by
Wing Kam Liu, Eduard G. Karpov, Harold S. Park



Time evolution of interacting atoms pursued by integrating the corresponding equations of motion



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






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)

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 >> m e
) 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.
( ,
1 2
,..., q s
)
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.
Examples:
Material point in xy -plane Pendulum Sliding suspension pendulum y x

 x q
1

,
2
 y q
1
  q
1

,
2
 
We are looking for a basic form of equation motion, which is valid for all coordinate systems.

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.
( ,
2
,..., q s
)
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.

Least Action (Hamilton’s) Principle
Lagrange function (Lagrangian)
Action integral
Trajectory variation
( ,
1 2
,..., q q q s
, ,
1 2
,..., q t s
, ) t
2
S

( , , ) t
1
 q t
  q t
 q t
1
  q t
2

0
Least action, or Hamilton’s, principle
Action is minimum along the true trajectory: S b
S a

S b
, S a

S c
( , , )

0 q t
1

S
   t
2 t
1 t
1
The main task of classical dynamics is to find the true trajectories
(laws of motion) for all degrees of freedom in the system.
S a
S c q t
2 t
2 t
Remark: in ADMD this principle is used to obtain the trajectories.

Lagrangian Equation of Motion
Lagrange function (Lagrangian)
Least action principle
( ,
1 2
,..., q q q s
; ,
1 2
,..., q t s
, )

S
  t
2  t
1
( , , )

0
Substitution of the Lagrange function into the action integral with further application of the least action principle yields the Lagrangian equation motion: d
 dt q
L



 q
L


0,
 
1, 2,..., s
Lagrangian equation is based on the least action principle only, and it is valid for all coordinate systems.

Lagrangian Equation of Motion: Derivation
Using the most general form of the Lagrange function, the least action principle gives

S
  t
2  t
1
  t
2 t
1

 t
2  t
1



L
 q
 q


L
 q

 q dt


L
 q
 q t
2 t
1
 t
2  t
1




L q
 d


L


 q dt

0 q t
1 t
1
S b
S a
S c q t
2 t
2 t
Variation of the coordinates cannot change the observable values q ( t
1
) and q ( t
2
):
 q : ( )
 
( )
  q t
  q t
Therefore,

L
 q
 q t
2 t
1

0 and the second term finally gives d


L



 q
L


0,
 
1, 2,..., s

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), and
2) 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 r
L
  r r
V t

,
L t
 t r r
' t
L r r

L '( r
 a r t
L r r

L r r t
 
)

Lagrange Function of a Material Point
Free material point
(generalized and
Cartesian coordinates)
Interacting material point
L
 m
2 q
2
, L
 m r
2
2
 m  x
2
 y
2
 z
2

2
L
 m
2 q
2 
( )
Conservative systems
L

( )

( )
E

2 T
 
Const
For conservative systems, kinetic energy depends only on velocities, and potential energy depends only on coordinates.

Lagrange Function: Examples
Pendulum q
1
 
 l
Bouncing ball q
1
 y
Particle in a circular cavity q
1

,
2
 y y y x
 m
L l
2
2

2  mgl
1) Kinetic energy; 2) Potential energy due to external gravitational field, where g is the acceleration of gravity).
L
 m
2 y
2  mgy
  e
  y
1) Kinetic energy; 2) Potential energy due to external gravitational field; 3) Potential energy of repulsion between the ball and floor.
L
 m
( x
2  y
2
)
  e
 

R
 x
2  y
2

2
1) Kinetic energy; 2) Potential energy of repulsion between the particle and cavity wall.

Pendulum: Lagrange Function Derivation
General form L

T
 
U

Kinetic energy tangential velocity
T v


 m
2 v

2 l
 m
2
  lim t 0
 t l
2

2
 l

 l v
τ
Potential energy U
 mgh
  mgl cos

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.
l cos
φ
The total Lagrangian
 m
L l
2
2

2  mgl
Parameters used: g
 2
9.8 m/s , l

2m

Pendulum: Equation of Motion and Solution
Lagrange function and equation motion:
 m
L l
2
2

2  mgl cos
  d dt


L




L


0

Initial conditions (radian):

(0)

0.3,

(0)

0

(0)

1.8,

(0)

0

Parameters: g

9.8 m/s , l

2m
 t
 g l sin
  t
 
0
(0)

3.12,

(0)

0

Bouncing Ball: Lagrange Function Derivation
General form L

( )

( )
Kinetic energy
Potential energy
The total Lagrangian:
T
 m y
2
2
U
 mgy
  e
  y y
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).
 m
2 y
2
Red solid line: the total potential.
β is a relative scaling factor: the ball-surface repulsive potential growths in e
β times for a unit length ball/surface penetration (from y = 0 to y =
–
1m).
 mgy
  e
  y
Parameters used: g



9.8 m/s ,
1J,
 
2 m
4m


1
1kg,

Bouncing Ball: Equation of Motion and Solution
L
Lagrange function and equation motion:
 m
2 y
2  mgy
  e
  y  d dt


L y



L y

0
Initial conditions:

( )
 
Parameters: g
 2
9.8 m/s , m

1kg,
 
1J,
 m e
 
 
4m

1

0 y (0)

5m y (0)

0 y y (0)

0 y y (0)

0

Particle in a Circular Cavity: Lagrange Function Derivation
General form L

( , )

( , ) y
R r
Kinetic energy
Potential energy
T
 m
( x
2
 y
2
)
2
U
  e
   
  e
 

R
 x
2  y
2
 x
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 e
β times between r = R and r = R +1.
The total Lagrangian L x y x y
 m
( , , , ) (
2 x
2  y
2
)
  e
 

R
 x
2  y
2


Particle in a Circular Cavity: Equation of Motion and Solution
Lagrangian function and equations:
L
 m
( x
2  y
2
)
  e
 

R
 x
2  y
2

2




   e
 
(

( ))
   e
 
(

( ))
  
27
10 J, ( )

2 x t

2 y t
Potential barrier:
Parameters:
  
1
4nm , R

10nm, m
 
9
10 kg
Initial conditions:

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.
Legendre’s transformation
(the passage from one set of independent variables to another)
    
Differential of the Lagrange function: L

( , ) dL
 


L
 q

 dq

 


L
 q
 dq

,
Generalized momentum (definition): p



L
 q

 dL
 
 p dq

 
 p dq

 
1, 2,..., s
A mass point in Cartesian coordinates: p x

, y
 m y , p z
 mz

Hamiltonian Equations of Motion dL
 
 p dq

 
 p dq
 dL
 
 p dq

 d






 
 q dp
 d



 p q

L


 
 q dp

 
 p dq

Hamiltonian of the system: dH

 
 p q

L

 q dp

 
 p dq

Equations of motion: q



H
 p

, p

 

H
 q


Hamiltonian Equations of Motion: Pendulum
Lagrange function and the generalized momentum:
 m
L l
2
2

2  mgl cos
  p



L

 m l
2

Hamiltonian of the system:
T p
2

2 ml
2
, U
  mgl cos
 
H p
2

2 ml
2
 mgl cos

Equations of motion:
 

H
 p
, p
 

H
 
   p ml
2
, p
  mgl sin

Conversion to the Lagrangian form (elimination of p): p
 ml
2
   mgl sin
    l g sin
 
0
 l

Hamiltonian Equations of Motion: Bouncing Ball
Lagrange function and the generalized momentum:
L
 m
2 y
2     y  p



L

 m y
Hamiltonian of the system:
T
 p
2
2 m
, U
 mgy
 e
  y 
H
 p
2
 mgy
 e
2 m
  y
Equations of motion: y


H
 p
, p
 

H
 y
 y
 p m
, p
  mg
  e
  y y
Conversion to the Lagrangian form (elimination of p): p
 my
  mg
  e
  y  y
 
 m e
  y 
0

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 2 s 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 trajectories in predictable systems q
         
C t
Variance of the trajectory depends linearly on time
Divergence of trajectories in chaotic (unpredictable) systems q
           e
 t
Variance of the trajectory depends exponentially on time (J.M. Haile,
Molecular Dynamics Simulation , 2002)
Trajectories in MD systems are unpredictable/unstable; they are characterized by a random dependence on initial conditions.

Example Quasiperiodic System: Particle in a Circular Cavity
Initial conditions x (0)
 
2.522
nm, x (0)

0 y (0)

0, y (0)

10 nm/s x y
(0)
(0)



0,
2.500
y (0) nm, x (0)

0

10 nm/s
Dependence of solutions on initial conditions is smooth and predictable in stable systems.

Example Unstable System: Discontinuous Boundary
Initial conditions x (0)
 
2.522
nm, x (0)

0 y (0)

0, y (0)

10 nm/s x y
(0)
 
2.500
nm, x (0)

0
(0)

0, y (0)

10 nm/s
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 divergence in 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 l
1
 
2
0.6m

1
(0)

1.9rad,
2
(0)

1
(0)
 
2
 
(0)

0
1.55rad

1
(0)

1.9rad,

1
(0)
 
2

2
(0)

(0)

0
1.7rad
Periodic motion Chaotic motion
A transition from a stable to chaotic motion can be observed in this system

The Phase Space Trajectory
The 2 s generalized coordinates represent a phase vector ( q
1
, q
2
, …, q s
, p
1
, p
2
, …, p s
) in an abstract 2 s -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 2 s degrees of freedom. q t
2 p t
2 q t
1 p t
1

The Phase Space Trajectories: Examples
Examples of phase space trajectories:
(Projection to the plane x , p x
)

• Newtonian dynamics
• Interatomic potentials
• Molecular dynamics simulations
• Boundary conditions
• Post-processing and visualization

Newtonian Dynamics
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
Z
L
  m i ( x i
2  y i
2  z i
2
)

U r
2 i r
 
1
 r i
  x y z i
 r j r ij r i
X
Y

Newtonian Dynamics
By utilizing the Lagrangian equation of motion at q
α
= x i
, q
α+ 1
= y i
, q
α+ 2
= z i
, d

 m x
L i





F
 q
L


0, d

L

 
1


L
 q
 
1

, m y i

F

0, d



L

2



L
 q
 
2
, m z
 f

0
This can be rewritten in the vector form:
F i
 m r , F i i i


F , F , F
, , ,

 

U
 r i

Interatomic Potential
The most general form of the potential is given by the series
U ( , ,..., r
N
)
  i
W
1 r
 

W
2 r r
 
,

,
 j
W
3 r r r k

The one-body potential W
1 describes external force fields (e.g. gravitational filed), and external constraining fields (e.g. the “wall function” for particles in a circular chamber) z r ij j
The two-body potential describes dependence of the potential energy on the distances between pairs of atoms in the system:
W
2 r r i j

W r
2
( ), r

| r ij
|

| r i
 r j
| x r j r i z
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
W
3 r r r

W
3
(cos
 ijk
), cos
 ijk
 r ji
 r jk r r x j r j i i r ji
θ ijk r k r jk k y y

Pair-Wise Potentials: Lennard-Jones
W
LJ
( )

4
 

 
 r
12


 
6
 


F
LJ
( )
 

W
LJ
( )
 r

24



 

2
  r
13

 7
 


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
F
LJ
 
0
   6 2

• 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
Another pair-wise potential model, effective for modeling crystalline solids, is the Morse potential ,
W
M
( )
 

 e
F
M
( )

2
 

 e
 r ) 
2 e
   r )  e
 r )


 r )


Here,
ε is the depth of the potential energy well and
β is a scaling factor; the equilibrium distance is given by
ρ
F
M
 
0
Similarly to the earlier example,
• The first term represents repulsive interaction
• The second term represents attractive interaction

Truncated Potential
• In system of N atoms
•
1 accumulates unique pair interactions
2

1 )
• 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 r c
•
Example: Lennard-Jones potential used in simulations
W
LJ
( )


 


 
4
 r
12
0

 6

 r
 r c r
 r c

Instability of Trajectories
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.

Fluctuations in MD Simulation
•
•
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 E k is preserved, and potential energy U fluctuate, the value of total energy E
E

E k
 

Fluctuations in MD Simulation…
4
3
2
1
6
5
4
3
2
1
6
5
0 2 4 time
6 8 0 2 4 time
6
Averaging of a fluctuating value F ( t ) over period of time t
1 to t
2 is accomplished according to

1 t
2
 t
1 t
2  t
1
8

Periodic Boundary Conditions
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
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:

Dynamic Molecular Modeling
Develop Model
Model Molecular
Interactions
Develop Equations of Motion
Molecular Dynamics Simulation
Initialization
Initialize
Parameters
Initialize Atoms Generate Trajectories
Analysis of Trajectories
Static and dynamic
(macro) properties
Predictions
Re-initialization

Initialization
•
Systems of units in which calculations will be carried out
•
Numerical algorithm to be used
•
Assignment of values to all free parameters
•
Assignment of initial positions
•
Assignment of initial velocities

Post-Processing
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.)

Temperature and Potential Energy
Absolute temperature is proportional to the average kinetic energy
T

2 f k
E kin
, E kin 
1
N
 i m x i
2
 y i
2
 z i
2
)
2 k Boltzman constant
N total number of atoms f number of DOF per atom
The time-averaged potential energy
U int
  t
1  t
0



 i
W r
1
 i
   i j i
W r
2 ij



...


 d

W
1
one-body potential, W
2
two-body potential, t total time

Pressure and Mean Square Force
• Pressure is defined by the atomic velocities
P x
 mN v x
2
V
P x
 mN
V v x
2 
V
N
E
V - volume, m - mass, N - number of particles kin x
• Mean square force is defined by the time-averaged derivative of the potential function
F
1
2  j


1


W r
1 j
)

2
,

 r

Entropy
For an adiabatically isolated system, the entropy is related to the phase volume integral:
S
 k ln
 k Boltzmann constant

phase volume integral (over all allowed states)
 
1 h 3 N N !
 


0,
 1,

E

E kin x

0 x

0 p h Planck's constant

U r
 d r
1
...
d r
N d p
1
...
d p
N
In greater detail, the properties and calculation of entropy for various systems will be discussed on weeks 3-5 lectures.

Thermodynamic Response Functions
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):
C v
 
E

T

V
• Thermal pressure (how pressure responds to an isometric change in temperature):
 v
 
P

T

V
• Adiabatic compressibility (how the system volume responds to an isentropic, change in pressure):
 s
1
V



V

P


S

Equilibration
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).

Adiabatic Example: Interactive Particles in a Circular Chamber
Repulsive interaction between the particles and the wall is described by the “wall function”, a one-body potential that depends on r i
– distance between the particle i and the chamber’s center):
W r wl i
  e
 
( R r i
)   e
 

R
 x i
2  y i
2
 y
Interaction between particles is modeled with the two-body Lennard-Jones potential ( r ij
– distance between particles i and j ):
W r
LJ ij

4



 ij
12

 r
12 r ij
6
6


The total potential:
U
  i
W r wl i
  i j i
W r
LJ ij r i
 x i
2  y i
2 r ij

( x i
 x j
) 2 
( y i
 y j
) 2 r ij r i r j
R x

Three Particles: Equation of Motion and Solution
The total potential:
U

W wl
( )
1

W wl
( )
2

W wl
( )
3

W
LJ
( r
12
)

W
LJ
( r
13
)

W
LJ
( r
23
), r i
 x i
2  y i
2 r ij

( x i
 x j
)
2 
( y i
 y j
)
2
Equations of motion: m x i
 

U
 x i
, m y i
 

U
 y i
, i

1, 2, 3
Parameters:
 
R

10nm, m

Initial conditions (nm, m/s): x
1
(0)

0, x
1
(0)

25, y
1
(0)
 
3.0, y
1
(0)

0 x
2
(0)

0, x
2
(0)

30, y
2
(0)
 
0.5, y
2
(0)

0 x
3
(0)

0, x
3
(0)

20, y
3
(0)

2.5, y
3
(0)

0

Five Particles: Equations of Motion and Solution
The total potential:
U

W wl r

W wl r

W wl r

W wl r

W wl r

W
LJ
( r
12
)

W
LJ r

W
LJ
( r
14
)

W
LJ
( r
15
)

W
LJ
( r
23
)

W
LJ
( r
24
)

W
LJ
( r
25
)

W
LJ
( r
35
)

W
LJ
( r
35
)

W
LJ
( r
45
)
Equations of motion: m x i
 

U
 x i
, m y i
 

U
 y i
, i

1, 2, 3, 4, 5
Parameters:
 
R

10nm, m

Initial conditions (nm, nm/s): x
1
(0)

0, x
1
(0)

25, y
1
(0)
 
5.6, y
1
(0)

0 x
2
(0)

0, x
2
(0)
 
30, y
2
(0)
 
3.0, y
2
(0)

0 x
3
(0)

0, x
3
(0)

20, y
3
(0)
 
0.5, y
3
(0)

0 x
4
(0)

0, x
4
(0)
 
24, y
4
(0)

2.5, y
4
(0)

0 x
5
(0)

0, x
5
(0)

22, y
5
(0)

4.9, y
5
(0)

0

Post-Processing: Kinetic Energy, Temperature and Pressure
Averaged kinetic energy vs. time (three particles)
Time averaged kinetic energy of particles is approaching the value
E kin  
25
3.057 10 J which corresponds to temperature
T

1 k
E kin 
3.057 10

25

23
0.02K
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:
P
 mN
V v 2 rad
, v rad
 v x cos
  v y sin
 
P
 
3.23 10 Pa
Kinetic energy, and therefore temperature and pressure are due to motion of the particles.

Post-Processing: Potential Energy
Averaged potential energy vs. time (three particles)
Time averaged potential energy of the system is approaching the value
U
 
25
0.480 10 J that gives the potential energy of the system.
Potential energy is due to interaction of particles with each other and with external constraining fields.

Post-Processing: Total Energy
Kinetic, potential and total energy vs. time (three particles)
E tot 
E kin 
E pot
E kin 
E pot 
U m
2 i
3 

1
( x i
2  y i
2
)
The total energy (solid red line): 9.65
x 10 -25 J. This value does not vary in time

Isothermal Example: 1D Lattice with a “Cold” Region y
“Cold” region
Total number of particles is large.
Interaction between particles is modeled with the two-body harmonic potential:
W r h ij

1
2 k r ij
2 here, r ij
= y i
– y j 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.

1D Lattice: Equations of Motion and Solution
Number of particles simulated: 50.
Boundary conditions are periodic , so that the coupling between the 50 th and 1 st particles is established. For a more symmetric view, the simulation shows the 1 st particle at both ends of the lattice.
The total potential (the last term is due to periodic boundary conditions):):
U
 i
49 

1 h
( i
 y i

1
)
 h
(
1
 y
50
), W y h i
 y j
)

1
2
( y i
 j
)
2
Equations of motion: m y i
 

U
 y i
, i

1, 2,...50
Parameters:
Initial conditions: m

10

21 kg, k

10

21
N/m middle atoms 22..27 : y i
(0)
 y i
(0)

0, y y i
 

Post-Processing: Kinetic Energy and Temperature
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
E kin  
21
0.286 10 J
0.4
0.3
which corresponds to temperature
T

2 k
E kin

21

1.38 10

23
41.4K
0.2
0.1
Time averaged
Equilibrium value
0 100 200
Time, s
300 400
In contrast to the adiabatic system example, the kinetic energy for the open isothermal subsystem both fluctuates and approaches asymptotically the statistical average.
500

Post-Processing: Potential Energy
Averaged potential energy vs. time (“cold” subsystem)
Time averaged potential energy of the “cold” subsystem is approaching the value
U
 
25
0.480 10 J
1
0.5
2
1.5
Time averaged
Internal energy asymptote
0 100 200
Time, s
300 400
In contrast to the adiabatic system example, the internal energy for the open isothermal subsystem both fluctuates and approaches asymptotically the statistical average.
500

Limitations of MD
•
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
• 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
•
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:
( )

A
0
 c
L n
• A
0
 lim
L
 estimate for true physical quantity

Tensile failure of a gold nanowire Dislocation dynamics of nanoindentation
Carbon nanotube immersed into monoatomic helium gas
Deposition of an amorphous carbon film

MD fracture simulations
Shear-dominant crack propagation