Nano Mechanics and Materials

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Nano Mechanics and Materials:

Theory, Multiscale Methods and Applications

by

Wing 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

>>

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

determine the state of the system at any given time.

set of

s

independent parameters that

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

,...,

S

t

2

t

1

 

q t

1

s

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

S

0

q t

1

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

,...,

s

;

1

,

2

,...,

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

L

L

q

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

1

q t

Therefore,

L

q

q t

2

t

1

0 and the second term finally gives

d

L

L

q

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

L

V

t

,

L

r r

t

L t

t

'

L

'(

r

a r

L t

)

Lagrange Function of a Material Point

Free material point

(generalized and

Cartesian coordinates)

Interacting material point

L

m q

2

,

2

L

m

r

2

2

L

m q

2

2

m

x

2

2

y

2

z

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

L

m 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

2

y

2

)

e

R

x

2

y

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

L

m 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:

L

m 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

l

2m

(0)

l g

sin

3.12,

0

(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

Initial conditions:

0

Parameters:

g

2

9.8 m/s ,

m

1kg,

1J,



e

m

4m

1

0

y

(0)

y

(0)

5m

0

y y

(0)

0

y y

(0)

0

Particle in a Circular Cavity: Lagrange Function Derivation

General form

L

 

The total Lagrangian

y

R r

Kinetic energy

Potential energy

T

m

(

x

2

2

y

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.

m

(

x

2

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

2

y

2

)

e

R

x

2

y

2

 



e

(

 



e

(

10

27

J,

2

( )

2

( )

Potential barrier:

Parameters:

1

4nm ,

R

10nm,

m

9

10 kg

Initial conditions:

Summary of the Lagrangian Method

1.

2.

The choice of

s

generalized coordinates (

s

– number of degrees of freedom).

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.

5.

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.

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

L

q dp

p dq

Hamiltonian of the system:

dH

q dp

L

p dq

Equations of motion:

q

H

p

,

p

 

H

q

Hamiltonian Equations of Motion: Pendulum

Lagrange function and the generalized momentum:

L

m

2

l

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

2

m

mgy

e

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.

2.

The choice of

s

generalized coordinates (

s

– number of degrees of freedom).

Derivation of the kinetic and potential energy in terms of the generalized coordinates.

3.

4.

Derivation of the generalized momenta.

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.

8.

Solution of the equations of motion, using a numerical time-integration algorithm.

Post-processing and visualization.

Predictable and Chaotic Systems

Divergence of trajectories in predictable systems

q

  

Variance of the trajectory depends linearly on time

 

C t

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,

y

(0)

0,

y

(0)

x

(0)

10 nm/s

0

x

(0)

y

(0)

0,

2.500

nm,

x

(0)

0

y

(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)

y

(0)

0,

2.522

nm,

x

(0)

0

y

(0)

10 nm/s

x

(0)

y

(0)

0,

2.500

nm,

x

(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,

1

(0)

2

2

(0)

(0)

0

1.55rad

1

(0)

1.9rad,

1

(0)

2

2

(0)

(0)

1.7rad

0

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

)

2.2 Molecular Forces

• 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

r

i

m i

2

(

x i

2

y i

2

z i

2

)

U

r

i

i i

,

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

L

L

q

0,

d

L

1

L

q

1

0,

d

L

2

 

L

q

2

m x i

F

,

i

F

,

m z

f

0

This can be rewritten in the vector form:

F

i

m

r

i i

,

F

i

F

,

F

,

F

 

U

r

i

Interatomic Potential

The most general form of the potential is given by the series

U

r r

2

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)

The

two-body potential

describes dependence of the potential energy on the distances between pairs of atoms in the system:

W

2

( , )

i j

W r

2

( ),

r

|

r

ij

|

|

r

i

r

j

|

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 x

j

r

j z

r

ij

r

i

i

y z

j

r

j

i

r

ji

θ ijk

r

k

r

jk

k

y

Pair-Wise Potentials: Lennard-Jones

W

LJ

F

LJ

4

 

12

 

W

LJ

r

24

6

2

 

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

F

M

e

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

 

 

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

const

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

t

2

1

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

• 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

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 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

W r

2

ij

...

d

W

1

one-body potential,

t

total time

W

2

two-body potential,

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

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

(

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

h

Planck's constant

U

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

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 particle

i r i

– distance between the and the chamber’s center):

W wl r i

e

(

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

LJ r ij

4

12

r

12

ij

r ij

6

6

The total potential:

U

i

W wl r i r i

r ij



i

W

LJ r ij x i

2

y i

2

(

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

W

LJ r

1

(

r

12

)

W wl

W

LJ r

2

(

r

13

W wl

)

W

LJ r

3

(

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:

i

 

U

x i

,

i

 

U

y i

,

i

1, 2, 3

Parameters:

R

10nm,

m

Initial conditions (nm, m/s):

x

1

(0)

x

2

(0)

x

3

(0)

0,

0,

0,

x

1

(0)

x

2

(0)

25,

30,

y

1

(0)

y

2

(0)

 

3.0,

 

0.5,

y

1

(0)

y

2

(0)

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

W wl

LJ r

(

r

12

)

W wl

W

LJ r r

W

LJ

W

LJ

W

LJ

(

(

r

23

r

35

)

(

r

45

)

)

W

LJ

W

LJ

(

r

24

(

r

35

)

)

W wl

W

LJ r

(

r

W wl

14

)

r

W

LJ

W

(

r

15

)

wl

W

LJ

(

r

25

)

r

Equations of motion:

i

 

U

x i

,

i

 

U

y i

,

i

1, 2, 3, 4, 5

Parameters:

R

10nm,

m

Initial conditions (nm, nm/s):

x x

2

(0)

x

1

3

(0)

(0)

0,

0,

0,

x

1

(0)

x

2

(0)

25,

 

30,

x

3

(0)

20,

x

4

(0)

0,

x

4

(0)

 

24,

y

1

(0)

y y y

2

3

4

(0)

(0)

(0)

 

5.6,

 

 

3.0,

0.5,

2.5,

y

1

(0)

y y y

2

3

4

(0)

(0)

(0)

0

0

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

J which corresponds to

temperature

T

1

E

kin

k

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

2

rad

,

V 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

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,

y r ij

=

y i

– y j

is the relative distance between particles

i

-axis) direction,

k

and

j

in the vertical

– 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 established. For a more symmetric view, the simulation shows the 1 st th and 1 st particles is 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

),

h i

y j

)

1

2

i

y 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

0.286 10

21

J

0.4

0.3

which corresponds to

temperature

T

2

E

kin

k

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

energy potential

of the “cold” subsystem is approaching the value

U

 

25

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

• 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:

A

0

c

L n

A

0

 lim

L

 estimate for true physical quantity

2.3 Molecular Dynamics Applications

Tensile failure of a gold nanowire Dislocation dynamics of nanoindentation

Carbon nanotube immersed into monoatomic helium gas

Deposition of an amorphous carbon film

Molecular Dynamics Applications (II)

MD fracture simulations

Shear-dominant crack propagation

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