Viscosities of hcp iron alloys under Earth’s inner core conditions

Abstract

Viscosity is critical for controlling the dynamics and evolution of the Earth’s inner core (IC). The viscosities of hexagonal close-packed (hcp) and body-centred cubic (bcc) Fe were studied experimentally and theoretically under Earth's core conditions. However, Earth’s inner core is mainly composed of Fe-Ni alloys with some light element impurities (Si, S, C, H, O), and the influence of impurities (Ni, Si, S, C, H, and O) on viscosity is still unknown. In this study, the diffusion coefficients of Fe, Ni, Si, S, C, H, and O were calculated under IC conditions using ab initio molecular dynamics (AIMD) and deep learning molecular dynamics (DPMD) methods. Among them, C, H, and O are highly diffusive like liquids in the lattice, while Fe, Ni, Si, and S diffuse through Fe site vacancies. In binary alloys, the influence of these impurities (Ni: 12.5%, S: 3.6%, Si: 3.1%, C: 1.3%, O: 1.7%, H: 0.4% by weight) on viscosity is insignificant. Based on the dislocation creep mechanism, the predicted viscosities of the hcp Fe alloys are 1 × 10¹⁴–2 × 10¹⁶ Pa·s, which is consistent with the values predicted by free inner core nutation and seismic wave attenuation observations.

Keywords

Earth’s inner core / High pressure and high temperature / Viscosity / Molecular dynamics

CRediT authorship contribution statement

Yunfan Xu: Writing – original draft, Validation, Software, Investigation, Formal analysis. Yu He: Writing – review & editing, Validation, Supervision, Resources, Investigation, Formal analysis, Conceptualization. Shichuan Sun: Software, Investigation. Wei Zhang: Validation, Software. Weiru Dai: Formal analysis, Writing – review & editing. Duck Young Kim: Writing – review & editing, Formal analysis. Heping Li: Writing – review & editing, Resources, Formal analysis.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

We acknowledge the support of the National Natural Science Foundation of China (42350002, 42074104), the CAS Youth Interdisciplinary Team (JCTD-2022-16), and the Youth Innovation Promotion Association of CAS (2020394). This study was also supported by the Guizhou Provincial 2020 Science and Technology Subsidies (No. GZ2020SIG). Numerical computations were performed at the Shanghai Supercomputer Center and National Supercomputer Center in Guangzhou. The computing resources used in this study were also provided by the Bohrium Cloud Platform (https://bohrium.dp.tech), which is supported by DP Technology.

Appendix A. Supplementary material

The following are the Supplementary data to this article:Download: Download Word document (17MB)

Supplementary Data 1.

References

[1] T. Alboussiere, R. Deguen, M. Melzani. Melting-induced stratification above the Earth's inner core due to convective translation. Nature, 466 (2010), pp. 744-U749,

[2] A.B. Belonoshko, T. Lukinov, J. Fu, J.J. Zhao, S. Davis, S.I. Simak. Stabilization of body-centred cubic iron under inner-core conditions. Nat. Geosci., 10 (2017), pp. 312-316,

[3] A.B. Belonoshko, J. Fu, T. Bryk, S.I. Simak, M. Mattesini. Low viscosity of the Earth’s inner core. Nat. Commun., 10, 2483 (2019),

[4] M.I. Bergman. Measurements of electric anisotropy due to solidification texturing and the implications for the Earth's inner core. Nature, 389 (1997), pp. 60-63,

[5] M.I. Bergman. Estimates of the Earth's inner core grain size. Geophys. Res. Lett., 25 (1998), pp. 1593-1596,

[6] M.I. Bergman, D.J. Lewis, I.H. Myint, L. Slivka, S.-I. Karato, A. Abreu. Grain growth and loss of texture during annealing of alloys, and the translation of Earth's inner core. Geophys. Res. Lett., 37 (22) (2010), p. L22313,

[7] J. Bloxham. The effect of thermal core-mantle interactions on the palaeomagnetic secular variation. Philos. Trans. r. Soc. A, 358 (2000), pp. 1171-1179,

[8] W. Blum, W. Maier. Harper-Dorn creep - a myth? Phys. Status Solidi A, 171 (1999), pp. 467-474,

[9] H. Brett, A. Deuss. Inner core anisotropy measured using new ultra-polar PKIKP paths. Geophys. J. Int., 223 (2020), pp. 1230-1246,

[10] B.A. Buffett. Geodynamic estimates of the viscosity of the Earth's inner core. Nature, 388 (1997), pp. 571-573,

[11] B.A. Buffett, J. Bloxham. Deformation of earth's inner core by electromagnetic forces. Geophys. Res. Lett., 27 (2000), pp. 4001-4004,

[12] B.A. Buffett, H.R. Wenk. Texturing of the Earth's inner core by Maxwell stresses. Nature, 413 (2001), pp. 60-63,

[13] D.M. Ceperley, B.J. Alder. Ground-state of the electron-gas by a stochastic method. Phys. Rev. Lett., 45 (1980), pp. 566-569,

[14] Creager, K.C., 1999. Large-scale variations in inner core anisotropy. J. Geophys. Res.: Solid Earth 104, 23127-23139. https://doi.org/10.1029/1999jb900162.

[15] C.J. Davies. Cooling history of Earth’s core with high thermal conductivity. Phys. Earth Planet. Inter., 247 (2015), pp. 65-79

[16] D.A. Frost, M. Lasbleis, B. Chandler, B. Romanowicz. Dynamic history of the inner core constrained by seismic anisotropy. Nat. Geosci., 14 (2021), pp. 531-535,

[17] R. Garcia, A. Souriau. Inner core anisotropy and heterogeneity level. Geophys. Res. Lett., 27 (2000), pp. 3121-3124,

[18] G.A. Glatzmaier, P.H. Roberts. A 3-dimensional self-consistent computer-simulation of a geomagnetic-field reversal. Nature, 377 (1995), pp. 203-209,

[19] A.E. Gleason, W.L. Mao. Strength of iron at core pressures and evidence for a weak Earth's inner core. Nat. Geosci., 6 (2013), pp. 571-574,

[20] J.I. Goldstein, R.E. Hanneman, R.E. Ogilvie. Diffusion in Fe-Ni system at 1 atm and 40 kbar pressure. Trans. Metall. Soc. AIME, 233 (1965), pp. 812-820

[21] Gomi, H., Hirose, K., 2023. Impurity resistivity of the Earth's inner core. J. Geophys. Res.: Solid Earth 128(11), e2023JB027097. https://doi.org/10.1029/2023jb027097.

[22] Y. He, S. Sun, D.Y. Kim, B.G. Jang, H. Li, H.-K. Mao. Superionic iron alloys and their seismic velocities in Earth's inner core. Nature, 602 (2022), pp. 258-262,

[23] Y. He, D.Y. Kim, V.V. Struzhkin, Z.M. Geballe, V. Prakapenka, H.K. Mao. The stability of FeHx and hydrogen transport at Earth's core mantle boundary. Sci. Bull., 68 (2023), pp. 1567-1573,

[24] Y. He, S. Sun, H. Li. Superionic iron alloys in Earth’s inner core and their effects. Chin. J. High Press. Phys., 38 (2024), pp. 132-141(in Chinese with English abstract)

[25] K. Hirose, B. Wood, L. Vocadlo. Light elements in the Earth's core. Nat. Rev. Earth Environ., 2 (2021), pp. 645-658,

[26] P. Hohenberg, W. Kohn. Inhomogeneous electron gas. Phys. Rev. B, 136 (1964), pp. B864-B871,

[27] Irving, J.C.E., Deuss, A., 2011. Hemispherical structure in inner core velocity anisotropy. J. Geophys. Res.: Solid Earth 116(B4), B04307. https://doi.org/10.1029/2010jb007942.

[28] M. Ishii, A.M. Dziewonski. The innermost inner core of the earth: evidence for a change in anisotropic behavior at the radius of about 300 km. Proc. Natl. Acad. Sci. USA, 99 (2002), pp. 14026-14030,

[29] Jackson, I., Fitz Gerald, J.D., Kokkonen, H., 2000. High-temperature viscoelastic relaxation in iron and its implications for the shear modulus and attenuation of the Earth's inner core. J. Geophys. Res.: Solid Earth 105, 23605-23634. https://doi.org/10.1029/2000jb900131.

[30] R. Jeanloz, H.R. Wenk. Convection and anisotropy of the inner core. Geophys. Res. Lett., 15 (1988), pp. 72-75,

[31] Z. Jing, T. Liu. Earth’s core composition constrained by H partitioning at the inner core boundary. Res. Square (2024), 10.21203/rs.3.rs-3974305/v1

[32] R. Jinnouchi, F. Karsai, C. Verdi, R. Asahi, G. Kresse. Descriptors representing two- and three-body atomic distributions and their effects on the accuracy of machine-learned inter-atomic potentials. J. Chem. Phys., 152 (2020), Article 234192,

[33] S. Karato. Inner-core anisotropy due to the magnetic-field-induced preferred orientation of iron. Science, 262 (1993), pp. 1708-1711,

[34] S. Karato. Seismic anisotropy of the Earth's inner core resulting from flow induced by Maxwell stresses. Nature, 402 (1999), pp. 871-873,

[35] W. Kohn, L.J. Sham. Self-consistent equations including exchange and correlation effects. Phys. Rev., 140 (1965), pp. 1133-1138,

[36] L. Koot, M. Dumberry. Viscosity of the Earth's inner core: constraints from nutation observations. Earth Planet. Sci. Lett., 308 (2011), pp. 343-349,

[37] G. Kresse. Ab-initio molecular-dynamics for liquid-metals. J. Non-Cryst. Solids, 193 (1995), pp. 222-229,

[38] Z. Li, S. Scandolo. Elasticity and vviscosity of hcp iron at Earth’s inner core conditions from machine learning-based large-scale atomistic simulations. Geophys. Res. Lett., 49 (24) (2022), Article e2022GL101161,

[39] K.H. Lythgoe, A. Deuss, J.F. Rudge, J.A. Neufeld. Earth's inner core: innermost inner core or hemispherical variations?. Earth Planet. Sci. Lett., 385 (2014), pp. 181-189,

[40] B. Militzer, H.R. Wenk, S. Stackhouse, L. Stixrude. First-principles calculation of the elastic moduli of sheet silicates and their application to shale anisotropy. Am. Mineral., 96 (2011), pp. 125-137,

[41] M. Monnereau, M. Calvet, L. Margerin, A. Souriau. Lopsided growth of Earth's inner core. Science, 328 (2010), pp. 1014-1017,

[42] A. Morelli, A.M. Dziewonski, J.H. Woodhouse. Anisotropy of the inner core inferred from Pkikp travel-times. Geophys. Res. Lett., 13 (1986), pp. 1545-1548,

[43] Y. Nishihara, S. Doi, N. Tsujino, D. Yamazaki, K.N. Matsukage, Y. Tsubokawa, T. Yoshino, A.R. Thomson, Y. Higo, Y. Tange. Rheology of hexagonal close-packed (hcp). Iron. J. Geophys. Res.: Solid Earth, 128(6), e2022JB026165 (2023),

[44] F.L. Niu, L.X. Wen. Hemispherical variations in seismic velocity at the top of the Earth's inner core. Nature, 410 (2001), pp. 1081-1084,

[45] S.I. Oreshin, L.P. Vinnik. Heterogeneity and anisotropy of seismic attenuation in the inner core. Geophys. Res. Lett., 31 (2) (2004), p. L02613,

[46] Y. Park, K. Yonemitsu, K. Hirose, Y. Kuwayama, S. Azuma, K. Ohta. Viscosity of Earth's inner core constrained by Fe-Ni interdiffusion in Fe-Si alloy in an internal-resistive-heated diamond anvil cell. Am. Mineral., 108 (2023), pp. 1064-1071,

[47] J.P. Perdew, A. Zunger. Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B, 23 (1981), pp. 5048-5079,

[48] T.-S. Pham, H. Tkalcic. Up-to-fivefold reverberating waves through the Earth's center and distinctly anisotropic innermost inner core. Nat. Commun., 14 (2023), p. 754,

[49] D.M. Reaman, G.S. Daehn, W.R. Panero. Predictive mechanism for anisotropy development in the Earth's inner core. Earth Planet. Sci. Lett., 312 (2011), pp. 437-442,

[50] D.M. Reaman, H.O. Colijn, F. Yang, A.J. Hauser, W.R. Panero. Interdiffusion of Earth's core materials to 65 GPa and 2200 K. Earth Planet. Sci. Lett., 349 (2012), pp. 8-14,

[51] S. Ritterbex, T. Tsuchiya. Viscosity of hcp iron at Earth's inner core conditions from density functional theory. Sci. Rep., 10 (2020), p. 6311,

[52] X.D. Song, D.V. Helmberger. Anisotropy of Earth’s inner-core. Geophys. Res. Lett., 20 (1993), pp. 2591-2594,

[53] X.D. Song, P.G. Richards. Seismological evidence for differential rotation of the Earth's inner core. Nature, 382 (1996), pp. 221-224,

[54] L. Stixrude, R.E. Cohen, R.J. Hemley. Theory of minerals at high pressure. Ultrahigh-pressure mineralogy: Physics and chemistry of the Earth's deep interior. R. J. Hemley., 37 (1998), pp. 639-671

[55] W.J. Su, A.M. Dziewonski. Inner-core anisotropy in 3 dimensions. J. Geophys Res. Solid Earth, 100 (1995), pp. 9831-9852,

[56] Sumita, I., Bergman, M.I., 2007. 8.10 - Inner-Core Dynamics. Treatise on Geophysics. G. Schubert. Amsterdam, Elsevier, pp. 299-318.

[57] S. Sun, Y. He, J. Yang, Y. Lin, J. Li, D.Y. Kim, H. Li, H.-K. Mao. Superionic effect and anisotropic texture in Earth's inner core driven by geomagnetic field. Nat. Commun., 14 (2023), p. 1656,

[58] X. Sun, X. Song. The inner inner core of the Earth: Texturing of iron crystals from three-dimensional seismic anisotropy. Earth Planet. Sci. Lett., 269 (2008), pp. 56-65,

[59] S. Tanaka, H. Hamaguchi. Degree one heterogeneity and hemispherical variation of anisotropy in the inner core from PKP(BC)-PKP(DF) times. J. Geophys Res. Solid Earth, 102 (1997), pp. 2925-2938,

[60] J. Tromp. Support for anisotropy of the Earth’s inner-core from free oscillations. Nature, 366 (1993), pp. 678-681,

[61] N. Tsujino, A. Marza, D. Yamazaki. Pressure dependence of Si diffusion in γ-Fe. Am. Mineral., 105 (2020), pp. 319-324,

[62] J.A. Van Orman. On the viscosity and creep mechanism of Earth's inner core. Geophys. Res. Lett., 31 (2004), p. L20606,

[63] Wang, H., Zhang, L.F., Han, J.Q., E, W.N., 2018. DeePMD-kit: A deep learning package for many-body potential energy representation and molecular dynamics. Comput. Phys. Commun. 228, 178-184. https://doi.org/10.1016/j.cpc.2018.03.016.

[64] L. Waszek, A. Deuss. Distinct layering in the hemispherical seismic velocity structure of Earth's upper inner core. J. Geophys. Res. Solid Earth, 116 (2011), p. B12313,

[65] L.X. Wen, F.L. Niu. Seismic velocity and attenuation structures in the top of the Earth's inner core. J. Geophys. Res. Solid Earth, 107 (B11) (2002), p. 2273,

[66] D. Yamazaki, N. Tsujino, A. Yoneda, E. Ito, T. Yoshino, Y. Tange, Y. Higo. Grain growth of ε-iron: Implications to grain size and its evolution in the Earth's inner core. Earth Planet. Sci. Lett., 459 (2017), pp. 238-243,

[67] Y. Yang, X. Song. Multidecadal variation of the Earth's inner-core rotation. Nat. Geosci., 16 (2023), p. 189,

[68] S. Yoshida, I. Sumita, M. Kumazawa. Growth model of the inner core coupled with the outer core dynamics and the resulting elastic anisotropy. J. Geophys. Res. Solid Earth, 101 (1996), pp. 28085-28103,

[69] W.C. Yu, L.X. Wen. Seismic velocity and attenuation structures in the top 400 km of the Earth's inner core along equatorial paths. J. Geophys. Res. Solid Earth, 111 (2006), p. B07308,

[70] L. Yuan, G. Steinle-Neumann. Hydrogen distribution between the Earth’s inner and outer core. Earth Planet. Sci. Lett., 609 (2023), Article 118084,

[71] M.L. Yunker, J.A. Van Orman. Interdiffusion of solid iron and nickel at high pressure. Earth Planet. Sci. Lett., 254 (2007), pp. 203-213,

[72] Y.Z. Zhang, H.D. Wang, W.J. Chen, J.Z. Zeng, L.F. Zhang, H. Wang, E. Weinan. DP-GEN: A concurrent learning platform for the generation of reliable deep learning based potential energy models. Comput. Phys. Commun., 253 (2020), Article 107206,

[73] Y.J. Zhang, Y. Wang, Y.Q. Huang, J.J. Wang, Z.X. Liang, L. Hao, Z.P. Gao, J. Li, Q. Wu, H. Zhang, Y. Liu, J. Sun, J.F. Lin. Collective motion in hcp-Fe at Earth’s inner core conditions. Proc. Natl. Acad. Sci. USA, 120 (41) (2023), Article e2309952120,