Complementary approaches to high T- high p crystal structure stability and melting!

Transcription

1 Complementary approaches to high T- high p crystal structure stability and melting! Dario ALFÈ Department of Earth Sciences & Department of Physics and Astronomy, Thomas Young Centre@UCL & London Centre for Nanotechnology University College London

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3 Phase diagram of Fe Picture adapted from: Nguyen and Holmes, Nature, 427, 339 (2004)

4 Transition metals Old DAC melting of transition metals appears to show low melting curves for W, Ta, Mo, V, Ti, which all flatten above GPa (Errandonea et al., PRB 63, (2001)). Shock data suggest high melting temperatures, with spectacular disagreement for example for Ta and Mo. New DAC data for Ta appear to resolve this disagreement (Dewaele et al. PRL 104, (2010)). Here present ab initio methods to compute melting curves and solid-solid phase transitions, and show applications for Fe, Ta and Mo.

5 Theory Statistical mechanics Free energies Coexistence of phases Z method Interatomic interactions Empirical potentials Density functional theory Quantum Monte Carlo

6 Phase stability

7 The Helmholtz free energy Solids: Low T F(V,T ) = F (V,T ) + F (V,T ) + F (V,T ) perf harm anharm F harm (V,T ) = 3k B T 1 ( " ln 2sinh ω (V,T )% q,s * $ ' N q,s q,s # 2k B T )* & + -,- Dynamical matrix: D(q) = 1 M R Φ(R) eiq R Force constant ma trix: F α ( R) = Φ αβ ( R R') u β ( R') R', β

8 Phonons Small displacement method: Iron: freely available at: D. Alfè Comp. Phys. Comm (2009)

9 Phonon dispersions of Ta and Mo at p=0 Tantalum Molybdenum

10 Stability of Mo with the harmonic approximation Temperature (K) bcc fcc bcc hcp bcc fcc [8] bcc hcp [8] FCC is the stable high temperature phase at high pressure. Possible explanation of the Hixson s data: P(GPa) [8] Belonoshko et al., Phys. Rev. Lett. 100, (2008) What about anharmonicity?

11 The Helmholtz free energy Solids: Low T High T F(V,T ) = F (V,T ) + F (V,T ) + F (V,T ) perf harm anharm F harm (V,T ) = 3k B T 1 ( " ln 2sinh ω (V,T )% q,s * $ ' N q,s q,s # 2k B T )* & + -,- Liquids: F(V,T ) = k B T ln 1 N!Λ 3N V dr e U ( R)/k BT

12 Thermodynamic integration U ref, F ref U λ = (1 λ)u ref + λu F λ = k B T ln 1 N!Λ 3N V dr e U λ ( R)/k BT F F ref 1 = dλ df λ dλ 0 df λ dλ = V dr U λ λ e U λ ( R)/k BT V dr e U λ ( R)/k B T = U λ λ λ = U U ref λ F = F ref + dλ U 1 0 U ref λ

13 Thermodynamic integration F = F ref + dλ U 1 0 U ref λ F = F ref + 1 T dλ U U ref λ = F ref + dt dλ U U dt ref 0 0 ( ) λ M. Watanabe and W. P. Reinhardt, Phys. Rev. Lett. 65, 3301 (1990)

14 Example: anharmonic free energy of solid Fe at ~350 GPa T=1000 K T=3000 K T=6500 K

15 Example: anharmonic free energy of solid Fe at ~350 GPa F = F harm + T 0 ( ) λ dt dλ U U dt harm U harm = 1 2 iα, jβ u iα Φ iα, jβ u jβ Anharmonic contribution to the free energy is important at high T

16 Stability of Mo including anharmonicity F bcc (9.64,T) F fcc (9.64,T) ev/atom Anharmonic AI Harmonic AI 325 < P < 345 GPa F bcc (9.20,T) F fcc (9.20,T) ev/atom Anharmonic AI Harmonic AI 390 < P < 415 GPa T (K) T (K) BCC is more stable than fcc at high temperature C. Cazorla, D. Alfè and M. J. Gillan, PRB 85,

17 New shock data (courtesy of Neil Holmes, Nguyen, Chao, Asimow, LLNL) Maybe there is no discontinuity after all

18 Improving the efficiency of TI F = F ref + dλ U 1 0 U ref λ F is independent on the choice of U ref, but for efficiency choose U ref such that: (U U ref U U ref ) 2 is minimum. For solid iron at Earth s core conditions a good U ref is: U ref = c 1 U harm + c 2 U IP U harm = 1 2 iα, jβ u iα Φ iα, jβ u jβ U IP = 1 A ; B = 5.86 B 2 i j r i r j

19 Improving the efficiency of TI (2) U ref = c 1 U harm + c 2 U IP At high temperature we find c 1 = 0.2, c 2 = 0.8

20 Melting

21 Liquid Fe U ref = 1 2 i j A r i r j B B = 5.86

22 Size tests ΔT 100 K ΔG 10 mev / atom

23 Hugoniot of Fe 1 p ( V V ) = E E H H H

24 The melting curve of Fe Belonoshko et al., PRL, 84, 3638 (2000) Nguyen and Holmes, Nature, 427, 339 (2004) Alfè, Price,Gillan, Nature, 401, 462 (1999); PRB, 64, (2001); PRB, 65, (2002); Alfè, PRB 79, (R) (2009) Ma et al., PEPI (2004) Laio et al., Science, 287, 1027 (2000) Boehler, Nature (1993)

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

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

27 The melting curve of Fe

28 Melting of Fe from QMC: Free energy corrections from DFT to QMC: δt m = ΔGls (T ref m ) ls S ref

29 QMC on Fe, technical details CASINO code: R. J. Needs, M. D. Towler, N. D. Drummond, P. Lopez-Rios, CASINO user manual, version 2.1, University of Cambridge, DFT pseudopotential, 3s 2 3p 6 4s 1 3d 7 (16 electrons in valence) Single particle orbitals from PWSCF (plane waves), 150 Ry PW cutoff. Then expanded in B-splines. Ψ T (R) = exp[j(r)]d {φ i (r j )}D {φ i (r j )} (D. Alfè and M. J. Gillan, Phys. Rev. B, 70, (R), (2004))

30 Thermodynamic integration, a perturbative approach: F = F ref + dλ U 1 0 U ref λ U U ref λ = U U ref λ=0 + λ U U ref λ λ λ=0 + o(λ 2 ) 1 dλ U U U U ref λ ref 0 λ=0 1 2k B T δδu 2 0 λ=0 δδu λ = U U ref U U ref λ

31 QMC correction to the DFT Fe melting curve ΔG ls (T m ref ) = 0.05 ± 0.02 ev/atom δt m = ΔGls (T ref m ) = 550 ± 250 ls S ref

32 QMC correction to the DFT Fe melting curve ΔG ls (T m ref ) = 0.05 ± ev/atom δt m = ΔGls (T ref m ) = 550 ± 300 ls S ref

33 Melting curve of Fe E. Sola and D. Alfè, Phys. Rev. Lett, 103, (2009)

34 Strategy for melting of Ta and Mo: Coexistence of phases with classical potential: U ref (r 1,r 2,,r N ) = 1 2 ε i j " $ # a r ij % ' & n Cε i + - -, j( i) " $ # a r ij % ' & m. 0 0 / 1/2 Free energy corrections: δt m = ΔGls (T ref m ) ls S ref

35 Melting curves of Ta and Mo Tantalum Molybdenum Hixson et al., PRL 62, 637 (1989) Errandonea et al., PRB 63, (2001) S. Taioli, C. Cazorla, M. J. Gillan, and D. Alfè, Phys. Rev B 75, (2007), J. Chem. Phys. 126, (2007)

36 Phase stability of Mo from highest melting curve T m (K) BCC (EAM), Cazorla et al. BCC (AI), Cazorla et al. FCC (EAM), present work FCC (AI), present work FCC, Belonoshko et al P(GPa) C. Cazorla, D. Alfè and M. J. Gillan, PRB 85,

37 Theory Statistical mechanics Free energies Coexistence of phases Z method Interatomic interactions Empirical potentials Density functional theory Quantum Monte Carlo

38 Homogeneous melting (Z method) Belonoshko et al., PRB 73, (2006) What is the maximum energy E LS a solid can have in a (N,V,E) simulation before melting? It is the lowest energy on the given isochore within the field of thermodynamic stability of the liquid. This energy E LS corresponds to a temperature T LS above which the solid will always melt. Since E LS is the lowest energy of the liquid on the given isochore, it should be the energy of the liquid in coexistence with the solid, so that it is associated to the melting temperature T m, E s (V,T LS )= E l (V,T m )

39 Fe, EAM, 7776 atoms D. Alfe`, C. Cazorla and M. J. Gillan, J. Chem. Phys. 135, (2011)

40 D. Alfe`, C. Cazorla and M. J. Gillan, J. Chem. Phys. 135, (2011)

41 D. Alfe`, C. Cazorla and M. J. Gillan, J. Chem. Phys. 135, (2011)

42 D. Alfe`, C. Cazorla and M. J. Gillan, J. Chem. Phys. 135, (2011)

43 Fe, DFT, 150 atoms, 50 ps D. Alfe`, C. Cazorla and M. J. Gillan, J. Chem. Phys. 135, (2011)

44 Conclusions Methods for phase stability: if applied consistently give the same answer. Free energy Small systems if reference potential is good Access to thermodynamics (Human) labour intensive Coexistence Z Computer does most of the work Large systems Only melting Simple (but really?) Hundreds (thousands) simulations required Only melting

45 Acknowledgments Simone Taioli (BKF, Trento) Claudio Cazorla (ICMAB- CSIC, Barcelona) Mike Gillan (UCL) EPSRC (HECToR time allocation, UK) ORNL (JaguarPF time allocation, USA)