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From Atoms to Materials: Predictive Theory and Simulations Introduction
Ale Strachan [email protected] School of Materials Engineering & Birck Nanotechnology Center Purdue University 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
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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 simula ons to enhance learning • Density func onal theory • Molecular dynamics
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Materials at molecular scales
Molecular materials Ceramics & semiconductors Metals
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Materials properties
Materials Selection in Mechanical Design (3rd edition) by MF Ashby, Butterworth Heinemann, 2005
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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.”
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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
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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
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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)
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Microscopic and macroscopic worlds
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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%
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Course resources
• Course webpage: • h ps://nanohub.org/courses/MSE697 • Course nanoHUB group: • h ps://nanohub.org/groups/atoms2materials
Books for Part 1 – electronic structure • The nature of the chemical bond, William A. Goddard, III h p://authors.library.caltech.edu/25022/ • Quantum Mechanics, Claude Cohen-Tannoudji, Bernard Diu, Frank Laloe • Atoms and Molecules: An Introduc on for Students of Physical Chemistry, • M. Karplus, Richard Needham Porter • Electronic Structure and the Proper es of Solids, Walter A. Harrison Computa onal Physics - J. M. Thijssen
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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
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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
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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)?
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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 Mul ply by r
! ! ! Momentum: p = ∇ i
Gradient
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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 posi on r
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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:
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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
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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
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