The shear viscosity of supercritical oxygen at high pressureE. H. Abramson Citation: The Journal of Chemical Physics 122, 084501 (2005); doi: 10.1063/1.1849166 View online: http://dx.doi.org/10.1063/1.1849166 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/122/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Reference Correlations for the Density and Viscosity of Squalane from 273 to 473 K at Pressures to 200 MPa J. Phys. Chem. Ref. Data 43, 013104 (2014); 10.1063/1.4863984 Crossover critical phenomena in an aqueous electrolyte solution: Light scattering, density and viscosity of the 3-methylpyridine + water + NaBr system J. Chem. Phys. 119, 4428 (2003); 10.1063/1.1594179 Density inhomogeneities and electron mobility in supercritical xenon J. Chem. Phys. 118, 706 (2003); 10.1063/1.1527944 Dielectric relaxation of supercritical water: Computer simulations J. Chem. Phys. 113, 3499 (2000); 10.1063/1.1289919 Density fluctuations and shear viscosity of molecular liquids: Carbon dioxide and nitrogen J. Chem. Phys. 112, 7118 (2000); 10.1063/1.481306

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The shear viscosity of supercritical oxygen at high pressureE. H. AbramsonDepartment of Earth and Space Sciences, University of Washington, Seattle, Washington 98195-1310

sReceived 5 October 2004; accepted 23 November 2004; published online 14 February 2005d

Shear viscosities of supercritical oxygen have been measured up to a pressure of 5.7 GPa at 294 K.A modified free-volume expression fits the data within 6% between the limits of the tenuous gas and4.8 times the critical density. Nitrogen viscosities were found to correspond to those of oxygenthrough a simple scaling by critical constants. Viscosities were measured in the high-pressurediamond-anvil cell with a rolling-ball technique. The dynamics of a sphere rolling on an inclinedplane were investigated in the context of these experiments. The effect of a second surface, situatedabove the sphere, was experimentally determined. ©2005 American Institute of Physics.fDOI: 10.1063/1.1849166g

I. INTRODUCTION

The question of how the shear viscosities of fluidsevolve with pressure and density, particularly at higher den-sities, has a history of intriguing hypotheses,1–6 and has en-gendered a rapidly increasing number of molecular-dynamicsimulations,7–13 but still there exist very few experimentaldata. A number of glassing systems have been studied to highpressure s8 GPa in the case of methanol14d using thediamond-anvil cell, however, the noble gases and varioussmall molecules such as nitrogen, oxygen, methane, carbondioxide, and water, which are not expected to glass, havebeen measured up to pressures of at most 1 GPa. These latterfluids, besides being of theoretical interest, are also of sig-nificance in the fields of planetary physics, explosives tech-nology, and high-pressure chemical synthesis. Here I reportthe shear viscosities of molecular oxygen, at room tempera-ture, to a pressure of 5.7 GPa and densities of 4.8 times thecritical value.

Viscosities were determined by observation of a sphererolling in a diamond-anvil cell. The detailed dynamics of asphere rollingsin a viscous fluid on a roughd plane are stillunder investigation, and the case of a sphere rolling betweentwo planes does not appear to have been previously consid-ered. Part of this paper concerns an attempt to gain an un-derstanding of these dynamics and the factors which governthem.

II. EXPERIMENT

As reported by King, Herbolzheimer, and Cook,15 shearviscosities may be measured in the high-pressure diamond-anvil cell by a technique in which a smalls30–60mmd metalsphere is allowed to roll down the inclined inner surface ofone of the two opposing diamond anvils. Videotaped imagesof the sphere are analyzed to give its speed, and this in turnis used to calculate the viscosity of the surrounding fluid.

In the experiments reported here, fluids were containedin a diamond-anvil cell of modified Merrill-Basset design,16

with culet diameters of 0.8 mm. A small piece of ruby in-cluded in each load allowed pressures to be measured with aprecision of 0.02 GPa.17 Angles from the horizontal were

determined with an electrolytic tilt gauge mounted next tothe cell. All measurements were taken at room temperaturewhich was measured to better than 0.5 K with a mercurythermometer. Images were recorded on videotape by use of alens sLeica Monozoom 7d coupled with a videocamerasSanyo D4 VCB-3524d. This combination gave a resolutionalong the path of the sphere’s travel of 1mm/pixel. The cellwas strobed at 60 Hz with 200ms pulses from a light-emitting diode.

The number of video images recorded for each run var-ies from as few as 6 for the lowest viscosities and largest tiltangles, to over 200. Analysis of the images yields the posi-tion of the sphere at 1/60 s intervalssFig. 1d. Terminal ve-locity is achieved in less than 1/30 s for all combinations offluids and tilt angles, and usually in less than the time reso-lution. Run to run variations in speed are typically less than4%, with standard deviations of,1%. Occasionally a jog isobserved in a trajectory due, presumably, to collision with asmall piece of debris or an irregularity on the sphere’s sur-face. The speeds before and after such jogs are found to bethe same.

FIG. 1. The position of a sphere is shown at intervals of 1/60 s as it rollsalong an innersdiamondd surface of a high-pressure cell. These calibrationruns were taken inn-hexane at 1 bar and tilt angles of 11°ssd, 14° shd,20°snd, 25° sLd, and 30°s* d. Straight lines fit to the data yield the rollingspeeds.

THE JOURNAL OF CHEMICAL PHYSICS122, 084501s2005d

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Spheres were made by brushing flaked platinum into amethane/oxygen flame, the resultant molten drops solidifyingin flight and being then collected on sheets of aluminum foil.The products of this process were typically smooth and ap-peared spherical over most of their surface with, however, acharacteristic small fringe where the solidification termi-nated. While occasional deviations from a straight trajectorywere observed in the sphere’s motion, due both to this fringeand to microscopic debris in the cell, it was always possibleto obtain data uninterrupted by such asperities.

Oxygen s99.999%d was loaded by condensation of thegas at 77 K. Argon was obtained and loaded directly as thecryogenic fluid. Organic solvents used for calibrationstolu-ene, n-pentane,n-hexane, andn-pentanold were purchasedfrom Sigma-Aldrich with a stated purity of.99% and usedwithout further processing.

III. DYNAMICS OF A ROLLING SPHERE

A. Observations in the diamond-anvil cell

As the dynamics of a sphere rolling along a surface in aviscous fluid are still under investigation, and as there ap-peared to be no published work concerning a sphere rollingbetween two walls, the first task was to investigate the be-havior of this system in greater detail. In previousstudies14,15,18–20it was assumed that the speedv of a rollingsphere is given by

v = F2gR2srS− rFdg9h

Gsinu, s1d

whereR, rS, rF, g, u, and h are, respectively, the sphere’sradius, sphere’s density, density of the fluid, gravitationalacceleration, incline angle to the horizontal, and shear vis-cosity, and the factorg is a correction for the increased dragdue to the surfacescompared with that for a freely fallingsphered. The factorg was reported to have been evaluated foreach combination of cell and sphere by calibration with afluid of known viscosity and was found15 for smallu to havea value of<0.14, reasonably independent of tilt angle, andassumed independent of viscosity.

In Fig. 2, speed of roll is plotted against the sine of thetilt angle for several fluids of known viscosities21,22 andwidely different chemical nature. Speed is not linear in sinsudas described by Eq.s1d; although the data fall on straightlines, there is always an additive componentC independentof angle, such that the equation might be rewritten as

v = F2gR2srS− rFdg9h

Gsinu − C. s2d

The magnitude ofC is not obviously affected by the chemi-cal nature of the fluid, being the same for a noble gas, anaromatic hydrocarbon, an alcohol, and water, but it is signifi-cantly less for the three fluids of lowest viscosity.

For the angles shown in Fig. 2 rolling is believed tooccur without slippage, but at largeru this is not the casessee belowd and recorded speeds lie above the extrapolationsof the lines. All data used here to determine viscosities weretaken at angles for which a straight line fit gives a good

account of the data. Reynolds numbers in these experiments,calculated as 2vrFR/h, range from 8310−4 to 3310−1.

Equations2d suggests that the slopes of the lines in Fig.2, modified by the factorsrS/ srS−rFd, should be propor-tional to the inverse viscosities of the fluids. A test of thisproposition is shown in Fig. 3sad where typical deviationsfrom a linear relation are less than 6% with a maximumdeviation of 13%. Calibration lines as in Fig. 3sad were pro-duced for each of four different spheres used in the measure-ments.

As it was practically impossible to disassemble the cellwithout loss of the included sphere, fluids were changed byopening a gap of a few microns between diamonds and gas-ket, allowing the contained fluid to evaporate and then add-

FIG. 2. Speed of roll is plotted against the sine of the tilt angle for aplatinum spheres37 mm diameterd rolling in fluids of viscositiessat 294 Kdbetween 0.25 and 3.8 mPa s.n-pentane svd; argon at 0.60 GPasxd;n-hexanes* d; tolueness, Ld; argon at 1.05 GPashd; waters,d; n-pentanolsnd. Except for argon, all fluids were used at 1 bar. Error bars are drawn for6 twice the root-mean-square deviations among runs if larger than the sym-bols. Straight lines are least-squares fits to the data. Points indicating speedsin n-hexane derive partly from the data shown in Fig. 1.

FIG. 3. sad Slopes of the lines in Fig. 2, multiplied by the factorsrS/ srS

−rFd, are plotted against the inverse viscosities of the fluids. A least squaresfit through the datasstraight lined passes through the origin with slope0.21 mPa cm and serves as a calibration line for oxygen viscosities mea-sured with the same sphere.sbd Slopes of calibration lines are plotted againstthe squares of the radii for four different spheresfthe slope of the line in Fig.3sad is designated by the filled circleg. In this case the fit to the datasstraightlined was forced through the origin. Error bars reflect an uncertainty of0.5 mm in the radii as measured from the video images.

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ing the new. From inspection of the cell it is clear that eachtime the cell was refilled there was an increase of debris onthe diamond surfaces and a corresponding increase in noisein the data. Further, it is probable that some of these particlesoccasionally stuck to the surface of the sphere; the two larg-est deviations from the line in Fig. 3sad s13% and 9%d areassociated with a series of runs inn-pentanol and a directlysubsequent series in argon. Debris is likely the largest sourceof uncertainty in the measurements and one which can besignificantly reduced in future experiments.

Equations2d also suggests that the slope of the calibra-tion line determined for each sphere should be given as2gR2rSg/9 and thus, ifg is constant over these experiments,be proportional to the square of the sphere’s radius. Confir-mation of this relation is shown in Fig. 3sbd where the cali-bration slopes are plotted for four spheres used in this study,ranging in diameter from 37 to 52mm. Values ofg calcu-lated from these data lie between 0.128 and 0.138, in goodagreement with those reported15 by King, Herbolzheimer,and Cooks0.14±0.02 foru,35°d and slightly larger thanthose of Carty23 s0.09–0.12d, the latter for much largerspheres.

B. Correspondence to theory

Smart, Beimfohr, and Leighton24 have published a treat-ment of the problem of a sphere rolling on a rough plane,which predicts the relation between speed and sinsud givenby Eq. s1d when below a limiting anglessuch that there willbe no slippage between sphere and planed, and departures tohigher speeds at higher angles. The present data are consis-tent with this theory after a minor modification to include aforce through the center of mass of the sphere with compo-nent −F in the direction of travel, independent of tilt angleand speed; the nature of this additional force has not beendetermined. The result, for a sphere rolling without slippage,is in the form of Eq.s2d with C=sgFd / s6phRd.

In this formulation g=fFt+Fr +43sTt+Trdg−1, where

Ft, Fr, Tt, andTr are approximations25 to the hydrodynamicforces and torques exerted on a sphere due to its translationand rotation near a plane. The factorg is then a function onlyof lnsd /Rd where d is the gap between sphere and planesassumed24 to correspond to the roughness of the latterd. Forthe experimentally derived valueg=0.13–0.14, the neces-sary gap would amount to,50 nm sd /R<10−3d, which isnot unreasonable for these surfaces. The fact that previousmeasurements ofg, for other systems,15,23 fall in a narrowrange s0.09–0.16d is attributable to its very slow variationwith d /R, e.g., a change ing from 0.11 to 0.14 requires afactor of 10 increase ind /R.

Integration of the equations of motionsin the case of noslippaged yields an exponential time constant for approach toterminal speed equal togsM + I /R2d / s6phRd, with M beingthe mass of the sphere andI being the moment of inertiaabout its diameter. The maximum time constant in these ex-periments, for the largest sphere and lowest viscosity, wouldthus be 9 ms, and the speed would be within 3% of its ter-minal value after 1/30 s, in agreement with our observations.

The angle above which slippage occurs, with speeds

then exceeding those given by Eq.s2d, depends both on thecoefficient of static friction and the component into the planeof the posited additional force. If we assume the latter to beroughly equal toF, then early data taken up tou=58° givean approximate value of 0.3–0.4 for the coefficient of frictionin the supercritical oxygen; the value has been reported26 as0.50 for a sslidingd platinum/diamond contact in highvacuum.

It appears that the model of Smart, Beimfohr, and Leigh-ton provides a good basis for understanding the details ofthis viscometry, with the exception that the origin and exactnature of the empirically necessary forceF is not known.

C. Effect of the second, upper diamond surface

In the case of a sphere settling through a fluid parallel toa nearby vertical wall the additional drag25 due to the wallexceeds 20% when the gap between sphere and wall is lessthan one diameter, and rises quickly with smaller gaps. Onemay then ask to what extent the upper diamond surface, typi-cally removed from the sphere by 1–2 diameters, will affectthe rolling speed, whether the small variations in the gaprequired to alter the pressure might also alter the calibration,and also whether this technique is suitable to the smallergaps that may be necessary to achieve higher pressures andcompressions.

In order to answer these questions a set of ancillary ex-periments was undertaken in which a 0.1 mm steel spherewas allowed to roll along the bottom wall of a methanol-filled cuvette, beneath an upper surface formed by a sectionof included glass plate. The cuvette was tilted at an angle of30° with respect to the horizontal and the slide was held at anangle of 10° with respect to the lower surface of the cuvettessee inset, Fig. 4d. The effect of proximity of the upper sur-

FIG. 4. The speed of a sphere rolling in contact with a lower surface isaffected by another, upper surface separated from the sphere by a variablegap. Speed, normalized by the measured speed in an infinite half space, isplotted sfilled circlesd against the ratio of the gap to the sphere’s diameter.sThe gap is that from the upper plane to the sphere’s surface.d The dataderive from a 100mm sphere rolling in methanol at a constant tilt angle ofu=30°. The inset figure depicts the experimental cell with the roll directionout of the page and the upper, glass plate drawn in solid black. For each run,the sphere was started at a different position along the long axis of the cell.Typical gap-to-diameter ratios in the diamond-anvil cell were between oneand two and, as seen here, the variations in rolling speed engendered byslight displacements of the upper diamond are not expected to be significant.

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face was explored by rolling the sphere at different distancesfrom the intersection of slide and cuvette wall.

The results, seen in Fig. 4, demonstrate that the frac-tional effect of thissupperd surface is much less than for asphere falling freely in a half space and for our purposes maybe considered constant; in these experiments the most sig-nificant variations in gap-to-diameter ratios, from 1.3 to 2.2,would have had an effect of only,2%. This conclusion isalso in accord with the accumulated data as there are nosystematic correlations between speed and gap which mightreasonably be ascribed to the effect of the upper surface. Thefact that the gap can be closed to less than one-tenth thesphere’s diameter with only minor effects allows extensionof this technique to more highly compressed fluids withdiamond-anvil separations significantly smaller than in thecurrent work.

IV. VISCOSITY OF FLUID OXYGEN

Experiments on the shear viscosity of oxygen have in thepast been restricted to pressures less than 0.08 GPa; the re-sults have been summarized by Lemmon and Jacobsen.27

Here, I report experiments conducted in supercritical oxygenat 294 K up to the freezing point at 5.7 GPa.

The measured values are given in Table I and are plottedagainst pressure in Fig. 5. Runs were made with five differ-ent spheressin three different cellsd. Four of the spheres werecalibratedin situ fas in Fig. 3sadg with fluids of known vis-cosity while the remaining sphere was calibrated through useof its measured diameter and Fig. 3sbd.

van der Gulik6 has pointed out that for simple fluidsabove, at most, a factor of 2.2 times their critical density, allpreviously measured viscosities yielded straight lines whenplotted against pressure. Here we see that at sufficiently highcompressions this relation no longer holds. Nor can the data

be fit to an Arrhenius expression with constant volume ofactivation, which would cause a severe overestimation in ex-trapolation to either higher or lower densities.

For an appreciable number of fluid systems, rangingfrom glass-forming organic molecules to metals and theoret-ical hard spheres, Doolittle’s28 empirical formula

lnshd = lnsAd + BF r

ro − rG s3d

gives a decent account of viscosities, withro a supposedmaximum density providing no “free-volume,” andA andBfitting constants. The formula was later interpreted by Cohenand Turnbull1 on the basis of a free-volume model for diffu-sivities, and extended to viscosities through the Stokes-Einstein relation. In this development,B is roughly the ratiobetween the minimum void necessary for effective diffusionand the minimum volume occupied by a molecule.

For comparison with Eq.s3d the data have been replottedsFig. 6d as lnshd against density,29 the latter reduced by itscritical value, rcrit=0.436 g cm−3. Points reported byHaynes,30 up to 0.035 GPa, define the viscosities at lowerdensities. In order to close the gap between the data ofHaynes and those taken in the diamond-anvil cell,sscaleddviscosities of nitrogen, taken up to 0.65 GPa, have been alsobeen plotted.31,32 Nitrogen and oxygen are known to exhibita corresponding states relationship to at least 1 GPa, both inthermodynamic variables and in thermal diffusivities.33 Here,nitrogen’s viscosities were scaled34 to those of oxygen,

hN2,scaled= hN2SVN2,crit

VO2,critD2/3SMO2

MN2

D1/2

, s4d

using the critical molar volumes and the molar masses. Itappears that the correspondence of states extends also to vis-cosities, to the highest density of the nitrogen data and withinthe agreement apparent in the figure.

TABLE I. Measured viscosities of oxygen, ordered from first to last ac-quired, and grouped according to the different spheres used.

P sGPad r sg cm−3d h smPa sd T sKd

1.53 1.584 0.602 294.53.60 1.911 2.00 294.52.09 1.703 0.888 294.5

5.54 2.099 4.53 294.64.45 2.001 2.78 294.33.70 1.922 2.06 294.22.89 1.823 1.38 294.22.07 1.698 0.858 294.21.13 1.490 0.460 294.3

5.73 2.114 5.05 294.4

3.32 1.878 1.53 294.41.47 1.577 0.583 294.60.93 1.427 0.367 295.2

2.31 1.737 0.947 295.25.72 2.113 4.38 295.05.03 2.055 3.32 294.41.90 1.667 0.757 294.51.00 1.451 0.369 294.2

FIG. 5. Viscosities of oxygen were measured at 294 K from 1 GPa up to thefreezing pressure of 5.7 GPa. The various unfilled symbols represent runsmade with five different spheressin three different cellsd. Spheres with di-ameters of 37.3, 40.7, 42.5, and 51.5mm sn, L, h, ands, respectivelydwere calibratedin situ fas in Fig. 3sadg while the remaining spheres,d,having been destroyed, was calibrated by using its measured diameter of47.6mm and the plot of Fig. 3sbd. Error bars represent 1s uncertainties.Data previously taken to 0.034 GPasRef. 30d s* d are shown in more detailin the inset. The curve is a fit to Eq.s5d.

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Equations3d fits the higher data wellsdotted line in Fig.6d although it fails at lower densities; the constants,B=9.1and ro=5.4 g cm−3, are not violently unreasonable if oneconsiders the many approximations in the interpretation ofDoolittle’s formula. In their derivation of Eq.s3d Cohen andTurnbull1 assumed the equivalent of a constant molecularfree path of<1 molecular diameter. In a later paper, Turn-bull and Cohen5 modified their treatment to allow a free pathinversely proportional to the density. While this variationmay give a better account of the viscosity at intermediatepressures, it will necessarily fail in the limit of zero density,where the viscosity remains constant even as the calculatedinverse diffusivity tends to zero. In order to force the correctbehavior in the limit of zero density we can use their modi-fied formula sin conjunction with the Stokes-Einstein rela-tiond while multiplying by a factor ofro/r to yield

lnshd = lnsAd + lnF Bro

sB − 1dr + roG + BF r

ro − rG . s5d

Equations5d gives a significantly better representation ofthe data, without requirement of an additional fitting con-stant. A fit through the points withr /rcrit.1.4 demonstratesa maximum deviation of 6% aroundr /rcrit=1 and appropri-ate behavior atr=0 ssolid line in Fig. 6d. Considering thatthe data cover the range from a tenuous gas to a fluid withalmost five times the critical density, the match between ex-periment and a three-parameter fit is remarkably good. Un-fortunately, the fitting parameters have less connection withthe theory than is desirable; for the modified equation weobtainB=17 andro=7.2. In regards to predictive capabilitiesin extrapolation to higher densities, a fit only of the scalednitrogen datasr /rcrit,3.1d, and forced through the 1 barvalue, underpredicts the viscosity at the highest oxygen den-sities sr /rcrit=4.8d by 20%.

Within the context of Eqs.s3d and s5d, viscosity is ex-plicitly dependent on temperature only through the termA,which is supposed proportional toÎT. Data taken byHaynes30 at lower temperatures fall considerably off the

curve of Fig. 6, whether scaled byÎT or not. A more oftensuccessful representation of the viscosity results from assum-ing it to be the sum of two terms, that of the gas in the limitof zero density and a “residual viscosity” which is a functiononly of densityssee, e.g., Ref. 35d. When plotted as residualviscosity the data from 300 K down to,150 K fall on asingle curve, independent of temperature, as shown in Fig. 7where I have now added both data taken at 289 KsRef. 36dand at 160 K.30

Below ,130 K, and between densities of 0.9 and1.2 g cm−3, viscosities30 diverge from the curve in Fig. 7,deviating rapidly to higher values and roughly defining anupper branch for the subcriticalsTcrit=155 Kd liquid. Thesame divergence between residual viscosities of the super-critical fluid and of the liquid occurs in the case of nitrogen;“the steep increase of viscosity at very high densities” notedby Stephan, Krauss, and Laesecke35 is due to the fact that, atthese higher densities, the data they were considering37 allderived from the liquid at subcritical temperatures. Thus, thevalues given for nitrogen’s residual viscosity,35 if scaled as inEq. s4d and plotted, would be seen to follow those of oxygenup tor /rcrit<2 and then to continue exactly along the upperbranch in Fig. 7.

In a more empirical approach to an adequate descriptionof small-molecule viscosities, use is made of astemperature-dependentd residual viscosity represented by a heavily pa-rametrized expansion in density and inverse temperature.The terms in the expansion are chosen to provide reasonableextrapolations, and the approach follows closely that whichhas been increasingly usedse.g., Ref. 38d for the thermody-namic Helmholtz function. An expansion of this form,27 fit-ted to the previously available data for oxygen, is shown asthe dashed line in Fig. 6. Experiments at temperatures nottoo far removed from 294 K had reached maximum densitiesof r /rcrit<1.5, and this is approximately where divergencebetween the results of the expansion and the rest of the plotis seen to commence.

FIG. 6. Oxygen viscosities shown in Fig. 5 are replotted against densityreduced by the critical densityfopen symbols, current data; * from HaynessRef. 30dg. The additional small, filled circles represent viscosities of nitro-gen sRef. 31d at 298 K, scaled as per Eq.s4d. The dotted line is a fit toDoolittle’s original formulafEq. s3dg, while the continuous curve is a fit toEq. s5d. A previously publishedsRef. 27d representation of the viscosity,evaluated at 294 K, is shown as a dashed curve.

FIG. 7. When plotted as residual quantities, viscosities at 160 Ksbd sRef.30d conform to the same curve as those at 294 Kfopen symbols, currentdata; * from HaynessRef. 30dg. Data sRef. 36d taken at 289 K up to0.07 GPa are also includedscd. Viscosities measuredsRef. 30d at lowertemperaturess130, 110, and 90 Kd, represented here by three line segments,deviate to higher values and define, roughly, a separate branch of the curvefor the subcritical liquid.

084501-5 Shear viscosity of supercritical oxygen J. Chem. Phys. 122, 084501 ~2005!

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V. CONCLUSIONS

The shear viscosity of supercritical oxygen has beenmeasured up to the freezing pressure of 5.7 GPa at 294 K.This is the highest pressure to which viscosities have beenmeasured in a nonglassing fluid. Viscosities are representedto better than 6% by a simple, three-parameter expressionfrom zero density to 4.8 times the critical value. The conceptof a temperature-independent residual viscosity gives a goodaccount of the data from 300 K down to 150 K, while atlower temperatures residual viscosities of the liquid differsubstantially from those of the supercritical fluid. Whenscaled according to a theory of corresponding states, viscosi-ties of nitrogen are found to fall on the same curves as thoseof oxygen, both for the liquid and for the supercritical fluid.

The model of a sphere rolling on a rough plane,24 pro-vides a useful description of the dynamics observed in thediamond-anvil cell. There is no reason that the techniqueshould not be applicable at the smaller anvil spacings neces-sary to higher compressions. Further development of the ex-perimental apparatus, in particular to allow for work at hightemperatures, will allow an extension of the data for oxygen,and the investigation of other important fluidsse.g., water,carbon dioxide, nitrogen, hydrogen, and heliumd at bothhigher densities and temperatures.

ACKNOWLEDGMENTS

B. Swanson is thanked for his loan of the necessaryvideo equipment. This work was supported by NSF GrantNo. EAR 0106683.

1M. H. Cohen and D. Turnbull, J. Chem. Phys.31, 1164s1959d.2J. H. Dymond, Int. J. Thermophys.18, 303 s1997d.3M. Dzugutov, NaturesLondond 381, 137 s1996d.4Y. Rosenfeld, Phys. Rev. A15, 2545s1977d.5D. Turnbull and M. H. Cohen, J. Chem. Phys.52, 3038s1970d.6P. S. van der Gulik, Physica A256, 39 s1998d.7S. Bastea, Phys. Rev. E68, 031204s2003d.8G. A. de Wijs, G. Kresse, L. Vocadlo, D. Dobson, D. Alfe, M. J. Gillan,and G. D. Price, NaturesLondond 392, 805 s1998d.

9J.-F. Dufreche and J. Clerouin, J. Phys. IV10, 303 s2000d.10R. Grover, W. G. Hoover, and B. Moran, J. Chem. Phys.83, 1255s1985d.11K. Meier, A. Laesecke, and S. Kabelac, Int. J. Thermophys.22, 161

s2001d.12R. L. Rowley and M. M. Painter, Int. J. Thermophys.18, 1109s1997d.13L. V. Woodcock and C. A. Angell, Phys. Rev. Lett.47, 1129s1981d.14C. A. Herbst, R. L. Cook, and H. E. King, Jr., J. Non-Cryst. Solids172,

265 s1994d.15H. E. King, Jr., E. Herbolzheimer, and R. L. Cook, J. Appl. Phys.71, 2071

s1992d.16L. Merrill and W. A. Bassett, Rev. Sci. Instrum.45, 290 s1974d.17H. K. Mao, J. Xu, and P. M. Bell, J. Geophys. Res.91, 4673s1986d.18C. A. Herbst, R. L. Cook, and H. E. King, Jr., NaturesLondond 361, 518

s1993d.19R. L. Cook, H. E. King, Jr., C. A. Herbst, and D. R. Herschbach, J. Chem.

Phys. 100, 5178s1994d.20K. U. Schug, H. E. King, Jr., and R. Bohmer, J. Chem. Phys.109, 1472

s1998d.21C. L. Yaws, Handbook of Transport Property Data: Viscosity, Thermal

Conductivity, and Diffusion Coefficients of Liquids and GasessGulf Pub-lishing, Houston, 1995d.

22N. J. Trappeniers, P. S. van der Gulik, and H. van den Hooff, Chem. Phys.Lett. 70, 438 s1980d.

23J. J. Carty, Jr., B.S. thesis, Massachusetts Institute of Technology, 1957.24J. R. Smart, S. Beimfohr, and D. T. Leighton, Jr., Phys. Fluids A5, 13

s1993d.25A. J. Goldman, R. G. Cox, and H. Brenner, Chem. Eng. Sci.22, 637

s1967d.26K. Miyoshi and D. H. Buckley, ASLE Trans.27, 15 s1984d.27E. W. Lemmon and R. T. Jacobsen, Int. J. Thermophys.25, 21 s2004d.28A. K. Doolittle, J. Appl. Phys.22, 1471s1951d.29E. H. Abramson, L. J. Slutsky, M. D. Harrell, and J. M. Brown, J. Chem.

Phys. 110, 10493s1999d.30W. M. Haynes, Physica A89, 569 s1977d.31J. Vermesse, Ann. Phys.sParisd 4, 245 s1969d.32R. Span, E. W. Lemmon, R. T. Jacobsen, W. Wagner, and A. Yokozeki, J.

Phys. Chem. Ref. Data29, 1361s2000d.33E. H. Abramson, L. J. Slutsky, and J. M. Brown, J. Chem. Phys.111, 9357

s1999d.34J. F. Ely and H. J. M. Hanley, Ind. Eng. Chem. Fundam.20, 323 s1981d.35K. Stephan, R. Krauss, and A. Laesecke, J. Phys. Chem. Ref. Data16,

993 s1987d.36I. F. Golubev,Viscosity of Gases and Gas Mixtures (A Handbook)sIsrael

Program for Scientific Translations, Jerusalem, 1970d.37D. E. Diller, Physica A119, 92 s1983d.38R. Span and W. Wagner, Int. J. Thermophys.18, 1415s1997d.

084501-6 E. H. Abramson J. Chem. Phys. 122, 084501 ~2005!

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