In situ measurement of viscosity of liquids in the Fe-FeS system at high pressures and temperatures.

David P Dobson, Wilson A Crichton, Lidunka Vo\o(c,ù)adlo, John P Brodholt, Adrian P Jones and Yanbin Wang*.
Department of Geological Sciences, University College London, London WC1E 6BT, UK.
*Department of Geosciences, University of Chicago, Chicago, USA.

Abstract

The viscosity of liquid FeS and Fe-FeS eutectic was measured at pressures between 0.5 and 5.0 GPa using a synchrotron-based falling sphere technique.  We obtain viscosities of between 2 x 10-2 to 4 x 10-3 Pas in FeS at 1450 to 1700 ¡C and 2 x 10-2 and 8 x 10-3 Pas in Fe-Seut at 1150 to 1380 ¡C.  These results are consistent with recent viscosity measurements in Fe-Seut at 5 to 7 GPa (Urakawa et al, 2000), measured diffusivities (Dobson 2000) and ab initio simulated viscosity (Vocadlo et al, 2000).  The results are also similar to the values for pure iron at low pressure (Shimoji and Itami, 1986).  A systematic increase in viscosity and activation energy is seen with increasing sulfur content.  Interpolation between the data presented yields a viscosity of 3 x 10-2 Pas for an outer core composition with ~10 wt % S at the melting temperature.  There is good evidence of homologous behaviour for Fe-S liquids which implies that the liquid alloy at the inner core boundary may have a similar viscosity.

Introduction

	A knowledge the viscosity of liquid iron alloys at high pressure is important for understanding the chemical, thermal and geomagnetic evolution of the Earth's outer core.  The ill constarined geophysical measurements of outer core viscosity, spanning several orders of magnitude (Poirier 1988), necessitate its estimation by other means.  Experimental measurements of liquid iron alloys are extremely technically demanding and it is only in the last decade that such measurements have been attempted with some surprising results.  Brazhkin's (1998) semi-empirical estimation of Fe viscosity, for example, based on quench crystallite size, yields extrapolations to outer core pressures of 1011 Pas suggesting that the outer core would be in the glassy state.  The Fe-FeS system has received considerable experimental attention for two reasons: (i) the low eutectic temperature renders it easily accessible to large volume high pressure devices and (ii) because sulfur has been widely suggested as the light element in the cores of the Earth and othe planetary bodies.  LeBlanc and Secco (1996) used a rising sphere Stokes viscometry method to measure viscosity of Fe-FeS eutectic compositions at 2 to 5 GPa.  They found Fe-Seut to have a surprisingly high viscosity of 101 to 102 Pas and a large activation energy, up to 3.5 cm3/mole.  

	By contrast, theoretical considerations (Poirier, 1988) and ab initio simulations (Vocadlo et al, 2000; AlfŽ and Gillan, 1998) suggest the viscosities of liquid Fe-S alloys are essentially independent of pressure along the melting path, with near-metallic viscosities around 10-2 Pas at the solidus (Tm).  This view is supported by recent self-diffusion measurements in eutectic Fe-S at 2 and 5 GPa (Dobson, 2000) which are similar to simulations results at outer core pressures (AlfŽ and Gillan, 1998).
	In order to resolve these discrepancies, we present results of viscosity measurements of liquid FeS and a eutectic composition, Fe0.63S0.37, using the synchrotron-based Stokes-falling sphere method.

Experimental Methods

	Samples consisted of synthetic 99.9% pure FeS or Fe-FeS powder mixtures,  pre-pressed into dense pellets of 1.6 mm diameter and 1 to 1.5 mm long.  Ir and W spheres were produced by passing an electric arc between two wires under nitrogen. Spheres between 40 and 120 mm diameter were placed in the top third of the sample and the whole was then loaded into a silica glass capsule and high pressure cell.  The silica glass capsules acted as a barrier to water diffusion from within the pressure medium and helped to  minimise oxidation and hydride formation in the sample.

	Experiments were performed in the 250 ton Dia-type multi anvil press installed on  bending magnet BM13 at the Advanced Photon Source, using the high pressure cell design shown in figure 1.  The cubic pyrophyllite pressure medium of 8.0 or 8.5 mm edge length was compressed on each face by 6 mm edge length tungsten carbide anvils.  The box-geometry of the graphite furnace and ZrO2 thermal insulation served to reduce thermal gradients, which are estimated to be <50 ¡C/mm in the central region of the furnace.  Tungsten-rhenium thermocouples were used with cold-reference junctions at room temperature and pressure; no pressure correction was applied.  The measured temperature at which the spheres first moved was typically 100 ¡C below the known melting temperature of FeS and quoted temperatures have been corrected to the published melting curve (Ryzhenko and Kennedy, 1973).

	Spheres were observed in situ using the synchrotron shadowgraphy technique described by Kanzaki et al (1987).  The detector consisted of a double-polished YAG single crystal wafer placed in the beam behind the press.  A video-camera detected optical fluorescence of the YAG crystal.  The camera was calibrated by moving the press in 50 mm steps and observing the position of an absorbant marker in the X-ray beam.  The spatial resolution of the camera-crystal arrangement depended on the thickness of the crystal and the resolving power of the camera and was approximately 4 mm.  Spheres smaller than 40 mm cast shadows too low in contrast to be detected in the current set up.

	Experiments were performed by compression to the desired load at room temperature to the desired load.  Pressure was calibrated against the equation of state of MgO.  The sample was heated slowly (~100 ¡C/min) until first movement of the sphere was observed upon which the temperature was rapidly increased by 100 to 200 ¡C.  After the spheres had fallen, the sample was quenched in temperature, recovered and sectioned for electron probe analysis.  One high temperature pressure calibration experiment was performed with MgO replacing the sample.  The pressure-temperature curve due to cell relaxation and thermal expansion was assumed to be reproducible between experiments.  Pressures are quoted to +/- 0.5 GPa.

Results

	Figure 2 shows distance-time graphs for experiments at 1.5 GPa.  The plots  have clear regions of constant fall velocity, shown with solid symbols.  For a solid sphere of density rs and radius rs falling in a liquid of density rl, the viscosity is given by:

				h = \f(2grs2(rs-rl)F,9vK)					(1)
where v is the terminal velocity, F is the finite cylinder correction and K is the finite length correction (Faxen,1925; Kingery, 1959):

			F = 1- 2.104(rs/rc) + 2.09(rs/rc)3 - 0.95(rs/rc)5		(2)
and
				K = 1 + 3.3(rs/h)					(3);
rc is the sample radius and h is the sample length, taken to be the entire fall distance.  The density of Ir has not been measured at high pressure and temperature, which for the purpose of the current viscosity measurements was assumed to be equal to the atmospheric pressure value of 22.4 g/cm3.  The density of liquid FeS was assumed to be equal to the solid calculated from published compressibility and thermal expansion data (King and Prewitt, 1982; Lennie et al, 1997).  Thus the high P-T liquid FeS density will be overestimated, and viscosity underestimated by a few percent.   The high P-T densities of Usselman (1975) were used for liquid Fe-FeS eutectic.  Measured terminal velocities and calculated viscosities are given in table 1.  These viscosity values of 10-2 to 10-3 Pas are in good agreement with those obtained by Urakawa et al (2000).  Errors in calculated viscosity arise from uncertainties in density of sphere and melt, sample length and sphere diameter.  Errors in sphere density are small by comparison to melt density which may be in error by as much as 15 %, resulting in 6% viscosity error.  Due to the thermal gradient in the cell, it is likely that the whole sample was not molten when the spheres fell, hence the fall distance provides the best estimate of the effective sample length, h.  Reaction between the melt and sphere material caused a reduction in sphere size during the experiment.  We estimate that the sphere diameter changed by less than 25% during the falling part of the experiment which would result in 40% overestimation  of viscosity.

Discussion

	Measured viscosity for liquid Fe0.63S0.37 and FeS are plotted in figure 2 with predicted viscosities from Fe self diffusion in Fe0.63S0.37 (Dobson, 2000).  The measured viscosity of Fe0.63S0.37 in this study agrees well with the values predicted from diffusion measurements and the higher activation energy for diffusion is also seen in liquid Fe (Yang et al, 1957; Shimoji and Itami, 1986).  Previous viscosity measurements by LeBlanc and Secco (1996) give values 2 to 3 orders of magnitude higher than the present study and that of Urakawa et al (2000).  This discrepancy needs some comment:  
	(i) convection may contribute to our underestimation and LeBlanc and Secco's overestimation of viscosity.  LeBlanc and Secco used a floating sphere technique, whereas we used a falling sphere technique, thus convection which would be downwards in the cooler centre of the sample and upwards at the warmer sample edges would accelerate falling and decelerate rising spheres.  It seems unlikely, however that convection is significant in the set-up of LeBlanc and Secco who had small thermal gradients, or in the present study, where our short sample lengths lead to a Raleigh number of 0.012 and insignificant convection.
	(ii) LeBlanc and Secco's electrodetection method involves passing a small electrical current through the sample and measuring the change in resistance during the passage of a corundum sphere past the electrodes.  The change in geometry during the sphere's passage leads to two effects.  First, as the sphere passes across the conducting region, charge must pass around either side of the sphere.  The mutual attraction between parallel conductors will tend to resist the spheres entry into the electrical flux, thereby reducing the measured sphere velocity and effectively increasing measured h.  Secondly, metal flowing around the sphere will experience electrical eddy currents induced by the probe current. These eddy currents will resist the motion of the flowing metal and sphere, once again reducing the measured velocity.  We note that the probe currents used by LeBlanc and Secco are small (approximately a few mA), but we also note that the buoyancy force on the sphere is small (~3 x10-6N) as well.
	(iii) our early experiments at the APS on Fe0.63S0.37 mixtures produced confusing results with high apparent viscosities.  It was found that if heating rates were too slow, the sphere fell through a liquid-crystal mixture, even at temperatures significantly above the solidus.  This can be explained if the density of the eutectic melt is intermediate to the densities of the FeS and Fe solid phases.  In this case, during slow heating, the solid phases can segregate, with the lighter FeS floating to the top and the denser Fe sinking to the bottom of the sample.  Once this density separation has occurred, the local composition may be a long way from the eutectic composition and so liquidus crystals persist for a long time.  We decided test this by performing experiments on end-member FeS and saw no experiments in this composition with anomalously high viscosities.
	(iv) electron probe analysis of sectioned samples after experiments revealed the presence of up to 4 at.% oxygen.  Oxygen may act to reduce viscosity in S-polymeriesd melts as discussed by Dobson (2000), however there is at present little evidence of S polymerisation in liquid Fe-S systems.

	The present viscosity measurements in liquid FeS and Fe0.63S0.37 are consistent with ab initio simulations values (Vocadlo et al, 2000), independent falling sphere measurements (Urakawa et al, 2000) and values derived from diffusivity measurements using the Stokes-Einstein relation.  They are also close to measured values for iron and FeS at atmospheric pressure.  Sulfur content has a large positive effect on activation energy with EhFe = 41.4 kJ/mol  (Shimoji and Itami, 1986), EhFe0.63S0.37 = 100 (5) kJ/mol and EhFeS = 255 (18) kJ/mol.  Sulfur is known to have a large effect on compressibility in the liquid Fe-S system (Nasch et al, 1994;1997), which may suggest that the large activation energy for viscosity is related to a large thermal expansion, although this has not been measured.  It might be reasonable to expect that the pressure dependences of viscosity in the Fe-S system would also be strongly dependent on sulfur content.  The present study covered too small a pressure range and errors were too large to resolve the activation volume for viscosity in liquid FeS and Fe0.63S0.37.  Other studies are currently under way to delineate this question (Urakawa et al., 2000).  

	The remarkable agreement between the measured viscosity in this study, that of Urakawa et al (2000) at 7 GPa, and the value obtained from simulations results at 15 and 330 GPa (Vocadlo et al, 2000; AlfŽ and Gillan, 1998) strongly implies that Fe-FeS compositions display homologous viscosity behaviour.  As previously noted (Poirier, 1988), such homologous behaviour implies that the viscosity of the liquid iron alloy at the inner core boundary (ICB) is the same as the ambient pressure Tm value.  Interpolating between our results and those for pure liquid Fe to the 10 wt % sulfur value required to fulfill outer core density requirements (assuming the light element is entirely sulfur), we obtain a Tm viscosity value of 3 x 10-2 Pas for outer core liquid alloy.  We therefore propose this value (3 x 10-2 Pas) for the liquid immediately above the inner core boundary which represents the system solidus at 330 GPa.

Acknowledgments

	DPD acknowledges his NERC research fellowship; JPB and LV acknowledge their Royal Society URFs.  WAC contributed to this study as part of a NERC funded PhD.  We thank S. Urakawa and R. Secco for helpful discussions, and M. Rivers in assistance in synchrotron experiments.  Use of the Advanced Photon Source was supported by the U.S. Department of Energy, Basic Energy Sciences, Office of Science, under Contract No. W-31-109-Eng-38.

References
AlfŽ, D. and Gillan, M.J.  First-principles simulations of liquid Fe-S under Earth's core conditions.  Phys. Rev. B, 58, 8248-8256, 1998.
LeBlanc, G. E.,  and Secco, R. A.,  Viscosity of and Fe-S liquid up to 1300 ¡C and 5 GPa.  Geophys. Res. Lett., 23, 213-216, 1996.
Brazhkin, V.V.,  Investigation of the crystallisation of liquid iron under pressure: extrapolation of melt viscosity into the megabar range.  JETP Letters, 68, 502-508, 1998
Dobson, D.P.,  57Fe and Co tracer diffusion in liquid Fe-FeS at 2 and 5 GPa.  in press Phys. Earth Planet. Inter.,  2000.
Faxen, H. Gegenseitige einwirkung zweier kugelen, die in einer zŠhen flŸssigkeit fallen.  Ark. Mat. Astron. Fys., 19, 1-8, 1925
Kanzaki M., Kurita, K., Fujii, T., Kato, T., Shimomura, O. and Akimoto, S.,  A new technique to measure the viscosity and density of silicate melts at high pressure.  In High-Pressure Research in Mineral Physics ed. M.H Manghnani and Y. Syono, AGU, Washington DC, 195-200, 1987.
King, H.E. and Prewitt, C.T., High-pressure and high-temperature polymorphism of iron sulfide (FeS).  Acta Cryst., B38, 1877-1887, 1982.
Kingery, W.D., Property measurements at high temperatures.  416pp., John Wiley and Sons Inc., New York, 1959.
Lennie, A.R., Redfern, S.A.T., Champness, P.E., Stoddart, C.P., Schofield, P.F. and Vaughan, D.J.,  Transformation of mackinawite to greigite: an in situ X-ray powder diffraction and transmission electron microscope study.  Am. Miner., 82, 302-309, 1997.
Nasch, P.M., Manghnani, M.H. and Secco. R.A., Sound velocity measurements in liquid iron by ultrasonic interferometry.  J. Geophys. Res., 99, 4285-4291, 1994.
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Poirier, J.P., Transport properties of liquid metals and viscosity of the Earth's core. Geophys. J., 92, 99-105, 1988
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 Table 1

P (GPa)	T ( ¡C)	sample	rs (mm)	v(mm/s)	h (mm)*	F/K	h(mPas)	
0.5	1450	FeS	33
33	1.77
2.30	0.4
0.3	0.718
0.596	16.9
19.3	
1.5	1580	FeS	33
55	6.3
7.99	0.60
	0.773
0.657	5.1
9.5 	
1.5	1517	FeS	33	3.91	0.80	0.804	8.6	
1.5	1473	FeS	33	2.66	0.65	0.782	12.2	
2.5	1547	FeS	55	5.50	0.65	0.669	13.9	
2.5	1552	FeS	22
44	2.55
4.51	0.75	0.859
0.741	6.2
12.0 	
4.8	1650à	FeS	28	5.8	0.7	0.818	4.2	
5.0	1707	FeS	22	3.84	0.50	0.823	3.9	
								
1.5	1290	Fe-Seut	50¤	5.5	1.2	0.649	11.8 	
2.2	1374	Fe-Seut	44	6.95	0.75	0.741	7.5	
5.5	1150à	Fe-Seut	250#	15¦	1.5	0.260	24.0 	
notes
* Total fall distance
  Terminal velocity not reached
à Temperature estimate based on power
¤ Platinum sphere
# Tungsten carbide sphere
¦ Sphere fell capsule length in one 0.1s frame.

Figure Captions

Figure 1.  High pressure cell for in situ high-pressure, high-temperature viscosity measurements.

Figure 2.  Distance-time graphs for three experiments.  The acceleration region and terminal velocity region (solid symbols) are clearly seen.  A region of deceleration is seen due in part edge effects and to thermal gradient effects is also seen.

Figure 3.  Ahrrenius plot for viscosity in FeS and Fe0.63S0.37.  The data for FeS at 0.5 to 2.5 GPa fall along the same curve, as do those for Fe0.63S0.37 at 1.5 to 5.5 GPa, suggesting the effect of pressure is small.  The error in the data is too large to estimate activation volumes over such a small pressure range.  The Fe0.63S0.37 viscosity is similar to that predicted from Fe self diffusion measurements, suggesting that the Stokes-Einstein relation holds in this system.

Fig. 1

Fig. 2

Fig. 3