3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction ... · II. Atomistic and molecular...
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3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modeling and Simulation Markus Buehler, Spring 2008
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1.021/3.021/10.333/18.361/22.00 Introduction to Modeling and Simulation
Part II - lecture 8
Atomistic and molecular methods
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Content overviewLectures 2-10February/March
Lectures 11-19March/April
Lectures 20-27April/May
I. Continuum methods1. Discrete modeling of simple physical systems:
Equilibrium, Dynamic, Eigenvalue problems2. Continuum modeling approaches, Weighted residual (Galerkin) methods,
Variational formulations3. Linear elasticity: Review of basic equations,
Weak formulation: the principle of virtual work, Numerical discretization: the finite element method
II. Atomistic and molecular methods1. Introduction to molecular dynamics2. Basic statistical mechanics, molecular dynamics, Monte Carlo3. Interatomic potentials4. Visualization, examples5. Thermodynamics as bridge between the scales6. Mechanical properties – how things fail7. Multi-scale modeling8. Biological systems (simulation in biophysics) – how proteins work and
how to model them
III. 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?
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Overview: Material covered Lecture 1: Introduction to atomistic modeling (multi-scale modeling paradigm, difference between continuum and atomistic approach, case study: diffusion)
Lecture 2: Basic statistical mechanics (property calculation: microscopic states vs. macroscopic properties, ensembles, probability density and partition function, solution techniques: Monte Carlo and molecular dynamics)
Lecture 3: Basic molecular dynamics (advanced property calculation, chemical interactions)
Lecture 4: Interatomic potential and force field (pair potentials, fitting procedure, force calculation, multi-body potentials-metals/EAM & applications, neighbor lists, periodic BCs, how to apply BCs)
Lecture 5: Interatomic potential and force field (cont’d) (organic force fields, bond order force fields-chemical reactions, additional algorithms (NVT, NPT), application: mechanical properties –basic introduction)
Lecture 6: Application to mechanics of materials-brittle materials (significance of fractures/flaws, brittle versus ductile behavior [motivating example], basic deformation mechanisms in brittle fracture, theory, case study: supersonic fracture (example for model building); case study: fracture of silicon (hybrid model), modeling approaches: brittle-pair potential/ReaxF, applied to silicon
Lecture 7: Application to mechanics of materials-ductile materials (dislocation mechanisms), case study: failure of copper nanocrystal, supercomputing/parallelization
Lecture 8: Review session
Lecture 9: QUIZ
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II. Atomistic and molecular methods
Lecture 8: Review session
Outline:1. Review of material covered in part II
1.1 Atomistic and molecular simulation algorithms1.2 Property calculation1.3 Potential/force field models1.4 Applications
2. Concluding remarks
Goal of today’s lecture: Review main concepts of atomistic and molecular dynamicsPrepare you for the quiz on Thursday
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1.1 Atomistic and molecular simulation algorithms
1.2 Property calculation1.3 Potential/force field models
1.4 Applications
Goals: Basic MD algorithm (integration scheme), initial/boundary conditions, numerical issues (supercomputing)
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Practice quiz II
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Basic concept: atomistic methods
)(),(),( tatvtr iii
PDE solved in MD:i
ii rd
rdUdt
rdm )(2
2
−= }{ jrr =
Ni ..1=
Ni ..1=)( Fma =
Results of MD simulation:
vi(t), ai(t)ri(t)
xy
z
N particles
mass mi
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Complete MD updating scheme
( ) ...)()(2)()( 20000 +Δ+Δ+Δ−−=Δ+ ttattrttrttr iiii
ijjij amf ,, =
Positions at t0
Accelerationsat t0
Positions at t0-Δt
jijij mfa /,, =
rx
Ff ijij
,, =
(4) Obtain force vectors from potential (sum over contributions from all neighbor of atom j)
Potential
(2) Updating method (integration scheme - Verlet)
(3) Obtain accelerations from forces
⎟⎟⎠
⎞⎜⎜⎝
⎛⎥⎦⎤
⎢⎣⎡−⎥⎦
⎤⎢⎣⎡=
612
4)(rr
r σσεφ
(1) Initial conditions: Positions & velocities at t0 (random velocities so that initial temperature is represented)
rrF
d)(dφ
−=
Nj ..1=∀
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Algorithm of force calculation
for i=1..N
for j=1..N (i ≠ j)
[add force contributions]
6 5
j=1
2 3
4
…
cutr
i
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Summary: Atomistic simulation – numerical approach “molecular dynamics – MD”
Atomistic model; requires atomistic microstructure and atomic positions at beginning (initial conditions), initial velocities from random distribution according to specified temperatureStep through time by integration scheme (Verlet)Repeat force calculation of atomic forces based on their positionsExplicit notion of chemical bonds – captured in interatomic potential
Loop
Set particlepositions
Assign particlevelocities
Calculate forceon each particle
Move particles bytimestep Dt
Save current positions and
velocities
Reachedmax. number of
timesteps?
Stop simulation Analyze dataprint results
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Practice quiz II
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Force calculation for N particles: pseudocode
Requires two nested loops, first over all atoms, and then over all other atoms to determine the distance
for i=1..N:
for j=1..N (i ≠ j):
determine distance between i and j
calculate force and energy (if rij < rcut, cutoff radius)
add to total force vector / energy
time ~ N2: computational disaster
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Strategies for more efficient computation
Two approaches
1. Neighbor lists: Store information about atoms in vicinity, calculated in an N2 effort, and keep information for 10..20 steps
2. Domain decomposition into bins: Decompose system into small bins; force calculation only between atoms in local neighboring bins
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Periodic boundary conditionsPeriodic boundary conditions allows studying bulk properties (no free surfaces) with small number of particles (here: N=3), all particles are “connected”
Original cell surrounded by 26 image cells; image particles move in exactly the same way as original particles (8 in 2D)
Particle leaving box enters on other side with same velocity vector.
Figure by MIT OpenCourseWare. After Buehler.
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Parallel computing – “supercomputers”
Supercomputers consist of a very large number of individual computing units (e.g. Central Processing Units, CPUs)
Images removed due to copyright restrictions
Photos of supercomputers.
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Domain decomposition
Each piece worked on by one of the computers in the supercomputer
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Modeling and simulation
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Modeling and simulation
Images removed due to copyright restrictions.
Actual map of Boston area subway lines compared to the simplified map found at mbta.com.
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Modeling and simulation
M
S
M
S
M
S
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Atomistic versus continuum viewpoint
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Diffusion: Phenomenological description
http://www.uic.edu/classes/phys/phys450/MARKO/N004.html
Ink dot
Tissue
Ink
conc
entra
tion
2
2
dxcdD
tc=
∂∂ 2nd Fick law (governing
equation)
BC: c (r = ∞) = 0IC: c (r=0, t = 0) = c0
Physical problem
Image removed due to copyright restrictions.
Please see http://www.uic.edu/classes/phys/phys450/MARKO/dif.gif.
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Continuum model: Empirical parameters
Continuum model requires parameter that describes microscopic processes inside the material
Need experimental measurements to calibrate
Microscopicmechanisms
???
Continuummodel
Length scale
Time scale“Empirical”or experimentalparameterfeeding
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Atomistic model of diffusion
Diffusion at macroscale (change of concentrations) is result of microscopic processes, random motion of particle
Atomistic model provides an alternative way to describe diffusion
Enables us to directly calculate the diffusion constant from thetrajectory of atoms
Follow trajectory of atoms and calculate how fast atoms leave their initial position
txpDΔΔ
−=2
Concept:follow this quantity overtime
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Atomistic model of diffusion
2rΔ
Average squares of displacements of all particles
( )22 )0()(1)( ∑ =−=Δi
ii trtrN
tr
Particles Trajectories
Mean Squared Displacement function
i
ii rd
rdUdt
rdm )(2
2
−= }{ jrr = Ni ..1=
Images courtesy of the Center for Polymer Studies at Boston University. Used with permission.
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2rΔ
Calculation of diffusion coefficient
1D=1, 2D=2, 3D=3
slope = D
( )22 )0()(1)( ∑ =−=Δi
ii trtrN
tr
Position of atom i at time t
Position of atom i at time t=0
( ))(lim21 2 tr
dtd
dD
tΔ=
∞→
Einstein equation
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???
SummaryMolecular dynamics provides a powerful approach to relate the diffusion constant that appears in continuum models to atomistictrajectories
Outlines multi-scale approach: Feed parameters from atomistic simulations to continuum models
MD
Continuummodel
Length scale
Time scale“Empirical”or experimentalparameterfeeding
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1.1 Atomistic and molecular simulation algorithms
1.2 Property calculation1.3 Potential/force field models
1.4 Applications
Goals: How to calculate “useful” properties from MD runs (temperature, pressure, RDF, VAF,..); significance of averaging; Monte Carlo schemes
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Property calculation: IntroductionHave:
“microscopic information”
Want: Thermodynamical properties (temperature, pressure, ..)Transport properties (diffusivities, shear viscosity, ..)Material’s state (gas, liquid, solid)
)(),(),( tatvtr iii Ni ..1=
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Micro-macro relation
)(131)(
1
2 tvmNk
tTN
iii
B∑=
=
)(tT
t
Which to pick?
Specific (individual) microscopic states are insufficient to relate to macroscopic properties
Courtesy of the Center for Polymer Studies at Boston University. Used with permission.
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Ergodic hypothesis: significance of averaging
Ergodic hypothesis:
Ensemble (statistical) average = time average
All microstates are sampled with appropriate probability density over long time scales
∑∑==
=>=<>=<tA Nit
TimeEnsNiA
iAN
AAiAN ..1..1
)(1)(1
Monte CarloAverage over Monte Carlo steps
NO DYNAMICAL INFORMATION
MDAverage over time steps
DYNAMICAL INFORMATION
drdprprpAAp r∫ ∫>=< ),(),( ρ property must
be properlyaveraged
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Monte Carlo schemeConcept: Find convenient way to solve the integral
Use idea of “random walk” to step through relevant microscopic states and thereby create proper weighting (visit states with higher probability density more often)
Monte Carlo schemes: Many applications (beyond what is discussed here; general method to solve complex integrations)
drdprprpAAp r∫ ∫>=< ),(),( ρ
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Monte Carlo scheme: area calculation
Ω
∫Ω
Ω= dxfA )(
Method to carry out integration (illustrate general concept)
Want:
E.g.: Area of circle (value of π)
4
2dACπ
=4π
=CA 1=d
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Monte Carlo schemeStep 1: Pick random point in Step 2: Accept/reject point based on criterion (e.g. if inside or outside of circle and if in area not yet counted)Step 3: If accepted, add to the total sum otherwise
Ω
1)( =ixf
ix
∫Ω
Ω= dxfAC )(
∑=i
iA
C xfN
A )(1
http://math.fullerton.edu/mathews/n2003/MonteCarloPiMod.html
AN : Attemptsmade
0)( =ixf
2/1=d
Courtesy of John H. Mathews. Used with permission.
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Area of Middlesex County (MSC)
∑=i
iA
SQMSC xfN
AA )(1
100 km
100
km 2km000,10=SQA
Fraction of points that lie within MS County
Expression provides area of MS County
x
y
Image courtesy of Wikimedia Commons.
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Detailed view - schematic
NA=55 points (attempts)12 points within
22 km8.2181km22.010000)(1=⋅== ∑
ii
ASQMSC xf
NAA
12
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Results U.S. Census Bureau
Taken from: http://quickfacts.census.gov/qfd/states/25/25017.html
2km8.2181=MSCAMonte Carlo result:
823.46 square miles = 2137 km2 (1 square mile=2.58998811 km2 )
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Analysis of satellite images
200 m
Spy Pond (Cambridge/Arlington)
Image removed due to copyright restrictions.
Google Satellite image of Spy Pond, in Cambridge, MA.
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Metropolis Hastings algorithm
drdprprpAAp r∫ ∫>=< ),(),( ρ
Images removed due to copyright restrictions. Images removed due to copyright restrictions.Please see: Fig. 2.7 in Buehler, Markus J. Atomistic Please see: Fig. 2.8 in Buehler, Markus J. AtomisticModeling of Materials Failure. New York, NY: Springer, 2008. Modeling of Materials Failure. New York, NY: Springer, 2008.
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Molecular Dynamics vs. Monte Carlo
MD provides actual dynamical data for nonequilibrium processes (fracture, deformation, instabilities)
Can study onset of failure, instabilities
MC provides information about equilibrium properties(diffusivities, temperature, pressure)
Not suitable for processes like fracture
∑∑==
=>=<>=<tA Nit
TimeEnsNiA
iAN
AAiAN ..1..1
)(1)(1
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Property calculation: Temperature and pressure
Temperature
Pressure( )>⋅−<
Ω= ∑
=
N
iiiii rfvmP
1
2
31
Distance vector multipliedby force vector (scalar product)
Volume
Kinetic contribution
><= ∑=
N
iii
B
vmNk
T1
2131
iii vvv ⋅=2
>⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
∂∂
−Ω
=< ∑ ∑ ∑= = ≠=
=3..1 ..1 ,..1,
,, |)(21
31
i N aNrri
iii r
rr
rrvvmP
α ββαααα αβ
φ
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41
Practice exam II
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42
Formal approach: Radial distribution function (RDF)
ρρ /)()( rrg =
The radial distribution function is defined as
Provides information about the density of atoms at a given radius r; ρ(r) is the local density of atoms
ρ1
)()()(
2
2r
r
rrNrg
Δ
Δ
±Ω>±<
=
=drrrg 22)( π Number of particles that lie in a spherical shellof radius r and thickness dr
Local density
Density of atoms (volume)
Volume of this shell (dr)
Number of atoms in the interval 2rr Δ±
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43
Radial distribution function: Solid versus liquid versus gas
Note: The first peak corresponds to the nearest neighbor shell, the second peak to the second nearest neighbor shell, etc.
In FCC: 12, 6, 24, and 12 in first four shells
5
4
3
2
1
00 0.2 0.4 0.6 0.8
g(r)
distance/nm
5
4
3
2
1
00 0.2 0.4 0.6 0.8
g(r)
distance/nm
Solid Argon
Liquid Argon
Liquid Ar(90 K)
Gaseous Ar(90 K)
Gaseous Ar(300 K)
Figure by MIT OpenCourseWare.
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RDF and crystal structure
Peaks in RDF characterize NN distance, can infer from RDF about crystal structure
http://physchem.ox.ac.uk/~rkt/lectures/liqsolns/liquids.html
1st nearest neighbor (NN)
2nd NN3rd NN
…
5
4
3
2
1
00 0.2 0.4 0.6 0.8
g(r)
distance/nm
Solid Argon
Liquid Argon
Figure by MIT OpenCourseWare.
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45
Practice exam II
X
1st nearest neighbor (NN)
2nd NN3rd NN
copper: NN distance > 2 Å
not a liquid (sharp peaks)
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Graphene/carbon nanotubes
Graphene/carbon nanotubes (rolled up graphene)NN: 1.42 Å, second NN 2.46 Å …
RDF
Images removed due to copyright restrictions. Please see:
http://weblogs3.nrc.nl/techno/wp-content/uploads/080424_Grafeen/Graphene_xyz.jpg
http://depts.washington.edu/polylab/images/cn1.jpg
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Practice quiz II
No
No
No
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Velocity autocorrelation function
http://www.eng.buffalo.edu/~kofke/ce530/Lectures/Lecture12/sld010.htm
∑=
Δ+>=Δ+=<N
iii tntvtv
Ntnttvv
1000 )()(1)()0(
VAF
t
Damping due to dissipation - solid
Courtesy of the Department of Chemical and Biological Engineering of the University at Buffalo. Used with permission.
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Velocity autocorrelation function: Solid, liquid, ideal gas
T
VAF
1
Liquid
Dense gas
Ideal gas
Solid
Figure by MIT OpenCourseWare.
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50
VAF used to calculate diffusion constant
')()0(31 '
0'
dttvvDt
t∫∞=
=
><=
Diffusion coefficient (see additional notes available on MIT Server“Additional notes re. velocity autocorrelation function”):
Diffusion coefficient can be obtained from both VAF and the MSD
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51
Practice quiz II
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Mean Squared Displacement function:transport of particles = diffusivities
( )2..1
2 )0()(1)( ∑=
=−>=Δ<Ni
ii trtrN
tr
Position of atom i at time t
Position of atom i at time t=0
Courtesy of Sid Yip. Used with permission.
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53
Practice quiz II
No
No
No
X
X
X *
* Based on autocorrelationfunction (time dependent)
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( )>⋅+<= ∑=
N
iiiii frvm
VP
1
2
31
><= ∑=
N
iii
B
vmNk
T1
2131
iii vvv ⋅=2
∑∑==
+>=<iN
kkiki
N
i i
ttvtvNN
tvv11
)()(11)()0(
>±Ω±
=<Δ
Δ
ρ)()()(
2
2r
r
rrNrg
( )22 )0()(1)( ∑ =−>=Δ<i
ii trtrN
tr
Temperature
Pressure
Stress
MSD
RDF
VAF
Direct
Direct
Direct
Diffusivity
Atomic structure (signature)
Diffusivity, phase state, transport properties
>⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
∂∂
+−<Ω
= ∑ ∑≠
=α ββα
ααα αβ
φσa
rrji
jiij rrr
rruum
,,,, |)(
211
Property Definition Application
Summary – property calculation
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Material properties: Classification
Structural – crystal structure, RDF, VAF
Thermodynamic -- equation of state, heat capacities, thermal expansion, free energies, use RDF, VAF, temperature, pressure
Mechanical -- elastic constants, cohesive and shear strength, elastic and plastic deformation, fracture toughness, use Virial Stress
Vibrational -- phonon dispersion curves, vibrational frequency spectrum, molecular spectroscopy, use VAF
Transport -- diffusion, viscous flow, thermal conduction, use MSD, temperature, VAF (viscosity, thermal conductivity)
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1.1 Atomistic and molecular simulation algorithms
1.2 Property calculation1.3 Potential/force field models
1.4 Applications
Goals: How to model chemical interactions between particles (interatomic potential, force fields); applications to metals, proteins
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Concept: Repulsion and attraction
“point particle” representation
Attraction: Formation of chemical bond by sharing of electronsRepulsion: Pauli exclusion (too many electrons in small volume)
r
Electrons
Core
e
Energy Ur
Repulsion
Attraction
1/r12 (or Exponential)
1/r6
Radius r (Distancebetween atoms)
Figure by MIT OpenCourseWare.
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Generic shape of interatomic potential
e
Energy Ur
Repulsion
Attraction
1/r12 (or Exponential)
1/r6
Radius r (Distancebetween atoms)
Attraction: Formation of chemical bond by sharing of electronsRepulsion: Pauli exclusion (too many electrons in small volume)
Figure by MIT OpenCourseWare.
Harmonic oscillatorr0
φ
r~ k(r - r0)2
Effective interatomic potential
Figure by MIT OpenCourseWare.
Many chemical bonds show this generic behavior
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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)
Wea
ker b
ondi
ng
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60
How to calculate forces from the “potential”or “force field”
Define interatomic potentials, that describe the energy of a set of atoms as a function of their coordinates
“Potential” = “force field”
)(rUU totaltotal =
NirUF totalri i..1)( =−∇=
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
∂∂
∂∂
=∇iii
r rrri,3,2,1
,,
Depends on position ofall other atoms
Change of potential energydue to change of position ofparticle i (“gradient”)
{ } Njrr j ..1==
Position vector of atom i
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61
Practice quiz II
Pair potentialNote cutoff!
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62
Pair potentials: energy calculation
Pair wise summation of bond energies
∑ ∑≠= =
=N
jii
N
jijrU
,1 121
tot )(φ
)( ijrφ
∑=
=N
jiji rU
1
)( φEnergy of atom i
Pair wiseinteraction potential
Simple approximation: Total energy is sum over the energy of all pairs of atoms in the system
=ijr distance betweenparticles i and j
avoid double counting
12 5
3 4
1 r12
2r25 5
3 4
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Total energy of pair potential
Assumption: Total energy of system is expressed as sum of the energy due to pairs of atoms
( )32312321131221 φφφφφφ +++++= totalU
∑ ∑≠= =
=N
jii
N
jijtotal rU
,1 121 )(φ
)( ijij rφφ =with
beyond cutoff=0
=0
12r 23r
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Force calculation – pair potential
ij
ij
rr
Fd
)(dφ−=
Forces can be calculated by taking derivatives from the potential function
Force magnitude: Negative derivative of potential energy with respect to atomic distance
To obtain force vector Fi, take projections into the three axial directions
x1
x2fij
ii r
xFf =
x1
x2
ijr
j
i
Component i of vector ijr
ijij rr =1f
2f
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Force calculation – pair potential
Forces on atom 3: only interactionwith atom 2 (cutoff)
23
23
d)(d
rrF φ
−=23rxFf i
i = 2323 rr =ijr
1. Determine distance between atoms 3 and 2, r232. Obtain magnitude of force vector based on derivative of potential3. To obtain force vector Fi, take projections into the
three axial directionsComponent i of vector
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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
( ) ( ))(exp2)(2exp)( 00 rrDrrDr ijijij −−−−−= ααφ
6
exp)( ⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
ij
ijij r
Cr
Ar σσ
φ
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎟⎠
⎞⎜⎜⎝
⎛=
612
4)(ijij
ij rrr σσεφ
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Lennard-Jones potential – example for copper
LJ potential – parameters for copper (Cleri et al., 1997)
ε
~σ
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Derivative of LJ potential ~ force
Image removed due to copyright restrictions.Please see: Fig. 2.14 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.
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Determination of parameters for atomistic interactions
Often, parameters are determined so that the interatomic potential reproduces experimental results / quantum mechanical results (part III)
Example – elastic properties
'')(2
2
φφ=
∂∂
=r
rk
Calculate K as a function of ε and σ (for LJ potential):
Then find two (or more) properties (experimental, for example), that can be used to fit the LJ/BP parameters (e.g. Young’s modulus E = 120 GPa)
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛=
612
4)(rr
r σσεφ
Shear modulusYoung’s modulus
μ3/8=E))21(3/( ν−= EK4/1=ν
Vkr /2/20=μ
4/30aV =
3/64 σε=K
6
exp)( ⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛−=
rCrAr σ
σφ
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Cutoff radius
∑=
=neighNj
iji rU..1
)(φ6 5
j=1
2 3
4
…
cutr
i φ
r
LJ 12:6potential
ε0
~σ
cutr
Cutoff radius = considering interactions only to a certain distanceBasis: Force contribution negligible (slope)
∑=
=N
jiji rU
1
)( φ
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Derivative of LJ potential ~ force
Beyond cutoff: Changes in energy (and thus forces) small
rrF
d)(dφ
−=
)(rφ
cutr
potentialshift
force not zero
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Crystal structure and potentialThe regular packing (ordering) of atoms into crystals is closely related to the potential detailsMany local minima for crystal structures exist, but materials tend to go to the structure that minimizes the energy; often this can be understood in terms of the energy per atomic bond and the equilibrium distance (at which a bond features the most potential energy)
φ
r
LJ 12:6potential
ε
~σ
2D example
Square lattice Hexagonal lattice
N=4 bonds N=6 bonds per atom
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Example
time
http://polymer.bu.edu/java/java/LJ/index.html
Initial: cubic lattice Transforms into triangular lattice
Courtesy of the Center for Polymer Studies at Boston University. Used with permission.
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74
Practice quiz II
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All bonds are not the same: metals
© 2006 Markus J. Buehler, CEE/MIT
Pair potentials: All bonds Reality: Have environment are equal! effects; it matter that there is
a free surface!
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Embedded-atom method (EAM)
)()(21
..1i
Njiji Fr
neigh
ρφφ += ∑=
Pair potential energy
Embedding energy
as a function of electron density
iρ Electron density at atom ibased on a pair potential:
∑=
=neighNj
iji r..1
)(πρ
new
Models other than EAM (alternatives):Glue model (Ercolessi, Tosatti, Parrinello)Finnis SinclairEquivalent crystal models (Smith and Banerjee)
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Effective pair interactions: EAM potential
Can describe differences between bulk and surface
∑∑∑ +=≠ i
ijii j
ij FrU )()(21
,
ρφ
Pair potential energy Embedding energy
as a function of electron density
Embedding term: depends on environment, “multi-body”
EAM=Embedded atom method
Bulk
Surface
Effe
ctiv
e pa
ir po
tent
ial (
eV )
1
0.5
0
-0.53 42
r (A)o
Figure by MIT OpenCourseWare.
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Summary: EAM method
State of the art approach to model metalsVery good potentials available for Ni, Cu, Al since late 1990s, 2000sNumerically efficient, can treat billions of particlesNot much more expensive than pair potential, but describes physics much better
Recommended for use!
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Model for chemical interactions
Similarly: Potentials for chemically complex materials – assume that total energy is the sum of the energy of different types of chemical bonds
bondHvdWMetallicCovalentElec −++++= UUUUUUtotal
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Concept: energy landscape for chemically complex materials
Different energy contributions from different kinds of chemical bonds are summed up individually, independently
Implies that bond properties of covalent bonds are not affected by other bonds, e.g. vdW interactions, H-bonds
Force fields for organic substances are constructed based on this concept:
water, polymers, biopolymers, proteins …
bondHvdWMetallicCovalentElec −++++= UUUUUUtotal
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Summary: CHARMM-type potential
Utotal = UElec +UCovalent +UMetallic +UvdW +UH−bond
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UCovalent =Ustretch +Ubend +U rot
1φ ( )2stretch = k
2 stretch r − r0
1φ ( )2bend = k
2 bend θ −θ0
1φrot = krot (1− cos(ϑ))
UElec : Coulomb potentialqiqφ(r j
ij ) = ε1rij
UvdW : LJ potential⎡⎛ ⎞
12⎛ ⎞
6⎤σ σφ(r ) = 4ε ⎢⎜ ⎟ − ⎜ ⎟ ⎥ij ⎜⎢⎝ r ⎟ ⎜⎠ r ⎟
⎣ ij ⎝ ij ⎠ ⎥⎦
UH−bond :⎡ ⎛ ⎞
12⎛ ⎞
10⎤Rφ(r ) = ⎢5⎜ H−bond RD ⎟ − 6⎜ H−bond−
⎟ ⎥ 4ij H bond cos (θ⎜ ⎟ ⎜ ⎟⎢ r r ⎥
DHA )⎣ ⎝ ij ⎠ ⎝ ij ⎠ ⎦
1φrot = k (1 cos( ))2 rot − ϑ
2
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Summary: CHARMM force field
Widely used and accepted model for protein structures
Programs such as NAMD have implemented the CHARMM force field
Problem set 2, GenePattern NAMD module, study of a protein domain part of human vimentin intermediate filaments
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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 III, methods limited to 100 atoms)
22
H
HCCHHCCH2
−=−→−≡−+
involves processeswith electrons
Reactive potentials ReaxFF
q q
q q
q
q
qq q
A
B
A--B
A
B
A--B
??
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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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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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Mg – water interaction – ReaxFF MD simulation
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Effect of Pt catalyst
MD simulation clearly proves the effect of the catalyst in greatly enhancing the reaction rate
It also leads to more controlled reaction conditions
5
4
3
2
1
0 0
( )
2
600 K with Pt
600 K without Pt
0.1 0.2 0.3 0.4 0.5 Time ns
Wat
er m
olec
ules
Number of H O molecules over time
Figure by MIT OpenCourseWare.
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•O2 close to Pt surface•Chemisorption of O2 (Pt-O-O)•Dissociation Pt-O and formation of Pt-O-H (stable)•Formation of Pt-O-H2 as another H2 approaches; thereby leads to water and H-O-O molecule
•Several water molecules interact via hydrogen bonds
H2O forms at the Pt (111) surface
1 2 3
Figure by MIT OpenCourseWare.
Formation mechanism
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Summary: ReaxFF force field
Widely used model to describe chemically complex materials and chemical reactions
Can describe covalent bonds, vdW bonds, H-bonds, electrostatic bonds, metallic bonds
Programs such as CMDF have implemented the ReaxFF force field
Problem set #2, GenePattern CMDF module, study of fracture of a silicon single crystal
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Summary: force fields discussed in IM/S
Pair potentials (LJ, Morse, harmonic potentials, biharmonic potentials), used for single crystal elasticity (pset #1) and fracture model (pset #2)Pair potential
EAM (=Embedded Atom Method), used for nanowire (pset #1)Multi-body potential
ReaxFF (reactive force field), used for CMDF/fracture model of silicon (pset#2)Multi-body potential (bond order potential)
Tersoff (nonreactive force field), used for CMDF/fracture model of silicon (note: Tersoff only suitable for elasticity of silicon)Multi-body potential (bond order potential)
CHARMM force field (organic force field), used for protein simulations (pset#2)Multi-body potential (angles, for instance)
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Practice quiz II
ReaxFF
EAM, LJ
CHARMM-type (organic)
ReaxFF
CHARMM
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f
x
Steered Molecular Dynamics (SMD) mimics AFM single molecule experiments
xk
fAtomic force microscope
k xf
pset #2, problem 1
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Comparison – CHARMM vs. ReaxFF
M. Buehler, JoMMS, 2007
Covalent bondsnever break in CHARMM
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1.1 Atomistic and molecular simulation algorithms
1.2 Property calculation1.3 Potential/force field models
1.4 Applications
Goals: Select applications of MD to address questions in materials science (ductile versus brittle materials behavior); observe what can be done with MD
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MD simulation: chemistry as bridge between disciplines
Images removed due to copyright restrictions.Please see: Fig. 2.1 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.
vi(t), ai(t)
xy
z
N particlesmass mi
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Application – protein folding
ACGT
Four letter code “DNA”
Combinationof 3 DNA letters equals a amino acid
E.g.: Proline –
CCT, CCC, CCA, CCG
Sequence of amino acids“polypeptide” (1D structure)
Transcription/translation
Folding (3D structure)
.. - Proline - Serine –Proline - Alanine - ..
Goal of protein folding simulations:Predict folded 3D structure based on polypeptide sequence
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Movie: protein folding with CHARMM
de novo Folding of a Transmembrane fd Coat Proteinhttp://www.charmm-gui.org/?doc=gallery&id=23
Quality of predicted structures quite good
Confirmed by comparison of the MSD deviations of a room temperature ensemble of conformations from the replica-exchange simulations and experimental structures from both solid-state NMR in lipid bilayers and solution-phase NMR on the protein in micelles)
Polypeptide chain
Images removed due to copyright restrictions.
Screenshots from protein folding video, which can be found here:http://www.charmm-gui.org/?doc=gallery&id=23.
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Application – mechanics of materialstensile test of a wire
Brittle Ductile
Strain
Stre
ss
BrittleDuctile
Necking
Figures by MIT OpenCourseWare.
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Practice quiz II
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Ductile versus brittle materials
Difficultto deform,breaks easily
Easy to deformhard to break
BRITTLE DUCTILE
Glass Polymers Ice...
Shear load
Copper, Gold
Figure by MIT OpenCourseWare.
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“Macro, global”
“Micro (nano), local”
)(rσ r
Deformation of materials:Nothing is perfect, and flaws or cracks matter
Griffith, Irwine and others: Failure initiates at defects, such as cracks, or grain boundaries with reduced traction, nano-voids, other imperfections
Failure of materials initiates at cracks
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Crack extension: brittle response
Image removed due to copyright restrictions.Please see: Fig. 1.4 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.
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Basic fracture process: dissipation of elastic energy
Image removed due to copyright restrictions.Please see: Fig. 6.5 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.
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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)
Images removed due to copyright restrictions.Please see: Fig. 6.56 and 6.90 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.
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Summary: mixed Hamiltonian approach
Tersoff
( )TersoffReaxFFReaxFFReaxFFTersoffReaxFF )1()( FwFxwF −+=−
ReaxFFw Tersoffw
xxwxw ∀=+ 1)()( TersoffReaxFF
Tersoff
xImage removed due to copyright restrictions.
See fig. 1 in Buehler, Markus J., Adri C. T. van Druin, and William Goddard III. "Multiparadigm Modeling of Dynamical Crack Propagation in Silicon Using a
Reactive Force Field." Phys Rev Lett 96 (2006): 095505.
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Atomistic fracture mechanism
M.J. Buehler et al., Phys. Rev. Lett., 2007
Images removed due to copyright restrictions.
Fig. 3 from Buehler, Markus, Harvey Tang, Adri C. T. van Duin, and William A. Goddard III. "Threshold Crack Speed Controls Dynamical Fracture of
Silicon Single Crystals." Phys Rev Letter 99 (2007): 165502.
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Lattice shearing: ductile response
Instead of crack extension, induce shearing of atomic latticeDue to large shear stresses at crack tipLecture 7
Figure by MIT OpenCourseWare.
Figure by MIT OpenCourseWare.
τ
τ
τ
τ
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Concept of dislocations
Localization of shear rather than homogeneous shear that requires cooperative motion
“size of single dislocation” = b, Burgers vector 54
additional half plane
Figure by MIT OpenCourseWare.
τ
τ
τ
τ
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A closer look: surface cracks
http://www2.ijs.si/~goran/sd96/e6sem1y.gif
Courtesy of Goran Drazic. Used with permission.
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Analysis of a one-billion atom simulation of work-hardening
Image removed due to copyright restrictions. Please see: Fig. 1c in Buehler, Markus J., et al. "The Dynamical Complexity of Work-hardening: A Large-scale Molecular Dynamics Simulation." Acta Mechanica Sinica 21 (2005): 103-111.
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Final sessile structure – “brittle”
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Practice quiz II
Copper, nickel (ductile) and glass, silicon (brittle)
For a material to be ductile, require dislocations, that is, shearing of lattices
Not possible here (disordered)
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Practice quiz II
Image removed due to copyright restrictions.Please see: Fig. 6.72 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.
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Suddenly stopping crack
Image removed due to copyright restrictions.Please see Fig. 6.57 and 6.74 in: Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.
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2. Concluding remarks
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Goals of part II - from lecture 1
You will be able to …
Carry out atomistic simulations of various processes (diffusion,deformation, fracture..)
Carbon nanotubes, nanowires, bulk metals, proteins, silicon crystals,…
Understand how to analyze atomistic simulations
Understand how to visualize (see snapshots on previous slide)
Understand how to link atomistic simulation results with continuum models
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Important terminology and concepts
Force field Potential / interatomic potentialReactive force field / nonreactive force fieldChemical bonding (covalent, metallic, ionic, H-bonds, vdW..)Time stepContinuum vs. atomistic viewpointsMonte Carlo vs. Molecular DynamicsProperty calculation (temperature, pressure, diffusivity [transport properties]-VAF,MSD, structural analysis-RDF)Choice of potential for different materials (pair potentials, chemical complexity)MD algorithm: how to calculate forces, energies Applications: brittle vs. ductile, data analysis