1.021 , 3.021, 10.333, 22.00 I ntroduc tion to Modeling and Simulation

Spring 2011

Part I – C ontinuum and partic le me thods

How to model chemical interactions II

Lecture 6

Markus J. Buehler

Laboratory for Atomistic and Molecular Mechanics Department of Civil and Environmental Engineering Massachusetts Institute of Technology

Content overview

I. Particle and continuum me thods

1. Atoms, molecul e s, chemistry

2. Continuum modeling approac hes and solution approaches

3. Statistical mechanics

4. Molecular dynamics, Monte Carlo

5. Visualization and data analysis

6. Mechanical proper ties – applic ation: how things fail (and how to prevent it)

7. Multi-scale modeling par adigm

8. Biological systems (simulation in biophysics) – h ow proteins work and how to model them

II. Quantum mechanical methods

1. It’s A Q uantum World: T he Theory of Quantum Mechanics

2. Quantum Mechanics: Practice Makes Perfect

3. The Many-Body Problem: Fr om Many-Body to Single- Particle

4. Quantum modeling of materials

5. From Atoms to Solids

6. Basic pr operties of mater i als

7. Advanced proper ties of materials

8. What else can we do?

Lectures 2-13

Lectures 14-26

2

Overview: Material covered so far…

 Lecture 1: Broad introduction to IM/S

 Lecture 2 : Introduction to atomistic and continuum modeling (multi-scale modeling paradigm, difference between continuum and atomistic approach, case study: diffusion)

 Lecture 3 : Basic statistical mechanics – p roperty calculation I (property calculation: microscopic states vs. macroscopic properties, ensembles, probability density and partition function)

 Lecture 4 : Property calculation II (Monte Carlo, advanced property calculation, introduction to chemical interactions)

 Lecture 5: How to model chemical interactions I (example: movie of copper deformation/dislocations, etc.)

 Lecture 6: How to model chemical interactions II 3

Lecture 6: How to model chemical interactions II

Outline:

1. Cas e study: Deformation of copper wire (cont’d)

2. How to model metals: Multi-body potentials

3. Brittle versus ductile materials

4. Basic deformation mechanism in brittle materials - c rack extension

Goal of today’s lecture:

 Complete example of copper deformation

 Learn how to build a model to describ e brittle fracture (from scratch)

 Learn bas ics in fracture of brittle materials

 Apply our tools to model a particu lar material phenomena – b rittle fracture ( useful for pset #2 )

1. Case study: Deformation of copper wire (cont’d)

A simulation with 1,000,000,000 particles Lennard-Jones - copper

Fi g. 1 c from Bu eh ler, M . , et al . "T he Dynami cal Complexity of Work - H arden i n g: A Large-Scale Molecu l a r Dynami cs Simu l a t i on." Ac ta Mech Sinica 21 (2 005): 103-11.

© S prin g e r- V e rlag. A ll r i gh t s r e s e rv ed. T h is c on t en t is ex cl u d e d f r om ou r C r ea t ive C o mm on s 6

lic e n s e. For more infor m at ion, s ee http:/ /ocw. m it.edu/fai ruse .

Image by MIT OpenCou r seWare.

Strengthening caused by hindering dis l ocation motion

If too difficult, ductile modes break down and material becomes brittle 7

Fig. 1 c from B u e h l e r, M. e t al. "The D y na mica l Co mp le xity o f Wo rk - H a rdening: A La rge- Sca l e Mo le cu la r D y na mics Simu la tion." Ac ta Me ch Sin i ca 21 (2005) : 10 3 - 111.

© S p r inge r - V e r lag . All r i ghts rese r v ed . Th is content is e x c l uded from ou r 8

C r eative Co mmons license . F o r mo re in fo r m a t ion , see h ttp ://ocw . mit.edu/fa iruse .

Parameters for Morse potential

(for reference)

Morse potential parameters for various metals

Morse Potential Parameters for 16 Metals

Pb	2.921	83.02	7.073	1.1836	3.733	0.2348
Ag	2.788	71.17	10.012	1.3690	3.1 15	0.3323
Ni	2.500	51.78	12.667	1.4199	2.780	0.4205
Cu	2.450	49.1 1	10.330	1.3588	2.866	0.3429
Al	2.347	44.17	8.144	1.1646	3.253	0.2703
Ca	2.238	39.63	4.888	0.80535	4.569	0.1623
Sr	2.238	39.63	4.557	0.73776	4.988	0.1513
Mo	2.368	88.91	24.197	1.5079	2.976	0.8032
W	2.225	72.19	29.843	1.41 16	3.032	0.9906
Cr	2.260	75.92	13.297	1.5721	2.754	0.4414
Fe	1.988	51.97	12.573	1.3885	2.845	0.4174
Ba	1.650	34.12	4.266	0.65698	5.373	0.1416
K	1.293	23.80	1.634	0.49767	6.369	0.05424
Na	1.267	23.28	1.908	0.58993	5.336	0.06334
Cs	1.260	23.14	1.351	0.41569	7.557	0.04485
Rb	1.206	22.15	1.399	0.42981	7.207	0.04644

Adapted from Table I in Girifalco, L. A., and V. G. Weizer. "Application of the Morse Potential Function to Cubic Metals." Physical Review 114 (May 1, 1959): 687- 690 .

Image by MIT OpenCou r seWare.

 ( r ij ) 

D exp   2  ( r ij

 r 0 )   2 D exp    ( r ij  r 0 ) 

Morse potential: application example (nanowire)

See: Komanduri , R., et al . " M o l ecu lar D y n a mic s (M D) Sim u la t i on of U n i a xi al Tension of Some Si ngle-Cr ystal Cubi c Metals at Nanolevel ." Int e rn ati o nal Journal of Mech anical Sci e n c es 43, no. 10 (2001): 2237-60.

Further Morse potential parameters:

Courte sy of El se vie r, Inc., h t t p ://www. scien c e dir ec t . c o m . Used wi th permission.

Cutoff-radius: saving time

Cutoff radius

U i 

  ( r ij )

j  1

6

…

5

j=1

i

4

2

3

r cut

U i 



  ( r ij )

j  1 .. N neigh

r

Cutoff radius = considering interact ions only to a certain distance Basis: Force contribution negligible (slope) 13

Derivative of LJ potential ~ force

0.2

0.1

F = _ d  ( r )

d r

0

P otential shift

-0.1

-0.2

2

r 0

3

4

5

r (A)

o

Image by MIT OpenCou r seWare.

Beyond cutoff: Changes in energy (and thus forces) small

Putting it all together…

15

MD updating scheme: Complete

(1) Updating method (integration scheme)

r i ( t 0   t )   r i ( t 0   t )  2 r i ( t 0 )  t  a i

( t 0

)   t  2

 ...

Positions at t 0 -  t

Positions at t 0

Accelerations at t 0

(2) Obtain accelerations from forces

(3) Obtain forces from potential

(4) Crystal (initial conditions)

Positions at t 0

F  

d  ( r )

d r

f i  F i

r

Potential

    12

   6 

 ( r ) 

      16

 

2.2 How to model metals: Multi-body

potentials

Pair potential : Total energy sum of all pairs of bonds

Individual bond contribution does not depend on other atoms

“ all b onds are the same ”

N N

Court e sy of the Cent er for Pol y mer St ud ies at Bost on Un iversity. Used with permis sion.

Is this a good assumption?

U total 1

   ( r ij )

i  1 , i  j j  1

17

Are all bonds the same? - v alency in hydrocarbons

Ethane C 2 H 6

(stable configuration)

H

All bonds are not the same! Adding another H is not favored

18

Are all bonds the same? – metallic systems

Surface

Bulk

stronger

+ different bond EQ distance

Pair potentials: All bonds are equal!

Reality: Have environment effects; it matter that there is a free sur f ace!

Bonds depend on the environment! 19

Are all bonds the same?

Bonding energy of red atom in is six times bonding energy in

This is in contradiction with bot h experiments and more accurate quantum mechanic al calc ulations on many materials

Bonding energy of atom i

U i 

  ( r ij )

j  1

Are all bonds the same?

Bonding energy of red atom in is six times bonding energy in

This is in contradiction with bot h experiments and more accurate quantum mechanic al calc ulations on many materials

For pair potentials ~ Z

For m e tals ~

Z : Coordination = how many immediate neighbors an atom has

Bond s get “weaker” as more atoms are added to central atom

Bond strength depends on coordination

energy per bond

potential

~

Nickel

Z

2 4 6 8 10 12 coordination

22

Da w, Foile s, Baskes, Mat. Science Reports , 1993

Transferability of pair potentials

 Pair potentials have limited transferability:

Parameters determined for molecules can not be used for crystals, parameters for specific types of crystals can not be used to describe range of crystal structures

 E.g. difference between FCC and BCC can not be captured using a pair potential

Metallic bonding: multi-body effects

 Need to consider more details of chemical bonding to understand environmental effects

Electron (q=-1) Ion core (q=+N)

Delocalized valence electrons moving between nuclei generate a binding force to hold the atoms together: Electron gas model ( positive ions in a sea of electrons )

Mostly non-directional bonding, but the bond strength indeed depends on the environment of an atom, precisely the electron density imposed by other atoms

Concept: include electron density effects

 

Each atom features a particular distribution of electron density

Concept: include electron density effects

Electron density at atom i

 i     , j ( r ij )

j  1 .. N neigh

r ij

 x j

 x i

j i

  , j ( r ij )

Atomic electron density contribution of atom j to atom i

Contribution to electron density at site i due to electron density of atom j evaluated at distance r ij 26

Concept: include electron density effects

 i  2

  ( r ij ) 

F (  i )

j  1 .. N neigh

embedding term F (how local electron density contributes to potential energy)

Electron density at atom i

  , j

( r ij )

 i     , j ( r ij )

j  1 .. N neigh

r ij

 x j

 x i

j i

x j x i

Embedded-atom method (EAM)

Atomic energy

1

new

Total energy

N

 i    ( r ij ) 

F (  i )

U total

   i

2 j  1 .. N

neigh

i  1

Pair potential energy Embedding ener gy

as a function of electron density

 i Electron density at atom i

based on a “pair potential”:

 i 

   , j ( r ij )

j  1 .. N neigh

First proposed by Finnis, Sinclair, Daw, Baskes et al. (1980s) 28

Embedding term: example

Embedding energy

   1  ( r )  F (  ) 

Image by MIT OpenCou r seWare.

Electron density

i   ij i 

j  1 .. N neigh

Pair potential energy

Embedding energy

as a function of electron dens ity 29

Pair potential term: example

Pair contribution

 ( r ij )

Image by MIT OpenCou r seWare.

   1  ( r )  F (  ) 

Distance

r ij

i   ij i 

j  1 .. N neigh

Pair potential energy

Embedding ener gy

as a function of electron density 30

Effective pair interactions

r

+ + + + + +

+ + + + + +

+ + + + + +

r

+ + + + + +

+ + + + +

+ + + + +

+

+

Image by MIT OpenCou r seWare.

Can describe differences between bulk and surface

31

See also: Daw, Foi l e s , Ba ske s, Mat. Science Reports , 1993

Comparison with experiment

Diffusion: Activation energies

(in eV)

Courte sy of El se vie r, Inc., h t t p ://www. scien c e dir ec t . c o m . Used wi th permission.

Comparison EAM model vs. experiment

Melting temperature (in K)

Summary: EAM method

 State of the art approach to model metals

 Very good potentials available for Ni, Cu, Al since late 1990s, 2000s

 Numerically efficient, can treat billions of particles

 Not much more expensive than pair potential (approximately three times), but describes physics much better

 Strongly recommended for use!

3. Brittle versus ductile materials

Brittle

Ductile

Strain

Tensile test of a wire

Image by MIT OpenCou r seWare. Image by MIT OpenCou r seWare.

Ductile versus brittle materials

BRITTLE

DUCTILE

Glass Polymers Ice...

Copper , Gold

Shear load

Difficult to deform,

breaks easily

Image by MIT OpenCou r seWare.

Easy to deform hard to break

43

Deformation of materials:

Nothing is perfect, and flaws or cracks matter

Failure of materials initiates at cracks

Griffith, Irwine and others: Failure initiates at defects, such as cracks, or

grain boundaries with reduced tracti on, nano-voids, other imperfections 39

SEM picture of material: nothing is perfect

40

Significance of material flaws

“global”

“local”

Fig. 1.3 in Buehler, Ma r k us J. Atom istic Modeling of Materials Failure . Spring er, 2008. © S pringer. All rig h ts r e s e rved. This

c o n t ent is e x clud ed from our C r ea t i ve C o mmons li cense. For more i n format ion, see ht tp:/ /ocw.mi t.edu/f ai ruse . 41

Stress concentrators: local stress >> global stress

Deformation of materials:

Nothing is perfect, and flaws or cracks matter

“Macro, global”

 ( r )

“M icro (nano), local”

r

Failure of materials initiates at cracks

Griffith, Irwine and others: Failure initiates at defects, such as cracks, or

grain boundaries with reduced tracti on, nano-voids, other imperfections 42

Cracks feature a singular stress field, with singularity at the tip of the crack

stress tensor

 ( r ) ~ 1

y

x

K I :

Stress intensity factor (function of geometry)

Image by MIT OpenCou r seWare.

43

Crack extension: brittle response

 ( r )

r

Large stresses lead to rupture of chemical bonds between atoms

Thus, crack extends

44

Image by MIT OpenCou r seWare.

Lattice shearing: ductile response

 Instead of crack extension, induce shearing of atomic lattice

 Due to large shear stresses at crack t ip

 Lecture 5









Image by MIT OpenCou r seWare.

Image by MIT OpenCou r seWare.

Brittle vs. ductile material behavior

 Whether a material is ductile or brittle depends on the material’s propensity to undergo shear at the crack tip , or to break atomic bonds that l e ads to crack extension

 Intimately link ed to the atomic structure and atomic bon d ing

 Related to temperature (activated process) ; some mechanism are eas ier accessible under higher/lower temperature

 Many materials show a propensity towards brittleness at low temperature

 Molecular dynamics is a quite suitable tool to study these mechanisms, that is, to find out what makes materials brittle or ductile

Historical example: significance of brittle vs. ductile fracture

 Liberty ships : cargo ships built in the U.S. during World War II (during 1930s and 40s)

 Eighteen U.S. shipy a rds built 2, 751 Liberties between 1941 and 1945

 Early Liberty ships suffered hull and de ck cracks, and several were lost to such structural defects

 Twelv e ships, including three of the 2710 Liberties built, broke in half without warning, including the SS John P. Gaines (sank 24 November 1943)

 Constance Tipper of Cambridge University demonstrated that the fractures were initiated by the grade of steel used whic h suffered from embrittlement .

 She discovered that the ships in t he North Atlantic were exposed to temperatures that could fall below a critical point when the mechanism of failure changed from ductile to brittle , and thus the hull could fracture relatively easily.

Liberty ships: brittle failure

48

Court e sy of the U.S. N a v y .

4. Basic deformation mechanism in brittle materials - crack extension

Introduction: brittle fracture

 Materials: glass, silicon, many ceramics, rocks

 At large loads, rather than accommodating a shape change, materials break

Imag e court e sy of quin n.anya . Li ce nse : CC - B Y .

Science of fracture: model geometry

 Typically consider a single crack in a crystal

 Remotely applied mechanic al load

 Follow ing dis c ussion focused on single cracks and their behav ior

remote load

a

remote load

51

Image by MIT OpenCou r seWare.

Brittle fracture loading conditions

 Commonly cons ider a single crack in a material geometry, under three types of loading: mode I, mode II and mode III

Mode I

Mode II

Mode III

T ensile load, focus of this lectur e

Image by MIT OpenCou r seWare.

Tens ile load, focus

of this lecture 52

Brittle fracture mechanisms: fracture is a multi- scale phenomenon, from nano to macro

Imag e removed du e t o copyrigh t re str i c t ion s . S e e Fig. 6.2 in Bu eh ler, Markus J.

At omistic M o deling of Materials Fa ilure . Sprin ger, 2008.

Focus of this part

 Basic fracture process : diss ipation of elastic energy

 Fracture initiation , that is, at what applied load to fractures initiate

 Fracture dynamics , that is, how fast can fracture propagate in material

Basic fracture process: dissipation of elastic energy

Undeformed Stretching=store elastic energy Release elastic energy

dissipated into breaking

chemical bonds 55

Elasticity = reversible deformation

Stress?

cross-sectional

area

A

Force per unit area

Elasticity = reversible deformation

Stress?

cross-sectional area

A

  F / A

Force per unit area

Elasticity = reversible deformation

cross-sectional area

  F / A

A

area under curve: stored energy

  E 

Young’s modulus

   u / L

58

Continuum description of fracture

 Fracture is a diss ipative process in which elastic energy is dissipated to break bonds (and to heat at large crack speeds)

 Energy to break bonds = surface energy  s (energy necessary to create new surface, dimens ions: energy/area, Nm/m 2 )

"Relaxed" element



"Strained" element

a ~

 a

a ~

a

(2)



(1)

Image by MIT OpenCou r seWare.



59

Continuum description of fracture

 Fracture is a diss ipative process in which elastic energy is dissipated to break bonds (and to heat at large crack speeds)

 Energy to break bonds = surface energy  s (energy necessary to create new surface, dimens ions: energy/area, Nm/m 2 )

"Relaxed" element

a ~

a

"Strained" element

 a a ~



2

2 E

1  2

(2)

1  2 ~

 (1)

Image by MIT OpenCou r seWare.

B out-of-plane thickness

W P ( 1 )  2

V 

E 2 E

 a B

W P ( 2 )  0 60

Continuum description of fracture

 Fracture is a diss ipative process in which elastic energy is dissipated to break bonds (and to heat at large crack speeds)

 Energy to break bonds = surface energy  s (energy necessary to create new surface, dimens ions: energy/area, Nm/m 2 )

1 

2

~

a B

"Relaxed" element



"Strained" element

a ~

 a

a ~

2 E





1  2

(1)

Image by MIT OpenCou r seWare.

V 

1  2

 a ~ B



a

(2)



2

2 E

W P ( 1 )  2 E 2 E

W P ( 2 )  W P ( 1 ) 

W P ( 2 )  0 61

Continuum description of fracture

 Fracture is a diss ipative process in which elastic energy is dissipated to break bonds (and to heat at large crack speeds)

 Energy to break bonds = surface energy  s (energy necessary to create new surface, dimens ions: energy/area, Nm/m 2 )