Langmuir 1997, 13, 5739-5750
5739
External Thermal Gradient Dependence of the Shear Viscosity of Helium in the Transition Regime German Urbina-Villalba,*,†,‡ Ma´ximo Garcı´a-Sucre,†,‡ Luis Araque-Lameda,§ and Rixio Parra† Instituto Venezolano de Investigaciones Cientı´ficas (IVIC), Centro de Fı´sica, Altos de Pipe, Carretera Panamericana, Km 11, Apdo. 21827, Caracas 1020A, Venezuela, Universidad Central de Venezuela, Ciudad Universitaria, Facultad de Ciencias, Escuela de Quı´mica, Postgrado, Caracas, Venezuela, and Universidad de Oriente, Departamento de Fı´sica, Nu´ cleo Sucre, Cumana´ , Edo Sucre, Venezuela Received October 15, 1996. In Final Form: July 2, 1997X The effective viscosity of helium in the transition regime has been measured at different external thermal gradients with a rotatory cylinder viscometer used as a damped oscillator. It was found that the viscosity of helium does not vary monotonically with the gradient of temperature, showing an appreciable change at G ) 6.9 ( 0.8 K/cm for average temperatures between 293 and 327 K and a distance between the fixed and moving plates of 2.0 ( 0.1 cm. Similar experiments held at constant thermal gradients and variable pressure indicated that under those conditions, and within the range 1-150 µmHg, the viscosity of helium presents two regions of rapid variation with the pressure whose positions depend on the magnitude of the thermal gradient. Some general characteristics of the behavior found can be predicted in terms of a simple theory based on the Boltzmann equation under the relaxation time approximation, although the detailed variation of the shear viscosity with the thermal gradients awaits a thorough explaination. It is clear, however, that higher order terms in the collision time are required for the theoretical consideration of this effect. The region of rapid variation of the shear viscosity at constant thermal gradient and varying pressure can be theoretically reproduced assuming a molecular diameter for helium of 3.1 ( 0.3 Å, while those at constant pressure and variable thermal gradient can be reproduced for a diameter of 1.9 ( 0.3 Å. The first of these values is close to the currently accepted diameter of helium gas deduced from theoretical computations and van der Waals excluded volume measurements. On the other hand, the second value is close to the shear viscosity and thermal conductivity measurements in the continuous regime.
Introduction The behavior of viscosity in the continuous regime is well-known. In 1972, Maitland and Smith1 published a critical reassessment of the viscosities of 11 common gases, including helium, for temperatures between 80 and 2000 K. Employing selected values of Kestin et al.,2-10 Guevara et al.,11-13 and Smith et al.14-18 as a body of reliable data, and discarding all other viscosity data differing more than 3% from the chosen one, they found that the gas viscosity † Instituto Venezolano de Investigaciones Cientı´ficas (IVIC). E-mail:
[email protected]. ‡ Universidad Central de Venezuela. § Universidad de Oriente. X Abstract published in Advance ACS Abstracts, September 15, 1997.
(1) Maitland, G. C.; Smith, E. B. J. Chem. Eng. Data 1972, 17, 150. (2) DiPippo, R.; Kestin, J. Proc. 4th Symposium on Thermo-Physical Properties, College Park, 1-4, April 1968. (3) DiPippo, R.; Kestin, J.; Oguchi, K. J. Chem. Phys. 1970, 52, 693. (4) DiPippo, R.; Kestin, J.; Whitelaw, J. H. Physica 1966, 32, 2064. (5) Kalelkar, A. S.; Kestin, J. J. Chem. Phys. 1970, 52, 4248. (6) Kestin, J.; Leidenfrost, W. Physica 1959, 25, 537 and 1033. (7) Kestin, J.; Nagashima, A. J. Chem. Phys. 1964, 40, 3648. (8) Kestin, J.; Ro, S. T.; Wakeham, W. A. Trans. Faraday Soc. 1971, 67, 2308. (9) Kestin, J.; Wakeham, J. H.; Watanabe, K. J. Chem. Phys. 1970, 53, 3773. (10) Kestin, J.; Yobayashi, T.; Wood, R. T. Physica 1966, 32, 1065. (11) Goldblatt, M.; Guevara, F. A.; McInteer, B. B. Phys. Fluids 1970, 13, 2873. (12) Guevara, F. A.; McInteer, B. B.; Otteson, D.; Hanley, H. J. M. Los Alamos Scientific Laboratory Rept., LA-4643-MS, University of California, 1971. (13) Guevara, F. A.; McInteer, B. B.; Wageman, W. E. Phys. Fluids 1969, 12, 2493. (14) Clarke, A. G.; Smith, E. B. J. Chem. Phys. 1968, 48, 3988. (15) Clarke, A. G.; Smith, E. B. J. Chem. Phys. 1969, 51, 4156. (16) Dawe, R. A.; Smith, E. B. J. Chem. Phys. 1970, 52, 693. (17) Dawe, R. A.; Maitland, G. C.; Rigby, M.; Smith, E. B. Trans. Faraday Soc. 1970, 66, 1955. (18) Maitland, G. C.; Smith, E. B. J. Chem. Phys. 1970, 67, 631.
S0743-7463(96)00988-2 CCC: $14.00
data could be suitable described through a simple monotonic function over a wide temperature range:
ln(η/S) ) A ln T + B/T + C/T2 + D
(1)
where η is the viscosity, S is a standard viscosity value defined with respect to the accepted viscosity value of nitrogen at 293.2 K (175.7 µP), and A, B, C, and D are constants which for the case of helium equal 0.71938, 12.451, -295.67, and -4.1249, respectively (S ) 196.0 µP). An extensive compilation of the currently accepted equilibrium and transport properties of noble gases and 26 binary and multicomponent mixtures was reported by Kestin et al.19 for “low densities” still belonging to the continuous regime. This type of experimental data has been employed to parametrize several intermolecular potentials for gases20-27 and test some other theoretical ones.28-29 A rough compilation of the molecular cross sections deduced from the transport coefficients data as well as a selected number of theoretical calculations are (19) Kestin, J.; Knierim, K.; Mason, E. A.; Najafi, B.; Ro, S. T.; Waldman, M. J. Phys. Chem. Ref. Data 1984, 13, 229-303. (20) de Boer, J.; Michels, A. Physica 1938, 6, 945. (21) Hirschfelder, J. O.; Bird, R. B.; Spotz, E. L. J. Chem. Phys. 1948, 205-231. (22) Kestin, J.; Whitelaw, J. H. Physica 1963, 29, 335-356. (23) Brunch, L. W.; McGee, I. J. J. Chem. Phys. 1967, 46, 2959. (24) Brunch, L. W.; McGee, I. J. J. Chem. Phys. 1970, 52, 5884. (25) Burgmans, A. L. J.; Farrar, Lee, Y. T. J. Chem. Phys. 1976, 64, 1345. (26) Kell, G. S.; McLaurin, G. E.; Whalley, J. Chem. Phys. 1978, 68, 2199. (27) Aziz, R. A.; Nain, V. P. S.; Carley, J. S.; Taylor, W. L.; McConville, G. T. J. Chem. Phys. 1979, 70, 4330-4342. (28) Aziz, R. A.; McCourt, F. R. W.; Wong, C. C. K. Molec. Phys. 1987, 61, 1487-1511. (29) Aziz, R. A.; Slaman, M. J.; Koide, A.; Allnatt, A. R.; Meath, W. J. Molec. Phys. 1992, 77, 321-337.
© 1997 American Chemical Society
5740 Langmuir, Vol. 13, No. 21, 1997
Urbina-Villalba et al.
Table 1. Some Molecular Diameters and Potential Parameters Reported for Heliuma potential Lennard-Jones 6-12 Lennard-Jones 6-12 Buckingham Exp-6 MDD-1 MDD-2 ESMMSV mLJ-D Lennard-Jones 6-12 HFDHE2 extended principle of corresponding states HFD-B(HE) XC-2 XCID-2 exact quantum Monte Carlo
/k 10.22 6.03 9.16 12.13 10.75 10.57 10.30 10.80 10.80 10.40 10.948 10.9853 10.9828 11.01
rm (Å)
3.160 2.975 3.0238 2.97 2.979 2.9673 2.963 2.9625 2.9638 2.9635
σ (Å)
ref
2.556 2.70 2.768 2.644 2.682 2.685 2.57 2.639 2.61
20 21 22 23 24 25 25 26 27 19
2.6369 2.6361 2.6374 2.646
28 29 29 30
a /k is the minimum value of the helium-helium potential in k units; rm is the internuclear distance at the minimum potential energy; and σ is the helium-helium internuclear distance for which the interatomic potential takes a zero value (Please note that here we have preserved the usual notation for the zero of the interatomic potential (σ), which should not be confused with the molecular cross section also denoted by sigma in the Results and Discussion section).
shown in Table 1 for the helium-helium case. An updated and complete compilation on the evolution of the theoretical intermolecular potential of helium was recently published by Anderson et al.30 At intermediate pressures,31 the theoretical expression for the torque of a fast moving viscometer (400-1700 rpm), valid in the free molecule (Knudsen) regime, does deviate from the experimental values of air for pressures between 1 and 200 µmHg. This discrepancy, as are most small deviations detected, has been systematically interpreted in terms of the slip of gases at solid surfaces. Due to the independence of the viscosity with pressure in the continuous regime, and the monotonic decrease of the coefficient with pressure in the Knudsen regime, the expected behavior in the intermediate regimesnaturally interpolated between those extreme casesshad mostly been corroborated by scattered experimental measurements. That intermediate regime is known as the transition regime, i.e., that range in pressures in which the mean free path of the gas is of the order of the size of its container (Kn ) l/L ∼ 1; where Kn is the Knudsen number, l is the mean free path, and L is the relevant macroscopic length of the container). Unlike the case for other transport coefficients,32-35 only few measurements of the shear viscosity of gases in the transition regime have been made.36-41 Since 1978, our group has reported evidence on a nonmonotonic variation of the effective shear viscosity of gases in the transition regime for He, Ar, CO2, and N2. In those experiments, (30) Anderson, J. B.; Trynor, C. A.; Boghosian, B. M. J. Chem. Phys. 1993, 99, 345-351. (31) Kuhlthau, A. R. J. App. Phys. 1949, 20, 217. (32) Teagan, W. P.; Springer, G. S. Phys. Fluids 1968, 11, 497. (33) Johnston, H.; Grilly, E. R. J. Chem. Phys. 1946, 14, 233. (34) Dybbs, A.; Springer, G. S. Phys. Fluids 1965, 8, 1946. (35) Kannuluik, W. G.; Carman, E. H. Proc. Phys. Soc. B 1952, 65, 701. (36) Moronta, D.; Garcı´a-Sucre, M. Phys. Rev. A 1978, 18, 756. (37) Garcı´a-Sucre, M., Moronta, D. Phys. Rev. A 1982, 26, 1713. (38) Garcı´a-Sucre, M.; Mata, G. Phys. Rev. A 1986, 34, 1591. (39) Garcı´a-Sucre, M.; Araque-Lameda, L.; Urbina-Villalba, G.; Parra, R. E. Proc. of the Third Caribbean Congress on Fluid Dynamics and Third latin-American Symposium on Fluid Mechanics, Caracas, Venezuela, 1995, Vol. 1, Section G, pp 1-8. (40) Garcı´a-Sucre, M.; Araque-Lameda, L.; Parra, R.; Urbina-Villalba, G. Ciencia 1995, 2, 109-116. (41) Garcı´a-Sucre, M.; Urbina-Villalba, G.; Araque-Lameda, L. A.; Parra, R. E. Ciencia 1996, 4, 313-322.
relative magnitudes of the shear viscosity are determined rather than absolute ones. Interestingly however, the most relevant evidence of the non monotonic variation of the viscosity of gases in the transition regime has been found for helium,36 which is the noble gas which presents the less pronounced variation of the viscosity with density at high pressures.42-45 In this paper the effect of the thermal gradient on the variation of the shear viscosity of helium in the transition regime is experimentally studied. We have explored two ways for doing this: Firstly, the dependence of the effective viscosity as a function of the external thermal gradient was measured for a given pressure of the gas corresponding to the transition regime. Secondly, the effective viscosity as a function of pressure was evaluated for different fixed values of the external thermal gradient. Within this last case, measurements for the particular zero thermal gradient case were also made, as we have already considered in several previous papers,36,38-41 but this time with a viscometer corresponding to a different value of L. The effect of the thermal gradient has been compared with the results of a simple theory previously developed for this case.37 The previous work of Gaeta, Migliardo, and Wanderlingh46 reported a relevant increase in the viscosity of liquids (ethyl cynnamate, formamide, and water) when an external gradient of temperature was located parallel to the gradient of shear, while no increase was observed when those gradients were mutually perpendicular. Although an opposite and much less pronounced dependence of the shear viscosity with the external thermal gradient is predicted for gases in the continuous regime, a quite strong variation is expected for gases under the transition regime.37 Although no complete theoretical formalism exists for explaining the nonmonotonic behavior of viscosity as a function of either pressure or thermal gradients, found for gases in the transition regime, our results had been justified on the grounds of a simple theory which employs the Boltzmann equation within the relaxation time approximation, keeping terms up to third order in the collision time.37 Under the relaxation time approximation a detailed analysis of the intermolecular forces involved in the collision process is avoided assuming that the effect of collisions is simply to restore the distribution function to one describing equilibrium conditions existing locally near a particular point.47 In this formulation the local distribution of velocities was assumed to be Maxwellian, an assumption that has been experimentally confirmed even for the Knudsen regime,48 employing static and dynamic gas expansion techniques. With these approximations, the theoretical model that is used to interpret our experimental results is sensitive to the effect of intermolecular forces through (1) the mean time between collisions and (2) the boundary conditions of the problem, which are related to the value of the fluid velocity and its spatial derivatives near the walls of the container. Concerning the first point, the mean time between collisions is influenced by the intermolecular potential, and therefore, the effective molecular cross section and the mean free path also depend on it. Thus, the region (42) Kestin, J.; Whitelaw, J. H. Physica 1963, 29, 335. (43) Flynn, G. P.; Hanks, R. V.; Lemaire, N. A.; Ross, J. J. Chem. Phys. 1963, 38, 154-162. (44) Gracki, J. A.; Flynn, G. P.; Ross, J. J. Chem. Phys. 1969, 51, 3856-3863. (45) Vermesse, J.; Vidal, D. C. R. Acad. Sci. Paris 1976, 282, B5-B7. (46) Gaeta, F. S.; Migliardo, P.; Wanderlingh, F. Phys. Rev. Lett. 1973, 31, 1181. (47) Reif, F. Fundamentos de Fı´sica estadı´stica y te´ rmica; McGrawHill: New York, 1968; Chapter 13, p 502. (48) Jitschin, W.; Reich, G. J. Vac. Sci. Techn. 1991, 9, 2752-2756.
Shear Viscosity of Helium
Langmuir, Vol. 13, No. 21, 1997 5741
where transition regime effects may be expected to occur (which in our model depend on the ratio Kn ) mean free path/distance between the solid surfaces) will also depend on the intermolecular forces. Regarding the second point, it is clear that the intermolecular potentials corresponding to either the gas-gas or the solid-gas molecular interactions will influence the values of the fluid velocity and its spatial derivatives near the solid surfaces. These values enter in the model as parameters, and the occurrence of nonmonotonic variations of the viscosity with density depends on them, though the main features related to the separation in density between the regions of rapid variation of this nonmonotonic dependence are not expected to vary strongly (see eq 8). The fact that a transported quantity depends on pressure in a nonmonotonic way in the transition regime is determined in our model by three factors:39 (1) The mathematical form of the equation of transport, which is a differential equation relating the flux (Ψ) to the transported quantity φ, whose nonuniform spatial distribution is related to the transport phenomena under consideration, i. e. Ψ ) F(∂φ/∂z, ∂3φ/∂z3), where F is a linear function and z is the direction along which the transport occurs; (2) The boundary values for φ and its spatial derivatives at the walls; and (3) the state in which the system is found in connection with its steadiness, which in our case includes the constancy of the flux Ψ of the transported quantity along z. According to the referred model37 employing constant boundary conditions (see Data Treatment and Error Evaluation) and assuming laminar flow, the shear viscosity is a periodic function of the density. The viscosity tends to either extremely high or low values whenever
tan(ωL/2) ) ωL/2
(2)
ω being the square root of the quotient between the firstand third-order coefficients of the tangential stress in terms of the spatial derivatives of the fluid velocity. From the referred relation it can be deduced37 that the molecular cross section (σ) is related to the separation in density of the regions of nonmonotonic variation of the shear viscosity, ∆n:
σ)
1 [4/(3π )] ∆nL 3 1/2
(3)
where “∆n” is expressed in number of molecules per unit volume and “L” is the distance between the relative moving plates of the viscometer. In relation to eq 3, it should be noticed that, with σ constant, the only way in which one could obtain a modification of ∆n for the same gas is by modifying the distance “L”. Thus, for a given L, the regions of nonmonotonic variation of the shear viscosity are separated in density by a fixed value (∆n), from which a measurement of the molecular cross section can be obtained. Within the transition regime the length of the mean free path is of the order of the size of the container. Therefore, the gas molecules have an approximately equal number of collisions with the walls of the container as they do with similar gas molecules. Thus, the nonmonotonic behavior of the shear viscosity of gases in the transition regime strongly depends on the structure of the walls. This case (case I) is similar although not equivalent to that case in which the shear viscosity of a fluid is measured employing an apparatus that restricts fluid to wall separations of the size of the molecular mean free path (case II). Both cases belongsby definitionsto the transition regime, but in case I, the transfer of
momentum between the molecules is due to nonlocal effects, since a fluid molecule travels a macroscopic length prior to the collision with either the wall or a similar molecule. In case II however, the fluid molecules undergo several collisions before they reach the walls, so that non local effects are much diminished and, instead, the behavior of the shear viscosity appears to be dominated by the possible geometric arrangements that those molecules can assume in the neighborhood of the walls. Moreover, case II may be expected to differ from case I because of (1) the perturbation of each boundary layer produced by the proximity of the other wall and (2) the sensitivity of the gas molecules to the details of the intermolecular potentials due to the fact that the mean free path of the molecules falls within the short and intermediate regions of the intermolecular forces. The fact there exist marked differences in the momentum transfer of case’s I and II had been evidenced recently by the application of a generalized Enskog’s kinetic theory of tracer diffusion for strongly inhomogeneous fluids49 to the problem of self-diffusion of fluids in slit pores. It was found from both theoretical calculations and molecular simulations that, in pores narrower than ten molecular diameters, the diffusivity deviates significantly from its bulk phase value at the same temperature and chemical potential. These deviations are due to the fact that, within the confining pore walls, the fluid tends to form layers. The pore diffusivity oscillates as a function of pore width, a local minimum occurring when the packing of a given number of fluid layers is favored and a local maximum occurring in the region of transition between these favored packing widths.49 Short range oscillatory solvation forces arise whenever liquid molecules are induced to order into quasi-discrete layers between two surfaces or within any highly restricted space.50 Such oscillatory forces are mainly due to geometric arrangements and not only depend on the properties of the intervening medium but also on the chemical and physical properties of the two boundary surfaces. They can be measured by means of the surface force apparatus (SFA),51 which employs smooth mica surfaces, a variety of force-measuring springs, and an optical technique based on multiple-beam interference fringes. According to the measured data gathered over almost two decades, the following points are now well established: 50-51 (1) In liquids where molecules are roughly spherical and fairly rigid, the periodicity of the oscillatory force with respect to the distance between the confining surfaces is nearly equal to the mean molecular diameter, d. (2) The peak to peak amplitude of the oscillations shows a roughly exponential decay with distance, with a characteristic decay length of 1.2d to 1.7d. (3) Irregularly shaped chain molecules with side groups or branching which lack a symmetry axis cannot easily order into discrete layers, producing disordered or amorphous liquid films that give rise to a monotonic rather than oscillatory force. (4) The effects of polydispersity and molecular polarity do not have large effects in the magnitude of the oscillatory forces. (5) The oscillatory force depends on the structure of the surface lattices. Employing the mean spherical approximation for liquid molecules52 and point charges for ions, Henderson et al.53 theoretically predicted an oscillatory force for the case of (49) Vanderlink, T. K.; Davis, H. T. J. Chem. Phys. 1987, 87, 17911795. (50) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, 1992. (51) Israelachvili, J. N. Chemtracts: Anal. Phys. Chem. 1989, 1, 1-12. (52) Carnie, S. L.; Chan, D. Y. C. J. Chem. Phys. 1980, 73, 29492957.
5742 Langmuir, Vol. 13, No. 21, 1997
two hard spheres immersed in an ion-dipole liquid. Within that scheme, they modeled the total force between such particles as a combination of (a) an attractive van der Waals force between the spheres, (b) a hard core exclusion term which depended on the radial distribution of the hard spheres, and (c) an electric term which resulted from the Poisson-Boltzmann distribution of point charges in a two-region dielectric medium surrounding the hard spheres. Canonical ensemble Monte Carlo simulations have also been used to study a simple system of a Lennard-Jones fluid adjacent to a rigid Lennard-Jones solid,54 which for a rigid close-packed face-centered cubic structure interacts through a 10-4-3 potential.55 Monte Carlo simulation was employed to calculate the force a fluid atom exerts on a solid, as well as the fluid density profile at various separations between two solid surfaces. It was found that the force was oscillatory with respect to the walls’ separation and that the period and number of oscillations reflected the behavior of the fluid density. As further NVT Monte Carlo calculations on the referred system showed,56-58 (a) the attractive component of the solidfluid potential dramatically increases the stratification of the fluid; (b) the effect of the purely repulsive wall is qualitatively similar to that of a hard wall; (c) the removal of the attractive component from the intermolecular potential in the fluid considerably increases the stratification of the fluid, particularly in the vicinity of the repulsive wall; and (d) neither the distribution nor the solvation force is very sensitive to changes in the chemical potential of the interstitial liquid. Measurements on the viscosity of thin liquid films by vibrating the upper curved mica surface of the SFA showed59 that when two surfaces are farther apart than 10 molecular diameters, a simple liquid in the gap retains its bulk Newtonian behavior and the shear plane remains coincident with the physical solid-liquid interface. However, as the use of a modified SFA with a lateral sliding mechanism specifically designed for viscosity measurements showed,60 the viscosity of thin films less than 10 molecular diameters apart rises considerably61 when two molecularly smooth mica surfaces slide past each other with one or two layers of cyclohexane in between them, the effective shear viscosity is 5-7 orders of magnitude higher than the bulk value, and the molecular relaxation times can be 1010 times slower. The films exhibit a yield point, and the shear stress no longer depends on the shear rate. The sliding now occurs with the surfaces separated by an integral number of liquid layers at surface separations coinciding roughly with the energy minima of the oscillatory force curves, and the shear stress is said to be “quantized” with the number of layers. These results have been confirmed by molecular dynamics and Monte Carlo calculations of shear stress62 over a Lennard-Jones fluid confined in between two parallel fcc Lennard-Jones walls (53) (a) Henderson, D.; Lozada-Cassou, M. J. Colloid Interface Sci. 1986, 114, 180-183. (b) Henderson, D.; Blum, L.; Lozada-Cassou, M. J. Electroanal. Chem. 1983, 150, 291-303. (c) Vericat, F.; Blum, L.; Henderson, D. J. Electronal. Chem. 1983, 150, 315-324. (54) Van Megen, W.; Snook, I. J. Chem. Soc., Faraday Trans. 1979, 2, 1095-1102. (55) Steele, W. A. Surf. Sci. 1973, 36, 317. (56) Snook, I. K.; van Megen, W. J. Chem. Phys. 1979, 70, 30993105. (57) Snook, K. I.; van Megen, W. J. Chem. Phys. 1980, 72, 29072913. (58) Van Megen, W.; Snook, I. K. J. Chem. Phys. 1981, 74, 14091411. (59) Israelachvili, J. N. J. Colloid Interface Sci. 1986, 110, 263-271. (60) Israelachvili, J. N.; McGuiggan, P. M.; Homola, A. M. Science 1988, 240, 189-191. (61) Gee, M. L.; McGuiggan, P. M.; Israelachvili, J. N. J. Chem. Phys. 1990, 93, 1895.
Urbina-Villalba et al.
representing a pore, one of them fixed and the other moving at a rate of 10-9 Å/ps. The following were found: (a) Besides the perpendicular order, which explains oscillations as a function of distance, there exists “transverse” order, which indicates that the pore “solid” forms epitaxially in successive layers of disordered fcc (100) structures. (b) As the walls are sheared pass each other, the stress rises approximately linearly up to a certain point, in which a maximum is attained. This is the critical stress required to initiate sliding. (c) Both the slope of the linear region and the critical stress decrease as the number of layers increases. (d) The range of values over which solid-like behavior persists increases with the number of layers; that is, the critical strain increases with the number of layers, being the critical strain per layer constant at about 0.08. For case I above, the nonmonotonic effect of the shear viscosity of gases in the transition regime is expected to be caused by the nonlocal character of the momentum transfer and not necessarily by layered molecular arrangements of the fluid molecules near the walls. However, the use of eq 2 for the comparable hypothetical experiments in which the mean free path is kept fixed (and so is the density) but the separation between the walls is continuously decreased leads to
∆L )
1 [4/(3π3)]1/2nσ
(4)
where ∆L is the separation between successive regions of rapid variation of the effective viscosity with L. Employing eq 4 for the same substances for which usual oscillatory force experiments had been made and assuming than the density of the thin films is approximately equal to that of the bulk liquids [water (nw ) 0.334 × 1023 molecules/cm3, diameter (d) ) 1.69 Å), cyclohexane (nc ) 0.056 × 1023 molecules/cm3, d ) 5.5 Å), octamethylcyclosiloxane (nOMCS ) 0.019 × 1023 molecules/cm3, d ) 8.5 Å), one finds that the oscillations should appear at wall separations of 9.5, 1.7, and 1.3 monolayers for water, cyclohexane, and OCMS, respectively. These results tend to indicate that the expected phenomenon would not be equal to that found by Israelachvili50,51,59 but is rather complementary. On the other hand, if the width of the monolayer is calculated from the bulk densityswhich tends to include the void spaces between molecules and disregards the inhomogeneities of the fluid density near the solid surfacessthen it can be shown from eq 4, that ∆L/d comes out constant (0.80) for liquids.63 This rather suggests that both phenomena (cases I and II) may be the result of transition regime effects.37 However, it should be mentioned here, first, that the use of the bulk density for the calculation of ∆L/d is a rough approximation and, second, that the simple model from which eq 4 is obtained is blind to the details of the intermolecular potential, which is taken into account in the model in an effective way through the mean molecular cross section and the boundary conditions. It is thus clear that the possible connections between cases I and II deserve further research. Experimental Section (1) General Characteristics of the Viscometer Used. A general scheme of the gas viscometer used is shown in Figure 1. This viscometer is a hybrid between the conventional rotatory cylinder viscometer,64 in which the external plate moves at a (62) Schoen, M.; Rhykerd, C. L.; Diestler, D. J.; Cushman, J. H. Science 1989, 245, 1223. (63) Garcı´a-Sucre, M.; Urbina-Villalba, G.; Araque-Lameda, L.; Parra, R. Ciencia 1997, 5, 51.
Shear Viscosity of Helium
Langmuir, Vol. 13, No. 21, 1997 5743 Table 2. Wall Temperature and Thermal Gradients of the Experiments Held at Constant Pressure exp no.
Tint (K)
Text (K)
G (K/cm)
1 2 3 4 5 6 7 8 9 10
38.1 36.1 34.4 31.0 29.1 26.2 23.7 21.9 19.1 17.8
70.4 64.2 60.9 55.1 52.4 46.1 41.8 37.7 34.3 30.3
9.5 8.3 7.8 7.1 6.8 5.8 5.3 4.6 4.5 3.7
Table 3. Wall Temperatures and Thermal Gradients for the Experiments Held at Variable Pressure
Figure 1. General scheme of the gas viscometer employed. constant speed and the deflection of the inner concentric cylinder is monitored, and the oscillating disk viscometer,22 in which the damping of an oscillating disk as a function of the gas friction is measured. The present viscometer consists of two cylindrical fixed walls (one internal and one external), between which a moving aluminum paper shell is located. The cylindrical aluminum shell is 70.0 ( 0.1 cm long, showing a radius of 4.5 ( 0.1 cm, a total area of 3958 ( 45 cm2, and a mass of 52.8 ( 0.1 g. That moving wall is suspended from a precision rotatory feed through (FPRM-275-38, Thermionics Northwest) by means of a quartz fiber 23.0 ( 0.1 cm long, with a diameter of 0.012 cm. Quartz has been used here because of its physical properties. Fused quartz presents (a) a low-temperature expansion coefficient (5.5 × 10-7 cm/(cm °C)),65 (b) one of the lowest monotonic temperature dependences of the shear modulus (Gt ) G15[ 1 + 0.00011(t - 15)], where Gt stands for the shear modulus at temperature “t”),66 (c) a very high tensile strength (∼359 462 mmHg (7000 psi),65 and (d) a relatively low density (2.2 g/cm3) with respect to the rigidity modulus (4.5 × 106 psi),65 which guarantees the generation of a very high velocity torsional wave inside the fiber (1.18 × 106 cm/s),67 unable to couple with the pendulum movement (pendulum period ∼23 s). The distance between the fixed and moving walls of the viscometer is 2.0 ( 0.1 cm. The torques are conveyed on the moving shell by means of a polyacrylamide and thread band, which connects the rotatory feed through to a stepping motor (Superior Electric). In all cases torques of 15° have been used. Despite the maximum angle swept, the oscillations have been (64) Vennard, J. K.; Street, R. L. CECSA, 3rd ed.; Trans-Editions Inc.: Mexico, 1989. (65) Lide, R. D., Ed. CRC Handbook of Chemistry and Physics, 74th ed., CRC Press: Boca Raton, FL, 1994. (66) Kaye, G. W. C.; Laby, T. H. Tables of Physical and Chemical Constants, 15th ed.; Longman Group Limited: New York, 1975. (67) Feynman, R. P. The Feynman Lectures on Physics; AddisonWesley Publishing Company, Inc.: Reading, MA, 1965; Vol. II, Chapter 38.
exp no.
Tint (K)
Text (K)
G (K/cm)
11 I 11 II 12 13 14
29.7 32.8 26.8 19.0 20.6
54.5 56.1 47.3 38.2 20.7
7.3 6.8 6.1 5.7 0.0
observed within an angle of 15° from the equilibrium position of the pendulum in order to avoid non linear effects in the fiber’s torsional motion. In order to measure the period of oscillation, a small mirror 4 × 3 mm2 has been attached to the quartz fiber. A 4 mW HeNe laser allowed us to monitor the cylinder movements. The bouncing signal is capture by two NTE-3002 phototransistors that are connected to a data acquisition card (AT-MIO-16, National Instruments), through appropriate termination breadboards. A signal was generated only when both sensors (separated by 1 cm) were activated. The data acquisition card mounted on a Kaypro 386 computer (30 MHz) provided an accuracy of 10-6 s in the time measurements. Besides, it allowed us to monitor the variations of temperature and pressure through the whole experiment. For that purpose, a C-coded program made with the Labwindows package (National Instruments) has been used. That program recorded the pressure, the temperature, and the period of oscillation of the viscometer at each laser pulse, allowing separate analysis of the variation of each period of oscillation with the pressure. In order to measure the temperature, two platinum resistance sensors (model 18642 A) as well as a HP-2802A Hewlett Packard thermometer have been used. The sensors (bottom of Figure 1) were appropriately calibrated with an ice bath and, in its final configuration, were separated by a distance of 3.4 cm, not touching either one of the fixed walls. No correction has been made to account for the difference between the temperature of the thermocouples and that of the gas. However, it should be kept in mind that a temperature jump between the gas and the walls is likely to exist, as well described in the work of Mandell and West.68 Since the viscometer possesses double internal and external walls, the external thermal gradient (G) was generated by means of two cryostats. The internal wall was always cooled with ethanol, while the external wall was heated with water. The thermal gradient vector always pointed from the inner fixed wall to the outer fixed wall, making angles of 0 and 180° with the molecular velocity gradient vectors correspondent to the subsystems: inner fixed wall-moving wall and moving wallexternal fixed wall, respectively. The average temperatures nearby the walls, as well as the thermal gradient obtained, are shown in Tables 2 and 3 for the experiments held at constant and variable pressure, respectively (see next section). The maximum vacuum obtained (7 × 10-4 mbar) was achieved with a mechanical 1376 B Welcher pump and a NCR-VHS-4 diffusion pump. The pressure was measured with TR-901 thermal conductivity sensors (from Leybold) and a Pirani PG-3 gauge controller. This instrument was connected to the ATMIO-16 card through an RS-232 port. The amplitude of the oscillatory motion of the viscometer is given by (68) Mandell, W.; West, J. Proc. Phys. Soc. 1924, 37, 20-41.
5744 Langmuir, Vol. 13, No. 21, 1997 Φ(t) ) Φ0 exp(-βt) cos(2πνt)
Urbina-Villalba et al. (5)
in which Φ is the torsion angle, ν is the pendulum frequency (1/T), and β is the damping parameter in s-1. This damping parameter β can be a function of both the pressure (p) and the thermal gradient (G) (β ) ηfA/(2ML))) essentially through the function f (see eq 7 below). As is shown in Figure 2, β increases with pressure in going from the Knudsen to the continuous regime. At 58.1 µmHg an anomalous variation of the parametersexpected for the transition regimestakes place (it corresponds to the highest point in Figure 2). In the continuous regime f ≈ constant, and β becomes reasonably constant. The parameter β progressively increases with pressure within the continuous regime (see Table 4). However, since the present viscometer was specifically designed for the transition regime, it is unable to measure viscosities above 700 mmHg due to the strong damping of the moving shell. Still, as it can be seen in Table 4, the damping parameter slightly increases as the atmospheric pressure is approached (showing a 7.84% variation in going from 5.26 to 625.80 Torr). Lindveit et al.69 and Copley et al.70 had described different ways to evaluate the damping parameter of a torsion pendulum employing optic detectors. In those evaluations the time interval between the luminic excitation of an even number of photocells symmetrically distributed around the equilibrium point of the pendulum was employed. In the present work however, we used again the logarithmic amplitude decrement method previously described in ref 36, employing the same curved screen of radius 2.10 m as before.36 For the present moving shell, an average value of β of (5.46 ( 0.03) × 10-3 s-1 has been found employing the last four measurements of Table 4. An approximate value was obtained employing a less accurate “angle” decrement method: β ) (6.1 ( 0.4) × 10-3 s-1 for pressures between 0.23 and 495.13 mmHg. Due to the usual corrections of ends and edges, our viscometer is not well adapted for absolute viscosimetric measurements but instead is well suited for relative viscosimetric measurements. According to Kobayashi,71 flat borders of the moving shell, low distances of separation between the plates, and low Reynolds numbers diminish the effect of the edge corrections. However, if it is true that our viscometer meets these three requirements (see below), it is also true that in our case (a) the length of the internal fixed wall is smaller than that of the moving shell; (b) the length of the moving wall is in turn smaller than that of the external fixed wall (see Figure 1); (c) the thermocouples are placed between the fixed and moving walls, increasing the nonideality of the instrument; (d) and the moving shell is made of aluminum paper, which does not form a completely uniform surface. For all the above mentioned reasons, we do not attempt to calibrate the instrument for absolute measurements. However, an approximate correction factor δ can be computed, equal to the quotient between the currently accepted viscosity of helium at STP conditions (194.1 µP) and the viscosity value deduced from the magnitude of the damping parameter in the continuous regime: β ) (5.46 ( 0.03) × 10-3 s-1 (292 ( 2 µP). For the present viscometer, δ came out to be 0.6647, considerably lower than the correction factor obtained for the viscometer previously used for helium measurements (δ ) 0.7477).36 This variation is expected due to the smaller radius of curvature of the present instrument. In principle, the computed correction factor can be applied to all viscosity measurements made with the present instruments. Yet, we are mainly interested in transition regime measurements where the function f varies appreciably and the effective viscosity depends on the flow geometry. Using the value of the damping parameter β ) 1.53 × 10-4 s-1, and the average period of oscillation 23.04 s, corresponding to the pressure of 0.0008 mmHg, a crude determination of the torsion constant (K) of the quartz fiber was made: 79.79 g cm2/s2. Since β
