Graeme Ackland March 2010 Molecular Dynamics "Everything is made of atoms." Molecular dynamics simulates the motions of atoms according to the forces between them 1. Define positions of atoms 2. Calculate forces between them 3. Solve Newtons equations of motion 4. Move atoms goto 1 Interatomic Potential – the only approximation

Graeme Ackland March 2010 Periodic Boundary Conditions 10 9 atoms - < micron Looking beyond the boundary see an image of the atoms in the other side. Least constraining boundary: all atoms are equivalent Infinitely repeated array of supercells

Graeme Ackland March 2010 Finding the neighbours Atoms interact with all others: time ~N 2 Atoms only interact with nearby: time: ~N A scheme for finding and maintaining neighbour lists. Neighbourlist moves with the atom Link cells are fixed in space

Graeme Ackland March 2010 Integration Schemes Lots of mathematical schemes… Runge-Kutta, Predictor- corrector, etc… (integration error pushed to arbitrary order) Generally use Verlet (integration error second order for trajectories) Time reversible, exact integral of some Hamiltonian: good energy conservation

Graeme Ackland March 2010 Ensembles Similar ensembles to Statistical Mechanics All with additional conservation of momentum Microcanonical NVE Canonical NVT Isobaric NPT, NPH Constant stress N T, N H Constant strain rate N(d dt)H

Graeme Ackland March 2010 Constant temperature Integrating Newtons equations – conserve E To conserve T (kinetic energy), need to supply energy. MATHS: Adjust velocities using to some scheme PHYSICS: Connect system to a heat bath Talk by Leimkuhler on Friday a.m. MOLDY example - Nosè

Graeme Ackland March 2010 Constant Stress Parrinello–Rahman: fictitious dynamics Cell parameters (h) have equations of motion r i = h.x i

Graeme Ackland March 2010 External strain For dislocation motion, may wish to apply a finite strain. PROBLEM: strain and release, atoms will simple return to unstrained state

Graeme Ackland March 2010 What can you measure with MD? ~ 10 6 atoms = 100x100x100 30nm, nanoseconds Defect motion (vacancy, dislocation) Segregaion Phase separation/transition Fracture, micromachining Microdeformation

Graeme Ackland March 2010 How much physics do we need? Interatomic potentials for metals and alloys

Graeme Ackland March 2010 Ab initio Dislocations in Iron: Epoch-making Simulation Earth Simulator Center Japan – 231 atoms iron – yield stress 1.1GPa This is pure iron, yet most steels have tensile stress less than this, and iron about 0.1GPa. Whats going on? Geometry – not quantum mechanics Density functional theory will not address this problem any time soon.

Graeme Ackland March 2010 What and Why? Classical molecular dynamics – Billions of atoms Proper quantum treatment of atoms requires optimising hundreds of basis functions for each electron We need to do this without electrons. There is no other way to understand high-T off-lattice atomistic properties of more then a few hundred atoms. Graeme Ackland

March 2010 What do we want? Use ab initio data in molecular dynamics. To describe processes occuring in the material i.e. formation/migration energies geometries of stable defects n.b. MD uses forces, but for thermal activation barrier energies are more important. Need both! Graeme Ackland

March 2010 Potentials -Functional Forms Must be such as to allow million atom MD Short-ranged (order-N calculation) Should describe electronic structure Motivated by DFT (a sufficient theory) Fitted to relevant properties (limited transferrability) Computationally simple Use information available in molecular dynamics EAM (simple DFT) Finnis-Sinclair (2 nd moment tight-binding) Graeme Ackland

March 2010 A Picture of Quantum Mechanics d-electrons form a band – only the lower energy states are filled => cohesion. Delocalised electrons, NOT pairwise bonds giving forces atom to atom Extension: Multiple bands – exchange coupling Graeme Ackland

March 2010 Whats wrong with pairwise bonds? Simplest model of interatomic forces is the pairwise potential : Lennard Jones Any pairwise potential predicts Elastic moduli C12=C44 Vacancy formation energy = cohesive energy In bcc titanium C12=83GPa, C44 = 37GPa In hcp titanium vacancy formation is 1.2eV, cohesive energy 4.85eV

Graeme Ackland March 2010 Close packing On Monday you saw that bcc, fcc and hcp Ti have very similar densities. Despite their different packing fractions i.e. Bonds in bcc are shorter than fcc General rule: lower co-ordinated atoms have fewer, stronger bonds (eg graphite bonds are 33% stonger than diamond)

Graeme Ackland March 2010 transition metal d-band bonding Second-moment tight-binding model: Finnis-Sinclair On forming a solid, band gets wider Electrons go to lower energy states Ackland GJ, Reed SK "Two-band second moment model and an interatomic potential for caesium" Phys Rev B

Graeme Ackland March 2010 EAM, Second Moment and all that In EAM ij j : it represents the charge density at i due to j In Friedel: F i = square root, ij is a hopping integral In Tersoff-Brenner, F i saturates once fully coordinated Details of ij labels: Irrelevant in pure materials Important in alloys measures local density

Graeme Ackland March 2010 Where do interatomic potentials come from? Functional form inspired by chemistry Parameters fitted to empirical or ab initio data Importance of fitting data weighted by intended application Interatomic potentials are not transferrable

Graeme Ackland March 2010 An MD simulation of a dislocation (bcc iron) Starting configuration Periodic in xy, fixed layer of atoms top and bottom in z Dislocation in the middle Move fixed layers to apply stress Final Configuration (n.b. periodic boundary) Dislocation has passed through the material many times: discontinuity on slip plane

Graeme Ackland March 2010 Molecular dynamics simulation of twin and dislocation deformation