Pressure Volume Temperature Equation of State

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1 Pressure Volume Temperature Equation of State S.-H. Dan Shim ( ) Acknowledgement: NSF-CSEDI, NSF-FESD, NSF-EAR, NASA-NExSS, Keck

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3 Equations relating state variables (pressure, temperature, volume, or energy).

4 Backgrounds Equations Limitations Applications

5 Ideal Gas Law PV = nrt

6 Ideal Gas Law Volume increases with temperature PV = nrt Volume decreases with pressure Pressure increases with temperature

7 Stress (σ) and Strain (ε)

8 Bridgmanite in the Mantle

9 Strain in the Mantle 20-30%

10 P V T EOS Bridgmanite

11 Energy

12 A Few Terms to Remember Isothermal Isobaric Isochoric Isentropic Adiabatic

13 Energy

14 Thermodynamic Parameters Isothermal bulk modulus

15 Thermodynamic Parameters Isothermal bulk modulus Thermal expansion parameter

16 Thermodynamic Parameters Isothermal bulk modulus Thermal expansion parameter Grüneisen parameter = V P = 1 P U V C V T V

17 P V T of EOS Bridgmanite KT α γ

18 P V T EOS

19 Shape of EOS

20 Shape of EOS Ptotal

21 Shape of EOS Pst Pth

22 Thermal Pressure F tot = F st + F b + F e ec P(V, T) =P st (V, T 0 )+ P th (V, T)

23 Isothermal EOS K = dp d ln V = dp d ln V = V 0 exp apple P K 0 Assumes that K does not change with P, T

24 Murnaghan EOS K = K 0 + K 0 0 P K = dp d ln V = dp d ln = 0 Ç 1 + K0 0 K 0 P å However, K increases nonlinearly with pressure

25 Birch-Murnaghan EOS F = + bƒ + cƒ 2 + dƒ V 0 V =(1 + 2ƒ )3/2 F : Energy (U or F) f : Eulerian finite strain Birch (1978)

26 Second Order BM EOS F = + bƒ + cƒ 2 P = 3K 0 ƒ (1 + 2ƒ ) 5/2 = 3K 0 2 ñ V0 V 7/3 V0 V 5/3 ô K = V dp dv = K 0 2 ñ 7 V0 V 7/3 5 V0 V 5/3 ô = K 0 (1 + 7ƒ )(1 + 2ƒ ) 5/2 Birch (1978)

27 Third Order BM EOS F = + bƒ + cƒ 2 + dƒ 3 P = 3K 0 2 ñ 7/3 V0 V V0 V 5/3 ô 1 ñ V0 V 2/3 1ô = 3 4 (4 K0 0 ) Birch (1978)

28 Truncation Problem F = + bƒ + cƒ 2 + dƒ V 0 V =(1 + 2ƒ )3/2 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

29 Helmholtz Free Energy df = SdT PdV P = df dv T Therefore, knowing the functional form of free energy with respect to volume change is important for EOS.

30 Vinet EOS Vinet et al. (1989)

31 Vinet EOS P = 3K 0(1 ) 2 exp[ (1 )] =(V/V 0 ) 1/3 = 3 2 (K0 0 1) Vinet et al. (1989)

32 Example: Isotherm Fitting SiC, Nisr et al., in prep.

33 Parameters to Fit V0, K0, K 0

34 Example: Isotherm Fitting SiC, Nisr et al., in prep.

35 Example: Isotherm Fitting Strong correlation between K0 and K0 SiC, Nisr et al., in prep.

36 Caution Fei et al. (2007)

37 Caution Do not mix equations and fitting results Fei et al. (2007)

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39 Shape of EOS Pst Pth

40 Thermodynamic Parameters Isothermal bulk modulus Thermal expansion parameter K T (V, T) = P T V = P th T V

41 Thermodynamic Approach P th = P th (V, T) P th (V, T 0 )= Z T T 0 [ K T ] V dt

42 Lattice Dynamic Approach P th = 1 V X E (V) V E th [ (V),T] Debye temperature E th = 9nR 3 Z 0 3 exp 1 d Debye model

43 Parameters to Fit V0, K0, K 0 γ0, q, θ0

44 Example: P-V-T Fitting SiC, Nisr et al., in prep.

45 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

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47 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 Ferropericlase CaSiO3 Pv 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.

48 Pressure Scale 660 Shim et al. 2001

49 Pressure Scale: Post-Spinel Ye et al. 2014

50 Pressure Scale: Post-Perovskite Shim 2008

51 So What is Problem? Stress conditions Temperature conditions Extreme thermal contribution electronic contribution in metal pressure standards

52 P(Au) P(Pt) P(MgO) Au/Pt (F07_BM) vs MgO (S01_BM) PAu - PMgO (GPa) PMgO (GPa) Ye et al. (2016) in prep.

53 Bridgmanite VIII Mg 2+VI Si 4+ O3 Fe 2+ Fe 3+ Al 3+

54 Bridgmanite Fe 2+ Lundin et al. 2008

55 Bridgmanite Fe 3+ Al 3+ Catalli et al. 2011

56 Ferropericlase Fei et al. 2007

57 Ferropericlase Wentzcovitch (2009)

58 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 + P V T S

59 Stishovite: Effect of Water Nisr et al., in prep.

60 Nexus for Exoplanet System Science

61 Mass-Radius Relations Hydrogen Water Silicate Iron Seager (2007)

62 Earth-Like Exoplanets Pepe et al. (2013) Nature

63 Elemental Abundances Bond et al. (2010)

64 Carbide Planets Nisr et al., in prep.

65 Further Readings Jackson and Rigden (1996) PEPI Anderson (2000) GJI Shim and Duffy (2000) AmMin

66 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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