Atomistic modeling of minerals and melts using advanced interatomic potentials

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Atomistic modeling of minerals and melts using
advanced interatomic potentials

• Sandro Jahn
• GeoForschungsZentrum Potsdam

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Atomistic simulations in Earth sciences

• minerals and melts of the Earths crust and
  mantle under extreme conditions of P/T
• complex oxides and silicates
• atomistic simulations allow
• direct access to the individual atoms and to the
  electronic structure
• easy access to extreme P/T conditions
• predictive power where experimental studies are
  difficult or not feasible
• need for accurate and transferable potentials

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Modeling ionic materials

• Coulomb interaction
• short-range repulsion
• dispersion (van der Waals)
• polarization
• spherical breathing, dipolar quadrupolar shape
  deformations

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Potential fitting

• Potential with 20-30 parameters
• Optimized by fitting to reference DFT calculations

Aguado et al, Faraday Discuss. 124, 171 (2003)
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Fit configurations

• Al2O3
• bixbyite, corundum, melt, ortho-perovskite,
  Rh2O3(II)
• MgO
• rocksalt, CsCl, sphalerite
• SiO2
• ?-quartz, ?-cristobalite, stishovite
• MgAl2O4
• spinel, Ca-ferrite, Ca-titanite
• MgSiO3
• ortho-perovskite, post-perovskite

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Potential fitting

• calculate ab initio forces, dipoles, quadrupoles
  and stress tensors for the different static
  configurations
• AIM potential with initial guess of parameters
• Minimize objective functions by variation of a
  set of parameters X and calculation of the AIM
  forces etc.
• A(X)1/N ?j (Yj(X)-YjAI)2 / (YjAI)2

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MD Simulations using the AIM

• potential contains 17 additional degrees of
  freedom from induced multipoles and ion shape
  deformations
• use conjugate gradient routines to search for
  ground state configuration of the electronic
  degrees of freedom before calculating the forces
  on the ions
• hence potential seen by the ions differs in each
  time step (many body character of interaction)

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Transferability test 1 Al2O3

• transferability of the potential to
• high pressures
• high temperatures
• different coordination environment
• examples
• bulk and surface properties of corundum
• high pressure polymorphs
• alumina melt

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P-V curve for corundum
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Thermal expansion of corundum
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Corundum phonons
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High pressure polymorphic phases
corundum stable up to 100 GPa
Rh2O3(II) structure (gt 100 GPa)
orthorhombic perovskite, possible high pressure
phase
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corundum (MgSiO3 akimotoite)
Rh2O3(II) structure (Al2O3)
orthorhombic perovskite (MgSiO3)
Jahn et al PRB 69, 020106 (2004)
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Enthalpy curves of Al2O3 polymorphsrelative to
corundum
DFT-GGA, AIM-GGA
DFT-LDA, AIM-LDA
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Corundum (0001) surface relaxation
before relaxation
after relaxation
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Corundum (0001) surface relaxation
atomic layer DFT-GGA (Ruberto et al, PRB 2003) AIM-LDA AIM-GGA
1st (Al-O) -85 -87 -85
2nd (O-Al) 3.2 5.8 5.0
3rd (Al-Al) -45 -42 -44
4th (Al-O) 20 20 19
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x-ray Ansell et al, PRL 78 (1997) 464 neutron
Landron et al, PRL 86 (2001) 4839
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Sound velocities in melts
Al2O3 melt full line simulation symbols
inelastic x-ray scattering experiment (Sinn et
al, Science, 299, 2047 (2003)
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Sound propagation in alumina melt
solid line experimental result vp7350
m/s (Sinn et al, Science, 299, 2047
(2003) symbols dispersions of longitudinal
modes from MD simulation
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5-6 coordinate 3-4 coordinate Presence of high
and low density liquid domains? Average
coordination number 4.5
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pressure dependence of g(r)
Rigid ion model Hoang et al, JPCM (2005)
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coordination under pressure
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self-diffusion under pressure
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Transferability test 2 Silicates

• transferability of the potential to
• different chemical compositions
• high P and T
• different coordination environment
• examples
• structure and elastic constants of spinels,
  Mg2SiO4 Al2SiO5 polymorphs
• SiO2 and MgSiO3 melts

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Elastic constants

• optimize cell parameters (T0)
• lattice parameters (a, b, c, ?, ?, ?)
• atomic positions
• finite cell strain
• calculate stress tensor in short MD simulation
  (T50 K)

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Lattice parameters and elastic constants of
spinel-type minerals
a/Å C11/GPa C12/GPa C44/GPa
MgAl2O4
Simulation 8.019 263 190 152
Experiment 8.083 282 154 154
?-Mg2SiO4
Simulation 7.982 357 128 130
Experiment 8.065 327 112 126
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Elastic constants of ?- and ?-Mg2SiO4
? (Simul.) ? (Exp.) ? (Simul.) ? (Exp.)
a/Å 4.732 4.753 5.637 5.698
b/Å 10.186 10.190 11.336 11.438
c/Å 5.947 5.978 8.235 8.257
C11/GPa 326 328 367 360
C22/GPa 188 200 367 383
C33/GPa 232 235 262 273
C44/GPa 62 67 95 112
C55/GPa 78 81 110 118
C66/GPa 82 81 104 98
C12/GPa 84 69 82 75
C13/GPa 82 69 103 110
C23/GPa 80 73 107 105
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Al2SiO5 polymorphs
andalusite
sillimanite
5 6 fold Al coordination
4 6 fold Al coordination
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Al2SiO5 polymorphs
Andalusite (Simul.) Andalusite (Exp.) Sillimanite(Simul.) Sillimanite (Exp.)
a/Å 7.662 7.798 7.42 7.488
b/Å 7.768 7.903 7.53 7.681
c/Å 5.472 5.557 5.73 5.777
C11/GPa 210 233 286 287
C22/GPa 262 289 214 232
C33/GPa 367 380 459 388
C44/GPa 82 100 94 122
C55/GPa 76 88 61 81
C66/GPa 108 112 73 89
C12/GPa 95 98 114 159
C13/GPa 119 116 108 83
C23/GPa 100 81 150 95
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Silica and silicate melt

• preliminary results
• SiO2 melt (1944 atoms, T2000K)
• molar volume (Vm40 Å3) about 11 lower than
  experimental value (45 Å3)
• MgSiO3 melt (2160 atoms, T2000K)
• molar volume 3 lower than experiment at T1913K

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Conclusion

• advanced interatomic potentials - a promising
  tool for atomistic simulation of minerals and
  melts
• defect structures in minerals at high P/T
• crystallization, melts, interfaces
• structural phase transitions, polymorphism
• transport properties, diffusion
• long term goal a set of accurate and
  transferable potentials for a wide range of
  minerals

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Acknowledgements

• Paul Madden (Edinburgh)
• Mark Wilson (London)
• Andres Aguado (Valladolid)
• Leonardo Bernasconi (Cambridge)

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MgO phonons at high P
B1 (NaCl) P400GPa B2 (CsCl)
P600GPa
lines DFT (A. R. Oganov et al, JCP
(2003)) symbols AIM (A. Aguado et al, PRB (2004))