Diffusion of Helium in Calcite and Aragonite:A First-Principles Study

LI Shuchen LIU Hong YANG Yaochun DING Jianhua LIU Lei LI Ying YI Li TIAN Hua

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Diffusion of Helium in Calcite and Aragonite:A First-Principles Study

    作者简介: LI Shuchen (1995-), female, master, major in condensed matter physics. E-mail: 642328905@qq.com;
    通讯作者: LIU Hong, liuh@cea-ies.ac.cn
  • 中图分类号: O521.2

氦在方解石和文石中的扩散:基于第一性原理的研究

    Author Bio: LI Shuchen (1995-), female, master, major in condensed matter physics. E-mail: 642328905@qq.com;
    Corresponding author: LIU Hong, liuh@cea-ies.ac.cn ;
  • CLC number: O521.2

  • 摘要: Helium diffusion in carbonate minerals is important for studying the physical and chemical properties and dynamic processes of Earth’s degassing. This paper discussed helium incorporation and diffusion mechanism in crystals of calcite and aragonite based on density functional theory calculations. The diffusion pathways, activation energies (Ea), and frequency factors (v) of helium under the surface and mantle condition were calculated. Calculations show an apperant anisotropy of helium diffusion in calcite, with more energetically favorable directions along a(b) axis. The moderate anisotropy of helium diffusion is showed in aragonite, in which the diffusion rate along c axis is slower than that along a axis. Under high pressure conditions, the activation energies of helium diffusion in aragonite increase with pressure. The closure temperature for calcite crystal varies from −54 ℃ to −25 ℃ in the direction [010], and for aragonite varies from −12 ℃ to 23 ℃ in [100]. Aragonite may be more retentive for helium than calcite under surface condition, which agrees well with previous experimental studies.
  • Figure 1.  Schematic diagrams showing unit cell of calcite ((a) and (b)) and aragonite ((c) and (d)) (The two structures both show layers of Ca2+ cations and layers of planar CO3 groups stacked perpendicular to the c-axis. Green: Ca, red: O, gray: C.)

    Figure 2.  Diffusion pathways of helium atom in calcite along [010] (a) and [001] via the S1′ site and reaching the S3 site (b); in aragonite along [100] (c) and [001] via the S1′ site and reaching the S1″ site (d)

    Figure 3.  Energy barriers of different paths for helium diffusion in calcite: (a) ${S_1^{\rm Cal}- S_2^{\rm Cal}}$ path in the [010] direction, (b) ${S_1^{\rm Cal}- S_{1'}^{\rm Cal}-S_3^{\rm Cal}}$ path in the [001] direction; in aragonite: (c) ${S_1^{\rm Arg}- S_2^{\rm Arg}}$ path in the [100] direction, (d) ${S_1^{\rm Arg}- S_{1'}^{\rm Arg}-S_{1''}^{\rm Arg}}$ path in the [001] direction

    Figure 4.  Comparisons of our Arrhenius relations for calcite (a) and aragonite (b) with the data of Cherniak et al.[4] (He diffusion in calcite displays marked anisotropy, while in aragonite shows moderate anisotropy.)

    Figure 5.  Effect of pressure on helium diffusion in aragonite up to 14 GPa in the [100] (a) and [001] (b) directions (The diffusion coefficients obviously decrease with pressure increasing in both directions.)

    Figure 6.  Calculated closure temperature (Tc) as a function of grain radius (a) along different directions in calcite and aragonite (Closure temperature are plotted for assuming spherical geometry (A=55) of the crystals. Helium in each carbonate composition using Dodson’s (1973) equation and a cooling rate of 10 ℃/Ma.)

    Table 1.  Calculated structural parameters of calcite and aragonite in comparison with previous theoretical and experimental values

    MineralData sourceUnit cell volume/nm3a/nmb/nmc/nmC-O bond distance/nmCa-O bond distance/nm
    Calcite This work 379.58 5.05 17.21 1.299 2.383
    Calculation[22] 383.20 5.05 17.33 1.291 2.397
    Experiment[23] 368.10 4.99 17.06 1.284 2.359
    Aragonite This work 232.58 5.01 8.01 5.79 1.291 2.469
    Calculation[24] 233.84 5.02 8.04 5.80 1.292 2.440
    Experiment[25] 226.65 4.96 7.96 5.74 1.284 2.414
    下载: 导出CSV

    Table 2.  Calculated parameters for helium diffusion in calcite and aragonite under ambient and high pressures

    Mineral Pressure/GPa Direction Ea/(kJ·mol–1) v/THz l/nm D0/(m2·s–1)
    Calcite 0 [010] 67.64 4.29 5.05 5.46×10–7
    0 [001] 97.36 4.19 4.71 4.65×10–7
    0 [100] 82.40 7.71 5.00 9.64×10–7
    0 [001] 96.00 6.34 5.79 1.06×10–6
    3 [100] 110.57 7.11 4.98 8.82×10–7
    3 [001] 125.43 6.57 5.68 1.06×10–6
    Aragonite 6 [100] 115.78 7.03 4.95 8.61×10–7
    6 [001] 133.63 6.85 5.85 1.17×10–6
    10 [100] 139.42 7.54 4.90 9.05×10–7
    10 [001] 160.17 7.56 5.47 1.13×10–6
    14 [100] 154.38 7.01 4.86 8.28×10–7
    14 [001] 174.45 8.41 5.36 1.21×10–6
    下载: 导出CSV

    Table 3.  Summary of the characteristic bond distances of activated states in calcite and aragonite under different pressure conditions (All bond distances are the smallest distances.)

    MineralPressure/GPaDirectionCa-O bond distance/nmC-O bond distance/nmHe-O bond distance/nm
    Calcite 0[010]2.2611.2992.033
    0[001]2.2411.2951.922
    0[100]2.3561.2962.141
    0[001]2.4121.2872.042
    3[100]2.3581.2942.067
    3[001]2.3791.2942.002
    Aragonite 6[100]2.3511.2892.026
    6[001]2.3171.2841.970
    10[100]2.3401.2781.982
    10[001]2.2901.2741.927
    14[100]2.3341.2771.951
    14[001]2.2651.2701.909
    下载: 导出CSV
  • [1] CHERNIAK D J, WATSON E B, THOMAS J B. Diffusion of helium in zircon and apatite [J]. Chemical Geology, 2009, 268(1): 155–166.
    [2] REICH M, EWING R C, EHLERS T A, et al. Low-temperature anisotropic diffusion of helium in zircon: implications for zircon (U–Th)/He thermochronometry [J]. Geochimica et Cosmochimica Acta, 2007, 71(12): 3119–3130. doi: 10.1016/j.gca.2007.03.033
    [3] REINERS P W. Zircon (U-Th)/He thermochronometry [J]. Reviews in Mineralogy Geochemistry, 2005, 58(1): 151–179. doi: 10.2138/rmg.2005.58.6
    [4] CHERNIAK D J, AMIDON W, HOBBS D, et al. Diffusion of helium in carbonates: effects of mineral structure and composition [J]. Geochimica et Cosmochimica Acta, 2015, 165: 449–465. doi: 10.1016/j.gca.2015.06.033
    [5] COPELAND P, WATSON E B, URIZAR S C, et al. Alpha thermochronology of carbonates [J]. Geochimica et Cosmochimica Acta, 2007, 71(18): 4488–4511. doi: 10.1016/j.gca.2007.07.004
    [6] COPELAND P, COX K, WATSON E B. The potential of crinoids as (U+Th+Sm) /He thermochronometers [J]. Earth and Planetary Science Letters, 2015, 42: 1–10.
    [7] CROS A, GAUTHERON C, PAGEL M, et al. 4He behavior in calcite filling viewed by (U-Th)/He dating, 4He diffusion and crystallographic studies [J]. Geochimica et Cosmochimica Acta, 2014, 125: 414–432. doi: 10.1016/j.gca.2013.09.038
    [8] AMIDON W H, HOBBS D, HYNEK S A, et al. Retention of cosmogenic 3He in calcite [J]. Quaternary Geochronology, 2015, 27: 172–184. doi: 10.1016/j.quageo.2015.03.004
    [9] BENGTSON A, EWING R C, BECKER U. He diffusion and closure temperatures in apatite and zircon: a density functional theory investigation [J]. Geochimica et Cosmochimica Acta, 2012, 86: 228–238. doi: 10.1016/j.gca.2012.03.004
    [10] WANG K, BRODHOLT J, LU X. Helium diffusion in olivine based on first principles calculations [J]. Geochimica et Cosmochimica Acta, 2015, 156: 145–153. doi: 10.1016/j.gca.2015.01.023
    [11] BALOUT H, ROQUES J, GAUTHERON C, et al. Helium diffusion in pure hematite (α-Fe3O3) for thermochronometric applications: a theoretical multi-scale study [J]. Computational and Theoretical Chemistry, 2017, 1099: 21–28. doi: 10.1016/j.comptc.2016.11.001
    [12] SONG Z, WU H, SHU S, et al. A first-principles and experimental study of helium diffusion in periclase MgO [J]. Physics and Chemistry of Minerals, 2018, 45(7): 641–654. doi: 10.1007/s00269-018-0949-y
    [13] DODSON M H. Closure temperatures in cooling geological and petrological systems [J]. Contributions to Mineralogy Petrology, 1973, 40(3): 259–274. doi: 10.1007/BF00373790
    [14] HOHENBERG P, KOHN W. Inhomogenous electron gas [J]. Physical Review, 1964, 136: 864–871. doi: 10.1103/PhysRev.136.B864
    [15] KOHN W, SHAM L J. Quantum density oscillations in an inhomogeneous electron gas [J]. Physical Review, 1965, 137: 1697–1705. doi: 10.1103/PhysRev.137.A1697
    [16] KRESSE G, FURTHMULLER J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set [J]. Computational Materials Science, 1996, 6(1): 15–50. doi: 10.1016/0927-0256(96)00008-0
    [17] KRESSE G, HAFNER J. Ab initio molecular dynamics for liquid-metals [J]. Physical Review B, 1993, 47(1): 558–561. doi: 10.1103/PhysRevB.47.558
    [18] BLÖCHL P E. Projected augmented-wave method [J]. Physical Review B, 1996, 50(24): 17953–17979.
    [19] KRESSE G, JOUBERT D. From ultrasoft pseudopotentials to the projector augmented-wave method [J]. Physical Review B, 1999, 59(3): 1758–1775. doi: 10.1103/PhysRevB.59.1758
    [20] PERDEW J P, BURKE K, ERNZERHOF M. Generalized gradient approximation made simple [J]. Physical Review Letters, 1996, 77(18): 3865–3868. doi: 10.1103/PhysRevLett.77.3865
    [21] CHADI D J. Special points for Brillouin-zone integrations [J]. Physical Review B, 1977, 16(4): 1746–1747. doi: 10.1103/PhysRevB.16.1746
    [22] BRIK M G. First-principles calculations of structural, electronic, optical and elastic properties of magnesite MgCO3 and calcite CaCO3 [J]. Physica B: Condensed Matter, 2011, 406(4): 1004–1012. doi: 10.1016/j.physb.2010.12.049
    [23] MALSEN E N, STRELTSOV V A, STRELTSOVA N R, et al. X-ray study of the electron density in calcite, CaCO3 [J]. Acta Crystallographica Section B: Structural Science, 1993, 49(4): 636–641. doi: 10.1107/S0108768193002575
    [24] OGANOV A R, GLASS C W, ONO S. High-pressure phases of CaCO3: crystal structure prediction and experiment [J]. Earth and Planetary Science Letters, 2006, 241(1): 95–103.
    [25] DICKENS B, BOWEN J S. Refinement of the crystal of the aragonite phase of CaCO3 [J]. Physics and Chemistry A, 1971, 75(1): 27–32.
    [26] HENKELMAN G. Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points [J]. Journal of Chemical Physics, 2000, 113(22): 9978–9985. doi: 10.1063/1.1323224
    [27] VINEYARD G H. Frequency factors and isotope effects in solid state rate processes [J]. Journal of Physics and Chemistry of Solids, 1957, 3(1/2): 121–127.
    [28] BENDER M L. Helium-uranium dating of corals [J]. Geochimica et Cosmochimica Acta, 1973, 37(5): 1229–1247. doi: 10.1016/0016-7037(73)90058-6
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出版历程
  • 收稿日期:  2018-12-12
  • 录用日期:  2019-01-03
  • 网络出版日期:  2019-09-10
  • 刊出日期:  2019-10-01

Diffusion of Helium in Calcite and Aragonite:A First-Principles Study

    作者简介:LI Shuchen (1995-), female, master, major in condensed matter physics. E-mail: 642328905@qq.com
    通讯作者: LIU Hong, liuh@cea-ies.ac.cn
  • 1. Institute of Earthquake Forecasting,China Earthquake Administration (CEA), Beijing 100036, China
  • 2. Dalian University of Technology, Dalian 116024, China
  • 3. Taiyuan University of Technology, Taiyuan 030024, China

摘要: Helium diffusion in carbonate minerals is important for studying the physical and chemical properties and dynamic processes of Earth’s degassing. This paper discussed helium incorporation and diffusion mechanism in crystals of calcite and aragonite based on density functional theory calculations. The diffusion pathways, activation energies (Ea), and frequency factors (v) of helium under the surface and mantle condition were calculated. Calculations show an apperant anisotropy of helium diffusion in calcite, with more energetically favorable directions along a(b) axis. The moderate anisotropy of helium diffusion is showed in aragonite, in which the diffusion rate along c axis is slower than that along a axis. Under high pressure conditions, the activation energies of helium diffusion in aragonite increase with pressure. The closure temperature for calcite crystal varies from −54 ℃ to −25 ℃ in the direction [010], and for aragonite varies from −12 ℃ to 23 ℃ in [100]. Aragonite may be more retentive for helium than calcite under surface condition, which agrees well with previous experimental studies.

English Abstract

  • Helium gas in minerals may provide plenty of useful information on Earth’s mantle evolution and geodynamic processes. There are many accessory minerals with high concentration of U and Th, such as apatite and zircon, have been widely used in (U-Th)/He dating method[1-3]. However, calcite, quartz, olivine, and other common minerals are less involved in this method. The application of carbonate (U-Th)/He method in thermo-chronometry has attracted more and more attention due to its ubiquity, large grain size, and extremely low closure temperature on the Earth[4-7]. 3He is attributed to the presence of primordial helium leaking from the mantle and 4He is produced by the decay of radioactive isotopes. High 3He/4He ratios are characteristic of samples of mantle origin. To understand dating and cooling histories of rocks, the diffusion mechanism of helium in carbonate minerals should be studied.

    Recently, the experimental approaches to determining He diffusivities in carbonates have obtained some new achievements[4-8]. Copeland et al.[5] undertook a series of bulk step-heating experiments on calcite. They suggested that the diffusion of helium in calcite has no connection with the origin of minerals or the source of helium. The potentiality of calcite (U-Th)/He dating was also investigated by Cros et al.[7] Later, Amidon et al.[8] explored the production and retention of helium in calcite samples at several different locations. Step-degassing experiments were also used to investigate the diffusion of He in calcite. All the above experiments indicate that helium diffusion in calcite is influenced by multiple diffusion domains (MDD). This factor has been attributed to the loss of He from small domains with a faster diffusion rate, making it difficult to speculate bulk He retention in natural samples[8]. To avoid these limitations of bulk degassing experiments, Cherniak et al.[4] performed ion implantation experiments and NRA (nuclear reaction analysis) measurements to study helium diffusion in calcite, dolomite, magnesite, and aragonite. This approach can be used to study the anisotropy of helium diffusion. They found that the diffusion is anisotropic in calcite, dolomite and magnesite, and is slowest along the [001] direction. They found that magnesite and calcite are unlikely to be retentive of He on the Earth’s surface conditions, while dolomite and aragonite can be retentive under cooler conditions[4].

    To better understand the diffusion mechanism and rate of helium in carbonates, we undertake a series of theoretical computations. The density functional theory (DFT) and climbing image nudged elastic band (CI-NEB) method are powerful for exploring the microscopic mechanism of He diffusion in minerals and have been applied to investigate the diffusion mechanism of helium in a few important minerals, such as zircon and apatite[9], olivine[10], hematite[11], preclase[12]. This method, using the microscopic atomic-scale calculations to elucidate He diffusion in perfect crystals without impurities, defects, or radiation damage, provides the basis for comparison of diffusion rate among different minerals. In this paper, the electronic nature, diffusion pathways, activation energies, and frequency factors of He diffusion in calcite and aragonite under ambient and high pressure conditions were investigated based on the DFT and CI-NEB method. After calculating the diffusion data, we discussed the anisotropy of helium diffusion in the two mineral phases. The closure temperature were also calculated using Dodson’s equation[13].

    • The structural properties and diffusion behavior of helium in calcite and aragonite were calculated based on the density functional theory and the plane-wave pseudo-potential approach[14-15] as implemented in the Vienna Ab Initio Simulation Package (VASP)[16-17]. The projector-augmented wave (PAW) potential was used to represent the interactions between ions and electrons[18-19]. We adopted the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerh of (PBE) functional[20] to describe the exchange-correlation interaction. The Brillouin zone was sampled by a 3×3×3 Monkhorst-Pack mesh[21] of k-points for the (2×2×1) calcite supercell. For aragonite, an 80-atom supercell formed by 2×1×2 unit cells was used and the Brillouin zone was sampled by a 2×2×2 Monkhorst-Pack mesh of k-points. The atomic coordinates of the two structures were fully optimized until the energy change on each atom was less than 10–4 eV and the force was less than 0.02 eV/nm. The cutoff energy was set to be 550 eV. The energy cutoff and k-points are sufficient to reach the required accuracy in the present work.

      We first study the structural properties of bulk calcite and aragonite. The relaxed lattice parameters, and several characteristic interionic distances are reported in Table 1. The other experimental and DFT data[22-25] are also presented as comparison. As seen from Table 1, good agreement is achieved between the calculated results in the present work and experimental data. The deviation between our calculated structural parameters and previous studies were less than 1.5% for calcite and aragonite.

      MineralData sourceUnit cell volume/nm3a/nmb/nmc/nmC-O bond distance/nmCa-O bond distance/nm
      Calcite This work 379.58 5.05 17.21 1.299 2.383
      Calculation[22] 383.20 5.05 17.33 1.291 2.397
      Experiment[23] 368.10 4.99 17.06 1.284 2.359
      Aragonite This work 232.58 5.01 8.01 5.79 1.291 2.469
      Calculation[24] 233.84 5.02 8.04 5.80 1.292 2.440
      Experiment[25] 226.65 4.96 7.96 5.74 1.284 2.414

      Table 1.  Calculated structural parameters of calcite and aragonite in comparison with previous theoretical and experimental values

      To study the He diffusion in calcite and aragonite, one He atom was placed on several different types of interstitial sites. We first found the most stable interstitial sites of helium in the two systems to evaluate He interstitial formation energies. The mechanical stability of He interstitials at each site is studied by inserting a helium atom at the site and then relaxing the lattice structure to check whether the He moved to another stable site or not. In addition, the thermodynamically stable sites are determined by the formation energy.

      The formation energy for a He atom at one interstitial site is defined as

      $ \Delta {E_{{\rm He}}} = E\left( {{\rm{CaC}}{{\rm{O}}_3} + {\rm{He}}} \right){\rm{ }} - E\left( {{\rm{CaC}}{{\rm{O}}_3}} \right) - E\left( {{\rm{He}}} \right) $

      where E(CaCO3+He) is the total energy of the relaxed atomic model containing a He atom, E(CaCO3) is the total energy without He atom, and E(He) is the energy of the isolated He in vacuum.

      The activation energy of helium in calcite and aragonite and the minimum energy paths (MEPs) were calculated using CI-NEB method[26]. All images were relaxed until the maximum force on each atom is less than 0.03 eV/nm. The diffusion rate is assumed to follow an Arrhenius form

      $ D = {D_0}{\rm{exp}}( - \Delta {E_{\rm a}}/RT) $

      where D is the diffusion rate, D0 is the pre-factor, $\Delta $Ea is the activation energy, R is the gas constant, and T is the temperature in Kelvin. The activation energy Ea can be obtained by the CI-NEB method, but the pre-factor D0 in Eq.(2) is also required to calculate the diffusion rates. In the harmonic transition state theory[27], all of the atoms in the lattice vibrate harmonically around their equilibrium positions and the diffusion rate relates to the atom vibration by

      $ {D_0} = \frac {1}{2} {l^2}v $

      where

      $ \nu =\frac {{\mathop \prod \nolimits_i^{3N} {\nu _i}}}{{\mathop \prod \nolimits_i^{3N - 1} {\nu _i'}}} $

      The factor 1/2 in Eq.(3) indicates that the helium atom is moving along the specific direction, l is the jump distance between two He atoms. The frequency, v, was calculated by displacing the helium atom in the activated and ground state[27], the value of vi (the ground state) to ${\nu _i'} $ (the activated state) for N He atoms (1 in this case).

    • Calcite belongs to the R3c space group with trigonal system. Fig.1(a) and Fig.1(b) show the unit cell of calcite at ambient pressure viewed from two different directions. To investigate the helium diffusion behavior in calcite crystal, we put a single helium atom in several possible interstitial occupation sites in calcite. We calculate the total energy of the whole system and find the most stable location with minimum energy of He in calcite. The computed formation energy $\Delta E_{\rm He}$ is 74.29 kJ/mol.

      Figure 1.  Schematic diagrams showing unit cell of calcite ((a) and (b)) and aragonite ((c) and (d)) (The two structures both show layers of Ca2+ cations and layers of planar CO3 groups stacked perpendicular to the c-axis. Green: Ca, red: O, gray: C.)

      First, we pay attention to the microscopic structure of calcite and find that in the [100] and [010] directions, the atomic arrangement and lattice position have no difference (see Fig.1(b)). Helium diffusion behaviors along a-axis and b-axis are exactly identical. Then, we choose two different possible diffusion pathways between the most energetically favorable sites and obtain the relative activation energies based on the CI-NEB method. One path is a direct hop along [010] direction between nearest-neighbor He atoms (S1S2) and parallel to the CO3 layers in calcite crystal structure (see Fig.2(a)). The jump distance and calculated activation energy Ea is 5.046 nm and 67.64 kJ/mol, respectively. For the direction [001], the helium diffusion behavior is much more complicated. The direct path between the nearest-neighbor He atoms along [001] direction (S1S3) is perpendicular to the CO3 layers but hindered by a Ca2+ cation, so we choose another possible pathway for helium diffusion: along S1S1′S3 (see Fig.2(b)). The jump distance and calculated activation energy Ea is 4.711 nm and 97.36 kJ/mol, respectively. The energy barriers curves for the helium atom moving along [010] and [001] directions are shown in Fig.3(a) and Fig.3(b). The CI-NEB calculations were performed by placing 5 images between neighboring stable sites for both directions. Then, we calculate the frequencies (v) and pre-factor (D0) of the two directions. The pre-factors are 5.46×10–7 m2/s (lg D0 = −6.26) and 4.65×10–7 m2/s (lg D0 = − 6.33) for [010] and [001] directions, respectively.

      Figure 2.  Diffusion pathways of helium atom in calcite along [010] (a) and [001] via the S1′ site and reaching the S3 site (b); in aragonite along [100] (c) and [001] via the S1′ site and reaching the S1″ site (d)

      Figure 3.  Energy barriers of different paths for helium diffusion in calcite: (a) ${S_1^{\rm Cal}- S_2^{\rm Cal}}$ path in the [010] direction, (b) ${S_1^{\rm Cal}- S_{1'}^{\rm Cal}-S_3^{\rm Cal}}$ path in the [001] direction; in aragonite: (c) ${S_1^{\rm Arg}- S_2^{\rm Arg}}$ path in the [100] direction, (d) ${S_1^{\rm Arg}- S_{1'}^{\rm Arg}-S_{1''}^{\rm Arg}}$ path in the [001] direction

      According to the parameters of Eq.(2), Eq.(3) and Eq.(4) as shown in Table 2, the calculated diffusive coefficients of helium in calcite are

      Mineral Pressure/GPa Direction Ea/(kJ·mol–1) v/THz l/nm D0/(m2·s–1)
      Calcite 0 [010] 67.64 4.29 5.05 5.46×10–7
      0 [001] 97.36 4.19 4.71 4.65×10–7
      0 [100] 82.40 7.71 5.00 9.64×10–7
      0 [001] 96.00 6.34 5.79 1.06×10–6
      3 [100] 110.57 7.11 4.98 8.82×10–7
      3 [001] 125.43 6.57 5.68 1.06×10–6
      Aragonite 6 [100] 115.78 7.03 4.95 8.61×10–7
      6 [001] 133.63 6.85 5.85 1.17×10–6
      10 [100] 139.42 7.54 4.90 9.05×10–7
      10 [001] 160.17 7.56 5.47 1.13×10–6
      14 [100] 154.38 7.01 4.86 8.28×10–7
      14 [001] 174.45 8.41 5.36 1.21×10–6

      Table 2.  Calculated parameters for helium diffusion in calcite and aragonite under ambient and high pressures

      $D_{\left[ {010} \right]\left( {\left[ {100} \right]} \right)} = 5.46 \times {10^{{\rm{ - }}7}}\;{\rm{exp}}\left( {\frac{{-67.64\;{\rm{kJ/mol}}}}{{RT}}} \right)( {{{\rm{m}}^{\rm{2}}}{\rm{/s}}} ) $

      $ D_{\left[ {001} \right]} = 4.65 \times {10^{ - 7}}\;{\rm{exp}}\left( {\frac{{-97.36\;{\rm{kJ}}/{\rm{mol}}}}{{RT}}} \right)( {{{\rm{m}}^{\rm{2}}}{\rm{/s}}} ) $

      Helium diffusion data from DFT calculations are reported in Table 2, and diffusion coefficients are plotted in Fig.4(a). Calculations show marked anisotropy of helium diffusion in calcite. Diffusion rate for [010] is about five orders of magnitude faster than [001] at ambient temperature (298 K) due to its lower activation energy. It can be explained by the different microscopic atomic structures. Table 3 shows the characteristic bond distances of activated states in calcite for [010] and [001] directions. The He-O distance in [010] is about 0.11 nm longer than that in [001], and the separation between two neighboring parallel CO3 layers in [010] is about 0.06 nm larger than that in [001], indicate a more narrow space when He moving along [001], making it difficult to go through. In contrast, the C-O and Ca-O bond distances have little difference between the two directions. As temperature increases, helium diffusion in calcite becomes more and more isotropic. At 800 K, the diffusion rate for [001] is only about 2 orders of magnitude slower than that for [010], indicating that anisotropy become insignificant at mantle temperature in calcite.

      Figure 4.  Comparisons of our Arrhenius relations for calcite (a) and aragonite (b) with the data of Cherniak et al.[4] (He diffusion in calcite displays marked anisotropy, while in aragonite shows moderate anisotropy.)

      MineralPressure/GPaDirectionCa-O bond distance/nmC-O bond distance/nmHe-O bond distance/nm
      Calcite 0[010]2.2611.2992.033
      0[001]2.2411.2951.922
      0[100]2.3561.2962.141
      0[001]2.4121.2872.042
      3[100]2.3581.2942.067
      3[001]2.3791.2942.002
      Aragonite 6[100]2.3511.2892.026
      6[001]2.3171.2841.970
      10[100]2.3401.2781.982
      10[001]2.2901.2741.927
      14[100]2.3341.2771.951
      14[001]2.2651.2701.909

      Table 3.  Summary of the characteristic bond distances of activated states in calcite and aragonite under different pressure conditions (All bond distances are the smallest distances.)

    • Aragonite is a high-pressure polymorph of calcite with orthorhombic symmetry and space group Pmcn, and consists of layers of Ca2+ cations with layers of planar CO3 groups stacked perpendicular to the c-axis. Fig.1(c) and Fig.1(d) show the unit cell of aragonite viewed from two different directions. In aragonite, we put a single helium atom in several possible interstitial occupation sites and find the most stable location with minimum energy of He. The computed formation energy $\Delta$EHe is 87.80 eV.

      We find two possible diffusion pathways of helium between the most energetically favorable sites in aragonite. Diffusion along [100] direction is a direct hop (S1S2) (see Fig.2(c)). The jump distance and calculated activation energy Ea is 5.00 nm and 82.40 kJ/mol, respectively. Diffusion along [001] is more complicated because Ca2+ cation will hinder the helium motion. We find another possible path for helium diffusion along [001] direction: S1S1′S1″ (see Fig.2(d)). The jump distance and calculated activation energy Ea is 5.79 nm and 96.00 kJ/mol, respectively. Then, we calculated the frequency factors (v) and pre-factor (D0) of these two directions. The pre-factors are 9.64×10–7 m2/s (lg D0 = −6.02) and 1.06×10–6 m2/s (lg D0= −5.97) for [100] and [001] directions, respectively.

      The helium diffusion in aragonite can be written in Arrhenius as follows

      $D_{\left[ {100} \right]} = 9.64 \times {10^{ - 7}}\;{\rm{exp }}\left(\frac {- 82.40\;{\mathop{\rm kJ}\nolimits} /{\rm{mol}}} {RT}\right)({{\rm{m}}^2}/{\rm{s}}) $

      $D_{\left[ {001} \right]} = 1.06 \times {10^{ - 6}}\;{\rm{exp }}\left(\frac {- 96.00\;{\rm{ kJ}}/{\rm{mol}}}{RT} \right)({{\rm{m}}^2}/{\rm{s}}) $

      Helium diffusion data from DFT calculations are list in Table 2, and diffusion coefficients are plotted in Fig.4(b). Helium diffusion shows moderate anisotropy in aragonite, with diffusive rate along [100] about 2 orders of magnitude faster than [001] due to its lower activation energy. As seen from Table 3, the He-O and Ca-O distances in [100] are about 0.10 nm and 0.14 nm longer than those in [001], respectively. The separation between two neighboring parallel CO3 layers in [100] is about 0.21 nm larger than that in [001] at ambient condition, which indicates the small space when He is moving along [001], making it difficult to go through and leading to great activation energy. In contrast, the C-O bond distances have little difference between the two directions.

    • To investigate the pressure effect on helium diffusion in aragonite, we performed calculation under high pressure up to 14 GPa. The high pressure data and diffusion coefficients are reported in Table 2 and Fig.5. For [100] direction, Ea increases from 82.40 kJ/mol at 0 GPa to 154.38 kJ/mol at 14 GPa. For [001] direction, Ea increases from 96.00 kJ/mol at 0 GPa to 174.45 kJ/mol at 14 GPa. The activation energies significantly increase with the pressure in both directions. The large differences in diffusion kinetics between ambient and high pressure conditions may due to the structural transformation of aragonite. We list the characteristic bond distances from activated states in aragonite under different pressure conditions (see Table 3). With increasing pressure, some characteristic bond distances become more and more short, creating more congested space to allow He atom to move. It can be noticed that the C-O bonds are hardly compressible in aragonite. For [100] direction, the He-O (Ca-O) bond distance decreases from 2.141(2.356) nm at 0 GPa to 1.951(2.334) nm at 14 GPa. For [001] direction, the He-O (Ca-O) bond distance decreases from 2.042(2.412) nm at 0 GPa to 1.909(2.265) nm at 14 GPa. Thus, the pressure effect will significantly influence the crystal structures and activation energies. With the increasing of pressure, helium diffusion rates in aragonite will become slower.

      Figure 5.  Effect of pressure on helium diffusion in aragonite up to 14 GPa in the [100] (a) and [001] (b) directions (The diffusion coefficients obviously decrease with pressure increasing in both directions.)

      As seen from Fig.5, the diffusion coefficients D[100] and D[001] obviously decrease with increasing pressure. When the pressure increases to 14 GPa, D[100] and D[001] nearly decline by 13 orders and 12 orders of magnitude at ambient temperature, and 5 orders and 6 orders of magnitude at 800 K.

    • In recent two decades, the diffusion mechanism of helium in natural carbonates samples has been studied by some experimental work (e.g. Refs.[45,78]). Copeland et al.[5] investigated the helium diffusion kinetics by soaking a wind range of carbonates in helium gas and conducting bulk step-degassing experiments. From the helium release behaviors, they constrained activation energies for calcite ranging from 121 to 170 kJ/mol, larger than those (67–97 kJ/mol) in the present work. The Arrhenius parameters show some high-temperature data have retrograde steps that do not fall on the initial array well documented as multiple diffusion domains (MDD) behaviors. He diffusion in saturated and natural samples both suggest the MDD behavior and the diffusion domain is smaller than the size of the sample analyzed, making some of the He release behaviors quite complicated[5]. The similar results were obtained by Cros et al.[7], they undertook step-degassing experiments on fault-filling calcite, and their results show a complex degassing behavior. Their Arrhenius parameters can be explained by the MDD model, or by the presence of fast paths in the crystals due to structural defects. The presence of defects creates the small domains in calcite crystals, making (U-Th)/He ages variable[7]. Amidon et al.[8] explored the potential of helium dating and evaluated He retention in natural sample under Earth’s surface condition. They constrained activation energies ranging from 101 to 113 kJ/mol by step-degassing experiments on calcite and suggested that He retention was controlled by MDD pattern in the crystals. All the above experiments indicate that the helium diffusion in calcite will be controlled by MDD, and these results are quite complex and difficult to investigate the real diffusion mechanism of helium in carbonates. In addition, the high activation energies they obtained may reflect the overprint of anisotropy of He diffusion in the calcite.

      To avoid the limitation of previous bulk degassing experiments and distinguish between the effects of diffusion kinetics, diffusion domain sizes and the anisotropy, Cherniak et al.[4] performed 3He ion implantation experiments and NRA measurements to study helium diffusion in calcite, dolomite, magnesite and aragonite. For calcite, they obtain the activation energies of 52 kJ/mol for direction perpendicular to [001], not far from our results (67 kJ/mol). However, they obtain the activation energies of 54 kJ/mol for the direction [001]. It is much smaller than our results (97 kJ/mol). Bengtson et al.[9] calculate the diffusion of He in apatite. They found the results that the activation barriers in experimental measurements are higher than calculation results, and it is likely explained by the presence of radiation damage. Even though calcite contain very low concentrations of radiation damage producing elements (U, Th, K), this effect may play an important role in the discrepancy between experimentally determined helium diffusion kinetics and the calculations presented here. In addition, natural calcite that undoubtedly contains abundant structural defects that disrupts the perfect CaCO3. Our calculation show He diffusion in calctie is very sensitive to local structure. This may be an important cause to leading the different activation energies and diffusivities between the results presented here and the experimental study by Cherniak et al.[4] In addition, the limited experiment conditions and our simulated conditions are quite different, which may cause the discrepancies in the results. As shown in Fig.4(a), Arrhenius relations for calcite between our data and Cherniak et al.[4] are similar. Their data showed that the He diffusion in calcite is anisotropic, and is slowest for diffusion along the [001] direction, which agree well with our calculations. This pattern of anisotropy was due to the smaller inter-atomic apertures along [001] direction, which may reduce the diffusion rate.

      For aragonite, they obtained the activation energies of 95 kJ/mol for [001] and 82 kJ/mol for [010], while our results are 96 kJ/mol for [001] and 82 kJ/mol for [100]. It is not clear why we obtain different anisotropy results, but the values of activation energy are identical. The findings of Cherniak et al.[4] indicate diffusivities are about 2 orders of magnitude lower than the results obtained in the present study (Fig.4(b)). The differences may due to the different research methods used, making relatively large discrepancy of “jump distance a” between experimental and theoretical value.

      In addition, we compare the helium diffusion rates between calcite and aragonite. For the convenience of comparison, we choose the faster diffusion paths [010](100) in calcite and [100] in aragonite. Helium diffuses faster in calcite, with diffusion rate about two orders of magnitude faster than in aragonite at ambient condition. The differences are due to the small activation energy of calcite, making it easier for helium atom to pass through. Thus, calcite may be less retentive for helium than aragonite under surface condition. Early work studied on calcite and aragonite suggest calcite is unlikely to be retentive of helium on the Earth’s surface, while aragonite may be retentive under cooler conditions[4,28], consistent with our study.

    • It is necessary to investigate detailed information about closure temperatures (Tc) for carbonates. For thermally activated diffusion D = D0 exp(−$\Delta E_{\rm a}/RT$), it is given by Dodson[13] as

      $ {T_{\rm c}} = \frac {E_{\rm a}}{R}\Bigg/{{\ln \left( {\dfrac{{ART_{\rm c}^2{D_0}}}{{{a^2}{E_{\rm a}}{\rm d}T/{\rm d}t}}} \right)}} $

      where Ea and D0 are from the current calculations. A cooling rate (dT/dt) of 10 ℃/Ma (3.168 88×10–13 K/s) (consistent with Bengtson et al.[9]) and spherical geometry (A = 55) were used[13]. Tc is plotted as a function of crystal size in Fig.6. The closure temperature of 0.2–2.0 mm grain size of calcite crystal varies from −54 ℃ to −25 ℃ in the direction [010] and 40 ℃ to 82 ℃ in the direction [001], and for aragonite varies from −12 ℃ to 23 ℃ in the direction [100] and 30 ℃ to 69 ℃ in the direction [001].

      Figure 6.  Calculated closure temperature (Tc) as a function of grain radius (a) along different directions in calcite and aragonite (Closure temperature are plotted for assuming spherical geometry (A=55) of the crystals. Helium in each carbonate composition using Dodson’s (1973) equation and a cooling rate of 10 ℃/Ma.)

      The low closure temperatures from our calculation are due to the relatively low activation energies. In calcite, the closure temperature are very low due to its rapid diffusion rate along [010] direction. Previous experimental study suggest that a good evaluation of the Tc of helium in calcite is around 60−80 ℃ and is independent of the size of the grain[5]. The high Tc in experimental samples are due to the high activation energies they obtained. In ideal aragonite, the calculated closure temperature is higher than that in calcite, indicating that helium will be more retentive in aragonite than in calcite. The temperature differences between aragonite and calcite were more pronounced in the study of Cherniak et al.[4] They suggest carbonate will retain initial He in their centers at temperatures up to −11 ℃ and 91 ℃ for calcite and aragonite in the case of 1 mm radius grains. The large difference is due to the significantly higher diffusivity in aragonite than in calcite they obtained.

    • Using first-principles calculations and CI-NEB method, we investigated the possible trapping sites and diffusion of He in the perfect host lattice of calcite and aragonite.

      He diffusion in calcite shows marked anisotropy, with more energetically favorable directions of [100] and [010]. Aragonite shows moderate anisotropy of He diffusion, with diffusion rate along [001] slower than [100]. The diffusivity of He is greater in calcite than that in aragonite. Aragonite has a higher closure temperature than calcite. Helium diffusion in aragonite is obviously influenced by pressure at low temperature. When the pressure reached up to 14 GPa, the activation energies increased with the pressure. The microscopic structure of minerals can significantly influence He diffusion.

      Previous step-degassing experiments suggested that the helium diffusion in calcite is controlled by MDD. The differences in diffusivity of He for computed structures and natural samples suggest natural samples are far from ideal in their structure, probably the result of varying sizes of diffusion domains.

      Our DFT calculations provide a theoretical research into the helium diffusion behavior in perfect crystals of calcite and aragonite under ambient and high pressure conditions. These results provide useful information for understanding dynamic and geochemical processes by investigating noble gases reserving in crust and mantle minerals.

      Acknowledgments: We thank the computational support from the Supercomputing Center of Dalian University of Technology and the Supercomputer Group of the Institute of Earthquake Forecasting.

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