Molecular Dynamics

1. 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 Newton’s equations of motion 4. Move atoms goto 1 Interatomic Potential – the only approximation

2. Periodic Boundary Conditions 109 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

3. Finding the neighbours • Atoms interact with all others: time ~N2 • 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

4. 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

5. Ensembles • Similar ensembles to Statistical Mechanics All with additional conservation of momentum • Microcanonical NVE • Canonical NVT • Isobaric NPT, NPH • Constant stress NsT, NsH • Constant strain rate N(de/dt)H

6. Constant temperature • Integrating Newton’s 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

7. Constant Stress • Parrinello–Rahman: fictitious dynamics Cell parameters (h) have equations of motion • ri = h.xi

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

9. What can you measure with MD? • ~ 106 atoms = 100x100x100 • 30nm, nanoseconds • Defect motion (vacancy, dislocation) • Segregaion • Phase separation/transition • Fracture, micromachining • Microdeformation

10. How much physics do we need?Interatomic potentials for metals and alloys

11. 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. What’s going on? Geometry – not quantum mechanics Density functional theory will not address this problem any time soon.

12. Graeme Ackland 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.

13. Graeme Ackland 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!

14. Graeme Ackland 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 (2nd moment tight-binding)

15. Graeme Ackland 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

16. What’s 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

17. 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)

18. 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 67 174108 2003

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

20. 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

21. 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

22. Molecular dynamics simulation of twin and dislocation deformation