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Pressure—Volume—Temperature Equation of State

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Pressure—Volume—Temperature Equation of State Pressure—Volume—Temperature Equation of State S.-H. Dan Shim (심상헌) Acknowledgement: NSF-CSEDI, NSF-FESD, NSF-EAR, NASA-NExSS, Keck Equations relating state variables (pressure, temperature, volume, or energy). • Backgrounds • Equations • Limitations • Applications Ideal Gas Law PV = nRT Ideal Gas Law • Volume increases with temperature • VolumePV decreases= nRTwith pressure • Pressure increases with temperature Stress (σ) and Strain (�) Bridgmanite in the Mantle Strain in the Mantle 20-30% P—V—T EOS Bridgmanite Energy A Few Terms to Remember • Isothermal • Isobaric • Isochoric • Isentropic • Adiabatic Energy Thermodynamic Parameters Isothermal bulk modulus Thermodynamic Parameters Isothermal bulk modulus Thermal expansion parameter Thermodynamic Parameters Isothermal bulk modulus Thermal expansion parameter Grüneisen parameter ∂P 1 ∂P γ = V = ∂U ρC ∂T ✓ ◆V V ✓ ◆V P—V—T of EOS Bridgmanite • KT • α • γ P—V—T EOS Shape of EOS Shape of EOS Ptotal Shape of EOS Pst Pth Thermal Pressure Ftot = Fst + Fb + Feec P(V, T)=Pst(V, T0)+ΔPth(V, T) Isothermal EOS dP dP K = = − d ln V d ln ρ P V = V0 exp − K 0 Assumes that K does not change with P, T Murnaghan EOS K = K0 + K00 P dP dP K = = − d ln V d ln ρ K00 ρ = ρ0 1 + P Ç K0 å However, K increases nonlinearly with pressure Birch-Murnaghan EOS 2 3 F = + bƒ + cƒ + dƒ + ... V 0 3/2 =(1 + 2ƒ ) V F : Energy (U or F) f : Eulerian finite strain Birch (1978) Second Order BM EOS 2 F = + bƒ + cƒ 3K V 7/3 V 5/3 5/2 0 0 0 P = 3K0ƒ (1 + 2ƒ ) = 2 V V ñ✓ ◆ − ✓ ◆ ô dP K V 7/3 V 5/3 0 0 0 5/2 K = V = 7 5 = K0(1 + 7ƒ )(1 + 2ƒ ) dV 2 V V − ñ ✓ ◆ − ✓ ◆ ô Birch (1978) Third Order BM EOS 2 3 F = + bƒ + cƒ + dƒ 7/3 5/3 2/3 3K0 V0 V0 V0 P = 1 ξ 1 2 V V V ñ✓ ◆ − ✓ ◆ ô® − ñ✓ ◆ − ô´ 3 ξ = (4 K00 ) 4 − Birch (1978) Truncation Problem 2 3 F = + bƒ + cƒ + dƒ + ... V 0 3/2 =(1 + 2ƒ ) V • Higher order terms can be large at high P • 2nd order BM assumes K0´ = 4 • 3rd order BM assumes complex relation among K0, K0´, and K0´´ Helmholtz Free Energy dF = SdT PdV − − dF P = dV − ✓ ◆T Therefore, knowing the functional form of free energy with respect to volume change is important for EOS. Vinet EOS Vinet et al. (1989) Vinet EOS 3K0(1 ) P = − exp[η(1 )] 2 − 3 1/3 η K 1 =(V/V0) = ( 00 ) 2 − Vinet et al. (1989) Example: Isotherm Fitting SiC, Nisr et al., in prep. Parameters to Fit V0, K0, K´0 Example: Isotherm Fitting SiC, Nisr et al., in prep. Example: Isotherm Fitting Strong correlation between K0 and K0´ SiC, Nisr et al., in prep. Caution Fei et al. (2007) Caution Do not mix equations and fitting results Fei et al. (2007) Shape of EOS Pst Pth Thermodynamic Parameters Isothermal bulk modulus Thermal expansion parameter ∂P ∂ΔPth αKT (V, T)= = ∂T ∂T ✓ ◆V ✓ ◆V Thermodynamic Approach T ΔPth = Pth(V, T) Pth(V, T0)= [αKT ]V dT − ZT0 Lattice Dynamic Approach 1 γ(V) ΔPth = γE ΔEth[θ(V),T] V ⇡ V X Debye temperature 9nR ξ3 E dξ Debye model th = 3 0 exp ξ 1 Z − Parameters to Fit V0, K0, K´0 γ0, q, θ0 Example: P-V-T Fitting SiC, Nisr et al., in prep. Derivation of Thermodynamic Parameters • Many useful parameters can be derived from Birch-Murnaghan-Debye and Vinet-Debye EOS. • See Jackson and Rigden (1996, PEPI) for detail • For example, K, α, (∂Κ/∂T)V, (∂Κ/∂T)S at any given pressure and temperature Lower Mantle Minerals Fiquet et al (2000), Shim and Duffy (2000), Shim et al. (2000), Stixrude et al. (2011) K0 (GPa) γ0 q θ0 (K) Bridgmanite 250-260 1.3-1.7 1.2-1.7 1000 Ferropericlase 160-165 1.4 1.3 673 CaSiO3 Pv 220-250 2 0.6 1000 • Density and elasticity of pyrolite agree reasonably well with those of PREM. • Density of MORB is 2-3% higher than pyrolite throughout the lower mantle. Pressure Scale 660 Shim et al. 2001 Pressure Scale: Post-Spinel Ye et al. 2014 Pressure Scale: Post-Perovskite Shim 2008 So What is Problem? • Stress conditions • Temperature conditions • Extreme thermal contribution — electronic contribution in metal pressure standards P(Au) — P(Pt) — P(MgO) Au/Pt (F07_BM) vs MgO (S01_BM) (GPa) MgO - P Au P PMgO (GPa) Ye et al. (2016) in prep. Bridgmanite VIII 2+VI 4+ Mg Si O3 Fe2+ Fe3+ Al3+ Bridgmanite Fe2+ Lundin et al. 2008 Bridgmanite Fe3+ Al3+ Catalli et al. 2011 Ferropericlase Fei et al. 2007 Ferropericlase Wentzcovitch (2009) Stishovite: Effect of Water • δ—AlOOH • Phase H (Nishi et al. 2014) • δ—H (Ohtani et al. 2014) • SiO2 (Spektor et al. 2011) ΔG = ΔU + PΔV TΔS − Stishovite: Effect of Water Nisr et al., in prep. Nexus for Exoplanet System Science http://www.nexss.io Mass-Radius Relations Hydrogen Water Silicate Iron Seager (2007) Earth-Like Exoplanets Pepe et al. (2013) Nature Elemental Abundances Bond et al. (2010) Carbide Planets Nisr et al., in prep. Further Readings • Jackson and Rigden (1996) PEPI • Anderson (2000) GJI • Shim and Duffy (2000) AmMin Future • Better description of thermal part • Better description of electronic contribution • New experimental techniques • Demand from exoplanet field • Database with community agreement (?) .
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