modeling’mechanics’of’materials’–’ part’1:’rela=on’to ... ·...

Izabela Szlufarska

Associate Professor

Department of Materials Science & Engineering

University of Wisconsin – Madison

[email protected]

Modeling mechanics of materials – Part 1: Rela=on to microstructure Part 2: Accelerated MD simula=ons

May 21, 2013 1

Experiments and simulations can be performed at matching length scales

Atomic Force Microscope (AFM)

Szlufarska, Chandross, Carpick, “Recent advances in single asperity nanotribology” J. Phys. D 41 123001 (2008) 2

Atomistic simulations

Rough engineering contacts Single asperity contacts

•  Roughness theories

•  Scaling models

•  Isolate mechanisms •  Quan=fy contribu=ons to fric=on

Challenge of MD =me scales

3

MD =me scale are set by laEce vibra=ons

•  Typical Gme step ∆t ≈ 10-‐15 s •  Carry out 106 – 108 steps ⇒  total simulated Gme 1 ns – 0.1 µs

Mechanics studies •  Typical sliding velociGes in mechanics

simulaGons: 0.1 m/s – 100 m/s •  Typical sliding velociGes in AFM

experiments: ~ 1µm/s •  Similar Gme scale discrepancies exist

for crack velociGes

High strain rates may lead to: •  Higher dislocaGon densiGes than in

experiments •  Change in mechanisms (temperature

acGvated fricGon vs. ballisGc fricGon) •  Exclusion of important phenomena

(e.g., diffusion and creep)

Mechanics and Microstructure

4

Mechanical properGes/mechanics   Surface/interfacial chemistry

  Environment   Materials microstructure

•  Grain boundary processes (migraGon, sliding, diffusion)

•  DislocaGon dynamics •  Cracks, voids •  Phase transformaGon •  Mechanically induced phase

segregaGon •  2nd phase precipitates •  etc

Materials design

NanoindentaGon, NanoTribology

Outline

1.  KineGc Monte Carlo: applicaGon to mechanics

2.  On the fly KineGc Monte Carlo: applicaGon to mechanics

3.  Green Kubo relaGons: fricGon at solid-‐liquid interfaces

4.  Parallel replica dynamics: applicaGon to mechanics

5

6

∞ Continuum Equations

kMC

MD

Time (s) 10-15 10-12 10-9 10-6 ∞

Leng

th (m

)

10-‐6

10-‐8

10-‐11

Kine=c Monte Carlo: A Coarse-‐Grained, Atomis=c, LaEce-‐Based Technique for Condensed-‐MaSer Dynamics

Slide: Adapted from Dane Morgan (UW)

7

MD of Co on Cu(001): The Whole Trajectory

kMC: Coarse-‐Grained Hops between minima

Coarse Graining

Kine=c Monte Carlo: Coarse-‐Graining MD

Slide: Adapted from Art Vother (LANL) and Dane Morgan (UW)

8

Coarse-‐Graining for Kine=c Monte Carlo

•  We must reduce our system to 2 key properGes – Minima: These are discrete regions of phase space that together cover the phase space in which the system spend almost all of its Gme (these will be local minima of the potenGal energy surface)

– Rates: The probability per unit Gme of changing from state i to state j, denoted Rij (this can be determined from knowledge of energy barriers)

9

FCC Atom Migration Barrier by Vacancy Mechanism

Atom Position → <110>

Ene

rgy

GA

Rate of transi=on/event: R =ν exp −GA

kBT"

#$

%

&'

Slide: Adapted from Dane Morgan (UW)

10

MD of Co on Cu(001): The Whole Trajectory kMC: Coarse-‐Grained

Hops between minima

Coarse Graining

Minima

Rates of transitions

Kine=c Monte Carlo: Coarse-‐Graining MD

•  Assume we know the minima, labeled i=1,2,3,... •  Assume we know the rates Rij •  KineGc Monte Carlo (KMC) allows one to evolve the system in Gme by

hopping from one minimum to another and correctly accounGng for Gme that passed

Slide: Adapted from Art Vother (LANL) and Dane Morgan (UW)

Fric=on of silica in aqueous environments

Photo of San Andreas fault hip://en.wikipedia.org/

Shallow tectonic earthquake = fric=onal slip Rate and state fric=on law

Two hypotheses: •  Contact area increase as a funcGon of Gme (Mechanical DeformaGon) •  Adhesion in the contact increases as a funcGon of Gme (Change of Chemistry)

What is the state variable θ? What fundamental process is responsible for =me

evolu=on of fric=on? 11

Earthquakes Wafer bonding

K. Turner (Upenn)

Ageing in Single Asperity Contacts

Ageing ∆f increases logarithmically with the holding =me

[1] Li, Q., T. E. Tullis, D. Goldsby, R. W. Carpick, (2011). Nature 480(7376): 233-‐236

•  Ageing observed in the absence of plas=c deforma=on

•  Support the hypothesis about chemical ageing

Slide-‐hold-‐slide experiments: staGc fricGon depends on the holding Gme

1. what molecular level processes are responsible for ageing and 2. why there is logarithmic dependence on holding =me?

12

Hypothesis: Siloxane Forma=on Across the Interface

2. Silica structures 3. Simula=on methods

-‐ MD simulaGon using LAMMPS based on Reaxff force field for amorphous interface -‐ Density Func2onal Theory simulaGon using VASP based on pseudo potenGals for quartz surface

Amorphous SiO2

Quartz

1. Simula=on approach

a. Bring two surfaces together b. Allow siloxane to form

13

Si-‐OH + Si-‐OH → Si-‐O-‐Si + H2O

Amorphous silica surfaces – ReaxFF MD

•  Interface of 5nm x 5nm in size, which is comparable to experimental contact area. •  ReacGons on different sites are not independent of each other

•  InteracGon strength is randomly distributed as a result of amorphous surface structure. •  InteracGon is strongly biased toward increase of energy cost.

EinteracGon = ∆E2 – ∆E1

O

Si

Si

O

Si

Si

Si

O

H

H O

Si

Si

O

H

H O

Si

Si

O

H

H O

Si

O

Si

Si

∆E1

Si

O

H

H O

Si

O

Si

Si

∆E2

EinteracGon > 0: barrier higher due to interacGons EinteracGon < 0: barrier lower due to interacGons

A

A

A

A

B B

B B

Si

Si

Si

Si

Si Si

Si Si

Si

Si

Si

Si

Si Si

Si Si

O O O

O

O

O

O O

O O

O

O H

H

H

H H

H

H

H

14

Quartz surfaces – Ab ini.o calcula=ons

“para”

① ②

1. Reac=on energy of the second siloxane is influenced by the existence and orienta=on of already formed siloxane.

“anG”

① ②

•  Pressure dependence of the reacGon energy • InteracGon is biased toward increase of energy cost. •  InteracGon is orientaGon dependent.

2. Characteris=cs:

3. What is the physics underlying the interac=on?   ElectrostaGc: Direct Coulomb or dipole interacGon between bridges   ElasGc : ReacGon is influenced by previously formed siloxane induced deformaGon

IndentaGon Depth [Å] In

teracGon

ene

rgy [eV]

Parallel

An=-‐parallel

O

Si

Si

O

Si

Si

O

Si

Si

O

Si

Si

O O O O

Si

Si Si

Si Si

Si Si

Si

15

Hypothesis 1: Electrosta=c Interac=on

WITH SURROUNDING BULK (ELASTIC + ELECTROSTATIC) E (parallel) – E(anGparallel) = 0.483 eV

Direct electrosta=c interac=ons excluded.

WITHOUT SURROUNDING BULK (ELECTROSTATIC) E (parallel) – E(anGparallel) = 0.003 eV


teracGon

ene

rgy [eV]

Parallel

An=-‐parallel

16

Hypothesis 2: Elas=c Deforma=on of Bulk

•  We calculated the elasGc energies using empirical potenGal developed by Leeuw [6].

•  ElasGc energy matches with ab ini2o calculated

energy difference

•  InteracGon is mediated by deformaGon of bond angles of tetrahedra in the SiO2 structure

•  FormaGon of one bridge on the average sGffens the structure and makes it harder for other bridges to form

Si

O O θ

EelasGc = f(θ-‐θ0)


teracGon

ene

rgy [eV]

Parallel

An=-‐parallel

∆ Int. en

ergy [e

V]

IndentaGon depth [Å]

17

Y. Liu and I. Szlufarska, Chemical origins of frictional ageing, Physical Review Letters (2012)

Rela=on between chemistry and fric=on

For ionic-‐covalent materials that from strong direcGon bonds, fricGon force is linear with the number of strong bonds formed across the interface

Y. Mo, K.T. Turner, I. Szlufarska, Nature 457, 1116-1119 (2009) Y. Mo, M. Muser, I. Szlufarska, Phys. Rev. B, 80, 155438 (2009)

C C C C C C C

C C C C C C C

H H H H H

H H H HH

Silica problem: •  MD is too short to simulate

chemical reacGons over ms, seconds and longer

•  FricGon is linearly dependent on the number of strong bonds formed across the interface

•  Determine reacGon rates – use analyGcal theory and KMC

18

The Mystery of Reac=on Rate

L. Prandtl, Z. Angew. Math. Mech. 8, 85 (1928).

Prandtl showed analyGcally that relaxaGon of interfaces for kineGc fricGon shows log behavior:

Fric=o

n

Log of =me [log(s)]

Sliding

•  Interfacial reac=on???

•  Simple reac=on (reactants concentra=on is constant):

[C]t = [C]0 + kt

•  Typical surface reac=on:

[C] vs =me is exponen=al

[A]t = [A]0 exp(−kt)

[C]t =1−[A]0 exp(−kt)

[C] vs =me is linear

k =ν exp −Eb

kBT"

#$

%

&'

19

•  First contact: high fric=on due to interlocking

•  Elas=c relaxa=on of asperi=es -‐> lower frc=on

The Mystery of Reac=on Rate Liu and Szlufarska, Chemical origins of fric2on aging, Phys. Rev. Lei. 109, 186102 (2012)

Prod

uc=o

n

Log of =me [log(s)]

•  Relaxa=on events = formaGon of chemical bonds •  First contact: Low fricGon (no bonding) •  With =me: bonding increase fricGon

20

If distribu=on of barriers is uniform

Then concentra=on of bonds (fric=on) increases logarithmically

with =me

The Mystery of Reac=on Rate Liu and Szlufarska, Chemical origins of fric2on aging, Phys. Rev. Lei. 109, 186102 (2012)

Prod

uc=o

n

Log of =me [log(s)]

21

DistribuGon of barriers from simulaGons

Is it valid?

•  Relaxa=on events = formaGon of chemical bonds •  First contact: Low fricGon (no bonding) •  With =me: bonding increase fricGon

If distribu=on of barriers is uniform

Then concentra=on of bonds (fric=on) increases logarithmically

with =me

Inside the black box: Kine=c Monte Carlo Model

1.  Every site starts with the energy barrier Eb that follows a certain distribuGon

2.  The interacGon between sites determined by ΔEb∈[e1, e2].

3.  We randomly choose ΔEb for each pair of sites

•  KMC model

3 2 3

2 1 2

3 2 3

When one siloxane is formed (at site 1 here)

3 2 3

2 1 2

3 2 3

Update surrounding barriers (increase/decrease)

3 2 3

2 1 2

3 2 3

Form another siloxane based on new barriers

22

Time evolu=on of the concentra=on of siloxane bridges

•  There is always a log regime, but it can be short •  Interac=ons extend the log regime •  Interac=ons flaSen the distribu=on of energy

barriers and shim it to larger barriers •  Within the range of physically jus=fiable

parameters, it is possible to match experimental data

Exp. Time

Pressure-‐contolled distribu=on

23

Distribu=on from simula=ons

Outline





24 End 3

No event list beforehand?

•  What if we couldn’t get the event for each KMC step beforehand? •  This is true for some complex systems: •  Ex1: simulaGon of amorphous material

  most atoms do not sit on a regular la�ce, which makes it difficult to define events beforehand

•  Ex2: migraGon of intersGGal clusters   Several intersGGals are strongly bonded to each other, so one single

step may require a coordinated movement of these atoms

•  Generate events on the fly! •  Several techniques have been developed to handle such problems,

and they can be classified into two categories: •  Open-‐ended techniques do not require knowledge of the final state

a propri: acGvaGon-‐relaxaGon technique(ART), dimer method;

25

Ac=va=on-‐Relaxa=on Technique (ART)

•  ART is an open-‐ended saddle point searching method. When combined with KMC algorithm, it can enable long Gme scale KMC simulaGons without la�ce approximaGon and predefined event list.

•  Given the iniGal minima, ART can search nearby saddle points by walking through the following 3 steps:

•  Step 1: leave local minima •  Step 2: find saddle point •  Step 3: relax into new minima

Mousseau, N. (2012). "The Activation-Relaxation Technique: ART Nouveau and Kinetic ART." Journal of Atomic, Molecular, and Optical Physics 2012: 925278. 26

ART Step 1: leave local minima

Make random displacement on selected atoms

Relax other atoms to avoid unphysical stress build up

Calculate the lowest eigenvalue of Hessian matrix

Nega=ve eigenvalue?

Lem local minima Go to Step 2

Hessian Matrix:

Yes

No

Hij =∂2E∂xi∂x j

i, j = 1,...,3N{ } 27

ART Step 2: find saddle point

Calculate the eigenvector of the lowest eigenvalue

Push selected atoms along this vector while relax others

Calculate the total force on the system

Force< threshold?

Found saddle point Go to Step 3

Yes

No

28

ART Step3: relax into new minima

Calculate the displacement from ini=al minima to saddle

Push atoms ~20% of this displacement further away

Relax system by any chosen relaxa=on algorithm

Force< threshold?

Relaxed into new minima End

Yes

No

29

Kine=c ART (K-‐ART)

•  A combinaGon of KMC algorithm and ART;

•  GeneraGng event list is the most expensive step in K-‐art;

•  Speed up: •  Can use empirical potenGal

instead of ab iniGal calculaGons to speed up.

•  Can apply topology analysis to categorize local configuraGons, so that ART could work only on few iniGal minima.

Ini=al configura=on

Generate event list by calling ART a certain =mes

Calculate rate of each event and the total rate

Enough steps?

End Yes

No

Select one event Update =me

30

Example: Relaxa=on of amorphous materials

•  Relaxa.on of an ion-‐bombarded c-‐Si •  SimulaGon se�ngs: 100,000 c-‐Si atoms bombarded by 3keV Si

atom at 300K, then annealed for 900ps by MD. •  The system is structurally too complex to know the events a priori

(for standard KMC). TradiGonal MD simulaGon cannot reach such a long Gme scale (µs or longer)

Ini2al configura2on of an ion-‐bombarded c-‐Si Energy of the system and number of topologies as a func2on of simula2on 2me

Jean-François Joly, L. K. B., Peter Brommer, Fedwa El-Mellouhi et Normand Mousseau (2012). "Optimization of the Kinetic Activation-Relaxation Technique, an off-lattice and self-learning kinetic Monte-Carlo method " Journal of Physics: Conference Series Series 341: 012007. 31

Outline





32 End

•  Slip velocity: the velocity difference at liquid/solid interface. •  Slip length: the extrapolated length where the velociGes of liquid and solid

match, also the inverse of fricGon coefficient.

Slip at the solid/liquid interface

33

0.1 nm 1 nm 10 nm 100 nm 1000 nm

Length scale

hydrophilic hydrophobic

Roach P. et al, Langmuir 23, 9823-9830 (2007)

l

0υx

z

( ) 0=− lυ

coefffrictionviscosity

==ηηl

•  Existence of slip is now accepted, but the effects of surface and liquid condiGons on slip are sGll poorly understood

•  Understanding slip is criGcal for micro and nanofluidics

Measuring slip is challenging Experiments: •  AFM: accurate but indirect •  Velocimetry technique: direct but

not accurate.

34 Peter A. Thompson & Sandra M. Troian NATURE |VOL 389 | 25 SEPTEMBER 1997

Simula=ons: •  Non-‐linear effect in non-‐equilibrium molecular dynamics (NEMD) simulaGons. •  Large staGsGcal error at low shear rates

lf ⇔*

Olga I. Vinogradova, Langmuir 1995,11, 2213-‐2220

Olga I. Vinogradova, et al, PRL 102, 118302 (2009)

Scaiered results in slip measurement

Mass transport in carbon nanotubes

35

Kannam et al. J. Chem. Phys. 138, 094701 (2013)

How to overcome the =me scale problem?

36

Goal: Develop a model that allows calculaGons of fricGon coefficient/slip length from MD simulaGons and that is not limited by the simulaGon Gme scales

Non-‐equilibrium MD: Simulate a dynamic response to an external force/perturbaGon

Equilibrium MD: Determine fluctuaGons of the

system at equilibrium

Linear response theory: One can calculate the transport coefficients of a slightly perturbed system from the fluctuaGons of the same system at its equilibrium state (Green Kubo relaGons).

Huang and Szlufarska, To be submiied (2013)

•  Diffusion/mobility velocity autocorrelaGon

External force/perturbaGon = concentraGon gradient

•  Liquid/solid fricGon coefficient (drag force) ???

Green-‐Kubo (GK) relaGons

37

Non-‐equilibrium MD: Simulate a dynamic response to an external force/perturbaGon

Equilibrium MD: Determine fluctuaGons of the

system at equilibrium

High Low

vi(0)

vi(t)

New Green-‐Kubo relaGon

η = −Fsu

ηtot =1A

ηii∑ηi =

Fi (0)Fi (t) ECdt

0

∞

∫

kBT − Fi (0)ui (t) ECdt

0

∞

∫

Coefficient of fricGon (non-‐equilibrium) can be related to fluctuaGons in force Fi

Equilibrium (new expression) Non-‐equilibrium sliding (standard MD simulaGons)

38

ValidaGon from MD simulaGons

39

•  LAMMPS •  Liquid: Hard-‐sphere and spring-‐bead flexible polymer (length 12, 100)

(48000 atoms) •  Solid: Lennard Jones potenGal fcc crystal •  10 independent runs to reduce noise

39

Equilibrium (new expression) Non-‐equilibrium sliding (standard MD simulaGons)

Comparison of NEMD and EMD results

40 40

N=12, ε=0.6

Equilibrium (GK relaGon)

Non-‐equilibrium (sliding)

•  Very efficient: equilibrium simulaGon is 500 Gmes faster than a single non-‐equilibrium calculaGon

•  This technique can be used for fast screening of surface chemistry to design opGmized interface

•  Excellent agreement validates the modified Green Kubo relaGon •  Agreement for a variety condiGons indicates that the relaGonship is

universal

Outline





41

Parallel Replica Dynamics Simula=ons of Fric=on

42

Li, Dong, Perez, MarGni, Carpick, Phys. Rev. Lei. 106, 126101 (2011)

•  The same materials and system geometry in simulaGons and experiments

•  SimulaGons with sliding velociGes up to mm/s, reducing the gap between simulaGons and experiments from 6 to 3 orders of magnitude

•  Technique applicable to crystalline well-‐defined interfaces (where it is easy to idenGfy a rare event/transiGon)

•  In the current form is not applicable to more complex interfaces (e.g., larger amorphous Gps)

Summary: Accelera=ng MD simula=ons for mechanics

43

•  Many techniques have been developed for acceleraGng MD simulaGons

•  A number of these techniques are applicable to mechanics problems

•  ApplicaGon of accelerated techniques to mechanics problems is not always possible

•  There is no one universal technique that can be applied to all problems

Thank You The END

44

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