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 Engineeringin Partial Fulfillment of the

Requirements of the

Degree of

MASTER OF SCIENCE

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

August 1982

Q Massachusetts Institute of Technology 1982

Signature of AuthorDepartment f Nuclear Engineering, August 19, 1982.

Certified by

Certified by

Accepted by

Thesis Supervisor

j -7- X10V T.-A Vr

'/rhesis SupervisorZ- ~~ tChairman, Department Committee on Graduate Students

ArchivesI.,SSACHUSETTS INSTITUTE

OF TECHNOLOGY

JaMis 1 1 i " . ,

I tnn A nI

MOLECULAR DYNAMICS SIMULATION STUDY OF STRUCTURAL STABILITYAND MELTING OF TWO-DIMENSIONAL CRYSTALS

by

Francisco Javier Carrion

Submitted to the Department of Nuclear Engineeringon August 19, 1982 in partial fulfillment of the

requirements for the Degree of Master of Science in

Nuclear Engineering.

AbstractA computer code for molecular dynamics simulation which

incorporates the recently developed technique of flexible periodicborders has been developed and used to study melting transitions andstructural stability of two-dimensional crystals with Lennard-Jonesinteratomic potential.

The structural transition of a perfect square lattice to aperfect triangular lattice in the case of 36-particle system has beenobserved to take place at various pressures with apparently little orno potential barrier. The simulation results, which confirm thestability analysis based on lattice dynamics, revealed clearly thatthe transition involved a shear mechanism and the stable lattice has a

higher density. Simulations using fixed borders showed also a squareto triangular 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 themelting temperature was determined to be approximately 0.165 inreduced units. Changes in internal energy, density, and enthalpyacross the transition obtained from the simulation runs are found tobe in good agreement with recent Monte Carlo calculations. It isfound that within the present accuracy flexible and fixed borders givethe same results, and that based on a simulation run using 400particles system size effect is not significant.

The crystalline order at elevated temperatures of a bicrystalwith two =7 coincidence tilt grain boundaries has been investigatedusing a system of 112 particles with two boundary periods. Theboundaries were observed to migrate and undergo a melting transition.From the variation of excess enthalpy with temperature the reducedmelting temperature was determined to be about 0.14 at a reducedpressure of 0.494 which is about 85% of the melting point of the

perfect lattice. These results are the first computer simulation dataproviding quantitative evidence that interfacial melting is distinctfrom normal crystal melting.

Thesis Supervisor: Sidney YipTitle: Professor of Nuclear Engineering.

Thesis Supervisor: Gretchen KalonjiTitle: Northon Professor of Material Processing.

3

Acknowledgements

My gratitude to Professor Sidney Yip who made possible this project.

His patience, advice and encouragement during the past two years are

fully appreciated. In particular, I want to thank him for his time,

corrections and comments during the writing of the thesis.

I also want to thank Professor Kalonji for her contribution, comments

and corrections 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 only for his support and

advice, but also for his frendship. The comments and observations of

Dr. R. Harrison are also appreciated. To all my friends at MIT,

specially to Ricardo and Pepe. Finally I want to thank the National

Council of Science and Technology of Mexico (CONACYT) for the

economical support during these past two years.

Funds for the simulation runs were provided by the Army Research

Office, Contract No. DAAG-29-78-C-0006.

4

To my parents

Blanca and Juan

5

List of Figures

No.

Page.

2.1 Simulation cell and its 8 images 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 cell according to the 27

vectors a and b.

2.5 Shape of the simulation cell to which the flexible 34

border technique is restricted.

2.6 Example of simulation cell in which the shear terms 35do not balance when using the flexible border technique.

2.7 Pair distribution function g(r) for the square lattice 39

and triangular lattice at the same density.

2.8 Asymptotic behavior of the orientational correlation 41function for the liquid (L) and the solid (S).

2.9 Typical behavior of the mean square displacement as a 45function of time in a solid phase.

2.10 General behavior of the mean square displacement as a 46function of time in a liquid phase.

3.1 Square lattice in two dimensions. 50

3.2 Functional behavior of the eigenvalues of the dynamical 52

matrix as a function of the lattice constant in a

square lattice with nearest neighbors interaction.

3.3 Triangular lattice in two dimensions. 53

3.4 Functional behavior of the eigenvalues of the dynamical 55matrix as a function of the lattice constant in a

triangular lattice with nearest neignbors interaction.

3.5 Scketch of the five regions found with the lattice 57dynamics calculation for the square and triangularlattices.

3.6 Temperature behavior in the transition of the square 59lattice to the triangular. The simulation started inregion and was induced by thermal perturbation.

6

3.7 Internal energy as a function of time during the 60transition from region 1 and with thermal perturbation.

3.8 Volume as a function of time during the transition 61from region 1 and with thermal perturbation.

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 64structural transition analysis.

3.11 Pair distribution function after 120 time steps 65

3.12 Pair distribution function after 200 time steps 663.13 Average of the pair distribution function g(r) 67

over the last 3000 time steps of the simulationafter the transition took place.

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 69time steps.

3.16 Instantaneous position of the particles after 120 70time steps.

3.17 Instantaneous position of the particles after 240 71time steps.

3.18 Instantaneous position of the particles after 360 72time steps.

3.19 Final structure after the transition took place and the 73system equilibrated.

3.20 Final structure after a transition from square lattice 75in. which the system was perturbed by shear.

3.21 Temperature behavior in the transition induced by shear 76

3.22 Potential Energy-behavior in the transition induced by 77shear.

3.23 Volume behavior in the transition induced by shear. 78

3.24 Behavior.of the temperature in a simulation where the 79initial square structure is in the stable region andthe system was perturbed by heat.

3.25 Behavior of the temperature in a simulation where the 80initial square structure is in the stable region and

7

the system was perturbed by shear deformation.

3.26 Final structure in a transition simualted with fixed 82borders.

4.1 Temperature behavior in a flexible border simulation 86with rescaling the first 500 time steps.

4.2 Potential energy behavior at =0.5820 and T0.137. 87

4.3 Volume behavior at =0.5820 and T0.137. 88

4.4 Enthalpy behavior at p-0.5820 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 as a function of T at =0.4936. 94

4.8 Enthalpy as a function of T at P=0.4936. 95

4.9 Internal energy as a function of T at =O.4936. 97

4.10 Density as a function of T at p=0.4936. 98

4.11 Temperature behavior in a simulation at P=0.4936 101and where rescaling was done up to 500 time steps,and then the system continued at constant enthalpy.

4.12 Snapshot of the MD cell and 3 of its images after 1500 102

time steps.

4.13 Snapshot of the MD cell and 3 of its images after 3000 103time steps.

4.14 Potential energy behavior for =0.4936 and T=0.17 with 104rescaling through all the simulation and with an initialperfect crystal structure.

4.15 Root mean square displacement for P=0.4936, T=0.17 105and an initial perfect crystal structure.

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 packed plane (111) in a fcc 3-D crystal. 113

8

5.2 Construction of the =7 2-D grain boundary. 114

5.3 Simulation cell with 3 of its images used for the 118

bicrystal study.

5.4 Enthalpy as a function of the temperature for 121

both the perfect crystal and the bicrystal.

5.5 Excess of enthalpy in the bicrystal with respect 122to the perfect crystal.

5.6 Excess of volume in the bicrystal with respect 123to the perfect crystal.

5.7 Excess of internal energy in the bicrystal with 124respect to the perfect crystal.

5.8 Migration of the grain boundary at T0.11 after 7000 126time steps and with p=0.4936.

5.9 Snapshot indicating the increase of disorder in the 128grain boundary region at T0.15.

5.10 Snapshot indicating a highly disordered structure at 129T=0.15 and =0.4936.

5.11 Resolidification to perfect crystal after 10000 time 130steps at T=0.15.sp

5.12 Potential energy behavior at T=O.15 and P=0.4936 131where the resolidification process is observed.

5.13 Scketch of how the resolidification process may take 132place from the datacalculated.

5.14 Potential energy behavior at T=0.16. 133

5.15 Snapshot of a high disordered structure at T=0.16 134after 5000 time steps.

9

Table of Contents

Page

ABSTRACT

ACKNOLEDGEMENTS

LIST OF FIGURES

TABLE OF CONTENTS

CHAPTER ONE. INTRODUCTION

CHAPTER TWO. MOLECULAR DYNAMICS SIMULATIONWITH FLEXIBLE BORDERS.

2.1 Brief Review of MD technique withfixed periodic borders

2.2 Lagrangian Formulation2.3 Calculation of thermodynamic and

and structural properties.2.4 Calculation of Dynamical Properties

CHAPTER THREE. STABILITY OF TWO-DIMENSIONAL LENNARD-JONES

CRYSTALS3.1 Lattice Dynamics Analysis3.2 MD Results

CHAPTER FOUR. MELTING AND PRE-MELTING STRUCTURAL DEFECTS

4.1 Thermodynamic Behavior

4.2 Onset of Structural Defects

CHAPTER FIVE. STUDY OF TWO-DIMENSIONAL BICRYSTAL

5.1 Construction of a =7 Grain Boundary

Model5.2 Thermodynamic Behavior5.3 Melting and Structural Stability

CHAPTER SIX DISCUSSIONS AND CONCLUSIONS

REFERENCES

2

9

10

15

16

26

36

42

47

4856

83

84

99

110

111

117125

137

139

10

Chapter One

Introduction

11

INTRODUCTION

Computer Molecular Dynamics (MD) is a well known technique that

has been used to study thermodynamic, structural and vibrational

properties of a system composed of a finite number of atoms. The main

information that is obtained from this calculation is the discrete

trajectory that each atom follows during the simulation. These

trajectories depend on the interatomic potential that has been

selected to describe the interaction between the atoms i.n the system.

This technique has the advantage that it can be applied to study

phenomena under 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. Some of these are the

diffusion mechanism 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 diffusional phase

transitions [6], calculation of phase diagrams [7,8], solidification,

nucleation [9] and melting phenomena [10-15], among others.

The traditional MD Technique is a formulation that simulates a

system under constant density or volume [3). Isobaric calculations

are possible by using special techniques that readjust the volume of

the system until the desired pressure is obtained. In general 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 or contract

according to the temperature and the difference between the internal

12

and external pressures. In this new technique the average of the

internal pressure is constant and equal to the external pressure.

A. Rahman and M. Parrinello have extended this technique and

applied it to study phase transitions under high pressure [6]. Some

preliminary studies of structural transitions in 2-D from a square

lattice to a triangular lattice have been also done by K. Touqan [5].

In the present work we study in more detail the structural

transition in 2-D of the square lattice to a triangular lattice using

both fixed and flexible border simulation techniques. The square

structure in 2-D has a higher potential energy than the triangular

structure. For that reason it was believed that the square lattice

was unstable and always will try to change to a triangular one.

Despite this higher potential energy, lattice dynamics calculations

predict a small range of density for the square lattice in which the

two frequency modes are real, that is to say, the structure is stable.

Although lattice dynamics can predict the stability of a structure, it

can not tell anything of the kind of transition that will take place.

At this point, MD is a more powerful tool, because it not only

confirms what lattice dynamics predicts, but also it can show what

kind of transition occurs and which mechanism the system follows. It

is also found that when the system can change its volume by using the

flexible border method,-the transition from a perfect square lattice

to a perfect triangular lattice takes place easily. However, when

using the fixed border technique the restriction of constant volume

prevents the system from accomplishing a perfect transition and it

will end in a triangular lattice with structural defects.

Melting is a very interesting phenomenon which can be studied at

13

constant pressure by using this flexible border technique. It is

known that melting in 3-D is a first order transition. Experiments

and studies 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 currently

studied very extensively. One of the models suggested by Kosterlitz

and Thouless [17,18] states that the creation of dislocations could be

the vehicle for the destruction of the lattice structure. On the

other hand, Halperin and Nelson [19,20) predict that an anisotropic

fluid phase called the hexatic phase occurs during the melting

transition. This second 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 simulations 8,14) there have

been some indications of the transition's mechanism, but yet, no

definitive conclusion has emerged.

In the present study the thermodynamic behavior of the (N,P,H)

ensemble is investigated using the flexible borders technique,

including the analysis of the formation of structural defects in the

premelted region, and the melting phenomenon itself. It is found that

the thermodynamic properties, the heat of fusion and the melting

temperature are in good agreement with previous results [8,14,21],

indicating that this new 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 have been

done in the past few years. Some of this work is concerned with the

14

dynamical behavior and migration of grain boundaries at high

temperatures [1,22,23]. Other studies concentrate on the vibrational

properties and the frequency modes at the boundary [24,25]. Despite

this, not enough work with this kind of structure has been done to

understand all its properties.

A study of a grain boundary =7 is done with the flexible

border technique. It is of interest to study the stability of the

grain boundary at high temperatures, as well as the determination of

the melting temperature, heat of fusion and other thermodynamic

properties. 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 and a

Lennard-Jones 6-12 pair potential. The crystals are composed of

identical particles and the range of interaction is carried up to

third neighbors.

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 cases consider a classical

system of N identical particles which obey Newton's equations of

motion,

m - = -ba R, )Lzi)..., N CI)

where (r,,...,rN) is the total potential energy of the system, which

in the case of central, conservative, pair potentials, can be written

in the following form:

@'jk. Orj- hkk (2)

For this case, if we substitute equation (2) into (1), the

resultant equation of motion is

dt -- ; r-i 3)tr., I

t m. . n 6)j,.rj

where 'Pj(rij) is a function of the magnitude of

vector r = - .

in this work we study rare gas solids which

the Lennard-Jones 6-12 potential [26]

the pair separation

can be described by

0 ~ Ij..)i b) =

17

[(2cDi(r'i)= L-

where E and a are the potential parameters.

Once we have determined the potential, we get a set of N-coupled

second order differential equations which can be solved with any of

the well known techniques used to solve these equations. The two most

common are predictor-corrector and finite difference methods. The

first technique is more accurate and is suitable for long time

simulations because it allows a larger time step to be used compared

to the finite difference method. The main 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 to be sufficiently

accurate if a small time step is used. Because of its simplicity it is

used in this work. For this case, the equation of motion (3) is

written as follows:

at~~~~~~~~~~~~~.

In this equation to calculate the subsequent time steps one only needs

two initial conditions which can be either r,(t-At) and ij (t), or

the condition ri(t-At) and vj(t-At). The second pair is a possible

18

choice because the position of the particle at the next time step can

be calculated from its velocity with the relation

(t) d B;(t-At) * 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 which the

simulation is going to be performed. The reason for this is because

almost half of the initial kinetic energy is- going to be transformed

to potential energy. We know that for an harmonic system the

equipartition theorem predicts that half of the energy we put into it

will be kinetic energy and the other half will be potential energy.

In our case we do not have an harmonic system but the anharmonicity

effect is 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 at twice the

temperature desired because the system may melt during equilibration.

For 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 by the

finite difference method, we get a set of N equations which can be

solved numerically. It is most convenient to express the variables in

dimensionless forms. For the Lennard-Jones 6-12 potential and 2-D

19

system, we use the following dimensionless units:

a) Distance ri* r//

b) Time t = /T

c) Energy E = E/4t

F'- F r/4 Ed) Force

e) Pressure

where

gases

units

p'= p /

T = 4 -M is called the characteristic

it is of the order of 1014 seconds.

we get the following equations:

time and for the rare

Using these dimensionless

IJ (r,) =

i (2.)

Ij 'i~,

... (", r .( --. [ t< ) , <'aY A t

note that in this case Q.; ( 4 E)= z Fy ( rj* ( ))j-4

(7

¢::)

20

In this case the kinetic energy of a particle is calculated from

where the dimensionless velocity for a particle is

V i - or a A (

In Molecular Dynamics simulation we are limited by the computer

size and we only can represent our system by a finite number of

particles, which for the best of the cases can go up to a few thousand

particles. To avoid surface effects in a small system it is

conventional to use the periodic border condition. The MD cell is

repeated periodically in all directions producing in 2-D 8 images and

a total of 9 simulation cells (see figure 2.1). Note that in

simulations using fixed borders the volume or density of the cell does

not change during the simulation, nor the vectors which describe the

cell and its images.

When we calculate the total force acting on a given particle, we

should include all the other particles in the simulation volume and

their images. 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. This

range is generally chosen to be between second and third neighbors.

In the cases of small simulation systems we have to be sure that a

Cell

Image

21

y

Cell

Image

.L~- ,-7 _ . ....... ..

S imulat ion.. ....-

=-Cl .g . .. ......17..-..

Cell

Image

Cell

Image

Cell

Image

Cell

Image

X

Figure 2.1

Simulation Cell and its 8 images in a two-dimensinal System

with periodic border condition.

Cell

Image

CellImage

[

I

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 calculation

of the forces on particles due to their neighbors. To make the

simulation as efficient as possible, it is useful to create the

neighbor table. This is a bookeeping device that keeps track of the

neighbors and their location of every particle in the system. As a

consequence of the movement of the particles we can have some of them

going in 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 of these

techniques updates the neighbor table after a certain number of time

steps [27]. Another method which is more precise [22] defines a

cut-off range RC , which is larger than the range of interaction. The

neighbor table is constructed up to this range, but the calculation of

the forces is still calculated up to the range of interaction. By

doing this, we 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 = (RC-RI) from a reference position we

know that updating becomes necessary. With this second method, the

choice of RC is very important because if it is very large we will

have a lot of neighbors per particle. On the other hand if it is close

to RI, then is very small and updating will be more frequent.

23

· · ·

* *

* 0 e@0

0 0* 0

00000l l

000· * ·* * ·

0 0 **

00* * * * * i

k· ·*0 0 0

0 0· 000*00 00

* 0 00· · ·

o * 9

* * *

o * *0 0

000000 E000 000·

000 000 000 000 000

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.

At time t particle j is out of the cut-off range of i.n

After an interval of time t, particles i and jhave moved

towards each other a distance such that j is now a

neighbor of i.

25

TABLE 1

Triangular Lattice

Number Accumulative NeighborNeighbors of number of distance

neighbors neighbors (RC/a)

1 6 6 1.02 6 12 r3=1.732

3 6 18 2.04 12 30 r7=2.6455 6 36 3.06 6 42 2r=3.4647 12 54 T3=3.6058 6 60 4.09 12 72 1-9=4.358

10 12 84 21~=4.58211 6 90 5.0

TABLE 2

Square Lattice

Number Accumulative NeighborNeighbors of number of distance

neighbors neighbors (RC/a)

1 4 4 1.02 4 8 /=1.4143 4 12 2.04 8 20 V=2.2365 4 24 2V\'=2.8286 4 28 3.07 8 36 iVT=3.1628 8 44 VX=3.6059 4 48 4.0

10 8 56 VT=4.12311 4 60 3V2=4.24212 8 68 20=4.47213 12 80 5.0

26

2.2 Lagrangian Formulation.

The idea of the (N,P,H) ensemble by H. C. Anderson [16) is the

result of the introduction of another degree of freedom to the (N,V,E)

ensemble. This degree of freedom allows the shape and the size of the

MD cell to change. These changes are determined by the difference

between the internal and external pressures. With this technique

isobaric processes can be studied without the artificial adjustment of

the volume that has to be done if the fixed border technique is used.

The most important feature of this method is that the system itself

determines the shape and volume it takes according to the given

temperature, pressure and structure. One of the first applications of

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 as

the position (x,y) of any particle can be written in terms of relative

coordinates (Eog) in the following way:

27

Y

A

Figure 2.4

Definition of the simulation cell according to the vectors-b

a and b.

28

r =~I- ¼ -s ~

where

I f i

and from equations (9-11)

E - £; - b

C1

range

square

( o)

(la)

early the values of the relative coordinates are numbers in the

O< ,q <1 . Considering two atoms i and j in the cell, the

of the distance between i and j is given by

.2 =Ii

s- GSU s..

where

=- T ( L)

( I 3)

29

With this transformation the number of dynamical variables is

2N+4, and the Lagrangian which gives the equations of motion [6 is

In this equation the first term represents the total 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 p is the hydrostatic pressure. W, which has dimensions

of mass, represents the inertia of the borders. This parameter

determines 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 = ~ uR E 0 ) (Si- SjB -G ; At; (17)

30

and

W k -M (5_

where the reciprocal

-1I(T Z2 SI

(18I

lattice is represented by the matrix

(I )

Note that for this case the velocity of the particles is

calculated from the equation

(zO)V; = < _I,

Here, the term h has been neglected in

contribution is small compared with

simplification is necessary in order to

resultant equations and make this technique

The internal pressure in this case

virial theorem,

virial theorem,

the assumption that its

the hs term. This

reduce the form of the

accessible.

can be calculated from the

(z2.)-a 17 = E m K \ Y .E E (. J)

P ; g

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 (17) and (18) are usually

the predictor-corrector or finite difference method. While using the

second method we find that in order to solve equation (17) we need h

, but when we want to solve the equation for h (eq. 18), we also

need s which we do not have. For that reason, we can not use a

central difference scheme in both equations. nstead, we use a

semi-implicit method in which we first calculate

s5; ( - s; (¢t., )At

Then the matrix is calculated to solve

we (,1) I k (4n) (-l) + I (t ) t

where

I (t _ W ( T - U

( 3)

(2 )

(zs)

.i_S.- (� -) =

32

..

Finally from equation (17) we solve for si(tn) to get i (t,+1)

from

This technique has several characteristics which have to be taken

into consideration. First, there are oscillations of the borders of

the MD cell about the equilibrium position. The amplitude of these

oscillations depends on the initial conditions, the temperature and

the size of the system. It has been observed that these oscillations

are not damped during the simulation, and thus the amplitudes of the

fluctuations in the thermodynamic properties (temperature, internal

energy, volume and internal pressure) are larger than the ones in a

fixed border simulation. 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. This decrease is due to an indirect coupling of equations

(24) and (26). At the same time, when the damping effect was turned

off, the oscillations of the border increased to almost the same

amplitude that it had before the damping started. Therefore, if

artificial damping to the borders is desired, it is suggested to

implement it with continuous rescaling.

33

A second characteristic of this technique is that it is

to MD cells with rectangular shape or with symmetry along the

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 by the following vectors,

( = b:

Suppose we are at T=O°K , p=O and we have

lattice. Then, from equations (19) and (21) we

diagonal matrix.

TF

TF =

O

and

a perfect triangular

find that will be a

0

(D)

-b ( < 2.~\ . K IX

(27)

(z 8)2

Then, from equations (18) and (27) we get a non diagonal matrix (eq.

29), where 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.

limited

x=y line

the a

-& CL'ACL =.

0

q =.fj

34

y

a

Figure 2.5 (a)

y

x=y

b

MD

a

X

Figure 2.5 (b)

Shapes of the simulation celltechnique is restricted.

to which the flexible border

b

MD Cell

- - - S b

35

Y

a

Figure 2.6

Example of simulation cell in which the shear terms do not

balance when using the flexible border technique.

y

36

2.3 Calculation of Thermodynamic and Structural Properties.

In a Molecular Dynamics Simulation the properties that are

calculated every time step are the positions of the particles and the

density of the MD cell. From this data it is possible to calculate

the thermodynamic and structural properties of the system. The

temperature which is identified with the time averaged mean 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~~> |j( j)2 ½ ;) l1> C Z

From the virial equation [28],

K E1 Fee ,> - pJ| AS - -aE) ('3)~~~~~S

37

An expression for the internal pressure in 2-D can be derived from

equation (33), which after integration becomes

In this equation the first term on the right hand side is called the

kinetic potential. The second term is the potential contribution due

to interatomic interactions.

In all these relations, the brackets < > refer to the average

over a finite interval of time. This average is assumed to be equal

to the appropriate ensemble average.

Other properties of the system can be calculated from these basic

thermodynamic quantities, such as the thermal expansion coefficient

a~p _- () , the specific heat C ; p ( ) thea P~~~~~~~~~~~

isothermal bulk modulus ?-V K'v)T etc.

To describe the structural characteristics of a system two basic

quantities are used: the pair distribution function g(r), and the

orientational correlation function g (r). These properties are

referred directly to the structure of the lattice itself and by

looking at them one can determine the phase and structure of the

system 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

(r) =

here n(r) is the time average number of particles situated at a

distance in the range r(Ar/2) from a given particle. Thus, g(r)

can be interpreted as the density in this ring divided by the total

density. Figures 2.7(a) and 2.7(b) represent the form of 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 in the cell as a function of the

distance at a given temperature. This function is defined as follows:

(r) =

where

I ) (Yr- r)....... I

C

I I,

eN?

(36)

Here j labels the 6 nearest neighbors for a given

angle eih is the angle between some fixed axis

particles i and j.

The importance of this function is that when

particle i and the

and the line joining

the system is in the

-CI n (Y.).2T-I r A�W

r) j* o

Y, (r) 77-

.,L,.~~-i--,, ''--"--c--i-- .......... c-- -.: ' : --- l--r - ~ --·- ~~.--':;_7_-: ~-- ;-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...__..~. _-::..... :.--,- ·- ·- ··--------- ;::':-_v :.:.~:'. :._.:" ....:?:2 ----":l---.-_,_ -_---?----i- _ :. '·- - -: :'_ r-: ----+-=~-Z.-s:--s--:--:r~-::::-:'. '.'L.---.....·--··--·- :-:'z;::---?----- ·...... - .... j;T--~'' ; -i:r---- : -~ ; ! i ;-; ?·- . : :--'- ' : : . : :".;··- --·

~_ -----:-,-; ·-~: -- i -- '--�- --t: , --:z':--- -' --------%~~-i- ']---' :_-[-- 1. -· L- Li- .':'4 :;:-........ ------TI..~ ............... ~ ' : -·---~~ --·'t - I-- ... ....~ .... " '......

-:-::-·' "-:Z ' -.'--'.": ::"-."E_--' --. :_ .Z~ "~-:' !"~Y--~---__ - --z ,..-_-.:-="'E::E-E:-.%-':k:iL':.i-:...:1_~ .... t---~--- .. . . . .- , "t.._ I.-....-. .,-.-. ...̂. ... .. ....----- - .- ," ._:'-. rt: .--~ '-~, __.--- --~.';- i . . ---i-- ··_ ,,-_;:- :::::::::::::.--:.:r..... · .. -----r...:_ .... -- ;--'; --'l-----, ...... ;.... :..... i ........ ...-- :- .....-

-'-- ...... "-::'-......~ .................T"+~'--- ,' : .....-- - ~ !! . . ..... e ..........................-··-·------~ ------ ---- ·- - ·-- :',- -- ....,.-.-....-..- -..-- .... .......

' -=:'* .....:....._..: :: ;'c.:*'-;-- _ ~', ~.... ' .-.......-.. ~....,... . ... ..._:.:.--.. ~~ .I-I+ .·.·-- · ·- · cccl-- ...-.. ~.. ~.---------~. ·

I

. i -. .-

3i:.:

- - -.. - -. - - ~ 7 - ~ _: -4

......... o . . . . . . . . . . . . . . . . . . .... . ......; .............. ........................

.............. -- 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

............... iT

..I~

I i ,

i =-: -:i, -i�--·----- ;------_1-_: r--_~:_-~ _ _- _ _ '_- f : I- - - ----- · r ··--------- r- - ------ ----- ·-·-- ·- ·- · · ·----- ·-·- · ·- -- -·- ·

I L A ~-L--- --

40

solid phase, the behavior of g 6 (r) is significantly different from

the one when it is in liquid phase. This characteristic makes this

correlation function more appropriate for melting and pre-melting

studies since the pair distribution function g(r) does not show this

kind of difference when changing from one phase to the other.

41

96(r)

0.5'

U.w -

Figure 2.8

Asymtotic behavior of the orientational correlation functionfor the liquid (L) and the solid (S).

A A

A

42

1.5 Calculation of Dynamic Properties.

The dynamic properties which are considered in a simulation are

the velocity autocorrelation function, the dynamic structure factor

and the mean square displacement.

The velocity autocorrelation function is defined as [3]

N

Y () < l 7.Vio

I V (O)

where vi(t) is the velocity of the ith particle at time t. Here,

this function is normalized to unity at time t=O and it decays

asymptotically to zero. The spectrum of phonon frequencies, P(w) can

be calculated by a Fourier transform of the velocity autocorrelation

function, i.e.,

o0p oU) 2. 9 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 body system 30).

Since this function can be measured by neutron scattering and laser

scattering, it has become of great interest in order to compare

experimental results with MD results. 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).

ae t-oO

- co

(c0 9A(c)

and p (t) is the density operator defined as:

The mean square displacement or the classical width function is

defined as:

This quantity is important because it tells us how much a given

particle in the system is moving.

(3q)

where

(40)

(41)

S(i� I ( t)

=. IN

-I (- J-

L -t

Forth caeo th soi te

44

motion of the particles is restricted to a certain range (see figure

2.9). For the case of liquids, we have particles diffusing so that

< 2 > grows linearly in time at long times (see figure 2.10).

45

Mean quare dp.^ IN_

1000 3000 5000 7000 90000 2000 4000e 6000 8000 10000

time tep

Figure 2.9

Typical behavior of the mean square displacement as a function

of time in a solid phase.

, C "

f

0.2

-0.1B

0.16

0.142

0.12

0.1

0.08

0.06

0.04

0 2

46

2 .25i

a2

1.75

1.5

1.2S

2 1u

0.75

e .s

0.25

e1000 3000 5000 7000 90000 2000 4000 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 means to

study the vibrational properties of the atoms or molecules in a

crystal. The fundamental assumption in a lattice dynamic calculation

are the adiabatic and harmonic approximations [31]. Generally

speaking the results are valid only for an infinite perfect crystal at

very low temperatures.

We consider a lattice dynamics model with a central pair

potential Vkk'(r) which is a function of the distance between the

interacting particles. For this case, the force constants for a pair

of atoms (K K) are

0{ (\\'} 0am' ()5k4<nol XrS j14 r)0 (3)

where the separation vector and its magnitude are

*r- 1f(k iA') \

and V'(r), V"(r) are the first and second derivatives of the

potential respectively.

The frequency modes and the wave vectors corresponding to these

modes are the eigenvalues and eigenvectors of the dynamical matrix.

This matrix by definition is written as

49

li'4)- )=a k i,) Q -(k)

where is the wave vector and M the mass of the particle.

To calculate this modes one has to solve the secular equation

(YS)

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:

D =

To

I,- Cols)

2 0A< I

(q(6)

v (v) _ I . . t I l -ck1=. r 2. IL v- 7 y.

In figure 3.1, we can see that for a square lattice there is

structural symmetry along the x and y axis. This symmetry is

where

( 8)

-) (-- - Li'I =

50

y

A

2

as

0

4

Figure 3.1

Square lattice in two dimensions.the nearest neighbors of 0.

Particles 1 to 4 are

13

II

iVIIIIII

-0~q

Al~~~~~~~~~~w

�CU:P -II: He Bye b

ANI&- M14M

%WI

I

I

I

I

I

I

I

I

I

I

I

I

I

I-----

51

responsible

obtain the

eigenvalues

that when we calculate the dynamical matrix (eq. 44) we

non diagonal terms equal to zero. In this case, the

are easily calculated from:

2.~~~~7 Z Z4 V 1 -Q+I'-kWO,, = Y, 1= [ (Z r l r

WL =r- Z( r - I

(5a)

Figure 3.2 is the plot of the eigenvalues wo as a function of

the interatomic distance a . In this case there are found three

regions which are defined by r,=1.112 and rL=1.244, which are the

values of a at which one of the frequencies of the modes is zero.

In region 1, 0 r r , the transvers modes is real and the

longitudinal mode is imaginary. The lattice is therefore unstable

because any excitation of the second mode will cause the oscillations

to grow in amplitude.

both modes are real and the

r -r , we find that the s)

is the transverse mode whil

If the same calculati

neighbors, we get the sarme

values for r, and r. I

that as the interaction r

(region 2) decreases. X

On the other hand, in region 2, r, r r ,

erefore the lattice is stable. In region 3

,stem is unstable but now the unbounded mode

e the longitudinal mode is real.

on is done taking into account second

general results but with slightly different

n this case r,=1.19417 and r=1. 2 35 14 , so

*ange is increased, the stability range

/hen the interaction range is taken to be

essentially infinite, we believe the stable region will

When the same analysis that is carried out for the

vanish.

square lattice

52

-L43a

-I0

-J.

-2

-3

-4-1

A 1. 1.4 1.6 1.8

Figure 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.

I

53

y

I

/3 \

!

',4 //

//

x\5

'__

at

*1

/2 '

! //

!/

/t

x1WI /

/

\ 6/

Figure 3.3

Triangular structure showing the six nearest neighbors

(particules 1 to 6) of particle 0 in the center.

I __

- - - - ~- R-

I I

W\ 0

54

is done for

interaction

D ()

the triangular lattice (figure 3.3), with a range of

up to first neighbors, the dynamical matrix is

7 3v -V to \.3V i C

(51)V/* +

It'1 I &O4 -

In this case again the symmetry reduces the calculation and the

eigenvalues are easily obtained from

dz - -- 138

)2.

'o (')'

D1 ° ( rY

For this lattice, the frequency modes or eigenvalues vanish at

the points r, =1.249 and r71.307 defining regions 4 and 5 (fig.

3.4). Region 4 , has two real frequency modes which make this

structure stable for interatomic distance in the range O0 r r,.

Region 5 which is in the range r, r , has at least one imaginary

mode which indicates that the triangular lattice is not stable.

In this lattice dynamics analysis for the two-dimensional

crystals we find the density at which the structure is stable, but we

cannot predict the transition that an unstable lattice will undergo

and how it will take place. To obtain such information it is

necessary to use other techniques such as molecular dynamics.

(S2)

I'LI

r

C.I I

- 2, �. V

55

6

4

a

e

-2

-4

-61.3 1.5 1.7 1.9

1.2 1.4 1.6 1.8 2

r

Figure 3.4

Functional behavior of the eigenvalues of the dynamical matrix

as a function of the lattice constant in a triangular lattice

with nearest neighbors interaction.

Ir,

56

3.2 Molecular Dynamics Results.

Molecular dynamics simulation is a powerful technique which can

be used to study the stability of two-dimensional crystals. This

technique has several advantages over lattice dynamics. First, 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 part of the work our main objective is to study the

mechanism of transition from a

and also to confirm the lattice

molecular dynamics technique.

particle system were done with

different regions that were

lattices in the last section (f

constant pressure using the

executed until equilibrium was

two different techniques. The f

square lattice to a triangular lattice

dynamics stability results with the

For this purpose simulations with a 36

initial conditions at each of the five

found for both square and triangular

ig. 3.5). Each simulation was done at

flexible border technique, and was

reached. To perturb the system we used

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. The interatomic

distance a was set to a=1.0977 which corresponds to a value in

region 1; the lattice is therefore unstable. The external pressure

was set to p=0.0, the range of interaction to R=2.5, the effective

mass of the border was W=4.0 (cf. Section 2.2), and the time step size

t=0o.005.

57

Region O(' 0

Region )

0

Figure 3.5

Scketch of the five regions found with the lattice dynamicscalculation for the square and triangular lattice.

D

-

58

The perturbation was induce by giving the system an initial

temperature of 0.05 (The melting temperature for the triangular

lattice has been found to be about 0.17, see chapter 4). Figure 3.6

shows the 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 instantaneous internal or potential

energy. 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 of the MD cell shows the same behavior (fig. 3.8). Here

its decrease is due to the fact that the triangular structure in 2-D

is a close packed plane, i.e., it has higher density for the same

value of lattice constant compared to the square lattice. In this

case it is found there is a small increase in the lattice constant but

even this, the volume decreases.

The structural analysis of the system was done by looking at the

pair distribution function g(r) during the simulation. Figures 3.9

to 3.13 represent this function during the transition from a square to

a triangular lattice. Figure 3.9 is the typical shape of g(r) for

the square lattice up to fourth neighbors. Once the planes start

59

o. -

04~00sea

*.~~ ·~~. .. e

* - - ~o0tA*~ 0.0

e 1000 e 3000ee 4000time ~ ~.,m

Figure 3.6

Temperature behvior in the transition of the square lattice

to the triangular. The simulation started in region 1 and

was induced by thermal perturbation.

60

-- 23.

-24_,- 4

- IV' - 5

.- as

20"

Figure 3.7

Internal energy as a function of time during the transition fromregion 1 and with thermal perturbation.

P

. E

-2.5.-5

61

. Volume

- e 0e0E@@ 3004 tIme -.4tep

Figure 3.8

Volume as a function of time during the transition fromregion 1 and with thermal perturbation.

62

sliding we observe that two of the nearest neighbors of a particle in

a square lattice will become first neighbors in a triangular lattice.

The other two will become second neighbors in the same triangular

lattice. In figure 3.10 we observe how the second nearest neighbors

peak starts to broad after 30 time steps. After 120 time steps this

peak is separating into two small peaks (fig. 3.11). By 200 time

steps the peak has disappeared (fig. 3.12). 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 taking

instantaneous pictures of the MD cell during the process. Figure 3.14

shows 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. In particular, figure 3.15 shows that the

second row from the top has a larger displacement from the initial

condition. Once the lattice has reached some point in which the

initial structure cannot be restored, the rest of the system follows

that initial deformation and propagates it through all the lattice

(figures 3.16-3.18). From these figures we conclude that the

transition takes place by sliding of the close packed planes of the

lattice.

The importance of this result is that MD not only confirms the

lattice dynamics calculation concerning instability, but also it shows

how a square structure in region 1 will go to a triangular structure

in region 4 (fig. 3.5).

A second simulation starting with the square structure with the

63

g(r)

r

6

5

4

3

2

1

00 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)

0 1 a 3

R

Figure 3.10

Instantaneous g(r) after 30 time steps in thestructural transition analysis.

S

5

-4

23a

I

0

65

g(r)

(

6

S

4

3

a

2I

0.5 1.50 .

R

Figure 3.11

Pair distribution function after 120 time steps.

3

66

g(r)

0. IR

Figure 3.12

2 3

Pair distribution function after 200 time steps.

S

4

3

a

I

0

67

g(r)

0.5 1 .5 265. 1 a 3.

R .

Figure 3.13

Average of the pair distribution function g(r) over thelast 3000 time steps of the simulation after the transitiontook place. -

: .- -: I.o 5

. . -S

# . I

-'' '" . . . .

.. 4

3 '5

.- . 3

: 2

.s

0

0

Ak . . . ..

4ft . z , .

68

6

S.1

4 ,

y3

2

I.

o 0 0 0 0 0

o 0 0 0 0 0

o o o 0 0 0

o 0 0 0 0 0

o 0 0 0 0 0oPI 0 0 0 00 0 0 0 0 0

i -- I i 3 5 7

0 2 4 6x

Figure 3.14

Initial structure of the MD cell used to studythe structural transition.

69

6

6

4.

I

3,

/

o /~~1

r~~~~~~

P' I 1

1

! I

;t I T I

. 1 I . , , , I

i 3 - 70 2 4 6

x

Figure 3.15

Instantaneous position of the particles after80 time steps

q

4

I �

70

- I,

-' -i 3 7 It,e0 2 4 6X

Figure 3x6

Figure 3.16

Instantaneous position of the particles after 120 time steps.

.~ I

5.5S

4.5

3.5S

2.S

1.5

0.5

__ C

71

5.5

4.5

3.5

2 .5

1.5

0 5

-0.5· 2 4 6

Figure 3.17

Figure 3.17

Instantaneous position-of the particles after 240 time steps.

0

72

5.5

4.S5

3.5

2.5

1.5

0.5

R& a

X

I -- - --0 2 4 6

Figure 3.18

Figure 38

Instantaneous position of the particles after 360 time steps.

73

ou

.4.5

&. .w

. 0 . 0 00~~~~~

I" 0 0 0 Ov~ o'0

0 0 0

.0 a 0 0

a _

0 . 0 0 0 00 0 0 -A 3 o.4 o O

5, * 3 :> ?~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

0 ... 4 _0 a P- 4 SXx

Figure 3.19

Final structure after the transition took place and the systemequilibrated.

0.5 -

74

interatomic distance in region was done; but in this case instead of

thermal perturbation we put a shear on the system and deformed the

lower row with a displacement 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. Here, the final structure is

also triangular (fig. 3.20) and the same thermodynamic behavior is

observed except temperature, initially zero, increases when the

transition takes place (fig. 3.21). The potential energy and the

volume remain constant (fig. 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 was p=-0.2655. Again two

ways of perturbing the system were used. In both cases, we found that

the system maintained its structure.

When heat was used to perturb the system, a 1000 time steps

simulation was done with an initial temperature T=0.0005. In this

case figure 3.24 shows a typical behavior of the temperature which

drops and- equilibrates at almost half its initial value. T.0005

and the its behavior can be seen in figure 3.24 where we have the

temperature during a 1000 time steps simulation. Figure 3.25 shows

the temperature behavior during a 4000 time step run where we started

at temperature T0.0 but with a shear of 1 on the lower row of the

lattice. Here it is observed that the temperature rises and

equilibrates at a low temperature. This increase is due to the energy

of deformation at the beginning of the simulation. From figures 3.24

75

3.5.

2 5

I .5

0 5

-0 5

-1.5

-1

o o 0

0

o

o

0 0

0 0

o

o

o

o

o o

0

o

0 0

o

01.

o

23

o

O

o

o

o

0 0

45

o

o

0

o

67

x

Figure 3.20

Final structure of the system after a transitionfrom square lattice in which the system wasperturbed by shear.

y

_ __ I_ __ _ ___ __ _

, -

bt .

76

Temperature

0 .0.09

0.08

0.06

T 0 0sm

0.03

0.02

0.el

0

0 1000 2C00 3000 4000

time atep

Figure 3.21

Temperature behavior in the transition induced by shear.

77

Potential Energy

0 1000 2000 3000time tep

Figure 3.22

Potential energy behavior in the transition induced by shear.

-12

-14

-16

-18

P

, E

4000

78

Volume

1500 2500 35001000 8000 300

time step

Figure 3.23

Volume behavior in the transition induced by shear.

39

39

3S .5S

38

37 S

37

36 5

36

35

0 4000

79

Temperature

C.c t1 o 47-

O. e004mo.ee42s

p

0. z0e325

C. e0e3

0.000275

e. c*.: n r111 ,, IV j ,_1 _ _ . - , I , - - - ,- , - - , , Or - -

lee 300 see 700 90k~~~~~~~~~801000 200 400 600 B0 1000

time step

Figure 3.24

Behavior of the temperature in a simulation where the initial squarestructure is in the stable region and the system was perturbed byheat.

I

80.

0.000003

2.5e-6

0.000002

.Te

mP 1.Se-6

0.000001

S.e-7

0

0500 1500 2500 3500

100.0 20ed 3000 4000time step

Figure 3.25

Behavior of the temperature in a simulation where the initialsquare structure is in the stable region and the system wasperturbed by shear deformation.

,, "�-. be-a,

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

cases the system was unable to maintain its structure and it

sublimated.

The transition from square to triangular lattice is not a

consequence of the flexible border technique. 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 square lattice

in region 1 was repeated using the fixed border technique. It was

found that in this case the system also tries to go to a lower

potential structure, but the constraint of the fixed volume and shape

of the D cell is an obstacle to accomplishing the transition, and the

system ends as a triangular lattice with structural defects (see

figure 3.26).

82

(1.

6

4'

3

2

2.

1 3 5 70 2 4 6

Figure 3x26

Figure 3.26

Final structure in a transition simulated with fixed borders.

I

I

I

I

11

II

at

83

Chapter Four

Melting and Pre-melting Structural Defects

84

4.1 Thermodynamic Behavior.

It is known that the 2-D melting transition does not behave in

the same way as in a 3-D system. Some experiments 363 and theories

[17-203 suggest that this might be a second order transition. On the

other hand, Monte-Carlo and molecular dynamics simulations

E8,11,14,213 give indications that this transition could be of first

order.

Our first objective is to demonstrate that the flexible border

technique is capable of describing the thermodynamics of the melting

process. Secondly, we hope to elucidate some of the structural

details associated with melting in 2-D. It should be noted at the

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 other techniques and its

thermodynamic properties are known. It was our interest to simulate a

point deep in the solid phase to avoid defect formation and

instabilities that appear in phase regions close to the melting

temperature.

A 56 particle system at a temperature of T=O.1375 and a pressure

of =0.58 2 0 was simulated for a total length of time of 5000 time

steps. The thermodynamic properties such 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 energy (fig. 4.2) shows an

increase due to the fact that we start with the 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. It should be noted that the initial volume was selected

to be close to the expected equilibrium value (fig. 4.3) to avoid

large transients at the beginning of the simulation. The enthalpy

which should be the constant of motion in this method does not remain

constant when the temperature is being rescaled, but if rescaling is

continuous through out the simulation it will oscillate around its

equilibrium value. If rescaling is stopped at a given time, it will

become a constant after this point (fig. 4.4). The root mean square

displacement (fig. 4.5) and the pair distribution function (fig. 4.6)

were calculated. From their behavior is clear that the system is in

the solid phase.

In the next table we compare the values calculated from our

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.

86

Temperature

1000 3000 4000 5000time step

Figure 4.1

Temperature behavior in a flexible border simulation with

rescaling the first 500 time steps.

0.17

0.16

0.14T

p

0Lii

0.1

e

O I I

e 1,

87

Potential Energy

-3Jb

-3?

-38

-3S;

-40

PP -41

E

-42

-43

-44

-4S

-46-

e 1ne0 2e00 3000 4e00 S000t me step

Figure 4.2

Potential energy behavior at p=0.582 and T=0.137.

88

' .0 '

. E- " G.' ·

:. .: A f

0 meoe 'E-O '' 3000.

Figure 4.3

Volume behavior at p=0.5820 and T=0.137.

4000 5eee

: : :

· o

4.

e.,!

89

Enthalpy

. .-

-¥_.I?

S,

4

3

i

d .e 1Z500 i e'X i e1ee , eee 3ee- 4000 S 0e0

tim .stepI . ' '

Figure 4.4

Enthalpy behavior at p=0.582 and T=0.137. Rescaling was doneduring the first 500 time steps.

H

·:- : ·

90

* 17

eviv

04. e | .

He 4Ies

2.00 e 3Se00 4500Dee 30ee., 4te .Se

Figure 4.5

Root mean square displacement for p=0.582 and T=0.137.

-� -- -1 �I -· ·

:i--:1::-

. . ..

.

DOQI4 00 ' 2

91

g(r)

4.S

4

* 2�S* ** C

a

i

0.S

0

.· -: . .

I.i ,. .i

I

R

Figure 4.6

Pair distribution function for p=0.582 and T=0.137.

2 3

. . ~

· `

.I . .

:: .l

I

92

The agreement of these results

conclude that this technique is

diagram under isobaric conditions.

Our next step was to obtain

constant pressure for points close

is sufficiently good that we can

suitable for mapping out the phase

the thermodynamic properties at

to the melting temperature in both

liquid and solid phases. The study of structure at different

temperatures is of particular interest, as well as the study of

melting through the thermodynamic data calculated by simulation.

The basic thermodynamic quantities calculated in these

simulations were the enthalpy, volume and internal energy. Table 4.1

contains the thermodynamic data at different temperatures for a

pressure p=O.4936. From this table we were able to plot the volume

as a function of the temperature (figure 4.7). In this plot we

clearly observe three temperature regions. The solid region which

goes up to a temperature of T=O.16, 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. From this curve we can calculate the

thermal expansion coefficient F for the solid and op(

for the liquid.

Figure 4.8 represents a plot of the enthalpy as a function of the

temperature. The specific heat at constant pressure CP tar)p was

calculated for both the solid and the liquid. We got the following,

cp=2.80 for the solid and cp=3.77 for the liquid. The potential

energy (fig. 4.9) and the density (fig. 4.10) as a function of the

temperature also show the same behavior as the volume and the

enthalpy. The three regions are also clearly observed.

From the thermodynamic behavior of the enthalpy, the volume, the

Table 4.1

Run T D/N P U/N H/N time steps

1 0.091 1.095 0.9132 -0.742 -0.1105 20002 0.094 1.099 0.9099 -0.739 -0.1025 20003 0.099 1.103 0.9066 -0.734 -0.0906 20004 0.107 1.112 0.8993 -0.722 -0.0661 30005 0.108 1.108 0.9025 -0.726 -0.0711 20006 0.118 1.123 0.8905 -0.708 -0.0357 30007 0.122 1.123 0.8905 -0.708 -0.0317 30008 0.130 1.134 0.8815 -0.694 0.0012 60009 0.132 1.139 0.8779 -0.691 0.0062 300010 0.147 1.150 0.8696 -0.679 0.0356 200011 0.148 1.152 0.8681 -0.674 0.0426 300012 0.153 1.157 0.8643 -0.668 0.0556 250013 0.156 1.162 0.8606 -0.664 0.0656 250014 0.160 1.164 0.8585 -0.660 0.0750 600015 0.162 1.182 0.8458 -0.642 0.1031 600016 0.164 1.185 0.8435 -0.640 0.1086 600017 0.170 1.251 0.7991 -0.578 0.2086 1000018 0.175 1.262 0.7919 -0.572 0.2253 1000019 0.185 1.277 0.7829 -0.560 0.2601 1000020 0.200 1.318 0.7582 -0.531 0.3197 10000

T, U, and H are in units of 4.The volume is in units of 2.

94

-...... ---- ------- - - --- - -- e S - _ _ _ _

- ......___A. .. . . . . ~ ~ ~ ~ ~ ~ ~ __- -:_.:-: ~~~~~~~~~~~~~~~~~~~~~~~...-.. -.-. .............e

i'. .

_,015

Figure 4.7

Volume as a function of the temparature at p=0.4936.

VET--

95

:! ,, I -,-.

I! --'- .

- � . I

i- :4 -- I I I i I~~~--~-7--, [ , , i -- -i' ½_...._iI ,__--4- .-- i i --- --l

______________.___... . ......... .. __ :----: _ ---.- ._

......................................................................... · ~"'~...--;.__/..'~: ..Z .... ~.. : -- -- ~ - '-- ~ '-------' ::'~~~~~~~~~~~~~~~~~~~~~~~~ ~---':-~-'----·--- ---: --·---- -~--- -'-- ~----~ -.---~---~-- : ....'--",'-''"..._.--':-!....--..... ..... ...--.-...; .-... ·----- '--i- .... !

...... ................................................ ..... ~ . . . ... ~ . . . . . . .......

i i ! ~ ~ ~ ~ ~ ~ ~ -

4: . .......

0.10 - 0.15 0.20

Figure 4.8

Enthalpy as a function of T at p=0.4936.

:-- H/t4-

0 .; 0--:': '-7:.

--- --- II

-... --- .--. 6 4--. - - -.- . , .-

- --------- t--- . -- I.--.- - . I - - - - - i

--- --

-- - -l··- --_----'-l-4 - -.-- ,--- I-I -- -`" '-I

--- 4--- 44.

- -4---·- --- --

-- -- ·-- - --·--- - ---. .-e- --~,. c.- .I; -.~-.-~..-.~- .- -...- - .- _ J __._~~_ I i

IT

--

i -- I . .

If-- ---------- --f ---. ·---------·---------------- -- i �------·-

`--------·----------�

--- �--· ..- --. L.---�-.---: --. I-;�.. '� ' -' 1'- --' -

---------------;;;---"~~~- I -- =':]-; -[,

i,,

�---------- �---------�-------- t------�------�----�-----�--;r =

96

density, and the internal energy, we are able to say that melting

occurs at approximately T.165. To support this conclusion we have

examined the root mean square displacement and the instantaneous

structure of the system.

With a melting temperature of TO.165 at a pressure of p=-O.4936,

and assuming that this is a first order transition, we can calculate

the step changes in thermodynamic properties. For the enthalpy we

found a change of H=O.1 and the density Ap =0.05. These data agree

with the results of by J. A. Barker et al. [8] in which for a melting

temperature of T=0.175 and a pressure of p=O.71 it was found that

AH=0.117 and Ap =0.051. On the other hand, if we take the data

reported by F. Abraham [21) for a pressure of p=O.0125 and a melting

temperature T=0.12 the change in the enthalpy is H=0.1 and the

change in the density is Ap 0.1. Here, 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. The simulations

were carried out with the corresponding volume obtained from the

flexible border technique. From the fixed border simulation we

calculate the internal pressure and energy. We found a very good

agreement of the internal pressure and the internal energy calculated

with fixed borders with those calculated with flexible borders. In

this case the agreement was within 3. All the previous results

indicate to us that, within the accuracy of 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 -- -

... . . . . .. . .. .

0.10 :.: : .0 5. ..Figure 4.9

Internal energy as a function of T at p=0.4936.

--.,b. ·

98

... : ' -7 : .':-

� --- -

I- - _. -

I

'' - -- ---- ~ -`- --' __ - - --- `- - __ . -~_ -7 '~-'-`- ~ ---- ~-----~ -- '-`-'- I-f . ·-- --

.:_- ------

-- -�----.--.---- --- ·----- ------ ·-----·- ----�---�--..... -.-I·---..- ---- ...--. ----.--- t..-- -- �. --. L--.-----�---- -..... .�..--��..--.- -- �-.--.·- · ----- · ---------- ------- -- -·- -- ·----- ------ ·- ·-·------------ -· -------- ------ �-..-.

-- ·-- --- ___----- ·- ·--- · ·-· ·- -- · ----·· --- -- · ·- ----- ------- ·-------- · ------ ·----- ------- -· -- ·- ·--- --· ---- ·-- ---· --- ··-- ·--·-·-·------------ ··- --- -- �.--c--- ----------- �--� I--.-I-- -L__,_.__�I_ �_----�--..----� --�--__..-�.�-- .

: ·· i _____.___·-----.-I 1 --.-- �.-...----. �.--�----^----�c�------.--�. ..-...-�--1.-.- -.. .- __ I-.--.-.- ---.-�.-- -.- --- 1.1··-· ·----- ·- -·.-..-...-.-..- .-...-... -. �.. ..-�.. ..��

li --: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~...........ee--.,.

t

Ii

--- 7-- -1.

_1I

lw~-

Figure 4.10

Density as a function of T at p=0.4936.

99

4.2 Onset of Structural Defects.

In the studies of the structural behavior on a 2-D system, two

particular cases which we consider of interest were found. 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, rescaling was not

continued and the system remained at constant enthalpy. This

simulation consisted of 8000 time steps. We observe from fig 4.11

that the system oscillates around its initial temperature for 2000

time steps. After that, the average temperature shows a significant

to a value of 0.125. This process was studied by taking snapshots of

the structure between 2000 and 3500 time steps. What we observed is

that at high temperatures the system had a perfect lattice structure

(see fig. 4.12). But, because of its high thermal motion, there is

always a particle in the lattice that causes deformation. Most of the

times the system can restore to its initial structure, but after 2000

time steps the deformation gets blocked into a new structure with

defects (fig. 4.13). The increase of internal energy because of the

defect formation, is the cause of the decrease of the temperature.

The second case which we are going to discuss is a simulation at

a 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 by saying that

the system tries 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 6000 time steps the behavior like in a

solid. After 6000 it shows an increase which is characteristic of

liquids. The sequence of transition in this case was studied by

looking at the snapshots at different times. Figure 4.16 represents

the structure of the system after 4000 time steps and clearly remains

in the solid state. Figure 4.17 is a snapshot at time 6000 and we

observe that the defects start to form in the system. By 7000 still

shows some ordering in the lattice but it almost reached 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

0.13

0.12

0.11

0.1

0.071000

0 20005000 7000

6000time .atop

Figure 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.

TemP

'Li

I S-

13

11

9

y 7

3

1-.

10.2

2 6 10 14 180 4 B 12 16

xFigure 4.12

Figure 4. 12

Snapshot of the MD cell and 3 of its images after 1500 time steps.

103

- .^ ~- 0

0 o 00 0

0 0 0

0 0

0 0

0 00 00000 0

O 0

0 O

s6

x

Figure 4.13

Snapshot of the MD and 3 of its images after 3000 time steps.

0000

13I E -

11

7

3

I

0

00

0

00

0

00

0

0

00

0 0

0-2

4*8 i0

1614

--- 4

18

_ 1 ·�C _IC_�C_ �C�C_·

A 7 _

IV_

D

I

104

Potential energy

-29

-i

20003e00

40005080

60007000

time step

Figure 4.14

Potential energy behaviorrescaling through all the

perfect crystal structure.

for p=0.4936 and T=0.17 withsimulation and with an initial

-31

-32

-33

-34

P-35

-37

-38

-39

-41

8000

I

10000

105

4 .'

1 ,4

I.e

1

m 0.8SD

0.6

0.4

0. a

1000 3000 500 7000 9000e ~ 200e 4000 6000 8000 10000

time 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

715

3

-i

0o 4 8

Fx

Figure 4.16

Snapshot at T=0.17 after 4000 time steps.

R_ _

107

4"0_I

is

14

10

y 8

6

4

P-

4 - 8 12 16

Figure 4.17

Snapshot at T=0.17 after 6000 time steps.

8

108

16

14

12

2

08v

6

4

_ ¢-_~~~~~~~~~~~~~~ I - - _- I

. 6 10 14 Is0 4 8 12 16 En

Figure 4.18

Snapshot at T=0.17 after 7000 time steps

10.9

1

I

I

E

.4

-a04

1e-T 8 14 i12 16 is

x

Figure 4.19

Snapshot at T-O.17 after 9000 time steps.

110

Chapter Five

Study of Two-Dimensional Bicrystal

111

5.1 Construction of =7 Grain Boundary.

A bicrystal is a structure composed of two crystals which exhibit

relative misorientation and/or translation but the material is

continuous across the interface that separates one from the other

L353. If the two crystals of which the bicrystal is composed are

identical materials, the interface between them is called a grain

boundary.

Our two dimensional model is constructed from the close packed

plane (111) in a fcc crystal (see figure 5.1). The grain boundary can

be obtained by Coincidence Site Lattice (CSL) construction if a

perfect crystal is rotated an angle about the [111] direction. The

relation that determines the value of this angle,

-t 'e - _ ( 2 __ _ (S)

where 1 and 1 are the coordinate numbers of the boundary.

A =7 grain boundary has coordinate numbers l, =2 and lz=1,

which from equation 54 gives a value for =38.21 . The value of

1=7 stems for the fact that 1/7 of the lattice sites of the two

crystals are coincident following the rotation.

Figures 5.2 represent the procedure of construction of a =7

grain boundary on the (213) plane. Starting from the perfect crystal

structure (fig. 5.2a), we rotate the crystal an angle about the

[111] direction. After this process, the rotated crystal overlaps the

non rotated part (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

oit-

4o

oo

z ,4

cx . be r-III.~~~~~~~~.

0 o

o O0

-,0) -I -i

M C-4'J -

LL r- 0

0) o5-'w

'4--i :o ~-W

°*0~ 0~ o $*~~~~~~~~~~~~H JJ0.00 w U)~~~~~~~~~~

* H5 -4 J *Ho

0 00 H--

CM

'Ir-!C~

115

I

L.j

4JU

D~ o 4.DU

0o 4

-W 0

0o -4o

03

4i 0,. >1 ON WH

4

U)

o >*00

1roI r-

t I,_J

4)CD

V4 0U 4-JU'*0 >la) U4

o o U

0 weQ4-

116

Io

c4

co(.)

I UCM X CZ

LL. i W

0r'--~U)

-0 0. )

o sk

4 4-o Ut. 4

co co0z o .,-IQ rx-

I Cco

*

117

5.2 Thermodynamic Behavior.

It is known that the thermodynamic behavior of a bicrystal is

different from the perfect crystal. In general, any kind of defect in

a crystal affects its thermodynamic properties. In the case of a

grain boundary at constant pressure these changes are reflected as an

increase in the potential energy, the volume and the enthalpy.

Furthermore, experimental work [32] and some theoretical models

[33,34] suggest that a grain boundary can melt at a lower temperature

than the bulk.

Studies on grain boundary melting phenomena using molecular

dynamics techniques have not been done in the past. One of our main

objectives is to study the thermodynamic properties of a grain

boundary and to demonstrate that molecular dynamics can give some

insight into the melting mechanism of the grain boundary. This

technique can be used to find out whether there is a melting

temperature for the grain boundary different from that of the bulk,

and in a more ambitious project, to elucidate more details of the

transition.

Our simulation cell of 112 particles represented a 2-D =7,

[111) tilt boundary (see fig. 5.3). This structure has been

previously determined to be stable. We calculated the thermodynamic

properties at different temperatures and constant pressure. The

external pressure was set equal to P=0.4936 which is the same

pressure used to calculate the thermodynamic properties of the perfect

crystal in the preceding chapter. Again, the flexible border

technique was used for these simulations.

118

4C

30

25

20

is

10

0.5

-15.5 -5.5 4.5 14.5 24.5x

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 volume

and the enthalpy of the system at different temperatures (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 which is lower than the melting temperature of

the perfect crystal at T=0.16. In the range of temperature between

0.14 and 0.16 a coexistance of liquid and solid phases appears after

the grain boundary melts. This coexistance permits the system to look

for a lower enthalpy state and it is observed that subsequently the

system resolidifies into a perfect crystal. If we plot from table 5.1

the enthalpy as a function of the temperature (fig. 5.4), we can

observe that a transition has already occurred around T=0.14. This

transition is more evident on the plots of the excess quantities of

the bicrystal He U, and V (figures 5.5 to 5.7).

120

Table 5.1

T

0.110

0.124

0.140

0.150

0.16

U/N

-0.6955

-0.6767

-0.6499

-0.6405

-0o.6304

Q/N

1.1357

1.1509

1.1777

1.1853

1.2009

H/N

-0.0239

0.0171

0.0714

0.0946

0.1224

U. /N

0.0245

0.0263

0.0341

0.0315

0.0296

Q /N

0.0217

0.0229

0.0327

0.0303

0.0359

H /N

0.032

0.034

0.047

0.044

0.044

The excess quantities are defined as follows:

He=(Hbc-HPc)

Ue = (Ubc UPC )

2e= (bC- f2PC)

The subscripts bc and pc stand for grain boundary and perfectcrystal,respectively.

121

H /N

iquid line

grain boundaryliquid

solid

perfect crystal s(olid line

I0 .15

Figure 5.4

Enthalpy as a function of the temperature forboth the perfect crystal and the grain boundary.

0.2 -

0.0 -

0.1

1 I

' T

122

H,/N

x

T

0.15

Figure 5.5

Excess of enthalpy inperfect crystal.

the bicrystal with respect to the

0.04 -

0.03

0.1

w

123

Qe/N

0.03

0.02 -

~i x

I~~~~~~~~

xx

I >_-·~~~~~~~~~~

0.1 0.15

Figure 5.6

Excess of volume in the bicrystal with respect to theperfect crystal.

124

U,/N

A

t(~~~~~~~~I

ICC~ ~~ " T

I- Tt

0.1 0 ,1$

Figure 5.7

Excess of internal energy in the bicrystal with respectto the perfect crystal.

0.03 '

0.0o2

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 analysis of the

perfect crystal, gives additional information regarding particle

localization in the system. n this case, the simulation cell was

divided into 4 regions and the mean square displacement was calculated

for each one (see figure 5.3). Regions and 3 correspond to the

grain boundary. Regions 2 and 4 correspond to the bulk.

A simulation at a temperature of T=O.11 demonstrated that the

grain boundary behaved as a solid structure. Although the boundary

maintained its initial structure, migration of the interface was

observed during the simulation. Figure 5.3 shows the initial

configuration of the system, and figure 5.8 is a snapshot after 7000

time steps. From these two figures it. is evident that grain boundary

migration is already taking place even at this temperature. The

dynamics of migration of grain boundaries has been studied using MD

simulation [22,23]. It is known that for a system like ours if the

temperature T is high enough and if we execute the simulation for a

long period of time, annihilation of grain boundaries will be

observed.

The stability behavior of the system above the melting

temperature of the grain boundary is found to be different from that

at lower temperatures. For example, at temperature of T=0.15 the

snapshots of the system show that the grain boundary melts and a

126

.a r-

40

35

30

20Is

15

10

S

0

-IS.5 -S. ; 4.5 14.5 24.Sx

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

is diffusing through out the cell

process. Our simulation cell has

two disordered regions after they mel

contact with each other (fig. 5.10),

reach its minimum enthalpy configur

perfect crystal structure (fig. 5.11)

by the potential energy, which shows

9000 time steps. If we calculat

decrease took place, we find out that

corresponds to the perfect crystal.

of T=O.16 shows a slightly different

0.15. The difference is probably

approaching the melting point of the

(fig. 5.9). The disorder

by a melting-resolidifi

two grain boundaries which

region

cation

create

t. Once these two regions get in

it is possible for the system to

ation, which for this case is the

. This observation is confirmed

a sharp decrease between 8000 and

:e the equilibrium state after the

: this value is the one which

The simulation at a temperature

behavior from the simulation at

due to the fact that we are

bulk. In general, though, the

process is 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 energy of the bicrystal

equilibrated first at point I (see figure 5.13). Then, it dropped

directly from this point to point II which correspond to a lower

potential energy point on the perfect crystal curve previously

calculated in chapter 4 (see fig. 5.12).

At T0.16 however, the system does not go directly from the

initial state (point a in fig. 5.13) to the lower potential energy

state (point d in fig. 5.13). Instead, before the resolidification

process takes place it spends some time at higher potential energy

states (points b and c in fig. 5.13). From the potential energy

behavior (fig. 5.14), the instantaneous structure of the system (fig.

128

O'O~

.S-5.5 14.S 24. S

Figure 5.9

Snapshot indicating the increase of disorder in the grain

boundary region at T=0.15 after 3000 time steps.

40

a25

'

S

0w

-IS-5

129

.. i. _ ..... . _ .. . .. ,. . .. .. .. , ., . . . _ _ _ ,, , ... _ . _ .. . I

Cob ~ ~ ~~~ . -1----

45

40

3S

30

Z5

20

10

5

-13 -3 7 17 27-18 -8 2 12 B 32

x

Figure 5.10

Snapshot indicating a highly disordered structure at thetime step 7000 for T=0.15 and p=O.4936. '

130

0

.0

-e-S 9.524.5

Figure 5.11

Resolidification to perfect crystalat T=O. 15.

after 10000 time steps

30

y2,0

10

0'

131

4 70002000 400e 6000 See 10000

time tep

Figure 5.12

Potential energy behavior at T=0.15 and p-=0.L4936 where theresolidification process is observed.

-74

-7E

P-78

-7

t00

132

Figure 5.13

Scketch of how the resolidification process may take placefrom the data calculated.

133

1000 300 0 5c00 7000 900L3 2000 4000 6000 8000 i0000

time top

Figure 5.14

Potential behavior at T=0.16 and p=0.4936

-06

-764-68

-?S

-70-84-76-72

-Be

-82

-84

- P r5

134

i T

4f

*3 , .3S-

3 0

20+

S

1510

At

I-- " L_- l _== 4 I ' 4 : ¢ 4

- 4 -4 6 16 2614

-1-I -g9 II !1 21 1

Figure 5.15

Snapshot of a high disordered structure at T=O.16 after5000 time steps.

135

5.15), and from what we know from the previous chapter, we conclude

that one of the states (point c) is the liquid phase. It is

interesting that the orientation of the perfect crystal which we

obtained in the T0.15 simulation corresponds to neither of the

starting orientations of our bicrystal (see figures 5.3 and 5.11).

Details of the mechanism of resolidification are difficult to

determine from the limited data we have collected to date. The

results obtained are not enough to obtain a concrete conclusion of

this process, and all the interpretation is just admittedly

speculative.

136

Chapter Six

Conclusions and Discussion

137

Conclusions and Discussion.

One of the main conclusions from our work is that we have been

able to calculate the thermodynamic properties of a 2-D system using

the flexible border technique. The agreement of our results is within

1t from those obtained by onte-Carlo simulation.

The flexible border technique has been successfully applied to

study melting at constant pressure through the thermodynamic

properties of a 56 particle system. The number dependence was

observed to be insignificant while simulating a 400 particle system.

No significant difference was found in the results calculated using

the fixed border technique, which in general was less than 3. By

looking at the instantaneous positions of the particles at different

times, we were able to follow the sequence of transition from the

initial state to the equilibrium state of the simulation. From these

snapshots we observe an 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 studied with the

technique of the flexible borders. It was possible to corroborate the

lattice dynamics analysis with 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 more appropriate

for this kind of study since the transition from a perfect square

crystal to a perfect triangular crystal is possible. By comparison,

in the fixed border technique the constant volume and shape represent

138

constraints that will impede such a transition. Although with the

fixed border technique the system will go from the square to

triangular, it can only reach a state which corresponds to a

triangular lattice with defects.

In our attempt to study grain boundary melting using MD, we found

a melting temperature for the interface which is about 85% of the

melting temperature of the perfect crystal. Once the grain boundary

melted, we observed a transient coexistance of disordered and ordered

regions. The disordered region could migrate and gives rise to a

melting-resolidification process. A resolidification process to a

perfect crystal from the transient state of coexistance was observed.

But it has not been determinated so far whether this phenomenon is

dependent on the particular grain boundary system we have studied, or

if it is a general characteristic of a bicrystal. From the

preliminary results we speculate that the melting mechanism for a

bicrystal could be significantly different from what is known for a

perfect crystal.

139

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