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3.22 Mechanical properties of materials

Review: Molecular Dynamics

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

© 2006 Markus J. Buehler, CEE/MIT

“atomistic” discrete “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

Motivation: Fracture

� Materials under high load are known to fracture

� MD modeling provides an excellent physical description ofthe fracture processes, as it can naturally describe theatomic bond breaking processes

� Other modeling approaches, such as the finite elementmethod, are based on empirical relations between loadand 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

U K E += N1K =

2 v m 2

j∑ j = 1

r U U = ( j ) Coupled system N-body

md 2 rj −∇ = r U j ) j = ..1 N problem, no exact(rjdt 2

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

i 0 Δ + ) = t t a 2Δr ( r t t ( )( 0

)( 0i

+ Δ t t v t

+ Δ t t v t

) + 1

i ( 0 )( ) + ...i 0 i 2 Δ − ) = t t a 2Δ+ r ( r t t ( ) −

1 i ( 0 )( ) + ...i 0 i 0 2

( ) ...)()(2)()( 2 0000 +Δ+rr iiii + Δ Δ − − = Δ + t t a t t r t t t t

Positions Positions Accelerations at t0 -Δ t at t0 at t0

“Verlet central difference method”

How to obtain i f = ima accelerations? m f a ii /= Need forces on atoms!

© 2006 Markus J. Buehler, CEE/MIT

Typical modeling procedure

Set particle positions

Assign particle velocities

Calculate force on each particle

Move particles by timestep Δt

Save current positions and

velocities

Reached max. number of

timesteps?

Stop simulation Analyze data print results

© 2006 Markus J. Buehler, CEE/MIT

Modeling vs. simulation

� Modeling: Building a mathematical or theoreticaldescription of a physical situation; maybe result in a set ofpartial 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 – post­processing (RDF, temperature…)

© 2006 Markus J. Buehler, CEE/MIT

Radial distribution function: Solid versus liquid versus gas

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

5

4

3

2

1

00 0.2 0.4 0.6 0.8

g(r)

distance/nm

5

4

3

2

1

00 0.2 0.4 0.6 0.8

g(r)

distance/nm

Solid Argon

Liquid Argon

Liquid Ar(90 K)

Gaseous Ar(90 K)

Gaseous Ar(300 K)

Figure by MIT OCW.

© 2006 Markus J. Buehler, CEE/MIT

Mean square displacement (MSD) function

Liquid Crystal

Δ < r 2 >= iN

Position of Position of

1 ∑( t r ) − t r = 0))i ( i (2

atom i at time t atom i at time t=0 Relation to diffusion constant:

lim d Δ < r 2 >

t→∞ dtlim d Δ < r 2 >= 2dD d=2 2D = D

t→∞ dt d=3 3D 2d

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

no p+ p+

p+

e -

e -e -

e -

e -

e -

no

no no

no

p+ no

p+

p+

V(t)

y x

r(t) a(t)

© 2006 Markus J. Buehler, CEE/MIT Figure by MIT OCW.

Figure by MIT OCW.

Pair interaction approximation

2 5

4

3

1

2 5

4

3

1

1Utotal = 2 ∑ ∑ | U ( rij ) i= ..1 N j= . . 1 N i≠ j

U ( rij ) Any function that expresses energy for atomic distance..

All pair interactions of atom 1 with neighboring atoms 2..5

All pair interactions of atom 2 with neighboring atoms 1, 3..5

Double count bond 1-2therefore factor

© 2006 Markus J. Buehler, CEE/MIT

Lennard-Jones potential

Units: Energy Attractive Units: Energy/length=force

φ 12 6 ⎞

weak (r) = 4ε⎜⎜⎛ .⎢⎡σ ⎤

⎥ − ⎡σ ⎤ ⎟ dV (r)⎢

⎝ ⎣ r ⎦ ⎣ r ⎦⎥ ⎟⎠ F = − Repulsive d r

r xFF i

i =

r

x1

x2F

© 2006 Markus J. Buehler, CEE/MIT

MD updating scheme: Complete

i (t r 0 Δ + t ) ( ) ...)()(2)( 2 000 +Δ+ ttt iii

(1) Updating method (integration scheme) + Δ Δ − − = t a t r t r

Positions Positions Accelerations at t0-Δ t at t0 at t0

(2) Obtain accelerations from forces “Verlet central difference method”

fi = ma ai = Fi / m (5) Crystal (initial conditions)i Positions at t0

(3) Obtain forces from potentialdV (r ) Fi = F xiF − =

d r r (4) Potential 12

⎟φ weak (r ) = 4ε⎜⎜⎛ .⎢⎡σ

⎥⎤ −

⎡σ ⎤ 6 ⎞

⎝ ⎣ r ⎦ ⎣⎢ r ⎦⎥ ⎠⎟ © 2006 Markus J. Buehler, CEE/MIT

© 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 F ripple “localized slip (ripple)”

FrippleLcrit ≈ τ

Beyond critical length L it is easer to have a localized ripple…

Ductile versus brittle materials

BRITTLE DUCTILE

Glass Polymers Ice... Copper, Gold

Shear load

© 2006 Markus J. Buehler, CEE/MIT

Figure by MIT OCW.