graeme ackland march 2010 molecular dynamics "everything is made of atoms." molecular...
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Graeme AcklandMarch 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 Newton’s equations of motion
4. Move atoms
goto 1
)( ijiii rUrmF
Interatomic Potential – the only approximation
Graeme AcklandMarch 2010
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
Graeme AcklandMarch 2010
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
Graeme AcklandMarch 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 AcklandMarch 2010
Ensembles
Similar ensembles to Statistical Mechanics
All with additional conservation of momentum
Microcanonical NVE Canonical NVT Isobaric NPT, NPH Constant stress NT, NH Constant strain rate N(ddt)H
Graeme AcklandMarch 2010
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
Graeme AcklandMarch 2010
Constant Stress
Parrinello–Rahman: fictitious dynamics
Cell parameters (h) have equations of motion
ri = h.xi
Graeme AcklandMarch 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 AcklandMarch 2010
What can you measure with MD?
~ 106 atoms = 100x100x100 30nm, nanoseconds
Defect motion (vacancy, dislocation) Segregaion Phase separation/transition Fracture, micromachining Microdeformation
Graeme AcklandMarch 2010
How much physics do we need?Interatomic potentials for metals and alloys
Graeme AcklandMarch 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. What’s going on? Geometry – not quantum mechanics
Density functional theory will not address this problem any time soon.
Graeme AcklandMarch 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
Graeme AcklandMarch 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 activationbarrier energies are more important. Need both!
Graeme Ackland
Graeme AcklandMarch 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 (2nd moment tight-binding)
Graeme Ackland
Graeme AcklandMarch 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
Graeme AcklandMarch 2010
What’s wrong with pairwise bonds?
Simplest model of interatomic forces is the pairwise potential : Lennard Jones
Any pairwise potential predicts Elastic moduli C12=C44Vacancy formation energy = cohesive energy
In bcc titanium C12=83GPa, C44 = 37GPa In hcp titanium vacancy formation is 1.2eV, cohesive
energy 4.85eV
Graeme AcklandMarch 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 fccGeneral rule: lower co-ordinated atoms
have fewer, stronger bonds (eg graphite bonds are 33% stonger than diamond)
Graeme AcklandMarch 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 67 174108 2003
Graeme AcklandMarch 2010
EAM, Second Moment and all that
])([)( j
ijijiij
ijij rFrV
In EAM ij j: it represents the charge density at i due to j
In Friedel: Fi = square root, ij is a hopping integral
In Tersoff-Brenner, Fi saturates once fully coordinated
Details of ij labels: Irrelevant in pure materials
Important in alloys measures local density
j
ijiji r )(
Graeme AcklandMarch 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 AcklandMarch 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 AcklandMarch 2010
Molecular dynamics simulation of twin and dislocation deformation