Equations of State. Tiziana Boffa Ballaran

Transcription

1 Equations o State iziana Boa Ballaran

2 Why EoS? he 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 o the constituent materials 4 v S G v S G S S v

3 Outline Overview o ew equations o state: rinciples and assumptions Hydrostaticity and pressure scales Working example

4 A Helmholtz ree energy, A A, A, c vib compression curve vibrational contribution electronic contribution el A,,, c vib el reerence curve thermal pressure Higher order derivatives o the energy unction etc...

5 Finite strain theory Ininitesimal elasticity: strains are determined by stresses and are reversible strains are small and their squares and products are negligible Reerence state: unstrained state (Lagrangian) deormed state, i.e. volume at inal compression (Eulerian) incremental strain (Hencky) hese EoS are empirical!

6 Birch-Murnaghan EoS Helmholtz ree energy: A A A A I order Birch-Murnaghan EoS Lagrangian Stacey et al. (98) Determination o the A coeicients at = with =, = and = 5/ 9 5 ' 4 ' " 4 ' d d d da

7 Birch-Murnaghan EoS Assumptions: Eulerian strains under hydrostatic compression strictly derived only or isotropic or cubic materials solid under compression is homogeneously strained EoS are continuously dierentiable higher order terms o the aylor expansion are negligible

8 Birch-Murnaghan EoS 5/ 9 5 ' 4 ' " 4 ' d da lausibility in the limit o large compression: " ' ' negative / " ' he behaviour o is a valuable test o the plausibility o a EoS

9 Logarithmic EoS Hencky strain: l d l ln l l he integration is possible only i the principal axes do not rotate during deormation H ln Hydrostatic compression oirier and arantola 998

10 Logarithmic EoS A A A A H ln with na d da d d d da N n n n ln oirier and arantola 998 Determination o the a coeicients at = with =, = and = ln ' ln III order

11 Interatomic potential Forces between atoms in the crystal are deined with assumed interatomic potential calculation o density as a unction o pressure EoS Empirical approach as or strain EoS

12 Born-Mie potential n m r b r a r E ) ( attractive orces repulsive orces m n m n oirier

13 Born-Mie potential Dierent values o m and n have been used in the literature (Stacey et al. 98) m n m n Special case: m = and n = II order Birch-Murnaghan EoS " ' ' lausibility:

14 inet EoS br e ar A Empirical potential: A, a, b constants depending on the material ' exp inet et al. 987, oirier Good or very high compression o metals and ionic solids

15 hermal pressure,, c vib

16 -- Birch-Murnaghan EoS 98,,... (),,... Isotherm at room temperature Isotherm at any given temperature and ( ) ( )exp ( ) d with ( ) a b ( ) ( ) linear variation with Usual assumption: is the same at all

17 hermal pressure th c,, th th th d,, (Jackson and Ridge, 996) (Anderson et al., 989) Simplest case: independent o temperature and volume th More generally: th ddt ln, 4 ln, a a a a th

18 hermodynamic EoS hermal expansion at atmospheric pressure: (, ) Compression at : (,) Compression at high temperature: (,),,, b X th i i, X i 5/ i, E E with b and b 9 ' 4 8 III order Birch-Murnaghan EoS or reerence isotherm X X 4 ln X X 5 6

19 Mie-Grüneisen EoS Lattice dynamics approach: the thermal pressure is related to the vibrational density o state th E th, E, th Grüneisen parameter: ( ) q q d d ln ln const. hermal ree energy: E th 9nR / e d ) exp ( ) ( q and independent o temperature, unctions o volume only

20 Experimental details

21 ressure medium Requirements Hydrostatic Does not dissolve sample Does not penetrate sample Non-hydrostatic stresses Cannot be measured accurately Reduce the data quality Result in dierent EoS aect structural phase transitions

22 Broadening o diraction peaks revious max Hydro ressure Diraction broadening (Ga) Non-hydrostatic stresses give Methanol:Ethanol.4 rise to broadening o the. diraction proiles Isopropanol 4..9 Silicone oil 5-7 <. Nitrogen. Argon 9 Fluorinert 5- Angel et al. 7 J Appl Cryst 4:6

23 Gas loading He and Neon No broadening observed so ar up to Ga with a point detector. BGI 75 Ga (ESRF)

24 ressure scales Single-crystal room up to Ga: quartz crystal (Angel et al. 997) Single-crystal and powder room any pressure: ruby luorescence owder H and H: EoS o metals like Au, t, Cu, Ag... EoS o non metal like MgO, NaCl... (Fei et al. 4) Single-crystal H and H:?... EoS o Neon and He

25 Ruby scales he ruby scales are based on the calibration o the shit o the ruby R luminescent line and the speciic volume o some standard material with increasing pressure Mao et al. (996): ruby vs Cu in Ar pressure medium Dorogokupets and Oganov (7): ruby vs semiempirical EoS o standard metals Jacobsen et al. 8: ruby vs MgO in He pressure medium

26 Ruby scales (Å ) (Å ) (Å ) Qz Mao et al. (986) Jacobsen et al. (8) Dorogokupets and Oganov (7) (Ga).8. 4 (Ga) (Ga) 6.9

27 In practice

28 Orthorhombic MgSiO perovskite bnm a - a - c + in-phase tilting around [] out-o-phase tilting along []

29 ..95 olume compressibility = 6.6 (4) Å = 5 () Ga = 4. (7) /.9.85 = 6.9 (6) Å = 5 () Ga =.99 (7) = 68.9 (5) Å = 4 () Ga.8 = 4. (8) 4 (Ga) 6 8 Boa Ballaran et al.

30 Bulk moduli 7 7 bc 6 6 (Ga) (Ga) ab initio BM-III BM-III (below 4 Ga) BM-I inet BM-III Bm-III (below 4 Ga) BM-I inet BM-III BM-III (below 4 Ga) BM-I inet ab initio calculation or Fe + AlO (Ga)

31 F- plot Angel Normalised stress vs inite strain isual diagnostic tool to determine which higher order term is signiicant in an EoS Can be applied to any isothermal EoS based upon inite strain

32 Birch-Murnaghan EoS I order Birch-Murnaghan EoS 5/ 9 5 ' 4 ' " 4 ' d d d da 5 F 9 5 ' 4 ' " 4 ' F Normalised stress

33 F- plot Normalised stress, F E (Ga) x - Eulerian strain, E

34 unit-cell axis / axis Axial compressibility a b c a b c (a) : (b) : (c).8 : :.8.8 : : : : (Ga) (Ga) 8 8

35 nma Lattice strain Z 55x - X b nma a nma c nma Y shear strain e e 4 c a a a a a Carpenter et al. 6 (Ga)

36 Reerences Angel, R.J. () Equations o state. Reviews in Mineralogy and Geochemistry, 4, Duy,.S., Wang, Y. (998) ressure-olume-emperature Equations o State. Reviews in Mineralogy, 7, Jakson, I., Ridgen, S.M. (996) Analysis o -- data: constraints on the thermoelastic properties o high-pressure minerals. EI, 96,85-. Stacey, F.D., Brennan, B.J., Irvine, R.D. (98) Finite strain theories and comparison with seismological data. Geophysical Surveys, 4, 89-. Fei, Y., Li, J., Hirose,., Minarik W., et al. (4) A critical evaluation o pressure scales at high temperatures by in situ X-ray diraction measurements. EI, 4-44,