Titanium diffusion in olivine

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ScienceDirect Geochimica et Cosmochimica Acta 147 (2014) 43–57 www.elsevier.com/locate/gca

Titanium diffusion in olivine Daniele J. Cherniak a,⇑, Yan Liang b a b

Department of Earth and Environmental Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, USA Department of Earth, Environmental and Planetary Sciences, Brown University, Providence, RI 02912, USA Received 26 June 2014; accepted in revised form 19 October 2014; Available online 24 October 2014

Abstract Diffusion of Ti has been characterized in natural olivine and synthetic forsterite. Experiments on the natural olivines were run under buffered conditions (IW and NNO), and those on synthetic forsterite were run in air. Titanium diffusion appears relatively insensitive to crystallographic orientation and oxygen fugacity under the range of investigated conditions, and diffusivities are similar for Fe-bearing olivine and forsterite. For Ti diffusion in synthetic forsterite, we obtain the following Arrhenius relation for diffusion over the temperature range 900–1400 °C: DForst ¼ 5:97  1014 expð203  19 kJ mol1 =RTÞ m2 s1 : For Ti diffusion in San Carlos olivine, we obtain the following “global” fit to all diffusion data (incorporating diffusion parallel to b and c-axes, and experiments run under IW and NNO buffers), over the temperature range 1050–1254 °C: Dol-SC ¼ 2:11  1014 expð195  32 kJ mol1 =RTÞ m2 s1 : Titanium diffusivities in olivine are similar to those of the trivalent REEs, but considerably slower than those of Cr, Ca, and Fe–Mg in olivine. Titanium diffusivities in olivine, diopside and orthopyroxene are comparable over investigated temperature ranges, differing by about half a log unit at 900 °C, but have increasing variance at lower temperatures given the higher activation energies (by 40–70 kJ mol1) for Ti diffusion in pyroxene compared with olivine. These large differences in cation mobility among Ti, Cr, Ca, Fe–Mg and the REE in pyroxene and olivine may allow us to distinguish dominant processes that give rise to the chemical disequilibria in olivine and pyroxene. With decreasing temperature, Ti is preferentially partitioned from olivine to pyroxene in ultramafic rocks, giving rise to characteristic reversed zoning in olivine and orthopyroxene and normal zoning in clinopyroxene. Diffusive exchange models with temperature-dependent diffusion and partition coefficients have been developed for the olivine–pyroxene bi-mineralic and tri-mineralic systems, allowing us to assess cooling rates and closure temperatures of the ultramafic and mafic rocks. Ó 2014 Elsevier Ltd. All rights reserved.

1. INTRODUCTION

⇑ Corresponding author at: Department of Earth and Environ-

mental Sciences, Science Center 1W19, Rensselaer Polytechnic Institute, 110 8th St., Troy, NY 12180, USA. Tel.: +1 518 276 8827; fax: +1 518 276 2012. E-mail address: [email protected] (D.J. Cherniak). http://dx.doi.org/10.1016/j.gca.2014.10.016 0016-7037/Ó 2014 Elsevier Ltd. All rights reserved.

Titanium is a trace element in olivine, which is the first mineral to crystallize from mantle-derived silicate melts at pressures below multiple saturation depth. Olivine coexists with at least one pyroxene in mafic and ultramafic rocks such as lherzolite, harzburgite, wehrlite, websterite, and dunite, are major constituents of the Earth’s upper mantle and lower crust. There is considerable interest in the

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geochemistry of Ti and other trace elements in olivine in mantle rocks and mantle derived igneous rocks (e.g., Milman-Barris et al., 2008; De Hoog et al., 2010; Foley et al., 2011, 2013). Under typical upper mantle and lower crust conditions, titanium is tetravalent (Ti4+) and has an ˚ , which is smaller than that of ionic radius of 0.605 A ˚ ) or Fe2+ (0.78 A ˚ ) at the M1 site (sixfold coorMg2+ (0.72 A dination), but larger than Si in fourfold coordination (0.26 ˚ for Si and Ti, respectively) (Shannon, 1976). vs. 0.42 A Under very reducing conditions, such as the lunar interior, a portion of titanium may be trivalent, although the proportion of Ti4+ and Ti3+ is still not well quantified. Titanium has been found to energetically favor the Si site in anhydrous forsterite, but under hydrous conditions may also be incorporated on the M1 site in substitution for Mg, with the most stable configuration involving edge sharing with a vacant Si site and OH groups formed to stabilize the underbonded O around the Si site (Berry et al., 2007). Schma¨dicke et al. (2013) have demonstrated that the coupled substitution Ti4+ + 2H+ ! Mg2+ + Si4+ is the most important means of incorporating water in olivine in upper mantle rocks, and thus a dominant means of water storage in the shallow upper mantle. Hence measurements of Ti contents of olivine can be a useful way to estimate the maximum amount of water incorporated in olivine through this coupled substitution and may also provide a means to approximate the extent of water loss, since water will be more mobile than Ti. Quantification of the diffusivity of Ti in olivine will provide additional constraints on these determinations. According to Foley et al. (2013), titanium abundances in olivine in kimberlites and basaltic rocks (up to 340 ppm) are typically higher than those in olivine in peridotites (typically <70 ppm) that have not been affected by mantle metasomatism. These authors also suggested that differences in Ti and other trace element (Al, Ca, and Ni) abundances in olivine can be used to discriminate between igneous and mantle olivine. One possible explanation of this dichotomy in Ti abundances in olivine in mantle rocks and mantle derived igneous rocks is subsolidus redistribution of Ti among olivine and pyroxenes. Titanium is incompatible in olivine relative to pyroxene, and its olivine–pyroxene partition coefficients decrease with decreasing temperature (e.g., Witt-Eickschen and O’Neill, 2005; Sun, 2014). Hence, during subsolidus re-equilibration, titanium diffuses out of olivine (and orthopyroxene) and into adjacent clinopyroxene. The extent of Ti redistribution depends on cooling rate and Ti diffusion rates in olivine and pyroxene, as well as the temperature-dependent olivine–pyroxene Ti partition coefficient. In a recent study, we measured tracer diffusion coefficients of Ti in natural orthopyroxene and clinopyroxene over a range of temperatures (900–1250 °C) and oxygen fugacity (IW, NNO, QFM, and air, Cherniak and Liang, 2012). We showed that the rates of Ti diffusion in pyroxene are comparable to the rates of REE diffusion in orthopyroxene and the rates of Al and middle REE diffusion in diopside at magmatic temperatures, but 1–3 orders of magnitude smaller than the rates of Fe and Mg diffusion in pyroxenes. During subsolidus re-equilibration, titanium dif-

fuses out of orthopyroxene and into coexisting clinopyroxene, producing characteristic “bell shaped” reversed Ti zoning in the former but “U shaped” normal Ti zoning in the latter. Such contrasting zoning patterns can be readily distinguished from simple normal or reversed zoning in coexisting pyroxenes produced by crystal-melt fractionation processes, and can be used to infer cooling rates of two-pyroxene bearing mafic and ultramafic rocks. In this study, we report results from an experimental investigation of Ti diffusion in natural olivine and synthetic forsterite over a range of temperatures (900–1400 °C), oxygen fugacity (IW, NNO, and in air) and major element compositions (Fo88.5–Fo100). Based on consideration of ionic charge and size, we would expect Ti4+ diffusion in olivine to be slower than those of divalent cations Ca and Ni, as well as Fe–Mg interdiffusion and Cr3+ diffusion, and perhaps comparable to Si, given that Ti may substitute on tetrahedral sites in olivine. However, other studies (Spandler et al., 2007; Spandler and O’Neill, 2010; Jollands et al., 2013) suggest that Ti diffusivities are comparable to those of Fe–Mg and Ni in olivine. Given the paucity of Ti diffusion data in olivine, we first focus our effort on anhydrous systems and at 1 bar. In a companion study, we will examine the effects of pressure and water content on Ti diffusion in olivine. Our measured Ti diffusion data for olivine complement earlier diffusion measurements of Ti in orthopyroxene and clinopyroxene, and permit us to assess the potential for diffusive fractionation of Ti among olivine and pyroxenes during magmatic and metamorphic processes under upper mantle and lower crust conditions. 2. EXPERIMENTAL PROCEDURE Titanium diffusion experiments were conducted with synthetic forsterite and natural olivine from two localities. The synthetic forsterite was from the Morion Corporation, and natural olivines from Kilbourne Hole, New Mexico (designated as KBH), and San Carlos, Arizona (SCO). Compositions of the olivines are presented in Table 1, which shows analyses for these materials originally reported by Thomas et al. (2008) and Morgan and Liang (2005). Samples selected for experiments were optically clear and free of inclusions. The synthetic forsterite samples were oriented to measure diffusion parallel to the c-axis, and to the a-axis, while the San Carlos olivine was oriented to measure diffusion parallel to the c- and b-axes. The Kilbourne Hole olivine, which was in small crystal fragments of a few mm in size, was not oriented prior to experiments given the size of the grains, but instead the largest faces of grains were polished. All specimens were polished with SiC paper, followed by polishing with 1.0 lm and 0.3 lm alumina powders, and finished with a chemical polish using a colloidal silica suspension. Following polishing, samples were cleaned ultrasonically in distilled water and ethanol. The source of diffusant for the experiments on the synthetic forsterite was a mixture of dried MgO, SiO2 and TiO2 powders (in molar ratio 20:9:1), combined and heated at 1250 °C overnight, then thoroughly mixed with synthetic forsterite powder (Alfa-Aesar, 99% purity) in 1:2 proportions

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45

Table 1 Compositions of olivines used in this study. San Carlos olivinea

KBH olivineb

Synthetic forsteritea

SiO2 TiO2 Al2O3 FeO MnO MgO NiO CaO Cr2O3

40.70 (0.26) tr. tr. 11.12 (0.23) 0.16 (0.05) 47.78 (0.25) 0.35 (0.08) 0.09 (0.02) 0.03(0.03)

41.02 (0.13) – – 8.66 (0.07) 0.14 (0.01) 50.13 (0.14) 0.34 (0.02) 0.31 (0.03) 0.12 (0.01)

42.96 (0.27) 0.02 (0.02) – – tr. 56.70 (0.26) – – tr.

Total Fo

100.22 88.5

100.72 91.2

99.68 100

a b

In wt% from Thomas et al. (2008); From Morgan and Liang (2005).

(by weight); this mixture was then heated for an additional two days at 1250 °C in a Pt crucible. To make the source for experiments on natural olivine, the pre-reacted MgO–SiO2– TiO2 powder mixture described above was thoroughly mixed in a 1:2 ratio (by weight) with finely ground San Carlos olivine. An overview of the use of powder sources as a means of introducing diffusants in experimental studies, along with a discussion of the consistency of diffusivities obtained by these methods and those from other types of experiments, is presented by Watson and Dohmen (2010). Experiments on synthetic forsterite were run in air, and those on the natural olivines were run under buffered conditions. For experiments run in air, the source and polished forsterite samples were placed in Pt capsules, which were then crimped shut. For buffered experiments, the source material and olivine were placed in crimped Pt capsules. The capsules were placed in silica glass capsules along with a crimped AgPd or Pt capsule containing either a mixture of iron flakes and wu¨stite powder (to buffer at IW) or a mixture of Ni and NiO powders (to buffer at NNO). Inside the silica glass capsules, silica glass chips were placed between the capsule containing the solid buffer and the capsule containing the source and sample to physically separate the sample and buffer. The assemblies in the silica glass tubes were then sealed under vacuum. The use of similar solid buffer methods to control oxygen fugacity in diffusion experiments has been discussed in previous work (Cherniak and Liang, 2012). Following diffusion anneals, the buffer materials were examined to check for the presence of buffer phases. Since Fe–Mg diffusion is relatively rapid in olivine (e.g., Chakraborty, 1997, 2010) and diffusion of some species in olivine may be affected by variations in Fe–Mg concentrations, we also conducted experiments to assess the effects on Ti diffusion in cases where Fe–Mg concentration gradients were induced in olivine. An experiment run with synthetic forsterite used the same source as described above for the experiments on San Carlos olivine, with Fe present in the source (run TiForst-15 in Table 2); an experiment on San Carlos olivine was run with a more Fe-rich source, which consisted of the source used for the other experi-

ments on San Carlos olivine mixed in a 3:1 ratio (by weight) with synthetic fayalite powder (run TiSCO-11 in Table 3). Both of these experiments were run with NNO buffers in sealed silica capsules, prepared in the manner described above for the experiments on Fe-bearing olivine. The capsules prepared as above for 1-atm experiments were annealed in 1-atm vertical tube furnaces for times ranging from a few hours to a few months at temperatures from 900 to 1400 °C. Temperatures were monitored with chromel–alumel (type K) thermocouples for temperatures below 1100 °C, and Pt-Pt10%Rh (type S) thermocouples for temperatures 1100 °C and above, with temperature uncertainties typically  ±2 °C in both cases. On completion of the diffusion anneals, samples were quenched by removing them from furnaces and permitting them to cool in air. Samples were then extracted from the capsules, freed of residual source material and cleaned ultrasonically in successive baths of distilled water and ethanol to prepare for analysis with Rutherford Backscattering Spectrometry (RBS). SEM images of samples (shown in Supplementary Fig. S1) following diffusion anneals and cleaning show isolated fragments of material clinging to sample surfaces, likely fragments of the source material or residual polishing compound. Given the distribution and composition of the material on the surface, it is unlikely to affect RBS spectra. Because the presence of hydrous species may affect the site preference for Ti in forsterite (e.g., Berry et al., 2007), which may influence diffusivities since Ti may migrate via different mechanisms, we also ran a Ti diffusion experiment in the presence of water vapor (run TiForst-16 in Table 2). For this experiment, polished forsterite and the Ti-doped source material were placed in a 5 mm Pt capsule, which was welded shut. The prepared capsule was placed in a small open silica tube in a horizontal tube furnace which maintained a temperature of 1100 °C and water vapor pressure of 50 torr. The atmosphere of water vapor pressure of 50 torr in the furnace was generated by passing air through a hot water bath kept at 80 °C; the air then flowed through the furnace tube. Although the sample was not directly exposed to the water vapor, the Pt capsule permits exchange of H from the atmosphere through the capsule

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Table 2 Ti diffusion in synthetic forsterite. T (°C) TiForst-9 TiForst-8c TiForst-8a TiForst-7 TiForst-6c TiForst-6a TiForst-4 TiForst-3 TiForst-10 TiForst-12 TiForst-15b TiForst-16a TiForst-1 TiForst-2 TiForst-5 TiForst-13 TiForst-11 TiForst-14 a b

1400 1351 1351 1302 1250 1250 1203 1148 1149 1151 1148 1100 1099 1100 1050 1000 950 895

D (m2 s1)

Time (s) 3

7.20  10 1.44  104 1.44  104 4.86  104 7.20  104 7.20  104 2.48  105 4.42  105 1.09  106 1.40  105 4.54  105 3.58  105 3.10  105 5.83x105 1.45  106 3.32  106 5.98  106 7.60  106

20

5.46  10 3.98  1020 4.82  1020 1.32  1020 6.29  1021 5.61  1021 4.76  1021 1.42  1021 1.13  1021 2.63  1021 1.48  1021 2.35  1021 9.28  1022 8.99x10–22 5.37  1022 4.32  1022 1.85  1022 8.43  1022

Log D

+/–

Orientation

Ti concentration (apfu)

19.26 19.40 19.32 19.88 20.20 20.25 20.32 20.85 20.95 20.58 20.83 20.63 21.03 21.05 21.21 21.36 21.73 22.07

0.39 0.36 0.21 0.21 0.19 0.34 0.11 0.14 0.17 0.30 0.22 0.21 0.31 0.15 0.22 0.29 0.24 0.32

[0 0 1] [0 0 1] [1 0 0] [0 0 1] [0 0 1] [1 0 0] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1]

0.0050 0.0054 0.0128 0.0071 0.0073 0.0055 0.0092 0.0082 0.0071 0.0061 0.0190 0.0074 0.0086 0.0103 0.0061 0.0045 0.0057 0.0099

Experiment run in water vapor furnace, 50 torr H2O pressure. Source with higher Fe concentration, buffered at NNO.

walls (e.g., Ebisuzaki et al., 1968; Sato and Tomozawa, 2004). 2.1. RBS analysis RBS has been used as an analytical method in several diffusion studies of olivine, including those for tungsten diffusion (Cherniak and Van Orman, 2014), REE diffusion (Cherniak, 2010), Ar diffusion (Watson et al., 2007; Thomas et al., 2008), and Fe–Mg interdiffusion (BertranAlvarez et al., 1992; Jaoul et al., 1995). The analytical approach used here is similar to that used in our work on REE diffusion in olivine (Cherniak, 2010) and Ti diffusion in pyroxene (Cherniak and Liang, 2012), using 4He+ beams with energies ranging between 2 and 3 MeV for analysis. Spectra were converted to Ti concentration profiles employing procedures comparable to those outlined in the former publication. The resultant profiles were fit with a model to determine the diffusion coefficient (D). Diffusion of Ti in olivine is modeled as simple one-dimensional, concentration independent diffusion in a semi-infinite medium with a source reservoir maintained at constant concentration (i.e., a complementary error function solution, C = C0erfc(x/(4Dt)1/2)). The rationale for the use of this model has been discussed in previous publications (e.g., Cherniak and Watson, 1992, 1994). Diffusivities are evaluated by plotting the inverse of the error function (i.e., erf1((C0  C(x,t))/C0)) vs. depth (x) in the sample. A straight line of slope (4Dt)1/2 results if the data satisfy the conditions of the model. C0, the surface concentration of diffusant, is independently determined by iteratively varying its value until the intercept of the line converges on zero. Typical diffusion profiles for olivines and their inversions through the error function are shown in Fig. 1. Titanium surface concentrations measured in diffusion profiles range from 0.0305 to 0.0045 Ti p.f.u., broadly similar

to Ti solubilities in olivine measured by Hermann et al. (2005) which ranged from 0.0230 to 0.0016 p.f.u., but are higher than concentrations in the range of a few tens to 100 weight ppm measured in the studies of Spandler et al. (2007) and Spandler and O’Neill (2010). The uncertainties in concentration and depth from each data point [mainly derived from counting statistics and background (in the natural olivine, primarily due to the presence of Fe) in the former and detector resolution in the latter] were used to evaluate the uncertainties in the diffusivities determined from the fits to the model. 3. RESULTS Results from the Ti diffusion experiments on synthetic forsterite are presented in Table 2 and plotted in Fig. 2a. The results from experiments measuring Ti diffusion in natural olivine are presented in Table 3 and plotted in Fig. 2b. Fig. 3 shows results from a time-series study of diffusion at 1150 °C, which yields similar diffusivities for Ti in synthetic forsterite for experimental durations ranging over about a factor of 8. The time-series study provides supporting evidence that the measured concentration profiles represent volume diffusion and are not a consequence of other phenomena such as surface reaction that may otherwise result in enhanced Ti concentrations in the near-surface region. For experiments run in air over the temperature range 900–1400 °C, we obtained the following Arrhenius relation for diffusion parallel to the c-axis of forsterite, DForst ¼ 5:97  1014 expð203  19 kJ mol1 =RT Þ m2 s1 ðlog D0 ¼ 13:22  0:71Þ; ð1Þ where R is the gas constant. Titanium diffusion parallel to the b-axis is similar. The diffusivity of Ti in forsterite under water vapor present is also consistent with the other diffusion data for forsterite (triangle in Fig. 2a).

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47

Table 3 Ti diffusion in natural olivine. D (m2 s1)

Log D

+/–

Buffer

Ti concentration (apfu)

Olivine, Kilbourne Hole, New Mexico TiKBHO-7 1254 1.65  105 TiKBHO-8 1251 1.44  105 TiKBHO-5 1200 2.70  105 TiKBHO-6 1200 2.70  105 TiKBHO-1 1149 4.25  105 TiKBHO-2 1149 4.25  105 TiKBHO-3 1101 5.83  105 TiKBHO-4 1101 5.83  105 TiKBHO-9 1050 1.19  106 TiKBHO-10 1050 1.13  106

2.56  1021 5.76  1021 1.53  1021 3.72  1021 1.08  1021 1.62  1021 1.39  1021 1.06  1021 2.24  1022 3.14  1022

20.59 20.24 20.82 20.43 20.97 20.79 20.86 20.97 21.65 21.50

0.23 0.24 0.29 0.38 0.26 0.30 0.27 0.35 0.49 0.42

IW NNO IW NNO NNO IW IW NNO NNO IW

0.0207 0.0184 0.0164 0.0146 0.0133 0.0186 0.0115 0.0142 0.0105 0.0114

Olivine, San Carlos, Arizona Diffusion parallel to b TiSCO-7b 1254 TiSCO-8b 1251 TiSCO-5b 1200 TiSCO-6b 1200 TiSCO-1b 1149 TiSCO-2b 1149 TiSCO-11a 1148 TiSCO-3b 1101 TiSCO-4b 1101 TiSCO-9b 1050 TiSCO-10b 1050

1.65  105 1.44  105 2.70  105 2.70  105 4.25  105 4.25  105 4.54  105 5.83  105 5.83  105 1.19  106 1.13  106

5.80  1021 4.35  1021 1.23  1021 3.33  1021 1.19  1021 1.33  1021 1.32  1021 9.19  1022 1.04  1021 3.97  1022 3.06  1022

20.24 20.36 20.91 20.48 21.92 20.88 20.88 21.04 20.98 21.40 21.51

0.21 0.29 0.29 0.23 0.30 0.32 0.29 0.24 0.24 0.34 0.25

IW NNO IW NNO NNO IW NNO IW NNO NNO IW

0.0120 0.0172 0.0215 0.0235 0.0134 0.0126 0.0305 0.0155 0.0151 0.0115 0.0269

Diffusion parallel to c TiSCO-7c 1254 TiSCO-8c 1251 TiSCO-6c 1200 TiSCO-5c 1200 TiSCO-1c 1149 TiSCO-2c 1149 TiSCO-3c 1101 TiSCO-4c 1101 TiSCO-9c 1050 TiSCO-10c 1050

1.65  105 1.44  105 2.70  105 2.70  105 4.25  105 4.25  105 5.83  105 5.83  105 1.19  106 1.13  106

4.71  1021 4.27  1021 1.94  1021 1.06  1021 1.69  1021 9.16  1022 7.90  1022 1.52  1021 3.26  1022 4.73  1022

20.33 20.37 20.71 20.97 20.77 21.04 21.10 20.82 21.49 21.33

0.22 0.22 0.24 0.46 0.31 0.46 0.28 0.32 0.33 0.37

IW NNO NNO IW NNO IW IW NNO NNO IW

0.0129 0.0218 0.0255 0.0120 0.0126 0.0164 0.0180 0.0134 0.0158 0.0141

T (°C)

a

Time (s)

Source with higher Fe concentration.

For diffusion under IW-buffered conditions, we obtain the following Arrhenius relation, from a regression of the data for the KBH olivine and the San Carlos olivine in both orientations: DIW ¼ 5:56  1015 expð180  37 kJ mol1 =RT Þ m2 s1 ðlog D0 ¼ 14:25  1:36Þ:

ð2Þ

For these olivines under NNO-buffered conditions, we obtain the Arrhenius relation: DNNO ¼ 2:98  1014 expð198  41 kJ mol1 =RT Þ m2 s1 ðlog D0 ¼ 13:53  1:51Þ ð3Þ Considering the effects of crystallographic orientation on the natural (San Carlos) olivine, we obtain the following Arrhenius relations for diffusion parallel to b and c, respectively, fitting data from both NNO- and IW-buffered experiments: DSCOb ¼ 6:77  1014 expð209  44 kJ mol1 =RTÞ m2 s1 ðlog D0 ¼ 13:17  1:62Þ; ð4Þ

DSCOc ¼ 4:80  1015 expð177  47 kJ mol1 =RTÞ m2 s1 ðlog D0 ¼ 14:32  1:73Þ: ð5Þ The Arrhenius relations in Eqs. ((2)–(5)) demonstrate that Ti diffusion in natural olivine is relatively insensitive to orientation and that Ti diffusion in olivine appears to show little dependence on oxygen fugacity (Fig. 2b and c). They are also within a half-order of magnitude of Ti diffusivities measured for forsterite in experiments run in air. Given the lack of significant dependence of diffusion on crystallographic orientation and oxygen fugacity, we have also performed “global” fits that incorporate all data obtained for the natural olivine: Dol-all ¼ 1:25  1014 expð189  28 kJ mol1 =RT Þ m2 s1 ðlog D0 ¼ 13:90  1:01Þ: ð6Þ A similar “global” fit for the San Carlos olivine only yields the following Arrhenius relation,   Dol-SC ¼ 2:11  1014 exp 195  32kJ mol1 =RT m2 s1 ðlog D0 ¼ 13:67  1:18Þ ð7Þ

D.J. Cherniak, Y. Liang / Geochimica et Cosmochimica Acta 147 (2014) 43–57

Ti concentration (x1020/cm3)

1.2

TiForst-6c

Ti concentration (x1020/cm3)

48

0.8

0.4

(a)

TiForst-7b

1.6

1.2

0.8

0.4

(c) 0.0

0.0 0

10

20

30

40

50

60

0

70

10

1.6

erf-1((Co - C)/Co)

erf-1((Co - C)/Co)

1.2

(b)

1.4

20

30

40

50

60

50

60

Depth (nm)

Depth (nm)

1.2 1.0 0.8 0.6 0.4

(d)

1.0 0.8 0.6 0.4 0.2

0.2

0.0

0.0 0

10

20

30

40

50

60

70

0

80

10

20

30

40

Depth (nm)

Depth (nm)

Fig. 1. Typical concentration profiles for Ti diffusion in synthetic forsterite (a, b) and natural (San Carlos) olivine (c, d), from runs TiForst6c and TiSCO7c, respectively. In (a) and (c), the measured diffusion profiles are plotted with complementary error function curves. In (b) and (d), the data are linearized by inversion through the error function. Slopes of the lines are equal to (4Dt)1/2. C is the Ti concentration at a given depth, and C0 is the Ti concentration at the olivine surface.

T (oC)

T (oC) 1400

1100

1200

1000

1400

900

-19

-19

1200

1100

1000

900

-19

Arrhenius relation for synthetic forsterite

diffusion llb in H2O vapor furnace

-21 Fe-bearing source

San Carlos IIb, NNO

-20

log D (m2sec-1)

log D (m2sec-1)

log D (m2sec-1)

diffusion llc

-20

San Carlos IIc, NNO

San Carlos IIb, IW

-21

KBH, IW

-20 o

1250 C -21

1050oC

San Carlos IIc, IW

-22

-22

(a) Forsterite 6.0

7.0

1/T (x104 K-1)

(b) Natural olivine 8.0

9.0

6.0

KBH, NNO

7.0

1/T (x104 K-1)

8.0

(c)

-22 9.0

0

-2

-4

-6

-8

-10 -12 - 4

116 -

log fO2

Fig. 2. Arrhenius plots of Ti diffusion data for forsterite (a) and natural olivine (b), run under a range of experimental conditions. Ti diffusivities for transport parallel to the b- and c-axes are similar for both natural olivine and synthetic forsterite, suggesting that anisotropy of Ti diffusion in olivine is not significant. For synthetic forsterite, Ti diffusion in the presence of water vapor is similar to that under dry conditions, indicating little effect of the presence of hydrous species on Ti diffusion. For natural olivine (Fig. 2b), Ti diffusivities are similar for experiments run with IW and NNO buffers (circles and squares, respectively), and fall along or near the line defining the Arrhenius relation for Ti diffusion in synthetic forsterite from experiments run in air (gray line), indicating that Ti diffusion is relatively insensitive to fO2 over the investigated range of conditions. This is also evident in Fig. 2c, where diffusivities for San Carlos olivine parallel to b and for the synthetic forsterite at two temperatures (1050 and 1250 °C) are plotted as a function of log fO2. Ti diffusivities from experiments run in air (white triangles), and under NNO-buffered (grey squares) and IW-buffered conditions (black circles) all agree within uncertainties at both temperatures. In the color Fig. 2b, data for San Carlos olivine parallel to c, parallel to b, and for the KBH olivine are plotted with red, blue, and green symbols, respectively. Experiments run with Fe-bearing and Fe-rich sources, respectively, for both forsterite and natural olivine, yield similar diffusivities to those measured in the other experiments, indicating that variations in Fe concentration have little effect on Ti diffusion over the compositional range investigated. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

D.J. Cherniak, Y. Liang / Geochimica et Cosmochimica Acta 147 (2014) 43–57

log D (m2sec-1)

-20 synthetic forsterite, IIc SCO, IIc, NNO SCO, llb, IW

-21 SCO, IIb, NNO

SCO, llc, IW

1150 °C

-22 0

50

100

150

200

250

300

time (h) Fig. 3. Time-series study of Ti diffusion in olivine at 1150 °C. Diffusivities of Ti in synthetic forsterite are quite similar over times differing by about a factor of 8, suggesting that volume diffusion is the dominant process being measured. Also plotted are Ti diffusivities in natural olivine for two different orientations [diffusion parallel to c- (squares) and b-axes (triangles)], and diffusivities for experiments run under IW-buffered and NNO-buffered conditions (black and white symbols, respectively), illustrating the relative insensitivity of Ti diffusion to both crystallographic orientation and oxygen fugacity.

The Arrhenius relations are generally quite similar, again pointing to the relative insensitivity of Ti diffusivities to oxygen fugacity, orientation, and composition within investigated ranges. Diffusivities in forsterite and San Carlos olivine from experiments run with more Fe-rich sources (TiForst15 and TiSCO-11, respectively; Tables 2 and 3) are similar to those found for the other samples, suggesting that differences in Fe concentration gradients have little effect on Ti diffusivities. Since Fe–Mg diffusion is rapid compared with that of Ti, near-surface Fe concentrations in both samples appear uniform over the depth accessible in the RBS spectra (with Fe constituting 7% and 20% of M cations in the near-surface regions in forsterite and San Carlos olivine from these experiments), and thus over the depth range of the Ti diffusion profiles. 4. DISCUSSION In this section, we discuss extant cation diffusion data in olivine in context with our new results for Ti (Fig. 4a and b). In considering these comparisons, it is important to bear in mind that different species may be preferentially sited on specific sites in the olivine structure. For example, larger cations such as the REE and Ca prefer the M2 site of olivine. Ti has been observed to substitute on the Si site in forsterite under anhydrous conditions (Berry et al., 2007), but is present on M1 (sixfold coordinated) sites under hydrous conditions, with compensation achieved through the presence of hydroxyl groups. At 900 °C, titanium diffusivities are about a half order of magnitude smaller than Dy (Cherniak, 2010), more than two orders of magnitude greater than W (Cherniak and Van Orman, 2014), and two orders of magnitude smaller than Cr (Ito and Ganguly, 2006) (Fig. 4a). The activation energy for Ti diffusion is considerably smaller than that for the REE (289 kJ mol1), chromium (299 kJ mol1 for diffusion parallel to c) and tungsten (365 kJ mol1). Compared with

49

divalent cations, titanium diffusivities are more than 3 orders of magnitude smaller than Ca (Coogan et al., 2005), and 4 orders of magnitude smaller than Fe–Mg (Dohmen and Chakraborty, 2007; Chakraborty, 1997) and Ni (Petry et al., 2004) diffusion in olivine. The comparatively slow diffusivity of Ti with respect to divalent elements is not surprising given its high charge. However, activation energies for diffusion of these divalent species, with values of 207, 201, and 220 kJ mol1 for Ca diffusion, Fe–Mg interdiffusion, and Ni diffusion, respectively, are comparable to those measured for Ti. Since Ti may substitute on the tetrahedral site in the olivine lattice (e.g., Hermann et al., 2005; Berry et al., 2007), it is useful to compare Si diffusivities with those of Ti (Fig. 4b). As in the case of Ti, there is little apparent dependence of Si diffusion on fO2 (Houlier et al., 1990; Costa and Chakraborty, 2008). Titanium diffusivities, however, are considerably faster than those of Si under dry conditions (Dohmen et al., 2002; Fei et al., 2012); but slower than Si diffusion under hydrous conditions (Costa and Chakraborty, 2008) at temperatures above 1000 °C. The transition from dry to wet diffusion mechanisms for Si in olivine has been proposed to take place with incorporation of relatively small concentrations of H (less than 10 ppm) into the olivine lattice (Chakraborty, 2010), so the data of Costa and Chakraborty (2008) may be most relevant for comparison. However, Ti4+, despite its larger size (0.42 ˚ in fourfold coordination, Shannon, 1976), has vs. 0.26 A a much lower activation energy for diffusion than measured for Si in olivine in all of these studies, where values are 410 ± 30 kJ mol1 for forsterite under dry conditions (Fei et al., 2012); 529 ± 41 kJ mol1 for diffusion in Fe-bearing olivine under dry conditions (Dohmen et al., 2002) and 358 ± 28 kJ mol1 for Fe-bearing olivine under hydrous conditions (Costa and Chakraborty, 2008). As noted above, the activation energy for Ti diffusion more closely approximates those for Fe–Mg interdiffusion (Chakraborty, 1997; Dohmen and Chakraborty, 2007) and diffusion of divalent elements Ni and Ca (Petry et al., 2004; Coogan et al., 2005), although Ti diffusivities are orders of magnitude slower. Si has been proposed to diffuse via a mechanism involving vacancy complexes (Costa and Chakraborty, 2008; Chakraborty, 2010), but it is problematic to speculate on a similar mechanism for Ti diffusion given the significant differences in activation energies and diffusivities compared with Si. That activation energies for Ti diffusion are comparable to those of the divalent cations might suggest the possibility of a similar diffusion mechanism. Ni, Fe–Mg and Ca diffusion have positive dependences on oxygen fugacity, and at oxygen fugacities above 1010 Pa, Dohmen and Chakraborty (2008) describe Fe–Mg interdiffusion as occurring via a fO2 dependent transition metal extrinsic (TaMED) mechanism, and a fO2 independent pure intrinsic (PED) mechanism at lower fO2 and temperature (below 800 °C; Chakraborty, 2010). Over the temperature range of our experiments, the NNO-buffered experiments will range up to 2 log units above an fO2 of 1010 Pa, while the IW-buffered experiments will all be below this value. We can also consider our data in light of other studies of Ti diffusion in olivine, which include Spandler et al. (2007)

50

D.J. Cherniak, Y. Liang / Geochimica et Cosmochimica Acta 147 (2014) 43–57 1400

-15

1200

800 T(°C)

1000

-14

1400

1200

Ti

Ni Fe Fe-Mg -M g

-16

Dy

-19

Ca Ca

-20

Cr Cr

log D (m2sec-1)

-18

-21 -22

FeMg Fe-Mg -18

Si (

CSi ) (C)

-20

Ti ((this work) Ti this wo rk)

-22

(b)

D)

Ti

WW

(a)

( Si

-23

T(°C)

-16

-17

log D (m2sec-1)

1000 Spandler et al. (2007); Spandler and O'Neill (2010)

Si ( Si (F)

Si (D)

F)

-24

-24 5

6

7

8

9

10

1/T (104/K)

6

7

8

9

1/T (104/K)

Fig. 4. Diffusion of Ti and other elements in olivine, compared with Ti diffusion measured in the present work. (a) Ti diffuses several orders of magnitude slower than divalent cations and Cr. (b) Ti diffusion is generally faster than Si diffusion under dry conditions, and slower than Si diffusion under hydrous conditions at higher temperatures (greater than 1000 °C), but has a considerably lower activation energy for diffusion than Si, closer in value to those for divalent cations. Ti diffusivities measured in this work are 4–5 orders of magnitude lower than those determined by Spandler et al. (2007) and Spandler and O’Neill (2010). Sources for data: Fe–Mg – Dohmen et al. (2007); Ca – Coogan et al. (2005); Cr – Ito and Ganguly (2006); REE – Cherniak (2010); W – Cherniak and Van Orman (2014); Ti – this study; Spandler et al. (2007), Spandler and O’Neill (2010); Si (D) – Dohmen et al. (2002); Si (F) – Fei et al. (2012); Si (C) – Costa and Chakraborty (2008).

and Spandler and O’Neill (2010) (Fig. 4b). Spandler et al. (2007) determined diffusion of Ti and other elements in olivine by measuring concentration gradients in grains of natural Fe-bearing olivine (Fo90) containing melt inclusions doped with several elements. Following annealing of these samples, they measured concentration gradients in the olivine extending out from the melt inclusions with laser ablation ICP-MS. Spandler et al. (2007) obtained a diffusivity for Ti of 1.0  1015 m2 s1 at 1300 °C (for log fO2 = 9.7 bar). Diffusion of Ti in natural (San Carlos) olivine at this temperature (1300 °C) but at a slightly higher oxygen fugacity log (fO2 = 8.3 bar [corresponding to QFM -1]) was measured by Spandler and O’Neill (2010). In this study, the diffusing species entered the olivine from a doped silicate melt, and concentration profiles were characterized with laser ablation ICP-MS. Spandler and O’Neill (2010) obtained Ti diffusivities of 5.1  1016, 4.4  1016, and 3.9  1015 m2 s1 for diffusion along [1 0 0], [0 1 0] and [0 0 1], respectively. These diffusivities are more than three orders of magnitude greater than those for Ti measured in the present work, but similar to rates of Fe–Mg interdiffusion in olivine (Chakraborty, 1997; Dohmen et al., 2007) (Fig. 4b). It is unclear why these large differences exist between Ti diffusivities in olivine measured in these studies and those in the present work, but similar differences have been noted for diffusion of REE in olivine, with REE diffusivities measured by Spandler et al. (2007) and Spandler and O’Neill (2010) orders of magnitude faster than those measured by Cherniak (2010) and Remmert et al., (2008). Recent work by Burgess and Cooper (2013) suggests that the comparatively rapid diffusivities measured for Ti, the REE and other highly-charged elements by Spandler and O’Neill (2010) are a consequence of the formation of extended defects in the olivine which provide fast paths for diffusion. The formation of these defects may be facili-

tated by the high concentration of TiO2 and other trace elements contained in the melts used in these experiments as a source of diffusant. 5. APPLICATIONS Diffusivities of Ti in olivine obtained from this study can be used to understand a number of geochemical problems involving olivine, and in Section 5.1 we will first discuss mean closure temperatures for Ti in olivine. Since olivine often coexists with pyroxene and we have previously measured Ti diffusion in pyroxene (Cherniak and Liang, 2012), we will also consider Ti diffusion and partitioning between olivine and pyroxene. In Fig. 5 we plot the Arrhenius relations for Ti diffusion in olivine, enstatite and diopside. Titanium diffusivities in pyroxene do not differ greatly from those for olivine over the investigated temperature range, but Ti diffusion in olivine will be faster than in pyroxene at temperatures below 900 °C, but within an order of magnitude at temperatures down to 700 °C. Because of the lower activation energy for Ti diffusion in olivine compared with pyroxene, titanium diffusivities in olivine will be slower than those in pyroxenes at temperatures above 1150 °C. We will explore this interesting feature in the Sections 5.2 and 5.3 below. 5.1. Calculated mean closure temperatures Dodson (1973) derived the classical expression for mean closure temperature (Tc) for slowly cooling systems, which takes the form:   E ART 2c D0 ; ð8Þ ¼ ln RT c Ea2 s_

D.J. Cherniak, Y. Liang / Geochimica et Cosmochimica Acta 147 (2014) 43–57

-19

1300

1000

1100

1400

T (°C)

ivi

ps

ne

id

e

-21

En

st

-22

-23 6.0

1100

7.0

T0 = 1100oC

1000 Dodson

o

T0 = 900 C

800

o

T0 = 800 C

700

at

7.5

o

T0 = 1000 C 900

ite

o

T0 = 700 C

600 500 10-5

6.5

T0 = 1200oC

o

Ol

T0 = 1300 C

1200

io

Closure T ( C)

log D (m2sec-1)

D

o

1300

Ti diffusion -20

51

8.0

8.5

1/T (x104 K-1) Fig. 5. Titanium diffusion in olivine compared with Ti diffusivities for diopside and enstatite measured by Cherniak and Liang (2012). Titanium diffusivities do not greatly differ over the investigated temperature range, but Ti diffusion in olivine is faster than in pyroxene at temperatures below 900 °C and within an order of magnitude at temperatures down to 700 °C. Because of the lower activation energy for Ti diffusion in olivine compared with pyroxene, these differences increase with decreasing temperature; for example, Ti diffusion at 500 °C will be two orders of magnitude faster in olivine than in orthopyroxene or clinopyroxene.

where E and D0 are the activation energy and pre-exponential factor for diffusion of the element of interest; s_ is the cooling rate at Tc; a is the effective diffusion radius; and A is a geometric factor. Dodson’s derivation of closure temperature is based on several assumptions (Dodson, 1973, 1986); among them the condition that the concentration of the diffusing species throughout the grain (even at its core) is significantly removed from its concentration at peak or initial temperature (T0). This assumption, which makes Tc independent of T0, is, as Ganguly et al. (1998) note, not satisfied for slowly diffusing species unless grain sizes are very small. Ganguly et al. (1998) and Ganguly and Tirone (1999) have modified Dodson’s expressions for closure temperature by including a correction term to the geometric factor in Eq. (8). Given that diffusion of Ti is relatively slow in olivine, we will use both the original and modified version of Dodson’s equation in the closure temperature calculations below. Fig. 6 displays mean closure temperatures for Ti in olivine as a function of effective diffusion radius for a cooling rate of 10 °C/Myr and several choices of initial temperature T0. (Additional examples with different values of cooling rates and a fixed grain size can be found in Supplementary Fig. S2.) In each case, curves for different values of T0 are shown (700–1300 °C), which have been calculated using the extension of the Dodson formulation from Ganguly and Tirone (1999) and software developed by these authors. These curves are plotted along with that calculated using the conventional Dodson (1973) expression (Eq. (8), heavy solid line). As shown in Fig. 6 (and Fig. S2), the closure temperature curves will converge upon the Dodson values for small grain radii and comparatively high peak temperatures, but significant departures may occur for larger grains

10-4

10-3

10-2

Effective diffusion radius (m)

Fig. 6. Variations of mean closure temperatures for Ti in olivine as a function of effective diffusion radius at a cooling rate of 10 °C/ Myr and several choices of peak temperatures T0, calculated using the adaptation of Dodson’s (1973) closure equation for a limited extent of diffusion, developed by Ganguly and Tirone (1999). Curves calculated using the classical Dodson (1973) equation are shown for comparison (heavy black line). See text for further discussion.

and lower initial temperatures. It is evident that effective mean closure temperatures for Ti in olivine may be significantly lower than Dodson values predict for all but the highest peak temperatures. For two elements in the same mineral, Eq. (8) or its modification by Ganguly and Tirone (1999) defines a “closure curve” in a closure temperature vs. closure temperature diagram (as the product a2 is canceled out). This is illustrated in Fig. 7, where we compare mean closure temperatures for Ti in olivine with those for REE, Cr, Ca, Ni, and Fe–Mg in olivine for an initial temperature of 1300 °C. (The same data are also plotted as a function of effective diffusion radius in Supplementary Fig. S3). The “closure curve” for a given element pair (e.g., Cr vs. Ti in Fig. 7) depends only on initial temperature and is a convenient way to compare closure temperatures of different elements in the same mineral. As shown in Fig. 7, there is crossover of Ti and REE closure temperatures (at 840 °C), with those for the REE higher than those for Ti for small grains (less than  a few hundred microns) and lower cooling rates (less than a few °C/Myr). Mean closure temperatures for Ti in olivine are generally higher than those of Cr, Ca, Ni, and Fe–Mg interdiffusion in olivine. Closure temperatures for Fe–Mg interdiffusion in olivine are considerably lower than those for the REE and Ti, not surprising given the comparatively faster diffusion rates of the former. Given an effective grain size a, it may be possible to better constrain cooling rates of olivine-bearing mafic and ultramafic rocks by comparing closure curves for elements with different diffusivities. We can also consider closure temperatures for Ti in olivine in comparison to those for Ti in pyroxene. In Fig. 8, we plot mean closure temperatures, calculated as above, using both Dodson’s classical equation and the modifications of Dodson’s formulation by Ganguly and Tirone (1999), using a peak temperature of 1300 °C and a cooling rate

52

D.J. Cherniak, Y. Liang / Geochimica et Cosmochimica Acta 147 (2014) 43–57

will be about 35 °C lower than that for diopside, and 60 °C lower than for enstatite.

Grain radius (mm) 0.1 1200

o

10

o

10 C/Myr o T0 = 1300 C

1100

Closure T ( C)

1

5.2. Time scale of diffusive re-equilibration between olivine and pyroxene

REE

1000 Cr

900 800

Ca

700 g Fe-M Fe-Mg

600 500 400 700

800

900

1000

1100

1200

o

Ti Closure T ( C)

Closure Temperature (oC)

Fig. 7. Comparison of mean closure temperatures for Ti in olivine with those for trivalent REE, Cr, Ca, Ni, and Fe–Mg interdiffusion in olivine. Curves were calculated using the adaptation of Dodson’s (1973) closure equation for a limited extent of diffusion, developed by Ganguly and Tirone (1999), using a peak temperature T0 of 1300 °C. Diffusion data sources: REE – Cherniak (2010); Cr – Ito and Ganguly (2006); Ni – Petry et al. (2004); Fe–Mg – Dohmen et al. (2007) for Fe–Mg. The dashed line is a 1:1 line. See text for additional discussion.

1300

T0 = 1300oC

1200

dT/dt = 10 C/Myr

o

1100 1000 REE - OL

Ti - Di

900 800

Ti - En

tD ¼

L2A ; bDA

Ti - OL Fe-Mg OL

600

10-4

10-3

10-2

10-1

Effective diffusion radius (m)

Fig. 8. Closure temperatures for Ti in olivine, compared with those for trivalent REE and Fe–Mg interdiffusion in olivine, and Ti diffusion in pyroxene. Curves were calculated using the adaptation of Dodson’s (1973) closure equation for a limited extent of diffusion, developed by Ganguly and Tirone (1999), using a peak temperature T0 of 1300 °C. Closure temperature calculations are made with a fixed cooling rate (10 °C/Myr), and variable grain radius. Closure temperatures for Ti in pyroxene will be higher than those for olivine except in the case of large grain sizes and rapid cooling rates. See text for additional discussion.

of 10 °C/Myr as a function of olivine grain size. (A similar figure but as a function of cooling rate and a fixed grain size of 1 mm can be found in Supplementary Fig. S4.) Curves are calculated for Ti in diopside and enstatite using the data of Cherniak and Liang (2012). Closure temperatures for Ti in pyroxene will be higher than those for olivine except in the case of large grain sizes and rapid cooling rates; for 1 mm grains at a cooling rate of 10 °C/Myr, Tc for olivine

ð9Þ

where DA is the diffusion coefficient for the element of interest in mineral A; b is a geometric factor, accounting for crystal shape. To achieve approximately 95% of equilibrium value in terms of mean concentration in the mineral grain, b takes on values of 1, 4, or 5 for plane sheet of half-length L, cylinder or sphere of radius L, respectively. For a closed system containing two minerals A and B, a diffusive reequilibration time can also be defined. In the absence of a fluid or melt phase, the diffusive re-equilibration time for the bi-mineralic system is a weighted sum of diffusive reequilibration times for the two minerals (Liang, 2014), viz.,  2   2  LA LB 1 þ w ð10aÞ tD ¼ w1 B A bDA bDB W1 A ¼

700

10-5

The time scale of diffusive re-equilibration is important in assessing the role of diffusion-limited processes in petrologic and geochemical systems (e.g., Watson and Baxter, 2007, and references therein). Longer diffusive time scales relative to petrologic processes of interest indicates the potentially important role of chemical disequilibrium, whereas shorter diffusive time scales allow one to make the assumption of local equilibrium at the mineral grain scale. For mono-mineralic systems, a diffusive reequilibration time (tD) is given by the simple expression

/A k AB /B ; w1 B ¼ /A k AB þ /B /A k AB þ /B

ð10bÞ

where /A and /A are volume fractions of the minerals A and B; kAB is the mineral A–B partition coefficient for the element of interest. Hence the time scale of diffusive reequilibration for the bi-mineralic system is bounded by the time scales of re-equilibration for the two minerals. Compared to mono-mineralic systems, the time scales of diffusive re-equilibration for bi-mineralic systems depend additionally on mineral modal abundances and mineral– mineral partition coefficients. Fig. 9 displays the Ti olivine–clinopyroxene and olivine–orthopyroxene partition coefficients as a function of temperature for a spinel lherzolite. Titanium is highly incompatible in olivine relative to pyroxene. From 1200 to 900 °C, Ti partition coefficients decrease from 0.028 to 0.012 for olivine–clinopyroxene and 0.078 to 0.047 for olivine–orthopyroxene. Fig. 10a and b display calculated diffusive re-equilibration times for Ti in olivine + orthopyroxene and olivine + clinopyroxene bi-mineralic aggregates as a function of olivine volume fraction at two selected temperatures. For purpose of illustration, a grain size of 1 mm is used for olivine and pyroxene. For both olivine + pyroxene bi-mineralic systems, the re-equilibration time is dominated by that of olivine unless the volume proportion of pyroxene is small (<20% at 1200 °C and <5% at 900 °C). This is due to the very small

D.J. Cherniak, Y. Liang / Geochimica et Cosmochimica Acta 147 (2014) 43–57

(Figs. 5, 9 and 10). This intriguing behavior is referred to as the minor’s rule: the mineral with lower equilibrium abundance of the element of interest contributes more to the overall diffusive re-equilibration time in a bi-mineralic system (Liang, 2014).

−1

Partition Coefficient

10

5.3. Kinetic fractionation of Ti in wehrlite and lherzolite during subsolidus re-equilibration

−2

10

olivine/cpx olivine/opx 900

1000

1100

1200

1300

1400

T (°C)

Fig. 9. Variations of olivine–clinopyroxene and olivine–orthopyroxene Ti partition coefficients as a function of temperature. For simplicity, we assume constant major element compositions in the olivine and pyroxene. From Sun (2014).

Ti olivine–pyroxene partition coefficients, which weight the time scales toward olivine (cf. Eqs. 10a and 10b). Since the partition coefficient is defined as the concentration ratio of the trace element in two minerals at final equilibrium, i.e., 1 k AB ¼ C 1 A =C B , the weighting factors in Eq. 10b can also be written as: /A C 1 /B C 1 1 A B ; 1 ; wB ¼ 1 þ /B C B /A C A þ /B C 1 B

ð10cÞ

/A C 1 A

which are mass fractions of the element of interest in mineral A and mineral B at equilibrium relative to its total mass in the bulk bi-mineralic aggregate. Since the final equilibrium concentration of Ti in olivine is considerably lower than that in orthopyroxene, and more so in clinopyroxene when the modal abundance of pyroxene is not too small, the bi-mineralic system approaches equilibrium mostly by adjusting Ti abundance in olivine even when the Ti diffusivity in olivine is smaller than that in pyroxene at 1200 °C

Cherniak and Liang (2012) discussed diffusive fractionation of Ti between coexisting clinopyroxene and orthopyroxene during subsolidus re-equilibration using a 1-D diffusion model with temperature dependent diffusion coefficients and orthopyroxene–clinopyroxene partition coefficients. Here we expand their diffusive re-equilibration model by considering olivine in a bi-mineralic wehrlite and a tri-mineralic lherzolite and by treating diffusion in a spherical geometry. The model setup includes diffusion in spherical geometry, local chemical equilibrium at the mineral–mineral interface, and conservation of total diffusive flux across mineral–mineral grain boundaries. For a system with N minerals, conservation of total diffusive flux in a local representative elementary volume (REV) takes the form, N X Dj /j @C j j ¼ 0; Rj @r r¼Rj j¼1

where Rj and /j are effective radius and volume fraction of mineral j in REV; Dj and Cj are diffusivity and concentration of Ti in mineral j, respectively. The diffusive flux in Eq. (11) for mineral j is evaluated at the surface of j. Eq. (11) is obtained by assuming fast grain boundary diffusion within the REV (e.g., Eiler et al., 1992). The diffusion equations are solved numerically using a finite difference method. Fig. 11 shows two examples of calculated Ti concentration profiles in clinopyroxene (left column) and olivine (middle column) in a wehrlite (65% olivine + 35% clinopyroxene) at five selected times (in Myr). Total Ti abundances

2.6

900

ol 800

2.2

Equilibration Time (Myr)

Equilibration Time (Myr)

2.4

2 1.8 1.6 1.4 1.2 1 0.8 0

ð11Þ

(a)1200°C olivine−opx olivine−cpx 0.2

0.4

opx

olivine−opx olivine−cpx

(b) 900°C 700 600 500 400

cpx

800

w1 A ¼

53

300 200

cpx 0.6

Olivine Fraction

0.8

1

100 0

0.2

0.4

0.6

0.8

1

Olivine Fraction

Fig. 10. Variations of diffusive re-equilibration times in bi-mineralic olivine + orthopyroxene and olivine + clinopyroxene aggregates at 1200 °C (a) and 900°C (b) as a function of olivine phase proportions in the systems. For reference, diffusive re-equilibration times for monomineralic olivine, orthopyroxene, and clinopyroxene in spherical geometry are also shown as circles.

54

D.J. Cherniak, Y. Liang / Geochimica et Cosmochimica Acta 147 (2014) 43–57

in the two cases are the same (600 ppm). Initially, the two minerals are in chemical equilibrium at 1300 °C. The system cools down to 800 °C in 100 Myr following the parabolic cooling path shown in Fig. 11c. In the first example (Fig. 11a and b), grain radii of olivine and clinopyroxene are the same (1 mm). In the second example (Fig. 11d–f), a fraction of clinopyroxene (10% modal abundance) has a smaller grain size (0.25 mm) and the remaining clinopyroxene (25% modal abundance) has the same grain size as olivine (1 mm). In Fig. 12 we further expand the previous example by replacing the 25% clinopyroxene in the wehrlite with 25% orthopyroxene (1 mm radius, left column), turning it into lherzolite with 65% olivine, 25% orthopyroxene, and 10% clinopyroxene. (Additional examples with a faster cooling rate can be found in the online Supplementary Fig. S5.) Three interesting observations can be readily made. During subsolidus re-equilibration, titanium diffuses out of olivine and orthopyroxene and into adjacent clinopyroxene, producing characteristic “bell shaped” reversed zoning in olivine and orthopyroxene and “U shaped” normal

60

(a)

55

Ti in olivine (ppm)

1680

1660

1640

50

1200

45

1150

40 35 30 25

t = 0.05 t=5 t = 25 t = 75 t = 100

20

1620

15 1600 −1

−0.5

0

0.5

10 −1

1

Radius (mm)

0

55

1700

Ti in olivine (ppm)

1680

1660

1640

0.5

1

13

-5

00

(t/

tm

ax

) 1/2

0.5

800 0

1

50

100

Time (Myr) 1720

(f) 1700

45 40 35 30 t = 0.05 t=5 t = 25 t = 75 t = 100

25

15 0

=

00

950

(e)

20

1620

Radius (mm)

T

1000

850

50

−0.5

1050

900

60

(d)

Ti in cpx (ppm)

−0.5

1100

Radius (mm)

1720

1600 −1

(c)

1250

Ti in small cpx (ppm)

Ti in cpx (ppm)

1700

1300

(b)

Temperature (°C)

1720

zoning in clinopyroxene. This is due mainly to the fact that Ti is (highly) incompatible in olivine and orthopyroxene relative to clinopyroxene (Fig. 9). Since Ti is an incompatible trace element in olivine and a minor element the pyroxenes, the contrasting zoning patterns in the three coexisting minerals cannot be produced by crystal-melt fractionation processes (e.g., melting, crystallization, or melt–rock reaction), and hence can be used as a telltale sign of diffusion-limited subsolidus re-equilibration. Cooling-induced reversed Ti (as well as Al, Cr, and Ca) zoning in orthopyroxene neoblasts was reported for a porphyroclastic mantle xenolith from West Eifel, Germany (Witt-Eickschen, 2007). Using their measured Ti diffusion data in pyroxenes, Cherniak and Liang (2012) modeled the “bell shaped” and “U shaped” zoning patterns of Ti in orthopyroxene and clinopyroxene in a lherzolite from the Trinity ophiolite and found their measured Ti concentration profiles can be explained by a cooling model whereby temperatures decrease from 1300 to 800 °C in 2.44 to 3.81 Myr, depending on cooling functions used in their modeling. With Ti diffusion data in olivine, it may be possible to further constrain cooling

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Fig. 11. Calculated Ti concentration profiles in clinopyroxene (first column) and olivine (second column) at 5 selected times (in Myr) during continuous cooling of a wehrlite (65% olivine and 35% clinopyroxene) for two choices of clinopyroxene grain sizes. (a) and (b) uniform grain size of 1 mm for both olivine and clinopyroxene. Grain sizes of clinopyroxene in the second example (d)–(f) have two populations: one with a radius of 1 mm (25% modal abundance) and the rest with a radius of 0.25 mm (10% modal abundance). Initially, the two minerals are in chemical equilibrium at 1300 °C. The system cools down to 800 °C following the parabolic cooling path shown in panel (c) where tmax = 109.1 Myr. For simplicity and the purpose of demonstration, we neglect major element variations in olivine and pyroxene during subsolidus re-equilibration. Temperature-dependent Ti diffusion and partition coefficients used in the calculations are shown in Figs. 7 and 8.

D.J. Cherniak, Y. Liang / Geochimica et Cosmochimica Acta 147 (2014) 43–57 3800

100

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t = 0.05 t=5 t = 25 t = 75 t = 100

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t = 0.05 t=5 t = 25 t = 75 t = 100

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Fig. 12. Calculated Ti concentration profiles in clinopyroxene (first column), olivine (second column), and orthopyroxene (third column) at 5 selected times (in Myr) during continuous cooling of a lherzolite (10% clinopyroxene, 25% Opx, and 65% olivine) for two choices of clinopyroxene grain sizes: 1 mm (a)–(c) and 0.25 mm (d)–(f). Initially, the three minerals are in chemical equilibrium at 1300 °C. The system cools down to 800 °C following the parabolic cooling path shown in Fig. 11d. The model setup is the same as in the cases shown in Fig. 11. Total Ti abundances in the two cases are the same (600 ppm).

paths in future studies by simultaneously modeling Ti zoning patterns in olivine in addition to the two pyroxenes in peridotites. Depending on cooling rate, the Ti abundance in olivine can be significantly lower than that at magmatic temperatures (cf. Figs. 11b, e, and 12b, e; see also Fig. S5). Hence, the very low Ti abundance in olivine in mantle rocks relative to that in mantle-derived igneous rocks is likely a result of subsolidus redistribution of Ti among olivine and pyroxenes in the former. Depending on their tectonic settings, cooling rates experienced by residual mantle rocks varies considerably, and are typically smaller than igneous rocks crystallized from mafic and ultramafic magmas at shallow depth. For mantle peridotites from stable cratonic lithosphere, REE and Ti in olivine, orthopyroxene and clinopyroxene are in chemical equilibrium at the hand specimen scale except for LREE in samples that were affected by (recent) mantle metasomatism (e.g., Witt-Eickschen and O’Neill, 2005; Lee et al., 2007; Agranier and Lee, 2007; Liang et al., 2013). For a wehrlite or lherzolite containing 600 ppm of Ti, the equilibrium Ti abundances in olivine at a given temperature can be read from the rim concentrations in Figs. 11 and 12, which range from 15 to 90 ppm.

Finally, modal abundance, grain size and grain size distribution, and hence texture, are also important in controlling the extent of element redistribution during diffusion-limited subsolidus re-equilibration. Titanium abundances in clinopyroxene range from 1660 to 1700 ppm in the lherzolite but 3200–3300 ppm in the wehrlite at 800 °C, in spite of the same bulk Ti abundance in the two rocks (600 ppm). Further, Ti abundances in the smaller clinopyroxene grains are higher than those in the larger clinopyroxene grains, even though clinopyroxene abundance remains the same (cf. Figs. 10d, f and 11a, d). Smaller grains simply reequilibrate with their surrounding minerals more efficiently, even though zoning is still present. If the olivine grain size is smaller than that of pyroxene, its Ti abundance at 800 °C would be lower, due to the faster diffusive exchange rate in olivine and smaller olivine-clinopyroxene partition coefficient at lower temperatures. Taken collectively, the numerical examples displayed in Figs. 11 and 12 (and Fig. S5) demonstrate that mineral grain scale heterogeneities in Ti abundances in peridotite samples may in part result from diffusion-limited subsolidus redistribution processes. This conclusion is likely true for other incompatible trace elements with slow diffusivities in pyroxene and olivine (e.g., REE).

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6. SUMMARY AND CONCLUSIONS

REFERENCES

Tracer diffusion coefficients of Ti in natural olivine and synthetic forsterite have been measured, with experiments at 900–1400 °C and 1 bar in air and under buffered conditions, and diffusion profiles measured using RBS. Within the temperature range and oxidation state explored in this study, we have found that the rate of Ti diffusion in olivine is not sensitive to either oxygen fugacity or crystallographic orientation. Diffusion rates of Ti in the San Carlos and Kilbourne Hole olivines are very similar, whereas diffusion rates of Ti in synthetic forsterite are about 2 times that in the natural olivine. This suggests that effects of major element composition on Ti diffusion in olivine are relatively small. Results from the present study, therefore, may be applicable to other olivine compositions. At magmatic temperatures, the rates of Ti diffusion in olivine are smaller than the rates of trivalent REE, Cr, Ca, Ni and Fe–Mg diffusion in olivine but faster than the rates of W and Si diffusion in olivine, consistent with cation size and charge considerations. Finally, the activation energies of Ti diffusion in the three olivines are similar to each other, 189–203 kJ mol1, but are lower than those of Ti in orthopyroxene and clinopyroxene (270–282 kJ mol1). The diffusion rates of Ti in olivine are lower than the diffusion rates of Ti in the pyroxenes at magmatic temperatures but are higher than those of Ti in the pyroxenes at temperatures below 950 °C. Diffusivities of Ti in olivine obtained from this study along with diffusivities of Ti in orthopyroxene and clinopyroxene that we reported previously can be used to understand a range of geochemical mass transfer problems involving olivine and pyroxene. The closure temperatures for Ti in olivine, orthopyroxene, and clinopyroxene are very similar to each other for comparable mineral grain sizes. The time scales of subsolidus re-equilibration between olivine and pyroxene are dominated by that of olivine unless the pyroxene volume fraction and/or grain size are relatively small compared to those of olivine. Subsolidus redistribution of Ti among olivine, orthopyroxene, and clinopyroxene results in characteristic reversed Ti zoning in olivine and orthopyroxene and normal Ti zoning in clinopyroxene. Such zoning patterns in coexisting olivine and pyroxene are measureable through modern analytical methods, and may be used to infer cooling rates of ultramafic and mafic rocks.

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ACKNOWLEDGEMENTS We wish to thank Bruce Watson for helpful discussion during the course of this study, Chenguang Sun for sharing his unpublished olivine–pyroxene Ti partitioning models, and Peter Lezzi for assistance with the experiment conducted in the H2O vapor furnace. We also thank Laurence Coogan and Jiba Ganguly for thoughtful review comments. This work was supported in part by NSF Grants EAR-0738734 (D.C.) and EAR-0738830 (Y.L.), and NASA Grant NNX13AH07G (Y.L.).

APPENDIX A. SUPPLEMENTARY DATA Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/ j.gca.2014.10.016.

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