Melting of Iron at Earth’s core conditions from quantum Monte Carlo free energy calculations

Department of Earth Sciences & Department of Physics and Astronomy,

Materials Simulation Laboratory & London Centre for Nanotechnology

University College London

Dario ALFÈ

• Birch (1952) - “The Core is iron alloyed with a small fraction of lighter elements”

2 4 6 8 10 12

4

6

8

10

Al Ti

Cr

Fe

Sn Ag

Mantle Core

V(km/s)

Density (g/cc)

• Nature of light element inferred from:

•Cosmochemistry

•Meteoritics

•Equations of state

Core composition

Temperature of the Earth’s core

• Exploit solid-liquid boundary

• Exploit core is mainly Fe• Melting temperature of Fe

Thermodynamic melting

F(V ,T ) = Fperf (V ,T) + Fharm(V ,T) + Fanharm(V ,T)

Fharm

(V ,T ) = 3kBT 1

Nq,s

ln 2sinhhωq,s(V ,T)

2kBT

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥q,s

∑

The Helmholtz free energy

Solids: Low T

Phonons of Fe

F(V ,T ) = Fperf (V ,T) + Fharm(V ,T) + Fanharm(V ,T)

Fharm

(V ,T ) = 3kBT 1

Nq,s

ln 2sinhhωq,s(V ,T)

2kBT

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥q,s

∑

The Helmholtz free energy

Solids:

Liquids:

F(V ,T ) = −kBT ln

1N!Λ3N

dR e−U (R)/kBT

V∫

Low THigh T

Fλ = −kBT ln

1N!Λ3N

dR e−Uλ (R)/kBT

V∫

U

ref, F

ref Uλ =(1−λ)Uref +λU

F − Fref = dλ

dFλ

dλ0

1

∫

dFλ

dλ=

dR ∂Uλ

∂λe−Uλ (R)/kBT

V∫

dR e−Uλ (R)/kBT

V∫

=∂Uλ

∂λλ

= U −Uref λ

Thermodynamic integration

F = Fref + dλ ⟨U0

1

∫ −Uref ⟩λ

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

F =Fharm+ dλ ⟨U

0

1

∫ −Uharm⟩λ F =Fharm+ dλ ⟨U

0

1

∫ −Uharm⟩λ =Fharm+ dtdλdt

U −Uharm( )λ0

T

∫

Improving the efficiency of TI

F = Fref + dλ ⟨U

0

1

∫ −Uref ⟩λ

F is independent on the choice of Uref , but for efficiency choose Uref such that:

⟨(U − Uref −⟨U −Uref ⟩)

2 ⟩

is minimum. For solid iron at Earth’s core conditions a good Uref is:

U

ref = c1Uharm + c2U IP

UIP=12

A

ri −rj

Bi≠j∑ ; B=5.86

U

harm =

12

uiαΦiα , jβujβiα , jβ∑

Improving the efficiency of TI (2)

U

ref = c1Uharm + c2U IP

At high temperature we find c1 = 0.2, c2 = 0.8

F independent on choice of Uref, but for efficiency choose Uref such that

is minimum.For liquid iron a good Uref is:

⟨(U − Uref −⟨U −Uref ⟩)

2 ⟩

Uref

=12

A

ri −rj

αi≠j∑

Free energy for liquid Fe:

F = Fref + dλ ⟨U

0

1

∫ −Uref ⟩λ

Liquid Fe

Uref

=12

A

ri −rj

Bi≠j∑

B=5.86

ΔT ≈100 K → ΔG ≈10 meV / atom

Size tests

0 0

1( )

2 H H Hp V V E E− = −

Hugoniot of Fe

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

NVE ensemble: for fixed V, if E is between solid and liquid values, simulation will give coexisting solid and liquid

Melting: coexistence of phases

QuickTime™ and aH.264 decompressor

are needed to see this picture.

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

Free energy approach and Coexistence give same result (as they should !)

Thermodynamic integration, a perturbative approach:

F = Fref + dλ ⟨U0

1

∫ −Uref ⟩λ

U −Uref λ= U −Uref λ=0

+λ∂ U −Uref λ

∂λλ=0

+o(λ2 )

∂ U −Uref λ

∂λ=

∂∂λ

dR U −Uref( )e−Uλ (R)/kBT

V∫

dR e−Uλ (R)/kBT

V∫

⎧

⎨

⎪⎪

⎩

⎪⎪

⎫

⎬

⎪⎪

⎭

⎪⎪

=

−1

kBT

dR U −Uref( )2e−Uλ (R)/kBT

V∫

dR e−Uλ (R)/kBT

V∫

−dR U −Uref( ) e

−Uλ (R)/kBT

V∫

dR e−Uλ (R)/kBT

V∫

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

2⎧

⎨

⎪⎪

⎩

⎪⎪

⎫

⎬

⎪⎪

⎭

⎪⎪

=−1

kBTδΔUλ

2

λ

U −Uref λ

= U −Uref λ=0−

λkBT

δΔU02

0+o(λ2 )

δΔUλ =U −Uref − U −Uref λ

dλ

0

1

∫ U −Uref λ; U −Uref λ=0

−1

2kBTδΔU0

2

0

Only need to run simulations with one potential (the reference potential for example).

Melting of Fe from QMC:

Free energy corrections from DFT to QMC:

δTm =ΔG ls (Tm

ref )

Srefls

QMC on Fe, technical details

• CASINO code: R. J. Needs, M. D. Towler, N. D. Drummond, P. Lopez-Rios, CASINO user manual, version 2.0, University of Cambridge, 2006.

• DFT pseudopotential, 3s23p64s13d7 (16 electrons in valence)

• Single particle orbitals from PWSCF (plane waves), 150 Ry PW cutoff. Then expanded in B-splines.

ΨT (R) =exp[J (R)]D↑{φi (r j )}D

↓{φi (r j )}

(D. Alfè  and M. J. Gillan, Phys. Rev. B, 70, 161101(R), (2004))

Blips

ψ n(r) = anj

j∑ Θ j (r)

Storing the coefficients: • avc(x,y,z,ib) [old]• avc(ib,x,y,z) [new]

New is faster on large systems, but slower on small systems (cutoff ~ 250 electrons)

Solid (h.c.p.) Fe, finite size Ester Sola

Solid Fe, equation of state at 300 K

QMC correction to the DFT Fe melting curve

δTm =ΔG ls (Tm

ref )

Srefls = 500 ± 200

ΔG ls (Tmref ) = 0.05 ± 0.02 eV/atom

QMC correction to the DFT Fe melting curve

δTm =ΔG ls (Tm

ref )

Srefls = 500 ± 250

ΔG ls (Tmref ) = 0.05 ± 0.025 eV/atom

Conclusions

• Melting temperature of Fe at 330 GPa = 6800 +- 400 K

• Melting point depression due to impurities ~ 800 K

• Probable temperature of the Earth’s core is ~ 6000 K