Pressure measurements at high temperature:

open issues and solutions

Peter I. DorogokupetsInstitute of the Earth’s Crust

SB RAS, Irkutsk, Russiador@crust.irk.ru

AcknowledgmentsArtem R. Oganov Lab. of Crystallography, ETH Zurich, Switzerland a.oganov@mat.ethz.chAgnes Dewaele CEA/DPTA Bruyeres-le-Chatel, France agnes.dewaele@cea.frPaul Loubeyre CEA/DPTA Bruyeres-le-Chatel, France

This work was supported by the Russian Foundation for Basic Research, Grant No. 05-05-64491.

Outline:

Intro.

Thermodynamics: EoS formulation

Best form of the ruby scale

EoS and thermodynamic behavior of Au, C, MgO, NaCl B1, NaCl B2, -Fe

Cross-check of EoS

Conclusion

IntroDorogokupets P.I., Oganov A.R. Ruby pressure scale: revision and alternatives // in Proceedings Joint 20th AIRAPT & 43th EHPRG Int. Conf. on High Pressure Science and Technology, June 27 to July 1, 2005, Karlsruhe, Germany (Forschungszentrum Karlsruhe, Karlsruhe, 2005). Дорогокупец П.И., Оганов А.Р. Уравнения состояния Al, Au, Cu, Pt, Ta и W и пересмотренная рубиновая шкала давлений // ДАН. 2006. Т. 410. № 2. 239–243. Dorogokupets P.I., Oganov A.R. Equations of State of Al, Au, Cu, Pt, Ta, and W and Revised Ruby Pressure Scale // Doklady Earth Scinces. 2006. V. 410. 1091-1095.Dewaele A., Loubeyre P., Occelli F., Mezouar M., Dorogokupets P.I., Torrent M. Quasihydrostatic equation of state of iron above 2 Mbar // Phys. Rev. Letters. 2006. V. 97. Art. No. 215504. Dorogokupets P.I., Oganov A.R. Ruby, metals, and MgO as alternative pressure scales: A semiempirical description of shock-wave, ultrasonic, x-ray, and thermochemical data at high temperatures and pressures // Phys. Rev. B 2007

ThermodynamicsHelmholtz free energy

),(

),(

),(

),(

)(0

TVF

TVF

TVF

TVF

VE

UF

def

el

anh

qh

U0 is the reference energy

E(V) is the cold part

Eqh(V,T) is the quasiharmonic part

Eanh(V,T) is the intrinsic anharmonicity

Eel(V,T) is the electronic contribution

Edef(V,T) is the thermal defects

Cold energy (Vinet form)

V

VPVKVK

V

VEVP

dP

dKxVVy

eyVKVE

T

y

d

)(d)(

d

)(d)(

12

3,)/(

,))1(1(11

9)(

3/13/10

)1200

Total quasi-harmonic energy:

Kut’in model

Einstein model

Kut’in model:see Kut’in et al.Rus. J. Phys. Chem. 72, 1567, 1998

11lnexp

11lnR

Tdd

TmFB

Bqh

0

0.5

1

1.5

2

2.5

3

0 50 100 150 200

Temperature, K

He

at

cap

aci

ty

d =1

d =10

d =3

Intrinsic anharmonicity(Oganov, Dorogokupets, 2004)

.

12

1

6R

22/

/2

2

/

,,2

1

,,

,

,,

,

Te

e

e

axmF

T

T

T

T

jiji

m

jijianh

ji

jiji

ji

Electronic contribution(Zharkov, Kalinin, 1971)

.R2

3

R2

3),(

20

2

Txen

eTnTVF

g

el

Thermal defects contribution

T

HxSxTnF

hf

def expR2

3

Thermodynamic functions

S = –(F/T)V, E=F + TS,

P = –(F/V)T, H=E+PV, G=F+PV,

CV = (E/T)V, KT = –V(P/V)T,

(P/T)V = KT,

CP=CV+2TVKT, KS=KT+VT(KT)2/CV,

Hugoniot pressure

xx

EVEV

VPPH

2)1(1

])([)( 0

We use input data are unbiased by calibration

22 parameters to fit!

At zero pressure:Heat capacity and enthalpy

Thermal expansion coefficient or volume

Adiabatic bulk modulus (from ultrasonic measurements)

Temperature interval:

from 10 K to melting temperature

At high P-T:Shock wave data

Room T isotherms obtained after fitting:

Compared with static compression data with Mao 86 ruby calibration(A=1904, B=7.665)

Compared with static compression data with new ruby calibration (A=1885, B=10.4)

Best ruby pressure scale

0

0

/5.51

/1884

P

Aleksandrov form

Use of all available data

At zero pressure:Heat capacity and enthalpy Thermal expansion coefficient or volumeAdiabatic bulk modulus (from ultrasonic measurements)

Temperature interval:from 10 K to melting temperature

At high P-T:Shock wave dataPV and PVT measurements (at later stages of refinement)

Results

With our formalism we carry out a simultaneous processing of all the available measurements of the Cp, α, V, Ks and KT at zero pressure, static measurements of V on a room-temperature isotherm and at higher temperatures, shock-wave data, and calculate thermodynamic functions vs. T and P.

Ag, Al, Au, Cu, Pt, Ta, W, Mo, Pb, Fe, MgO, diamond, NaCl EoS have been calculated.

See Dorogokupets, Phys. Rev B, 2007

Comparison of calculated EoS and thermodynamic parameters with data

Au, heat capacity

-2

-1

0

1

2

0 50 100 150 200 250 300Temperature, K

(CP

exp-

CP

cal)/

CP

exp*

10

0, %

Geballe & Giauque (1952) Touloukiand & Buyco (1970)

S298=47.35 J/(mol K)AuB

Au, thermal expansion

3.E-05

4.E-05

5.E-05

6.E-05

7.E-05

100 300 500 700 900 1100 1300

Temperature, K

, 1/K

Touloukian et al (1975)

Novikova (1974)

AuC

Au, bulk moduli

120

130

140

150

160

170

180

0 300 600 900 1200

Temperature, K

Bul

k m

odul

i, G

Pa

Neighbours & Alers (1958) Chang & Himmel (1966) Collard & McLellan (1991) Shim et al (2002) Holzapfel et al (2001)

KS

KT

AuD

Au, 300 KK0=166.7 GPa, K′=6

-7

-5

-3

-1

1

3

0 20 40 60 80 100 120

Pressure (GPa)

P

(G

Pa

)

Dew aele et al (2004) Heinz & Jeanloz (1984) Takemura (2001) d111 Takemura (2001) d200 Chijioke et al. (2005) H2 Yagi et al. (2004)

Au

Au, 300 KK0=166.7 GPa, K′=6

-7

-5

-3

-1

1

3

0 50 100 150Pressure (GPa)

P

(G

Pa

)

Jamieson et al (1982) SWRI Anderson et al (1989) Chijioke et al (2005) SWRI Wang et al (2002) SWRI Shim et al (2002)Tsuchiya, 2003Boettger, 2003 Greeff & Graf (2004) Souvatzis et al. (2006)

Diamond, 300 K K0=443.16 GPa, K′=3.777

Diamond

-12

-10

-8

-6

-4

-2

0

2

0 50 100 150 200Pressure, GPa

P, G

Pa

Gillet et al (1999), original

Occelli, original, Mao ruby

Occelli, original, DO ruby

Diamond, heat capacity

5

10

15

20

25

30

300 700 1100 1500 1900 2300 2700

Temperature, K

CP,

CV,

J/(m

ol K

)

Gurvich et al. (1979) Robie et al. (1978) Victor (1962) Reeber & Wang (1996), Cv

Diamond, bulk moduli

350

370

390

410

430

450

0 500 1000 1500 2000 2500 3000Temperature, K

Bul

k m

odul

i, G

Pa

Zouboulis et al (1998)

McSkimin & Andreatch (1972)

KSKT

4.1

4.3

4.5

4.7

4.9

5.1

5.3

5.5

5.7

50 100 150 200 250 300 350Pressure (GPa)

Vol

ume

(cm

3 )Jephcoat et al. (1986)

Brown et al. (2000)

Mao et al. (1990)

Dewaele et al. (2006)

7000 K

5000 K

iron

-5

0

5

10

15

0 100 200 300Pressure, GPa

P

, GP

aDew aele 2006 He ruby DODew aele 2006 He W DODew aele 2006 Ne W DOBrow n, linearBrow n, quadraticBrow n, exp

iron

MgO, 300 K K0=160.3 GPa, K′=4.18

-3

-2

-1

0

1

2

3

0 10 20 30 40 50 60

Pressure, GPa

P

, GP

aSpeziale et al (2001) Mao rubyDewaele et al (2000) 300 KDewaele et al (2000) T>1750 KSpeziale et al (2001) DO rubyLi et al. (2006)

MgO, bulk moduli

-3

-2

-1

0

1

2

0 10 20 30 40 50 60

Pressure, GPa

K

S, %

Merkel et al (2002)

Zha et al (2000)

Sinogeikin and Bass (2000)

Li et al. (2006)

MgO, bulk moduli

80

100

120

140

160

0 500 1000 1500 2000 2500 3000

Temperature, K

Bu

lk n

od

uli,

GP

a

Isaak et al. (1989)Sumino et al. (1983)Anderson and Andreath (1966)Sinogeikin et al. (2000)Zouboulis & Grimsditch (1991)

KSKT

MgO, K0=160.3 GPa, K′=4.18

-2

-1.5

-1

-0.5

0

0.5

0 2 4 6 8 10Pressure (GPa)

P (

GP

a)

300 K 373 K 473 K

573 K 673 K 773 K

873 K 973 K 1073 K

1273 K 1673 K

Utsumi et al. (1998)

MgO, Zhang data fittedK0=161 GPa, K′=1.84

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

300 600 900Temperature (K)

P

(G

Pa

)

2.6 GPa

5.4 GPa

8.2 GPa

Zhang (2000)

NaCl B1, RT-isothermK0=23.9 GPa, K′=5.13

-1.5

-1

-0.5

0

0.5

1

0 5 10 15 20 25 30

Pressure, GPa

P

(G

Pa)

Decker (1971) Brown, 1999 Fei, 1999, from MgOShock data Vaidya, Kennedy, 1971 Fritz et al. (1971) 300 K Birch, 1986Fritz et al 1971Ahrens & Johnson 1995 H

NaCl B1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 5 10 15 20 25 30

Pressure, GPa

P

, GP

a

Decker (1972), 1073K Brown, 1999, 1100 K Birch, 1986, 773 K Fei, 1999, from MgO, 1100 К

NaCl B1

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0 1 2 3

Pressure, GPa

P

(G

Pa)

298 373473 573

673 773

Boehler & Kennedy (1980)

NaCl B1, bulk moduli

10

15

20

25

0 300 600 900Temperature, K

Bul

k m

odul

i, G

Pa

Yamamoto et al., 1987Fugate, Schuele, 1966Spetzler et al., 1972Slagle, McKinstry, 1967

KS

KT

NaCl B1, bulk moduli

2535455565758595

105115125135

0 5 10 15 20 25 30Pressure, GPa

KS,

GP

a

Kinoshita et al., 1979Mueller et al., 2003Frankel et al., 1976thisMorris et al., 1976

NaCl B2, RT-isothermK0=37.04 GPa, K′=4.99

10

20

30

40

50

60

70

80

13 15 17 19 21

V, A

Pre

ssu

re (

GP

a)

Sata MgO SpezialeSata Pt HolmesOno Au AndersonOno Au DOHJ1984FritzAltshulerThielMarsh 1Marsh 2TruninFei (2006) AGU

B1 Hugoniot

B2 Hugoniot

B2 300 K

B1 300 K

NaCl B2, RT-isothermK0=37.04 GPa, K′=4.99

0

20

40

60

80

100

120

140

11 13 15 17 19 21 23 25 27V, A

Pre

ssu

re (

GP

a)

Sata MgOSpezialeSata Pt Holmes

Ono Au Anderson

Ono Au DO

Hugoniot

300 K

Altshuler

Thiel

Marsh 1

Marsh 2

Trunin

Fei (2006) AGU

Two materials are compressed together in a high pressure/high temperature apparatus and their V is measured

Pressure given by their EoS are compared

If same pressure, validation of the EoS

Cross-check between EoS at high T

Comparison NaCl B2 and -Fe

20

30

40

50

60

70

80

20 30 40 50 60 70 80Pressure (GPa) e-Fe Dewaele 2006

Pre

ssu

re (

GP

a)

Na

Cl B

2

Seagle et al 2006 300 K

Seagle et al 2006 ~2000 KSeagle 2006 eFe Mao NaCl Fei

Seagle 2006 original

Within ~7GPa

Au-MgO: Inoue et al. (2006) Phys. Chem. Minerals 33, 106.

10

15

20

25

10 15 20 25

Pressure MgO (GPa)

Pre

ssu

re A

u (

GP

a)

300 K

1473 K

1573 K

1673 K

1773 K

K. Litasov et al. EPSL 238 (2005) 311

21

22

23

24

25

21 22 23 24 25

Pressure MgO (GPa)

Pre

ss

ure

Au

(G

Pa

) 1273

1473

1673

1873

20

22

24

26

20 22 24 26Pressure MgO (GPa)

Pre

ssu

re A

u (

GP

a)

Matsui and Nishiyama (2003)

Nishiyama et al (2004)

b

1873 K

Fei et al. (2004). PEPI, 143-144, 515

8

10

12

14

16

18

20

22

24

26

8 10 12 14 16 18 20 22 24 26

Pressure MgO (GPa)

Pre

ss

ure

Au

(G

Pa

)

1273 1473

1673 1873

2073 2173

21

22

23

24

25

26

21 22 23 24 25 26

Pressure MgO (GPa)

Pre

ssu

re A

u (

GP

a)

1673

1873

2023

2173

10

15

20

25

30

10 15 20 25 30

Pressure Pt (GPa)

Pre

ssu

re A

u (

GP

a)

300

1473

1673

1873

MgO and Au EoS are within ~1 GPa at P<30 GPa, T<2200K

Hirose et al. (2006). Geophys. Res. Lett. 33, L01310.

90

95

100

105

110

115

120

90 95 100 105 110 115 120

Pressure MgO Speziale et al. (2001) (GPa)

Pre

ssu

re A

u (

GP

a) MgO and Au DO

Au Tsuchiya (2003)

Au Fei et al. (2004) Au Shim et al. (2002)

2330 K

300 K

1340 Kb

Hirose et al. (2006)

MgO and Au EoS are within ~3 GPa at P<120 GPa, T<2300K

ConclusionsWe have proposed a ruby pressure scale based on precise measurements of Dewaele et al. [2004, 2006]. The obtained ruby pressure scale agrees within 2% with the most recent ruby pressure scales.Our EoSs of Al, Au, Cu, Pt, Ta, W, MgO, C, NaCl are consistent with shock-wave and X-ray data and with numerous measurements of the heat capacity, volume, adiabatic bulk moduli, etc. at zero pressure.The EoSs of Au and Pt agree with the EoSs of Ag and MgO, constructed on independent measurements. The obtained P-V-T EoSs enable consistent pressure measurement using EoSs of any of the reference substances (Ag, Al, Au, Cu, Pt, Ta, W, MgO). This solves problems of inconsistency between different pressure scales and enables accurate pressure measurement at elevated temperatures, where the ruby scale cannot be used.