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

Product design optimization Process optimization Reduced experimentation Physical system Process model Product model Product Market need Multiscale Modeling Methods

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)

Macroscopic (meter, hour) Mesoscopic Kinetics Energetics Atomic Electronic (Å, fs) Bottom-up approach Theoretical approach based on: 1)Fundamental laws of physics 2)Computer modeling and simulations

Multi-scale modeling (fs) (ps) (ns) (  s) (ms) (nm) (  m) LENGTH (m) TIME (s) Mesoscale methods Atomistic Simulation Methods Semi-empirical methods Ab initio methods Monte Carlo Molecular dynamics tight-binding Continuum Finite elements methods Methods Based on SDSC Blue Horizon (SP3) processors Tflops peak performance CPU time = 1 week / processor

The Building Blocks Electronic Structure Calculations Solve Schrodinger’s equation for ground states of electrons: Affinities SensorsSolid state lighting LED Band gap calculations Band gap Scale ~ 0.1nm

The Building Blocks Atomistic Simulations Molecular dynamics Monte Carlo Nanostructured materialsThin film growth Polymer nanocomposites Nanocrystalline materials Discrete model of island nucleation Mapping to continuum Scale ~ 10nm Enzyme in octane

The Building Blocks Discrete Mesoscale Simulations Coarse grained polymer models Discrete dislocation dynamics (metals) Discrete dislocation dynamics Polymer models Continuum Atomistically informed constitutive equations Scale ~ 1  m Polycrystal plasticity AtomisticCoarse grained

The Building Blocks Continuum Simulations Single scale models – Integrate the relevant system of PDEs. Multiscale models – Sequential methods: Variational multiscale Time/space assymptotic expansion – Embedded methods: Multigrid Domain decomposition Scale > 0.1  m (system specific)

Linking the Building Blocks Across Scales Electronic structure Atomistics Mesoscale Continuum micro Discrete models Continuum models Coupled atomistic-continuum Interatomic potentials Calibration of higher order continuum based on atomistics and Continuum macro Calibration of continuum constitutive laws based on discrete models Continuum multiscale models

Modeling Challenges 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. The time scale linking problem is much more difficult; consistent procedures with high degree of generality are lacking. The various physical phenomena are intimately coupled at the atomic scale. They are usually treated as being decoupled in continuum models. Temporal scale linking Spatial scale linking Multiple physical phenomena

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 Applications:…numerous (some include)  Semiconductors, synthetic light weight materials  Drug discovery, protein structure prediction  Energy: alternative fuels (nanotubes, etc)

Mathematical Modelling: Wave Function We seek to find the steady state of the electron distribution  Each electron e i is described by a corresponding wave function ψ i …  ψ i is a function of space (r)…in particular it is determined by  The position r k of all particles (including nuclei and electrons)  It is normalized in such a way that  Max Bohr’s probabilistic interpretation: Considering a region D, then …describes the probability of electron e i being in region D. Thus: the distribution of electrons e i in space is defined by the wave function ψ i

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… H nucl Kinetic energy of the nuclei H e Kinetic energy of electrons U nucl Interaction energy of nuclei (Coulombic repulsion) V ext Nuclei electrostatic potential with which electrons interact U ee Electrostatic repulsion between electrons

Mathematical Modelling: Schrödinger's Equation Let the columns of Ψ: hold the wave functions corresponding the electrons…Then it holds that  This is an eigenvalue problem…that becomes a usual…  “algebraic” eigenvalue problem when we discretize  i w.r.t. space (r)  Extremely complex and nonlinear problem…since  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

Materials simulation Empirical Potentials Cascade + Surfaces + dynamics Ab initio H-FHartreeDFT LDAGGA Pseudopotentials Full potentials FFTReal spaceWavelets Quantum Monte Carlo 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 The Ground State

Schrödinger's Equation: Basic Approximations Multiple interactions of all particles…result to extremely complex Hamiltonian…which typically becomes huge when we discretize Thus…a number of reasonable approximations/simplifications have been considered…with negligible effects on the accuracy of the modeling:  Born-Oppenheimer: Separate the movement of nuclei and electrons…the 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 U ee is approximated by assuming that density is locally uniform

Density Functional Theory High complexity is mainly due to the many-electron formulation of ab initio calculations…is 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 theorem…the density functional always exists  …but the theorem does not prescribe a way to compute it…  This energy functional is highly complicated…  Thus approximations are considered…concerning:  Kinetic energy and  Exchange-Correlation energies of the system of electrons

Density Functional Theory: Formulation (1/2) Equivalent eigenproblem: Kinetic energy of electron e i Total potential that acts on e i at position rEnergy of the i-th state of the systemOne electron wave function Charge density at position r

Density Functional Theory: Formulation (2/2) Furthermore: Exchange-Correlation potential…also a function of the charge density  Coulomb potential form valence electrons Potential due to nuclei and core electrons 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!

Self Consistent Iteration

S.C.I: Computational Considerations

Conventional approach:  Solve the eigenvalue problem (1)…and compute the charge densities…  This is a tough problem…many of the smallest eigenvalues…deep 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…!

LAPW

Lennard-Jones potential V(R) =  i<j v(r i -r j ) v(r) = 4  [(  /r) 12 - (  /r) 6 ]  = well depth  = wall of potential Reduced units: –Energy in  –Lengths in  Phase diagram is universal! 

Morse potential Like Lennard-Jones Repulsion is more realistic-but attraction less so. Minimum at r=r 0 Minimum energy is  An extra parameter “a” which can be used to fit a third property: lattice constant, bulk modulus and cohesive energy.

Various Potentials a) Hard sphere b) Hard sphere square well c) Coulomb (long- ranged) d) 1/r 12 potential (short ranged)

Fit for a Born (1923) potential EXAMPLE: NaCl Obviously Z i =  1 Use cohesive energy and lattice constant (at T=0) to determine A and n  n=8.87 A=1500eVǺ 8.87 Now we need a check. The “bulk modulus”. –We get 4.35 x dy/cm 2 –experiment is 2.52 x dy/cm 2 You get what you fit for! Attractive charge-charge interaction Repulsive interaction determined by atom core.

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 r<a Minimum at 109 o riri rkrk rjrj ii

Protein potential Empirical potentials to describe interactions between moleculations AMBER potential is: –Two-body Lennard-Jones+ charge interaction –Bonding potential: k r (r i -r j ) 2 –Bond angle potential k a (  -  0 ) 2 –Dihedral angle: v n [ 1 - cos(n  )] –All parameters taken from experiment. –Rules to decide when to use which parameter. Many other “force fields” commercially available.

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