^{1}

### Abstract

Shear viscosities of supercritical oxygen have been measured up to a pressure of at . A modified free-volume expression fits the data within 6% between the limits of the tenuous gas and 4.8 times the critical density. Nitrogen viscosities were found to correspond to those of oxygen through a simple scaling by critical constants. Viscosities were measured in the high-pressurediamond-anvilcell with a rolling-ball technique. The dynamics of a sphere rolling on an inclined plane were investigated in the context of these experiments. The effect of a second surface, situated above the sphere, was experimentally determined.

B. Swanson is thanked for his loan of the necessary video equipment. This work was supported by NSF Grant No. EAR 0106683.

I. INTRODUCTION

II. EXPERIMENT

III. DYNAMICS OF A ROLLING SPHERE

A. Observations in the diamond-anvilcell

B. Correspondence to theory

C. Effect of the second, upper diamondsurface

IV. VISCOSITY OF FLUID OXYGEN

V. CONCLUSIONS

### Key Topics

- Viscosity
- 39.0
- Diamond anvil cells
- 15.0
- High pressure
- 11.0
- Viscosity measurements
- 11.0
- Shear rate dependent viscosity
- 9.0

## Figures

The position of a sphere is shown at intervals of as it rolls along an inner (diamond) surface of a high-pressure cell. These calibration runs were taken in -hexane at and tilt angles of 11° (엯), 14° (◻), 20°(▵), 25° (◇), and 30° (*). Straight lines fit to the data yield the rolling speeds.

The position of a sphere is shown at intervals of as it rolls along an inner (diamond) surface of a high-pressure cell. These calibration runs were taken in -hexane at and tilt angles of 11° (엯), 14° (◻), 20°(▵), 25° (◇), and 30° (*). Straight lines fit to the data yield the rolling speeds.

Speed of roll is plotted against the sine of the tilt angle for a platinum sphere ( diameter) rolling in fluids of viscosities (at ) between 0.25 and . -pentane (◁); argon at (▷); -hexane (*); toluene (엯, ◇); argon at (◻); water (▿); -pentanol (▵). Except for argon, all fluids were used at . Error bars are drawn for ± twice the root-mean-square deviations among runs if larger than the symbols. Straight lines are least-squares fits to the data. Points indicating speeds in -hexane derive partly from the data shown in Fig. 1.

Speed of roll is plotted against the sine of the tilt angle for a platinum sphere ( diameter) rolling in fluids of viscosities (at ) between 0.25 and . -pentane (◁); argon at (▷); -hexane (*); toluene (엯, ◇); argon at (◻); water (▿); -pentanol (▵). Except for argon, all fluids were used at . Error bars are drawn for ± twice the root-mean-square deviations among runs if larger than the symbols. Straight lines are least-squares fits to the data. Points indicating speeds in -hexane derive partly from the data shown in Fig. 1.

(a) Slopes of the lines in Fig. 2, multiplied by the factors , are plotted against the inverse viscosities of the fluids. A least squares fit through the data (straight line) passes through the origin with slope and serves as a calibration line for oxygen viscosities measured with the same sphere. (b) Slopes of calibration lines are plotted against the squares of the radii for four different spheres [the slope of the line in Fig. 3(a) is designated by the filled circle]. In this case the fit to the data (straight line) was forced through the origin. Error bars reflect an uncertainty of in the radii as measured from the video images.

(a) Slopes of the lines in Fig. 2, multiplied by the factors , are plotted against the inverse viscosities of the fluids. A least squares fit through the data (straight line) passes through the origin with slope and serves as a calibration line for oxygen viscosities measured with the same sphere. (b) Slopes of calibration lines are plotted against the squares of the radii for four different spheres [the slope of the line in Fig. 3(a) is designated by the filled circle]. In this case the fit to the data (straight line) was forced through the origin. Error bars reflect an uncertainty of in the radii as measured from the video images.

The speed of a sphere rolling in contact with a lower surface is affected by another, upper surface separated from the sphere by a variable gap. Speed, normalized by the measured speed in an infinite half space, is plotted (filled circles) against the ratio of the gap to the sphere’s diameter. (The gap is that from the upper plane to the sphere’s surface.) The data derive from a sphere rolling in methanol at a constant tilt angle of . The inset figure depicts the experimental cell with the roll direction out 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 one and two and, as seen here, the variations in rolling speed engendered by slight displacements of the upper diamond are not expected to be significant.

The speed of a sphere rolling in contact with a lower surface is affected by another, upper surface separated from the sphere by a variable gap. Speed, normalized by the measured speed in an infinite half space, is plotted (filled circles) against the ratio of the gap to the sphere’s diameter. (The gap is that from the upper plane to the sphere’s surface.) The data derive from a sphere rolling in methanol at a constant tilt angle of . The inset figure depicts the experimental cell with the roll direction out 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 one and two and, as seen here, the variations in rolling speed engendered by slight displacements of the upper diamond are not expected to be significant.

Viscosities of oxygen were measured at from up to the freezing pressure of . The various unfilled symbols represent runs made with five different spheres (in three different cells). Spheres with diameters of 37.3, 40.7, 42.5, and (▵, ◇, ◻, and 엯, respectively) were calibrated *in situ* [as in Fig. 3(a)] while the remaining sphere (▿), having been destroyed, was calibrated by using its measured diameter of and the plot of Fig. 3(b). Error bars represent uncertainties. Data previously taken to (Ref. 30) (*) are shown in more detail in the inset. The curve is a fit to Eq. (5).

Viscosities of oxygen were measured at from up to the freezing pressure of . The various unfilled symbols represent runs made with five different spheres (in three different cells). Spheres with diameters of 37.3, 40.7, 42.5, and (▵, ◇, ◻, and 엯, respectively) were calibrated *in situ* [as in Fig. 3(a)] while the remaining sphere (▿), having been destroyed, was calibrated by using its measured diameter of and the plot of Fig. 3(b). Error bars represent uncertainties. Data previously taken to (Ref. 30) (*) are shown in more detail in the inset. The curve is a fit to Eq. (5).

Oxygen viscosities shown in Fig. 5 are replotted against density reduced by the critical density [open symbols, current data; * from Haynes (Ref. 30)]. The additional small, filled circles represent viscosities of nitrogen (Ref. 31) at , scaled as per Eq. (4). The dotted line is a fit to Doolittle’s original formula [Eq. (3)], while the continuous curve is a fit to Eq. (5). A previously published (Ref. 27) representation of the viscosity, evaluated at , is shown as a dashed curve.

Oxygen viscosities shown in Fig. 5 are replotted against density reduced by the critical density [open symbols, current data; * from Haynes (Ref. 30)]. The additional small, filled circles represent viscosities of nitrogen (Ref. 31) at , scaled as per Eq. (4). The dotted line is a fit to Doolittle’s original formula [Eq. (3)], while the continuous curve is a fit to Eq. (5). A previously published (Ref. 27) representation of the viscosity, evaluated at , is shown as a dashed curve.

When plotted as residual quantities, viscosities at (◀) (Ref. 30) conform to the same curve as those at [open symbols, current data; * from Haynes (Ref. 30)]. Data (Ref. 36) taken at up to are also included (▶). Viscosities measured (Ref. 30) at lower temperatures (130, 110, and ), represented here by three line segments, deviate to higher values and define, roughly, a separate branch of the curve for the subcritical liquid.

When plotted as residual quantities, viscosities at (◀) (Ref. 30) conform to the same curve as those at [open symbols, current data; * from Haynes (Ref. 30)]. Data (Ref. 36) taken at up to are also included (▶). Viscosities measured (Ref. 30) at lower temperatures (130, 110, and ), represented here by three line segments, deviate to higher values and define, roughly, a separate branch of the curve for the subcritical liquid.

## Tables

Measured viscosities of oxygen, ordered from first to last acquired, and grouped according to the different spheres used.

Measured viscosities of oxygen, ordered from first to last acquired, and grouped according to the different spheres used.

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