1.021, 3.021, 10.333, 22.00 Introduction to Modeling and Simulation
Part I – Continuum and particle methods
Review session—preparation quiz I
Lecture 12
Markus J. Buehler
Laboratory for Atomistic and Molecular Mechanics
Department of Civil and Environmental Engineering
Massachusetts Institute of Technology
1
Content overview
I. Particle and continuum methods
1.
2.
3.
4.
5.
6.
7.
8.
Atoms, molecules, chemistry
Continuum modeling approaches and solution approaches
Statistical mechanics
Molecular dynamics, Monte Carlo
Visualization and data analysis
Mechanical properties – application: how things fail (and
how to prevent it)
Multi-scale modeling paradigm
Biological systems (simulation in biophysics) – how
proteins work and how to model them
II. Quantum mechanical methods
1.
2.
3.
4.
5.
6.
7.
8.
Lectures 2-13
Lectures 14-26
It’s A Quantum World: The Theory of Quantum Mechanics
Quantum Mechanics: Practice Makes Perfect
The Many-Body Problem: From Many-Body to SingleParticle
Quantum modeling of materials
From Atoms to Solids
Basic properties of materials
Advanced properties of materials
What else can we do?
2
Overview: Material covered so far…
ƒ
Lecture 1: Broad introduction to IM/S
ƒ
Lecture 2: Introduction to atomistic and continuum modeling (multi-scale modeling paradigm,
difference between continuum and atomistic approach, case study: diffusion)
ƒ
Lecture 3: Basic statistical mechanics – property calculation I (property calculation: microscopic
states vs. macroscopic properties, ensembles, probability density and partition function)
ƒ
Lecture 4: Property calculation II (Monte Carlo, advanced property calculation, introduction to
chemical interactions)
ƒ
Lecture 5: How to model chemical interactions I (example: movie of copper
deformation/dislocations, etc.)
ƒ
Lecture 6: How to model chemical interactions II (EAM, a bit of ReaxFF—chemical reactions)
ƒ
Lecture 7: Application to modeling brittle materials I
ƒ
Lecture 8: Application to modeling brittle materials II
ƒ
Lecture 9: Application – Applications to materials failure
ƒ
Lecture 10: Applications to biophysics and bionanomechanics
ƒ
Lecture 11: Applications to biophysics and bionanomechanics (cont’d)
3
ƒ
Lecture 12: Review session – part I
Check: goals of part I
You will be able to …
ƒ Carry out atomistic simulations of various processes
(diffusion, deformation/stretching, materials failure)
Carbon nanotubes, nanowires, bulk metals, proteins, silicon
crystals, …
ƒ Analyze atomistic simulations
ƒ Visualize atomistic/molecular data
ƒ Understand how to link atomistic simulation results with
continuum models within a multi-scale scheme
4
Topics covered – part I
Atomistic &
molecular
simulation
algorithms
Potential/
force field
models
Property
calculation
Applications
5
Lecture 12: Review session – part I
1.
2.
Review of material covered in part I
1.1 Atomistic and molecular simulation algorithms
1.2 Property calculation
1.3 Potential/force field models
1.4 Applications
Important terminology and concepts
Goal of today’s lecture:
ƒ
Review main concepts of atomistic and molecular
dynamics, continuum models
ƒ
Prepare you for the quiz on Thursday
6
1.1 Atomistic and molecular simulation
algorithms
1.2 Property calculation
1.3 Potential/force field models
1.4 Applications
Goals: Basic MD algorithm (integration scheme), initial/boundary
conditions, numerical issues (supercomputing)
7
Differential equations solved in MD
8
Basic concept: atomistic methods
G
d ri
dU ( r )
mi 2 = − G
dt
dri
2
PDE solved in MD:
(ma = F )
G
r = {rj }
i = 1.. N
Results of MD simulation:
G
G
G
ri (t ), vi (t ), ai (t )
i = 1.. N
9
Complete MD updating scheme
(1) Initial conditions: Positions & velocities at t0 (random velocities so that initial
temperature is represented)
(2) Updating method (integration scheme - Verlet)
ri (t0 + Δt ) = −ri (t0 − Δt ) + 2ri (t0 )Δt + ai (t0 )(Δt ) + ...
2
Positions
at t0-Δt
Positions
at t0
Accelerations
at t0
(3) Obtain accelerations from forces
f j ,i = m j a j ,i
a j ,i = f j ,i / m j
∀j = 1.. N
(4) Obtain force vectors from potential (sum over contributions from all neighbor
of atom j)
d φ (r)
F =−
dr
f j ,i = F
x j ,i
r
⎛ ⎡ σ ⎤12 ⎡ σ ⎤ 6 ⎞
⎟
Potential φ ( r ) = 4ε ⎜
−
⎢⎣ r ⎥⎦ ⎟
⎜ ⎢⎣ r ⎥⎦
⎝
⎠ 10
Algorithm of force calculation
6
…
5
for i=1..N
for j=1..N (i ≠ j)
[add force
contributions]
i
j=1
2
4
3
rcut
11
Summary: Atomistic simulation – numerical
approach “molecular dynamics – MD”
ƒ Atomistic model; requires atomistic microstructure and atomic positions at
beginning (initial conditions), initial velocities from random distribution
according to specified temperature
ƒ Step through time by integration scheme (Verlet)
ƒ Repeat force calculation of atomic forces based on their positions
ƒ Explicit notion of chemical bonds – captured in interatomic potential
Loop
12
Scaling behavior of MD code
13
Force calculation for N particles: pseudocode
ƒ Requires two nested loops, first over all atoms, and then
over all other atoms to determine the distance
for j=1..N (i ≠ j):
determine distance between i and j
Δt1~NΔt0
calculate force and energy (if
rij < rcut, cutoff radius)
Δt0
Δttotal~N·NΔt0=N2Δt0
for i=1..N:
add to total force vector / energy
time ~ N2: computational disaster
14
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
Concept: Store information of neighbors of each atom within
vicinity of cutoff radius (e.g. list in a vector); update list only every
10..20 steps
2.
Domain decomposition into bins: Decompose system into
small bins; force calculation only between atoms in local
neighboring bins
Concept: Even overall system grows, calculation is done only in a
local environment (have two nested loops but # of atoms does not
15
increase locally)
Force calculation with neighbor lists
ƒ Requires two nested loops, first over all atoms, and then
over all other atoms to determine the distance
for i=1..Nneighbors,i:
Δttotal~NΔt0
for j=1..N (i ≠ j):
determine distance between i and j
Δt1~NΔt0
calculate force and energy (if
rij < rcut, cutoff radius)
Δt0
add to total force vector / energy
time ~ N: computationally tractable even for large N
16
Periodic boundary conditions
ƒ Periodic boundary conditions allows studying bulk properties (no free
surfaces) with small number of particles (here: N=3), all particles are
“connected”
ƒ 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.
Image by MIT OpenCourseWare. After Buehler.
17
Parallel computing – “supercomputers”
Images removed due to copyright restrictions. Please see
http://www.sandia.gov/ASC/images/library/ASCI-White.jpg
Supercomputers
consist of a very large
number of individual
computing units (e.g.
Central Processing
Units, CPUs)
18
Domain decomposition
Each piece worked on by one of the computers in the supercomputer
Fig. 1 c from Buehler, M., et al. "The Dynamical Complexity of Work-Hardening: A Large-Scale Molecular Dynamics Simulation.”
Acta Mech Sinica 21 (2005): 103-11. © Springer-Verlag. All rights reserved. This content is excluded from our Creative Commons license.
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19
Modeling and simulation
20
Modeling and simulation
What is a model? What is a simulation?
Modeling vs Simulation
• Modeling: developing a mathematical representation
of a physical situation
•Simulation: solving the equations that arose from the
development of the model.
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Creative Commons license. For more
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Transportation Authority. All rights
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license. For more information, see
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21
Modeling and simulation
M
S
M
S
M
22
Atomistic versus continuum viewpoint
23
Diffusion: Phenomenological description
Physical problem
Ink dot
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is excluded from our Creative Commons license. For
more information, see http://ocw.mit.edu/fairuse
d 2c
∂c
=D 2
dx
∂t
2nd Fick law (governing
equation)
Tissue
BC: c (r = ∞) = 0
IC: c (r=0, t = 0) = c0
24
Continuum model: Empirical parameters
ƒ Continuum model requires parameter that describes microscopic
processes inside the material
ƒ Need experimental measurements to calibrate
Time scale
Continuum
model
Microscopic
mechanisms
???
“Empirical”
or experimental
parameter
feeding
Length scale
25
How to solve continuum problem: Finite difference
scheme
d 2c
∂c
=D 2
dx
∂t
Concentration at i
at old time
i=
j=
space
time
“explicit” numerical scheme
δt
ci , j +1 = ci , j + 2 D (ci +1, j − 2 ci , j − ci −1, j ) (new concentration directly from
δx
concentration at earlier time)
Concentration at i
at old time
Concentration at i
at new time
Concentration at i-1
at old time
Concentration at i+1
at old time
26
Other methods: Finite element method
Atomistic model of diffusion
ƒ Diffusion at macroscale (change of concentrations) is result of
microscopic processes, random motion of particle
ƒ Atomistic model provides an alternative way to describe diffusion
ƒ Enables us to directly calculate the diffusion constant from the
trajectory of atoms
ƒ Follow trajectory of atoms and calculate how fast atoms leave their
initial position
Concept:
follow this quantity over
time
Δx 2
D = −p
Δt
27
Atomistic model of diffusion
G
G
d ri
dU ( r )
r = {rj } i = 1.. N
mi 2 = −
G
dt
dri
2
Images courtesy of the Center for Polymer
Studies at Boston University. Used with
permission.
Δr 2
Trajectories
Particles
Mean Squared
Displacement function
Δr 2 (t ) =
1
N
Average squares of
displacements of all
particles
G
G
2
(
)
r
(
t
)
−
r
(
t
=
0
)
∑ i
i
i
28
Calculation of diffusion coefficient
1
Δr (t ) =
N
2
∑ (r (t ) − r (t = 0))
2
i
i
i
Position of
atom i at time t
Position of
atom i at time t=0
(nm)
Einstein equation
Δr 2
slope = D
1
d
D=
lim (Δr 2 (t ) )
2d t →∞ dt
(ps)
1D=1, 2D=2, 3D=3
29
Summary
ƒ Molecular dynamics provides a powerful approach to relate the
diffusion constant that appears in continuum models to atomistic
trajectories
ƒ Outlines multi-scale approach: Feed parameters from atomistic
simulations to continuum models
Time scale
Continuum
model
“Empirical”
or experimental
parameter
feeding
MD
???
Length scale
30
1.1 Atomistic and molecular simulation
algorithms
1.2 Property calculation
1.3 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
31
Property calculation: Introduction
Have:
G
G
G
ri (t ), vi (t ), ai (t )
i = 1.. N
“microscopic information”
Want:
ƒ Thermodynamical properties (temperature,
pressure, ..)
ƒ Transport properties (diffusivities, shear viscosity, ..)
ƒ Material’s state (gas, liquid, solid)
32
Micro-macro relation
T (t )
Which to pick?
1 1
T (t ) =
3 Nk B
Images courtesy of the Center for Polymer Studies
at Boston University. Used with permission.
G2
∑ mi vi (t )
N
t
i =1
Specific (individual) microscopic states are insufficient to relate to
33
macroscopic properties
Ergodic hypothesis: significance of averaging
< A >= ∫ ∫ A( p, r ) ρ ( p, r )drdp
p r
ƒ Ergodic hypothesis:
property must
be properly
averaged
Ensemble (statistical) average = time average
ƒ All microstates are sampled with appropriate probability density over
long time scales
1
1
A(i ) =< A > Ens =< A >Time =
∑
N A i =1.. N A
Nt
Monte Carlo
Average over Monte Carlo steps
NO DYNAMICAL INFORMATION
∑ A(i )
i =1.. N t
MD
Average over time steps34
DYNAMICAL INFORMATION
Monte Carlo scheme
ƒ Concept: Find convenient way to solve the integral
< A >= ∫ ∫ A( p, r ) ρ ( p, r )drdp
p r
ƒ Use idea of “random walk” to step through relevant
microscopic states and thereby create proper weighting
(visit states with higher probability density more often)
ƒ Monte Carlo schemes: Many applications (beyond what
is discussed here; general method to solve complex
integrations)
35
Monte Carlo scheme: area calculation
Method to carry out integration (illustrate general concept)
Want:
A= ∫
G
f ( x ) dΩ
Ω
E.g.: Area of circle
(value of π)
AC =
πd 2
4
AC =
π
4
d =1
Ω
36
Monte Carlo scheme
G
ƒ Step 1: Pick random point xi in Ω
ƒ Step 2: Accept/reject point based on criterion (e.g. if
inside or outside of circle and if in area not yet counted)
G
ƒ Step 3: If accepted, add f ( xi )G = 1
to the total sum otherwise f ( xi ) = 0
AC = ∫
G
f ( x ) dΩ
Ω
N A : Attempts
d = 1/ 2
made
G
1
AC =
f ( xi )
∑
NA i
Courtesy of John H. Mathews. Used with permission.
See also: http://math.fullerton.edu/mathews/n2003/MonteCarloPiMod.html
37
Area of Middlesex County (MSC)
100 km
100 km
ASQ = 10,000 km2
y
x
Image courtesy of Wikimedia Commons.
Fraction of points that lie within MS County
AMSC = ASQ
G
1
f ( xi )
∑
NA i
Expression provides area of MS County
38
Detailed view - schematic
NA=55 points (attempts)
12 points within
AMSC = ASQ
12
G
1
2
2
f
(
x
)
=
10000
⋅
0
.
22
km
=
2181
.
8
km
∑ i
39
NA i
Results U.S. Census Bureau
823.46 square miles = 2137 km2 (1 square mile=2.58998811 km2 )
Taken from: http://quickfacts.census.gov/qfd/states/25/25017.html
Monte Carlo result:
AMSC = 2181.8 km2
40
Analysis of satellite images
Spy Pond
(Cambridge/Arlington)
Image removed due to copyright restrictions.
Google Satellite image of Spy Pond, in Cambridge, MA.
200 m
41
Metropolis Hastings algorithm
Images removed due to copyright restrictions.
Images removed due to copyright restrictions.
Please see: Fig. 2.7 in Buehler, Markus J. Atomistic
Modeling of Materials Failure. Springer, 2008.
Please see: Fig. 2.8 in Buehler, Markus J. Atomistic Modeling
of Materials Failure. Springer, 2008.
< A >= ∫ ∫ A( p, r ) ρ ( p, r )drdp
p r
42
Molecular Dynamics vs. Monte Carlo
ƒ MD provides actual dynamical data for nonequilibrium
processes (fracture, deformation, instabilities)
Can study onset of failure, instabilities
ƒ MC provides information about equilibrium properties
(diffusivities, temperature, pressure)
Not suitable for processes like fracture
1
1
A(i ) =< A > Ens =< A >Time =
∑
N A i =1.. N A
Nt
∑ A(i )
i =1.. N t
43
Property calculation: Temperature and pressure
ƒ Temperature
N
G2
1 1
T=
< ∑ mi vi >
3 Nk B i =1
G2 G G
vi = vi ⋅ vi
T ~ v2
v~ T
44
Temperature control in MD (NVT ensemble)
Berendsen thermostat
ri (t0 + Δt ) = 2 ri (t0 ) − ri (t0 − Δt ) + ai (t0 )Δt 2 + ...
Calculate temperature
N
G2
1 1
T=
< ∑ mi vi >
3 Nk B i =1
v~ T
Calculate ratio of current vs. desired temperature
Rescale velocities
45
Interpretation of RDF
46
Formal approach: Radial distribution
function (RDF)
The radial distribution function is defined as
Density of atoms (volume)
g (r ) = ρ (r ) / ρ
Local density
Provides information about the density of atoms at a given
radius r; ρ(r) is the local density of atoms
Number of atoms in the interval r ±
Δr
2
< N ( r ± Δ2r ) > 1
g (r) =
Ω( r ± Δ2r ) ρ
Volume of this shell (dr)
g (r )2πr 2 dr = Number of particles that lie in a spherical shell
of radius r and thickness dr
47
Radial distribution function:
Solid versus liquid versus gas
5
5
4
4
3
g(r)
g(r)
Solid Argon
2
Gaseous Ar
(90 K)
Liquid Ar
(90 K)
3
Gaseous Ar
(300 K)
2
Liquid Argon
1
0
1
0
0.2
0.4
distance/nm
0.6
0.8
0
0
0.2
0.4
0.6
0.8
distance/nm
Image by MIT OpenCourseWare.
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
48
RDF and crystal structure
1st nearest neighbor (NN)
5
3rd NN
4
g(r)
Solid Argon
2nd NN
…
3
2
Liquid Argon
1
0
0
0.2
0.4
0.6
0.8
distance/nm
Image by MIT OpenCourseWare.
Peaks in RDF characterize NN distance,
can infer from RDF about crystal structure
49
http://physchem.ox.ac.uk/~rkt/lectures/liqsolns/liquids.html
Images courtesy of Mark Tuckerman. Used with permission.
50
Interpretation of RDF
1st nearest neighbor (NN)
2nd NN
3rd NN
copper: NN distance > 2 Å
X
not a liquid (sharp peaks)
51
Graphene/carbon nanotubes
RDF
Images of graphene/carbon nanutubes:
http://weblogs3.nrc.nl/techno/wpcontent/uploads/080424_Grafeen/Graphene_xyz.jpg
http://depts.washington.edu/polylab/images/cn1.jpg
Graphene/carbon nanotubes (rolled up graphene)
NN: 1.42 Å, second NN 2.46 Å …
52
Summary – property calculation
Property
Definition
Temperature
N
G
1 1
T=
< ∑ mi vi2 >
3 Nk B i =1
MSD
1
< Δr (t ) >=
N
RDF
2
Application
G2 G G
vi = vi ⋅ vi
∑ (r (t ) − r (t = 0))
Direct
2
i
i
Diffusivity
i
N ( r ± Δ2r )
>
g ( r ) =<
Δr
Ω( r ± 2 ) ρ
Atomic structure
(signature)
53
Material properties: Classification
ƒ Structural – crystal structure, RDF
ƒ Thermodynamic -- equation of state, heat capacities, thermal
expansion, free energies, use RDF, temperature, pressure/stress
ƒ Mechanical -- elastic constants, cohesive and shear strength, elastic
and plastic deformation, fracture toughness, use stress
ƒ Transport -- diffusion, viscous flow, thermal conduction, use MSD,
temperature
54
Bell model analysis (protein rupture)
55
Bell model analysis (protein rupture)
a=86.86 pN
f (v; xb , Eb ) = a ⋅ ln v + b
b=800 pN
kB T
a=
(1)
xb
kB=1.3806505E-23 J/K
⎛ E b ⎞⎞
kb T ⎛
b=−
ln⎜ω 0 x b exp⎜ −
⎟⎟ (1)
xb ⎝
⎝ kb T ⎠⎠
ω0 = 1 × 1013 1 / sec
Get a and b from fitting to the
graphs
Solve (1) for a, use (2) to solve
for Eb
a=(800-400)/(0-(-4.6))) pN
=86.86 pN Æ xb=0.47 Å
56
Determining the energy barrier
⎛ Eb ⎞ ⎞
kbT ⎛⎜
kbT
⎟
⎜
⎟
ln⎜ ω0 xb exp⎜ −
=−
b=−
⎟
⎟
xb ⎝
xb
⎝ kbT ⎠ ⎠
Eb
bxb
−
= ln(ω0 xb ) −
kbT
kbT
⎡
Eb ⎤
⎥
⎢ln(ω0 xb ) −
kbT ⎦
⎣
bxb
Eb
= ln(ω0 xb ) +
kbT
kbT
⎡
bxb ⎤
Eb = ⎢ln(ω0 xb ) +
⎥ kbT
kbT ⎦
⎣
b=800 pN Æ Eb≈9 kcal/mol
1 kcal/mol=6.9477E-21 J
Possible mechanism: Rupture of 3 H-bonds (H-bond energy 3 kcal/mol)
57
Scaling with Eb : shifts curve
f ( v; xb , Eb ) = a ⋅ ln v + b
Eb ↑
kB ⋅ T
a=
xb
kB ⋅ T
b=−
⋅ ln v0
xb
⎛ Eb ⎞
⎟⎟
v0 = ω0 ⋅ xb ⋅ exp⎜⎜ −
kb ⋅ T ⎠
58 ⎝
Scaling with xb: changes slope
f (v; xb , Eb ) = a ⋅ ln v + b
xb ↓
kB ⋅ T
a=
xb
kB ⋅ T
b=−
⋅ ln v0
xb
⎛ Eb ⎞
⎟⎟
v0 = ω0 ⋅ xb ⋅ exp⎜⎜ −
kb ⋅ T ⎠
59 ⎝
1.1 Atomistic and molecular simulation
algorithms
1.2 Property calculation
1.3 Potential/force field models
1.4 Applications
Goals: How to model chemical interactions between particles (interatomic
potential, force fields); applications to metals, proteins
60
Concept: Repulsion and attraction
Electrons
Energy U
r
1/r12 (or Exponential)
Core
Repulsion
r
Radius r (Distance
between atoms)
e
Attraction
1/r6
Image by MIT OpenCourseWare.
“point particle” representation
Attraction: Formation of chemical bond by sharing of electrons
Repulsion: Pauli exclusion (too many electrons in small volume)
61
Generic shape of interatomic potential
φ
Energy U
r
Effective
interatomic
potential
1/r12 (or Exponential)
Repulsion
Radius r (Distance
between atoms)
e
r0
Harmonic oscillator
~ k(r - r0)2
Attraction
r
1/r6
Image by MIT OpenCourseWare.
Image by MIT OpenCourseWare.
Many chemical bonds show
this generic behavior
Attraction: Formation of chemical bond by sharing of electrons
Repulsion: Pauli exclusion (too many electrons in small volume)
62
Weaker bonding
Atomic interactions – different types of chemical bonds
ƒ Primary bonds (“strong”)
ƒ Ionic (ceramics, quartz, feldspar - rocks)
ƒ Covalent (silicon)
ƒ Metallic (copper, nickel, gold, silver)
(high melting point, 1000-5,000K)
ƒ Secondary bonds (“weak”)
ƒ Van der Waals (wax, low melting point)
ƒ Hydrogen bonds (proteins, spider silk)
(melting point 100-500K)
ƒ Ionic: Non-directional (point charges interacting)
ƒ Covalent: Directional (bond angles, torsions matter)
ƒ Metallic: Non-directional (electron gas concept)
Know models for all types of chemical bonds!
63
How to choose a potential
64
How to calculate forces from the “potential”
or “force field”
ƒ Define interatomic potentials, that describe the energy of a set of atoms
as a function of their coordinates
ƒ “Potential” = “force field”
G
r = {rj } j = 1.. N
U total = U total (r )
G
Fi = −∇ rKiU total ( r )
Position
vector of
atom i
Depends on position of
all other atoms
i = 1.. N
⎛ ∂
∂
∂ ⎞
⎟
∇ rGi = ⎜⎜
,
,
⎟
∂
∂
∂
r
r
r
⎝ 1,i 2,i 3,i ⎠
Change of potential energy
due to change of position of
65
particle i (“gradient”)
Force calculation
Pair potential
Note cutoff!
66
Pair potentials: energy calculation
Pair wise
interaction
potential
Simple approximation: Total energy is sum over the
energy of all pairs of atoms in the system
φ ( rij )
Pair wise summation of bond energies
r12
r25
avoid double counting
U tot =
N
1
2
N
∑ ∑φ (r )
ij
i =1,i ≠ j j =1
N
rij =
distance between
particles i and j
Energy of atom i
U i = ∑ φ ( rij )
j =1
67
Total energy of pair potential
ƒ Assumption: Total energy of system is expressed as sum of the energy
due to pairs of atoms
U total =
r12
r23
N
1
2
N
∑ ∑φ (r )
i =1,i ≠ j j =1
ij
φij = φ ( rij )
with
U total
=0
1
= (φ12 + φ13 + φ21 + φ23 + φ31 + φ32 )
2
=0
beyond cutoff
68
Force calculation – pair potential
Forces can be calculated by taking derivatives from the potential function
Force magnitude: Negative derivative of potential energy with
respect to atomic distance
F =−
d φ ( rij )
d rij
To obtain force vector Fi, take projections into the
three axial directions
Component
G i of
vector rij
xi
fi = F
rij
G
rij = rij
j
rij
i
f
f1
x2
f2
x1
x2
x1
69
Force calculation – pair potential
Forces on atom 3: only interaction
with atom 2 (cutoff)
1. Determine distance between atoms 3 and 2, r23
2. Obtain magnitude of force vector based on derivative of potential
3. To obtain force vector Fi, take projections into the
three axial directions
Component
G i of
vector rij
d φ ( r23 )
F =−
d r23
xi
fi = F
r23
G
r23 = r23
70
Interatomic pair potentials: examples
φ ( rij ) = D exp(− 2α ( rij − r0 ) ) − 2 D exp(− α ( rij − r0 ) )
⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤
φ ( rij ) = 4ε ⎢⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ ⎥
rij ⎠ ⎥
⎢⎝ rij ⎠
⎝
⎦
⎣
⎛σ ⎞
⎛ rij ⎞
φ ( rij ) = A exp⎜⎜ − ⎟⎟ − C ⎜⎜ ⎟⎟
⎝ σ⎠
⎝ rij ⎠
Morse potential
Lennard-Jones 12:6
potential (excellent model
for noble gases, Ar, Ne,
Xe..)
6
Buckingham potential
Harmonic approximation
71
Lennard-Jones potential – example for copper
0.2
φ (eV)
0.1
0
σ6 2=r
ε
-0.1
-0.2
0
2
r0
3
4
5
o
r (A)
Image by MIT OpenCourseWare.
LJ potential – parameters for copper (e.g.: Cleri et al., 1997)
72
Derivative of LJ potential ~ force
0.1
dφ(r)
= -φ'
F=_
dr
ο
-dφ/dr (eV/A)
0.2
0
Fmax,LJ
-0.1
-0.2
φ(r)
2
r0
3
4
5
o
r (A)
Image by MIT OpenCourseWare.
EQ r0
73
Cutoff radius
N
U i = ∑ φ ( rij )
j =1
6
…
i
j=1
Ui =
5
4
∑φ (r )
j =1.. N neigh
φ
~σ
2
ij
LJ 12:6
potential
3
rcut
ε0
rcut
Cutoff radius = considering interactions only to a certain distance
Basis: Force contribution negligible (slope)
r
74
Derivative of LJ potential ~ force
rcut
0.1
ο
-dφ/dr (eV/A)
0.2
dφ(r)
F = _ _____
dr
0
force not zero
Potential shift
-0.1
-0.2
2
r0
3
4
5
o
r (A)
Image by MIT OpenCourseWare.
Beyond cutoff: Changes in energy (and thus forces) small 75
Crystal structure and potential
ƒ The regular packing (ordering) of atoms into crystals is closely related to the
potential details
ƒ Many 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)
2D example
LJ 12:6
potential
N=4 bonds
φ
N=6 bonds per atom
~σ
ε
r
Square lattice
76
Hexagonal lattice
Example
Initial: cubic lattice
Images courtesy of
the Center for
Polymer Studies at
Boston University.
Used with permission.
Transforms into triangular lattice
time
http://polymer.bu.edu/java/java/LJ/index.html
77
All bonds are not the same – metals
stronger
+ different
bond EQ
distance
Surface
Bulk
Pair potentials: All bonds are equal!
Reality: Have environment
effects; it matter that there is a
free surface!
78
Embedded-atom method (EAM): multi-body potential
new
1
φi = ∑
φ ( rij ) + F ( ρ i )
j =1.. N neigh 2
Pair potential
energy
ρi
ρi =
Embedding
energy
as a function of
electron density
Electron density at atom i
based on a pair potential:
∑ π (r )
j =1.. N neigh
ij
Models other than EAM (alternatives):
Glue model (Ercolessi, Tosatti, Parrinello)
Finnis Sinclair
79
Equivalent crystal models (Smith and Banerjee)
Effective pair interactions: EAM potential
Can describe
differences between
bulk and surface
1
Effective pair potential (eV )
EAM=Embedded
atom method
Bulk
0.5
Surface
0
-0.5
2
3
4
r (A)
o
Image by MIT OpenCourseWare.
Pair potential
energy
Embedding energy
as a function of electron density
1
U = ∑∑ φ ( rij ) + ∑ F ( ρ i )
2 i ,i ≠ j j
i
Embedding term: depends on
environment, “multi-body”
80
Effective pair interactions: EAM
r
Effective pair potential (eV )
1
++++++
++++++
++++++
Bulk
0.5
Surface
Change in
NN distance
r
0
-0.5
2
3
4
r (A)
o
Image by MIT OpenCourseWare.
Can describe differences between bulk and surface
Daw, Foiles, Baskes, Mat. Science Reports, 1993
++++++
++++++
++++++
81
Model for chemical interactions
ƒ Similarly: Potentials for chemically complex materials – assume that total
energy is the sum of the energy of different types of chemical bonds
U total = U Elec + U Covalent + U Metallic + U vdW + U H − bond
82
Concept: energy landscape for chemically complex
materials
U total = U Elec + U Covalent + U Metallic + U vdW + U H − bond
ƒ Different energy contributions from different kinds of chemical bonds
are summed up individually, independently
ƒ 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 …
83
Summary: CHARMM-type potential
=0 for proteins
U total = U Elec + U Covalent + U Metallic + U vdW + U H − bond
U Elec :
Coulomb potential
φ ( rij ) =
qi q j
ε1rij
1
φstretch = kstretch ( r − r0 ) 2
2
1
φbend = k bend (θ − θ 0 ) 2
U Covalent = U stretch + U bend + U rot
2
1
φrot = k rot (1 − cos(ϑ ))
2
⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤
U vdW : LJ potential φ ( rij ) = 4ε ⎢⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ ⎥
rij ⎠ ⎥
⎢⎝ rij ⎠
⎝
⎣
⎦
12
10
⎡ ⎛R
⎤
⎞
⎛
⎞
R
U H − bond : φ ( rij ) = DH −bond ⎢5⎜⎜ H −bond ⎟⎟ − 6⎜⎜ H −bond ⎟⎟ ⎥ cos4 (θ DHA )
rij ⎠ ⎥
⎢ ⎝ rij ⎠
⎝
⎣
⎦
84
Summary of energy expressions: CHARMM, DREIDING, etc.
δ−
Bond stretching
δ+
δ+
Angle Bending
Bond Rotation
Electrostatic
interactions
δ−
vdW Interactions
vdW ONLY
between atoms of
different
molecules
Image by MIT OpenCourseWare.
….
85
Identify a strategy to model the change of an interatomic bond under a
chemical reaction (based on harmonic potential) – from single to double
bond
φ
Energy φ
φstretch
1
= kstretch ( r − r0 ) 2
2
Effective
interatomic
potential
r0
Harmonic oscillator
~ k(r - r0)2
r
Image by MIT OpenCourseWare.
r
Energy
φstretch
focus on C-C bond
Transition point
sp3
sp2
BO=1
C-C distance
BO=2
r
sp2
sp3
Solution sketch/concept
φstretch
kstretch = kstretch (BO)
1
= kstretch ( r − r0 ) 2
2
r0 = r0 (BO)
Make potential parameters kstretch and r0 a function of the chemical state of
the molecule (“modulate parameters”)
E.g. use bond order as parameter, which allows to smoothly interpolate
between one type of bonding and another one dependent on geometry of
See lecture notes (lecture 7): Bond order
molecule
potential for silicon
Potential energy (eV)
5
0
-5
-10
0.5
1
1.5
2
2.5
3
3.5
o
Distance (A)
Fig. 2.21c in Buehler, Markus J. Atomistic Modeling
of 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.
Triple
S.C.
Double
F.C.C.
Single
Effective pair-interactions for various C-C (Carbon) bonds
Image by MIT OpenCourseWare.
Summary: CHARMM force field
ƒ Widely used and accepted model for protein structures
ƒ Programs such as NAMD have implemented the CHARMM force
field
E.g., problem set 3, nanoHUB stretchmol/NAMD module, study
of a protein domain part of human vimentin intermediate
filaments
88
Overview: force fields discussed in IM/S
ƒ
Pair potentials (LJ, Morse, harmonic potentials, biharmonic potentials), used
for single crystal elasticity (pset #1) and fracture model (pset #2)
Pair potential
ƒ
EAM (=Embedded Atom Method), used for nanowire (pset #1), suitable for
metals
Multi-body potential
ƒ
ReaxFF (reactive force field), used for CMDF/fracture model of silicon (pset
#2) – very good to model breaking of bonds (chemistry)
Multi-body potential (bond order potential)
ƒ
Tersoff (nonreactive force field), used for CMDF/fracture model of silicon
(note: Tersoff only suitable for elasticity of silicon, pset #2)
Multi-body potential (bond order potential)
ƒ
CHARMM force field (organic force field), used for protein simulations (pset
#3)
Multi-body potential (angles, for instance)
89
How to choose a potential
ReaxFF
EAM, LJ
CHARMM-type (organic)
ReaxFF
CHARMM
90
SMD mimics AFM single molecule experiments
G
v
Atomic force microscope
k
pset #3
G
v
x
k
x
f
x
91
Steered molecular dynamics (SMD)
pset #3
Steered molecular
dynamics used to apply
forces to protein
structures
Virtual atom
moves w/ velocity
f = k (v ⋅ t − x )
G
v
G
v
G
G
v ⋅t − x
SMD spring constant
f
k
G
x
end point of
molecule
G
G
G
f = k (v ⋅ t − x )
SMD
deformation
speed vector
time
Distance between end
point of molecule and
virtual atom
92
Comparison – CHARMM vs. ReaxFF
Covalent bonds
never break
in CHARMM
93
M. Buehler, JoMMS, 2007
1.1 Atomistic and molecular simulation
algorithms
1.2 Property calculation
1.3 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
94
Application – mechanics of materials
tensile test of a wire
Brittle
Stress
Ductile
Necking
Strain
Image by MIT OpenCourseWare.
Brittle
Ductile
Image by MIT OpenCourseWare.
95
Brittle versus ductile materials behavior
96
Ductile versus brittle materials
BRITTLE
Glass Polymers
Ice...
DUCTILE
Copper, Gold
Shear load
Difficult
to deform,
breaks easily
Image by MIT OpenCourseWare.
Easy to deform
hard to break
97
Deformation of materials:
Nothing is perfect, and flaws or cracks matter
“Micro (nano), local”
“Macro,
global”
σ (r )
r
Failure of materials initiates at cracks
Griffith, Irwine and others: Failure initiates at defects, such as cracks, or
grain boundaries with reduced traction, nano-voids, other imperfections 98
Crack extension: brittle response
σ (r )
r
Large stresses lead to
rupture of chemical
bonds between atoms
Thus, crack extends
99
Image by MIT OpenCourseWare.
Basic fracture process: dissipation of elastic energy
δa
Undeformed
Stretching=store elastic energy
Release elastic energy
dissipated into breaking
chemical bonds
100
Limiting speeds of cracks: linear elastic
continuum theory
Mother-daughter mechanism
SuperSub-Rayleigh Rayleigh
Intersonic
Supersonic
Mode II
Mode I
Cr
Cs
Subsonic
Cl
Limiting speed v
Supersonic
Mode III
Cs
Linear
Cl
Limiting speed v
Nonlinear
Image by MIT OpenCourseWare.
• 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)
101
Summary: mixed Hamiltonian approach
Wi
100%
0%
W1
W2
R-Rbuf
ReaxFF
R+Rtrans R+Rtrans +Rbuf
R
Transition
layer
x
Tersoff
Transition region
Tersoff ghost atoms
ReaxFF ghost atoms
Image by MIT OpenCourseWare.
FReaxFF −Tersoff = (wReaxFF ( x ) FReaxFF + (1 − wReaxFF ) FTersoff )
wReaxFF ( x ) + wTersoff ( x ) = 1
∀x
102
Atomistic fracture mechanism
Image by MIT OpenCourseWare.
103
Lattice shearing: ductile response
ƒ Instead of crack extension,
induce shearing of atomic lattice
ƒ Due to large shear stresses at
crack tip
ƒ Lecture 9
Image by MIT OpenCourseWare.
τ
τ
τ
τ
Image by MIT OpenCourseWare.
104
Concept of dislocations
additional half plane
τ
τ
τ
τ
b
Image by MIT OpenCourseWare.
Localization of shear rather than homogeneous shear that requires
cooperative motion
“size of single dislocation” = b, Burgers vector
105
Final sessile structure – “brittle”
Fig. 1 c from Buehler, M., et al. "The Dynamical Complexity of Work-Hardening: A Large-Scale Molecular Dynamics
Simulation." Acta Mech Sinica 21 (2005): 103-11.
© Springer-Verlag. All rights reserved. This content is excluded from our Creative Commons license. For more information,
see http://ocw.mit.edu/fairuse.
106
Brittle versus ductile materials behavior
Copper, nickel (ductile) and glass, silicon (brittle)
For a material to be
ductile, require
dislocations, that is,
shearing of lattices
Not possible here
(disordered)
© source unknown. All rights reserved. This content is excluded
from our Creative Commons license. For more information, see
http://ocw.mit.edu/fairuse.
107
Crack speed analysis
crack has stopped
slope=equal to
crack speed
108
MD simulation: chemistry as bridge between
disciplines
MD
109
Application – protein folding
Combination
of 3 DNA
letters equals
a amino acid
ACGT
Four letter
code “DNA”
E.g.: Proline –
.. - Proline - Serine –
Proline - Alanine - ..
CCT, CCC,
CCA, CCG
Transcription/
translation
Sequence of amino acids
“polypeptide” (1D
structure)
Folding
(3D structure)
Goal of protein folding simulations:
Predict folded 3D structure based on polypeptide sequence
110
Movie: protein folding with CHARMM
ƒ de novo Folding of a Transmembrane fd Coat Protein
http://www.charmm-gui.org/?doc=gallery&id=23
Polypeptide chain
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
111
solution-phase NMR on the protein in micelles)
2. Important terminology and concepts
112
Important terminology and concepts
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
Force field
Potential / interatomic potential
Reactive force field / nonreactive force field
Chemical bonding (covalent, metallic, ionic, H-bonds, vdW..)
Time step
Continuum vs. atomistic viewpoints
Monte Carlo vs. Molecular Dynamics
Property calculation (temperature, pressure, diffusivity [transport
properties]-MSD, structural analysis-RDF)
Choice of potential for different materials (pair potentials, chemical
complexity)
MD algorithm: how to calculate forces, energies
MC algorithm (Metropolis-Hastings)
Applications: brittle vs. ductile, crack speeds, data analysis
113
3. Continuum vs. atomistic approaches
114
Relevant scales in materials
λ (atom)
<<
ξ (crystal)
<< d (grain)
<<
<< dx, dy, dz
y
Ay
σyy
ny
H ,W , D
σyx
σyz
σxz
σxy
nx
σxx
x
Image from
Wikimedia
Commons,
http://commons
.wikimedia.org.
Courtesy of Elsevier, Inc.,
http://www.sciencedirect.com.
Used with permission.
Bottom-up
Fig. 8.7 in: Buehler, M.
Atomistic Modeling of 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.
z
Image by MIT
OpenCourseWare.
Ax
Top-down
Atomistic viewpoint:
•Explicitly consider discrete atomistic structure
•Solve for atomic trajectories and infer from these about material properties &
behavior
•Features internal length scales (atomic distance)
116
“Many-particle system with statistical properties”
Relevant scales in materials
Separation of scales
λ (atom)
<<
ξ (crystal)
RVE
<< d (grain)
y
σyx
σyz
σxz
H ,W , D
Ay
σyy
ny
Image from
Wikimedia
Commons,
http://commons
.wikimedia.org.
<<
<< dx, dy, dz
σxy
nx
σxx
x
Courtesy of Elsevier, Inc.,
http://www.sciencedirect.com.
Used with permission.
Fig. 8.7 in: Buehler, M.
Atomistic Modeling of 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.
z
Image by MIT
OpenCourseWare.
Ax
Top-down
Continuum viewpoint:
•Treat material as matter with no internal structure
•Develop mathematical model (governing equation) based on representative
volume element (RVE, contains “enough” material such that internal structure
can be neglected
•Features no characteristic length scales, provided RVE is large enough 117
“PDE with parameters”
Example – diffusion
118
Example solution – 2nd Fick’s law
∂c
d 2c
=D 2
dx
∂t
c0
Concentration
c0
x
time t
c( x, t )
BC: c (x = 0) = c0
IC: c (x > 0, t = 0) = 0
x
Need diffusion coefficient to solve for distribution!
119
2. Finite difference method
How to solve a PDE
∂c
d 2c
=D 2
∂t
dx
Note: Flux
+ BCs, ICs
dc
J = −D
dx
120
Finite difference method
d 2c
∂c
=D 2
dx
∂t
Idea: Instead of solving for continuous field
solve for
i=
space
j=
time
c(ti , x j )
c(ti , x j ) = ci , j
c(t , x )
discrete grid
(space and time)
i
i = 1..Nx
t
j
δt
δx
121
Finite difference method
d 2c
∂c
=D 2
dx
∂t
Idea: Instead of solving for continuous field
solve for
c(t , x )
c(ti , x j )
Approach: Express continuous derivatives as discrete differentials
∂c Δc
≈
∂t Δt
122
Finite difference method
d 2c
∂c
=D 2
dx
∂t
Idea: Instead of solving for continuous field
solve for
c(t , x )
c(ti , x j )
Approach: Express continuous derivatives as discrete differentials
∂c Δc
≈
∂t Δt
∂c Δc
≈
∂x Δx
⎛ Δc ⎞
Δ⎜ ⎟
2
∂c
Δt ⎠
⎝
≈
2
Δx
∂x
123
Forward, backward and central difference
∂c
∂x
c
Forward difference
xi ,t j
forward
Δc
ci +1, j
ci , j
ci −1, j
backward
Backward difference
x
xi −1
xi
c
Central difference
δx
xi +1
ci +1, j
ci , j
ci −1, j
central
x
xi −1
xi
xi +1
124
How to calculate derivatives with discrete
c
differentials
∂c Δc
≈
∂t Δt
d 2c
∂c
=D 2
dx
∂t
ci , j +1
ci , j
ci , j +1 − ci , j ci , j +1 − ci , j
∂c
≈
=
t j +1 − t j
δt
∂t xi ,t j
δt
δt
Δc
(1)
discrete time step
tj
t j +1
∂c
∂t xi ,t j
125
t
How to calculate derivatives with discrete
differentials
∂c Δc
≈
∂x Δx
∂c
∂x
≈
x
i+ 1
2
∂c
∂x x
i− 1
2
,t j
∂c
d 2c
⎛ Δc ⎞ ∂t = D dx 2
Δ⎜ ⎟
2
∂c
Δt ⎠
⎝
≈
2
Δx
∂x
ci +1, j − ci , j
xi +1 − xi
=
ci +1, j − ci , j
δx
∂c
c ∂x
x
i+ 1
2
∂c
∂x
,t j
x
i+ 1
2
,t j
ci +1, j
ci , j
ci −1, j
central
x
xi −1
xi
xi +1
xi + 1
,t j
ci , j − ci −1, j ci , j − ci −1, j
≈
=
xi − xi −1
δx
2
xi − 1
2
126
How to calculate derivatives with discrete
differentials ∂c
2
∂c
d 2c
⎛ ∂c ⎞ ∂t = D dx 2
Δ⎜ ⎟
2
∂c
⎝ ∂x ⎠
≈
Δx
∂x 2
∂c Δc
≈
∂x Δx
∂x
∂c
∂x 2
xi ,t j
xi ,t j
c 'i + 1 , j
2
∂c
∂x
≈
x
i+ 1
2
∂c
∂x x
i− 1
2
∂c
∂x 2
,t j
,t j
2
xi +1 − xi
=
ci +1, j − ci , j
2
∂c
∂x
x
i+ 1
2
,t j
∂c
−
∂x
xi − 12 ,t j
xi + 1 − xi − 1
2
2
central
x
δx
xi − 1
2
ci , j − ci −1, j ci , j − ci −1, j
≈
=
xi − xi −1
δx
≈
xi ,t j
ci +1, j − ci , j
c 'i − 1 , j
=
xi
xi + 1
2
Apply central difference
method again…
ci +1, j − ci , j − (ci , j − ci −1, j )
δx
2
(2)
=
ci +1, j − 2ci , j − ci −1, j
δx 2
127
Complete finite difference scheme
∂c
d 2c
=D 2
∂t
dx
∂ 2c
∂x 2
≈
xi ,t j
ci +1, j − 2ci , j − ci −1, j
δx 2
ci , j +1 − ci , j
∂c
≈
δt
∂t xi ,t j
ci , j +1 − ci , j
δt
⎛ ci +1, j − 2ci , j − ci −1, j ⎞
= D ⎜⎜
⎟⎟
2
δx
⎝
⎠
128
Complete finite difference scheme
∂c
d 2c
=D 2
∂t
dx
∂ 2c
∂x 2
≈
xi ,t j
ci +1, j − 2ci , j − ci −1, j
δx 2
ci , j +1 − ci , j
∂c
≈
δt
∂t xi ,t j
Want
ci , j +1 − ci , j
δt
⎛ ci +1, j − 2ci , j − ci −1, j ⎞
= D ⎜⎜
⎟⎟
2
δx
⎝
⎠
δt
ci , j +1 = ci , j + 2 D (ci +1, j − 2ci , j − ci −1, j )
δx
129
Complete finite difference scheme
∂c
d 2c
=D 2
∂t
dx
Concentration at i
at old time
i=
j=
space
time
“explicit” numerical scheme
δt
ci , j +1 = ci , j + 2 D (ci +1, j − 2 ci , j − ci −1, j ) (new concentration directly from
δx
concentration at earlier time)
Concentration at i
at old time
Concentration at i
at new time
Concentration at i-1
at old time
Concentration at i+1
at old time
130
Summary
ƒ Developed finite difference approach to solve for diffusion
equation
ƒ Use parameter (D), e.g. calculated from molecular
dynamics, and solve “complex” problem at continuum
scale
ƒ Do not consider “atoms” or “particles”
131
Summary
Multi-scale approach:
Feed parameters from atomistic simulations to continuum models
Time scale
Continuum
model
MD
“Empirical”
or experimental
parameter
feeding
Quantum
mechanics
Length scale
132
MIT OpenCourseWare
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3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modeling and Simulation
Spring 2012
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