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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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Concept: include electron density effects
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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Concept: include electron density effects
)()(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
Key features of reactive potentials
q q
q q
q
q
q q q
A
B
A--B
A
B
A--B
??
Key features of reactive potentials
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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)
Key features of reactive potentials
Theoretical basis: bond order potential
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Theoretical basis: bond order potential
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)
Image by MIT OpenCourseWare.
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
Theoretical basis: bond order potential
Image by MIT OpenCourseWare.
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
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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
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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
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