Fom Atoms to Mateials: Pedictive Theoy and Simulations Intoduction Ale Stachan stachan@pudue.edu School of Mateials Engineeing & Bick Nanotechnology Cente Pudue Univesity West Lafayette, Indiana USA Mateials ae eveywhee Stuctual mateials http://www.boeing.com/commecial/787family/ Phamaceuticals Kwong, Kauffman, Hute & Muelle Natue Biotechnology, 9, 993 (011) Nanoelectonics The High-k Solution, Boh, Chau, Ghani, and Misty http://www.spectum.ieee.og/oct07/5553 Ale Stachan Atoms to Mateials 1
Leaning objectives In Atoms to Mateials you will: Lean the basics physics that goven mateials at atomic scales Relate these pocesses to the macoscopic wold Use online simula@ons to enhance leaning Density func@onal theoy Molecula dynamics Ale Stachan Atoms to Mateials 3 Mateials at molecula scales Molecula mateials Ceamics & semiconductos Metals Ale Stachan Atoms to Mateials 4
Mateials popeties Mateials Selection in Mechanical Design (3d edition) by MF Ashby, Buttewoth Heinemann, 005 Ale Stachan Atoms to Mateials 5 Pedictive science of mateials The fundamental laws necessay fo the mathematical teatment of a lage pat of physics and the whole of chemisty ae thus completely known, and the difficulty lies only in the fact that application of these laws leads to equations that ae too complex to be solved. Ale Stachan Atoms to Mateials 6 3
Fundamental physics & appoximations Quantum mechanics: A goup of atoms can be fully descibed by thei wavefunction The time evolution of this wavefunction is given by the Schödinge equation i d dt Ψ,t ( ) = HΨ (,t ) Too complex to be solved even with supecomputes Electons: Time independent Schödinge Eq. Ions: Classical (Newton s) mechanics Hψ = Eψ F = ma Ale Stachan Atoms to Mateials 7 Electonic and atomic pocesses Initial condition { } { R i } V i Time evolution R i = V i V i = F i M i Enegy & foces Hψ = Eψ F i = Ri E ({ R i }) Ale Stachan Atoms to Mateials 8 4
Molecula dynamics Initial conditions { R i } V i { } Compute enegy & foces Hψ = Eψ F i = Ri E ({ R i }) Integate Eqs. of Motion R i V i ( t) R i ( t + Δt) ( t) V i ( t + Δt) Ale Stachan Atoms to Mateials 9 Micoscopic and macoscopic wolds Ale Stachan Atoms to Mateials 10 5
Couse outline The quantum mechanics of bonding and electonic stuctue Atoms, molecules and cystals Electonic stuctue calculations Hatee-Fock & post-hatee-fock methods Density functional theoy Beyond density functional theoy Popety pedictions Classical and statistical mechanics Hamilton s fomalism of classical mechanics Nomal modes and phonons Statistical mechanics (classical, Bose-Einstein, and Femi-Diac) Molecula dynamics simulations Inteatomic potentials fo vaious classes of mateials Computing the themo-mechanical esponse of mateials Kinetic theoy and Boltzmann equation Dynamics with Implicit Degees of Feedom Coase gained simulations of molecula mateials Two tempeatue model and electonic themal tanspot Atomistic simulations Ale Stachan of electochemical Atoms to Mateials eactions 11 Gading policy Gading Policy Poject Repot: 40% Quizzes: 0% Final Exam (take home): 40% Ale Stachan Atoms to Mateials 1 6
Couse esouces Couse webpage: hops://nanohub.og/couses/mse697 Couse nanohub goup: hops://nanohub.og/goups/atomsmateials Books fo Pat 1 electonic stuctue The natue of the chemical bond, William A. Goddad, III hop://authos.libay.caltech.edu/50/ Quantum Mechanics, Claude Cohen- Tannoudji, Benad Diu, Fank Laloe Atoms and Molecules: An Intoduc8on fo Students of Physical Chemisty, M. Kaplus, Richad Needham Pote Electonic Stuctue and the Pope@es of Solids, Walte A. Haison Computa6onal Physics - J. M. Thijssen Ale Stachan Atoms to Mateials 13 Fom Atoms to Mateials: Pedictive Theoy and Simulations Why Quantum Mechanics? Ale Stachan stachan@pudue.edu School of Mateials Engineeing & Bick Nanotechnology Cente Pudue Univesity West Lafayette, Indiana USA 7
The simplest atom: hydogen What if we teat it with classical mechanics? Poton (e) State of the system (assume massive poton is fixed in space) Position and velocity of electon t ( ) υ t ( ) Enegy (equilibium will coespond to minimum enegy) Electon (- e) Potential: V (! ) = Potential enegy distance Linea momentum Kinetic Enegy: K = 1 m v p = m Alejando Stachan Atoms to Mateials 1 Classical Hydogen poton Electon (e - ) What if we teat it with classical mechanics? Enegy: E = e + p m State with minimum enegy (equilibium)? Alejando Stachan Atoms to Mateials 8
Ty Quantum Mechanics poton Electon (- e) State of the system wave function (a function of position () that descibes the pobability of finding the electon at position ) ψ ψ ( ) Physical Obsevable opeato (a mathematical object that acts on a function) Χ Position: Mul@ply by Momentum:! p =!! i Gadient Alejando Stachan Atoms to Mateials 4 Quantum mechanics poton Electon (e - ) Expeimental measuements: expectation value of an opeato Integal ove all space: Example:! = O = ψ ( )Oψ ( )dx dy dz Integal ove all space dx. dy dz WF squaed is the pobability density of finding the electon aound posi@on 3 d Alejando Stachan Atoms to Mateials 5 9
Quantum hydogen poton Electon (e - ) H = $ Ψ & % ( ) m e ' )Ψ( )d 3 ( = m ( ) Ψ ( ) Ψ Ψ( )d 3 e d 3 To minimize the enegy: Kinetic enegy: Potential enegy: Alejando Stachan Atoms to Mateials 6 Quantum hydogen poton Electon (-e) ( ) H = m Ψ ( ) Ψ Ψ( )d 3 e d 3 WF WF WF x Potential Potential Potential Alejando Stachan Atoms to Mateials 7 10
Summay Classical mechanics State of the system:!! ( t) p( t) Enegy: qiq j e V = = p 1 K = = mv m Gound state (minimum enegy): = 0 E = Atoms do not exist!! poton Electon (e - ) Classical mechanics fails: quantum mechanics State: wave function: ( ) E =! Enegy: m! ψ e Gound state: finite size The kinetic enegy makes atoms stable Alejando Stachan Atoms to Mateials 8 11