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Eosfit7c and a Fortran Module (Library) for Equation of State Calculations

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Eosfit7c and a Fortran Module (Library) for Equation of State Calculations DOI 10.1515/zkri-2013-1711 Z. Kristallogr. 2014; 229(5): 405–419 Ross J. Angel*, Javier Gonzalez-Platas and Matteo Alvaro EosFit7c and a Fortran module (library) for equation of state calculations Abstract: The relationship between linear elasticity theo- and expansion. Equations of state therefore provide not ry of solids and their equations of state (EoS) is reviewed, only fundamental thermodynamic data that is required, along with the commonly-used types of isothermal EoS, for example, for the calculation of equilibrium phase dia- thermal expansion models, and P-V-T EoS. A new con- grams (e.g. Berman, 1988; Gottschalk, 1997; Holland & sole program, EosFit7c, is presented. It performs EoS cal- Powell, 1998; Stixrude & Lithgow-Bertelloni, 2005; Hol- culations and fitting for both volume and linear isother- land & Powell, 2011) but also give insights in to the de- mal data, isobaric data and P-T data. Linear data is tails of interatomic interactions within the solid state, as handled by cubing the quantities and treating them as it is these that resist the externally-applied compressive volumes in all EoS formulations. Least-squares fitting of stresses (e.g. Brown, Klages & Skowron, 2003; Zhao, EoS to data incorporates the option to weight the fit with Ross & Angel, 2004; Fabbiani & Pulham, 2006) and con- the measurement uncertainties in P, V and T simulta- trol the dynamics that lead to thermal expansion (e.g. neously. The EosFit7c program is built with a new library Willis & Pryor, 1975). of subroutines for EoS calculations and manipulation, The majority of equation of state studies measure the written in Fortran. The library has been incorporated as a volume or unit-cell parameter variations with pressure module, cfml_eos, in the publicly-available CrysFML li- (and/or temperature), with the aim of deriving elastic brary. The module handles Murnaghan, Tait, Birch-Mur- parameters that are derivatives of the data. For example, naghan, Vinet, and Natural Strain EoS. For P-V-T calcula- the bulk modulus that defines the instantaneous volume tions any of these isothermal EoS can be combined with variation with pressure is K ¼Vð@P=@VÞ. The bulk a variety of published thermal expansion models, includ- moduli of most inorganic solids range from ~40 to ing a model of thermal pressure. The entire library has 400 GPa. Therefore the volume changes induced by com- been revalidated against other software and against an pression over the relatively-easily accessible experimen- ab-initio re-derivation of the EoS, which identified a tal pressure range (0–10 GPa) are only larger than the ex- number of small errors in published formulae for some perimental uncertainties by 1 or 2 orders of magnitude. EoS. The need to obtain parameters that are derivatives of the original data, in combination with the small data range, Keywords: equations of state, thermal expansion, EosFit clearly makes the reliable determination of the para- meters difficult to achieve. Strong correlations between parameters, which include the pressure derivatives of K, *Corresponding Author: Ross J. Angel, Dipartimento di Geoscienze, exacerbate the problems. These issues were previously Università di Padova, Via G. Gradenigo 6, Padova, 35131, Italy, e-mail: [email protected] addressed by the development of the EosFit program Matteo Alvaro: Dipartimento di Geoscienze, Università di Padova, (Angel, 2000a) which provided the capability of fitting Via G. Gradenigo 6, Padova, 35131, Italy P-V (and unit-cell parameter data) with various EoS, with Javier Gonzalez-Platas: Departamento de Física Fundamental II. the correct algebra and with options to fully weight the Servicio de Difracción de Rayos X, Universidad de La Laguna, data with the measurement uncertainties. The Eosfit pro- La Laguna, Tenerife 38206, Spain gram includes utilities to do further EoS calculations be- yond just fitting EoS parameters to P-V data. The ready availability of a variety of spread sheets Introduction and algebraic software makes it relatively easy, in princi- ple, to fit any EoS formulation to data. However, the An equation of state (EoS) describes how the volume or complex nature of some equation of state functions has density of a material varies with changes in pressure and led to their algebraic forms being mis-stated in the litera- temperature. It also defines how some of the elastic prop- ture. The additional possibility of errors in coding the erties of the material change in response to compression complex equations, and the requirement to implement Brought to you by | Universita degli Studi di Padova Authenticated | [email protected] author's copy Download Date | 5/9/14 9:37 AM 406 R. J. Angel et al., EosFit7c and equations of state specific (non-standard) weighting methods to overcome presentation of the compliance tensor of the material correlation problems, makes imperative the provision of (Nye, 1957). a validated set of publicly-available self-consistent algo- Hydrostatic pressure is a special stress state in which rithms for EoS calculations. We have now translated the the normal stresses are all equal (σ1 ¼ σ2 ¼ σ3 ¼ P) and original code of EosFit to Fortran-95, revalidated it, and there are no shear stresses (σ4 ¼ σ5 ¼ σ6 ¼ 0). Thus, at have built it in to a module that is part of the larger Crys- any pressure P, the strains caused by an infinitesimal in- tallographic Fortran Modules Library CrysFML (Rodri- crease in pressure δP can be calculated through linear guez-Carvajal & Gonzalez-Platas, 2003). This will ensure elasticity theory by setting σ1 ¼ σ2 ¼ σ3 ¼δP. (Note the long-term survival and availability to programmers of that while pressure is considered to be a positive quan- a validated set of easy-to-use procedures for EoS calcula- tity, compressive stresses are by convention considered tions. to be negative; Nye, 1957). Therefore each of the resulting In this paper we first briefly review the relationship strain elements is given by: between linear elasticity theory of solids and their equa- ε ¼ð σ þ σ þ σ Þ¼ð þ þ Þ δ : tions of state, and then present the types of isothermal i si1 1 si2 2 si3 3 si1 si2 si3 P EoS, thermal expansion models, and thus P-V-T EoS that are implemented within the new EoS module of CrysFML. The sum of the three normal strains is, in the infinitesi- The handling of linear data (e.g. cell parameters) in a mal limit, equal to the fractional change in the volume manner consistent with both volume EoS and linear elas- @V=V, thus: ticity theory is discussed, prior to a description of the À@ = ¼ðε þε þε Þ¼½ þ þ þ ð þ þ Þ δ : implementation of these approaches in the CrysFML V V 1 2 3 s11 s22 s33 2 s12 s13 s23 P module. A new program, EosFit7c, that performs EoS cal- culations and fits P-V, V-T and P-V-T data (and linear Re-arrangement of this last equation shows that the bulk equivalents), is presented as an example of the new fea- modulus for hydrostatic compression at any pressure is tures that are available in the module. defined by six of the elements of the compressibility ma- trix at that same pressure: À1 Isothermal equations of state K ¼VδP=δV ¼½s11 þ s22 þ s33 þ 2 ðs12 þ s13 þ s23Þ : Basis in linear elasticity theory This bulk modulus for hydrostatic compression of a so- lid, whether a powder or single crystal, is therefore equal The variation of the volume of a solid with hydrostatic to the Reuss bound on the bulk modulus of a polycrystal pressure at fixed temperature is termed its ‘isothermal made of the same material where it represents the vol- equation of state’. It is characterised by the bulk modu- ume response when every constituent grain is subject to ¼ ð@ =@ Þ lus of the material, K V P V T , which is a func- the same stress. tion of both temperature and pressure. For infinitesimal While linear elasticity thus defines the bulk modulus changes in pressure which give rise to infinitesimal of a material under hydrostatic compression at any pres- changes in volume, the bulk modulus can also be de- sure, it cannot define an equation of state which de- fined in terms of the elastic tensor of the material by ap- scribes the large (finite) changes in volume due to large plying linear elasticity theory, or ‘Hooke’s law’. In linear (finite) changes in pressure. In this sense an equation of elasticity it is assumed that the strains εi of a solid are state is an extension of linear elasticity; although nor- linearly related to the magnitude of the applied stress mally defined in terms of the volume variation with pres- field σj by the matrix equation εi ¼ sijσj. The suffixes run sure, it can also be seen as a definition of the variation from 1 to 6, with i,j ¼ 1, 2, 3 referring to normal stresses of bulk modulus with pressure. Because there is no abso- or strains along orthogonal axes, and i,j ¼ 4, 5, 6 refer- lute thermodynamic basis for specifying how the bulk ring to shear stresses or strains (e.g. Nye, 1957; Angel, modulus K varies with pressure, all EoS that have been Jackson, Reichmann & Speziale, 2009). The elastic prop- developed and are in widespread use are based upon a erties of the material are represented by the values of the number of assumptions (e.g. Anderson, 1995; Duffy & elements of the compliance matrix sij which is symmetric Wang, 1998; Holzapfel, 2001). The validity of such as- and can contain up to 21 independent elements for tricli- sumptions can only be judged in terms of whether the nic crystals, less for crystals and materials with higher derived EoS reproduces experimental data for volume or symmetries.
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