Atomistic Simulation of Materials

Mostafa Youssef Department of Nuclear Science and Engineering Massachusetts Institute of Technology

NPSS - IEEE Alex SC Live Webinar February 14, 2014

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Outline • Why modelling and simulation? • Are we modelling or simulating? • Methods of atomistic simulation of materials - (1) Classical Molecular Dynamics - (2) Metropolis Monte Carlo - (3) Density Functional Theory

• Case studies - (1) Calcium-Silicate-Hydrate structure (Civil + Nuclear) - (2) Designing a catalyst (chemical)

• How to get involved?

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Why modelling and simulation?

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(1) Because of theoretical difficulties Interaction

Given initial and boundary conditions, the two body problem can be solved analytically

Three body problem cannot be solved analytically

N-body problem !!

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(2) Because of experimental difficulties • Extreme or uncontrollable conditions (very high T, very high P as in earth core)

Spin transition in FeCO3

Courtesy of MIT OCW H. Shi et al, Phys. Rev. B 78, 155119 (2008)

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(2) Because of experimental difficulties • Expensive and time consuming (example phase diagrams of alloys)

B. Puchala and A. Van der Ven, Phys. Rev. B (2013)

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(2) Because of experimental difficulties • Restricted/ Dangerous materials (UO2, H2S,…)

On computers, nothing is dangerous or restricted.

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Is Computational science the magic solution? No! Synergistic inquiry is needed

Theory

Computation

Experiment

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Are we modelling or simulating?

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Modelling the interaction between atoms

Short range repulsion  density Long range attraction cohesion

Lennard-Jones potential for ideal gas crystals

Buckingham potential for ionic crystals

Back to first-principles (quantum mechanics) 10

Simulating the interaction between atoms Running the fundamental equations that govern the system using the models as input. (In addition to boundary conditions and initial conditions)

Molecular dynamics, Monte Carlo, Density functional theory, ….

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Methods of atomistic simulation of materials

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Multi-scale Modelling and Simulation

Understanding and Prediction

Applications

Methods of atomistic simulation of materials (1) Classical Molecular Dynamics

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Definition of Molecular Dynamics Simulation

Process by which one generates the atomic trajectories of a system of N atoms by direct numerical integration of Newton’s equation of motion. (with appropriate specification of interatomic potential, thermodynamic ensemble and initial and boundary conditions.) Goal of performing Molecular Dynamics Simulation

Understanding and analyzing complex phenomena on atomic level such as: -Strength –Plastic Deformation – Fracture – Fluid Dynamics – Transport –phonons and vibrations… etc

Process by which one generates the atomic trajectories of a system of N atoms by direct numerical integration of Newton’s equation of motion. (with appropriate specification of interatomic potential,thermodynamics ensemble and initial and boundary conditions.)

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Integration: Various Standard algorithms exist such as: Verlet, Velocity Verlet, Leapfrog Verlet.

- The integration of the coupled equations requires specification of the time step,∆t. Typical time steps 0.1-5 femto-second (10-15 sec). This limits the total simulation time to the order of micro-second as an upper limit!!

Process by which one generates the atomic trajectories of a system of N atoms by direct numerical integration of Newton’s equation of motion. (with appropriate specification of interatomic potential, thermodynamic ensemble and initial and boundary conditions.) -The initial conditions are the number of atoms, their positions and their velocities at the time origin. - Typical number of atoms 103 -104. -Positions can be the crystal structure sites (solids), random (liquids)

-Velocities are sampled from Maxwell-Boltzmann distribution at the desired simulation temperature.

The crystal structure of Tobermorite 9 A

Maxwell-Boltzmann Distribution

Process by which one generates the atomic trajectories of a system of N atoms by direct numerical integration of Newton’s equation of motion. (with appropriate specification of interatomic potential, thermodynamic ensemble and initial and boundary conditions.)

-Simulating solids and liquids of large size requires the use of periodic boundary conditions which renders the system infinite.

Process by which one generates the atomic trajectories of a system of N atoms by direct numerical integration of Newton’s equation of motion. (with appropriate specification of interatomic potential, thermodynamic ensemble and initial and boundary conditions.)

-Several thermodynamic ensembles can be modeled by adopting a Lagrangian formulation for the equations of motion of the particles. The possibilities include: NVE NVT NPT NσT NPH

N≡ number of particles V ≡ volume T ≡ Temperature P ≡ pressure H ≡ enthalpy σ ≡ stress E ≡ total energy

Process by which one generates the atomic trajectories of a system of N atoms by direct numerical integration of Newton’s equation of motion. (with appropriate specification of interatomic potential, thermodynamic ensemble and initial and boundary conditions.)

Two fundamental assumptions are inherent in classical molecular dynamics:

1- Newtonian (Lagrangian/Hamiltonian) mechanics. 2- Born-Oppenheimer approximation: All electrons are in the ground state when the nuclei moves. This also applies to quantum mechanical methods.

Interatomic potential The expansion of the potential, U, in terms of one-body, two-body,…., N-body interactions

Always absent unless external field is applied.

For many materials this term is sufficient

N-body interactions are important for metals Important for elements that favor directional bonding such as Silicon

What is the output of a molecular dynamics simulation? The position, velocity and force on each atom at each time step. From these we can calculate, in principle, everything within the assumptions inherent in classical molecular dynamics.

Let’s Review the molecular dynamics algorithm…..

Choose thermodynamic ensemble, interatomic potential and initial and boundary conditions

Integrate equations of motion

Output r,v,F

Stop

Limitations of Molecular Dynamics: 1- The two assumpations (Born Oppenheimer and Newtonian Mechanics). 2- System size is limited (N<109). 3- Time is limited to the micro-second order of magnitude. Not enough to model rare events.

Ice Melting Movie M. Mochizuki et al, Nature 498, 350-354, (2013)

Methods of atomistic simulation of materials (2) Metropolis Monte Carlo

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Monte Carlo Methods Monte Carlo: Any computational methods that relies on generating random numbers

W. Krauth, Statistical Mechanics: Algorithms and Computations

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Conceptual difference between Monte Carlo and Molecular Dynamics

Interdiffusion is an example of a very slow process

Weinan E et al., J. Phys. Chem. B (2005)

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Metropolis Monte Carlo

Basic Idea: Do not follow the dynamics of the system (very slow), instead sample the important configurations

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Metropolis Monte Carlo

Courtesy of MIT OCW

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Example

B. Puchala and A. Van der Ven, Phys. Rev. B (2013) 30

Scales of modelling and simulation of materials (3) Density Functional Theory

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Why Quantum Mechanics? All electronic, magnetic, chemical, optical requires dealing with electrons. Electrons are quantum species!

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Kinetic Energy Potential Energy

Total Energy

Solving Shrödinger equation for N-electrons is impossible, instead we solve 1-electron problem in the field of N-1 electrons.

   (r1 ,..., rN )   ( x, y, z ) Density Functional Theory is an exact reformulation of the quantum mechanics problem from a 3N wave function to a 3-dimensional charge density 33

Case studies (1) Calcium Silicate Hydrate structure (Civil + Nuclear)

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Calcium-Silicate-Hydrate (The glue of cement)

Neutron scattering experiment

-. Richardson,I. G. Cem. Concr. Res. 2004, 34, 1733. -. Allen ,A. J.; Thomas, J.; Jennings, H. Nature Materials 2007, 6, 311. -.Pellenq, R.; Kushima, A.; Shahsavari, A.; Van Vliet, K.; Buehler, M.; Yip, S.; Ulm, F.-J. PNAS 2009, 106, 16102.

Transition Electron Microscope image of C-S-H

Computational recipe of constructing a C-S-H structure

Mol. Dyn. + Monte Carlo

Experimental guidance

Pellenq, R.; Kushima, A.; Shahsavari, A.; Van Vliet, K.; Buehler, M.; Yip, S.; Ulm, F.-J. PNAS 2009, 106, 16102.

Storing radioactive 90Sr in cement (nuclear waste)

Sr favorably replaces calcium in the interlayer space of C-S-hydrates (which turns out to degrade mechanical properties)

M. Youssef et al. , J. Phys. Chem. Earth(2014)

Case studies (2) Designing a catalyst (Chemical)

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CO dissociation on metals to produce CH4 Activation Energy

Adsorption Energy

These energies has to be calculated by Density Functional Theory because they involve breaking chemical bonds 39

CO dissociation on metals to produce CH4 Hard to dissociate

Activation Energy

Adsorption Energy

Too Sticky

J. Norskov et al. , Nat. Chem. 2009

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CO dissociation on metals to produce CH4

This alloy was commercialized. J. Norskov et al. , Nat. Chem. 2009

Experimental Validation 41

CO dissociation on metals to produce CH4

This alloy was commercialized. J. Norskov et al. , Nat. Chem. 2009

Experimental Validation 42

How to get involved?

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Prerequisites Computational Materials Scientist Computational skills (on demand) - Programming (Fortran, C,..) - Linux scripting - Parallel computing - Numerical Methods

Math and Physics (on demand) - Classical and Quantum Mech. - Solid State Physics - Thermodynamics/ Stat. Mech. - Linear Algebra

- STUDY THE LITERATURE OF YOUR PROBLEM!

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Most of simulation software are free • Molecular Dynamics LAMMPS, DL_POLY, GULP, …

• Monte Carlo GULP, DL_MONTE, ….

• Density Functional Theory Quantum Espresso, ABINIT, SIESTA,…. And of course you can write your own software!

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Resources • Video lectures of MIT OCW (class 3.320 ) will be posted on NPSS-IEEE Facebook page Recommended Reading For Molecular dynamics and Monte Carlo: *Allen and Tildesley, Computer Simulation of Liquids (Oxford). *Frenkel and Smit, Understanding Molecular Simulations (Academic Press). * A Rahman Phys. Rev. 1964, 136, A405. * http://www.fisica.uniud.it/~ercolessi/md/ (Free)

Recommended Reading For Density Functional Theory: * D. Sholl, J. Steckel, Density Functional Theory: A Practical Introduction (WILEY)

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