3.22 Mechanical properties of materials
Review: Molecular Dynamics
xxx
Markus J. Buehler
Outline: 4 Lectures on Molecular Dynamics (=MD)
„
Lecture 1: Basic Classical Molecular Dynamics
General concepts, difference to MC methods, challenges, potential and
implementation
„
Lecture 2: Introduction to Interatomic Potentials
Discuss empirical atomic interaction laws, often derived from quantum
mechanics or experiment
„
Lecture 3: Modeling of Metals
Application of MD to describe deformation of metals, concepts:
dislocations, fracture
„
Lecture 4: Reactive Potentials
New frontier in research: Modeling chemistry with molecular dynamics
using reactive potentials
© 2006 Markus J. Buehler, CEE/MIT
Elasticity and atomistic bonding
“atomistic”
discrete
“continuum”
© 2006 Markus J. Buehler, CEE/MIT
Molecular dynamics versus Monte Carlo
„
MD is an alternative approach to MC by sampling phase and
state space, but obtaining actual deterministic trajectories;
thus:
Full dynamical information
„
In long time limit - for equilibrium properties - the results of
MC correspond to results obtained by MD
„
MD can model processes that are characterized by extreme
driving forces and that are non-equilibrium processes
Example: Fracture
© 2006 Markus J. Buehler, CEE/MIT
Motivation: Fracture
„
Materials under high load are known to fracture
„
MD modeling provides an excellent physical description of
the fracture processes, as it can naturally describe the
atomic bond breaking processes
„
Other modeling approaches, such as the finite element
method, are based on empirical relations between load
and crack formation and/or propagation; MD does not
require such input
„
What “is” fracture?
© 2006 Markus J. Buehler, CEE/MIT
Ductile versus brittle materials
BRITTLE
DUCTILE
Glass Polymers
Ice...
Copper, Gold
Shear load
Figure by MIT OCW.
(a)
(b)
© 2006 Markus J. Buehler, CEE/MIT
Molecular dynamics
Total energy of system
E = K +U
1 N 2
K = m∑ v j
2 j =1
U = U (rj )
m
d 2 rj
dt
2
= −∇ r j U (rj )
j = 1..N
Coupled system N-body
problem, no exact
solution for N>2
System of coupled 2nd order nonlinear differential equations
Solve by discretizing in time (spatial discretization given by
“atom size”)
© 2006 Markus J. Buehler, CEE/MIT
Solving the equations
+
1
2
ri (t0 + Δt ) = ri (t0 ) + vi (t0 )Δt + ai (t0 )(Δt ) + ...
2
1
2
(
)
ri (t0 − Δt ) = ri (t0 ) − vi (t0 )Δt + ai (t0 ) Δt + ...
2
ri (t0 + Δt ) = −ri (t0 − Δt ) + 2ri (t0 )Δt + ai (t0 )(Δt ) + ...
2
Positions
at t0-Δt
Positions
at t0
Accelerations
at t0
“Verlet central difference method”
How to obtain
accelerations?
f i = mai
ai = f i / m
Need forces on atoms!
© 2006 Markus J. Buehler, CEE/MIT
Typical modeling procedure
Set particle
positions
Calculate force
on each particle
Move particles by
timestep Δt
Stop simulation
Assign particle
velocities
Save current
positions and
velocities
Reached
max. number of
timesteps?
Analyze data
print results
© 2006 Markus J. Buehler, CEE/MIT
Modeling vs. simulation
„
Modeling: Building a mathematical or theoretical
description of a physical situation; maybe result in a set of
partial differential equations
For MD: Choice of potential, choice of crystal structure,
write down F=ma…
„
Simulation: Numerical solution of the problem at hand
(code, infrastructure..)
Solve the equations – e.g. Verlet method, parallelization (later)
„
Simulation is usually followed by analysis methods – post­
processing (RDF, temperature…)
© 2006 Markus J. Buehler, CEE/MIT
5
5
4
4
Solid Argon
3
g(r)
g(r)
Radial distribution function: Solid versus liquid versus gas
2
0
0
0.2
Liquid Ar
(90 K)
3
Gaseous Ar
(300 K)
2
Liquid Argon
1
Gaseous Ar
(90 K)
1
0.4
distance/nm
0.6
0.8
0
0
0.2
0.4
distance/nm
0.6
0.8
Figure by MIT OCW.
Note: The first peak corresponds to the nearest
neighbor shell, the second peak to the second
nearest neighbor shell, etc.
In FCC: 12, 6, 24, and 12 in first four shells
© 2006 Markus J. Buehler, CEE/MIT
Mean square displacement (MSD) function
Liquid
1
< Δr >=
N
2
Crystal
∑ (r (t ) − r (t = 0))
2
i
i
i
Position of
atom i at time t
Relation to diffusion constant:
d
lim < Δr 2 >= 2dD
t→∞ dt
d=2 2D
d=3 3D
Position of
atom i at time t=0
d
lim < Δr 2 >
t→∞ dt
=D
2d J. Buehler, CEE/MIT
© 2006 Markus
Atomic scale
„
„
Atoms are composed of electrons, protons, and neutrons.
Electron and protons are negative and positive charges of
the same magnitude, 1.6 × 10-19 Coulombs
Chemical bonds between atoms by interactions of the
electrons of different atoms
(see QM part
later in IM/S!)
“Point” representation
e-
ep o
p+ no +n
p+ p o
no no n+
no + p
p
V(t)
+
e-
e-
r(t)
a(t)
y
x
eFigure by MIT OCW.
eFigure by MIT OCW.
© 2006 Markus J. Buehler, CEE/MIT
Pair interaction approximation
U total =
1
5
2
1
2
∑ ∑
| U (rij )
i=1.. N j=1.. N i≠ j
Any function that expresses
U (rij )
energy for atomic distance..
3
All pair interactions of atom 1 with
neighboring atoms 2..5
4
1
5
2
All pair interactions of atom 2 with
neighboring atoms 1, 3..5
3
4
Double count bond 1-2
therefore factor © 2006 Markus J. Buehler, CEE/MIT
Lennard-Jones potential
Attractive
Units: Energy
⎛ ⎡σ ⎤12 ⎡σ ⎤ 6 ⎞
φweak (r) = 4ε ⎜ .⎢ ⎥ − ⎢
⎥ ⎟
⎜ ⎣r⎦
⎟
r
⎣
⎦
⎝
⎠
Repulsive
Units: Energy/length=force
dV (r)
F =−
dr
xi
Fi = F
r
r
x2
F
x1
© 2006 Markus J. Buehler, CEE/MIT
MD updating scheme: Complete
(1) Updating method (integration scheme)
2
ri (t0 + Δt ) = −ri (t0 − Δt ) + 2ri (t0 )Δt + ai (t0 )(Δt ) + ...
Positions
at t0-Δt
Positions
at t0
(2) Obtain accelerations from forces
f i = mai
ai = Fi / m
Accelerations
at t0
“Verlet central difference method”
(5) Crystal (initial conditions)
Positions at t0
(3) Obtain forces from potential
dV (r )
F =−
dr
xi
Fi = F
r
(4) Potential
⎛ ⎡σ ⎤12 ⎡σ ⎤ 6 ⎞
φweak (r ) = 4ε ⎜ .⎢ ⎥ − ⎢ ⎥ ⎟
⎜ ⎣r⎦
⎟
r
⎣
⎦
⎝
⎠
© 2006 Markus J. Buehler, CEE/MIT
Deformation of crystals
„
Deformation of a crystal is similar to pushing a sticky tape
across a surface:
F~ τ ⋅ L
“homogeneous shear”
F ≈ Fripple
“localized slip (ripple)”
Lcrit ≈
Fripple
τ
Beyond critical length L it is easer to have a localized
© 2006 Markus ripple…
J. Buehler, CEE/MIT
Ductile versus brittle materials
BRITTLE
Glass Polymers
Ice...
DUCTILE
Copper, Gold
Shear load
Figure by
MIT OCW.
© 2006 Markus J. Buehler, CEE/MIT