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Universal Equation of State for Elastic Solids; Mgsio3 Perovskite Is an Example

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Universal Equation of State for Elastic Solids; Mgsio3 Perovskite Is an Example Universal equation of state for elastic solids; MgSiO3 perovskite is an example József Garai Department of Earth Sciences, Florida International University, Miami, FL 33199, USA Abstract Universal (P-V-T) equation of state is derived from theoretical considerations. Correlation coefficients, root-main-square-deviations, and Akaike Information Criterias are used to evaluate the fitting to the experiments of perovskite 0-109 GPa and 293-2000 K. The proposed equation remains valid through the entire pressure and temperature range and has superior fitting parameters in comparison to the Birch-Murnaghan, Vinet, and Roy & Roy equation of states. Contents 1. Introduction 2. The equation of state 2.1 Thermal EoS 2.2 The temperature effect on the bulk modulus 2.3 Isothermal EoS 2.3.1 Finite-strain EoS 2.3.2 Inter-atomic potential EoSs 2.3.3 Empirical EoSs 3. Fundamental components of the volume in solid phase 4. New description for the pressure-temperature-volume relationship 4.1 The affect of pressure and temperature 5 Testing the EoS to experiments of perovskite 5.1 Fitting criterias 5.2 Fitting parameters 6. Conclusions Acknowledgement References - 2 - 1. Introduction The relationships among the pressure, the volume, and the temperature are described by the Equation of State (EoS). The volume-temperature relationship is described by the definition of the volume coefficient of expansion [ αV ] 1 ∂V α V p ≡ (1) V ∂T p The relationship between the pressure and the volume is given by the isothermal bulk modulus [BT ] ∂p BT ≡ −V . (2) ∂V T For the validity of equation (2) it is assumed that the solid is homogeneous, isotropic, non- viscous and has linear elasticity. It is also assumed that the stresses are isotropic; therefore, the principal stresses can be identified as the pressure p = σ1 = σ2 = σ3 . The schematic relationships between the thermodynamic quantities are shown on Fig. 1-a. Experiments show that both the volume coefficient of expansion and the isothermal bulk modulus are pressure and temperature dependent; therefore, it is necessary to know the derivatives of these parameters. ∂αV ∂αV ∂BT ∂BT ; ; ; (3) ∂T p ∂p T ∂T p ∂p T A universal EoS must cover the entire pressure and temperature range; therefore, it is necessary to incorporate all of the derivatives of the volume coefficient of expansion and the isothermal bulk modulus. There is no single expression known for universal (P-V-T) EoS (MacDonald, - 3 - 1965; Baonza et al., 1996). An attempt is made here to derive and test the first universal EoS for elastic solids. 2. The equation of state In order to overcome the complexity of the EoS, the common practice is that the temperature of the substance is raised first and then the substance is compressed along the isotherm of interest (Duffy and Wang, 1998; Angel, 2000). The relevant equations are called the thermal and the isothermal equation of state (EoS) respectively. The thermal EoS is used to calculate the volume at atmospheric pressure and temperature T []V0,T . It is also necessary to know the temperature affect on the bulk modulus []B0 ()T . Using the values of the volume and the bulk modulus at the corresponding temperature the isothermal EoS calculates the affect of pressure by using the first and the second derivates of the bulk ∂B ∂ 2 B modulus, and at the given temperature. 2 ∂p T ∂p T The simplest complete thermodynamic description of a single component solid then requires a minimum of four parameters: ∂B ∂B ; B T ; ; α V 0 ()0 (4) ∂T p=0 ∂p T a more precise description would require six parameters ∂α ∂B ∂B ∂ 2 B α ; V ; B T ; ; ; and . V 0 ()0 2 (5) ∂T p=0 ∂T p=0 ∂p T ∂p T - 4 - 2.1 Thermal EoS The simplest thermal equation of state is derived by the integration of the thermodynamic definition of the volume coefficient of thermal expansion Eq.(1) T α ()T dT ∫ Vp (6) T0 V0 ()T = V0 (T0 )e . If a wider temperature range is considered then the temperature dependence of the volume coefficient of thermal expansion should be known. Knowing the first derivative of this parameter allows one to calculate the high temperature values: ∂α V α V p ()T = α V p (T0 )+ ()T − T0 . (7) ∂T p The thermodynamic Gruneisen-Anderson parameter [δT ] is defined as (Anderson, 1987): ∂ln K T 1 ∂ln K T δT = = − (8) ∂lnρ p α V ∂T p Assuming that the solid at higher temperatures follows classical behavior, then the product of α K is constant and the Gruneisen-Anderson parameter is independent of temperature, Vp T Anderson et al. (1992) and Shanker (1993) proposed the following isobaric EoS: 1 − V = V 1− α δ T − T δ0 (9) 0 []V0 0 ()0 where the subscript zero values of the parameters refers to the initial temperature of T.0 Assuming that the product of α K is constant and the Gruneisen-Anderson parameter Vp T changes linearly with the volume, the following EoS has been proposed by Kumar (2002) and Kushwah et al. (1996) - 5 - 1 V= V0 1− ln[]1− α V A()T − T0 (10) A 0 where A = δ0 +1. Thermal EoS have been suggested by Akaogi and Navrotsky (1984; 1985), assuming that the thermal expansion is quadratic in the temperature, and independent of pressure V = V 1+ α T − T + α ' T − T 2 , p 0 [ V0 ()()0 V0 0 ] (11) where α ' is the temperature derivative of α at temperature T = T . Taking into consideration V0 V 0 the affect of the pressure the equation can be written as: −1 ' K' B ()p − p 0 2 V = V 1+ 0 0 1+ α T − T + α ' T − T , (12) p,T 0 []V0 ()()0 V0 0 B0 Fei and Saxena (1986) revised the quadratic relationship of Eq. (12) and proposed the following empirical expression: 1 ' 2 −1 V.p = V0 1+ α V ()T − T0 + α V ()T − T0 − α V ()T − T0 (13) 0 2 0 0 Assuming linear change as a function of temperature in the volume coefficient of thermal expansion leads to the following expression: 1 ' 2 αV ()T−T0 + αV ()T−T0 0 2 0 (14) Vp = V0e The proposed general expression of Eq. (14) for bcc iron is: −1 ' 1 K0 α ()T−T + α' ()T−T 2 ' ∂B p − p V0 0 V0 0 V = V 1+ B B − T − T 0 e 2 (15) p 0 0 0 ()0 2 ∂T 0 B0 An empirical expression has been given by Plymate and Stout (1989) - 6 - 1 2 2 − ∂BT 1 ' ∂BT 1 ()T−T0 ' αV + ()T−T0 + αV + B0 0 ∂T ' 0 ∂T 2 ' 2 ∂B T − T 0 B0B0 0 B B0 V = V 1+ T 0 e 0 0 (16) ∂T 0 B0 2.2 The temperature effect on the bulk modulus The temperature has an effect not only on the volume and the volume coefficient of thermal expansion but on the bulk modulus as well. In order to use the isothermal EoS it is necessary to know the value of the bulk modulus at the temperature of interest, which can be obtained from ∂BT BT0 ()T = BT0 (T0 )+ ()T − T0 . (17) ∂T p Theoretically the temperature dependence of the elastic constants can be determined as the sum of the anharmonic terms (Kittel, 1968; Levy, 1986). At sufficiently low temperatures the elastic constant should vary as T4 (Born & Huag, 1956, p 437). Contrary to this suggestion some metallic substances have been found to show a T2 rather than a T4 dependence at low temperatures (Alers, 1961; Chang. & Graham, 1966). There is no general prediction for higher temperatures. Experiments on refractory oxides, conducted at higher than room temperature, show a linear relationship between the bulk modulus and the temperature (Wachtman, 1959). The third law of thermodynamics requires that the derivative of any elastic constant with respect to the temperature must approach zero as the temperature approaches absolute zero. Combining this criterion with the observed linear relationship at higher temperatures, Wachtman et al. (1961) suggested an equation in the form of T − 0 T (18) B.= B0 − b1Te - 7 - where K0 is the bulk modulus at absolute zero, and b1 and T0 are arbitrary constants. Theoretical justification for the Wachtman’s Equation was suggested by Anderson (1966). Based on shock-wave and static-compression measurements on metals, a linear relationship between the logarithm of the bulk modulus and the specific volume has been detected for metals (Grover et al., 1973): ∆V ln BT = ln B0 + α , (19) V where α is a constant depending on the material. The linear correlation is valid up to 40% volume change. Using this linear correlation Jacobs and Oonk (2000) proposed a new equation of state. They rewrite equation (19) as B0 (T) V 0 (T) = V 0 (T ) + b ln , m m 0 0 (20) B (T0 ) 0 where Vm denotes molar volume, T0 the reference temperature and the superscript “0” refers to standard pressure (1 bar). Equation (20) successfully reproduces the available experimental data for MgO, Mg 2SiO 4 , and Fe2SiO 4 (Jacobs and Oonk, 2000; Jacobs et al. 2001; Jacobs and Oonk, 2001). Assuming that the product of the volume coefficient thermal expansion and the bulk modulus is constant at temperatures higher than the Debye temperature analytical solution for the temperature dependence of the bulk modulus was derived (Garai and Laugier, 2006). T − δTαVdT ∫T=0 (21) BT = e K T=0 . where δT is the isothermal Anderson-Grüneisen parameter given by: - 8 - 1 ∂BT δT = − . (22) α B ∂T Vp T p Equation (21) was able to mimic experiments with high accuracy for the investigated substances.
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