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Diffusion-Controlled Kinetic Evolution of Radiation-Induced Point Defects in Polycrystalline Uo2 from Atomic-Level Simulation

Permanent Link: http://ufdc.ufl.edu/UFE0042191/00001

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Title: Diffusion-Controlled Kinetic Evolution of Radiation-Induced Point Defects in Polycrystalline Uo2 from Atomic-Level Simulation
Physical Description: 1 online resource (88 p.)
Language: english
Creator: Tsai, Kun-Ta
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: Materials Science and Engineering -- Dissertations, Academic -- UF
Genre: Materials Science and Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: DIFFUSION-CONTROLLED KINETIC EVOLUTION OF RADIATION-INDUCED POINT DEFECTS IN POLYCRYSTALLINE UO2 FROM ATOMIC-LEVEL SIMULATION Kun-Ta Tsai (352)871-2382 kunta0803@gmail.com Materials Science and Engineering Simon R. Phillpot Master of Science Generating nuclear energy is one of the solutions to the increasing demand for energy, and it is getting more reliable with modern design of nuclear reactors. In addition, the superiority of non-CO2-emitting source makes nuclear energy more promising in the future. In most operating nuclear reactors, fluorite-structured UO2 is used as the fuel. After irradiation, UO2 undergoes structural damage by forming high concentration of point defects, which can limit the lifetime of the fuel. In order to enhance the radiation tolerance of UO2, it is of great interest to understand the fundamental mechanism of intrinsic point defects evolution under different conditions. Rather than a single crystal, this thesis focuses on the more representative polycrystalline UO2, and uses molecular dynamics (MD) as the modeling tool. This study provides a better understanding of defects evolution in the kinetic phase at the atomistic scale. Meanwhile, this thesis contributes to the exploration of radiation damage for nuclear fuel which is widely used but least understood.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Kun-Ta Tsai.
Thesis: Thesis (M.S.)--University of Florida, 2010.
Local: Adviser: Phillpot, Simon R.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042191:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042191/00001

Material Information

Title: Diffusion-Controlled Kinetic Evolution of Radiation-Induced Point Defects in Polycrystalline Uo2 from Atomic-Level Simulation
Physical Description: 1 online resource (88 p.)
Language: english
Creator: Tsai, Kun-Ta
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: Materials Science and Engineering -- Dissertations, Academic -- UF
Genre: Materials Science and Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: DIFFUSION-CONTROLLED KINETIC EVOLUTION OF RADIATION-INDUCED POINT DEFECTS IN POLYCRYSTALLINE UO2 FROM ATOMIC-LEVEL SIMULATION Kun-Ta Tsai (352)871-2382 kunta0803@gmail.com Materials Science and Engineering Simon R. Phillpot Master of Science Generating nuclear energy is one of the solutions to the increasing demand for energy, and it is getting more reliable with modern design of nuclear reactors. In addition, the superiority of non-CO2-emitting source makes nuclear energy more promising in the future. In most operating nuclear reactors, fluorite-structured UO2 is used as the fuel. After irradiation, UO2 undergoes structural damage by forming high concentration of point defects, which can limit the lifetime of the fuel. In order to enhance the radiation tolerance of UO2, it is of great interest to understand the fundamental mechanism of intrinsic point defects evolution under different conditions. Rather than a single crystal, this thesis focuses on the more representative polycrystalline UO2, and uses molecular dynamics (MD) as the modeling tool. This study provides a better understanding of defects evolution in the kinetic phase at the atomistic scale. Meanwhile, this thesis contributes to the exploration of radiation damage for nuclear fuel which is widely used but least understood.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Kun-Ta Tsai.
Thesis: Thesis (M.S.)--University of Florida, 2010.
Local: Adviser: Phillpot, Simon R.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042191:00001


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DIFFUSION-CONTROLLED KINETIC EVOLUTION OF RADIATION-INDUCED
POINT DEFECTS IN POLYCRYSTALLINE UO2 FROM ATOMIC-LEVEL SIMULATION




















By

KUN-TA TSAI


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2010

































2010 Kun-Ta Tsai





















To my family with love and gratitude









ACKNOWLEDGEMENTS

First of all, I would like to express my sincerest gratitude to my advisor Prof. Simon

R. Phillpot for his invaluable guidance and encouragement. In the beginning, for me as a

non-native speaker of English, the Santa Claus' British accent distances us. Nonetheless,

throughout my master's study, I am subtly but deeply influenced by his wisdom and

enthusiasm towards science blended with a great sense of humor. He likes to share his

knowledge by throwing open questions and to give us the liberty to brainstorm. It is my

greatest honor to have him as my advisor. I would also like to express my deepest

appreciation to my committee members Prof. Susan B. Sinnott and Prof. Juan C. Nino

for providing valuable suggestions to my research work.

I feel extremely fortunate to work with a wonderful team, the Computational

Materials Science Focus Group. I want to specially thank Dr. Dilpuneet Aidhy for his kind

help on my research work. This thesis benefited immensely and would not be

accomplished without him. I am also thankful to Dr. Tao Liang, Dr. Alex Chernatinsky, Dr.

Rakesh Behera, Dr. Haixuan Xu, Mr. Donghwa Lee, Mr. Priyank Shukla, Mr. Chan-woo

Lee, Mr. Donghyun Kim, and Mr. Eric Bucholz for their friendly assistance. I want to

mention my two dearest senior friends Tze-Ray Shan and Yu-Ting Cheng. Many thanks

to both of them for constant support and encouragement in my life inside and outside the

school.

Finally, I would like to thank my family for continuous support with unconditional

love. This thesis is dedicated to them.









TABLE OF CONTENTS
page

A C KN O W LEDG EM ENTS ......... .................. .......... ....................... ............... 4

LIS T O F TA B LE S .................................................................................................. 7

LIS T O F F IG U R E S .................................................................. 8

A B ST R A C T .................... ........................................................................... ...... 10

CHAPTER

1 IN T R O D U C T IO N ..................................................................................................... 1 2

1.1 Sources of Energy ................................. ............................. 12
1.2 Nuclear Energy .............................................................................. ...... .......... 14
1.3 Radioactive Decay and Nuclear Fission ................ ................................. 18
1.4 Motivation and Objective ......... ........... ....................... 22
1.5 Beauty of Simulation .................... ............ ........ ......... 23

2 D E FEC TS IN SO LID S ........... ......... ......... ............................... ............... 24

2.1 Introduction ............. .. ......... ................................................................ 24
2.2 Intrinsic Point Defects .................. ..... ........................... ................... 24
2 .2 .1 S chottky D efect.................................................................... ......... 25
2.2.2 Frenkel Defect ................................ .. ........... ..... .............. 25
2.2.3 Thermodynamics of Point Defects: Equilibrium Concentration .............. 26
2.3 G rain Boundary...................................................... ............... 27
2.4 Atom Movements in Materials................... ...... .................... 28
2.4.1 Diffusion Mechanisms................. .......................... 28
2.4.2 Atomic Theory of Diffusion ................. ............... .................... .. 30
2.5 Crystallography of UO2 ............... ..... ........................... 34
2.6 R radiation D am age ........... ........ ......... .......... ............... .............. 36

3 SIMULATION METHODOLOGY. ................. ... ....................................... 39

3.1 Simulation Methodologies......................... ....... ......... 39
3.2 Molecular Dynamics Simulation ........................ ........ ................ 41
3.3 General MD Algorithm ............... .... ........................ 41
3.4 Periodic Boundary Condition .................. ............................ 43
3.5 Interatomic Interactions ................ ............................... 44
3.5.1 Long-Range Interactions....................... .... .......................... 45
3.5.2 Short-Range Interactions................................................................ 46
3.5.3 Thermodynamic Conditions (Ensembles) for Simulations ....................... 47
3 .5 .4 T herm ostat ... ............................................................................. 49
3 .5 .5 B a ro stat .......................................................... ............................. .. 4 9
3.6 Conventional Radiation Damage Method by MD Simulations......................... 50









4 GRAIN BOUNDARY INFLUENCE ON RADIATION DAMAGE IN UO2..................... 52

4.1 Kinetically-Evolving Irradiation-Induced Defects Method by MD Simulation..... 52
4.2 Simulation Methodology ............ ............. ........................ 52
4.2.1 MD Simulation ......... .................... ......................... 52
4.2.2 Defects Analysis M ethods............................................... ......... ...... 54
4 .3 R e s u lts ................................ ... .................................................. 5 6
4.3.1 Frenkel-Pair Defects in Single-Crystal UO2 ................ ......... ......... 56
4.3.2 Frenkel-Pair Defects in Polycrystalline U02..................... .... .......... 58
4.3.3 Grain Boundary Source/Sink Behaviors for Point Defects ................... 63
4.3.4 Vacancy Clustering: V-40 Clusters ............. ........... .............. 77
4.3.5 Vacancy Clustering: Schottky Defects ............ .......... .......... ..... 74
4.3.6 Interstitial Clustering: Cuboctahedral Clusters ............. ..... .......... 75

5 CONCLUSIONS AND FUTURE WORK............................................. 79

5 .1 C o nclusio ns ..... ................................................. ................ 79
5.2 Future W ork........................................... .......... 82

LIST O F R EFER EN C ES .................................................. ....................... ............... 83

BIOGRAPHICAL SKETCH ...... .................. .................. 88









LIST OF TABLES

page

Potential parameters used in the present work................................................... 48


Table

3-1









LIST OF FIGURES


Figure page


1-1 1973 and 2007 fuel shares of total primary energy supply ...... ....................... 12

1-2 Evolution from 1971 to 2007 of world total primary energy supply by fuel.......... 13

1-3 World net nuclear electric power generation (1980-2006) .............................. 16

1-4 W orld map for commercial nuclear power plants ........................................... 17

2-1 Point defects: (a) vacancy, and (b) interstitial .................................. ............... 25

2-2 Point defects in (a) Schottky defect, and (b) Frenkel defect ...... ........................ 26

2-3 Schematic representation of three diffusion mechanisms.............................. 29

2-4 Fluorite crystal structure. ............... ........ .. .. .... ..... ................ ............... 35

2-5 Three different symmetry operations for the fluorite crystal structure ................ 36

2-6 Defect evolution during ballistic and kinetic phases of the radiation damage..... 37

3-1 Hierarchical multi-scale approach to computational material engineering. ......... 40

3-2 Periodic boundary conditions of a fluorite structure supercell........................... 44

3-3 Inter-ionic potential as a function of distance between U+2.4 and 0-1.2 ions.... 48

4-1 Fully equilibrated polycrystalline UO2 microstructure. ............. ...... ........ 53

4-2 Schematic representations of the common neighbor analysis (CNA)................. 54

4-3 Schematic representations of the lattice matching analysis (LMA)..................... 55

4-4 Evolution of O FPs in the singel crystal. ................................. .. ... ............. 57

4-5 Snapshot of evolution of point defect in polycrystal ................. .......... ......... 62

4-6 Evolution of O FPs in the polycrystal ........................................... 63

4-7 Snapshots showing the GB being the source for the O vacancy..................... 65

4-8 Close-up views of the GB being the source for the O vacancy...................... 66

4-9 Progressive snapshots showing the GB being the sink for the O vacancy......... 67

4-10 Close-up views of the GB being the sink for the O vacancy............................. 68









4-11 S ingle point alignm ent offsets................................................. .... .. ............... 69

4-12 Snapshots showing the GB being the sink for the O interstitial ........... .......... 71

4-13 Close-up views of the GB being the sink for the O interstitial........................... 72

4-14 Three different configurations of Schottky defects ................. ........ ........... 74

4-15 Snapshot showing Schottky defects ...... .... ............... ............. 75

4-16 Formation process of COT clusters............................... ............. 76

4-17 Snapshot showing COT clusters .............................................. 77

4-18 C onfigurations of V -40 clusters............................................... .... ................. 78

4-19 Snapshot showing V-40 oxygen clusters................ ..... ................. 78









Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science



DIFFUSION-CONTROLLED KINETIC EVOLUTION OF RADIATION-INDUCED
POINT DEFECTS IN POLYCRYSTALLINE UO2 FROM ATOMIC-LEVEL SIMULATION

By

Kun-Ta Tsai

August 2010

Chair: Simon R. Phillpot
Major: Materials Science and Engineering

Nuclear energy is capable of providing people with energy for some billions of

years if used properly. In most operating nuclear reactors, fluorite-structured UO2 is used

as the fuel. After irradiation, UO2 undergoes structural damage by forming high

concentration of point defects, which can limit the lifetime of the fuel. In order to enhance

the radiation tolerance of UO2, it is of great interest to understand the fundamental

mechanism of intrinsic point defects evolution under different conditions.

Rather than a single crystal, this thesis focuses on the more representative

polycrystalline UO2, and uses molecular dynamics (MD) as the modeling tool. This study

provides a better understanding of defects evolution in the kinetic phase at the atomistic

scale.

In addition to spontaneous vacancy-interstitial recombination events, some of the

point defects are found to have interactions with GBs. These source/sink behaviors take

place by a set of interstitialcy diffusion processes. The diffusion direction is ascertained

to follow the primary diffusion direction <001> for the fluorite-based structure with the

reasonable oxygen vacancy diffusivity on the order of 10-6 cm2/s (10-4 A2/s). A mean









square displacement calculation confirms that atoms may realign after the diffusion

activity, supporting the concept that damage can be healed by GBs in a macroscopic

sense.

It is also found that the presence of defects on the U sub-lattice can lead to the

formation of clusters, which make U02 less radiation tolerant. In addition, the equilibrium

concentration of O defects is much larger in polycrystal than that in single crystal. This

can be attributed to GBs possibly supplying O defects into the bulk.








CHAPTER 1
INTRODUCTION

1.1 Sources of Energy
"Energy can be neither created nor destroyed," say our science books, "but can

only be transferred from one form to another." Mankind has been engaged with this

transformation since life began on the planet. Since industrialization, humans have

utilized a variety of energies and explored more. With growth accelerating in developing

countries, demands on energy resources are being stretching to new limits. In recent

decades, the urgent need for energy has created a global movement to exploit all

potential materials. On the other hand, materials are used to produce energy or enable

energy to be converted into other useful forms. Energy and materials therefore have a

continual and mutually enriching relationship.

The choice of materials for energy generation depends on the availability and

accessibility of the source. Fig. 1-1 gives pie charts of the fraction of the total primary

1973 2007
Combustible Combustible
Hydro renewables renewables
1.8% & waste Other Hydro & waste
10.6% 2.2% 9.8% Other*
Nuclear 0.1% Nuclear 0.70
0.9% Coal/peat 5.9% Coal/peat
24.5% 26.5%
Gas
16.0% s2 G 34 %
Oil 20.9% Oil
--- 46.1% r 34.0%

6 115 Mtoe 12 029 Mtoe

Figure 1-1. 1973 and 2007 fuel shares of total primary energy supply (Mtoe, million tonne
of oil equivalent). (*Others include geothermal, solar, wind, heat, etc.)1









energy from various sources supply in 1973 and 2007.1 It was estimated by the

International Energy Agency (IEA) that in 2007 primary sources of energy consisted of oil

34.0 %, coal/peat 26.5 %, and natural gas 20.9 %, amounting to an 81.4 % share of

fossil fuels in primary energy consumption in the world. This indicates the world's

significant dependence on fossil fuels due to their continuing widespread availability and

the large developed infrastructure. Yet fossil fuels are non-renewable resources and their

production raises many environmental concerns; thus there has been a gradual

movement towards cleaner fuels. Fig. 1-2 shows the evolution from 1971 to 2007 of

world total primary energy supply by fuel.1 It should be noted that hydro power was

surpassed by nuclear power in total electrical generation in 1984, and that nuclear the

share has increased to 5 % of the total energy supply over the last 34 years. Along with

increasing awareness of environmental issues, nuclear energy is expected to play an

important role in the energy portfolio in the coming decades.2

14000
12000
10000
8000

4 000
4000
2000
0
1971 1975 1979 1983 1987 1991 1995 1999 2003 2007

Coal/peat M Oil I Gas Nuclear
-1 Hydro Combustible renewables & waste Other*

Figure 1-2. Evolution from 1971 to 2007 of world total primary energy supply by fuel
(Mtoe).1









1.2 Nuclear Energy

Ernest Rutherford is one of the central figures in the exploration of nuclear physics.

He was awarded the 1908 Nobel Prize in Chemistry for investigations into the

disintegration of the elements and the chemistry of radioactive substances.3 ".... Taking

the atomic weight of the emanation as 222, one gram of the emanation emits during its

life 2x 109 gram calories of heat. This evolution of heat is enormous compared with that

emitted in any known chemical reaction. .", excerpted from Rutherford's article in

Encyclopedia Britannica (1910).4 The basic energy fact is that the fission of a uranium

atom generates 10 million times the energy as that produced by burning an atom of

carbon from coal. This concept that a large amount of energy can be released by the

disintegration of an atom was soon brought to public attention.

The first time that electricity was generated by a nuclear reactor occurred in an

Idaho experimental station.5 The world's first nuclear power plant was operational in

Russia in 1954 generating around 5 megawatts (MW) of electric power.6 In the following

years, other nuclear stations were opened in the U.S. and the U.K., etc. In the early

years, nuclear capacity increased rapidly, rising from less than 1 gigawatt (GW) in 1960

to 100 GW in the late 1970s, and 300 GW in the late 1980s.7 Since then, the increase in

worldwide capacity started to slow down due to consecutive major nuclear accidents in

Three Mile Island (1979) and Chernobyl (1986). More and more anti-nuclear movements

arose based on the fear of possible nuclear disasters, radiation, and waste production.

Despite those controversies, many countries remain active in developing nuclear power,

including China, India, Japan, and Republic of Korea.8 With modern designs to meet

strict safety standards, nuclear reactors are becoming much more reliable. In addition,

the superiority of the non-C02-emitting source makes nuclear energy more promising.









Nuclear power energy can be generated by two types of nuclear reactions: fission

and fusion. Fission-based nuclear technology is well established, whereas fusion

technology is expected to become commercially available in the second half of the

century. The most widely used material for fission is uranium. Natural uranium is almost

entirely composed of two isotopes, U-238 (99.3 %) and U-235 (0.7 %), indicating that

U-238 is about 140 times more prevalent than U-235. However, only U-235 can directly

sustain the fission chain reaction as a nuclear fuel source; U-238 is not fissionable with

thermal neutrons.

A nuclear power reactor has a large number of fuel rods in the core. Every rod

contains pellets of uranium oxide. When an atom of U-235 is hit by a neutron, it can

undergo a fission reaction, producing two fission fragments and other free neutrons with

a very high speed. In steady-state operation of the reactor, the free neutron is absorbed

by another U-235 atom leading to another fission. The kinetic energy of other fission

products is converted to thermal energy when they are stopped by nearby atoms. The

heat then is absorbed by water, which becomes steam to drive the turbines to generate

the electricity. The power level of the reactor can be controlled by the amount of steam

withdrawn. In addition, the control rods which absorb neutrons can also be moved in and

out of the reactor to avoid the reactor temperature getting too high.

Some analysts suggest that oil wells might be depleted within 70-80 years.9 Natural

gas might run out a little later. Yet present reactors that use only the U-235 in natural

uranium will likely have fuel supplies for some hundreds of years. Bernard Cohen, a

Newton scholar in history of science, even predicts that with breeder reactors, we can

have plenty of energy for some billions of years.10









If more efficient usage of neutrons in the reactor is designed, U-238 could be

converted to plutonium, which produces more fissionable materials. This kind of design

is called a breeder reactor.7 Thus, with the usage of U-238 as the fuel in breeder reactors,

fuel resources based on breeder reactors could provide adequate energy for billions of

years. Commercial breeder reactors have been deployed in several countries, such as

the U.S., U.K., France, Russia, Japan, and India.11 Breeder reactors can be regarded as

a kind of resource instead of just a reserve. Because their cost is a lot higher than

present reactors, and because large uranium reserves have also been discovered in

recent years, it is perhaps unlikely that we will see breeder reactors all over the world

soon. Nevertheless, in the very long term, breeder reactors will be used since they

supply about 100 times as much energy from a kilogram of uranium as do present

reactors. This will make the present stock of uranium go much farther.



20000 25
I ITotal
Nuclear

15000- E E u 20


0 CD
l _-- -

S10000- 15


S5000 10
+s.


on.1.1


1a111


10E1MRMMM11 1


1980 1985 1990 1995 2000 2005
Year


Figurel-3. World net nuclear electric power generation (1980-2006).12









Fig. 1-3 demonstrates the world net nuclear electric power generation from 1980 to

2006. The nuclear share of electricity net generation increased from 4.5 % in 1973 to

around 20 % in 1990, since then it has been approximately constant.12 According to

International Atomic Energy Agency (IAEA), there are 436 commercial nuclear power

plants, operating in 31 countries, as illustrated in Fig. 1-4.13 Not all the world is employing

nuclear energy. France generates 78 % of its electricity from nuclear reactors, most of

the rest being hydroelectric. Japan is close to 30 % and increasing steadily due to their

lack of domestic coal and oil.14 Ten African countries are now in pursuit of their own

nuclear plants because wind and solar solutions are not reliable enough.















Operating reactors, building new
SOperating reactors, planning for new
No reactors, building new
,No reactors, planning for new
|Operating reactors, stable
Operating reactors, considering phase-out
Civil nuclear power is illegal
No reactors


Figurel-4. World map for commercial nuclear power plants (research reactors not
included).13

To sum up, in the short term, probably the next hundred years, there is so much

uranium that no one can profitably prospect for more. In the medium term, breeder









reactors will extend the energy obtained per kilogram of uranium by a factor of about 100.

In the very long term, plenty of uranium can be extracted from seawater for a few billion

years.

After being no longer efficient in sustaining a nuclear reaction in the nuclear reactor

(usually at the nuclear power plant), spent nuclear fuel is put in storage condition. Spent

nuclear fuel is highly radioactive and potentially very harmful, such as causing

incapacitation and death. The radioactivity of nuclear wastes decreases with the

passage of time through radioactive decay. For radioactive materials, the amount of time

needed to decrease the radioactivity to one-half the original amount is called the

radioactive half-life.15 Many of the radioactive elements in spent fuel have long-lives. For

example, U-235 has the half-life of 713,000,000 years, and U-238 has the half-life of

4500,000,000 years.16 With these long-lived radioactive elements, nuclear fuel must be

isolated and controlled for thousands of years by a barrier or radiation protection shield.

Department of Energy (DOE) is developing plans for a permanent disposal facility for

spent fuel at Yucca Mountain and Nevada.15 This has aroused some controversy,

particularly with state and local authorities. Several complementary measures are still

required for the long-term development of nuclear energy.

1.3 Radioactive Decay and Nuclear Fission

Spontaneous radioactivity was first recognized by a French physicist, Henri

Becquerel, in 1896 while investigating phosphorescence in uranium salt. Along with

Marie Curie and Pierre Curie, who jointly found another two radioactive elements, radium

and polonium in 1898, they won the 1903 Noble Prize in Physics.17

The main processes of radioactivity involve four types of decay; alpha ( a) decay,

beta ( 3) decay, gamma ( y) decay, and neutron capture.









Alpha (a) decay: Alpha decay is the process in which a nucleus ejects an alpha particle,

which is essentially the helium-4 nucleus (4He2). For example, Eq. 1-118 represents the

daughter nucleus (234Th) emitted by the parent atom (2 2U), producing an alpha

particle.

23U234 Th+H (1-1)

Alpha decay is the most common type of nuclear decay for elements with atomic

number greater than 82. This is because the greatest bonding energy is lost per nucleon

and the alpha particle is an especially low energy arrangement of two protons and two

neutrons. The typical kinetic energy of alpha particles is about 5 MeV with a speed of

15,000 km/s (i.e. 5% of the speed of light). Being relatively slow, heavy, and positively

charged, alpha particles have a short free path in length that is so small that they easily

interact with other atoms by losing their kinetic energy within a very short distance,

usually a few centimeters of air. Therefore, external alpha radiation is generally not

harmful.

Beta (,p) decay: Beta decay is the transformation of a proton into a neutron with

emission of a positron, or the transformation of a neutron into a proton with emission of

an electron. It occurs when there are too many protons or neutrons in a nucleus.19 There

are three kinds of beta decay, beta plus ( P/) decay, beta minus ( / ) decay, and electron

capture (K-capture).

In p+ decay, the weak nuclear force converts a proton into a neutron while emitting

a positron ( e+) and an electron neutrino ( ve):

p-n + e + ve (1-2)

Neutrinos are electrically neutral elementary particles that travel at the speed of light.









In /- decay, a neutron transforms into a proton by emitting an electron ( e-) and an

electron antineutrino (Ve):

0n->lp+e +Ve (1-3)

Antineutrinos are antiparticles of neutrinos.

The third type of beta decay is electron capture, which is where 8+ decay

accompanies a shell electron throughout the decay process. It can be expressed as:

,p+e ->n +ve (1-4)

This decay is also called K-capture because the innermost electron lies in the K-shell,

which has the highest probability to interact with the nucleus.

To sum up, within all types of beta decay, the atomic number of the parent nucleus

is different from that of the daughter nucleus while the mass number is the same. All

three general types of beta decay of 43Tc20 and |9Rh21 are represented in Eq. 1-5:
99M o 6+ 99Tc 6- 499Rue EC 99Rh
Mo Tc^ -^Ru Rh (1-5)

The kinetic energy of beta particles depends on the parent and daughter nuclear states

in the decay chain, typically ranging from a few keV to a few tens of MeV. The most

energetic beta particles can reach speeds approaching the speed of light. In addition,

because they are less charged and much smaller than alpha particles, beta particles

generally travel further into tissues, which can cause cells damage at the molecular level.

Gamma (y) decay: Gamma decay refers to as a nucleus jumping down from a higher

energy state to a lower one by emitting electromagnetic radiation (photons), which is

known for the gamma ray:

2Ni -28Ni+ y (1-6)









There is no change in the number of protons and neutrons between the parent and

daughter nuclei throughout the process, but the emitted gamma rays are characteristic

for each decay. Gamma rays typically have energies in the range from 100k keV to 10

MeV and wavelength less than 10 pm, which is often smaller than an atom. Therefore,

they are usually more penetrating, and can cause biological damage.

Neutron capture: Neutrons are electrically neutral, so more easily interact with a

nucleus than charged particles. In neutron capture, a nucleus collides with a neutron to

merge into a heavier one with a higher energy state.22 The excited nucleus quickly

decays to the ground state through emission of gamma rays:

on+ 9Au-198Au* (1-7)
o 79 79

198Au* -19Au +y (1-8)


In this type of decay, the mass number rises by one, which is crucial for the formation of

heavier elements in the cosmos. Nuclei with mass numbers greater than 56 can be

created by neutron capture but are unlikely to be formed by thermonuclear reactions (i.e.

nuclear fusion).

Nuclear fission: Nuclear fission is a form of nuclear transmutation in which an atom is

induced to split into two lighter elements with emission of free neutrons:

on+ 2 U--4Ba+38Kr +3on (1-9)

In the event, two more new neutrons are produced, which make the reaction be a

self-sustaining chain reaction in the nuclear reactor. The energy released by the fission

fragments and neutrons can be roughly approximated from the mass differences21:

Q =[m(235U)-m(144Ba)- m(89Kr) 2m]c2 = 173 MeV (1-10)

This simple calculation is very close to real results (180.5 MeV), including the kinetic









energy of the fission fragments (168.2 MeV) and the neutrons (4.8 MeV), as well as the

energy carried off by gamma rays (7.5 MeV). Taking into account other contributions by

following decay reactions, the sum of above process gives an average total energy of

195 MeV.23 Compared with this high energy, a typical chemical reaction usually only

involves energy changes less than 10 eV.

1.4 Motivation and Objective

This thesis focuses on nuclear materials, especially on understanding the

fundamental mechanism of intrinsic point defects evolution, which plays a crucial role in

the material performance. In nuclear reactors, the uranium dioxide (U02) fuel undergoes

structural damage (formation of defects) after irradiation. Radiation damage (RD) due to

high concentration of point defects can limit the lifetime of the fuel. In order to enhance

the radiation tolerance of U02, it is of great interest to accurately predict the long term

behavior under different conditions.

Real nuclear fuels are made of several U02 crystals separated by interfaces or

grain boundaries (GBs). To determine the GB effect, the results obtained in a single

crystal are generally used to compare with those in polycrystalline materials. Therefore,

building on the previous work by Aidhy et al. in single-crystal U0224, this thesis

investigates more realistic polycrystalline U02. It has been justified that defect evolution

is largely independent of the initial damage state on MgO simulations.25 Therefore,

conventional molecular dynamics (MD) simulations using primary knock-on atom (PKA)

approach are not capable of capturing the long-term evolution of point defects, mainly

due to lack of complete development of complex defect structures. Moreover, in order to

elucidate the interplay of U and O point defects, evolution from initially different defect

conditions has to be studied. This is also not possible for standard MD using PKA









approach. As a result, a new methodology is developed to perform such atomic

simulation to characterize detailed physical processes associated with GBs and to

indicate to what extent GBs interact with the defects.

1.5 Beauty of Simulation

Although U02 has been extensively studied in the past, there is little understanding

of its microscopic behavior during irradiation. This is because experiments are expensive

and have little access to the defect length scale and irradiation events time scale. In a

nuclear reactor, radiation events occur on a very short time scale (~nanoseconds), so

observing them by experiment is not possible. Hence, atomic simulation can be a

complementing alternative to obtain valuable information and widen the scope of

knowledge in this area.

Understanding the basic diffusion mechanism at the atomic scale is a key

ingredient in the development of realistic macroscopic models. Moreover, GBs in

polycrystalline materials are known to largely influence these events, which take place

simultaneously with vacancy-interstitial recombination in the bulk. Only simulations are

able to directly distinguish the process and to determine the dominant mechanism. On

the other hand, it is also not experimentally possible to capture a material's responses to

the presence of different types of point defects, which might reveal its ability to maintain

the desired properties.









CHAPTER 2
DEFECTS IN SOLIDS

2.1 Introduction

The scientific study of crystalline solids is better developed than that of amorphous

materials due to their regularity and symmetry. On the atomic scale, crystalline solids are

composed of a set of atoms arranged in a particular way with repeating pattern

throughout all three spatial dimensions. There are a large number of crystal structures,

from the simple ones of elemental metals to complex ceramics. It is impossible to

maintain the infinite long-range order in all three dimensions. In general, the regularity is

often disrupted by defects. These imperfections may have a profound effect on the

properties of materials. Defects can be classified into three categories by their

dimensions: three-dimensional bulk defects (e.g. voids or precipitates), two-dimensional

planar defects (e.g. grain boundaries or interfaces), one-dimensional line defects (e.g.

dislocations), and zero-dimensional point defects (e.g. vacancies or interstitials). In this

work, the behavior of point defects is of our focus.

2.2 Intrinsic Point Defects

From the Nernst theorem it is known that a crystal may have a perfect structure

only at the absolute zero temperature.2 Therefore, a small number of intrinsic point

defects are always present at any temperature above absolute zero. Imperfections

involving atoms only native to the crystal without any external intervention (e.g. Na or Cl

in NaCI crystals) are called "intrinsic" defects. They do not require the introduction of

impurities or aliovalent ions and are intrinsically related to the structure of the compound.

The vacancy is simply an atom missing from a lattice site, which would be occupied in a

perfect lattice as in Fig. 2-1 (a). The self-interstitial is an atom crowded in a void space









between lattice atoms as in Fig. 2-1 (b). These point defects in the lattice create varying

separations between the neighboring atoms, which hence generate lattice strains.








ol.








(a) (b)

Figure 2-1. Point defects: (a) vacancy, and (b) interstitial.27

2.2.1 Schottky Defect

A Schottky defect is unique to ionic materials and is commonly found in ceramics. It

forms when stoichiometric numbers of ions are missing from the crystal lattice, leaving

behind vacancies on the cation and anion sites in order to maintain overall electrical

neutrality in the material, as shown in Fig. 2-2 (a). The defect is named after the German

Scientist Walter H. Schottky. The following is the chemical reaction in Kroger-Vink

notation for the formation of the Schottky defect in U02:

nil Vu +2Vo (2-2)

2.2.2 Frenkel Defect

In crystals containing more than one ion, vacancies and interstitials may occupy

either anion or cation sites. A Frenkel defect (or Frenkel pair (FP)) is a vacancy-interstitial







pair created by thermal vibration when an ion hops from a normal lattice site to a nearby
interstitial site, leaving behind a vacancy. It was named after the Russian scientist Yakov
Frenkel, who discovered the phenomenon in 1926.28 Fig. 2-2 (b) is a schematic
representation of a Frenkel defect for an ionic material. Frenkel defects can also be
found in metals and covalent compounds.
Taking oxides as an example, the Frenkel defect on the oxygen anion sub-lattice
site can be described using Kroger-Vink notation as:
Oo0 <-> O, + V (2-1)




00 o0o o /
0O00 Oo dO0t)


00



(a) (b)

Figure 2-2. Point defects in (a) Schottky defect, and (b) Frenkel defect.27
There are also other types of point defects, both intrinsic and extrinsic. However,
only Frenkel defects are under study in this thesis.
2.2.3 Thermodynamics of Point Defects: Equilibrium Concentration
The formation of intrinsic point defects is governed by the principle of minimizing
the total Gibbs free energy. Any destruction of the perfect lattice may increase the lattice
energy. However, when point defects are present, the overall randomness or entropy of









the material increases, and hence can decrease the total Gibbs free energy, which can

increase the thermodynamic stability of a crystalline material. Vacancies play an

important role in determining the rate at which atoms or ions can move around, or diffuse

in a solid material, especially in pure metals.

At room temperature, the concentration of point defects is small, but the equilibrium

numbers of point defects nv and ni increases exponentially as the temperature is

increased, as shown by the following Arrhenius eqations:29


n=N. exp-kTY (2-3)


n, N=. exp -Q1 (2-4)
\ kT J

where nv and ni are the number of vacancies and interstitials per cm3; N is the number of

atoms per cm3; Qv and Qi are the energy required to produce a vacancy and interstitial,

in eV/mol; k is the Boltzmann constant, 8.617x10-5 eV/K; T is the temperature in Kelvin.

2.3 Grain Boundary

Grain boundaries (GBs) are the lattice defects which have been longest known but

least understood. Most engineering materials are polycrystalline in nature in that they are

made of many single crystals which are misoriented with respect to each other. In a

polycrystalline solid, GBs are the internal interfaces that separate neighboring regions of

the same crystal structure but of different orientation. These interfaces, which are usually

planar, have a two-dimensional periodic atomic structure. For example, considering

grains in 100 diameter in a polycrystalline cube with 1 cm on edge, there would be more

than 1015 grains, with a GB area of several square meters. Thus, the details of the atomic

structures of GBs play an important role in controlling the properties of the polycrystalline









solid. According to the misorientation between adjacent grains, low-angle grain

boundaries (LAGBs) are those with a slight orientation mismatch, which can be

described in terms of dislocation arrays.29 For high-angle grain boundaries (HAGBs), the

properties are normally independent of the misorientation angle. The transition angle

varies from 10-20 degrees, but is not known exactly.30 Tilt boundaries are formed when

angle of misorientation lies in the plane of boundaries, which can be described by an

array of edge dislocations. When the angle of misorientation is parallel to the plane of

boundaries, twist boundaries result, which can be described by means of screw

dislocation arrays. Symmetric boundaries bisect the angle of rotation. A twin boundary is

a special type of GB in a mirror-symmetric manner. In general, GBs have very different

atomic configurations and local atomic densities from those of the perfect crystal. The

deficiency of fundamental knowledge of GBs is mainly due to their complex structure,

which requires extensive description for their macroscopic characterization.

2.4 Atom Movements in Materials

2.4.1 Diffusion Mechanisms

Understanding of diffusion is based on two important fundamentals31: (1) the

continuum theory of diffusion originated from work of the German scientist Adolf Fick,

and (2) the Brownian motion detected by the Scottish botanist Robert Brown.

Vacancy diffusion: As mentioned in 2.2.3, in thermodynamic equilibrium at elevated

temperature, a specific number of vacancies are present in a crystal. Vacancy diffusion

involves the movement of an atom from a regular lattice position into an adjacent vacant

lattice site; thus the atom and the vacancy move in opposite directions (Fig. 2-3 (a)). The

number of vacancies, which increases as the temperature increases, determines the

extent of this mode of diffusion.








0000 0000
0000 0000
oo o 000oo
0000 00Y00
(a) I I (b)

0000 A





(c) (d)
Figure 2-3. Schematic representation of three diffusion mechanisms: (a) vacancy
diffusion mechanism, (b) interstitial diffusion mechanism, (c) interstitialcy
mechanism, and (d) enthalpy barrier LH, required to make a jump for the
interstitial atom shown in (b).32
Interstitial diffusion: This type of diffusion involves an interstitial atom directly migrating
to another nearby interstitial site without causing the net motion of any other atom (Fig.
2-3 (b)).
In some modes, to make the jump, an enthalpy barrier has to be overcome for the
atom to squeeze through a narrow passage. This enthalpy barrier is known for the
migration energy and is shown schematically in Fig. 2-3 (d).
The migration energy of the vacancy is larger than that of the interstitial, so the
pre-exponential diffusivity of the vacancy is much smaller. There are two main reasons
why interstitial diffusion takes place much faster than diffusion by vacancy mode.29 First,









interstitial atoms are usually smaller and thus more mobile. Second, there are more

interstitial positions than vacant sites, so the probability of interstitial atomic movement is

greater than vacancy diffusion.

Interstitialcy diffusion: A less common mechanism is the interstitialcy mechanism,

where an interstitial atom pushes an atom into an interstitial site and move into the lattice

site itself (Fig. 2-3 (c)). This mechanism of diffusion will be observed in this research.

2.4.2 Atomic Theory of Diffusion

Phenomenological equations: Diffusion is the phenomenon of material transport by

atomic motion.29 In solids, diffusion is a very slow process. The rate at which atoms

diffuse in a material can be measured by the flux ( J), defined as the number of atoms

passing through a cross-sectional unit area per unit time. If the diffusion is steady-state

in one dimension (x-direction), which means flux does not change with time, the net flux

of atoms can be described by Fick's first law:

dc
J = -D- (2-5)
dx

where D is the self-diffusion coefficient, which is a measure of the ease and frequency

with which that atom jumps around in a crystal lattice in the absence of external forces,

i.e., in a totally random fashion. Experimentally, the temperature dependence of the

diffusivity is:


D = Do exp HmAH (2-6)
RT

where AH, is the enthalpy barrier for diffusion independent of temperature; Do is a

temperature-independent pre-exponential parameter. Note that the diffusion rate largely

decreases with decreasing temperature.








Atomistics of solid state diffusion: In general, diffusion in solids occurs in the

presence of point defects.33 Ion irradiation results in the formation of vacancy-interstitial

pairs.34 If the defects are produced at temperatures where they are mobile, they can

partly anneal out by diffusion. Therefore, the balance between the formation rate and the

annihilation rate leads to a steady state of the excess concentration of defects. Atoms

usually move by interchanging positions between atoms and neighboring vacant sites.

For such a diffusion process, the relationship relating the diffusion coefficient ( D ) of an

atom to the diffusion jump distance ( A ) in a solid is:32

D = aDA2 (2-7)

where a is a geometric constant depending on the crystal structure, and is the

frequency of successful jumps.

The jump frequency is the product of the probability of an atom having sufficient

energy to make a jump ( v) and the probability of available adjacent vacant sites ( e ):

0 = v (2-8)

where v can be further expressed by:

-A nH ,k, (2-9)
v = vo exp C m (2-9)


where AH~, is the enthalpy barrier required to make a jump, and vo is vibrational

frequency of the atom.

Defect diffusivity: As noted previously, the two major defects responsible for the

movement of atoms are vacancies and interstitials. For both cases at low concentrations,

the sites adjacent to the defect will be always available, so e 1.33

An interstitial will make a jump with the rate only depending on its frequency of









successful jumps. By combining Eq. (2-7), (2-8), and (2-9), the interstitial diffusivity Di is

given by:


D, =aA2o exp AHm (2-10)


On the other hand, for a vacancy, the probability of successful jumps is increased

c-fold, where c is the coordination number of the vacancy. Thus, the vacancy diffusivity

Dv is expressed by:


D, = aA2Vo .exp _AHm (2-11)


Atom diffusivity: In contrast to the vacancies and interstitials, most atoms have few

available sites nearby into which they can diffuse, so e << 1.32 The probability of a site

being vacant is simply equal to the vacancy fraction in the solid, denoted by A. Therefore,

the atomic diffusivity DA is given by the following expression, which resembles Eq. (2-6),

namely:


DA = A 2 Avo ex = DH o exp H (2-12)
kBT kBT

Comparing Eq. (2-11) and (2-12) reveals an important relationship between

vacancy and atom diffusivity:

DA =ADv (2-13)

We assume that A f, C,/cA << 1, where fv is a correlation factor, and CA, Cv are the

concentrations of atoms and vacancies, respectively. A more useful expression is thus

given by:34

DACA =fvDvC or DACA =fiDiC (2-14)

Correlation factors depend on the crystal structure and diffusion mechanism. The









theoretical values for fluorite structures have been reported for the three mechanisms:

0.653 for the free vacancy, 1 for the interstitial, and 0.739 for the indirect non-collinear

interstitialcy.35 The physical picture of Eq. (2-14) is easy to grasp: the defects (vacancies

or interstitials) move often (high D) but are not that numerous, while the atoms move less

frequently (low D) but are much more in number.

With CA 1, rearranging Eq. (2-14) yields:

D4
D, =A (2-15)


D
D, =A (2-16)
cif,

Diffusivity in simulations: In atomic simulations, the self-diffusivity is determined based

on the time evolution of the mean square displacement (MSD) under zero-stress

conditions. The quantity can be easily calculated by recording the atomic positions

during the simulation:

1 N 2
MSD(t)= I [r,- ] -2) (2-17)


where N is the total number of ions; t is the simulation time; r, and Fo are atomic

positions at time = t and 0, respectively; the angular brackets here denote an average

over the square of the displacement that the atom has undergone during a time interval t.

Since the thermal energy is involved in the MSD, atomic positions can be simply related

with the initial thermal velocity v,(0):

f t)= (0)+ v ()t (2-18)

Inserting Eq. (2-18) into (2-17) gives:









(,-r-Fo2)=_ [,(0)] t2 (2-19)

From kinetic-molecular theory, the average kinetic energy is proportional to the absolute

temperature in the following relation:

1 3
-mV = kBT (2-20)
2 2

where Vrms is the root mean square velocity. Inserting Eq. (2-20) into (2-19) gives:

i* o2)=t2 3k 2 t (2-21)

For liquids, the MSD grows linearly with time, while for solids, the MSD saturates to a

finite value. It is useful to characterize the diffusive motion in terms of the slope, which is

the self-diffusion coefficient of the atoms ( DA), namely:
1
(Ifr_,f-r 6Dt

DA \,-, 2) (2-22)

2.5 Crystallography of U02

In this thesis, the central theme is on the evolution of point defects in the fluorite-

based U02. As shown in Fig. 2-4, the fluorite structure can be viewed as a face-centered

cubic (FCC) array of cations, represented by blue spheres, with the anions residing in all

the tetrahedral sites, represented by red spheres.

A fluorite crystal unit cell consists of four U atoms and eight 0 atoms. The

coordination number of the cation is eight and that of the anion is four. The cations

occupy the Wyckoff site 4a and the anions occupy 8b. The space group of fluorite crystal

is Fm3m (number 225 and point group symmetry m3m36). Fm3m can be fully written
















O







Figure 2-4. Fluorite crystal structure. The FCC network of cations (blue spheres) with
anions (red spheres) occupying the tetrahedral sites.

4-2
as F 3 The first letter refers to the lattice type. Here, F stands for face-center cubic
mm

(FCC). The following three sets of symbols denote the three kinds of symmetry directions

of the lattice relative to the space group, naming primary, secondary, and tertiary


directions, respectively. For the FCC unit cell, the primary symmetry directions are <100>,

i.e., [100], [010], and [001]; the secondary symmetry directions are <111>; the tertiary

symmetry directions are <110>.

42 4
In the F 32 symmetry, represents four-fold rotational symmetry along the
mm m

principle axis with a mirror plane perpendicular to the axis. 3 refers to the rotoinversion

along the diagonal (i.e. secondary) direction of the unit cell. stands for two-fold
m

rotation along the edge of the unit cell (i.e. tertiary direction) with a mirror plane

perpendicular to the edge. Fig. 2-5 schematically shows all three symmetry operations

mentioned above.










0


(a) (b) (c)

Figure 2-5. Graphic representation of three different symmetry operations for space
4-2
group F 3 :(a) four-fold rotation along <100>, (b) three-fold rotation
mm
along <111>, and (c) two-fold rotation along <110>.

2.6 Radiation Damage

Nuclear energy is part of solution to the increasing global energy demand and

concerns about greenhouse gas emissions. A fundamental understanding of defect

formation, accumulation, and annihilation in irradiated ceramics is required to develop


novel radiation tolerant ceramics for improving the performance of the fuel, and for

developing long-lasting storage materials.

When materials are bombarded with radiation, electrons may come out of their

original orbitals, and atoms may displace from their original sites. In this process, defects

are also introduced, either by bombarding ions residing in the material or following

nuclear reactions. Therefore, the properties of the material are altered, sometimes

heavily, sometimes just slightly. These spontaneous processes can lead to the gradual

and ultimately complete amorphization of the ceramic crystalline structure, which is

undesirable for nuclear application because it is often accompanied by degradation of

thermo-mechanical properties and reduced chemical durability. However, sometimes








defects can be annealed out, so the material will remain crystalline. The radiation

response of ceramics relies on a dynamic balance between damage by defect

accumulation and recovery from thermal annealing.

The radiation damage phenomena take place in a very short time, and include two

important phases (Fig. 2-6).37 The "ballistic phase" involves the atomic collision cascade

arising from the high-energy "primary knock-on atom" (PKA). Atoms are displaced from

their original lattice sites, thereby producing a large number of point defects and small

clusters. The "ballistic phase" usually lasts only a few picoseconds. By contrast, the

"kinetic phase" lasts longer, and captures the mainly time-dependent diffusion-controlled

kinetic evolution of these defects, including the annihilation and formation of larger

long-lasting defect clusters. Some defects may recombine or be eliminated at

microstructural sinks (GBs, dislocations, or external surface); others cluster to form voids

or interstitial loops. Annihilation of point defects promotes radiation tolerance of the


-- Ballistic




1 Kinetic




I \



0 5 10 15
Time (ps)

Figure 2-6. Defect evolution during ballistic and kinetic phases of the radiation damage.37









material while clusters lead to long-lived radiation damage. As a result, the "kinetic phase"

process determines the performance and lifetime of the material. Understanding the

defect evolution during the "kinetic phase" is thus crucial for engineering radiation

tolerant ceramics, and in U02 it is also the central problem of this thesis.









CHAPTER 3
SIMULATION METHODOLOGY

Computer simulations are carried out to solve material science problems related to

the design a material with specified properties and functionalities. They have high

potential to impact real industrial research and development. As a matter of fact,

simulations serve as a complement to conventional experiments, enabling us to integrate

computational and theoretical methods with experimental analysis and characterization

methods.

3.1 Simulation Methodologies

To characterize various materials behaviors over wide length and time scales, a

wide set of simulation methodologies are used, as show in Fig. 3-1.

Starting at the highest level, engineering design is basically the primary analytical

tool for relatively large-scale manufacturing systems in industry, varying from integrated

circuits to distillation columns and engines. At the level of continuum models, gradual

transitions without abrupt changes or discontinuities are described, such as macroscopic

stresses, large temperature gradients. Below this level, the meso-scale modeling

includes kinetic Monte Carlo (KMC) models38 and phase-field methods39 as approaches

to modeling and predicting morphological and microstructure evolution. A further level

down, atomic phenomena are accurately captured by molecular dynamics (MD)

simulation with interatomic potentials40 and force fields41. Details of this method will be

discussed in the following section. At the lowest level, computational quantum chemical

methods are characterized by quantum mechanics at the level of the electronic structure.

Electronic states are described in terms of the Schrodinger equation:42

HW = EW (3-1)









where E is the total system energy (eigenvalue); H is the Hamiltonian operator; u is the

wave function (eigenfunction). There are two main approximation methods to solve the

equation: semi-empirical methods and ab initio methods. Semi-empirical methods use

experimentally derived parameters to strive for accuracy, but ab initio methods use

calculations to evaluate all integrals. With correlation, ab initio methods can even have

accuracy comparable with experiments in structure and energy predictions. However,

calculations become extremely demanding in computer resources.

This work focuses on phenomena at the atomistic length scale and picosecond

time scale; therefore, MD simulation is chosen here.


E0

0
TO
(I pml
-5
CD
0)
-j,,


105 1-2 1" 10 1 -1 1 1 103 106
fs ps ns ps ms s min hr day yr
Time scale (s)
Figure 3-1. Hierarchical multi-scale approach to computational material engineering.









3.2 Molecular Dynamics Simulation

The MD simulation is a technique that computes time-dependent behaviors, such

as structures, dynamics, and thermodynamics. It enables us to probe features of the

motion of atoms, and to take "still snapshots" of crystal structures. Now it is frequently

used in the study of materials science because no conventional experiment allows

access to all the time scale of motion with the atomic resolution. The accuracy of the

calculated properties relies entirely on the fidelity of the interatomic potential, which is

typically defined by quantum based methods or experimental data.

3.3 General MD Algorithm

In MD simulations, all atoms are given initial positions theoretically, which is defined

by the crystal structure of the material and the microstructure of interest. The atoms are

assigned random velocities ( v) according to the simulation temperature ( T):

-my =kBT (3-2)
2 2

where m is the mass of the atom; kB is the Boltzman's constant.

The motion of the atom can be described by solving Newton's second law:

d2r
F =ma= m-- (3-3)
dt2

where F is force exerted in the atom; m is its mass; a is its acceleration. The force can

also be calculated from the gradient of the interatomic potential energy V:

F = -VV (3-4)

Numerous numerical algorithms have been developed for integrating the equations

of motion, such as Verlet43, leap-frog44'45, Beeman's46, and predictor-corrector47. They

differ in accuracy and computational load. In this thesis, a fifth-order Gear predictor-








corrector method is used. This method consists of three steps: prediction, evaluation,

and correction.

In the first prediction step, a high-order Taylor expansion is performed to estimate

the atom positions around time t to predict the positions at t+At:
df~t) 1 d27(t) 1 d3J(t) 1 d47(t) 1 d(F(t)
S(t +At)= f(t)+ d At+ t)2 + ()3 + t) +
dt 2 dt2 3! dt3 4! dt4 5! dt5

d(t) d2 (t) 1 d3 ) d47(t) (t)4
VP (t +At) = d(t)+ d2f(At nt+ d3f(A 1 t (At)3 (t)
dt dt2 2 dt3 3! dt4 4! dt5

d27(t) d37(t)At I d 4f(t) t)2 d5 F(t) 3
a(t+At)= + At(A)t)3
dt2 dt3 2 dt4 3! dt5

bP(t+t)= d3f(t) d4'(t) 1 d (t) (At)2
dt3 dt4 2 dt5

(t+At) d4r(t) d5r(t)At
cd (Q +At) = + At
dt4 dt5

dP(t+At)= d) (3-5)
dt5

The superscript refers to predicted values; r, v ,a, b, cand d denotes positions,

velocities, accelerations, and third, fourth, and fifth time derivative of position,

respectively.

Newton's second law is introduced. Therefore, the correct accelerations can be

obtained:

S dV(r) (3-6)
m m dr

where F is the force exerted on the atom; m is its mass; a is its acceleration. The

acceleration can be evaluated by comparing with the predicted accelerations from Eq.








(3-5), to estimate the error in the prediction step:

La(t + At)= ac (t + At)- A (t + At) (3-7)

This error and the results of the prediction step are fed into the correction step:

C (t + At) (t t) Co (tA (t + At)

vc(t + At) (t + At)-c,A(t + At)


ac (t + t) = ba (t +t)-c,a(t + at)
bc(t+At)= bP(t+At)-c3Aa(t+At)

c(t+ At)= cP (t +At)-c4A(t+At)

dc (t + At)= d (t + At)- cAa(t + At) (3-8)

where the Gear corrector coefficients are co= 3/20, cl = 251/360, c2 = 1, 3 = 11/18, c4

1/6, and C5= 1/60, respectively.48

3.4 Periodic Boundary Condition

In MD simulations, the simulation box size must be large enough to avoid boundary

condition artifacts. There are two main kinds of boundary conditions49: isolated boundary

conditions (IBCs) and periodic boundary conditions (PBCs). In IBCs, the system is

surrounded by vacuum, so atoms only interact among themselves. Therefore, IBCs are

ideally suited for the study of clusters and molecules. On the other hand, in PBCs, the

system is surrounded by infinitely image of the supercell itself in all three dimensions of

space, as shown in Fig. 3-2. Hence, an atom may interact not only with atoms in the

same supercell but also with atoms in neighboring supercells. In other words, one side of

the simulation loops back to the other side, mimicking a bulk phase. In this way, PBCs

are suitable for investigating bulk liquids and solids.
















I go o o oo



a P










Figure 3-2. Schematic representation of periodic boundary conditions of a fluorite
structure supercell. The center box is the actual simulation box.

3.5 Interatomic Interactions
For different types of materials, different kinds of interatomic attractions are needed

to correctly describe the system. For any given functional form, different parameters can

be chosen so as to describe different materials. Therefore, it is essential to define the

interatomic interactions before any practical calculations can be performed.

In ionic systems, the potential energy consists of both long-range and short-range

interactions. Long range interactions are attractive interactions caused by Columbic

forces. Short-range interactions are mainly repulsive due to overlapping electron orbitals,

but there are also some small van der Waals attractive forces. Both types of interactions

will be discussed in detail in the following section.
'w A f a ^ '
1^9e32.Shmti ersetto o | pro ^^ bo _[a onios ofa lort





















will be discussed in detail in the following section.









3.5.1 Long-Range Interactions

For inorganic materials, especially oxides, long-range forces come from

electrostatic interactions, which determine to a large extent the thermodynamic and

transport properties. Therefore, precisely evaluating the Coulombic force is essential. In

ionic systems, cations and anions are arranged in a defined order. The energy

associated with this specific periodicity is the electrostatic attraction energy between two

charged ( qi and qj) ions. The magnitude of the energy ( Elong-range) can be calculated by

Coulomb's law:


E long-range rij- (3-9)


where rjy is the distance between ions i andj. A negative potential represents an attractive

interaction while a positive potential represents a repulsive interaction. Despite the

simple formula given above, this is actually the most difficult interaction to evaluate for

periodic system.

Since long-range Coulomb's interactions are responsible for the stability of

crystalline structures, a critical approximation has to be made to develop an infinite

system. It is computationally convenient to have a cutoff truncation beyond a radius or

the periodicity of the lattice. Among several methods50-60, the Ewald summation50 and the

direct summation50 are the two most common methods. Though the Ewald summation is

more accurate in calculation, it is more computationally intensive than the direction

summation. According to the Ewald summation, the high computational load has scaling

with system size of N2, or better N3/2, or even at best N log(N).56-59 In the present study,

we are observing the equilibration evolution of point defects, in which the large system

requires a larger MD computational cost. As a result, the computationally more efficient









method, direct summation, is applied here. This method has been validated by

simulations not only of crystalline but also arbitrarily disordered, charged or neutral ionic

systems.60

Direction summation method: When performing a spherically truncated pairwise r1

sum in a crystal, the system summed over is never electrically neutral, which essentially

leads to non-convergence of the summation. This long-range electrostatic concern can

be resolved by an assumption that no basis molecule may be broken up over the entire

Bravais lattice.60 For example, rather than regarding the rock salt (or NaCI) lattice as an

FCC lattice with a dipolar molecule, one can choose the simple cubic unit with the

octopolar (NaCI)4 basis.61 This tetra-unit basis is kept together while applying the

truncation with the cutoff radius. The direct summation may therefore achieve charge

neutrality by introducing additional charge to the truncation sphere. This is not just a

mathematical concept; a reconstruction of the polar NiO surface has been observed62

and this experimentally upholds the validity of the energy convergence approach.

3.5.2 Short-Range Interactions

From Eq. (3-9), the long-range attractive potential between two oppositely charged

atoms should bring them closer. In order not to cause structure collapse of the lattice,

some repulsive potential must be present to keep the two atoms apart.

The short-range interactions can be expressed by the Buckingham potential:


Eshort-range (rJ)= A,= .exp (3-10)


where A#, py, and Ci are empirical parameters; rj is the distance between ions i and j.

The first term describes the short-range repulsive interactions derived from overlapping

electron orbitals, which decrease exponentially with increasing distance between ions i








and j; the second term represents the van der Waals attractive contribution due to dipole

interactions.

Throughout this study, the short-range interactions are characterized by using the

Bushing-Ida63 type. In addition to the Buckingham potential, a Morse term is included to

introduce the covalentt" contribution:

Eshort-range rij)= Aj exp -
P ii ) ij 6

+ D,[exp(- 2,(r, r,))- 2exp(p,(r, r,))\ (3-11)

where Di and pi3 are empirical parameters; ri is the anion-cation i-j bond length. In a

mixed ionic covalent system, the ionic charges are given non-formal values with partial

covalence to consider charge transfer between the ions.

Hence, taking into account all interactions acting on an atom associated with

surrounding atoms, the energy can be given by:

Etotali(r) = + q A j .exp r 6
rj pij ) rij

+ D,[exp(- 2,, (ri, r,))- 2 -exp(,,j(r, r, )) (3-12)

In this thesis, this classical pairwise potential energy form is used. Parameters are taken

from Basak et al.64, as listed in Table 3-1. Fig. 3-3 illustrates the inter-ionic potential

energy as a function of distance between U+24 and 0-1.2 ions.

3.5.3 Thermodynamic Conditions (Ensembles) for Simulations

An ensemble is a collection of all possible environmental conditions which have

different microscopic states but have an identical macroscopic or thermodynamic state.

In MD simulations, certain conditions are assigned in order to obtain desired properties








Table 3-1. Potential parameters used in the present work (Eq. (3-12))64


Parameter Basak et al.
qu(e) 2.40
qo(e) -1.20
Au-o (eV) 693.9297
pu-o (A) 0.3270(22)
Cu-o (eV A) 0.0
Du-o (eV) 0.577(45)
3u-o (A-1) 1.65
ru-0 (A) 2.369
Ao-o (eV) 1633.666(6)
po-o (A) 0.327(022)
Co-0 (eV A6) 3.950(63)
Au-u (eV) 294.759(3)
pu-u (A) 0.327(022)
Cu-u (eV A) 0.0


0 0.5 1 1.5 2 2.5 3 3.5 4
rij (A)


Figure 3-3. Inter-ionic potential as a function of distance between U+24 and O-1.2 ions.
The total potential (Etotal) is the sum of the long-range Coulombic (Elong-range)
abd short-range (Eshort-range) potentials.









of the material, such as total number of atoms, volume, temperature, pressure, etc. A

thermodynamic state with fixed number of atoms ( N ), a constant volume ( V ), and a

fixed energy ( E ) corresponds to an isolated system, that is, the isochoric micro-

canonical ensemble ( NVE ). Similarly, NVT stands for the canonical ensemble and NPT

refers to the isobaric-isothermal ensemble. These three are the most widely used

ensembles, and NPT is performed throughout this study. In addition, there are also other

types of ensembles, such as the grand-canonical ensemble ( pVT) and the isobaric-

isoenthalpic ensemble ( NPH ). H and p represent enthalpy and chemical potential,

respectively.48

3.5.4 Thermostat

In order to control the system simulating at the desired temperature, several

approaches have been employed. The simplest one is probably the velocity-rescaling

thermostat48. Recalling Eq. (3-1), the target velocity ( Vt) can be rescaled by the

following relation:


vt(t)= ,(t) (3-13)

where T1 is the initial temperature; Vi is the initial velocity; Tt is the target temperature.

Other popular methods to control the system temperature are Nose-Hoover65'66

Berendsen67, and generalized Langevin68

3.5.5 Barostat

In addition to the temperature, the volume is another thermodynamic condition

needed to be controlled. Various algorithms can be applied to achieve a constant

pressure. For example, the idea of the Andersen barostat is to assume that the system is

simulated in a container, which is able to be compressed by a piston with certain mass









m.69 The Lagrangian takes the form:

1 2N N 1 .2
L pN,pNV, V, V3mi p2- U V3pj +-mV -poV (3-14)
2 pip i
1
where p is the scaled coordinates (= r / V3; r is the Cartesian coordinates and V is the

volume); po is the external pressure. The first two terms on the right are just the

Lagrangian of the unsealed system. The third term represents a kinetic energy for the

piston of mass m with volume of V, and the fourth term stands for a potential energy

derived from po acing on the piston. However, the Andersen method assumes that the

external pressure is hydrostatic, which is not sufficient in all cases. Therefore, Parrinello

and Raham further extended this method to anisotropic coupling by allowing the

simulation box to change its shape.70' 71 In this thesis, the P-R method is used.

3.6 Conventional Radiation Damage Method by MD Simulations

Atomic-level simulations have long been employed to investigate the fundamental

phenomena in radiation damage (RD).72-76 MD simulations are capable of capturing two

major phases: the ballistic phase and the kinetic phase.

Standard MD simulation of radiation damage has been combined with temperature

accelerated dynamics (TAD) simulations to achieve the longer and experimental time

scale. It has been shown that the defect evolution during the kinetic phase is largely

independent of the initial damage created during the ballistic phase.77'78 Therefore, it can

be inferred that it is not necessary to perform the simulation from the beginning because

the main job of the ballistic phase is to introduce non-equilibrium point-defects into the

system. In other words, starting the simulation directly from the kinetic phase, which is

most interesting to us, seems to be more efficient. In addition, since much attention will









be drawn on the kinetic evolution of these point-defects, the recombination and

clustering mechanism for each separate kind of point-defect will provide more basic

information in the study of RD, which is not accessible for conventional RD MD

simulations. As a result, another kinetically-evolving irradiation-induced point- defects

method will be introduced in this thesis.









CHAPTER 4
GRAIN BOUNDARY INFLUENCE ON RADIATION DAMAGE IN UO2

4.1 Kinetically-Evolving Irradiation-Induced Defects Method by MD Simulation

This kinetically-evolving irradiation-induced defects approach does not replace

cascade simulations targeting on elucidating the effect of experimental conditions, such

as the dose or the energy of incident species. Instead, this method, which bypasses the

ballistic phase, allows a much wider range of different defect conditions to be explored.

In this approach, a specific number of Frenkel pairs (FPs) are randomly introduced to the

system which is equilibrated at high-temperature (1000 K) for a sufficiently long time.

Creating FPs by hand makes it possible to generate defects on a selective sub-lattice or

both sub-lattices. To avoid uninteresting but spontaneous recombination events, all

vacancies are separated from their counterpart interstitials by a distance greater than the

recombination radius. Then the system is equilibrated at 1000 K. The defect structure,

unlike rapidly recombining point defects, can have a more significant role in determining

the true radiation performances of the material. This study will discuss the following

kinetic evolution of the FPs.

4.2 Simulation Methodology

4.2.1 MD Simulation

Using the interatomic interaction described in 3.5, the melting point of UO2 and

oxygen sub-lattices were determined to be around 3450 50 K79 and 2200 K79, which

are in fair agreement with the experimental value of 3100 K80 and 2600K81, respectively.

In this work, all simulations have been carried out at 1000 K, which is well below the

melting temperature of both the material and sub-lattices. At this temperature, the

oxygen interstitials and vacancies have high mobility, whereas the uranium counterparts









have very high migration energy and therefore diffuse less. In addition, this temperature

is also in the range of 800-1600 K, typical working temperature in the fuel pellet.

The single-crystal simulation supercell contains 20x20x20 cubic fluorite unit cells

with 96,000 atoms. The polycrystalline microstructure is a large rectangular box (51.7x

30.2x5.5 nm) consisting of 611,010 atoms in six hexagonal equal-sized (d = 20 nm)

grains (Fig. 4-1). The crystallographic misorientation between three adjacent grains are 0,

30, and 60 degrees with respect to each other, then the GBs are asymmetric high-angle

tilt boundaries. Periodic boundary conditions are applied in all three dimensions. The

time step of 0.5 fs is chosen as it gives good energy conservation for several thousand

steps in NVE test runs. In order to capture as much of the defect evolution as possible,

all simulations have been carried out for more than 1 ns. Two analysis methods to

identify point defects will be discussed in the following section.








20 300 600
nm






y 0 60 30
xtz

Figure 4-1. Snapshot of the fully equilibrated polycrystalline U02 microstructure with d =
20 nm at 1000 K prior to introduction of defects. The viewing direction is
along [100] columnar axis. Three crystalline orientations are 0, 30, and 60
degrees relative to the columnar axis.








4.2.2 Defects Analysis Methods
To observe the kinetic evolution point defects, it is very important to capture the
precise number, type, and position of all atoms, especially in the presence of grain
boundaries. In fact, the grain boundary is a kind of a defect, so it is relatively valueless to
count point defects in the grain boundary.
Here are two methods to perform the analysis. One is the common neighbor
analysis; the other is the lattice matching analysis. The two methods are used in
complementary manner here.
Common neighbor analysis (CNA): As implied in the name, the CNA82 utilizes the
coordination number (CN) of neighboring atoms to identify defects. When it comes to the
CN, the cutoff radius plays an important role to determine how far the nearby atom is
treated as the neighbor. In UO2, CNs of U atoms are eight and of O atoms are four. As




o o'
.0 @0 @

.0 K114i i P


' -
o*0 o o


mI

.0
p.
^-


^ -4 -^ ^ ^ ^-
(a) (b)

Figure 4-2. Schematic representations of the common neighbor analysis (CNA). (a)
Every atom is normally coordinated in the perfect structure (b) There are
different coordination numbers (CNs) for the neighboring atoms when
defects are present (dark color for higher CNs, light color for lower CNs).








shown in Fig. 4-2 (a), every atom is normally coordinated in the perfect structure. When

defects are present (Fig. 4-2 (b)), CNs near interstitials will increase (in darker color)

while near vacancies will decrease (in lighter color). In terms of the miscoordination of

the defect atom itself and the neighboring atoms, the defect position is identified.

However, one big problem is that if there is more than one type of defects close to each

other, such as clusters containing both vacancies and interstitials, the CNs will be very

complicated to analyze.

Lattice matching analysis (LMA): In view of the drawbacks mentioned above, here a

second method is also used to identify defects. Considering atoms diffusion at high

temperature, the average travelling distance can be calculated. Therefore, a cutoff radius

( rctoff) of 0.19ao (about 1.039 A) is chosen with regard to all lattice sites of the reference
structure (Fig. 4-3 (a)), which has been equilibrated for a long time. As a result, if there is



A, v" ..-4" .. ) ..











(a) (b)

Figure 4-3. Schematic representations of the lattice matching analysis (LMA). (a) A cutoff
radius ( rcutof) is assigned to all lattice sites of the reference structure. (b) If
-- '. ....... ;-" '







there is no atom within the assigned radius, it is a vacancy. If an atom is not

within any assigned radius, it is an interstitial.
.( ., __ (, ;i '_ ,t _, E ,



(a) (b)









no atom within the assigned cutoff radius of the original lattice site, the site is regarded

as the vacancy. But if there is an atom sitting out of the cutoff circle, the atom will be

defined as the interstitial (Fig. 4-3 (b)). Another rare case is that if there is more than one

atom within the cutoff circle, all atoms will be reported, such as dumbbell interstitials.

With the LMA, we can therefore determine the exact number, type, and position of every

point defect.

4.3 Results

This work starts with the simpler case, the single crystal, to confirm that simulation

methods and tools are correctly implemented. Then the more complicated case, the

polycrystal, is investigated.

4.3.1 Frenkel-Pair Defects in Single-Crystal U02

The same simulations are run to reproduce previous work by Aidhy et al. in single-

crystal U02.24'37 FPs are created by randomly picking up an ion from its cubic sub-lattice

site and placing it at an empty octahedral interstitial site. A defect concentration about

0.21 % was chosen, so 200 was picked up out of 96,000 atoms as defects. Two

conditions are performed: (1) 200 FPs only on the U sub-lattice, and (2) 200 FPs on both

U and O sub-lattices. Due to the high migration energies of both the U interstitial and

vacancy, U FPs have lower diffusivity. On the other hand, both the O interstitial and

vacancy have relatively low migration energies, which make them more mobile.

Therefore, evolution of the number of U FPs will be less substantial. The results are

shown in Fig. 4-4.

FPs only on the U sub-lattice: Due to the high migration energy of U interstitials and

vacancies, recombination events are less frequent. Out of the initial 200 FPs, 190 U FPs

still remain after 1 ns. However, the high concentration of defects on the U sub-lattice









nucleates 0 FPs. The number of nucleated 0 FPs is shown by circles in Fig. 4-4. Similar

phenomena were also observed by Aidhy et a/.24'37 Different from that work, some data

within the first 10 ps were analyzed in this work. Interestingly, rather than a gradual

increase, the number of 0 FPs increases abruptly to more than 50, and then decreases

to 37 after 10 ps, finally an equilibrium number around 35.


500
350 --U & 0 (Aidhy et al.)
450 320
290 -U (Aidhy et al.)
400
260 -B-U & 0 (This work)
350 230
350 200 --U (This work)
2
LL 300 0 5 10 15 20
O
S250 60
0 50
200 40
E30
S150 20
020
Z 10
100 o(
0 5 10 15 20
50

0
0 200 400 600 800 1000 1200 1400 1600

Time (ps)


Figure 4-4. Comparison of results from Aidhy et al. (red) and this work (blue) for total
number of O FPs in two initial conditions: (1) 200 FPs only on the U sub-
lattice (circles), and (2) 200 FPs on both U and O sub-lattices (squares).
Inset figures are close-up views within the first 10 ps for both conditions.

FPs on both U and O sub-lattices: This is a more realistic case in the study of radiation

damage. 200 FPs are simultaneously created on both U and 0 sub-lattices. Based on

simulations on the U sub-lattice alone, most of U FPs remain at their initial sites because

of their low diffusivities. In contrast, a considerable increase in the number of 0 FPs









takes place within the first 200 ps, as shown by the squares in Fig. 4-4. Afterwards, the

number is followed by a decline to the equilibrium number to around 250. Instead of

complete annihilation of 0 FPs, the number is actually larger than that of the initial

concentration (200 FPs).

From the above simulation run, it can be indicated that these signatures are

consistent with the previous results from Aidhy et al. In addition, more information taking

place within the first few picoseconds shows that the high concentration of defects on the

U sub-lattice nucleates 0 FPs in a very fast manner.

4.3.2 Frenkel-Pair Defects in Polycrystalline U02

For a more representative of the real material, polycrystalline U02 is investigated.

Before generating the defects, the polycrystalline structure is first equilibrated at the

working temperature (1000 K) for 1 ns to ensure no spontaneous formation of defects.

Given that the formation energies for 0 FPs and U FPs are 6.0 and 17.0 eV83 with the

Basak potential64. the defect concentration of 5.8x10-31 and 2.1 x10-86 would be expected

in this simulation. Consistent with this, no defects are present in the bulk grain region.

This equilibrated structure is used as the initial structure for rest of the simulations and as

the reference structure in the LMA. In order to perform a simulation under similar

conditions to the previous single-crystal work, 185 FPs are introduced to selected grains

of the polycrystal. The defect concentration relative to the bulk (eliminating atoms in GBs)

corresponds to 200 FPs in the single crystal containing 96,000 atoms. Three different

scenarios are analyzed for each of the single grains: (1) 185 FPs only on the U

sub-lattice, (2) 185 FPs only on the 0 sub-lattice, and (3) 185 FPs on both U and 0

sub-lattices. All FPs are introduced to the bulk region. All types of vacancies are









separated from their counterpart interstitials by 1.5ao.The structure is then equilibrated at

1000 K with the NPT ensemble for 1 ns.

FPs only on the O sub-lattice: A snapshot of the initial structure is shown in Fig. 4-5 (a)

(left grain, Grain 1). Due to the low migration energies of both O interstitials and

vacancies, O FPs are more mobile, which means they have higher diffusivities. The

number of O defects is shown in Fig. 4.6 (green triangles). After the first 25 ps, only 44

out of 185 O FPs are left (Fig. 4-6 (b)). Most of the O FPs are annihilated by

vacancy-interstitial recombination mechanism. After 400 ps, fewer than 10 O FPs are

sporadically present (Fig. 4-6 (d)). Nevertheless, after 100 ps the number of O vacancies

is 3 greater than the number of O interstitials because a V-40 vacancy-interstitial

clusters forms. This kind of cluster will be discussed in detail later. However, no

interstitial-interstitial or vacancy-vacancy cluster is observed, which indicates that in the

absence of defects on U (or cation) sub-lattice, the defects on O sub-lattice (or anion) are

almost healed within a few picoseconds. As a result, there will be no long-lasting damage

to the material.

FPs only on the U sub-lattice: The initial snapshot is shown in Fig. 4-5 (a) (right grain,

Grain 3). Within first 400 ps, only 27 U FPs out of 185 U FPs have been recovered.

Similar to the case in single crystal, the high migration energy of U interstitials and

vacancies makes U FPs unable to recombine. Therefore, the remaining high

concentration of defects on the U sub-lattice nucleates new O FPs. The nucleation

processes seem to occur rapidly right at the onset of the equilibration. As shown in Fig.

4-6 (red circles), the number of O interstitials and vacancies increases with a very steep

slope in the beginning. Different from the case in the single crystal, it is followed by an









equilibrium plateau rather than a slow decrease. What should be noted here is that the

equilibrium number of O defects is much larger (about three to four times) than that in

single crystal. The equilibrium number of O defects is determined by competition

between two phenomena taking place simultaneously: (1) random diffusion of O FPs

leading to annihilation by recombination, and (2) GBs being the source and sink of O FPs.

In the single crystal, there are no GBs, so the number of O FPs solely depends on the

recombination rate. Nevertheless, with the presence of GBs in polycrystal, the

source/sink behavior will influence the defect concentration. In addition, there are some

clusters being observed, such as V-40 clusters, Schottky defects, and cuboctahedral

(COT) clusters. This will also be discussed in detail later.

FPs on both U and O sub-lattices: Fig. 4-5 (a) (center grain, Grain 2) shows the

snapshot of the initial structure with 185 FPs on both U and O sub-lattices

simultaneously. As observing in the previous simulation of defects on the U sub-lattice

alone, there is no distinct change in the number of U FPs, but there is a substantial

increase of O FPs, as shown in Fig. 4-6 (blue squares). However, it has to be noted that

the number of O interstitials is always smaller than that of O vacancies by about 30 after

first 100 ps. A similar phenomenon is also observed in the case of FPs on the U

sub-lattice alone. This is because GBs can behave as a source or sink for vacancies but

only as a sink for interstitials, which will be discussed in detail in the next section. In both

these two conditions, the total number of clusters is larger due to higher initial defect

concentration on the U-sub-lattice.

In the single crystal, the number of vacancies and interstitials has to be equal in

order to achieve electroneutrality. In the polycrystal, the overall system still requires















: 1 ;:. '*..':..'.


', .
'" ":'"' 41 '
* *. 4 4,. .r .. *
," .`. ..: "4 C: t'. ., O: .' .'- _. _4 C"
.4.4 *. -r O



.. .. ... .. .. Q .. ,
























*. ... I r p Q -I .. Cr o
*. 7 o .: .... Q. .v. .
.* t ** .' .. .. *. 4' ., 4 i




















** 6
.4. k .,-. : :
,: ': ,., .
., ,,:: ', ,' ., : ', ,' .









;:~~ ~ .:. .a. vp ,r, ,-j ... t" ,.._ .
..~ ~ ~ ~~~~~ cr.:! 4. "7 :; -- .".
:' ~ ~ ~~ ~~ ~ ~ 06 .. ..; _,{r, .''".d C I .:.













(C) I


': -" "' ""-.-; -. "-, .
..-.... .. ...'" ....* f .
-. ....:.a M ^ -. y



*^ ^ ^ ^^^ t Cd e .<
'.*'.I ZE Vi.'.... ".. .. .- .. .
,i ^ ..**^. .^.^ s
:. .-** 'EC ~ ^ c ^ *** *
:.-.:.'.^:: :: T; .:'-.*: : *


-. "
:.Cd". .4j~' -'. c.2.



:: .. .
'* *Moc S ofc" C Ct '
q7'; ot$ C J0 -*O '' ,rC
0 %
.iQ ^ t'; -& & -


rue. ^ ....
*'V o t: r
:-* CI idy.^ q G
,-."-:.^ ~ ~ i e ~ (C ..r;^<*''.**


IC.
:"'i,

* ,
,


'''`'
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,, ,`'' >,_,~






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:; .. .: .:..; !,-~:~.: -~
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1-:~':'4: .~ ">,?-i:'.-
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L. ; .. K ,"
.'" ., .... :'"'
'* -.. .* ^ ,' ,. ^ Ef "-,[. ... .

-. ''y "t



-f?;*:*r rE ** ,.^"E ^
:-. .. ,



<*- & *,.. .& .- *- < .
.. :- .' :... f .. f *^ ?
a&. o% Cd0.OS:



ji .. *.':",; *., -. ..,:.
."' ..,* *^*\** .*t' ; p ^, *..*"',
':!'V:~ .>.:"".v'Y "iI


Figure 4-5.


Evolution of point defect (small circles in green and red for oxygen interstitials
and vacancies, large circles in yellow and grey for uranium interstitials and
vacancies). Snapshots taken at (a) 0 ps, (b) 25 ps, (c) 50 ps, and (d) 400 ps.
All defects are in superposed on the initial structure without introduction of
any defects.


_____i_ __ j:l1 jL_


..i:


:


",.,


:

~.':~.-.'~':::1


-
`
.::::-:_.:..'.::

.::~.;.:~.
'
~. I .
.:
.. ...


;;













250

S200 --*-Grain 1
185
0 ---Grain 1
150
0 ---Grain 2
S--D-Grain 2
.0 100
E --Grain 3
Z 50 ----Grain 3

0
0 100 200 300 400 500 600
Time (ps)

Figure 4-6. Number of O defect showing up in the bulk grain in three initial conditions: (1)
185 FPs only on the U sub-lattice (circle), (2) 185 FPs only on the O sub-
lattice (triangle), and (3) 185 FPs on both U and O sub-lattices (square).
Solid and open notations stand for interstitials and vacancies, respectively.

charge neutrality; however, the results show that there are some differences between the

number of vacancies and interstitials. It can be assumed that when some of the defects

form clusters, the bulk region may not be electroneutral. Therefore, GBs will play the role

of maintaining the charge neutrality in the bulk by modulating the defect concentration

via source/sink behaviors of point defects. Hence, we will take a closer look to see how

GBs interact with these defects.

4.3.3 Grain Boundary Source/Sink Behaviors for Point Defects

The major difference in the defect evolution between the single crystal and the

polycrystal is the presence of GBs. Here specific interaction mechanisms between GBs

and point defects will be characterized. Since uranium vacancies and interstitials have

low migration energies and are less mobile, attention will be focused only on the more

interesting oxygen vacancies and interstitials.









GBs as vacancy sources: The emission of a vacancy from the GB is presumably a

thermally-activated process that is associated with an attempt frequency.84 GBs may

produce many vacancies and reabsorb them before a successful emission occurs in

which the vacancy can sufficiently distance itself into the grain.85 An example is

demonstrated in Fig. 4-7; the red circle indicates the location of the oxygen vacancy

because an oxygen atom is missing from the column which should have specific number

of oxygen atoms. In Fig. 4-7 (a), a snapshot taken at 20.5 ps, the circled vacancy is one

of many vacancies produced by the GB, and now it is ready to go into the interior grain.

As time progresses, the vacancy diffuses further from the GB (Fig. 4-7 (b)-(e)). Fig.4-8

gives close-up views of each snapshot in Fig. 4-7, illustrating the mechanism for the

vacancy moving into the grain. The viewing direction is also along [100]; larger spheres

are uranium atoms; smaller spheres are oxygen atoms; the purple area (defined by the

presence of many miscoordinated atoms) refers to the GB region shown in Fig.4-7; the

big red circle sitting on the oxygen column describes a missing oxygen atom in that

column, i.e. the approximate vacancy position; small black circles represent positions of

specific oxygen atoms involved in this diffusion activity (here, the small black circle 1 in

the figure will be notated as 0-1 in the following text for convenience). From Fig. 4-8 (a)

to (b), 0-1 goes into the GB to the original site of the vacancy, followed by 0-2 filling in

the vacant site of 0-1. Therefore, a macroscopic view showing that the vacancy emits

from the GB to where 0-2 was. This diffusion mechanism is referred to the interstitialcy

mechanism, in which the moving direction of the defect and the atom are opposite.

Similar diffusion mechanism takes place from (c) to (e), so the vacancy finally resides in

the original site of 0-7.
















*** .. .
*, .. .

*^*,'.r .* -. .
.* .** .


.* ** .


*.*,* ... -.. 9


****** ..**. 9* *e .* .% *
."' .. *..
... ... ..** *





.9** '. .


... .. .". .









: :. :. : e ; '.**
S* *
\ .. .. ... .






.. .* *



....... ... .. .. .-.*. ..
; t
9; o* 0 9 9
*. .** .* .


.00. .. 0. *. S .


**.. ... .*.**.(.*.m 0 ; ** .



i 9 9* *


.* **
9** *. 9 9 9 9 9


Are; .


%.. "* 4 *- 0.'. 9 4 .6



* *.*..* *
** **.***.**.*,* 0 *
..... *. *


... .*.... .* .
.* 9 V -9' 9 .
S* *** **....1% t .o .

. '. % o


9 .9. 9 99,9..
.. ...*.. 9.*... .. ,*
.... *.... :..,






--. .9 9 9 0i. *

-.. t
r ~r~~~~~~~~~~~~~ to;
~~~e*


S- *** 9 -

9 '. .
e .. ."* .
*. 9 9 *9
...0 *.






.. ...... .
59 *. .O

.* .. .. .. 'ef **. .. *
***. ***. ***.. *.'. .*







.. *; .*



..... ...... 9. ..o i.- ..*
99.e.e .. -9* .e 9 9O

.... .....;. .. '. 9


Figure 4-7. Progressive snapshots showing the GB being the source for the O vacancy

(red) taken at (a) 20.5 ps, (b) 21 ps, (c) 21.5 ps, (d) 23.5 ps, and (e) 24.5 ps.





65










(a)
4 7
S"' '
5 v
e04



a, "







&



a o .
3 'S
l, m "--
d. .6 .






.4A 03
r- *4 .
*0
a; ;ag :W
"*^. 1< `** s ^ :"


(b)


.'. i 4'
e



463
> & 5 e 3





". '" -" .
". ~" .








() 5



0. 3 ,i

2 ;0
0 1 ..r

.. .... ,
11 (c b ': B


(e) T. it"

7,30






S .0
1e a' -<
3 "
.-




Figure 4-8. Close-up views of the GB being the source for the O vacancy taken at (a)
20.5 ps, (b) 21 ps, (c) 21.5 ps, (d) 23.5 ps, and (e) 24.5 ps. The purple area
refers to the GB region; the labeled oxygen atoms are involved in this activity;
the large red circle indicates the location of the vacancy.










GBs as vacancy sinks: GBs, in addition to being vacancy sources, can also operate as

sinks for both vacancies and interstitials. Fig. 4-9 demonstrates an example of the GB

being the vacancy sink. The snapshot taken at 9 ps shows that the vacancy is still in the

grain but near the GB (Fig. 4-9 (a)). After 0.5 ps, the vacancy is closer to the GB (Fig.

4-9(b)) and finally absorbed by the GB at 10 ps (Fig. 4-9(c)). This vacancy does not

re-emerge after 12.5 ps. The interstitialcy diffusion mechanism is illustrated in close-up

views of Fig. 4-10 with respect to Fig. 4-9. The original vacancy site is filled by 0-1,

followed by 0-2 taking the place of the vacant site of 0-1, so the vacancy is

macroscopically absorbed by the GB. However, it turns out that 0-2 is initially not quite in


Figure 4-9. Progressive snapshots showing the GB being the sink for the O vacancy (red)
taken at (a) 9 ps, (b) 9.5 ps, (c), and 10 ps.


* .."..' -.',,'*I *r* *- -.. ..
.'. .'.'. i. .'. '..' .p
0.*
* ..g .
** ", ".*". Og

-. e -
............... *
.'.f -'* **
*............o.. ... .......: ....***.

.* ,**.**.-.*'.**.* ..
..... .-....
S*.










(a) Y ..,. (b): -


S* o. ,0. ol J -
L 2b)
it 20.
2

01 ..













01 100
() 9 10 1



...a (t. .. ..O ..





02s o



0 1 ..,.; 0 1. :


F igr 4-10. .,Con the 0" vcn take a (
( p (..5o 'af p 1. p
S(C ) _: ai ;- .




01 o01 .




Figure 4-10. Close-up views of the GB being the sink for the 0 vacancy taken at (a) 9 ps,
(b) 9.5 ps, (c) 10 ps, (d) 10.5 ps, (e) 11 ps, (f) 11.5 ps, (g) 12 ps, and (h)
12.5 ps. The purple area refers to the GB region; the labeled oxygen atoms
are involved in this activity; the large green circle indicates the location of
the vacancy.

equilibrium in that vacant site. Fig. 4-10 (d) to (h) exhibits 0-2 moving back and forth

between the GB and the grain, but finally sitting in the site that 0-1 originally was. This

indicates that the thermally-activated diffusion process is not a one-time and one-step


I _^ _












process. Instead, the process requires some attempts to find the most favorable site

before the "successful" interstitialcy diffusion.

It also has to be noted that atoms look to be better aligned in the columnar fashion

at position 0-2 in the Fig. 4-10 (c) than the same column in (b). Therefore, Fig. 4-11

shows single point alignment offsets at positions 0-1 (blue triangles) and 0-2 (red

squares) in Fig. 4-10 (c) calculated from 9 ps to 12.5 ps, referring to Fig. 4-10 (a) to (h).

The offset is with respect to the oxygen columnar axis viewing along [100] direction.

From Fig. 4-11, the highest offset of position 0-1 is at 9.5 ps, when 0-1 is participating in

the vacancy diffusion; the highest offset of position 0-2 is at 11 ps, when 0-2 is also

taking part in the vacancy diffusion. In addition, the offset of position 0-2 is always larger

than that of position 0-1 because position 0-2 is closer to the GB area, which is

characteristically disorder. Offsets for both positions finally decrease after the vacancy is

"successfully" absorbed by the GB, which also indicates that the damage is healed in a

more macroscopic sense.

0.5
-- position 0-2
0.45 --position 0-1
0 0.4


5 0.35 -
0.3
O 0.3
0.25

0.2
8.5 9 9.5 10 10.5 11 11.5 12 12.5 13
Time (ps)

Figure 4-11. Single point alignment offsets at positions 0-1 and 0-2 in Fig. 4-10 (c) with
respect to the time from Fig. 4-10 (a) to (h).









So far, evidence has been shown that GBs can nucleate and annihilate vacancies.

As for interstitials, there are some differences from vacancies.

GBs as interstitial sinks: Annihilation of interstitials by the GB is also observed, as

shown in Fig.4-12. In a similar manner, the snapshot taken at 6.5 ps displays that the

circled interstitial (green) is far from the GB and is not ready to diffuse (Fig. 4-12 (a)). At

10 ps, the interstitial is still in the same site, but atoms between it and the GB are no

longer in a columnar alignment (Fig. 4-12 (b)). This disorder is thermally activated by

some atoms acquiring enough energy to make jumps, and this actually paves the way for

the interstitial to move towards the GB, finally disappearing in the GB, as shown in Fig.

4-12 (c) to (e). These phenomena can be further focused in close-up views in Fig. 4-13.

Especially in (c), a maximum randomness appears for the column occupied by 0-3, 0-4,

0-5, 0-6, and 0-7. It turns out that only three of them can stay, so 0-6 moves into the

GB and 0-7 goes back to its original site, leaving 0-3, 0-4, and 0-5 behind. Similarly, the

rearrangement of atoms makes them more aligned in Fig. 4-13 (e) than in (c).

From the study of defects evolution above, interstitials diffusing into the GB,

vacancies coming out of the GB and diffusing back to the GB are observed. It should be

noted that no interstitials are observed to come out of the GB. Therefore, it can be

inferred that GBs can be sources or sinks for vacancies but only sinks for interstitials. It

has been reported by a radiation damage simulation research in copper that the

formation energy of interstitials at the GB is smaller than in the bulk. Therefore, the

formation energy of oxygen interstitials at the GB can also be smaller than in the bulk.

With this assumption, there is a barrier for interstitials to re-enter the perfect bulk. As a

result, no interstitials will be emitted from GBs.



















r r irr C*wr-* -. '. *
R .' R.c *' *
*0*
S...' -.: .. ... a a.










(C) ** c. ...:..^*


0 .y *.R;. a ..



R .^ .. *. .
'




* c.. *
.o e I e I o
r eo C"o
,oj~, e
r '. Ye lo e +0











el llI I"~j
L
-+-L ) ..


.-: 'o* -.-'.. '*.** ". -* -o'. "*
R.. *' *R ..
.. ..'..b-. .-. .* *
S
.r. .



::I*.RR&i.*. *Y*\*
.. .,



R.% ** ***%*..*.- *. *
"'e R 0





** ** ** ^1 ol-* 1 *,*f *.-* c



. :::* **^c y
R.cR .R pr. *'
.."
.o" 0'. ,'I


.* .. ..










:*:^ & ^ ;. RC..**.-.*/*.*
*.- *. *


.* **.*.* *
Ri Jgg !IJ RhpcJ
(d) [. -.




::::: c. R .o** *
e+0*' .* R* 0 C C

i-iil^ ^
,. R.
,.-. ',.'-... '. ... .


.
". *.* -.
*C .



S.R... ... :. ..



.. .. %
R R .Ri
I .R c... Re I I ..RR I
CR....... .** *1
c Re .. uq.i
.~ ~r' Ce


Figure 4-12. Progressive snapshots showing the GB being the sink for the O interstitial
(green) taken at (a) 6.5 ps, (b) 10 ps, (c) 10.5 ps, (d) 11 ps, and (e) 15 ps.


*.. I
,i .

R^-. .. ** *.*.* *~ ** R. ,
.."' .. '' ...

< R c.* R. .*..
E R i o I


* R* R% ~ a ~ *&345R "r *" C'*- R *
c.- < RC R.. 0. 0
*' RRR RO C R -.* i .^f'^ r ^s~ -, .R*1 *"
R R. R .R .I S 5
......:


..... ...:, .. -,. .

.. ...... +: ."
i~~~~~~ io i













20

3
30
4
50
a 06
." : 07


r **. *


> oo1 <




06o
4


4 : 7 ." .
f *_ .^ o "


(b)K,



L 20














2
.. ..

4 6*
0 0







I.* 70


Figure 4-13. Close-up views of the GB being the sink for the O interstitial taken at (a) 6.5
ps, (b) 10 ps, (c) 10.5 ps, (d) 11 ps, and (e) 15 ps. The purple area refers to
the GB region; the labeled oxygen atoms are involved in this activity; the
large green circle indicates the location of the interstitial.


S7
2* 60
7 '
.. .,- 0
* *'6` __ ^__









The diffusion mechanism was qualitatively described above, and now some

quantitative analyses will be performed in order to fully characterize the diffusion.

To take a specific example, in the second case of GBs being the vacancy sink, it

can be seen from Fig. 4-9 (b) to (c) that the vacancy travel about 1.6 A along [010]

direction within 0.5 ps. This diffusion distance is less than the first nearest neighbor

distance (2.7 A) in the single crystal.

To ascertain if this rate of vacancy diffusion is reasonable or not, recall Eq. 2-6 that

the diffusivity has a temperature-dependence:


D = Do exp -H (2-6)


using the diffusion parameters from Govers et al.86 for UO2-x with the Basak potential.

The pre-exponential diffusivity ( Do) is 0.00045 cm2/s and the migration enthalpy barrier

(AH^,) is 0.45 eV. With the equilibration under 1000 K, the atom diffusivity ( DA ) can be

estimated to be about 2.4x10-2 A2/ps.

The atom diffusivity is therefore calculated form the Eq. 2-14 with the correlation

factor f, = 0.653 for the free vacancy, c,= 1.5x10-3, and CA' 1.

DACA = fvDvC = fiDiCi (2-14)

The calculation gives the atom diffusivity ( Dv) to be about 24.8 A2/ps.

In simulations, the self-diffusion distance of the vacancy can be evaluated by Eq.

2-22:

112
D,v =, Fo (2-22)

So the diffusion distance is determined to be about 8.6 A. Compared with 1.6 A from the

observation, it is of the same order of magnitude. In addition, the diffusion direction is








also consistent with the primary oxygen diffusion pathway, <001> direction, for the

fluorite-based material.87 The discrepancy of the value may come from evaluation by an

only single diffusion event in a very short observation time. In order to obtain high

accuracy, the diffusion rate should be estimated by an average of multiple diffusion

events for a longer time period.

4.3.4 Vacancy Clustering: Schottky Defects

A Schottky defect is composed of one U vacancy and two O vacancies. The

formation of Schottky defects are observed in the presence of U FPs, that is, the

condition of defects on the U sub-lattice alone and on both U and O sub-lattices.

Evidence has been shown that the VJ Vu -VJ* Schottky (neutral tri-vacancies)

defect cluster is more stable than either the V'V Vu di-cluster or entirely separated

(a ^ ~& (hi- (b i W _. (








(c)t


,. ... ..L





Figure 4-14. Schematic representation of three different configurations of Schottky
defects: (a) <100>, (b) <110>, and (c) <111>. Grey spheres stand for
vacancies.



































Figure 4-15. Snapshot of selective area showing Schottky defects taken at t = 250 ps: (a)
<111> Schottky, (b) <110> Schottky, and (c) <100> Schottky.

vacancies.88 Hence, even though the Schottky defect is created by diffusion, it is

relatively stable. According to the relative positions of the two oxygen vacancies, there

are three types of Schottky defects (Fig. 4-14), <100>, <110>, and <111> Schottky,

namely, among which <110> and <111> Schottky clusters are also observed by Aidhy et

al. in the single-crystal U02. In this work24'37, <100> Schottky is also observed. Three

kinds of Schottky defects are simultaneously shown in Fig. 4-15 taken at 250 ps. The

formation of the Schottky defect anchors 0 vacancies, which sequesters them from the

vacancy-interstitial recombination. As a result, this leads to their many counterpart 0

interstitials developing other stable complex clusters.

4.3.5 Interstitial Clustering: Cuboctahedral Clusters

The presence of Schottky defects sequesters 0 vacancies from their counterparts,

so these interstitials develop a relatively complex structure, namely, cuboctahedral


S. .

. ( .. .. .. .m
. .



...o.ooo.o.I
.. : o .Z : o: o o*, .. o o o .o o. .



.- o e : .V-. .,,.. E. *.*, ,
j "'>*'' t.^. r.F i ioc r s Sa
D .o \\ Qo-- \.i \ ^ \ cs .-
~ ~ ~ ~ ~ "~ .....,, WK. ....^.,f
~ ~ ~ ~ o o Q .c^
o o o C -/'l* < C CJC
*~ ~ ~ *~ *^ ** O**








(COT) clusters. Discussion about COT clusters has been well described by Aidhy et al. in

the work of single-crystal U02.24'37 The progressive formation process is recalled in Fig.

4-16. The relative energy is decreasing step by step, indicating that the system tends to

form clusters. There are three types of COT clusters defined by the occupancy of the

octahedral site (green circle). It can be a U interstitial (COT-u), an O interstitial (COT-o),

or a vacancy (COT-v). In this work, three kinds of COT clusters are simultaneously

shown in the snapshot taken at 250 ps (Fig 4-17). In addition to COT clusters (a) and (b)

already in complete shape, COT clusters (c) and (d) are found to be still under

construction, which are found to correspond to the step (c) and (d) in Fig. 4-16. Hence,

(a) *(b) (C)









(d) e (e)









Figure 4-16. Schematic representations of the formation process of COT clusters (blue
for U, red for 0, pink for O interstitial, grey for O vacancy, and green for
either U, O, or vacancy). (a) Fluorite unit cell. (b) The first O interstitial
enters the unit cell by knocking two O ions off their original lattice sites,
leading to the formation of two vacancies and three interstitials. (c), (d), and
(e) are snapshots for the second, third, and forth O interstitials coming into
the unit cell.





























Figure 4-17.


Snapshot of selective area showing COT clusters taken at t = 250 ps: (a)
COT-v, (b) COT-o, (c) COT-u, and (d) COT-u. It should be noted that both
(COT-u)s are not fully constructed.


with the complex structure, the aggregation of interstitials into COT clusters is

ascertained to be a rapid but not one-step process. For every complete COT cluster,

eight more 0 FPs will be introduced to the system. This clustering mechanism accounts

for the sharp increase in both cases initially with the presence of U FPs.

4.3.6 Interstitial Clustering: V-40 Clusters

In the V-40 cluster, an 0 vacancy is surrounded by four 0 interstitials in a

pyramidal fashion (Fig. 4-18). The same kind of oxygen cluster is also reported in

relaxation of Willis clusters89'90 by Geng et al. using first-principle calculations in UO2+x.91

According to their results, this kind of cluster is the minority among all defects. In the

condition of defects on the 0 sub-lattice alone, most of the 0 FPs are healed after 400 ps,

but two V-40 clusters form. Therefore, from Fig. 4-6, the number of 0 interstitials is a


~d~a~3









little fewer than the number of 0 vacancies, and the difference is 6. A snapshot taken at


400 ps is shown in Fig. 4-19.


Figure 4-18. Schematic representation of the configurations of V-40 clusters. The 0
vacancy is surrounded by four 0 interstitials (grey) in a pyramidal fashion.


Figure 4-19. Snapshot of selective area showing V-40 oxygen clusters taken at t = 400
ps. In each cluster, an 0 vacancy is surrounded by four tetrahedral-sited 0
interstitials.









CHAPTER 5
CONCLUSIONS AND FUTURE WORK

5.1 Conclusions

In terms of analysis tools, the lattice matching analysis (LMA) method is

complementary to the conventional common neighbor analysis (CNA) method. The

cutoff radius of 0.19ao (about 1.039 A) is tested in the single crystal to identify the same

amount and spot of defects as the CNA. The LMA is then applied to polycrystal to

determine the exact number, type, and position of every point defect. A big advantage is

that it can easily tell the cluster structure.

As for MD simulations, evolution of the defects basically has the same trend as in

the single crystal. Similarly, U FPs are seldom annihilated during equilibration while of O

FPs are mostly healed except those which form clusters. Three kinds of initial conditions

are performed. For FPs created on the O sub-lattice alone, the number of O interstitials

is slightly larger than that of O vacancies because some charged pyramidal clusters form,

in which an O vacancy is surrounded by four O interstitials. In cases of FPs created on

the U sub-lattice alone and on both U and O sub-lattices, the number of O vacancies is

larger than that of O interstitials. In addition to spontaneous vacancy-interstitial

recombination events, point defects have another choice, either diffusing into or emitting

from the highly energetic GB. When some of the defects form clusters, one grain bulk

region may lose its local electroneutrality. Therefore, GBs maintain the local charge

neutrality in the bulk by modulating the defect concentration via source/sink behaviors of

point defects. Nevertheless, GBs can be the source and sink for vacancies but only the

sink for interstitials. It has been reported by a radiation damage research in copper that

formation energy of interstitials at the GB is smaller than in the bulk.93 Therefore, in UO2,









the formation energy of oxygen interstitials at the GB can also be smaller than in the bulk.

Thus, there is a barrier for interstitials to re-enter the perfect bulk. As a result, no

interstitials will be emitted from GBs. In addition, for the case of creating defects on only

the U sub-lattice, the equilibrium concentration of O defects is much larger in the

polycrystal than that in the single crystal. This can also be explained by the source/sink

behaviors with GBs. In the single crystal, the concentration of newly nucleated O defects

solely depends on the number of the U FPs. However, in the polycrystal, more O defects

can be possibly provided by GBs. Once they successfully enter the interior grain, they

will be trapped by forming clusters, such as Schottky or cuboctahedral (COT) clusters.

With respect to the diffusion mechanism, these source/sink activities take place

with a set of chain reactions via the vacancy and interstitialcy diffusion. That is, not

always the same single atom is moving; instead, many atoms are involved in one

diffusion event. The equilibration process is diffusion-controlled, which highly depends

on the formation and migration energies of the defects. The diffusion path is ascertained

to follow the primary diffusion direction <001> for the fluorite-based structure with the

reasonable oxygen vacancy diffusion distance.87 The arrangement of atoms is focused in

close-up views to become disordered when the diffusion activity is about to occur;

subsequently, all the atoms come back to lattice sites and realign in the normal fashion.

In addition, the single point alignment offsets give further quantitative analysis to confirm

the phenomena being observed.

Regarding clusters, there are many kinds of clusters formed during equilibration.

First, a minor kind of cluster, V-40, is observed, and the appearance of V-40 clusters

accounts for the number of O interstitials is fewer than the number of O vacancies in the









condition of defects on the O sub-lattice alone. Compared with V-40 clusters, Schottky

defects and COT clusters are much more prevalent. Schottky defects can be divided into

three types by the relative positions of oxygen vacancies: <100>, <110>, and <111>

Schottky. Among them, <111> Schottky has the lowest formation energy, followed by

<110>, and then <100> with the Basak potential.83 Therefore, <111> Schottky is the most

common type in our simulations. As for COT clusters, there are also three kinds defined

by what occupies the octahedral site: COT-v, COT-u, and COT-o. In literature, COT-o

has lower formation energy than COT-v, both negative;93 nevertheless, there is no

information about COT-u. Our results show that COT-u is the most common. This may

be due to the initial high concentration of defects on the U-sub-lattice. Therefore, it can

be expected that the formation energy of COT-u will be the lowest among these three

clusters.

In conclusion, the diffusion-controlled kinetic evolution of defects method is

complementary to conventional collision cascade simulations to provide a better

understanding of the kinetic phase of defects evolution at the atomistic scale. Extending

from the work on the single crystal, this work focuses on the more representative

polycrystalline UO2. Differences in the numbers of radiation-induced point defects

(vacancies and interstitials) are attributed to the presence of GBs, which can modulate

the defect concentration via source/sink behaviors. These activities involving the

vacancy and interstitialcy diffusion mechanism are evidentially witnessed by

atomistic-scale snapshots, along with three types of clusters, V-40 clusters, Schottky

defects, and COT clusters. However, with the presence of GBs, whether the radiation

tolerance can be enhanced or not is still not determined.









5.2 Future Work

This study has been performed on polycrystalline U02 elucidating qualitative GB

source/sink strengths. A corresponding work is required to provide more quantitative

information. For example, calculating the total number of clusters is needed to identity if

GBs essentially enhance or impede the formation of clusters. Actually, GBs open up the

possibility of variety of mechanisms that are still not well understood, such as (1)

diffusivities in the bulk and GB regions, (2) diffusivities in high- and low-angle

misorientation. An understanding of differences of diffusivities in these cases allows us to

precisely predict the defect evolution.









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BIOGRAPHICAL SKETCH

Kun-Ta Tsai was born in Hsinchu, Taiwan. He received his bachelor's degree in

chemical engineering from National Cheng-Kung University, Tainan, Taiwan, in 2007.

Kun-Ta came to the United States to pursue higher education and entered the

Department of Materials Science and Engineering at University of Florida, Gainesville, in

Fall 2008. He joined the Computational Materials Science Focus Group with Prof. Simon

R. Phillpot in Spring 2009. Kun-Ta is expecting his master's degree in Summer 2010.

After joining the Gator Nation, Kun-Ta watched his first football game in his life, and

started to become crazy about it. In the same academic year, the Florida Gators won the

Bowl Championship Series (BCS) National Championships. He is proud to say "Go

Gators".





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1 DIFFUSION CONTROLLED KINETIC EVOLUTION OF RADIATION INDUCED POINT DEFECT S IN POLYCRYSTALLINE UO 2 FROM ATOMIC LEVEL SIMULATION By KUN TA TSAI A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2010

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2 2010 Kun Ta Tsai

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3 To my family with love and gratitude

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4 ACKNOWLEDG E MENTS First of all, I would like to express my sincer est gratitude to my advisor Prof. Simon R. Phillpot for his invaluable guidance and encouragement In the beginning, for me as a non native speaker of English, the Santa Claus British accent distances us. Nonetheless, throughout my master s study, I am subtly but deeply influenced by his wisdom and enthusiasm towards science blended with a great sense of humor. He likes to share his knowledge by throwing open questions and to give us the libe rty to brainstorm. It is my greatest honor to have him as my advisor. I would also like to express my deepest appreciation to my committee members Prof. Susan B. Sinnott and Prof Juan C. Nino for providing valuable suggestions to my research work. I feel extremely fortunate to work with a wonderful team the Computational Materials Science Focus Group. I want to specially thank Dr. Dilpuneet Aidhy for his kind help on my research work. This thesis benefit ed immensely and would not be accomplished without h im. I am also thankful to Dr. Tao Liang, Dr Alex Chernatinsky Dr. Rakesh Behera, Dr. Haixuan Xu, Mr. Donghwa Lee Mr. Priyank Shukla Mr. Chan woo Lee, Mr. Donghyun Kim and Mr. Eric Bucholz for their friendly assistance. I want to mention my two dearest senior friends Tze Ray Shan and Yu Ting Cheng. Many thanks to both of them for constant support and encouragement in my life inside and outside the school. Finally, I would like to thank my family for continuou s support with unconditional love. This thesis is dedicated to them.

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5 TABLE OF CONTENTS page ACKNOWLEDG E MENTS ................................ ................................ ............................... 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 8 A BSTRACT ................................ ................................ ................................ ................... 10 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ ..... 12 1.1 Sources of Energy ................................ ................................ ............................ 12 1.2 Nuclea r Energy ................................ ................................ ................................ 14 1.3 Radioactive Decay and Nuclear Fission ................................ ........................... 18 1.4 Motivation and Objective ................................ ................................ .................. 22 1.5 Beauty of Simulation ................................ ................................ ......................... 23 2 DEFECTS IN SOLIDS ................................ ................................ ............................. 24 2.1 Introduction ................................ ................................ ................................ ....... 24 2.2 Intrinsic Po int Defects ................................ ................................ ....................... 24 2.2.1 Schottky Defect ................................ ................................ ........................ 25 2.2.2 Fren kel Defect ................................ ................................ ......................... 25 2.2.3 Thermodynamics of Point Defects: Equilibrium Concentration ................ 26 2.3 Grain Boundary ................................ ................................ ................................ 27 2.4 Atom Movements in Materials ................................ ................................ ........... 28 2.4.1 Diffusion Mechanisms ................................ ................................ .............. 28 2.4.2 Atomic Theory of Diffusion ................................ ................................ ....... 30 2.5 Crystallography of UO 2 ................................ ................................ ..................... 34 2.6 Ra diation Damage ................................ ................................ ............................ 36 3 SIMULATION METHODOLOGY ................................ ................................ .............. 39 3.1 Simulation Methodologies ................................ ................................ ................. 39 3.2 Molecular Dynamics Simulation ................................ ................................ ........ 41 3.3 General MD Algorithm ................................ ................................ ...................... 41 3.4 Periodic Boundary Condition ................................ ................................ ............ 43 3.5 Interatomic Interactions ................................ ................................ .................... 44 3.5.1 Long Range Interactions ................................ ................................ .......... 45 3.5.2 Short Range Interactions ................................ ................................ ......... 46 3. 5.3 Thermodynamic Conditions (Ensembles) for Simulations ........................ 47 3.5.4 Thermostat ................................ ................................ ............................... 49 3.5.5 Barostat ................................ ................................ ................................ ... 49 3.6 Conventional R adiation Damage Method by MD Simulations ........................... 50

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6 4 GRAIN BOUNDARY INFUENCE ON RADIATION DAMAGE IN UO 2 ..................... 52 4.1 Kinetically Evolving Irradiation Induced Defects Method by MD Simulation ..... 52 4.2 Simulation Methodology ................................ ................................ ................... 52 4.2.1 MD Simulation ................................ ................................ ......................... 52 4.2.2 Defect s Analysis Methods ................................ ................................ ........ 54 4.3 Results ................................ ................................ ................................ .............. 56 4.3.1 Frenkel Pair Defects in Single Crystal UO 2 ................................ ............. 56 4.3.2 Frenkel Pair Defects in Polycrystalline UO 2 ................................ ............. 5 8 4.3.3 Grain Boundary Source/Sink Behaviors for Point Defects ....................... 63 4.3.4 Vacancy Clustering: V 4O Clusters ................................ ......................... 77 4.3.5 Vacancy Clustering: Schottky Defects ................................ ..................... 74 4.3.6 Interstitial Clustering: Cuboctahedral Clusters ................................ ......... 75 5 CONCLUSIONS AND FUTURE WORK ................................ ................................ .. 79 5.1 Conclusions ................................ ................................ ................................ ...... 79 5.2 Future Work ................................ ................................ ................................ ...... 82 LIST OF REFERENCES ................................ ................................ ............................... 83 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 88

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7 LIST OF TABLES Table page 3 1 Potential parameters used in the present work ................................ ................... 48

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8 LIST OF FIGURES Figure page 1 1 1973 and 2007 fuel shares of total primary energy supply ................................ 12 1 2 Evolution from 1971 to 2007 of world total primary energy supply by fuel .......... 13 1 3 World net nuclear electric power generation (1980 2006) ................................ .. 16 1 4 World map for commercial nuclear power plants ................................ ................ 17 2 1 Point defects: (a) vacancy, and (b) interstitial ................................ ..................... 25 2 2 Point defects in (a) Schottky defect, and (b) Frenkel defect ............................... 26 2 3 Schematic representation of three diffusion mechanisms ................................ ... 29 2 4 Fluorite crystal structure ................................ ................................ .................... 35 2 5 T hree different symmetry operations for the f luorite crystal structure ................. 36 2 6 Defect evolution during ballistic and kinetic phases of th e radiation damage ..... 37 3 1 Hierarchical multi scale approach to computational material engineering ......... 40 3 2 P eriodic boundary conditions of a fluorite structure supercell ............................. 44 3 3 Inter ionic potential as a function of distance between U+2.4 and O 1.2 ions .. 48 4 1 F ully equilibrated polycrystalline UO2 microstructure ................................ ........ 53 4 2 Schematic representations of the common neighbor analysis (CNA) ................. 54 4 3 Schematic representations of the lattice matching analysis (LMA) .................... 55 4 4 Evolution of O FPs in the singel crystal ................................ ............................. 57 4 5 Snapshot of e volution of point defect in polycrystal ................................ ............ 62 4 6 Evolution of O FPs in the polycrystal ................................ ................................ .. 63 4 7 S napshots showing the GB being the source for the O vacancy ........................ 65 4 8 Close up views of the GB being the source for the O vacancy. .......................... 66 4 9 Progressive snapshots showing the GB being the sink for the O vacancy ......... 67 4 10 Close up views of the GB being the sink for the O vacancy ............................... 68

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9 4 11 S ingle point alignment offset s ................................ ................................ ............. 69 4 12 S napshots showing the GB being the sink for the O interstitial .......................... 71 4 13 Close up views of the GB being the sink for the O interstitial ............................. 72 4 1 4 T hree different configurations of Schottky defects ................................ .............. 74 4 1 5 Snapshot showing Schottky defects ................................ ................................ ... 75 4 1 6 F ormation process of COT cluster s ................................ ................................ ... 76 4 1 7 Snapshot showing COT clusters ................................ ................................ ........ 77 4 1 8 C onfigurations of V 4O clusters ................................ ................................ .......... 78 4 1 9 Snapshot showing V 4O oxygen clusters ................................ .......................... 78

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10 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science DIFFUSION CONTROLLED KINETIC EVOLUTION OF RADIATION INDUCED POINT DEFECT S IN POLYCRYSTALLINE UO 2 FROM ATOMIC LEVEL SIMULATION By Kun Ta Tsai August 2010 Chair: Simon R. Phillpot Major: Materials Science and Engineering Nuclear energy is capable of provid ing people with energy for some billions of years if used properly. In most operating nuclear reactors, fluorite structured UO 2 is used as the fuel. After irradiation, UO 2 undergoes structural damage by forming high concentration of point defects, which can l imit the lifetime of the fuel. In order to enhance the radiation tolerance of UO 2 it is of great interest to understand the fundamental mechanism of intrinsic point defects evolution under different conditions. Rather than a single crystal, t his thesis fo cuses on the more representative polycrystalline UO 2 and uses molecular dynamics (MD) as the modeling tool. This study provides a better understanding of defects evolution in the kinetic phase at the atomistic scale. I n addition to spontaneous vacancy interstitial recombination events, some of the point defects are found to have interactions with GBs. These source/sink behaviors take place by a set of interstitialcy diffusion processes The diffusion direction is ascertained to follow the primar y diffusion direction <001> for the fluorite based structure with the reasonable oxygen vacancy diffusivity on the order of 10 6 cm 2 /s ( 10 4 2 /s ) A mean

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11 square displacement calculation confirms that atoms may realign after the diffusion activity, supporting the concept that damage can be heal ed by GBs in a macroscopic sense. It is also found that the presence of defects on the U sub lattice can lead to the formation of clusters, which make UO 2 less radiation tolerant. In addition t he equilibrium concentration of O defects is much larger in polycrystal than that in single crystal. This can be attributed to GBs possibly supplyi ng O defects into the bulk.

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12 CHAPTER 1 INTRODUCTION 1.1 Sources of Energy Energy can be neither created nor destroyed, say our science books, but can only be transferred from one form to another. Mankind has been engaged with this transformation since life began on the planet. Since industrialization, humans have utilized a variety of energies and explored more. With growth accelerating in developing countries, demands on energy resources are being stretching to new limits. In recent decades, the urgent need for energy has create d a global movement to exploit all potential materials. On the other hand, materials are used to produce energy or enable energy to be converted into other useful forms. Energy and materials therefore have a continual and mutually enriching relationship. The c hoice of materials for energy generation depends on the availability and accessibility of the source. Fig. 1 1 gives pie charts of the fraction of the total primary Figure 1 1. 1973 and 2007 fuel shares of total primary energy supply (Mtoe, million ton n e of oil equivalent). (*Others include geothermal, solar, wind, heat etc. ) 1

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13 energy from various sources supply in 1973 and 2007. 1 It was estimat ed by the International Energy Agency (IEA) that in 2007 primary sources of energy consisted of oil 34.0 %, coal /peat 26.5 %, and natural gas 20.9 %, amounting to an 81.4 % share of fossil fuels in primary energy consumption in the world. This indicates the world s significant dependence on fossil fuels due to their continuing widespread availability and the large d eveloped infrastructure. Yet fossil fuels are non renewable resources and their production raises many environmental concerns ; thus there has been a gradual movement towards cleaner fuels. Fig 1 2 shows the evolution from 1971 to 2007 of world total prima ry energy supply by fuel. 1 It should be noted that hydro power was surpassed by nuclear power in total electrical generation in 1984 and that nuclear the share has increased to 5 % of the total energy supply over the last 34 years. Along with increasing awareness of environmental issues, nuclear energy is expected to play an important role in the energy portfolio in the coming decades. 2 Figure 1 2. Evolution from 1971 to 2007 of world tota l primary energy supply by fuel (Mtoe). 1

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14 1.2 Nuclear Energy Ernest Rutherford is one of the central figures in the exploration of nuclear physic s. He was awarded the 1908 Nobel Prize in Chemistry for investigations into the disintegration of the elements and the chemistry of radioactive substances. 3 . Taking the atomic weight of the emanation as 222, one gram of the eman ation emits during its life 2 10 9 gram calories of heat. This evolution of heat is enormous compared with that emitted in any known chemical reaction . excerpted from Rutherford s article in Encyclopedia Britannica (1910) 4 The basic ener gy fact is that the fission of a uranium atom g enerates 10 million times the energy as that produced by bu r ning an atom of carbon from coal. This concept that a large amount of energy can be released by the disintegra tion of an atom was soon brought to public attention. The first time that electricity was ge nerated by a nuclear reactor occurred in an Idaho experimental station. 5 The world s first nuclear power plant was operational in Russia in 1954 generating around 5 megawatts (MW) of electric power. 6 In the following years, other nuclear stations were opened in the U.S. and the U.K., etc. In the early years, nuclear capacity increased rapidly, rising from less than 1 gigawatt (GW) in 1960 to 100 GW in the late 1970s, and 300 GW in the late 1980s. 7 Since then, the increase in worldwide capacity started to slow down due to consecutive major nuclear accidents in Three Mile Island (1979) and Chernobyl (1986). More and more anti nuclear movements arose based on the fear of possible nuclear disasters, radiation, and waste production. Despite those controversi es many countries remain active in developing nuclear power, including China, India, Japan, and Republic of Korea. 8 With modern designs to meet strict safety standards, nuclear reactors are becoming much more reliable In ad dition, the superiority of the non CO 2 emitting source makes nuclear energy more promising.

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15 Nuclear power energy can be generated by two types of n uclear reactions: fission and fusion. Fission based nuclear technology is well established, whereas fusion technology is expected to become commercially available in the second half of the century. The most widely used material for fission is uranium. Natu ral uranium is almost entirely composed of two isotopes, U 23 8 (99.3 %) and U 23 5 (0.7 % ) indicating that U 238 is about 140 times more prevalent than U 235 However, only U 235 can directly sustain the fission chain reaction as a nuclear fuel source ; U 238 is not fissionable with thermal neutrons. A nuclear power reactor has a large number of fuel rods in the core. E very rod contains pellets of uranium oxide. When an atom of U 235 is hit by a neutron, it can undergo a fission reaction, producing two f ission fragments and other free neutrons with a very high speed. In steady state operation of the reactor, the free neutron is absorbed by another U 235 atom leading to another fission. T he kinetic energy of other fission products is converted to thermal e nergy when they are stopped by nearby atoms. The heat then is absorbed by water, which becomes steam to drive the turbines to generate the electricity. T he power le vel of the reactor can be controlled by the amount of steam withdrawn. In addition, the cont rol rods which absorb neutrons can also be moved in and out of the reactor to avoid the reactor temperature getting too high. Some analysts suggest that oil wells might be depleted within 70 80 years. 9 Natural gas might run out a little later. Yet p resent reactors that use only the U 235 in natural urani um will likely have fuel supplies f or some hundreds of years. Bernard Cohen a Newton scholar in history of science, even predicts that wi th breeder reactors, we can have plenty of energy for some billions of years. 10

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16 If more efficient usage of neutrons in the reactor is designed, U 238 could be converted to plutonium which produces more fissionable material s. This kind of design is called a breeder reactor. 7 Thus, with the usage of U 238 as the fuel in breeder reactors, fuel res ources based on breeder reactors could provide adequate energy for billions of years. Commercial breeder reactors have been deployed in several countries, such as th e U.S., U.K., France, Russia, Japan, and India. 11 Breeder reactors can be regarded as a kind of resource instead of just a reserve. Because their cost is a lot higher than present reactors, and because large uranium reserves have also been discovered in recent years it is perhaps unlikely that we will see breeder reactors all over the world soon Nevertheless, i n the very long term breeder reactors will be used since they supply about 100 times as much energy from a kilogram of uranium as do present rea ctors. This will make the present stock of uranium go much farther Figure 1 3. World n et n uclear e lectric p ower g eneration (1980 2006) 12

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17 Fig 1 3 demonstrates the world net nuclear electric power generation from 1980 to 2006. The nuclear share of electricity net generation increased from 4.5 % in 1973 to around 20 % in 1990 since th en it has been approximately constant 12 According to Internat ional Atomic Energy Agency (IAEA), there are 436 commercial nuclear power plants, operating in 31 countries, as illustrated in Fig 1 4. 13 Not all the world is employing nuclear energy. France generates 78 % of its electricity from nuclear reactors, most o f the rest being hydroelectric. Japan is close to 30 % and increasing steadily due to their lack of domestic coal and oil. 14 Ten African countries are now in pursuit of their own nuclear plants because wind and solar solutions are not reliable enough. Figure 1 4. World map for commercial nuclear power plants (research reactors not included). 13 To sum up, i n the short term, probably the next hundred years, there is so much uranium that no one can profitably prospect for more. In the medium term breeder

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18 reactors will extend the energy obtained per kilogram of uranium by a factor of about 100. In the very long term plenty of uranium can be extracted from seawater for a few billion years. After being no longer efficient in sustaining a nuclear reaction in the nuclear reactor (usually at the nuclear power plant), spent nuclear fuel is put in storage condition. Spent nuclear fuel is highly radioactive and potentially very harmful, such as causing incapacitation and death. The radioactivity of nuclear waste s decreases with the passage of time through radioactive decay For radioactive material s, t he amount of time needed to decrease the radioactivity to one half the original amount is called the radioactive half lif e 15 Many of the radioactive elements in spe nt fuel have long lives. For example, U 235 has the half life of 713,000,000 years, and U 238 has the half life of 4500,000,000 years. 16 With these long lived radioactive elements, nuclear fuel must be isolated and controlled for thousands of years by a ba rrier or radiation protection shield. Department of Energy (DOE) is developing plans for a permanent disposal facility for spent fuel at Yucca Mountain and Nevada 15 This has aroused some controversy, particularly with state and local authorities. Several complementary measures are still required for the long term development of nuclear energy. 1.3 Radioactive Decay and Nuclear Fission Spontaneous radioactivity was first recognized by a French physicist, Henri Becquerel, in 1896 while investigating phosphoresce nce in uranium salt. Along with Marie Curie and Pierre Curie, who jointly found another two radioactive elements, radium and polonium in 1898, they won the 1903 Noble Prize in Physics. 1 7 The main processes of radioactivity involve four types of decay; alpha ( ) decay beta ( ) decay gamma ( ) decay, and neutron capture.

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19 Alpha ( ) decay : A lpha decay is the process in which a nucleus ejects an alpha particle, which is essentially the helium 4 nucle us ( ). For example E q 1 1 1 8 represents the daughter nucleus ( ) emitted by the parent atom ( ), producing an alpha particle. ( 1 1) Alpha decay is the most common type of nuclear decay for elements with atomic number greater than 82. This is because the greatest bonding energy is lost per nucleon and the alpha particle is an especially low energy arrangement of two protons and two neutrons. The typical kinetic energy of alpha particles is about 5 MeV with a speed of 15,000 km/s (i.e. 5% of the speed of light). Being relatively slow, heavy, and positively charged, alp ha particl es have a short free p ath in length that is so small that they easily interact with other atoms by losing their kinetic energy within a very short distance, usually a few centimeters of air. Therefore, external alpha radiation is generally not harmful. Bet a ( ) deca y : Beta decay is the transformation of a proton into a neutron with emission of a positron, or the transformation of a neutron into a proton with emission of an electron It oc curs when there are too many protons or neutrons in a nucleus 1 9 There are three kinds of beta decay, beta plus ( + ) decay, beta minus ( ) decay, and electron capture ( K capture). In + decay, the weak nuclear force convert s a proton into a neutron while emitting a positron ( e + ) and a n electron neutrino ( e ): ( 1 2 ) Neutrinos are electrically neutral elementa ry particles that travel at the speed of light.

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20 In decay, a neutron transforms into a proton by emitting an electron ( e ) and an electron anti neutrino ( ): ( 1 3 ) Anti neutrinos are antiparticles of neutrinos. The third type of beta decay is electron capture, which is where + decay accompanies a shell electron throughout the decay process. It can be expressed as: ( 1 4 ) This decay is also called K capture because the innermost electron lies in the K shell, which has the highest probability to interact with the nucleus. To sum up, within all types of beta decay, the atomic number of the parent nucleus is different from that of the daughter nucleus while the mass number is the same. All three general types of beta decay of 20 and 21 are represent ed i n E q 1 5: ( 1 5 ) The kinetic energy of beta particles depends on the parent and daughter nuclear states in the decay chain typically ranging from a few keV to a few tens of MeV. The most energetic beta particle s can reach speed s approaching the speed of light. In addition, because they are less charged and much smaller than alpha particles, beta particles generally travel fu rther into tissues, which can cause cells damage at the molecular level. Gamma ( ) decay : Gamma decay refers to as a nucleus jumping down from a higher energy state to a lower one by emitting electromagnetic radiation (photons), which is known for the gamma ray: ( 1 6 )

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21 There is no change in the number of protons and neutrons between the parent and daughter nucle i throughout the process, but the emitted gamma rays are characteristic for each decay. Gamma rays typically have energies in the range from 100k keV to 10 MeV and wavelength less than 10 pm, which is often smaller than an atom. Therefore, they are usually more penetrating, and can cause b iological damage. Neutron capture : Neutrons are electrically neutral, so more easily interact with a nucleus than charged particles. In neutron capture, a nucleus collides with a neutron to merge into a heavier one with a higher energy state. 2 2 The excited nucleus quickly decays to the ground state through emission of gamma rays: ( 1 7 ) ( 1 8 ) In this type of decay, the mass number rises by one, which is crucial for the formation of heavier elements in the cosmos. Nuclei with mass numbers greater than 56 can be created by neutron capture but are unlikely to be formed by thermonuclear reactions (i.e nuclear fusion). Nuclear fission : Nuclear fission is a form of nuclear transmutation in which an atom is induced to split into two lighter elements with emission of free neutrons : ( 1 9 ) In the event, two more new neutrons are produced, which make the reaction be a self sustaining chain reaction in the nuclear reactor. The energy released by the fission fragments and neutrons can be roughly approximated from the mass differences 21 : ( 1 10 ) This simple calculation is very close to real results (180.5 MeV), including the kinetic

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22 energy of the fission fragments (168.2 MeV) and the neutrons (4.8 MeV), as well as the energy carried off by gamma rays (7.5 MeV). Taking into account other contributions by following decay reactions, the sum of above process gives an average total energy of 195 MeV. 2 3 Compared with this high energy, a typical chemical reaction usually only involves energy change s less than 10 eV. 1.4 Motivation and Objective T his thesis focuses on nuclear materials, especially on understanding the fundamental mechanism of intrinsic point defects evolution, which plays a crucial role in the material performance. In nuclear reactors, the uranium dioxide (UO 2 ) fuel undergoes struc tural damage (formation of defects) after irradiation. Radiation damage (RD) due to high concentration of point defects can limit the lifetime of the fuel. In order to enhance the radiation tolerance of UO 2 it is of great interest to accurately predict th e long term behavior under different conditions. Real nuclear fuels are made of several UO 2 crystals separated by interfaces or grain boundaries (GBs). To determine the GB effect, the results obtained in a single crystal are generally used to compare with those in polycrystalline materials. Therefo re, building on the previous work by Aidhy et al. in single crystal UO 2 2 4 this thesis investigates more realistic polycrystalline UO 2 It has been justified that defect evolution is largely independent of the initial damage state on MgO simulations. 2 5 Therefore, conventional molecular dynamics (MD) simulations using primary knock on atom (PKA) approach are not capable of capturing the long term evolution of point defects, mainly due to lack of complete developm ent of complex defect structures. Moreover, in order to elucidate the interplay of U and O point defects, evolution from initially different defect conditions has to be studied. This is also not possible for standard MD using PKA

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23 approach. As a result, a n ew methodology is developed to perform such atom ic simulation to characterize detailed physical processes associated with GBs and to indicate to what extent GBs interact with the defects. 1.5 Beauty of Simulation Although UO 2 has been extensively studied in the past, there is little understanding of its microscopic behavior during irradiation. This is because experiments are expensive and have little access to the defect length scale and irradiation events time scale In a nuclear reactor, radiati on events occur on a very short time scale (~nanoseconds), so observing them by experiment is not possible. Hence atomic simulation can be a complementing alternative to obtain valuable information and widen the scope of knowledge in this area. U nderstand ing the basic diffusion mechanism at the atomic scale is a key ingredient in the development of realistic macroscopic models. Moreover, GBs in polycrystalline materials are known to largely influence these events, which take place simultaneously with vacan cy interstitial recombination in the bulk. Only simulations are a ble to directly distinguish the process and to determine the dominant mechanism. On the other hand, it is also not experimentally possible to capture a material s responses to the presence of different types of point defects, which might reveal its ability to maintain the desired properties.

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24 CHAPTER 2 DEFECTS IN SOLIDS 2.1 Introduction The scientific study of crystalline solids is better developed than that of amorphous materials due to their regularity and symmetry. On the atomic scale, crystalline solids are composed of a set of atoms ar ranged in a particular way with repeating pattern throughout all three spatial dimensions. There are a large number of crystal structures, from the simple on es of elemental metals to complex ceramics. It is impossible to maintain the infinite long range order in all three dimensions. In general, the regularity is often dis rupted by defects. These imperfections may have a profound effect on the properties of ma terials. Defects can be classified into three categories by their dimensions: three dimensional bulk defects (e.g. voids or precipitates), two dimensional planar defects (e.g. grain boundaries or interfaces), one dimensional line defects (e.g. dislocations ), and zero dimensional point defects (e.g. vacancies or interstitials). In this work, the behavior of point defects is of our focus. 2.2 Intrinsic Point Defects From the Nernst theorem it is known that a crystal may have a perfect structure only at the absolu te zero temperature. 2 6 Therefore, a small number of intrinsic point defects are always present at any temperature above absolute zero. Imperfections involving atoms only native to the crystal without any external intervention (e.g. Na or Cl in NaCl crystals) are called intrinsic defects. They do not require the in troduction of impurities or aliovalent ions and are intrinsically related to the structure of the compound. The vacancy is simply an atom missing from a l attice site, which would be occupied in a perfect lattice as in Fig 2 1 (a). The self interstitial is an atom crowded in a void space

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25 between lattice atoms as in Fig 2 1 (b). These point defects in the lattice create varying separations between the neighboring atoms, which hence generate lattice strains. Figure 2 1. Poi nt defects: (a) vacancy, and (b) interstitial. 2 7 2.2.1 Schottky Defect A Schottky defect is unique to ionic materials and is commonly found in ceramics. It forms when stoichiometric number s of ions are missing from the crystal lattice, leaving behind vacancies on the cation and anion sites in order to maintain overall electrical neutrality in the material, as shown in Fig 2 2 ( a ). The defect is named after the German Scientist Walter H. Schottky. The following is the chemical reaction in K rger V ink notation for the formation of the Schottky defect in UO 2 : (2 2) 2.2.2 Frenkel Defect In crystals containing more than one ion, vacancies and interstitials may occupy either anion or cation sites. A Frenkel defect (or Frenkel pair (FP)) is a vacancy interstitial

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26 pair created by thermal vibration when an ion hops from a normal lattice site t o a nearby interstitial site, leaving behind a vacancy. I t was named after the Russian scientist Yakov Frenkel, who discovered the phenomenon in 1926. 28 Fig 2 2 (b) is a schematic representation of a Frenkel defect for an ionic material. Frenkel defects c an also be found in metals and covalent compounds. Taking oxides as an example, the Frenkel defect on the oxygen anion sub lattice site can be described using K rger V ink notation as: (2 1) Figure 2 2. Point de fects in (a) Schottky defect, and (b) Frenkel defect. 2 7 There are also other types of point defects, both intrinsic and extrinsic. However only Frenkel defects are under study in this thesis. 2.2.3 Thermodynamics of Point Defects: Equilibrium Concentra tion The formation of intrinsic point defects is governed by the principle of minimizing the tota l Gibbs free energy. Any destruction of the perfect lattice may increase the lattice energy. However, when point defects are present, the overall randomness or entropy of

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27 the material increases, and hence can decrease the total Gibbs free energy, which can increase the thermodynamic stability of a crystalline material Vacancies play an important role in determining the rate at which atoms or ions can move around, or diffuse in a solid material, especially in pure metals. At room temperature, the concentration of point defects is small, but the equilibrium numbers of point defects n v and n i increases exponentially as the temperature is increased, as shown by the following Arrhenius eqations: 2 9 (2 3 ) (2 4) where n v and n i are the number of vacancies and interstitials per cm 3 ; N is the number of atoms per cm 3 ; Q v and Q i are the energy required to produce a vacancy and interstitial, in eV/mol; k is the Boltzmann constant, 8.617 10 5 eV/K; T is the temperature in Kelvin. 2.3 Grain Boundary Grain boundaries (GBs) are the lattice defects which have been longest known but least understood. Most engineering materials are polycrystalline in nature in that they are made of many single crystals which are misoriented with respect to each other. In a polycrystalline solid, GBs are the internal interfaces that separate neighboring regio ns of the same crystal structure but of different orientation. These interfaces, which are usually planar, have a two dimensional periodic atomic structure. For example, considering grain s in 100 diameter in a polycrystalline cube with 1 cm on edge, there would be more than 10 1 5 grains, with a GB area of several square meters. Thus, the details of the atomic structures of GBs play an important role in controlling the properties of the polycrystall ine

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28 solid. According to the misorientation between adjacent grains, low angle grain boundaries (LAGBs) are those with a slight orientation mismatch, which can be described in terms of dislocation arrays. 29 For high angle grain boundaries (HAGBs), the prope rties are normally independent of the misorientation angle. The transition angle varies from 10 20 degrees, but is not known exactly. 30 Tilt boundaries are formed when angle of misorientation lies in the plane of boundaries, which can be described by an ar ray of edge dislocations. When the angle of misorientation is parallel to the plane of boundaries, twist boundaries result, which can be described by means of screw dislocation arrays. Symmetric boundaries bisect the angle of rotation A twin boundary is a special type of GB in a mirror symmetric manner. In general, GBs have very different atomic configurations and local atomic densities from those of the perfect crystal. The deficiency of fundamental knowledge of GBs is mainly due to their complex structur e, which requires extensive description for their macroscopic characterization. 2.4 Atom Movements in Materials 2.4.1 Diffusion Mechanisms Understanding of diffusion is based on two important fundamentals 31 : (1) the continuum theory of diffusion originated from work of the German scientist Adolf Fick, and (2) the Brownian motion detected by the Scottish botanist Robert Brown. Vacancy diffusion: As mentioned in 2.2.3, in thermodynamic equilibrium at elevated temperature, a specific number of vacancies are present in a crystal. Vacancy diffusion involves the movement of an atom from a regular lattice position into an adjacent vacant lattice site; t hus the atom and the vacancy move in opposite directions (Fig 2 3 (a)). The number of vacancies, which increases as the temperature increases, determines the extent of this mode of diffusion.

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29 Figure 2 3. Schematic representation of three di ffusion mech anisms: (a) vacancy diffusion mechanism, (b) interstitial diffusion mechanism, (c) interstitialcy mechanism, and (d) enthalpy barrier required to make a jump for the interstitial atom shown in (b). 32 Interstitial diffusion: This type of diffusion involves an interstitial atom directly migrating to another nearby interstitial site without causing the net motion of any other atom (Fig. 2 3 (b)). In some modes, to make the jump, an enthalpy barrier has to be overcome for the ato m to squeeze through a narrow passage. This enthalpy barrier is known for the migration energy and is shown schematically in Fig 2 3 (d). The migration energy of the vacancy is larger than that of the interstitial, so the pre exponential diffusivity of th e vacancy is much smaller. There are two main reasons why interstitial diffusion takes place much faster than diffusion by vacancy mode. 29 First,

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30 interstitial atoms are usually smaller and thus more mobile. Second, there are more interstitial positions tha n vacant sites, so the probability of interstitial atomic movement is greater than vacancy diffusion. Interstitialcy diffusion: A less common mechanism is the interstitialcy mechanism, where an interstitial atom pushes an atom into an interstitial site and move into the lattice site itself (Fig 2 3 (c)). This mechanism of diffusion will be observed in this research 2.4.2 Atomic Theory of Diffusion Phenomenological equations: Diff usion is the phenomenon of material transport by atomic motion. 2 9 In solids, diffusion is a very slow process. The rate at which atoms diffus e in a material can be measured by the flux ( J ), defined as the number of atoms passing through a cross sectional unit area per unit time. If the diffusion is steady state in one dimension ( x direction), which means flux does not change with time, the net flux of atoms can be described by Fick s first law: (2 5) where D is the self diffusion coefficient, which is a measure of the ease and frequency with which that atom jumps around in a crystal lattice in the absence of external forces, i.e., in a totally random fashion. Experimentally, the temperature dependence of the diffusivity is: (2 6) where is the enthalpy barrier for diffusion independent of temperature; D 0 is a temperature independent pre exponential parameter. Note that the diffusion rate largely decreases with decreasing temperature.

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31 Atomistics of solid state diffusion: In general, diffusion in solids occurs in the presence of point defects. 33 Ion irradia tion results in the formation of vacancy interstitial pairs. 3 4 If the defects are produced at temperatures where they are mobile, they can partly anneal out by diffusion. Therefore, t he balance between the formation rate and the annihilation rate leads to a steady state of the excess concentration of defects. Atoms usually move by interchanging positions between atoms and neighboring vacant sites. For such a diffusion process, the relationship relating the diffusion coefficient ( D ) of an atom to the diffu si on jump distance ( ) in a solid is: 32 (2 7) where geometric constant depending on the crystal structure and is the frequency of successful jumps. The jump frequency is the product of the probability of an atom having sufficient energy to make a jump ( ) and the probability of available adjacent vacant site s ( ) : (2 8) where can be further e xpressed by: (2 9) where is the enthalpy barrier required to make a jump and 0 i s vibration al frequency of the atom. Defect diffusivity : As noted previously, the two major defects responsible for the movement of atoms are vacancies and interstitials. For both cases at low concentrations, the sites adjacent to the defect will be always available, so 1. 33 An interstitia l will make a jump with the rate only depending on its frequency of

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32 successful jumps. By combining Eq. (2 7), (2 8), and (2 9), the interstitial diffusivity D i is given by: (2 10) On the other hand, for a vacancy, the probability of successful jumps is increased fold, where is the coordination number of the vacancy. Thus, the vacancy diffusivity D v is expressed by: (2 11) Atom diffusivity : In contrast to the vacancies and interstitia ls, most atoms have few available sites nearby into which they can diffuse, s o << 1. 32 The probability of a site being vacant is simply equal to the vacancy fraction in the solid, denoted by Therefore, the atomic diffusivity D A is given by the following expression, which resembles Eq. (2 6), namely: (2 12) Comparing Eq. (2 11) and (2 12) reveals an important relationship between vacancy and atom diffusivity: (2 13) We assume that << 1, where f v is a correlation factor, and c A c v are the concentrations of atoms and vacancies, respectively. A more useful expression is thus given by: 3 4 o r (2 14) C orrelation factors depend on the crystal structure and diffusion mechanism. The

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33 theoretical values for fluorite structures have been reported for the three mechanisms: 0.653 for the free vacancy, 1 for the interstitial, and 0.739 for the indirect non collinear interstitialcy. 3 5 The physical picture of Eq. (2 14) is easy to grasp: the defects (vacancies or interstitials) move often (high D ) but are not that numerous, while the atoms move less frequently ( low D ) but are much more in number. With c A 1, rearranging Eq. (2 14) yields: (2 15) (2 16) D iffusi vity in simulations: In atomic simulations the self diffusivity is determined based on the time evolution of the mean square displacement (MSD) under zero stress conditions. The quantity can be easily calculated by recording the atomic positions during the simulation: (2 17) where N is the total number of ions; t is the simulation time; and are atomic positions at time = t and 0 respectively; the angular brackets here denote an average over the square of the displacement that the atom has undergone during a time interval t Since the thermal energy is involved in the MSD, atomic positions can be simply relate d with the initial t hermal velocity : (2 18) Inserting Eq. (2 18) into (2 17) gives:

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34 (2 19) From kinetic molecular theory, the average kinetic energy is proportional to the absolute temperature in the following relation: (2 20) where v rms is the root mean square velocity. Inserting Eq. (2 20) into (2 19) gives: (2 21) For liquids, the MSD grows linearly with time, while for solids, the MSD saturates to a finit e value. I t is u seful to characterize the diffusive motion in terms of the slope, which is the self diffusion coefficient of the atoms ( D A ), namely: (2 22) 2.5 Crystallography of UO 2 In this thesis, the central theme is on the evolution of point defects in the fluorite based UO 2 As shown in Fig 2 4 the fluori t e stru cture can be viewed as a face centered cubic (FCC) array of cations, represented by blue spheres, with the anions residing in all the tetrahedral sites, represented by red spheres. A fluorite crystal unit cell consists of four U atoms and eight O atoms. The coordination number of the cation is eight and that of the anion i s four. The cations occupy the Wyckoff site 4 a and the anions occupy 8 b The space group of fluorite crystal is (number 225 and point group symmetry 36 ). can be fully written

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35 Figure 2 4 Fluorite cr ystal structure. The FCC network of cations (blue spheres) with anions (red spheres) occupying the tetrahedral sites. as The first letter refers to the lattice type. Here, F stands for face center cubic (FCC). T he following three sets of symbols denote the three kinds of symmetry directions of the lattice relative to the space group, naming primary, secondary, and tertiary directions, respectively. For the FCC unit cell, the primary symmetry directions are <100>, i.e., [100], [010], and [001]; the secondary symmetry directions are <111>; the tertiary symmetry directions are <110>. In the symmetry, represent s four fold rotational symmetry along the principle axis with a mirror plane perpendicular to the axis. refers to the rotoinversion along the diagonal (i.e. secondary) direction of the unit cell. stands for two fold rotation along the edge of the unit cell (i.e. tertiary direction) with a mirror plane perpendicular to the edge. Fig 2 5 schematically shows all three symmetry operations mentioned above.

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36 Figure 2 5 Gr aphic representation of three different symmetry operations for space group : (a) four fold rotation along <100>, (b) three fold rotation along <111>, and (c) two fold rotation along <110> 2.6 Radiation Damage Nuclear energy is part of solution to the increasing global energy demand and concerns about greenhouse gas emissions. A fundamental understanding of defect formation, accumulation, and annihilation in irradiated ceramics is required to develop novel radiation tolerant ceramics for improving the performance of the fuel, and for developing long lasting storage materials. When materials are bombarded with radiation, electrons may come out of their original orbital s, and atoms may displace from their original sites. In this process, defects are also introduced, either by bombarding ions residing in the material or following nuclear reactions. Therefore, the properties of the material are altered, sometimes heavily, sometimes just slightly. These spontaneous processes can lead to the gradual and ultimately complete amorphization of the ceramic crystalline structure, which is undesirable for nuclear application because it is often accompanied by degradation of thermo mechanical properties and reduced chemical durability However, sometimes

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37 defects can be annealed out, so the material will remain crystalline. The radiation response of ceramics relies on a dynamic balance between damage by defect accumulation and recovery from thermal annealing. The radiation damage phenomena take place in a very short time, and include two important phases (Fig. 2 6) 3 7 arising from the high their original lattice sites, there by producing a large number of point defect s and s mall lly lasts only a few picoseconds. By contrast the lasts longer and captures the mainly time dependent diffusion controlled kinetic evolution of these defects, including the annihilation and formation of larger long lasting defect clusters Some defects may recombine or be eliminated at microstructural sinks (GBs, dis locations, or external surface); others cluster to form voids or interstitial loops. Annihilation of point defects promotes radiation tolerance of the Figure 2 6 Defect evolution during ballistic and kinetic phases of the radiation damage. 3 7

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38 material while clusters l ead to long process determines the performance and lifetime of the material. Understanding the tolerant ceramic s and in UO 2 it is also the central problem of this thesis.

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39 CHAPTER 3 SIMULATION METHODOLOGY Computer simulations are carried out to solve material science problems related to the design a material with specified properties and functionalities. They have high potential to impact real industrial research and development. As a matter of fact, simulations serve as a com plement to conventional experiments, enabling us to inte grate computational and theoretical methods with experimental analysis and characterization methods 3.1 Simulation Methodologies To characterize various materials behaviors over wide length and time scal es, a wide set of simulation methodologies are used, as show in Fig. 3 1. Starting at the highest level, engineering design is basically the primary analytical tool for relatively large scale manufacturing systems in industry, varying from integrated circu its to distillation columns and engines. At the level of continuum models, gradual transitio ns without abrupt changes or discontinuities are described such as macroscopic stresses, large temperature gradients. Below this level, the meso scale modeling inc ludes kinetic Monte Carlo (KMC) models 3 8 and phase field method s 3 9 as approaches to modeling and predicting morphological and microstructure evolution. A further level down, atomic phenomena are accurately captured by molecular dynamics (MD) simulation with i nteratomic potential s 40 and force fields 41 Details of this method will be discussed in the following sec tion. At th e lowest level computational quantum chemical methods are characterized by quantum me chanic s at the level of the electronic structure. E lectr onic states are described in terms of the Schr di nger equation: 42 (3 1)

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40 where E is the total system energy (eigenvalue); H is the Hamiltonian operator; is the wave function (eigenfunction). There are two main approximation methods to solve the equation : semi empirical methods and ab initio methods. Semi empirical methods use experimentally derived parameters to strive for accuracy, but ab init io methods use calculations to evaluate all integrals. With correlation, ab initio methods can even have accuracy comparable with experiments in structure and energy predictions. However, calculation s become extremely demanding in computer resources. This work focuses on phenomena at the atomistic length scale and picosecond time scale; therefore, MD simulation is chosen here. Figure 3 1. Hierarchical multi scale approach to computational material engineering.

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41 3.2 Molecular Dynamics Simulation The MD simulation is a technique that computes time dependent behaviors, such as structures, dynamics, and thermodynamics. I t enables us to probe features of the motion of atoms, and to take still snapshots of crystal structures. Now it is frequently use d in the study of materials science because no conventional experiment allows access to all the time scale of motion with the atomic resolution. The accuracy of the calculated properties relies entirely on the fidelity of the i nteratomic potential, which i s typically defined by quantum based methods or experimental data. 3.3 General MD Algorithm In MD simulations, all atoms are given initial positions theoretically, which is defined by the crystal structure of the material and the microstructure of interest Th e atoms are assigned random velocities ( v ) according to the simulation temperature ( T ): (3 2) where m is the mass of the atom; k B is the Boltzman s constant. The motion of the atom can be described by solving Newton s second law: (3 3) where F is force exerted in the atom; m is its mass; is its acceleration. The force can also be calculated from the gradient of the i nteratomic potential energy V : (3 4) Numerous numerical algorithms have been developed for integrating the equations of motion, such as Verlet 43 leap frog 4 4 ,4 5 Beeman s 4 6 and predictor corrector 4 7 They differ in accuracy and computational load. In this thesis, a fifth order Gear predictor

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42 corrector method is used. This method consists of three steps: prediction, evaluation, and correction. In the first prediction step, a high order Taylor expansi on is performed to estimate the atom positions around time t to predict the positions at : (3 5) The superscript refers to predicted values; , and denotes positions, velocities, accelerations and third, fourth, and fifth time derivative of position, respectively. Newton s second law is introduced. Therefore, the correct accelerations can be obtained: (3 6) where F is the force exerted on the atom; m is its mass; a is its acceleration. The acceleration can be evaluated by comparing with the predicted accelerations from Eq.

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43 (3 5), to estimate the error in the prediction step: (3 7) This error and the results of the prediction step are fed into the correction step: (3 8) where the Gear corrector coefficients are c 0 = 3/20, c 1 = 251/360, c 2 = 1, c 3 = 11/18, c 4 = 1/6, and c 5 = 1/60, respectively. 4 8 3.4 Periodic Boundary Condition In MD simulations, the simulation box size must be large enough to avoid boundary condition artifacts. There are two main kinds of boundary conditions 4 9 : isolated boundary conditions (IBCs) and periodic boundary conditions (PBCs). In IBCs, the system is surrounded by vacuum, so atoms only interact among themselves. Therefore, IBCs are ideally sui ted for the study of clusters and molecules. On the other hand, in PBCs, the system is surrounded by infinitely image of the supercell itself in all three dimensions of space, as shown in Fig. 3 2. Hence, an atom may interact not only with atoms in the sam e supercell but also with atoms in neighboring supercells. In other words, one side of the simulation loops back to the other side, mimicking a bulk phase. In this way, PBCs are suitable for investigating bulk liquids and solids.

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44 Figure 3 2. Schematic representation of periodic boundary conditions of a fluorite structure supercell. The center box is the actual simulation box. 3.5 Interatomic Interactions For different types of materials, different kinds of interatomic attractions are needed to correctly describe the syste m. For any given functional form, different parameters can be chosen so as to describe different materials. Therefore, it is essential to define the interatomic interactions before any practical cal culations can be per formed. In ionic systems, the potential energy consists of both long range and short range interactions. Long range interactions are attractive interactions caused by Columbic forces. Short range interactions are mainly repulsive due to overlapping electro n orbitals, but there are also some small van der Waals attractive forces. Both types of interactions will be discussed in detail in the following section.

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45 3.5.1 Long R ange Interactions For inorganic materials, especially oxides, long range forces come from electrostatic interactions, which determine to a large extent the thermodynamic and transport properties. Therefore, precisely evaluating the Coulombic force is essential. In ionic systems, cations and anions are arrang ed in a defined order. The energy associated with this specific periodicity is the electrostatic attraction ene rgy between two charged ( q i and q j ) ions. The magnitude of the energy ( E long range ) can be calculated by Coulomb s law: (3 9 ) where r ij is the distance between ions i and j A negative potential represent s an attractive interaction while a positive potential represent s a repulsive interaction. Despite the simple formula given above, this is actually the most difficult interaction to evaluate for periodic system. Since long range Coulomb s interactions are responsible for the stability of crystalline structures, a critical approximation has to be made to develop an infinite system. It is computationally convenient to have a cutoff truncation beyond a radius or the periodicity of the lattice. Among several methods 50 6 0 the Ewald summation 50 and the direct summation 50 are the two most common methods. Though the Ewald summation is more ac curate in calculation, it is more computationally intensive than the direction summation. According to the Ewald summation, the high computational load has scaling with system size of N 2 or better N 3/2 or even at best N log ( N ). 5 6 5 9 In the present study, we are observing the equilibration evolution of point defects, in which the large system requires a larger MD computational cost. As a result, the computationally more efficient

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46 method, direct summation, is applied here. This method has been validated by simulations not only of crystalline but also arbitrarily disordered, charged or neutral ionic systems. 6 0 Direction summation method : When performing a spherically truncated pairwise r 1 sum in a crystal, the sys tem summed over is never electrically neutral, which essentially leads to non convergence of the summation This long range electrostatic concern can be res olved by an assumption that no basis molecule may be broken up over the entire Bravais lattice. 60 Fo r example, rather than regarding the rock salt (or NaCl) lattice as an FCC lattice with a dipolar molecule, one can choose the simple cubic unit with the octopolar (NaCl) 4 basis. 61 This tetra unit basis is kept together while applying the truncation with the cutoff radius. The direct summation may therefore achieve charge neutrality by introducing additional charge to the truncation sphere. This is not just a mathematical concept; a reconstruction of the polar NiO surface has been observed 62 and this experimentally upholds the validity of the energy convergence approach. 3.5.2 Short R ange Interactions F r om Eq. (3 9), the long range attractive potential between two oppositely charged atoms should bring them closer. In order not to cause structure collapse of the lattice some repulsive potential must be present to keep the two atoms apart. The short range interactions can be expressed by the Buckingham potential: (3 10 ) where A ij ij and C ij are empirical parameters; r ij is the distance between ions i and j The first term describes the short range repulsive interactions derived from overlapping electron orbitals, which decrease exponentially with increasing distance between ions i

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47 and j ; the second term represents the van der Waals attractive contribution du e to dipole interactions. Throughout this study, the short range interactions are characterized by using the Bushing Ida 63 type In addition to the Buckingham potential, a Morse term is included to introduce the contribution: (3 11 ) where D ij a nd ij are empiri cal parameters; is the anion cation i j bond length. In a mixed ionic covalent system, the ionic charges are given non formal values w ith partial covalence to consider charge transfer between the ions. Hence, taking into account all interactions acting on an atom associated with surrounding atoms, the energy can be given by: (3 12 ) In this thesis, this classical pair wise potential energy form is used P arameters are taken from Basak et al 6 4 as listed in Table 3 1. Fig. 3 3 illustrates the inter ionic potential energy as a function of distance between U +2.4 and O 1.2 ions. 3.5.3 Thermodynamic Conditions (Ensembles) for Simulation s An ensemble is a collection of all possible environmental conditions which have different microscopic states but have an identical macroscopic or thermodynamic state. In MD simulations, certain conditions are assigned in order to obtain desired properties

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48 Table 3 1. P otential p arameters used in the present work (Eq. (3 12)) 6 4 Parameter Basak et al q U ( e ) 2.40 q O ( e ) 1.20 A U O (eV) 693.9297 U O () 0.3270(22) C U O (eV 6 ) 0.0 D U O (eV) 0.577(45) U O ( 1 ) 1.65 ( ) 2.369 A O O (eV) 1633.666(6) O O () 0.327(022) C O O (eV 6 ) 3.950(63) A U U (eV) 294.759(3) U U () 0.327(022) C U U (eV 6 ) 0.0 Figure 3 3. Inter ionic potential as a function of distance between U + 2. 4 and O 1. 2 ions. The total potential (E t otal ) is the sum of the long range Coulombic (E long range ) abd short range (E short range ) potentials. 30 20 10 0 10 20 30 0 0.5 1 1.5 2 2.5 3 3.5 4 E(r ij ) (eV) r ij ( ) E short range E long range E total

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49 of the material, such as total number of atoms, volume, temperature, pressure, etc. A thermodynamic state with fixed number of atoms ( N ), a constant volume ( V ), and a fixed energy ( E ) corresponds to an isolated system, that is, the isochoric micro c anonical ensemble ( NVE ). Similarly, NVT stands for the canonical ensemble and NPT refers to the isobaric isothermal ensemble. These three are the most widely used ensembles, and NPT is performed throughout this study. In adition, there are also other typ es of ensembles, such as the grand canonical ensemble ( VT ) and the isobaric isoenthalpic ensemble ( NPH ). H and represent enthalpy and chemical potential, respectively. 48 3.5.4 Thermo stat In order to control the system simulating at the desired temperatur e, several approaches have been employed The simplest one is probably the velocity rescaling thermostat 4 8 Recalling Eq. (3 1), the target velocity ( V t ) can be rescaled by the following relation: (3 13 ) where T i is the initial temperature; V i is the initial velocity; T t is the target temperature. Other popular methods to control the system temperature are Nos Hoover 6 5 ,6 6 Berendsen 6 7 and generalized Langevin 6 8 3.5.5 Bar o stat In addition to the temperature, the volume is another thermodynamic condition needed to be controlled. Various algorithms can be applied to achieve a constant pressure. For example, the idea of the Andersen barostat is to assume that the system is simulate d in a container which is able to be compressed by a piston with certain mass

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50 m 6 9 The Lagrangian takes the form: (3 14 ) where is the scaled coordinates ( ; r is the Cartesian coordinates and V is the volume ); p 0 is the external pressure. The first two terms on the right are just the Lagrangian of the unscaled system. The third term represents a kinetic energy for the piston of mass m with volume of V a nd the fourth term stands for a potential energy derived from p 0 acing on the piston. However, the Andersen method assumes that the external pressure is hydrostatic, which is not sufficient in all cases. Therefore, Parrinello and Raham further extended this method to anisotropic coupling by allowi ng the simulation box to change its shape. 70 7 1 In this thesis, the P R method is used. 3.6 Conventional Radiation Damage Method by MD Simulations Atomic level simulations have long been employed to investigate the fundamental phenomena in radiation damage (RD). 72 7 6 MD simulations are capable of capturing two major phases: the ballistic phase and the kinetic phase. Standard MD simulation of radiation damage has been combined with temperature accelerated dynamics (TAD) simulations to ach ieve the longer and experimental time scale. It has been shown that the defect evolution during the kinetic phase is largely independent of the initial damage created during the ballistic phase. 7 7 ,7 8 Therefore, it can be inferred that it is not necessary t o perform the simulation from the beginning because the main job of the ballistic phase is to introduce non equilibrium point defects into the system. In other words, starting the simulation directly from the kinetic phase, which is most interesting to us, seems to be more efficient. In addition, since much attention will

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51 be drawn on the kinetic evolution of these point defects, the recombination and clustering mechanism for each separate kind of point defect will provide more basic information in the study of RD, which is not accessible for conventional RD MD simulations. As a result, another kinetically evolving irradiation induced point defects method will be introduced in this thesis.

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52 CHAPTER 4 GRAIN BOUNDARY INFUENCE ON RADIATION DAMAGE IN UO 2 4.1 Kinetically Evolving I rradiation Induced Defects Method by MD Simulation This k inetic ally e vol ving irradiation induced d efects appro ach does not replace cascade simulations targeting on elucidating the effect of experimental conditions, such as the dose or the energy of incident species. Instead, this method, which bypasses the b allistic phase, allows a much wider range of different defect conditions to be explored. In this approach, a specific number of Frenkel pairs (FPs) are randomly intro duced to the system which is equilibrated at high temperature (1000 K) for a sufficiently long time. Creating FPs by hand makes it possible to generate defects on a selective sub lattice or both sub lattices To avoid un interesting but spontaneous recombin ation events, all vacancies are separated from their counterpart interstitials by a distance greater than the recombination radius. Then the system is equilibrated at 1000 K. T he defect structure, unlike rapidly recombining point defects, can have a more s ignificant role in determining the true radiation performances of the material. This study will discuss the following kinetic evolution of the FPs. 4.2 Simulation Methodology 4.2.1 MD Simulation U sing th e interatomic interaction descri b ed in 3.5 the melting point o f UO 2 and oxygen sub lattices were determined to be around 3450 50 K 7 9 and 2200 K 7 9 which are in fair agreement with the experimental value of 3100 K 80 and 2600K 81 respectively. In this work, all simulations have been carried out at 1000 K, which is well below the melting temperature of both the material and sub lattices At this temperature, the oxygen interstitials and vacancies have high mobility, whereas the uranium counterpa rts

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53 have very high migration energy and therefore diffuse less In addition this temperatu re is also in the range of 800 1600 K, typical working temperature in the fuel pellet. The single crystal simulation supercell contains 202020 cubic fluorite unit cells with 96,000 atoms. The polycrystalline micro structure is a large rectangular box (51.7 30.2 5.5 nm) consisting of 611,010 atoms in six hexagonal equal sized ( d = 20 nm) grains (Fig. 4 1). The c rystallographic mis orientation between three adjacent grains are 0, 30, and 60 degrees with respect to each other, then the GBs are asymmetric high angle tilt boundaries Perio dic boundary conditions are applied in all three dimensions The time step of 0.5 fs is cho sen as i t give s goo d energy conservation f or several thousand steps in NVE test runs. In order to capture as much of the defect evolution as possible, all simulations have been carried out for more than 1 ns. Two analysis methods to identify point defects will be discussed in the following section Figure 4 1. Snapshot of the fully equilibrated polycrystalline UO 2 microstructure with d = 20 nm at 1000 K prior to introduction of defects. The viewing direction is along [100] columnar axis. T hree crystalline orientations are 0, 30, and 60 degrees relative to the columnar axis.

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54 4.2.2 Defects Analysis Methods T o observe the kinetic evolution point defects, it is very important to capture the precise number, type, and position of all atoms especially in the presence of grain boundari e s. In fact, the grain boundary is a kind of a defect, so it is relatively valueless to count point defects in the grain boundary. Here are two methods to perform the analysis. One is the common neighbor analysis; the other is the lattice matching analysis. The two methods are used in complementar y manner here Common neighbor analysis (CNA) : As implied in the name, the CNA 82 utilizes the coordination number (CN) of neighboring atoms to iden tify defects. When it comes to the CN, the cutoff radius plays an important role to determine how far the nearby atom is treated as the neighbor. In UO 2 CNs of U atoms are eig ht and of O a toms are four. As Figure 4 2. Schematic representations of the common neighbor analysis (CNA). (a) E very atom is normally coordinated in the perfect structure (b) There are different coordination numbers (CNs) for the neighboring atoms when defects are present (dark color for higher CNs, light color for lower CNs).

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55 shown in Fig. 4 2 (a), every atom is normal ly coordinated in the perfect structure. When defects are present (Fig. 4 2 (b)), CNs near interstitials will increase (in darker color) while ne ar vacancies will decrease (in lighter color). In terms of the miscoordination of the defect atom itself and th e neighboring atoms, the defect position is identified. However, one big problem is that if there is more than one type of defects close to each other, such as clusters containing both vacancies and interstitials, the CNs will be very complicated to analyz e. Lattice matching analysis (LMA) : In view of the drawbacks mentioned above, here a second method is also use d to identify defects. Considering atoms diffusion at high temperature, the average travelling distance can be calculated. Therefore, a cutoff rad ius ( r cutoff ) of 0.19a o (about 1.039 ) is chosen wi th regard to all lattice sites of the reference structure (Fig. 4 3 (a)), which has been equilibrated for a long time As a result if there is Figure 4 3. Schematic representations of the lattice matching analysis (LMA). (a) A cutoff radius ( r cutoff ) is assigned to all lattice sites of the reference structure. (b) If there is no atom within the assigned radius, it is a vacancy. If an atom is not within any assigned radius, it is an interstitial.

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56 no atom within the assigned cutoff radius of the original lattice site, the site is regarded as the vacancy. But if there is an atom sitting out of the cutoff circle, the atom will be defined as the interstitial (Fig. 4 3 (b)). Another rare case is that if there is more than one atom within the cutoff circle, all atoms will be reported, such as dumbbell interstitials. With the LMA, we can therefore determine the exact number, type, and position of every point defect. 4.3 Results This work starts with the simple r case, the single crystal, to confirm that simulation methods and tools are correctly implemented Then the more complicated case, the polycrystal, is investigated. 4.3.1 Frenkel Pair Defects i n S ingle C rystal UO 2 The same simulations are run to reproduce previous work by Aidhy et al. in single crystal UO 2 24 ,3 7 FPs are created by randomly picking up an ion from its cubic sub lattice site and placing it at an empty octahedral interstitial site. A defect concentrati on about 0.21 % was chosen so 200 was picked up out of 96,000 atoms as defects Two conditions are performed: (1) 200 FPs only on the U sub lattice and (2) 200 FPs on both U and O sub lattices. D ue to the high migration energies of both the U interstitial and vacancy, U FPs have lower diffusivity. On the other hand, both the O interstitial and vacancy have relatively low migration energies, which make them more mobile. Therefore, evolution of the number of U FPs will be less substantia l The results are shown in Fig 4 4. FPs only on the U sub lattice: Due to the high migration energy of U interstitials and vacancies, recombination events are less frequent. Out of the initial 200 FPs, 190 U FPs still remain after 1 ns. However, the high concentration of defects on the U sub lattice

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57 nucleates O FPs. The number of nucleated O FPs is shown by circle s in Fig. 4 4. Similar phenomen a we re also observed by Aidhy et al 24,37 Different from that work, som e data within the first 10 ps we re analyzed in this work. Interestingly, rather than a gradual increase, the number of O FPs increases abruptly to more than 50, and then decrease s to 3 7 af ter 10 ps, finally an equilibrium number around 3 5 Figure 4 4. Comparison of results from Aidhy et al. (red) and this work (blue) for total number of O FPs in two initial conditions: (1) 200 FPs only on the U sub lattice (circles), and (2) 200 FPs on both U and O sub lattices (squares). Inset figures are close up views within the first 10 ps for both conditions. FPs on both U and O sub lattices: This is a more realistic case in the study of radiation damage. 200 FPs are simultaneously created on both U and O sub lattices. Based on simulations on the U sub lattice alone, most of U FPs remain at their initial sites because of their low diffusivities. In contrast, a considerable increase in the number of O FPs 0 50 100 150 200 250 300 350 400 450 500 0 200 400 600 800 1000 1200 1400 1600 Number of O FPs Time (ps) U & O (Aidhy et al.) U (Aidhy et al.) U & O (This work) U (This work) 0 10 20 30 40 50 60 0 5 10 15 20 200 230 260 290 320 350 0 5 10 15 20

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58 takes place within the first 200 ps, as shown by the squares in Fig 4 4. Afterwards, the number is followed by a decline to the equilibrium number to around 250. Instead of complete annihilation of O FPs, the number is actually larger than that of the initial concentration (200 FPs). From the above simulation run, it can be indicated that these signatur es are consistent with the previous results from Aidhy et al In addition, more information taking place within the first few picoseconds shows that the high concentration of defects on the U sub lattice nucleates O FPs in a very fast manner. 4.3.2 Frenkel Pair Defects i n Polycrystalline UO 2 For a more representative of the real material, polycrystalline UO 2 is investigated. Before generating the defects, the polycrystalline structure is first equilibrated at the working temperature (1000 K) for 1 ns to ensure no spontaneous formation of defects. Given that the formation energies for O FPs and U FPs are 6.0 and 17.0 eV 83 with the Basak potential 6 4 the defect concentration of 5.8 1 0 31 and 2.1 1 0 86 would be expected in this simulation. Consistent with this, no defects are present in the bulk grain region This equilibrated structure is used as the initial structure for rest of the simulations and as the reference structure in the LMA In order to perform a simulation under similar conditions to the previous single crystal work, 185 FPs are introduced to selected grains of the polycrystal. The defect concentrat ion relative to the bulk (eliminating atoms in GBs) correspond s to 200 FPs in the single crystal co ntaining 96,000 atoms. Three different scenarios are analyzed for each of the single grains: (1) 185 FPs only on the U sub lattice (2) 185 FPs only on the O sub lattice and (3) 185 FPs on both U and O sub lattices. All FPs are introduced to the bulk regi on. All types of vacancies are

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59 separated from their counterpart interstit ials by 1.5a o .Th e structure is then equilibrated at 1000 K with the NPT ensemble f or 1 ns. FPs only on the O sub lattice: A snapshot of the initial structure is shown in Fig 4 5 (a) (left grain Grain 1 ). Due to the low migration energies of both O interstitials and vacancies, O FPs are more mobile, which means they have higher diffusivities. The number of O defects is shown in Fig. 4.6 (green triangles). After the first 25 p s, only 44 out of 185 O FPs are left (Fig 4 6 (b)). Most of the O FPs are annihilated by vacancy interstitial recombination mechanism. After 400 ps, fewer than 10 O FPs are sporadically present (Fig 4 6 (d)). Nevertheless, after 100 ps the number of O va cancies is 3 greater than the number of O interstitials because a V 4O vacancy interstitial clusters f o r m s This kind of cluster will be discussed in detail later. However, no interstitial interstitial or vacancy vacancy cluster is observed, which indicate s that in the absence of defects on U (or cation) sub lattice, the defects on O sub lattice (or anion) are almost healed within a few picoseconds. As a result, there will be no long lasting damage to the material. FPs only on the U sub lattice: The initial snapshot is shown in Fig 4 5 (a) (right grain Grain 3 ). Within first 400 ps, only 27 U F Ps out of 185 U FPs have been recovered. Similar to the case in single crystal, the high migration energy of U interstitials and vacancies makes U FPs unable to recombine. Therefore, the remaining high concentration of defects on the U sub lattice nucleate s new O FPs. The nucleation processes seem to occur rapidly right at the onset of the equilibration. As shown in Fig. 4 6 (red circles), the number of O interstitials and vacancies increases with a very steep slope in the beginning. Different from the case in the single crystal, it is followed by an

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60 equilibrium plateau rather than a slow decrease. What should be noted here is that the equilibrium number of O defects is much larger (about three to four times) than that in single crystal. The equilibrium numb er of O defects is determined by competition between two phenomena taking place simultaneously: (1) random diffusion of O FPs leading to annihilation by recombination, and (2) GBs being the source and sink of O FPs. In the single crystal, there are no GBs, so the number of O FPs solely depends on the recombination rate. Nevertheless, with the presence of GBs in polycrystal, the source/sink behavior will influence the defect concentration. In addition, there are some clusters being observed, such as V 4O clu sters, Schottky defects, and cuboctahedral (COT) clusters. This will also be discussed in detail later. FPs on both U and O sub lattices: Fig 4 5 (a) (center grain Grain 2 ) shows the snapshot of the initial structure with 185 FPs on both U and O sub latt ices simultaneously. A s observing in the previous simulation of defects on the U sub lattice alone there is no distinct change in the number of U FPs, but there is a substantial increase of O F Ps, as shown in Fig 4 6 (blue squares). However, it has to be noted that the number of O interstitials is always smaller than that of O vacancies by about 30 after fi rst 100 ps. A similar phenomenon is also observed in the case of FPs on the U sub lattice alone. This is because GBs can behave as a source o r sink for vacancies but only as a sink for interstitials, which will be discussed in detail in the next section. In both these two conditions, the total number of clusters is larger due to high er initial defect concentration on the U sub lattice. In the si ngle crystal, the number of vacancies and interstitials has to be equal in order to achieve electroneutrality. In the polycrystal, the overall system still requires

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61

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62 Figure 4 5. Evolution of point defect (small circles in green and red for oxygen interstitials and vacancies, large circles in yellow and grey for uranium interstitials and vacancies). Snapshots taken at (a) 0 ps, (b) 25 ps, (c) 50 ps, and (d) 400 ps. All defects are in superposed on the initial structure without introduction of any defects.

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63 Figure 4 6. Number of O defect showing up in the bulk grain in three initial conditions: (1) 185 FPs only on the U sub lattice (circle), (2) 185 FPs only on the O sub lattice (triangle), and (3) 185 FPs on both U and O sub lattices (square). Solid and open notations stand for interstitials and vacancies, respectively. charge neutrality; however, the results show that there are some differences between the number of vacancies and interstitials. It can be assumed that when some of the defects form clusters, the b u lk region may not be electroneutral. Therefore, GBs will play the role of maintain ing the charge neutrality in the bulk by modulating the defect concentration via sourc e/sink behavior s of point defects. Hence, we will take a closer look to see how GBs interact with these defects. 4.3.3 Grain Boundary Source/Sink Behavior s for Point Defects The major difference in the defect evolution between the single crystal and the polycrystal is the presence of GBs. H ere specific interaction mechanisms between GBs and point defects will be characterized. Since uranium vacancies and interstitials have low migration energies and are less mobile, attention will be focused only on the more interesting oxygen vacancies and interstitials. 0 50 100 150 200 250 300 0 100 200 300 400 500 600 Number of O defects Time (ps) Grain 1 Grain 1 Grain 2 Grain 2 Grain 3 Grain 3 185

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64 GB s as vacancy source s : The emission of a vacancy from the GB is presumably a thermally activated process that is associated with an attempt frequency 8 4 GBs may produce many vacancies and reabsorb them before a successful emission occurs in which the vacancy can sufficiently distance itself into the grain. 8 5 An example is demonstrated in Fig. 4 7; the red circle indicates the location of the oxygen vacancy because an oxygen atom is missing from the column which should have specific number of oxygen atoms In Fig. 4 7 (a), a snapshot taken at 20.5 ps, the circ led vacancy is one of many vacancies produced by the GB, and now it is ready to go into the interior grain. As time progresses, the vacancy diffuses further from the GB (Fig. 4 7 (b) (e)). Fig.4 8 gives close up views of each snapshot in Fig. 4 7, illustra ting the mechanism for the vacancy moving into the grain. The vie wing direction is also along [100]; large r spheres are uranium atoms; smaller spheres are oxygen atoms; the purple area (defined by the presence of many miscoordinated atoms) refers to the GB region s hown in Fig.4 7; the big red circle sitting on the oxygen column describes a missing oxygen atom in that column, i.e. the a pproximate vacancy position; small black circles represent positions of specific oxygen atoms involved in this diffusion act ivity (here, the small black circle 1 in the figure will be notated as O 1 in the following text for convenience). From Fig. 4 8 (a) to (b), O 1 goes into the GB to the original site of the vacancy, followed by O 2 filling in the vacant site of O 1. Theref ore, a macroscopic view showing that the vacancy emits from the GB to where O 2 was. This diffusion mechanism is referred to the interstitialcy mechanism, in which the moving direction of the defect and the atom are opposite. Similar diffusion mechanism ta kes place from (c) to (e), so the vacancy finally resides in the original site of O 7.

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65 Figure 4 7. Progressive snapshots showing the GB being the source for the O vacancy (red) taken at (a) 20.5 ps, (b) 21 ps, (c) 21.5 ps, (d) 23.5 ps, and (e) 24.5 ps.

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66 Figure 4 8. Close up views of the GB being the source for the O vacancy taken at (a) 20.5 ps, (b) 21 ps, (c) 21.5 ps, (d) 23.5 ps, and (e) 24.5 ps. The purple area refers to the GB region; the labeled oxygen atoms are involved in this activity ; the large red circle indicates the location of the vacancy.

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67 GB s as vacancy sinks : GBs, in addition to being vacancy sources, can also operate as sinks for both vacancies and interstitials. Fig. 4 9 demonstrates an example of the GB being the vacancy sink The snapshot taken at 9 ps shows that the vacancy is still in the grain but near the GB (Fig. 4 9 (a)). After 0.5 ps, the vacancy is closer to the GB (Fig. 4 9(b)) and finally absorbed by the GB at 10 ps (Fig. 4 9(c)). This vacancy does not re emerge aft er 12.5 ps. The interstitialcy diffusion mechanism is illustrated in close up views of Fig. 4 10 with respect to Fig. 4 9. The original vacancy site is filled by O 1, followed by O 2 taking the place of the vacant site of O 1, so the vacancy is macroscopic ally absorbed by the GB. However, it turns out that O 2 is initially not quite in Figure 4 9. Progressive snapshots showing the GB being the sink for the O vacancy (red) taken at (a) 9 ps, (b) 9.5 ps, (c), and 10 ps.

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68 Figure 4 10. Close up views of the GB being the sink for the O vacancy taken at (a) 9 ps, (b) 9.5 ps, (c) 10 ps, (d) 10.5 ps, (e) 11 ps, (f) 11.5 ps, (g) 12 ps, and (h) 12.5 ps. The purple area refers to the GB region; the labeled oxygen atoms are involved in this activity ; the large green circle indicates the location of the vacancy. equilibrium in that vacant site. Fig. 4 10 (d) to (h) exhibits O 2 moving back and forth between the GB and the grain, but finally sitting in the site that O 1 originally was. This indicates that the thermally activated diffusion process is not a one time and one step

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69 process. Instead, the process requires some attempts to find the most favorable site before the uccessfu l" interstitialcy diffusion. It also has to be noted that atoms look to be better aligned in the columnar fashion at position O 2 in the Fig. 4 10 (c) than the same column in (b). Therefore, Fig. 4 11 shows single point alignment offsets at positi ons O 1 (blue triangles) and O 2 (red squares) in Fig. 4 10 (c) calculated from 9 ps to 12.5 ps, referring to Fig. 4 10 (a) to (h). The offset is with respect to the oxygen columnar axis viewing along [100] direction. From Fig. 4 11, the highest offset of position O 1 is at 9.5 ps, when O 1 is participating in the vacancy diffusion; the highest offset of position O 2 is at 11 ps, when O 2 is also taking part in the vacancy diffusion. In addition, the offset of position O 2 is always larger than that of position O 1 because position O 2 is closer to the GB area, which is characteristically disorder. Offset s for both positions finally decrease after the vacancy is successfully absorbed by the GB, which also indicates that the damage is healed in a more m acroscopic sense. Figure 4 11. Single point alignment offsets at positions O 1 and O 2 in Fig. 4 10 (c) with respect to the time from Fig 4 10 (a) to (h). 0.2 0.25 0.3 0.35 0.4 0.45 0.5 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 Offset ( 2 ) Time (ps) position O 2 position O 1

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70 So far, evidence has been shown that GBs can nucleate and annihilate vacancies As for interstiti a ls, there are some differences from vacancies. GB s as interstitial sinks : Annihilation of interstitials by the GB is also observed as shown in Fig.4 12. In a similar manner, the snapshot taken at 6.5 ps displays that the circled interstitial (green) is far from the GB and is not ready to diffuse (Fig. 4 12 (a)). At 10 ps, the interstitial is still in the same site, but atoms between it and the GB are no longer in a columnar alignment (Fig. 4 12 (b)). This disorder is thermally activated by some atoms acquiring enough energy to make jumps, and this a ctually paves the way for the interstitial to mov e towards the GB finally disappearin g in the GB, as shown in Fig. 4 12 (c) to (e). These phenomena can be furt her focused in c lose up views in Fig. 4 13. E specially in (c), a maximum randomness appears for the column occupied by O 3, O 4, O 5, O 6, and O 7. It turns out that only three of th em can stay, so O 6 moves into the GB and O 7 goes back to its original site, leaving O 3, O 4, and O 5 behind. Similarly, the rearrangement of atoms makes them more aligned in Fig. 4 13 (e) than in (c). From the study of defects evolution above, interstit ials diffusing into the GB, vacancies coming out of the GB and diffusing back to the GB are observed. It should be n ote d that no interstitials are observed to com e out of the GB. Therefore, it can be inferred that GB s can be source s or sink s for vacancies but only sink s for interstitials. It has been reported by a radiation damage simulation research in copper that the formation energy of interstitials at the GB is smaller than in the bulk. Therefore, the formation energy of oxygen interstit ials at the GB can also be smaller than in the bulk. With this assumption, there is a barrier for interstitials to re enter the perfect bulk. As a result, no interstitials will be emitted from GBs.

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71 Figure 4 12. Progressive snapshots showing the GB being the sink for the O interstitial (green) taken at (a) 6.5 ps, (b) 10 ps, (c) 10.5 ps, (d) 11 ps, and (e) 15 ps.

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72 Figure 4 13. Close up views of the GB being the sink for the O interstitial taken at (a) 6.5 ps, (b) 10 ps, (c) 10.5 ps, (d) 11 ps and (e) 15 ps. The purple area refers to the GB region; the labeled oxygen atoms are involved in this activity; the large green circle indicates the location of the interstitial.

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73 The diffusion mechanism was qualitatively described above, and now some qua ntitative analyses will be performed in order to fully characterize the diffusion. To take a specific example, in the second case of GBs being the vacancy sink, it can be s e e n from Fig. 4 9 (b) to (c) that the vacancy travel about 1.6 along [010] direction within 0.5 ps. This diffusion distance is less than the first nearest neighbor distance (2.7 ) in the single crystal. To ascertain if thi s r ate of va cancy diffusion is reasonable or not, recall Eq. 2 6 that the diffusivity has a tem perature dependence: (2 6) using the diffusion parameters from Govers et al 8 6 for UO 2 x with the Basak potential. The pre exponential diffusivity ( D 0 ) is 0.00045 cm 2 /s and the migration enthalpy barrier ( ) is 0.45 eV. With the equilibration under 1000 K, the atom diffusivity ( D A ) can be estimated to be about 2.410 2 2 /ps The atom diffusivity is therefore calculated form the Eq. 2 14 with the correlation factor f v = 0.653 for the free vacancy, c v = 1.5 10 3 and c A 1. (2 14) The calculation gives the atom diffusivity ( D V ) to be about 24.8 2 /ps In simulations, the self diffusion distance of the vacancy can be evaluated by Eq. 2 22: (2 22) So the diffusion distance is determined to be about 8.6 Compared with 1.6 from the observation, it is of the same order of magnitude. In addition, the diffusion direction is

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74 also consistent with the primary oxygen diffusion pathway, <001> direction, for the fluorite based material. 8 7 The discrepancy of the value may come from e valuation by an only single diffusion event in a very short observation time. In order to obtain high accuracy, the diffusion rate should be estimated by an average of multiple diffusion events for a longer time period. 4.3.4 Vacancy Clustering: Schottky Defects A Schottky defect is composed of one U vacancy and two O vacancies. The formation of Schottky defects are observed in the presence of U FPs, that is, the condition of defects on the U sub lattice alone and on both U and O sub lattices. Evidence has been show n that the Schottky (neutral tri vacancies) defect cluster is more stable than either the di cluster or entirely separated Figure 4 1 4 Schematic representation of three different configurations of Schottky defect s: (a) <100>, (b) <110>, and (c) <111>. Grey spheres stand for vacancies

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75 Figure 4 1 5 Snapshot of selective area showing Schottky defects taken at t = 250 ps: (a) <111> Schottky, (b) <110> Schottky, and (c) <100> Schottky. vacancies. 8 8 Hence, even though the Schottky defect is created by diffusion, it is relatively stable. According to the relative positions of the two oxygen vacancies, there are three types of Schottky defects (Fig. 4 1 4 ), <100>, <110>, and <111> Schottky, namely among which <110> and <111> Schottky clusters are also observed by Aidhy et al in the single crystal UO 2 In this work 24,37 <100> Schottky is also observed. T hree kinds of Schottky defects are simultaneously shown in Fig 4 1 5 taken at 250 ps The formation of the Schottky defect anchors O vacancies which sequesters them from the vacancy interstitial recombination. As a result, this leads to their many counterpart O interstitials developing other stable complex clusters. 4.3.5 Interstitial Clusteri ng: Cuboctahedral Clusters The presence of Schottky defects sequesters O vacancies from their counterpart s, so these interstitials develop a relatively complex structure, namely, c uboctahedral

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76 (COT) clusters. Discussion about COT clusters has been well de scribed by Aidhy et al in the work of single crystal UO 2 24,37 The progressive formation process is recalled in Fig. 4 1 6 The relative energy is decreasing step by step, indicating that the system tends to form clusters. There are three types of COT clusters defined by the occupancy of the octahedral site (green circle). It can be a U interstitial (COT u ), an O interstitial (COT o ), or a vacancy (COT v ). In this work, three kinds of COT clusters are simultaneously shown in the snapshot taken at 250 ps ( Fig 4 1 7 ) In addition to COT clusters (a) and (b) already in complete shape, COT clusters (c) and (d) are found to be still under construction, which are found to correspond to the step (c) and (d) in Fig. 4 1 6 Hence, Figure 4 1 6 Schematic representations of the formation process of COT cluster s (blue for U, red for O, pink for O interstitial, grey for O vacancy, and green for either U, O, or vacancy) (a) Fluorite unit cell. (b) The first O interstitial enters the unit cell by knocking two O ions off their original lattice sites, leading to the formation of two vacancies and three interstitials. (c), (d), and (e) are snapshots for the second, third, and forth O interstitials coming into the unit cell.

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77 Figure 4 1 7 Snapshot of selective are a showing COT clusters taken at t = 250 ps: (a) COT v (b) COT o (c) COT u and (d) COT u It should be noted that both (COT u )s are not fully constructed. with the complex structure, the aggregation of interstitials into COT clusters is ascertained to be a rapid but not one step process. For every complete COT cluster, eight more O FPs will be introduced to the system. This clustering mechanism accounts for the sharp increa se in both cases initially with the presence of U FPs. 4.3.6 Interstitial Clustering : V 4O Clusters In the V 4O cluster, an O vacancy is surrounded by four O interstitials in a pyramid al fashion (Fig. 4 1 8 ). The same kind of oxygen cluster is als o reported in relaxation of Willis clusters 89 9 0 by Geng et al using first principle calculations in UO 2+x 9 1 According to their results, this kind of cluster is the minority among all defects. In the condition of defects on the O sub lattice alone, most of the O FPs are healed after 400 ps, but two V 4O clusters form. Therefore, from Fig. 4 6, the number of O in terstitials is a

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78 little fewer than the number of O vacancies, and the difference is 6. A snapshot taken at 400 ps is shown in Fig. 4 1 9 Figure 4 1 8 Schematic representation of the configurations of V 4O clusters. The O vacanc y is surrounded by four O interstitials ( grey ) in a pyramid al fashion Figure 4 1 9 Snapshot of selective area showing V 4O oxygen clusters taken at t = 400 ps. In each cluster, an O vacancy is surrounded by four tetrahedral sited O interstitials.

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79 CHAPTER 5 CONCLUSIONS AND FUTURE WORK 5.1 Conclusions In terms of analysis tool s the lattice matching analysis (LMA) method is complementary to the conventional common neighbor analysis (CNA) method. The cutoff radius of 0.19a o (about 1.039 ) is tested in the single crystal to identify the same amount and spot of defects as the C N A T he LMA is then applied to polycrystal to determine the exact number, type, and position of every point defect. A big advantage is that it can easily tell the clu ster structure. As for MD simulations, evolution of the defects basically has the same trend as in the single crystal. Similarly U FPs are seldom annihilated during equilibration while of O FPs are mostly healed except those which form clusters. Three kin ds of initial conditions are performed. For FPs created on the O sub lattice alone, the number of O interstitials is slightly larger than that of O vacancies because some charged pyramidal clusters form, in which an O vacancy is surrounded by four O inters titials. In cases of FPs created on the U sub lattice alone and on both U and O sub lattices, the number of O vacancies is larger than that of O interstitials. In addition to spontaneous vacancy interstitial recombination events, point defects have another choice, either diffus ing into or emit ting from the highly energetic GB. When some of the defects form clusters, one grain b u lk region m ay lose its local electroneutrality. Therefore, GBs maintain the local charge neutrality in the bulk by modulating the d efect concentration via source/sink behavior s of point defects. Nevertheless, GBs can be the source and sink for vacancies but only the sink for interstitials. It has been reported by a radiation damage research in copper that formation energy of interstit ials at the GB is smaller than in the bulk 93 T herefore in UO 2

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80 the formation energy of oxygen interstitials at the GB can also be smaller than in the bulk Thus there is a barrier for interstitials to re enter the perfect bulk. As a result, no interstit ials will be emitted from GBs. In addition, for the cas e of creating defects on only the U sub lattice, the equilibrium concentration of O defects is much larger in the polycrystal than that in the single crystal. This can also be explained by the source/sink behavior s with GBs. In the single crystal, the concentration of newly nucleated O defects solely depends on the number of the U FPs. H owever, in the polycrystal, more O defects can be possibly p rovided by GBs. Once they successfully enter the interior grain, they will be trapped by form ing clusters, such as Schottky or c uboctahedral (COT) clusters. With respect to the diffusion mechanism, t hese source/sink activities take place with a set of chain reactions via the vacancy and interstitialcy diffusion. That is, not always the same single atom is moving; instead, many atoms are involve d in one diffusion event T he equilibration process is diffusion controlle d which highly depend s on the formation and migration energies of the defects. The diffusion path is ascertained to follow the primary diffusion direction <001> for the fluorite based structure with the reasonable oxygen vacancy diffusi on distance. 87 T he arrangement of atoms is focus ed in close u p views to become disorder ed when the diffusion activity is about to occur; subsequently all the atoms come back to lattice sites and realign in the n ormal fashion. In addition, the single point alignment offsets give furth er quantitative analysis to confirm the phenomena being observed. Regarding clusters, there are many kinds of clusters formed during equilibration First, a minor kind of cluster, V 4O, is observed, and the appearance of V 4O clusters accounts for the number of O interstitials is fewer than the number of O vacancies in the

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81 condition of defects on the O sub lattice alone. Compared with V 4O clusters Schottky defects and COT clusters are much more prevalen t Schottky defects can be divided into three types by the relative positions of oxygen vacancies: <100>, <110>, and <111> Schottky. Among them, <111> Schottky has the lowest formation energy, followed by <110>, and then <100> with th e Basak potential. 83 T herefore, <111> Schottky is the most common type in our simulations. A s for COT clusters, there are also three kinds defined by what occupies the octahedral site: COT v COT u and COT o In literature, COT o has lower formation energy than COT v both negative; 9 3 neverthe less, there is no information about COT u Our results show that COT u is the most common This may be due to the initial high concentration of defects on the U sub lattice. Therefore, i t can be expected that the formation energy of COT u will be the lowest among these three clusters. In conclusion, t he diffusion controlled kinetic evolution of defects method is complementary to conventional collision cascade simulations to provide a better understanding of the kinetic phase of defect s evolution at the atomistic scale Extending from the work o n the single crystal, this work focuses on the more representative polycrystalline UO 2 Differences in the numbers of radiation induced point defects (vacancies and interstitials) are attributed to the presence of GBs, which can modulate the defect concentration via source/sink behavior s. These activities involving the vacancy and interstitialcy diffusion mechanism are evidentially witnessed by atomistic scale snapshots, along with three types of clusters, V 4O clusters, Schottky defects, and COT clusters However, with the presence of GBs whether the radiation tolerance can be enhance d or not is still not determined.

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82 5.2 Future Work This study has been performed on polycrystalline UO 2 elucidating qualitative GB source/sink strengths. A corresponding work is required to provide more quantitative information. For example, calculating the total number of clusters is needed to identity if GBs essentially enhance or impede the formation of clusters. Actually, GBs open up the possibility of variety of mechanisms that are still not well understood, such as (1) diffusivities in the bulk and GB regions, (2) diffusivities in high and low angle misorientation. An understanding of differences of d iffusivities in these cases allows us to precisel y predict the defect evolution.

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88 BIOGRAPHICAL SKETCH Kun Ta Tsai was born in Hsinchu, Taiwan He received his bachelor s degree in c hemical e ngineering from National Cheng Kung University, Tainan, Taiwan, in 2007. Kun Ta came to the United States to pursue higher education and entered the Department of Materials Science and Engineering at Universi ty of Florida, Gainesville, in F all 2008. He joined the Computational Materials Science Focus Group w ith Prof. Simon R. Phillpot in S pring 2009. Kun Ta is expecting his master s degree in S ummer 2010. After joining the Gator Nation, Kun Ta watched his first football game in his life, and started to bec o me crazy about it. In the same academic year, the Florida Gators won the Bowl Championship Series ( BCS ) National Championships. He is proud to say Go Gators