- Email: [email protected]
Shear viscosity of simple liquids measured at low shear rates near the triple point
Volume 80A, number 2,3
PHYSICS LETTERS
24 November 1980
SHEAR VISCOSITY OF SIMPLE LIQUIDS MEASURED AT LOW SHEAR RATES NEAR THE TRIPLE POINT H. ABACHI and J. MOLENAT Laboratoire des InteractionsMol&ulaires, Groupe de Dynamique des Phases Condenskes I, USTL, 34060 Montpellier Cidex, France
and P. MALBRUNOT Laboratoire de Physique Appliquie, CEES 2, USTL, 34060 Montpellier Cidex, France Received 5 September 1980
Shear viscosity measurements performed at low shear rates near the triple point of argon, krypton and xenon give results which agree with those obtained at higher shear rates. The non-newtonian behavior found in non-equilibrium molecular dynamics models seems to have no counterpart in the real fluid.
Nonequilibrium molecular dynamics simulations of Couette flow give the following results: - for higher densities, near solidification, shear viscosity depends on shear rate whether it be for a hardsphere potential [l-3], a soft-sphere potential [4], or Lennard-Jones potential [4-71; - the limit viscosity values for a zero shear rate lead to the values obtained via equilibrium molecular dynamics for hard-sphere [8] and Lennard-Jones potentials [91; - for lower densities no variation with shear rate was observed; - finally this non-newtonian effect depends on the system size; it tends to decrease as the number of particles increases [2-41, In order to be certain that this non-newtonian behavior is only due to molecular dynamics, it is necessary to test simple liquids since they are the better adapted for comparison with theoretical models; hitherto, no viscosity measurements in simple liquids at different shear rates have been performed. For this purpose, we have used a vibrating wire viscometer under very small amplitude conditions [lo]. When using such 1 Laboratoire associkau CNRS no. 233. 2 Laboratoire associe au CNRS no. 21.
a viscometer the fluid undergoes an alternative shear rate of decreasing amplitude. Therefore a quantitative rheological study is not possible (an average time shear rate has no physical significance). On the other hand, when working at very small wire amplitudes as is possible with our device [lo], we can measure viscosity in the neighborhood of zero shear rate. We obtained viscosity data under the following experimental conditions: - the frequency, f, was of the order of 2800 Hz. A lower frequency would diminish the shear rate, however, this working frequency was kept in order to insure good experimental conditions (i.e. the wire tensile strength high enough to eliminate mechanical parasites and obtain a sufficient number of oscillations); - the maximum voltage, V,, of 1 W, corresponded to a maximum wire velocity, r/, = 2 V,/m, of the order of 0.025 cm/s and to a maximum shear rate, D, = um (nflv)U2,of the order of 50 s-l ,I being the wire length (5 cm), B the magnetic field (0.17 Wb/m2) and u the kinematic viscosity (about 2 X low3 St); - the avaible part of the decreasing exponential curve was obtained with a signal to noise ratio better than 35 dB; - experiments were performed in an accurate cryostat 171
PHYSICS LETTERS
Volume 80A, number 2,3
24 November 1980
Table 1 Shear viscosity of simple liquids measured at the triple point region under several shear rate conditions. n: viscosity; D: shear rate and D,: maximum shear rate T WI
II
(1 o-5 P)
D= 1750(~-‘)~)
D = 3000 (s-l) b,
D max = 1500 (s-l) c,
D max = 50 (s-l) d,
argon triple point at 83.78 (K)
83.794 83.893 84 85 88 90
_ _ 289.5 280.75 255.5 _
_ _ _ 281.5 255 241.3
310e) 300.5 294 284.7 260.2 244
307 e, 302 291 285 259 243
krypton triple point at 115.777 (K)
115.8 116 117 118 120 125
_ 440.4 425.4 416.7 400.8 __
_ 437 _ 397 355.2
462 e, 448 440.6 423 401.2 361.6
456 e, 451 440 424 403 361
161.41 161.61 162 163 165 170
_
540 _ 534.2 525 507 462
572 e, 560.5 551.7 540 530 492
574 e, 560 555 540 528 495
xenon triple point at 161.396 (K)
_ _ 501.2 _
a) Boon et al.‘s results (ref. [ 13]), we calculated D fromthe diameter of the capillary given in ref. [ 131. b) Sluysar et al.‘s results (ref. [ 14]), we calculated D from data given in ref. [ 151. c) Results obtained with vibrating wire viscometer under usual conditions. d) Results obtained with vibrating wire viscometer working at very small amplitudes. e) Uncertain value because at this temperature a beginning of solidification could arise.
[ 111 and temperature was measured within an error of kO.02 K by a standard resistance thermometer [ 121 immersed in the liquid sample; - the final accuracy was +2%. Table 1 presents our results and those of authors who have also measured viscosity of argon, krypton and xenon very near the triple point [ 13,141. Their results have been obtained with shear rates at least fifty times greater than ours. Boon et al. [ 131 used a capillary viscometer (accuracy f 1%) and Sluysar et al. [ 141 a falling cylinder viscometer (accuracy ?4%). Comparison of these several data leads to an obvious conclusion: within the limits of our respective errors, there is no effect due to the different conditions of shear rate. The small systematic deviation between our respective values is partly due to the use of slightly different density data [ 16,171 for viscosity determinations. Thus, using the same density values would reduce the apparent discrepancies to about 1.5% for argon, 1% 172
for krypton and 0.5% for xenon. Table 2 shows an evaluation of viscosity values which should be obtained at low shear rates if real fluids behaved as simulated fluids; these values are at least 30% higher than the present experimental results at low shear rates reported in the last column. So, the pseudo-plastic behavior observed in molecular dynamic fluids near solidification seems to have no correspondence in real fluids. This conclusion leads to several comments. Firstly, the shear rate values of computer simulations are very different from those of real experiments. The first ones, applied to a small number of particles each of them submitted to high velocities *l , are of the order of 101’ s-l, while the second ones, applied to a con-
* ’ These velocities are in a Maxwell distribution when viscosity calculations are performed and so, large shear rates are required to ensure an average shear stress higher than natural pressure fluctuations and so ensure a linear velocity profile.
Volume 80A, number 2,3
PHYSICS LETTERS
Table 2 Evaluation of the low shear rate viscosity of real fluids near the triple point if a non-newtonian behavior is supposed. These values were obtained with the two relations used in nonequillbrium molecular dynamics to determine zero shear rate viscosities: Ree-Eyring hyperbolic arcsine relation and KawasakiGunton square root relation. These relations were scaled on Sluysar’s data considered as obtained in the middle of the shear rates range which is experimentally possible in laminar flow conditions. n(Q)RB, zero shear rate viscosity obtained with the Ree-Eyring relation; n(Q)KG, zero shear rate vlscosity obtained with the Kawasaki-Gunton relation; vex, present low shear rate results. n(Q)RE
-
argon krypton xenon
385 580 718 ______
~(Q)KG
nex
432 652 805
302 451 560
tinuous medium at rest without any flow are in the range of lo3 s-l. However in the two cases, shear rate values agree with the larninar character of the considered flow. Secondly, to be absolutely certain that no effect exists in real fluids, experimental data obtained at higher shear rates would be desirable. In fact, this is very limited as Sluysar’s data were obtained with a Reynolds number of 800 which does not differ a great deal from the maximum admissible value of 1500 for laminar flow conditions. Two possibilities are then offered: (i), any comparison between real fluids and non-equilibrium molecular dynamics is rejected because of the very high values of shear rates required by computer simulations of Couette Bow. Another question is then brought up: what is the validity of viscosity coefficients obtained in such non-realistic physical conditions?; (ii), the model of non-equilibrium molecular dynamics is valuable on a phenomenological point of view. The comparison with real experiments is then possible because, in the two cases, viscosities are obtained in similar hydrodynamical conditions: laminar flow and shear rates as low as possible (even if their absolute values are very different). What then is the explanation of the pseudo-plastic behavior only exhibited by the simulated fluid? One reason which can always be advanced in such a comparison with real fluids is the approximative character of the model of potential (hard sphere, soft sphere or Lennard-Jones) to represent molecular interactions in noble gas liquids. In the present case this reason is important as Evans [7] showed that the viscosi-
24 November 1980
ty variation with shear rate is only due to the potential contribution to the pressure tensor. On the other hand, it must be noted that this pseudo-plastic behavior decreases when the number of particles in the system is increased [2,3,5]. Therefore, as viscosity is a physical property which takes into account all of the particles in the system,it could be reasonably supposed that the non-newtonian behavior is limited to the model of the non-equilibrium molecular dynamics and must consequently vanish at the macroscopic hydrodynamics limit. From this point of view, calculations with a greater number of particles would be desirable more especially as the computation time could then be increased to obtain a linear velocity profile in a lower range of shear rates. In fact investigation of this problem continues: Evans [7] found that this shear rate dependence of the shear viscosity coefficient is related to the “long time tail” phenomena; Zwanzig [ 181 showed that this dependence can also be obtained in classical kinetic theory from a Boltzmann type equation applied to such a Couette ilow velocity field. In table 1, we did not mention dimensional shear viscosities obtained from molecular dynamics as it would necessitate a particular study on parameter values of interaction potentials. We will soon publish such a comparison of molecular dynamic viscosities with our experimental data.
The authors are grateful to D. Levesque (Laboratoire de Physique Theorique et Hautes Energies, Orsay) for some helpful discussions. References [ I] T. Naitoh and S. Ono, Phys. Lett. 57A (1976) 448. [2] T. Naitohand
S. Ono, Phys. Lett. 69A (1978) 125.
[ 31 T. Naitoh and S. Ono, J. Chem. Phys. 70 (1979) 4515. [4] W.T. Ashurst and W.G. Hoover, Phys. Rev. Al 1 (1975) 658. [5] W.T. Ashurst and W.G. Hoover, Phys. Rev. Lett. 31 (1973) 206. [6] W.T. Ashurst and W.G. Hoover, Phys. Lett. 61A (1977) 175. [7] D.J. Evans, Phys. Lett. 74A (1979) 229. [ 81 B.J. Alder, D.M. Gass and T.E. Wainwright, J. Chem. Phys. 53 (I 970) 3813. [9] D. Levesque, L. Verlet and J. Kurkijarvi, Phys. Rev. A7 (1973) 1690.
173
Volume
[lo]
80A, number
2,3
PHYSICS
J. Molenat et al., Experimental approach of the zero shear rate viscosity, to be published. [l 11 H. Abachi, J. Molenat and P. Malbrunot, J. Phys. El 2 (1979) 706. [I 2) Type 8164-B built and calibrated by Leads and Northrup, USA. [I 31 J.P. Boon, J.C. Legros and G. Thomaes, Physica 33 (1967) 547.
174
LETTERS [14]
24 November
1980
V.P. Sluysar, N.S. Rudenko and V.M. Tret’yakov, Ukr. Fiz. Zh. USSR 17 (1972) 1257. [IS] N.S. Rudenko and V.P. Sluysar, Ukr. Fiz. Zh. USSR 13 (1968) 917. [ 161 International thermodynamic tables of the fluid state (Argon 1971), ed. HJPAC (Butterworths, London) [ 171 M.J. Terry, J.T. Lynch, M. Bunclark, K.R. Manse11 and L.A.K. Staveley, J. Chem. Thermodyn. 1 (1969) 413. [18] R. Zwanzig, J. Chem. Phys. 71 (1979) 4416.
PHYSICS LETTERS
24 November 1980
SHEAR VISCOSITY OF SIMPLE LIQUIDS MEASURED AT LOW SHEAR RATES NEAR THE TRIPLE POINT H. ABACHI and J. MOLENAT Laboratoire des InteractionsMol&ulaires, Groupe de Dynamique des Phases Condenskes I, USTL, 34060 Montpellier Cidex, France
and P. MALBRUNOT Laboratoire de Physique Appliquie, CEES 2, USTL, 34060 Montpellier Cidex, France Received 5 September 1980
Shear viscosity measurements performed at low shear rates near the triple point of argon, krypton and xenon give results which agree with those obtained at higher shear rates. The non-newtonian behavior found in non-equilibrium molecular dynamics models seems to have no counterpart in the real fluid.
Nonequilibrium molecular dynamics simulations of Couette flow give the following results: - for higher densities, near solidification, shear viscosity depends on shear rate whether it be for a hardsphere potential [l-3], a soft-sphere potential [4], or Lennard-Jones potential [4-71; - the limit viscosity values for a zero shear rate lead to the values obtained via equilibrium molecular dynamics for hard-sphere [8] and Lennard-Jones potentials [91; - for lower densities no variation with shear rate was observed; - finally this non-newtonian effect depends on the system size; it tends to decrease as the number of particles increases [2-41, In order to be certain that this non-newtonian behavior is only due to molecular dynamics, it is necessary to test simple liquids since they are the better adapted for comparison with theoretical models; hitherto, no viscosity measurements in simple liquids at different shear rates have been performed. For this purpose, we have used a vibrating wire viscometer under very small amplitude conditions [lo]. When using such 1 Laboratoire associkau CNRS no. 233. 2 Laboratoire associe au CNRS no. 21.
a viscometer the fluid undergoes an alternative shear rate of decreasing amplitude. Therefore a quantitative rheological study is not possible (an average time shear rate has no physical significance). On the other hand, when working at very small wire amplitudes as is possible with our device [lo], we can measure viscosity in the neighborhood of zero shear rate. We obtained viscosity data under the following experimental conditions: - the frequency, f, was of the order of 2800 Hz. A lower frequency would diminish the shear rate, however, this working frequency was kept in order to insure good experimental conditions (i.e. the wire tensile strength high enough to eliminate mechanical parasites and obtain a sufficient number of oscillations); - the maximum voltage, V,, of 1 W, corresponded to a maximum wire velocity, r/, = 2 V,/m, of the order of 0.025 cm/s and to a maximum shear rate, D, = um (nflv)U2,of the order of 50 s-l ,I being the wire length (5 cm), B the magnetic field (0.17 Wb/m2) and u the kinematic viscosity (about 2 X low3 St); - the avaible part of the decreasing exponential curve was obtained with a signal to noise ratio better than 35 dB; - experiments were performed in an accurate cryostat 171
PHYSICS LETTERS
Volume 80A, number 2,3
24 November 1980
Table 1 Shear viscosity of simple liquids measured at the triple point region under several shear rate conditions. n: viscosity; D: shear rate and D,: maximum shear rate T WI
II
(1 o-5 P)
D= 1750(~-‘)~)
D = 3000 (s-l) b,
D max = 1500 (s-l) c,
D max = 50 (s-l) d,
argon triple point at 83.78 (K)
83.794 83.893 84 85 88 90
_ _ 289.5 280.75 255.5 _
_ _ _ 281.5 255 241.3
310e) 300.5 294 284.7 260.2 244
307 e, 302 291 285 259 243
krypton triple point at 115.777 (K)
115.8 116 117 118 120 125
_ 440.4 425.4 416.7 400.8 __
_ 437 _ 397 355.2
462 e, 448 440.6 423 401.2 361.6
456 e, 451 440 424 403 361
161.41 161.61 162 163 165 170
_
540 _ 534.2 525 507 462
572 e, 560.5 551.7 540 530 492
574 e, 560 555 540 528 495
xenon triple point at 161.396 (K)
_ _ 501.2 _
a) Boon et al.‘s results (ref. [ 13]), we calculated D fromthe diameter of the capillary given in ref. [ 131. b) Sluysar et al.‘s results (ref. [ 14]), we calculated D from data given in ref. [ 151. c) Results obtained with vibrating wire viscometer under usual conditions. d) Results obtained with vibrating wire viscometer working at very small amplitudes. e) Uncertain value because at this temperature a beginning of solidification could arise.
[ 111 and temperature was measured within an error of kO.02 K by a standard resistance thermometer [ 121 immersed in the liquid sample; - the final accuracy was +2%. Table 1 presents our results and those of authors who have also measured viscosity of argon, krypton and xenon very near the triple point [ 13,141. Their results have been obtained with shear rates at least fifty times greater than ours. Boon et al. [ 131 used a capillary viscometer (accuracy f 1%) and Sluysar et al. [ 141 a falling cylinder viscometer (accuracy ?4%). Comparison of these several data leads to an obvious conclusion: within the limits of our respective errors, there is no effect due to the different conditions of shear rate. The small systematic deviation between our respective values is partly due to the use of slightly different density data [ 16,171 for viscosity determinations. Thus, using the same density values would reduce the apparent discrepancies to about 1.5% for argon, 1% 172
for krypton and 0.5% for xenon. Table 2 shows an evaluation of viscosity values which should be obtained at low shear rates if real fluids behaved as simulated fluids; these values are at least 30% higher than the present experimental results at low shear rates reported in the last column. So, the pseudo-plastic behavior observed in molecular dynamic fluids near solidification seems to have no correspondence in real fluids. This conclusion leads to several comments. Firstly, the shear rate values of computer simulations are very different from those of real experiments. The first ones, applied to a small number of particles each of them submitted to high velocities *l , are of the order of 101’ s-l, while the second ones, applied to a con-
* ’ These velocities are in a Maxwell distribution when viscosity calculations are performed and so, large shear rates are required to ensure an average shear stress higher than natural pressure fluctuations and so ensure a linear velocity profile.
Volume 80A, number 2,3
PHYSICS LETTERS
Table 2 Evaluation of the low shear rate viscosity of real fluids near the triple point if a non-newtonian behavior is supposed. These values were obtained with the two relations used in nonequillbrium molecular dynamics to determine zero shear rate viscosities: Ree-Eyring hyperbolic arcsine relation and KawasakiGunton square root relation. These relations were scaled on Sluysar’s data considered as obtained in the middle of the shear rates range which is experimentally possible in laminar flow conditions. n(Q)RB, zero shear rate viscosity obtained with the Ree-Eyring relation; n(Q)KG, zero shear rate vlscosity obtained with the Kawasaki-Gunton relation; vex, present low shear rate results. n(Q)RE
-
argon krypton xenon
385 580 718 ______
~(Q)KG
nex
432 652 805
302 451 560
tinuous medium at rest without any flow are in the range of lo3 s-l. However in the two cases, shear rate values agree with the larninar character of the considered flow. Secondly, to be absolutely certain that no effect exists in real fluids, experimental data obtained at higher shear rates would be desirable. In fact, this is very limited as Sluysar’s data were obtained with a Reynolds number of 800 which does not differ a great deal from the maximum admissible value of 1500 for laminar flow conditions. Two possibilities are then offered: (i), any comparison between real fluids and non-equilibrium molecular dynamics is rejected because of the very high values of shear rates required by computer simulations of Couette Bow. Another question is then brought up: what is the validity of viscosity coefficients obtained in such non-realistic physical conditions?; (ii), the model of non-equilibrium molecular dynamics is valuable on a phenomenological point of view. The comparison with real experiments is then possible because, in the two cases, viscosities are obtained in similar hydrodynamical conditions: laminar flow and shear rates as low as possible (even if their absolute values are very different). What then is the explanation of the pseudo-plastic behavior only exhibited by the simulated fluid? One reason which can always be advanced in such a comparison with real fluids is the approximative character of the model of potential (hard sphere, soft sphere or Lennard-Jones) to represent molecular interactions in noble gas liquids. In the present case this reason is important as Evans [7] showed that the viscosi-
24 November 1980
ty variation with shear rate is only due to the potential contribution to the pressure tensor. On the other hand, it must be noted that this pseudo-plastic behavior decreases when the number of particles in the system is increased [2,3,5]. Therefore, as viscosity is a physical property which takes into account all of the particles in the system,it could be reasonably supposed that the non-newtonian behavior is limited to the model of the non-equilibrium molecular dynamics and must consequently vanish at the macroscopic hydrodynamics limit. From this point of view, calculations with a greater number of particles would be desirable more especially as the computation time could then be increased to obtain a linear velocity profile in a lower range of shear rates. In fact investigation of this problem continues: Evans [7] found that this shear rate dependence of the shear viscosity coefficient is related to the “long time tail” phenomena; Zwanzig [ 181 showed that this dependence can also be obtained in classical kinetic theory from a Boltzmann type equation applied to such a Couette ilow velocity field. In table 1, we did not mention dimensional shear viscosities obtained from molecular dynamics as it would necessitate a particular study on parameter values of interaction potentials. We will soon publish such a comparison of molecular dynamic viscosities with our experimental data.
The authors are grateful to D. Levesque (Laboratoire de Physique Theorique et Hautes Energies, Orsay) for some helpful discussions. References [ I] T. Naitoh and S. Ono, Phys. Lett. 57A (1976) 448. [2] T. Naitohand
S. Ono, Phys. Lett. 69A (1978) 125.
[ 31 T. Naitoh and S. Ono, J. Chem. Phys. 70 (1979) 4515. [4] W.T. Ashurst and W.G. Hoover, Phys. Rev. Al 1 (1975) 658. [5] W.T. Ashurst and W.G. Hoover, Phys. Rev. Lett. 31 (1973) 206. [6] W.T. Ashurst and W.G. Hoover, Phys. Lett. 61A (1977) 175. [7] D.J. Evans, Phys. Lett. 74A (1979) 229. [ 81 B.J. Alder, D.M. Gass and T.E. Wainwright, J. Chem. Phys. 53 (I 970) 3813. [9] D. Levesque, L. Verlet and J. Kurkijarvi, Phys. Rev. A7 (1973) 1690.
173
Volume
[lo]
80A, number
2,3
PHYSICS
J. Molenat et al., Experimental approach of the zero shear rate viscosity, to be published. [l 11 H. Abachi, J. Molenat and P. Malbrunot, J. Phys. El 2 (1979) 706. [I 2) Type 8164-B built and calibrated by Leads and Northrup, USA. [I 31 J.P. Boon, J.C. Legros and G. Thomaes, Physica 33 (1967) 547.
174
LETTERS [14]
24 November
1980
V.P. Sluysar, N.S. Rudenko and V.M. Tret’yakov, Ukr. Fiz. Zh. USSR 17 (1972) 1257. [IS] N.S. Rudenko and V.P. Sluysar, Ukr. Fiz. Zh. USSR 13 (1968) 917. [ 161 International thermodynamic tables of the fluid state (Argon 1971), ed. HJPAC (Butterworths, London) [ 171 M.J. Terry, J.T. Lynch, M. Bunclark, K.R. Manse11 and L.A.K. Staveley, J. Chem. Thermodyn. 1 (1969) 413. [18] R. Zwanzig, J. Chem. Phys. 71 (1979) 4416.