1.021, 3.021, 10.333, 22.00 Introduction to Modeling and Simulation
Spring 2011
Part I – Continuum and particle methods
Basic molecular dynamics
Lecture 2
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.
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?
Lectures 2-13
Lectures 14-26
2
Goals of part I (particle methods)
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, etc.
ƒ Analyze atomistic simulations (make sense of all the
numbers)
ƒ Visualize atomistic/molecular data (bring data to life)
ƒ Understand how to link atomistic simulation results with
continuum models within a multi-scale scheme
3
Lecture 2: Basic molecular dynamics
Outline:
1. Introduction
2. Case study: Diffusion
2.1 Continuum model
2.2 Atomistic model
3. Additional remarks – historical perspective
Goals of today’s lecture:
ƒ
Through case study of diffusion, illustrate the concepts of a
continuum model and an atomistic model
ƒ
Develop appreciation for distinction of continuum and atomistic
approach
ƒ
Develop equations/models for diffusion problem from both
perspectives
ƒ
Develop atomistic simulation approach (e.g. algorithm,
pseudocode, etc.) and apply to describe diffusion (calculate
diffusivity)
ƒ
Historical perspective on computer simulation with MD, examples
from literature
4
1. Introduction
5
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)
6
“Many-particle system with statistical properties”
Relevant scales in materials
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
7
“PDE with parameters”
2. Case study: Diffusion
Continuum and atomistic modeling
8
Goal of this section
ƒ Diffusion as example
ƒ Continuum description (top-down approach), partial
differential equation
ƒ Atomistic description (bottom-up approach), based on
dynamics of molecules, obtained via numerical
simulation of the molecular dynamics
9
Introduction: Diffusion
ƒ Particles move from a domain with high concentration to an area of
low concentration
ƒ Macroscopically, diffusion measured by change in concentration
ƒ Microscopically, diffusion is process of spontaneous net movement
of particles
Result of random motion of particles (“Brownian motion”)
Low
concentration
High
concentration
c = m/V = c(x, t)
10
Ink droplet in water
hot
cold
© source unknown. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/fairuse.
11
Microscopic observation of diffusion
12
Microscopic mechanism: “Random walk” –
or Brownian motion
ƒ Brownian motion was first observed (1827) by the British botanist Robert
Brown (1773-1858) when studying pollen grains in water
ƒ Initially thought to be sign of life, but later confirmed to be also present in
inorganic particles
ƒ The effect was finally explained in 1905 by Albert Einstein, who realized it
was caused by water molecules randomly smacking into the particles.
Public domain image.
13
Brownian motion leads to net particle
movement
time
Particle “slowly” moves away from its initial position
14
Robert Brown’s microscope
Robert Brown's Microscope
1827
Instrument with which Robert Brown studied Brownian
motion and which he used in his work on identifying the
nucleus of the living cell
Instrument is preserved at the Linnean Society in London
It is made of brass and is mounted onto the lid of the box in
which it can be stored
http://www.brianjford.com/pbrownmica.jpg
15
Simulation of Brownian motion
ƒ http://www.scienceisart.com/A_Diffus/Jav1_2.html
16
Macroscopic observation of diffusion
17
Macroscopic observation: concentration change
Ink dot
Tissue
See also:
http://www.uic.edu/classes/phys/phys450/MARKO/N004.html
© source unknown. All rights reserved. This content is excluded from our
Creative Commons license. For more information, see http://ocw.mit.edu/fairuse.
18
Brownian motion leads to net particle movement
Time t
c1=m1/V (low)
c2=m2/V (high)
© source unknown. All rights reserved. This content is
excluded from our Creative Commons license. For more
information, see http://ocw.mit.edu/fairuse.
© The McGraw Hill Companies. All rights reserved. This content is excluded
from our Creative Commons license. For more information, see
http://ocw.mit.edu/fairuse.
19
Diffusion in biology
Image removed due to copyright restrictions. See the image now: http://www.biog1105-1106.org/demos/105/unit9/media/diffusion-after-act-pot.jpg.
20
http://www.biog1105-1106.org/demos/105/unit9/roleofdiffusion.html
2.1 Continuum model
How to build a continuum model to describe the
physical phenomena of diffusion?
21
Approach 1: Continuum model
ƒ Develop differential equation based on differential
element
mR
mL
J
W=1
H=1
x
Δx
Δx
J: Mass flux (mass per unit time per unit area)
Concept: Balance mass [here], force etc. in a differential volume
element; much greater in dimension than inhomogeneities
(“sufficiently large RVE”)
22
Approach 1: Continuum model
ƒ Develop differential equation based on differential
element
mR
mL
J
W=1
H=1
x
Δx
Probability of mass m
crossing the central boundary
during a period Δt is equal to p
Δx
23
Approach 1: Continuum model
ƒ Develop differential equation based on differential
element
mR
mL
J
W=1
H=1
x
Δx
Probability of mass m
crossing the central boundary
during a period Δt is equal to p
Δx
1 p
mL Mass flux from left to right
1 × 1 Δt
1 p
mR Mass flux from right to left
JR =
1 × 1 Δt
JL =
[J] = mass per unit time per
unit area
24
Approach 1: Continuum model
ƒ Develop differential equation based on differential
element
mR
mL
J
W=1
H=1
x
Δx
Probability of mass m
crossing the central boundary
during a period Δt is equal to p
Δx
1 p
mL Mass flux from left to right
1 × 1 Δt
1 p
mR Mass flux from right to left
JR =
1 × 1 Δt
JL =
More mass, more flux (mL is ~ to number of particles)
Effective mass flux
1 p
(mL − mR )
J=
1 × 1 Δt
25
Continuum model of diffusion
Express in terms of mass concentrations
p
m
(mL − mR )
J
=
m = cV
c=
Δt
V
cR
cL
J
W=1
H=1
x
Δx
Δx
26
Continuum model of diffusion
Express in terms of mass concentrations
p
m
(mL − mR )
J
=
m = cV
c=
Δt
V
J=
1 p
(cL − cR ) ⋅ Δx × 1 × 1
1 × 1 Δt
= − Δc
=V
cR
cL
J
W=1
H=1
x
Δx
Δx
27
Continuum model of diffusion
Express in terms of mass concentrations
m
V
1 p
(cL − cR ) ⋅ Δx × 1 × 1
J=
1 × 1 Δt
= − Δc
=V
c=
Concentration
gradient
J =−
Δc
p
p
ΔcΔx = − Δx 2
Δt
Δt
Δx
cR
cL
J
W=1
H=1
x
Δx
Δx
expand Δx
28
Continuum model of diffusion
Express in terms of mass concentrations
m
V
1 p
(cL − cR ) ⋅ Δx × 1 × 1
J=
1 × 1 Δt
= − Δc
=V
c=
Concentration
gradient
J =−
Δc
p
p
ΔcΔx = − Δx 2
Δx
Δt
Δt
cR
cL
J
W=1
H=1
x
Δx
Δx
p
dc
2 dc
J = − Δx
= −D
Δt
dx
dx
Δx 2 Parameter that measures how “fast” mass
29
D= p
Δt moves (in square of distance per unit time)
Diffusion constant & 1st Fick law
Reiterate: Diffusion constant D describes the how much mass moves
per unit time
Movement of mass characterized by square of displacement from initial
position
Flux
J =−
p
dc
dc
Δx 2
= −D
Δt
dx
dx
Δx 2
D= p
Δt
1st Fick law
(Adolph Fick, 1829-1901)
D~ p
Diffusion constant relates to the ability of mass to move a distance
Δx2 over a time Δt (strongly temperature dependent, e.g. Arrhenius)
30
2nd Fick law (time dependence)
c
J = −D
dc
dx
1st Fick law
Δc ( J 1 − J 2 ) × 1 × 1
=
Δt
Δx × 1 × 1
J2
J1
x0
Δx
x
x1
J: Mass flux (mass per unit time per unit area)
31
2nd Fick law (time dependence)
c
J = −D
dc
dx
1st Fick law
J2
J1
x0
Δx
x
x1
dc
dc ⎤ ⎞
1 ⎛
Δc J1 − J 2
⎡
⎜⎜ − D
| − ⎢− D
| ⎥ ⎟⎟
=
=
dx x = x0 ⎣
dx x = x1 ⎦ ⎠
Δt
Δx
Δx ⎝
dc
J1 = J ( x = x0 ) = − D
|
dx x = x0
32
2nd Fick law (time dependence)
c
J = −D
dc
dx
1st Fick law
ΔJ
J2
J1
x0
Δx
x
x1
dc
Δc 1 ⎛
dc
⎞
|
|
=
−
+
D
D
⎜
⎟
dx x = x1 ⎠
Δt Δx ⎝
dx x = x0
∂c
d
d ⎛
dc ⎞
(
)
=−
J = − ⎜− D ⎟
∂t
dx
dx ⎝
dx ⎠
Change of concentration in
time equals change of flux with
x (mass balance)
33
2nd Fick law (time dependence)
c
J = −D
dc
dx
1st Fick law
J2
J1
ΔJ
x0
Δx
x
x1
dc
Δc 1 ⎛
dc
⎞
|
|
=
−
+
D
D
⎜
⎟
dx x = x1 ⎠
Δt Δx ⎝
dx x = x0
∂c
dc ⎞
d ⎛
d
(
)
=−
J = − ⎜− D ⎟
∂t
dx ⎠
dx ⎝
dx
d 2c
∂c
=D 2
dx
∂t
Change of concentration in
time equals change of flux with
x (mass balance)
2nd Fick law
Solve by applying ICs and BCs…
PDE
34
Example solution – 2nd Fick’s law
Concentration
d 2c
∂c
=D 2
dx
∂t
c0
c0
x
time t
c( x, t )
BC: c (x = 0) = c0
IC: c (x > 0, t = 0) = 0
Need diffusion coefficient to solve for distribution!
x
35
How to obtain diffusion coefficient?
ƒ Laboratory experiment
ƒ Study distribution of concentrations (previous slide)
ƒ Then “fit” the appropriate diffusion coefficient so that the
solution matches
ƒ Approach can then be used to solve for more complex
geometries, situations etc. for which no lab experiment
exists
ƒ “Top down approach”
36
Matching with experiment (parameter
identification)
Concentration
c0
time t0
experiment
continuum model solution
c( x, t0 )
x
fit parameter D
37
Summary
ƒ Continuum model requires parameter that describes microscopic
processes inside the material
ƒ Typically need experimental measurements to calibrate
Time scale
Continuum
model
Microscopic
mechanisms
???
“Empirical”
or experimental
parameter
feeding
Length scale
38
2.2 Atomistic model
How to build an atomistic bottom-up model to
describe the physical phenomena of diffusion?
39
Approach 2: Atomistic model
ƒ Atomistic model provides an alternative approach to describe
diffusion
ƒ Enables us to directly calculate the diffusion constant from the
trajectory of atoms (“microscopic definition”)
ƒ Approach: Consider set of atoms/molecules
ƒ Follow their trajectory and calculate how fast atoms leave their initial
position
Follow this quantity over
time
Δx 2
D= p
Δt
Recall: Diffusion constant relates to the “ability” of particle to move a
distance Δx2 over a time Δt
40
Molecular dynamics – simulate trajectory of atoms
Goal: Need an algorithm to predict positions, velocities, accelerations as
41
function of time
Solving the equations: What we want
To solve those equations: Discretize in time (n steps), Δt time step:
ri (t0 ) → ri (t0 + Δt ) → ri (t0 + 2Δt ) → ri (t0 + 3Δt ) → ... → ri (t0 + nΔt )
42
Solving the equations
To solve those equations: Discretize in time (n steps), Δt time step:
ri (t0 ) → ri (t0 + Δt ) → ri (t0 + 2Δt ) → ri (t0 + 3Δt ) → ... → ri (t0 + nΔt )
Recall: Taylor expansion of function f around point a
43
Solving the equations
To solve those equations: Discretize in time (n steps), Δt time step:
ri (t0 ) → ri (t0 + Δt ) → ri (t0 + 2Δt ) → ri (t0 + 3Δt ) → ... → ri (t0 + nΔt )
Recall: Taylor expansion of function f around point a
Taylor series expansion ri (t ) around
a = t0
x = t0 + Δt
x − a = t0 + Δt − t0 = Δt
44
Solving the equations
To solve those equations: Discretize in time (n steps), Δt time step:
ri (t0 ) → ri (t0 + Δt ) → ri (t0 + 2Δt ) → ri (t0 + 3Δt ) → ... → ri (t0 + nΔt )
Recall: Taylor expansion of function f around point a
Taylor series expansion ri (t ) around
a = t0
x = t0 + Δt
x − a = t0 + Δt − t0 = Δt
1
ri (t0 + Δt ) = ri (t0 ) + vi (t0 )Δt + ai (t0 ) Δt 2 + ...
2
45
Taylor expansion of ri (t )
1
ri (t0 + Δt ) = ri (t0 ) + vi (t0 )Δt + ai (t0 )Δt 2 + ...
2
1
ri (t0 − Δt ) = ri (t0 ) − vi (t0 ) Δt + ai (t0 )Δt 2 + ...
2
a = t0
x = t0 + Δt
x − a = t0 + Δt − t0 = Δt
a = t0
x = t0 − Δt
x − a = t0 − Δt − t0 = −Δt
46
Taylor expansion of ri(t)
1
ri (t0 + Δt ) = ri (t0 ) + vi (t0 )Δt + ai (t0 ) Δt 2 + ...
2
1
2
r
(
t
−
Δ
t
)
=
r
(
t
)
−
v
(
t
)
Δ
t
+
a
(
t
)
Δ
t
+ ...
i 0
i 0
i 0
+ i 0
2
ri (t0 − Δt ) + ri (t0 + Δt ) = 2ri (t0 ) − vi (t0 ) Δt + vi (t0 )Δt + ai (t0 )Δt 2 + ...
47
Taylor expansion of ri(t)
1
ri (t0 + Δt ) = ri (t0 ) + vi (t0 )Δt + ai (t0 ) Δt 2 + ...
2
1
2
r
(
t
−
Δ
t
)
=
r
(
t
)
−
v
(
t
)
Δ
t
+
a
(
t
)
Δ
t
+ ...
i 0
i 0
i 0
+ i 0
2
ri (t0 − Δt ) + ri (t0 + Δt ) = 2ri (t0 ) − vi (t0 ) Δt + vi (t0 )Δt + ai (t0 )Δt 2 + ...
ri (t0 + Δt ) = 2 ri (t0 ) − ri (t0 − Δt ) + ai (t0 )Δt 2 + ...
48
Taylor expansion of ri(t)
1
ri (t0 + Δt ) = ri (t0 ) + vi (t0 )Δt + ai (t0 ) Δt 2 + ...
2
1
2
r
(
t
−
Δ
t
)
=
r
(
t
)
−
v
(
t
)
Δ
t
+
a
(
t
)
Δ
t
+ ...
i 0
i 0
i 0
+ i 0
2
ri (t0 − Δt ) + ri (t0 + Δt ) = 2ri (t0 ) − vi (t0 ) Δt + vi (t0 )Δt + ai (t0 )Δt 2 + ...
ri (t0 + Δt ) = 2 ri (t0 ) − ri (t0 − Δt ) + ai (t0 )Δt 2 + ...
Positions
at t0
Positions
at t0-Δt
Accelerations
at t0
49
Physics of particle interactions
Laws of Motion of Isaac Newton (1642 – 1727):
1.
2.
3.
Every body continues in its state of rest, or of uniform motion
in a right line, unless it is compelled to change that state by
forces impressed upon it.
The change of motion is proportional to the motive force
impresses, and is made in the direction of the right line in
which that force is impressed.
To every action there is always opposed an equal reaction: or,
the mutual action of two bodies upon each other are always
equal, and directed to contrary parts.
d 2x
f = m 2 = ma
dt
2nd law
50
Verlet central difference method
ri (t0 + Δt ) = 2 ri (t0 ) − ri (t0 − Δt ) + ai (t0 )Δt 2 + ...
Positions
at t0
How to obtain
accelerations?
Positions
at t0-Δt
f i = mai
ai = f i / m
Accelerations
at t0
Need forces on atoms!
51
Verlet central difference method
ri (t0 + Δt ) = 2ri (t0 ) − ri (t0 − Δt ) + f i (t0 ) / mΔt 2 + ...
Positions
at t0
Positions
at t0-Δt
Forces
at t0
52
Forces on atoms
ƒ Consider energy landscape due to chemical bonds
Energy U
r
1/r12 (or Exponential)
Repulsion
Radius r (Distance
between atoms)
e
Attraction
1/r6
Image by MIT OpenCourseWare.
Attraction: Formation of chemical bond by sharing of electrons
Repulsion: Pauli exclusion (too many electrons in small volume)
53
How are forces calculated?
Force magnitude: Derivative of potential energy with respect to
atomic distance
d U (r)
f =−
dr
To obtain force vector fi, take projections into the
three axial directions
x
fi = f i
r
r
x2
f
x1
Often: Assume pair-wise interaction between atoms
54
Note on force calculation
ƒ Forces can be obtained from a variety of models for
interatomic energy, e.g.
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
Pair potentials (e.g. LJ, Morse, Buckingham)
Multi-body potentials (e.g. EAM, CHARMM, UFF, DREIDING)
Reactive potentials (e.g. ReaxFF)
Quantum mechanics (e.g. DFT) – part II
Tight-binding
…
ƒ …will be discussed in next lectures
55
Molecular dynamics
Follow trajectories of atoms
“Verlet central difference method”
ri (t0 + Δt ) = 2 ri (t0 ) − ri (t0 − Δt ) + ai (t0 )Δt 2 + ...
ai = f i / m
Positions
at t0
Positions
at t0-Δt
Accelerations
at t0
56
Summary: Atomistic simulation – numerical
approach “molecular dynamics – MD”
ƒ Atomistic model; requires atomistic microstructure and atomic
position at beginning
ƒ Step through time by integration scheme
ƒ Repeated force calculation of atomic forces
ƒ Explicit notion of chemical bonds – captured in interatomic potential
57
Pseudocode
Set particle positions (e.g. crystal lattice)
Assign initial velocities
For (all time steps):
Calculate force on each particle (subroutine)
Move particle by time step Δt
Save particle position, velocity, acceleration
Save results
Stop simulation
58
Atomic positions
(initial conditions)
ƒ Typically, have cubical cell in which particles are places in
a regular or irregular manner
“gas (liquid)”
“solid - crystal”
59
Atomistic description
Back to the application of diffusion problem…
ƒ
ƒ
ƒ
ƒ
Atomistic description provides alternative way to predict D
Simple solve equation of motion
Follow the trajectory of an atom
Relate the average distance as function of time from initial point to
diffusivity
ƒ Goal: Calculate how particles move “randomly”, away from
initial position
60
JAVA applet
Courtesy of the Center for Polymer Studies at Boston University. Used with
permission.
URL http://polymer.bu.edu/java/java/LJ/index.html
61
Link atomistic trajectory with diffusion constant (1D)
Diffusion constant relates to the “ability” of a particle to move a distance Δx2
(from left to right) over a time Δt
Δx 2
D= p
Δt
Δx 2
Idea – Use MD simulation to measure square of displacement from initial
position of particles, Δr 2 (t ) :
time
62
Link atomistic trajectory with diffusion constant (1D)
Diffusion constant relates to the “ability” of a particle to move a distance Δx2
(from left to right) over a time Δt
Δx 2
D= p
Δt
Δx 2
MD simulation: Measure square of displacement from initial position
of particles, Δr 2 (t ) :
Δr 2
t
63
Link atomistic trajectory with diffusion constant (1D)
Diffusion constant relates to the “ability” of a particle to move a distance Δx2
(from left to right) over a time Δt
Δx 2
D= p
Δt
Δx 2
MD simulation: Measure square of displacement from initial position
of particles, Δr 2 (t ) and not Δx 2 (t ) ….
Replace
Δx 2
D= p
Δt
1 Δr 2
D=
2 Δt
Δr 2
Factor 1/2 = no directionality in (equal
probability to move forth or back)
64
Link atomistic trajectory with diffusion constant (1D)
MD simulation: Measure square of displacement from initial position
of particles, Δr 2 (t ) :
Δr 2
Δr 2
D=
2t
Δr 2 = 2 Dt
R~ t
t
65
Link atomistic trajectory with diffusion constant (2D/3D)
Δx 2
D= p
Δt
Higher dimensions
1 1 Δr 2
D=
2 d Δt
Factor 1/2 = no directionality in (forth/back)
Factor d = 1, 2, or 3 due to 1D, 2D, 3D
(dimensionality)
Since:
2dDΔt ~ Δr 2
2dDΔt + C = Δr 2
C = constant (does not affect D)
66
Example: MD simulation
D=
slope = D
Δr 2
1
d
lim (Δr 2 )
2d t →∞ d t
1D=1, 2D=2, 3D=3
source: S. Yip, lecture notes
1
d
lim
D=
Δr 2
2d t →∞ d t
.. = average over all particles
67
Example molecular dynamics
Courtesy of the Center for Polymer Studies at Boston University. Used with permission.
Δr 2
Trajectories
Particles
Mean Square
Displacement function
Δr 2 (t ) =
1
N
G
G
2
(
)
r
(
t
)
−
r
(
t
=
0
)
∑ i
i
i
Average square of
displacement of all
particles
68
Example calculation of diffusion coefficient
1
Δr (t ) =
N
2
∑ (r (t ) − r (t = 0))
2
i
i
Position of
atom i at time t
Δr 2
i
Position of
atom i at time t=0
slope = D
D=
1
d
lim
Δr 2
2d t →∞ d t
1D=1, 2D=2, 3D=3
69
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
MD
“Empirical”
or experimental
parameter
feeding
Quantum
mechanics
70
Length scale
Multi-scale simulation paradigm
Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
71
3. Additional remarks
72
Historical development of computer simulation
ƒ Began as tool to exploit computing machines developed during World
War II
ƒ MANIAC (1952) at Los Alamos used for computer simulations
ƒ Metropolis, Rosenbluth, Teller (1953): Metropolis Monte Carlo method
ƒ Alder and Wainwright (Livermore National Lab, 1956/1957): dynamics
of hard spheres
ƒ Vineyard (Brookhaven 1959-60): dynamics of radiation damage in
copper
ƒ Rahman (Argonne 1964): liquid argon
ƒ Application to more complex fluids (e.g. water) in 1970s
ƒ Car and Parrinello (1985 and following): ab-initio MD
ƒ Since 1980s: Many applications, including:
ƒ Karplus, Goddard et al.: Applications to polymers/biopolymers,
proteins since 1980s
ƒ Applications to fracture since mid 1990s to 2000
ƒ Other engineering applications (nanotechnology, e.g. CNTs,
nanowires etc.) since mid 1990s-2000
73
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