Molecular Dynamics of Disordered Ice

by

Hafiz Ghulam Abbas

MS Thesis

May 2015

Department of Physics

LUMS Syed Baber Ali School of Science and Engineering

LAHORE UNIVERSITY OF MANAGEMENT SCIENCES

Department of Physics

CERTIFICATE

I hereby recommend that the thesis prepared under my supervision by: —— Hafiz

Ghulam Abbas —— on title: —— Molecular dynamics of disordered ice ———– be

accepted in partial fulfillment of the requirements for the MS degree.

Dr. Fakhar ul Inam

——————————————-

Advisor (Chairperson of Defense Committee)

Recommendation of Thesis Defense Committee :

Dr. Muhammad Faryad ——————————————-

Name Signature Date

———————————————————————————-

Name Signature Date

2

I would like to dedicate this thesis to my mother and respected teachers

3

ACKNOWLEDGMENT

I would never be able to finish my dissertation without the guidance of my advisor, help

from friends, and support of my family. I would like to express my sincere gratitude to for

supervisor Dr. Fakhar ul Inam, for his excellent guidance, encouragement, support and

providing me an opportunity to do my research work under his supervision.

Moreover I am thankful to all of my friends at LUMS Lahore, particularly Mr.

Muhammad Umer, and Mr. Arshad Marral for all kind of support. Uniquely I am

thankful to Mr. Irtaza Hussain of LUMS who was always helping me in thesis work.

Finally my parents were always supporting and encouraging with their best wishes.

Hafiz Ghulam Abbas

4

ABSTRACT

We have studied the structural properties of coarse grained model of Low Density

Amorphous (LDA) ice. Using calculation techniques used in molecular dynamics, the

average radial distribution function (RDF) of coarse grained model of LDA ice was

calculated at varying sizes. The average RDF of coarse grained model of ice was also

simulated to the second shell of experimental RDF of LDA ice at 80 K. The accuracy of

the simulation was improved by increasing the size of coarse grained model of LDA ice. A

comparison was carried out between different models of water using simulation accuracy

on the experimental RDF. Phase transition of coarse grained model of LDA ice was also

studied and it was observed that by using Stilling–Weber potential, there was no phase

transition of coarse grained model of LDA ice at different pressures.

5

List of Figures

1.1 Radial distribution function of amorphous ice. First shell represents first

nearest neighbours from central atom. At long distance radial distribution

function show absence of long range order. . . . . . . . . . . . . . . . . . . 3

1.2 Radial distribution function of amorphous silicon. First shell represents first

nearest neighbours from central atom. At long distance radial distribution

function show absence of long range order. . . . . . . . . . . . . . . . . . . 3

1.3 Comperision of radial distribution function of amorphous silicon and ice.

Black line show radial distribution function of amphrous silicon and blue

line show radial distribution function of amphrous ice. . . . . . . . . . . . . 4

1.4 Tetrahedral geometry of amorphous silicon. Pink sphere show the silicon

atom. Bond angle between silicon atoms approximate to tetrahedral angle. 5

1.5 Tetrahedral structure of coarse grained LDA ice. Red sphere represents the

hydrogen atom and blue sphere represents the oxygen atom. Bond angle

between oxygen atoms close to tetrahedral angle. . . . . . . . . . . . . . . 6

1.6 Tetrahedral structure of water molecule. Red sphere shows oxygen atom and

blue sphere represents hydrogen atom. . . . . . . . . . . . . . . . . . . . . 7

6

1.7 Structure of amorphous ice. Red sphere show oxygen atom and blue sphere

show hydrogen atom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.8 Phase diagram for amorphous ice. The line R shows first order transition,

line P shows metastabelity limit for LDA ice and Q line shows metastabelity

limit for HDA ice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.9 Radial distribution function (RDF) of different ice phases. Pink line show

RDF of LDA ice, blue line show RDF of HDA ice and purple line show RDF

of VHDA ice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Steps of molecular dynamics algorithm. . . . . . . . . . . . . . . . . . . . . 17

4.1 Optimized lattice constant and total energy of coarse grained model of LDA

ice. Otimized lattice constant 6.20◦A which is close to experimental value

6.36◦A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 RDF of different coarse grained models of LDA ice at zero pressure and

matched with refrence LDA ice model. . . . . . . . . . . . . . . . . . . . . 32

4.3 Bond length distribution of LDA ice. Maximum bond length between two

oxygen atoms is 2.68◦A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.4 Bond angle distribution of LDA ice. Tetrahedral angle is 109.5 ◦. . . . . . . 33

4.5 Average RDF of different size coarse grained LDA model of ice at 80 K and

compression to the second shell of experimental LDA ice. . . . . . . . . . . 35

4.6 Density of LDA ice at different pressures during compression and decom-

pression. HDA ice does not sustain their phase during decompression. . . . 35

7

4.7 RDF of LDA ice different models and comperision to the second shell of

experimental RDF of LDA ice model at 80 K. . . . . . . . . . . . . . . . . 36

4.8 RDF of LDA ice during compression at different pressures. Pressure increase

from down to upward direction. At 18 Kbars, LDA ice change into HDA ice. 37

4.9 RDF of LDA ice during Decompression at different pressures. Pressure de-

crease from top to bottom direction. At 0 Kbars, HDA ice change into LDA

ice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

8

List of Tables

2.1 Most satisfactory parameter set for water and use input data file of LAMMPS. 24

4.1 Optimized lattice constant and bond length of LDA ice. . . . . . . . . . . . 34

4.2 Volume and number density of LDA ice coarse grained models at 80 K.

Experimental volume density of LDA ice is 0.943 gcm−3 at 80 K. Compare

experimental value of volume density with calculated value. . . . . . . . . . 34

9

Table of Contents

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Importance of Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Hydrogen Bond in Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Crystalline Ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Amorphous Ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6 Low Density Amorphous Ice . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.7 High Density Amorphous Ice . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.8 Very High Density Amorphous Ice (VHDA) . . . . . . . . . . . . . . . . . 11

1.8.1 Radial Distribution Function (RDF) . . . . . . . . . . . . . . . . . 12

2 Classical Molecular Dynamics 15

2.1 Basic Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

10

2.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Thermodynamic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.1 Isothermal Isobaric Ensemble (NPT) . . . . . . . . . . . . . . . . . 20

2.4 Constant Temperature (NVT) . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Coarse Grained Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6 Stilling–Weber Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 LAMMPS Molecular Dynamics Simulation Details 25

3.1 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Atom Defination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 Settings of Force field coefficients . . . . . . . . . . . . . . . . . . . . . . . 28

3.5 Settings Fixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.6 Settings Output Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.7 Running Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Results and Disscussion 31

4.1 Structural Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Radial Distribution at Temperature 80 K . . . . . . . . . . . . . . . . . . . 33

11

4.3 Low Density Amorphous Ice of Different Models . . . . . . . . . . . . . . . 35

4.4 Low Density Amorphous Ice at High Pressures . . . . . . . . . . . . . . . . 36

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

12

Chapter 1

Introduction

1.1 Motivation

Silicon like water forms a tetrahedral crystal at room pressure and has amorphous phases.

Tetrahedral structure of coarse grained amorphous ice and silicon are shown in Figs. 1.4

and 1.5. We summarize some similarities between amorphous water and silicon below,

1. Low density amorphous ice and silicon have disordered structure with the

tetrahedral coordination.

2. High density glasses of these substance have similar structure [1].

3. Density of water is maximum at 277 K and sharply decreases in super cooled region.

Silicon also displays maximum density, deep in super cooled regime [1].

4. Dynamics of these liquids are similar, while the viscosity of normal liquid increase

with pressure, liquid silicon and water become more fluid on compression. This

1

irregularity is more important in super cooled regime, and disappears at high

temperature [1].

We calculated radial distribution function and numbers of neighbours within a distance s

from a central atom numerically in FORTRAN, by following equations and derivation

described in section 1.8.1,

gαβ(s) =1

4πs2ρNCαCβ

∑i 6=j

(~s− ~sij),

n(s) = 4πρ

∫ smin

0

s2g(s)ds,

where n(s) represents number of neighbours within a distance s from a central atom and

g(s) partial radial distribution function. When g(s) of oxygen and silicon are evaluated at

first minimum, it gives a four nearest neighbour in the first shell. Second shell minima of

nearest neighbours of oxygen and silicon atoms are at 5.3◦A and 4.5

◦A. The Behavior of

radial distribution function of oxygen and silicon are shown in Figs. 1.1, 1.2 and 1.3.

First transition occur in amorphous ice, low density amorphous (LDA) ice transform into

high density amorphous (HDA) ice. Similarly during first transition in amorphous silicon,

LDA silicon transforms into HDA silicon. These similarities between these tetrahedral

liquid suggest that water may be modeled on a similar way as amorphous silicon, with

only short-ranged interactions. This does not mean that electrostatic interactions are

irrelevant in determining water structure and thermodynamic, but that their effect may

be effectively produced with a monoatomic short ranged potential. Due to these

similarities between amorphous silicon and ice, we have used Stilling–Weber potential for

coarse grained structure of LDA ice.

2

0 2 4 6 80

1

2

3

4

5

A

g oo(r

)

O−O

Figure 1.1: Radial distribution

function of amorphous ice. First

shell represents first nearest neigh-

bours from central atom. At long

distance radial distribution function

show absence of long range order.

0 2 4 6 80

5

10

15

A

g oo(r

)

Si−Si

Figure 1.2: Radial distribution func-

tion of amorphous silicon. First shell

represents first nearest neighbours

from central atom. At long distance

radial distribution function show ab-

sence of long range order.

1.2 Importance of Water

Water is the most abundant substance on earth. It has many physical properties which it

more distinctive. We summarize some of its properties below,

1. Water plays an important role in all aspects of life. Water has 160◦ higher boiling

point than H2S, due to strong hydrogen bonding. As a result our planet is bathed

in liquid water.

2. The large heat capacity of oceans and seas makes them heat reservoirs which

moderate our atmospheric conditions, resulting relatively small temperature

fluctuations.

3. Water has high surface tension and expands on freezing, which provides abrasion of

rocks to make soil.

3

Figure 1.3: Comperision of radial distribution function of amorphous silicon and ice.

Black line show radial distribution function of amphrous silicon and blue line show radial

distribution function of amphrous ice.

4. Water is excellent solvent for ionic compounds due to high dielectric constant, small

size and polarity.

5. Water has maximum density at 4◦ C. Water density is minimum at 0◦ C, molecules

are for away from each other, and when temperature increase from 0 to 4◦ C

molecules come close to each other and density become maximum. Due to this

property rivers ,oceans and lakes freeze from top to bottom. This insulates the

water from further freezing, reflects back sunlight into space and allows rapid

melting, hence permitting survival of ecology at the bottom of water bodies [2].

6. It contributes to the ionic interactions in biological systems.

7. Water contributes to the thermal regulation. It resists local temperature

fluctuations and stabilizes our body temperature due to the large heat capacity and

thermal conductivity.

4

Figure 1.4: Tetrahedral geometry of amorphous silicon. Pink sphere show the silicon atom.

Bond angle between silicon atoms approximate to tetrahedral angle.

1.3 Hydrogen Bond in Water

The distinctive properties of water can be attributed to hydrogen bonding [3]. In water

the bond between H-atom and oxygen atom is polar in nature, in which the oxygen atom

attract the bond pair with greater force. Due to polarity, slight negative charge induces

on oxygen atom and slight positive charge on H-atom. Due to polarity, each water

molecule attract its neighbour molecule, which cause H-bonding.

Polarity and hydrogen bonding are important features in water. This molecule forms an

angle of 104.5◦ with two hydrogen and oxygen atom. This angle is opposed to a typical

tetrahedral angle of 109◦ because oxygen atom is more electro negative than hydrogen

atom. Oxygen atom has two lone pairs who repel the electrons of hydrogen atom when

they form bond with oxygen atom. Dipole moment points from oxygen atom toward

hydrogen atom due to charge difference, water molecules attract to each other and other

polar molecules. This attraction contributes to hydrogen bonding. Water molecule can

form a maximum four hydrogen bonds. It can accept and donate two hydrogen atoms.

Water molecule forms tetrahedral order due to four hydrogen bonds. Amorphous

5

Figure 1.5: Tetrahedral structure of coarse grained LDA ice. Red sphere represents the

hydrogen atom and blue sphere represents the oxygen atom. Bond angle between oxygen

atoms close to tetrahedral angle.

structure of water can be formed by contribution of oxygen. It has maximum density at

4◦ C and minimum density at 0◦ C. Tetrahedral geometry of water is shown in Figs. 1.6.

1.4 Crystalline Ice

With tetrahedral arrangement of oxygen atoms, hexagonal ice (Ih) ice has most stable

phase at atmospheric pressure below melting point temperature Tm and above 72 K [4].

It can be regarded as an C.D.C.D stacking of layer consisting of network of open packed

hexagonal rings. Its unit cell has dimensions a = 4.498◦A and c = 7.338

◦A at 98 K with

density 0.924 gcm−3 [5]. The cubic ice phase is formed when water freezes below 190 K.

Cubic ice Ic is identical to hexagonal ice, when C.D.E.C.D.E stacking is same to

diamond cubic system [6, 7]. Pseudo-hexagonal unit cell has dimensions a = 4.495◦A and

c = 11.012◦A at 88 K with density of 0.923 gcm−3 [5]. Cubic ice Ic has distance 50

◦A

more than for crystalline ice . Local tetrahedral structure of first hydration shell is

identical to layer spacing and equal to the phases of ice, cubic ice, Ih ice and stacking

6

Figure 1.6: Tetrahedral structure of water molecule. Red sphere shows oxygen atom and

blue sphere represents hydrogen atom.

disordered ice [8, 9].

1.5 Amorphous Ice

Amorphous ice consist of water molecules that are randomly arranged. In crystalline form

of ice, water molecules are regularly arranged. Amorphous ice is distinguished from

crystalline ice due to lack of long range order. This ice formed by compressing ordinary

ice at low temperature. Water ice on Earth is common crystalline Ih ice. A particularly

striking feature of amorphous ice has two distinct amorphous forms. The two observed

forms of amorphous ice differ significantly in their density. LDA ice can be formed by

vapor deposition at a temperature below 77 K and other observed form is HDA ice, and

can be produced through pressure induced amorphization of Ih ice at 77 K. LDA ice is

isothermally compressed at 77 K and at pressure 600 MPa, it transform to HDA ice. The

7

HDA ice can be recovered at pressure 1 atm and this transform to LDA ice when heated

above 120 K. At pressure 1 atm HDA ice has high density 1.17 g/cm−3 and LDA ice has

0.934 g/cm−3. Pressure induced LDA to HDA ice transition is a first order phase

transition, this separates two amorphous forms of ice. This propose a phase diagram

relating to LDA and HDA ice, and consistent with experimental data. This phase

diagram propose first order phase transition that separates LDA and HDA ice. The line P

is the metastability limit for LDA ice and Q is metastability limit for HDA ice. The LDA

ice is more stable below first order transition line R. In region above R and below P, LDA

is metastable with respect to HDA ice. The LDA ice becomes unstable above line P. The

HDA ice is more stable from above line R, this is metastable with respect to LDA ice

between R and Q and unstable below Q [10]. Structure of amorphous ice and phase

diagram are shown in Figs. 1.7 and 1.8.

1.6 Low Density Amorphous Ice

Amorphous solids are formed by cooling the liquid below glass transition temperature.

Near atmospherics pressure, if this performed with water, it becomes amorphous solids

hyper quenched glassy water (HGW) or amorphous solids water (ASW). The HGW and

ASW have low density 0.94 gcm−3 after heating at temperature 77 K. The LDA, ASW

and HGA ice have similar pair correlation function. Their behavior change with heating.

Pair correlation function of low density amorphous ice is shown in Fig. 1.9, which show a

clear separation between first and second hydration shells located at 3.2◦A and 5

◦A with

low probability of interstitial molecules. It exhibits a local tetrahedral behavior.

8

Figure 1.7: Structure of amorphous ice. Red sphere show oxygen atom and blue sphere

show hydrogen atom.

1.7 High Density Amorphous Ice

In 1984, Mishima et al [11] found something surprising, Ih ice changes to HDA ice, when

it is compressed above 10 Kbars and at 77 K. At zero pressure, HDA ice has 24 percent

higher density than LDA ice [11]. Second hydration shell of HDA ice collapses at 3.6◦A as

compared to LDA ice. Interstitial molecule is not directly H-bonded to centeral molecule

inside 3.3◦A distance [12]. It transforms irreversibly into LDA ice with heat evolution of

42± 8 Jg−1, if heated above 117 K and at atmospheric pressure [11]. A sharp transition

occurs at 6± 0.5 Kbars to HDA ice, when it is pressurized at 77 K, this transition is

irreversible [11]. This effect can be reversible at elevated temperature of 135 K. Abrupt

volume change of about 0.20± 0.01 gcm−3 occurs at 2 Kbars [13]. This suggest the

9

Figure 1.8: Phase diagram for amorphous ice. The line R shows first order transition, line

P shows metastabelity limit for LDA ice and Q line shows metastabelity limit for HDA ice.

existance of first order transition between two glassy phases of water. Annealing at

atmospheric pressure, results in a structural transformation from HDA to LDA ice [14].

LDA to HDA reversible transition is not conclusive evidence of first order phase

transition. It should be possible to map out coexistence line of two phases. We observe

nucleation of one phase growing out of other. Microscopically, this has not been possible

[15]. The LDA and HDA ice have been prepared, both with macroscopic samples [16, 17].

At ambient pressure, thermal stability of HDA ice varies strongly with method of

preparation [15]. Therefore HDA ice is classified into two classes, expanded HDA ice and

unannealed HDA ice [12, 18]. These transform into LDA ice upon heating at 117 K and

pressure 131, 134 Kbars respectively [16, 17]. The LDA and HDA ice has two different

glassy phases of water when they are heated above their glass transition temperature.

They transform into two different liquids which are thermodynamically connected.

10

0 2 4 6 80

1

2

3

4

5

A

g oo(r

)

LDAHDAVHDA

Figure 1.9: Radial distribution function (RDF) of different ice phases. Pink line show RDF

of LDA ice, blue line show RDF of HDA ice and purple line show RDF of VHDA ice.

1.8 Very High Density Amorphous Ice (VHDA)

If HDA ice is annealed above 0.8 GPa and 130 K, then amorphous solid with density

higher than HDA ice is formed [19]. This has density of 1.25 gcm−3 at 1 bar and 77 K,

and have been named VHDA ice [19]. Radial distribution function of very VHDA ice is

shown in Fig. 1.9, which shows stronger collapse of second hydration shell as compared to

LDA ice. HDA ice to VHDA ice transformation is continuous and reversible [20, 21]. Two

distinct phases of HDA and VHDA ice are connected thermodynamically with two

separate liquids.

11

1.8.1 Radial Distribution Function (RDF)

The pair distribution function describes the probability of finding a particle at a

separation s from another particle. RDF is a useful tool to describe the structure of

system. At long separation, RDF approaches to one which indicates that there is no long

range order [23]. Suppose we have a system containing N -number of particles at position

s1, s2, s3, ....sN , the joint probability distribution for finding the particle 1 at position s1

and particle 2 at position s2, is given by the following equations,

P ( 2N

)(s1, s2) =

∫ds3

∫ds4....

∫dsNP (sN),

P (sN) =exp(−βU(sN))

Z,

where β is a Boltzmann constant, Z partition function and U potential energy. The joint

distribution function for finding a particle at position s1 and another particle at s2, so

joint distribution function can be written by following equation,

ρ( 2N

)(s1, s2) = N(N − 1)P ( 2N

)(s1, s2).

There are N number of possible ways of picking the particle 1 and N − 1 possible ways of

picking the particle number 2. The reduced distribution function for two particles can be

write by following equations,

ρ(2)(s1, s2) = ρ(s1)ρ(s2)g(2)(s1, s2)

in which ρ(s) is one particle density function and g(2)(s1, s2) is the two particle

correlation function. For homogeneous, system the two body density function is reduced

to following form of equation,

ρ(2)(s1, s2) = ρ2g(2)(s1, s2),

12

ρ = N/V,

g(2)(s1, s2) =N(N − 1)

ρ2P (2)(s1, s2),

=N(N − 1)

ρ2Z

∫ds3

∫ds4

∫ds5....

∫dsN exp(−βU(s1, s2, ...sN)),

=N(N − 1)

ρ2Z

∫dr1δ(s1 − s′1)

∫ds2δ(s2 − s′2)

∫ds3

∫ds4

∫dr5....

∫dsN exp(−βU(s1, s2, ...sN)),

=N(N − 1)

ρ2〈δ(s1 − s′1)〉〈δ(s2 − s′2)〉.

We define new variables S and s as,

S =s1 + s2

2,

s = s1 − s2,

so that,

s1 = S +s

2,

s2 = S − s

2.

g(2)(S, s) =N(N − 1)

ρ2〈δ(S +

1

2s− s′1)〉〈δ(S − 1

2s− s′2)〉,

g(2)(s) =1

V

∫dSg(S, s),

=N(N − 1)

V ρ2〈∫dSδ(S +

s

2− s′1)δ(S +

s

2− s′2)〉,

=N(N − 1)

ρ2V〈δ(s− s12).

This is average of all possible distances between the two particles.

ρ(n)(s1, s2, s3...sn) = ρ(s1)ρ(s2)ρ(s3)...ρ(sn)g(s1, s2...sn).

g(n)(s1, s2, s3, ..., sn) =1

ρnρn(s1, s2, s3, ...sn),

=V n

ZNNn

N !

(N − n)!

∫e−βUN (s′1,s

′2,...,s

′N )δ(s1 − s′1)...δ(sn − s′n)ds′1, ..., ds

′N ,

=V n

N

N !

(N − n)!〈n∏i=1

δ(si − s′i)〉,

13

g(2)(s) =N(N − 1)

ρ2V

1

N(N − 1)

∑i 6=j

δ(s− sij),

g(2)(s) =1

Nρ

∑i 6=j

δ(s− sij).

Normalized RDF of particles can be calculate by following equations,

gαβ(s) =1

4πs2ρNCαCβ

∑i 6=j

(~r − ~sij),

gαβ(s) =∑αβ

CαCβgαβ(s),

where α and β represents particle 1 and particle 2. Coordination number can be calculate

by following equation,

n(s) = 4πρ

∫ smin

0

s2g(s)ds,

where n(s) is the numbers of neighbours within a distance s from central atom.

Thermodynamic properties can be studied by calculating radial distribution function. We

consider spherical shell of volume 4πs3δs that contains 4π2ρg(s)δs number of particles. If

pair potential at a distance s has value U(s), energy of interaction between particle in

shell and central particle is 4πs3ρg(s)δsU(s). Total potential energy is obtained by

integrating s from zero to infinity and multiplying by N/2 [23]. The total energy can be

write by following equation:

E =3

2NkBT+ 2πNρ

∫ ∞0

s3U(s)g(s)ds.

14

Chapter 2

Classical Molecular Dynamics

2.1 Basic Idea

Molecular dynamics is used to study equilibrium properties of many body systems.

Classical means nuclear motion of constituent particles which obeys laws of classical

mechanics. What is the reason to treat nuclei as a classical particles?. Electronic and

nuclei energies are comparable. Adiabatic approximation separate the electronic and

nuclei degrees of freedom, this separate the bonding of electrons with nuclei through

electronic potential. This is a best approach for wide range of materials. When we

consider or the transnational motion, rotational motion and vibrational motion of light

atoms with frequency ν. We select a model system consisting of N particles while

considering molecular dynamics simulation. Then we solve Newton’s equations of motion

for this system until properties of system does not change with time [24]. In molecular

15

dynamics technique, time evaluation of interacting atoms can be followed by integrating

the equation of motion,

Fi = miai, (2.1)

here mi is mass of atom, ai the acceleration of atoms and Fi is the force acting on it due

to interaction with other atoms. The forces acting on atoms are obtained from classical

potentials after performing actual quantification. During computation in molecular

dynamics simulation, we must be able to express an observable as a function of momenta

and positions of particles in the system. Temperature in a many body system makes use

of the equipartition energy over all degrees of freedom, which enter quadratically in

Hamiltonian H of a system. Average kinetic energy per degrees of freedom is useful

following form:

〈12mv2〉 =

1

2kBT (2.2)

The temperature is measured by dividing total kinetic energy of system with number of

degrees of freedom. Kinetic energy of the system has direct relation with temperature.

Temperature can be writen by following equation,

T (t) =N∑i=1

miv2i (t)

kBNf

, (2.3)

where Nf is the number of degrees of freedom. Relative fluctuations in temperature will

be order of the 1√Nf

[24].

2.2 Algorithm

The steps of algorithm are described below:

1. Initialization

16

Figure 2.1: Steps of molecular dynamics algorithm.

In order to start a simulation, initial positions are assigned to all particles and

particles are put on this lattice site in the system. Certain value is attributed to

each velocity component of every particle, that is drawn from uniform distribution

in time interval [−0.5, 0.5]. In thermal equilibrium, following relation should hold

〈v2〉 =kbT

m(2.4)

Where v is component of velocity in x, y and z direction of a given particle. The

instantaneous temperature at time t is given by following equation:

T (t)kB =N∑i=1

miv2i (t)

Nf

(2.5)

We adjust instantaneous temperature T (t) to match desired temperature by scaling

all velocities with factor [(T/T (t))]12 . Initial setting of temperature is not critical

and T will change during equilibrium [24].

2. Force Calculation

Calculation of force acting on every particle is most time consuming part of all

molecular dynamics simulation. If model system has pairwise additive interactions,

17

we consider contribution to the force on particle (i) due to all its neighbours. If we

consider interaction between particle and nearest neighbours of another particle,

then we must evaluate N(N − 1)/2 pair distance. If pair of a particle is close

enough to interact, compute force between the particles and contribution to the

potential energy by following equation,

F (r) = −∂U(r)

∂r(2.6)

3. Integrating Equation of Motion

We have all particles and can integrate equation of motion with Verlet algorithm.

To derive Verlet algorithm, we start with Taylor expansion of coordinates of a

particle around a time t

r(t+ ∆t) = r(t) + v(t)∆t+F (t)

2m∆t2 +

∆t3

6r... +O(∆t4),

r(t−∆t) = r(t)− v(t)∆t+F (t)

2m∆t2 − ∆t3

6r... +O(∆t4),

The adding of above equations gives:

r(t+ ∆t) + r(t−∆t) = 2r(t) +F (t)

m∆t2 +O(∆t4),

if we know positions at time t and acceleration at time ∆t2, calculate the position

at a time r(t+ ∆t) + r(t−∆t). Verlet algorithm is use to compute new positions.

We derive velocity from trajectory of a particle, so we write by following equation,

v(t) =r(t+ ∆t)− r(t−∆t)

2∆t+O(∆t2)

This velocity expression is only accurate to the second order in ∆t2. We have

computed new position at a time (t−∆t). Current positions become old positions

and new position becomes current positions. After each time step, we compute

current potential energy, pressure, temperature and total energy in current force

loop. The total energy should be conserved. Verlet algorithm is fast but is not good

for long time steps. It requires as little memory as possible [24].

18

4. Compute Average Quantities

Calculate average of all force, energy, velocity, position, density, volume, pressure

and radial distribution function.

2.3 Thermodynamic Potentials

Why free energies are important when we are interested in relative stability of phases? In

the second law of thermodynamics a closed system is in equilibrium. At corresponding

equilibrium conditions system can exchange heat or volume with reservoir. If a system is

in contact with heat bath, its temperature, volume and number of particles are fixed.

Gibbs free energy G = F + PV is at minimum for a system of N particles at constant

temperature and pressure. If we determine which of the two phases is stable at desired

temperature and density, we should compare Helmholtz free energy of these phases.

Entropy, free energy and other related thermodynamic quantities are not average

functions of phase space coordinates of the system. Free energy is related to the partition

function Z(N, V, T ) and can be written as

F = −kBT lnZ(N, V, T ),

Z(N, V, T ) = ln(

∫dpNdrnexp−βH(PN ,rN )

ΛdNN !),

F = −kBT ln(

∫dpNdrnexp−βH(PN ,rN )

ΛdNN !),

Where d is dimensionality of the system. Thermal quantities can not be measured

directly in simulation. Similar problem occurs in real world: these quantities can not be

measured directly in real experiment. As pressure and energy are mechanical quantities,

they can be measured in a simulation [24]. By using energy, we determine the heat

19

capacity of the system by using following equation,

CV =∂E

∂T|V ,

The above relation use to calculate heat capacity in NVT ensemble at different

temperature.

2.3.1 Isothermal Isobaric Ensemble (NPT)

In isothermal isobaric ensemble, number of atoms, pressure and temperature are constant.

It describes a system that is in contact with a barostat P and a thermostat T . The

system exchanges heat with thermostat and exchanges volume with barostat. Total

number of particles N are constant, total energy E and volume V fluctuate at thermal

equilibrium. This ensemble used for computing equilibrium under isobaric condition. It is

used to study structural phase transition. NPT ensemble are most difficult to generate as

compared to other ensembles due to requirement that the total energy, volume and

pressure must fluctuate according to ensemble distribution [25]. Volume fluctuations can

be written by the following equation

∆V

V=

1√V.

If V approaches to infinity in thermodynamic limit, relative fluctuations in volume is

negligible and difference between NPT ensemble and NV T ensemble vanishes. In NPT

ensemble, volume can fluctuate. We introduce a potential function U(s, V ) that confines

the position s within volume V . Partition function in NV T ensemble can be written by

following equation,

Z(N, V, T ) =V N

N !h3N(2πmkBT )

32 ,

20

where T and V are applied external temperature and volume of system. The partition

function of NPT ensemble is given as,

Q(N,P, T ) =

∫ ∞0

Z(N, V, T )dV exp(−βPV ),

β =1

KbT,

Q(N,P, T ) =1

N !λ3N

∫ ∞0

V NdV exp(−βPV ),

Q(N,P, T ) =1

N !λ3N(βP )N+1

∫ ∞0

xNdx exp(−x),

Q(N,P, T ) =1

N !λ3N(βP )N+1N !,

Q(N,P, T ) =1

λ3N(βP )N+1,

Q(N,P, T ) partition function of isothermal isobaric ensemble.

2.4 Constant Temperature (NVT)

In molecular dynamics, we often encounter limitations and inconsistencies which arise

from the use of the micro-canonical ensemble corresponding to simulations at constant

energy. In particular, ordinary laboratory experiments are performed at constant P and

constant T but many molecular dynamics simulations are done at constant E and V .

However, the temperature can be related to the average of the kinetic energy

n∑1

p2i

2mi

=3

2NKT

In conventional constant-energy molecular dynamics, the T can only be obtained after

carrying out the simulations and calculating the average kinetic energy. To resolve this

situation, constant T and constant V (NVT) simulation methods have been developed

[25].

21

2.5 Coarse Grained Model

One atom represents a group of atoms in coarse grained model. In Coarse grained model

of water, hydrogen atoms ignore explicitly. In coarse grained modal of water, inter

molecular interaction are represents with spherically symmetric potential without

electrostatic interactions and hydrogen atoms. Isotropic potential cannot reproduce

energetic structure of water. Coarse grained modal reproduce phase behavior and

structure of water without using electrostatic interaction and hydrogen atoms. Its have

short range interactions. To study a large system, large computational resources and

increased timescale are required because they require many time steps. Coarse grained

model of water is used to reduce computational cost. Stilling–Weber potential has best

suited to simulate water as a coarse grained model, because they reduce computational

cost and improve simulation accuracy.

2.6 Stilling–Weber Potential

Stilling–Weber potential is a sum of two body term and three body term potential. It was

introduced in 1985 and gained significant popularity, Stilling–Weber potential is one of

first potential to be used for diamond lattice (e.g Si,GaAs,Ge,C) [27]. Description of

bonding in silicon requires that potential predicts that each atom has four neighbours in a

tetrahedral structure as most stable atomic configuration. Directional bonding is

introduced in Stilling–Weber potential through an explicit three body term of potential

energy expansion [1]. Three body term corrects that configuration when angles are not

tetrahedral. Stilling–Weber potential can be written as,

U(~r1, ~r2, .., ~rn) =∑ij

U2(~ri, ~rj) +∑i,j,k

∑i<j<k

U3(~ri, ~rj, ~rk),

22

Two dimensional potential is a pair potential and consist of three body term, electrostatic

repulsion due to ionic sizes, Coulomb interaction and charge dipole interactions to include

effects of electronic polarization [28]. General form of two body interactions can be

written by following equation,

U2(r) =Hij

rη+ZiZjr−

(αiZ2j + αjZ

2i )

r4exp

−ra ,

where Hij and η are strength and exponents of electrostatic repulsion. The Zi and αi

represent effective charge and electronic polarization of ion [28].

U2(~rij) = εf2(~rijσ

),

U3(~ri, ~rj, ~rk) = εf3(~riσ,~rjσ,~rkσ

)

where ε and σ are energy and length unit. The ε give function f2 depth, σ vanish function

f2(21/6) term, f2 is a function of scalar distance and function of f3 must posses full

transnational and rotational symmetry of potential. Reduced form of pair potential can

be written as,

f2(r) = {A(Br−P − r−q exp(r−a)−1

, r < a,

f2(r) = {0, r ≥ a,

where A, B, P, and a are positive constant. At r = a, f2(r) goes to zero, which is a

distinct advantage in any molecular dynamics simulation. Three body part of potential

can be written as,

f3(~ri, ~rj, ~rk) = h(~rij, ~rik, θjik) + h(~rji, ~rjk, θijk) + h(~rki, ~rkj, θikj), (2.7)

where θijk is the angle between ~rkj and ~rki subtended at vertex j, θikj is the angle

between ~rki and ~rkj at vertex k and θjik is the angle between ~rij and ~rik at vertex i.

Distance between the atoms ~rij and ~rik are less than cutoff distance a. Above equation

2.7 consist of two types of terms, radial and angular. Radial part represents bond

23

Table 2.1: Most satisfactory parameter set for water and use input data file of LAMMPS.

Parameters Values

A 7.049556277

B 0.6022245584

p 4

q 0

a 1.80

λ 23.15

ε 6.189 K

calmole−

σ 2.3925◦A

stretching and angular part shows bond bending. In three body interactions, radial part

remains same but angular part is different due to different values of angles [28]. Function

h can be written as,

h(~rij, ~rik, θjik) = {λ expΓ(~rij−a)−1+γ(~rik−a)−1 ×(cos θjik − cos θt)2, For : (~rij < a,~rik < a),

h(~rij, ~rik, θjik) = 0, For : (~rij ≥ a, ~rik ≥ a),

θt is a tetrahedral angle [27]. Parameter λ tunes the strength of tetrahedral drawback.

Parameters A, B, p, q, a, λ, and Γ to identify choice of f2 and f3. The parametrization of

λ take place in water on the basic of tetrahedral ordering. Tetrahedral order of water is

intermediate in between carbon and silicon. Tetrahedral strength of water is higher than

silicon, so we take higher value of λ for water. Most satisfactory parameter set for water

are presented in Table 2.1.

24

Chapter 3

LAMMPS Molecular Dynamics

Simulation Details

Simulation performed by using LAMMPS daily “Large scale Atomic Molecular Massively

Parallel Simulator” classical molecular dynamics program by Sandia National

Laboratories. It is capable of running on either single or parallel processor. Two inputs

are required in LAMMPS: input data file and input script. Input data file containing

information about atom types, bonds, angles, mass, simulation box and initial

coordinates [29]. Input script is divided into four parts: initialization, atom definition,

settings and run simulations. These four parts are explained below:

1. In initialization, parameters are set that atoms can be read from input data file.

2. In atom definition, initial trajectories, atom types, and molecular structure

information are read from input file.

3. In settings, molecular topology is defined. The simulation parameters, fixes, dump

25

and compute style.

4. In last step, force is minimizing to 0.005 (eV/◦A) and energy is minimizing to

0.0001 (eV).

3.1 Initialization

• The command “Units” set the style of units for simulations. This command

determines units of all quantities those specified in data file, input script, log file,

dump file and quantities output on the screen. Unit style chosen was metal as the

parameters for Stilling–Weber potential file provided with LAMMPS are

parameterized with metal unit. This unit style uses molecular dynamics units given

as,

• mass = gmole−1

• distance =◦A

• time = ps

• energy = eV

• temperature = K

• force = eV/◦A

• pressure = bars

• The command “Atom-style” define the style of atoms in simulation. Atom style

chosen for simulation of coarse grained model of ice is atomic. It does not read

26

information about angles and bonds from input data file. Stilling–Weber potential

can determine angles and bonds itself.

• The command “Dimension” set dimension of simulation which is three.

• The command “Boundary” set boundary conditions for simulation box in each

dimension. The p p p style is chosen that indicating box is periodic in x, y and z

dimensions. Periodic boundary condition means that particles enter one end of box

and renter from other end [29].

3.2 Atom Defination

1. The command “Read data” reads information from data file to run simulation in

LAMMPS. The data file for coarse grained model of ice contains masses, initial

coordinates, simulation box, bonds, bond types, angles and angles types. These

parameters define geometry of molecules.

2. The command “Group” identifies a collection of atoms belonging to a group. It

assigned a ID to each atom. Group ID used in fix, velocity and compute command

to act on those atom together. Atom of same type belongs to same type of

molecules and grouped together [29].

3.3 Settings

1. The command “Neighbour and neighb modify” set the parameters and affects

buildings and use pairwise neighbour list. LAMMPS employs to keep track nearby

atoms for computational efficiency. Within neighbour cuttoff distance all atoms

27

equal to their force cuttoff and skin distance in neighbour list. Bin style creates

neighbour list. Skin distance chose 2.0◦A, which avoid the dangerous builds that

may indicate problems in neighbor list. Neighbour modify command are “delay 10

and check 0 yes” which preserves the warning of resetting neighboring criteria

during energy and force minimization. It instructs LAMMPS to build neighbor list

on every step, if some atom has moved more than half skin distance at last build.

These parameters iteratively chosen to avoid dangerous build from occurring.

2. The command “Time step” set the time-step for molecular dynamics simulation as

0.002 pico second (ps). This time step must be so small, but avoid discretization

errors. This time large enough for total simulation duration to access desired

phenomena in useful way.

3. The command “Min style” specifies as force and energy minimization algorithm to

use. The conjugate gradient descent algorithm for energy minimization [29].

3.4 Settings of Force field coefficients

1. The command “Pair style and pair coefficient” set formula and coefficients that

LAAMPS use to calculate pairwise interaction. Pair potential defined between pairs

that are within cuttoff distance. The style chosen for simulation is “sw” and

location of Stilling–Weber parameter file is specified as an arguments for “pair

coeff” command, with that information identifies atom types in LAMMPS input

trajectory file [29].

28

3.5 Settings Fixes

Fix is any operation on in LAMMPS that applied to system during minimization,

thermostat, applying constraints to the atoms and enforcing boundary conditions.

1. The command “Fix npt” performs time integration on Nose Hoover Hamiltonian

equation of motions. It designed to generate velocities and positions sampled from

NPT ensemble. Thermostatting is gained by adding dynamics variables those are

coupled to particles velocities. LAMMPS creates a chain of three thermostats that

coupled to particles thermostat for which equation of motion describes in Sandia.

Thermostat applied only transnational degrees of freedom by using press and

temperature argument, with desired pressure and temperature at each Molecular

dynamics step corresponding to ramped value during run from Pstart to Pstop and

Tstart to Tstop. Damping parameter Tdamp, Pdamp determine how temperature and

pressure is relaxed. Desired temperature and pressure is constant with Tstart, Tstop

and Pstart, Pstop. Temperature is specified at 80 K in simulation for coarse grained

model of ice [29].

3.6 Settings Output Options

1. The command “Thermo and thermostyle” use to compute and print

thermodynamics information on time steps are multiple of 5000000 at start and end

of simulation. Thermostyle used to printing thermodynamic data to log file. Custom

format set essential data in a specified order, step, etotal, ftotal, temp, press, and

prints time steps total energy, total forces, temperature and pressure respectively.

2. The command “Dump” used to dump of specific atom quantities to an input file at

29

every time step. Custom style maintain in a specified order, id, type, x, y, and z

print the coordinates. These trajectories and types of atoms are visualized in Visual

molecular dynamic [29].

3.7 Running Simulation

1. The command “Minimize” perform energy and force minimization on system by

iteratively adjusting atom trajectories. For minimize command exp−6, exp−6, 2000,

2000 refer to stopping tolerance for force, energy, maximum number of iterations

and force evaluations. When first criteria is matched energy change between

successive iterations divided by energy magnitude is less than or equal to tolerance.

Second criteria is matched when final force on any component of any atom does not

exceed 10−6 eV/◦A.

2. The command “Run” run molecular dynamics simulations for a specific number of

time steps. Simulation run from 50000000 time step and depends upon number of

atoms [29].

30

Chapter 4

Results and Disscussion

4.1 Structural Properties

We have characterized structural properties of our models through lattice optimization,

partial radial distribution function, bond length distribution and bond angle distribution

(BAD). Structural properties are very important for understanding the material

properties at microscopic level.

Energy is a function of the lattice constant, so energy should change when the lattice

constant varied. By using this procedure, we obtained an optimized lattice constant at

which energy becomes minimum as shown in Fig. 4.1. Optimized values of lattice

constant and bond length are presented in Table 1 for the coarse grained model of ice.

The exact minimum value of energy is found by least square fit method in gnu-plot

software by using following equation.

f(x) = ax2 + bx+ c

31

Figure 4.1: Optimized lattice constant and total energy of coarse grained model of LDA

ice. Otimized lattice constant 6.20◦A which is close to experimental value 6.36

◦A.

Why we are using least square equation to fit parabola and get minimum value of energy.

When we expand Taylor series to the second order term and get symmetric form of

energy and value of a lattice constant that agrees minimum value of energy. If we expand

to the first order term in the Taylor series expansion, we obtain a straight line shape and

do not get minimum value of energy corresponding optimized lattice constant.

Figure 4.2: RDF of different coarse grained models of LDA ice at zero pressure and matched

with refrence LDA ice model.

Simulations of our reference coarse grained LDA ice model at zero pressure agrees well

32

the obtained my Average radial distribution function (RDF) of different models.

Similarly, excellent agreement is obtained for average bond length, volume density and

number density of different LDA models. Volume and number density of different models

are presented in Table 2. The RDF of different models are shown in Fig. 4.2, heights of

first and second shells of 216, 512 and 1000 atom models are matched to our reference

LDA ice 216 atoms model.

Figure 4.3: Bond length distribution

of LDA ice. Maximum bond length

between two oxygen atoms is 2.68◦A.

Figure 4.4: Bond angle distribution

of LDA ice. Tetrahedral angle is

109.5 ◦.

Bond length distribution (BLD) weas calculated for all models which reveals that in

coarse grained model of LDA ice, the mean bond length was 2.68◦A. Similarly mean bond

angle was found to be close to a tetrahedral angle 104◦ in all models through bond angle

distribution. BLD and BAD of our models are shown in Figs. 4.3 and 4.4.

4.2 Radial Distribution at Temperature 80 K

At 80 K, average RDF of different models of LDA ice were simulated to the result were

compared with the experimental RDF of LDA ice. The effects of size of coarse grained

33

Table 4.1: Optimized lattice constant and bond length of LDA ice.

Structure Experimental

lattice

constant(◦A)

Optimized lattice

constant (◦A)

Experimental

Bond length (◦A)

Optimized Bond

length (◦A)

Oxygen

(H2O)

6.36 6.20 ± 0.3154 2.75 2.68 ± 2.005

Table 4.2: Volume and number density of LDA ice coarse grained models at 80 K. Experi-

mental volume density of LDA ice is 0.943 gcm−3 at 80 K. Compare experimental value of

volume density with calculated value.

No. of atoms Compound Calculated Volume

Density (g/cm3)

Number density

(m−3)

216 H2O 0.954 9.667 ×1028

512 H2O 0.966 9.700 ×1028

1000 H2O 0.990 9.946 ×1028

8000 H2O 0.982 9.862 ×1028

27000 H2O 0.973 9.770 ×1028

34

Figure 4.5: Average RDF of different size

coarse grained LDA model of ice at 80 K

and compression to the second shell of

experimental LDA ice.

0 10 20 30 40 500.9

1

1.1

1.2

1.3

1.4

P (Kbars)

ρ (g

cm−

3 )

CompressionD−compressionD−Compression

Figure 4.6: Density of LDA ice at differ-

ent pressures during compression and de-

compression. HDA ice does not sustain

their phase during decompression.

models are important in Stilling–Weber potential, by increasing size of coarse grained

model of LDA ice, accuracy of simulation to the experimental RDF of LDA ice also

improved. The RDF of 1000, 8000 and 27000 of atoms gave us the most accurate results

with their RDF overlapping to the second shell of experimental LDA ice. Average RDF of

our different models of LDA ice are shown in Fig. 4.5.

4.3 Low Density Amorphous Ice of Different Models

The average RDF of different models of water at 80 K was calculated. First minima of

oxygen-oxygen (O-O) mean RDF of TIP4P model of water was found to be at 3.06◦A and

the average RDF of coarse grained model of water, ab-initio model of water and

experimental average RDF of water had their first minima at 3.37◦A. The simulation of

second shell of ab-initio model of water was more coordinate with the second shell of

experimental O-O average RDF. The simulation of average RDF of the coarse grained

35

Figure 4.7: RDF of LDA ice different models and comperision to the second shell of exper-

imental RDF of LDA ice model at 80 K.

model of water to the second shell of experimental average RDF was less matched than

ab-initio model of water. The simulation of ab-initio model of water was more emulate

than TIP4P model of water. Stilling–Weber potential improved the simulation perfection

to the second shell of experimental average RDF as compared to other classical potentials.

4.4 Low Density Amorphous Ice at High Pressures

Density and volume have an inverse relation. By increasing pressure, volume should

decrease and mass density increases. At zero pressure, structure of LDA ice remains

stable. Gradually compressing the LDA ice by increasing pressure from 0 Kbars to 22

Kbars, we observed abrupt increase in density of ice at 18 Kbars are shown in Fig. 4.6.

Due to compression of unit cell, O-O distance becomes slightly shorter, but coordination

number of first shell remains unchanged from ideal value of 4. This observation indicates

that local geometry of our coarse grained model of LDA ice is not affected significantly,

36

only long range order has been partially disrupted. In the new structure, the center of

second shell at 3.4◦A and effect of pressure reveals at average angle uniformly in second

and third shell of models pair correlation function that we have considered are shown in

Fig. 4.7. Pressure effects appears in second and third shell due to the angular part, while

the radial part remain the same in three body interaction of Stilling–Weber potential. We

do not obtain high density amorphous ice phase and density by slowly releasing pressure

to 0 Kbars as shown in Figs. 4.6, and 4.8.

Figure 4.8: RDF of LDA ice during com-

pression at different pressures. Pressure

increase from down to upward direction.

At 18 Kbars, LDA ice change into HDA

ice.

Figure 4.9: RDF of LDA ice during De-

compression at different pressures. Pres-

sure decrease from top to bottom direc-

tion. At 0 Kbars, HDA ice change into

LDA ice.

4.5 Conclusions

LDA coarse grained models of ice have resemblance with single specie model of silicon.

We have studied structural properties of coarse grained model of LDA ice and observed

37

that mean bond length was 2.68◦A between O-O atoms and that bond angle was

approximately equal to tetrahedral angle. In average RDF of coarse grained model of

LDA ice at 80 K, heights of first and second shell exactly match to the experimental LDA

ice. Increasing the size of coarse grained models lead to the simulation perfection being

improved to the second shell. Coarse grained model of water had more perfection than

TIP4P model of water and less emulate as compared to ab-initio model of water. In

coarse grained model of LDA ice, the phase transitions of ice cannot be observed by using

Stilling–Weber potential at different pressures.

38

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