MOLECULAR DYNAMICS SIMULATION STUDY OF STRUCTURAL STABILITY
AND MELTING OF TWO-DIMENSIONAL CRYSTALS
by
Francisco Javier Carrion
//
Lic. Instituto Politecnico Nacional, Mexico
(1980)
Submitted
to the Department
of
Nuclear Engineering
in Partial Fulfillment of the
Requirements of the
Degree of
MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
August
Q
MassachusettsInstitute of Technology 1982
Signature of Author
Department
Certified
1982
f Nuclear Engineering, August 19, 1982.
by
Thesis Supervisor
Certified
by
j
X10V
-7T.-A
Vr
Z
'/rhesis Supervisor
Accepted by
-
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t
Chairman, Department Committee on Graduate Students
Archives
I.,SSACHUSETTSINSTITUTE
OFTECHNOLOGY
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.
,
MOLECULAR DYNAMICS SIMULATION STUDY OF STRUCTURAL STABILITY
AND MELTING OF TWO-DIMENSIONAL CRYSTALS
by
Francisco Javier Carrion
Submitted to the Department of Nuclear Engineering
on August 19, 1982 in partial fulfillment of the
requirements
for the Degree
of Master
of Science
in
Nuclear Engineering.
Abstract
which
dynamics
simulation
A computer code for molecular
incorporates the recently developed technique of flexible periodic
borders has been developed and used to study melting transitions and
structural stability of two-dimensional crystals with Lennard-Jones
interatomic potential.
The structural transition of a perfect square lattice to a
perfect triangular lattice in the case of 36-particle system has been
observed to take place at various pressures with apparently little or
no potential barrier.
The simulation results, which confirm the
based
on
lattice dynamics, revealed clearly that
stability analysis
the transition
higher
involved
density.
to triangular
a shear
mechanism
and the stable
lattice
has a
Simulations using fixed borders showed also a square
transition
but defects
will
be present
because
of
the
rigid system borders.
Melting behavior at constant pressure in a triangular lattice of
56 particles has been studied. At a reduced pressure of 0.494 the
in
melting temperature was determined to be approximately 0.165
reduced units.
Changes in internal energy, density, and enthalpy
across the transition obtained from the simulation runs are found to
It is
be in good agreement with recent Monte Carlo calculations.
found that within the present accuracy flexible and fixed borders give
the same results, and that based on a simulation run using 400
particles system size effect is not significant.
The crystalline order at elevated temperatures of a bicrystal
with two
=7 coincidence tilt grain boundaries has been investigated
The
using a system of 112 particles with two boundary periods.
boundaries were observed to migrate and undergo a melting transition.
From the variation of excess enthalpy with temperature the reduced
melting temperature was determined to be about 0.14 at a reduced
pressure
of 0.494
which
is about
85%
of
the
melting
point
of
the
perfect lattice. These results are the first computer simulation data
providing quantitative evidence that interfacial melting is distinct
from normal crystal melting.
Thesis Supervisor: Sidney Yip
Title:
Professor of Nuclear Engineering.
Thesis Supervisor: Gretchen Kalonji
Title:
Northon Professor of Material Processing.
3
Acknowledgements
My gratitude to Professor Sidney Yip who made possible
His
patience,
this
project.
advice and encouragement during the past two years are
In particular, I want to thank him for
fully appreciated.
his
time,
corrections and comments during the writing of the thesis.
I also want to thank Professor Kalonji for her contribution,
and
corrections
comments
to this work. The economical support from the Norton
Company for the last term is really appreciated since without it
this
work would have not been finished.
I am fully indebet to Khalid Touqan, not
advice,
but also for his frendship.
Council
to
of
Ricardo and Pepe.
Science
and
for
his
support
and
The comments and observations of
Dr. R. Harrison are also appreciated.
specially
only
To
all
my
friends
at
MIT,
Finally I want to thank the National
Technology
of
Mexico
(CONACYT)
for
the
Research
the
economical support during these past two years.
Funds for the simulation runs
were
provided
Office, Contract No. DAAG-29-78-C-0006.
by
Army
4
To my parents
Blanca and Juan
5
List of Figures
No.
Page.
2.1
Simulation
its 8 images
cell and
in a two-dimensional
21
system with periodic border condition.
2.2
Definition of range of interaction.
23
2.3
Criterion for updating the neighbor table
24
2.4
Definition of the simulation
vectors a and b.
2.5
Shape of the simulation
to the
cell according
cell to which
the flexible
27
34
border technique is restricted.
2.6
Example of simulation cell in which the shear terms
do not balance when using the flexible border technique.
35
2.7
Pair distribution function g(r) for the square lattice
and triangular lattice at the same density.
39
2.8
Asymptotic behavior of the orientational correlation
41
for the liquid
function
(L) and the solid
(S).
2.9
Typical behavior of the mean square displacement as a
function of time in a solid phase.
45
2.10
General behavior of the mean square displacement as a
function of time in a liquid phase.
46
3.1
Square lattice in two dimensions.
50
3.2
of the dynamical
Functional behavior of the eigenvalues
matrix as a function of the lattice constant in a
52
square lattice with nearest neighbors interaction.
3.3
3.4
Triangular
lattice
53
in two dimensions.
Functional behavior of the eigenvalues of the dynamical
matrix
as a function
of the
lattice
constant
55
in a
triangular lattice with nearest neignbors interaction.
3.5
Scketch of the five regions found with the lattice
dynamics calculation for the square and triangular
lattices.
57
3.6
Temperature behavior in the transition of the square
lattice to the triangular. The simulation started in
region
and was induced by thermal perturbation.
59
6
3.7
Internal energy as a function of time during the
transition from region 1 and with thermal perturbation.
60
3.8
Volume as a function of time during the transition
from region 1 and with thermal perturbation.
61
3.9
Initial
form of the pair distribution
function
g(r)
63
in the structural transition simulation.
3.10
Instantaneous g(r) after 30 time steps in the
structural transition analysis.
64
3.11
Pair distribution function after 120 time steps
65
3.12
3.13
Pair distribution function after 200 time steps
Average of the pair distribution function g(r)
over the last 3000 time steps of the simulation
after the transition took place.
66
67
3.14
Initial
structure
of the MD cell used
to study
the
68
the structural transition.
3.15
Instantaneous position of the particles after 80
69
time steps.
3.16
Instantaneous position of the particles after 120
time steps.
70
3.17
Instantaneous position of the particles after 240
time steps.
71
3.18
Instantaneous position of the particles after 360
time steps.
72
3.19
Final structure after the transition took place and the
system equilibrated.
73
3.20
Final structure after a transition from square lattice
in.which the system was perturbed by shear.
75
3.21
Temperature
3.22
Potential Energy-behavior in the transition induced by
shear.
77
3.23
Volume behavior in the transition induced by shear.
78
3.24
Behavior.of the temperature in a simulation where the
initial square structure is in the stable region and
the system was perturbed by heat.
79
3.25
Behavior of the temperature in a simulation where the
initial square structure is in the stable region and
80
behavior
in the transition
induced
by shear
76
7
the system was perturbed by shear deformation.
3.26
Final structure in a transition simualted with fixed
borders.
82
4.1
Temperature behavior in a flexible border simulation
with rescaling the first 500 time steps.
86
4.2
Potential energy behavior at
87
4.3
Volume behavior at
4.4
Enthalpy behavior at p-0.582 0 and T=0.137.
89
4.5
Root mean square displacement for P=0.5820 and T=0.137.
90
4.6
Pair distribution function for p=0.5820 and T=0.137.
91
4.7
Volume
4.8
Enthalpy as a function of T at P=0.4936.
4.9
Internal
4.10
Density
4.11
Temperature behavior in a simulation at P=0.4936
and where rescaling was done up to 500 time steps,
and then the system continued at constant enthalpy.
101
4.12
Snapshot
=0.5820 and T0.137.
as a function
energy
=0.5820 and T0.137.
of T at
as a function
as a function
88
=0.4936.
of T at
94
95
=O.4936.
97
98
of T at p=0.4936.
of the MD cell and 3 of
its images
after
1500
102
of the MD cell and 3 of its images after
3000
103
time steps.
4.13
Snapshot
time steps.
4.14
Potential energy behavior for =0.4936 and T=0.17 with
rescaling through all the simulation and with an initial
perfect crystal structure.
104
4.15
Root mean square displacement for P=0.4936, T=0.17
and an initial perfect crystal structure.
105
4.16
Snapshot at T=0.17 after 4000 time steps.
106
4.17
Snapshot at T=0.17 after 6000 time steps.
107
4.18
Snapshot at T=0.17 after 7000 time steps.
108
4.19
Snapshot at T=0.17 after 9000 time steps.
109
5.1
Closed
113
packed
plane
(111)
in a fcc 3-D crystal.
8
5.2
Construction of the
5.3
Simulation
cell with
=7 2-D grain boundary.
3 of its images used
114
for the
118
for
121
bicrystal study.
5.4
Enthalpy
as a function
of the temperature
both the perfect crystal and the bicrystal.
5.5
Excess of enthalpy in the bicrystal with respect
to the perfect crystal.
122
5.6
Excess of volume in the bicrystal with respect
to the perfect crystal.
123
5.7
Excess of internal energy in the bicrystal with
respect to the perfect crystal.
124
5.8
Migration of the grain boundary at T0.11
time steps and with p=0.4936.
126
5.9
Snapshot indicating the increase of disorder in the
grain boundary region at T0.15.
128
5.10
Snapshot indicating a highly disordered structure at
T=0.15 and =0.4936.
129
5.11
Resolidification to perfect crystal after 10000 time
steps at T=0.15.sp
130
5.12
Potential energy behavior at T=O.15 and P=0.4936
where the resolidification process is observed.
131
5.13
Scketch of how the resolidification process may take
place from the datacalculated.
132
5.14
Potential energy behavior at T=0.16.
133
5.15
Snapshot of a high disordered structure at T=0.16
after 5000 time steps.
134
after 7000
9
Table of Contents
Page
ABSTRACT
2
ACKNOLEDGEMENTS
LIST OF FIGURES
TABLE OF CONTENTS
9
CHAPTER ONE.
INTRODUCTION
10
CHAPTER TWO.
MOLECULAR DYNAMICS SIMULATION
WITH FLEXIBLE BORDERS.
15
2.1
16
2.2
2.3
2.4
CHAPTER THREE.
CHAPTER FOUR.
CHAPTER FIVE.
CHAPTER SIX
REFERENCES
Brief Review of MD technique with
fixed periodic borders
Lagrangian Formulation
Calculation of thermodynamic and
and structural properties.
Calculation of Dynamical Properties
26
36
42
STABILITY OF TWO-DIMENSIONAL LENNARD-JONES
47
CRYSTALS
3.1 Lattice Dynamics Analysis
3.2 MD Results
56
MELTING AND PRE-MELTING STRUCTURAL DEFECTS
83
4.1
4.2
84
99
48
Thermodynamic Behavior
Onset of Structural Defects
STUDY OF TWO-DIMENSIONAL BICRYSTAL
=7 Grain
110
5.1
Construction
5.2
5.3
Model
Thermodynamic Behavior
Melting and Structural Stability
of a
DISCUSSIONS AND CONCLUSIONS
Boundary
111
117
125
137
139
10
Chapter One
Introduction
11
INTRODUCTION
Computer Molecular Dynamics (MD) is a well known
has
been
used
to
thermodynamic,
study
technique
that
structural and vibrational
properties of a system composed of a finite number of atoms.
The main
information that is obtained from this
discrete
trajectory
that
trajectories
selected
each
depend
atom
on
the
follows
calculation
during
interatomic
the
is
the
simulation.
potential
that
These
been
has
to describe the interaction between the atoms i.nthe system.
This technique has the advantage that
phenomena
under
it
can
be
applied
to
study
extreme conditions of pressure and temperature where
experimental observations are difficult or impossible.
The method of MD simulation has been applied to study problems in
solid state physics and materials science.
diffusion
mechanism
Some
of
these
are
the
in grain boundaries [1], the fracture and plastic
deformation phenomena in solids [2], the thermodynamic and vibrational
properties of solids
[3,4],
structural
[5]
and
[6], calculation of phase diagrams [7,8],
transitions
diffusional
phase
solidification,
nucleation [9] and melting phenomena [10-15], among others.
The traditional MD Technique is a formulation
system
under
constant
density or volume [3).
that
simulates
Isobaric calculations
are possible by using special techniques that readjust the
the
system
until
the desired
pressure
a
is obtained.
volume
In general
of
it has
been found that this kind of adjustment is not easy and requires a lot
of effort.
For that reason a new MD technique has been proposed by H.
C. Anderson
[16]
in which
the system
is allowed
to expand
according to the temperature and the difference between
or contract
the
internal
12
and
external
internal
pressures.
pressure
In
is constant
this
and equal
A. Rahman and M. Parrinello
applied
new technique the average of the
have
to the external
pressure.
extended
technique
this
to study phase transitions under high pressure [6].
it
preliminary studies of structural transitions in
2-D
from
a
and
Some
square
lattice to a triangular lattice have been also done by K. Touqan [5].
In the present work
transition
we
study
in
more
detail
the
structural
in 2-D of the square lattice to a triangular lattice using
both fixed and flexible
border
structure
a higher potential energy than the triangular
in
structure.
was
2-D
has
simulation
For that reason it was believed that
unstable
and
always
will
try
Despite this higher potential energy,
predict
techniques.
a
small
two frequency
range of density
modes
are real,
that
to
the
The
square
square
lattice
change to a triangular one.
lattice
dynamics
for the square
is to say,
calculations
lattice
the structure
in which
the
is stable.
Although lattice dynamics can predict the stability of a structure, it
can not tell anything
of the kind of transition
At
is
this
point,
MD
a
more
powerful
that will
tool,
confirms what lattice dynamics predicts, but also
kind
because
it
can
take
place.
it not only
show
what
of transition occurs and which mechanism the system follows.
It
is also found that when the system can change its volume by using
flexible
border
method,-the transition from a perfect square lattice
to a perfect triangular lattice takes
using
the
fixed
the
place
easily.
However,
when
border technique the restriction of constant volume
prevents the system from accomplishing a
perfect
transition
and
it
Melting is a very interesting phenomenon which can be studied
at
will end in a triangular lattice with structural defects.
13
constant
pressure
by
using
flexible border technique.
this
known that melting in 3-D is a first
and
studies
order
transition.
It is
Experiments
on 2-D Lennard-Jones systems indicate that melting might
be a higher order transition [17-20,36].
Nevertheless, there are some
who claim that this is also a first order transition [Il].
The mechanism through which 2-D melting takes place is
studied
very
extensively.
currently
One of the models suggested by Kosterlitz
and Thouless [17,18] states that the creation of dislocations could be
for the destruction
the vehicle
other
hand,
Halperin
of
the
structure.
lattice
On
the
and Nelson [19,20) predict that an anisotropic
fluid phase
called
the
hexatic
transition.
This
second
phase
occurs
during
the
melting
model suggests that melting is a two steps
process, first to the hexatic phase and the second to the liquid.
Some attempts have been made to determine how 2-D melting occurs.
In recent works based on Monte-Carlo
been
some
indications
of
the
simulations
transition's
8,14)
there
have
mechanism, but yet, no
definitive conclusion has emerged.
In the present
ensemble
is
study
the thermodynamic
investigated
using
the
behavior
flexible
of
borders
the
technique,
including the analysis of the formation of structural defects
premelted
the
region,
thermodynamic
and the melting
properties,
the
temperature are in good agreement
indicating
that
this
new
phenomenon
heat
with
itself.
of
(N,P,H)
in
It is found
the
that
fusion and the melting
previous
results
[8,14,21],
technique can be used for a more thorough
study of melting or any other problem with this kind of ensemble.
A few studies on grain boundaries using MD techniques
done
in
the past few years.
have
been
Some of this work is concerned with the
14
dynamical
behavior
temperatures
and
migration
[1,22,23].
of
grain
boundaries
not
understand
enough
all
work
[24,25].
its properties.
technique.
It
is
=7
is
done
with
melting
properties.
temperature,
the
flexible
of interest to study the stability of the
grain boundary at high temperatures, as well as the
the
Despite
with this kind of structure has been done to
A study of a grain boundary
border
high
Other studies concentrate on the vibrational
properties and the frequency modes at the boundary
this,
at
heat
of
fusion
and
determination
of
other thermodynamic
In particular the motive of this study is to find whether
the grain boundary region melts at the same temperature
as
the
bulk
crystal.
For all the simulations we use periodic border conditions
Lennard-Jones
6-12
pair
potential.
identical particles and the range of
third neighbors.
The
crystals
interaction
is
and
a
are composed of
carried
up
to
15
Chapter Two
Molecular Dynamics Simulation with Flexible Borders
16
2.1 Brief review of MD technique with fixed periodic borders.
Molecular Dynamics studies in most
system
of
N
identical
particles
cases
which
consider
a
classical
obey Newton's equations of
motion,
= -ba
m-
where
R,
)
Lzi)..., N
CI)
(r,,...,rN) is the total potential energy of the system, which
in the case of central, conservative, pair potentials, can be
in the following
0Ij..)i
~
b)
For this
written
form:
@'jk.Orj-
=
case,
if
we
(2)
hkk
substitute
equation
(2)
into
(1),
the
resultant equation of motion is
dt--
;
t
where
r-i
I
tr.,
6)j,.rj
. . mn
'Pj(rij) is a function
vector
r
=
-
3)
of the magnitude
of
the
pair
.
in this work we study rare gas solids which can be
the Lennard-Jones
separation
6-12 potential
[26]
described
by
17
[(2
cDi(r'i)=
where
L-
and a are the potential parameters.
E
Once we have determined the potential, we get a set
second
order
differential
of
equations which can be solved with any of
the well known techniques used to solve these equations.
common are predictor-corrector and
first
technique
is
more
finite
accurate
and
difference
is
The two most
methods.
suitable
the
finite
difference
method.
The
main
The
for long time
simulations because it allows a larger time step to be
to
N-coupled
used
compared
disadvantages of the
predictor-corrector are that it requires more memory and is more
time
consuming per time step.
The finite difference scheme has been found
accurate
used
in
if a small time
this
work.
step
For
is used.
this
Because
case,
to
be
sufficiently
of its simplicity
the equation
of motion
it is
(3) is
written as follows:
at~~~~~~~~~~~~~.
In this equation to calculate the subsequent time steps one only needs
two initial
the
conditions
condition
which
ri(t-At) and
can be either
r,(t-At)
and
ij (t),
or
vj(t-At). The second pair is a possible
18
can
choice because the position of the particle at the next time step
be calculated from its velocity with the relation
(t)
B;(t-At)
d
V(f-t4 At
*
In a typical simulation
we
(')
start
with
the
minimum
potential
energy structure and a velocity distribution sampled from a Maxwellian
distribution.
This
distribution
is sampled at a given temperature,
which in most cases is chosen to be twice the temperatureat
simulation is going to be performed.
almost
half
The reason for this
energy.
equipartition
theorem
We
know
predicts
that
that
for
an
effect
case
is
harmonic
half of the energy
will be kinetic energy and the other half will
our
is
because
of the initial kinetic energy is-going to be transformed
to potential
In
which the
be
system
we put into
potential
the
it
energy.
we do not have an harmonic system but the anharmonicity
expected
to
be
small
at
low
temperatures
so
the
equipartition theorem should be a good approximation.
Sometimes
it is not possible
to start
the simulation
temperature desired because the system may melt during
For
at twice
the
equilibration.
this reason temperature rescaling techniques are used in order to
heat
the
crystal
gradually
during
the
initial
stages
of
the
simulation.
Once the N differential equations have been approximated
finite
difference
solved numerically.
method,
dimensionless forms.
we
by
the
get a set of N equations which can be
It is most convenient to express the variables in
For the Lennard-Jones
6-12
potential
and
2-D
19
system, we use the following dimensionless units:
a) Distance
ri*
b) Time
t =
r//
/T
E = E/4t
c) Energy
r/4
F
d) Force
F'-
e) Pressure
p'= p /
where
T=
4
-M is
E
called the characteristic
gases it is of the order of
1014
time
and
for
the
rare
Using these dimensionless
seconds.
units we get the following equations:
i
IJ
(r,)
(2.)
(7
=
Ij
...
r (",
.( --.[
'i~,
)t<,
A t
<'aY
¢::)
note that in this case
Q.;
( 4
z
E)=Fy ( rj*( ))
j-4
20
In this case the kinetic energy of a particle is calculated from
where the dimensionless velocity for a particle is
Vi
-
aA
or
(
In Molecular Dynamics simulation we are limited by
size
and
we
only
can
represent
our
the
computer
system by a finite number of
particles, which for the best of the cases can go up to a few thousand
particles.
To
conventional
avoid
to
use
surface
the
effects
periodic
in
a
small
system
it
border condition. The MD cell is
repeated periodically in all directions producing in 2-D 8 images
a
total
of
9
simulation
cells
is
(see
figure
2.1).
Note
and
that
in
simulations using fixed borders the volume or density of the cell does
not change during the simulation, nor the vectors which
cell and
describe
the
its images.
When we calculate the total force acting on a given particle,
should
include
their images.
all
we
the other particles in the simulation volume and
However, because the force derived from the interatomic
potential has a finite range, only the contribution from the particles
within a certain range of
interaction RI needs to be considered.
range is generally chosen to be between second
In
the
cases
of
and
third
This
neighbors.
small simulation systems we have to be sure that a
21
y
I
Cell
Cell
Cell
Image
Image
Image
,-7
_ . ....... ..
.L~-
Cell
S imulat ion
..
Image
Cell
....-
=-Cl
.g . .. ......
Image
17..-..
[
X
Cell
Cell
Cell
Image
Image
Image
Figure 2.1
Simulation Cell and its 8 images in a two-dimensinal System
with periodic border condition.
22
particle does not interact with its own image, or with a particle
and
its image simultaneously.
Once the value of RI has been selected, the interaction with
any
other particle at a distance larger than this limit will be considered
negligible
or
zero.
The
particles that are within this range from
particle i, are called neighbors of i.
The most time consuming' part of MD simulation is the
of
the
forces
on
particles
simulation as efficient as
neighbor
table.
due
to
possible,
their neighbors.
it
is
useful
to
going in
To make the
create
the
is a bookeeping device that keeps track of
This
neighbors and their location of every particle in the
consequence
calculation
system.
the
As
a
of the movement of the particles we can have some of them
and
out
of
the
range
of
interaction
RI
during
the
simulation. When this happens we have to update the neighbor table. To
do
so,
several
techniques
have
been
developed.
One
techniques updates the neighbor table after a certain number
steps
[27].
Another
cut-off range
method
which
is
more
of
these
of
time
precise [22] defines a
RC , which is larger than the range of interaction. The
neighbor table is constructed up to this range, but the calculation of
the forces
doing
is still calculated
this,
we
up to the
range
of
interaction.
By
always make sure that in every time step we consider
all the particles within-the range of interaction, and also whenever a
=
particle moves more than
know
that
updating
(RC-RI) from a
becomes necessary.
reference
a lot of neighbors
to RI, then
is very
per particle.
small
very
On the other
and updating
we
With this second method, the
choice of RC is very important because if it is
have
position
will
hand
be more
large
we
will
if it is close
frequent.
23
·
·
·
00
** 0 00
*
·
·
·
*
000
*@00 e
E000
*
0 0
*
*
*
00
*
*
i
000·
* *
000
*0
0 0 000
*000
* *
00
000l
l
000
· 000
* · 0 0·000
000
*0000
* * ·
*
0
k·
·
o 0
* 09
o
Figure 2.2
Definition of the range of interaction. Particle i only
interacts with particles in the circle of radius RI
(particle j). The force between i and any other particle
outside from this range (particle k) is set to zero.
24
I
Figure 2.3
Criterion for updating the neighbor table.
particle j is out of the cut-off range of i.
n
After an interval of time t, particles i and jhave moved
At time
t
towards each other a distance
neighbor
of i.
such that j is now a
25
TABLE
1
Triangular Lattice
Neighbors
1
2
Number
of
neighbors
Accumulative
number of
neighbors
Neighbor
distance
(RC/a)
6
6
6
12
1.0
r3=1.732
3
6
18
4
5
12
6
30
36
r7=2.645
3.0
6
7
6
42
2r=3.464
12
54
2.0
8
9
6
60
12
72
T3=3.605
4.0
1-9=4.358
10
12
84
21~=4.582
11
6
90
TABLE
5.0
2
Square Lattice
Neighbors
Number
of
neighbors
Accumulative
number of
neighbors
Neighbor
distance
(RC/a)
1
4
4
2
4
8
3
4
12
2.0
4
5
6
7
8
9
10
8
4
20
24
4
28
8
8
4
8
36
44
48
56
V=2.236
2V\'=2.828
3.0
iVT=3.162
11
12
4
8
60
68
13
12
80
1.0
/=1.414
VX=3.605
4.0
VT=4.123
3V2=4.242
20=4.472
5.0
26
2.2
Lagrangian Formulation.
The
result
idea of the
of the
introduction
of another
degree
of freedom
[16)
is
to the
the
(N,V,E)
This degree of freedom allows the shape and the size of the
ensemble.
MD
by H. C. Anderson
(N,P,H) ensemble
cell
to
These
change.
between the internal and
changes are determined by the difference
pressures.
external
With
this
technique
isobaric processes can be studied without the artificial adjustment of
the
volume
that
has to be done
if the fixed
border
The most important feature of this method is that
determines
the
shape
and
volume
it
takes
is used.
technique
the
system
itself
according to the given
One of the first applications of
temperature, pressure and structure.
this technique was done by M. Parrinello and A. Rahman
[6]
and
they
found that this is suitable for isobaric transitions.
The MD cell
in this
technique
is defined
in 2-D by two vectors
)L~~~~~(~
If the
matrix
h
is defined
the position
(x,y) of any particle
coordinates
(Eog)
in the following
as
can be written
way:
in terms
of relative
27
Y
A
Figure 2.4
Definition of the simulation cell according to the vectors
-b
a and b.
28
r =~I-¼ s- ~
o)
(
(la)
where
I
i
f
and from equations (9-11)
E- £;
C1
range
b
-
early the values of the relative coordinates are numbers in the
O<
.
,q <1
Considering two atoms i and j
square of the distance
.2
Ii
=
sSU
G
between
s..
i and j
is given
in
the
cell,
the
by
( I 3)
where
=- T
( L)
29
With this transformation the number
of
dynamical
variables
2N+4, and the Lagrangian which gives the equations of motion [6
In this equation the first term
represents
the
total
is
is
kinetic
energy of the system. The second represents the total potential energy
of the N particles.
The third term represents the kinetic energy that
the borders of the MD cell have, and the final term is the hydrostatic
energy.
Here
of mass,
represents
determines
p is the hydrostatic pressure. W, which has dimensions
the
inertia
of
the
borders.
This
parameter
the relaxation time for recovery from an imbalance between
the external and the internal pressures, and
is
the volume
of
the
MD cell.
¢,,)
The equations of motion derived from
this
Lagrangian
(eq.
15)
are:
Si
G =~
; uRAt;
E 0 ) (SiSjB-
(17)
30
and
P; g
k -M(5_
W
(18I
where the reciprocal lattice is represented by the matrix
I-1
(TZ2
SI
Note
(I
that
for
this
case
the
velocity
of
the
particles
)
is
calculated from the equation
(zO)
V; = <I, _
Here,
the term
h
has been
contribution
is
simplification
is necessary
small
neglected
compared
in
order
in
the
the
hs
reduce
the
with
to
assumption
that
term.
form
its
This
of
the
resultant equations and make this technique accessible.
The internal pressure
virial
in this case can be calculated from the
virial
theorem,
theorem,
-a 17 = E m K \
Y
E E
(.
J)
.
(z2.)
31
For this system,
when
there
is a large
number
of particles,
it is
found that the constant of motion is the enthalpy
Numerical schemes to solve equations
the
(17) and
(18)
predictor-corrector or finite difference method.
are
usually
While using the
second method we find that in order to solve equation (17) we need
,
but
need
when
s
central
we
want to solve the equation for
which we do not have.
difference
scheme
in
For that reason,
both
h
we
equations.
h
(eq. 18), we also
can
nstead,
not
we
use
a
use a
semi-implicit method in which we first calculate
.i_
S.- ( -) =
s5;
( - s; (¢t., )
Then the
we(,1)
( 3)
At
matrix is calculated to solve
I k (4n)
(-l)
+ I (t )
t
(2 )
where
I (t
_ W ( T -
U
(zs)
32
..
Finally from equation
(17) we solve for
si(tn) to get
(t,+1 )
i
from
This technique has several characteristics which have to be taken
into consideration. First, there are oscillations of
the
MD
cell
about the equilibrium position.
oscillations depends on the initial conditions,
the
size of the system.
the
borders
The amplitude of these
the
temperature
in
the
of
the
thermodynamic properties (temperature, internal
energy, volume and internal pressure) are larger than the
fixed
and
It has been observed that these oscillations
are not damped during the simulation, and thus the amplitudes
fluctuations
of
border simulation.
ones
in
a
A main consequence of these fluctuations is
that the number of time steps has to be increased in
order
to
allow
the system to equilibrate and then to achieve a better time average of
the properties of the system.
Artificial damping implemented in equation (24) has been tried to
decrease the oscillations of the borders.
But it was found
that
the
temperature was decreasing steadily as well as the internal energy and
the volume.
(24)
off,
and
This decrease is due to an indirect coupling of equations
(26).
At the same time, when the damping effect was turned
the oscillations
amplitude
that
artificial
damping
it
of the
had
border
before
to the borders
increased
the
to
almost
the
same
Therefore, if
desired,
suggested
is
damping started.
implement it with continuous rescaling.
it
is
to
33
A second
characteristic
of this
technique
is that
it
limited
is
to MD cells with rectangular shape or with symmetry along the x=y line
in the xy plane
(fig. 2.5).
The main reason of this limitation can be seen if we take
MD cell
(fig. 2.6) defined
-&
a
vectors,
CL'A
CL =.
=
(
have
a
0
,
Suppose we are at T=O°K
lattice.
by the following
the
p=O
and
we
b:
perfect
triangular
will
Then, from equations (19) and (21) we find that
be a
diagonal matrix.
0
TF
TF
(27)
=
b
O
and
q=.fj
(D)
( < 2I
.~\
. K
2
(z8)
X
Then, from equations (18) and (27) we get a non diagonal
29),
where
matrix
(eq.
the non diagonal term will produce an artificial shear to
the MD cell which will
result
in a rotation
over
the
(0,0)
point
of
the
cell and a continuous increase of energy due to the external work
done
by this shear.
34
y
b
MD Cell
-
-
S
-
b
a
Figure 2.5
(a)
y
x=y
b
MD
a
X
Figure
Shapes
of the simulation
technique is restricted.
2.5 (b)
cell to which the flexible border
35
Y
y
a
Figure 2.6
Example
of simulation
cell
in which
the shear
terms
balance when using the flexible border technique.
do not
36
2.3
Calculation of Thermodynamic and Structural Properties.
In a
Molecular
calculated
density
the
Dynamics
Simulation
the
properties
that
are
every time step are the positions of the particles and the
of the MD cell.
thermodynamic
From this data
and
structural
it is
possible
properties
temperature which is identified with the time
of
the
averaged
to
calculate
system.
mean
The
kinetic
energy is calculated from the relation:
where the total kinetic energy is,
K;
-~
V
(31)
The internal energy of the system can be calculated from the time
average
of
the
total
potential
energy.
For
our
Lennard-Jones
monoatomic system this relation is:
u=K~~>
½;) l1>
|j(
j)2
C
Z
From the virial equation [28],
KE1 Fee~~~~~S
,>
pJ|
-
AS - -aE)
('3)
37
An expression for the internal pressure in 2-D
equation
can
be
derived
(33), which after integration becomes
In this equation the first term on the right hand side is
kinetic
from
potential.
called
the
The second term is the potential contribution due
to interatomic interactions.
In all these relations, the brackets
over
a
finite
interval
of time.
This
< >
average
refer to
the
is assumed
average
to be equal
to the appropriate ensemble average.
Other properties of the system can be calculated from these basic
thermodynamic quantities, such as the
()
a~p _-
,
the
a
specific
isothermal bulk modulus
?-V
heat
thermal
expansion
;
C
K'v)T
coefficient
)
(
p
the
P~~~~~~~~~~~
etc.
To describe the structural characteristics of a system two
quantities
are
used:
the
orientational correlation
referred
directly
to
pair
g
structure
looking at them one can determine
system
distribution function g(r), and the
function
the
basic
the
(r).
of
the
phase
These
properties
lattice
and
are
itself and by
structure
of
the
once it reaches equilibrium. Also, if during the equilibration
period there has been a phase transition, it is possible to study
the
process that is taking place.
The pair distribution function is the conventional
quantity
for
representing the equilibrium structure of the system under simulation.
This function is defined as:
38
n (Y.)
-CI
=
(r)
.2T-Ir A
W
here
is the time average number
n(r)
distance
can be
in
the range
interpreted
density.
r(Ar/2)
as the density
Figures
of
particles
situated
from a given particle.
in this
ring divided
Thus,
by
g(r)
the
2.7(a) and 2.7(b) represent the form of
a
at
total
g(r)
for
the square and triangular lattices at the same density.
The orientational
correlation
function
g
(r)
[19,29],
is
a
quantity which is defined for hexagonal or triangular lattices in 2-D,
and
is
used
to measure
the disorder
distance at a given temperature.
in the cell as a function
of the
This function is defined as follows:
r) j*o
(r) =
where
e
C
(r)
Y,
77-
I
.......
(Yr- r)
)
I
I
I,
(36)
N?
Here j
labels the 6 nearest neighbors for a given particle i
angle
eihis
particles
the
and
the
angle between some fixed axis and the line joining
i and j.
The importance of this function is that when the system
is in the
~~.--':;_7_-:
--- l--r -;-~--·-
-.
c-- :
:
~-'..........
...
;::':-_v
:.:.~:'.
:._.:"
_-::.....
:.--,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...__..~
·- ·-··--------'
.,L,.~~-i--,,
''--"--c--i--
'·--~ :'_ ----::.:+-=~-Z.-s:--s--:--:r~-::::-:'.
":l---.-_,_
---- -_---?----i.:?:2
: --·
. : :".;··' : :--'?·- -; .
; ! -i ;.'L.---.....·--··--·; i:r---j;T--~''
.·.....
- ....- _r-:
:-:'z;::---?-----
']---'
-- -' --------%~~-i·-~:
---- i'-----,t: :z':---
:-,-;
-----
~_
:;::_-[--· 1. L-Li- .':'4
... .... "'......
--·'t ~....
I-~ ' : -·---~~
-----........
TI..~...............
--. :_.Z~ "~-:' !"~Y--~---__ - - z ,..-_-.:-="'E::E-E:-.%-':k:iL':.i-:
' "-:Z
-.'--'.": ::"-."E_--'
'-~, __.---~.';-
rt: .--~
,"._:'-.
------ .-
.-
,
. . .
...:1_~
.... ..t---~---:-::-·'
-'-- ......"-::'-......~
T"+~'---.................
,' : .....-- -
- ·--
- ·-··-·------~
----- ----
--. :::::::::::::.--:.:r
---i-- ·_
.,,-_;:-
;....
......
'l-----,;--'; --
r...:_ ....
.. -..... · -----
.,-.-. ...^..........
"t.._
I.-....-.
i
~
:',-
!!
.-.......-..
~.... '
' -=:'*
.....:....._..:
:: ;'c.:*'-;-- _ ~',
~~
.--..
· ·- ·
.·.·--.I-I+
T
...............
I~
i
i
I
=-:
i
r
·
·---------
.
i
-
-
I
A
~-L--L
-:i
,
-i--·----·
--
r-
_ __ _ '_- f : I
r--_~:_-~ --- ;------_1-_:
----·-·-- ·- ·- · · ·----- ·-·- · ·- -- -·- ·
------
-
.
- -.
.
.o .
.
.
.
.
.
.
.
.
.
.; .
. . .
....
~
--
- -.
- - -..
..........
...-..
...........
~....,...
. ... ..._:.:
~..~.---------~.
3i:.:
..
,
I
--
e..........................
-- ....,.-.-....-..- -..--
·
cccl--
:-.....:.....i ........ ...--
. . .....
.
7- ~ _: -4
....................
........................
:.
..............
-- f:..":-_: .:.:_:':.:::
:.::::.:.:.:::::-.A
.
. . . . . . . . . . ...............
7....':::-:"___ragur:Ltie
Pair distribution function g(r) for the square and triangular
lattice at the same density. In this case the lattice constant
for the square lattice is set to
a =1.0 .
S
40
solid phase, the behavior of
the
one
when
g
6
(r)
is significantly
it is in liquid phase.
correlation function more
appropriate
different
from
This characteristic makes this
for
studies since the pair distribution function
melting
g(r)
and
pre-melting
does not show this
kind of difference when changing from one phase to the other.
41
96(r)
A
0.5'
U.w
A
-
A
Figure 2.8
Asymtotic behavior of the orientational correlation function
for the liquid
(L) and the solid
(S).
42
1.5
Calculation of Dynamic Properties.
The dynamic properties which are considered in a
the
velocity
autocorrelation
simulation
are
function, the dynamic structure factor
and the mean square displacement.
The velocity autocorrelation function is defined as [3]
N
Y ()
< l
7.Vio
I
where
vi(t)
this
function
V (O)
is the velocity
is
normalized
calculated
function,
to
unity
at
at
time
t.
t=O and
time
Here,
it decays
The spectrum of phonon frequencies, P(w)
asymptotically to zero.
be
of the ith particle
can
by a Fourier transform of the velocity autocorrelation
i.e.,
p oU)
9
2.
o0
( )
t O-C:3(Wt
TT
The dynamic structure factor
S(Q,w)
is the fundamental quantity
in studies of dynamics and correlations in a many
Since
this
function
results
system
30).
can be measured by neutron scattering and laser
scattering, it has become
experimental
body
great
of
with
MD
interest
results.
in
order
to
compare
This function describes the
density fluctuations in a system as reflected in the vibrational modes
in a crystal.
The dynamical structure
factor
is
calculated
by
the
Fourier
43
transform of the intermediate scattering function
I (Q,t).
-oO
ae
S(i
t
I(
t)
(3q)
- co
where
-I
and
(-
=.
J-
(c0
I
9A(c)
(40)
N
p (t)
is the density operator defined as:
L
-t
(41)
The mean square displacement or the classical width
function
is
defined as:
This quantity is important because
particle
in
the
system
is
it
moving.
tells
us
how
much
a
given
Forth caeo th soi te
44
motion of the particles is restricted to a certain range
2.9).
<
2
For
the
case
(see
figure
of liquids, we have particles diffusing so that
> grows linearly in time at long times (see figure 2.10).
45
Mean quare dp.
f
^
IN_
,
C "
0.2
-0.1B
0.16
0.14
2
0.12
0.1
0.08
0.06
0.04
02
0
1000
2000
3000
5000
6000
4000e
time tep
Figure
7000
8000
9000
10000
2.9
Typical behavior of the mean square displacement as a function
of time
in a solid phase.
46
2 .25i
a
2
1.75
1.5
1.2S
u
1
2
0.75
e .s
0.25
0
e1000
2000
4000
9000
7000
5000
3000
6000
8000
10000
time step
Figure 2.10
General behavior of the mean square displacement as a function
of time in a liquid phase.
47
Chapter Three
Stability of Two-Dimensional Lennard-Jones Crystals
48
3.1
Lattice Dynamics Analysis.
The theory of lattice dynamics provides the most basic
study
the
crystal.
are
vibrational
properties
of
the
atoms
adiabatic
and
harmonic
approximations
to
or molecules in a
calculation
The fundamental assumption in a lattice dynamic
the
means
[31].
Generally
speaking the results are valid only for an infinite perfect crystal at
very low temperatures.
We
consider
potential
a
lattice
which
Vkk'(r)
(K
a
model
with
a
central
pair
function of the distance between the
For this case, the force constants for a
interacting particles.
of atoms
is
dynamics
pair
are
K)
XrS j14 r)0
(\\'} ()5k4<nol
0{0am'
(3)
where the separation vector and its magnitude are
*r- 1f(kiA')\
and
V'(r),
V"(r)
are the first
and
second
derivatives
of
the
potential respectively.
The frequency modes and the wave vectors corresponding
modes
are
the
to
these
eigenvalues and eigenvectors of the dynamical matrix.
This matrix by definition is written as
49
)=a
-
k i,)
Q
-(k)
li'4)
is the wave vector and M the mass of the particle.
where
To calculate this modes one has to solve the secular equation
(YS)
-)(--- Li'I =
If the dynamical matrix is evaluated for a perfect square lattice
(fig. 3.1), considering interaction only with first neighbors
through
a Lennard-Jones pair potential, we get:
I,- Cols)
D =
To
(q(6)
2 0A<I
where
(
v (v)
ck1 =.
r
_ I .
2.
.
IL
t
7
v-
In figure 3.1, we can see that for
structural
symmetry
along
I
the
x
and
a
l -
y.
square
y
8)
axis.
lattice
This
there
is
symmetry
is
50
y
ANI&
- M14M
%W
I
I
2
I
I
I
I
I
I
I
I
I
a
I
I
s
I
Al~~~~~~~~~~w
CU:P
A
-II:
13
b
Bye
He
0
I
I
iV
I
4
I-----
I
I
I
I
I
-0~q
Figure
3.1
Square lattice in two dimensions. Particles
the nearest neighbors of 0.
1 to
4 are
51
responsible that when we calculate the dynamical matrix
obtain
non
the
diagonal
terms
equal
to
zero.
(eq.
44)
we
In this case, the
eigenvalues are easily calculated from:
1
Z4
V
[ (Z l -Q+I'-k
r
r
2.~~~~7
= ZY, 1=
WO,,
(5a)
WL=r-
Figure
the
r
Z(
distance
of
a
1,
longitudinal
at which
0
r
mode
r
is
.
a
regions which are defined by
In region
I
3.2 is the plot of the eigenvalues
interatomic
values
-
In
wo
and
transvers
imaginary.
function
rL=1.244, which
one of the frequencies
the
a
grow
in
both modes
amplitude.
are
of the modes
modes
is
the
is zero.
real
and
the
The lattice is therefore unstable
because any excitation of the second mode will cause the
to
of
this case there are found three
r,=1.112
,
as
On the other
hand,
are real and theerefore the lattice
in region
oscillations
r
,
In region
3
2,
is stable.
r, r
, we find that the s),stem is unstable but now the unbounded mode
r -r
is the transverse
If the same
calculation
is
done
mode
taking
is real.
into
account
second
we get the sarme general results but with slightly different
neighbors,
values
mode whil e the longitudinal
for
r,
and
r.
that as the
interaction
(region
decreases.
2)
In this
r*ange
X/hen
case
r,=1.19417
is
increased,
the
interaction
and
the
r=1.
2 35
stability
14 , so
range
range is taken to be
essentially infinite, we believe the stable region will vanish.
When the same analysis that is carried out for the square lattice
52
I
-L
4
3
a
-I0
-J.
-2
-3
-4
A
-1
1.
1.6
1.4
Figure
1.8
3.2
Functional behavior of the eigenvalues of the dynamical matrix
as a function of the lattice constant in a square lattice
with nearest neighbors interaction.
53
y
a
I
I
t
/3
I
*1
__
I
/2 '
\
t
!
-
-
',4
/
! /
W/
/
/
-
!
~-R-
-
1W
\ 0
I /
x
/
/
/
/
\
x\5
6/
'__
Figure 3.3
Triangular structure showing the six nearest neighbors
(particules
1 to 6) of particle
0 in the center.
54
for the triangular lattice
is done
(figure
3.3),
with
a
range
of
interaction up to first neighbors, the dynamical matrix is
7 3vV
- to
.3Vi
\
C
D ()
(51)
+
V/*
It'1
I
&O4
-
In this case again the symmetry reduces the calculation
and
the
eigenvalues are easily obtained from
I'L
I
dz -
-- 138 r
'o
(')'
(S2)
C.
I
)2.
-
D1° ( rY
I
2, . V
For this lattice, the frequency modes or
the
r, =1.249
points
Region 4 ,
3.4).
structure
Region
stable
5 which
eigenvalues
r71.307 defining
and
has
two
real
for
interatomic
is in the range
frequency
regions
modes
r, r
distance
vanish
4 and
which
, has
at
least
5 (fig.
make
in the range
O0 r
one
at
this
r,.
imaginary
mode which indicates that the triangular lattice is not stable.
In
crystals
this
lattice
dynamics
analysis
for
how
two-dimensional
we find the density at which the structure is stable, but we
cannot predict the transition that an unstable
and
the
it
will
take
place.
To
obtain
lattice
such
will
undergo
information it is
necessary to use other techniques such as molecular dynamics.
55
Ir,
6
4
a
e
-2
-4
-6
1.2
1.6
1.4
1.9
1.7
1.5
1.3
1.8
2
r
Figure
3.4
Functional behavior of the eigenvalues of the dynamical matrix
as a function
of the lattice
constant
with nearest neighbors interaction.
in a triangular
lattice
56
3.2
Molecular Dynamics Results.
Molecular dynamics simulation is a powerful technique
be
used
to
study
the
which
stability of two-dimensional crystals.
can
technique has several advantages over lattice dynamics. First,
This
MD
is
not restricted to harmonic systems. Second, it is not only possible to
find out if a lattice is stable under certain conditions, but also, it
is possible to follow the thermodynamic and structural behavior of the
system during a transition.
In this
mechanism
part of the work
our
main
objective
is
to
study
the
of transition from a square lattice to a triangular lattice
and also to confirm the lattice dynamics stability
results
with
the
simulations
with
a 36
particle system were done with initial conditions at each of the
five
molecular
different
dynamics technique.
regions
that
were
For this purpose
found
for
both square and triangular
lattices in the last section (f ig. 3.5).
Each simulation was done
constant
border
pressure
using
the
flexible
executed until equilibrium was reached.
technique,
and
at
was
To perturb the system we used
two different techniques. The f irst was to heat up the system slightly
and the second was to give the MD simulation cell a small deformation.
The first simulation was done starting with a square lattice with
the particles at the minimum potential energy sites.
distance
a
was
set
to
a=1.0977
set
to
t=0o.005.
The
external
p=0.0, the range of interaction to R=2.5,
mass of the border was W=4.0
interatomic
which corresponds to a value in
region 1; the lattice is therefore unstable.
was
The
pressure
the effective
(cf. Section 2.2), and the time step size
57
D
Region
0 0
O('
-
Region
)
Figure 3.5
Scketch of the five regions found with the lattice dynamics
calculation for the square and triangular lattice.
58
The perturbation was induce
temperature
of
0.05
by
giving
(The melting
the
temperature
system
for
an
the triangular
lattice has been found to be about 0.17, see chapter 4).
shows
the
initial
Figure
3.6
temperature behavior of the system during a 4000 time step
simulation.
It
is
important
to
notice
that
in
this
case
the
temperature at the beginning drops to almost half the initial value of
0.05,
but
after
a
few
ime steps, the system can not maintain its
structure and undergoes a transition.
Since the square structure is a
higher potential energy structure compared with the triangular, when a
transition takes place most of the difference in potential energy will
be
transformed
into
kinetic
energy
producing
the
increase
of
temperature that we see in figure 3.6 after 500 time steps.
In figure 3.7 we show the
energy.
instantaneous
internal
or
potential
Here there is an increase at the beginning of the simulation
due to the heat that was put into the system, but after the transition
takes place the system goes to a lower potential energy configuration.
The volume
its
of the MD cell
decrease
packed
value
lattice
case
the same
behavior
(fig.
3.8).
Here
is due to the fact that the triangular structure in 2-D
is a close
of
shows
it is found
plane,
i.e.,
constant
there
it has
higher
density
for
the
compared to the square lattice.
is a small
increase
in the lattice
same
In this
constant
but
even this, the volume decreases.
The structural analysis of the system was done by looking at
pair distribution function
g(r)
during the simulation.
the
Figures 3.9
to 3.13 represent this function during the transition from a square to
a
triangular
lattice.
Figure
the square lattice up to fourth
3.9
is the typical
neighbors.
Once
shape
the
of
g(r) for
planes
start
59
-
o.
~~.
*.~~
..
A*~
*
- -
· e
04
0.0
sea
~o0t
~00
e
1000
3000ee
e
~ ~.,m
time
Figure
4000
3.6
Temperature behvior in the transition of the square lattice
to the triangular.
The simulation
started
was induced by thermal perturbation.
in region
1 and
60
-- 23.
-24
P
_,- 4
. E
- IV'- 5
.- as
-2.5.-5
20"
Figure 3.7
Internal energy as a function of time during the transition from
region 1 and with thermal perturbation.
61
.
-
e
Volume
0e0E@@
3004
tIme -.4tep
Figure
3.8
Volume as a function of time during the transition from
region 1 and with thermal perturbation.
62
sliding
we observe
that two of the nearest
neighbors
of a particle
a square lattice will become first neighbors in a triangular
The
other
lattice.
peak
two
the
peak
into two small
has
lattice.
second neighbors in the same triangular
to broad after 30 time steps.
is separating
steps
become
In figure 3.10 we observe how the second
starts
peak
will
in
peaks
disappeared
neighbors
After 120 time steps this
(fig.
(fig.
nearest
3.11).
3.12).
By
200
time
In the same set of
figures we can see how the third and fourth neighbors
peaks
intermix
and finally form the second and third neighbors peak of the triangular
lattice.
The structural transition was followed
very
closely
by
instantaneous pictures of the MD cell during the process.
shows
taking
Figure 3.14
the initial structure of the system. Figure 3.19 represents the
structure at the end of the simulation, and figures 3.15-3.18 show the
mechanism of transition.
second
row
from
condition.
initial
that
top has a larger
Once the lattice has
structure
initial
(figures
the
In particular, figure 3.15
cannot
deformation
3.16-3.18).
and propagates
From
displacement
reached
be restored,
these
shows
some
from
that
point
the
the initial
in
which
the
the rest of the system
follows
it
lattice
through
figures
we
all
the
conclude
transition takes place by sliding of the close packed
that
the
of
the
confirms
the
planes
lattice.
The importance of this result is that MD not
only
lattice dynamics calculation concerning instability, but also it shows
how
a
square
in region
structure
in region
1 will
go to a triangular
structure
4 (fig. 3.5).
A second simulation starting with the square structure
with
the
63
g(r)
r
6
5
4
3
2
1
0
0
1
2
3
R
Figure 3.9
Initial form of the pair distribution function g(r) in the
simulation of the structural transition.
This function clearly represents the square lattice with
which we start.
64
g(r)
S
5
-4
2
3
a
I
0
0
a
1
3
R
Figure 3.10
Instantaneous g(r) after 30 time steps in the
structural transition analysis.
65
g(r)
(
6
S
4
3
a
2I
1.5
0.5
0
.
3
R
Figure 3.11
Pair distribution function after 120 time steps.
66
g(r)
S
4
3
a
I
0
0.
I
2
R
Figure 3.12
Pair distribution function after 200 time steps.
3
67
g(r)
:
.-
-:
I.
o 5
-S
.
.
.
-'''"
.
.
.
4
..
Ak
.
.
.
I
.
#
..
3'5
.- .
:
4ft
.
z,
3
2
.
.s
0
0
0.5
.
1
1 .5
R
a
265
3.
.
Figure 3.13
Average of the pair distribution function g(r) over the
last 3000 time steps of the simulation after the transition
took place.
68
6
o
0
0
0
0
0
S.1
o
0
0
0
0
0
o
o
o
0
0
0
o
0
0
0
0
0
o
oPI
0
0
0
0
0
0
0
0
0
0
0
4,
y
3
2
I.
i
0
-- I
i
0
2
0
0
5
3
4
0
7
6
x
Figure 3.14
Initial structure of the MD cell used to study
the structural transition.
69
6
/
6
4.
/~~1
o
r~~~~~~
I
q
P'
I
1
1
;t
4
3,
I
!I
.
i
0
1
.
I
I
T
I
,
I,
3
4
2
,
7
6
x
Figure 3.15
Instantaneous position of the particles after
80 time steps
70
.~
I
5.5S
4.5
3.5S
2.S
1.5
-
I,
0.5
__
C
e0
-'
i-
2
3 7
4
6
It,
X
Figure 3x6
Figure
3.16
Instantaneous position of the particles after 120 time steps.
71
5.5
4.5
3.5
2 .5
1.5
0 5
-0.5
0
·
2
4
6
Figure 3.17
Figure 3.17
Instantaneous position-of the particles after 240 time steps.
72
5.5
X
4.S5
3.5
2.5
1.5
0.5
R&a
I
0
--
-
2
-4
6
Figure3.18
Figure38
Instantaneous position of the particles after 360 time steps.
73
ou
.0
.4.5
I"
.
v~
0
&.
.w
0
o'0
0
0
0~~~~~
0
0
O
0
.0
a
0
0
a
0.5 -
0
A
. 0
o.4
5,
0 .
0..
*
0
:>
3
0
_
-
3
0
0
o
0
O
?~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
44
aP-
_
S
X
x
Figure 3.19
Final structure after the transition took place and the system
equilibrated.
74
interatomic distance in region
thermal
perturbation
lower row with
we
put
was done; but in this case instead of
a displacement
a shear on the system and deformed the
of 1
from
its
original
position.
In
this run the initial temperature was zero but all the other parameters
were
the
same as
in the previous
case.
also triangular (fig. 3.20) and the
observed
except
temperature,
transition takes place (fig.
volume
same
the final
thermodynamic
initially
3.21).
remain constant (fig.
Here,
zero,
The
structure
is
behavior
is
increases
potential
when
the
and
the
energy
3.22,3.23) for the first 250 time steps,
and then by the 400 time step they drop to a new value.
The study of
square lattices in region 2 was our
next
step.
For
this case we used the same system with interatomic distance a=1.15 and
interaction
with
first
neighbors only (RI=1.4).
The time step used
was
t=0.002 and the external pressure
ways
of perturbing the system were used. In both cases, we found that
the system maintained
was
done
the
system,
and- equilibrates
at
of
during
Again
a
1000
T=0.0005.
the
3.24
temperature
where
1000 time steps simulation.
temperature
lattice.
Here
equilibrates
T0.0
it
is
but with a shear of
observed
that
the
1
steps
In this
which
T.0005
we
have
the
Figure 3.25 shows
the temperature behavior during a 4000 time step run where we
at
two
time
almost half its initial value.
and the its behavior can be seen in figure
temperature
a
with an initial temperature
case figure 3.24 shows a typical behavior
drops
p=-0.2655.
its structure.
When heat was used to perturb
simulation
was
started
on the lower row of the
temperature
rises
and
at a low temperature. This increase is due to the energy
of deformation at the beginning of the simulation.
From figures
3.24
75
,
bt .
-
o
o0
0
3.5.
o
0
0
2 5
o
o
o
o
0
o
o
0
o
y
o
O
0
o
o
o
o
o
I .5
0
0 5
-0
o
0
o
o
0
0
0
o
5
_
-1.5
__
0
-1
1.
I_
__
_
___ __
3
_
6
4
2
5
7
x
Figure 3.20
Final structure of the system after a transition
from square lattice in which the system was
perturbed by shear.
76
Temperature
0.
0.09
0.08
0.06
T
0 0s
m
0.03
0.02
0.el
0
0
1000
2C00
time atep
Figure
3000
4000
3.21
Temperature behavior in the transition induced by shear.
77
Potential Energy
-12
-14
-16
-18
P
,E
0
1000
2000
time
3000
4000
tep
Figure 3.22
Potential energy behavior in the transition induced by shear.
78
Volume
39
39
3S
.5S
38
37 S
37
36 5
36
35
2500
1500
0
1000
8000
time
300
3500
step
Figure 3.23
Volume behavior in the transition induced by shear.
4000
79
Temperature
C.c
t147o
O. e004
mo.ee42s
p
0. z0e325
C.
e0e3
0.000275
e.
111
IV
,,
c*.:n r
j
,_1
_
_
0
.
-
,
lee
I
200
,
300
-
-
-
,-
-
-,
see
400
600
,
-
,
-
Or
I
700
90k
~~~~~~~~~80100
B0
1000
time step
Figure 3.24
Behavior of the temperature in a simulation where the initial square
structure is in the stable region and the system was perturbed by
heat.
80.
,,
"
-. be-a,
0.000003
2.5e-6
0.000002
.T
e
m
P
1.Se-6
0.000001
S.e-7
0
500
0
1500
100.0
2500
20ed
3000
3500
4000
time step
Figure 3.25
Behavior of the temperature in a simulation where the initial
square structure is in the stable region and the system was
perturbed by shear deformation.
81
and 3.25 it can be seen that the system responds faster to the thermal
perturbation than to the shear perturbation.
Simulations of
the
square
lattice
in
region
3
and
of
the
triangular lattice in region 5 were done and it was found that in both
the
cases
system
unable
was
to
its
maintain
and
structure
it
sublimated.
The transition
consequence
from
square
to
triangular
of the flexible border technique.
lattice
is
not
a
For that reason it was
of interest to if the transition can take place using the fixed border
technique.
The previous simulation starting with the
in
1
region
was
of the
lattice
repeated using the fixed border technique.
found that in this case the
potential
square
system
also
tries
to
go
to
It was
a
lower
structure, but the constraint of the fixed volume and shape
D cell
is an obstacle
system ends as a
figure 3.26).
triangular
to accomplishing
lattice
with
the transition,
structural
and the
defects
(see
82
(1.
6
I
I
4'
I
3
I
2
2. 11
I
I
at
5
3
1
0
2
4
6
7
Figure 3x26
Figure 3.26
Final structure in a transition simulated with fixed borders.
83
Chapter Four
Melting and Pre-melting Structural Defects
84
4.1
Thermodynamic Behavior.
It is known that the 2-D melting transition does
the
way as in a 3-D system.
same
not
behave
363 and theories
Some experiments
On
[17-203 suggest that this might be a second order transition.
hand,
other
and
Monte-Carlo
dynamics
molecular
in
the
simulations
E8,11,14,213 give indications that this transition could be
of
first
order.
the
Our first objective is to demonstrate that
technique
is
process.
details
flexible
border
capable of describing the thermodynamics of the melting
Secondly, we
associated
to
hope
with
elucidate
melting
some
of
the
structural
It should be noted at the
in 2-D.
outset that we do not expect to make any contribution to
the
current
question on whether melting in 2-D is a second order transition.
The first step was to simulate a point in the phase diagram which
has
been
already
calculated
with
thermodynamic properties are known.
point
deep
in
the
phase
solid
instabilities that appear
in
other
and
techniques
its
It was our interest to simulate a
to
phase
avoid
regions
defect
close
formation
to
the
and
melting
temperature.
A 56 particle
system
at a temperature
=0.58 2 0 was simulated
of
steps.
The
for a total
thermodynamic
of
T=O.1375
length of
properties
such
time
and a pressure
of
5000
time
as temperature, volume,
enthalpy, and internal energy were calculated during
the
simulation.
The temperature which was rescaled through the first 500 time steps is
shown
in
figure
4.1.
The
potential
increase due to the fact that we
start
energy
with
the
(fig.
4.2) shows an
minimum
potential
85
energy
structure.
The volume we start with is held constant for the
first 750 time steps and then is released after the initial transition
takes place.
to
be
It should be noted that the initial volume was
close
to
the
expected equilibrium value
large transients at the beginning of
which
the
selected
(fig. 4.3) to avoid
simulation.
The
enthalpy
should be the constant of motion in this method does not remain
constant when the temperature is being rescaled, but if
continuous
through
equilibrium
value.
become
a
out
is
simulation it will oscillate around its
If rescaling
is stopped
at a given
constant after this point (fig. 4.4).
displacement
(fig. 4.5) and the pair
distribution
time,
it
will
The root mean square
function
(fig.
4.6)
From their behavior is clear that the system is in
calculated.
were
the
rescaling
the solid phase.
In the next table we
compare
the
values
calculated
from
simulation with the data calculated with Monte-Carlo technique
7].
Monte-Carlo
Flexible Borders
% Difference
p
0.5820
0.5820
0.0
T
0.1375
0.1370
0.36
Q/N
1.1249
1.1235
0.13
U/N
-0.7026
-0.6971
0.45
*
Values without long range correction.
our
86
Temperature
O
I
I
0.17
0.16
T
0.14
p
e 1,
0Lii
0.1
e
1000
3000
time step
Figure
4000
5000
4.1
Temperature behavior in a flexible border simulation with
rescaling
the first
500 time steps.
87
Potential Energy
-3Jb
-3?
-38
-3S;
-40
P
P
-41
E
-42
-43
-44
-4S
-46e
1ne0
2e00
3000
4e00
t me step
Figure 4.2
Potential energy behavior at p=0.582 and T=0.137.
S000
88
:
·
:
:
o
4.
' .0
. E
- " G.'
:. .: A
f
'
·
e.,!
0
meoe
'E-O ''
3000.
Figure 4.3
Volume behavior
at p=0.5820 and T=0.137.
4000
5eee
89
Enthalpy
.
·:-
: ·
..I?
-¥_
S,
4
H
3
i
d
X
.ee' 1Z500
i e1ee
i
, eee
I
3ee-
tim
. .step
4000
S 0e0
'
'
Figure 4.4
Enthalpy behavior at p=0.58 2 and T=0.137. Rescaling was done
during the first 500 time steps.
90
:i--
:1::-
17*
eviv
e | 04.
.
He 4Ies
-
--
-1
I4 00 '
.
.
I
.
2 Dee
Figure
.
-·
.
2.00
e
3Se00
4te
30ee.,
·
4500
4.5
Root mean square displacement for p=0.58 2 and T=0.137.
.Se
DOQ
91
g(r)
.
.
~
4.S
-:
.·
4
.
.
I.
·
`
.I . .
.i
i ,.
l
2S
*
*
**
C
a
I
:: .
i
0.S
0
2
I
R
Figure 4.6
Pair distribution function for p=0.58 2 and T=0.137.
3
92
The agreement of these results is sufficiently good that
conclude
that
this
technique
we
can
is suitable for mapping out the phase
diagram under isobaric conditions.
Our next step was
constant
to
obtain
the
thermodynamic
properties
at
liquid
pressure for points close to the melting temperature in both
and
solid
temperatures
is
phases.
of
The
study
particular
of
structure
interest,
as
well
at
different
as the study
of
melting through the thermodynamic data calculated by simulation.
The
basic
simulations
thermodynamic
p=O.4936.
as a function
clearly
goes
calculated
in
to
a
data
at
different
these
Table 4.1
temperatures
for
a
From this table we were able to plot the volume
of the
observe
up
quantities
were the enthalpy, volume and internal energy.
contains the
pressure
thermodynamic
temperature
three
(figure
temperature
temperature
of
4.7).
regions.
T=O.16,
In
this
plot
we
The solid region which
the
liquid
region
for
temperatures over T=0.17, and the transition region which is the range
between
T=0.16
and
T=0.17.
thermal expansion coefficient
for the
From this curve we can calculate the
for the solid and
F
op(
liquid.
Figure 4.8 represents a plot of the enthalpy as a function of the
temperature. The specific
calculated
for
both
(fig.
4.9)
temperature also
at
constant
pressure
CP tar)p
was
the solid and the liquid. We got the following,
cp=2.80 for the solid and
energy
heat
and
show
cp=3.77
for
the density
the
same
the
liquid.
(fig. 4.10)
behavior
as
The
potential
as a function
the
volume
of the
and
the
From the thermodynamic behavior of the enthalpy, the volume,
the
enthalpy. The three regions are also clearly observed.
Table 4.1
Run
T
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0.091
0.094
0.099
0.107
0.108
0.118
0.122
0.130
0.132
0.147
0.148
0.153
0.156
0.160
0.162
0.164
0.170
0.175
0.185
0.200
D/N
1.095
1.099
1.103
1.112
1.108
1.123
1.123
1.134
1.139
1.150
1.152
1.157
1.162
1.164
1.182
1.185
1.251
1.262
1.277
1.318
P
U/N
H/N
0.9132
0.9099
0.9066
0.8993
0.9025
0.8905
0.8905
0.8815
0.8779
0.8696
0.8681
0.8643
0.8606
0.8585
0.8458
0.8435
0.7991
0.7919
0.7829
0.7582
-0.742
-0.739
-0.734
-0.722
-0.726
-0.708
-0.708
-0.694
-0.691
-0.679
-0.674
-0.668
-0.664
-0.660
-0.642
-0.640
-0.578
-0.572
-0.560
-0.531
-0.1105
-0.1025
-0.0906
-0.0661
-0.0711
-0.0357
-0.0317
0.0012
0.0062
0.0356
0.0426
0.0556
0.0656
0.0750
0.1031
0.1086
0.2086
0.2253
0.2601
0.3197
T, U, and H are in units of 4.
2
The volume
is in units of
.
time
steps
2000
2000
2000
3000
2000
3000
3000
6000
3000
2000
3000
2500
2500
6000
6000
6000
10000
10000
10000
10000
94
-......
------------ - ---
- -- e S -
- -:_.:-: ~~~~~~~~~~~~~~~~~~~~~~~...-..
e
___A.
.
..
_ _ _ _
.
.
.
-~ ......
~ ~-.-.
~ ~ ~ ~
.............
VET--
i'.
.
_,015
Figure
4.7
Volume as a function of the temparature at p=0.4936.
__
95
---
---
I
I
---
.
.
---
-
.--.
I.--.-
:!
,,
I
---- 44-
-,-.
--
6
- . I - -
---------
4-- -.-
.
t---
- --l··--_----'-l-4`"
,--'-I
-.--
- -
-
.
i
,
4---·----
- .- _ J
__._~~_
I!
.-
--'- .
-
I-I
--
-
:-- H/t4-
-
f--
·----------------
I
T
---
--
I
.
i-I
'
-----------------
______________.___...
.
:4 --
, i[
----------
----------- i
--f
---.
I
·--------------·-
I
`--------·----------
i
I
-i
.
.-e---. -- ~,.c.- .I; -.~-.-~..-.~.- -...-
----·--- --·---
4.-
-
-..
½_...._iI
4.--
i
t-----------------------;r
--·
..-
--.
L.----.---:
--.
I-;..
'
I-- =':]-;
;;;---"~~~--------------I,
,__--
I
i,
i
-- i --=
l
'
-'
'-
1
--'
-
,
-[i,,
I~~~--~-7--
--
-'::'~~~~~~~
~---':-~-'
~---~--:-·----~-:-'-~---.-....
'-_
.
.........
:----:
..
__
---.-
._
.........................................................................
· ~"'~...--;.__/..'~:
..Z.... ~.. : -- --
~ - '--
~ '------
-.....
.....
...--.-...
; .-...·----- '--i- .... !
......................................................
~ . . . ... ~. . . . . . .......
.....
0:':.;'-7:.
0--
i
4:
0.10
i
!
~
~
~
~
~
~
~
........
0.15
-
Figure 4.8
Enthalpy
as a function
of T at p=0.4936.
0.20
-
96
density,
and
internal
the
energy,
occurs at approximately T.165.
examined
the
root
mean
we are able to say that melting
To support this conclusion
square
we
have
and the instantaneous
displacement
structure of the system.
With
a melting
temperature
of
TO.165
at a pressure
and assuming that this is a first order transition, we
the
step
in
changes
with
the results
of by J. A. Barker
reported
and
Ap =0.051.
by F. Abraham
temperature
T=0.12
On
the
p=O.71
other
[21) for a pressure
the
change in the density is
it
0.1.
agree
for a melting
was
found
that
hand, if we take the data
of
p=O.0125
change in the enthalpy is
Ap
calculate
These data
et al. [8] in which
temperature of T=0.175 and a pressure of
AH=0.117
can
For the enthalpy we
thermodynamic properties.
H=O.1 and the density Ap =0.05.
found a change of
of p=-O.4936,
Here,
the
and a
melting
H=0.1
and the
disagreement
may
be
attributed to the fact that the pressure difference is quite large.
To check our results with the fixed border technique, we choose 2
points in the solid phase and 2 points in the liquid.
were carried out with
flexible
border
the
corresponding
technique.
From
the
volume
fixed
calculate the internal pressure and energy.
agreement
of the
internal
pressure
this
case
the
agreement
was
obtained
border
We
found
and the internal
with fixed borders with those calculated with
within
3.
indicate to us that, within the accuracy of
The simulations
the
simulation we
a
energy
flexible
from
very
good
calculated
borders.
In
All the previous results
our
data,
the
flexible
border technique gives the same results as the fixed borders technique
and both are in agreement with Monte-Carlo results.
97
-4
--
-
--.
,b.
·
...
.
.
.
.
..
. ..
.
.0:.: ..:
0.10
5.
Figure 4.9
Internal
energy
as a function
of T at p=0.4936.
98
----
...:
I
: -7
' .':I-
''
-
-- -`-
-'
----
~
--- `- __
- -
- -
__ .
~
-~_ '~-'-`-
----
~-----~
-7 -- '-`-'-
I-f
- _.
-
.
·--
--
.:_-------
-- -----.--.------ ·---------- ·-----·----------.....
-.-I·---....-- -- . --. L--.---------..... ...--..--.-
·- ·
-- ·-------·- ·--- · ·-··----- · -----·-----
-----------.__I_
··-----.-I
:
-----
· ----------
-------
-·
1i
·
.------^----c------.--.
-.-...----.
--.-.- ---.-.-- -.---- 1.1··-· ·-----
---- ...--.
-------- -·- -- ·----- -----·- ·-·------------- ·
----··
--- -- · ·------ ·- ·----·
---- ·-- ---· --- ··-- ·--·-·-·
----.--- -- -.--.t
-· --------
------
-..-.
___
------- ·---
··---- -- _------..---.--c-------------- I--.-I-- .
---__..-.--
_____.___ --.-..-...--1.-.·- -·.-..-...-.-..- .-...-... -. .. ..-.. ..
__ I-.
-L__,_
-.. .
t
I
i
--: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~...........ee--.,.
li
---
7--
-1.
_1
lw~-
Figure 4.10
Density as a function of T at p=0.4936.
I
99
4.2
Onset of Structural Defects.
In the studies of the structural behavior on a
particular
2-D
system,
cases which we consider of interest were found.
two
The first
is a simulation in which the temperature was rescaled during the first
500 time steps to a value of T=O.16, after
that,
continued
constant
and
the
system
remained
at
simulation consisted of 8000 time steps.
that
the
system
time steps.
to
oscillates
We
observe
from
This
fig
4.11
a
significant
This process was studied by taking snapshots of
What we
observed
is
high temperatures the system had a perfect lattice structure
(see fig. 4.12).
But, because
of
its high
thermal
motion,
always a particle in the lattice that causes deformation.
times
not
around its initial temperature for 2000
the structure between 2000 and 3500 time steps.
at
was
enthalpy.
After that, the average temperature shows
a value of 0.125.
that
rescaling
there
is
Most of the
the system can restore to its initial structure, but after 2000
time steps the deformation gets blocked
defects
(fig.
defect
formation,
4.13).
The
increase
is the cause
of
into
a
internal
of the decrease
new
structure
energy
because
with
of the
of the temperature.
The second case which we are going to discuss is a simulation
a
at
temperature of T=0.17. In here, we start with a solid structure and
the
temperature
was
rescaled
during
all
the
simulation.
The
particularity of this simulation can be seen from the potential energy
behavior
(fig.
4.14).
In this case, the system shows a drop between
time steps 3000 and 6500. This drop can be explained
the
system
tries
by
saying
that
to go to the solid state, but after all, it cannot
remain in this phase and it melts.
To support this argument the
root
100
mean square displacement is plot as a function of time (fig. 4.15). In
this
figure
for
the
first
solid. After 6000 it shows an
liquids.
The
sequence
of
6000
time steps the behavior like in a
increase
which
transition
in
looking at the snapshots at different times.
the
is
characteristic
of
this case was studied by
Figure
4.16
represents
structure of the system after 4000 time steps and clearly remains
in the solid state.
observe
that
Figure 4.17 is a snapshot at
time
the defects start to form in the system.
6000
shows some ordering in the lattice but it almost
and
we
reached
By 7000 still
the
liquid
phase (fig. 4.18). After 9000 the system already melted and it shows a
large disorder
(fig. 4.19).
101
Temp
O.1S
0.16
0.14
T
e
m
P
0.13
0.12
0.11
0.1
0.07
5000
1000
0
2000
6000
7000
time .atop
Figure
4.11
6
4
Temperature behavior in a simulation at p=0. 93 and where
rescaling
was done
up to 500 time steps,
system continued at constant enthalpy.
and then
the
10.2
'Li
I S-
13
11
9
y
7
3
-.1
2
0
6
10
B
4
18
14
12
16
4.12 xFigure
Figure
Snapshot
4. 12
of the MD cell and 3 of
its images after
1500 time steps.
103
A7
_
~-
.^
-
IV
0
_
13
E I
00
0
11
0 0
0
o
0
D
0
0
0
0
0 0
0 0
0
0
00
0
3
0
0
7
00
0
000
0
O
I
0 0
0
0
I
0 0
0
0
0
0
0
O
0
_
1
-2
·C
4
0
*8
_IC_C_
s
CC_·
16
i0
14
its images after
3000
6
--- 4
18
x
Figure 4.13
Snapshot
of the MD and 3 of
time steps.
104
Potential energy
-29
-31
-32
-33
-34
P
-35
-37
-38
-39
-41
-i
2000
I
7000
5080
3e00
6000
4000
time step
8000
10000
Figure 4.14
Potential energy behavior for p=0.4 936 and T=0.17 with
rescaling
through
all the simulation
perfect crystal structure.
and with
an
initial
105
4
.'
1 ,4
I.e
1
m
0.8
S
D
0.6
0.4
0. a
1000
e
~
3000
200e
7000
500
4000
time
6000
9000
8000
10000
stbep
Figure 4.15
Root mean square displacement for p=0.4936 and T=0.17, and
an initial perfect crystal structure.
106
e
-7
I$
13
11
9
7
15
3
-i
R
0o
4
8
Fx
Figure 4.16
Snapshot at T=0.17 after 4000 time steps.
_ _
107
4"0_I
is
14
10
y
8
6
4
P-
4 -
8
12
Figure 4.17
Snapshot at T=0.17 after 6000 time steps.
16
8
108
16
14
12
08
v
6
2
4
_
0
¢-_~~~~~~~~~~~~~~
4
.6
10
8
I
-
-
_-
14
12
Figure 4.18
Snapshot at T=0.17 after 7000 time steps
I
Is
16
En
10.9
1
I
I
E
.4
-a
04
-T
8
12
1e
14
x
Figure 4.19
Snapshot
at T-O.17
after
9000
time steps.
is
16
i
110
Chapter Five
Study of Two-Dimensional Bicrystal
111
5.1
Construction of
A bicrystal
relative
=7
is a structure composed of two crystals which exhibit
misorientation
continuous
Grain Boundary.
across
the
and/or
translation
interface
that
L353. If the two crystals of which
identical
materials,
the
the
interface
but
the
material
is
separates one from the other
bicrystal
between
is
composed
are
them is called a grain
boundary.
Our two dimensional model is constructed from
plane (111)
be
in a fcc crystal (see figure 5.1).
obtained
by
Coincidence
Site
Lattice
the
packed
The grain boundary can
(CSL)
construction if a
about the [111]
perfect crystal is rotated an angle
close
direction.
The
relation that determines the value of this angle,
-t
1
where
A
which
'e
-
and
1
_
(
2
__
(S)
_
are the coordinate numbers of the boundary.
=7 grain boundary has coordinate numbers
from
=38.21 .
equation 54 gives a value for
1=7 stems for the fact that 1/7
of
the
l, =2
lattice
and
The
sites
lz=1,
value of
of
the
two
a
=7
crystals are coincident following the rotation.
Figures 5.2 represent the procedure of
grain
boundary on the (213) plane.
structure (fig. 5.2a), we rotate the
[111]
non
direction.
rotated
construction
of
Starting from the perfect crystal
crystal
an
angle
about
the
part
After this process, the rotated crystal overlaps the
(see
figure
5.2b).
The bicrystal with the grain
112
boundary
in the
(213) plane
is obtained
by retaining
on
one
side
of
this plane the atoms of the non rotated crystal, and on the other side
the atoms of the rotated crystal (see fig. 5.2c).
113
[001]
[010]
[100]
-10
Figure 5.1
Closed packed plane (111) in a fcc 3-D crystal.
The closed circles represent atoms on the front sides of
the lattice, and the open circles represent atoms on the
opposite side.
114
'I
it-o
4
o
o
o
III
cx .~~~~~~~~.
.
be rz,4
0
o
o
CM
-,
0)
I
r'I
M
!C~
LL
o
O0
-i
C-4'J
-
r-
0
0)
5-'w
'4--i
o
0.0
*
:
~-W
*~~~~~~~~~~~~H
o °$
0
w
*0~ 0~
JJ
U)~~~~~~~~~~
5 -4 H
*H
J
o
0
0
0
H--
115
I
L.j
4J
U
D~
4
o .DU
0
o
4
1ro
I rt I
,_J
4)
,.4i
1
CD
0
>
W
0O
N
-W
H
o
V4
U
'*0
a)
0
4-J
U
>l
U4
0-4
o
4
03
U)
o
o
0
w
eQ4-
U
o
00
>*
116
Io
*
c4
co
(.)
I C
co
CM
LL.
IX
U
CZ
i
W
0r'--~
U)
-0
0
.
)
o
sk
4o
t.
44U
co
0z
co
o
.,-I
Q
rx-
117
5.2
Thermodynamic Behavior.
It is known
that the thermodynamic
different from the perfect crystal.
a
crystal
affects
its
behavior
of
a
bicrystal
is
In general, any kind of defect in
thermodynamic
properties.
In the case
of a
grain boundary at constant pressure these changes are reflected as
increase
in
the
Furthermore,
[33,34]
potential
experimental
energy,
work
the
[32]
volume
and
some
and
an
the enthalpy.
theoretical
models
suggest that a grain boundary can melt at a lower temperature
than the bulk.
Studies on
dynamics
grain
techniques
boundary
to
study
boundary
to
demonstrate
insight into the
technique
can
the
melting
be
phenomena
using
have not been done in the past.
objectives is
and
melting
used
thermodynamic
that
find
One of our main
properties
of
out
the
grain
in
a
more
ambitious
a
grain
whether
boundary.
there
temperature for the grain boundary different from that
and
of
molecular dynamics can give some
mechanism
to
molecular
is
of
This
a melting
the
bulk,
project, to elucidate more details of the
transition.
Our simulation
[111)
tilt
cell of
boundary
112
(see
particles
fig.
5.3).
previously determined to be stable.
properties
at
different
external pressure was set
represented
This
equal
structure
We calculated
temperatures
to
and
a
the
constant
P=0.4936
2-D
=7,
has
been
thermodynamic
pressure.
which
is
the
The
same
pressure used to calculate the thermodynamic properties of the perfect
crystal
in
the
preceding
chapter.
technique was used for these simulations.
Again,
the
flexible
border
118
4C
30
25
20
is
10
0
-5.5
-15.5
4.5
14.5
24.5
.5
x
Figure 5.3
Simulation
cell with
3 of
its images used
for the bicrystal
study. Regions 1 and 3 correspond to the grain boundary,
regions
2 and 4 correspond
to the perfect
crystal.
119
From our simulations we obtained the internal energy, the
and
the enthalpy
of the system
at different
temperatures
volume
(table 5.1).
From the data of the perfect crystal we were able to calculate surface
excess thermodynamic properties of the grain boundary.
These
results
indicate that the melting temperature for the grain boundary occurs at
approximately
T=0.1
the perfect crystal at
0.14
and
which
is lower than the melting temperature of
T=0.16.
In the range of
temperature
between
0.16 a coexistance of liquid and solid phases appears after
the grain boundary melts. This coexistance permits the system to
for
a
lower
enthalpy
state
and it is observed
system resolidifies into a perfect crystal.
the enthalpy
observe
that
as a function
a
of
the
He
temperature
(fig.
5.4),
we
transition has already occurred around T=0.14.
U,
and
V
(figures 5.5
the
If we plot from table 5.1
transition is more evident on the plots of the
the bicrystal
that subsequently
look
excess
to 5.7).
quantities
can
This
of
120
Table 5.1
T
U/N
Q/N
H/N
U. /N
Q /N
0.110
-0.6955
1.1357
-0.0239
0.0245
0.0217
0.032
0.124
-0.6767
1.1509
0.0171
0.0263
0.0229
0.034
0.140
-0.6499
1.1777
0.0714
0.0341
0.0327
0.047
0.150
-0.6405
1.1853
0.0946
0.0315
0.0303
0.044
0.16
-0o.6304
1.2009
0.1224
0.0296
0.0359
0.044
boundary
and
H /N
The excess quantities are defined as follows:
He=(Hbc-HPc)
Ue = (Ubc
2e= (bC-
UPC )
f2PC)
The subscripts bc and
crystal,respectively.
pc
stand
for
grain
perfect
121
H /N
1
I
iquid line
0.2 liquid
grain boundary
solid
0.0 -
perfect crystal s(olid line
I
0.1
0 .15
Figure
' T
5.4
Enthalpy as a function of the temperature for
both the perfect crystal and the grain boundary.
122
H,/N
0.04 -
x
0.03
w
0.1
T
0.15
Figure 5.5
Excess of enthalpy in the bicrystal with respect to the
perfect crystal.
123
Qe/N
~i
x
I~~~~~~~~
0.03
x
x
0.02 -
I
>_-
·~~~~~~~~~~
0.1
0.15
Figure 5.6
Excess of volume in the bicrystal with respect to the
perfect crystal.
124
U,/N
A
0.03 '
t(~~~~~~~~I
0.0o2
ICC~
~~
T
"
I-
T
t
0 ,1$
0.1
Figure 5.7
Excess of internal
energy in the bicrystal
to the perfect crystal.
with respect
125
5.3
Melting and Structural Stability.
The analysis of melting for the grain boundary is not based
only
on the thermodynamic properties of the system. It is also based on the
analysis
of snapshots of the instantaneous position of the particles.
The mean square displacement, as in the case of the
perfect
crystal,
gives
additional
localization in the system.
information
n this case,
analysis
of
the
regarding
simulation
the
particle
cell
was
divided into 4 regions and the mean square displacement was calculated
for
each
one
(see
grain boundary.
figure
5.3).
Regions
boundary
behaved
structure,
observed
simulation.
during
the
T=O.11
demonstrated
as a solid structure.
maintained its initial
migration
Figure
steps.
of
simulation
migration
[22,23].
temperature
long
the
of
the
5.3
interface
shows
the
was
initial
after
7000
From these two figures it. is evident that grain boundary
migration is already taking
dynamics
that
Although the boundary
configuration of the system, and figure 5.8 is a snapshot
time
to the
Regions 2 and 4 correspond to the bulk.
A simulation at a temperature of
grain
and 3 correspond
T
period
is
of
place
even
at
this
temperature.
The
of grain boundaries has been studied using MD
It is known
that for a system
like
ours
if
the
high enough and if we execute the simulation for a
time,
annihilation
of
grain
boundaries
will
be
observed.
The
stability
temperature
at lower
snapshots
of
behavior
the
system
above
the
melting
the grain boundary is found to be different from that
temperatures.
of
of
the
For example,
system
show
that
at
temperature
of
T=0.15
the
the grain boundary melts and a
126
.a
r-
40
35
30
20
Is
15
10
S
0
-IS.5
-S.
;
14.5
4.5
24.S
x
Figure 5.8
Migration of the grain boundary at T=O.11 after 7000 time
steps and with p=0.4936. The arrow indicates the distance
that the grain boundary has migrated.
127
coexistance of two phases takes place (fig.
is diffusing
process.
through
Our
out
the
simulation
cell
by
5.9).
a
The disorder region
melting-resolidifi cation
cell has two grain boundaries which create
two disordered regions after they mel t.
Once these two regions get in
contact with each other (fig. 5.10), it is possible
reach
its
minimum enthalpy configur ation, which
perfect crystal structure (fig. 5.11) .
for the system
for this case
This observation
is
to
is the
confirmed
by the potential energy, which shows a sharp decrease between 8000 and
9000
time
steps.
If
calculat:e the equilibrium state after the
we
decrease took place, we find out that: this
corresponds
to
the perfect crystal.
value
is
the
simulation
due
to
the
approaching the melting point of the bulk.
In
general,
The
process
difference
is
is
which
The simulation at a temperature
of T=O.16 shows a slightly different behavior from the
0.15.
one
probably
fact
at
that we are
though,
the
the same in the sense that the grain boundary melts first
and at the end the system resolidifies.
At a temperature of 0.15 the potential
equilibrated
first
at
point
I
energy
II which
potential
energy
perfect
calculated
in chapter
At T0.16
initial
state
on
the
a
takes
place
it
states (points b and c in
behavior
(fig. 5.14),
does
in fig. 5.13)
state (point d in fig. 5.13).
process
bicrystal
correspond
crystal
to
curve
a
lower
previously
4 (see fig. 5.12).
however, the system
(point
the
(see figure 5.13). Then, it dropped
directly from this point to point
point
of
Instead,
not
to the
go
directly
from
lower potential
before
the
the
energy
resolidification
spends some time at higher potential energy
fig.
5.13).
the instantaneous
From
the
structure
potential
of the system
energy
(fig.
128
O'
40
O~
a25
'
S
0
w
-IS-5
-5.5
.S
14.S
Figure
24. S
5.9
Snapshot indicating the increase of disorder in the grain
after 3000 time steps.
boundary region at T=0.15
129
... ,i. _........
... _ ,...
Cob
~ ~ ~~~
_,.
. . ....
_
_
,,
.
,
...
_
.
_
..
.
I
-1----
.
45
40
3S
30
Z5
20
10
5
-18
7
-3
-13
-8
2
27
17
12
B
32
x
Figure 5.10
Snapshot indicating a highly disordered structure at the
'
time step 7000 for T=0.15 and p=O.4936.
130
0
.0
30
y
2,0
10
0'
-e-S
9.5
24.5
Figure
5.11
Resolidification to perfect crystal after 10000 time steps
at T=O. 15.
131
-7
-74
-7E
P
-78
4
2000
400e
6000
7000
See
t00
10000
time tep
Figure 5.12
Potential energy behavior at T=0.15 and p-=0.L4936
where the
resolidification process is observed.
132
Figure 5.13
Scketch of how the resolidification process may take place
from the data calculated.
133
-06
-68
-764
-?S
-70
-84
-76
-72
-Be
-82
-84
- P r5
300 0
1000
L3
2000
900
7000
5c00
4000
6000
8000
time top
Figure 5.14
Potential behavior at T=0.16 and p=0.4 93 6
i0000
134
i
T
4f
*3, .
3S-
30
20+
S
15
10
At
-- "
I
L_-
l _==
-1-I
4'
4
- 4
-4
-g9
14
:I
¢
6
16
II
4
26
21
!1
1
Figure 5.15
Snapshot of a high disordered structure at T=O.16 after
5000 time steps.
135
5.15), and from what we know from the previous
that
one
of
the
states
(point
interesting that the orientation
obtained
in
the
T0.15
c)
of
is
the
chapter,
the
liquid
perfect
to
conclude
phase.
crystal
corresponds
simulation
we
It is
which
we
neither of the
starting orientations of our bicrystal (see figures 5.3 and 5.11).
Details of the mechanism of
determine
from
the
limited
resolidification
data
we
results obtained are not enough to obtain
this
process,
speculative.
and
all
the
have
a
are
difficult
collected to date.
concrete
interpretation
is
conclusion
just
to
The
of
admittedly
136
Chapter Six
Conclusions and Discussion
137
Conclusions and Discussion.
One of the main conclusions from our work is that
able
to
1t from those obtained by
been
The agreement of our results is within
onte-Carlo simulation.
The flexible border technique has been
study
melting
properties of
observed
to
at
a
constant
56
particle
pressure
system.
successfully
through
The
fixed
the
applied
number
thermodynamic
dependence
border
results
calculated
were
able
to
at
we
observe
3.
By
different
follow the sequence of transition from the
initial state to the equilibrium state of the simulation.
snapshots
was
using
technique, which in general was less than
looking at the instantaneous positions of the particles
we
to
be insignificant while simulating a 400 particle system.
No significant difference was found in the
times,
have
calculate the thermodynamic properties of a 2-D system using
the flexible border technique.
the
we
an
From
these
increase of the number of dislocations and
disclinations when the system goes from. a ordered state (solid)
to
a
less ordered state (liquid).
The structural transition on a 2-D system was
technique of the flexible borders.
lattice
dynamics
analysis
with
studied
with
the
It was possible to corroborate the
out simulation results; in addition
using molecular dynamics it was possible to
study
the
mechanism
of
transition and the final state after the transition.
The flexible borders technique was found to be
for
this
kind
of
study
appropriate
since the transition from a perfect square
crystal to a perfect triangular crystal is possible.
in
more
By
comparison,
the fixed border technique the constant volume and shape represent
138
constraints that will impede such a
fixed
border
triangular,
technique
it
can
the
only
transition.
system
reach
a
will
go
state
Although
from
which
the
with
the
square
corresponds
to
to
a
triangular lattice with defects.
In our attempt to study grain boundary melting using MD, we found
a melting
temperature
melting
temperature
for the interface
which
of the perfect crystal.
is
about
85%
of
Once the grain boundary
melted, we observed a transient coexistance of disordered and
regions.
The
disordered
region
melting-resolidification process.
perfect
dependent
it
A
ordered
migrate and gives rise to a
resolidification
process
to
a
crystal from the transient state of coexistance was observed.
But it has not been determinated
if
could
the
is
preliminary
so far
whether
this
phenomenon
is
on the particular grain boundary system we have studied, or
a
general
results
characteristic
we
speculate
of
that
a
bicrystal.
the
the melting mechanism for a
bicrystal could be significantly different from what is
perfect crystal.
From
known
for
a
139
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