Atomistic potentials for metallic systems Recent advances and existing challenges

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Atomistic potentials for metallic systemsRecent
advances and existing challenges

• Y. Mishin
• George Mason University, Virginia, USA

CECAM Workshop Ab initio meets classical
simulations (Lyon, France) 10/18/2005
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Outline

• What is EAM?
• Functional form, properties and areas of
  application
• Potential generation procedures science or art?
• EAM success stories
• FCC metals
• Intermetallic compounds
• EAM challenges
• Difficult properties
• BCC transition metals
• Thermodynamic properties and phase diagrams
• Angular-dependent potentials
• MEAM
• Angular-dependent potentials (ADP)
• Future work

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Embedded-atom potential (EAM)
Daw and Baskes (1984) and Finnis and Sinclair
(1984)

• Potential functions contain adjustable parameters
• Classical (Newtonian) forces ? Energy
  minimization and MD
• Expression for unrelaxed Evf
• Expressions for elastic constants cij
• Expressions for the dynamical matrix ? Phonon
  dispersion and DOS
• Expressions for the stress tensor
• Fast calculation of ?E in Monte Carlo simulations

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The workhorse of atomistic simulations
Extremely fast calculation of energies and
forces!
Fast MD and Monte Carlo simulations (106 atoms,
10 ns)

• Mechanical properties
• Dislocations (core structure, Peierls stresses,
  dynamics)
• Grain boundaries (structure, energy, motion,
  segregation)
• Fracture cracks
• Diffusion
• Direct MD simulations (especially grain
  boundaries and surfaces)
• Combine harmonic TST with KMC (barriers by NEB)
• Thermodynamics
• Quasi-harmonic thermodynamics
• Grain-canonical Monte Carlo
• Calculation of phase diagrams etc, etc, etc

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Potential generation procedures Parameterization
of functions

• Elemental metal 3 functions, binary system 7
  functions, etc
• No physical meaning (Any function is good as
  long as it works)
• Forms of functions Vij(r) and ?i(r)
• Analytical smooth and reliable but lack
  flexibilitity
• Cubic splines very flexible but can give
  surprises
• Smooth cutoff with Rc covering 3-5 coordination
  shells
• Direct fit of F(?) or inversion of (modified)
  Roses UEOS E(V)
• Fitting minimization of the weighted mean
  squared deviation from target properties
• Weights give a powerful tool for controlling the
  quality of potential
• Multi-dimensional minimization (simplex method)
• Simulated annealing

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Potential generation procedures Fitting and
testing for elemental metals

• Experimental data
• E0, a0, cij, Evf, SF energy
• Sometimes Evm, Eif, surface energies, phonons,
  thermal expansion
• Ab initio data
• Structural energies E(V) for alternative
  structures
• Homogeneous deformation paths e.g. Bain path,
  trigonal path (FCC-SC-BCC), twinning deformation
  path (FCC-FCC or BCC-BCC)
• Forces in snapshots drawn from MD (solid and
  liquid). Force matching method (Ercolessi and
  Adams, 1994)

In modern potentials the ab initio part of the
database strongly dominates over experiment. Some
potentials use only 2-3 experimental numbers and
the rest ab initio
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Transferability of potentials
Ability to give reasonable results between and
beyond fitted points in configuration space.
This is the most meaningful measure of quality
of potentials
Equilibrium
???
????
??
Configuration space
Fitted points

• Ab initio data sample a larger area of
  configuration space than experiments can do
• Split the database into the fitting and testing
  sets!
• The more properties you test the more flaws you
  find

Potential generation is largely based on
experience, intuition, tricks of the trade, and
luck. It is currently art rather then science
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Transferability of potentials
Ability to give reasonable results between and
beyond fitted points in configuration space.
This is the most meaningful measure of quality
of potentials
Equilibrium
???
????
??
Configuration space
Fitted points

• Ab initio data sample a larger area of
  configuration space than experiments can do
• Split the database into the fitting and testing
  sets!
• The more properties you test the more flaws you
  find

Potential generation is largely based on
experience, intuition, tricks of the trade, and
luck. It is currently art rather then science
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Potential generation procedures Fitting and
testing for binary systems

• Typical scheme
• Take/generate accurate potentials for metals A
  and B
• Fit the cross-interaction function VAB(r)
• Use transformation coefficients as additional
  parameters
• For systems with compounds fit to (test against)
• Experimental properties of a chosen compound (E0,
  a0, cij, planar faults sometime surface
  energies, phonons, etc)
• Ab initio E(V) functions for a set of compounds
  with different structures and compositions across
  the diagram. Most of them do not exist on the
  diagram. One compound is not enough!
• Other ab initio data (e.g. point defect formation
  energies)
• For systems without intermediate phases (e.g.
  Cu-Ni, eutectic systems)
• Fit to ab initio E(V) functions for a set of
  non-existent compounds with different structures
  and stoichiometries across the diagram

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EAM success stories
Excellent potentials exist for several FCC metals
(e.g. Cu, Ag, Ni and Al) Accurately reproduce E0,
a0, cij, phonon dispersion curves, thermal
expansion, Evf, Evm, interstitials, SF energy,
gamma surfaces, structural energies,
high-pressure p(V), etc.
Melting properties (not used in the fit!) Cu Tm
1327 K (exper. 1357 K) Hm 12.1 kJ/mol
(exper. 11.5 kJ/mol) Ag Tm 1265 K (exper. 1235
K) Hm 12.2 kJ/mol (exper. 11.7 kJ/mol)
Thermal expansion
Cu
Ag
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Phonon dispersion curves for FCC metals
Cu
Ag
Similarly good agreement for Ni and Al
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EAM potentials for intermetallic compounds
Ti-Al L10-TiAl (Phys.Rev. B 68, 024102
(2003)) Ni-Al B2-NiAl (Phys.Rev. B 65, 224114
(2002)) Ni-Al Ni3Al (Acta Mater. 52, 1451
(2004))
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EAM potential for Ni3Al
Ni-Al phase diagram

• Accurate lattice properties of Ni3Al, B2-NiAl
• Thermodynamics and phase diagram
• Point defects and diffusion in Ni3Al
• Generalized stacking faults in Ni3Al
• ?/? interphase boundaries
• ? particles in ? matrix
• Dislocations in Ni3Al

110 screw dislocation in Ni3Al
?/? alloy at T 700 K
(100) ?/? interface at T 700 K
Nye tensor distribution
Y. Mishin, Acta Mater. 52, 1451 (2004)
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EAM challenges

• There are difficult properties which
  consistently defy EAM
• Surface energies are always too low (10-20)
• Vacancy migration energy is too low (unless
  included in the fit)
• Melting points are often way off experiment
  (unless liquid included in the fit)
• Binary phase diagrams are often incorrect even
  qualitatively. There have been only a few
  attempts to compute them

Cu-Ag interactions were fit to ab initio E(V)
functions of several compounds with different
structures and stoichiometries. No experimental
data! The diagram has been calculated by
grand-canonical Monte Carlo simulations
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EAM challenges (continued)
EAM does not work well for BCC transition metals.
This central-force model cannot capture the
covalent component of bonding

• Modified EAM (MEAM) (Baskes, 1987)
• Tensor electron density
• Short-range interactions (1-2 coordination
  shells)
• Screening procedure
• In principle can work for transition metals and
  even covalent solids (Si, Ge)
• Much slower than EAM
• No extensive use of ab initio data
• No extensive testing like for EAM-FCC

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Angular-dependent potentials (ADP)

• An extension of EAM to include non-central
  interactions
• Can apply to BCC transition metals
• Energy is penalized for dipole and quadrupole
  distortions

Angular-dependent terms
Regular EAM
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ADP method detail

• Potential functions
• EAM Vij, ?i, Fi
• metal 3 functions, binary 7 functions
• ADP Vij, ?i, Fi, Uij, Wij
• metal 5 functions, binary 13 functions
• Must be fit to a large ab initio database
• History
• First proposed for the Fe-Ni system (Acta Mater.
  53, 4029 (2005) )
• More general than the embedded-defect method
  (Pasianot et al, 1991)
• Similar to but much faster than the MEAM (Baskes,
  1987)
• Computational overhead about a factor of 2
  slower than EAM

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ADP fitting database Fe-Ni system

• Pure Ni and Fe
• Experimental a0, E0,cij, Ev, ?SF(Ni), ?s, for
  FCC-Ni and BCC-Fe
• Ab initio energies E(V) of FCC, BCC, SC and
  Diamond
• Fe-Ni alloys
• Ab initio energies E(V) of L12-Fe3Ni, D03-Fe3Ni,
  L10-FeNi, B2-FeNi, B1-FeNi, L12-Ni3Fe. No
  experimental data!

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Structural energies of Fe-Ni alloys
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ADP potential for BCC Ta

• Accurate lattice properties, including elastic
  constants, thermal expansion, high-pressure
  behavior
• Defects vacancy formation and migration, surface
  energies (!)
• Structural energies, including A15, omega, ?-U
• Reasonable agreement with ab initio for
• homogeneous deformation paths
• Gamma surfaces

High-pressure EOS
Homogeneous twinning path
With A.Y. Lozovoi (QUB)
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Gamma surfaces for BCC Ta
lt100gt110
lt110gt110
lt111gt211
lt111gt110
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½lt111gt screw dislocation core in Ta
Contour plot of the screw component of the Nye
tensor
Computed with the new ADP potential
The compact core with mirror symmetry across
(211) planes is in agreement with ab initio
calculations (Rao and Woodward, 2002)
Working on the Cu-Ta system
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Future work

• Application of EAM potentials for binary phase
  diagram construction. This might be more accurate
  than CE calculations
• Development of ADP potentials for binary systems
  involving BCC transition metals
• Potentials for ternary systems almost
  unexplored area
• Global potential database (at least for metallic
  systems). Possible problems
• Uniform format of potential files
• Uniform testing procedures (which properties to
  test, which agreement is considered good?)
• Quality control only high-quality potentials
  can be included