MIT OpenCourseWare http://ocw.mit.edu 3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modeling and Simulation Markus Buehler, Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 1.021/3.021/10.333/18.361/22.00 Introduction to Modeling and Simulation Part II - lecture 4 Atomistic and molecular methods 1 Content overview I. Continuum methods 1. Discrete modeling of simple physical systems: Lectures 2-10 Equilibrium, Dynamic, Eigenvalue problems February/March 2. Continuum modeling approaches, Weighted residual (Galerkin) methods, Variational formulations 3. Linear elasticity: Review of basic equations, Weak formulation: the principle of virtual work, Numerical discretization: the finite element method II. Atomistic and molecular methods 1. Introduction to molecular dynamics 2. Basic statistical mechanics, molecular dynamics, Monte Carlo Lectures 11-19 3. Interatomic potentials 4. Visualization, examples March/April 5. Thermodynamics as bridge between the scales 6. Mechanical properties – how things fail 7. Multi-scale modeling 8. Biological systems (simulation in biophysics) – how proteins work and how to model them III. Quantum mechanical methods 1. It’s A Quantum World: The Theory of Quantum Mechanics 2. Quantum Mechanics: Practice Makes Perfect 3. The Many-Body Problem: From Many-Body to Single-Particle Lectures 20-27 4. Quantum modeling of materials 5. From Atoms to Solids April/May 6. Basic properties of materials 7. Advanced properties of materials 8. What else can we do? 2 Overview: Material covered Lecture 1: Introduction to atomistic modeling (multi-scale modeling paradigm, difference between continuum and atomistic approach, case study: diffusion) Lecture 2: Basic statistical mechanics (property calculation: microscopic states vs. macroscopic properties, ensembles, probability density and partition function, solution techniques: Monte Carlo and molecular dynamics) Lecture 3: Basic molecular dynamics (advanced property calculation, chemical interactions) Lecture 4: Interatomic potential and force field (pair potentials, fitting procedure, force calculation, multi-body potentials-metals/EAM & applications, neighbor lists, periodic BCs, how to apply BCs) Lecture 5: Interatomic potential and force field (cont’d) (organic force fields, bond order force fields-chemical reactions, additional algorithms (NVT, NpT), application: mechanical properties – basic introduction, Cauchy-Born rule as link between chemistry and mechanics – impose deformation field) Lecture 6: Application to mechanics of materials-ductile materials (significance of fractures/flaws, brittle versus ductile behavior [motivating example], basic deformation mechanisms (cracking, dislocations), modeling approaches: metals-EAM, brittle-pair potential/ReaxFF (silicon)) Lecture 7: Application to mechanics of materials-brittle materials; case study: supersonic fracture (example for model building); case study: fracture of silicon (hybrid model) Lecture 8: Review session Lecture 9: QUIZ 3 Important dates Problem set 1: handout: March 31, due Monday April 7 Problem set 2: handout: Monday April 7, due Monday April 14 Review lecture: Tuesday April 15 Quiz: Thursday April 17 4 II. Atomistic and molecular methods Lecture 4: Interatomic potential and force field Outline: 1. MD algorithm - overview 2. How to model chemical interactions 2.1 Pair potentials & how to define a pair potential 2.2 Multi-body potentials-metals/EAM & applications 2.3 Bookkeeping matters: neighbor lists, periodic BCs, how to apply BCs 1. Goal of today’s lecture: How to model chemical interactions in MD and how to develop the models (parameter determination) How to model crystalline materials, in particular metals How to implement force calculation in a MD code (algorithm, bookkeeping, efficient approaches) 5 1. MD algorithm - overview 6 Molecular dynamics: A “bold” idea Particles with mass mi N particles ri(t) z vi(t), ai(t) y x Generate new positions G G G G 2 +Δ=−rt(i ) t0 ( rti −Δ+0 ) t 2i rttatt (0 ) Δ+i 0 ( () ) Δ+ ... Positions Positions Accelerations at t0-Δt at t0 at t0 Based on atomic accelerations, which can be obtained from interatomic forces G G ai= f i/ m i 7 Forces between atoms… how to obtain? Recall: what we need for MD Initial positions of atoms (e.g. crystal structure) Initial velocities (assign random velocities according to a distribution of velocities, e.g. Maxwell-Boltzmann distribution, so that it corresponds to a certain temperature) Time stepping method (e.g. Verlet central difference method) Model for chemical bonds (potentials or force fields, to be covered today) 8 2. How to model chemical interactions Concept: Define energy landscape of chemical interactions, then take spatial derivatives to obtain forces, to be used in MD algorithm 9 Atomic scale – how atoms interact atom core Atoms are composed of electrons, protons, and neutrons. Electron and protons are negative and positive charges of the same magnitude, with 1.6 × 10-19 Coulombs Chemical bonds between atoms by interactions of the electrons of different atoms Distribution of electrons is described by “orbitals” “Point” representation e- e- + V(t) p o + no n p + + p o - o p n e - n o e no n p+ r(t) a(t) p+ y x e- e- Figure by MIT OpenCourseWare. After Buehler. 10 Figure by MIT OpenCourseWare. Atomic interactions – quantum perspective Density distribution of electrons around a H-H molecule How electrons from different atoms interact defines nature of chemical bond Images remove due to copyright restrictions. Please see http://winter.group.shef.ac.uk/orbitron/MOs/H2/1s1s-sigma/index.html Much more about it in part III 11 http://winter.group.shef.ac.uk/orbitron/MOs/H2/1s1s-sigma/index.html Concept: Repulsion and attraction Electrons Energy U r 1/r12 (or Exponential) Core Repulsion Radius r (Distance between atoms) r e Attraction 1/r6 Figure by MIT OpenCourseWare. “point particle” representation Attraction: Formation of chemical bond by sharing of electrons Repulsion: Pauli exclusion (too many electrons in small volume) 12 Generic shape of interatomic potential φ Energy U r Effective Effective 1/r12 (or Exponential) interatomic interatomicpotential Repulsion potential Radius r (Distance between atoms) Harmonic oscillator e r0 Attraction 2 ~ k(r - r0) 1/r6 r Figure by MIT OpenCourseWare. Many chemical bonds show Figure by MIT OpenCourseWare. this generic behavior Attraction: Formation of chemical bond by sharing of electrons Repulsion: Pauli exclusion (too many electrons in small volume) 13 Atomic interactions – different types of chemical bonds Primary bonds (“strong”) Ionic (ceramics, quartz, feldspar - rocks) Covalent (silicon) Metallic (copper, nickel, gold, silver) (high melting point, 1000-5,000K) Secondary bonds (“weak”) Van der Waals (wax, low melting point) Hydrogen bonds (proteins, spider silk) (melting point 100-500K) Ionic: Non-directional (point charges interacting) Covalent: Directional (bond angles, torsions matter) Metallic: Non-directional (electron gas concept) Difference of material properties originates from different atomic interactions 14 Wax Courtesy of Ruth Ruane. http://www.whitewitch.ie. Used with permission. 15 Soft, deformable, does not break under deformation Rocks Image courtesy of Wikimedia Commons. 16 Rocks and sand on Mars 17 Image courtesy of NASA. Gold Image courtesy of Wikimedia Commons. 18 Silicon Image courtesy of NASA. 19 Spider web Image courtesy of U.S. Fish and Wildlife Service. 20 Tree’s leaf Image courtesy of Wikimedia Commons. 21 2.1 Pair potentials & how to define a pair potential 22 Models for atomic interactions: pair potential Atom-atom interactions are necessary to compute the forces and accelerations at each MD time integration step: Update to new positions! Usually define interatomic potentials, that describe the energy of a set of atoms as a function of their coordinates: Depends on position of all other atoms Utotal = Utotal() r G r= r j=1 .. N G { j } F= −∇ UK ( ) r = i 1 .. N i ri total ⎛ ∂ ∂ ∂ ⎞ Change of potential energy ∇G =⎜ ,, ⎟ due to change of position of ri ⎜ ⎟ 23 ⎝1r∂ ,i∂ r 2 i ,∂ r 3i⎠ , particle i (“gradient”) Pair potentials: energy calculation Simple approximation: Total energy is sum over the Pair wise energy of all pairs of atoms in the system interaction potential 1 φ()rij 2 5 3 4 Pair wise summation of bond energies r avoid double counting 1 12 N N r25 2 5 1 Utotal = 2 ∑∑φ()rij i==11, i≠ jj 3 4 N rij = distance between Energ of atom yi Ui = φ()ij r ∑ 24 particles i and j j=1 Interatomic pair potentials: examples ( )φr expij = D 2( r −α (ij r− 0 )− D 2(−α expij − r0 ) r Morse ( potential ) 12 6 ⎡⎛ ⎞ ⎛ ⎞ ⎤ Lennard-Jones 12:6 σ⎜ ⎟ ⎜ σ⎟ φ(rij )= 4 ε⎢ − ⎥ potential ⎢⎜r⎟ ⎜ r⎟ ⎥ ⎣⎝ ij ⎠ ⎝ ij ⎠ ⎦ (excellent model for noble Gases, Ar, Ne, Xe..) 6 ⎛ r ⎞ ⎛ σ ⎞ (φr )= A exp⎜ −ij ⎟ − C⎜ ⎟ ij ⎜ ⎟ ⎜ ⎟ Buckingham potential ⎝ σ ⎠ ⎝ rij ⎠ Harmonic approximation 25 Lennard-Jones potential: schematic φ LJ 12:6 ~σ potential ε : well depth (energy per bond) ? ? σ : force vanishes (EQ r ε distance between atoms) r Parameters 12 6 ⎛σ⎡ ⎤ ⎡ σ⎤ ⎞ φ(r )= 4 ε⎜ − ⎟ ⎜ ⎢ ⎥ ⎢ ⎥ ⎟ ⎝ ⎣r⎦ ⎣ r⎦ ⎠ E. Sir J. Lennard-Jones (Cambridge UK) Lennard-Jones 12:6 26 Lennard-Jones potential – example for copper LJ potential – parameters for copper (Cleri et al., 1997) 27 Paper’s introduction removed due to copyright restrictions. 28 Paper posted on MIT Server Force calculation – pair potential Forces can be calculated by taking derivatives from the potential function Force magnitude: