Simulating structure and physical properties of complex metallic alloys

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Simulating structure and physical properties of
complex metallic alloys

• Hans-Rainer Trebin
• Peter Brommer
• Michael Engel
• Franz Gähler
• Stephen Hocker
• Frohmut Rösch
• Johannes Roth

Institut für Theoretische und Angewandte Physik
der Universität Stuttgart Euroschool Ljubljana
26 May 2007
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Outline

• Numerical simulations of matter
• Classical molecular dynamics simulations and IMD
• Model potentials
• Realistic potentials
• potfit for EAM
• Simulations of physical properties (also CaCd6!)

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1. Numerical simulations of matter
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Basic equation for a solid
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Basic equations continued
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Ab-initio molecular dynamics
d-Al-Cu-Co
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2. Classical molecular dynamics simulations and
IMD
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Intention

• Calculate
• Motion of particles in a many-body system under
  specified interactions in classical approximation

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Step 1

• Fix structure in the form of particle coordinates
• Random
• Model structure
• From real experiment

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Step 2

• Determine interactions
• From electronic structure calculations
  (ab-initio, force matching)
• Model potentials
• Potentials two-, three-, many-body

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Step 3

• Fix boundary conditions
• Open boundaries
• Periodic boundaries
• Fixed boundaries
• Other (spherical, twisted, Lees-Edwards)

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Step 4

• Solve Newtons equations
• Discretize in time
• Choose integrator

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Atomistic Simulations
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Verlet and Leap-frog algorithm
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Step 5

• Influence from outside
• Temperature (numerical thermostats)
• Pressure (numerical barostats)
• Stress, strain, flow

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Nosé-Hoover thermostat
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Volume control barostat
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Lees-Edwards boundary conditions
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Step 6

• Extract data
• Potential energy, kinetic energy, total energy
• Free energy
• Total and local stress
• Displacement fields
• Elastic coefficients
• Transport coefficients (diffusion, heat)
• Correlation functions (static, dynamic)
• Diffraction pattern (static, dynamic)

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Step 7

• Visualize data
• Direct plot of atoms
• Color code for observables
• Selective visualization in 3d
• Animations

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Equilibrium problems

• Grain boundary structures

• Phason dynamics

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Nonequilibrium problems

• Plastic deformation
• Fracture
• Shock waves

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IMD (ITAP Molecular Dynamics)

• Molecular dynamics program package
• Established 1996, continuously improved and
  extended since
• Easily portable and extendable
• Workstations, clusters, massively parallel
  supercomputers
• Parallelized with Message Passing Interface
• Many effective potentials applicable
• Simple integrators (Verlet, Leap-frog) with
  energy stability over long times
• Timesteps a few fs, computation time a few µs
  for each time step and atom
• Scalable up to thousands of CPUs
• Available at
• http//www.itap.physik.uni-stuttgart.de/imd

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World records in particle numbers
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World records in particle numbers
IBM BlueGene/L, 65.536 nodes with 2 IBM PowerPC
440 processors, 360 Tflop/s
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3. Model potentialsa) Lennard-Jones
potentialb) Dzugutov potentialc)
Lennard-Jones-Gauss potential
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Lennard-Jones pair potential
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Structures with LJ potentials
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Ground state for LJ-potentials
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Dzugutov potential
bcc fcc s-phase dodecagonal qc - glass
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Lennard-Jones-Gauss potential
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Phase diagram with LJG potential
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Diffracton pattern decagonal random tiling
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Phase transition quasicrystal - crystal
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4. Realistic potentialsa) Advanced pair
potentialsb) Embedded Atom potentialsc) MEAM,
ADP etc.
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Advanced two-body potentials
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Embedded-atom potentials (EAM)
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5. potfit for EAM
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Embedded-atom potentials (EAM)
potfit Fi, fj, Fij arbitrary
spline-interpolated, parameterized functions with
about 10 nodes
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potfit

• Select reference configurations of a small system
• Determine forces between particles, energies and
  stresses from quantum mechanics (VASP)
• Calculate same data from MD with EAM potentials
• Sum squares of differences
• Minimize squares by parameter fits Force
  matching
• Minimization first by Simulated Annealing, then
  by Conjugate Gradients

P. Brommer and F. Gähler 2007 Potfit effective
potentials from ab-initio dataModelling Simul.
Mater. Sci. Eng. 15 (3) 295-304
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6. Simulations of physical propertiesa)
Diffusion in d-Al-Ni-Cob) Dynamical structure
factor for Zn2Mgc) Cracks in NbCr2d)
Order-disorder transition in CaCd6
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i) Diffusion in d-AlNiCo
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EAM potential for Al-Ni-Co
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Moriarty-Widom pair potentials Al-Ni-Co und
Al-Cu-Co
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Time averaged probability density maps for
d-Al-Ni-Co
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Diffusion channels in d-Al-Ni-Co
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Diffusion in d-Al-Ni-Co
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Arrhenius diagram d-Al-Ni-Co
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ii) Dynamical structure factor for Zn2Mg
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Dynamical structure factor from MD model
calculations
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Dynamical structure factor Zn2Mg
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iii) Crack propagation in NbCr2
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EAM potentials for NbCr2
Lattice constant 6.94 Å
ab-initio 6.97 Å C11, C12, C44
300, 181, 55 GPa ab-initio
309, 198, 69 GPa 24000 atoms kBTmelt
0.17 eV experimental kBTmelt 0.176
eV Surfaces stable, no evaporation of atoms
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Surface reconstruction NbCr2
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Crack propagation NbCr2
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iv) Order-disorder transition in CdCa6
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Cluster structure of CaCd6
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Multiscale algorithm

• Quantum mechanics
• Forces, energies, stresses of 34 reference
  configurations (phases with less atoms in the
  unit cell, expanded, heated). EAM potentials
  fitted thus that they reproduce the data.
• Molecular dynamics
• Structure calculations with different
  orientations of the tetrahedra, deformation of
  the cluster shells, cluster binding energies Ea
  (26 types of two-fold, 16 of three-fold bonds),
  250 clusters (5x5x5 cubic cells).
• Monte-Carlo simulations
• Effective Hamiltonian as sum of the binding
  energies. Calculation of equilibrium structures
  of a system of 128 clusters for different
  temperatures. Sharp jump of internal energy at 89
  K! ?S 1 kB per cluster.

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Order-disorder transition in CaCd6
P. Brommer, F. Gähler, and M. Mihalkovic
2007 Ordering and correlation of cluster
orientations in CaCd6Phil. Mag. in press
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Order-disorder transition
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Summary

• MD method allows to simulate a large variety of
  material properties and dynamical processes at
  the atomic level
• Large systems accessible Complex Metallic
  Alloys!
• Necessary Realistic potentials derived from
  quantum mechanical calculations
• Extensions
• in space by hybrid methods
• in time by accelerators (MC, HMC, DPD, SRD, )

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Time scales
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Cutoff and neighbour lists

• Force calculation O(N2), 80 runtime
• Short range potentials allow decomposition into
  cells
• Only atoms in adjacent cells interact
• Each cell pair only once considered
• Run time O(N)

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Stillinger-Weber potentials
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Microconvergence

• Algorithm for energy relaxation
• Rapid cooling mechanism

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Integration of motion
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Three-body Tersoff potentials
attractive
repulsive
Complicated function of number and type of k atoms
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Explicit form of Tersoff-potential
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Fracture planes perpendicular to fivefold axes
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Long-range interactions(Coulomb- and polar)
By Greengard and Rokhlin Short range
interactions are calculated directly. For long
range interactions a multipole expansion is
performed on an octal tree
By Eastwood Short range interactions are
calculated directly. For long range interactions
Poisson equation is solved on a mesh

• Ewald sum method
• Reaction field method
• Particle-particle/particle-mesh (PPPM)
• Fast multipole method

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Dynamical structure factor Zn2Mg
Longitudinal phonons in the hexagonal plane
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Simulationskontrolle

• Gleichgewicht-Ensembles NVE, NVT, NPT
• Nichtgleichgewicht Scherfluss, Dehnung,
  Plastische Verformung, Rissausbreitung mit
  Stadion-Dämpfung
• Steuerung einzelner Atome (Fixierung,
  Sonderkräfte)

Datenausgabe

• Nur Atome mit besonderen Eigenschaften Energie,
  Koordination, Auslenkung aus der
  Referenzkonfiguration
• Mittelwerte über kleine Raumbereiche

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Skalierung

• Hochleistungsnetzwerk mit niedriger Latenz nötig
• Tests auf IBM BlueGene/L Lineare Skalierung bis
  zu tausenden von CPUs

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Elaboration of potentials

• Force Matching Method
• Select typical configurations
• Calculate wavefunctions, forces on ions
• Fit parametrized potentials (e.g. EAM)