# Kinetic and equilibrium Ca isotope effects in high-T rocks and minerals

Michael Antonelli, Martin Schiller, Edwin Schauble, Tushar Mittal, Donald J.

Depaolo, Thomas Chacko, Edward Grew, Barbara Tripoli

► To cite this version:

Michael Antonelli, Martin Schiller, Edwin Schauble, Tushar Mittal, Donald J. Depaolo, et al. Kinetic and equilibrium Ca isotope effects in high-T rocks and minerals. Earth and Planetary Science Letters, 2019, 517, pp.71-82. 10.1016/j.epsl.2019.04.013. insu-02912507

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# Kinetic and equilibrium Ca isotope effects in high-T rocks and minerals

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**Authors:** Michael A. Antonelli<sup>a,1</sup>, Martin Schiller<sup>b</sup>, Edwin A. Schauble<sup>c</sup>, Tushar Mittal<sup>a</sup>, Donald

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J. DePaolo<sup>a,d</sup>, Thomas Chacko<sup>e</sup>, Edward S. Grew<sup>f</sup>, Barbara Tripoli<sup>a</sup>

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**Affiliations:** <sup>a</sup>Department of Earth and Planetary Science, University of California - Berkeley, Berkeley,

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CA 94720, USA. <sup>b</sup>Natural History Museum of Denmark, Øster Voldgade 5-7, 1350 København K,

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Denmark. <sup>c</sup>Department of Earth and Space Sciences, University of California – Los Angeles, USA.

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<sup>d</sup>Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA. <sup>e</sup>Department of Earth and

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Atmospheric Sciences, University of Alberta. <sup>f</sup>School of Earth and Climate Sciences, University of

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Maine. <sup>1</sup>Corresponding author. E-mail address: Mantonelli@berkeley.edu (M.A. Antonelli).

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**Abstract:** Calcium isotope data (δ<sup>44</sup>Ca, µ<sup>42/44</sup>Ca, and µ<sup>48/44</sup>Ca) are reported for high-temperature

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metamorphic rocks and minerals, and compared with density-functional theory (DFT) estimates

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of equilibrium Ca isotope fractionation factors for plagioclase, garnet, clinopyroxene,

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orthopyroxene, olivine, and apatite. The data and calculations are used to evaluate equilibrium

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and kinetic fractionation effects that apply to high-temperature metamorphism, where extended

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residence at high temperature should promote equilibration, but where centimeter-to-meter scale

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Ca transport could produce diffusive kinetic effects. At upper-granulite facies conditions (T ≥

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~900°C), DFT-predicted equilibrium fractionations between minerals are ≤0.8‰, decreasing to

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ca. 0.6‰ at 1100°C. We find much larger δ<sup>44</sup>Ca variations in both whole-rock samples (range of

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~4‰) and individual minerals (range of ~8‰), and large variations in the Ca isotope

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fractionation between mineral pairs (e.g. Δ<sup>44</sup>Ca<sub>grt-plag</sub> from -1.5 to +1.5 ‰). Deviations from

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equilibrium tend to be larger in concert with indications of higher temperature, such as

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increasing whole-rock Mg#, plagioclase anorthite content, orthopyroxene Ca/Mg, and garnet
Mg#. These large variations are inferred to be due to intragranular or grain-boundary diffusion during metamorphism, as this is the only mechanism that can produce such large isotopic variations. We can confirm the kinetic origin of the variations using measurements of µ$^{48/44}$Ca by MC-ICP-MS to distinguish kinetic from equilibrium fractionation processes using a triple-isotope approach. A new variable (Δ$^{48}$Ca') quantifies deviations from the Ca-isotope equilibrium slope on a plot of $^{48}$Ca/$^{44}$Ca vs. $^{42}$Ca/$^{44}$Ca. Available geochronological constraints and numerical modeling indicate that observed kinetic isotope fractionations between adjacent high and low Ca rock layers require effective Ca diffusivities of 10$^{-10}$ to 10$^{-7}$ m$^2$/yr, and a ratio of Ca isotope diffusivities of D$_{44}$/D$_{40}$ ≈ 0.99. The diffusivities are consistent with Ca transport by volume- and grain-boundary diffusion. The apparent contrast in isotopic diffusivities is large and more consistent with silicate liquids than aqueous fluids. This study confirms that kinetic Ca isotope effects are abundant in nature and can overwrite small equilibrium effects, even at high temperatures and even when other techniques (such as Fe-Mg exchange and Ca-in-Opx thermometry) suggest the establishment of chemical equilibrium. Our results imply that kinetic fractionation effects may complicate the use of δ$^{44}$Ca measurements for geothermometry or as a tracer of carbonate recycling into the mantle.

**Keywords:** Calcium isotopes; granulites-facies; lower crust; density-functional theory; diffusion; disequilibrium.
43 **1. Introduction**

Over the last two decades, precise stable Ca isotope measurements in high-temperature rocks and minerals have started to yield important constraints on solid-Earth processes (e.g. DePaolo, 2004; Schiller et al., 2016). Isotopic variations are generally reported in per mil notation as δ<sup>44</sup>Ca, defined 10<sup>-3</sup> [(<sup>44</sup>Ca/<sup>40</sup>Ca<sub>sample</sub>)/(<sup>44</sup>Ca/<sup>40</sup>Ca<sub>standard</sub>) - 1], where it is important to note that a partially radiogenic isotope of Ca (<sup>40</sup>Ca) is used in the denominators and that stable isotope fractionations must be either corrected for potential radiogenic <sup>40</sup>Ca ingrowth (this study) or measured relative to a non-radiogenic isotope of Ca (e.g. <sup>42</sup>Ca). Recent studies have mostly focused on δ<sup>44</sup>Ca variations in mantle xenoliths and bulk lavas (Amini et al., 2009; Chen et al., 2018; Huang et al., 2011; S. Huang et al., 2010; Kang et al., 2017, 2016; Liu et al., 2017; Zhao et al., 2017), as well as *ab-initio* predictions for Ca isotope equilibrium fractionation factors between clinopyroxene and orthopyroxene at various temperatures (Feng et al., 2014; Wang et al., 2017). Results from these studies have been used to suggest that equilibrium Ca isotope fractionation between minerals, which depends mainly on strength and coordination of Ca-O bonds in various mineral structures, can be used for isotopic thermometry. Several studies also use trends in δ<sup>44</sup>Ca for estimating fractions of recycled carbonate in mantle xenoliths and ocean island basalts (e.g. Chen et al., 2018; Huang et al., 2011; S. Huang et al., 2010; Kang et al., 2017, 2016; Liu et al., 2017), but the appropriateness of δ<sup>44</sup>Ca as a tracer of carbonate recycling has been questioned by more recent observations (Ionov et al., 2019).

The applicability of equilibrium predictions to natural samples needs to be evaluated in context of potential kinetic fractionations that could affect inter-mineral fractionation, Δ<sup>44</sup>Ca<sub>A-B</sub>, defined as δ<sup>44</sup>Ca<sub>A</sub> - δ<sup>44</sup>Ca<sub>B</sub> for minerals A and B. Rayleigh-fractionation models of carbonate precipitation within mantle fluids have been invoked to explain positive δ<sup>44</sup>Ca shifts in mantle xenolith
samples (Kang et al., 2017). Fluids enriched in δ⁴⁴Ca, potentially from carbonate precipitation, have also been suspected as a potential culprit for δ⁴⁴Ca enrichments in blueschist-hosted eclogite veins (John et al., 2012). Negative δ⁴⁴Ca shifts in mantle xenoliths, on the other hand, are often interpreted as the result of direct mixing with subducted carbonates (e.g. Chen et al., 2018) without any associated fractionations; one study, however, has identified relatively small negative kinetic δ⁴⁴Ca effects (<1 ‰) in mantle olivine (Zhao et al., 2017). Other than these limited studies, kinetic Ca isotope fractionation at high temperatures has remained largely unexplored in natural samples.

Kinetic effects at high-temperatures have been studied in the laboratory, however, and a large-body of experimental work has shown that Ca diffusion in molten silicates can result in much larger Ca isotope fractionations than have been documented so far in natural rocks and minerals (Watkins et al., 2017). Due to large mass differences between isotopes of calcium (up to ~20% for ⁴⁸Ca vs. ⁴⁰Ca), there are large differences in average speed for different isotopic species at a given kinetic energy, resulting in separation of isotopes during diffusive transport. In general, diffusivity of ⁴⁴Ca relative to ⁴⁰Ca (D₄₄/D₄₀, equal to square-root of the inverse mass ratio raised to an empirically-derived exponent) depends most significantly on diffusion medium (Richter et al., 1999). Large δ⁴⁴Ca effects (up to ~10‰) have been documented for Ca isotopes undergoing simple chemical diffusion (e.g. Richter et al., 2009, 2003, 1999, Watkins et al., 2014, 2011, 2009), thermal (Soret) diffusion (F. Huang et al., 2010; Richter et al., 2009), and isotopic self-diffusion in silicate liquids (Watkins et al., 2014), yet these effects remain underexplored in nature [see (Chopra et al., 2012) for an example using Mg isotopes]. However, Ca diffusion in solid minerals (i.e., ‘volume diffusion’) and along grain-boundaries, applicable to a majority of metamorphic settings, are many orders of magnitude slower than in silicate liquids (Dohmen and
Milke, 2010), and several orders of magnitude slower than Fe and Mg diffusion in analogous processes (Grant, 1988; Hudgins et al., 2011; Vielzeuf et al., 2007). Due to the time-scales required, Ca isotope fractionation during grain-boundary and volume-diffusion are much more difficult to evaluate experimentally.

In order to begin addressing the many open questions regarding Ca isotope fractionation in high-temperature rocks and minerals, and to gain insight into Ca isotope fractionation during lower-crustal metamorphic processes, we compare equilibrium estimates from density-functional theory (DFT) to Ca isotope measurements in high-temperature metamorphic mineral-separates and whole-rocks from various locations.

## 2. Samples and Methods

We report $\delta^{44}$Ca measurements, corrected for radiogenic $^{40}$Ca excesses [as reported in (Antonelli et al., 2019), see Fig. S1], along with bulk-rock and mineral chemical compositions for twenty-one whole-rock samples and their major minerals (including plagioclase, K-feldspar, garnet, clinopyroxene, orthopyroxene, hornblende, and apatite) from four different localities. Modal mineralogies, whole-rock compositions, in-situ mineral compositions, and mineral separate compositions are reported in Tables S1 to S4. Measured $\delta^{44}$Ca for samples and standards are reported in Tables S5 and S6, respectively.

We also present multiple Ca isotope results, including $\mu^{48}$Ca {defined as $10^6 \cdot [(^{48}\text{Ca}/^{44}\text{Ca})_{\text{sample}}/(^{48}\text{Ca}/^{44}\text{Ca})_{\text{standard}} - 1]$} for a select group of whole-rocks and mineral separates (including plagioclase, garnet, and orthopyroxene) measured by MC-ICP-MS (Table S7), DFT estimates for equilibrium fractionation in a variety of minerals with variable solid-solution compositions (including plagioclase, garnet, clinopyroxene, orthopyroxene, olivine,
apatite, and several others) and numerical modelling results for diffusive Ca isotope fractionation in metamorphic whole-rocks (see Supplementary Materials, SM, for details).

Samples in this study are mostly granulite and ultrahigh-temperature (UHT) granulite-facies rocks with granoblastic to porphyroblastic textures (n = 17), but also include several amphibolites (n = 3) and a single eclogite [the same samples as studied in (Antonelli et al., 2019)]. Twelve granulite samples come from the northern part of the Napier Complex (NC), Antarctica; four lower-crustal granulite-facies xenoliths come from Diavik Diamond Mine, Slave Province (SP), Canada; four amphibolite to granulite-facies samples come from the Ivrea-Verbano Zone (IVZ), Italy; and the eclogite sample (Sumdo Eclogite, SE) comes from Lhasa Province, Tibet, China. Petrographic descriptions for samples, along with previous work highlighting geochronologic and thermobarometric constraints for localities, are available in the Supplementary Information of (Antonelli et al., 2019).

## 3. Results

### 3.1. Predicted Equilibrium Fractionations

The results of our DFT predictions, in the form of reduced partition function ratios (RPFR, 1000lnβ), and comparisons between different calculation methods are tabulated in Table 1. We find that equilibrium δ⁴⁴Ca signatures are expected to increase in the general order: plagioclase < apatite < clinopyroxene < orthopyroxene ≤ garnet < olivine. We also confirm the previously proposed relationship (e.g. Feng et al., 2014; S. Huang et al., 2010; Magna et al., 2015; Wang et al., 2017) between 1000lnβ and Ca-O bond lengths (and coordination) in the studied minerals, where minerals with shorter bonds favor the heavy isotope (Fig. 1). Although minerals with eight-fold coordination (e.g. clinopyroxene & garnet) generally have longer Ca-O bond lengths
than minerals with six-fold coordination (e.g. orthopyroxene & olivine, Table S8), they are predicted to have heavier δ⁴⁴Ca at a given bond-length. However, some of the more complex structures have extra Ca-O bonds with different lengths than the others, as tabulated in Table S8, and it is not always straightforward to disentangle coordination and average bond length for these irregular sites.

<table><caption>Table 1. Calculated reduced partition-function ratios (1000ln⁴⁴/⁴⁰β) from DFT modeling.</caption><thead><tr><th rowspan="3"></th><th rowspan="3"></th><th colspan="6">Reduced partition function ratios, 1000•ln(⁴⁴/⁴⁰β)</th></tr><tr><th colspan="2">Phonon-based (PBE*)</th><th colspan="2">Force-constants (PBE*)</th><th colspan="2">Force-constants (LDA)</th></tr><tr><th>298.15 K</th><th>1000 K</th><th>298.15 K</th><th>1000 K</th><th>298.15 K</th><th>1000 K</th></tr></thead><tbody><tr><td rowspan="3">Diopside</td><td>-</td><td>14.26</td><td>1.32</td><td>13.77</td><td>1.23</td><td>14.61</td><td>1.31</td></tr><tr><td>Doubled c</td><td>-</td><td>-</td><td>14.26</td><td>1.28</td><td>-</td><td>-</td></tr><tr><td>Full C2/c cell, 2c</td><td>-</td><td>-</td><td>14.65</td><td>1.31</td><td>-</td><td>-</td></tr><tr><td rowspan="4">Enstatite</td><td>Ca:Mg = 1:15</td><td>-</td><td>-</td><td>19.20</td><td>1.72</td><td>-</td><td>-</td></tr><tr><td>Ca:Mg = 1:15, 2c</td><td>-</td><td>-</td><td>19.28</td><td>1.73</td><td>-</td><td>-</td></tr><tr><td>Ca:Mg = 1:31, 2c</td><td>-</td><td>-</td><td>20.50</td><td>1.84</td><td>-</td><td>-</td></tr><tr><td>Ca:Mg = 1:63, quad cell</td><td>-</td><td>-</td><td>20.58</td><td>1.85</td><td>-</td><td>-</td></tr><tr><td rowspan="4">Forsterite</td><td>Ca:Mg = 1:7, M2 sub</td><td>-</td><td>-</td><td>20.30</td><td>1.82</td><td>-</td><td>-</td></tr><tr><td>Ca:Mg = 1:15, 2a</td><td>-</td><td>-</td><td>23.35</td><td>2.10</td><td>-</td><td>-</td></tr><tr><td>Ca:Mg = 1:31, 2a & c</td><td>-</td><td>-</td><td>23.11</td><td>2.08</td><td>-</td><td>-</td></tr><tr><td>Ca:Mg = 1:63, 4a & 2c</td><td>-</td><td>-</td><td>23.48</td><td>2.11</td><td>-</td><td>-</td></tr><tr><td rowspan="5">Garnet</td><td>Grossular</td><td>18.63</td><td>1.74</td><td>19.39</td><td>1.74</td><td>20.09</td><td>1.80</td></tr><tr><td>Ca:Mg = 1:11 (pyr)</td><td>-</td><td>-</td><td>22.32</td><td>2.00</td><td>-</td><td>-</td></tr><tr><td>Ca:Mg = 1:23 (pyr, FC)</td><td>-</td><td>-</td><td>22.60</td><td>2.03</td><td>-</td><td>-</td></tr><tr><td>Ca:Fe = 1:11 (alm, FM)</td><td>-</td><td>-</td><td>20.60</td><td>1.85</td><td>-</td><td>-</td></tr><tr><td>Ca:Mn = 1:11 (sps, FM)</td><td>-</td><td>-</td><td>20.43</td><td>1.83</td><td>-</td><td>-</td></tr><tr><td rowspan="4">Plagioclase</td><td>Anorthite</td><td>11.41</td><td>1.06</td><td>12.06</td><td>1.08</td><td>14.06</td><td>1.26</td></tr><tr><td>An50 (v1)</td><td>-</td><td>-</td><td>10.93</td><td>0.98</td><td>-</td><td>-</td></tr><tr><td>An50 (v2)</td><td>-</td><td>-</td><td>11.53</td><td>1.03</td><td>-</td><td>-</td></tr><tr><td>An50 (v3)</td><td>-</td><td>-</td><td>11.27</td><td>1.01</td><td>-</td><td>-</td></tr><tr><td>Apatite</td><td>Fluorapatite</td><td>13.10</td><td>1.21</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Lime</td><td>(CaO)</td><td>16.43</td><td>1.51</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Fluorite</td><td>(CaF<sub>2</sub>)</td><td>14.69</td><td>1.34</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></tbody></table>

<footnote>*Assumed scale-factor of 1.06, LDA are unscaled. Preferred values are in bold. '2c' – doubled c axis, '2a' – doubled a axis, '4a' – quadrupled a axis; 'FC' – full cubic cell, 'FM' – Ferromagnetic; 'v1' through 'v3' are using ab-initio constraints obtained from (Kaercher et al., 2014).</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 1. Reduced partition-function ratios for <sup>44</sup>Ca/<sup>40</sup>Ca (1000lnβ) vs. mean Ca-O bond lengths from our DFT simulations (n=23). Bond lengths and coordination are tabulated in Table S8. Triangles and circles represent minerals with ~eight-fold and ~six-fold coordination, respectively.</figcaption></figure>

Our calculated equilibrium Δ<sup>44</sup>Ca values between orthopyroxene and clinopyroxene are in general agreement with published values (Feng et al., 2014; Wang et al., 2017), and confirm that decreasing Ca/Mg in enstatite from 1/16 to 1/32 leads to increasing RPFRs, and that further decreases (to 1/64) do not lead to significantly larger effects. Our models, however, show less sensitivity to Ca isotope fractionation due to Ca dilution effects in enstatite (~0.1‰, at 1000 K) than reported in previous studies (~0.3‰, Feng et al., 2014; Wang et al., 2017).

Applying the same exercise to forsterite and pyrope with variable Ca/Mg (ranging from 1/8 to 1/64, and 1/12 to 1/24, respectively) we find that dilution does not significantly affect results for olivine (in agreement with Huang et al., 2019) or garnet. We do find, however, a small yet potentially measurable difference in predicted equilibrium Ca isotopic compositions for anorthite and labradorite (An50), for all three labradorite structures used in our simulations (Kaercher et
al., 2014) (Table 1). We find that labradorite has the lowest 1000lnβ of all our results and is predicted to be the lightest at equilibrium conditions. The fluorite structure (CaF₂) leads to similar results as diopside, while fluorapatite [Ca₅(PO₄)₃F] lies between anorthite and fluorite, and lime (CaO) is predicted to have slightly heavier δ⁴⁴Ca than fluorite. Although we focus on Mg-bearing end-members for the majority of our DFT simulations, incorporation of Fe into our mineral structures would probably lead to slight decreases in expected inter-mineral fractionations (by increasing Ca-O bond lengths in Fe-bearing minerals), as suggested for orthopyroxene (Wang et al., 2017). A recently published DFT study using LDA functionals for several Ca-bearing mineral species (including forsterite, grossular, pigeonite, and anorthite) is in good agreement with our results (Huang et al., 2019); LDA predictions, however, appear to generally yield slightly higher 1000lnβ values than our preferred estimates (PBE functionals, see Table 1).

Assuming that temperature dependence of 1000lnβ can be well represented at high temperature by the equation [1000lnβ(T) = 1000lnβ(1000 K) x (10⁶/T²), where temperature is in Kelvin], we find that equilibrium Ca isotope fractionations between minerals do not exceed ~0.8‰ at temperatures relevant to the samples in our study (greater than ~800-900 °C, Fig. 2). Fractionation factors (1000lnα), which represent the prediction for Δ⁴⁴Ca between minerals, can be approximated visually from Fig. 2 (using 1000lnβ_a – 1000lnβ_b), but are calculated for our purposes using α_a-b = β_a/β_b. Note that the fractionation factor between two phases, a and b, is the ratio of their β values (α_a-b = β_a/β_b) at a given temperature, and that Δ⁴⁴Ca_a-b ≈ 1000lnα_a-b = 1000lnβ_a – 1000lnβ_b.
<figure><img src="image_3.png" /><figcaption>Fig. 2. Predicted temperature- variations for DFT-based $^{44}$Ca/$^{40}$Ca reduced partition-function ratios (1000lnβ) in the studied silicate mineral structures. Lines are labeled by mineral-type and include (Ca:Mg), (Ca:Fe), or (Ca:Mn), as applicable to the various species. Expanded legend (*) indicates the order of results with similar RPFRs in the region indicated by the asterisk. Double-headed red arrow represents the approximate range in Δ$^{44}$Ca$_{A-B}$ predicted at 900 °C. Fo = forsterite, Pyr = pyrope, En = enstatite, Alm = almandine, Sps = spessartine, Grs = grossular, Di = diopside, Ap = fluorapatite, An = anorthite, An50 = labradorite. Fractionation factors (1000lnα$_{A-B}$) and inter-mineral fractionations (Δ$^{44}$Ca$_{A-B}$) are ≈1000lnβ$_{A}$-1000lnβ$_{B}$.</figcaption></figure>

**3.2. Bulk-rock and Mineral δ<sup>44</sup>Ca Variations**

The δ<sup>44</sup>Ca of whole-rocks and minerals analyzed in the present study, corrected for radiogenic ingrowth of <sup>40</sup>Ca, are displayed in Fig. 3. We find that mafic rocks generally have more restricted whole-rock compositions compared to metapelites, and that minerals other than clinopyroxene exhibit large ranges in δ<sup>44</sup>Ca (greater than ~3‰). Orthopyroxene, and in a few instances, feldspar and garnet, are found to be the most <sup>44</sup>Ca-enriched mineral phases, whereas clinopyroxene and hornblende are most similar to bulk-silicate Earth (BSE) and garnet and K-feldspar are the most <sup>44</sup>Ca depleted. Our results are in general agreement with previous δ<sup>44</sup>Ca measurements in granitic minerals from a single whole-rock sample (Ryu et al., 2011). Given that our orthopyroxene
separates were not completely pure, we used unmixing calculations to estimate isotopic
composition of pure orthopyroxene (Opx*, SM and Fig. S2 for details). The single analyzed
apatite separate was found to be the most ⁴⁴Ca-depleted mineral phase in our eclogite sample.
We show our δ⁴⁴Ca results, organized by whole-rock sample, in Fig. S3.

<figure><img src="image_4.png" /></figure>

Fig. 3. Plot of δ⁴⁴Ca for the mineral separates and whole-rocks analyzed in this study (n=66). Mafic (n=12) and pelitic (n=10) whole-rocks, orthopyroxene (n=9), pure-orthopyroxene (Opx*, n=6), plagioclase (n=10), garnet (n=14), K-feldspar (n=3), clinopyroxene (n=6), hornblende (n=1), and apatite (n=1). 2SD error bars on most measurements (< ±0.14‰) are smaller than the symbols, except for Opx*, where uncertainties range from ±0.2 to ±1.5‰ (with an average of ±0.5‰), depending on the unmixing parameters (Fig. S2). Data are relative to bulk-silicate Earth (BSE), which is +0.96 ± 0.05‰ (2SE) relative to SRM915a, in agreement with previous estimates (e.g. Foley et al., 2019; Kang et al., 2017; Simon and DePaolo, 2010).

Comparing our results with whole-rock chemical compositions, we find that mafic and pelitic
samples with the lowest CaO and highest Mg# [defined as molar MgO/(MgO+FeO), using
FeO<sub>total</sub>] have the highest variability in bulk-rock δ<sup>44</sup>Ca (Fig. 4). Interlayered whole-rocks
(connected by colored arrows in Fig. 4) are found to exhibit the largest whole-rock δ<sup>44</sup>Ca
fractionations. Minerals from within analyzed whole rocks generally show increasing δ<sup>44</sup>Ca with increasing whole-rock Mg# and decreasing whole-rock and mineral CaO (wt%) (Fig. S4). As discussed further below, these patterns in whole rock chemistry and Ca isotopic composition can be explained by diffusive transport of Ca between high-Ca and low-Ca zones in layered gneisses of the NC.

<figure><img src="image_5.png" /><figcaption>Fig. 4a,b. Plots of radiogenic-corrected whole-rock δ<sup>44</sup>Ca versus (a) whole-rock CaO (wt%) and (b) whole-rock Mg# [molar MgO/(MgO+FeO)]. Colored arrows point in the direction of decreasing CaO between whole-rock samples collected from units with shared interfaces; 2035A-2035B (orange arrow), 2032A-2032I (red arrow), 2036A-2036C-2036E (purple arrows). Error bars are smaller than the symbols. Bulk-silicate Earth is +0.96 ± 0.05‰ (2SE) relative to SRM915a, in agreement with previous estimates (e.g. Foley et al., 2019; Kang et al., 2017; Simon and DePaolo, 2010).</figcaption></figure>

**3.3. Inter-mineral δ<sup>44</sup>Ca fractionations (Δ<sup>44</sup>Ca)**
Many of our samples have inter-mineral Δ<sup>44</sup>Ca fractionations significantly different from equilibrium predictions (Fig. 5a-f). Within a single sample, δ<sup>44</sup>Ca differences (Δ<sup>44</sup>Ca) between minerals can be as large as ~2‰, exceeding the range that we predict at temperatures relevant to our samples (Fig. 2). Differences in δ<sup>44</sup>Ca between whole-rocks and their constituent minerals can also exceed 2‰, and certain samples [e.g. 2035B (δ<sup>44</sup>Ca = 3.3‰), consisting of ~80% orthopyroxene (measured δ<sup>44</sup>Ca = +5.2‰) and ~15% plagioclase (δ<sup>44</sup>Ca = +4.66‰)] suggest that unidentified minor phases can have highly-fractionated δ<sup>44</sup>Ca. The modal mineralogy, however, was estimated from a single thin-section and it is likely that our estimates (which have a large influence on predicted δ<sup>44</sup>Ca of unidentified minerals) for orthopyroxene and plagioclase are slightly high. In accordance with our whole-rock (Table S2) and mineral measurements (Table S3), Ca mass balance for the rock can be achieved using reasonable values for clinopyroxene (i.e., similar to BSE) if modal mineralogy is assumed to contain slightly higher clinopyroxene (e.g. ~5%) and slightly lower plagioclase (e.g. ~10%) than estimated for the thin-section in Table S1.

At granulite-facies temperatures (~800-900 °C), Δ<sup>44</sup>Ca<sub>grt-cpx</sub> and Δ<sup>44</sup>Ca<sub>grt-fsp</sub> are expected to be positive and in the range of 0.3-0.6‰ and 0.5-0.9‰, respectively (depending on compositions of garnet and feldspar), but are observed to vary between -1.5 and +1.5‰. At the same temperatures, fractionation between low-Ca orthopyroxene and plagioclase is expected to range from ~0.6 to 0.8‰, but Δ<sup>44</sup>Ca<sub>opx-fsp</sub> in our samples range from 0.2 to ~2.0‰. Analogously, Δ<sup>44</sup>Ca<sub>opx-cpx</sub> equilibrium should be approximately +0.3 to +0.5‰, but measured values can exceed ~1‰. Δ<sup>44</sup>Ca<sub>grt-opx</sub> (predicted to be -0.1 to +0.2‰) is observed to range between -1.6 and +0.3‰.
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As shown in Fig. 5a-f, garnet-mineral and orthopyroxene-mineral $\Delta^{44}$Ca deviate further from predicted equilibrium with increases in several whole-rock and mineral-composition proxies. Garnet fractionations ($\Delta^{44}$Ca$_{grt-fsp}$, $\Delta^{44}$Ca$_{grt-cpx}$, $\Delta^{44}$Ca$_{grt-opx}$, & $\Delta^{44}$Ca$_{grt-wr}$) grow increasingly negative with increasing whole-rock Mg# and garnet Mg# (Fig. 5a,b), and orthopyroxene-mineral fractionations grow increasingly positive with increasing whole-rock Mg#, feldspar anorthite content, orthopyroxene Ca/Mg, and modal orthopyroxene content (Fig. 5c-f). The observation of increasing $\Delta^{44}$Ca$_{opx-cpx}$ with increasing orthopyroxene Ca/Mg (Fig. 5e) is the opposite of what has been so far observed in peridotitic minerals (Chen et al., 2018; S. Huang et al., 2010; Kang et al., 2016; Zhao et al., 2017) and to the dilution-effects predicted at equilibrium (Feng et al., 2014; Wang et al., 2017; this study). Assuming equilibrium, increased Ca concentrations in orthopyroxene are associated with increasing temperatures (Brey and Kohler, 1990), and should therefore be accompanied by decreasing isotopic fractionation between minerals.

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

257
Fig. 5a-f. Plots of mineral Δ<sup>44</sup>Ca versus whole-rock and mineral compositions. Garnet-mineral and garnet-whole rock Δ<sup>44</sup>Ca versus (a) whole rock Mg# and (b) garnet Mg#. Orthopyroxene-mineral and orthopyroxene-whole rock Δ<sup>44</sup>Ca versus (c) whole-rock Mg#, (d) feldspar anorthite content, (e) Ca/Mg of orthopyroxene, and (f) modal orthopyroxene content. Panels (d) and (f) also include measurements of Δ<sup>44</sup>Ca<sub>cpx-fsp</sub>. For non-orthopyroxene mineral-pairs, error bars are smaller than the symbols, for data including Opx*, uncertainties range from 0.2 to 1.5‰, varying with unmixing parameters (Fig. S2). Colored bands indicate predicted equilibrium at 800-900 °C: Grt-Opx (purple), Grt-Cpx (green), Grt-Fsp (blue), Opx-Fsp (yellow), Opx-Cpx (pale green), and Cpx-Fsp (cyan). Orthopyroxene predictions are based on Ca:Mg of 1:63.

Based on single analyses, Δ<sup>44</sup>Ca<sub>hbl-plag</sub> (+0.28‰) is similar to Δ<sup>44</sup>Ca<sub>cpx-plag</sub> in other rocks, and Δ<sup>44</sup>Ca<sub>cpx-ap</sub> is slightly higher (+0.41‰, suggesting disequilibrium when compared to DFT). No trends between Δ<sup>44</sup>Ca<sub>cpx-plag</sub> and whole-rock or mineral compositions are clearly distinguishable. Half of the measured Δ<sup>44</sup>Ca<sub>cpx-plag</sub> values are similar to equilibrium predictions (+0.2 to +0.3‰), and the other half have slightly smaller Δ<sup>44</sup>Ca<sub>cpx-plag</sub> (0.0 to +0.1‰) that may suggest temperatures higher than ~900°C, assuming equilibrium. However, these isotopic differences are at the limit of our analytical uncertainty, so it is hard to distinguish whether these values represent equilibrium at different temperatures or whether they result from small kinetic effects.

## 3.4. Δ<sup>48</sup>Ca' and deviations from Ca-isotope Equilibrium

Although previously untested, a possibility exists for distinguishing kinetic from equilibrium fractionation by measuring <sup>48</sup>Ca/<sup>44</sup>Ca. Given sufficient precision, this can be achieved by exploiting slopes between samples in triple-isotope space (e.g. <sup>48</sup>Ca/<sup>44</sup>Ca vs. <sup>42</sup>Ca/<sup>44</sup>Ca). The approach exploits differences in the kinetic-limit and equilibrium-limit for mass-dependence during stable-isotope fractionation (e.g. Young et al., 2002), and similar approaches have been used to distinguish kinetic from equilibrium effects in isotopic measurements of Fe (McCoy-West et al., 2018) and Mg (Olsen et al., 2013; Young and Galy, 2004). For Ca isotopes, this difference can be manifested using any two of the five total isotope ratios. Due to their relatively
high abundances, we use µ⁴⁸Ca/µ⁴²Ca to evaluate slopes in triple isotope space, where µ⁴⁵Ca (in 284 ppm) is equal to 10⁶·[(⁴⁵Ca/⁴⁴Ca)sample/(⁴⁵Ca/⁴⁴Ca)standard – 1] and has a factor of 10³ difference 285 with regular δ-notation. 286

The equilibrium-limit, describing a µ⁴⁸Ca/µ⁴²Ca slope of -1.751, is defined by the relationship 287 [(1/m₁)-(1/m₃)] ÷ [(1/m₁)-(1/m₂)], where m₁, m₂, & m₃ are masses of ⁴⁴Ca, ⁴²Ca, and ⁴⁸Ca, 288 respectively (Young et al., 2002); the kinetic limit (exponential) is described by 289 ln(m₁/m₃)/ln(m₁/m₂), and results in a slope of -1.872. The difference in these slopes leads to a 290 µ⁴⁸Ca difference of ~200 ppm (equivalent to ~0.2‰) between equilibrium and kinetic 291 fractionations at a corresponding δ⁴⁴Ca fractionation of ~3‰. Using the method of (Schiller et 292 al., 2012) and ⁴⁴Ca signals of ~10-30 V, however, the precision of our MC-ICP-MS 293 measurements is sufficient to distinguish between equilibrium and kinetic µ⁴⁸Ca fractionations 294 for corresponding δ⁴⁴Ca fractionations as small as ~0.5‰ (SM). 295

Similar to notation for other elements with triple-isotope systems, we propose a new variable 296 (Δ⁴⁸Ca') in order to easily quantify deviations from Ca-isotope equilibrium. This variable 297 corresponds to excesses or deficits in µ⁴⁸Ca from predicted equilibrium at a given µ⁴²Ca, and can 298 be reported as: 299

$$ \Delta^{48}Ca' = \mu^{48}Ca_{measured} - \mu^{48}Ca_{predicted} \quad (1) $$

$$ \Delta^{48}Ca' = \mu^{48}Ca_{measured} - 10^6 \left[ \left( \frac{\mu^{42}Ca_{measured}}{10^6} + 1 \right)^{-1.751} - 1 \right] \quad (2) $$

In order to test whether dominantly-kinetic vs. dominantly-equilibrium Ca isotope fractionation 302 can be successfully identified using Δ⁴⁸Ca', we start by analyzing two igneous end-member 303 samples, that are unrelated to the study at hand, but that are further explored in upcoming work. 304 The first sample, in which fractionations are strongly kinetic in origin, comes from experimental 305
plagioclase phenocryst growth in a simplified tonalite melt. In this sample, the δ<sup>44</sup>Ca difference between plagioclase separates and glass (Δ<sup>44</sup>Ca) exceeds 2‰ (Table S7). The second end-member sample, a cumulate gabbro from Stillwater igneous complex, Montana, where orthopyroxene and plagioclase separates have Δ<sup>44</sup>Ca<sub>opx-fsp</sub> of ~0.7‰, represents slow-crystallization at magmatic temperatures over probably hundreds to thousands of years.

We find that the end-member samples have clearly distinguishable Δ<sup>48</sup>Ca' (Fig. 6), with μ<sup>48</sup>Ca/μ<sup>42</sup>Ca slopes closely matching the equilibrium- and kinetic-limits described above. The experimental plagioclase phenocrysts show a strong depletion in Δ<sup>48</sup>Ca'plag [-153 ± 15 ppm (2SE)] relative to Δ<sup>48</sup>Ca'glass [-15 ± 17 ppm (2SE)], while the Stillwater orthopyroxene and plagioclase separates have Δ<sup>48</sup>Ca' indistinguishable from each other [30 ± 8 ppm and 24 ± 9 ppm (2SE), respectively]. Given this promising result, we analyze several contrasting samples from our dataset in order to establish whether (i) Δ<sup>44</sup>Ca significantly-different from our DFT predictions (e.g. negative Δ<sup>44</sup>Ca<sub>grt-plag</sub>) truly represent kinetic effects, and whether (ii) Δ<sup>44</sup>Ca in apparent agreement with our DFT predictions are corroborated by equilibrium Δ<sup>48</sup>Ca' between minerals (≈ 0).

We analyze garnet-feldspar pairs from two whole-rock samples, and find that (i) negative Δ<sup>44</sup>Ca<sub>grt-plag</sub> (-1.6‰, DDM303) is accompanied by a large Δ<sup>48</sup>Ca' depletion, and that (ii) positive Δ<sup>44</sup>Ca<sub>grt-plag</sub> (+0.6‰, IV-16-19) is not accompanied by measureable Δ<sup>48</sup>Ca' fractionation (Fig. 6).
This suggests that our DFT predictions for garnet and plagioclase are approximately correct and that deviations from these values are the result of kinetic processes.
<figure><img src="image_9.png" /></figure>

Fig. 6. Plot of Δ<sup>48</sup>Ca' vs. δ<sup>42/44</sup>Ca for select samples and for kinetic and equilibrium end-members, measured by MC-ICP-MS. Equivalent δ<sup>44</sup>Ca values [converted using a factor of -2.05 (Fantle and Tipper, 2014)] are also included for reference. Data were measured relative to SRM915b, which has δ<sup>44</sup>Ca<sub>BSE</sub> of -0.1‰ and δ<sup>42/44</sup>Ca<sub>BSE</sub> of +0.05‰ (Schiller et al., 2012).

In a similar fashion, the large δ<sup>44</sup>Ca difference between whole-rock samples from the center (2035A, -0.2‰) and edge (2035B, +3.0‰) of a metamorphosed ultramafic dike, is accompanied by a strong Δ<sup>48</sup>Ca' enrichment [+199 ± 8 ppm in 2035B vs. +9 ± 12 ppm (2SE) in 2035A], which is in good agreement with the predicted kinetic slope for μ<sup>48</sup>Ca/μ<sup>42</sup>Ca (Fig. 6). In concert with our experimental phenocryst constraints, this serves to re-inforce applicability of the kinetic-limit for mass-dependence during diffusion (μ<sup>48</sup>Ca/μ<sup>42</sup>Ca = -1.872), for both positive and negative kinetic fractionations in significantly different high-T systems.

We also confirm that the largest measured Δ<sup>44</sup>Ca<sub>opx-plag</sub> (sample 2030B) indeed represents kinetic enrichment (Fig. 6, SM). In sum, these novel constraints suggest that our DFT calculations are approximately correct, and that the large deviations we observe from predicted equilibrium are
341 the manifestation of kinetic processes affecting both whole-rock and inter-mineral Ca isotope
342 variability.

343 **3.5. Whole-rock δ<sup>44</sup>Ca variations and diffusion modeling**

344 Given that large whole-rock δ<sup>44</sup>Ca variations are of kinetic origin, and that they are found to vary
345 most dramatically when coming from units with shared interfaces, it is likely that they result
346 from isotopic fractionation during diffusion of Ca. Paired with geochronological constraints for
347 duration of metamorphism in the NC [~100 Myr (e.g. Harley, 2016)], δ<sup>44</sup>Ca in adjacent high-Ca
348 and low-Ca samples (colored arrows in Fig. 4), can be used to gain information on Ca diffusivity
349 and mechanisms attending long-lived high-temperature metamorphism in lower-crustal settings.

350 In order to successfully model Ca diffusion between whole-rock units with different bulk-
351 compositions, without assuming a single diffusivity for all considered layers, we use a Python-
352 based finite-difference diffusion model with a time-stepping grid, based on the method described
353 in (Langtangen and Linge, 2017) for diffusion in heterogeneous media. The model has tunable
354 input parameters for each layer, including (i) Ca concentration (wt%), (ii) Ca diffusivity (m²/yr),
355 and (iii) isotopic fractionation factor (D₄₄/D₄₀), and runs for any pre-specified amount of time
356 (details in SM).

357 Using spatial constraints from the field, a diffusion time-scale of 100 Myr, and our chemical and
358 isotopic measurements as final conditions, we generate models required to explain the sample-
359 pairs high-lighted in Fig. 4 (SM). We focus our discussion on ultramafic dike samples (2035A &
360 2035B) because they have the best spatial constraints: 2035A was collected from the center of a
361 1.7m thick dike, and 2035B was collected ~8cm from the edge.
<figure><img src="image_10.png" /></figure>

Fig. 7a,b. Numerical diffusion model results for spatial- variations in Ca concentration and change in δ<sup>44</sup>Ca (Δ<sup>44</sup>Ca) over time, showing the effect of variable dike diffusivities, as relevant to the center (2035A, δ<sup>44</sup>Ca = -0.09‰) and edge (2035B, δ<sup>44</sup>Ca = +3.22‰) of our metabasic dike samples from the NC. Dike (Dk) width is 1.7 m with initial [Ca] = 3.5 wt%; quartzitic country-rock (CR) has initial [Ca] = 0.5 wt%. In (a) dike diffusivity (D<sub>Dk</sub>) is 10<sup>-10</sup> m<sup>2</sup>/yr and D<sub>CR</sub>/D<sub>Dk</sub> = 10<sup>3</sup>, in (b) D<sub>Dk</sub> is 10<sup>-9</sup>m<sup>2</sup>/yr and D<sub>CR</sub>/D<sub>Dk</sub> = 10<sup>2</sup>. Other parameters are listed in the figure. Dashed lines indicate the approximate location of sample 2035B. Diffusivities are in m<sup>2</sup>/yr.

Using a country-rock Ca content, [Ca]CR, of 0.5 wt% (average for quartzites in our study), and an
initial dike Ca content, [Ca]Dk, of 3.5 wt%, we find that diffusivities (D Dk) of ~10-10 m²/yr with
D₄₄/D₄₀ ~0.990 ± 0.002 are required to explain our results for metabasic dike samples 2035A
and 2035B (Fig. 7a, SM). For these simulations, a Ca diffusivity ratio in country rock relative to
dike (DCR/D_Dk) lower than ~10³ inhibits sufficient diffusion from taking place within 100 Myr,
while higher dike diffusivities (e.g. 10⁻⁹ m²/yr) overstep the conditions represented by our
samples after ~30 Myr (Figs. 7b and S5), even with lower DCR/D_Dk. There are several reasons,
377 however, to believe that pelitic/quartzitic units would have higher effective Ca diffusivities
378 compared to mafic units, as discussed in the following section. Diffusion model results for our
379 pelitic sample pairs are discussed in the SM and depicted in Fig. S6.

380 **4. Discussion**

381 **4.1. Implications for diffusion**

382 Based on our diffusion models, we find that effective Ca diffusivities in UHT granulites from the
383 NC, a metamorphic terrane that represents slow-cooling over ~100 Myr at lower-crustal depths
384 (Harley, 2016), are likely to range from ~10⁻⁷ to ~10⁻¹⁰ m²/yr. These estimates compare favorably
385 to experimental and natural estimates for Ca diffusion in minerals (aka. volume-diffusion) and
386 via grain-boundaries (e.g. Dohmen and Milke, 2010; Farver and Yund, 1995; Giletti, 1994;
387 Keller et al., 2008).

388 Our observations suggest that effective Ca diffusivities in mafic units (~10⁻¹⁰ m²/yr, similar to
389 volume-diffusion in feldspar at 900°C) are less affected by grain-boundary diffusion, and that
390 pelitic units (Ca diffusivities up to ~10⁻⁷ m²/yr) are more consistent with Ca transport dominated
391 by grain-boundary diffusion (Farver and Yund, 1995). Pelitic and felsic units generally have
392 lower liquidus temperatures and are more likely to contain higher melt-fractions at a given
393 temperature (compared to mafic units), which may serve to enhance effective Ca diffusivities.
394 Effective diffusivity in polycrystalline materials also increases with decreasing grain sizes
395 (increasing grain-boundary densities) and with increasing [Ca] in the grain boundaries (GB) vs.
396 solids ([Ca]GB/[Ca]solid) (Dohmen and Milke, 2010), both of which are higher in metapelitic
397 samples.
The D₄₄/D₄₀ ratios required by our models are generally lower (~0.990) in mafic units and higher
(as high as ~0.999) in pelitic units, but poor spatial constraints for pelitic units lead to much
larger uncertainties for associated D₄₄/D₄₀ estimates (SM). Although isotopic diffusivities of Ca
have not yet been investigated in partially-solid or fully-solid systems, D₄₄/D₄₀ in experimental
silicate liquids is observed to range from ~0.980 to ~0.997 (e.g. Watkins et al., 2017). In
magmatic systems, where Ca diffusivities are many orders of magnitude larger than in our
models, diffusive Ca isotopic fractionation increases with magma viscosities, and there is a
positive correlation between D₄₄/D₄₀ and SiO₂ + Al₂O₃ (Watkins et al., 2017); in basaltic melts
(SiO₂ + Al₂O₃ = ~50 wt%) D₄₄/D₄₀ is ~0.997, which progresses to ~0.980 in rhyolitic melts
(SiO₂ + Al₂O₃ = ~85 wt%). Within silicate melts, Ca may diffuse as an individual species or as a
monomer with the approximate mass of plagioclase (Watkins et al., 2014), but it is uncertain
what species are applicable to volume and grain-boundary diffusion. Taken at face-value,
however, the best-constrained D₄₄/D₄₀ from our models (~0.990) is the same as estimated for an
anhydrous dacitic melt (SiO₂ + Al₂O₃ = 70 wt%), which implies that associated grain-boundary
fluids (e.g. inter-granular fluids present during granulite-facies metamorphism) are likely to be
silicate melts.

4.2. Implications for Ca isotope thermometry

Based on our Δ<sup>44</sup>Ca and Δ<sup>48</sup>Ca' measurements, many of our inter-mineral samples are indicative
of disequilibrium Ca isotope fractionation and cannot be reliably used for Ca isotope
geothermometry. Excluding samples with obvious disequilibrium Δ<sup>44</sup>Ca (details in SM), we use
our DFT results to calculate 1000lnα (equivalent to Δ<sup>44</sup>Ca) for mineral-pairs at various
temperatures and compare them to our Ca isotope measurements. Given that Ca diffusion in
minerals is several orders of magnitude slower than Fe-Mg exchange (Carlson, 2006, 2002;
Hudgins et al., 2011; Vielzeuf et al., 2007), and therefore has higher closure temperatures (e.g.
Hudgins et al., 2011 and references therein), it is expected that Δ⁴⁴Ca temperatures will be
slightly higher than traditional Fe-Mg exchange temperatures. We find, however, that most of
our samples give unreliable temperatures (described in SM, and tabulated in Table S9), even
after removing those in obvious disequilibrium.

Ca-in-opx temperatures in our samples, although not specifically developed for application to
granulites (Brey and Kohler, 1990), appear to be more reliable than our Δ⁴⁴Ca based estimates in
mafic samples, agreeing with other temperature estimates for the samples and localities (Table
S9). This indicates that calcium concentrations in Opx can be approximately at equilibrium, but
still retain a large degree of isotopic disequilibrium (e.g. sample 2030B has Ca-in-Opx
temperature of 899°C but Δ⁴⁴Caopx-plag >1.66‰), which suggests that kinetic processes influenced
the Ca pool available to orthopyroxene (further discussed in Section 4.4.2). Previous work has
also found that orthopyroxene in peridotites can be at approximate chemical equilibrium (with
clinopyroxene) but can still retain measureable disequilibrium Ca isotope compositions (Zhao et
al., 2017). Samples DDM303 and DDM342, which have obvious kinetic Δ⁴⁴Cagrt-fsp (~ -1‰), are
also approximately in Fe-Mg chemical equilibrium (Table S9).

All in all, it appears that only ~5 of our samples give Δ⁴⁴Ca-based temperatures that are
approximately in the correct range, suggesting that even samples with near-equilibrium chemical
compositions can be affected by disequilibrium isotopic effects during metamorphism. Our
dataset was obtained mostly from granulite-facies samples, and only a few lower-grade
metamorphic rocks. However, available data suggest that Ca isotope thermometry in these lower-
grade samples yields better results than in granulites (Table S9).

4.3. Implications for magmatic differentiation
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Given that plagioclase represents the isotopically lightest silicate phase at equilibrium, enrichment of plagioclase in melt (e.g. during partial-melting) would lead to negative Δ⁴⁴Caₘelt-residue, if occurring at equilibrium. This could help explain rough trends observed in igneous rocks, where more evolved melts may have slightly lower δ⁴⁴Ca (Amini et al., 2009; Schiller et al., 2016). An incipiently-melted amphibolite from IVZ (IZ408), seems to indicate that there may be a negative δ⁴⁴Ca shift between melt (leucosome, -0.22‰) and residue (melanosome, -0.06‰).

The melt phase from this sample has essentially the same δ⁴⁴Ca as measured in feldspathic dikes from the NC (2038 and 2044, both -0.35‰). Garnet fractionation in magma sources has also been suggested to potentially explain δ⁴⁴Ca vs. REE trends in mafic whole-rocks (He et al., 2017; Kang et al., 2017; Magna et al., 2015), which we find to be in agreement with our DFT estimates for the four major garnet species (combined with our observations suggesting that magmatic garnet crystallization tends to occur closer to equilibrium than in metamorphic rocks).

Models for kinetic isotope fractionation during crystallization depend most significantly on (i) distribution coefficients for Ca in minerals *vs.* in melt and (ii) crystal growth rates relative to Ca diffusivity in the magma (Watkins et al., 2017; Watson and Müller, 2009). Generally, crystals with larger growth rates are expected to have larger kinetic deviations from equilibrium, which we confirm based on our Δ⁴⁴Ca and Δ⁴⁸Ca' measurements in a fast-cooled experimental tonalite sample (Fig. 6).

Slower crystallization rates are expected to result in isotopic fractionations closer to predicted equilibrium (Watson and Müller, 2009), which we confirm by measuring equilibrium Δ⁴⁸Ca' in slow-cooled mineral-separates from a cumulate gabbro sample (Stillwater Complex, Montana, Fig. 6, Table S7). A recent study of crystallization in a Hawaiian magma chamber suggests that
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slow crystallization of plagioclase + clinopyroxene, however, does not lead to significant
mineral-melt Δ⁴⁴Ca fractionation due to small competing fractionations (Zhang et al., 2018).

468

**4.4. Mineral disequilibria and possible paragenetic constraints**

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Although most of our samples cannot be used for thermometry, the kinetic effects contain information regarding mineral growth/paragenesis during high-T metamorphism. The triple-isotope slopes between mineral separates from these samples are approximately the same as measured for whole-rocks and for the fast-cooled experimental glass-plagioclase separates (Fig. 6), which suggests that diffusive processes are responsible for both whole-rock and inter-mineral disequilibria.

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Based on the trends in Fig. 4, it appears that many garnet and orthopyroxene Δ⁴⁴Ca measurements deviate progressively further from our DFT predictions in a systematic fashion. Specifically, deviations increase with increasing whole-rock and garnet Mg#, feldspar An content, Ca/Mg of orthopyroxene, and modal orthopyroxene content. Interestingly, all of these parameters tend to get larger in higher-temperature rocks. Thus, unlike equilibrium fractionation effects, which generally decrease with increasing temperature, increasing temperatures may play a role in variably activating diffusion pathways for Ca in different minerals and/or along grain-boundaries. Activation energies for grain-boundary diffusion are generally 40-60% lower than those required for volume diffusion (Dohmen and Milke, 2010), potentially implying that activation of volume-diffusion at higher temperatures may be responsible for greater Ca isotope fractionations with increasing temperatures. Changes in temperature also affect the composition and assemblage of stable metamorphic phases.

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488

In typical mafic composition rocks, the high-temperature side of granulite-facies mineral reactions generally include plagioclase + orthopyroxene + clinopyroxene ± garnet (Turner,
1981). High-temperature pelitic and semi-pelitic composition rocks may contain plagioclase + garnet ± orthopyroxene ± sillimante/kyanite assemblages, but in high-Mg pelites (such as 2032A) can also include sapphirine. Many metamorphic reactions in both bulk compositions have garnet on the higher-pressure side and orthopyroxene and plagioclase on the lower-pressure, higher-temperature side. As such, different pressure-temperature-time (P-T-t) paths and their associated reaction histories may influence inter-mineral Δ⁴⁴Ca fractionations and whether kinetic or equilibrium processes control Ca isotope partitioning.

### 4.4.1. Garnet Δ⁴⁴Ca disequilibria

In the NC samples, garnet grains contain inclusions of several high-temperature minerals and appear to have generally formed during isobaric cooling (e.g. 'retrograde garnet growth'), which is consistent with inferred P-T-t paths for the region (e.g. Harley, 2016). Growing garnet grains would thus have to incorporate Ca originally-sourced from other minerals, which could be subjected to isotopic fractionation during diffusion along grain boundaries and/or through minerals. All but one of our garnet-bearing samples from the NC have negative Δ⁴⁴Ca_grt-fsp and/or Δ⁴⁴Ca_grt-WR (discussed in SM). Slave Province xenoliths have variable Δ⁴⁴Ca_grt-fsp which range from strongly negative to slightly positive and are also consistent with isobaric cooling pathways [Supplementary Information of (Antonelli et al., 2019)]. We also note that only samples with low modal clinopyroxene (< ~10%) have large positive and negative Δ⁴⁴Ca_grt-mineral (Fig. S7).

These observations suggest that retrograde transformation of clinopyroxene (+ plagioclase) to garnet does not lead to significant disequilibrium Δ⁴⁴Ca_grt-mineral, but that transformation of orthopyroxene (or sapphirine) and plagioclase to garnet is accompanied by larger disequilibrium effects. This suggests that Ca diffusion length-scales and accompanying isotopic fractionations increase as Ca becomes locally depleted, or that garnet sourced from clinopyroxene is not
subjected to significant Ca isotope kinetic effects. In a general sense, isotopic fractionation
during retrograde garnet growth could happen during (i) Ca diffusion out of other minerals, (ii)
transport of Ca towards growing garnet porphyroblasts, and/or (iii) attachment or removal,
respectively, of Ca from growing garnet or resorbing mineral surfaces.

All three garnet-bearing samples from the IVZ, which has a clockwise PTt path (e.g. Redler et al., 2012), are likely to represent prograde garnet growth during metamorphism (excluding IV-16-19) and have positive Δ⁴⁴Ca<sub>grt-fsp</sub>. Sample IV-16-19, which was shown to contain equilibrium Δ⁴⁸Ca' signatures and reasonable equilibrium Δ⁴⁴Ca<sub>grt-fsp</sub> temperatures (800-1000 °C), represents magmatic garnet formation in a crustally-contaminated late-stage lower-crustal gabbro intrusion, as opposed to prograde metamorphism (SM). The Ca isotope differences between the contrasting garnet formation pathways are potential manifestations of diffusion limitation during retrograde growth, because prograde garnet formation is attended by fluids, whereas volatiles have been mostly lost from the system prior to retrograde garnet formation.

In sum, our observations suggest that magmatic garnet formation may occur close to equilibrium
in igneous rocks, whereas garnet formation in high-grade metamorphic rocks depends on PTt
paths, where retrograde garnet formation leads to large negative kinetic effects and prograde
garnet formation occurs closer to equilibrium. The rough <sup>40</sup>K-<sup>40</sup>Ca isochrons presented in
(Antonelli et al., 2019) for NC complex samples (average of 2.3 ± 0.2 Ga) are in agreement with
more accurate ages for granulite-facies metamorphism in the area, suggesting that our samples
are not likely to have been isotopically reset by later events. Our data also appear to suggest that
resorption of garnet in partial melts (as evidenced by NC sample 2038, SM) potentially leads to
large positive Δ<sup>44</sup>Ca<sub>grt-fsp</sub>, similar to the interpretation for disequilibrium Fe isotope fractionations
in xenocrystic olivine grains (McCoy-West et al., 2018).
535 **4.4.2. Orthopyroxene Δ<sup>44</sup>Ca disequilibria**

536 Large disequilibrium Δ<sup>44</sup>Ca<sub>opx-mineral</sub>, like large disequilibrium Δ<sup>44</sup>Ca<sub>grt-mineral</sub>, is limited to rocks
537 with not only low modal clinopyroxene (< ~15%), but also low feldspar (<20%), and no apparent
garnet (Fig. S7). Correspondingly, we find that rocks with greater modal orthopyroxene
538 generally have Δ<sup>44</sup>Ca<sub>opx-mineral</sub> further from equilibrium (Fig. 4f), along with other proxies
539 consistent with increasing temperatures (Fig. 4c-e). This suggests that large positive
540 orthopyroxene Ca isotope fractionations occur at high-temperatures when Ca is generally sparse
541 in rocks, implying that (i) large positive Δ<sup>44</sup>Ca<sub>opx-mineral</sub> could be limited to samples where Ca has
542 to be sourced from minerals with low [Ca] (e.g. albitic plagioclase or K-feldspar) and/or that (ii)
543 new pathways for loss of light Ca isotopes from orthopyroxene (relative to other mineral phases)
544 are activated with increasing metamorphic temperatures.
545

546 Although Ca concentration gradients do not generally favor diffusive loss from orthopyroxene,
547 Fe and Mg, which are strongly concentrated in opx, may serve to carry Ca outwards and lead to
548 enhanced isotopic fractionation. In olivine, Mg and Fe have been shown to diffuse 4-6 times
549 faster than Ca during high-T metamorphism (corona-formation), and can serve to carry Ca (even
550 up its own concentration gradients) (Grant, 1988). Other studies have also shown that Fe and Mg
551 diffusivities are orders of magnitude faster than Ca diffusion in olivine, clinopyroxene, and
552 orthopyroxene (Hudgins et al., 2011), which may be one of the reasons that samples with Fe-Mg
553 exchange temperatures in apparent equilibrium can still have disequilibrium Δ<sup>44</sup>Ca (Table S9).
554

554 Ca could also be lost from orthopyroxene during cooling, as it becomes less compatible with
555 decreasing temperatures (Brey and Kohler, 1990). Our Ca-in-Opx temperature estimates, based
556 on the thermometer of (Brey and Kohler, 1990), appear to reflect near-equilibrium [Ca]
557 concentrations in orthopyroxene at appropriate temperatures (except in 2040C, Table S9), yet we
observe that these samples can still contain excessively high or excessively low $\Delta^{44}\text{Ca}_{\text{opx-mineral}}$.

This implies that Ca in orthopyroxene can exhibit isotopic disequilibrium while simultaneously reflecting Ca chemical equilibrium. This combination suggests that either (i) isotopic rate differences ($D_{44}/D_{40}$) are high relative to Ca diffusivities (otherwise diffusive fractionation in grain-boundary fluids or in mineral grains would be accompanied by depletions in [Ca]), or that (ii) the effects are due to isotopic rate differences during attachment/detachment at crystal surfaces ['surface-kinetic effects' (e.g. DePaolo, 2011)] which are currently unexplored in silicates. On the other hand, several samples with $\Delta^{44}\text{Ca}_{\text{opx-mineral}}$ too small to be explained by equilibrium at relevant temperatures could be a manifestation of the same processes envisaged for large negative $\Delta^{44}\text{Ca}_{\text{grt-mineral}}$ (e.g. fractionation during diffusion towards growing minerals) but would suggest fractionation during prograde as opposed to retrograde metamorphism, due to increasing stability of Opx with increasing temperatures.

## 5. Conclusions

Using TIMS to measure radiogenic-corrected $\delta^{44}\text{Ca}$ in a large number of high-temperature metamorphic rocks and mineral separates (spanning amphibolite-facies, granulite-facies, UHT, and eclogite-facies) from various localities, and comparing them to $\mu^{48/44}\text{Ca}$ and $\mu^{42/44}\text{Ca}$ measurements by MC-ICP-MS, ab-initio predictions from DFT, and numerical diffusion models, we have demonstrated that:

(1) At temperatures greater than ~800°C, equilibrium $\Delta^{44}\text{Ca}$ fractionations do not exceed ~1‰, and $\delta^{44}\text{Ca}$ is expected to decrease in the general order: olivine > garnet ≥ orthopyroxene > clinopyroxene > apatite > plagioclase. At ~900°C, average analytical uncertainties in measured $\delta^{44}\text{Ca}$ in minerals and, in turn, inter-mineral $\Delta^{44}\text{Ca}$ fractionations translate into isotopic temperature uncertainties of ~ ±50°C for Opx/Ol-Plag, ~ ±100°C for Opx/Ol-Cpx,
and larger for minerals with smaller predicted $\Delta^{44}$Ca (e.g. Cpx-Plag, ~ ±250°C). The best mineral pairs for mantle thermometry, assuming equilibrium, would thus be olivine-plagioclase and/or olivine-clinopyroxene.

(2) Variations in whole-rock and mineral $\delta^{44}$Ca in high-temperature metamorphic samples are found to span a large range of ~4‰ and ~8‰, respectively. $\Delta^{44}$Ca between adjacent whole-rock samples can exceed 3‰, and inter-mineral fractionations (e.g. $\Delta^{44}$Ca$_{grt-fsp}$) are observed to vary by a similar amount. We find that inter-mineral $\Delta^{44}$Ca fractionations deviate progressively further from predicted equilibrium in conjunction with chemical and mineralogical parameters traditionally associated with increasing temperatures (e.g. Mg#, feldspar anorthite content, & Opx Ca/Mg), and when samples have low modal clinopyroxene fractions (< ~15%). Only a small percentage of our samples have Ca-isotope signatures that agree with our equilibrium predictions at relevant temperatures.

(3) Kinetic and equilibrium Ca isotope fractionations greater than ~0.5‰ can be distinguished using the parameter $\Delta^{48}$Ca', which corresponds to $\mu^{48/44}$Ca deviations from predicted equilibrium values in triple-isotope space. We show that the largest positive and negative $\Delta^{44}$Ca fractionations in our data set (including whole-rocks, Grt-Plag, and Opx-Plag pairs), along with plagioclase crystallization in a fast-cooled experimental tonalite, are consistent with (non-zero) $\Delta^{48}$Ca' for kinetic-fractionation, generally falling along the slope predicted by the kinetic-limit for mass-dependent fractionation (Young et al., 2002). Samples from a slowly-crystalized cumulate gabbro sample (Stillwater-complex), along with garnet and feldspar separates from the metamorphic sample which best-corroborates our DFT predictions, were found to have equilibrium $\Delta^{48}$Ca'.
603

(4) Ca diffusivities required to model our data from the Napier Complex (10⁻¹⁰ to 10⁻⁷ m²/yr) are much slower than in silicate liquids (Watkins et al., 2017), but are consistent with estimates for Ca diffusion within minerals (volume-diffusion) and faster Ca diffusion via grain boundary fluids (e.g. Dohmen and Milke, 2010). The best-constrained isotopic diffusivity differences (D₄₄/D₄₀, ~0.990) are approximately the same as determined for anhydrous silicate liquids with dacitic bulk compositions (e.g. Watkins et al., 2017), which implies that silicate melts are likely to be present as intergranular fluids during granulite-facies metamorphism.

604

605

606

607

608

609

610

611

(5) Inter-mineral Δ⁴⁴Ca disequilibria are likely to be related to variations in mineral paragenesis along different PTt paths for various localities. Our data suggest that retrograde garnet formation (e.g. during slow isobaric cooling from high-temperatures) leads to negative disequilibrium Δ⁴⁴Ca₉₄⁻mineral, which is likely a result of diffusive limitations and associated kinetic isotope fractionations during garnet growth. Magmatic and prograde garnet formation, on the other hand, appear to occur closer to equilibrium. Large positive Δ⁴⁴Ca₉₄⁻mineral (observed in a single feldspathic dike sample from the NC) may be the result of outward Ca diffusion (aided by Fe and Mg diffusion) from garnet entrained in partial-melts.

612

613

614

615

616

617

618

619

(6) Kinetic Ca isotope effects in orthopyroxene are capable of generating Δ⁴⁴Ca₉₄⁻mineral both much higher and much lower than predicted equilibrium at relevant temperatures, but they are not as easily linked to paragenetic processes. Both Fe-Mg exchange and Ca-in-Opx thermometry yield temperatures in agreement with previous work for the different regions, even for samples with obvious disequilibrium Δ⁴⁴Ca₉₄⁻mineral. Thus, Ca, Fe, and Mg concentrations in orthopyroxene can be in approximate chemical equilibrium, but Ca isotopic compositions (including Δ⁴⁸Ca') can still be significantly out of equilibrium. This implies

620

621

622

623

624

625
626

that the relevant processes have large isotopic rate differences ($D_{44}/D_{40}$) relative to their Ca
diffusivities. Hence, disequilibrium orthopyroxene signatures are not likely to be generated
627
628
by isotopic fractionation during Ca diffusion from far-field sources, and may instead result
629
from isotopic fractionation during nearby volume diffusion and/or during
630
attachment/detachment at mineral surfaces (e.g. DePaolo, 2011).

627

628

629

630

631

(7) In summary, we have shown that kinetic Ca isotope effects are abundant in natural samples, even at very high-temperatures (>900°C). Disentangling equilibrium effects from kinetic effects, and understanding the mechanisms/pathways responsible for Ca isotopic disequilibria in nature, however, would be further enhanced by similar studies focused on higher-resolution sampling, and/or by development of in-situ δ<sup>44</sup>Ca analysis techniques. Although disequilibrium effects may be more pervasive in metamorphic rocks than in igneous rocks (e.g. due to higher associated viscosities and/or longer time-scales), we have shown that a large degree of caution should be taken when assuming Ca isotope equilibrium (e.g. for Ca isotope thermometry), or if using stable Ca isotope variations to trace complicated multi-step processes, such as carbonate recycling into the mantle.

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641

Acknowledgements

642

643

644

645

646

647

648

We thank N. Botto, S. Matveev, A. Locock, W. Yang, T. Teague, and S.T. Brown for their
technical expertise, and Daniela Rubatto for Ivrea samples. This research was primarily
supported by a U.S. National Science Foundation (EAR100500) grant to D.J.D. T.C.
acknowledges funding from a Natural Sciences and Engineering Research Council of Canada
(NSERC) discovery grant, and M.A.A. acknowledges NSERC post-graduate funding that aided
in supporting this work. E.A.S. acknowledges support from NSF grant EAR1530306. E.S.G.
thanks the Australian Antarctic Division for the opportunity to participate in the 1977-1978
649 Australian National Antarctic Research Expedition and for logistic support, and acknowledges
650 NSF grant DPP 76-80957 to UCLA.

651 Author contributions

652 M.A.A. designed research; M.A.A. performed TIMS and chemical analyses; M.S. performed
653 MC-ICP-MS analyses; E.A.S. performed ab-initio modeling; T.M. and M.A.A. performed
654 diffusion modeling, M.A.A., D.J.D., and T.C. analyzed data; E.S.G, T.C., and D.J.D. provided
655 samples collected in the field; B.T. provided high-T experimental samples; M.A.A. wrote the
656 paper with input from T.C., M.S., E.A.S., E.S.G., and D.J.D.

657 Competing financial interests

658 The authors declare no competing financial interests.

659
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