32S isotopic fractionation in different calcium carbonates

Chemical Geology 374–375 (2014) 84–91

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First-principles modeling of sulfate incorporation and 34S/32S isotopic fractionation in different calcium carbonates Etienne Balan ⁎, Marc Blanchard, Carlos Pinilla, Michele Lazzeri Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie (IMPMC), Sorbonne Universités, UPMC Univ Paris 06, UMR CNRS 7590, Muséum National d'Histoire Naturelle, IRD UMR 206, 4 Place Jussieu, F-75005 Paris, France

a r t i c l e

i n f o

Article history: Received 17 January 2014 Received in revised form 25 February 2014 Accepted 1 March 2014 Available online 13 March 2014 Editor: Michael E. Böttcher Keywords: CAS Ab initio Sulfur isotope fractionation

a b s t r a c t Incorporation mechanism of sulfate groups in major calcium carbonates (calcite, aragonite and vaterite) is investigated using first-principles quantum-mechanical calculations. For each mineral, the stable structure of the substituted site is determined. In calcite and aragonite, a tilting of the sulfate-group with respect to the orientation of planar carbonate groups is observed. The theoretical vibrational properties of the sulfate-bearing carbonates are determined together with those of anhydrite (CaSO4) and isolated SO2 molecule. Comparison with experimental data supports the substitution of sulfate for carbonate groups in carbonate minerals. The energetic of sulfate incorporation is increasingly unfavorable in the order vaterite, calcite, and aragonite. Regarding isotopic fractionation properties, our calculations suggest that the equilibrium 34S/32S isotopic fractionation factor between vaterite and calcite is small (~0.5‰ at 0 °C). A slightly larger fractionation factor (~1‰ at 0 °C) is expected between aragonite and calcite; whereas a negative fractionation (~−1.5‰ at 0 °C) is expected between anhydrite and calcite. Finally, it is suggested that the isotopic fractionation between calcite and aqueous sulfate is smaller than 4‰. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Incorporation of trace elements in authigenic minerals provides key information to unravel past chemical–physical conditions of Earth's surface environments. This is particularly true for the isotopic composition of molecular anions, such as sulfates and borates, incorporated at small concentration levels in carbonate group minerals (e.g. Burdett et al., 1989; Hemming and Hanson, 1992). More specifically, the carbonate-associated sulfate (CAS) is considered as an efficient proxy of the sulfur isotope composition of ancient oceans (e.g. Burdett et al., 1989; Kampschulte & Strauss, 2004). Experimental observations indeed show that the sulfur isotopic composition of sulfate groups in modern biogenic carbonate minerals is very similar to that of dissolved seawater sulfate (e.g. Kampschulte & Strauss, 2004). As a consequence, the CAS sulfur isotope composition is assumed to reflect that of the seawater; which is controlled by the balance between oxidized sulfate and reduced forms of sulfur. These latter sulfur forms display a lighter sulfur isotope composition and are predominantly buried in sediments (e.g. Halevy et al., 2012). Its determination bears on our understanding of global environmental changes, including the snowball Earth hypothesis, the cause of mass extinctions and the factors controlling the atmospheric oxygen concentration over geological times (e.g. Hurtgen et al., 2002; Berner, 2005; Gellatly and Lyons, 2005; Riccardi et al., 2006; Newton ⁎ Corresponding author. E-mail address: [email protected] (E. Balan).

http://dx.doi.org/10.1016/j.chemgeo.2014.03.004 0009-2541/© 2014 Elsevier B.V. All rights reserved.

and Bottrell, 2007). Compared with evaporitic sulfate deposits, CAS is considered to bring a more continuous and representative time record of the seawater sulfate isotope composition (Strauss, 1999). The CAS sulfur isotope composition also seems relatively resistant to meteoric diagenesis processes leading to the replacement of aragonite by calcite (Gill et al., 2008). In calcium carbonate minerals precipitated from seawater, sulfate concentrations range from 1000 to 10,000 ppm in modern samples and decrease to 0 to 1000 ppm in geological samples (Gill et al., 2008, and references therein). Infrared and X-ray absorption spectroscopic investigations of sulfur-bearing carbonate minerals consistently indicate that sulfate is the predominant sulfur speciation and that sulfur does not occur as gypsum or anhydrite inclusions (Takano et al., 1980; Pingitore et al., 1995; Riccardi, 2007; Fernández-Díaz et al., 2010). An increase in the cell parameters of synthetic calcite samples is observed as a function of their sulfate concentration (Kontrec et al., 2004). A control of the CaCO3 polymorphism and modifications of carbonate-minerals solubility by sulfate groups have been reported from precipitation or dissolution experiments (Busenberg and Plummer, 1985; Fernández-Díaz et al., 2010; Bots et al., 2011). In situ atomic-force microscopic observations have also revealed significant modifications of the calcite growth morphology in presence of sulfate (Vavouraki et al., 2008). These observations suggest that the sulfate groups occur at a crystallographic site of the carbonate structure, most probably replacing carbonate groups. Consistently, Reeder et al. (1994) found direct evidences for the substitution of the larger SeO2− ions at the carbonate site of calcite using X-ray 4

E. Balan et al. / Chemical Geology 374–375 (2014) 84–91

absorption fine structure spectroscopy, although some uncertainties remain on the atomic-scale model of the substituted site. The replacement of a triangular molecule by a tetrahedral one implies significant structural modifications that can be determined using atomistic modeling tools. The study of Fernández-Díaz et al. (2010) suggests that in calcite and aragonite, three of the sulfate oxygen atoms occupy a position close to that of the carbonate oxygen atoms in non-substituted sites; whereas major structural distortions occur around the apical sulfate oxygen atom. In vaterite, a different configuration of the sulfate group is observed, the S\O bonds tending to be parallel to the C\O bonds (Fernández-Díaz et al., 2010). These different configurations are associated with different capacity of the polymorphs to incorporate sulfate groups. A stabilizing effect of a small sulfate concentration is expected for the vaterite structure; whereas the sulfate incorporation is less favorable in calcite, and even less in aragonite (Fernández-Díaz et al., 2010). Considering the significant changes in the sulfate local environment between solid phases and aqueous solution, their potential effect on equilibrium fractionation factors also deserves a special attention. In the present study, we use a first-principles modeling approach to analyze the incorporation mechanism of sulfate groups in major calcium carbonates (calcite, aragonite and vaterite) and its relation with the isotopic composition of sulfate-sulfur. Compared with the previous study of Fernández-Díaz et al. (2010), which is based on an empirical description of the chemical bonds, the explicit treatment of the electronic structure enables the computation of their vibrational and infrared absorption properties; improving the comparison of the atomicscale models with experimental spectroscopic observations. A similar approach has been previously used to validate atomic-scale models of OH-bearing defects in nominally anhydrous minerals (e.g. Balan et al., 2011) or carbonate defects in fluorapatite (Yi et al., 2013). Using this theoretical approach, one can also determine reliable isotopic fractionation factors (e.g. Schauble et al., 2006; Méheut et al., 2007, 2009, 2010; Blanchard et al., 2009, Rustad and Dixon, 2009; Javoy et al., 2012), thus providing important constraints on the magnitude of equilibrium isotopic fractionation between sulfate forms.

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(Pmcn, 20 atoms per unit cell), and vaterite. The periodic models of CAS were built by substituting one sulfate group for one carbonate group in the unit cell. In order to minimize as much as possible spurious interactions between the periodic images of the sulfate defects, 2 × 2 × 2 super-cells were used to model CAS in calcite (rhombohedral cell, 80 atoms) and aragonite (160 atoms). For vaterite, the primitive cell (90 atoms) of the most stable model (space group P3221) previously proposed by Demichelis et al. (2012) was used. The structural and vibrational properties of pure anhydrite (CaSO4, 12 atoms per primitive unit cell) and those of an isolated SO2 molecule were also calculated for comparison. The SO2 molecule was modeled using a cubic periodic model of 12.8 Å edge. Calculations were done within the density functional theory framework, using periodic boundary conditions and the generalized gradient approximation (GGA) to the exchange-correlation functional as proposed by Perdew, Burke and Ernzerhof (PBE; Perdew et al., 1996). The ionic cores were described by ultra-soft pseudopotentials from the GBRV library (Garrity et al., 2014). The electronic wave-functions and charge density were expanded in plane-waves with 60 and 300 Ry cutoffs, respectively, leading to a convergence of the total energy of ~ 1 mRy/atom. The structure relaxations were done using the PWscf code of the Quantum Espresso package (Giannozzi et al., 2009; http:// www.quantum-espresso.org). The forces on atoms were minimized to less than 10−4 Ry/a.u. The Brillouin zone sampling of CAS models and SO2 molecule was restricted to a single k-point; which is appropriate to treat such systems with large unit-cell. Anhydrite properties were determined using a denser 3 × 3 × 3 k-point grid. The harmonic dynamical matrix and dielectric quantities, such as the Born effective charges and the electronic dielectric tensor, were calculated at the Brillouin zone center (Γ point), using the linear response theory (Baroni et al., 2001) as implemented in the PHonon code (Giannozzi et al., 2009; http:// www.quantum-espresso.org). 3. Results and discussion 3.1. Structure of sulfate-bearing models

2. Theoretical methods Calculations were performed on the three common CaCO3 polymorphs: calcite (R3c, 10 atoms per rhombohedral unit cell), aragonite

Models of CAS were obtained by substituting one sulfate group for one carbonate group in the carbonate mineral super-cells. This leads to SO24 −/CO23 − ratio of 6.66, 3.22 and 5.88% for calcite, aragonite and

Table 1 Cell parameters and S\O bond length of sulfate-bearing minerals. The theoretical cell parameters of the constant volume models are those of the pure carbonate phases. Experimental cell parameters and selected bond lengths of the pure phases are given in parenthesis. Constant volume models (SO4/CO3) (%)

Relaxed volume models d S\O (Å)

S-calcite

6.66

a (Å) α (°)

12.87 (12.75) 46.17 (46.06)

S\O1 S\O2 S\O3 S\O4

1.47 1.50 1.49 1.50

S-aragonite

3.22

a (Å) b− c−

10.03 (9.92) 16.03 (15.93) 11.64 (11.48)

S\O1 S\O2 S\O3 S\O4

1.48 1.48 1.49 1.48

S-vaterite

5.88

a (Å) c−

7.26 25.47

S\O1 S\O2 S\O3 S\O4

1.47 1.49 1.49 1.51

Anhydrite

–

d S\O (Å) a (Å) b− c− α (°) β− γ− a (Å) b− c− α (°) β− γ− a (Å) b− c− α (°) β− γ− a (Å) b− c−

13.03 13.00 13.00 45.6 45.7 45.6 9.99 16.07 11.83 90.0 90.0 90.1 7.32 7.27 25.47 90.1 89.7 119.9 7.10 (6.993) 7.11 (6.995) 6.29 (6.245)

Experimental data: calcite (Markgraf and Reeder, 1985), aragonite (De Villiers, 1971), anhydrite (Hawthorne and Ferguson, 1975).

S\O1 S\O2 S\O3 S\O4

1.48 1.50 1.49 1.50

S\O1 S\O2 S\O3 S\O4

1.48 1.48 1.49 1.49

S\O1 S\O2 S\O3 S\O4

1.47 1.49 1.49 1.51

S\O1 S\O2

1.492 (1.473) 1.494 (1.474)

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vaterite, respectively. We note here that compared with a modeling approach based on empirical potentials (e.g. Fernández-Díaz et al., 2010), the improved transferability and predictability related to a first-principles approach is obtained at the expense of the unit-cell size of the periodic system that can be realistically handled. This limits the investigated sulfate concentration to relatively large values. For each model, two types of relaxation were performed. In the first relaxation scheme (here referred to as constant-volume relaxation), only the atomic positions are relaxed whereas the cell parameters are fixed to those of the theoretically relaxed pure carbonate minerals. This is the approach usually followed to model isolated point defects in minerals using periodic boundary conditions (e.g. Balan et al., 2011). The cell parameters of the pure carbonate minerals slightly

overestimate their experimental counterparts (Table 1); an overestimation usually observed for calculations performed at the GGA level. In the second relaxation scheme, both the atomic positions and cell parameters are optimized under zero pressure. The cell volume changes by + 1.1, + 1.5 and + 0.9% for the calcite, aragonite and vaterite CAS model, respectively (Table 1). In calcite and aragonite, a dominant elongation of the cell dimension along the direction perpendicular to the CO3 groups is observed, which is consistent with the results of X-ray diffraction experiments on natural and synthetic sulfate-bearing calcite (Takano, 1985; Kontrec et al., 2004). However, both relaxation schemes lead to a very similar local structure of the defects (Table 1) and only the results of the constant-volume relaxation will be discussed in further details.

Fig. 1. Theoretical structure of sulfate-bearing calcium carbonates (right side): (a) calcite (b) aragonite (c) vaterite. For comparison, the structure of the pure carbonate phase viewed along the same direction is reported on the left side. The black triangles represent the carbonate groups. The yellow sphere is the sulfur atom. The oxygen (red sphere) labeled “O1” is the one defined as apical in the text.

E. Balan et al. / Chemical Geology 374–375 (2014) 84–91

Following Fernández-Díaz et al. (2010), the starting models of CAS in calcite and aragonite were built by setting one of the triangular face of the sulfate tetrahedron at the position of the triangular carbonate group and the oxygen atom at the remaining apex in interstitial position. After relaxation, a tilting of the sulfate group is observed. In calcite (Fig. 1a), the apical S\O1 bond forms an angle of 16° with the three-fold axis of the rhombohedral cell, i.e. with the direction perpendicular to the triangular carbonate groups, and the triangle formed by the basal oxygen atoms is no more parallel to the carbonate groups. The SO24 − group is distorted with a slightly shorter apical S\O1 bond (Table 1). The apical oxygen atom enters the coordination sphere of two neighboring Ca atoms with O1\Ca distances of 2.48 and 2.57 Å. In aragonite (Fig. 1b; Table 1), the tilting of the apical S\O1 bond increases to 34° and the basal oxygen triangle is inclined by 26° from the (a,b) plane. As in calcite, the O1 atoms also contribute to the neighboring Ca coordination with O1\Ca distances of 2.46 and 2.54 Å. In calcite and aragonite, the incorporation of sulfate thus leads to a significant structural distortion. The tilting of SO24 − groups in the relaxed theoretical model suggests that between the two coordination schemes of the larger selenate ions in calcite proposed by Reeder et al. (1994), the model with a non-axial position of SeO2− groups is more favorable. 4 For vaterite, several structural models have been proposed to account for the disordering of carbonate groups and a significant variability of the vaterite structure is experimentally observed. Recent TEM investigations suggest that vaterite consists in two interspersed crystal structures; the major one exhibiting a hexagonal structure and the other one remaining undetermined (Kabalah-Amitai et al., 2013). The vaterite structure was approximately described using partial carbonate occupancies in highly symmetric hexagonal models (Kamhi, 1963), or misaligned domains with orthorhombic cell (Le Bail et al., 2011). Models with a larger hexagonal cell account for some degree of longrange ordering and have been recently investigated by theoretical modeling (Wang and Becker, 2009; Demichelis et al., 2012). Significant uncertainties thus exist on the disordered structure of vaterite and no model can be considered as definitely accepted. However, the limited departure from a highly symmetric model suggests that a relevant information on the local arrangement of SO4 groups in vaterite can be obtained by simply substituting one of the CO3 group of the most stable structure (with space group P3221) proposed by Demichelis et al. (2012); which displays four nonequivalent CO3 groups. Thereby, it is assumed that the magnitude of the structural distortions induced by the substitution of SO4 for CO3 groups overpasses those related to the disordering of CO3 groups in pure vaterite. In this structure, the carbonate ions are nearly parallel to the c axis. After relaxation, one edge of the sulfate tetrahedron is horizontally oriented whereas the opposite edge is almost parallel to the c axis (Fig. 1c). The distortion of the tetrahedron is slightly larger than that observed for calcite and aragonite, with S\O distances ranging between 1.47 and 1.51 Å (Table 1). The orthorhombic anhydrite structure (CaSO4, space group Amma) displays chains of alternating edge-sharing CaO8 and SO4 polyhedron extended along the c axis (Hawthorne and Ferguson, 1975). The theoretical cell parameters and the S\O distances are slightly larger than their experimental counterparts (Table 1). Finally, the theoretical S\O distance in the SO2 molecule is 1.45 Å with a O\S\O angle of 119.2°, to be compared with experimental values of 1.431 and 119°, respectively.

3.2. Energetic of sulfate incorporation in carbonate-group minerals Information on the energetic of sulfate incorporation in carbonate minerals can be obtained by comparing the energy of the CAS model with that of a chemically equivalent mechanical mixture of pure carbonate and anhydrite (CaSO4). The corresponding mixing energy is: Emix ¼ EðCaðCO3 Þð1−xÞ ðSO4 ÞðxÞ Þ−xEðCaSO4 Þ−ð1−xÞEðCaCO3 Þ

ð1Þ

87

where E(Ca(CO3) (1 − x)(SO4) (x)), E(CaCO3) and E(CaSO4) are the energy of one mole of the corresponding phase, and x the sulfate fraction in the CAS model. In the dilute limit (x → 0), the ratio Emix / x can be considered as an estimate of the single parameter describing the excess Gibbs free energy of a hypothetical regular solid solution between calcium carbonate and calcium sulfate (see e.g. Prieto et al., 2000; Vinograd et al., 2013). According to the present calculations (Table 2), sulfate incorporation is moderately less favorable in calcite than in vaterite, and significantly less favorable in aragonite than in calcite; leading the following order for sulfate compatibility: vaterite N calcite ≫ aragonite. This order is not modified for the relaxed volume models, even though Emix / x tends to be smaller because of the minimization of residual elastic interactions between the image defects. It is consistent with the order previously determined by Fernández-Díaz et al. (2010) using empirical potentials. The Emix / x parameters presently determined for calcite and aragonite (~100 and ~150 kJ/mol, respectively) are however smaller than those assessed from the data of Fernández-Díaz et al. (2010) (~150 and ~400 kJ/mol, respectively). For vaterite, Fernández-Díaz et al. (2010) observed a linear variation of the mixing energy over the whole concentration range investigated (as for calcite and aragonite) but obtained negative mixing energies at low sulfate concentrations (b~ 3 mol%) and positive energies at larger concentrations. The results were similar for the two investigated models; which display orthorhombic (space group Pbnm) or hexagonal (space group P63/mmc) cell-geometry. As the mixing energy should vanish in the infinite dilution limit (i.e. the pure phase), the behavior observed by Fernández-Díaz et al. (2010) is unexpected. Calculations were thus performed using a 2 × 2 × 1 vaterite unit cell of the presently used P3221 model (360 atoms), which corresponds to a sulfate concentration of 1.4 mol%. The local structure of the defect is not significantly modified with a variation of S\O bond lengths smaller than 0.01 Å. In contrast to the findings of Fernández-Díaz et al. (2010), the corresponding mixing energy is still positive at this lower concentration. Noteworthy, the Emix / x parameter is 15 kJ/mol smaller for a sulfate concentration of 1.4 mol% than for a concentration of 5.5 mol%, indicating that the interaction between substituted sulfate groups is of repulsive nature. In fact, as previously observed by Demichelis et al. (2012), the orthorhombic Pbnm model used by Fernández-Díaz et al. (2010) is metastable compared with the P3221 presently used (by ~3.8 kJ/mol, according to our calculations). The optimization of the cell geometry and atomic positions of a Pbnm 2 × 2 × 2 super-cell of pure vaterite (160 atoms; Meyer, 1959), performed after a small randomization of the initial atomic positions, decreases the system energy by 2.7 kJ/mol with respect to the optimized symmetric structure. Thus, the introduction of a sulfate group in a Pbnm super-cell relaxes the symmetry constraints and drives the system toward a more stable structure, explaining why a linear extrapolation to infinite dilution of the results of Fernández-Díaz et al. (2010) leads to a negative energy value. A similar conclusion most likely holds for the P63/mmc model, which also differs from the minimal energy structure found by Demichelis et al. (2012). 3.3. Vibrational properties of sulfate-bearing calcite, anhydrite and sulfur dioxide The zone-center harmonic vibrational modes are determined by diagonalizing the dynamical matrix of the system. This leads to thirtyTable 2 Energetic of sulfate for carbonate substitution in carbonate-group minerals. The parameter n is the number of Ca atoms in the unit-cell of the model and x the corresponding sulfate molar concentration (see text). “Relaxed” corresponds to the relaxed volume models.

S-calcite S-aragonite S-vaterite S-vaterite

n

x (%)

Emix / x (kJ/mol)

Emix / x (relaxed) (kJ/mol)

16 32 18 72

6.25 3.125 5.55 1.39

103 146 93 78

97 126 90 –

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Fig. 2. Theoretical infrared absorption spectrum of sulfate-bearing calcite. The bands related to the internal vibrational modes of sulfate are indicated. Calculations are done for spherical sulfate-bearing calcite particles inserted in a homogeneous KBr matrix as in Balan et al. (2008).

three vibrational modes for the anhydrite crystal and three for the isolated SO2 molecule. For the sulfate-bearing calcite model, 240 vibrational modes are computed. The theoretical vibrational properties of the CAS models are investigated in order (i) to compare them with available spectroscopic data and (ii) to determine the corresponding sulfur isotopic fractionation properties. We therefore focus on the vibrational modes specifically involving the SO4 groups. These vibrational modes can be categorized as internal, i.e. corresponding to a distortion of the molecular group, or external, i.e. involving a rigid displacement, rotation or translation, of the molecular group. The internal modes of tetrahedral molecules correspond to one symmetric stretching (ν1), three anti-symmetric stretching (ν3), and two sets consisting in two (ν2) and three (ν4) bending modes (Farmer, 1974). The theoretical IR spectrum of the sulfatebearing calcite (Fig. 2) is computed following the approach exposed in Balan et al. (2008) and Yi et al. (2014). The nine internal modes of the sulfate group lead to well-defined absorption bands, which are distinct from the calcite vibrational modes, even though the intensity of the ν2 bending modes is much too weak to be experimentally observed.

Fig. 3. Theoretical vs. experimental vibrational frequencies. Squares: sulfate-bearing calcite; circles: anhydrite; triangles: SO2 molecule; open crosses: aqueous sulfate. Calculations for the filled symbols are from the present work. Calculations for the open crosses are from Otake et al. (2008). The dashed line is a linear fit to the filled symbols.

The internal-mode frequencies of the sulfate-bearing calcite (ν1, ν3 and ν4 modes), anhydrite and SO2 molecule can be compared to available experimental data (Table 3; Fig. 3). For the SO2 molecule, the symmetric stretching, anti-symmetric stretching and bending modes are computed at 1310, 1113 and 493 cm−1, respectively. They underestimate the corresponding experimental frequencies (1362, 1151 and 518 cm−1) by ~ 3.7%. Comparable underestimations of 5.2% and 4.5% are observed for anhydrite and sulfate-bearing calcite, respectively. We note that the splitting of the three ν3 modes in calcite is larger than that reported from experimental observations (Takano et al., 1980; Riccardi, 2007; Fernández-Díaz et al., 2010). However, the three expected contributions of ν3 modes are poorly resolved in the experimental infrared spectra (Riccardi, 2007; Fernández-Díaz et al., 2010). The small shift (5 cm−1) to a higher frequency of the ν1 mode observed in sulfate-bearing calcite compared with anhydrite, is correctly reproduced by the theoretical models. Concerning the ν4 modes, they lead to three well-defined infrared absorption bands (Fig. 2). They are clearly observed in the infrared spectra reported by Takano et al. (1980). The two components at a lower-frequency are up-shifted in S-bearing calcite compared with anhydrite, both in the experimental observations and theoretical data (Table 3). This is a further evidence for the

Table 3 Frequencies (cm−1) of internal vibrational modes of sulfate groups: present theoretical values (PBE), experimental values (exp.). The theoretical (HF/6-31-G(d)) and experimental frequencies of aqueous sulfate are from Otake et al. (2008).

ν2 ν4

ν1 ν3

S-calcite PBE (loc.)

S-calcite PBE (full)

S-calcite (exp.)⁎

S-arag. PBE

S-vaterite PBE

Anhydrite PBE

Anhydrite (exp.)⁎⁎

428 468 585 591 629 961 1070 1107 1153

431 469 587 592 631 961 1070 1106 1152

– – 610 627 674 1018 1143 1160 1180

411 493 562 623 641 976 1092 1143 1150

427 473 582 600 626 959 1075 1112 1177

395 477 557 571 636 956 1036 1075 1099

– 515 592 612 671 1013 1095 1126 1149

exp: ⁎ Takano et al. (1980). ⁎⁎ Farmer (1974).

SO42−(aq) HF/6-31-G(d)

SO42−(aq) (exp.)

422

452

582

617

950 1084

982 1105

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substitutional position of sulfate groups in the travertine samples investigated by Takano et al. (1980).

Table 4 Polynomial expansion of 1000ln(β) for 34S/32S ratio: 1000ln(β) = A 103/T + B 106/T2 + C 108/T3 + D 1011/T4.

3.4. Isotopic fractionation properties of CAS models The isotopic fractionation coefficient of an element Y between two phases a and b, referred to as α(a,b,Y) can be related to the reduced partition function ratios (also called β-factors) of the two phases by: ln α ða; b; Y Þ ¼ ln βða; Y Þ− ln βðb; Y Þ

89

S-calcite S-aragonite S-vaterite Anhydrite SO2

A

B

C

D

−1.296 −1.325 −1.307 −1.262 −0.736

12.007 12.256 12.133 11.727 6.965

−16.427 −17.101 −16.798 −15.753 −11.487

0.889 0.950 0.924 0.831 0.758

ð1Þ

in which the reduced partition function ratio β(a,Y) corresponds to the isotope fractionation coefficient between the phase a and its classical counterpart (Bigeleisen and Mayer, 1947; Richet et al., 1977). In the present study, the reduced partition function ratios were obtained from the harmonic vibrational frequencies using the Teller–Redlich rule and the expressions previously given by, e.g., Méheut et al. (2007). For anhydrite, the phonon frequencies were determined on a 3 × 3 × 3 q-point grid, corresponding to 8 non-equivalent q-points. The uncertainty of theoretical isotopic fractionation coefficients related to the use of harmonic PBE frequencies is not expected to exceed 10 rel.% (Méheut et al., 2007, 2009). For isolated molecules and isolated defects, such as sulfate groups in the structure of carbonate minerals, there is no need to compute the dispersion of phonon and the required vibrational frequencies can be obtained at a single q-point. However, the calculation of the complete dynamical matrix for a large-cell model, such as that of the sulfatebearing aragonite (161 atoms), is highly time-consuming. An approximate approach (here referred to as the localized mode approximation) was thus used by restricting the computation to the dynamical matrix coefficients involving a displacement of the S or O atoms belonging to the sulfate group. This approximation was checked on calcite by comparing the results of the diagonalization of the partial dynamical matrix (leading to 15 vibrational modes), with those of the complete dynamical matrix (243 modes). The frequency of the SO4 internal modes is modified by less than 3 cm−1 (Table 3); which is of the order of the expected precision of the theoretical model. This demonstrates that these internal modes correspond to localized vibrations that are weakly

Fig. 4. Theoretical 34S/32S reduced partition function ratio of sulfate-sulfur in calcite computed from the full-dynamical matrix (circles), the localized mode approximation (solid line) and the force constants acting on the S atom (straight line).

coupled to those of the crystalline carbonate host. The internal modes of the SO4 groups can thus be accurately determined using a reducedsize dynamical matrix. The localized mode approximation is not valid for the external modes, which are observed below 250 cm− 1 and display significant contributions from the crystal host. According to the present calculations, such low frequency vibrational modes do not significantly contribute to the sulfur isotopic fractionation of the sulfate defect. The 34S/32S reduced partition function ratio computed using the full dynamical matrix or the localized mode approximation differ by less than 0.3 per mil at 0 °C and the two values merge at high temperatures (Fig. 4). For comparison, we also determined the reduced partition function ratio by restricting the calculation to the force constants acting on the S atom (e.g. Bigeleisen and Mayer, 1947). This approximation turns out to be valid only at temperatures above ~ 800 K because of the high frequency of the S\O stretching modes (Fig. 4). The dependence of 34S/32S reduced partition function ratio as a function of temperature has been fitted using a polynomial expansion in powers of 1/T (Table 4). The 34S/32S reduced partition function ratio of the three calcium carbonates and anhydrite reaches ~ 90‰ at 0 °C (Fig. 5). They are however relatively close to each other for the calcium carbonates, leading to equilibrium fractionation factors not exceeding 3‰ at 0 °C (Fig. 6). In particular, the equilibrium fractionation between vaterite and calcite is smaller than 0.5‰ at 0 °C and the two phases are therefore hardly distinguishable within the present approximation level. In contrast, a higher 34S/32S reduced partition function ratio is

Fig. 5. Theoretical 34S/32S reduced partition function ratio of sulfate-sulfur in calcite, aragonite and vaterite compared with that of anhydrite and SO2 molecule. Values reported by (1) Otake et al. (2008) and (2) Richet et al. (1977) are reported for the SO2 molecule. Theoretical values for gas-phase and aqueous sulfate molecules of Otake et al. (2008) are also reported (dotted lines). The inset is an enlarged view of the low-temperature part of the figure clarifying the relative position of the curves.

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observed for aragonite, suggesting that under equilibrium conditions sulfate-sulfur composition in aragonite would be ~ 1‰ heavier than that in calcite. Compared with anhydrite, calcite displays an increase in 34S/32S reaching ~1.5‰ at 0 °C. The theoretical 34S/32S reduced partition function ratio can also be compared to previous determinations (Fig. 5). For the SO2 molecule, our approach leads to values close to those calculated by Richet et al. (1977) using experimental spectroscopic data. In comparison, higher values of the 34S/32S reduced partition function ratio of SO2 have been determined at the Hartree–Fock level by Otake et al. (2008). This difference can be traced back to the overestimation of the force constants related to the Hartree–Fock approximation. In contrast, the vibrational frequencies of the aqueous sulfate molecule reported by Otake et al. (2008) underestimate their experimental counterparts by ~ 3% (Table 3). This underestimation is thus similar to that observed for the frequencies calculated in the present study using the PBE functional (Fig. 3). Accordingly, it is reasonable to compare the theoretical fractionation properties of aqueous sulfate molecule reported by Otake et al. (2008) to those obtained in the present study on the solid phases. The corresponding isotopic fractionation between calcite and aqueous sulfate is smaller than 4‰. Considering the uncertainty related to the fact that these two theoretical results have been obtained using different theoretical approaches (density functional theory for the present study, Hartree–Fock theory for Otake et al., 2008), the small magnitude of the theoretical calcite–aqueous sulfate fractionation is consistent with the experimental observation that the sulfur isotopic composition in modern carbonate samples (δ34S = 21.2 ± 0.8‰) is nearly identical to the isotopic composition of corresponding sea water (20.9 ± 0.5‰) (Kampschulte and Strauss, 2004). Burdett et al. (1989) also did not find any significant difference between the sulfur isotopic composition of modern carbonates and seawater. Finally, we note that an equilibrium fractionation factor of −1.5‰ is expected between anhydrite and calcite at 0 °C. However, this fractionation factor cannot be directly applied to observations made on evaporitic samples, for which the sulfate is almost conservatively transferred from the seawater to the solid phase.

Fig. 6. Equilibrium 34S/32S isotopic fractionation factor between models of substituted sulfate in calcium carbonates and anhydrite.

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