1. Composition of the Earth’s core from ab-initio calculation of chemical potentials Department of Earth Sciences & Department of Physics and Astronomy, Thomas Young Centre@UCL & London Centre for Nanotechnology University College London Dario ALFÈ, M. J. Gillan, G. D. Price

4. The Earth’s core is the seat of major global processes. Convection in the outer core generates the Earth’s magnetic field. Heat flow from the core helps drive Mantle convection. The importance of the core Temperature of the core ? Composition of the core ?

5. Birch (1952) - “The Core is iron alloyed with a small fraction of lighter elements” Nature of light element inferred from: Cosmochemistry Meteoritics Equations of state Core composition

6. As pointed out by Poirier (1994), the favoured light element has varied with time, and the strength of the personalities involved! On this basis, S, O, Si, H and C are the primary candidates. Ni, K, etc also possible in core. Which light elements?

7. Density change at ICB ~ 5 % (seismological data). Density change on melting for Fe ~ 1.7 % (from ab- initio calculations).  Partition of light elements. Strategy to constrain the composition of the Earth’s core

8. Solid-liquid equilibrium Binary mixture, solvent A, solute X Equality of chemical potentials Note: T m is different, in general, from the melting temperature T m 0 of the pure solvent

9. Low concentration To obtain an equation for T m we need the corresponding expansion for  A. Gibbs-Duhem:

10. Shift of melting temperature:

11. Summary of equations:

12. Calculating chemical potentials Awkward to add one atom. Quantities to compute: Easier to transmute one solvent atom into a solute atom. This provides difference of chemical potentials:

13. Density Functional Theory: Schrödinger equation: ? Quantum mechanics

14. Calculating  0 for the pure solvent Solids: Low T

15. Example: phonons of Fe PHON code, freely available at: http://chianti.geol.ucl.ac.uk/~dario D. Alfè Comp. Phys. Comm. 180 2622 (2009)

16. The Helmholtz free energy Solids: Liquids: Low T High T

17. Thermodynamic integration

18. Example: anharmonic free energy of solid Fe at ~350 GPa

19. F independent on choice of U ref, but for efficiency choose U ref such that is minimum. For liquid iron a good U ref is: Free energy for liquid Fe:

20. Liquid Fe

21. Improving the efficiency of TI F is independent on the choice of U ref, but for efficiency choose U ref such that: is minimum. For solid iron at Earth’s core conditions a good U ref is:

22. Improving the efficiency of TI (2) At high temperature we find c 1 = 0.2, c 2 = 0.8

23. Size tests

24. Hugoniot of Fe

25. Melting

26. Alfè, Price,Gillan, Nature, 401, 462 (1999); Phys. Rev. B, 64, 045123 (2001); Phys. Rev. B, 65, 165118 (2002); J. Chem. Phys., 116, 6170 (2002 ) The melting curve of Fe

27. NVE ensemble: for fixed V, if E is between solid and liquid values, simulation will give coexisting solid and liquid Melting: coexistence of phases

28. D. Alfè, Phys. Rev. B, 79, 060601(R) (2009)

29. Alfè, Price,Gillan, Nature, 401, 462 (1999); PRB, 64, 045123 (2001); PRB, 65, 165118 (2002); Alfè, PRB 79, 060601(R) (2009) The melting curve of Fe Laio et al., Science, 287, 1027 (2000) Belonoshko et al., PRL, 84, 3638 (2000)

30. Quantities to compute computed next...

31. “Alchemy”  Calculating  0l XA (liquid)

32. Example: Fe/O

33. Dependence on concentration: Monte Carlo simulation Calculating  0s XA (solid)

34. Results

36. SolidLiquid S/Si 8.5 ± 2.5%10 ± 2.5 % O 0.2 ± 0.1 %8 ± 2.5 % Alfè, Price, Gillan, Nature, 405, 172 (2000); GRL, 27, 2417 (2000); EPSL, 195, 91 (2002); JCP, 116, 7127 (2002) O helps compositional convection… Composition of the Earth’s core