Multi Scale Computational Challenges

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Multi Scale Computational Challenges in
Materials Science
H. Aourag
LEPM, URMER University of Tlemcen, Algeria
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Multiscale Modeling Methods
Product

Product design optimization
Process optimization
Process model
Product model
Physical system
Reduced experimentation
Market need
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Parallel computing and materials simulations
Water-metal interface
Dynamics of electron excitation/transfer
Biomembrane Aquaporin water channel in
membrane K. Murata et al, Nature, 407, 599 (2002)
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Bottom-up approach
Macroscopic (meter, hour)
Mesoscopic
Kinetics
Energetics
Atomic
Electronic (Å, fs)

• Theoretical approach based on
• Fundamental laws of physics
• Computer modeling and simulations

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Multi-scale modeling
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The Building Blocks Electronic Structure
Calculations

• Scale 0.1nm

Solve Schrodingers equation for ground states of
electrons
Affinities
Band gap calculations
Band gap
LED
Sensors
Solid state lighting
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The Building Blocks Atomistic Simulations
Scale 10nm
Molecular dynamics
Monte Carlo

Enzyme in octane
Discrete model of island nucleation
Polymer nanocomposites
Mapping to continuum
Nanocrystalline materials
Nanostructured materials
Thin film growth
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The Building Blocks Discrete Mesoscale
Simulations
Scale 1mm
Coarse grained polymer models
Discrete dislocation dynamics (metals)

Discrete dislocation dynamics
Atomistic
Coarse grained
Atomistically informed constitutive equations
Continuum


Polymer models
Polycrystal plasticity
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The Building Blocks Continuum Simulations
Scale gt 0.1mm (system specific)
Single scale models Integrate the relevant
system of PDEs.

Multiscale models Sequential methods
Variational multiscale
Time/space assymptotic expansion
Embedded methods Multigrid
Domain decomposition
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Linking the Building Blocks Across Scales
Continuum multiscale models
Calibration of continuum constitutive laws based
on discrete models

Calibration of higher order continuum based on
atomistics
and
Continuum macro
Coupled atomistic-continuum
Interatomic potentials
Continuum micro
Continuum models
Mesoscale
Atomistics
Discrete models
Electronic structure
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Modeling Challenges
Spatial scale linking

• Usually, no more than 2 scales are linked. Most
  models refer to a single spatial scale.
• This requires assumptions to be made about the
  gross behavior (constitutive laws) of the smaller
  scale.

Temporal scale linking

• The time scale linking problem is much more
  difficult consistent procedures with high degree
  of generality are lacking.

Multiple physical phenomena

• The various physical phenomena are intimately
  coupled at the atomic scale. They are usually
  treated as being decoupled in continuum models.

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Introduction and Motivation

• Computational Materials Science Target Problem
• predict the properties of materials
• How?
• AB INITIO calculations Simulate the behavior of
  materials at the atomic level, by applying the
  basic laws of physics (Quantum mechanics)
• What do we (hope to) achieve?
• Explain the experimentally found properties of
  materials
• Engineer new materials with desired properties
• Applicationsnumerous (some include)
• Semiconductors, synthetic light weight materials
  
• Drug discovery, protein structure prediction
• Energy alternative fuels (nanotubes, etc)

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Mathematical Modelling Wave Function
We seek to find the steady state of the electron
distribution

• Each electron ei is described by a corresponding
  wave function ?i
• ?i is a function of space (r)in particular it
  is determined by
• The position rk of all particles (including
  nuclei and electrons)
• It is normalized in such a way that
• Max Bohrs probabilistic interpretation
  Considering a region D, then
• describes the probability of electron ei being
  in region D. Thus the distribution of electrons
  ei in space is defined by the wave function ?i

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Mathematical Modelling Hamiltonian

• Steady state of the electron distribution
• is it such that it minimizes the total energy of
  the molecular system(energy due to dynamic
  interaction of all the particles involved
  because of the forces that act upon them)

• Hamiltonian H of the molecular system
• Operator that governs the interaction of the
  involved particles
• Considering all forces between nuclei and
  electrons we have

Hnucl Kinetic energy of the nuclei He Kinetic
energy of electrons Unucl Interaction energy of
nuclei (Coulombic repulsion) Vext Nuclei
electrostatic potential with which electrons
interact Uee Electrostatic repulsion between
electrons

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Mathematical Modelling Schrödinger's Equation
Let the columns of ? hold the wave functions
corresponding the electronsThen it holds that

• This is an eigenvalue problemthat becomes a
  usual
• algebraic eigenvalue problem when we
  discretize ?i w.r.t. space (r)
• Extremely complex and nonlinear problemsince
• Hamiltonian and wave functions depend upon all
  particles
• We can very rarely (only for trivial cases)
  solve it exactly

Variational Principle (in simple terms!) Minimal
energy and the corresponding electron
distribution amounts to calculating the smallest
eigenvalue/eigenvector of the Schrödinger
equation

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The Ground State

• How to minimize in such a large space
• Methods of Quantum Chemistry- expand in extremely
  large bases - Billions - grows exponentially with
  size of system
• Limited to small molecules
• Quantum Monte Carlo - statistical sampling of
  high-dimensional spaces
• Exact for Bosons (Helium 4)
• Fermion sign problem for Electrons

Quantum Monte Carlo
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Schrödinger's Equation Basic Approximations

• Multiple interactions of all particlesresult to
  extremely complex Hamiltonianwhich typically
  becomes huge when we discretize
• Thusa number of reasonable approximations/simplif
  ications have been consideredwith negligible
  effects on the accuracy of the modeling
• Born-Oppenheimer Separate the movement of
  nuclei and electronsthe latter depends on the
  positions of the nuclei in a parametric
  way(essentially neglect the kinetic energy of
  the nuclei)
• Full Potential or Pseudopotential
  approximation (FP-LAPW, FP-LMTO) accurate and
  slow or (VASP, CPMD, PWSCF) Nucleus and
  surrounding core electrons are treated as one
  entity , fast but with uncertainty
• Local Density Approximation If electron density
  does not change rapidly w.r.t. sparse (r)then
  electrostatic repulsion Uee is approximated by
  assuming that density is locally uniform

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Density Functional Theory

• High complexity is mainly due to the
  many-electron formulation of ab initio
  calculationsis there a way to come up with an
  one-electron formulation?
• Key Theory
• DFT Density Functional Theory (Hohenberg-Kohn,
  64)
• The total ground energy of a system of electrons
  is a functional of the electronic density(number
  of electrons in a cubic unit)
• The energy of a system of electrons is at a
  minimum if it is an exact density of the ground
  state!
• This is an existence theoremthe density
  functional always exists
• but the theorem does not prescribe a way to
  compute it
• This energy functional is highly complicated
• Thus approximations are consideredconcerning
• Kinetic energy and
• Exchange-Correlation energies of the system of
  electrons

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Density Functional Theory Formulation (1/2)
Equivalent eigenproblem

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Density Functional Theory Formulation (2/2)
Furthermore
Potential due to nuclei and core electrons
Coulomb potential form valence electrons
Exchange-Correlation potentialalso a function of
the charge density ?
Non-linearity The new Hamiltonian depends upon
the charge density ? while ? itself depends upon
the wave functions (eigenvectors) ?i Thus some
short of iteration is required until convergence
is achieved!

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Self Consistent Iteration

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S.C.I Computational Considerations

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S.C.I Computational Considerations

• Conventional approach
• Solve the eigenvalue problem (1)and compute the
  charge densities
• This is a tough problemmany of the smallest
  eigenvaluesdeep into the spectrum are required!
  Thus
• efficient eigensolvers have a significant impact
  on electronic structure calculations!

• Alternative approach
• The eigenvectors ?i are required only to compute
  ?k(r)
• Can we instead approximate charge densities
  without eigenvectors?
• Yes!

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LAPW
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Lennard-Jones potential

• V(R) ?iltjv(ri-rj) v(r) 4?(?/r)12-
  (?/r)6
• ? well depth
• ? wall of potential
• Reduced units
• Energy in ?
• Lengths in ?
• Phase diagram is universal!

?
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Morse potential

• Like Lennard-Jones
• Repulsion is more realistic-but attraction less
  so.
• Minimum at rr0
• Minimum energy is ?
• An extra parameter a which can be used to fit a
  third property lattice constant, bulk modulus
  and cohesive energy.

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Various Potentials

• a) Hard sphere
• b) Hard sphere square well
• c) Coulomb (long-ranged)
• d) 1/r12 potential (short ranged)

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Fit for a Born (1923) potential

• Attractive charge-charge interaction
• Repulsive interaction determined by atom core.

• EXAMPLE NaCl
• Obviously Zi?1
• Use cohesive energy and lattice constant (at T0)
  to determine A and n
• ? n8.87 A1500eV?8.87
• Now we need a check. The bulk modulus.
• We get 4.35 x 1011 dy/cm2
• experiment is 2.52 x 1011 dy/cm2
• You get what you fit for!

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Silicon potential

• Solid silicon can not be described with a pair
  potential.
• Tetrahedral bonding structure caused by the
  partially filled p-shell.
• Very stiff potential, short-ranged caused by
  localized electrons
• Stillinger-Weber (1985) potential fit from
• Lattice constant,cohesive energy, melting point,
  structure of liquid Si
• for rlta
• Minimum at 109o

rk
ri
?i
rj
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Protein potential

• Empirical potentials to describe interactions
  between moleculations
• AMBER potential is
• Two-body Lennard-Jones charge interaction
• Bonding potential kr(ri-rj)2
• Bond angle potential ka(?- ?0)2
• Dihedral angle vn 1 - cos(n?)
• All parameters taken from experiment.
• Rules to decide when to use which parameter.
• Many other force fields commercially available.

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Metallic potentials

• Have a inner core valence electrons
• Valence electrons are delocalized. Hence pair
  potentials do not work very well. Strength of
  bonds decreases as density increases because of
  Pauli principle.
• EXAMPLE at a surface LJ potential predicts
  expansion but metals contract
• Embedded atom (EAM) or glue models work better.
• Daw and Baskes, PRB 29, 6443 (1984).
• Embedding function electron density pair
  potential
• Good for spherically symmetric atoms Cu, Al, Pb

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