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 = V = U C 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)+Pth(V, T) Isothermal EOS

dP dP K = = d ln V d ln

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

K00 = 0 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 1 2 V V V ñ✓ ◆ ✓ ◆ ô® ñ✓ ◆ ô´

3 = (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[(1 )] 2

3 1/3 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 Pth KT (V, T)= = T T ✓ ◆V ✓ ◆V Thermodynamic Approach

T Pth = Pth(V, T) Pth(V, T0)= [KT ]V dT ZT0 Lattice Dynamic Approach

1 (V) Pth = E Eth[(V),T] V ⇡ V X

Debye temperature

9nR 3 E d Debye model th = 3 0 exp 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)

G = U + PV TS 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 (?)