Introduction to Multiscale Modeling with applications

1. Introduction to Multiscale Modeling with applications

3. Overview • This course will will provide a broad survey of modern theory and modeling methods for predicting and understanding the properties of materials. • In particular we will cover: • Commonly used theoretical and simulation methods for the modeling of materials properties such as structure, electronic behavior, mechanical properties, dynamics, etc. from atomistic, electronic models up to macroscale continuum simulations • Quantum methods both at the first principles and semi-empirical level • Classical molecular modeling: molecular dynamics and Montecarlo methods • Solid defect theory • Continuum modeling approaches • Lectures will be complemented by hands-on computing using publicly available or personally developed scientific software packages

4. Goals and objectives • Provide you with a basic background and the skills needed to • Appreciate and understand the use of theory and simulation in research on materials science and the physics of nano- to macroscale systems • Be able to read the simulation literature and evaluate it critically • Identify problems in materials science/condensed matter physics amenable to simulation, and decide on appropriate theory/simulation strategies to study them • Acquire the basic knowledge to perform a materials modeling task using state-of-the-art scientific software, analyze critically the results and draw scientific conclusions on the basis of the computational experiment.

5. Modeling and scientific computing • Continuous advances in computer technology make possible the simulation approach to scientific investigation as a third stream together with pure theory and pure experiment • Experiment: primary concerned with the accumulation of factual information • Theory: mainly directed towards the interpretation and ordering of the information in coherent patters to provide with predictive laws for the behavior of matter through mathematical formulations • Computation: push theories and experiments beyond the limits of manageable mathematics and feasible experiments • Properties of materials under extreme conditions (temperature, pressure, etc.) • Study of properties of complex systems - a solid crystal is already an unmanageable system for a microscopic mathematical model! • Testing of theories vs. experimental observation • Suggestion of experiments for validation of the theory

6. Modeling and scientific computing • Steps to set up a meaningful computational model: • Individuate the physical phenomenon to study • Develop a theory and a mathematical model to describe the phenomenon • Cast the mathematical model in a discrete form, suitable for computer programming • Develop and/or apply suitable numerical algorithms • Write the simulation program • Perform the computer experiment • A good computational scientist has to be a little bit of: • A theorist, to to develop new approaches to solve new problems • An applied mathematician, to be able to translate the theory in a mathematical form suitable for computation • A computer scientist/programmer, to write new scientific codes or modify existing ones to fit the needs and deal with the always changing world of advanced and high-performance computing • An experimentalist, to be able to define a meaningful path of computer experiments that should lead to the description of the physical phenomenon A very demanding task!

8. 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 • Challenge: modeling a physical phenomenon from a broad range of perspectives, from the atomistic to the macroscopic end

10. Multi-scale modeling • Ab initiomethods: calculate materials properties from first principles, solving the quantum-mechanical Schrödinger (or Dirac) equation numerically • Pros: • Give information on both the electronic and structural/mechanical • behavior • Can handle processes that involve bond breaking/formation, or electronic rearrangement (e.g. chemical reactions). • Methods offer ways to systematically improve on the results, making it easy to assess their quality. • Can (in principle) obtain essentially exact properties without any input but the atoms conforming the system. • Cons: • Can handle only relatively small systems, about O(102) atoms. • Can only study fast processes, usually O(10) ps. • Numerically expensive!

11. Multi-scale modeling Density Functional Theory methods (DFT) Different approximations

12. Multi-scale modeling • Semi-empirical methods: use simplified versions of equations from ab initio methods, e.g. only treat valence electrons explicitly; include parameters fitted to experimental data. • Pros: • Can also handle processes that involve bond breaking/formation, or electronic rearrangement. • Can handle larger and more complex systems than ab initio methods, often of O(103) atoms. • Can be used to study processes on longer timescales than can be studied with ab initio methods, of about O(10) ns. • Cons: • Difficult to assess the quality of the results. • Need input from experiments or ab initio calculations and large parameter sets.

13. Multi-scale modeling • Atomistic methods: use empirical or ab initio derived force fields, together with semi-classical statistical mechanics (SM), to determine thermodynamic (MC, MD) and transport (MD) properties of systems. SM solved ‘exactly’. • Pros: • Can be used to determine the microscopic structure of more complex systems, O(104-6) atoms. • Can study dynamical processes on longer timescales, up to O(1) s • Cons: • Results depend on the quality of the force field used to represent the system. • Many physical processes happen on length- and time-scales inaccessible by these methods, e.g. diffusion in solids, many chemical reactions, protein folding, micellization.

14. Multi-scale modeling Classical molecular dynamics (MD) the potential energy U can be written as: where Rij=Ri−Rj are interatomic separations; U2 represents interactions between pairs of atoms; U3 depends on the relative orientations of triplets of atoms, etc.

15. Multi-scale modeling Monte Carlo (MC) Methods MC methods are stochastic algorithms for exploring the system phase space although their implementation for equilibrium and non-equilibrium calculations presents some differences. The Metropolis algorithm in which the probabilities pi=exp(−Ei) / kBT The rate of transitions TRij from state i to state j satisfies the relation: A random number rε(0,1) is then selected and the system is moved to state j only if rbexp[−(Ej−Ei) / kBT]. This is the socalled Metropolis algorithm. Kinetic Monte Carlo (KMC) simulations Master equation

16. Multi-scale modeling • Mesoscale methods: introduce simplifications to atomistic methods to remove the faster degrees of freedom, and/or treat groups of atoms (‘blobs of matter’) as individual entities interacting through effective potentials. • Pros: • Can be used to study structural features of complex systems with O(108-9) atoms. • Can study dynamical processes on timescales inaccessible to classical methods, even up to O(1) s. • Cons: • Can often describe only qualitative tendencies, the quality of quantitative results may be difficult to ascertain. • In many cases, the approximations introduced limit the ability to physically interpret the results.

17. Multi-scale modeling • Continuum methods: Assume that matter is continuous and treat the properties of the system as field quantities. Numerically solve balance equations coupled with phenomenological equations to predict the properties of the systems. • Pros: • Can in principle handle systems of any (macroscopic) size and dynamic processes on longer timescales. • Cons: • Require input (elastic tensors, diffusion coefficients, equations of state, etc.) from experiment or from a lower-scale methods that can be difficult to obtain. • Cannot explain results that depend on the electronic or molecular level of detail.

18. Multi-scale modeling • Connection between the scales: • “Upscaling” • Using results from a lower-scale calculation to obtain parameters for a higher-scale method. This is relatively easy to do; deductive approach. Examples: • Calculation of phenomenological coefficients (e.g. elastic tensors, viscosities, diffusivities)from atomistic simulations for later use in a continuum model. • Fitting of force-fields using ab initio results for later use in atomistic simulations. • Deriving potential energy surface for a chemical reaction, to be used in atomistic MD simulations • Deriving coarse-grained potentials for ‘blobs of matter’ from atomistic simulation, to be used in meso-scale simulations

19. Multi-scale modeling • Connection between the scales: • “Downscaling” • Using higher-scale information (often experimental) to build parameters for lower-scale methods. This is more difficult, due to the non-uniqueness problem. For example, the results from a meso-scale simulation do not contain atomistic detail, but it would be desirable to be able to use such results to return to the atomistic simulation level. Inductive approach. Examples: • Fitting of two-electron integrals in semiempirical electronic structure methods to experimental data (ionization energies, electron affinities, etc.) • Fitting of empirical force fields to reproduce experimental thermodynamic properties, e.g. second virial coefficients, saturated liquid density and vapor pressure

22. Multi-scale modeling: an example Behavior of carbon nanotubes under mechanical deformations • Carbon nanotubes are an excellent example of a physical system whose properties can be described at multiple length- and time-scales • An example: mechanical properties of nanotubes under deformation or tension (Ruoff, PRL, 2000) (Postma et al, Science 2001)

23. Multi-scale modeling: an example • Carbon nanotubes under tension: the ab initio results: • A relatively small nanotube • A very short simulation time (~ 1 ps) Nanotubes break by first forming a bond rotation 5-7-7-5 defect. Buongiorno Nardelli, Yakobson, Bernholc PRL 81, 4656 (1998)

24. Multi-scale modeling: an example • Carbon nanotubes under tension: the atomistic results: • Expansion to a larger system and longer simulation times allows exploration and discovery of new behaviors For low strain values and high temperatures the (5775) defect behaves as a dislocation loop made up of two edge dislocations: (57) and (75). The two dislocations can migrate on the nanotube wall through a sequence of bond rotations  PLASTIC BEHAVIOR Buongiorno Nardelli, Yakobson, Bernholc PRL 81, 4656 (1998)

25. Multi-scale modeling: an example • Carbon nanotubes under tension: the atomistic results: • Expansion to a larger system and longer simulation times allows exploration and discovery of new behaviors Under high tension and low temperature conditions, additional bond rotations lead to larger defects and cleavage  BRITTLE BEHAVIOR Buongiorno Nardelli, Yakobson, Bernholc PRL 81, 4656 (1998)

26. Multi-scale modeling: an example

29. Modèle de percolation dynamique Modèle valable aussi pour un système statique

30. Modèle de percolation dynamique

33. Modèle de percolation dynamique • déplacement des ions sur un diagramme de réseau • La migration des ions est interdite ou permise • Attributions (interdite ou permise) changent en fonction du temps de renouvellement, ren. • Le comportement est toujours diffuse D = A < r2 () > / ren