8/25/15

From Atoms to Materials: Predictive Theory and Simulations Introduction

Ale Strachan [email protected] School of Materials Engineering & Birck Center West Lafayette, Indiana USA

Materials are everywhere

Structural materials http://www.boeing.com/commercial/787family/

Pharmaceuticals Kwong, Kauffman, Hurter & Mueller Nature Biotechnology, 29, 993 (2011)

Nanoelectronics “The High-k Solution”, Bohr, Chau, Ghani, and Mistry http://www.spectrum.ieee.org/oct07/5553

Ale Strachan – Atoms to Materials 2

1 8/25/15

Learning objectives

In Atoms to Materials you will:

• Learn the basics physics that govern materials at atomic scales

• Relate these processes to the macroscopic world

• Use online simulaons to enhance learning • Density funconal theory • Molecular dynamics

Ale Strachan – Atoms to Materials 3

Materials at molecular scales

Molecular materials Ceramics & Metals

Ale Strachan – Atoms to Materials 4

2 8/25/15

Materials properties

Materials Selection in Mechanical Design (3rd edition) by MF Ashby, Butterworth Heinemann, 2005

Ale Strachan – Atoms to Materials 5

Predictive science of materials

“The fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry are thus completely known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved.”

Ale Strachan – Atoms to Materials 6

3 8/25/15

Fundamental physics & approximations Quantum mechanics: • A group of atoms can be fully described by their wavefunction • The time evolution of this wavefunction is given by the Schrödinger equation d i Ψ(r,t) = HΨ(r,t) dt Too complex to be solved even with supercomputers

Electrons: Ions: Time independent€ Schrödinger Eq. Classical (Newton’s) mechanics

Hψ = Eψ F = ma

Ale Strachan – Atoms to Materials 7 € €

Electronic and atomic processes Initial condition

{Ri } {Vi }

Time evolution  Ri = Vi

 Fi Vi = Mi

Energy & forces Hψ = Eψ

F = −∇ E R i Ri ({ i }) Ale Strachan – Atoms to Materials 8

4 8/25/15

Molecular dynamics Initial conditions

{Ri } {Vi }

Compute energy & forces Hψ = Eψ F = −∇ E R i Ri ({ i })

Integrate Eqs. of Motion

Ri (t) → Ri (t + Δt)

Vi (t) → Vi (t + Δt)

Ale Strachan – Atoms to Materials 9

Microscopic and macroscopic worlds

Ale Strachan – Atoms to Materials 10

5 8/25/15

Course outline

• The quantum mechanics of bonding and electronic structure • Atoms, molecules and crystals • Electronic structure calculations • Hartree-Fock & post-Hartree-Fock methods • Density functional theory • Beyond density functional theory • Property predictions • Classical and statistical mechanics • Hamilton’s formalism of classical mechanics • Normal modes and phonons • Statistical mechanics (classical, Bose-Einstein, and Fermi-Dirac) • Molecular dynamics simulations • Interatomic potentials for various classes of materials • Computing the thermo-mechanical response of materials • Kinetic theory and Boltzmann equation • Dynamics with Implicit Degrees of Freedom • Coarse grained simulations of molecular materials • Two temperature model and electronic thermal transport • Atomistic simulationsAle Strachan of electrochemical – Atoms to Materials reactions 11

Grading policy

Grading Policy Project Report: 40% Quizzes: 20% Final Exam (take home): 40%

Ale Strachan – Atoms to Materials 12

6 8/25/15

Course resources

• Course webpage: • hps://nanohub.org/courses/MSE697 • Course nanoHUB group: • hps://nanohub.org/groups/atoms2materials

Books for Part 1 – electronic structure • The nature of the chemical bond, William A. Goddard, III hp://authors.library.caltech.edu/25022/ • Quantum Mechanics, Claude Cohen-Tannoudji, Bernard Diu, Frank Laloe • Atoms and Molecules: An Introducon for Students of Physical Chemistry, • M. Karplus, Richard Needham Porter • Electronic Structure and the Properes of Solids, Walter A. Harrison Computaonal Physics - J. M. Thijssen

Ale Strachan – Atoms to Materials 13

From Atoms to Materials: Predictive Theory and Simulations Why Quantum Mechanics?

Ale Strachan [email protected] School of Materials Engineering & Birck Nanotechnology Center Purdue University West Lafayette, Indiana USA

7 8/25/15

The simplest atom: hydrogen r Electron (-e) Proton (e) What if we treat it with classical mechanics? State of the system (assume massive proton is fixed in space) Position and velocity of electron   r (t) υ (t) Energy (equilibrium will correspond to minimum energy) ! • Potential: V (r) =

• Kinetic Energy: distance  2 energy p 1  2 K = m v = 2 2m

Potential Linear momentum

Alejandro Strachan – Atoms to Materials 1

proton Electron (e-) Classical Hydrogen r

What if we treat it with classical mechanics?  2 e2 p Energy: E = −  + r 2m

State with minimum energy (equilibrium)?

Alejandro Strachan – Atoms to Materials 2

8 8/25/15

proton Electron (-e) Try Quantum Mechanics r

State of the system → wave function (a function of position (r) that describes the probability of finding the electron at position r)  ψ ψ r ( ) Χ

Physical Observable → operator (a mathematical object that acts on a function)

  Position: r Mulply by r

! ! ! Momentum: p = ∇ i

Gradient

Alejandro Strachan – Atoms to Materials 4

proton Electron (e-) Quantum mechanics r

Experimental measurements: expectation value of an operator Integral over all space:   O = ∫ ψ (r)Oψ (r)dx dy dz Integral over all space Example: dx. dy dz≡ d3 r ! r = WF squared is the probability density of finding the electron around posion r

Alejandro Strachan – Atoms to Materials 5

9 8/25/15

proton Electron (e-) Quantum hydrogen r  $ 2 2 ' 2 ∇ e 3   2 3 2 Ψ (r) 3 H = ∫ Ψ(r)&− − )Ψ(r)d r = − ∫ Ψ(r)∇ Ψ(r)d r − e ∫ d r % 2m r ( 2m r

To minimize the energy:

Kinetic energy:

Potential energy:

Alejandro Strachan – Atoms to Materials 6

Electron (-e) proton Quantum hydrogen r

2   2 3 2 Ψ (r) 3 H = − Ψ(r)∇ Ψ(r)d r − e d r 2m ∫ ∫ r

WF WF WF x

Potential Potential Potential

Alejandro Strachan – Atoms to Materials 7

10 8/25/15

Summary proton Electron (e-) r Classical mechanics Classical mechanics fails: quantum mechanics ! State of the system: State: wave function: ψ (r ) ! ! r(t) p(t) 2 2 ! 2 e Energy: Energy: E = − ∇ − 2m r q q e2 V = i j = − r r 2 p 1 2 Ground state: finite size K = = mv 2m 2 Ground state (minimum energy): r = 0 E = −∞ Atoms do not exist!! The kinetic energy makes atoms stable

Alejandro Strachan – Atoms to Materials 8

11