Nano Mechanics and Materials: Theory, Multiscale Methods and Applications

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Nano Mechanics and MaterialsTheory, Multiscale
Methods and Applications

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

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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
  1970s
• Digital computer become powerful and affordable

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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

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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

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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)

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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
ltlt c) and heavy (m gtgt 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.
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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.
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.
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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.
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Least Action (Hamiltons) Principle
Lagrange function (Lagrangian) Action
integral Trajectory variation Least action,
or Hamiltons, principle
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.
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Lagrangian Equation of Motion
Lagrange function (Lagrangian) Least action
principle
Substitution of the Lagrange function into the
action integral with further application of the
least action principle yields the Lagrangian
equation motion
Lagrangian equation is based on the least action
principle only, and it is valid for all
coordinate systems.
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Lagrangian Equation of Motion Derivation
Using the most general form of the Lagrange
function, the least action principle gives
Variation of the coordinates cannot change the
observable values q(t1) and q(t2) Therefore,
and the second term finally gives
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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
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Lagrange Function of a Material Point
Free material point (generalized and Cartesian
coordinates) Interacting material
point Conservative systems
For conservative systems, kinetic energy depends
only on velocities, and potential energy depends
only on coordinates.
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Lagrange Function Examples
Pendulum Bouncing ball Particle
in a circular cavity
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.
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Pendulum Lagrange Function Derivation
General form Kinetic energy
tangential velocity Potential
energy The total Lagrangian
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.
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Pendulum Equation of Motion and Solution
Lagrange function and equation motion Initia
l conditions (radian)
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Bouncing Ball Lagrange Function Derivation
General form Kinetic energy
Potential energy
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. ß is a relative
scaling factor the ball-surface repulsive
potential growths in eß times for a unit length
ball/surface penetration (from y 0 to y
1m).
The total Lagrangian
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Bouncing Ball Equation of Motion and Solution
Lagrange function and equation motion Initial
conditions
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Particle in a Circular Cavity Lagrange Function
Derivation
General form Kinetic energy
Potential energy The total Lagrangian
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 lt R, U does
not alter the trajectory. ß is a relative
scaling factor the potential energy growths in
eß times between r R and r R1.
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Particle in a Circular Cavity Equation of Motion
and Solution
Lagrangian function and equations
Potential
barrier
Initial conditions
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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.

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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.
Legendres transformation (the passage from one
set of independent variables to another)
Differential of the Lagrange function
Generalized momentum (definition) A mass
point in Cartesian coordinates
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Hamiltonian Equations of Motion
Hamiltonian of the system
Equations of motion
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Hamiltonian Equations of Motion Pendulum
Lagrange function and the generalized momentum
Hamiltonian of the system
Equations of motion
Conversion to the Lagrangian form (elimination of
p)
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Hamiltonian Equations of Motion Bouncing Ball
Lagrange function and the generalized momentum
Hamiltonian of the system
Equations of motion
Conversion to the Lagrangian form (elimination of
p)
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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.

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Predictable and Chaotic Systems
Divergence of trajectories in predictable systems
Divergence of trajectories in chaotic
(unpredictable) systems
Variance of the trajectory depends linearly on
time
Variance of the trajectory depends exponentially
on time (J.M. Haile, Molecular Dynamics
Simulation, 2002)
Trajectories in MD systems are unpredictable/unsta
ble they are characterized by a random
dependence on initial conditions.
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Example Quasiperiodic System Particle in a
Circular Cavity
Initial conditions
Dependence of solutions on initial conditions is
smooth and predictable in stable systems.
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Example Unstable System Discontinuous Boundary
Initial conditions
No predictable dependence on initial conditions
exists unstable systems.
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Comparison for Longer-Time Simulation
Stable quasiperiodic trajectory
Unstable chaotic trajectory
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Divergence of Trajectories
Slow and predictable divergence
Quick and random divergence in stable
systems
in unstable systems Relative
variance of x(0)
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Transition from Stable to Chaotic Motion Example
Initial conditions
Periodic motion
Chaotic motion
A transition from a stable to chaotic motion can
be observed in this system
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The Phase Space Trajectory
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.
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The Phase Space Trajectories Examples
Examples of phase space trajectories
(Projection to the plane x, px)
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2.2 Molecular Forces

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

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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 Newtons 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
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Newtonian Dynamics
By utilizing the Lagrangian equation of motion at
qa xi, qa1 yi, qa2 zi,
This can be rewritten in the vector form
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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
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Pair-Wise Potentials Lennard-Jones

• Here, e is the depth of the potential energy well
  and s is the value of r where that potential
  energy becomes zero the equilibrium distance ?
  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

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Pair-Wise Potentials Morse

• Another pair-wise potential model, effective for
  modeling crystalline solids, is the Morse
  potential,
• Here, e is the depth of the potential energy well
  and ß is a scaling factor the equilibrium
  distance is given by ?
• Similarly to the earlier example,
• The first term represents repulsive interaction
• The second term represents attractive
  interaction

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Truncated Potential

• 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

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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.

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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 Ek and potential
  energy U fluctuate, the value of total energy E
  is preserved,

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Fluctuations in MD Simulation
Averaging of a fluctuating value F(t) over period
of time t1 to t2 is accomplished according to
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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

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Periodic Boundary Conditions
Translation image boxes
Original box
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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
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Dynamic Molecular Modeling
Develop Model
Molecular Dynamics Simulation
Model Molecular Interactions
Develop Equations of Motion
Initialization
Initialize Parameters
Generate Trajectories
Initialize Atoms
Static and dynamic (macro) properties
Predictions
Analysis of Trajectories
Re-initialization
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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

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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.)

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Temperature and Potential Energy
Absolute temperature is proportional to the
average kinetic energy
The time-averaged potential energy
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Pressure and Mean Square Force

• Pressure is defined by the atomic velocities
• Mean square force is defined by the
  time-averaged derivative of the potential function

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Entropy
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.
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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)
• Thermal pressure (how pressure responds to an
  isometric change in temperature)
• Adiabatic compressibility (how the system
  volume responds to an isentropic, change in
  pressure)

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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).
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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 ri distance
between the particle i and the chambers
center) Interaction between particles is
modeled with the two-body Lennard-Jones potential
(rij distance between particles i and j) The
total potential
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Three Particles Equation of Motion and Solution
The total potential Equations of
motion Parameters Initial conditions (nm,
m/s)
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Five Particles Equations of Motion and Solution
The total potential Equations of
motion Parameters Initial conditions (nm,
nm/s)
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Post-Processing Kinetic Energy, Temperature and
Pressure
Averaged kinetic energy vs. time (three 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
Kinetic energy, and therefore temperature and
pressure are due to motion of the particles.
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Post-Processing Potential Energy
Averaged potential energy vs. time (three
particles)
Time averaged potential energy of the system is
approaching the value that gives the
potential energy of the system.
Potential energy is due to interaction of
particles with each other and with external
constraining fields.
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Post-Processing Total 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
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Isothermal Example 1D Lattice with a Cold
Region
Cold region
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.
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1D Lattice Equations of Motion and Solution
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)) Equat
ions of motion Parameters Initial conditions
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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
which corresponds to temperature
In contrast to the adiabatic system example, the
kinetic energy for the open isothermal subsystem
both fluctuates and approaches asymptotically the
statistical average.
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Post-Processing Potential Energy
Averaged potential energy vs. time (cold
subsystem)
Time averaged potential energy of the cold
subsystem is approaching the value
In contrast to the adiabatic system example, the
internal energy for the open isothermal subsystem
both fluctuates and approaches asymptotically the
statistical average.
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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

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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

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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

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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

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Molecular Dynamics Applications (II)

• MD fracture simulations

• Shear-dominant crack propagation