Direct MD simulations is nanostructured materials
2
T = 0.94 Tm
Laser melting of thin Au films: The kinetics of melting are related to time-resolved electron
diffraction measurements. Lin, Bringa, Leveugle, to Atomistic Simulations, C 114, 5686 (2010)
Uni
Pressure in Molecular Dynamics I
In order to introduce pressure, let us consider a system of N atoms that is developing in
a finite space and let us introduce a function that is called Clausius virial function:
v
Tot r
W ( r1 ,., rN ) =
N
i =1
r r Tot
ri
Term project
Objective: To get experience in designing and performing computer simulations.
Parts of the project:
Design (or adapt an idea from literature) a simulation that is of scientific or
computational interest to you.
Choose and justify computati
University of Virginia, Department of Materials Science and Engineering
Fall 2011, Tuesday and Thursday, 8:00 - 9:15 pm, Thornton Hall A 119
MSE 4270/6270: Introduction to Atomistic Simulations
Instructor: Leonid V. Zhigilei
Office: Wilsdorf Hall 303D
Off
Homework #1 (page 1 of 2)
Simple MD code with Velocity Verlet algorithm
Write the simplest possible one-dimensional molecular dynamics code for two particles
connected by a spring (Force = k(x2-x1-x0) where x=x2-x1 is the distance between particles,
x0
MSE 4270/6270: Introduction to Atomistic Simulations, Fall 2011
Homework #2 Running a simulation with MSE627-MD
Objective: Become familiar with the MSE627-MD code, pick an appropriate timestep for
integration, understand the partitioning of the thermal en
Homework #3 (150 points), page 1 of 2
Objective: Building initial system with MSE627-CG code. Understand the connection
between the temperature and velocity distribution. Analysis of melting and phase
transformations in terms of evolution of kinetic and p
Homework #4
Objective: Getting experience with Metropolis Monte Carlo simulations, using
Ising model to study compositional ordering and segregation in binary alloys.
1. Review computer code that implements a simple Ising model, mse627-mc.f90,
(the code w
Homework #5 (page 1 of 2)
Objective: Understanding the relation between microscopic mechanisms and
continuum description of diffusion.
1. Using the same FCC crystal that you used in homework #3 (5x5x5 unit cells, 500
atoms, afcc = 5.78 ), perform two simu
Data analysis for different types of simulations
We may be interested in:
Equilibrium properties of the model system.
Structure and properties of the system in a metastable state.
Dynamic processes in the system far from equilibrium.
Issues relevant to th
Embedded-atom and related methods for metallic systems (IV)
The development of EAM-type potentials for BCC and HCP metals is more difficult and the
progress in this direction is slower.
HCP metals:
In fitting potentials for HCP metals one should make sure
Embedded-atom and related methods for metallic systems (I)
As we discussed above, pair potentials, even with an additional density-dependent term
cannot provide an adequate description of metallic systems. An alternative simple but rather
realistic approa
Potential cut-off (I)
The potential functions like L-J have an infinite range of interaction. In practice a cutoff
radius Rc is established and interactions between atoms separated by more than Rc are
ignored. There are two reasons for this:
1. The number
Control Data Corporation (CDC) 3600 at NCAR (1963-1969)
beautiful computer with smoked glass panels and a solid and stunning look
from http:/www.cisl.ucar.edu/computers/gallery/cdc/3600.jsp
several million dollars
University of Virginia, MSE 4270/6270: In
University of Virginia, Department of Materials Science and Engineering
Fall 2011, Tuesday and Thursday, 8:00 - 9:15 pm, Thornton Hall A 119
MSE 4270/6270: Introduction to Atomistic Simulations
Instructor: Leonid V. Zhigilei
Office: Wilsdorf Hall 303D
Off
Review of classical mechanics used in following pages
Lagrangian:
&
L = K ( q i , q i , t ) U ( q i , t ),
i = 1, 2 ,., g
K kinetic energy, U potential energy, qi coordinates,
&
q i velocities, g - number of degrees of freedom
Equations of motion:
d L
L
=
The extended system method for constant pressure V. Summary
Equations of motion:
&
MV = P int P ext
m i & i = V 1 / 3 Fi
d
2
&
m iVdi
3V
The equation of motion for the volume is defined so that deviation of the
internal pressure from its average works as
The extended system method for constant temperature IV
Q& = s
s
i
&
m i x i2
&
Qs 2
sgkT +
s
All derivatives here are
derivatives by real time.
&
m i & i = Fi s 1 m i s x i
x
Hoover [Phys. Rev. A 31, 1695 (1985)] proposed to use a new variable
&
= d(ln
Formulation for constant temperature and pressure II
It could be a good idea to rewrite the equations of motion in real time:
s2
s2
M&
2
&+
& = s P +
&
MV
mi d i
F d + Vs
1/ 3
2/3 i i
3V
3V
s
i
i
2
Qs 2
2/3
& 2 sgkT + &
Q& = sV m i d i
s
s
i
& = 1 F 2 m
Limitations of simulations in the microcanonical ensemble
In a classical mechanical system free from an external force the total
energy E is conserved => simulations are performed at a constant
(E,V,N) condition => this correspond to the microcanonical en
Introduction to interatomic potentials (I)
In order to use Molecular Dynamics or Monte Carlo methods we have to define the rules that
are governing interaction of atoms in the system. In classical and semi-classical simulations
these rules are often expre
Fitting the parameters of the potential to experimental data
interstitials
equilibrium
vacancies
Thermal expansion
Elastic and vibrational properties
High pressure measurements
University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations,
Numerical solution of differential equations
Deriving and solving differential equations (DE) is a common task in
computational research.
Many physical laws/relations are formulated in terms of DE.
Most continuum simulation methods are based on solution
Building initial configuration. Ideal crystals (I)
Any crystalline solid can be defined in terms of Bravais lattice, which specifies the
periodic array in which the repeated units
of the crystal are arranged. Bravais lattice is
r r r
defined by three prim
Homework #2: Extend your MD code to many particles and Lennard-Jones potential
(1) Write a simple two-dimensional (2D) MD code for particles interacting through Lennard-Jones
potential. In the 2D system all atoms are moving on the same plane you create th
Computer Physics Communications 119 (1999) 1351148
www.elsevier.nl/locate/cpc
Near-neighbor calculations
using a modified cell-linked list method
William Mattson 1 , Betsy M. Rice 2
The U.S. Army Research Laboratory, Aberdeen Proving Ground, MD 21005, USA
JOURNAL OF APPLIED PHYSICS 104, 083503 2008
Interfacial thermal conductance between silicon and a vertical carbon
nanotube
Ming Hu,1,a Pawel Keblinski,1,2,a Jian-Sheng Wang,3 and Nachiket Raravikar4
1
Rensselaer Nanotechnology Center, Rensselaer Polytechn
Introduction to interatomic potentials (I)
In order to use Molecular Dynamics or Monte Carlo methods we have to define the rules that
are governing interaction of atoms in the system. In classical and semi-classical simulations
these rules are often expre
Mobility of atoms and diffusion. Einstein relation.
In MD simulation we can describe the mobility of atoms through the
mean square displacement that can be calculated as
r
r
1 N r
2
2
MSD r ( t )
(
r
(
t
)
r
(
0
)
i
i
N i =1
The MSD contains information o
week ending
5 DECEMBER 2014
PHYSICAL REVIEW LETTERS
PRL 113, 238701 (2014)
Universal Power Law Governing Pedestrian Interactions
1
Ioannis Karamouzas,1 Brian Skinner,2 and Stephen J. Guy1
Department of Computer Science and Engineering, University of Minne