Molecular Dynamics of Disordered Ice by Ha z Ghulam Abbas · Molecular Dynamics of Disordered Ice...
Transcript of Molecular Dynamics of Disordered Ice by Ha z Ghulam Abbas · Molecular Dynamics of Disordered Ice...
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
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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
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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.
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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
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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
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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
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4.3 Low Density Amorphous Ice of Different Models . . . . . . . . . . . . . . . 35
4.4 Low Density Amorphous Ice at High Pressures . . . . . . . . . . . . . . . . 36
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
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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
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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.
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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.
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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.
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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
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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
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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.
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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
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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.
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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.
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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),
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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)〉,
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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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