Equations of State

Tiziana Boffa Ballaran Why EoS ?

The Earth‘s interior is divided globally into layers having distinct seismic properties

Speed with which body waves travel through the Earth‘s interior are simply related to the density and elastic moduli of the constituent materials

2  4  v P   KS  GS    3  2 v S  GS 

v    K  Outline

 Overview of few equations of state: Principles and assumptions

 Hydrostaticity and pressure scales

 Working example Helmholtz free energy

AV ,T  Ac V  Avib V ,T  Ael V ,T 

compression curve vibrational contribution

electronic contribution

 A  P    PV ,T  Pc V  Pvib V ,T  Pel V ,T   V T

reference curve thermal pressure

Higher order derivatives of the energy function

etc... KT  V P V T KT  KT PT Finite strain theory

Infinitesimal elasticity:

 strains are determined by stresses and are reversible  strains are small and their squares and products are negligible

Reference state:

 unstrained state (Lagrangian)  deformed state, i.e. volume at final compression (Eulerian)  incremental strain (Hencky)

These EoS are empirical ! Birch-Murnaghan EoS

Stacey et al. (1981)

2 3 4 Helmholtz free energy: A  A2 f  A3 f  A4 f  ...

 2 3  2 3 1  V  1   V   f      1 Lagrangian f    1    2 V  2 V   0     0  

Determination of the A coefficients at P = 0 with V = V0, K = K0 and K‘ = K0‘

dA df 5/ 2  3 3  35  2  P    3K0 f 1 2 f 1 K0 '4 f   K0 K0 "K0 '4 K0 '3   f  df dV  2 2  9  

IV order Birch-Murnaghan EoS Birch-Murnaghan EoS

Assumptions:

 Eulerian strains under hydrostatic compression

 strictly derived only for isotropic or cubic materials

 solid under compression is homogeneously strained

 EoS are continuously differentiable

 higher order terms of the Taylor expansion are negligible Birch-Murnaghan EoS

ThedA behaviour of K‘5/ 2is a3 valuable test3  of the plausibility35 2  P     3K0 f 1 2 f 1 K0 '4 f   K0K0"K0 '4 K0 '3   f   ofdV a PEoS0  2 2  9  

V Plausibility in the limit of large compression: 0   V

K‘ negative

2 3 3  V   K' K ' K K "  1 0 2 0 0 V    0   3/ 2 V0  2K0 '   1  V   3K0 K0 " Logarithmic EoS

 l l Hencky strain:   d  ln l0 l0 The integration is possible only if the principal axes do not rotate during deformation

Hydrostatic compression

Poirier and Tarantola 1998

1 V  H  ln 3 V0 Logarithmic EoS

2 3 4 1 V0 A  A f  A f  A f  ... with f    ln 2 3 4 H 3 V

Poirier and Tarantola 1998

dA df 1 dA 1 V N na  V  P     0 n ln 0  n   df dV 3V df V0 V n2 3  V   Determination of the a coefficients at P = 0 with V = V , K = K and K‘ = K ‘  0 0 0 

  K0 '2    P  K0 ln 1  ln  III order 0 0   2  0  Interatomic potential

Forces between atoms in the crystal are defined with

assumed interatomic potential

calculation of density as a function of pressure

EoS

Empirical approach as for strain EoS Born-Mie potential

a b E(r)    rm rn

attractive forces repulsive forces

Poirier 2000

1n 3 1m 3 3K V  V   P  0  0    0   n  m  V   V   Born-Mie potential

1n 3 1m 3 3K V  V   P  0  0    0   n  m  V   V  

Different values of m and n have been used in the literature (Stacey et al. 1981)

Special case: m = 2 and n = 4 P Plausibility: K'  K 'K K " 0 0 0 K  7 3 5 3  3 V0  V0  P  K0       II order Birch-Murnaghan EoS 2  V   V   Vinet EoS

br Empirical potential: A1 are

A, a, b constants depending on the material

Vinet et al. 1987, Poirier 2000

2 3 1 3  1 3   V    V   3   V             P  3K0   1   exp K0 '1 1    V  V  2  V   0    0      0  

Good for very high compression of metals and ionic solids Thermal pressure

PV ,T  Pc V  Pvib V ,T  P-V-T Birch-Murnaghan EoS

Isotherm at room temperature 298 K V0, K0, K0‘ ...

T (K) VT0, KT0, KT0‘ ... Isotherm at any given temperature

T V (T )  V (T )exp (T )dT 0 0  with (T )  a  bT T0 and  K  K(T )  K0 (T0 )    T  T0 linear variation with T  T P

Usual assumption: K‘ is the same at all T Thermal pressure (Jackson and Ridge, 1996) (Anderson et al., 1989)

PV ,T  Pc V  Pth V ,T

 P     T   V T  P  P V ,T  P V ,T   K dT th th th 0  T V T0

T T T  K    V   P  K P ,T T   T   ln T  T  dTdt More generally:Simplest case:th  T 0 Pth  K T T  0    T V V0  T0  T00T  K independent of temperature and volume T V  2 3 Pth V ,T  a1 T  T0  a2 ln T  T0  a3 T  T0  a4 T  T0  V0  „Thermodynamic“ EoS

 Thermal expansion at atmospheric pressure: V(T,P0)

 Compression at 300 K: V(T0,P)  Compression at high temperature: V(T,P)

V  PV ,T  P V ,T0  P th V ,T  bi X i  ,T   V0 

V  i 5/ 2 b  9 K K '  4  8 X i  ,T   2 f E 1 2 f E with b1  3KT 2 and 2 T 0 T 0  V0  0

III order Birch-Murnaghan EoS for reference isotherm

X  T  T X  T  T lnV V  2 3 3 0 4 0 0 X 5  T  T0  X 6  T  T0  Mie-Grüneisen EoS

Lattice dynamics approach: the thermal pressure is related to the vibrational density of state  V  P  E V ,T  E V ,T   th V th th 0

q   V  d ln Grüneisen parameter: (V )     q   const. 0   d lnV V0 

  T  9nRT 3 Thermal free energy: E  d th 3  /T 0 e 1

(V )  0 exp 0  (V ) q

 and  independent of temperature, functions of volume only Experimental details Pressure medium

 Requirements – Hydrostatic – Does not dissolve sample – Does not penetrate sample

 Non-hydrostatic stresses – Cannot be measured accurately – Reduce the data quality – Result in different EoS – affect structural phase transitions Broadening of diffraction peaks

Previous max Diffraction “Hydro broadening Pressure” Non-hydrostatic(GPa) stresses give rise to broadening of the Methanol:Ethanol 10.4 10.1 diffraction profiles Isopropanol 4.3 3.9

Silicone oil 5-7 < 2.0

Nitrogen 13.0 3

Argon 9 2

Fluorinert 5-10 1

Angel et al. 2007 J Appl Cryst 40:26 Gas loading

He and Neon No broadening observed so far up to 22 GPa with a point detector.

BGI 75 GPa (ESRF) Pressure scales

 Single-crystal room T up to 10 GPa: quartz crystal (Angel et al. 1997)

 Single-crystal and powder room T any pressure: ruby fluorescence

 Powder HT and HP: EoS of metals like Au, Pt, Cu, Ag ... EoS of non metal like MgO, NaCl ... (Fei et al. 2004)

 Single-crystal HT and HP: ? ... EoS of Neon and He Ruby scales

The ruby scales are based on the calibration of the shift of the ruby R1 luminescent line and the specific volume of some standard material with increasing pressure

 Mao et al. (1996): ruby vs Cu in Ar pressure medium

 Dorogokupets and Oganov (2007): ruby vs semiempirical EoS of standard metals

 Jacobsen et al. 2008: ruby vs MgO in He pressure medium Ruby scales

167.54 )

3 167.52 168 (Å

V 167.50

) 167.48

3 167 (Å

V 164.44 2.00 2.04 2.08 2.12

166 ) P (GPa) 3

Qz (Å 164.40

Mao et al. (1986) V 165 Jacobsen et al. (2008) Dorogokupets and Oganov (2007)164.36

0 1 2 3 4 5 6.606 6.70 6.80 6.90 P (GPa) P (GPa) In practice Orthorhombic MgSiO3 perovskite Pbnm a- a- c+

in-phase tilting around [001]

out-of-phase tilting along [110] Volume compressibility 1.00 V0 = 162.36 (4) Å

K0 = 251 (2) GPa 0.95 K‘ = 4.11 (7)

V0 = 163.09 (6) Å 0 0.90 K0 = 253 (2) GPa V / V K‘ = 3.99 (7)

0.85 V0 = 168.93 (5) Å

K0 = 240 (2) GPa 0.80 K‘ = 4.12 (8) 0 20 40 60 80 Boffa Ballaran et al. 2012 P (GPa) Bulk moduli

700700 bc

600600 (GPa) (GPa) (GPa) K

K 500500

K 500

BM-III ab initio BM-III Bm-III BM-III (below 40 GPa) BM-III BM-III (below (below40 GPa) 40 GPa) 400400 BM-IV BM-IV Vinet BM-IV Vinet Vinet 3+ ab initio calculation for Fe AlO3

404040 606060 8080 100100100 120120120 P P(GPa) (GPa) F-f plot

Angel 2000

Normalised stress vs finite strain

 Visual diagnostic tool to determine which higher order term is significant in an EoS

 Can be applied to any isothermal EoS based upon finite strain Birch-Murnaghan EoS

2 3 1  V   f      1 IV order Birch-Murnaghan EoS 2 V   0  

dA df 5/ 2  3 3  35  2  P    3K0 f 1 2 f 1 K0 '4 f   K0K0"K0 '4 K0 '3   f  df dV  2 2  9  

P Normalised stress F  3 f 1 2 f 5 2

3 3  35 2 F  K0  K0 K0 '4 f  K0 K0K0"  K0 '4 K0 '3  f 2 2  9  F-f plot

260 (GPa) E F 250

240

230 Normalised stress,

-3 10 20 30 40 50 60 70x10

Eulerian strain, fE Axial compressibility

1.001.00 a b a K0 (a) : K0 (b) : K0 (c) c b c 0.980.98 0.81 : 1 : 0.82 0 0 s

0.960.96 0.83 : 1 : 0.69 unit-cell axis / axis unit-cell axis / axi 0.940.94 0.84 : 1 : 0.85

0.92 0.92 0 20 40 60 80 0 20 40 60 80 P (GPa) P (GPa) Lattice strain Pnma

-3 Z 55x10

a Pnma 50

bPnma 45 Y c X Pnma 40 shear strain e4 35

c 2  a0 a 2  a0 30 e4   a0 a0 0 20 40 60 80 Carpenter et al. 2006 P (GPa) References

 Angel, R.J. (2000) Equations of state. Reviews in Mineralogy and Geochemistry, 41, 35-59.  Duffy, T.S., Wang, Y. (1998) Pressure-Volume-Temperature Equations of State. Reviews in Mineralogy, 37, 425-457.  Jakson, I., Ridgen, S.M. (1996) Analysis of P-V-T data: constraints on the thermoelastic properties of high-pressure minerals. PEPI, 96,85-112.  Stacey, F.D., Brennan, B.J., Irvine, R.D. (1981) Finite strain theories and comparison with seismological data. Geophysical Surveys, 4, 189-232.  Fei, Y., Li, J., Hirose, K., Minarik W., et al. (2004) A critical evaluation of pressure scales at high temperatures by in situ X-ray diffraction measurements. PEPI, 143-144, 515-526.