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Meteoritics & Planetary Science 42, Nr 9, 1597–1613 (2007) Abstract available online at http://meteoritics.org

Diffusion-driven kinetic isotope effect of Fe and Ni during formation of the Widmanstätten

Nicolas DAUPHAS

Origins Laboratory, Department of the Geophysical Sciences and Enrico Fermi Institute, The University of Chicago, 5734 South Ellis Avenue, Chicago, Illinois 60637, USA E-mail: [email protected] (Received 10 October 2006; revision accepted 02 March 2007)

Abstract–Iron meteorites show resolvable Fe and Ni isotopic fractionation between taenite and kamacite. For Toluca (IAB), the isotopic fractionations between the two phases are around +0.1‰/amu for Fe and −0.4‰/amu for Ni. These variations may be due to i) equilibrium fractionation, ii) differences in the diffusivities of the different isotopes, or iii) a combination of both processes. A computer algorithm was developed in order to follow the growth of kamacite out of taenite during the formation of the Widmanstätten pattern as well as calculate the fractionation of Fe and Ni isotopes for a set of cooling rates ranging from 25 to 500 °C/Myr. Using a relative difference in diffusion coefficients of adjacent isotopes of 4‰/amu for Fe and Ni (β = 0.25), the observations made in Toluca can be reproduced for a cooling rate of 50 °C/Myr. This value agrees with earlier cooling rate estimates based on Ni concentration profiles. This supports the idea that the fractionation measured for Fe and Ni in iron meteorites is driven by differences in diffusivities of isotopes. It also supports the validity of the value of 0.25 adopted for β for diffusion of Fe and Ni in Fe-Ni alloy in the range of 400–700 °C.

INTRODUCTION Surprisingly, little work has been done on modeling isotopic fractionation associated with this phenomenon (Jambon Diffusion is a key process in earth sciences because it can 1980). Frank (1950) and Berner (1968) showed that during provide constraints on geological time scales (e.g., what the growth of planar and spherical concretions in a semi-infinite closure temperature of a mineral to a specific medium, the system reaches a self-similar asymptotic regime radiochronometer is, how fast a cooled after whereby the interface moves as the square root of time and the solidification from a melt). In geological systems, diffusion is rate of growth depends on the degree of supersaturation at most often revealed by concentration gradients, which reflect infinity relative to the surface. Dauphas and Rouxel (2006) differences in chemical potentials and nonequilibrium extended this work to isotopes and showed that when the conditions. Diffusion may also leave its imprint on stable degree of supersaturation at infinity is high, the grows isotope ratios. Indeed, isotopes, which have identical rapidly, and isotopic fractionation due to diffusion is chemical properties but different masses, diffuse at different maximal, but any equilibrium fractionation at the interface is speeds in (light isotopes diffuse faster than heavy ones) expressed in the surrounding medium. On the contrary, when (e.g., Chapman and Cowling 1952; Mullen 1961; Coleman the degree of supersaturation is low, the phase grows slowly, et al. 1968; Heumann and Imm 1968; Graham 1969; Walter there is no fractionation due to diffusion, but any equilibrium and Peterson 1969; Fishman et al. 1970; Oviedo de González fractionation at the interface is expressed in the growing and Walsöe de Reca 1971; Irmer and Feller-Kniepmeier 1972; phase. Iijima et al. 1988; Richter et al. 1999, 2003; Rodushkin et al. of undifferentiated precursors and subsequent 2004; Lundstrom et al. 2005; Beck et al. 2006; Roskosz et al. segregation of from silicate is responsible for the 2006; Teng et al. 2006). formation of iron meteorites. Some of these meteorites have Phase growth in a diffusion-limited regime is a peculiar textures known as the Widmanstätten pattern. At high ubiquitous process in earth sciences. It can occur during rapid temperature, the stable Fe-Ni phase is taenite (γ, face-centered cooling and from a melt, during concretion cubic). When meteorite parent bodies cool through the growth in aqueous systems, and during subsolidus reactions. temperature range of 500–800 °C, they enter the two phase

1597 © The Meteoritical Society, 2007. Printed in USA. 1598 N. Dauphas

Fig. 1. Iron and nickel isotope measurements in taenite and kamacite of IAB iron meteorites. The left panel shows Fe isotopic composition in ‰/amu deviation relative to IRMM-014 (Poitrasson et al. 2005; Cook et al., Forthcoming; Horn et al. 2006a). All studies agree that there is little or no significant Fe isotopic fraction between taenite and kamacite (<0.1‰/amu). The results of Ko¢sler et al. (2005) were not included because analytical artifacts may be present (Horn et al. 2006b; however, see Ko¢sler et al. 2006). The right panel shows Ni isotopic composition in ‰/amu relative to SRM-986 (Cook et al. 2006, Forthcoming). The Ni isotopic fractionation between kamacite and taenite is ~0.4 ‰/amu. The vertical, heavy lines are the average Fe and Ni isotopic compositions of all iron meteorites measured so far. field of kamacite (α, body-centered cubic) + taenite. In most Recently, several groups have reported Fe and Ni isotope instances, the growth of kamacite out of taenite is diffusion- variations in adjacent taenite and kamacite in some iron limited, which results in Fe-Ni concentration gradients in the meteorites (Fig. 1) (Poitrasson et al. 2005; Horn et al. 2006a; two phases that can be used to retrieve cooling rates of iron Cook et al. 2006, Forthcoming). It is not clear at present what meteorite parent bodies (Wood 1964; Goldstein and Ogilvie the mechanism responsible for this fractionation is. It could 1965). Knowing the cooling rate of iron meteorites is crucial be due to i) equilibrium fractionation at the interface between because it can help restrain the nature of the heat source the two phases, ii) differences in the diffusion coefficients of responsible for differentiation of planetesimals (decay of the different isotopes, or iii) a combination of both processes. short-lived nuclides 26Al and 60Fe, collisions between Whatever the mechanism is, it should depend on and planetesimals, electromagnetic induction) as well as the size therefore record cooling rates. Indeed, if equilibrium of asteroids present in the inner part of the early solar system fractionation is involved, diffusion will control how the signal (2–100 km) (Chabot and Haack 2006). Following the seminal propagates on each side of the interface and that will depend studies of Wood (1964) and Goldstein and Ogilvie (1965), a on how fast the meteorite cooled. Similarly, if differences in great deal of work has been done on modeling the formation diffusion coefficients are involved, the resulting isotopic of the Widmanstätten pattern. This involved i) developing fast profile will depend on the cooling rate. Thus, isotopic ratios and stable computer algorithms for simulating phase growth provide us with a means of testing the models of formation of in a finite system (Wood 1964; Goldstein and Ogilvie 1965; the Widmanstätten pattern. In order to understand the Goldstein and Short 1967; Willis and Wasson 1978; implications of the isotopic variations that have been Rasmussen 1982; Saikumar and Goldstein 1988), ii) documented so far in taenite and kamacite and guide future measuring Fe-Ni interdiffusion coefficients in taenite and analyses, one must be able to simulate such fractionation. At kamacite as a function of temperature and composition of the the annual Meteoritical Society meeting in 2006, Bourdon alloy (Hirano et al. 1961; Borg and Lai 1963; Goldstein et al. et al. reported preliminary results of modeling of isotopic 1964; Dean and Goldstein 1986; Cermak et al. 1989; Meibom fractionation due to differences of diffusivities of isotopes. et al. 1994; Yang and Goldstein 2004), iii) refining the Fe-Ni They used an approximate solution initially derived by Pawel in the temperature range 300–900 °C, (1973/1974) in the context of isothermal metal oxidation for including the influence of phosphorus on the positions of the diffusion in a finite system with a moving boundary. phase boundaries (Goldstein and Ogilvie 1965; Doan and Although this is useful as a first-order approach, one cannot Goldstein 1970; Willis and Wasson 1978; Moren and make the economy of a numerical model, taking into account Goldstein 1979; Romig and Goldstein 1981; Meibom et al. the complexity of phase relationships and interdiffusion 1994; Yang et al. 1996; Yang and Goldstein 2006), and iv) coefficients as a function of temperature and composition, in identifying the conditions for nucleation of kamacite, which order to compare measurements made on natural samples and controls the degree of undercooling that affected the meteorite theoretical predictions. This is the topic of the present paper, (Narayan and Goldstein 1985; Rasmussen et al. 1995; Yang which builds on preliminary results reported at the same and Goldstein 2005, 2006). meeting (Dauphas 2006). Diffusion-driven kinetic isotope effect during formation of the Widmanstätten pattern 1599

OBSERVATIONS possibly be attributed to differences in sampling. As shown in this study, in a single meteorite sample, the Fe isotopic The primary motivation for undertaking this work was composition should depend on the location of sampling and the discovery by several groups of Ni and Fe isotopic on the distance of nucleation between kamacite crystals. A fractionation between adjacent taenite and kamacite (Fig. 1). more practical complication arises from the fact that none of In the following, we shall use the FFe and FNi notations, which the crystals were oriented in any of the three studies. It is are deviations in parts permil per atomic mass unit relative to possible that during sampling of one phase, the drill or the the composition of a reference material (IRMM-014 for Fe, spot penetrated into the other phase, resulting in mixing SRM-986 for Ni, or bulk meteorites when specified): of the isotope signal. More work is required but it is clear from the studies available that the fractionation between i j i j FA = []()AA⁄ sample/A()⁄ Areference – 1 (1) taenite and kamacite sampled at a scale of 35–100 m in Toluca × 103 ⁄ ()ij– is small, presumably lower than 0.1‰/amu. Ko¢sler et al. (2005) reported large Fe isotopic fractionation between where i and j are two stable isotopes of element A. As taenite and kamacite of up to 1‰/amu by LA-ICPMS but discussed by Dauphas and Rouxel (2006), the virtues of this these results may be plagued by analytical problems (Horn notation are that it does not depend on the pair of isotopes that et al. 2006b; however, see Ko¢sler et al. 2006) and will not be are used (as long as isotopic variations are mass-dependent) discussed any further. and it allows direct comparison between the isotopic Bourdon et al. (2006) reported Ni isotope measurements compositions of different elements (here Fe and Ni). of iron meteorites. Kamacite is enriched in the heavy isotopes Commonly used pairs of isotopes for calculation of FFe and of Ni relative to taenite by ~+0.5‰/amu. This is similar to the 56 54 61 58 FNi are Fe/ Fe and Ni/ Ni, respectively. value documented by Cook et al. (2006, Forthcoming) for The database of bulk Fe isotope measurements published Toluca of ~0.4‰/amu. The latter authors analyzed Ni to date contains 36 analyses representing six iron meteorite isotopes in the same aliquots that were microdrilled for Fe, groups. The average of these data is 0.031‰/amu relative to allowing direct comparison of the behavior of the two IRMM-014 (Zhu et al. 2001; Kehm et al. 2003; Poitrasson elements. et al. 2005; Schoenberg and von Blanckenburg 2006; Cook The observations made of Fe and Ni isotope fractionation et al., Forthcoming). There is no clear evidence for Fe isotope between taenite and kamacite lend support to the idea that heterogeneity at a bulk scale in iron meteorites (Dauphas and differences in diffusivities played a role in explaining the Rouxel 2006). Note that Cook et al. (Forthcoming) have not isotopic fractionation. Indeed, during formation of the substantiated the results of Mullane et al. (2005), who found Widmanstätten pattern, growth of kamacite is mainly limited variable Fe isotopic compositions in IIIAB iron meteorites. by Ni diffusion in taenite. Because light isotopes diffuse faster The database of bulk stable isotope measurements of Ni in iron than heavy ones, one would expect the center of taenite to meteorites is limited and the results available seem to suggest have light Ni and heavy Fe isotopic compositions relative to that there are small yet resolvable Ni isotope variations in iron adjacent kamacite. The second observation that supports the meteorites (Moynier et al. 2005; Cook et al. 2006, idea that the isotopic fractionation is predominantly kinetic is Forthcoming). Overall, iron meteorites have heavy Ni isotopic the fact that the fractionation of Fe is smaller than that of Ni compositions relative to SRM-986 of around +0.2‰/amu (<0.1‰/amu for Fe versus >0.4‰/amu for Ni). At 500 °C in (Moynier et al. 2005; Cook et al. 2006, Forthcoming; Bourdon Toluca, the equilibrium Ni concentration in kamacite is et al. 2006). 7.6 wt% and that in taenite is 26.6 wt%. In contrast, the Poitrasson et al. (2005) first reported Fe isotope concentrations of Fe in kamacite and taenite are 92.4 wt% and measurements using multicollector–inductively coupled 73.4 wt%, respectively. Because of dilution effects, one mass spectrometry (MC-ICPMS) of associated taenite would expect lower kinetic isotope fractionation for Fe and kamacite in Toluca and Cranbourne, both IAB meteorites. compared to Ni, as observed. However, at 500 °C, one cannot Sampling was done by microdrilling; the results represent completely rule out the possibility that equilibrium bulk measurements of individual phases. In both meteorites, fractionation contributed to the overall isotopic fractionation taenite was enriched in the heavy isotopes of Fe by ~0.05– that is measured (e.g., Polyakov and Mineev 2000; Poitrasson 0.1‰/amu relative to kamacite. Horn et al. (2006a) analyzed et al. 2005; Williams et al. 2006; Schuessler et al. 2007). taenite and kamacite using laser-ablation (LA) MC-ICPMS. Here a numerical model is developed to address more They also found that taenite was enriched in the heavy quantitatively the question of what caused Fe and Ni isotopic isotopes of Fe relative to kamacite by ~+0.15‰/amu. Cook fractionation in iron meteorites as well as to evaluate the et al. (Forthcoming) measured the Fe isotopic composition of potential of this system as a proxy for estimating cooling rates Toluca using computer-assisted sample microdrilling; they in iron meteorites. Because Toluca has been extensively could not resolve any fractionation between taenite and studied for both Ni concentration profiles (Wood 1964; kamacite outside of error bars. The difference in the Saikumar and Goldstein 1988; Meibom et al. 1995; Hopfe magnitude of the fractionation in Toluca (or lack thereof) can and Goldstein 2001; Yang and Goldstein 2003) and Fe-Ni 1600 N. Dauphas

Fig. 2. Phase diagram and interdiffusion coefficients at the interface between taenite and kamacite in Toluca (IAB). The phase diagram (a) was constructed from the polynomial fits to the data of Dean and Goldstein (1996) given by Hopfe and Goldstein (2001). The influence of P was taken into account using the pivot method described by Moren and Goldstein (1979). Aun and aln (gun and gln) are the α/α + γ (α + γ/γ)phase boundaries for P-saturated and P-free systems. Above P-saturation (T ≈ 520 °C), the phase boundaries (heavy gray lines) lie between P-free and P-saturated curves. After saturation, the phase boundaries follow P-saturated curves. During cooling, Toluca (8.14 wt% Ni, 0.16 wt% P) crosses the stability field of kamacite at 727.43 °C. In model simulations, nucleation is not allowed to occur until the temperature reaches 700 °C. The interdiffusion coefficients (b) were calculated for Toluca at the interface between the two phases (the values depend on the Ni concentration and the degree of saturation in P). Dγ is always lower than Dα by many orders of magnitude. isotope systematics (Poitrasson et al. 2005; Horn et al. 2006a; The interdiffusion coefficients in taenite and kamacite Cook et al. 2006, Forthcoming), it provides a good case depend on the degree of saturation in P. They increase linearly example for the present study. with the degree of saturation when P is undersaturated and plateau at a value higher than in the P-free system when P is INPUT PARAMETERS saturated (Equations 7 and 9 in Hopfe and Goldstein 2001). The interdiffusion coefficients also depend on the Ni A number of parameters go into the computer model used concentrations (Equations 6 and 8 in Hopfe and Goldstein to simulate growth of kamacite out of taenite and to calculate 2001, note that there is a sign error in Equation 6 before isotopic fractionation associated with this process. The 116.112). Another parameter that comes into play is the present paper follows Hopfe and Goldstein (2001) for the degree of undercooling that affected the meteorite. Given the phase diagram and the diffusion coefficients. Yang and bulk Ni and P concentrations, it is possible to calculate the Goldstein (2003, 2005, 2006) developed a model with temperature at which the meteorite enters the stability field of improved treatment of nucleation and updated interdiffusion kamacite and to specify a temperature below this point where coefficients. This may become important as the quality of kamacite is allowed to nucleate. In the following, an isotopic measurements improves in terms of precision and undercooling of ~27 °C is used for Toluca, corresponding to a spatial resolution. The polynomials given by Hopfe and nucleation temperature of 700 °C. This is the value used by Goldstein (2001; their Table 1) are used for the phase Saikumar and Goldstein (1988) and Hopfe and Goldstein boundaries. The Ni and P concentrations in Toluca are (2001) in their models. 8.14 wt% and 0.16 wt%, respectively (Buchwald 1975). The ratio of the diffusivities of two isotopes (i and j) is Wasson and Kallemeyn (2002) revised the Ni concentration often parameterized as follows: of Toluca to 8.02 wt%, but the earlier value of Buchwald (1975) was adopted to facilitate comparison with the results D M β -----i = ⎛⎞------j (2) of Hopfe and Goldstein (2001). The influence of phosphorus Dj ⎝⎠Mi on the phase diagram is taken into account by using the pivot method described by Moren and Goldstein (1979). As long as where Mi and Mj are the masses of the two isotopes and β is an P is undersaturated (T > 520 °C in Toluca), the phase empirical parameter. Isotopic fractionation during diffusion in boundaries lie between the phase boundaries for P-saturated metal has been extensively studied because it is useful in and P-free systems. When phosphides precipitate (T ≤ elucidating diffusion mechanisms (e.g., Schoen 1958; 520 °C), the phase boundaries follow the P-saturated curves. Tharmalingam and Lidiard 1959; Mullen 1961). The β-values The phase diagram computed for Toluca and used in this calculated for diffusion of Fe in pure α (bcc, <910 °C), (fcc, study is shown in Fig. 2. 910 to 1390 °C), δ (bcc 1390 to melting at 1534 °C) iron, pure Diffusion-driven kinetic isotope effect during formation of the Widmanstätten pattern 1601

V, Cu, Ag, Pt, and equiatomic Fe-Co (Mullen 1961; Coleman et al. 1968; Heumann and Imm 1968; Graham 1969; Walter and Peterson 1969; Fishman et al. 1970; Oviedo de González and Walsöe de Reca 1971; Irmer and Feller-Kniepmeier 1972; Iijima et al. 1988; Roskosz et al. 2006) are compiled in Fig. 3. Although the β values of Fe vary with temperature and composition of the alloy, the range is small (0.15 to 0.35). A review of literature data also shows that for a wide range of and elements, β in metal is within 0.1 to 0.5. It is therefore reasonable to take 0.25 as a starting point for Fe and Ni in order to compute isotopic fractionation associated with the formation of the Widmanstätten pattern. A β of 0.25 corresponds to a difference of ~4‰/amu in diffusion coefficient for adjacent isotopes in the Fe-Ni mass region. Diffusion-driven kinetic isotope fractionation should scale linearly with β (e.g., if β doubles, then the isotopic fractionation that is produced should also double). There is no constraint on what the equilibrium fractionation between taenite and kamacite might be at the temperatures relevant to the formation of the Widmanstätten pattern. Another complication arises due to the fact that, if resolvable, the equilibrium fractionation will vary with temperature. In the present contribution, an exploratory calculation is presented for Ni with arbitrary constant equilibrium fractionation at the interface of 0.4‰/amu. Fig. 3. β-values (Equation 2) calculated from published data on the Fe isotope effect during diffusion (Mullen 1961; Coleman et al. NUMERICAL MODEL 1968; Heumann and Imm 1968; Graham 1969; Walter and Peterson 1969; Fishman et al. 1970; Oviedo de González and Walsöe de Reca The problem at hand shares some similarities with the 1971; Irmer and Feller-Kniepmeier 1972; Iijima et al. 1988; Roskosz case studied by Dauphas and Rouxel (2006). The task is to et al. 2006). The top panel is for diffusion in pure Fe and Fe + 3% Si (the vertical lines mark transitions between α, γ, δ, and Fe). develop a numerical model that follows isotopes in a finite The bottom panel is for diffusion in other (V, Cu, Ag, Pt, and system with a moving boundary. In essence, the model equi-atomic Fe-Co). presented here follows the front-tracking, fixed finite- difference grid method described by Crank (1984). We forward time centered space (FTCS) scheme. If i is a grid consider an infinite system where kamacite plates nucleate point and n is a time step, Equation 3 translates into: at regular intervals. The half-distance between two nucleation sites is denoted as L and is called the C – C 1 ------in, + 1 ------in, = ------()D – D (4) impingement length (Fig. 4). Because identical are δt 4δx2 i + 1, n i – 1, n repeated infinitely, one can only consider a subunit D ()C – C + ------in, ()C + C – 2C comprising half of a taenite-kamacite pattern with no flux- i + 1, n i – 1, n δx2 i + 1, n i – 1, n in, boundary conditions on each side. The general equation governing changes in concentration as a function of time is For kamacite, the first term on the right side of the equation Fick’s second law: disappears because Dα is constant at a given temperature. All properties are symmetric along the left and right boundaries ∂C ∂D∂C ∂2C ------= ------+ D------(3) (i = 1 and i = l), so we have at these grid points: ∂t ∂x ∂x ∂x2 C – C 2Dα ------1, n + 1 1, n = ------1, n-()C – C (5) The derivative of the diffusion coefficient is included because δt δx2 2, n 1, n in taenite it depends on the Ni concentration and hence on x. The scheme used for differentiating Equation 3 is the Iterated and Crank-Nicholson (ICN), which is a popular algorithm in numerical relativity (Teukolsky 2000). Its virtue is that it is γ easy to implement and yet possesses some of the stability Cln, + 1 – Cln, 2Dln, ------= ------()Cl – 1, n – Cln, (6) features of implicit methods. The first step in the ICN is the δt δx2 1602 N. Dauphas

C – C 2Dγ ------k + 1, n + 1------k + 1, n- = ------k + 1, n (8) δt δx2 Cγ C C ------n -–------k + 2, n – ------k + 3, n ()p – 2 ()p – 3 2 – p p – 3 γ 1 –3C ()1 + p C pC + ------n - ++------k + 2, n------k + 3, n- δx2 ()p – 2 ()p – 3 2 – p p – 3 –3Dγ ()1 + p Dγ pDγ × ------kpn------+ , - ++------k + 2, n ------k + 3, n ()p – 2 ()p – 3 2 – p p – 3

At each time step, the displacement of the interface allowed by diffusion is calculated. If we note ξ the absolute position of the boundary, ξ = (k + p − 1)δx, we have:

∂ξ α γ γ∂C+ α∂C– ------()C – C = D ------– D ------(9) ∂t ∂x ∂x

where Cα and Cγ are the concentrations at the interface given by the phase diagram (at the interface the two phases Fig. 4. Schematic diagram showing the spatial discretization used for are in equilibrium) and the derivatives of the concentration calculation of partial derivatives. The top panel shows how a system are taken at the right (+) and the left (−) of the interface. with kamacite nucleation at periodic intervals can be reduced to a These are estimated using Lagragian interpolations between subunit with no-flux conditions at the boundaries. The width of this subunit, L, is called the impingement length. In the bottom panel, k is the grid points k − 2, k − 1, and k + p, and k + p, k + 2, and the integer number for the grid point at the left of the interface k + 3: between taenite and kamacite, p is the fraction of δx where the interface is located, and ξ = (k + p − 1)δx is the absolute position of γ γ ξn + 1 – ξn α γ Dkpn+ , ()2p – 5 Cn the interface. These grid points are used for computation of first and ------()Cn – Cn = ------+ (10) second derivatives at the interface using interpolations of Lagrangian δt δx ()p – 2 ()p – 3 type (Crank 1984; see the Numerical Model section). ()p – 3 C ()p – 2 C Dα ------k + 2, n- + ------k + 3, n- – ------kpn+ , - p – 2 3 – p δx For the two grid points located on each side of the α ()p + 1 C ()p + 2 C ()2p + 3 C interface between kamacite and taenite, the calculation is ------k – 2, n- – ------k – 1, n- + ------n more involved. The interface is located between k and k + 1, p + 2 p + 1 ()p + 2 ()p + 1 at a fraction p of the distance δx between adjacent grid points (0 < p < 1) (Fig. 4). Around the interface, the points are For computing isotopic fractionation, one cannot simply unequally spaced and one can use interpolation formulae of independently calculate the concentration profiles of two Lagrangian type to calculate first and second derivatives isotopes and ratio them. Indeed, doing so would result in (Crank 1984). In this study, the grid points k − 2, k − 1, and k computing distinct interface positions (ξ) for the two isotopes + p, and k + p, k + 2, and k + 3 were used at the left and the while this value is uniquely defined. In addition, the right of the interface, respectively (instead of k − 1, k, and k + concentrations on both sides of the interface are given by the p, and k + p, k + 1, and k + 2 as in Crank 1984, because phase diagram; independently calculating the profiles for two otherwise instability problems arose for p close to 0 and 1). In isotopes would artificially anchor the isotopic composition at kamacite (index k), the interdiffusion coefficient is constant the interface to 0‰/amu. and we have: In the following, we consider two isotopes referred to as leading (setting the pace for migration of the interface) and α C – C 2D trailing (reacting in response to migration of the interface). ------kn, + 1 kn, - = ------kn, (7) δt δx2 For all grid points away from the interface (1, …, k − 1, k + 2, α …, l), the calculation for the trailing isotope is identical to that Ck – 2, n Ck – 1, n Cn ------–------+ ------described for the leading isotope (see Equations 4 to 6). At the p + 2 p + 1 ()p + 1 ()p + 2 interface, we use the displacement calculated for the leading isotope (Equation 10) to calculate the concentrations of the α where Cn is the Ni concentration in kamacite given by the trailing isotope at and around the interface. The concentration phase diagram. In taenite (index k + 1), other terms must be of the trailing isotope is denoted as C* and its diffusion added because the diffusion coefficient depends on the Ni coefficient as D*. The ratio of the diffusion coefficients of the concentration: two isotopes: Diffusion-driven kinetic isotope effect during formation of the Widmanstätten pattern 1603

* 3. New values at t + δt are computed and inserted back into rD= ⁄ D (11) Equation 1. is given by Equation 2. When r is equal to 1 (β = 0), then there This iteration can be repeated an infinite number of times and is no kinetic isotope fractionation. The equilibrium converges to the Crank-Nicholson method. In the present fractionation at the interface is: contribution, only two iterations after the FTCS step were performed because Teukolsky (2000) showed that in most Cγ* ⁄ Cγ cases this is enough to ensure stability of the algorithm. After λ = ------(12) Cα* ⁄ Cα computation of ξ at t + δt, p is calculated (p = ξ/δx + 1 − k). If it is higher than 1, then k increases by 1, p decreases by 1, and When λ is equal to 1, then there is no equilibrium the point near the interface that was in taenite moves into fractionation. The objective at this point is to estimate Cα∗ and kamacite. If p is lower than 0, then k decreases by 1, p Cγ∗ using Equations 10–12. As discussed previously, the increases by 1, and the point near the interface that was in position of the interface is uniquely defined. We can therefore kamacite moves into taenite. Thus, the algorithm allows the apply Equation 10 to the trailing isotope: interface to move in both directions. At each time step, the concentrations at the interface in taenite and kamacite, which γ γ* ξn + 1 – ξn α* γ* rDkpn+ , ()2p – 5 Cn depend on the temperature, are updated. As the temperature ------()C – C = ------+ (13) δt n n δx ()p – 2 ()p – 3 decreases, the diffusion coefficients decrease and the ()p – 3 C* ()p – 2 ()C* rDα parameter that governs stability of the algorithm (∝δx2/Dδt) ------k + 2, n- + ------k + 3, n – ------kpn+ , p – 2 3 – p δx increases. This makes it possible to refine the mesh size to a α* final spatial resolution of δx ≈ 1 μm. The code was written in ()p + 1 C* ()p + 2 C* ()2p + 3 C ------k – 2, n – ------k – 1, n + ------n R and the simulations were performed on a dual 2 GHz p + 2 p + 1 ()p + 2 ()p + 1 PowerPC G5 with 6 GB of RAM.

TEST OF THE ALGORITHM We have two equations (12 and 13) in two unknowns (Cα∗ and Cγ∗) and the system can be solved: Modeling growth of the Widmanstätten pattern involves following a moving boundary in a finite system with time- * * γ* r γ ()p – 3 C ()p – 2 C dependent conditions at the interface. Few research groups C = ----- D ------k + 2, n- + ------k + 3, n- (14) n δx kpn+ , p – 2 3 – p have developed such algorithms (Wood 1964; Goldstein and Ogilvie 1965; Goldstein and Short 1965; Willis and Wasson ()p + 1 C* ()p + 2 C* – Dα ------k – 2, n- – ------k – 1, n- ⁄ 1978; Rasmussen 1982; Saikumar and Goldstein 1988), so kpn+ , p + 2 p + 1 care was taken to validate the numerical code used in this α γ study. It was tested against Ni profiles given by Hopfe and ξ – ξ ⎛⎞1C rD 2p – 5 ------n + 1 n-⎜⎟------n – 1 – ------kpn+ , ------+ Goldstein (2001) and also against the analytical solution of a δt λ γ δx ()p – 2 ()p – 3 ⎝⎠Cn problem that shares similarities with the problem at hand. α α The problem treated hereafter has no physical rD 2p + 3 1C ------kpn+ , ------n foundation. Its only purpose is to provide a basis for testing δx ()p + 2 ()p + 1 λ γ Cn the finite difference algorithm (see the Numerical Model section) against a problem that has an exact analytical The value of Cγ∗ thus calculated is inserted in Equation solution. Let us consider a system with an element present at α∗ 12 to calculate C , which is in turn used in Equations 7 and 8 concentration C(x,0) = C0 in a semi-infinite phase I. At time 0, to update the concentrations of the trailing isotope at the grid the concentration at the surface (x = 0) is lowered and points k and k + 1. It is then straightforward to calculate maintained at a constant value afterward, C(0,t) = Cs. As a isotopic fractionation using the computed concentration result, a second phase (II) grows at the surface. The phase profiles of the trailing and leading isotopes. If there is no diagram for I and II gives equilibrium concentrations at the equilibrium or kinetic isotope fractionation, then Equations interface between the two phases, CII,I and CI,II (in II and I, 10 and 14 give Cγ∗ = Cγ, as expected. The FTCS step just respectively). After some time, the system converges toward a presented is the first step of the ICN method. The ICN is self-similar asymptotic regime whereby all properties centered in time and space and proceeds as follows: propagate in space with t1/2 (regardless of the initial 1. The values calculated at t + δt are averaged with those at conditions, if one waits long enough and takes a snapshot of t to get an estimate at t + δt/2. the system at a given time, the concentration plotted as a 2. In finite difference equations (e.g., Equations 3 and 9), function of the distance divided by the square root of time will the space derivatives (right side) are computed using be very similar to the profile shown in Fig. 5). Jost and estimates at t + δt/2. Wagner (Jost 1952) presented the analytical solution for this 1604 N. Dauphas

Fig. 5. Comparison between the analytical self-similar asymptotic solution of the problem presented in the Test of the Algorithm section (Jost 1952) with the numerical result obtained after convergence using the Iterated Crank-Nicholson algorithm (see the Numerical Model section). regime. As before, ξ is the absolute position of the interface between the two phases. We have for the concentration in phase II (0 < x < ξ):

()C – C x CC= – ------s II,I-erf⎛⎞------(15) s erfγ ⎝⎠ Fig. 6. Comparison between Ni concentration profiles in Toluca 2 DIIt computed by Hopfe and Goldstein (2001) and this study (cooling rate the concentration in phase I (ξ < x): of 50 °C/Myr and impingement length of 700 μm). Note that in Hopfe and Goldstein (2001), the label of Fig. 6 indicates that it C – C x corresponds to a cooling rate of 25 °C/Myr, but the taenite half-width CC= + ------0 ------I,II e r f c ⎛⎞------(16) (31.9 μm) and central Ni (23.96 wt%) fall on the 50 °C/Myr curve in 0 erfc()γD ⁄ D ⎝⎠ II I 2 DIt Fig. 7, so Fig. 6 was presumably mislabelled (25 instead of 50 °C/Myr). The different curves correspond to snapshots taken at and the position of the interface: different temperatures during cooling of the meteorite. The results of the two studies, which used identical phase diagrams and ξ = 2γ D t (17) interdiffusion coefficients, are very similar (the final taenite half- II width and central Ni calculated here are 27.2 μm and 23.93 wt%, where γ is the root of the transcendental equation: respectively). The only notable difference is the position of the interface. The reason for this small discrepancy is that in the present C – C study, a kamacite nucleus of 25 μm was introduced at the beginning s II,I 2 of the simulation to initiate the differentiation scheme. CII,I – CI,II = ------exp()–γ – (18) πγerfγ CI,II – C0 2 Equation 18 is 0.446973. The algorithm developed for ------exp()–γ DII ⁄ DI πD ⁄γD erfc()γ D ⁄ D following the growth of kamacite from taenite was modified II I II I to accommodate the conditions set in this section. The Note that Equation 1.315 of Jost (1952) contains an error analytical solution is compared with that obtained from the (DII/DI was not included in the last term of Equation 18). If ICN after convergence in Fig. 5. As shown, there is perfect one introduces u = x/t0.5, the functions given above only agreement between the two approaches, which supports the depend on this variable. In order to compare these analytical validity of the code used. expressions with the numerical treatment described before, The algorithm was also tested against Ni concentration one must choose numerical values for the parameters. The profiles simulated by Hopfe and Goldstein (2001) for following values were adopted: CS = 5 wt%, CII,I = 10 wt%, Toluca using a nucleation temperature of 700 °C, an 2 CI,II = 15 wt%, C0 = 20 wt%, DII = 3 μm /yr, and DI = impingement length of 700 μm, and a cooling rate (CR) of 2 0.3 μm /yr (the values for DII and DI correspond to the 50 °C/Myr (the present best estimate for the cooling rate of interdiffusion coefficients in kamacite and taenite at 700 °C in this meteorite). The phase diagram and diffusion a P-free system). The value of γ obtained by solving coefficients used in the present study are identical to those Diffusion-driven kinetic isotope effect during formation of the Widmanstätten pattern 1605

Fig. 7. Nickel concentration and kinetic isotope fractionation (β = 0.25 in Equation 2 and λ = 1 in Equation 12) for three cooling rates, 25 (left), 100 (center), and 500 °C/Myr (right). The three colored curves in each panel correspond to snapshots taken at different temperatures (650 °C, red; 550 °C, orange; 450 °C, blue). Below 450 °C, the system is almost frozen for all cooling rates, except for small variations at the interface. The top panels show Ni concentration profiles (wt%). The middle panels show Ni isotope profiles (‰/amu deviation relative to the bulk). The bottom panels show the product of the Ni concentrations and isotopic compositions. The later curves help to visually evaluate how isotopic compositions are balanced between positive and negative values (the area below the curve must be equal to 0). used by Hopfe and Goldstein (2001), so the results of the RESULTS two studies can be directly compared. Figure 6 shows snapshots taken at different temperatures during cooling of Several computer simulations were performed in order to the meteorite. The simulations in this study were stopped at understand what causes isotopic variations in iron meteorites 450 °C because below this temperature, the diffusion scale and to evaluate the potential of Fe and Ni isotopes to refine is so small that the system is almost frozen, except for cooling rate estimates. Because these calculations are variations right at the interface. As can be seen, there is computer-intensive, only a small fraction of the parameter excellent agreement between the two studies. The only space was explored. As discussed previously, the case noticeable difference lies in the position of the interface, example and primary focus of this study is Toluca, an IAB which is shifted toward the right by ~5–25 μm in this study iron meteorite (see Choi et al. 1995 and Benedix et al. 2000 compared to Hopfe and Goldstein (2001). This is due to the for details on the petrogenesis of this type of meteorite). fact that in the present work, a fixed-grid method with an The influence of cooling rates on isotopic fractionation initial spacing of 12.5 μm was used and a seed of kamacite due to diffusion was explored by fixing the impingement comprising 2 grid points (25 μm) was used to initiate the length at 700 μm and varying the cooling rate over an interval differentiation. Again, the agreement between the present (25 to 500 °C/Myr) that encompasses the range of study and Hopfe and Goldstein (2001) supports the validity metallographic cooling rates estimated for IAB meteorites by of the algorithm. Herpfer et al. in 1994 (30–70 °C/Myr; however, Rasmussen Finally, two consistency checks were performed for all 1989 and Meibom et al. 1995 estimated lower values of 1 to simulations. The no flux-boundary conditions at the left and 15 °C/Myr). the right of the system impose that mass conservation must be respected. For all simulations, the total Ni content did not Kinetic Isotope Fractionation of Nickel vary beyond 1 part per thousand (given by the area below CNi) and the Ni isotopic composition of the system remained Kinetic isotope fractionation was investigated by setting constant within ~0.01‰/amu (given by the area below FNi × β = 0.25 (Equations 2 and 11) and λ = 1 (Equation 12). At CNi). 650 °C (50 °C after nucleation), the isotopic composition of 1606 N. Dauphas

influence on the magnitude of the fractionation. At 25 °C/Myr, the maximum difference in isotopic composition between taenite and kamacite (0.18‰/amu) is smaller than at 100 °C/Myr (1.10‰/amu) and 500 °C/Myr (0.98‰/amu). At low cooling rates, the system tends toward equilibrium at all time steps with little kinetic isotope fractionation between the two phases. As the temperature decreases to 550 °C, a new feature becomes apparent. At 650 °C, the isotopic composition of kamacite was homogeneous due to high diffusion rate. At 550 °C, transport of Ni in kamacite becomes diffusion- limited; this phase shows a gradient in Ni isotopic composition. The center of kamacite becomes enriched in the light isotopes relative to the interface. This is due to the fact that as the temperature decreases, the Ni concentration of kamacite at the interface increases relative to the center, which causes diffusion and hence isotopic fractionation. The fractionations between the center of kamacite and the interface at 550 °C are 0.07‰/amu, 0.15‰/amu, and 0.19‰/amu for cooling rates of 25, 100, and 500 °C/Myr, respectively. At around 450 °C, the largest isotopic fractionation in taenite does not occur at the center but at some distance away from the boundary. This was already visible at 650 °C for the highest cooling rate (500 °C/Myr), but at 450 °C, all profiles show such a feature on a scale commensurate with the width of the taenite crystal. The isotopic fractionation between the center of kamacite and the interface is higher at 450 °C than it is at 550 °C. For the low cooling rate (25 °C/Myr), the fractionation within kamacite is almost as high as the fractionation within taenite (0.23 and 0.29‰/amu, respectively). For higher cooling rates, kamacite has more homogeneous isotopic composition than taenite. These results show that one cannot directly compare interdiffusion Fig. 8. Dimensionless parameter governing diffusion in taenite and coefficients to argue for diffusion-limited transport in one kamacite (η = w2cr/DΔT, where w is the half-width of the phase, cr phase or another. The other parameter that comes into play is is the cooling rate, D is the interdiffusion coefficient in taenite or the distance over which must diffuse. As kamacite kamacite taken at the interface, and ΔT is a characteristic temperature grows, the width of taenite decreases and this balances part of interval used for calculating diffusion distances, 20 °C following Wood 1964). For η << 1, diffusion is fast enough to equilibrate the the contrast between the interdiffusion coefficients of the two phase on a time scale relevant to cooling. For η >> 1, the diffusion phases. The dimensionless parameter that is relevant for length scale is small compared to the size of the phase and the system discussing these issues is: stops evolving. For intermediate values, diffusion-limited growth and kinetic isotope fractionation can occur. w2cr η = ------(19) DΔT Ni in kamacite is almost homogeneous for all cooling rates (Fig. 7). Taenite already shows the imprint of diffusion, with where w is the half-width of either taenite or kamacite, cr is overall light Ni isotopic composition relative to kamacite. At the cooling rate, D is the interdiffusion coefficient in taenite moderately slow cooling rates (25 and 100 °C/Myr), taenite or kamacite at the interface, and ΔT is a characteristic shows zonation from light cores to heavy interfaces. At a high temperature interval used for calculating diffusion distances cooling rate (500 °C/Myr), the profile is different. The (20 °C, following Wood 1964; note that Rasmussen et al. meteorite cooled so fast that the diffusion signal did not have 2001 adopted a slightly higher value of 50 °C). If η << 1, then time to propagate all the way to the center of the crystal. As a diffusion has time to homogenize the phase on the time scale result, the center of taenite has undisturbed Ni isotopic relevant to cooling. If η >> 1, then the system is frozen composition and the point that shows the largest effect is because the diffusion signal does not have time to propagate located ~300 μm off center. The cooling rate also has a strong across the phase. For values around unity, diffusion can Diffusion-driven kinetic isotope effect during formation of the Widmanstätten pattern 1607

Fig. 9. Iron concentration and kinetic isotope fractionation (β = 0.25 in Equation 2 and λ = 1 in Equation 12) for three cooling rates, 25 (left), 100 (center), and 500 °C/Myr (right). The three colored curves correspond to snapshots taken at different temperatures (650 °C, red; 550 °C, orange; 450 °C, blue). The top panels show Fe concentration profiles (wt%). The middle panels show Fe isotope profiles (‰/amu deviation relative to the bulk). The bottom panels show the product of the Fe concentrations and isotopic compositions. These curves help to visually evaluate how isotopic compositions are balanced between positive and negative values (the area below the curve must be equal to 0). produce isotopic fractionation. The values of η calculated for and sign of the fractionation. Concentration gradients for Fe Toluca with an impingement length of 700 μm and a are similar in magnitude but opposite in sign compared to Ni. nucleation temperature of 700 °C are reported in Fig. 8. For a For this reason, one would expect Fe to be isotopically light cooling rate of 25 °C/Myr, the value of η for kamacite is when Ni is heavy and vice-versa. This is what is observed in higher or similar to that of taenite for temperatures below model results. Also, because the absolute concentration of Fe ~550 °C. This explains why the two phases show similar is higher than that of Ni in both taenite and kamacite, isotopic fractionation. At 100 and 500 °C/Myr, η in kamacite diffusing Fe atoms are diluted to a greater extent by normal Fe is always lower than in taenite. As a result, kamacite is more than is the case for Ni. This explains why for the most part the isotopically homogeneous than taenite. Below 450 °C, the isotopic fractionation of Fe is smaller than that of Ni. diffusion length is significantly reduced at all cooling rates The exact relationship between Fe and Ni isotopic and the system stops evolving. fractionation deserves special attention because it is a key observation that may help us distinguish between kinetic Kinetic Isotope Fractionation of Iron and equilibrium isotope fractionation (Fig. 10). At all cooling rates, the isotopic compositions relative to the bulk, The algorithm developed for calculating Ni isotope taken at the centers of both phases, are more fractionated in fractionation during growth of kamacite out of taenite can absolute values for Ni than they are for Fe. Isotopic easily be modified to investigate Fe isotope fractionation fractionation does not only depend on concentration (Fig. 9). A set of simulations were run for Fe for cooling rates gradients, which are opposite in sign but equal in of 25, 100, and 500 °C/Myr, assuming that β of Fe is identical magnitude, but also on absolute concentrations. For a given to that of Ni (0.25) and λ = 1 (Equation 12). As expected, the concentration gradient, the fractionation will be large for Ni calculated Fe concentrations exactly mirror Ni concentrations and small for Fe if the Ni concentration is low and the Fe ([Fe] + [Ni] = 100) and the position of the interface based on concentration is large (and vice-versa). There are large Fe is the same as that based on Ni (Fig. 9, top panels). The concentration differences in Ni between taenite and main features that can be identified in Fe isotope profiles are kamacite and within taenite. These dilution effects are the very similar to those already presented for Ni and will not be main cause for the non-linearity in the relationship between repeated hereafter. The main differences are the magnitude Fe and Ni isotopic fractionation. 1608 N. Dauphas

factor. At a slow cooling rate (25 °C/Myr), taenite and kamacite have almost homogeneous isotopic compositions. If taenite and kamacite were in complete equilibrium, the isotopic compositions of each phase could be calculated by mass-balance:

α t γ γ C ⎛⎞C – C F = –Δ------⎜⎟------(20) Ct ⎝⎠Cα – Cγ

Fα = Δ + Fγ

where Δ = (1/λ−1)*103 is the kamacite-taenite equilibrium fractionation (arbitrary, 0.4‰/amu), Ct is the bulk Ni concentration (8.14 wt% for Toluca), and the other concentrations are equilibrium values given by the phase diagram (Fig. 2). The values calculated at 650, 550, and 450 °C for kamacite in equilibrium with taenite (Fγ) are 0.23, 0.09, and 0.03‰/amu, respectively. The results of the simulations for 25 °C/Myr taken at the center of kamacite are 0.22, 0.11, and 0.07‰/amu at the same temperatures. These values are very close to the equilibrium values, and it shows that throughout the thermal history of the meteorite, the kamacite-taenite fractionation at each point will remain close to what is expected for bulk equilibrium. As the temperature cools to 450 °C, most of the Ni is in kamacite, the concentration in this phase is close to the bulk concentration (Cα ≈ Ct), and it follows from Equation 20 that Fα ≈ 0 and Fγ ≈ −Δ. At higher cooling rates (100 and 500 °C/Myr), large isotopic gradients develop in the two phases. For positive kamacite-taenite isotopic fractionation, the centers of taenite and kamacite always display heavy Ni isotopic compositions relative to the compositions at the interface. As the temperature decreases, the isotopic fractionation at the interface is expressed predominantly in taenite, and the largest gradients are expressed in this phase. As discussed previously (Fig. 8), at 100 and 500 °C/Myr, growth of kamacite crystals is limited by Fe-Ni diffusion in taenite. At any temperature, and for any time interval, the distance over which atoms can diffuse in kamacite is higher than in taenite. If this scale is larger than crystal sizes, then the system is in Fig. 10. Fe-Ni isotope correlations in Toluca at 450 °C for diffusion- bulk equilibrium, and Equation 20 applies. Otherwise, one driven kinetic isotope fractionation (β = 0.25). The points along the should write the mass-balance over the diffusion scale. For a curves correspond to individual grid points. Kamacite is in black; while taenite is in red. The centers of the two phases are shown as subsystem with isotopic composition equal to the bulk (FNi = large squares. The dashed curves are point-to-point linear 0‰/amu), the mass-balance equation takes the form: interpolations and have no other virtue than guiding the . α α* γ C w Fixed Equilibrium Fractionation of Nickel F = –Δ------(21) Cαwα* + Cγwγ* The difficulty with equilibrium fractionation is that we Fα = Δ + Fγ do not know what the value is and how it varies with temperature. For illustration purposes, an arbitrary value of where wα∗ and wγ∗ are the diffusion scales in taenite and +0.4‰/amu between kamacite and taenite was adopted kamacite. Because Cγwγ∗ << Cαwα∗ at the interface for high (Fig. 11). Because the outcome of the simulations scales with cooling rates, we have Fα ≈ 0 and Fγ ≈ −Δ. This is the primary this value, it is easy to predict what the effect would be if reason why for 100 and 500 °C/Myr, isotopic fractionation at equilibrium fractionation was reduced or increased by a given the interface is expressed predominantly in taenite. Diffusion-driven kinetic isotope effect during formation of the Widmanstätten pattern 1609

Fig. 11. Nickel isotopic fractionation during formation of Widmanstätten pattern in Toluca for an arbitrarily fixed equilibrium fractionation at the interface between kamacite and taenite of +0.4 ‰/amu (λ = 1.004 in Equation 12). No fractionation due to diffusion was included in this calculation (r = 1 in Equation 11). The three columns correspond to different cooling rates (25, 100, and 500 °C/Myr) and the different curves in each panel are snapshots taken at different temperatures (650, 550, and 450 °C). The top panels show Ni isotope profiles (‰/amu deviation relative to the bulk). The bottom panels show the product of the Ni concentrations and isotopic compositions. These curves help to visually evaluate how isotopic compositions are balanced between positive and negative values (the area below the curve must be equal to 0).

DISCUSSION respectively, a cooling rate of 50 °C/Myr reproduces the observations for both Fe and Ni (Fig. 12). Using Wood’s As discussed previously, a principal issue with Fe and Ni method of matching measured and modeled taenite central isotopic fractionation measured in taenite and kamacite is Ni content and half-width, Hopfe and Goldstein (2001) determining whether it reflects equilibrium, kinetic, or a concluded that Toluca had cooled at a rate of 25 °C/Myr. combination of both processes. Although critical These authors also concluded that a cooling rate of measurements and experiments are missing to definitively 50 °C/Myr was within model uncertainties. This speaks answer this question, model results presented here give strongly in favor of the possibility that the fractionation important clues on which process may dominate. For all measured for Fe and Ni in iron meteorites was produced by measurements available, there are either no constraints on partial separation of isotopes through diffusion. It also where the material was sampled or the sampling scale is too supports the validity of the value of 0.25 adopted for β. large and isotopic compositions were integrated over a wide An additional constraint on whether kinetic or area (Poitrasson et al. 2005; Cook et al. 2006, Forthcoming; equilibrium effects control the fractionation comes from the Horn et al. 2006). difference between the isotopic compositions of kamacite and the bulk. Overall, the general pattern for Fe isotopes is to have Kinetic or Equilibrium Fractionation? taenite heavy relative to kamacite. If this were due to equilibrium, then the taenite-kamacite fractionation should be One must bear in mind that as far as the magnitude of positive. Simulations with fixed equilibrium isotope the isotopic fractionation is concerned, there is some fractionation at the interface show that for all points along the degeneracy between the cooling rate and the relative profile, the isotopic composition of kamacite and taenite diffusion coefficients of neighbor isotopes. Thus the should have light and heavy Fe isotopic compositions, discussion that follows critically depends on the value respectively. This is not true for diffusion-driven kinetic adopted for β, which is still uncertain (0.25) (Fig. 3) (Mullen isotope fractionation with slow cooling. Indeed, for cooling 1961; Coleman et al. 1968; Heumann and Imm 1968; rates of 25 to 50 °C/Myr, transport of Fe is diffusion-limited Graham 1969; Walter and Peterson 1969; Fishman et al. in kamacite to the same extent as it is in taenite (Fig. 8). This 1970; Oviedo de González and Walsöe de Reca 1971; Irmer results in Fe isotope variations within kamacite characterized and Feller-Kniepmeier 1972; Iijima et al. 1988; Roskosz by a heavy center and a light interface. If kinetic fractionation et al. 2006). While the simulations at 25 °C/Myr and were the controlling factor, one would expect to find both 100 °C/Myr give too small and too large fractionations, light and heavy Fe isotopic compositions relative to the bulk 1610 N. Dauphas

Fig. 13. Thought experiment illustrating how isotopes can remove degeneracy in model parameters (see the Discussion section for details). Phase II grows at the expense of semi-infinite, well-mixed phase I. The concentration in II at the interface is noted CII,I. The left graph in the two panels show two possible time evolutions for CII,I (governed by a hypothetical phase diagram and time-temperature evolution). In the top panel, CII,I does not change much and the diffusion coefficient in II is very small. In the bottom panel, CII,I increases dramatically, but this is mitigated by diffusion in II. The two scenarios would create similar chemical profiles (C versus x) but very distinct isotopic profiles (F versus x).

parameters (if these are not perfectly known) and reveal problems that had not been appreciated before. Figure 13 shows, using a thought experiment, how isotopic fractionation is not redundant with chemical zoning. Let us consider a system whereby a finite phase II grows at the expense of a semi-infinite phase I. The time scale for advection in I is assumed to be lower than diffusion, so that I Fig. 12. Comparison between predicted (diffusion-driven kinetic keeps constant and homogeneous isotopic and chemical isotope fractionation with β = 0.25) and measured Fe and Ni isotopic compositions. In the first case (Fig. 13, top), the concentration fractionation in Toluca for a cooling rate of 50 °C/Myr and an in II at the interface (CII,I, given by a hypothetical phase impingement length of 700 μm. The top two panels show Ni and Fe diagram) remains almost constant during the experiment and isotope profiles at 450 °C (for the Ni concentration profile, see Fig. 6). The bottom panel shows the Fe-Ni isotopic correlation. The the diffusion in II is negligibly small. Little concentration values taken at each grid point along the profile are shown. The gradient will develop (CII,I remains almost constant) and the symbols and notations are identical to Fig. 10. The range of measured isotopic composition will remain unmodified (no diffusion values is reported as hatched regions relative to the bulk Fe and Ni takes place). In the second case (Fig. 13, bottom), the isotopic compositions of all iron meteorites measured so far (thin concentration in II at the interface increases dramatically, but horizontal and vertical lines). As illustrated, there is good agreement between measured and calculated isotopic fractionations for both Fe there is significant diffusion in II. The final concentration and Ni in taenite and kamacite. gradient will be small because of diffusion, but significant isotopic fractionation will be produced in II due to diffusive in a single phase. As shown in Fig. 1, available data suggests separation of isotopes. This thought experiment illustrates that this may be the case. that during phase growth, two different sets of conditions for the phase diagram and the interdiffusion coefficients can Perspectives in Stable Isotope Thermometry create identical chemical zoning, which can be distinguished on the basis of diffusion-driven kinetic isotope effect. Kinetic isotope fractionation associated with the Thus, isotopes can be used to test the validity of the formation of the Widmanstätten pattern is an extension of parameters and assumptions that go into modeling the chemical zoning and cannot be used as an independent formation of the Widmanstätten pattern (Wood 1964; thermochronometer. However, it provides additional Goldstein and Ogilvie 1965; Goldstein and Short 1965; Willis constraints that may help remove degeneracy in model and Wasson 1978; Rasmussen 1982; Saikumar and Goldstein Diffusion-driven kinetic isotope effect during formation of the Widmanstätten pattern 1611

1988). The present study shows that Toluca, the meteorite for equilibrium fractionation at the interface and the other with which the largest database of isotope measurements exists, kinetic isotope fractionation driven by diffusion. Using a β of has passed this test. An obvious extension of this work is the 0.25, which corresponds to a difference in diffusivities of analysis of meteorites from the group IVA of iron meteorites. adjacent isotopes of 4‰/amu in the Fe-Ni mass region, the These meteorites are related genetically by a trend of taenite-kamacite isotopic fractionation measured by several magmatic differentiation (Wasson and Richardson 2001). If workers in Toluca can be reproduced with a cooling rate of all IVA meteorites were derived from a single core, one would 50 °C/Myr. This is in the range of values previously estimated expect isothermal cooling (e.g., Haack and Scott 1992). The for Toluca (Hopfe and Goldstein 2001). Other methods are reason is that in a metallic core is much available for testing the validity of the assumptions and higher than in the overlying silicate mantle, so one would not parameters used in models of formation of the Widmanstätten expect large temperature gradients between samples from the pattern, including analysis of trace highly siderophile element central and peripheral regions of the core. However, several distribution (Goldstein 1967; Watson and Watson 2003; groups have shown that for IVA, the computed cooling rates Righter et al. 2005). The main virtue of Fe and Ni isotopes is vary from one meteorite to another and correlate with bulk Ni that they provide a direct test that only involves the concentrations (Goldstein and Short 1967; Moren and determination of an extra parameter, β. If this value is known Goldstein 1979; Rasmussen 1982; Rasmussen et al. 1995; for Fe and Ni, then calculating isotopic fractionation follows Yang et al. 2007). Haack et al. (1996a) suggested that after in a straightforward way from modeling concentration crystallization, the core was disrupted by impact and the profiles. Obvious targets for study are IVA iron meteorites, resulting debris were reassembled by gravitation. According where simulations indicate that meteorites from this group to this model, the correlation between cooling rates and Ni cooled at different rates, which contradicts the idea that they concentrations would be purely fortuitous, reflecting limited formed from differentiation of an asteroid core. This may sampling of this asteroid. More recently, Yang et al. (2007) reflect a complex thermal evolution such as breakup- argued that the core preserved its integrity but that the parent reassembly of a planetary core or removal of the overlying body of IVAs was stripped of part of its mantle by collisions. mantle. It may also be due to inaccuracies in the assumptions Others (Willis and Wasson 1978; Wasson and Richardson and parameters used in the simulations; this could be tested 2001) have argued that the correlation between computed using isotopes. cooling rates and Ni concentrations must reflect systematic errors in model assumptions or parameters. If the later Acknowledgments–I wish to thank Joseph I. Goldstein and interpretation is correct, then one may find that the Fe and Ni Jijin Yang for the clear and diligent answers that they gave isotopic compositions that are predicted based on models of to the numerous questions I had on modeling the formation growth of Widmanstätten pattern in IVA are inconsistent with of the Widmanstätten pattern. Constructive comments by measured compositions. associate editor Edward R. D. Scott and reviewers Henning At this stage, several experiments and analyses must be Haack and Jijin Yang substantially improved the manuscript performed to put the study of Fe and Ni isotope fractionation and were greatly appreciated. I am grateful to David L. on foundations. The equilibrium fractionation between Cook, Frank M. Richter, Fang-Zhen Teng, Mathieu taenite and kamacite must be documented. This can be done Roskosz, Bernard Bourdon, John T. Wasson, Richard experimentally, but one could also resort to natural samples. If Dahringer, and Ghylaine Quitté for assistance and there is equilibrium fractionation, then the compositions discussions. Anders Meibom suggested the use of immediately at the left and at the right of the interface should mesosiderites to estimate the equilibrium fractionation show a jump corresponding to the equilibrium fractionation at between taenite and kamacite. 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