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

PV = nRT• Volume increases with temperature

• Volume decreases with pressure

• Pressure increases with temperature

Stress (σ) and Strain (𝜖)

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 ParametersIsothermal bulk modulus

Thermodynamic ParametersIsothermal bulk modulus

Thermal expansion parameter

Thermodynamic ParametersIsothermal bulk modulus

Thermal expansion parameter

Grüneisen parameter

� = V✓ �P�U

◆

V=

1

�CV

✓ �P�T

◆

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 + F��b + Fe�ec

P(V, T) = Pst(V, T0) + �Pth(V, T)

Isothermal EOS

K = �dP

d lnV=

dP

d ln�

V = V0 exp�PK0

�

Assumes that K does not change with P, T

Murnaghan EOS

However, K increases nonlinearly with pressure

K = K0 + K 00P

K = �dP

d lnV=

dP

d ln�

� = �0

Ç1 +

K 00K0

P

å

Birch-Murnaghan EOS

F : Energy (U or F)f : Eulerian finite strain

F = � + bƒ + cƒ2 + dƒ3 + ...

V0V= (1 + 2ƒ )3/2

Birch (1978)

Second Order BM EOS

F = � + bƒ + cƒ2

P = 3K0ƒ (1 + 2ƒ )5/2 =3K02

ñ✓V0V

◆7/3�✓V0V

◆5/3ô

K = �VdP

dV=K02

ñ7✓V0V

◆7/3� 5✓V0V

◆5/3ô= K0(1 + 7ƒ )(1 + 2ƒ )5/2

Birch (1978)

Third Order BM EOS

F = � + bƒ + cƒ2 + dƒ3

P =3K02

ñ✓V0V

◆7/3�✓V0V

◆5/3ô®1 � �ñ✓V0

V

◆2/3� 1ô´

� =3

4(4 � K 00)

Birch (1978)

Truncation Problem

• 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´´

F = � + bƒ + cƒ2 + dƒ3 + ...V0V= (1 + 2ƒ )3/2

Helmholtz Free Energy

dF = �SdT � PdV

P = �✓ dFdV

◆

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

Vinet et al. (1989)

P =3K0(1 � �)

�2exp[�(1 � �)]

� = (V/V0)1/3 � =3

2(K 00 � 1)

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

SiC, Nisr et al., in prep.

Strong correlation between K0 and K0´

Caution

Fei et al. (2007)

Caution

Do not mix equations and fitting results

Fei et al. (2007)

Shape of EOS

Pst Pth

Thermodynamic ParametersIsothermal bulk modulus

Thermal expansion parameter

�KT (V, T) =✓ �P�T

◆

V=✓��Pth

�T

◆

V

Thermodynamic Approach

�Pth = Pth(V, T) � Pth(V, T0) =Z T

T0[�KT ]VdT

Lattice Dynamic Approach

�Pth =1

V

X

���E� ⇡

�(V)

V�Eth[�(V), T]

Debye temperature

Eth =9nR

�3

Z �

0

�3

exp� � 1d�Debye model

Parameters to Fit

V0, K0, K´0γ0, 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

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

Fiquet et al (2000), Shim and Duffy (2000), Shim et al. (2000), Stixrude et al. (2011)

• 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

Shim et al. 2001

660

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)PA

u - P

MgO

(GPa

)

PMgO (GPa)

Au/Pt (F07_BM) vs MgO (S01_BM)

Ye et al. (2016) in prep.

Bridgmanite

VIIIMg2+VISi4+O3

Fe2+Fe3+Al3+

Bridgmanite

Lundin et al. 2008

Fe2+

Bridgmanite

Catalli et al. 2011

Fe3+Al3+

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 + P�V � T�S

Stishovite: Effect of Water

Nisr et al., in prep.

Nexus for Exoplanet System Science

http://www.nexss.io

Seager (2007)

IronSilicateWater

Hydrogen

Mass-Radius Relations

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 (?)