MIT3_021JS11_P1_L8

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1.021, 3.021, 10.333, 22.00 Introduction to Modeling and Simulation

Spring 2011

Part I – Continuum and particle methods

Markus J. Buehler

Laboratory for Atomistic and Molecular Mechanics

Department of Civil and Environmental Engineering

Massachusetts Insti tute of Technology

Reactive potentials and applicationsLecture 8

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Content overview

I. Particle and continuum methods1. Atoms, molecules, chemistry

2. Continuum modeling approaches and solution approaches

3. Statistical mechanics

4. Molecular dynamics, Monte Carlo

5. Visualization and data analysis6. Mechanical properties – application: how things fail (and

how to prevent it)

7. Multi-scale modeling paradigm

8. Biological systems (simulation in biophysics) – how

proteins work and how to model them

II. Quantum mechanical methods1. It’s A Quantum World: The Theory of Quantum Mechanics

2. Quantum Mechanics: Practice Makes Perfect

3. The Many-Body Problem: From Many-Body to Single-Particle

4. Quantum modeling of materials

5. From Atoms to Solids

6. Basic properties of materials

7. Advanced properties of materials8. What else can we do?

Lectures 2-13

Lectures 14-26

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Overview: Material covered so far…

Lecture 1: Broad introduction to IM/S

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

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

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

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

Lecture 6: How to model chemical interactions II (EAM, a bit of ReaxFF— chemical reactions)

Lecture 7: Application – MD simulation of materials failure

Lecture 8: Appl ication – Reactive potentials and applications

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Lecture 8: Reactive potentials and applications

Outline:

1. Bond order force fields - how to model chemical reactions

1.1 EAM potential for metals1.2 ReaxFF force field

2. Hybrid multi-paradigm fracture models

Goal of today’s lecture:

Learn new potential: ReaxFF, to describe complex chemistry (bondbreaking and formation)

Application in hybrid simulation approaches (combine different forcefields)

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1. Bond order force fields - how tomodel chemical reactions

Potential energy expressions for more

complex materials/chemistry, including

bond formation and breaking

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Review: atomic interactions – different types of

chemical bonds

Primary bonds (“ strong” )

Ionic (ceramics, quartz, feldspar - rocks)

Covalent (silicon)

Metallic (copper, nickel, gold, silver)(high melting point, 1000-5,000K)

Secondary bonds (“ weak” )

Van der Waals (wax, low melting point) Hydrogen bonds (proteins, spider silk)

(melting point 100-500K)

Ionic: Non-directional (point charges interacting)

Covalent: Directional (bond angles, torsions matter)

Metallic: Non-directional (electron gas concept)

Difference of material properties originates from different atomic

interactions

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Interatomic pair potentials: examples

Morse potential

Lennard-Jones 12:6

potential

(excellent model for nobleGases, Ar, Ne, Xe..)

Buckingham potential

Harmonic approximation(no bond breaking)

)(exp2)(2exp)( 00 r r Dr r Dr ijijij −−−−−= α α φ

6

exp)( ⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

−⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

−=ij

ij

ijr C

r

Ar σ

σ φ

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛ =

612

4)(

ijij

ij

r r

r σ σ

ε φ

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Another example: harmonic and harmonic bond

snapping potential (see lecture 7)

200 )(2

1r r k −=φ

φ

φ

r

r

break r

⎪

⎪

⎩

⎪⎪⎨

⎧

≥−

<−=

break

2

0 break 0

break

2

00

)(2

1

)(2

1

r r r r k

r r r r k

φ

'φ

0r

0r

~ force

~ force

'φ

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Example: calculation of total energy

“simply” the sum of all energies of pairs of atoms

( ) N N N N totalU ,1223211141312 .........21

−+++++++++= φ φ φ φ φ φ φ φ

∑ ∑≠= =

= N

jii

N

jijtotal r U ,1 1

2

1

)(φ

)( ijij r φ φ =

with

12r

25r

two “loops” over pairs of all

particles

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Are all bonds the same? – metallic systems

Bonds depend on the environment!

Pair potentials: All bonds are equal!Reality: Have environment

effects; it matter that there is a

free surface!

+ different

bond EQ

distance

stronger

Surface

Bulk

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Are all bonds the same?

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

This is in contradiction with both experiments and more accurate quantummechanical calculations on many materials

∑=

= N

j

iji r U 1

)( φ Bonding energy of atom i

∑=

=6

1

)( j

iji r U φ )( iji r U φ =

After: G. Ceder

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Are all bonds the same?

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

This is in contradiction with both experiments and more accurate quantummechanical calculations on many materials

For pair potentials

For metals

E.g. in metals, bonds get “ weaker” as more atoms are added to central

atom

Z ~

Z ~

: Z Coordination = how many

immediate neighbors an atom has

After: G. Ceder

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Bond strength depends on coordination

2 4 6 8 10 12 coordination

energy per bond

Z ~

Z

Z ~

pair potential

Nickel

Daw, Foiles, Baskes, Mat. Science Reports, 1993

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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 cannot be used to describe range of crystal structures

E.g. difference between FCC and BCC can not becaptured using a pair potential

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1.1 EAM potential for metals

Note: already used in pset #1,

nanowire simulation

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Metallic bonding: multi-body effects

Need to consider more details of chemical bonding tounderstand 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

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Concept: include electron density effects

)(, ij j r ρ π

Each atom features a particular distribution of electron density

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Electron density at atom i

Atomic electrondensity of atom j

i j

)(..1

, ij

N j

ji r

neigh

∑=

= ρ π ρ

)(, ij j r ρ π

ijr

Contribution to electron density at site i due to electrondensity of atom j evaluated at correct distance (r ij)

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)()(21

..1

i

N j

iji F r neigh

ρ φ φ += ∑=

Electron density at atom i

Atomic electrondensity of atom j

Embedding term F

(how local electrondensity contributes to

potential energy)

i j

)(..1

, ij

N j

ji r neigh

∑=

= ρ π ρ

)(, ij j r ρ π

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Embedded-atom method (EAM)

)()(2

1

..1

i

N j

iji F r neigh

ρ φ φ += ∑=

Pair potential energy Embedding energy

as a function of electron

density

i ρ Electron density at atom i

based on a “pair potential”:

∑=

=neigh N j

ij ji r ..1

, )( ρ π ρ

new

Atomic energy

∑=

= N

i

itotalU 1

φ

Total energy

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

Physical concept: EAM potential

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Physical concept: EAM potential

Describes bonding energy due to electron delocalization

As electrons get more states to spread out over their kinetic

energy decreases

When an impurity is put into a metal its energy is lowered

because the electrons from the impurity can delocalize intothe solid.

The embedding density (electron density at the embeddingsite) is a measure of the number of states available to

delocalize onto.

Inherently MANY BODY effect!

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Effective pair interactions

Can describe differences between bulk and surface

r

+ + + + + ++ + + + + ++ + + + + +

r

+ + + + + ++ + + + + +

+ + + + + +

See also: Daw, Foiles, Baskes, Mat. Science Reports, 1993

Image by MIT OpenCourseWare.

Bulk

Surface

E f f e c t

i v e p a i r p o t e n t i a l ( e V

)

1

0.5

0

-0.53 42

r (A)o

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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!

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Another challenge: chemical reactions

Simple pair potentials can not describe chemical reactions

Energy

C-C distance

Transition point ???

sp3sp2 r

stretchφ sp3

sp2

Why can not model chemical reactions with spring-

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Why can not model chemical reactions with spring-

like potentials?

2

0stretchstretch )(2

1r r k −=φ

Set of parameters only valid for particular

molecule type / type of chemical bond

Reactive potentials or reactive force fields overcome these limitations

32 stretch,stretch, spsp k k ≠

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1.2 ReaxFF force field

For chemical reactions, catalysis,

etc.

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Key features of reactive potentials

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How can one accurately describe the transition energies during chemicalreactions?

Use computationally more efficient descriptions than relying on purely

quantum mechanical (QM) methods (see part II, methods limited to 100atoms)

22

H

HCCHHCCH2

−=−→−≡−

+

involves processes

with electrons


q q

q q

q

q

q q q

A

B

A--B

A

B

A--B

??


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Molecular model that is capable of describing chemical reactions

Continuous energy landscape during reactions (key to enableintegration of equations)

No typing necessary, that is, atoms can be sp, sp2, sp3… w/o

further “tags” – only element types

Computationally efficient (that is, should involve finite range

interactions), so that large systems can be treated (> 10,000 atoms)

Parameters with physical meaning (such as for the LJ potential)


Theoretical basis: bond order potential

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Modulate strength of

attractive part

(e.g. by coordination,or “bond order”)

Abell, Tersoff

Changes in spring constant as function of bond order

Continuous change possible

= continuous energy landscape during chemical reactions

Concept: Use pair potential that depends on atomic environment

(similar to EAM, here applied to covalent bonds)


5

0

-5

-100.5 1 1.5 2 2.5 3 3.5

TripleDouble

Single

S.C.F.C.C.

P o t e n t i a l e n e r g y

( e V )

Distance (A)o

Effective pair-interactions for various C-C (Carbon) bonds

Th ti l b i b d d t ti l

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31D. Brenner, 2000



5

0

-5

-100.5 1 1.5 2 2.5 3 3.5

Triple

Double

Single

S.C.

F.C.C.

P o t e n t i a l e n

e r g y

( e V )

Distance (A)o

Effective pair-interactions for various C-C (Carbon) bonds

Concept of bond order (BO)

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Concept of bond order (BO)

sp3

sp2

sp

r

BO

1

2

3

Bond order based energy landscape

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Bond order based energy landscape

Bond length

Bond order

Energy

Bond length

Energy

Bond order potential Allows for a more general

description of chemistry

All energy terms dependent

on bond order

Conventional potential

(e.g. LJ, Morse)

Pauling

Historical perspective of

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p p

reactive bond order potentials

1985: Abell: General expression for binding energy as a sum of nearnieghbor pair interactions moderated by local atomic environment

1990s: Tersoff, Brenner: Use Abell formalism applied to silicon(successful for various solid state structures)

2000: Stuart et al.: Reactive potential for hydrocarbons

2001: Duin, Godddard et al.: Reactive potential for hydrocarbons“ReaxFF”

2002: Brenner et al.: Second generation “REBO” potential forhydrocarbons

2003-2005: Extension of ReaxFF to various materials including

metals, ceramics, silicon, polymers and more in Goddard‘s group

Example: ReaxFF reactive force field

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Example: ReaxFF reactive force field

William A. Goddard IIICalifornia Institute of Technology

Adri C.T. v. Duin

California Institute of Technology

Courtesy of Bill Goddard. Used with permission.

ReaxFF: A reactive force field

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ReaxFF: A reactive force field

under over

torsanglevalCoulombvdWaalsbond system

E E

E E E E E E

++

++++=,

2-body

multi-body

3-body 4-body

Total energy is expressed as the sum of various terms describing

individual chemical bonds

All expressions in terms of bond order

All interactions calculated between ALL atoms in system…

No more atom typing: Atom type = chemical element

Example: Calculation of bond energy

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Example: Calculation of bond energy

Bond energy between atoms i and j does not depend on bond distance

Instead, it depends on bond order

under over torsanglevalCoulombvdWaalsbond system E E E E E E E E ++++++=,

( ) be ,1

bond e be,1BO exp 1 BO pij ij E D p⎡ ⎤= − ⋅ ⋅ −⎣ ⎦

Bond order functions

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38 All energy terms are expressed as a function of bond orders

(1)

(2) (3)

0 0 0

BO exp exp expij ij ij

ij

r r r

r r r

β β β π ππ σ π ππ

σ π ππ α α α

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎢ ⎥ ⎢ ⎥⎢ ⎥= ⋅ + ⋅ + ⋅⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦ ⎣ ⎦

BO goessmoothly

from 3-2-

1-0

Characteristic bond distance

Fig. 2.21c in Buehler, Markus J. Atomistic Modelingof Materials Failure. Springer, 2008. © Springer. All rightsreserved. This content is excluded from our Creative Commonslicense. For more information, see http://ocw.mit.edu/fairuse.

(1)

(2)(3)

Illustration: Bond energy

http://ocw.mit.edu/fairuse


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Illustration: Bond energy

Image removed due to copyright restrictions.Please see slide 10 in van Duin, Adri. "Dishing Out the Dirt on ReaxFF."

http://www.wag.caltech.edu/home/duin/FFgroup/Dirt.ppt.

vdW interactions

http://www.wag.caltech.edu/home/duin/FFgroup/Dirt.ppt


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under over torsanglevalCoulombvdWaalsbond system E E E E E E E E ++++++=,

Accounts for short distance repulsion (Pauli principle

orthogonalization) and attraction energies at large distances(dispersion)

Included for all atoms with shielding at small distances

Image removed due to copyright restrictions.Please see slide 11 in van Duin, Adri. "Dishing Out the Dirt on ReaxFF."http://www.wag.caltech.edu/home/duin/FFgroup/Dirt.ppt.

Resulting energy landscape



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Resulting energy landscape

Contribution of Ebond and vdW

energy

Source: van Duin, C. T. Adri, et al. "ReaxFF: A Reactive ForceField for Hydrocarbons." Journal of Physical Chemistry A 105(2001). © American Chemical Society. All rights reserved. Thiscontent is excluded from our Creative Commons license. For moreinformation, see http://ocw.mit.edu/fairuse.

Current development status of ReaxFF



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: not currently described by ReaxFF

p

A

B

A--B

Allows to interface metals, ceramics

with organic chemistry: Key for

complex materials, specifically

biological materials

Periodic table courtesy of Wikimedia Commons.

Mg-water interaction: How to make fire with water

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Mg water interaction: How to make fire with water

http://video.google.com/videoplay?docid=4697996292949045921&q=magnesium+water&total=46&start=0&num=50&so=0&type=search&plindex=0

Mg

Images removed due to copyright restrictions.Images from video showing explosive reaction of magnesium, silver nitrate, and water,which can be accessed here: http://www.youtube.com/watch?v=QTKivMVUcqE.

Mg – water interaction – ReaxFF MD simulation


http://www.youtube.com/watch?v=QTKivMVUcqE

http://www.youtube.com/watch?v=QTKivMVUcqE


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htt // it d

http://ocw.mit.edu/

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3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modeling and SimulationSpring 2011

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MIT3_021JS11_P1_L8