Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T06:08:59.321Z Has data issue: false hasContentIssue false

Simulations of H–He mixtures using the van der Waals density functional

Published online by Cambridge University Press:  29 June 2018

Manuel Schöttler*
Affiliation:
University of Rostock, Institute of Physics, D-18059 Rostock, Germany
Ronald Redmer
Affiliation:
University of Rostock, Institute of Physics, D-18059 Rostock, Germany
*
Email address for correspondence: [email protected]

Abstract

We show results on the high-pressure equation of state of hydrogen–helium mixtures obtained from finite-temperature density functional theory molecular dynamics simulations using the van der Waals density functional. We discuss the calculation of non-ideal entropies based on different methods and show how nuclear quantum corrections influence the free enthalpy of mixing. Furthermore, we calculate a Saturn isentrope based on our new equation of state data.

Type
Research Article
Copyright
© Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baldereschi, A. 1973 Mean-value point in the Brillouin zone. Phys. Rev. B 7 (12), 52125215.Google Scholar
Becker, A., Lorenzen, W., Fortney, J. J., Nettelmann, N., Schoettler, M. & Redmer, R. 2014 Ab initio equations of state for hydrogen (H-REOS.3) and helium (He-REOS.3) and their implications for the interior of brown dwarfs. Astrophys. J. Suppl. Ser. 215 (2), 21.Google Scholar
Berens, P. H., Mackay, D. H. J., White, G. M. & Wilson, K. R. 1983 Thermodynamics and quantum corrections from molecular dynamics for liquid water. J. Chem. Phys. 79 (5), 23752389.Google Scholar
Bethkenhagen, M., Meyer, E. R., Hamel, S., Nettelmann, N., French, M., Scheibe, L., Ticknor, C., Collins, L. A., Kress, J. D., Fortney, J. J. et al. 2017 Planetary ices and the linear mixing approximation. Astrophys. J. 848 (1), 67.Google Scholar
Clay, R. C. III, Holzmann, M., Ceperley, D. M. & Morales, M. A. 2016 Benchmarking density functionals for hydrogen–helium mixtures with quantum Monte Carlo: Energetics, pressures, and forces. Phys. Rev. B 93 (3), 035121.Google Scholar
Conrath, B. J. & Gautier, D. 2000 Saturn helium abundance: a reanalysis of voyager measurements. Icarus 144 (1), 124134.Google Scholar
Davis, P., Döppner, T., Rygg, J. R., Fortmann, C., Divol, L., Pak, A., Fletcher, L., Becker, A., Holst, B., Sperling, P. et al. 2016 X-ray scattering measurements of dissociation-induced metallization of dynamically compressed deuterium. Nat. Commun. 7, 11189.Google Scholar
Desjarlais, M. P. 2013 First-principles calculation of entropy for liquid metals. Phys. Rev. E 88 (6), 062145.Google Scholar
Dias, R. P. & Silvera, I. F. 2017 Observation of the Wigner–Huntington transition to metallic hydrogen. Science 355, 715.CrossRefGoogle ScholarPubMed
Dion, M., Rydberg, H., Schröder, E., Langreth, D. C. & Lundqvist, B. I. 2004 Van der Waals density functional for general geometries. Phys. Rev. Lett. 92 (24), 246401.Google Scholar
Earl, B. L. 1989 The method of intercepts: alternative derivation. J. Chem. Educ. 66 (1), 56.Google Scholar
Fortney, J. J. & Hubbard, W. B. 2003 Phase separation in giant planets: inhomogeneous evolution of Saturn. Icarus 164 (1), 228243.Google Scholar
Fortney, J. J., Ikoma, M., Nettelmann, N., Guillot, T. & Marley, M. S. 2011 Self-consistent model atmospheres and the cooling of the solar system’s giant planets. Astrophys. J. 729 (1), 32.CrossRefGoogle Scholar
Fortney, J. J. & Nettelmann, N. 2010 The interior structure, composition, and evolution of giant planets. Space Sci. Rev. 152 (1), 423447.CrossRefGoogle Scholar
French, M., Desjarlais, M. P. & Redmer, R. 2016 Ab initio calculation of thermodynamic potentials and entropies for superionic water. Phys. Rev. E 93 (2), 022140.Google ScholarPubMed
Gastine, T., Wicht, J., Duarte, L. D. V., Heimpel, M. & Becker, A. 2014 Explaining Jupiter’s magnetic field and equatorial jet dynamics. Geophys. Res. Lett. 41 (15), 54105419.Google Scholar
Glenzer, S. H., Fletcher, L. B., Galtier, E., Nagler, B., Alonso-Mori, R., Barbrel, B., Brown, S. B., Chapman, D. A., Chen, Z., Curry, C. B. et al. 2016 Matter under extreme conditions experiments at the linac coherent light source. J. Phys. B: At. Mol. Opt. Phys. 49 (9), 092001.Google Scholar
Guillot, T., Atreya, S., Charnoz, S., Dougherty, M. K. & Read, P. 2009 Saturn’s exploration beyond Cassini–Huygens. In Saturn from Cassini–Huygens, 1st edn. chap. 23, pp. 745761. Springer.Google Scholar
Guillot, T., Miguel, Y., Militzer, B., Hubbard, W. B., Kaspi, Y., Galanti, E., Cao, H., Helled, R., Wahl, S. M., Iess, L. et al. 2018 A suppression of differential rotation in Jupiters deep interior. Nature 555, 227.Google Scholar
Hicks, D. G., Boehly, T. R., Celliers, P. M., Eggert, J. H., Moon, S. J., Meyerhofer, D. D. & Collins, G. W. 2009 Laser-driven single shock compression of fluid deuterium from 45 to 220 GPa. Phys. Rev. B 79, 014112.Google Scholar
Hoover, W. G. 1985 Canonical dynamics: equilibrium phase-space distributions. Phys. Rev. A 31 (3), 16951697.CrossRefGoogle ScholarPubMed
Hubbard, W. B., Dougherty, M. K., Gautier, D. & Jacobson, R. 2009 The interior of Saturn. In Saturn from Cassini–Huygens, 1st edn. chap. 4, pp. 7581. Springer.Google Scholar
Hubbard, W. B. & Militzer, B. 2016 A preliminary Jupiter model. Astrophys. J. 820 (1), 80.Google Scholar
Hummer, G. & Szabo, A. 1996 Calculation of free-energy differences from computer simulations of initial and final states. J. Chem. Phys. 105 (5), 20042010.Google Scholar
Kirkwood, J. G. 1935 Statistical mechanics of fluid mixtures. J. Chem. Phys. 3 (5), 300313.Google Scholar
Klepeis, J. E., Schafer, K. J., Barbee, T. W. & Ross, M. 1991 Hydrogen–Helium mixtures at megabar pressures – implications for Jupiter and Saturn. Science 254 (5034), 986989.Google Scholar
Klimeš, J., Bowler, D. R. & Michaelides, A. 2010 Chemical accuracy for the van der Waals density functional. J. Phys.: Condens. Matter 22 (2), 022201.Google Scholar
Klimeš, J., Bowler, D. R. & Michaelides, A. 2011 Van der Waals density functionals applied to solids. Phys. Rev. B 83 (19), 195131.Google Scholar
Knudson, M. D. & Desjarlais, M. P. 2017 High-precision shock wave measurements of deuterium: evaluation of exchange-correlation functionals at the molecular-to-atomic transition. Phys. Rev. Lett. 118 (3), 035501.Google Scholar
Knudson, M. D., Desjarlais, M. P., Becker, A., Lemke, R. W., Cochrane, K. R., Savage, M. E., Bliss, D. E., Mattsson, T. R. & Redmer, R. 2015 Direct observation of an abrupt insulator-to-metal transition in dense liquid deuterium. Science 348 (6242), 14551460.Google Scholar
Kresse, G. & Furthmüller, J. 1996a Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 6 (1), 1550.Google Scholar
Kresse, G. & Furthmüller, J. 1996b Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54 (16), 1116911186.Google Scholar
Kresse, G. & Hafner, J. 1993 Ab initio molecular dynamics for liquid metals. Phys. Rev. B 47 (1), 558561.Google Scholar
Lai, P.-K., Hsieh, C.-M. & Lin, S.-T. 2012 Rapid determination of entropy and free energy of mixtures from molecular dynamics simulations with the two-phase thermodynamic model. Phys. Chem. Chem. Phys. 14 (43), 1520615213.Google Scholar
Leconte, J. & Chabrier, G. 2013 Layered convection as the origin of Saturn’s luminosity anomaly. Nat. Geosci. 6, 347350.Google Scholar
Lin, S.-T., Blanco, M. & Goddard, W. A. III 2003 The two-phase model for calculating thermodynamic properties of liquids from molecular dynamics: validation for the phase diagram of lennard-jones fluids. J. Chem. Phys. 119 (22), 1179211805.Google Scholar
Lin, S. T., Maiti, P. K. & Goddard, W. A. III 2010 Two-phase thermodynamic model for efficient and accurate absolute entropy of water from molecular dynamics simulations. J. Phys. Chem. B 114 (24), 81918198.Google Scholar
Lorenzen, W., Holst, B. & Redmer, R. 2009 Demixing of hydrogen and helium at megabar pressures. Phys. Rev. Lett. 102 (11), 115701.Google Scholar
Lorenzen, W., Holst, B. & Redmer, R. 2011 Metallization in hydrogen–helium mixtures. Phys. Rev. B 84 (23), 235109.Google Scholar
Loubeyre, P., LeToullec, R. & Pinceaux, J. P. 1991 A new determination of the binary phase-diagram of H2–He mixtures at 296-K. J. Phys.: Condens. Matter 3 (18), 31833192.Google Scholar
Mazzola, G., Helled, R. & Sorella, S. 2018 Phase diagram of hydrogen and a hydrogen–helium mixture at planetary conditions by quantum Monte Carlo simulations. Phys. Rev. Lett. 120, 025701.Google Scholar
McMahon, J. M., Morales, M. A., Pierleoni, C. & Ceperley, D. M. 2012 The properties of hydrogen and helium under extreme conditions. Rev. Mod. Phys. 84, 16071653.Google Scholar
Meyer, E. R., Ticknor, C., Kress, J. D. & Collins, L. A. 2016 Alternative first-principles calculation of entropy for liquids. Phys. Rev. E 93 (4), 042119.Google Scholar
Miguel, Y., Guillot, T. & Fayon, L. 2016 Jupiter internal structure: the effect of different equations of state. Astron. Astrophys. 596, A114.Google Scholar
Militzer, B. 2013 Equation of state calculations of hydrogen–helium mixtures in solar and extrasolar giant planets. Phys. Rev. B 87 (1), 014202.Google Scholar
Militzer, B. & Hubbard, W. B. 2013 Ab initio equation of state for hydrogen–helium mixtures with recalibration of the giant-planet mass–radius relation. Astrophys. J. 774 (2), 148.Google Scholar
Morales, M. A., Hamel, S., Caspersen, K. & Schwegler, E. 2013 Hydrogen–helium demixing from first principles: from diamond anvil cells to planetary interiors. Phys. Rev. B 87 (17), 174105.Google Scholar
Morales, M. A., Schwegler, E., Ceperley, D., Pierleoni, C., Hamel, S. & Caspersen, K. 2009 Phase separation in hydrogen–helium mixtures at Mbar pressures. Proc. Natl Acad. Sci. USA 106 (5), 13241329.CrossRefGoogle ScholarPubMed
Nettelmann, N., Fortney, J. J., Moore, K. & Mankovich, C. 2015 An exploration of double diffusive convection in Jupiter as a result of hydrogen–helium phase separation. Mon. Not. R. Astron. Soc. 447 (4), 34223441.Google Scholar
Nettelmann, N., Püstow, R. & Redmer, R. 2013 Saturn layered structure and homogeneous evolution models with different EOSs. Icarus 225 (1), 548557.Google Scholar
Ohta, K., Ichimaru, K., Einaga, M., Kawaguchi, S., Shimizu, K., Matsuoka, T., Hirao, N. & Ohishi, Y. 2015 Phase boundary of hot dense fluid hydrogen. Sci. Rep. 5 (1), 16560.CrossRefGoogle ScholarPubMed
Perdew, J. P., Burke, K. & Ernzerhof, M. 1996 Generalized gradient approximation made simple. Phys. Rev. Lett. 77 (18), 38653868.Google Scholar
Pfaffenzeller, O., Hohl, D. & Ballone, P. 1995 Miscibility of hydrogen and helium under astrophysical conditions. Phys. Rev. Lett. 74 (13), 25992602.Google Scholar
Preising, M., Lorenzen, W., Becker, A., Redmer, R., Knudson, M. D. & Desjarlais, M. P. 2018 Equation of state and optical properties of warm dense helium. Phys. Plasmas 25 (1), 012706.Google Scholar
Püstow, R., Nettelmann, N., Lorenzen, W. & Redmer, R. 2016 H/He demixing and the cooling behavior of Saturn. Icarus 267, 323333.CrossRefGoogle Scholar
Redlich, O. & Kister, A. T. 1948 Algebraic representation of thermodynamic properties and the classification of solutions. Ind. Engng Chem. 40 (2), 345348.Google Scholar
Salpeter, E. E. 1973 Convection and gravitational layering in Jupiter and in stars of low mass. Astrophys. J. 181 (2), L83L86.Google Scholar
Saumon, D., Chabrier, G. & van Horn, H. M. 1995 An equation of state for low-mass stars and giant planets. Astrophys. J. Suppl. Ser. 99, 713.Google Scholar
Schöttler, M. & Redmer, R. 2018 Ab initio calculation of the miscibility diagram for hydrogen–helium mixtures. Phys. Rev. Lett. 120, 115703.Google Scholar
Smith, P. E. & Gunsteren, W. F. v. 1994 Predictions of free energy differences from a single simulation of the initial state. J. Chem. Phys. 100 (1), 577585.Google Scholar
Smoluchowski, R. 1967 Internal structure and energy emission of Jupiter. Nature 215, 691.Google Scholar
Stevenson, D. J. 1975 Thermodynamics and phase separation of dense fully ionized hydrogen–helium fluid mixtures. Phys. Rev. B 12 (10), 39994007.Google Scholar
Tubman, N. M., Liberatore, E., Pierleoni, C., Holzmann, M. & Ceperley, D. M. 2015 Molecular-atomic transition along the deuterium Hugoniot curve with coupled electron–ion Monte Carlo simulations. Phys. Rev. Lett. 115, 045301.Google Scholar
Wahl, S. M., Hubbard, W. B., Militzer, B., Guillot, T., Miguel, Y., Movshovitz, N., Kaspi, Y., Helled, R., Reese, D., Galanti, E. et al. 2017 Comparing Jupiter interior structure models to Juno gravity measurements and the role of a dilute core. Geophys. Res. Lett. 44 (10), 46494659.Google Scholar
Zaghoo, M., Salamat, A. & Silvera, I. F. 2016 Evidence of a first-order phase transition to metallic hydrogen. Phys. Rev. B 93 (15), 155128.Google Scholar
Zahn, U., Hunten, D. M. & Lehmacher, G. 1998 Helium in Jupiter’s atmosphere: results from the Galileo probe helium interferometer experiment. J. Geophys. Res. Planets 103 (E10), 2281522829.Google Scholar