Diffusive fractionation of carbon isotopes in γ-Fe: Experiment, models and implications for early solar system processes

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Diffusive fractionation of carbon isotopes in c-Fe: Experiment, models and implications for early solar system processes Thomas Mueller a,b,⇑, E. Bruce Watson a, Dustin Trail a,c, Michael Wiedenbeck d, James Van Orman e, Erik H. Hauri f a

New York Center for Astrobiology, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180, USA b Institut fu¨r Geologie, Mineralogie und Geophysik, Ruhr-Universita¨t Bochum, D-44801 Bochum, Germany c Department of Earth and Environmental Sciences, University of Rochester, Rochester, NY 14627, USA d Helmholtz Zentrum Potsdam, GFZ German Research Centre for Geosciences, 14473 Potsdam, Germany e Dept. of Earth, Environmental and Planetary Sciences, Case Western University, Cleveland, USA f Carnegie Institution of Washington, Washington, DC 20015, USA

Received 27 March 2013; accepted in revised form 13 November 2013; available online 21 November 2013

Abstract Carbon is an abundant element of planets and meteorites whose isotopes provide unique insights into both organic and inorganic geochemical processes. The identities of carbonaceous phases and their textural and isotopic characters shed light on dynamical processes in modern Earth systems and the evolution of the early solar system. In meteorites and their parent bodies, reduced carbon is often associated with Fe–Ni alloys, so knowledge of the mechanisms that fractionate C isotopes in such phases is crucial for deciphering the isotopic record of planetary materials. Here we present the results of a diffusioncouple experiment in which cylinders of polycrystalline Fe containing 11,500 and 150 lg/g of C were juxtaposed at 1273 K and 1.5 GPa for a duration of 36 min. Diffusion profiles of total C concentration and 13C/12C were measured by secondary ion mass spectrometry (SIMS). The elemental diffusivity extracted from the data is 3.0  1011 m2 s1, where 13C/12C was observed to change significantly along the diffusion profile, reflecting a higher diffusivity of 12C relative to 13C. The maximum isotopic fractionation along the diffusion profile is 30–40&. The relative diffusivities (D) of the carbon isotopes can be related to their masses (M) by D13 C =D12 C ¼ ðM 12 C =M 13 C Þb ; the exponent b calculated from our data has a value of 0.225 ± 0.025. Similarly high b values for diffusion of other elements in metals have been taken as an indication of interstitial diffusion, so our results are consistent with C diffusion in Fe by an interstitial mechanism. The high b-value reported here means that significant fractionation of carbon isotopes in nature may arise via diffusion in Fe(–Ni) metal, which is an abundant component of planetary interiors and meteorites. Ó 2013 Elsevier Ltd. All rights reserved.

1. INTRODUCTION The ubiquity of carbon in natural systems and its numerous chemical forms (organics, carbonates, carbides, ⇑ Corresponding author. Present address: Institut fu¨r Geologie, Mineralogie und Geophysik, Ruhr-Universita¨t Bochum, Unversita¨tsstr. 150, Geba¨ude NA 03/586, D-44780 Bochum, Germany. Tel.: +49 234 322 3522. E-mail address: [email protected] (T. Mueller).

0016-7037/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.gca.2013.11.014

graphite, diamond, etc.) make this element a valuable source of information bearing on natural chemical processes occurring on different scales in space and time. Many studies have shed light on bio(geo)chemical processes that modify the isotopic composition of carbonaceous matter on modern Earth. Nonetheless, a detailed understanding of the origin and evolution of the isotopic fingerprint of carbon-rich compounds in the early solar system or its behavior during differentiation processes of the Early Earth is yet to be achieved.

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T. Mueller et al. / Geochimica et Cosmochimica Acta 127 (2014) 57–66

For instance, carbonaceous phases in chondrites can have different origins such as soluble or insoluble organics, carbonates or graphite with different degrees of crystallinity, all of them occurring as primary species in carbonaceous chondrites (CCs), and all of which have very different d13C-values ranging from 65& to +25& relative to PDB standard (Mostefaoui et al., 2000). Ordinary chondrites (OCs), which form the largest class of meteorites, experienced various degrees of thermal metamorphism accompanied by textural observations that can be related to increasing temperatures. The abundance of carbon in these chondrites and other meteorites such as primitive achondrites or iron meteorites is relatively low (0.2– 4 wt%; Kerridge, 1985). However, in addition to organics, carbonates, and primary elemental carbon, the occurrence of carbon is frequently associated with Fe–Ni metal; the carbon is either dissolved in the metal or found as isolated inclusions in metal grains (so-called MAC – metal-associated carbon after Mostefaoui et al. (2000)). High resolution ion microprobe analysis of the graphite inclusions in the Acapulco meteorite reveal a systematic difference between the measured d13C-values of the bulk carbon and the graphite inclusions which exhibit systematically lighter carbon isotope ratios and multiple graphite inclusions associated with the same metal grain show isotopic heterogeneity, which is commonly as large as several percent (El Goresy et al., 1995; Mostefaoui et al., 2000, 2005). These isotopic variations were preserved although the meteorites were subjected to periods of intensive heating up to 1200 °C (Zipfel et al., 1995). The lack of homogenization is interpreted as retention of the isotopic composition of graphite inclusions from the precursor material. However, the physical process of equilibrating the isotope composition of graphite inclusions within a metal grain requires substantial mass transport which in turn may be limited by diffusion of the isotopic species (which is equally temperature dependent). In addition, equilibrium partitioning of isotopes between phases involved in the exchange process (here graphite and Fe-metal) and their modal abundance have to be taken into account as shown in the theoretical models of diffusive exchange between co-existing phases for various boundary conditions (Lasaga, 1986; Eiler et al., 1994; Dohmen and Chakraborty, 2003). Deines and Wickman (1975) studied the isotope composition of co-existing graphite and cohenite/taenite in iron meteorites and found systematic lighter d13C-signatures in the metal phases of up to 15&. In summary, these data imply that a record of incomplete carbon isotope equilibration has been preserved, and furthermore, that carbon isotopes in these materials might be fractionated by diffusive transport during heating. As a result, scenarios describing the evolution of the early Earth might produce distinct signatures in the isotopic record perhaps to an extent that could affect the carbon isotope ratio of the bulk silicate Earth. Given the variety of terrestrial and planet-forming scenarios in which Fe-rich metal (solid or liquid) might interact diffusively with silicates or sulfides, significant potential exists for diffusive fractionation of carbon isotopes during these processes. Consequently, knowledge on the mass-dependent transport properties of carbon

isotope species in Fe–(Ni)-metal is crucial to evaluate the extent of isotope fractionation recorded on different scales. Various theories of diffusion suggest that the migration rate of a molecular or atomic species through a given medium should be affected by its mass—a prediction that logically extends to different isotopes of the same element (Richter et al., 2003). In gases at low pressure, the average translational speed of a molecule is inversely proportional to the square root of its mass, which leads to a simple equation relating the diffusivities of two atoms to the inverse of their mass ratio; i.e.,  12 DII MI ¼ ; ð1Þ DI M II where M and D represent the masses and diffusivities, respectively, of molecules I and II, which may differ only in their isotopic make-up [Note that Eq. (1) applies strictly to low-density gases in which collisions between molecules are infrequent; in denser gases, the mean molecular weight of the gas medium also enters in]. In metals, the diffusivities of two isotopes of a given solute element are expected to differ because of subtle differences in the vibrational characteristics of the bonds they form with their neighbors (Glicksmann, 2000, Sections 15.2 and 16.2). The physical principles are different from those governing diffusion in gases because the atoms are either bound in lattice sites or sit in interstitial sites and are thus constrained to discrete jumps from one lattice location to another. There is a substantial amount of literature on diffusive isotope fractionation in solids, in particular metals (e.g., Schoen, 1958; Le Claire, 1966; Rothman and Peterson, 1969) and the isotope effect on non-metal liquids such as methanol has also been studied (Weingartner et al., 1989; Holz et al., 1996). Richter et al. (1999, 2003) were the first to characterize the isotopic effect in a geologically relevant context for diffusion in molten silicates, and recent studies of Watkins et al. (2009, 2011) investigated the influence of melt composition and melt structures on the b-value. Melt structures are by far more complex than solid crystal lattices or gases. As a result, the deviation from the square root value is often described by a single parameter, where the experimentally observed mass dependence is expressed through the value of the exponent b on the mass ratio:  b DII MI ¼ : DI M II

ð2Þ

The summary by Richter et al. (2009) of existing data on the isotope effect in metals reveals that b ranges from essentially zero (no isotope effect) to 0.45, which approaches the theoretical limit of 1/2 for diffuse gases (Eq. (1)). Some of the highest b values (0.45) are for Li diffusion in various host metals (Si, W), but the uncertainties are large (±0.1). Despite the importance of C in steel manufacturing, and the utilization of isotope fractionations by materials scientists to characterize diffusion mechanisms, no data exist for C isotope diffusion in Fe. In this paper we present a laboratory quantification of the isotope mass dependence of

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59

carbon diffusion in c-Fe-metal. The experimental results are placed in a general context using the theoretical concept of diffusion which allows the extrapolation of the isotope effect to different carbon concentrations and isotope compositions. We are presenting a simplified model to illustrate the effect of mass-dependent carbon isotope fractionation through diffusion in a spherical object. The modeling results emphasize the potential of using the spatial variations in the carbon isotope signature within metal phases (e.g., OCs, primitive or iron meteorites) to decipher thermal histories and/or timescales of alteration processes. 2. MATERIALS AND EXPERIMENTAL/ANALYTICAL METHODS We sought to characterize the isotopic effect of C diffusion in c-Fe (austenite, FCC) because this is a suitable analog for meteorite and planetary core-forming materials over a range of conditions, and C is also soluble in c-Fe at the 1 wt% level. The Fe–C phase diagram has not been characterized at 1.5 GPa, but many determinations have been made at atmospheric pressure, and the locations of key phase boundaries at 5 GPa are also known (Lord et al., 2009; see Fig. 1). These data provide an adequate guide to the phase relations at the experimental conditions of 1.5 GPa. The diffusion couple was constructed by juxtaposing two polycrystalline Fe cylinders containing different initial concentrations of C. These were intended to have the same 13 C/12C ratio, but turned out to be slightly different due to the presence of impurity C in the starting materials. The high-carbon half of the couple was prefabricated by tumble-mixing Fe powder (Alfa Aesar 99.9%; lot No. J15H04; found to already contain 1500 lg/g carbon during reconnaissance experiments) with 1 wt% of high-purity graphite powder. An aliquot of the mixture was then loaded into a silica glass container and placed in the pressure cell of a solid-media, piston-cylinder apparatus (Fig. 2). The charge was contained in a silica glass capsule because—unlike the MgO filler pieces of the pressure cell—it is impervious to C (Hayden and Watson, 2008). The Fe–C mixture

Fig. 2. Experimental setup for pre-synthesis and carbon diffusion at elevated pressures and temperatures. Fe-metal powder was mixed with high purity graphite powder and the resulting Fe–C mixture was then pressurized and heated to 1.5 GPa and 1000 °C for 3 h in a silica tube. Silica tubes were found to be impervious to C at least at the detection limit of C by electron probe, so contamination of the sample from the graphite heating element was prevented. See Hayden and Watson (2008) for details. Diffusion couples were made by placing the pre-synthesized Fe-steel with 1 wt% carbon against a Fe-rod with 150 lg/g C. Diffusion couples were placed into a silica glass capsule to prevent carbon contamination from the heating furnace and annealed at 1.5 GPa and 1000 °C for 36 min. Experimental run products were cut and polished before analyzing profiles for total carbon and carbon isotope composition.

was pressurized to 1.5 GPa, heated at 100 °C/min to 1000 °C, held at these conditions for 3 h, and quenched by turning off the power to the heater. Cooling from 1000° to 100° occurred in 20 s, with a simultaneous decrease in pressure to 1 GPa. The C-doped Fe sample was recovered from the pressure assembly as a steel cylinder 2.5 mm dia.  7 mm long and with a slightly irregular shape. This cylinder was machined

Fig. 1. Fe–C phase diagram for 1 bar (left) and 5 GPa (right) after Lord et al. (2009). Austenite (c-iron) should be the only stable phase at the experimental conditions (1.5 GPa, 1000 °C) used in this study. Note the formation of ferrite (a-iron) as secondary quench phases during pressure and temperature release.

T. Mueller et al. / Geochimica et Cosmochimica Acta 127 (2014) 57–66 0.00974 0.00972 0.00970 0.00968

12

C/ C

(without lubricant) to a uniform 2-mm diameter, and cut into two shorter cylinders—one to be used as an analytical control, the other as the high-carbon half of the subsequent diffusion-couple experiment, containing 11,500 lg/g of C. Our initial intent had been to conduct the experiments at C levels as high as possible in order to minimize the analytical uncertainty in measuring the isotopes. The maximum solubility of C in c-Fe at both 1 atm and 5 GPa is 2 wt%, which occurs at 1150 and 1300 °C, respectively (Fig. 1). However, when our early experiments revealed that C levels in excess of 1% result in very heterogeneous distribution of C in the quenched run products (a-Fe (ferrite) + Fe3C), the idea of working at maximum C levels was abandoned. Limiting C content to a maximum of 1 wt% did not completely solve this problem, but the experimental products were more homogenous on the scale required for the analyses. The low-concentration half of the diffusion couple did not require prefabrication. This was machined directly from high-purity Fe rod (Aesar 99.95%; lot No. C09W037) to match the 2 mm diameter of the pre-synthesized high-concentration half. In exploratory experiments leading up to the successful diffusion couple, the Fe rod was discovered to contain 150 lg/g of impurity C. Although unanticipated, this “contamination” turned out to be acceptable because the 13C/12C ratio is sufficiently similar to that in the high-concentration half. The diffusion couple was assembled by placing the 11,500 and 150 lg/g C halves end-to-end in a silica glass container (Fig. 2). The experiment reported here was conducted in the same solid-media apparatus as the synthesis experiments, and was run at 1000 °C and 1.5 GPa, for 36 min, using the same ramp-up and quench procedures. After depressurization and removal of adhering silica glass, the diffusion couple was press-fit into machined slots in a 2.5-cm dia. fully dense aluminum disk. It was then ground to near-axial sections with SiC paper and polished with 1lm alumina. Total carbon diffusion profiles were obtained by electron microprobe to confirm expected behavior based on known diffusion laws for C in c-Fe. Detailed analytical traverses for 13C and 12C were conducted with the Cameca ims 6f ion microprobe at the Helmholtz Zentrum Potsdam (GFZ), Germany. Prior to the SIMS analysis, the sample was cleaned ultrasonically with distilled water and ethanol and coated with a 35 nm thick layer of high-purity gold. A Cs+ primary beam with nominally 10 kV and 1 nA was used in raster mode (to minimize local heterogeneities due to quenching) over an area of 50  50 lm on the sample surface. Each analysis consisted of a 3 min pre-sputter (to remove the gold coating and establish stable sputtering conditions) and time series of 50 scans for 12C (1 s), 13C (10 s) and 56Fe (1 s), resulting in a total time of 24 min per analysis. The mass resolution was set to M/DM  3800 and typical count rates of 200 kHz for 12C, 2 kHz for 13C and 0.8 kHz for 56Fe were measured for steel with 1 wt% carbon using the described conditions. The standard deviation for a single analysis (1r) was typically < 2& for high carbon concentration (1 wt%) and increased to 5–6& for low carbon concentrations (0.15 wt%).

0.00966

13

60

0.00964 0.00962 0.00960 0.00958

0

50

100

150 12

13

200

250

300

350

400

56

( C+ C)/ Fe [cps] Fig. 3. Measured carbon isotope ratios of three certified eurostandards for Fe metals with different carbon concentrations from the Bundesanstalt fu¨r Materialforschung (BAM), Germany.

To the best of our knowledge there is no standard reference material (RM) for carbon isotope composition in Fe-metal. In order to test the quality of the analytical procedure and the extent of instrumental mass-fractionation, we used RMs from the Bundesanstalt fu¨r Materialforschung (BAM), Germany. These euro-standards are certified for their bulk carbon content (B090-1 = 1 wt%, B0551 = 0.5 wt%, B061-1 = 0.2 wt%). Unfortunately, only bulk analysis with large sample amounts had been used for the certification, which does not take into account spatial variations by local heterogeneities. However, multiple analysis as well as isotope maps of the standards revealed a heterogeneous distribution of C similar to the material for the diffusion couple. Hence, the internal reproducibility of multiple analyses on a single reference was only 8& (1r). A series of 3–4 measurements on each RM were averaged, resulting in a reproducibility of 3& (1r) (Fig. 3), which is our best estimate of the analytical uncertainty of our method when not measuring very low C content materials. Accordingly, we combined 3–6 analyses (perpendicular to the diffusion direction) on the unknown sample to represent a single point in the profile shown in Fig. 4 (see Supplementary material for more detailed information). In addition, multiple analysis of the carbon isotope composition of each RM were measured at the University of Mu¨nster using a conventional elemental analyzer coupled to a Finnigan Mat DeltaPlus mass spectrometer for comparison and to determine the instrumental mass-fractionation. Results are listed in Table 1. These data reveal no effect of carbon concentration on the measured isotope composition within the limits of the analytical uncertainty. 3. EXPERIMENTAL RESULTS The result of the ion microprobe analyses of the diffusion experiment is shown in Fig. 4 and summarized in Table 2 (see also Supplementary material for details of each analyses). It includes the diffusion profile for total carbon (i.e., the sum of both isotopes; Fig. 4A) as well as the measured isotope ratio along the diffusion traverse (Fig. 4B and C). The total carbon profile was fit to the standard solution

Total carbon [spc]

T. Mueller et al. / Geochimica et Cosmochimica Acta 127 (2014) 57–66

2x10

5

2x10

5

1x10

5

5x10

4

11-FeC-09 1000°C, 1.5 GPa, 36min

C

-7

2

D =3.0x10 cm /s

A 12C

13C/12C

0.0098

-7

2

D = 3x10 cm /s β = 0.225

0.0096

β=0.2

0.0094

0.0092

β=0.25

B low C site 13

13C/12C

0.0098

12

C/ C + 10‰

0.0096 low C site 13

0.0094

0.0092

13

12

C/ C - 10‰

12

C/ C boundary conditions as measured

C -1500

-1000

-500

0

500

1000

1500

Distance [µm] Fig. 4. (A) Concentration profile for total (12C + 13C) carbon show the expected shape of typical diffusion profile without significant concentration dependence. The best fit using Eq. (5) yield a nominal diffusivity of carbon in c-Fe of 3.0  1011 m2 s1 at 1000 °C and 1.5 GPa, which agrees well with the 1-atmosphere values of Agren (1986) and Agarwala et al. (1970). (B) Simulated profiles of the expected carbon isotope fractionation along the advancing diffusion front. The extracted diffusion coefficient from the total carbon diffusion profile was assigned to 12C and the diffusion coefficient of 13C was adjusted corresponding to different b-values. The best fit was obtained for b = 0.225. Note that changes for b > 0.025 result in fractionation profiles that do not fit the maximum measured fraction within its error limits. (C) Variations of model fits with changing initial carbon isotope composition at the low carbon concentration side. The modeled fraction only changes within the limits of the analytical uncertainty for variations of ±10&.

of the non-steady state diffusion equation for an infinite diffusion couple (Crank, 1975):     C 1  C þ1 x 1  erf pffiffiffiffiffi ð3Þ Cðx; tÞ ¼ C þ1 þ 2 2 Dt where C(x,t) is carbon concentration at distance x and time t (original interface at x = 0), D is the diffusivity, and the

61

subscripts + and  refer to the concentrations at the ends of the couple (at x = + and x = ), which do not change with time. Implementation of Eq. (3) implicitly assumes that D is independent of carbon concentration. This may not be strictly the case (Ernst et al., 2009), but our data conform reasonably well to Eq. (3). Use of a concentrationdependent D thus seems unwarranted, at least on the scale of the overall profile for total carbon. The diffusivity was determined by fitting the experimental data to Eq. (3) with D as a fit parameter, using the iterative Levenberg–Marquardt algorithm. The smooth curve in Fig. 4A show the bestfit to the experimental profile. Our total carbon profiles yield a nominal diffusivity of C in c-Fe of 3.0  1011 m2 s1 at 1000 °C and 1.5 GPa, which agrees fairly well with the 1-atmosphere values of 2.2  1011 m2 s1 of Agren (1986) and 1.3  1011 m2 s1 reported by Agarwala et al. (1970; this value is for 304 stainless steel). The 13C/12C profile (Fig. 4B and C) shows a well-defined “dip” centered at x  700 lm having a “depth” of 30&. The variation in the carbon isotope ratio is about one order of magnitude larger than the analytical uncertainty and is thus the unequivocal signature of an isotope effect on diffusion (Richter et al., 1999). The observed feature develops because the heavier isotope (13C) diffuses more slowly than 12C. A 13C “deficit” thus becomes apparent at the leading edge of the diffusion profile. The isotope ratio profile was modeled essentially by trial and error: the nominal diffusivity (3.0  1011 m2 s1) was assigned to the abundant isotope (12C) and trial values for 13C values were explored by specifying b (see Eq. (2)). As shown in Fig. 4B, the 13C/12C profiles are in good agreement with a model based on b = 0.225 ± 0.025. The ratio for the diffusivities of the two isotopes is therefore D13C/D12C  0.9822 resulting in a nominal diffusivity for 13C of 2.95  1011 m2 s1. Calculating b requires the following input parameters: (i) the initial 13C/12C-ratio of each half of the diffusion couple; (ii) the diffusivity of 12C; and (iii) the final 13C/12C-ratio along the diffusion profile. The diffusivity of total carbon that is assigned to the dominating 12C species can be extracted with little uncertainty (Fig. 4A). The initial isotope ratios are constrained with less accuracy due to the relatively large analytical scatter on the carbon-rich half (±2.6& accuracy, 1r), and the large uncertainty due to poor counting rates on the low carbon end (±10&, precision 1r), respectively (see also Supplementary material). The presence of some local C heterogeneity in the high-C half is apparent, because the scatter in the profile is substantially larger than the analytical uncertainty. This is believed to be a consequence of the c ! a transformation and concomitant Fe3C precipitation that occurs upon quenching (see Section 2.1 and Shewmon, 1989, ch. 6). Back-scattered electron imaging reveals locally patchy variation in mean Z, suggesting a heterogeneous distribution of carbon atoms on a scale of 10–50 lm (Fig. 5). This length scale is much smaller than that of overall diffusive transport during the experiment (2 mm), so it seems clear that the heterogeneity developed during quench. The effect of such heterogeneity was aimed to be suppressed by rastering the primary beam over the sample and then subsequently averaging multiple analyses (n = 3 for the profile, n = 6–9 for the boundaries)

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Table 1 Isotope analysis of certified BAMS standards. Standard

B061-1 (0.2 wt%) B055-1 (0.5 wt%) B090-1 (1 wt%)

Element analyzer (Univ Muenster)

D13C instrumental mass fractionation [&]

SIMS (GFZ Potsdam)

# of analysis

dI3C vs PDB [&]

%C

# of analysis

Measured I3 C/I2C ratio

dI3C vs PDB [&]

Carbon signal [cps]

4 4 3

26.56 23.69 26.32

0.19 0.51 1.09

4 4 3

0.00967 0.00965 0.00969

139 141 138

90 156 304

113 118 111

Table 2 Summary of SIMS analyses.

Low carbon concentration

High carbon concentration

Distance from interface [lm]

# of averaged analyses

13

r

C/Fe [cps]

C total [cps]

1500 1000 700 500 300 150 150 300 700 1100 1500

9 3 3 3 3 3 3 3 3 3 3

0.00979 0.00950 0.009247 0.009321 0.00937 0.009509 0.00957 0.009575 0.009646 0.009556 0.009528

9.36E05 6.88E05 4.08E05 3.17E05 2.17E05 1.82E05 1.23E05 1.28E05 1.07E05 1.19E05 9.37E06

3 8 22 42 78 82 107 135 170 173 182

1070 3631 8890 19,247 39,162 50,738 104,805 118,294 188,806 174,091 202,961

Fig. 5. BSE image showing the patchy texture of cohenite as exsolution phases during quenching of the experiment. Three raster craters are visible with mixtures of c-iron (bright phase) and exsolution of cohenite (dark phase). Variations in the 13C/12C isotopic signature are modeled to not exceed a few & (Fig. 8). To avoid a large scatter to due mixed analyses, a raster beam of 50  50 lm has been used during SIMS data acquisition.

to produce a single point in the profile (Fig. 4; see also Supplementary information). Determination of the initial carbon isotope composition in the low carbon half of the diffusion couple was also evaluated using this model (Fig. 4C). This is because the measured carbon signal was too low to obtain an averaged analysis with a precision better than ±10&. For this reason, we modeled the predicted fractionation profiles with isotope ratios for 13C/12C varying by ±10& for the initial boundary conditions in the

C/12C

low carbon half of the diffusion couple (Fig. 4C). The resulting maximum fractionation is only affected by 6&, which is smaller than the analytical uncertainty. The best-fit for b using the parameters described above is 0.225 (Fig. 4B). Note that a change in b produces a corresponding change in the amplitude of fractionation; for example, a b value of 0.2 yields a fractionation amplitude of about half that resulting from b = 0.4. Upper and lower bounds for the value of b (±0.025) were determined using the analytical uncertainty of the average of multiple spots resulting in the single point showing the maximum fractionation (Fig. 4). Variation of b between 0.2 and 0.25 causes an overall change of the predicted maximum isotope fractionation of 10&. The position of the minimum on the isotope interdiffusion profile is insensitive to changes of the fractionation factor within the chosen limits. Taken together, these observations substantiate our estimate of b = 0.225 ± 0.025 as a robust number within the limits of the experimental and analytical uncertainties. 4. DISCUSSION 4.1. Origin of observed isotope fractionation The presence of heterogeneous carbon distribution in the experiment raises the question whether the observed variation in the isotope composition is due to mass-dependent diffusion along the concentration gradient. If this is the case, then it is necessary to evaluate to what extent the formation of the quench phases can affect the measured isotope signature. The possibility of diffusive C-isotope fractionation during exsolution of carbide in iron meteorite parent bodies is discussed in quantitative terms at the end of Section 5, but

T. Mueller et al. / Geochimica et Cosmochimica Acta 127 (2014) 57–66 1e cm /s

0 1e

5e

cm /s

-4

/s 1e cm

D( C) = 1e cm /s (800 °C) β = 0.225 K = 7.41

5e

cohenite Fe C

cm

Δ13C [‰]

-2

cm /s

/s

-6 1e cm /s

-8 0.0

γ-iron

0.1

0.2

0.3

Distance from cohenite core [cm] Fig. 6. Kinetically controlled carbon isotope fractionation by growth of cohenite as exsolution phase from c-iron during quench using the model of Watson and Mueller (2009). The diffusion coefficient for 12C was calculated from Agren (1986) and different growth rates (see labels on the profiles) applied. Observed fractionation of 6–8& can be expected for either large crystals growing at a similar rate compared to the diffusivity or by growth rates being significantly faster as it can be expected for quench products.

the same phenomenon could occur over much shorter time and length scales in our experiments. In a recent study, Watson and Mueller (2009) demonstrated that rapid growth of a phase can result in fractionation of the isotopes of any element that is strongly partitioned between the precipitating phase and the growth medium. We evaluated the C isotopic composition of a spherical cohenite precipitate in iron metal using the conceptual model of Watson and Mueller (2009), assuming growth of a spherical crystal from an infinite, initially homogeneous reservoir of c-Fe at 800 °C (Fig. 6). Applied to the growth of a quench phase in our experiments, results of the numerical modeling approach of Watson and Mueller (2009) suggests that kinetically controlled fractionations exceeding 5& could be achieved either by growing a large Fe3C particle (>1 mm) at a rate comparable with the diffusion rate of C in the host Fe phase, or by growth of smaller Fe3C particles at rates significantly faster than C diffusion in the host phase. In our experiment, 20 lm Fe3C particles formed during the 30 s. required to cool our experiment to <100 °C, indicating high growth rates. Our modeling results reveal that rapidly-grown carbide crystals smaller than 50 lm acquire a 13C/12C value which is ca. <10& lower than that from the surrounding diffusive boundary layer (Fig. 6), which is in agreement with the scatter in our measurements. Thus, aside from making the profiles “noisier” at high carbon concentrations, the exsolution of Fe3C during quench does not seriously affect the observed isotope effect. Hence, we interpret the observed spatial variation in the carbon isotope ratio to be uniquely the consequence of the mass-dependent difference in the diffusivity of both carbon species. 4.2. Theoretical constraints on b using the diffusion mechanism Despite the extensive metallurgical literature on the isotope mass effect for diffusion in metals, to our knowledge

63

no data exist for C in Fe. Carbon is believed to diffuse by an interstitial mechanism in a- and c-Fe based on the large difference in ionic radius between carbon and iron atoms as well as the fact that carbon diffuses much faster compared to Fe self-diffusion rates (Shewmon, 1989). Diffusion by an interstitial mechanism is expected to yield a b-value approaching 0.5. Our experiment, in contrast, yields a b value of 0.225 ± 0.025, suggesting a significant deviation from the square root value. The so-called isotope mass effect (E) for self- or impurity diffusion of tracer elements in solids was derived in seminal studies by Mullen (1961) and Le Claire (1966). These authors have shown that mass-dependent difference in the diffusivity and thus the isotope fractionation can be related to two fundamental parameters f and DK by: DI 1 DII E ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ f  DK mII =mI  1

ð4Þ

The symbol f refers to the Bardeen-Herring correlation factor (<1), that corrects for non-random aspects of atomic jumps that arise, for example, in vacancy-controlled diffusion. For diffusion of interstitial solutes f is generally assumed to be close to unity (e.g. Le Claire, 1966). The factor DK accounts for a kinetic energy term that is associated with the diffusive jump. Conceptually, it describes the vibrational modes of an atom when it passes through the saddle point during displacement. It can be also visualized as the kinetic energy related to the degree of coupling between the motion of a moving atom and the surrounding atoms. To that end, Le Claire (1966) suggested a relation of DK with the deformation of the crystal lattice that accompanies the diffusive displacement and it is therefore related to the activation volume (i.e. a measure of displacement of neighboring atoms during the diffusive jump) and/ or the dilation/relaxation around an impurity for solute diffusion. In any case, the product f * DK describes the deviation of the isotope effect from a square root of mass dependence; both f and DK will be positive numbers 6 1 and so will be their product. Characterization of the experimentally determined isotope effect (DII/DI) can, in principle, discriminate between diffusion mechanisms if the extracted value for the product of f * DK falls within a narrow range that corresponds to a specific diffusion mechanism. For example, in simple interstitial diffusion the correlation factor f becomes unity (in contrast to vacancy-controlled diffusion), and thus all deviations must be accounted for by distortion of the lattice (which, however, is generally supposed to be close to one for interstitial diffusion). On the other hand, if vacancies are involved, then the measured deviation will always be the product of f * DK. Numerous measurements of the diffusivity ratio and its dependence on temperature were made in the 1960s and 70s as a means of gaining insight into diffusion mechanisms in metals (see summary in Richter et al., 2009). Our experimental results suggest a value of f * DK = 0.445 for carbon diffusion on c-Fe-metal (Eq. (4)) which is far away from unity. The apparent deviation may be due to carbon diffusion by a mixed interstitialvacancy controlled mechanism through the Fe-lattice. To

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this end, it is important to keep in mind that every crystal lattice – no matter if metal, metalloid or mineral phase – contains a temperature dependent (i.e., intrinsic) concentration of vacant lattice sites as the increase in entropy exceeds the change of enthalpy and thus lowers the Gibbs free energy of the phase. Hence, there will always be some interaction of the solute atom with vacant sites in the lattice. It is possible to calculate the equilibrium distribution of carbon on the interstitial and Fe-vacancy sites, which is related to the concentration using an Arrhenius relation of the form KD ¼

C interstitial b 0 ¼ ae½DE DEv =kT C Fevacancy

ð5Þ

In this equation, k is the Boltzmann constant and the factor a accounts for certain entropy terms; however, this is on the order of unity and can be ignored for our purpose (Borg and Dienes, 1988, ch. 4). We calculated the distribution coefficient (KD) using published data for the enthalpy of vacancy formation [DE0v from McLellan (1988)] and the binding energy of a carbon atom to a Fe-vacancy [DEb – using data for ferrite (McLellan, 1988)]. This calculation points to several tens of thousands atoms sitting on interstitials for each atom located on a vacant site at 1000 °C, which suggests that the diffusion mechanism is overwhelmingly interstitial controlled. This, in turn, indicates that the correlation factor should be very close to unity and thus the entire isotope effect has to be attributed to DK in the present case. In general it is believed that interstitial diffusion is characterized by values for f and DK to be close to one. In this light, our derived small value (DK  0.445) is very intriguing. On the one hand, this kinetic energy term may be useful to obtain information on the activation volume as DK is related to the relaxation of the crystal lattice around impurities on the interstitial site. On the other hand, Le Claire (1966) suggested that quantum corrections are necessary to fully evaluate the isotope effect for light atoms. Quantum effects have a very important effect on diffusion of H or D in metals at low temperatures, but it is not clear whether quantum effects are also important for carbon atoms (which are heavier) at the temperatures of our experiments. It is beyond the scope of this contribution to investigate the nature of DK in more detail. However, more experiments and modeling effort could potentially use DK in order to decipher more detailed, quantitative information on the diffusion mechanism and thus being able to predict the isotope effect for different systems in the future. 5. IMPLICATIONS FOR FRACTIONATION OF CARBON ISOTOPES IN METEORITES AND DURING CORE-FORMING PROCESSES As discussed in the previous section, knowledge of b may yield clues about the diffusion mechanism, but it is also useful for estimating the extent of isotopic fractionation in natural systems. Analytical solutions for diffusion in spheres can assess both the fractionation of carbon isotopes in meteorites or other bodies formed by accretion as well as in objects that have been subsequently exposed to a period of high temperature (such as metal-magma interaction or any metamorphic overprint) sufficient to mobilize C. Such

simple kinetic calculations can also be used to extract timescales for the formation or the maximum time an object of a given size can be exposed to a certain temperature revealing the measured isotopic heterogeneity. Analytical solutions for diffusion in various geometries are well known (e.g., Crank, 1975). The extent of diffusive equilibration of an element or isotope in a spherical body (from core to rim) is best captured by the parameter D * t/r2, which can be thought of as dimensionless parameter related to time. This parameter provides an indication of the effectiveness of transport processes in terms of the effective diffusion coefficient (D) of the species of interest (which is assumed to be constant over the duration (t) of the diffusion event) and the radius r of the body under consideration. Small values of Dt/r2 (e.g., <0.01) result in partial equilibration of the outer part of the sphere; the effects of diffusion are first felt at the center once Dt/r2 P 0.03, and full equilibration can be considered for values >0.5 (Crank, 1975; Fig. 6.1). The slight difference in the mass-dependent diffusivity of the different carbon isotopes results in increasing isotope fractionation. Once both isotopes reach the center, the observed isotopic fractionation decreases again as the slower isotope (13C) “catches up” during the final stage of homogenizing the sphere. Thus, a maximum kineticallycontrolled isotope fractionation, caused by diffusive transport, can be estimated as shown in Fig. 7. Inspection of this figure reveals that the position of maximum isotope fractionation is located at a value of Dt/a2 of about 0.07, being a consequence of the delay of 13C reaching the center relative to the faster 12C species. The maximum isotopic fractionation that can be induced by diffusive transport in a sphere is estimated to be around 27&. The same set of simple calculations can be used to estimate the timescales necessary for either partial or total equilibration. For example, if C is diffusing into a 1 mm Fe-metal grain at 1000 °C, the grains will show the highest isotope “anomaly” after about 10 min, and will be completely homogenized after 70 min. Diffusion is not very efficient at large length scales, so for objects such as the cores of small planetesimals (30 km in diameter), timescales of equilibration can exceed the age of the Earth (in the absence advective transport). Diffusive fractionation of C isotopes might also occur on a small scale as a consequence of phase separation during the cooling of an iron meteorite parent body; the exsolution for Fe3C during rapid quench in our laboratory experiments provide evidence for this possibility in natural systems. The classical separation of kamacite from taenite is one possibility, but in this case the lamellar growth of kamacite may be slow enough (relative to the rapid diffusion of C) to preclude any record of non-equilibrium fractionation of C isotopes. Growth of cohenite (Fe,Ni,Co)3C may be another matter, as C diffusion in carbides is expected to be slow (Agren, 1986). The differing diffusivities of 12C and 13C in the c-Fe matrix as they diffuse toward the interface of the growing cohenite induces changes in 13C/12C value near the interface relative to the initial matrix. Such isotopic shifts would be recorded in the growing carbide phase as a consequence of local equilibrium being maintained at the interface (see Watson

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meteorites by measuring carbon isotope ratios in metal grains and graphite inclusions combined with numerical modeling tools. ACKNOWLEDGEMENTS This work was supported by the NASA Astrobiology Institute under Grant No. NNA09DA80A to Rensselaer Polytechnic Institute. We thank Harald Strauss from University of Muenster for the conventional carbon isotope measurements of the reference materials. Constructive reviews from Dominik Hezel, James Watkins and an anonymous reviewer are greatly acknowledged and helped significantly to improve the manuscript.

APPENDIX A. SUPPLEMENTARY DATA Fig. 7. Expected amount of carbon isotope fraction for isothermal diffusion in a sphere. Note, that a maximum fractionation of 30& is obtained for any combination of time and size of a spherical body that leads to a value of 0.08 for the dimensionless parameter Dt/r2 (for a given diffusion coefficient of total carbon diffusion at a given temperature).

Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/ j.gca.2013.11.014.

and Mueller, 2009 for details of the model). A graphical presentation of the model and the resulting radial isotope profiles in the cohenite precipitate and the host Fe metal for different growth rates are shown in Fig. 6.

Agarwala R., Naik M., Anand M. and Paul A. (1970) Diffusion of carbon in stainless steels. J. Nucl. Mater. 36, 41–47. Agren J. (1986) A revised expression for the diffusivity of carbon in binary Fe–C austenite. Scr. Metall. 20, 1507–1510. Borg R. J. and Dienes G. J. (1988) An Introduction to Solid State Diffusion. Academic Press, San Diego. Crank J. (1975) The Mathematics of Diffusion. Oxford University Press, New York. Deines P. and Wickman F. E. (1975) A contribution to the stable carbon isotope geochemistry of iron meteorites. Geochimica Et Cosmochimica Acta 39, 547–557. Dohmen R. and Chakraborty S. (2003) Mechanism and kinetics of element and isotopic exchange mediated by a fluid phase. Am. Miner. 88, 1251–1270. Eiler J. M., Baumgartner L. P. and Valley J. W. (1994) Fast grainboundary – a FORTRAN-77 program for calculating the effects of retrograde interdiffusion of stable isotopes. Comput. Geosci. 20, 1415–1434. El Goresy A., Zinner E. and Martl K. (1995) Survival of isotopically heterogeneous graphite in a differentiated meteorite. Nature 373, 496–499. Ernst O. G., Powell C. E., Silvester D. J. and Ullmann E. (2009) Efficient solvers for a linear stochastic Galerkin mixed formulation of diffusion problems with random data. SIAM J. Sci. Comput. 31, 1424–1447. Glicksmann M. E. (2000) Diffusion in Solids: Field Theory, Solid State Principles and Applications. Wiley, New York. Hayden L. A. and Watson E. B. (2008) Grain boundary mobility of carbon in Earth’s mantle: A possible carbon flux from the core. Proc. Nat. Acad. Sci. 105, 8537–8541. Holz M., Mao X. -A., Seiferling D. and Sacco A. (1996) Experimental study of dynamic isotope effects in molecular liquids: Detection of translation-rotation coupling. Journal of Chemical Physics 104, 669–679. Kerridge J. F. (1985) Carbon, hydrogen and nitrogen in carbonaceous chondrites – abundances and isotopic compositions in bulk samples. Geochim. Cosmochim. Acta 49, 1707–1714. Lasaga A. C. (1986) Metamorphic reaction rate laws and development of isograds. Mineral. Mag. 50, 359–373. Le Claire A. (1966) Some comments on the mass effect in diffusion. Phil. Mag. 14, 1271–1284.

6. CONCLUDING REMARKS AND FUTURE PROSPECTS The isotope mass effect on C diffusion through polycrystalline c-Fe was evaluated directly in a high P–T diffusioncouple experiment that was analyzed by ion microprobe. Robust diffusion parameters for total carbon diffusion in c-Fe (3.0  1011 m2 s1) and the fractionation factor b (0.225 ± 0.025) were obtained at 1.5 GPa and 1000 °C, despite small-scale exsolution of carbide during the quench. The b-value lies well within the range of observed b-values for other isotopic systems that undergo mass-dependent isotopic fractionation in metals. Diffusive isotope fractionation may therefore be responsible for some of the carbon isotopic variations measured in meteorites; the processes of diffusion may have even influenced the carbon isotope distribution of the bulk Earth during its formative stages. Careful inspection of Fig. 8 in the review of Richter et al. (2009) reveals that there are significant variations in b values determined for various isotope systems at the same temperature. These authors concluded that “the fractionation factor is not obviously dependent on the temperature, the specific diffusing element, or the diffusing medium”. However, in the light of the arguments presented in this study, it is possible that the fractionation factor b may be a function of both the diffusion mechanism and, more subtly, on temperature as well. That said, the predicted effect on b would be a useful parameter to be quantified in future experiments in which different temperatures and different starting isotopic compositions are used. This study highlights the potential of unraveling thermal histories and timescales of heating and cooling events in

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