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Interdiffusion of Earth's core materials to 65 GPa and 2200 K
Earth and Planetary Science Letters 349-350 (2012) 8–14
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Earth and Planetary Science Letters journal homepage: www.elsevier.com/locate/epsl
Letters
Interdiffusion of Earth’s core materials to 65 GPa and 2200 K Daniel M. Reaman a,n, Hendrik O. Colijn b, Fengyuan Yang c, Adam J. Hauser c, Wendy R. Panero a a b c
The Ohio State University, School of Earth Sciences, 275 Mendenhall Lab, 125 S. Oval Mall, Columbus, OH 43210, USA Department of Materials Science and Engineering, The Ohio State University, USA Department of Physics, The Ohio State University, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 October 2011 Received in revised form 25 June 2012 Accepted 27 June 2012 Editor: L. Stixrude Available online 25 July 2012
Diffusion of iron and nickel in an Fe64Ni36 alloy was measured via the use of focused-ion-beam (FIB) milling and energy dispersive x-ray spectroscopy (EDX) to 65 GPa and 2200 K. The experiments were performed using a combination of micro-fabricated samples in the laser-heated diamond-anvil cell (DAC), increasing the pressure range for experimentally determined interdiffusion coefficients in FeNi alloys by 4 40 GPa. Diffusivity in the iron–nickel system follows a homologous temperature relationship D ¼D0 exp( 19.3(2.7)Tm/T) where D0 is 4.6(7) 10 4 m2 s 1, with zero-pressure activation energy, 301(40) kJ mol 1, consistent with measurements at lower pressures. At inner core conditions, this predicts an inner core solid-state diffusivity of 1 10 14 m2 s 1 (Tm/T¼ 0.95), with an inner core viscosity of 1020–1022 Pa s assuming power-law creep (n ¼ 3) with a deviatoric stress of 103–104 Pa. & 2012 Elsevier B.V. All rights reserved.
Keywords: FeNi alloys diffusion high pressure high temperature diamond-anvil cell inner core
1. Introduction Planetary cores are composed primarily of Fe–Ni alloys; thus, the self-diffusion rates of these alloys have important implications for the physical and chemical processes of deep planetary interiors. Specifically, debate persists regarding the rate and style of deformation in Earth’s inner core (Jeanloz and Wenk, 1988; Karato, 1999; Buffett and Wenk, 2001; Yoshida et al., 1996; Van Orman, 2004) which is ultimately responsible for some, if not all, of its observed anisotropic seismic properties. A first-order, ratelimiting constraint on deformation of Earth’s inner core is diffusion, the value of which can be used to constrain other properties, such as solid-state viscosity. Diffusion rates in Fe–Ni alloys have been measured to 23 GPa (Yunker and Van Orman, 2007; Goldstein et al., 1965). The extreme conditions of planetary interiors require that the pressure range be extended in order to isolate the effect of pressure on diffusion rates in these alloys and the subsequent effect on physical and chemical properties under inner core conditions. While diffusivity in the FeNi system varies with composition, in Fe-rich FeNi alloys the differences in intrinsic diffusion between Fe and Ni diffusion are small (Goldstein et al., 1965). We therefore treat the diffusion of Fe and Ni as a proxy for the self-diffusion of
n Corresponding author. Present address: University of Chicago, Dept. of the Geophysical Sciences, 5734 S. Ellis Ave., Chicago, IL 60637, USA. Fax: þ773 702 9505. E-mail address: [email protected] (D.M. Reaman).
0012-821X/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.epsl.2012.06.053
iron, required for modeling of grain growth and deformation of solid inner core materials. Here, we present experimental data on diffusion of Fe and Ni in FeNi alloys at approximately 65 GPa, creating samples with microfabrication techniques and measuring diffusion ex situ with ion-beam liftout and TEM methods, extending the pressure range of Fe–Ni diffusion experiments through the use of the diamondanvil cell. The results are extrapolated to inner-core conditions to estimate viscosity and deformation within the Earth’s inner core.
2. Experimental methods and analyses Micro fabrication of controlled-geometry samples for use in the diamond-anvil cell (DAC) allows chemically marked crystalline layers to be created with initial vertical compositional discontinuities (Fig. 1). The initial concentration discontinuity allows for precise measurement of material transport properties subsequent to compression and heating through solution of the diffusion equation: @C ¼ r ð Dr C Þ @t
ð1Þ
where D is the diffusion coefficient and C is the concentration as a function of time, t, and space. Diffusivity as a function of pressure, P, temperature, T, activation energy, EA, and activation volume, VA, can be expressed as D ¼ D0 exp½ðEA þ PV A Þ=RT
ð2Þ
D.M. Reaman et al. / Earth and Planetary Science Letters 349-350 (2012) 8–14
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Fig. 1. Sample geometry pictured with Gaussian temperature profile representing the temperature distribution across the surface of the sample and temperature distribution through the sample.
in which D0 is the frequency factor. In many materials, the pressure dependence of the diffusivity can be directly related to the melting temperature at a given pressure using the homologous temperature relation: D ¼ D0 expðAT m =RTÞ
ð3Þ
where A/R is an empirical constant and Tm is the melting temperature at that pressure. 2.1. Sample fabrication 300 nm films of nickel or iron were grown on 5–10 mm thick Fe64Ni36 foil substrates (Fig. 1) in an ultrahigh vacuum (UHV) magnetron sputtering system with base pressure 5 10 10 Torr. Films were grown in a 3 m Torr ultrapure Ar sputtering gas (99.9995%) further purified by a Matheson NanoChem Purifilter to achieve a specified impurity level of one part per billion. The substrate–film interface gives a well-defined vertical discontinuity of iron and nickel concentration 300 nm below the sample surface. Growing thicker layers was hampered by adhesion of the layers and complicates the quantification of the temperature of the boundary in laser heating due to the axial temperature gradient. Samples were stored under vacuum to prevent oxidation, then cut to the appropriate size for experiments. 2.2. Experiment Samples were loaded into diamond-anvil cells (DAC) cryogenically in an Ar pressure and insulating medium. Ruby grains were used as the pressure standard and to elevate the sample above the bottom diamond surface. After room-temperature compression, the samples were heated with a TEM00 40 W diode-pumped Photonics YLF laser (l ¼1064 nm) on the side with the pure metal layer. To reduce complications from a variable temperature history for the sample, the desired temperature was reached within 3 s through rapid increase in the laser power, and the shutter closed to quench the experiment after 120 s at high temperature. The effect of the ramp-up time is considered
negligible due to the exponential temperature dependence of diffusivity. For metal foils 10 mm thick with 10 mm of insulating media, the axial temperature gradient in the metal is 50 K mm 1 (Manga and Jeanloz, 1996). This axial gradient is less than the inherent uncertainty in the measured temperature. We adopt a standard uncertainty in the temperature of 7100 K (Benedetti and Loubeyre, 2004). Temperatures were measured by fitting the Planck radiation function to the spectra generated by the hotspot. An Acton SP 150 imaging spectrometer was used to record spectra at a magnification of 10. Temperature gradients were determined by fitting spectra at successive distances from the hotspot, which were recorded with an imaging spectrometer, where the Gaussian temperature distribution is consistent with the Gaussian laser source. Diffusion profiles were measured in quenched samples using STEM energy-dispersive X-ray spectroscopy (EDX) after sample preparation with a focused-ion beam (FIB). Multiple compositional profiles from the surface were measured as a function of distance from the location of the peak temperature. The temperature of the compositional profile is then assigned according to the measured distance from the peak temperature correlated to the measured radial temperature profile. A Ni–Fe64Ni36 couple were heated at 65 GPa and a Fe–Fe64Ni36 couple at 75 GPa. The composition of the Ni–Fe64Ni36 couples ensures that the sample remains in the fcc phase throughout the range of pressures and temperatures of these experiments. The Fe–Fe64Ni36 couples will have a mixture of hcp and fcc phases at pressures above the e to g phase boundary. Because increasing Ni content stabilizes the fcc phase, as Ni diffuses upward, the fcc þhcp phase mixture will migrate upwards during the diffusion process. 2.3. Analysis Samples were recovered from the DAC and mounted on SEM stubs using adhesive carbon film and stored in an oxygen-free environment (Fig. 2a). After recovery and mounting, the samples were prepared in a Helios 600 dual-beam FIB (focused ion-beam system). An electron beam was used to locate the center of the
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Fig. 2. (a) Sample after recovery from DAC clearly showing laser-heated spot. (b) Milled and undercut sample before extraction and final thinning.
heated spot, and a Ga þ beam was used to mill out a vertical cross section through the hotspot (Fig. 2b). In situ lift out was used to mount the sample to a grid for final thinning and cleaning, in which the sample was thinned to 100–200 nm. Iron and nickel concentration profiles were collected vertically from the sample surface using STEM/EDX at regular intervals from the center of the hotspot. Each composition profile reflects diffusion at different temperatures as a consequence of the radial temperature gradients arising from the Gaussian laser profile (Fig. 1). The STEM EDX profiles were collected and analyzed using the FEI TIA (aka ESVision) software. The standardless Cliff–Lorimer quantification was found to be accurate for these current samples, consistently resulting in the Fe64Ni36 alloy measurements within 5%. The sample thicknesses were such that absorption corrections were not necessary. This was also confirmed by modeling spectra using the DTSA-II and WinXRay Monte Carlo software (Ritchie, 2009; Gauvin et al., 2006). For validation of the plausibility of measuring diffusion coefficients in the short scales required by the diamond-anvil cell, one experimental run with a Ni–Fe64Ni36 couple was encased in MgO and loaded into the hydrothermal diamond-anvil cell (HDAC) at 1 GPa and heated at 1063 K for 122 min, thereby eliminating the temperature gradients or uncertainty in location of the peak temperature. The sample was then extracted and analyzed using a physical electronics time-of-flight secondary-ionization mass spectrometer via depth profiling.
2.4. Calculation of diffusivity For one-dimensional diffusion where diffusivity is independent of Ni concentration, Eq. (1) can be written as @C @2 C ¼D 2 @t @x
ð4Þ
Eq. (4) can be solved analytically assuming diffusion between two semi-infinite half spaces, x pffiffiffiffiffiffi C ðx,t Þ ¼ C 0 erf ð5Þ 2 Dt where C0 is the concentration of the upper layer (Fig. 1), x is the distance from the discontinuity, parallel to the axis of the heating laser. This solution describes the behavior of the solute by the pffiffiffiffiffiffi single dimensionless parameter x=2 Dt . With the 300 nm surface layer of Ni or Fe, the samples are in the semi-infinite approximation for D o10 16 m2 s 1. For diffusivities 410 16 m2 s 1 we fit the data with a thin-film solution
to Eq. (4), assuming constant D, according to (Crank, 1970) 2 bC 0 x cðx,t Þ ¼ pffiffiffiffiffiffi exp 4Dt 2 Dt
ð6Þ
where b is the thickness of the solute and c0 is the concentration of the solute. As a cross check to the thin-film solution, we also numerically model the diffusion via a Taylor series expansion of Eq. (1), C ðx,t Þ ¼
DDt C ðx þ Dx,t Þ þC ðxDx,t Þ2C ðx,t Þ þT ðx,tDt Þ Dx2
ð7Þ
where the time step Dt and spatial resolution Dx are 0.01 s and 50 nm respectively. The diffusivity is determined by minimizing the w2 between the model and the measured profiles. The diffusivities predicted by the thin-film solution are within a factor of three of those predicted using Eq. (6), well within the uncertainty of the interdiffusion coefficients.
3. Results The samples were imaged using high-angle annular dark field scanning transmission electron microscopy (HAADF-STEM), allowing EDX compositional profiles to be taken perpendicular to the sample surface at regular intervals as well as providing excellent Z contrast (Fig. 3). Diffusion profiles as a function of distance from the center of the hotspot show clear decrease in diffusion rate (Fig. 4), consistent with temperature gradients as determined by radial and axial heat flow and the shape of the heating laser. Cross sections from Ni–Fe64Ni36 couples at 1 and 65 GPa yield 6 independent measurements of FeNi diffusivity between 1063 K and 2200 K (Table 1, Fig. 5). From these data we calculate an activation volume using homologous temperature scaling according to (Sammis et al., 1977)
EA dT m =dP
VA ¼ ð8Þ T m P dT m =dP where we predict an activation volume of 2.58(34) cm3 mol 1 using melting data on pure iron, with dTm/dP from Anderson and Isaak (2000) and Tm of 2900 K assuming a 100 K melting point depression of the 36% Ni alloy (Nguyen and Holmes, 2004; Yang et al., 1996). We used the data sets of Yunker and Van Orman (2007) and Goldstein et al. (1965) combined with these data (Fig. 6) to further constrain the activation volume for Fe and Ni interdiffusion on the Fe-rich side (uncertainties in the diffusion data of Goldstein et al. (1965) were not provided). These results fit to Eq. (2) result in D0 ¼4.6(7) 10 4 m2 s 1, EA ¼301(40) kJ
D.M. Reaman et al. / Earth and Planetary Science Letters 349-350 (2012) 8–14
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with Fe depleted (55%) to a depth of 1.8 mm (Fig. 7a), below which point the concentration of Fe increases to 70%. Examination of the microstructure of this sample reveals a dominance of grain boundaries oriented at 601 (Fig. 7b), which we suggest is responsible for much of the mass transport. This phenomenon and microstructure suggests a fast path for diffusion, which can be attributed to grain-boundary diffusion. Since the sample was not melted, it further supports our conclusion of grain-boundary diffusion in the sample in Fig. 7b.
4. Discussion
Fig. 3. Thinned foil recovered from heating to 2200 K at 65 GPa, showing lines (red) along which EDX profiles were collected relative to the center of the laserheated spot. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. Example of diffusion profiles recorded from Fig. 3 as a function of distance from the hotspot with associated fits to Eq. (5). Note how the shape of the diffusion curve changes as a function of distance from the hotspot.
mol 1, and A/R¼19.3(2.7) (Table 2). A linear fit to the Arrhenius curve (D as a function of 1/T) for a zero-pressure EA of 301 kJ mol 1 yields an activation volume of 2.62(3) cm3 mol 1. Results from the Fe–Fe64Ni36 coupled experiment at 75 GPa do not exhibit a diffusion profile consistent with lattice diffusion. While the substrate remained in the fcc phase, the Fe cap underwent a phase transition to hcp upon compression. During heating, a fcc þhcp phase boundary migrated upwards according to the increasing Ni content as a result of diffusion. The resulting composition after 120 s of heating remains relatively unchanged
Diffusion measurements in the laser-heated diamond-anvil cell yield diffusivities that are consistent with previously published results on pure Fe and Ni (Brown and Ashby, 1980) and Fe90Ni10 alloys (Goldstein et al., 1965; Yunker and Van Orman, 2007) for a given homologous temperature, Tm/T (Fig. 5). Goldstein et al. (1965) found a pressure dependence of diffusion in the Fe–Ni system (Fe90Ni10) to 4 GPa at temperatures approaching 1573 K. Yunker and Van Orman (2007) extrapolated these results to a common temperature of 1873 K (T/Tm ¼0.874), and combined with their data for an identical composition found no pressure dependence to 23 GPa. Tripling the pressure range in these experiments, we find the pre-exponential factor, D0, is invariant with pressure (Fig. 8), with overlapping uncertainties for the free fit parameter A/ R, confirming a homologous temperature relationship. The substrate of these experiments consisted of a Fe64Ni36 alloy, in to which a pure Fe/Ni solute diffused. Due to the concentration gradient in the samples we were able to measure diffusion without measurable deviation from a constant diffusivity at a given temperature. For these reasons, the diffusivities measured in this study represent diffusivities that span a wide range of composition and thus can be interpreted as an average diffusivity. The primary source of uncertainty is the measured temperature, 7100 K. For metal foils 10 mm thick with 10 mm of insulating media, the axial temperature gradient in the metal is 50 K mm 1. This axial gradient is less than the inherent uncertainty in the measured temperature. Least-square fits of the concentration data to Eqs. (5) or (7) yield uncertainties in the range of 0.1–0.2 log units of diffusivity. The value and uncertainty in the parameter A/R as derived from a least-squares fit to Eq. (3), 19.3(2.7), therefore incorporates the uncertainties in temperature and diffusivity, for a net uncertainty of 1.1 log units in diffusivity at a fixed temperature and pressure. We assume diffusion coefficients are independent of concentration, as the nature of our samples does not permit quantitative analysis of concentration dependence; i.e. the diffusion lengths are too short and essentially represent average diffusivities over a range of compositions. It is clear that these experiments suffer from limitations of sample size and range of measurement. The concentration dependence is not resolvable in the short diffusion profiles; as such, we must assume that diffusion coefficients are independent of concentration. Axial ( 50 K mm 1) and radial (300 K mm 1 6 mm from the center of the hotspot) temperature gradients present in laserheated diamond-anvil cell experiments introduce (Heinz and Jeanloz, 1987; Sinmyo and Hirose, 2010) concentration gradients that arise from the response of different chemical components to thermal gradients. We approximate the Soret effect in the axial direction to determine the relative contribution of thermally driven diffusion to the measured composition profiles. Since the species are similar in mass and diffuse at approximately the same rate in a metallic continuum, a reasonable approximation of the
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Table 1 Experimentally determined diffusivities from this study, with homologous temperatures and distances from the laser-heated spot. The melting temperature of the alloy is estimated from Nguyen and Holmes (2004) ( 3000 K) with a melting point depression of 100 K for 36% Ni (Yang et al., 1996). Diffusivity (Fe) (m2 s 1)
Diffusivity (Ni) (m2 s 1)
1.26 10 15 8.70 10 15 15
1.67 10 6.40 10 15 1.63 10 15 9.00 10 16
1.49 10 15 6.33 10 15 6.50 10
16
8.88 10 16 1.28 10 16 6.66 10 16 4.13 10 16 4.30 10 17 3.80 10 17 1.99 10 17
T (K)
Tm/T
Distance from hotspot (mm)
Pressure (GPa)
2200(100)
1.32(6)
0
65
2200(100)
1.32(6)
0
65
2180(100)
1.33(6)
1
65
2180(100)
1.33(6)
1
65
2130(100) 2130(100) 2040(100) 2040(100)
1.36(6) 1.36(6) 1.42(7) 1.42(7)
2 2 3 3
65 65 65 65
1760(100) 1760(100) 1100(5)
1.65(10) 1.65(10) 1.7
6 6 0
65 65 1
Table 2 Experimentally derived parameters for the studies of Goldstein et al. (1965), Yunker and Van Orman (2007) and this study. Parameter
This study 36% Ni
Activation energy (kJ/mol) Empirical constant, A/R Pre exponential factor, D0 (m2/s) Activation volume (cm3/mol)
Yunker and Van Orman 10% Ni
Goldstein et al. 10% Ni
301(40) 300 19.3(2.7) 20.4 4.6(7) 10 4 2.7 10 4
318 Not given 5.3 10 4
2.62(3)
6
3.1(0.7)
Soret coefficient is (Jost, 1952) s¼ Fig. 5. Diffusivity as a function of homologous temperature for pure Fe and Ni (0 GPa, solid and dotted lines, Brown and Ashby, 1980), Fe90Ni10 (1 bar (larger dotted line) and 4 GPa (dot and dashed), Goldstein et al., 1965; Yunker and Van Orman, 2007, 1 GPa (circles), 12 GPa (diamonds), 23 GPa (squares)). The Goldstein et al. (1965) data are lower because they are not scaled to a common temperature—details of this scaling can be found in Yunker and Van Orman, 2007.
Fig. 6. Measured diffusion coefficients as a function of pressure referenced to 1900 K assuming a homologous temperature relationship for the studies of Goldstein et al. (1965) and Yunker and Van Orman (2007) in Fe90Ni10 and this study in Fe64Ni36.
Dth 1 dN Ni ¼ DC NNi NFe dT
ð9Þ
where Nx are the mole fractions of the appropriate species. Given the axial temperature gradient of 50 K mm 1, we find, for the hotspot profile, that Dth/DC ¼ 0.04, such that this is not expected to be significant. We can also approximate the Soret effect through comparison of diffusion along the radial and axial directions. The compositional profiles plotted in Fig. 4 are made up of points taken at identical depths into the sample; thus, we are able to use them in the radial direction to extract an effective radial diffusion coefficient. For example, if we consider the data at a fixed depth, the apparent lateral diffusivity is more than an order of magnitude less than the axial diffusion coefficients at identical distances. Therefore, the thermal diffusivity along steeper temperature gradients (radially vs. axially) is considerably less than the chemical diffusion. Thus, we can ignore it to that approximation, where we limit the Soret effect to o10% overestimate of our reported diffusivities. Diffusion in the Fe–Ni system occurs by a vacancy mechanism, and solutions for Fe-rich fcc Fe–Ni alloys are nearly ideal (Rammensee and Fraser, 1981). Additionally, Brown and Ashby (1980) have shown that a wide variety of hcp and fcc metals have similar diffusivities at the same homologous temperatures. Therefore, we expect the difference in Fe–Ni diffusivity and Ni or Fe self-diffusion to be small; indeed, the differences are indistinguishable within uncertainty (Fig. 5) consistent across compositions up to 40% Ni (Goldstein et al., 1965). While the alloy used in these experiments is in the fcc phase throughout the pressure
D.M. Reaman et al. / Earth and Planetary Science Letters 349-350 (2012) 8–14
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Fig. 7. (a, b) Depth profile and image of sample exhibiting grain-boundary diffusion.
high pressure will result in a non-linear relationship between the log of the diffusion coefficient and the pressure, which subsequently will overestimate the value of the interdiffusion coefficient. Conversely, assuming a large activation volume, such as that found in Goldstein et al. (1965), results in a predicted interdiffusion coefficient of 5.6 10 25 m2 s 1. Thus, the value of the activation volume has drastic effects on the predicted diffusivity and should be carefully considered when making any predictions.
5. Conclusions
Fig. 8. Experimentally determined pre exponential factor, D0, compared with that of Goldstein et al. (1965) and Yunker and Van Orman (2007) as a function of pressure.
range, we assume that the high-pressure close-packed fcc structure follows the same homologous temperature relationship for the hcp melting curve. The homologous temperature relation has been shown to hold for a wide range of metals (Brown and Ashby, 1980; Sammis et al., 1981;) and at pressures up to 23 GPa (Yunker and Van Orman, 2007). The results of this study confirm the accuracy of the homologous temperature relationship in Fe–Ni alloys to 65 GPa. Fig. 6 shows a plot of the diffusion coefficient as a function of pressure for the studies of Goldstein et al. (1965) and Yunker and Van Orman (2007) in Fe90Ni10 and that predicted by the homologous temperature relationship in Fe64Ni36 at 1900 K. Any curvature in Fig. 6 would indicate that the activation volume changes with pressure, which is not resolvable to 65 GPa. The homologous temperature relationship predicts diffusivities at ICB conditions in the range of 5.7 10 14–6.9 10 13 m2 s 1. Likewise, if the activation volume found in this study is assumed to remain constant at 2.62(3) cm3 mol 1 to inner core conditions, this predicts what is likely an extreme lower limit of 1 10– 14 m2 s 1, consistent with values predicted by the homologous temperature relationship. However, as VA likely decreases at large compressions, it will have a significant effect on the interdiffusion coefficient. Using Eq. (8) and assuming the hcp value of dTm/dP is 7.9 K GPa 1 from the melting curve at 330 GPa (Anderson and Isaak, 2000) and EA ¼301 kJ (this study), the 330 GPa activation volume is predicted to be 1 cm3 mol 1, reflecting a 50% reduction between 65 and 330 GPa. The predicted diffusivity for these parameters is 7.5 10 10 m2 s 1 assuming a T of 5700 K (Anderson and Isaak, 2000; Yunker and Van Orman, 2007) and a P of 330 GPa. While this is a reasonable way to estimate the activation and resulting diffusivity, the shallower melting curve at
Interdiffusion coefficients in the fcc FeNi system determined to 65 GPa via the use of FIB and EDX analysis extend the pressure range of existing diffusion data by approximately a factor of three. The diffusivities for Fe and Ni are indistinguishable within uncertainty and support the homologous temperature assumption in extrapolating diffusion data in Fe–Ni alloys to higher pressures. Though the studies of Goldstein et al. (1965) and Yunker and Van Orman (2007) concern the full range of Fe–Ni alloy compositions, the analyses of Yunker and Van Orman (2007) focused primarily on the Fe90Ni10 composition because it is likely close to the composition of Earth’s core; the diffusivities measured in this study on a Fe64Ni36 alloy assuming a homologous temperature relationship are indistinguishable from those in Fe90Ni10 alloys. The diffusion of Fe and Ni can be described by the homologous temperature relationship, D ¼D0 exp( ATm/RT), where D0 ¼4.6 10 4 m2 s 1 and the dimensionless parameter A/R¼19.3(2.7). Combining the data from this study with those of Yunker and Van Orman (2007) to 23 GPa predicts diffusivities in the range of 10 12–10 14 m2 s 1 at conditions of the inner core. Though the diffusivities are expected to be similar between hcp and fcc metals and alloys at the same homologous temperature, other factors such as pressure-dependent activation volume, altered diffusion pathways due to compression, and the effects of dissolved light elements may significantly alter the diffusive behavior of diffusion of iron and nickel in close-packed alloys. Further study of diffusion in hcp Fe to higher pressures is required to further constrain these values. Given a diffusivity of 10 14 m2 s 1 under inner core conditions, we can approximate the solid-state viscosity of the inner core adopting a power-law creep model (n ¼3) based on dislocation link climb (Reaman et al., 2011). This creep mechanism suggests significantly greater viscosities than when assuming Harper–Dorn creep (e.g. Van Orman, 2004) which is a consequence of transient deformation in low strain-rate measurements (Kumar and Kassner, 2009). Therefore, assuming a deviatoric stress of 103–104 Pa resulting from anisotropic growth (Yoshida et al., 1996) or response to mantle mass variations (Buffett, 1997),
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we predict an inner core viscosity of 1020–1022 Pa s (Reaman et al., 2011). Diffusivity can be measured at extreme conditions using micro-fabricated samples in the diamond-anvil cell to 1.1 log unit accuracy. These experiments then provide a basis for extending diffusion experiments into the pressure and temperature regime of the diamond-anvil cell.
Acknowledgments The authors would like to thank Daniel Huber and Jonathan Orsborn for their assistance with sample preparation and to Leonard Brillson and Philip Smith for assistance with NanoSIMS. The authors are indebted to Jim Van Orman and two anonymous reviewers for their thorough and helpful comments which vastly improved the flow and understanding of the paper. Funding was provided by NSF (EAR 0955647) and CDAC (4-3253-04). References Anderson, O.L., Isaak, D.G., 2000. Calculated melting curves for phases of iron. Am. Mineral. 85, 376–385. Benedetti, L.R., Loubeyre, P., 2004. Temperature gradients, wavelength-dependent emissivity, and accuracy of high and very-high temperatures measured in the laser-heated diamond cell. High Pressure Res. 24 (4), 423–445. Brown, A.M., Ashby, M.F., 1980. Correlations for diffusion constants. Acta Metall. 28, 1085–1101. Buffett, B.A., 1997. Geodynamic estimates of the viscosity of the Earth’s inner core. Nature 388, 571–573. Buffett, B.A., Wenk, H.R., 2001. Texturing of the Earth’s inner core by Maxwell stresses. Nature 413, 60–63. Crank, J., 1970. Mahematics of Diffusion. Clarendon Press, Oxford. Goldstein, J.J., Hanneman, R.E., Ogilvie, R.G., 1965. Diffusion in the Fe–Ni system at 1 atm and 40 kbar pressure. Trans. AIME 233, 812–820.
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Earth and Planetary Science Letters journal homepage: www.elsevier.com/locate/epsl
Letters
Interdiffusion of Earth’s core materials to 65 GPa and 2200 K Daniel M. Reaman a,n, Hendrik O. Colijn b, Fengyuan Yang c, Adam J. Hauser c, Wendy R. Panero a a b c
The Ohio State University, School of Earth Sciences, 275 Mendenhall Lab, 125 S. Oval Mall, Columbus, OH 43210, USA Department of Materials Science and Engineering, The Ohio State University, USA Department of Physics, The Ohio State University, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 October 2011 Received in revised form 25 June 2012 Accepted 27 June 2012 Editor: L. Stixrude Available online 25 July 2012
Diffusion of iron and nickel in an Fe64Ni36 alloy was measured via the use of focused-ion-beam (FIB) milling and energy dispersive x-ray spectroscopy (EDX) to 65 GPa and 2200 K. The experiments were performed using a combination of micro-fabricated samples in the laser-heated diamond-anvil cell (DAC), increasing the pressure range for experimentally determined interdiffusion coefficients in FeNi alloys by 4 40 GPa. Diffusivity in the iron–nickel system follows a homologous temperature relationship D ¼D0 exp( 19.3(2.7)Tm/T) where D0 is 4.6(7) 10 4 m2 s 1, with zero-pressure activation energy, 301(40) kJ mol 1, consistent with measurements at lower pressures. At inner core conditions, this predicts an inner core solid-state diffusivity of 1 10 14 m2 s 1 (Tm/T¼ 0.95), with an inner core viscosity of 1020–1022 Pa s assuming power-law creep (n ¼ 3) with a deviatoric stress of 103–104 Pa. & 2012 Elsevier B.V. All rights reserved.
Keywords: FeNi alloys diffusion high pressure high temperature diamond-anvil cell inner core
1. Introduction Planetary cores are composed primarily of Fe–Ni alloys; thus, the self-diffusion rates of these alloys have important implications for the physical and chemical processes of deep planetary interiors. Specifically, debate persists regarding the rate and style of deformation in Earth’s inner core (Jeanloz and Wenk, 1988; Karato, 1999; Buffett and Wenk, 2001; Yoshida et al., 1996; Van Orman, 2004) which is ultimately responsible for some, if not all, of its observed anisotropic seismic properties. A first-order, ratelimiting constraint on deformation of Earth’s inner core is diffusion, the value of which can be used to constrain other properties, such as solid-state viscosity. Diffusion rates in Fe–Ni alloys have been measured to 23 GPa (Yunker and Van Orman, 2007; Goldstein et al., 1965). The extreme conditions of planetary interiors require that the pressure range be extended in order to isolate the effect of pressure on diffusion rates in these alloys and the subsequent effect on physical and chemical properties under inner core conditions. While diffusivity in the FeNi system varies with composition, in Fe-rich FeNi alloys the differences in intrinsic diffusion between Fe and Ni diffusion are small (Goldstein et al., 1965). We therefore treat the diffusion of Fe and Ni as a proxy for the self-diffusion of
n Corresponding author. Present address: University of Chicago, Dept. of the Geophysical Sciences, 5734 S. Ellis Ave., Chicago, IL 60637, USA. Fax: þ773 702 9505. E-mail address: [email protected] (D.M. Reaman).
0012-821X/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.epsl.2012.06.053
iron, required for modeling of grain growth and deformation of solid inner core materials. Here, we present experimental data on diffusion of Fe and Ni in FeNi alloys at approximately 65 GPa, creating samples with microfabrication techniques and measuring diffusion ex situ with ion-beam liftout and TEM methods, extending the pressure range of Fe–Ni diffusion experiments through the use of the diamondanvil cell. The results are extrapolated to inner-core conditions to estimate viscosity and deformation within the Earth’s inner core.
2. Experimental methods and analyses Micro fabrication of controlled-geometry samples for use in the diamond-anvil cell (DAC) allows chemically marked crystalline layers to be created with initial vertical compositional discontinuities (Fig. 1). The initial concentration discontinuity allows for precise measurement of material transport properties subsequent to compression and heating through solution of the diffusion equation: @C ¼ r ð Dr C Þ @t
ð1Þ
where D is the diffusion coefficient and C is the concentration as a function of time, t, and space. Diffusivity as a function of pressure, P, temperature, T, activation energy, EA, and activation volume, VA, can be expressed as D ¼ D0 exp½ðEA þ PV A Þ=RT
ð2Þ
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Fig. 1. Sample geometry pictured with Gaussian temperature profile representing the temperature distribution across the surface of the sample and temperature distribution through the sample.
in which D0 is the frequency factor. In many materials, the pressure dependence of the diffusivity can be directly related to the melting temperature at a given pressure using the homologous temperature relation: D ¼ D0 expðAT m =RTÞ
ð3Þ
where A/R is an empirical constant and Tm is the melting temperature at that pressure. 2.1. Sample fabrication 300 nm films of nickel or iron were grown on 5–10 mm thick Fe64Ni36 foil substrates (Fig. 1) in an ultrahigh vacuum (UHV) magnetron sputtering system with base pressure 5 10 10 Torr. Films were grown in a 3 m Torr ultrapure Ar sputtering gas (99.9995%) further purified by a Matheson NanoChem Purifilter to achieve a specified impurity level of one part per billion. The substrate–film interface gives a well-defined vertical discontinuity of iron and nickel concentration 300 nm below the sample surface. Growing thicker layers was hampered by adhesion of the layers and complicates the quantification of the temperature of the boundary in laser heating due to the axial temperature gradient. Samples were stored under vacuum to prevent oxidation, then cut to the appropriate size for experiments. 2.2. Experiment Samples were loaded into diamond-anvil cells (DAC) cryogenically in an Ar pressure and insulating medium. Ruby grains were used as the pressure standard and to elevate the sample above the bottom diamond surface. After room-temperature compression, the samples were heated with a TEM00 40 W diode-pumped Photonics YLF laser (l ¼1064 nm) on the side with the pure metal layer. To reduce complications from a variable temperature history for the sample, the desired temperature was reached within 3 s through rapid increase in the laser power, and the shutter closed to quench the experiment after 120 s at high temperature. The effect of the ramp-up time is considered
negligible due to the exponential temperature dependence of diffusivity. For metal foils 10 mm thick with 10 mm of insulating media, the axial temperature gradient in the metal is 50 K mm 1 (Manga and Jeanloz, 1996). This axial gradient is less than the inherent uncertainty in the measured temperature. We adopt a standard uncertainty in the temperature of 7100 K (Benedetti and Loubeyre, 2004). Temperatures were measured by fitting the Planck radiation function to the spectra generated by the hotspot. An Acton SP 150 imaging spectrometer was used to record spectra at a magnification of 10. Temperature gradients were determined by fitting spectra at successive distances from the hotspot, which were recorded with an imaging spectrometer, where the Gaussian temperature distribution is consistent with the Gaussian laser source. Diffusion profiles were measured in quenched samples using STEM energy-dispersive X-ray spectroscopy (EDX) after sample preparation with a focused-ion beam (FIB). Multiple compositional profiles from the surface were measured as a function of distance from the location of the peak temperature. The temperature of the compositional profile is then assigned according to the measured distance from the peak temperature correlated to the measured radial temperature profile. A Ni–Fe64Ni36 couple were heated at 65 GPa and a Fe–Fe64Ni36 couple at 75 GPa. The composition of the Ni–Fe64Ni36 couples ensures that the sample remains in the fcc phase throughout the range of pressures and temperatures of these experiments. The Fe–Fe64Ni36 couples will have a mixture of hcp and fcc phases at pressures above the e to g phase boundary. Because increasing Ni content stabilizes the fcc phase, as Ni diffuses upward, the fcc þhcp phase mixture will migrate upwards during the diffusion process. 2.3. Analysis Samples were recovered from the DAC and mounted on SEM stubs using adhesive carbon film and stored in an oxygen-free environment (Fig. 2a). After recovery and mounting, the samples were prepared in a Helios 600 dual-beam FIB (focused ion-beam system). An electron beam was used to locate the center of the
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Fig. 2. (a) Sample after recovery from DAC clearly showing laser-heated spot. (b) Milled and undercut sample before extraction and final thinning.
heated spot, and a Ga þ beam was used to mill out a vertical cross section through the hotspot (Fig. 2b). In situ lift out was used to mount the sample to a grid for final thinning and cleaning, in which the sample was thinned to 100–200 nm. Iron and nickel concentration profiles were collected vertically from the sample surface using STEM/EDX at regular intervals from the center of the hotspot. Each composition profile reflects diffusion at different temperatures as a consequence of the radial temperature gradients arising from the Gaussian laser profile (Fig. 1). The STEM EDX profiles were collected and analyzed using the FEI TIA (aka ESVision) software. The standardless Cliff–Lorimer quantification was found to be accurate for these current samples, consistently resulting in the Fe64Ni36 alloy measurements within 5%. The sample thicknesses were such that absorption corrections were not necessary. This was also confirmed by modeling spectra using the DTSA-II and WinXRay Monte Carlo software (Ritchie, 2009; Gauvin et al., 2006). For validation of the plausibility of measuring diffusion coefficients in the short scales required by the diamond-anvil cell, one experimental run with a Ni–Fe64Ni36 couple was encased in MgO and loaded into the hydrothermal diamond-anvil cell (HDAC) at 1 GPa and heated at 1063 K for 122 min, thereby eliminating the temperature gradients or uncertainty in location of the peak temperature. The sample was then extracted and analyzed using a physical electronics time-of-flight secondary-ionization mass spectrometer via depth profiling.
2.4. Calculation of diffusivity For one-dimensional diffusion where diffusivity is independent of Ni concentration, Eq. (1) can be written as @C @2 C ¼D 2 @t @x
ð4Þ
Eq. (4) can be solved analytically assuming diffusion between two semi-infinite half spaces, x pffiffiffiffiffiffi C ðx,t Þ ¼ C 0 erf ð5Þ 2 Dt where C0 is the concentration of the upper layer (Fig. 1), x is the distance from the discontinuity, parallel to the axis of the heating laser. This solution describes the behavior of the solute by the pffiffiffiffiffiffi single dimensionless parameter x=2 Dt . With the 300 nm surface layer of Ni or Fe, the samples are in the semi-infinite approximation for D o10 16 m2 s 1. For diffusivities 410 16 m2 s 1 we fit the data with a thin-film solution
to Eq. (4), assuming constant D, according to (Crank, 1970) 2 bC 0 x cðx,t Þ ¼ pffiffiffiffiffiffi exp 4Dt 2 Dt
ð6Þ
where b is the thickness of the solute and c0 is the concentration of the solute. As a cross check to the thin-film solution, we also numerically model the diffusion via a Taylor series expansion of Eq. (1), C ðx,t Þ ¼
DDt C ðx þ Dx,t Þ þC ðxDx,t Þ2C ðx,t Þ þT ðx,tDt Þ Dx2
ð7Þ
where the time step Dt and spatial resolution Dx are 0.01 s and 50 nm respectively. The diffusivity is determined by minimizing the w2 between the model and the measured profiles. The diffusivities predicted by the thin-film solution are within a factor of three of those predicted using Eq. (6), well within the uncertainty of the interdiffusion coefficients.
3. Results The samples were imaged using high-angle annular dark field scanning transmission electron microscopy (HAADF-STEM), allowing EDX compositional profiles to be taken perpendicular to the sample surface at regular intervals as well as providing excellent Z contrast (Fig. 3). Diffusion profiles as a function of distance from the center of the hotspot show clear decrease in diffusion rate (Fig. 4), consistent with temperature gradients as determined by radial and axial heat flow and the shape of the heating laser. Cross sections from Ni–Fe64Ni36 couples at 1 and 65 GPa yield 6 independent measurements of FeNi diffusivity between 1063 K and 2200 K (Table 1, Fig. 5). From these data we calculate an activation volume using homologous temperature scaling according to (Sammis et al., 1977)
EA dT m =dP
VA ¼ ð8Þ T m P dT m =dP where we predict an activation volume of 2.58(34) cm3 mol 1 using melting data on pure iron, with dTm/dP from Anderson and Isaak (2000) and Tm of 2900 K assuming a 100 K melting point depression of the 36% Ni alloy (Nguyen and Holmes, 2004; Yang et al., 1996). We used the data sets of Yunker and Van Orman (2007) and Goldstein et al. (1965) combined with these data (Fig. 6) to further constrain the activation volume for Fe and Ni interdiffusion on the Fe-rich side (uncertainties in the diffusion data of Goldstein et al. (1965) were not provided). These results fit to Eq. (2) result in D0 ¼4.6(7) 10 4 m2 s 1, EA ¼301(40) kJ
D.M. Reaman et al. / Earth and Planetary Science Letters 349-350 (2012) 8–14
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with Fe depleted (55%) to a depth of 1.8 mm (Fig. 7a), below which point the concentration of Fe increases to 70%. Examination of the microstructure of this sample reveals a dominance of grain boundaries oriented at 601 (Fig. 7b), which we suggest is responsible for much of the mass transport. This phenomenon and microstructure suggests a fast path for diffusion, which can be attributed to grain-boundary diffusion. Since the sample was not melted, it further supports our conclusion of grain-boundary diffusion in the sample in Fig. 7b.
4. Discussion
Fig. 3. Thinned foil recovered from heating to 2200 K at 65 GPa, showing lines (red) along which EDX profiles were collected relative to the center of the laserheated spot. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. Example of diffusion profiles recorded from Fig. 3 as a function of distance from the hotspot with associated fits to Eq. (5). Note how the shape of the diffusion curve changes as a function of distance from the hotspot.
mol 1, and A/R¼19.3(2.7) (Table 2). A linear fit to the Arrhenius curve (D as a function of 1/T) for a zero-pressure EA of 301 kJ mol 1 yields an activation volume of 2.62(3) cm3 mol 1. Results from the Fe–Fe64Ni36 coupled experiment at 75 GPa do not exhibit a diffusion profile consistent with lattice diffusion. While the substrate remained in the fcc phase, the Fe cap underwent a phase transition to hcp upon compression. During heating, a fcc þhcp phase boundary migrated upwards according to the increasing Ni content as a result of diffusion. The resulting composition after 120 s of heating remains relatively unchanged
Diffusion measurements in the laser-heated diamond-anvil cell yield diffusivities that are consistent with previously published results on pure Fe and Ni (Brown and Ashby, 1980) and Fe90Ni10 alloys (Goldstein et al., 1965; Yunker and Van Orman, 2007) for a given homologous temperature, Tm/T (Fig. 5). Goldstein et al. (1965) found a pressure dependence of diffusion in the Fe–Ni system (Fe90Ni10) to 4 GPa at temperatures approaching 1573 K. Yunker and Van Orman (2007) extrapolated these results to a common temperature of 1873 K (T/Tm ¼0.874), and combined with their data for an identical composition found no pressure dependence to 23 GPa. Tripling the pressure range in these experiments, we find the pre-exponential factor, D0, is invariant with pressure (Fig. 8), with overlapping uncertainties for the free fit parameter A/ R, confirming a homologous temperature relationship. The substrate of these experiments consisted of a Fe64Ni36 alloy, in to which a pure Fe/Ni solute diffused. Due to the concentration gradient in the samples we were able to measure diffusion without measurable deviation from a constant diffusivity at a given temperature. For these reasons, the diffusivities measured in this study represent diffusivities that span a wide range of composition and thus can be interpreted as an average diffusivity. The primary source of uncertainty is the measured temperature, 7100 K. For metal foils 10 mm thick with 10 mm of insulating media, the axial temperature gradient in the metal is 50 K mm 1. This axial gradient is less than the inherent uncertainty in the measured temperature. Least-square fits of the concentration data to Eqs. (5) or (7) yield uncertainties in the range of 0.1–0.2 log units of diffusivity. The value and uncertainty in the parameter A/R as derived from a least-squares fit to Eq. (3), 19.3(2.7), therefore incorporates the uncertainties in temperature and diffusivity, for a net uncertainty of 1.1 log units in diffusivity at a fixed temperature and pressure. We assume diffusion coefficients are independent of concentration, as the nature of our samples does not permit quantitative analysis of concentration dependence; i.e. the diffusion lengths are too short and essentially represent average diffusivities over a range of compositions. It is clear that these experiments suffer from limitations of sample size and range of measurement. The concentration dependence is not resolvable in the short diffusion profiles; as such, we must assume that diffusion coefficients are independent of concentration. Axial ( 50 K mm 1) and radial (300 K mm 1 6 mm from the center of the hotspot) temperature gradients present in laserheated diamond-anvil cell experiments introduce (Heinz and Jeanloz, 1987; Sinmyo and Hirose, 2010) concentration gradients that arise from the response of different chemical components to thermal gradients. We approximate the Soret effect in the axial direction to determine the relative contribution of thermally driven diffusion to the measured composition profiles. Since the species are similar in mass and diffuse at approximately the same rate in a metallic continuum, a reasonable approximation of the
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Table 1 Experimentally determined diffusivities from this study, with homologous temperatures and distances from the laser-heated spot. The melting temperature of the alloy is estimated from Nguyen and Holmes (2004) ( 3000 K) with a melting point depression of 100 K for 36% Ni (Yang et al., 1996). Diffusivity (Fe) (m2 s 1)
Diffusivity (Ni) (m2 s 1)
1.26 10 15 8.70 10 15 15
1.67 10 6.40 10 15 1.63 10 15 9.00 10 16
1.49 10 15 6.33 10 15 6.50 10
16
8.88 10 16 1.28 10 16 6.66 10 16 4.13 10 16 4.30 10 17 3.80 10 17 1.99 10 17
T (K)
Tm/T
Distance from hotspot (mm)
Pressure (GPa)
2200(100)
1.32(6)
0
65
2200(100)
1.32(6)
0
65
2180(100)
1.33(6)
1
65
2180(100)
1.33(6)
1
65
2130(100) 2130(100) 2040(100) 2040(100)
1.36(6) 1.36(6) 1.42(7) 1.42(7)
2 2 3 3
65 65 65 65
1760(100) 1760(100) 1100(5)
1.65(10) 1.65(10) 1.7
6 6 0
65 65 1
Table 2 Experimentally derived parameters for the studies of Goldstein et al. (1965), Yunker and Van Orman (2007) and this study. Parameter
This study 36% Ni
Activation energy (kJ/mol) Empirical constant, A/R Pre exponential factor, D0 (m2/s) Activation volume (cm3/mol)
Yunker and Van Orman 10% Ni
Goldstein et al. 10% Ni
301(40) 300 19.3(2.7) 20.4 4.6(7) 10 4 2.7 10 4
318 Not given 5.3 10 4
2.62(3)
6
3.1(0.7)
Soret coefficient is (Jost, 1952) s¼ Fig. 5. Diffusivity as a function of homologous temperature for pure Fe and Ni (0 GPa, solid and dotted lines, Brown and Ashby, 1980), Fe90Ni10 (1 bar (larger dotted line) and 4 GPa (dot and dashed), Goldstein et al., 1965; Yunker and Van Orman, 2007, 1 GPa (circles), 12 GPa (diamonds), 23 GPa (squares)). The Goldstein et al. (1965) data are lower because they are not scaled to a common temperature—details of this scaling can be found in Yunker and Van Orman, 2007.
Fig. 6. Measured diffusion coefficients as a function of pressure referenced to 1900 K assuming a homologous temperature relationship for the studies of Goldstein et al. (1965) and Yunker and Van Orman (2007) in Fe90Ni10 and this study in Fe64Ni36.
Dth 1 dN Ni ¼ DC NNi NFe dT
ð9Þ
where Nx are the mole fractions of the appropriate species. Given the axial temperature gradient of 50 K mm 1, we find, for the hotspot profile, that Dth/DC ¼ 0.04, such that this is not expected to be significant. We can also approximate the Soret effect through comparison of diffusion along the radial and axial directions. The compositional profiles plotted in Fig. 4 are made up of points taken at identical depths into the sample; thus, we are able to use them in the radial direction to extract an effective radial diffusion coefficient. For example, if we consider the data at a fixed depth, the apparent lateral diffusivity is more than an order of magnitude less than the axial diffusion coefficients at identical distances. Therefore, the thermal diffusivity along steeper temperature gradients (radially vs. axially) is considerably less than the chemical diffusion. Thus, we can ignore it to that approximation, where we limit the Soret effect to o10% overestimate of our reported diffusivities. Diffusion in the Fe–Ni system occurs by a vacancy mechanism, and solutions for Fe-rich fcc Fe–Ni alloys are nearly ideal (Rammensee and Fraser, 1981). Additionally, Brown and Ashby (1980) have shown that a wide variety of hcp and fcc metals have similar diffusivities at the same homologous temperatures. Therefore, we expect the difference in Fe–Ni diffusivity and Ni or Fe self-diffusion to be small; indeed, the differences are indistinguishable within uncertainty (Fig. 5) consistent across compositions up to 40% Ni (Goldstein et al., 1965). While the alloy used in these experiments is in the fcc phase throughout the pressure
D.M. Reaman et al. / Earth and Planetary Science Letters 349-350 (2012) 8–14
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Fig. 7. (a, b) Depth profile and image of sample exhibiting grain-boundary diffusion.
high pressure will result in a non-linear relationship between the log of the diffusion coefficient and the pressure, which subsequently will overestimate the value of the interdiffusion coefficient. Conversely, assuming a large activation volume, such as that found in Goldstein et al. (1965), results in a predicted interdiffusion coefficient of 5.6 10 25 m2 s 1. Thus, the value of the activation volume has drastic effects on the predicted diffusivity and should be carefully considered when making any predictions.
5. Conclusions
Fig. 8. Experimentally determined pre exponential factor, D0, compared with that of Goldstein et al. (1965) and Yunker and Van Orman (2007) as a function of pressure.
range, we assume that the high-pressure close-packed fcc structure follows the same homologous temperature relationship for the hcp melting curve. The homologous temperature relation has been shown to hold for a wide range of metals (Brown and Ashby, 1980; Sammis et al., 1981;) and at pressures up to 23 GPa (Yunker and Van Orman, 2007). The results of this study confirm the accuracy of the homologous temperature relationship in Fe–Ni alloys to 65 GPa. Fig. 6 shows a plot of the diffusion coefficient as a function of pressure for the studies of Goldstein et al. (1965) and Yunker and Van Orman (2007) in Fe90Ni10 and that predicted by the homologous temperature relationship in Fe64Ni36 at 1900 K. Any curvature in Fig. 6 would indicate that the activation volume changes with pressure, which is not resolvable to 65 GPa. The homologous temperature relationship predicts diffusivities at ICB conditions in the range of 5.7 10 14–6.9 10 13 m2 s 1. Likewise, if the activation volume found in this study is assumed to remain constant at 2.62(3) cm3 mol 1 to inner core conditions, this predicts what is likely an extreme lower limit of 1 10– 14 m2 s 1, consistent with values predicted by the homologous temperature relationship. However, as VA likely decreases at large compressions, it will have a significant effect on the interdiffusion coefficient. Using Eq. (8) and assuming the hcp value of dTm/dP is 7.9 K GPa 1 from the melting curve at 330 GPa (Anderson and Isaak, 2000) and EA ¼301 kJ (this study), the 330 GPa activation volume is predicted to be 1 cm3 mol 1, reflecting a 50% reduction between 65 and 330 GPa. The predicted diffusivity for these parameters is 7.5 10 10 m2 s 1 assuming a T of 5700 K (Anderson and Isaak, 2000; Yunker and Van Orman, 2007) and a P of 330 GPa. While this is a reasonable way to estimate the activation and resulting diffusivity, the shallower melting curve at
Interdiffusion coefficients in the fcc FeNi system determined to 65 GPa via the use of FIB and EDX analysis extend the pressure range of existing diffusion data by approximately a factor of three. The diffusivities for Fe and Ni are indistinguishable within uncertainty and support the homologous temperature assumption in extrapolating diffusion data in Fe–Ni alloys to higher pressures. Though the studies of Goldstein et al. (1965) and Yunker and Van Orman (2007) concern the full range of Fe–Ni alloy compositions, the analyses of Yunker and Van Orman (2007) focused primarily on the Fe90Ni10 composition because it is likely close to the composition of Earth’s core; the diffusivities measured in this study on a Fe64Ni36 alloy assuming a homologous temperature relationship are indistinguishable from those in Fe90Ni10 alloys. The diffusion of Fe and Ni can be described by the homologous temperature relationship, D ¼D0 exp( ATm/RT), where D0 ¼4.6 10 4 m2 s 1 and the dimensionless parameter A/R¼19.3(2.7). Combining the data from this study with those of Yunker and Van Orman (2007) to 23 GPa predicts diffusivities in the range of 10 12–10 14 m2 s 1 at conditions of the inner core. Though the diffusivities are expected to be similar between hcp and fcc metals and alloys at the same homologous temperature, other factors such as pressure-dependent activation volume, altered diffusion pathways due to compression, and the effects of dissolved light elements may significantly alter the diffusive behavior of diffusion of iron and nickel in close-packed alloys. Further study of diffusion in hcp Fe to higher pressures is required to further constrain these values. Given a diffusivity of 10 14 m2 s 1 under inner core conditions, we can approximate the solid-state viscosity of the inner core adopting a power-law creep model (n ¼3) based on dislocation link climb (Reaman et al., 2011). This creep mechanism suggests significantly greater viscosities than when assuming Harper–Dorn creep (e.g. Van Orman, 2004) which is a consequence of transient deformation in low strain-rate measurements (Kumar and Kassner, 2009). Therefore, assuming a deviatoric stress of 103–104 Pa resulting from anisotropic growth (Yoshida et al., 1996) or response to mantle mass variations (Buffett, 1997),
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we predict an inner core viscosity of 1020–1022 Pa s (Reaman et al., 2011). Diffusivity can be measured at extreme conditions using micro-fabricated samples in the diamond-anvil cell to 1.1 log unit accuracy. These experiments then provide a basis for extending diffusion experiments into the pressure and temperature regime of the diamond-anvil cell.
Acknowledgments The authors would like to thank Daniel Huber and Jonathan Orsborn for their assistance with sample preparation and to Leonard Brillson and Philip Smith for assistance with NanoSIMS. The authors are indebted to Jim Van Orman and two anonymous reviewers for their thorough and helpful comments which vastly improved the flow and understanding of the paper. Funding was provided by NSF (EAR 0955647) and CDAC (4-3253-04). References Anderson, O.L., Isaak, D.G., 2000. Calculated melting curves for phases of iron. Am. Mineral. 85, 376–385. Benedetti, L.R., Loubeyre, P., 2004. Temperature gradients, wavelength-dependent emissivity, and accuracy of high and very-high temperatures measured in the laser-heated diamond cell. High Pressure Res. 24 (4), 423–445. Brown, A.M., Ashby, M.F., 1980. Correlations for diffusion constants. Acta Metall. 28, 1085–1101. Buffett, B.A., 1997. Geodynamic estimates of the viscosity of the Earth’s inner core. Nature 388, 571–573. Buffett, B.A., Wenk, H.R., 2001. Texturing of the Earth’s inner core by Maxwell stresses. Nature 413, 60–63. Crank, J., 1970. Mahematics of Diffusion. Clarendon Press, Oxford. Goldstein, J.J., Hanneman, R.E., Ogilvie, R.G., 1965. Diffusion in the Fe–Ni system at 1 atm and 40 kbar pressure. Trans. AIME 233, 812–820.
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