1

Molecular Dynamics for Beginnners

MSM, TS, CTW, UTwente, NL

Stefan Luding, s.luding@utwente.nl

Stefan Luding

MSM, TS, CTW, UTwente, NL

PDFs of the presentationsRennes1 -> entespace documentsgestion des mes espacesModSim2Luc OgerStefan_Luding_Cours

Graphics xballs:/home/public/modsim/xballs

/home/public/modsim/md1.cc

2

Granular Materials

Real:

• sand, soil, rock,

• grain, rice, lentils,

• powder, pills, granulate,

• micro- and nano-particles

Model Granular Materials

• steel/aluminum spheres

• spheres with dissipation/friction/adhesion

• Introduction

• Single Particles

• Particle Contacts/Interactions

• Many particle cooperative behavior

• Applications/Examples

• Conclusion

Single particle

Contacts

Many

particle simulation

Continuum Theory

Approach philosophy

3

Deterministic Models …

Navier Stokes

Kinetic Theory

Stat. Phys.

(Kinetic Theory)

…

Theory

LB

DSMC

MC

ED

MD

Abbrev.

Lattice (Boltzmann) Models

Direct Simulation Monte Carlo

Monte Carlo (random motion)

Event Driven (hard particles)

Molecular dynamics (soft particles)

Method

DCCSE – steps in simulation

1. Setting up a model

2. Analytical treatment

3. Numerical treatment

4. Implementation

5. Embedding

6. Visualisation

7. Validation

1. Particle model

2. Kinetic theory

3. Algorithms for MD

4. FORTRAN or C++/MPI

5. Linux – research codes

6. xballs X11 C-tool

7. theory/experiment

4

What is Molecular Dynamics ?

1. Specify interactions

between bodies (for example:

two spherical atoms)

2. Compute all forces

3. Integrate the equations

of motion for all particles (Verlet,

Runge-Kutta, Predictor-Corrector, …)

with fixed time-step dti j i

j i

m →≠

=∑x f&&

j i→f

Equations of motion2

2

ii i

d rm

dtf=rr

Overlap ( ) ( )1

2i j i jd d r r nδ = + − − ⋅

r r r

$ ( )i j

ij

i j

r rn n

r r

−= =

−

r rr

r r

c w

ii i ic wf f f m g= + +∑ ∑

rr r

Forces and torques:

Normal

Contacts

Many particle

simulation

Discrete particle model

Contact if Overlap > 0

5

i ijf m kδ δ γδ= − = +&& &

- really simple ☺☺☺☺

- linear, analytical

- very easy to implement

Linear Contact modeli

f

δ

overlap ( ) ( )1

2i j i jd d r r nδ = + − − ⋅

r r r

rel. velocity ( )i jnδ = − − ⋅

r r r& v v

acceleration ( )δ = − − ⋅r r r&&

i ja a n

http://www.ica1.http://www.ica1.uniuni--stuttgartstuttgart.de/~.de/~luilui/PAPERS/coll2p./PAPERS/coll2p.pdfpdf

i ijf m kδ δ γδ= − = +&& &

- really simple ☺☺☺☺

- linear, analytical

- very easy to implement

Linear Contact modeli

f

δ

overlap ( ) ( )1

2i j i jd d r r nδ = + − − ⋅

r r r

rel. velocity ( )i jnδ = − − ⋅

r r r& v v

acceleration

http://www.ica1.http://www.ica1.uniuni--stuttgartstuttgart.de/~.de/~luilui/PAPERS/coll2p./PAPERS/coll2p.pdfpdf

( )( )

1if f

j

jii j i

i jij

ffa a n n f n

m m mδ

=−

= − − ⋅ = − − ⋅ = − ⋅

r r

rrrr r r r r&&

6

i ijf m kδ δ γδ= − = +&& &

- really simple ☺☺☺☺

- linear, analytical

- very easy to implement

Linear Contact model

elastic freq.0

ij

km

ω =

eigen-freq.

visc. diss.

0ijk mδ γδ δ+ + =& &&

2 02

ij ij

k

m m

γδ δ δ+ + =& &&

2

0 2 0ω δ ηδ δ+ + =& &&

2 2

0ω ω η= −

2ij

m

γη =

http://www.ica1.http://www.ica1.uniuni--stuttgartstuttgart.de/~.de/~luilui/PAPERS/coll2p./PAPERS/coll2p.pdfpdf

i ijf m kδ δ γδ= − = +&& &

- really simple ☺☺☺☺

- linear, analytical

- very easy to implement

Linear Contact model

elastic freq.0

ij

km

ω =

eigen-freq.

visc. diss.

( ) exp( )sin( )t t tδ η ωω

= −0v

0ijk mδ γδ δ+ + =& &&

2 02

ij ij

k

m m

γδ δ δ+ + =& &&

2

0 2 0ω δ ηδ δ+ + =& &&

2 2

0ω ω η= −

2ij

m

γη =

[]

( ) exp( ) sin( )

cos( )

t t t

t

δ η η ωω

ω ω

= − −

+

& 0v

contact duration ct π

ω=

restitution coefficient( )

exp( )

c

c

tr

tη

= −

= −0

v

v

http://www.ica1.http://www.ica1.uniuni--stuttgartstuttgart.de/~.de/~luilui/PAPERS/coll2p./PAPERS/coll2p.pdfpdf

7

Linear Contact model (mw=∞)

elastic freq.0

ij

km

ω =

eigen-freq.

visc. diss.

2 2

0ω ω η= −

2ij

m

γη =

contact duration ct π

ω=

restitution coeff. exp( )c

r tη= −

particle-particle particle-wall

00

2

wall

i

km

ωω = =

2 2

0 2 4wallω ω η= −

2 2

wall

im

γ ηη = =

wallwallc c

t tπω

= >

exp( )wall wall wall

cr tη= −

http://www.ica1.http://www.ica1.uniuni--stuttgartstuttgart.de/~.de/~luilui/PAPERS/coll2p./PAPERS/coll2p.pdfpdf

Time-scales

contact duration ct π

ω= wall

wallc ct tπ

ω= >

time-step50

ct

t∆ <=

time between contacts

n ct t<

n ct t>

sound propagation ... with number of layers L c L

N t N

experiment T

http://www.ica1.http://www.ica1.uniuni--stuttgartstuttgart.de/~.de/~luilui/PAPERS/coll2p./PAPERS/coll2p.pdfpdf

8

Time-scales

contact duration ct π

ω= argl e small

c ct t>

time-step50

ct

t∆ <=

different sized particlesn c

t t<

n ct t>

sound propagation ... with number of layers L c L

N t N

experiment T

http://www.ica1.http://www.ica1.uniuni--stuttgartstuttgart.de/~.de/~luilui/PAPERS/coll2p./PAPERS/coll2p.pdfpdf

time between contacts

Equations of motion2

2

ii i

d rm

dtf=rr

Overlap ( ) ( )1

2i j i jd d r r nδ = + − − ⋅

r r r

$ ( )i j

ij

i j

r rn n

r r

−= =

−

r rr

r r

c w

ii i ic wf f f m g= + +∑ ∑

rr r

Forces and torques:

Normal

Contacts

Many particle

simulation

Discrete particle model

Contact if Overlap > 0

9

Convection

Segregation

10

Applications of Molecular Dynamics

• Gas-/Liquid-simulations

• Granular Materials

• Electro-spray

• Polymers, Membranes, …

• Process/Battlefield-simulations

• … and many others …

Molecular Dynamics examplefrom astrophysics

11

Algorithmic trick(s) for speed-up

• Linked cells neighborhood search O(1) (short range forces)

• Linked cells update after 10-100 time-steps O(N )

What is Molecular Dynamics ?

1. Specify interactions

between bodies (for example:

two spherical atoms)

2. Compute all forces

3. Integrate the equations

of motion for all particles (Verlet,

Runge-Kutta, Predictor-Corrector, …)

with fixed time-step dti j i

j i

m →≠

=∑x f&&

j i→f

12

Rigid interaction (hard spheres)

Stiff (rigid) interactions require dt=0

Events (=collisions) occur in zero-time(instantaneously)

that means: Integration is impossible !

1. Propagate particles between collisions

2. Identify next event (collision)

3. Apply collision matrix

Why use hard spheres ?

+ advantages

• Event driven (ED) is faster than MD

• Analytical kinetic theory is available

(with 99.9% agreement)

– drawback

• Implementation of arbitrary forces is expensive

• Parallelization is less successful

13

Why use hard spheres ?

+ advantages

• Event driven (ED) is faster than MD

• Analytical kinetic theory is available

(with 99.9% agreement)

– drawback

• Implementation of arbitrary forces is expensive

• Parallelization is less successful

Algorithm (serial)

0. Initialize

• Compute all forces O(1 )

• Integrate equations of motion t+dt

• O(N ) – goto 1.

Total effort: O(N )

14

Rigid interaction (hard spheres)

0. Stiff (rigid) interactions require dt=0

Events (=collisions) occur in zero-time (instantaneously)

Integration is impossible !

1. Propagate particles between collisions

2. Identify next event (collision)

3. Apply collision matrix

Rigid interaction (hard spheres)

1. Stiff (rigid) interactions require dt=0

Events (=collisions) occur in zero-time(instantaneously)

15

Rigid interaction (hard spheres)

2. Solve equation of motion between collisions

- trajectory

- contact

- event-time

( ) ( ) ( ) 212

0 0i i it t t= + +x x v g

( ) ( ) 1 2ij i jt t r r∆ = − = +x x x

( ) ( )( ) ( )2 2

1 20 0ij ij t r r∆ + ∆ = +x v

( ){

22 2 2

1 2 0ij ij ij ij

bc

t

a

r r t∆ − + + ∆ ⋅∆ + ∆ =x x v v1442443 14243

2

1,2

4

2

b bt

ca

a

− ± −=

Time evolution

position

tn

time

16

Rigid interaction (hard spheres)

( )1,2 1,2 1,21 2P mr= ± ∆+′v v

Collision rule (translational)

Momentum conservation + dissipation

with restitution coefficient (normal): r

Rigid interaction (hard spheres)

Collision rule (translational and rotational)

Restitution coefficient (normal): r (tangential) rt

( )1,2 1,2 1,21 2P mr= ± ∆+′v v ( )1,2 1,2 1,21 2t Lr Iω ω′ ± ∆= +

17

Time evolution

position

time

tn tn+1 tn+2

Algorithm (ED serial)

0. Initialize

• Propagate particle(s) to next event O(1)

• Compute event (collision or cell-change)

• Calculate new events and times O(1)

• Update priority queue (heap tree) O(logN )

• O(N ) – goto 1.

Total effort: O(N logN )

18

Algorithmic trick(s) for speed-up

• Linked cells neighborhood search O(1) (short range forces)

• Linked cells update not needed !

Performance

• Short range contacts

• Linked cells neighbourhood search

Cells per particle

low density high density

19

Algorithm (parallel)

0. Initialize

• Communication between processors

• Process next events tn to tn+m (see serial)

• Send and receive border-particle info

• If causality error then rollback goto 2.

• Synchronisation (for load-balancing and I/O)

• goto 1.

Parallelization – communication

processor 1 processor 2

border zone

20

Parallelization – load balancing

processor 1

processor 4

processor 3

processor 2

Parallelization – load balancing

processor 1

processor 4

processor 3

processor 2

21

Parallelization – load balancing

processor 1

processor 4

processor 3

processor 2

Performance (fixed N)

• Required memory per processor [MByte]1 2

33

c cN c

P P

∝ + +

22

Performance (3D fixed N)

• Fixed density and number of particles

1/ 2P∝

62.10N C= =

Performance (2D fixed N)

• Fixed density and number of particles

1/ 2P∝

62 10N C= =

23

Performance (3D fixed N/P)

• Fixed number of particles per processor

1/ 2P∝

4/ 4.10N P =

Two-phaseFlows

24

Two phaseFlows

Two phaseFlows withAggregation

25

From Boltzmann (low density) …

• binary collisions

• successive collisions are uncorrelated

• neglect boundary effects

• collision rate & pressure increase with density

• add coll.-transport of energy and momentum

… to Chapman-Enskog (high density)

Elastic Hard Sphere Model

… plot streaklines

simulate 2000 particlesin a peridoc box

26

Hard Sphere Modelν=0.01ν=0.25ν=0.70ν=0.80

r=0.95

ν=0.70

27

r =0.90

r =0.80

28

Collision parameter

Contact probability – correlation function

29

Collision rate – time scale

Pressure (Equation of State – 2D)

fluid, disordered

solid, ordered

phase transition

at critical density

PV/E-1=2ννννg(νννν)

30

Pressure (Equation of State – 3D)

fluid, disordered

solid, ordered

phase transition

at critical density

Pressure (Equation of State – 3D)

solid, ordered

phase transitionat critical density

grow … shrink …P P

d

d

a

t=

31

Pressure (Equation of State – 3D)

fluid, disordered

solid, ordered

phase transitionat critical density

slow … slower …

P Pd

d

a

t=

Elastic hard spheres in gravity

• N particles

• Kinetic Energy

• What is the density profile ?

gravity

32

Elastic hard spheres in gravity

• N particles

• Kinetic Energy

• What is the density profile ?

gravity

Elastic hard spheres in gravity

• N particles

• Kinetic Energy

• What is the density profile ?

gravity

33

Elastic hard spheres in gravity

• Pressure P global equation of state

• Shear Stress

• Energy Dissipation Rate I=0

0t

∂=

∂

0 i

i

P gx

ρ∂

= − +∂

elastic steady state:

mass & energy conservation – OK

momentum balance:

dev 0ijσ =

0iu I= =

Hard sphere gas in gravity

fluid, disorderedexponential tail

solid, ordered

phase transitionat critical density

34

Shear (energy and rate)

Shear (pressure and viscosity)

p µ

35

Shear (viscosity at high density)

homogeneous

inhomogeneous

⇒ dilatancy

critical density

Structure formation

Low density -> linear velocity profile

High density -> shear localization

36

Structure formation

Shear (first normal stress difference)

37

Equations of motion2

2

ii i

d rm

dtf=rr

Overlap ( ) ( )1

2i j i jd d r r nδ = + − − ⋅

r r r

$ ( )i j

ij

i j

r rn n

r r

−= =

−

r rr

r r

c w

ii i ic wf f f m g= + +∑ ∑

rr r

Forces and torques:

Normal

Contacts

Many particle

simulation

Discrete particle model – gravity ?!

Contact if Overlap > 0

Astrophysics

38

Experiment & Model for Sprays

• EHDA = ElectroHydroDynamic Atomisation

• DCT/PART – Marijnissen, Geerse, Winkels, …

liquid

high voltage

Experiment

•• Deposition analysisDeposition analysis: : TinopalTinopal visible with UVvisible with UV--lightlight

•• Applications: Applications:

GreenGreen--house sprays, medical inhalation,house sprays, medical inhalation,

Spray painting, Spray painting, nanonano--particle production particle production

39

Model

•• Solve NewtonSolve Newton’’s second law for every droplet s second law for every droplet ii

•• Forces acting on a dropletForces acting on a droplet

–– electric field forceelectric field force

–– gravitation forcegravitation force

riq E

im gr

ii i

d( v )F m

dt=∑

rr

–– drag forcedrag force

–– Coulomb interactionsCoulomb interactions

2( )

8D air i air i air iC d v v v v

πρ − −

r r r r

3

04

Nj ij

i

j i ij

q rq

rπε≠

∑r

Model

40

•• Real number of droplets is impossibleReal number of droplets is impossible

•• ScaleScale--up method to simulate the spray up method to simulate the spray

–– use few droplets with increased Coulomb interactionsuse few droplets with increased Coulomb interactions

,i Coulomb q iq f q=3

04

Nq j ij

q i

j i ij

f q rf q

rπε≠

∑r

Model

•• ScaleScale--up correlation between up correlation between ffqq and and ttprodprod

•• validity scalevalidity scale--upup 0 57

1 06(1) (1)

(2) (2)

.

.q prod

q prod

f t

f t

=

0

5

10

15

20

25

30

0 0.005 0.01 0.015 0.02 0.025

droplet production time tprod (s)

Co

ulo

mb

ch

arg

e f

ac

tor

f q

(-)

<rdep> = 0.0041

stdev = 5 . 10-5

Results

41

Electro-spray Model

•• DepositionDeposition

–– experimentexperiment –– simulationsimulation

y

x

Results

42

•• Density, shape and velocityDensity, shape and velocity

0.5 m/s

Results

•• SummarySummary

–– ScaleScale--up method worksup method works

–– EHDA spray can be qualitatively simulatedEHDA spray can be qualitatively simulated

–– Some simplificationsSome simplifications in the in the modelmodel

Results

43

•• Model including scaleModel including scale--up method worksup method works

–– validity has to be checked with other experimentsvalidity has to be checked with other experiments

•• MainMain challenges (for challenges (for modelingmodeling))

–– aeroaero--//hydrohydro--dynamics coupling dynamics coupling

–– more (more (chargedcharged) ) particlesparticles (2000 (2000 toto 101044--101055) )

Open questions

• Introduction

• Single Particles

• Particle Contacts/Interactions

• Many particle cooperative behavior

• Applications/Examples

• Conclusion

Single particle

Contacts

Many

particle simulation

Continuum Theory

Approach philosophy

E x p e r i

m e n t s

44

0.0 0.2 0.4 0.6 0.80.0

0.1

0.2

0.3

0.4

0.5

Norm

aliz

ed a

dhes

ion

forc

e (m

N/m

)

Normalized load (mN/m)

Contact force measurement (PIA)

Contact Force Measurement

45

Adhesion and Friction

20 30 40 50 60 70

65

70

75

80

85

Adh

äsio

ns [n

N]

Relative Luftfeuchte (%)

-1000-750 -500 -250 0 250 500 7500

100

200

300

400

500

Reibungskraft (nN)

Andruckkraft (nN)

Friction force (Friction force (nNnN))Adhesion force (Adhesion force (nNnN))

Normal force (Normal force (nNnN))relrel. humidity (%). humidity (%)

0 50 100 150 200 250-5

0

5

10

15

20Polystryrene

Defo

rmation

Hys

tere

sis

in n

m

Force in nN

0 1000 2000 3000-5

0

5

10

15

20

Borosilicate glass

Defo

rmation H

yste

resis

in n

m

Force in nN

Hysteresis (plastic deformation)

Collaborations:Collaborations:

MPIMPI--Polymer Science (Butt et al.)Polymer Science (Butt et al.)

Contact properties via AFMContact properties via AFM

46

Equations of motion2

2

ii i

d rm

dtf=rr

Overlap ( ) ( )1

2i j i jd d r r nδ = + − − ⋅

r r r

$ ( )i j

ij

i j

r rn n

r r

−= =

−

r rr

r r

c w

ii i ic wf f f m g= + +∑ ∑

rr r

Forces and torques:

Normal

Contacts

Many particle

simulation

Discrete particle model

Contact if Overlap > 0

( )1

2 0

for loading

for un-/reloading

- for unloading

hys

i

c

k

f k

k

δ

δ δ

δ

= −

( )max

0 1 2 max

2 1min max

2

min min

1 /

c

c

k k

k k

k k

f k

δ

δ δ

δ δ

δ

= −

−=

+

= −

Maximum overlap

stress-free overlap

strong attraction

attractive for

est at:

the max. : ce

- (too) simple ☺☺☺☺

- piecewise linear

- easy to implement

Contact model

47

1 for un-/re-loadinghys

i

k

f

δ

=

- (really too) simple ☺☺☺☺

- linear

- very easy to implement

Linear Contact model

3/ 2

1 for un-/re-loadinghys

i

k

f

δ

=

- simple ☺☺☺☺

- non-linear

- easy to implement

Hertz Contact model

48

3D

Anisotropy 3D

Sound3-dimensional modeling of sound propagation

Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft

Stefan Luding, s.luding@tnw.tudelft.nl

P-wave shape and speed

49

•• AgglomerationAgglomeration

–– population balance population balance –– cluster evolutioncluster evolution

–– phase transitions, cooperative phase transitions, cooperative behaviorbehavior

•• MainMain challenges (for challenges (for modelingmodeling))

–– aeroaero--//hydrohydro--dynamics coupling dynamics coupling

–– cluster cluster stabilitystability, , statisticsstatistics, , sinteringsintering, ..., ...

Open questions