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1.021, 3.021, 10.333, 22.00 Introduction to Modeling and SimulationSpring 2011
Part I – Continuum and particle methods
Markus J. BuehlerLaboratory for Atomistic and Molecular MechanicsDepartment of Civil and Environmental EngineeringMassachusetts Institute of Technology
Application to modeling brittle materialsLecture 7
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Content overview
I. Particle and continuum methods1. Atoms, molecules, chemistry2. Continuum modeling approaches and solution approaches 3. Statistical mechanics4. Molecular dynamics, Monte Carlo5. Visualization and data analysis 6. Mechanical properties – application: how things fail (and
how to prevent it)7. Multi-scale modeling paradigm8. 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 Mechanics2. Quantum Mechanics: Practice Makes Perfect3. The Many-Body Problem: From Many-Body to Single-
Particle4. Quantum modeling of materials5. From Atoms to Solids6. Basic properties of materials7. 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-scale modeling paradigm, difference between continuum and atomistic approach, case study: diffusion)
Lecture 3: Basic statistical mechanics – property 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 (pair potentials, fracture – introduction)
Lecture 7: Application – MD simulation of materials failure
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Lecture 7: Application to modeling brittle materials
Outline:1. Basic deformation mechanism in brittle materials - crack extension2. Atomistic modeling of fracture
2.1 Physical properties of atomic lattices 2.2 Application
3. Bond order force fields - how to model chemical reactions
Goal of today’s lecture: Apply our tools to model a particular material phenomena – brittle fracture (useful for pset #2)Learn basics in fracture of brittle materialsLearn how to build a model to describe brittle fracture (from scratch)
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1. Basic deformation mechanism in brittle materials - crack extension
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Brittle fracture
Materials: glass, silicon, many ceramics, rocks
At large loads, rather than accommodating a shape change, materials break
Image courtesy of quinn.anya. License: CC-BY.
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Basic fracture process: dissipation of elastic energy
δa
Undeformed Stretching=store elastic energy Release elastic energydissipated into breaking chemical bonds
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Continuum description of fractureFracture is a dissipative process in which elastic energy is dissipated to break bonds (and to heat at large crack speeds)Energy to break bonds = surface energy γs (energy necessary to create new surface, dimensions: energy/area, Nm/m2)
aBs~2γ=
!Ba
E~
21 2
ξσ=
energy to create surfaces
ξγσ Es4
=
E
2
21 σ
σ
ε
change of elastic (potential) energy = G
(1)(2)
a
δa
~
a~
"Relaxed"element
"Strained"element
ξ
a
σ
σ
Image by MIT OpenCourseWare.
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Griffith condition for fracture initiationEnergy release rate G, that is, the elastic energy released per unit crack advance must be equal or larger than the energy necessary to create new surfaces
Provides criterion to predict failure initiation
Calculation of G can be complex, but straightforward for thin strips as shown above
Approach to calculate G based on “stress intensity factor” (see further literature, e.g. Broberg, Anderson, Freund, Tada)
syE
G 221:
2
== ξσ
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Brittle fracture mechanisms
Once nucleated, cracks in brittle materials spread rapidly, on the order of sound speeds
Sound speeds in materials (=wave speeds):
Rayleigh-wave speed cR (speed of surface waves)shear wave speed cs (speed of shear waves)longitudinal wave speed cl (speed of longitudinal waves)
Maximum speeds of cracks is given by sound speeds, depending on mode of loading (mode I, II, III)
Linear elastic continuum theory
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Brittle fracture loading conditionsCommonly consider a single crack in a material geometry, under three types of loading: mode I, mode II and mode III
Image by MIT OpenCourseWare.
Mode I Mode II Mode III
Tensile load, focus of this lecture
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Limiting speeds of cracks: linear elastic continuum theory
• Cracks can not exceed the limiting speed given by the corresponding wave speeds unless material behavior is nonlinear• Cracks that exceed limiting speed would produce energy (physically impossible - linear elastic continuum theory)
Image by MIT OpenCourseWare.
Linear Nonlinear
Mode I
Mode II
Mode III
Limiting speed v
Limiting speed v
Cr Cs
Cs
Cl
Cl
Subsonic Supersonic
SupersonicIntersonicSub-RayleighSuper-
Rayleigh
Mother-daughter mechanism
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Sound speeds in materials: overview
Wave speeds are calculated based on elastic properties of material
μ = shear modulus μ3/8=E E8/3=μ
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Brittle fracture mechanisms: fracture is a multi-scale phenomenon, from nano to macro
Image removed due to copyright restrictions. Fig. 6.2 in Buehler, Markus J. Atomistic Modeling of Materials Failure. Springer, 2008.
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Physical reason for crack limiting speedPhysical (mathematical) reason for the limiting speed is that it becomes increasingly difficult to increase the speed of the crack by adding a larger loadWhen the crack approaches the limiting speed, the resistance to fracture diverges to infinity (=dynamic fracture toughness)
divergence
Driving force (applied load)
EG
vyfE
G s
2
2
~
),(21
σ
ξσ==
Image removed due to copyright restrictions.Please see: Fig. 6.15 in Buehler, Markus J. Atomistic Modeling of Materials Failure. Springer, 2008.
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2. Atomistic modeling of fracture
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What is a model?Mike Ashby (Cambridge University):
A model is an idealization. Its relationship to the real problem is like that of the map of the London tube trains to the real tube systems: a gross simplification, but one that captures certain essentials.
“Physical situation” “Model”© Massachusetts Bay Transportation Authority. All rights
© Google, Inc. All rights reserved. This content is reserved. This content is excluded from our Creative Commons excluded from our Creative Commons license. For more license. For more information, see http://ocw.mit.edu/fairuse.information, see http://ocw.mit.edu/fairuse. 17
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Pair potential to describe atomic interactionsConfine crack to a 1D path (weak fracture layer): Define a pair potentialwhose bonds never break (bulk) and a potential whose bonds break
stable configuration of pair potential
A “simple” atomistic model: geometry2
00 )(21 rrk −=φ
⎪⎪⎩
⎪⎪⎨
⎧
≥−
<−=
break2
0break0
break2
00
)(21
)(21
rrrrk
rrrrkφ
bond can break
2D triangular (hexagonal) latticeelasticity (store energy)
Image by MIT OpenCourseWare.
X
YV
31/2rora
Iy
Ix
Weak fracture layer
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Harmonic and harmonic bond snapping potential
200 )(
21 rrk −=φ
φ
φ
r
r
breakr
⎪⎪⎩
⎪⎪⎨
⎧
≥−
<−=
break2
0break0
break2
00
)(21
)(21
rrrrk
rrrrkφ
'φ
0r
0r
~ force
~ force
'φ
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2.1 Physical properties of atomic lattices
How to calculate elastic properties and fracture surface energy – parameters to link with continuum
theory of fracture
εεψσ
∂∂
=)(
2
2 )(ε
εψ∂
∂=EStress Young’s modulus
free energy density (energy per unit volume)
εσ E=
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1D example – “Cauchy-Born rule”
Impose homogeneous strain field on 1D string of atomsThen obtain from that
))1((1)(1)( 000
rDr
rDr
⋅+⋅
=⋅
= εφφεψ
(1+ε) r01 32
r0
Dr ⋅0 atomic volume
Strain energy densityfunction
interatomicpotential
out-of-plane area
0)1( rr ⋅+= ε
εεψσ
∂∂
=)(
2
2 )(ε
εψ∂
∂=E
εσ E=
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2D hexagonal lattice
( )21221
2222211
211
'' )(32383)( εεεεεεφεψ ++++=ij
ij
ijij ε
εψσ
∂∂
=)(
klij
ijijklc
εεεψ
∂∂∂
=)(2
k=''φ Image by MIT OpenCourseWare.
r2
[010]
[100]
y
x
r2 r2
r2
r1r1
r2
r2 r2
r2
r1r1
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2D triangular lattice, LJ potential
ε = 1σ = 1
12:6 LJ potential
Image by MIT OpenCourseWare.
Image by MIT OpenCourseWare.
r2
[010]
[100]
y
x
r2 r2
r2
r1r1
r2
r2 r2
r2
r1r1
5
4
3
2
1
0 0
00
5
4
3
2
1
0 0
�
� �
�
00
�
�
[ ] (x) S [ ] (y)
0.1 0.2
0.1 0.2
0.1 0.2
20
40
60
80
Strain xx
Stre
ss
Stre
ss
Strain xx
0.1 0.2
20
40
60
80
Strain yy
Strain yy
Tangent modulus Exx
Strain in 100 -direction train in 010 -direction
Tangent modulus Eyy
LJ Solid Harmonic Solid Poisson ratio LJ solid
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2D triangular lattice, harmonic potential
Image by MIT OpenCourseWare.
00
σ xx,
σyy
, and
νx,
νy
1
2
3
4
0.02 0.04Strain
Stress vs. Strain and Poisson Ratio
0.06 0.08
Stress σxxPoisson ratio νx
Poisson ratio νy
Stress σyy
025
E x, E
y 30
35
40
45
50
0.02 0.04Strain
Tangent Modulus
0.06 0.08
ExEy
Elastic properties of the triangular lattice with harmonic interactions, stress versus strain (left) and tangent moduli Ex and Ey (right). The stress state is uniaxial tension, that is the stress in the direction orthogonal to the loading is relaxed and zero.
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Elastic properties
Enables to calculate wave speeds:
k=''φ
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Surface energy calculation
Note: out-of-plane unity thickness
Harmonic potential with bond snapping distance rbreak
Image by MIT OpenCourseWare.
31/2ro
ro
fracture plane
fracture plane
x
y
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2.2 Application
Focus: effects of material nonlinearities (reflected in choice of model)
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Coordinate system and atomistic model
Pair potential to describe atomic interactionsConfine crack to a 1D path (weak fracture layer)
Image by MIT OpenCourseWare.
Y
X
V
31/2ro
Ixa r
Weak fracture layer
Iy
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Linear versus nonlinear elasticity=hyperelasticity
Linear elasticity: Young’s modulus (stiffness) does not changewith deformationNonlinear elasticity = hyperelasticity: Young’s modulus (stiffness) changes with deformation
Image by MIT OpenCourseWare.
Str
ess
Strain
Stiffen
ing
Softening
Hyperelasticity
Line
ar the
ory
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Subsonic and supersonic fracture
Under certain conditions, material nonlinearities (that is, the behavior of materials under large deformation = hyperelasticity)becomes important
This can lead to different limiting speeds than described by the model introduced above
large deformationnonlinear zone
“singularity”
Deformation field near a crack
small deformation
rr 1~)(σ
Image by MIT OpenCourseWare.
Strain
Str
ess
Line
ar e
last
ic
clas
sica
l the
ories
Nonlinear real
materials
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Limiting speeds of cracks
• Under presence of hyperelastic effects, cracks can exceed the conventional barrier given by the wave speeds• This is a “local” effect due to enhancement of energy flux• Subsonic fracture due to local softening, that is, reduction of energy flux
Image by MIT OpenCourseWare.
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Stiffening vs. softening behavior
Increased/decreased wave speed
E
2
21 σ
σ
ε
“linear elasticity”
real materials
Image by MIT OpenCourseWare.
Str
ess
Strain
Stiffen
ing
Softening
HyperelasticityLine
ar the
ory
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MD model development: biharmonic potential
Polymers, rubber
Metals
• Stiffness change under deformation, with different strength
• Atomic bonds break at critical atomic separation
• Want: simple set of parameters that control these properties (as few as possible, to gain generic insight)
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Biharmonic potential – control parameters
breakε
0k
1k
01ratio / kkk =
breakronr r 0
0
rrr −
=ε
'φ
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Biharmonic potential definition
The biharmonic potential is defined as:
where is the critical atomic separation for the onset of the hyperelastic effect, and
and
are found by continuity conditions of the potential at . The values and refer to the small- and large-strain spring constants.
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Energy filtering: visualization approach
Only plot atoms associated with higher energyEnables determination of crack tip (atoms at surface have higherenergy, as they have fewer neighbors than atoms in the bulk)
∑=
=N
jiji rU
1
)( φEnergy of atom i
Image by MIT OpenCourseWare.
a
x-coordinate
z-co
ord
inat
e
200
180
160
140
120
100
80
60
40
20
050 100 150 200 250 300 350 4000
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a (t)
v(t)=da/dt
crack speed=time derivative of a
MD simulation results: confirms linear continuum theory
Image by MIT OpenCourseWare.
0 100 200 300 400 500 600
500
1000
1500
2000
2500
0Rel
ativ
e cr
ack
exte
nsio
n ∆
a
0 100 200 300 400 500 600
1
2
3
4
5
6
0
Tip
spee
d V
t
Nondimensional time t
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Virial stress field around a crack
large stressesat crack tip - induce bond failure
Images removed due to copyright restrictions.Please see: Fig. 6.30 in Buehler, Markus J. Atomistic Modeling of Materials Failure. Springer, 2008.
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Biharmonic potential – bilinear elasticity
Image by MIT OpenCourseWare.
StiffeningSoftening
2
1.5
1
0.5
00 0.01 0.02 0.03
Str
ess
(σ)
Strain (ε)
Et=66
Et=33
Harm
onic
solid
Hypere
lastic
solid
2
1.5
1
0.5
00 0.01 0.02 0.03
Str
ess
(σ)
Strain (ε)
Et=66
Et=33
Harmon
ic soli
dHype
relastic solid
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Harmonic system
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Subsonic fractureSoftening material behavior leads to subsonic fracture, that is, the crack can never attain its theoretical limiting speedMaterials: metals, ceramics
< 3.4 = cR
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Supersonic fractureStiffening material behavior leads to subsonic fracture, that is, the crack can exceed its theoretical limiting speedMaterials: polymers
> 3.4 = cR
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0.7
0.9
1.1
1.3
1.5
1.7
1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2
k2/k1=1/4k2/k1=1/2k2/k1=2k2/k1=3k2/k1=4
Different ratios of spring constants
Stiffening and softening effect: Increase or reduction of crack speed
ron
Shear wave speed
Intersonic fracture
Sub-Rayleigh fracture
Reduced crack speed c/cR
Softening
Stiffening
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Physical basis for subsonic/supersonic fracture
Changes in energy flow at the crack tip due to changes in local wave speed (energy flux higher in materials with higher wave speed)Controlled by a characteristic length scale χ
Reprinted by permission from Macmillan Publishers Ltd: Nature. Source: Buehler, M., F. Abraham, and H. Gao. "Hyperelasticity Governs Dynamic Fracture at a Critical Length Scale." Nature 426 (2003): 141-6. © 2003.
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Supersonic fracture: mode II (shear)
Please see: Buehler, Markus J., Farid F. Abraham, and Huajian Gao. "Hyperelasticity Governs Dynamic Fracture at a Critical Length Scale." Nature 426 (November 13, 2003): 141-6.
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Theoretical concept: energy flux reduction/enhancement
Energy flux related to wave speed: high local wave speed, high energy flux,crack can move faster (and reverse for low local wave speed)
Image by MIT OpenCourseWare.
Classical
New
Fracture process zoneHyperelastic zone
K dominance zone Characteristic energy length
Energy Flux Related to Wave Speed: High local wave speed, high energy flux, crack can move faster (and reversefor low local wave speed).
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3. Bond order force fields - how to model chemical reactions
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Challenge: chemical reactions
CHARMM-type potential can not describe chemical reactions
Energy
C-C distance
Transition point ???
sp3sp2 r
stretchφsp3
sp2
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Why can not model chemical reactions with |spring-like potentials?
20stretchstretch )(
21 rrk −=φ
20bendbend )(
21 θθφ −= k
Set of parameters only valid for particularmolecule type / type of chemical bond
Reactive potentials or reactive force fields overcome these limitations
32 stretch,stretch, spspkk ≠
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How can one accurately describe the transition energies during chemical reactions?
Use computationally more efficient descriptions than relying on purely quantum mechanical (QM) methods (see part II, methods limited to 100 atoms)
22
H
HCCHHCCH2
−=−→−≡−+
involves processeswith electrons
Key features of reactive potentials
q q
q q
q
q
qq 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 enable integration 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
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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 orderContinuous 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
Triple
Double
Single
S.C.
F.C.C.
Pote
ntial
ener
gy
(eV)
Distance (A)o
Effective pair-interactions for various C-C (Carbon) bonds
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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.
Pote
ntial
ener
gy
(eV)
Distance (A)o
Effective pair-interactions for various C-C (Carbon) bonds
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Concept of bond order (BO)
sp3
sp2
sp
r
BO
1
2
3
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Bond order based energy landscape
Bond length
Bond order
Energy
Bond length
Energy
Bond order potentialAllows for a more general description of chemistryAll energy terms dependent on bond order
Conventional potential(e.g. LJ, Morse)
Pauling
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Historical perspective of reactive bond order potentials
1985: Abell: General expression for binding energy as a sum of near nieghbor 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 for hydrocarbons
2003-2005: Extension of ReaxFF to various materials including metals, ceramics, silicon, polymers and more in Goddard‘s group
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Example: ReaxFF reactive force field
William A. Goddard IIICalifornia Institute of Technology
Adri C.T. v. DuinCalifornia Institute of Technology
Courtesy of Bill Goddard. Used with permission.
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ReaxFF: A reactive force field
underover
torsanglevalCoulombvdWaalsbondsystem
EE
EEEEEE
++
++++= ,
2-body
multi-body
3-body 4-body
Total energy is expressed as the sum of various terms describingindividual 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
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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
underovertorsanglevalCoulombvdWaalsbondsystem EEEEEEEE ++++++= ,
( )be,1bond e be,1BO exp 1 BO p
ij ijE D p⎡ ⎤= − ⋅ ⋅ −⎣ ⎦
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Bond order functions
(1)
All energy terms are expressed as a function of bond orders
(2) (3)
0 0 0
BO exp exp expij ij ijij
r r rr r r
β ββ π ππσ π ππ
σ π ππα α α⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎢ ⎥ ⎢ ⎥⎢ ⎥= ⋅ + ⋅ + ⋅⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦ ⎣ ⎦
BO goes smoothly from 3-2-1-0
Characteristic bond distance
Fig. 2.21c in Buehler, Markus J. Atomistic Modelingof Materials Failure. Springer, 2008. © Springer. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse.
(2)(3)
(1)
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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.
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underovertorsanglevalCoulombvdW aalsbondsystem EEEEEEEE ++++++= ,
vdW interactions
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.
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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. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse.
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LaAc
: not currently described by ReaxFF
Current development status of ReaxFF
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.
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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
Video stills removed due to copyright restrictions; watch the video now: http://www.youtube.com/watch?v=QTKivMVUcqE.
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Mg – water interaction – ReaxFF MD simulation
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