Multi Scale Computational Challenges in Materials Science

1. Multi Scale Computational Challenges in Materials Science

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

3. 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)

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

5. Continuum Methods TIME (s) Based on SDSC Blue Horizon (SP3) 512-1024 processors 1.728 Tflops peak performance CPU time = 1 week / processor 100 Atomistic SimulationMethods Mesoscale methods (ms) 10-3 Finite elements methods (s) 10-6 Semi-empirical (ns) 10-9 methods Monte Carlo Molecular dynamics (ps) 10-12 Ab initio methods tight-binding (fs) 10-15 10-10 10-9 10-8 10-7 10-6 10-5 10-4 (nm) (mm) LENGTH (m) Multi-scale modeling

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

7. 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

8. 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

9. The Building Blocks Continuum Simulations Scale > 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

10. 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

11. 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.

12. 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)

13. Mathematical Modelling: Wave Function We seek to find the steady state of the electron distribution • Each electron eiis described by a corresponding wave function ψi … • ψiis 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 Bohr’s 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 ψi

14. 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) HamiltonianH 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 UnuclInteraction energy of nuclei (Coulombic repulsion) VextNuclei electrostatic potential with which electrons interact UeeElectrostatic repulsion between electrons

15. 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

16. Materials simulation Empirical Potentials Ab initio H-F Hartree DFT Cascade + Surfaces LDA GGA + dynamics Full potentials Pseudopotentials Real space FFT Wavelets 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

17. 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 Uee is approximated by assuming that density is locally uniform

18. 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

19. Kinetic energy of electronei Total potential that acts on ei at position r Energy of the i-th state of the system One electron wave function Charge density at position r Density Functional Theory: Formulation (1/2) Equivalent eigenproblem:

20. Density Functional Theory: Formulation (2/2) Furthermore: Potential due to nuclei and core electrons Coulomb potential form valence electrons Exchange-Correlation potential…also 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!

21. Self Consistent Iteration

22. S.C.I: Computational Considerations

23. 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…!

24. LAPW

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

26. Morse potential • Like Lennard-Jones • Repulsion is more realistic-but attraction less so. • Minimum at r=r0 • Minimum energy is  • An extra parameter “a” which can be used to fit a third property: lattice constant, bulk modulus and cohesive energy.

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

28. 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 T=0) to determine A and n  n=8.87 A=1500eVǺ8.87 • 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!

29. 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 109o rk ri i rj

30. 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.

31. 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