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Equation of state of boron nitride combining computation, modeling, and experiment

Shuai Zhang, Amy Lazicki, Burkhard Militzer, Lin H. Yang, Kyle Caspersen, Jim A. Gaffney, Markus W. Däne, John E. Pask, Walter R. Johnson, Abhiraj Sharma, Phanish Suryanarayana, Duane D. Johnson, Andrey V. Smirnov, Philip A. Sterne, David Erskine, Richard A. London, Federica Coppari, Damian Swift, Joseph Nilsen, Art J. Nelson, and Heather D. Whitley
Phys. Rev. B 99, 165103 – Published 3 April 2019
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  • INTRODUCTION
  • FIRST-PRINCIPLES SIMULATION METHODS
  • SHOCK HUGONIOT EXPERIMENT
  • CONSTRUCTION OF EOS MODELS FOR BN
  • RESULTS AND DISCUSSION
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • Supplemental Material
    References

    Abstract

    The equation of state (EOS) of materials at warm dense conditions poses significant challenges to both theory and experiment. We report a combined computational, modeling, and experimental investigation leveraging new theoretical and experimental capabilities to investigate warm-dense boron nitride (BN). The simulation methodologies include path integral Monte Carlo (PIMC), several density functional theory (DFT) molecular dynamics methods [plane-wave pseudopotential, Fermi operator expansion (FOE), and spectral quadrature (SQ)], activity expansion (actex), and all-electron Green's function Korringa-Kohn-Rostoker (mecca), and compute the pressure and internal energy of BN over a broad range of densities and temperatures. Our experiments were conducted at the Omega laser facility and the Hugoniot response of BN to unprecedented pressures (1200–2650 GPa). The EOSs computed using different methods cross validate one another in the warm-dense matter regime, and the experimental Hugoniot data are in good agreement with our theoretical predictions. By comparing the EOS results from different methods, we assess that the largest discrepancies between theoretical predictions are ≲4% in pressure and ≲3% in energy and occur at 106K, slightly below the peak compression that corresponds to the K-shell ionization regime. At these conditions, we find remarkable consistency between the EOS from DFT calculations performed on different platforms and using different exchange-correlation functionals and those from PIMC using free-particle nodes. This provides strong evidence for the accuracy of both PIMC and DFT in the high-pressure, high-temperature regime. Moreover, the recently developed SQ and FOE methods produce EOS data that have significantly smaller statistical error bars than PIMC, and so represent significant advances for efficient computation at high temperatures. The shock Hugoniot predicted by PIMC, actex, and mecca shows a maximum compression ratio of 4.55±0.05 for an initial density of 2.26g/cm3, higher than the Thomas-Fermi predictions by about 5%. In addition, we construct tabular EOS models that are consistent with the first-principles simulations and the experimental data. Our findings clarify the ionic and electronic structure of BN over a broad range of temperatures and densities and quantify their roles in the EOS and properties of this material. The tabular models may be utilized for future simulations of laser-driven experiments that include BN as a candidate ablator material.

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    • Received 2 February 2019

    DOI:https://doi.org/10.1103/PhysRevB.99.165103

    ©2019 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Electronic excitation & ionizationElectronic structureHigh-energy-density plasmasInertial confinement fusionShock waves
    1. Physical Systems
    Warm-dense matter
    1. Techniques
    Density functional calculationsFirst-principles calculationsMolecular dynamics
    Interdisciplinary PhysicsPlasma PhysicsCondensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Shuai Zhang1,*, Amy Lazicki1,†, Burkhard Militzer2,3,‡, Lin H. Yang1, Kyle Caspersen1, Jim A. Gaffney1, Markus W. Däne1, John E. Pask1, Walter R. Johnson4, Abhiraj Sharma5, Phanish Suryanarayana5, Duane D. Johnson6,7, Andrey V. Smirnov6, Philip A. Sterne1, David Erskine1, Richard A. London1, Federica Coppari1, Damian Swift1, Joseph Nilsen1, Art J. Nelson1, and Heather D. Whitley1,§

    • 1Lawrence Livermore National Laboratory, Livermore, California 94550, USA
    • 2Department of Earth and Planetary Science, University of California, Berkeley, California 94720, USA
    • 3Department of Astronomy, University of California, Berkeley, California 94720, USA
    • 4Department of Physics, 225 Nieuwland Science Hall, University of Notre Dame, Notre Dame, Indiana 46556, USA
    • 5College of Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
    • 6Division of Materials Science & Engineering, Ames Laboratory, Ames, Iowa 50011, USA
    • 7Department of Materials Science & Engineering, Iowa State University, Ames, Iowa 50011, USA

    • *zhang49@llnl.gov
    • †jenei2@llnl.gov
    • ‡militzer@berkeley.edu
    • §whitley3@llnl.gov

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    Issue

    Vol. 99, Iss. 16 — 15 April 2019

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    Images

    • Figure 1
      Figure 1

      Temperature-density diagram showing the parameter regions where the methods in this paper are used for calculating the EOS of BN.

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    • Figure 2
      Figure 2

      (a) Experimental configuration (not drawn to scale), (b) image of a typical c-BN crystal glued to the quartz plate, viewed from the perspective of the VISAR diagnostic, and (c) image of the VISAR data from shot 75265, with the analyzed velocities shown as red and blue traces (corresponding the two interferometer legs). The dashed traces are the apparent velocities and the solid traces are corrected for the index of refraction in quartz and c-BN.

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    • Figure 3
      Figure 3

      (a) Pressure- and (b) temperature-compression Hugoniot of BN predicted by different LEOS models in comparison with PIMC and DFT-MD (PAWpw). The initial density of all Hugoniot curves is set to be 2.15g/cm3. Note that the deviations at above 106 GPa and 2×107K are due to the electron relativistic effect, which is included in the Purgatorio tables (thus fully in X2152 and partially in X2151) but not in L2150 or PIMC.

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    • Figure 4
      Figure 4

      Comparison of the pressure and the energy terms of the Hugoniot function along the 2×106K isotherm, which is near the compression maximum. Shaded areas denote the error bar of the PIMC data.

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    • Figure 5
      Figure 5

      Comparison of the pressure-temperature profiles of BN along several isochores from PIMC, DFT-MD (PAW, frozen 1s), DFT-MD (ONCV, frozen 1s), FOE (all electron), SQ (all electron), actex, mecca, and X2152. Subplot (b) is a zoom-in version of (a).

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    • Figure 6
      Figure 6

      Percent difference in internal energy of BN between actex and X2152 along several isochores. The compression ratio (with respect to ρ0=2.258g/cm3) are labeled at the top of the plotting area. The reference points for actex and X2152 are both at ρ0 and ambient temperature.

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    • Figure 7
      Figure 7

      Percent contributions of the ion-thermal (left) and electron-thermal (right) terms to the total pressure of BN. The remaining contributions are from the cold curve. The temperature-density conditions corresponding to several isochores along which we performed EOS calculations are shown with “+” symbols.

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    • Figure 8
      Figure 8

      Comparison of the nuclear pair correlation function obtained from DFT-MD (PAWpw) for BN using 24-atom (red) and 96-atom (dark) cells at two different densities and three temperatures. The reference density ρ0 is 2.26g/cm3. The peaks at 104K indicate a polymeric structure of the liquid. Differences between small and large cells are evident at 4000 K, indicating a significant finite-size effect. This effect is stronger at higher densities and becomes negligible at temperatures higher than 5×104K.

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    • Figure 9
      Figure 9

      EOS differences of PIMC (red), FOE (black), and mecca (blue) relative to SQ along two isotherms (1.01×106 and 1.35×106K). Because of the different references chosen in the EOS data sets, all energies have been shifted by the corresponding value at 15.80g/cm3 and 1.35×106K. The energy differences are normalized by the corresponding ideal-gas values (21kBT per BN). The statistical error bars correspond to the 1σ uncertainty of the FOE and PIMC data.

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    • Figure 10
      Figure 10

      Comparison of the Hugoniot of BN from experiment to predictions from PIMC and DFT-MD (PAWpw) simulations and the X2152 model in (a) pressure-density and (b) pressure-compression ratio representations. The initial densities of corresponding Hugoniots are shown in the legend. In (a), equal-temperature conditions along the two Hugoniot curves are connected with lines (as guides to the eyes) to approximate the location of isotherms. The corresponding temperatures are labeled in colored texts. Note that the deviation between PIMC and X2152 curves at above 106 GPa is due to the electron relativistic effect, which is considered in X2152 but not in PIMC.

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    • Figure 11
      Figure 11

      Comparison of the pressure-compression Hugoniot of BN from different theories and LEOS models. The initial density of every Hugoniot curve is 2.26g/cm3. Two sets of DFT-MD (PAWpw) Hugoniots constructed with a difference of the cohesive energy (Ecoh∼7.1 eV/atom [127]) in the initial energy are also shown for comparison. Note that all mecca pressures in the EOS have been shifted relative to the value at the initial density and 300 K. The deviations at pressures above 106 GPa and 2×107K are due to the fully relativistic treatment of the free electrons in Purgatorio and actex.

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    • Figure 12
      Figure 12

      (a) Pressure- and (b) temperature-density Hugoniot of BN in comparison with C. The electron-thermal contribution to both tables is based on Purgatorio. The initial density of both materials is set to be 2.26g/cm3.

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    • Figure 13
      Figure 13

      Zero-point motion effects on the pressure of BN as a function of density along several isotherms. The inset shows the percent increase in pressure for the EOS (black) and along the Hugoniot (red) and percent decrease in compression ratio along the Hugoniot (blue).

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    • Figure 14
      Figure 14

      Percent pressure difference between calculations using ONCV all-electron pseudopotentials and regularized Coulomb potentials for BN in the cubic phase. For most of the phase points examined in this study, the difference is within 1% except a few cases where the difference is slightly greater.

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