ladd A J C and Moran B 1982 Phys. Rev. Lett Kawasaki K and Gunton J D 1973 Phys. Rev. A Kirkpatrick T R 1984 Phys. Rev. Len Kirkpatrick T R and Nieuwondt J 1986 Phys. Rev. A Landolt-Bornstein 1969 Zahlenwerte und Funktionem 5 Tiel (Berlin: Springer) Lees A W and Edwards S F 1972 J. Phys. C: Solid State Phys Morriss G P and Evans D E 1985 Mol. Phys Price D L 1971 Phys. Rev. A Price D L, Singwi K S and Tosi M P 1970 Phys. Rev. B Yamada T and Kawasaki K 1975 Prog. Theor. Phys