Lawrence Berkeley National Laboratory Recent Work Title Evaluation of thermodynamic equations of state across chemistry and structure in the materials project Permalink https://escholarship.org/uc/item/0v70b5jx Journal npj Computational Materials, 4(1) ISSN 2057-3960 Authors Latimer, K Dwaraknath, S Mathew, K et al. Publication Date 2018-12-01 DOI 10.1038/s41524-018-0091-x Peer reviewed eScholarship.org Powered by the California Digital Library University of California www.nature.com/npjcompumats ARTICLE OPEN Evaluation of thermodynamic equations of state across chemistry and structure in the materials project Katherine Latimer1, Shyam Dwaraknath2, Kiran Mathew3, Donald Winston2 and Kristin A. Persson2,3 Thermodynamic equations of state (EOS) for crystalline solids describe material behaviors under changes in pressure, volume, entropy and temperature, making them fundamental to scientific research in a wide range of fields including geophysics, energy storage and development of novel materials. Despite over a century of theoretical development and experimental testing of energy–volume (E–V) EOS for solids, there is still a lack of consensus with regard to which equation is indeed optimal, as well as to what metric is most appropriate for making this judgment. In this study, several metrics were used to evaluate quality of fit for 8 different EOS across 87 elements and over 100 compounds which appear in the literature. Our findings do not indicate a clear “best” EOS, but we identify three which consistently perform well relative to the rest of the set. Furthermore, we find that for the aggregate data set, the RMSrD is not strongly correlated with the nature of the compound, e.g., whether it is a metal, insulator, or semiconductor, nor the bulk modulus for any of the EOS, indicating that a single equation can be used across a broad range of classes of materials. npj Computational Materials (2018) 4:40 ; doi:10.1038/s41524-018-0091-x INTRODUCTION metrics that define EOS applicability? Are the optimal DFT-based Thermodynamic equations of state (EOS) for crystalline solids EOS consistent with experimental values, and are therefore likely describe material behaviors under changes in pressure, volume, to reflect our physical reality? What are the limitations of these entropy and temperature; making them fundamental to scientific EOS? research in a wide range of fields, including geophysics, energy storage and development of novel materials.1–3 Despite over a century of theoretical development and experimental testing of RESULTS AND DISCUSSIONS EOS for solids,4,5 there is still a lack of consensus on the most Equations of state appropriate EOS under various conditions or even the metric to The equations investigated in this study are described in detail evaluate appropriateness. Previous attempts6,7 have focused on below, and summarized in Table 1. comparison with the experimental data, which limits the range of the accessible data. By contrast, computational studies can span Birch (Euler and Lagrange) chemical and structural space in a consistent and methodical Both forms of the Birch equation were derived by Francis Birch in manner, and are often able to probe extreme conditions beyond 1947 for crystalline solids of cubic symmetry.10 The derivation was the range of current experimental techniques. Density-functional based on Francis Murnaghan’s extensive tensor formalism for theory (DFT) is gaining grounds as a first principles methodology fi 11 fi fi analyzing nite strains. The rst Birch equation uses the Eulerian that can accurately and ef ciently search material space for strain metric; the second form is based on the Lagrangian metric complex functionality. In well-characterized systems, DFT calcula- 8 (roughly the inverse of the Eulerian). As noted by both Birch and tions have been shown to be precise and reliable. In particular, Murnaghan, the Eulerian metric is generally considered to be a Lajaeghere et al. recently benchmarked results for DFT-calculated more apt description of elastic behavior for non-infinitesimal thermodynamic properties of the elements, and found overall strains, since it treats the final (rather than initial) coordinates as satisfactory agreement between prediction and experiment the independent variable. The energy–volume relation arising (excepting cases where magnetic, correlative, or relativistic effects from the Eulerian viewpoint is: are significant; or in cases where van der Waals forces are dominant).9 While direct calculations of systems under pressure À2 2 C À2 3 are possible in most DFT codes, the equations of state provide a E ¼ Eo þ BVo ν 3 À 1 þ ν 3 À 1 ; (1) simple mathematical handle with minimum computational cost. 2 These considerations pave the way for high-throughput studies that probe extreme conditions using DFT generated EOS. where ν V , and E and V are the calculated equilibrium energy Vo o o There are three key questions to answer in order to utilize high- and volume, respectively, at zero pressure and absolute zero throughput DFT-based EOS. Which EOS best encapsulates the temperature. As for all subsequent formulations, the values of bulk fundamental physics presented by DFT and are there rules or modulus (K) and its pressure derivative (K′)atν = 1 may be 1Department of Physics, University of California Berkeley, Berkeley, CA 94720, USA; 2Energy Storage and Distributed Resources Division, Lawrence Berkeley National Lab, Berkeley, CA 94720, USA and 3Department of Materials Science, University of California Berkeley, Berkeley, CA 94720, USA Correspondence: Kristin A. Persson ([email protected]) Received: 3 November 2017 Revised: 4 June 2018 Accepted: 11 June 2018 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences Evaluation of thermodynamic equations of state across chemistry and K Latimer et al. 2 Table 1. Theoretical isothermal EOS investigated in this study Equation E(νa) K(ν = 1) K′(ν = 1) Ref 2 3 À2 À2 10 = b þ ν 3 À þ C ν 3 À 8B + Birch (Euler) E Eo BVo 1 2 1 9 C 4 2 2 2 2 10 = þ À ν3 ð À Þ À ν3 þ À ν3 þ 16B − Birch (Lagrange) E Eo BVoC BVo C 2 1 C 1 C 9 C 2 BV BV À1 ÀC þ 6 – = þ o À o ν 3 À 1 ν 3 B 7 C Mie Gruneisen E Eo C CÀ1 C 9 3 νÀC À = þ BVo 1 þ ν À + 13 Murnaghan E Eo ðCþ1Þ C 1 BC1 1 BV ð Àν3 Þ 1 14 – – = þ o 1 3C 1 À À À ν3 + Pack Evans James E Eo C C e 1 31 BC1 2 15 Poirier–Tarantola E = Eo + BVo(ln(ν)) (3 − C(ln(ν))) 6BC+ 2 ÀÁÀÁ = þ BVo ν À þ 1 Cð1ÀνÞ À − 16 Tait E Eo C 1 C e 1 BC1 1 BV 1 À ðν3À Þ 18 = þ o À þ ν3 À C 1 B 2 þ Vinet E Eo C2 1 1 C 1 e 9 3 C 1 aν ¼ V V , where Vo is the volume at zero pressure b o Eo = E(ν = 1) expressed in terms of the fitting parameters B and C: respect to pressure.13 This hypothesis leads to the equation: 8B BV νÀC À 1 K ¼ (2) E ¼ E þ o þ ν À 1 ; (11) 9 o ðC þ 1Þ C 1234567890():,; K0¼C þ 4 (3) where: ¼ Birch’s equation from the Lagrangian viewpoint is: K B (12) 2 2 2 2 K0¼C þ 1 (13) E ¼ Eo þ BVoC À BVoν3 ðC À 2Þ 1 À ν3 þC 1 À ν3 þ C ; (4) Pack–Evans–James where: Pack et al. posited an exponential variation of pressure with 16B respect to changes in volume (based on quantum mechanics), and K ¼ (5) 9 adjusted their expression to ensure proper limiting behavior as pressure approaches zero or infinity:14 0¼ À K C 2 (6) 1 BVo 1 3Cð1Àν3Þ 1 E ¼ E þ e À 1 À 31À ν3 ; (14) o C C – Mie Gruneisen where: The Mie–Gruneisen EOS is based on the well-established empirical ¼ form of a general interatomic potential:4,5 K B (15) m n E ¼ αr þ βr (7) K0¼C þ 1 (16) Setting m = −1 gives a reasonable representation of the Coulombic interaction between two atoms;12 under the restriction Poirier–Tarantola that E = Eo at r = ro reduces the four parameters in Eq. (7) to two to give: Poirier and Tarantola proposed that pressure is best expressed in terms of a logarithmic strain metric (as opposed to the Eulerian or 15 BVo BVo À1 1 ÀC Lagrangian metrics of Birch’s equations). The resulting E ¼ E þ À ν 3 À ν 3 ; (8) o C C À 1 C energy–volume relationship is: 2 where: E ¼ Eo þ BVoðÞlnðνÞ ðÞ3 À CðÞlnðνÞ ; (17) B K ¼ (9) where: 9 K ¼ 6B (18) 7 þ C K0¼ (10) 0¼ þ 3 K C 2 (19) Murnaghan Tait Interestingly, the Murnaghan equation is based not on Francis The Tait equation was developed nearly 150 years ago based Murnaghan’s prodigious 1937 paper, but rather on a short entirely on the empirical observations of Peter Tait, who was communication from 1944 in which he proposed a relatively investigating the compressibility of seawater, and modified a few simple EOS based on a linear variation of the bulk modulus with decades later by Gustav Tammann from a linear average to a npj Computational Materials (2018) 40 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences Evaluation of thermodynamic equations of state across chemistry and K Latimer et al. 3 Fig. 1 RMSrD values for theoretical EOS fitted to DFT calculations of the elements differential form,16 leading to the relation: BV 1 E ¼ E þ o ν À 1 þ eCð1ÀνÞ À 1 ; (20) o C C where: K ¼ B (21) K0¼C À 1 (22) Vinet The Vinet equation was shown by Stacey to be equivalent in 7,17 formulation to the Rydberg potential: Fig.