Title: Molecular Dynamics
1Molecular 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
2Periodic Boundary Conditions
109 atoms - lt 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
3Finding 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
4Integration 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
5Ensembles
- 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
6Constant 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
7Constant Stress
- ParrinelloRahman fictitious dynamics
- Cell parameters (h) have equations of motion
- ri h.xi
8External strain
For dislocation motion, may wish to apply a
finite strain. PROBLEM strain and release, atoms
will simple return to unstrained state
9What can you measure with MD?
- 106 atoms 100x100x100
- 30nm, nanoseconds
- Defect motion (vacancy, dislocation)
- Segregaion
- Phase separation/transition
- Fracture, micromachining
- Microdeformation
10How much physics do we need?Interatomic
potentials for metals and alloys
11Ab 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.
12What and Why?
Graeme Ackland
- 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.
13What do we want?
Graeme Ackland
- 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!
14Potentials -Functional Forms
Graeme Ackland
- 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)
15A Picture of Quantum Mechanics
Graeme Ackland
- d-electrons form a band only the lower energy
states are filled gt cohesion. - Delocalised electrons, NOT pairwise bonds giving
forces atom to atom - Extension Multiple bands exchange coupling
16Whats wrong with pairwise bonds?
- Simplest model of interatomic forces is the
pairwise potential Lennard Jones - Any pairwise potential predicts
- Elastic moduli C12C44
- Vacancy formation energy cohesive energy
- In bcc titanium C1283GPa, C44 37GPa
- In hcp titanium vacancy formation is 1.2eV,
cohesive energy 4.85eV -
17Close 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)
18transition 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
19EAM, 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
20Where 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
21An 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
22Molecular dynamics simulation of twin and
dislocation deformation