Post on 28-Dec-2015
Nano Mechanics and Materials:Theory, Multiscale Methods and Applications
byWing Kam Liu, Eduard G. Karpov, Harold S. Park
2. Classical Molecular Dynamics
A computer simulation technique Time evolution of interacting atoms pursued by
integrating the corresponding equations of motion
Based on the Newtonian classical dynamics
Method received widespread attention in the 1970`s Digital computer become powerful and affordable
Molecular Dynamics Today
Liquids Allows the study of new systems, elemental and multicomponent Investigation of transport phenomena i.e. viscosity and heat flow
Defects in crystals Improved realism due to better potentials constitutes a driving force
Fracture Provides insight on ways and speeds of fracture process
Surfaces Helps understand surface reconstructing, surface melting,
faceting, surface diffusion, roughening, etc. Friction
Atomic force microscope Investigates adhesion and friction between two solids
Molecular Dynamics Today
Clusters Corporation of many atoms (few-several thousand) Comprise bridge among molecular systems and solids Many atoms have comparable energies producing a difficulty in
establishing stable structures
Biomolecules Dynamics of large macromolecules
(proteins, nucleic acids (DNA, RNA), membranes)
Electronic properties and dynamics Development of Car-Parrinello method Forces on atoms gained through solving electronic structure
problem Allows study of electronic properties of materials and their
dynamics
MD System
Subdomain of a macroscale object
Manipulated and controlled form the environment via interactions
Various kinds of boundaries and interactions are possible
We consider:
Adiabatically isolated systems that can exchange neither matter nor energy with their surroundings
Non-isolated systems that can exchange heat with the surrounding media (the heat bath, and the multiscale boundary conditions developed at Northwestern)
2.1 Mechanics of a System of Particles
Particle, or material point, or mass point is a mathematical model of a body whose dimensions can be neglected in describing its motion.
Particle is a dimensionless object having a non-zero mass.
Particle is indestructible; it has no internal structure and no internal degrees of freedoms.
Classical mechanics studies “slow” (v << c) and “heavy” (m >> me) particles.
Examples:
1) Planets of the solar system in their motion about the sun
2) Atoms of a gas in a macroscopic vessel
Note: Spherical objects are typically treated as material points, e.g. atoms comprising a molecule. The material point points are associated with the centers of the spheres. Characteristic physical dimensions of the spheres are modeled through particle-particle interaction.
Generalized Coordinates
Generalized coordinates are given by a minimum set of independent parameters (distances and angles) that determine any given state of the system.
Standard coordinate systems are Cartesian, polar, elliptic, cylindrical and spherical.Other systems of coordinates can also be chosen. We will be are looking for a basic form of the equation, which invariant for all coordinate systems.
1 2( , ,..., )sq q q
Examples:
Material point in xy-plane Pendulum Sliding suspension pendulum
We are looking for a basic form of equation motion, which is valid for all coordinate systems.
x
1q 1 2,q x q
y
x
1 2,q x q y
Generalized Coordinates
One distinctive feature of the classical Lagrangian mechanics, as compared with special theories (e.g., Newtonian dynamics and continuum mechanics), is that the choice of coordinates is left arbitrary.
Number s is called the number of degrees of freedom in the system.
The mathematical formulation will be valid for any set of s independent parameters that determine the state of the system at any given time.
Solution of a problem begins with the search for such a set of parameters.
1 2( , ,..., )sq q q
Least Action (Hamilton’s) Principle
Lagrange function (Lagrangian)
Action integral
Trajectory variation
Least action, or Hamilton’s, principle
2
1
( , , ) 0t
t
S L q q t dt
1 2 1 2( , ,..., , , ,..., , )s sL q q q q q q t
,a b a cS S S S
2( )q t
1( )q t
bS
aS cS
1tt
2t
( )q t
2
1
( , , )t
t
S L q q t dt
1 2( ) ( ), ( ) ( ) 0q t q t q t q t
Action is minimum along the true trajectory:
The main task of classical dynamics is to find the true trajectories (laws of motion) for all degrees of freedom in the system.
Remark: in ADMD this principle is used to obtain the trajectories.
Lagrangian Equation of Motion
Lagrange function (Lagrangian)
Least action principle
2
1
( , , ) 0t
t
S L q q t dt
0, 1,2,...,d L L
sdt q q
1 2 1 2( , ,..., ; , ,..., , )s sL q q q q q q t
Lagrangian equation is based on the least action principle only, and it is valid for all coordinate systems.
Substitution of the Lagrange function into the action integral with further application of the least action principle yields the Lagrangian equation motion:
Lagrangian Equation of Motion: Derivation
Using the most general form of the Lagrange function, the least action principle gives
0, 1,2,...,d L L
sdt q q
2 2
1 1
22 2
1 11
( , , ) ( , , )
0
t t
t t
tt t
t tt
S L q q t dt L q q t dt
L L L L d Lq q dt q q dt
q q q q dt q
1 2: ( ) ( ) ( ) ( )q q t q t q t q t
2( )q t
1( )q t
bS
aS cS
1tt
2t
( )q t
Variation of the coordinates cannot change the observable values q(t1) and q(t2):
Therefore,
and the second term finally gives
2
1
0t
t
Lq
q
Lagrange Function in Inertial Coordinate Systems
General form of the Lagrangian function is obtained based on these arguments:In inertial coordinate systems, equations of motion are:
1) Invariant as to the choice of a coordinate system (frame invariance), and2) Compliant with the basic time-space symmetries.
Frame invariance:
Galilean coordinate transformation
Galilean relativity principle
Time-space symmetries
Homogeneity and isotropy of space
Homogeneity of time
( , , ) '( ', ', )L t L tr r r r
' , 't t t r r V
( , , ) '( , , )L t L t r r r a r
( , , ) '( , , )L t L t r r r r
Lagrange Function of a Material Point
Free material point(generalized and Cartesian coordinates)
Interacting material point
Conservative systems
2 2 2 2 2,2 2 2
m m mL q L x y z r
2 ( )2
mL q U q
( ) ( )
2
L T q U q
E T L Const
For conservative systems, kinetic energy depends only on velocities, and potential energy depends only on coordinates.
Lagrange Function: Examples
2 2 cos( )2
mL l mgl
Pendulum
Bouncing ball
Particle in a circular cavity
2
2ym
L y mgy e y
2 22 2( )
2
R x ymL x y e
1) Kinetic energy; 2) Potential energy due to external gravitational field, where g is the acceleration of gravity).
1) Kinetic energy; 2) Potential energy due to external gravitational field; 3) Potential energy of repulsion between the ball and floor.
1) Kinetic energy; 2) Potential energy of repulsion between the particle and cavity wall.
l
y
x
1q
1q y
1 2,q x q y
Pendulum: Lagrange Function Derivation
( ) ( )L T U General form
Kinetic energy
tangential velocity
Potential energy
The total Lagrangian
l
vτ
l cos φ2 2 2
2 2
m mT v l
0lim
t
lv l
t
cosU mgh mgl
Potential energy depends on the height h with respect to a zero-energy level. Here, such a level is chosen at the suspension point, i.e. below the suspension level, the height is negative.
2 2 cos( )2
mL l mgl 29.8m/s , 2mg l
Parameters used:
Pendulum: Equation of Motion and Solution
Lagrange function and equation motion:
Initial conditions (radian):
(0) 0.3, (0) 0
2 2 cos 0 ( ) sin ( ) 02
m d L L gL l mgl t t
dt l
(0) 1.8, (0) 0 (0) 3.12, (0) 0
29.8m/s , 2mg l Parameters:
Bouncing Ball: Lagrange Function Derivation
( ) ( )L T y U y General form
Kinetic energy
Potential energy
2
2
mT y
yU mgy e
Purple dashed line: the first term (gravitational interaction between the ball and the Earth). Blue dotted line: the second term (repulsion between the ball and the bouncing surface).
Red solid line: the total potential.
β is a relative scaling factor: the ball-surface repulsive potential growths in eβ times for a unit length ball/surface penetration (from y = 0 to y = – 1m).
y
2( , )2
ymL y y y mgy e The total Lagrangian:
2
19.8m/s , 1kg,1J, 4m
g m
Parameters used:
Bouncing Ball: Equation of Motion and Solution
Lagrange function and equation motion:
Initial conditions:
(0) 5m(0) 0
yy
2 ( )0 ( ) 02
y y tm d L LL y mgy e y t g e
dt y y m
(0) 10m(0) 0
yy
(0) 15m(0) 0
yy
2 19.8m/s , 1kg, 1J, 4mg m Parameters:
Particle in a Circular Cavity: Lagrange Function Derivation
( , ) ( , )L T x y U x y General form
Kinetic energy
Potential energy
The total Lagrangian
2 2( )2
mT x y
2 2R x yR rU e e
The potential energy grows quickly and becomes larger than the typical kinetic energy, when the distance r between the particle and the center of the cavity approaches value R.
R is the effective radius of the cavity. At r < R, U does not alter the trajectory.
β is a relative scaling factor: the potential energy growths in eβ times between r = R and r = R+1.
2 22 2( , , , ) ( )
2
R x ymL x y x y x y e
y
xr
R
Particle in a Circular Cavity: Equation of Motion and Solution
Lagrangian function and equations: Potential barrier:
27 2 210 J, ( ) ( ) ( )r t x t y t
1 94nm , 10nm, 10 kgR m Parameters:
Initial conditions:
2 2 ( ( ))2 2
( ( ))
( ) ( )/ ( )( )
2 ( ) ( )/ ( )
R r tR x y
R r t
m x t e x t r tmL x y e
m y t e y t r t
Summary of the Lagrangian Method
1. The choice of s generalized coordinates (s – number of degrees of freedom).
2. Derivation of the kinetic and potential energy in terms of the generalized coordinates
3. The difference between the kinetic and potential energies gives the Lagrangian function.
4. Substitution of the Lagrange function into the Lagrangian equation of motion and derivation of a system of s second-order differential equations to be solved.
5. Solution of the equations of motion, using a numerical time-integration algorithm.
6. Post-processing and visualization.
Hamiltonian Mechanics
Description of mechanical systems in terms of generalized coordinates and velocities is not unique.Alternative formulation formulation in terms of generalized coordinates and momenta can be utilized. This formulation is used in statistical mechanics.
, ,q q q p
( , )
, 1, 2,...,
L L q q
L LdL dq dq s
q q
Lp dL p dq p dq
q
Generalized momentum (definition):
A mass point in Cartesian coordinates:
, ,x y zp mx p m y p mz
Legendre’s transformation(the passage from one set of independent variables to another)
Differential of the Lagrange function:
Hamiltonian Equations of Motion
dL p dq p dq
d p q L q dp p dq
Hamiltonian of the system:
dL p dq d p q q dp
( , , )H p q t p q L
dH q dp p dq
,H H
q pp q
Equations of motion:
Hamiltonian Equations of Motion: Pendulum
Hamiltonian of the system:
2, , sin
H H pp p mgl
p ml
Equations of motion:
Lagrange function and the generalized momentum:
2 2 2cos2
m LL l mgl p ml
2 2
2 2, cos cos
2 2
p pT U mgl H mgl
ml ml
2 sin sin 0g
p ml mgll
Conversion to the Lagrangian form (elimination of p):
l
Hamiltonian Equations of Motion: Bouncing Ball
Hamiltonian of the system:
, , yH H py p y p mg e
p y m
Equations of motion:
Lagrange function and the generalized momentum:
2
2ym L
L y mgy e p my
2 2
,2 2
y yp pT U mgy e H mgy e
m m
0y yp my mg e y g em
Conversion to the Lagrangian form (elimination of p):
y
Summary of the Hamiltonian Method
1. The choice of s generalized coordinates (s – number of degrees of freedom).
2. Derivation of the kinetic and potential energy in terms of the generalized coordinates.
3. Derivation of the generalized momenta.
4. Expression of the kinetic energy in terms of the generalized momenta.
5. The sum the kinetic and potential energies gives the Hamiltonian function.
6. Substitution of the Hamiltonian function into the Lagrangian equation of motion and derivation of a system of 2s first-order differential equations to be solved.
7. Solution of the equations of motion, using a numerical time-integration algorithm.
8. Post-processing and visualization.
Predictable and Chaotic Systems
Divergence of trajectoriesin predictable systems
Divergence of trajectoriesin chaotic (unpredictable) systems
Trajectories in MD systems are unpredictable/unstable; they are characterized by a random dependence on initial conditions.
, (0) : , (0) , (0)q q t q q t q q t q C t
, (0) : , (0) , (0) tq q t q q t q q t q e
Variance of the trajectory depends linearly on time
Variance of the trajectory depends exponentially on time (J.M. Haile, Molecular Dynamics Simulation, 2002)
Example Quasiperiodic System: Particle in a Circular Cavity
Initial conditions
(0) nm, (0) 0
(0) 0, (0) 10nm/s
x x
y y
2.522
(0) nm, (0) 0
(0) 0, (0) 10nm/s
x x
y y
2.500
Dependence of solutions on initial conditions is smooth and predictable in stable systems.
Example Unstable System: Discontinuous Boundary
Initial conditions
(0) nm, (0) 0
(0) 0, (0) 10nm/s
x x
y y
2.522
(0) nm, (0) 0
(0) 0, (0) 10nm/s
x x
y y
2.500
No predictable dependence on initial conditions exists unstable systems.
Comparison for Longer-Time Simulation
Stable quasiperiodic trajectory Unstable chaotic trajectory
Divergence of Trajectories
Slow and predictable divergence Quick and random divergencein stable systems in unstable systems
Relative variance of x(0):
(0) 0.00 to 0.06 (0) 0.0000 to 0.0003
Transition from Stable to Chaotic Motion: Example
Initial conditions
1 2
1 2
1.55rad(0) 1.9rad, (0)(0) (0) 0
1 2 0.6ml l
1 2
1 2
1.7rad(0) 1.9rad, (0)(0) (0) 0
A transition from a stable to chaotic motion can be observed in this system
Periodic motion Chaotic motion
The Phase Space Trajectory
1 1( ), ( )q t p t( )q t
( )p t
2 2( ), ( )q t p t
The 2s generalized coordinates represent a phase vector (q1, q2, …, qs, p1, p2, …, ps) in an abstract 2s-dimensional space, the so-called phase space.
A particular realization of the phase vector at a given time provides a phase point in the phase space. Each phase point uniquely represents the state of the system at this time.
In the course of time, the phase point moves in phase space, generating a phase trajectory, which represents dynamics of the 2s degrees of freedom.
The Phase Space Trajectories: Examples
Examples of phase space trajectories:
(Projection to the plane x, px)
2.2 Molecular Forces
• Newtonian dynamics
• Interatomic potentials
• Molecular dynamics simulations
• Boundary conditions
• Post-processing and visualization
Molecular forces and positions change with time.
In principle, an MD simulation is a solution of a system of Newtonian equations of motion.
The Newtonian equation (the Newton’s second law) is a special case of the Lagrangian equation of motion for mass points in a Cartesian system.
For such a system, the Lagrangian function is given by
2 2 2
1 2
( ) ( )2
, ,... , , ,
ii i i
i
i i i i
mL x y z U
x y z
r
r r r r
X
Y
Z
ri
rj
rij
Newtonian Dynamics
By utilizing the Lagrangian equation of motion at qα= xi, qα+1= yi, qα+2= zi,
, , ,
( ), , ,i i i i x i y i z i
i
Um F F F
r
F r Fr
1 1 2 2
, ,
0, 0, 0
, , i i x i i i y i
d L L d L L d L L
dt q q dt q q dt q q
m x F m y F
, i i z im z f
This can be rewritten in the vector form:
Newtonian Dynamics
Interatomic Potential
The most general form of the potential is given by the series
The one-body potential W1 describes external force fields (e.g. gravitational filed), and external constraining fields (e.g. the “wall function” for particles in a circular chamber)
The two-body potential describes dependence of the potential energy on the distances between pairs of atoms in the system:
The three-body and higher order potentials (often ignored) provide dependence on the geometry of atomic arrangement/bonding.For instance, a dependence on the angle between three mass points is given by
1 2 1 2 3, , ,
( , ,..., ) ( ) ( , ) ( , , ) ...N i i j i j ki i j i i j i k j
U W W W
r r r r r r r r r
2 2( , ) ( ), | | | |i j ij i jW W r r r r r r rx
y
z
rj
rij
ri
i
j
3 3( , , ) (cos ), cos| | | |
ji jki j k ijk ijk
ji jk
W W
r r
r r rr r x
y
z
rj rk
θijk
i
k
j
rji
rjk
Pair-Wise Potentials: Lennard-Jones
12 6
13 7
( ) 4
( )( ) 24 2
LJ
LJLJ
W rr r
W rF r
r r r
6( ) 0 2LJF
Here, ε is the depth of the potential energy well and σ is the value of r where that potential energy becomes zero; the equilibrium distance ρ is given by
• The first term represents repulsive interaction • At small distances atoms repel due to quantum effects (to be discussed in a later lecture)
• The second term represents attractive interaction • This term represents electrostatic attraction at large distances
Pair-Wise Potentials: Morse
2 ( ) ( )
2 ( ) ( )
( ) 2
( ) 2
r rM
r rM
W r e e
F r e e
Another pair-wise potential model, effective for modeling crystalline solids, is the Morse potential,
Here, ε is the depth of the potential energy well and β is a scaling factor; the equilibrium distance is given by ρ
Similarly to the earlier example,
• The first term represents repulsive interaction
• The second term represents attractive interaction
( ) 0MF
• In system of N atoms
• accumulates unique pair interactions
• if all pair interactions are sampled, the number increases with square of the number of atoms
• Saving computer time
• Neglect pair interactions beyond some distance
• Example: Lennard-Jones potential used in simulations
)1(2
1NN
12 6
4( )
0
cLJ
c
r rW r r r
r r
cr
Truncated Potential
Due to essential non-linearity of the molecular interaction, classical equations of motion yield non-stable (non-predictable) trajectories.
Therefore, the results of MD simulations are intriguing.
Instability of Trajectories
Macroscopic properties are determined by behavior of individual molecules, in particular:
• Any measurable property can be translated into a function that depends on the positions of phase points in phase space
• The measured value of a property is generated from finite duration experiments
As a phase point travels on a hypersurface of constant energy, most quantities are not constant, but fluctuating (discussed in more detail Week 3 lecture).
While kinetic energy Ek and potential energy U fluctuate, the value of total energy E is preserved,
constkE E U
Fluctuations in MD Simulation
0 2 4 6 8time
1
2
3
4
5
6
elbavresbo
0 2 4 6 8time
1
2
3
4
5
6
degarevaelbavresbo
Averaging of a fluctuating value F(t) over period of time t1 to t2 is accomplished according to
2
12 1
1( ) ( )
t
t
F t F t dtt t
Fluctuations in MD Simulation…
Simulated system encompasses boundary conditions
Periodic Boundary conditions for particles in a box • Box replicated to infinity in all three Cartesian directions;
Motivation for periodic boundary conditions: domain reduction and analysis of a representative substructure only.
Particles in all boxes move simultaneously, though only one modeled explicitly, i.e. represented in the computer code • Each particle interacts with other particles in the box and with images in
nearby boxes • Interactions occur also through the boundaries • No surface effects take place
Periodic Boundary Conditions
Original box
Translation image boxes
Periodic Boundary Conditions
Example simulation of an atomic cluster with periodic boundary conditions.A particles, going through a boundaries returns to the box from the opposite side:
This model is equivalent to a larger system, comprised of the translation image boxes:
Periodic Boundary Conditions: Example
Develop Model Molecular Dynamics Simulation
Model Molecular Interactions
Develop Equations of Motion
Generate Trajectories
Analysis of Trajectories
Initialization
Static and dynamic (macro) properties
Predictions
Initialize Parameters
Initialize Atoms
Re-initialization
Dynamic Molecular Modeling
• Decisions concerning preliminaries
• Systems of units in which calculations will be carried out
• Numerical algorithm to be used
• Assignment of values to all free parameters
• Initialization of atoms
• Assignment of initial positions
• Assignment of initial velocities
Initialization
After simulation is completed, macroscopic properties of the system are evaluated based on (microscopic) atomic positions and velocities:
1. Macroscopic thermodynamic parameters - temperature - internal energy - pressure - entropy (will be discussed in week 3 lecture)
2. Thermodynamic response functions, e.g. heat capacity
3. Other properties (e.g. viscosity, crack propagation speed, etc.)
Post-Processing
Absolute temperature is proportional to the average kinetic energy
int1 2
0
1( ) ( ) ...
t
i iji i j i
U U W r W r dt
1 2 one-body potential, two-body potential,
total time
W W
t
2 2 2kin kin ( )2 1
,2
i i i
i
m x y zT E E
f k N
The time-averaged potential energy
Boltzman constant
total number of atoms
number of DOF per atom
k
N
f
Temperature and Potential Energy
• Pressure is defined by the atomic velocities
• Mean square force is defined by the time-averaged derivative of the potential function
2 2 kin
- volume, - mass, - number of particles
2,x x x x x
mN mN NP v P v E
V V VV m N
221 1
1
( )( ) ,j
j
u rF W r W
r
Pressure and Mean Square Force
ln
Boltzmann constant
phase volume integral (over all allowed states)
S k
k
kin1 13
1( ) ( ) ... ...
!0, 0
( )1, 0
Planck's constant
N NNE E U d d d d
h Nx
xx
h
p r r r p p
For an adiabatically isolated system, the entropy is related to the phase volume integral:
In greater detail, the properties and calculation of entropy for various systems will be discussed on weeks 3-5 lectures.
Entropy
The TD response functions reveal how simple thermodynamic quantities respond to changes in measurables, usually either pressure or temperature. Thus, they are derivative quantities (coefficients):
• Heat capacity (how the system internal energy responds to an isometric change in temperature):
• Thermal pressure (how pressure responds to an isometric change in temperature):
• Adiabatic compressibility (how the system volume responds to an isentropic, change in pressure):
vV
EC
T
1s
S
V
V P
vV
P
T
Thermodynamic Response Functions
In order to evaluate the averaged macroscopic parameters, the simulated system must achieve a thermodynamic equilibrium. Indeed, the thermodynamic parameters, such as temperature, internal energy, etc., are defined for equilibrium systems.
In the equilibrium:
1) the macroscopic parameters fluctuate around their statistically averaged values;
2) the property averages are stable to small perturbations;
3) different parts of the system yield the same averaged values of the macroscopic parameters.
Equilibration of a macroscopic parameter is achieved in distinct ways in closed adiabatic and isothermal systems (surrounded by a heat bath).
Closed systems: value of the macroscopic parameter fluctuates about the averaged value with a decaying fluctuation amplitude.
Isothermal systems: value of the macroscopic parameter both fluctuates, and also asymptotically approaches the averaged statistical value (examples are to follow).
Equilibration
Repulsive interaction between the particles and the wall is described by the “wall function”, a one-body potential that depends on ri – distance between the particle i and the chamber’s center):
Interaction between particles is modeled with the two-body Lennard-Jones potential (rij – distance between particles i and j):
The total potential:
12 6
12 6( ) 4LJ ij
ij ij
W rr r
2 2
2 2
( ) ( ),
( ) ( )
wl i LJ iji i j i
i i i
ij i j i j
U W r W r
r x y
r x x y y
2 2)(( )
i iiR x y
wl i
R rW r e e
y
x
ri
R
rij
rj
Adiabatic Example: Interactive Particles in a Circular Chamber
The total potential:
Equations of motion:
Parameters:
Initial conditions (nm, m/s):
1 2 3
12 13 23
2 2
2 2
( ) ( ) ( )
( ) ( ) ( ),
( ) ( )
wl wl wl
LJ LJ LJ
i i i
ij i j i j
U W r W r W r
W r W r W r
r x y
r x x y y
, , 1, 2,3i i i ii i
U Um x m y i
x y
1 94nm , 10nm, 10 kgR m
1 1 1 1
2 2 2 2
3 3 3 3
(0) 0, (0) 25, (0) 3.0, (0) 0
(0) 0, (0) 30, (0) 0.5, (0) 0
(0) 0, (0) 20, (0) 2.5, (0) 0
x x y y
x x y y
x x y y
Three Particles: Equation of Motion and Solution
The total potential:
Equations of motion:
Parameters:
Initial conditions (nm, nm/s):
1 2 3 4 5
12 13 14 15
23 24 25
35 35
45
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( )
wl wl wl wl wl
LJ LJ LJ LJ
LJ LJ LJ
LJ LJ
LJ
U W r W r W r W r W r
W r W r W r W r
W r W r W r
W r W r
W r
, , 1, 2,3,4,5i i i ii i
U Um x m y i
x y
1 94nm , 10nm, 10 kgR m
1 1 1 1
2 2 2 2
3 3 3 3
4 4 4 4
5 5 5 5
(0) 0, (0) 25, (0) 5.6, (0) 0
(0) 0, (0) 30, (0) 3.0, (0) 0
(0) 0, (0) 20, (0) 0.5, (0) 0
(0) 0, (0) 24, (0) 2.5, (0) 0
(0) 0, (0) 22, (0) 4.9, (0) 0
x x y y
x x y y
x x y y
x x y y
x x y y
Five Particles: Equations of Motion and Solution
Averaged kinetic energy vs. time (three particles)
Kinetic energy, and therefore temperature and pressure are due to motion of the particles.
Time averaged kinetic energy of particles is approaching the value
which corresponds to temperature
Note: a low temperature system was chosen in order to observe the real-time atomic motion.
Pressure in the system is due to the radial components of velocities:
kin 253.057 10 JE
25kin
23
1 3.057 100.02K
1.38 10T E
k
2 9, cos sin 3.23 10 Parad rad x y
mNP v v v v P
V
Post-Processing: Kinetic Energy, Temperature and Pressure
Averaged potential energy vs. time (three particles)
Potential energy is due to interaction of particles with each other and with external constraining fields.
Time averaged potential energy of the system is approaching the value
that gives the potential energy of the system.
250.480 10 JU
Post-Processing: Potential Energy
Kinetic, potential and total energy vs. time (three particles)
The total energy (solid red line): 9.65x10-25 J. This value does not vary in time
tot kin pot
3kin 2 2
1
pot
( )2 i i
i
E E E
mE x y
E U
Post-Processing: Total Energy
Total number of particles is large.
Interaction between particles is modeled with the two-body harmonic potential:
here, rij = yi – yj is the relative distance between particles i and j in the vertical (y-axis) direction, k – linear interaction coefficient (similar to spring stiffness).
Several atoms in the middle of the chain (between the yellow dashed lines) represent the initially “cold” region. Initial velocities and displacements for these atoms are zero. The remaining atoms have randomly distributed initial velocities and displacements.
21( )
2h ij ijW r kr
y “Cold” region
Isothermal Example: 1D Lattice with a “Cold” Region
Number of particles simulated: 50.
Boundary conditions are periodic, so that the coupling between the 50th and 1st particles is established. For a more symmetric view, the simulation shows the 1st particle at both ends of the lattice.
The total potential (the last term is due to periodic boundary conditions):):
Equations of motion:
Parameters:
Initial conditions:
492
1 1 501
1( ) ( ), ( ) ( )
2h i i h h i j i ji
U W y y W y y W y y k y y
, 1, 2,...50ii
Um y i
y
21 2110 kg, 10 N/mm k
middle atoms 22..27 : (0) (0) 0,
other atoms : (0), (0) random within [ 1,1]i i
i i
y y
y y
1D Lattice: Equations of Motion and Solution
Averaged kinetic energy per particle vs. time (for the initially “cold” subsystem)
Time averaged kinetic energy of particles in the “cold” subsystem is approaching the value
which corresponds to temperature
kin 210.286 10 JE
kin
21
23
2
2 0.286 1041.4K
1.38 10
T Ek
0 100 200 300 400 500
Time, s
0.1
0.2
0.3
0.4
citeniK
ygrene,
01
12J
Equilibrium value
Time averaged
In contrast to the adiabatic system example, the kinetic energy for the open isothermal subsystem both fluctuates and approaches asymptotically the statistical average.
Post-Processing: Kinetic Energy and Temperature
Averaged potential energy vs. time (“cold” subsystem)
Time averaged potential energy of the “cold” subsystem is approaching the value
250.480 10 JU
In contrast to the adiabatic system example, the internal energy for the open isothermal subsystem both fluctuates and approaches asymptotically the statistical average.
0 100 200 300 400 500
Time, s
0.5
1
1.5
2
laitnetoP
ygrene,
01
12J
Internal energy asymptote
Time averaged
Post-Processing: Potential Energy
• Realism of forces
• Simulation imitates the behavior of a real system only to the extend that interatomic forces are alike to those that real nuclei would experience
when arranged in same configuration
• In simulation forces are obtained as a gradient of a potential energy function, which depends on the positions of the particles
• Realism depends on the ability of potential chosen to replicate the conduct of the material under the circumstance at which the simulation is governed
Limitations of MD
• Time Limitations
• Simulation is “safe” when duration of the simulation is much greater than relaxation time
• Systems have a propensity to become slow and sluggish near phase transitions
• Relaxation times order of magnitude larger than times reachable by simulation
Limitations of MD
• System Size Limitations
• Correlation lengths can increase or deviate near phase transitions when comparing the size of the MD cell with one of the spatial correlation
functions
• Partially assuaged by Finite Size Scaling
• Compute physical property, A, using many different size boxes, L, then relating the results to:
and using as fitting parameters
• where , and should be taken as the best estimate for true physical quantity
0( )n
cA L A
L
0 , ,A c n
0 lim ( )LA A L
Limitations of MD
2.3 Molecular Dynamics Applications
Tensile failure of a gold nanowire
Carbon nanotube immersed into monoatomic helium gas
Dislocation dynamics of nanoindentation
Deposition of an amorphous carbon film
Molecular Dynamics Applications (II)
MD fracture simulations
Shear-dominant crack propagation