BINARY COLLISION AND MOLECULAR DYNAMICS SIMULATION OF Fe-
Ni-Cr ALLOYS AT SUPERCRITICAL WATER CONDITION
A THESIS PRESENTED TO DEPARTMENT OF NUCLEAR ENGINEERING,
SCHOOL OF NUCLEAR AND ALLIED SCIENCES
COLLEGE OF BASIC AND APPLIED SCIENCES,
UNIVERSITY OF GHANA
BY
COLLINS NANA ANDOH (ID: 10443957)
B.Sc. (CAPE COAST), 2010
IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF
MASTER OF PHILOSOPHY
IN
COMPUTATIONAL NUCLEAR SCIENCES AND ENGINEERING
JULY, 2015
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DECLARATION
I hereby declare that with the exception of references to other people’s work which has
been duly acknowledged, this thesis is the result of my own research work and no part of
it has been presented for another degree in this University or elsewhere.
…………………………………… Date…………………………….
COLLINS NANA ANDOH
(Student)
We hereby declare that the preparation of this thesis was supervised in accordance with the
guidelines of the supervision of Thesis work laid down by University of Ghana.
…………………………… ……………………………….
Dr. G.K. BANINI NANA (Prof.) A. AYENSU GYEABOUR I
(PRINCIPAL SUPERVISOR) (Co-SUPERVISOR)
Date…………………… Date……………………
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ABSTARCT
Fe-Ni-Cr alloys are commonly used as pressure vessel (in-core) materials for nuclear
reactors and have been classified as candidate materials for Supercritical Water-Cooled
Reactors (SCWR). In service, the in-core materials are exposed to harsh environments:
intense neutron irradiation, mechanical and thermal stresses, and aggressive corrosion
prone environment which all contribute to the components’ deterioration. For better
understanding of the mechanisms responsible for degradation of the Fe-Ni-Cr alloys
(SS304, SS308, SS309 and SS316) under high neutron irradiation dose, pressure and
temperature conditions as pertains in SCWR conditions, these alloys were examined using
Binary collision and molecular dynamics simulations using (SRIM-TRIM code and
LAMMPS, VMD codes) respectively. The neutron irradiation damage assessments were
conducted under irradiation doses of 30 dpa (thermal neutron spectrum) and 150 dpa (fast
neutron spectrum). The results indicated that more defects were generated in the fast
neutron spectrum SCWR than in the thermal neutron spectrum, and the depth of penetration
of neutron in the fast spectrum was (32.3 µm) about three times that of the thermal
spectrum (~ 11.3 µm). The work revealed that there was a marginal difference of 97.18 %
of the neutron energy loss in SS308 compared to 97.14 % in SS316 and SS309.
The evaluation of mechanical deterioration revealed that Young’s Modulus, Ultimate
Tensile Strength and the Breaking/Fracture Strength decreased with increasing
temperature. The SS308 and SS304 two materials had very high ultimate tensile strengths
and breaking strengths even at the temperature of 500 ºC. By linking the neutron damage
assessment and the mechanical evaluation, SS304 and SS308 could be considered in the
design of the SCWR pressure vessel and couplings since the SS308 was found to be least
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damaged by the neutron irradiation whiles SS304 had high breaking strength. However,
further research is recommended on the two Fe-Ni-Cr alloys SS308 and SS304 on
hydrogen embrittlement, swelling, creep, as well as corrosion studies upon interactions
with supercritical water environment; an extensive testing and evaluation program is
required to assess the corrosion effects on the material properties of these two materials.
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DEDICATION
I dedicate this thesis work to my father, Mr. Paul Nana Andoh, who after all his health
problems stood firm to help me finish this Master’s Degree, his brother Peter Adansi
Andoh, and my step mother Victoria Okyere whose encouragement has brought me this
far. Finally, to my son Allswel Nana Andoh and my two grandmothers Asi Kumiwaa and
Esi Asiedua (Esi Kakraba) who have been on their knees praying for my success
throughout my education.
.
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ACKNOWLEDGEMENTS
I am very thankful to God Almighty for giving me the strength to undertake this study. My
sincere gratitude goes to my supervisors, Nana (Prof) A. Ayensu Gyeabour I and Dr. G. K.
Banini for their tolerance, encouragement, priceless advice, constructive criticisms and
kind supervision.
I am also thankful to all my siblings, colleagues (especially my roommates Maruf
Abubakar and Ernest Kwame Ampomah) and friends for their guidance, fruitful
discussions and outstanding assistance offered me in completing this thesis.
I am also grateful to Mr. Raymond Oteng-Appiagyei for his advice and to Mr. Isaac Benkyi,
University of Helsinki, Finland for guidance in the applications of the LAMMPS code
which is being used the first time at Graduate School of Nuclear and Allied Sciences,
University of Ghana.
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TABLE OF CONTENTS Page No.
DECLARATION ................................................................................................................ ii
ABSTRACT ....................................................................................................................... iii
DEDICATION .....................................................................................................................v
ACKNOWLEDGEMENTS ............................................................................................... vi
TABLE OF CONTENTS .................................................................................................. vii
LIST OF FIGURES ......................................................................................................... xiii
LIST OF TABLES ......................................................................................................... xviii
ABBREVIATIONS ......................................................................................................... xix
LIST OF SYMBOLS .........................................................................................................xx
CHAPTER ONE: INTRODUCTION ............................................................................1
1.1 RESEARCH BACKGROUND ......................................................................1
1.2 RESEARCH PROBLEM STATEMENT .......................................................2
1.3 RESEARCH JUSTIFICATION .....................................................................2
1.4 RESEARCH GOAL .......................................................................................3
1.5 RESEARCH OBJECTIVES ...........................................................................3
1.6 SCOPE OF RESEARCH ................................................................................4
CHAPTER TWO: LITERATURE REVIEW ..............................................................5
2.1 SUPERCRITICAL WATER CONDITION ..................................................5
2.2 DESIGN PARAMETERS FOR PRESSURE VESSEL OF SCWR ...............6
2.3 MATERIALS CHALLENGES WITH SCWR ...............................................8
2.3.1 Material Needs of SCWR Design .....................................................8
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2.3.2 Candidate In-core Structural Materials of SCWR ...........................9
2.3.3 Structural Integrity of Irradiated Materials .....................................10
2.4 IRRADIATION DAMAGE ASESSMENT ................................................10
2.4.1 Irradiation Damage Events and Mechanisms……………………. 10
2.4.2 Rate of Production of Displacement…………………….………..13
2.4.3 Kinchin – Pease Model of Radiation Damage.. .............................16
2.4.3.1 Displacement Probabilty……………………………… 16
2.4.3.2 Displacement Energy ......................................................18
2.4.3.3 Radiation Damage ...........................................................18
2.4.4 Norgett-Robinson-Torrens Model of Radiation Damage……….. 24
2.4.5 Radiation Damage Defects .............................................................25
2.5 MECHANICAL PROPERTIES OF PRESSURE VESSEL MATERIALS....27
2.5.1 Stainless Steels Alloys for Pressure Vessel and In-core
Structure………………………………………………………….28
2.5.2 Austenitic Stainless Steel ..............................................................28
2.5.3 Effects of Irradiation on Physical Properties of Steels ................29
2.5.4. Mechanical Strength Parameters....................................................30
2.5.4.1 Youngs Modulus……………………………………… 31
2.5.4.2 Tensile Strenth ................................................................31
2.5.4.3 Fracture or BreakingStrength ..........................................32
2.5.4.4 Yield Strength…………………….……………………32
2.6. COMPUTER SIMULATION CODES FOR IRRADIATION DAMAGE
AND INDUCED MECHANICAL DEGRADATION…………………..33
2.6.1. Binary Collision Approximation (BCA) Method…..……………33
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2.6.2. Molecular Dynamics (MD) Method .............................................36
2.6.3. Kinetic Monte Carlo (KMC)… ......................................................39
2.6.4 Computer Simulation codes ...........................................................39
2.6.4.1 SRIM-TRIM ...................................................................39
2.6.4.2. Large –Scale Atomic/Molecular Massively Parallel
Simulator (LAMMPS) ....................................................41
2.6.4.3 Visual Molecular Dynamics (VMD) ..............................42
CHAPTER THREE: RESEARCH METHODOLOGY ...........................................43
3.1 SELECTION OF Fe-Ni-Cr ALLOYS .......................................................43
3.1.1 Structural and In-core Materials ....................................................43
3.1.2 Characterization and Properties of Selected Alloys.......................45
3.2 NEUTRON IRRADIATION DAMAGE ASSESSMENT BY BCA .......46
3.2.1 Estimation of Energy Level at 30 dpa of Thermal Neutrons
Spectrum ........................................................................................46
3.2.2 Estimation of Energy Level at 150 dpa of fast Neutrons
Spectrum ........................................................................................47
3.2.3 SRIM-TRIM Setup and Input Requirements ................................47
3.2.4 SRIM-TRIM code Simulation Algorithm and Flowchart ..............53
3.2.5 SRIM – TRIM Simulation Implementation ...................................55
3.2.6 SRIM-TRIM Output files ..............................................................56
3.3 EVALUATION OF MECHANICAL DEGRADATION OF ALLOYS...57
3.3.1 LAMMPS Setup and Input Requirements .....................................57
3.3.2 Interatomic Potential Developed for MD Simulation ....................59
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3.3.3 LAMMPS Simulation Algorithm ..................................................63
3.3.4 Implementation of LAMMPS Simulation .....................................65
3.3.5 Output of LAMMPS Simulation ....................................................66
3.3.6 Visualization of Output of Simulation by VMD and MATLAB ...67
CHAPTER FOUR: RESULTS AND DISCUSSIONS ..................................................68
4.1. THERMAL AND FAST NEUTRON IRRADIATION DAMAGE IN 304
Fe-Ni-Cr ALLOY .....................................................................................68
4.1.1 Collision Cascade............................................................................68
4.1.2 Projected Neutron Range Distribution ............................................69
4.1.3 Lateral Neutron Range Distribution................................................70
4.1.4 Ionization Energy Distribution .......................................................71
4.1.5 Phonons ...........................................................................................73
4.1.6 Neutron Energy to Recoil Dsitribution ...........................................74
4.1.7 Collision Events ...............................................................................75
4.1.8 Sputtering Yield ...............................................................................77
4.2. EVALUATION OF MECHANICAL DETORIORATION OF Fe-Ni- Cr
ALLOYS ....................................................................................................78
4.2.1 Cohesive Energy of the Fe-Ni-Cr Potential File .............................78
4.2.2 VMD Output for the Tensile Deformation .....................................79
4.2.3 Stress-Strain Plots at Ambient and Supercritical Conditions ........80
4.2.4 Mechanical Properties of Fe-Ni-Cr Alloy.......................................83
4.3. DISCUSSION ............................................................................................85
4.3.1 General Discussion .........................................................................85
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4.3.2 Discussion on Neutron Irradiation Damage ....................................87
4.3.2.1 Projected Neutron Range .................................................... 87
4.3.2.2 Energy Loss to Ionization .................................................... 87
4.3.2.3 Fe-Ni-Cr alloy’s Energy Loss to Phonon ............................. 88
4.3.2.4 Energy Loss to Vacancy creations in Fe-Ni-Cr alloys ........ 89
4.3.2.5 Energy to Recoil Cascade ................................................... 89
4.3.2.6 Sputtering Yield ................................................................... 90
4.3.3 Discussion on Mechanical Detorioration ............................................ 91
4.3.3.1 Cohesive Energy .................................................................. 91
4.3.3.2 Young’s Modulus ................................................................. 91
4.3.3.3 Yield Strength ..................................................................... 92
4.3.3.4 Ultimate Tensile Strength ................................................... 92
4.3.3.5 Breaking or Fracture Strength .............................................. 93
4.3.4 Discussion on Linking of the Neutron Irradiation Damage and
Mechanical Detorioration .................................................................. 94
CHAPTER FIVE: CONCLUSIONS AND RECOMENDATIONS ............................95
5.1 CONCLUSIONS .........................................................................................95
5.2 RECOMENDATIONS ................................................................................97
REFERENCES .................................................................................................................98
APPENDICES ................................................................................................................106
APPENDIX I: SRIM–TRIM simulation: input and output spectra from neutron
Irradiation Damage Assessment of Fe-Ni-Cr Alloys.................................106
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APPENDIX II: Input files for Molecular Dynamics Simulation of Mechanical Damage
Assessment …………………………….…………………………...…107
APPENDIX III: Algorithm for Animation of Tensile Deformation Using VMD .......110
APPENDIX IV: SRIM–TRIM Simulation Output Spectra for Neutron Irradiation
Damage Assessment of Fe-Ni-Cr Alloys..........................................111
APPENDIX V: Comparison of the Neutron Irradiation Damage Assessment of Fe-Ni-Cr
Alloys under Thermal and Fast Neutron Spectrum of the SCWR……131
APPENDIX VI: Output files of Molecular Dynamics Simulation of Mechanical
Damage Assessment of Fe-Ni-Cr Alloys……………………......…132
APPENDIX VII: Mechanical Properties of the Fe-Ni-Cr Alloys under Ambient
Temperature and Supercritical Water Conditions….……...........…139
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LIST OF FIGURES
Title Page No.
Figure 2.1: Phase Diagram of SCW condition 5
Figure 2.2: Conceptual Design of SCWR 6
Figure 2.3: Mechanism of irradiation damage in the nuclear reactor system
12
Figure 2.4: The displacement probability Pd (T) as function of the
transferred kinetic energy, assuming (a) a sharp or (b) a
smoothly varying displacement threshold
17
Figure 2.5: Average number of displacements v(T) produced by a PKA, as a
function of the recoil energy T according to the model of
Kinchin -Pease
23
Figure 2.6: Radiation damage defects processes 26
Figure 2.7: Defects in the lattice structure of materials that can change their
material properties
27
Figure 2.8: A typical Stress-Strain Curve for Fe-Ni-Cr Alloys 30
Figure 2.9: The trajectory of two particles interacting according to a
conservative central repulsive force in the laboratory system
34
Figure 2.10: Molecular Dynamics Simulation flow chart 37
Figure 2.11: Molecular Dynamics Simulation of a unit cell of the material 38
Figure 3.1: Phase diagram of Stainless Steel Alloys 44
Figure 3.2: TRIM Input Parameter Window showing all inputs for Stainless
Steel grade 316 assessment
48
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Figure 3.3: SRIM – TRIM Code flowchart for simulation 55
Figure 3.4: Steps followed in designing LAMMPS input file 58
Figure 3.5. Graphical representation of the periodic boundary conditions.
The arrows indicate the velocities of atoms. The atoms could
interact with atoms in the neighboring boxes without having
any boundary effects.
62
Figure 3.6: Command prompt loop for LAMMPS Simulation 63
Figure 3.7: On screen view of Output values from Simulation
64
Figure 3.8: (a) The crystal structure of an FCC lattice and (b) the (100)
orientation in the x, y and z direction where the uniaxial
deformation was applied.
65
Figure 4.1: Collision Cascade window for (a) thermal and (b) fast neutron
irradiation damage in Fe-Ni-Cr alloy SS304
68
Figure 4.2: Projected Range of (a) thermal and (b) fast neutrons in Fe-Ni-Cr
Alloy SS304
70
Figure 4.3: The lateral Range distribution of the (a) thermal and (b) fast
neutrons in Fe-Ni-Cr Alloy SS304
71
Figure 4.4: 2D view of the Ionization energy distribution of (a) thermal
neutrons as compared with the (b) fast neutrons (c) and (d) 3D
view of the Ionization energy distribution in the Fe-Ni-Cr alloy
SS304
72
Figure 4.5: Distribution of (a) thermal and (b) fast neutrons energy loss to 74
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Fe-Ni-Cr Alloy SS304 phonons
Figure 4.6: Distribution of Energy absorbed by the SS304 Fe-Ni-Cr Alloys
elements in the two different spectra.
75
Figure 4.7: 2D view of the collision events of (a) thermal and (b) fast
neutron spectrum, (c) and (d) gives 3D view of the collision
events
77
Figure 4.8: Distribution of integral sputtering yield of Fe-Ni-Cr Alloy
SS304 in (a) thermal and (b) fast neutron spectrum
78
Figure 4.9: VMD Snapshot showing Fe-Ni-Cr alloy model of size 10 Å x
10 Å x 10 Å (No. of atoms 4000)
79
Figure 4.10: Stress- Strain curve showing all the Mechanical Properties of
the Fe-Ni-Cr Alloy, SS 304 under Ambient Conditions
80
Figure 4.11: Stress-Strain plot for Fe-Ni-Cr Alloys at Ambient Condition
and Supercritical Water Condition at strain rate of 5x1010 s-1 for
(a) SS304 (b) SS308 (c) SS309 and (b) SS316
81
Figure 4.12: Variation of (a) Young’s Modulus (b)Yield Strength (c)
Ultimate Tensile Strength and (d) Breaking or Fracture Strength
of the alloys with respect to ambient and SCW condition
83
Figure 4.13: Diagram on the Collision Cascade for (a) thermal and (b) fast
neutron damage in Fe-Ni-Cr alloy SS308
111
Figure 4.14: Projected Range Distribution of (a) thermal and (b) fast neutrons
in Fe-Ni-Cr Alloy SS308
112
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Figure 4.15: Lateral Range Distribution of (a) thermal and (b) fast neutron in
the Fe-Ni-Cr Alloy SS308
112
Figure 4.16: 2D and 3D view of Ionization energy distribution of the Fe-Ni
Cr alloy SS308 in both thermal and fast neutron spectrum
113
Figure 4.17: Distribution of Energy Loss as Phonons by Fe-Ni-Cr Alloy
SS308 in the (a) thermal and (b) fast neutron spectrum
114
Figure 4.18: Distribution of Energy Absorbed by each elements in the SS308
in the (a) thermal and (b) fast neutron spectrum
115
Figure 4.19: Collision events of SS308 in 2D and 3D view respectively in the
(a) thermal and (b) fast neutron spectrum
116
Figure 4.20: Integral sputtering yield of SS308 for (a) thermal and
(b) fast neutron spectrum
117
Figure 4.21: Diagram on the Collision Cascade for (a) thermal and (b) fast
neutron damage in Fe-Ni-Cr alloy SS309
117
Figure 4.22: Projected Range Distribution of (a) thermal and (b) fast neutron
in the Fe-Ni-Cr Alloy SS309
118
Figure 4.23: Lateral Range Distribution of (a) thermal and (b) fast neutron in
the Fe-Ni-Cr Alloy SS309
119
Figure 4.24: 2D and 3D view of Ionization energy distribution of the Fe-Ni-
Cr alloy SS309 in the(a) thermal and (b)fast neutron spectrum
119
Figure 4.25: Distribution of Energy Loss as Phonons by the Fe-Ni-Cr Alloy 121
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SS309 in the (a) thermal and (b) fast neutron spectrum
Figure 4.26: Distribution of Energy Absorbed by each elements in the SS309
in the (a) thermal and (b) fast neutron spectrum.
121
Figure 4.27: Collision events of SS309 in (a) 2D and 3D view respectively
in the thermal and fast neutron spectrum
122
Figure 4.28: Plot of SS309 for the both the thermal and fast neutron spectrum 123
Figure 4.29: Diagram on the Collision Cascade for (a) thermal and (b) fast
neutron damage in Fe-Ni-Cr alloy SS316
124
Figure 4.30: Projected Range Distribution of (a) thermal and (b) fast neutrons
in the Fe-Ni-Cr alloy SS316
125
Figure 4.31: Lateral Range Distribution of (a) thermal and (b) fast neutron in
the Fe-Ni-Cr alloy SS316
125
Figure 4.32: 2D and 3D view of Ionization energy distribution of the Fe-Ni-
Cr alloy SS316 in the both thermal and fast neutron spectrum
126
Figure 4.33: Distribution of Energy Loss as Phonons by the Fe-Ni-Cr Alloy
SS316 in the (a) thermal and (b) fast neutron spectrum
127
Figure 4.34: Distribution of Energy Absorbed by each elements in the SS316
Fe-Ni-Cr alloys in (a) thermal and (b) fast neutron spectrum.
128
Figure 4.35: Collision events of SS316 in 2D and 3D view respectively in
the thermal and fast neutron spectrum
128
Figure 4.36: Integral sputtering yield of SS316 for the both the thermal
and fast neutron spectrum
130
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LIST OF TABLES Page No.
Table 2.1: SCWR reference design power and coolant conditions 7
Table 2.2: Reference reactor pressure vessel design for SCWR 8
Table 2.3: Material property and their Damage effects on Microstructure 29
Table 3.1: Composition and Fe-Ni-Cr alloys selected for Damage 45
assessment at SCW condition
Table 3.2: Ion Data and Input parameters used in the SRIM-TRIM code 49
Table 3.3: Target Data and Input parameters in SRIM-TRIM code 49
Table 3.4: TRIM.IN setup parameters for TRIM simulation of SS304 50
Table 3.5: Lattice Parameters used for the LAMMPS 59
simulation
Table 4.1: Summary of Equilibrium Lattice Constant and Cohesive
energy from the simulation compared with theoretical 79
value
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ABBREVIATIONS
LWR Light Water Reactor
SCWR Supercritical Water-Cooled Reactor
SRIM Stopping and Range of Ion in Matter
TRIM Transport of Ion in Matter
LAMMPS Large-scale Atomic/Molecular Massively Parallel Simulator
VMD Visual Molecular Dynamics
SCC Stress Corrosion Cracking
IASCC Irradiation Assisted Stress Corrosion Cracking
UTS Ultimate Tensile Stress
PKA Primary Knock-on-Atom
BCA Binary Collision Approximation
MD Molecular Dynamics
KMC Kinetic Monte Carlo
DPA Displacement per atom
TRIM.DAT Transport of Ion in Matter Data
SCW Supercritical Water Condition
.txt Text File
NRT Norgett Robinson Torrens
KP Kinchin – Pease
SBE Surface Binding Energy
SS Stainless Steel
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LIST OF SYMBOLS
NOMENCLATURE MEANING
eV Electron Volt
keV kilo-electron volt
MeV Mega-Electron Volt
GeV Giga-Electron Volt
g/cm3 Gram per meters cube
dpa/s Displacement per Second
MPa Mega-Pascal
GPa Giga-Pascal
T Kinetic Energy of a PKA
E Energy
Tdam Damage Energy
Tmax Maximum damage energy
Emax, Emin Minimum and Maximum displacement energy
dEe Differential of electron Energy
f(T) Probability function
ED or Ed Displacement Energy
VD(T) Average number of displaced atoms created in a cascade
𝜎D(Ee) Displacement of cross section
Ф(Ee) Electron energy flux
Ee Electron Energy
Rd Damage rate
?̂? , ?̌? Maximum and Minimum transferred energy
?̂?, ?̌? Maximum and Minimum threshold energy
Elatt, Esurf Lattice Binding Energy , Surface Binding Energy
ɛ , σ Strain , Stress
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CHAPTER ONE: INTRODUCTION
1.1. BACKGROUND
Supercritical Water-Cooled Reactor (SCWR) is a Light Water Reactor (LWR) operating at
higher pressure and temperatures with a direct, once-through cycle [1-3]. The coolant of
the reactor remains single-phase throughout the system since it operates above the critical
pressure and temperature (374 ºC, 22.5 MPa) and hence eliminates coolant boiling [4, 5].
The neutron radiation is anticipated to be 10–30 dpa (displacement per atom) and 100 –
150 dpa for the thermal and fast spectrum at energy of 365 MeV and 1.5 GeV respectively
[1, 6].
Research is keenly needed in the design of the SCWR since no nuclear reactors that uses
supercritical water as its coolant have so far been built, though it is promising, but however
demonstration or experimental reactors of very closely related concepts have already been
built for the other Generation IV concepts [7, 8].
Development of fission reactor critically depends on advances made in nuclear fuels and
also in their systems and structural materials which may have to withstand the severe
environmental conditions (such as high temperatures, neutron irradiation and strong
corrosive environments) in combination with complex loading and operational cycles and
longer design life requirements [9].
Searching for new materials and tailoring them to the desired system properties and
operational requirements is therefore central to the reactor developments to establish the
optimal materials operational parameters range for SCWR for the selection of structural
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and cladding materials that will maintain reliable operation of a SCWR power plant for its
design life of 60 years [10, 11].
Some candidate materials such as ferritic-martensitic steels and low-swelling austenitic
steels have been identified [12]. These materials are not proven [7, 13] and hence global
on-going research to characterize their physical, nuclear and mechanical properties [14].
1.2. RESEARCH PROBLEM STATEMENT
To examine the irradiation resistance and mechanical integrity of Fe-Ni-Cr based alloys as
candidate materials for Supercritical Water (SCW) condition pressure vessel and in-core
structural components through Binary Collision Approximation and Molecular Dynamics
simulations respectively on stainless steels (SS) categories 304, 308, 309 and 316.
1.3. RESEARCH JUSTIFICATION
Neutron irradiation of in-core materials creates point defects [15] which results in
significant modifications in physical dimensions, strength and hardness, thermal and
electrical conductivity, resistance to corrosion, etc. A nuclear reactor operates within very
stringent requirement throughout the working life of the reactor and so structural materials
must maintain their mechanical properties and dimensional stability.
Hence incremental changes in materials properties during steady state reactor operations
must stay within specifications and all materials must be able to perform throughout the
reactor’s life as required under all postulated accident conditions.
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Reactor safety, material degradation and failure are areas of critical importance as high
neutron flux could lead to high irradiation dose [6, 16, 17]. There is the need for research
to investigate the Fe-Cr-Ni alloy as possible candidate for SCWR in-core structural
materials, especially the austenitic steel which are durable and have good mechanical
strength.
1.4. RESEARCH GOAL
The goal is to examine neutron irradiation damage and mechanical degradation of the Fe-
Cr-Ni alloys (SS304, SS308, SS309 and SS316) by high neutron dose loading of 10 – 30
dpa and 100 – 150 dpa respectively. Both thermal and fast neutron bombardment in high
pressure and temperature conditions as would pertain in SCW conditions are of interest
and hence the above materials were is to be examined using Binary Collision
Approximation and Molecular Dynamics simulations SRIM-TRIM, LAMMPS code
respectively along with VMD and MATLAB.
1.5. RESEARCH OBJECTIVES
The main objectives of the research were to:
Simulate thermal and fast neutron irradiation damage of Fe-Cr-Ni alloys at 30 dpa
and 150 dpa to determine the Depth of penetration, Ionization energy, Energy to
Phonons, Energy to recoils, and Vacancy production for current SCWR design.
Evaluate the mechanical behavior and dimensional stability of the Fe-Ni-Cr as a
function of high pressure and temperature using LAMMPS and VMD codes
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Compare suitability of the four Fe-Ni-Cr alloys as in-core structural materials and
for pressure vessel design and make selection by TRIM code and mechanical
integrity assessment by the LAMMPS and VMD code.
1.6. SCOPE OF THE RESEARCH
The thesis is divided into five Chapters.
Chapter One provide the background to the research work, research problem statement,
relevance and justification of the research, the research goal, objectives and the scope of
the research work.
Chapter Two deals with the review of relevant Literature on materials challenges for
SCWR, irradiation damage mechanisms and induced mechanical deterioration, current
materials selection for SCWR vessel, mechanical integrity evaluation and Computer
Simulation code of radiation damage and mechanical degradation.
Chapter Three presents the Research Methodologies employed relating to the irradiation
damage assessment of the materials using the SRIM-TRIM code and the mechanical
damage evaluation using LAMMPS and VMD codes
In Chapter Four, the Results obtained, interpretation of the data and discussion of the
research findings are presented.
Chapter Five provides the Conclusions, Recommendations and Suggestions for future
research work. Also the references cited are listed and Appendices are also presented.
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CHAPTER TWO: LITERATURE REVIEW
2.1 SUPERCRITICAL WATER CONDITION
Supercritical Water-Cooled Reactors are high temperature, high-pressure, light water
reactors that operate above the thermodynamic critical point of water (374 °C, 22.1 MPa).
The reactor core may have a thermal or a fast-neutron spectrum, depending on the core
design [15]. Figure 2.1 shows two different SWC regimes, the US and CANDU design.
The US design which operates at temperature range of 280°C to 500°C and at a constant
pressure of 25 MPa was chosen for the research work [6].
Fig 2.1: Phase Diagram of SCW condition [18]
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2.2 DESIGN PARAMETERS FOR STRUCTURAL AND PRESSURE VESSEL
OF SCWR
The concept of SCWR as shown in Fig. 2.2, would either be based on current pressure-
vessel or on pressure-tube reactors, and hence may use light water or heavy water as a
moderator. SCWR is being researched into in countries like Canada, China, EU, Japan,
Korea, Russia and US and out of all these countries, its only Canada that has a reactor
based on pressure-tube concept [3, 8]. The SCWR coolant will experience a higher
enthalpy rise in the core which will then reduce the core mass flow for a given thermal
power compared to the current LWR reactors. For both pressure-vessel and pressure-tube
designs, a once-through steam cycle has been envisaged, omitting any coolant recirculation
inside the reactor [18].
Fig 2.2: Conceptual Design of SCWR [19]
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The superheated steam will be supplied directly to the high pressure steam turbine and the
feed water from the steam cycle will be supplied back to the core. SCWR concepts combine
the design and operation experience gained from hundreds of water-cooled reactors and
the experience from hundreds of fossil-fired power plants operated with supercritical water.
In contrast to some of the other Generation IV nuclear systems, the SCWR can be
developed step-by-step from current water-cooled reactors [20, 21].
The design parameters of the SCWR considered for the research are given in Table 2.1 and
2.2
Table 2.1: SCWR reference design power and coolant conditions [6, 22-25].
Parameters Value
Thermal power 3,575 MWt
Net electric power 1,600 MWe
Net thermal efficiency 44.8 %
Operating pressure 25 MPa
Reactor inlet coolant temperature 280 ºC
Reactor outlet temperature 500 ºC
Plant lifetime 60 years
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Table 2.2: Reference reactor pressure vessel design for SCWR [6, 22-25].
Parameters Value
Type PWR with CRD
Height 12.40 m
Operating/design pressure 22.0/27.5 MPa
Operating/design temperature 280/371 ºC
Number of cold/hot nozzles 2/2
Inside diameter of shell 5.322 m
Thickness of shell 0.46 m
Inside diameter of head 5.352 m
Thickness of head 0.305 m
Peak fluence (> 1 MeV) < 5 x 1019 n/cm2
Core Structural materials considered SS304, SS308, SS309, SS316
*PWR-Pressurized Water Reactor, CRD-Control Rod Driven
2.3 MATERIAL CHALLENGES WITH SCWR
2.3.1. Material Needs of SCWR
Some of the material challenges associated with the SCWR are
Higher pressure combined with higher temperature and also a temperature rise
across the core result in increased mechanical and thermal stresses on the vessel
materials [5, 14].
Extensive material development (i.e. high irradiation and mechanical resistant
ones) and research on supercritical water chemistry under radiation are needed [8,
24, 26].
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The identification of appropriate materials for the pressure vessel and core structure, and
understanding of supercritical water (SCW) chemistry are two of the main challenges for
the development of SCWR [27, 28]. Zirconium-based alloys, may not be a viable material
without some sort of thermal and/or corrosion-resistant barrier [29]. Although there is
considerable experience with fast reactors and supercritical-water-cooled fossil fueled
plants (FFPs), little or no data on the in-flux behavior of these materials at the temperature
of 500 ºC and pressure of 25 MPa exists [25]. The understanding of the primary radiation
damage in Fe-based alloys is of interest for the use of advanced steels in future fusion and
fission reactors [30].
2.3.2. Candidate In-core Structural Materials of SCWR
Based on experiences from LWRs, fast reactors, and SWC Fossil Fire Plants (FFPs), Fe-
Ni-Cr austenitic stainless steels (e.g., 304, 316) with higher Cr contents, corrosion-resistant
ferritics (e.g., HT-9), and advanced ferritic/martensitic (e.g., 9 to 14% Cr), are being
considered as materials for core internal components [8]. Precipitation-hardened Ni-based
alloys (e.g., 718, 625) have also received attention for applications where dose rates are on
the lower end of the projected range.
In structures where temperatures will be significantly above 300 ºC, or irradiation dosses
above 30 dpa, candidate structural materials will be primary ferritic or martensitic steels
and low swelling austenitic stainless steels. Fe-Ni-Cr alloys with acceptable mechanical
behavior and dimensional stability are also possible candidates, though, there is currently
insufficient technical knowledge and data for predicting Fe-Ni-Cr alloy behavior under
supercritical water condition [9].
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2.3.3. Structural Integrity of Irradiated materials
Irradiation-induced changes to the cladding and structural materials especially the pressure
vessel due to swelling, helium-bubble formation and growth, and microstructure
precipitation are being investigated to overcome any compromise to the irradiation resistant
and mechanical properties of the components for the design life of the reactor [31-34]. Also
He segregation will be an important consideration because of the greater relative
production of He/dpa (displacement per atom) at thermal neutron energies. For
temperatures between 280 °C and 350 °C, the irradiation damage behavior for 304, 308,
309 and 316 Fe-Ni-Cr Alloys has been studied [35] since such materials have been used in
the existing Light Water Reactors (LWR). The viability of a SCWR will also depend on
mechanical behavior of both in-core and out-core materials.
2.4. IRRADIATION DAMAGE ASSESSMENT
2.4.1. Irradiation Damage Events and Mechanisms
Effects of radiation on solids have been studied extensively. Of much more interest to the
present research, however, are the high energy radiation fields in reactors. That the success
of reactor technology would depend critically on the choice of high-temperature material
with satisfactory neutronic properties was pointed out by Fermi in 1946 [37].
The radiation damage event occurs by transfer of energy from a high energy incident
particle to the solid and the resulting distribution of target atoms in the lattice after
completion of the event. The displacement of the host lattice atom in the Coulomb collision
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and the consequent production of point defects marks the final stage of the damage
sequence. The radiation damage event processes that occur are [38],
Transfer of kinetic energy to the lattice atom leading to a Primary-Knocked-on-
Atom (PKA);
Displacement of a primary-knocked-on atom (PKA) from its lattice site;
The passage of the displaced atom through the lattice and the accompanying
creation of additional knock-on atoms;
Production of a displacement cascade; and
Termination of the PKA as creating interstitials, vacancies, and Frankel pairs.
When energy transferred to a lattice atom is larger than the energy binding the atom in the
lattice site, the lattice atom is displaced from its original position. The displaced atom might
carry high enough kinetic energy to create a series of lattice displacements before
finally coming to rest. The displaced atom eventually appears in the lattice as an interstitial
atom leading to vacancies generations. Collection of point defects created by a single
primary knock-on atom is known as a displacement cascade [41-43].
Neutron irradiation mechanism begins with the unstable radionuclide atom given off
gamma (γ) rays and fissile particles such as alpha (α), beta (β), neutrons, ions, electrons,
and other fission products to materials [24]. Figure 2.3 shows that exposure of matter to
highly energetic radiations or particles results in changes in the physical, chemical,
biological or mechanical properties, which start from the microscopic state to the
macroscopic or observable state. In a nuclear reactor, thermal and fast neutrons are released
depending on the energy spectrum generated, in addition to alpha particles, beta particles,
gamma rays and other fission products.
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Fig. 2.3: Mechanisms of irradiation damage in the nuclear reactor system by thermal and
fast neutrons[12]
Gamma, beta, and alpha are classified as ionizing radiation because they interact only with
the electrons surrounding the nuclei of the material which normally destroys the atomic
bonding of the material and thereby causing damage [28] .But the damage in metals is
sometimes not significant since metals have a relative immunity to ionization radiation
unlike nonmetallic substances, such as water and other organic compounds [44].
Thermal neutrons are absorbed or captured as upon interaction with the nuclei of non-fuel
material, thereby leaving these nuclei in excitation or high energy state. The excess energy
(being the gamma radiation and the kinetic energy of the recoiled nuclei) is released by
emitting high energy gamma rays with the result that these emitting nuclei recoil [29]. If
the kinetic energy exceeds a certain minimum value called the displacement energy, which
is ranges from about 25 to 30 eV for most metals, then the recoiling (“knock-on”) atom is
displaced from its equilibrium position in the crystal lattice. The released high energy
Thermal Neutrons
Excited
Compound
Neutrons
Impurity
Atoms From
(n,p),(n,α)
Gamma Rays
Displaced
Atoms
(Interstitials
and vacancies)
Thermal
Spikes
Ionization and
Excitation
Energetic Recoil
Nuclei
Fast Neutrons
Dissipation
of energy in
the form of
heat
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gamma radiation however undergoes ionization and excitation and is converted into kinetic
energy of electrons or positrons which dissipates heat over a short distance. The excited
compound nuclei could also be transmuted to other nuclei which may be radioactive hence
leads to impurities deposition in a form of alpha particles (helium) and protons (hydrogen)
which are neutralized in the material of passage [45, 46].
Fast neutrons, due to their high energy undergo elastic collisions with the atomic nucleus
of the material resulting in production of alpha and beta particles and also transfer of their
kinetic energy to the recoil nucleus. The highly energetic recoil nucleus undergoing
ionization and excitation dissipates electron energy in a form of heat. Also due to the high
kinetic energy of the recoil nucleus, the recoiling atom (also known as primary knock-on
atom) could be displaced from its equilibrium position in the crystal lattice [46, 47]. Since
the PKA now possesses substantial kinetic energy, it becomes energetic particle in its own
right and it’s capable of creating additional lattice displacement which continues until the
displaced atom has insufficient energy to eject another atom. These subsequent generations
of displaced lattice atoms are known as secondary or higher order knock-ons. Finally when
the fast neutron is slowed down to the point where it can no longer cause atomic
displacement, much of its remaining energy will be dissipated within a short distance as
vibrational (heat) energy of the target atom. A thermal spike, in which high local
temperatures are attained, may then be formed [12, 48].
2.4.2. Rate of Production of Displacements ( dN E )
The rate of production of displacement, dN E is the number of vacancy and interstitial
pairs (Frankel pairs) produced per second by an incident particle of energy E per second
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of neutron radiation and is given as the product of effective/macroscopic displacement
cross section (Nσd(E) ) and neutron flux E [12, 15, 45, and 49]:
d dN E = N σ E E (2.1)
where dσ E is the microscopic displacement cross section for neutron radiation (i.e. an
incident particle) of energy E per second, E is the neutron flux of energy E and N is
the atom density of the target material in which the displacements occur.
The microscopic displacement cross section for neutron with energy E per second is
defined by [12, 46, 47- 48]:
m
d
T E
d 2Eσ E = v T σ E, T dT
(2.2)
where v T , σ E, T and mT E are related by the equations:
v(T) ≈ CT ≈
T
2Ed
(2.3)
sm
σ Eσ E, T = T E
(2.4)
m 2
4A 4T E = 1 - α E = E EAA+1
(2.5)
where, 𝛼 is a property of the scattering nucleus related to its mass and is defined by 𝛼 =
(
𝐴−1
𝐴+1
)
2
, and equation (2.5) is the maximum energy loss in a collision that can be transferred
to a knock-on atom by a neutron, v T is the mean number of displacements in a cascade
originating from the primary knock-on, T is the amount of energy transferred to the atom
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ejected from the lattice, C is the knock-on energy at which atom displacement is
terminated, σ E, T is the differential cross section (per unit energy) for the transfer of
kinetic energy T to a knock-on atomic in an elastic collision with energy of E, sσ E is
the elastic scattering cross section for the target material of neutron of energy E, mT E is
the maximum energy that can be transferred to a knock-on atom by a radiation particle of
energy E and A is the mass number of the target nucleus of the reactor core material
Substituting equations (2.3), (2.4) and (2.5) into (2.2) resulted in
m
d
T E s
d 2E d m
σ ETσ E = . dT2E T E
(2.6)
m
d
T Es
d 2Ed m
σ Eσ E = T dT2E . T E
(2.7)
Integrating equation (2.7) from d m2E to T E leads to
d s
d
Eσ E σ E . AE
(2.8)
Hence, substituting equation (2.8) into equation (2.1) became
d s
d
EN E N E . σ E . AE
(2.9)
Equation (2.9) indicates that rate of production of atomic displacement defects produced
in a nuclear material exposed to a constant neutron flux with time which has a relation with
atomic density of the target material, constant neutron flux, and total elastic scattering cross
section of the target material, neutron energy (E), atomic number of the target material and
threshold displacement energy [50].
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The fluence is the product of the constant neutron flux and the exposure time Ф (E). t.
Multiplying equation (2.9) by time resulted in the total number of displaced atom as:
sd
d
σ EN E . t N E . t . EAE
(2.10)
The number of displacements per atom (dpa) is given by [49,51]
d s
d
N E . t σ E dpa = E . t . EN AE
(2.11)
where E . t is the fluence of neutron radiation measured in neutrons/cm
2.
2.4.3. Kinchin-Pease Model of Radiation Damage
The Kinchin – Pease Model assumes that between a specified threshold energy and an
upper energy cut-off, there is a linear relationship between the number of Frenkel pair
produced and the PKA energy. Below the threshold, no new displacements would be
produced. Above the high energy cut-off, it was assumed that the additional energy was
dissipated in electronic excitation and Ionization [15, 31, and 52].
2.4.3.1 Displacement probability:
The Displacement probability is defined as the probability that a struck atom is displaced
upon receipt of energy T. This is due to exchange in energy during a Coulomb collision
between an electron and a lattice nucleus of the target material. The simplest model for the
displacement probability is a step function, with a sharp displacement energy value Ed,
shown in Figure 2.4 (a) below and expressed by [15, 46, and 48]:
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Pd(T) = {
0 for T < Ed
.
1 for T ≥ Ed
(2.12)
This model is constant for all collisions because it neglects the thermal atomic vibration of
the lattice, which introduces a width of the order of kT in the displacement probability.
A more accurate model is shown by a function in which the energy threshold is not sharp,
but it goes from 0 to 1 with a smooth curve, as it is shown in Figure 2.2b. The corresponding
mathematical formulation [15, 48] and where f T a function varying smoothly between
[0, 1]:
min
d min max
max
0 for T < E
P T f T for E < T < E
1 for T E
(2.13)
Fig. 2.4: The displacement probability Pd (T) as function of the transferred kinetic energy,
assuming (a) a sharp or (b) a smoothly varying displacement threshold [31, 48]
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2.4.3.2 Displacement Energy
A lattice atom must receive a minimum amount of energy in the collision in order to be
displaced. The struck lattice atom of energy T, is referred to as a primary knock-on atom
(PKA), displacement energy Ed, threshold displacement energy or displacement threshold
energy [53]. The magnitude of Ed is dependent upon the crystallographic structure of the
lattice, the direction of the incident PKA, the thermal energy of the lattice atom, etc. If the
energy transferred, T is less than Ed, the struck atom will vibrate about its equilibrium
position but will not be displaced. These vibrations diffuse through the lattice and
transform the absorbed kinetic energy into heat. On the contrary, if
dT E the PKA starts
travelling into the solid structure with kinetic energy
dT E [32]. Maximum kinetic
energy transferred in an elastic collision of particle is given by:
Max 2
4MmT = E m + M
(2.14)
Where E is kinetic energy, M is mass of the incident particle and m is mass of the material
atom (target atom)
2.4.3.3 Radiation Damage
The theoretical basis of calculating the total number of displaced atoms resulting from a
single PKA of energy E is now considered. The number of displaced atoms is denoted by
v E . Simplest theory of displacement cascade is that of Kinchin-Pease [31, 32, and 52]
based on the assumptions that:
1. The cascade is created by a sequence of two-body elastic collisions between atoms
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2. The displacement probability is 1 for
dT>E as given by equation (2.12)
3. When an atom with initial energy T emerges from a collision with energyT and
generates a new recoil with energy ε , it is assumed that no energy passes to the
lattice andT=T +ε
4. Energy loss by electron stopping is treated by the cut-off energy 3cE ~10 eV . If the
PKA energy is greater that
cE no additional displacements occur until electronic
energy losses reduce the PKA energy to
cE . For all energies less than cE electronic
stopping is ignored and only atomic collisions take place
5. The energy transfer cross section is given by the hard sphere model
6. The arrangement of the atoms in the solid is random effects due to the crystal
structure are neglected
The cascade is initiated by a single PKA of energy T, which eventually produces v T
displaced atoms. If PKA of energy E transfer energy T to the struck atom and leaves the
collision with energy dεE , the PKA has residual energyT-ε , so that:
dv T = v T - ε v ε E (2.15)
where
dE is the energy consumed in the reaction. If ε ≫ Ed according to assumption 3, then
equation (2.15) simplifies to:
v T = v T - ε v ε (2.16)
Equation (2.16) is not sufficient to determine v T because the energy transfer ε may lie
anywhere between 0 and T. However, if we know the probability of transferring energy in
the range ε ,dε in a collision then multiplying equation (2.16) by this probability and
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integrate over all allowable values of ε will yield the average number of displacements,
𝑣(T).
By Kinchin and Pease model of the displacement, the energy transfer cross section 𝜎, by
hard-sphere assumption 5 is [48],
σ T σ Tσ T,ε = = γT T
(for like atoms, 𝛾 = 1) (2.17)
The probability of a PKA energy T transfers energy in range ε ,dε to the struck atom is
[15, 48]:
p̂ =
σ T,ε dε dε = σ T T
(2.18)
Hence, the average number of displacement, given by
v̂ (T) = ∫ v T
𝑇
0
x p̂ = ∫
v(T)
T
dε
T
0
(2.19)
v̂(T) =
1
T
∫ [v(T − ε) + v(ε)]dε
T
0
(2.20)
v̂(T) =
1
T
[∫ v(T − ε)dε + ∫ v(ε)dε
T
0
T
0
] (2.21)
Setting εε′ = T − ε, then dT − d𝜀 = dε′ in the first integral and equation (2.21) became
v̂(T) =
1
T
∫ v(ε′)dε′ +
1
T
∫ v(ε)dε
T
0
T
0
(2.22)
v̂(T) =
2
T
∫ v(ε
T
0
)dε since εε′ (2.23)
By examining the following relations
v̂(T) = 0 for 0 < T < Ed (2.24)
v̂(T) = 1 for 0 < T < 2Ed (2.25)
For if
dT<E there will be no displacements; if dT>E and dT<2E , two outcomes are
possible. The first is that, the struck atom is displaced from its lattice site and the PKA now
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leaves with energy less than
dE , falls into its place. However, if the original PKA does not
transfer
dE , the struck atom remains in place and no displacement occurs. In either case,
only one displacement in total is possible from a PKA with energy between
dE and d2E .
By splitting and integrating equation (2.23) with equations (2.24) and (2.25) as limits and
evaluate:
v̂(T) =
2
T
[∫ (0)dε + ∫ (1)dε + ∫ v(ε)dε
T
2Ed
2Ed
Ed
Ed
0
] (2.26)
then
v̂(T) =
2Ed
T
+
2
T
∫ v(ε)dε
T
2Ed
(2.27)
v̂(T) =
2
T
[0 + Ed + ∫ v(ε)dε
T
2Ed
] (2.28)
Multipling Equation (2.28) by T and then differentiating with respect to T
v̂(T) x T = 2Ed + 2∫ v(ε)dε
T
2Ed
(2.29)
and
d
dT
(v̂(T) x T) = 2
d
dT
∫ v(ε)dε
T
2Ed
(2.30)
Since T = ε + ε′ =≫ T ≈ ε and dT = dε for ε′ ≪ ε
d
dT
(v̂(T) x T) = 2
d
dT
∫ v(T)dT
T
2Ed
(2.31)
and
v̂(T) + T
dv̂(T)
dT
= 2 v(T) (2.32)
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implying:
T
dv(T)
dT(T)
= v(T) (2.33)
or:
dv(T)
v(T)
=
dT
T
(2.34)
with solution:
lnv(T) = lnT + lnC = ln(CT) (2.35)
v̂(T) = CT (2.36)
where value of C, obtained by putting equation (2.36) into (2.27)
𝐶𝑇 =
2Ed
T
+
2
T
∫ (CT) dT
T
2Ed
=
2Ed
T
+
2C
T
⌈
𝑇2
2
⌉
2𝐸𝑑
𝑇
(2.37)
and:
𝐶𝑇 =
2Ed
T
+
2C
T
[
𝑇2
2
−
4𝐸𝑑
2
2
] =
2Ed
T
+ CT −
4𝐶𝐸𝑑
2
T
(2.38)
or:
4𝐶𝐸𝑑
2
T
=
2Ed
T
(2.39)
Therefore:
C =
1
2Ed
(2.40)
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and therefore substituting (2.40) into (2.36) gives:
v̂(T) =
T
2Ed
for
d c2E <T<E (2.41)
By assumption 4, the upper limit is set by
cE . When a PKA is born with cT>E , the number
of displacements is v̂(T) = Ec 2Ed⁄ for cT>E . Hence the full Kinchin – Pease (K – P)
model equations are
v̂(T) =
{
0 ; for T < Ed
1 ; for Ed < T < 2Ed
T
2Ed
; for 2Ed < T < Ec
Ec
2Ed
; for T > Ec
(2.42)
The graphical description of the average number of displacements,v1(T) of equation
(2.30) is shown in Fig. 2.5.
Fig. 2.5: Average number of displacements 𝑣(T) produced by a PKA, as a function of the
recoil energy T according to the model of Kinchin – Pease [15, 48 ].
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The energy ranges are explainable as follows. First of all, there can be no displacement if
the recoil energy T is lower than
dE , because of the sharp displacement threshold
assumption 5. Secondly, when
d dE <T<2E , displacement occurs and a PKA starts
travelling through the lattice of the vessel, but with kinetic energy dT - E , since dE was
necessary to overwhelm the lattice potential. Therefore, in this energy range the PKA does
not have enough energy to produce further displacements. Above
d2E the behavior is
linear, until the electron energy loss limit
cE is reached.
2.4.4. Norgett-Robinson-Torrens Radiation Damage Model
The Norgett-Robinson-Torrens (NRT) model, was developed as secondary displacement
model, (also known as modified K-PM) for computing the number of displacements per
atom (dpa) for a PKA with a given energy. The NRT model was broadly adopted by the
International Radiation Effects Community and continues to be the internationally-
recognized standard method for computing atomic displacement rate [31, 32, and 48].
The NRT displacement model gives the total number of stable Frenkel pair produced by a
PKA with kinetic energy E as [34, 41]:
e dam
dam d
d d
NRT
d dam d
dam d
k T-E kT
= , T 2E
2E 2E
v =
1, E <T 2E
0, 0<T E
(2.43)
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where
NRTv , k, T, eE , dE , and damT are the number of displaced atoms produced by PKA,
damage efficiency, recoil energy of a PKA, total energy lost by electron excitation,
threshold displacement energy and damage energy available for elastic collisions
respectively. The damage efficiency k holds a value of 0.8 and accounts for the fact that
not all collisions are for ideal hard spheres [48, 49, and 54].
If
dam dT < E there will be no damage energy to cause displacements and also if dam dT > E and
dam dT < 2E , two effects are possible as shown in equation (2.43).
2.4.5. Radiation Damage Defects
The radiation damage event takes a very short time of the order of 10−11 s, and produces a
collection of point defects (i.e. vacancies and interstitials) and the formation of defect
clusters. Figure 2.6 shows defects in the lattice structure of materials that can change the
material properties and can affect the performance [55, 56].
The Radiation Damage Event may be followed by a cluster of phenomena which can be
classified as radiation damage effects:
1. migration of point defects and clusters, according to their mobility inside the crystal
structure, with possible growth of the cluster or recombination of a vacancy-
interstitial couple
2. modification of the material composition due to a varied mobility of impurities
contained in alloys
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3. possible degradation of structural properties and hindering of the component
purposes.
Fig 2.6: Radiation damage defects creation processes [55]
Irradiation as shown in Fig. 2.6 produces interstitial and vacancy point defects in the lattice
shown in Fig. 2.7 which often leads to changes in properties of the material. Interstitial and
vacancy are point defects which often have high mobility and readily diffuse through the
crystal lattice, and the freely-migrating point defects combine to form higher-order point
defect complexes. This point defect condensation process progresses continuously into a
nucleation and growth process of extended defect clusters. Interstitial clusters become
interstitial loops. Vacancy clusters become either vacancy loops or voids. This
microstructural evolution leads to property changes such as embrittlement and macroscopic
swelling and hence effects on the performance of the nuclear materials. Also, changes in
properties are proportional to radiation flux, particle energy, irradiation time and
temperature.
Diffusion of
vacancies (V),
interstitials (I)
and Solutes/
Impurities
Particle-atom
Collision
(displacement
production in
cascade)
V/I clustering:
formation of
(bubble, void,
loop)
V/I absorption at
sinks (e.g.
dislocations,
clusters
V-I
Recombination
Damage
Annihilation
Damage
accumulation
(dimensional
and
mechanical
property
changes
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Fig. 2.7: Defects in the lattice structure of materials [57]
2.5. MECHANICAL PROPERTIES OF STRUCTURAL MATERIALS
A nuclear reactor system must include a containing pressure vessel for the core infrastructure,
mechanical support for the core components, piping for the coolant, and cladding for the fuel
elements. The requirements for these materials to serve the different purposes will vary with
the reactor type, but some general characteristics may be noted, Mechanical properties, such
as tensile strength, yield strength, ductility, impact strength, fatigue and creep, must be
adequate for the operating conditions [48].
The choice of material for fast reactors is less dependent on neutron absorption cross sections
than for thermal reactors. However, the four elements with low thermal-neutron absorption
cross sections (less than 0.24 b) and reasonably high, melting points; are aluminum, beryllium,
magnesium, and zirconium. Of these, aluminum, magnesium, and zirconium have been utilized
in fuel-element cladding, while beryllium has been used for reflectors Although the major
constituents of stainless steels, namely, iron, chromium, and nickel, have relatively high
thermal-neutron absorption cross section of 2.6, 3.1 and 4.4 b, respectively, these steels also
have good mechanical properties and are resistant to corrosion by water at temperatures more
than 300 ºC. Consequently, stainless steels are commonly used for structural components in
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water-cooled power reactors. They are also utilized in fuel element cladding as well as
structural material in fast reactors. Pressure vessels are made of low-alloy, carbon steels lined
with corrosion resistant alloy. There are several potential uses for nickel-base alloys. The
properties of these stainless steel materials are described below [12].
2.5.1. Stainless Steels alloys for pressure vessels and in-core structural materials
Stainless steel contains chromium which ensures resistant to corrosion, tarnishing and rust.
Stainless steels vary widely in composition and are classified according to the metallurgical
phase produced on solidification of the metal and there are more than 250 different stainless
steels [58]. The main grades of stainless are divided into four major groups/classes namely
austenitic, ferritic, martensitic and duplex (austenite and ferrite). The high corrosion resistance
of these steels is derived from the ability of the alloy to form a protective, self-repairing oxide
film, subject to the availability of oxygen in the environments surrounding the alloy [59].
Stainless steels possess strength and toughness at both extremes of the temperature scale, yet
can be fabricated into intricate shapes for many uses [60]. The corrosion resistance of stainless
steel is the result of the addition of minimum 11% Cr. In addition, steels with the minimum Cr
content and too high carbon content (≥ 0.03%) are susceptible to precipitation when the metal
is welded or heat treated in the temperature range 500–800 ºC [58].
2.5.2. Austenitic Stainless Steel
Austenitic stainless steels have excellent formability, corrosion resistance and also offer rather
easy processing as well as good forming ability and corrosion with a great structure stability,
which allows their use in a large temperature range. They are not tempered but cold-work
strengthened and has a good long term mechanical behaviour. Austenitic steels contain 15 %
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to 30 % chromium, 2 % to 20 % nickel and the rest is Iron (for enhanced surface quality,
formability and increased corrosion and wear resistance) [60, 61].
2.5.3. Effect of Radiation Damage on Mechanical Properties of Stainless Steel
Stainless steels are selected for corrosion resistance, high-temperature strength, ductility
and toughness as illustrated in Table 2.3.
Table 2.3: Material property and their Damage effects on Microstructure [12, 62]
STRUCTURAL AND
MECHANICAL
RADIATION DAMAGE EFFECTS
Hardness and strength Increases in radiation hardening due to increase in
formation and growth of defects cluster and mainly
dislocation loops
Ductility Decreases on radiation embrittlement
Creep Enhances on radiation-induced and radiation-enhanced
due to increases defects and diffusion-rates
Fatigue Low cycle fatigue life decreases due to embrittlement;
High cycle fatigue life increases due to hardening
Density Decreases in swelling on irradiation due to formation
and growth of voids and gas bubbles, cavities, depleted
zones
Electrical resistivity Increases on irradiation due to increased defect
concentration
Conductivity Decreases on irradiation due to increased defects
concentration
Thermal conductivity Decreases on irradiation due to increased defects
concentrations
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2.5.4. Mechanical Strength Parameters
Mechanical testing plays an important role in evaluating fundamental properties of engineering
materials as well as in developing new materials and in controlling the quality of materials for
use in design and construction. If a material is to be used as part of an engineering structure
such as the nuclear reactor that will be subjected to a load (i.e thermal loading), it is important
to know the material’s strength to aid in deciding on the part of the reactor that such material
could function very well, since different parts of the reactor are subjected to different loads.
The most common type of test used to measure the mechanical properties of a material is the
Tension Test. Tension test is widely used to provide a basic design information on the strength
of materials and is an acceptance test for the specification of materials. The major parameters
that describe the stress-strain curve obtained during tension test are the tensile strength (UTS),
yield strength or yield point (YS), Young’s or Elastic modulus (E), percent elongation (ΔL %)
and the Breaking or fracture Strength. Bulk Modulus can also be found by the use of this testing
technique.[63] The parameters are illustrated in σ - ɛ plot of Figure 2.8.
Figure 2.8: A typical Stress-Strain Curve for Fe-Ni-Cr Alloys [63]
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2.5.4.1 Young’s Modulus
Young’s Modulus also known as elastic Modulus or modulus of elasticity is the measure
of the stiffness of an elastic material and is a quantity used to characterize materials. It is
defined as a ratio of the stress (force per unit area) along an axis to strain (ratio of the
deformation over initial length) along that axis in the range of stress in which Hooke’s law
holds. It is therefore the slope of the part of the strain- stress curve that obeys the Hooke’s
Law [64].
Young’s Modulus is mathematically expressed as:
E ≡
Tensile Stress
Extensional Strain
=
σ
ε
=
F
A0
⁄
∆L
L0
⁄
=
FL0
A0∆L
(2.44)
where E is the Young’s modulus (modulus of elasticity), F the force exerted on the metal
under tension, 𝐴0 the original cross-sectional area through which the force is applied, ∆𝐿
the amount by which the length of the object changes, and 𝐿0 the original length of the
material.
2.5.4.2 Tensile Strength
The tensile strength which normally called ultimate tensile strength is the stress obtained
at the highest applied force and it is the maximum stress on the engineering stress – strain
curve. In many ductile materials such as the Face Centered Cubic (FCC) material under
consideration, deformation does not remain uniform. At some point, one region deforms
more than other areas and a large local decrease in the cross-sectional area occurs. It is also
the point at which necking begins. It is useful in comparing materials and it permit the
estimation of other properties, which are more difficult to measure. [64]
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2.5.4.2 Fracture or Breaking Strength
Fracture strength, also known as breaking strength, is the engineering stress at which a
material fails through fracture. This is usually determined for a given specimen by a tensile
test, which charts the stress-strain curve (see Fig. 2.8). The final recorded point is the
fracture strength.
Ductile materials have a fracture strength lower than the ultimate tensile strength (UTS),
whereas in brittle materials the fracture strength is equivalent to the UTS. If a ductile
material reaches its ultimate tensile strength in a load-controlled situation, it will continue
to deform, with no additional load application, until it ruptures. However, if the loading is
displacement-controlled, the deformation of the material may relieve the load, preventing
rupture. [64]
2.5.4.4 Yield Strength
Yield strength is an indication of maximum stress a material can take without a plastic
deformation. Beyond the yield point of the material, it exhibits a specified permanent
deformation and is practically expressed as an elastic limit approximation [63].
Yield is very important in engineering structural design such as component that must
support loads in operation so that the component will not deform plastically. Therefore,
materials with sufficient yield strength are normally selected for design purposes [8].
In reactor design or any design applications, the yield strength is often used as an upper
limit for allowable stress that can be applied so that that material will work with a precise
dimensional tolerance. Hence, materials without clear distinct yield point, yield strength is
usually stated as stress at which permanent deformation of 0.2 % of the original dimension
will result (usually termed as 0.2 % offset).
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2.5 COMPUTER SIMULATIONS OF IRRADIATION DAMAGE AND
INDUCED MECHANICAL DEGRADATION
Three principal methods are used to simulate the behavior of atoms in a displacement
material. These are Binary Collision Approximation (BCA) Method, Molecular Dynamics
(MD) Method and the Kinetic Monte Carlo (KMC) Method [48].
2.6.1.1 Binary Collision Approximation (BCA) Method
The binary collision approximation is a method used in ion irradiation physics to enable
efficient computer simulation of the penetration depth and defect production by energetic
ions (>keV) ions in solids. [65].
A binary collision between a projectile and a target atom is calculated in the BCA
simulation and the final position and velocity of the projectile and the target atom at each
collision are obtained analytically in a two-body interatomic potential V(r), expressed as:
𝑉(𝑟) =
𝑍1𝑍2𝑒
2
𝑟
Ф(
𝑟
𝑎
) (2.45)
where 𝑍1 and 𝑍2 are the atomic numbers of the energetic ion and the target atoms, e is the
electric charge, r the distance and a is an empirical screening length which depends on the
atomic numbers of the two atoms by the semi-empirical formula[49]:
𝒂 =
𝟎.𝟖𝟖𝟓𝟒𝒂𝑩𝒐𝒉𝒓
𝒁𝟏
𝟎.𝟐𝟑+𝒁𝟐
𝟎.𝟐𝟑 (2.46)
where 𝑎𝐵𝑜ℎ𝑟 is the Bohr radius (the radius of the hydrogyen atom 0.53Å). Ф is the
“universal” screening function determined by exact fitting formula of the calculated
interatomic potentials of 521 randomly selected element combinations given by:
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Ф[
𝑟
𝑎
] = ∑ 𝐴𝑖
4
𝑖=1 𝑒𝑥𝑝 [−𝐵𝑖 (
𝑟
𝑎
)] (2.47)
In this research, BCA simulation was implemented using SRIM-TRIM code and the
Moliere approximation to the Thomas-Fermi potential [66] was employed. Figure 2.9
shows the trajectory of two particles interacting according to a conservative central
repulsive force.
Fig 2.9: The trajectory of two particles interacting according to a conservative central repulsive
force in the laboratory system showing the positions of the projectile and the target atom
correspond to the apsis of the collision [67].
The scattering angle in the center-of-mass system (CM-system) is
𝛩𝐶𝑀 = 𝜋 − 2𝑏 ∫
1
𝑟2𝑔(𝑟)
𝑑𝑟,
∞
𝑟0
(2.48)
𝑔(𝑟) = √(1 −
𝑏2
𝑟2
−
𝑉(𝑟)
𝐸𝑟
) (2.49)
b is the impact parameter, 𝐸𝑟 = 𝐸0 𝑚1 / (𝑚1+𝑚2) is the relative kinetic energy, 𝐸0 is the
incident kinetic energy of the projectile, r0 is the solution of g(r) = 0, 𝑚1 and 𝑚2 are the
mass of the projectile and the target atom, respectively. The trajectories of particles are
approximated as the asymptotes of them in the laboratory system (L-system). So they
consist of linkage of straight-line segments. The starting point of the projectile and the
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35
recoil atom after a collision is given by ∆𝑥1 and ∆𝑥2, which are the shifts from the initial
position of the target atom shown in Fig. 2.9:
∆𝑥1 =
2𝜏+(𝐴−1)𝑏𝑡𝑎𝑛(𝛩 2⁄ )
1+𝐴
(2.50)
∆𝑥2 = 𝑏𝑡𝑎𝑛(
𝛩
2⁄ ) − ∆𝑥1 (2.51)
where 𝜏 = √𝑟02 − 𝑏2 − ∫ {
1
𝑔(𝑟)
−
𝑟
√𝑟2−𝑏2
}
∞
𝑟0
𝑑𝑟 (2.52)
and the mass ratio A = m2 / m1; r is the atomic distance
The SRIM-TRIM code evaluates the energy loss by electron excitation for each collision.
The four-parameter fitting formula (eqn 2.50) for hydrogen of the electronic stopping cross
section which was originally proposed by Varelas and Biersack [68, 69] was employed.
𝑆𝑒
𝐻(𝐸)−1 = (𝑆𝐿𝑂𝑊
𝐻 )−1 + (𝑆𝐻𝐼𝐺𝐻
𝐻 )−1 (2.53)
where
𝑆𝐿𝑂𝑊
𝐻 = 𝐴1𝐸
0.45 (2.54)
𝑆𝐻𝐼𝐺𝐻
𝐻 =
𝐴2
𝐸
ln [1 +
𝐴3
𝐸
+ 𝐴4] (2.55)
and four parameters A1, A2, A3, and A4 are derived from fitting experimental data [68]. The
symbol S in Equation (2.53-2.55) is the Bethe's stopping-power formula [33, 70].
The BCA approach provides a good approximation to the collision stage, since the
neglected many-body interactions make little contribution to the atom trajectories at
collision energies well above the atom displacement energy. At energies near or even less
than the displacement energy, ballistic features of cascades such as replacement-collision
sequences and focused-collision sequences can be reasonably captured by BCA
calculations. At primary recoil energies above approximately 20 keV, cascades may have
more than one damage region. Because the mean free path between high-energy collisions
of a recoil atom increases with energy, higher energy cascades will consist of multiple
damage regions or sub-cascades that are well separated in space due to high-energy
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collisions. Channeling of primary or high-energy secondary recoils also contributes to sub-
cascade formation when the channeled recoils lose energy and de-channel.
2.6.2. Molecular Dynamics (MD) Method
MD simulation is based on Newton’s second law of motion with total force on an N-atom
system as
𝐹(𝑟1, 𝑟2, … . 𝑟𝑁) = ∑ 𝑚𝑖𝑎𝑖𝑖 = ∑ 𝑚𝑖
𝑑2𝑟
𝑑𝑡2𝑖
(2.56)
where 𝑭𝒊 is the force exerted on the particle 𝑖, 𝑚𝑖 is the mass of particle 𝑖, 𝑎𝑖 is the
acceleration of particle 𝑖, 𝑟𝑖 is position and t is time-step.[71]
MD simulation generally proceeds as
Given the initial positions and velocities of every atom, and using the provided
interatomic potential, the forces on each atom were calculated.
Using the information gathered from above, the initial positions are advanced
toward lower energy states through a small time interval (called a time-step,∆𝑡),
resulting in new positions, velocities, etc.
With these new data as inputs, the above steps are repeated, for more than thousands
of such time-steps until an equilibrium was reached, and the system properties do
not change with time.
During and after equilibration, various raw data are stored for each or some time-steps that
include atomic properties, energies, forces, etc. Properties that are calculated directly or
via statistical analysis from these data are
Basic Energetics, structural and mechanical properties (Note that some of these data
are used to fit the potentials empirically,)
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Thermal expansion coefficient, melting point, and phase diagram in terms of
pressure and volume [71].
The simulation flow diagram of the MD run is as shown in Fig. 2.10;
Fig. 2.10: Molecular Dynamics Simulation flow chart [71]
MD is a computationally intensive method for modelling atomic systems on the appropriate
scale for the simulation of displacement cascades and provides a realistic description of
atomic interactions in cascades [72]. Molecular Modeling is concerned with the description
of the atomic and molecular interactions that govern microscopic and macroscopic
behaviors of physical systems [73]. Using realistic interatomic potentials and appropriate
boundary conditions, the fate of all atoms in a volume containing the cascade can be
described through the various stages of cascade development. The analytical interatomic
potential functions describe the force on an atom as a function of the distance between it
and the other atoms in the system. It account for both attractive and repulsive forces in
order to obtain stable lattice configurations.
Static and Dynamic Properties such as temperature, stress, strain etc
PBC, 𝑟𝑐𝑢𝑡, 𝑟𝑛𝑒𝑖, ∆𝑡
Initialization
Initial Position
Initial Velocities
Structure
Integration/Equilibration
Force Calculations
Solutions for 𝑭 = 𝒎𝒂
𝑡 → 𝑡 + ∆𝑡
Data Production
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In MD simulations as shown in Figure 2.11, the total energy of the system of atoms being
simulated is calculated by summing over all the atoms. The forces on the atoms are used
to calculate acceleration according to 𝐹 = 𝑚𝑎, yielding the equations of motions for the
atoms. The computer code solves these equations numerically over very small time steps,
and then recalculates the forces at the end of the time step, to be applied in the calculations
in the next time step. The process is repeated until the desired state is achieved
Fig 2.11: Molecular Dynamics Simulation of a unit cell of the material [74]
Time steps in MD simulation must be very small (5.0 – 10.0 ×10−15 s), so MD simulations
are generally run for not more than 100 ps. As the initial primary kinetic energy E increases,
larger and larger numerical crystallites are required to contain the event. The size of the
crystallite is roughly proportional to E, and E2 is the required computing time-scales. The
demand on computing time limits the statistical capabilities of MD simulation, which
provides a detailed view of the spatial extent of the damage process on an atomic level that
is not afforded by other simulation method. A cascade simulation begins by thermally
equilibrating a block of atoms that constitutes the system being studied, and the process
allows the determination of the lattice vibrations for the simulated temperature. Next, the
cascade simulation is initiated by giving one of the atoms a specified amount of kinetic
energy and an initial direction. Several cascades must be run in order to obtain results that
can be used to represent the average behaviour of the system at any energy and temperature
[72].
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2.6.3. Kinetic Monte Carlo (KMC) Method
Kinetic Monte Carlo attempts to overcome the limitation of time-scale problem associated
with MD by exploiting the fact that the long-time dynamics of this kind of system typically
consists of diffusive jumps from state to state. Rather than following the trajectory through
every vibrational period, these state-to-state transitions are treated directly. Given a set of
rate constants connecting states of a system, KMC offers a way to propagate dynamically
correct trajectories through the state space. If the rate catalog is constructed properly, the
easily implemented KMC dynamics can give exact state-to-state evolution of the system,
in the sense that it will be statistically indistinguishable from a long term molecular
dynamics simulation. KMC is the most powerful approach available for making dynamical
predictions at the mesoscale timestep without resorting to model assumptions. It can also
be used to provide input to and/or verification for higher-level treatments such as rate
theory models [34, 75].
2.6.4. Computer Simulation Codes
2.6.4.1 SRIM – TRIM Code
TRIM is the acronym of Transport of Ions in Matter, while SRIM is stopping and range of
ions in matter [45]. TRIM is a BCA code that uses Monte Carlo techniques to describe the
trajectory of the incident particle and the damage created by that particle in amorphous
solids and has all the properties or theoretical background of the BCA method. TRIM uses
a maximum impact parameter set by the density of the medium and a constant mean free
path between collisions which is related to this. Stochastic methods are used to select the
impact parameter for each collision and to determine the scattering plane. ALICE, PHITS,
MARS etc. are other BCA codes which are used for damage assessments.
TRIM is a Monte Carlo simulation code embedded in SRIM code and widely used to
compute a number of parameters relevant to radiation damage exposure calculation to
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nuclear structural material and has the capability to compute a common radiation damage
exposure unit known as the atomic displacements per atom (dpa) and other defects,
interstitials and vacancies.. TRIM code can calculate the stopping and range of ions from
10 eV to 2 GeV into any kind of matter using a quantum mechanical treatment of ion-atom
collision [75].
The moving atom is referred to as an "ion", and all target atoms as "atoms". The calculation
is made very efficient by the use of statistical algorithms which allow the ion to make jumps
between calculated collisions, and then averaging the collision results over the intervening
gap. During the collisions, the ion and atom have a screened Coulomb collision, including
exchange and correlation interactions between the overlapping electron shells. The ion has
long range interactions creating electron excitations and Plasmon within the target. These
are described by including a description of the target's collective electronic structure and
interatomic bond structure when the calculation is setup [76].
SRIM is a group of computer programs which calculate interaction of ions with matter and
it consists of two main program modules and several programs for specialized tasks. The
core modules are the Tables of Stopping and Ranges and Monte Carlo Transport
Calculation. The code also contains tables and plots concerning experimentally determined
ranges for the most common materials [77].
The TRIM window is used to input data on the ion, target layers and the type of TRIM
calculation that is needed. The output lists or plots the following:
Three-dimensional distribution of the ions in the solid and its parameters, such as
penetration depth, spread along the ion beam (called straggle) and perpendicular to
it, all target atom cascades in the target are followed in detail concentration of
vacancies, sputtering rate, ionization and phonon production in the target material.
Energy partitioning between the nuclear and electron losses, energy deposition rate.
The programs can be interrupted at any time, and then resumed later, have a very
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easy-to-use user interface and built-in default parameters for all ions and materials.
These features have made SRIM immensely popular. However, it doesn't take
account of the crystal structure nor dynamic composition changes in the material
that severely limits its usefulness in some cases.
Other approximation of the program include:
The electronic stopping power:- an averaging fit to the large number of
experiments.
The interatomic potential:- a universal form which is an averaging fit to the
quantum mechanical calculations.
The target atom:- which reaches the surface and then causes surface sputtering if it
has momentum and energy to pass the surface barrier. The system is layered, i.e.
simulation of materials with composition differences in 2D or 3D is not possible
[77, 78].
2.6.4.2 Large-Scale Atomic/Molecular Massively Parallel Simulator (LAMMPS)
LAMMPS is a parallel general purpose particle simulation code developed at Sandia
National Laboratories, USA, with contributions from many labs throughout the world [74].
LAMMPS integrates Newton’s equations of motion for collections of atoms, molecules, or
macroscopic particles that interact via short or long range forces with a variety of boundary
conditions. LAMMPS performs structural optimization of the atomic positions and cell
parameters as well as molecular dynamics calculations. Applications are manifold, and
researchers use LAMMPS to predict a variety of phenomena and properties, such as
diffusion of molecules in polymer matrices, solubility parameters and miscibility, surface
adhesion, viscosity, friction, density etc. for both inorganic and organic systems. LAMMPS
runs efficiently on single-processor desktop or laptop machines, but is also designed for
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parallel computers, hence it can model systems with only a few particles up to millions or
billions [79, 80]. Other codes aside LAMMPS for evaluation of mechanical damage are
AMBER and CHARMM.
2.6.4.3 Visual Molecular Dynamics (VMD)
Visual Molecular Dynamics is a code designed to visualize dump or output files from
LAMMPS and is useful in visualizing the stresses surrounding each atom. VMD can also
be used to animate and analyze the trajectory of a molecular dynamics (MD) simulation.
In particular, VMD can act as a graphical front end for an external MD program by
displaying and animating a molecule undergoing simulation on a remote computer. This
molecular graphics program is designed for interactive visualization and analysis and runs
on all major Unix workstations, Apple MacOS X, and Microsoft Windows [81].
VMD provides a wide variety of methods for rendering and coloring a molecule and can
act as a graphical front end for an external MD program by displaying and animating a
molecule undergoing simulation on a remote computer [82]. The Ovito code is also a
visualization code for MD simulations.
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CHAPTER THREE: RESEARCH METHODOLOGY
In this chapter, the research methodologies adopted in assessing the effects of neutron
irradiation on microstructural damage and mechanical degradation of grades of Fe-Ni-Cr
stainless steels which might be candidate alloys for SCWR Pressure Vessel design.
Simulations of irradiation damage in the steels were implemented by SRIM–TRIM Code,
whereas the mechanical degradation was simulated using LAMMPS along with the VMD
and MATLAB.
3.1. SELECTION OF Fe-Ni-Cr ALLOYS
3.1.1 Structural and In-Core Reactor Materials
The qualification of materials for the SCWR design has been one of the key challenges
associated with the design and development of the nuclear power plant.
In deciding on the optimum materials for the design of SCWR structures where the
temperatures will be significantly above 300 °C, or irradiation doses 10 – 150 dpa,
candidate structural materials might be primarily ferritic or martensitic steels and low
swelling austenitic stainless steels [17]. For Fe-Cr-Ni alloys acceptable mechanical
behaviour and dimensional stability is also possible though there is currently an insufficient
knowledge base for predicting Stress Corrosion Cracking (SCC) or Irradiation Assisted
Stress Corrosion Cracking (IASCC) behaviour under supercritical water conditions. Some
austenitic stainless steel alloys have demonstrated low swelling in doses of up to 10-30 dpa
in thermal neutron spectrum in the temperature regime of 200-500 ºC [12, 47].
Ferritic/ Martensitic (F/M) as well as austenitic stainless steels have been used throughout
the first through to third generation as in-core materials as well as pressure vessels; and as
such has led to them being classified as Generation IV fission reactors candidate materials
[9].
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But due to the harsh environmental conditions of high stress cracking corrosion and high
temperature the Generation IV in-core materials are to be exposed to, low swelling
austenitic stainless steels with high Ni and Cr components as shown in the phase diagram
shown in Figure. 3.1 , which are basically alloying elements excellent for corrosion
resistance and high mechanical strength [9].
Hence in assessing materials for the pressure vessel design, austenitic stainless steels
grades SS304, SS316, SS308 and SS309 were considered in the research work based on
the above evidence and also because of their excellent corrosion and high temperature
strength although their void swelling is inferior to that of Ferritic/Martensitic stainless
steels.
Figure 3.1: Phase diagram of Stainless Steel Alloys [33]
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3.1.2 Characterization and Properties of Selected Alloys
The austenitic stainless steels have oxidation limit of 900-1100 ºC, Specific heat capacity
of 510 J/Kg/ ºC, Thermal conductivity, of 16 W/M/ ºC and thermal expansion 16 x 10-6 ºC
[61]. Table 3.1 shows the grades which after thorough literature review were considered in
the research. Austenitic (fcc) has good corrosion resistance and ductility over wide
temperature range, depending on precise composition. No ductile-brittle transition so used
for cryogenic applications such as food production tools [84].
Table 3.1: Composition for Austenitic grades of Fe-Ni-Cr alloys selected for Damage
Assessment at SCW conditions [84 -88], designated by AISI and BS codes
Coding
System
% Fe %Ni %Cr %C
%
Mn
Other
Element
Tensile
Strength
(MPa)
Youngs
Modulus
(GPa)
AISI BS
304 304S15 69.42 9.50 19.00 0.08 2.0 - 580
193
308 - 68.92 10.00 19.00 0.08 2.0 - 515 193
309 - 61.30 13.50 23.00 0.20 2.0 - 620 200
316 316S16
66.92 12.00 17.00 0.08
2.0 Mo(2%)
627 193
* American Iron and Steel Institute (AISI) and British Standard (BS) System
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3.2. NEUTRON IRRADIATION DAMAGE ASSESSMENT BY BCA
Neutron Irradiation damages in the Fe-Ni-Cr alloys SS304, SS308, SS309 and SS316 were
assessed using SRIM-TRIM code which relied on the universal potential of Zeigler et al.
and Robinson were used [78, 89]. The 2013 version of SRIM-TRIM code was employed
in the radiation damage assessment along with the required input file. The setup window
and input parameters used for the simulation exercise and output files are presented in
Sections 3.2.1, 3.2.2 and 3.2.3.
3.2.1 Estimation of Energy Level at 30 dpa of Thermal Neutrons
Spectrum
Thermal neutron spectrum for Supercritical Water- Cooled Reactor was assumed to
produce a dose of 10-30 dpa [6, 9] Hence, for a stainless steel (with iron as major
component) exposed to neutron fluence of 5 x 1019 neutrons/cm2, the neutron energy was
calculated from the rate equation (2.11)
𝑑𝑝𝑎 = [
Ṅd(E).t
N
] ≈ [Ф(𝐸). 𝑡].
𝜎𝑠(𝐸)
𝐴𝐸𝑑
. 𝐸 (3.1)
making E the subject of the equation (3.1) gives
𝐸 =
𝑑𝑝𝑎 𝑥 𝐴𝐸𝑑
[Ф(𝐸).𝑡] 𝑥 𝜎𝑠(𝐸)
(3.2)
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where Ф(𝐸). 𝑡 is the fluence = 5 x 1019 neutrons/cm2, dpa = 30, A = 56 for iron (main alloy
component) and 𝐸𝑑 = 25 eV. The scattering cross section, 𝜎𝑠(𝐸) for neutrons was roughly
3𝜋R2, where R = 1.3 x 10-15 A1/3 m is the radius of the nucleus; hence, 𝜎𝑠(𝐸) ≈2.3 x 10
-24
cm2. Consequently,
E =
30 𝑥 56 𝑥 25
5 𝑥 1019𝑥 2.3 𝑥10−24
= 365 𝑀𝑒𝑉
3.2.2 Estimation of Energy Level at 150 dpa of Fast Neutrons Spectrum
Fast neutron spectrum for Supercritical Water- Cooled Reactor was also assumed to
produce a dose of 150 dpa [6, 9]. Hence, for a stainless steel (iron exposed to neutron
fluence of 5 x 1019 neutrons/cm2, using equation (3.52) above
E =
150 𝑥 56 𝑥 25
5 𝑥 1019𝑥 2.3 𝑥10−24
= 1.82 𝐺𝑒𝑉
3.2.3 SRIM-TRIM Setup and Input Requirements
TRIM was accessed from the main menu of SRIM. The TRIM Setup Window was used to
input the data on the ion, target, and the type of TRIM calculation desired.
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Figure 3.2: TRIM Input Parameter Window showing all inputs for Stainless Steel grade
316 assessment
The ion data and input parameters used to calculate the radiation damage by the SRIM –
TRIM Code is shown in Table 3.2, while Table 3.3 show the target data and the input
parameters.
As shown in Table 3.1 only 4 alloys (SS304, SS308, SS309 and SS316) were bombarded
with 100 Uranium neutron Ions (incident projectiles are considered by TRIM code as
Ions and also neutrons cannot be entered straight into the setup but elements that its
neutrons are being considered) of energy 365 MeV for the thermal neutron spectrum and
1.82GeV for the fast neutron spectrum. The “Ions with specific energy/angle/depth (full
cascade) using TRIM.DAT” damage type [78], in the 2013 version of SRIM-TRIM code
menu, was employed. (See Appendix I)
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Table 3.2: Ion Data and input parameters used in the SRIM – TRIM Code
Ion Data Name/Value
Incident Ion type Thermal and Fast Neutrons from Uranium
Symbol for the incident ion U
Atomic number 92
Atomic mass 238.051 amu
Incident Ion Energy
365 MeV (thermal neutrons) and 1.82 GeV
(fast neutrons)
Damage Type
Ions with specific energy/angle/depth (full
cascade) using TRIM.DAT (Appendix I)
Angle of incident 0
Table 3.3: Target Data and input parameters in the SRIM – TRIM Code
Target data Name/Value
Layer name Stainless Steel (304, 308, 309 and 316)
Compound Correction 1
Width 0.46 m (equivalent RPV thickness)
Density 7.83977236 g/cm3
Atomic density 6.031x1022 atom/cm3
Depth 2.0 x10-5 m and 4.0 x10-5 m (Set up)
Calculated Ions 100
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Table 3.4: TRIM.IN contains setup parameters for TRIM Simulation of type 304
1. SRIM - 2013.00version:- This file however controls TRIM Calculations.
2. Ion: Z1 , M1, Energy (keV), Angle, Number, Bragg Corr, AutoSave Number.
92 238 365000 0 100 1 10
3. Cascades(1=No;2=Full;3=Sputt;4-5=Ions;6-7=Neutrons), Random Number Seed, Remin.
5 0 0
4. Diskfiles (0=no,1=yes): Ranges, Backscatt, Transmit, Sputtered, Collisions (1=Ion;2=
Ion + Recoils), Special EXYZ.txt file
1 1 1 1 2 100000
5. Target material : Number of Elements & Layers
"U (365000) into Fe-Ni-Cr Alloy (SS304) " 5 1
6. PlotType (0-5); Plot Depths: Xmin, Xmax(Ang.) [=0 0 for Viewing Full Target]
0 0 200000
7. Target Elements: Z Mass(amu)
Atom 1 = Fe = 26 55.847
Atom 2 = Ni = 28 58.69
Atom 3 = Cr = 24 51.996
Atom 4 = C = 6 12.011
Atom 5 = Mn = 25 54.938
8. Layer Layer Name / Width Density Fe(26) Ni(28) Cr(24) C(6) Mn(25)
Numb. Description(Ang) (g/cm3) Stoich Stoich Stoich Stoich Stoich Stoich
1 "Stainless Steel(304)" 4600000000 7.82399572 .6942 .095 .19 .0008 .02 .02
9. Target layer phases (0=Solid, 1=Gas)
0
10. Target Compound Corrections (Bragg)
1
11. Individual target atom displacement energies (eV)
25 25 25 28 25
Individual target atom lattice binding energies (eV)
3 3 3 3 3
Individual target atom surface binding energies (eV)
4.34 4.46 4.12 7.41 2.98
12. Stopping Power Version (1=2011, 0=2011)
0
Note: the numbering of lines was done for easy description of the input parameters
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The TRIM.IN file in Table 3.4 contained information, which are in two-line increments,
with explanation in the first line which does not contain data, and data values in the next
line.
The first line (labelled 1) of TRIM.IN contained the version number of the SRIM code
being used.
The 2nd and 3rd lines (labelled 2) contained information about the ion (atomic number,
mass, energy and incident angle to the target), the total ions to calculate (20,000), a term
called “Bragg Corr” (not used), and the AutoSave number (the TRIM calculation was
automatically saved after this number of ions).
The next two lines (labelled 3) contained parameters for calculating: (a) the Cascades
number declares the type of Damage Calculation (upper right menu of the TRIM Setup
window), and (b) a random number seed (zero for the default value) and (c) Reminder
The next two lines (labelled 4) contained instructions about any datafiles that should be
created when TRIM starts (these datafiles are described in the menu at the bottom of the
TRIM Setup window). Note that the explanation line is quite long and is duplicated here
as two lines of text. In the data line, a “0” means no file, and a “1” asks that this file be
created. The EXYZ parameter was a number such as 10000 which gives results at every
10000 eV of energy increment.
The data line labelled 5 contained (a) a description of the calculation that would be
included in every datafile produced by TRIM (in quotes), and (b) the number of elements
in the target and the number of layers in the target. This latter information is necessary
because the TRIM.IN file may have several lines of data depending on the number of
elements and layers in the target.
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The data line labelled 6 gave the type of initial plot that TRIM should display (use “0”
for no plot), and the depths of the Viewing Window. This window could cover the entire
target depth, or could blow up a small segment of the target so interactions may be seen
with greater detail. To default the Viewing Window to the total target depth, you can use
“0 0” as the depths.
The data lines labelled 7 defined the elements in the target. First is the chemical symbol,
then the atomic number and finally the mass of each target atom. The number of lines
must agree with the number of target elements declared above. The first 15 characters in
each line were ignored (e.g. “Atom 1 = Fe = ”) and were only included to make the file
readable. Only the Z and Mass are used by TRIM.
The section labelled 8 described each layer of the target. Note that the explanations takes
up two lines of the TRIM.IN datafile. The “Layer Name” was a description (in quotes)
that would appear in all the plots. Next to the layer name was the layer width (Ǻ), the
layer density (g/cm3), and the relative concentration of each of the elements in that layer
of the target.
The next two lines (labelled 9) gave the phase state of the layers, i.e. either a solid or a
gas layer.
The next two lines (labelled 10) gave the bonding corrections needed to be applied to the
electronic stopping powers of the ion. A “1” meant no special bonding correction.
The next section (labelled 11) gave the damage parameters for the target layers. Each
atom had a Displacement Energy, Binding Energy and Surface Binding Energy for each
layer. All energies were in units of eV. You can input “0” for the absence of a layer. The
final input (labelled 12) was a declaration of which version of SRIM’s Stopping Powers
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to use. About every 5 years, the complete stopping theory of SRIM was revisited and all
the experimental data of that period is added to the database. Any new ideas on stopping
theory were also included. This results in new stopping power concepts, and variations
in stopping powers from earlier versions of SRIM.
3.2.4 SRIM – TRIM code Simulation Algorithm and Flowchart
The time taken for the completion of the simulation depended on the following; type of
calculation and the number of Ions being dealt with. The procedure for the simulation in
SRIM-TRIM version 2013 with input file were:
1. Choosing the type of TRIM Damage Calculation: At the damage calculation
corner of the TRIM window, the “Ions with specific energy/angle/depth were
selected (full cascade) using TRIM.DAT” type, since only neutrons were
considered.
2. Selection of Ion type: Uranium was selected as the Ion (Only elements were
allowed to be selected by the program instead of neutrons particles required for the
research hence the need for the type of damage selected.
3. Insertion of Ion Parameters: The energy of 365 MeV (365000 i.e. converted to
keV before entering since the setup only takes values in keV) was inserted into the
energy column
4. Selection of Target Material: Stainless Steel (a default compound) was selected
from the Compound Dictionary and was then changed to 304, 308, 309 and 316
based on their elemental composition /Atomic Stoichiometry, respectively.
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5. Insertion of Atomic Stoichiometry: The elemental compositions of those four
alloys were entered into their respective columns (e.g Stainless Steel (304)
composed of Fe-0.6942, Cr-0.19, Ni-0.095, C-0.0008, Mn-0.02)
6. Insertion of Target Name: The name of the target was inserted at its column in
the windows setup (e.g. Stainless Steel (304))
7. Insertion of Target Thickness: The thickness of the material was entered. (i.e. 0.46
m representing PV thickness)
8. Setting Special Parameters: The total number of ions for the simulation was
entered as 100 (since the output files of the collision history of ions greater than
100 from the TRIM.DAT file was above 2GB). Also the plotting window depth
was set to 200000Å since that only works with the default unit (angstrom).
9. Selection of Output files: The output files required for the damage assessment such
as vacancies, sputtering etc. were selected.
10. Saving of input & Running the TRIM: The Save Input & Run TRIM button was
clicked to start the simulation.
The flowchart of the simulation for the assessment of irradiation damage of Fe-Ni-Cr to
alloy for designing the reactor internals and pressure vessel using the SRIM-TRIM code is
shown in Figure 3.3
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Figure 3.3: SRIM – TRIM Code flowchart for simulation
3.2.5 Implementation of SRIM – TRIM Simulation
In simulating the neutron irradiation damage, the selected Fe-Ni-Cr alloys SS304, SS308,
SS309, and SS316 were each subjected to 100 neutrons exposure from Uranium (proposed
fuel of the SCWR) at neutron particle energy of 365 MeV. Following procedures outlined
under the structural algorithm stage, the simulation was performed, each simulation lasting
about 7 hours to complete. The output files were then saved with their corresponding data
files.
Choose Type of TRIM Calculation (Ions with specific energy/angle/depth
(full cascade) using TRIM.DAT)
Incre
ase
the
numb
er of
ions
Monte Carlo Calculations of ion-target collisions and associated
damage
Total number of Ions reached?
End
Save Input Data and Run TRIM
User Ion Data and input Parameters (Table 3.1) and
Target Data and Input Parameters (Table 3.2)
Start
Yes No
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3.2.6 SRIM – TRIM Output files
After a successful simulation of the TRIM program for each alloy, the output plots and
data files obtained were [77, 78, and 90]:
Ion Range– Determined the final distribution of ions directed into the Fe-Ni-Cr
alloys
Lateral Ion Range Distribution – For ion beams incident perpendicular on
target, the ion distribution spread out with azimuthal symmetry, and the Lateral
Range was merely the average final y-z displacement of the ions assuming a
perpendicular incidence of the ion beam.
Ionization Energy Losses – The energy loss of ions to the target electrons. Upon
penetrating into the stainless steel the ions interacted immediately with the
electrons, both single electrons and collectively; which led to energy loss by the
ion to the electrons, represented by Ionization Energy Loss.
Phonons – Described the ion’s energy loss to the Target Phonons. Thus when
ion collides with a nucleus and the energy of the incident ion was less than the
displacement energy (energy required to eject an atom from the lattice site) the
atom returned to the lattice site and the recoil energy was transferred into target
phonons.
Energy to Recoil – Represented by the plot of Ion energy transferred to the
stainless steel and the percentage of energy absorbed by each element.
Damage Events – A collision event (comprising of Displacement, Replacement
and Vacancies) and its 3D plot are required. The plot gave the total number of
atoms displaced by both the ion and the PKA and all the recoiling target atoms.
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Sputtering – An integral sputtering plot which was useful to estimate how many
atoms reach the surface and did not possess enough energy to escape.
3.3 EVALUATION OF MECHANICAL INTEGRITY OF MATERIALS
The mechanical properties of Fe-Ni-Cr alloys SS304, SS308, SS309 and SS316 were
evaluated at SCW conditions by Molecular Dynamics Simulation method employing
LAMMPS and VMD. The 30 Sep 2014 edition of LAMMPS, VMD version 1.9.2 and
version R2013a of MATLAB were employed. The setup window, input parameters and the
output files used for the simulation exercise are presented in the subsequent sections.
3.3.1 LAMMPS Setup and Input Requirements
In running the LAMMPS program, three (3) program files required were
In.file (input script to create models and for calculation involved in the simulation,
(See Appendix II)
Potential file (contains data about the interatomic bond between atoms, (See
Appendix II)
.exe file (required to run the commands in the in.file)
The LAMMPS input consisted of Initiation, Atom definition, Settings, and Running
simulation [80], where the
Initiation – Setting parameters to be defined before atoms were created, such as
units, dimension, boundary condition, and atom type.
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Atom definition – i.e. atom or molecular topology information required for the
simulation.
Settings –specified as force field, various simulation parameters, and output
options, etc.
Run – The time step and the number of runs for the simulation were stated at the
final stage.
The flowchart for designing the LAMMPS input file is indicated in Fig. 3.4 and a
copy of the files used for this simulation in appendix II.
Figure 3.4: Steps followed in designing LAMMPS input file
The Lattice Parameters which were used for the simulation were tabulated in Table 3.5.
Set up units, dimension, atom
type, boundary, processors, etc.
Define atom/molecular topology
information
Atom Definition
Specify force field (eg.
interatomic potential type used),
various simulation parameters,
and output options, etc.
Settings
Run a simulation
Initialization
Specify the time-step and number
of runs required for the simulation
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Table 3.5: Lattice Parameters used for the LAMMPS Simulations [91].
TREATMENT
LATTICE PARAMETERS
SS304 SS308 SS309 SS316
Ambient
Temperature 3.5918 3.5907 3.5930 3.5935
300 ºC
3.5919 3.5914 3.5942 3.5975
400 ºC
3.5613 3.5663 3.5781 3.5864
500 ºC 3.5628 3.5642 3.5613 3.5664
3.3.2 Interatomic Potential Adapted for the MD Simulation
The critical part of MD simulation was the force calculation, which depended on
interatomic potentials. The net force acting on each atom in the system was the result of
the interactions with all other atoms, which amounted to a set of rules known as a force
field or interaction potential. Accurate, robust, and transferable force fields were critical
to perform physically realistic molecular simulations.
The sum of all forces acting on an atom, F,
𝐹 = ∑𝑚𝑎 = ∑𝑚
𝑑𝑣
𝑑𝑡
= ∑𝑚
𝑑2𝑟
𝑑𝑡2
=
𝑑𝒑
𝑑𝑡
(3.3)
where a is acceleration, v is velocity, t is time, r is position, and p is momentum. Since
position r is a vector, the first and second derivatives, v and a, and corresponding p and F,
are also vectors. For a constant total Energy E in time (dE/dt =0), which was the case of
isolated system for MD simulations, F was related to the negative gradient of potential with
respect to position, i.e.
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𝐹 = −∇𝑈 (3.4)
𝐹𝑖 = 𝑚𝑖
𝑑2𝑟𝑖
𝑑𝑡2
= −
𝑑𝑼(𝒓𝒊)
𝑑𝒓𝒊
(3.5)
where U is potential. Knowing the potential of a system as a function of interatomic
distance, the force on atoms could be obtained for time evolution of the system [71].
The four common types of potentials mostly used in MD simulations are Pair Potentials,
potentials by Embedded Atom Method, Tersoff Potentials, and Potentials for Ionic Solids.
The Pair Potential, Tersoff and Potential for Ionic Solids are generally meant for noble gas
(i.e. Ar, Ne, Kr, etc.), covalent solids and ionic solids respectively, whereas the embedded
atom method (EAM) potential is appropriate for metals and transition metals, such as FCC
metals. Hence for the Austenitic Stainless Steel, the EAM was adopted for the simulation.
The EAM potential, 𝑈𝐸𝐴𝑀, consisted of a term for pair interaction and another for
embedding energy as a function of electron density 𝜌𝑖 at atom 𝑖:[71]
𝑈𝐸𝐴𝑀 = ∑ 𝑈𝑖𝑗(𝑟𝑖𝑗) +𝑖≠𝑗 ∑ 𝐹𝑖(𝜌𝑖)𝑖 (3.6)
where 𝐹𝑖(𝜌𝑖) is the embedding energy function, 𝑟𝑖𝑗is the scalar distance between atom 𝑖
and 𝑗, and atom-atom distances in the x-, y-, and z- axis as
𝑟𝑖𝑗 = |𝑟𝑖 − 𝑟𝑗| = √(𝑥𝑖 − 𝑥𝑗)
2
+ (𝑦𝑖 − 𝑦𝑖)2 + (𝑧𝑖 − 𝑧𝑗)2 (3.7)
In equation (3.6), the first summation notation represented a sum of all unique pair
interactions excluding any double counting:
1
2
∑ 𝑈𝑖𝑗(𝑟𝑖𝑗)
𝑁
𝑖≠𝑗 = 𝑈12 + 𝑈13 +⋯+ 𝑈23 +⋯+ 𝑈34 + 𝑈35 +⋯+ 𝑈45 + 𝑈46 + (3.8)
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The electron density at site 𝑖 was the linear superposition of valence-electron clouds from
all other atoms:
𝜌𝑖 =
1
2
∑ 𝜌𝑖(𝑟𝑖𝑗)𝑗(≠𝑖) (3.9)
The net force acting on the atom at a given time was then obtained exactly from the
interatomic potential, as a function only of the positions of all atoms.
In calculating the net force, common algorithms for the numerical integration of the
Newton’s equations of motion and calculation of atomic trajectories are Verlet algorithm,
the Velocity Verlet Algorithm, the Predictor-Corrector Algorithm, nth order Runge-Kutta,
and the Gear Algorithm [71].
Velocity Verlet algorithm was implemented for the numerical integration despite the
algorithm though has poor accuracy for large time step (hence ∆𝑡 must be small), has
proved to be fast, simple, stable, time reversible, required low memory, and symplectic
(phase, space, volume and energy conserving). In the Verlet scheme, the positions,
velocities, and acceleration at time 𝑡 + ∆𝑡 were obtained from the corresponding quantities
at time t by advancing the velocity, 𝑣, by a half step, the position 𝑟 by a full step using the
half-step and then the acceleration, a, by full step from the potential relationship such that
𝑣 (t +
∆t
2
) = 𝑣(𝑡) +
1
2!
𝑎(𝑡)∆𝑡 (3.10)
𝑟(𝑡 + ∆𝑡) = 𝑟(𝑡) + 𝑣(𝑡)∆𝑡 +
1
2!
𝑎(𝑡)∆𝑡2 = 𝑟(𝑡) + 𝑣(𝑡 +
∆𝑡
2
)∆𝑡 (3.11)
𝑎(𝑡 + ∆𝑡) = −(
1
𝑚
)
𝑑𝑈[𝑟(𝑡+∆𝑡)]
𝑑𝑡
=
𝐹(𝑡+∆𝑡)
𝑚
(3.12)
The velocity term 𝑣 was advanced by a full step from 𝑎 at the previous and current
timesteps and expressed it using the half-step advanced 𝑣 and full-step advanced 𝑎 as
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𝑣(𝑡 + ∆𝑡) = 𝑣 (𝑡 +
∆𝑡
2
) +
1
2
𝑎(𝑡 + ∆𝑡)∆𝑡 (3.13)
and gave the atomic coordinates and velocities at time t and with the force field F=ma, the
entire future position of the atoms were determined by the F = ma. The Verlet algorithm
was also useful in calculating some thermophysical properties, such as temperature, stress,
strain, pressure, volume etc. of the materials at any point in time [71, 92].
The effects of edges in 2-D systems such as graphene and surfaces in 3-D systems such as
graphite were eliminated in the MD simulations in order to obtain the bulk properties of
these systems by simulating an extremely large system to ensure that the surfaces and edges
have only a small influence on the properties. However, this approach is computationally
expensive. The most efficient way to simulate an indefinitely large system was using
periodic boundary conditions (PBC). In PBC, the cubical simulation box was replicated
throughout space to form an infinite lattice as shown for a 2-D case in Fig. 3.5. During the
simulation, when a molecule moved in the central box, the periodic images in every other
box also move in exactly the same way. Thus, as a molecule leaves the central box, one of
its images would enter through the opposite face. Therefore the system had no edges [93].
Figure 3.5 Graphical representation of the periodic boundary conditions. The arrows
indicate the velocities of atoms. The atoms could interact with atoms in the neighboring
boxes without having any boundary effects [93].
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The potential on the Fe-Ni-Cr alloys was modified from Bonny G. et al. [94] for the
research work that is shown in Appendix II.
3.3.3 LAMMPS Simulation Algorithm
A directory was created for the two files: LAMMPS’s script file, Potential file; with
the executable, lmp_mpi (parallel executable type lmp_mpi.exe).
Then the command prompt screen was opened by typing “cmd” in the start menu
of the computer.
In the displaying screen, the default directory was displayed as “C:\Users\name of
computer> “. Since the above program files were not in the directly, the path was
changed by typing in the screen as “C:\Users\name of the computer> cd directory
name: then enter. The desired directory was gotten as “C:\Users\name of the
computer\directory name>”. The screen now displayed the same path address as
required as seen in Fig. 3.6.
Figure 3.6: Command prompt look for LAMMPS Simulation
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The “Path address>lmp_mpi<in.file name” was typed, enter key was pressed and
automatically the in.file was then executed by the .exe file and the simulation
continued till the output values were displaced, indicating the end of the program
as shown in the Fig 3.7.
Figure 3.7: On screen view of Output values from Simulation
3.3.4 Implementation of LAMMPS Simulation
To get a reliable results from the modified potential file, an initial simulations was
performed to determine the equilibrium lattice constant and cohesive energy of the Fe-Ni-
Cr alloys, and the values compared with published data in order to verify the interatomic
potential adapted [80] for the main bulk properties calculations proven.
The MD simulation with LAMMPS were conducted by the procedures;
A simulation cell with Face Centered Cubic (FCC) atoms with (100) orientations
in the x, y, and z directions was generated as in Fig. 3.8 with a simulation cell size
of 10 lattice units in each directions (10 × 10 × 10 Å) to ensure the simulation
converged.
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Fig 3.8 (a) Fig 3.8 (b)
Fig 3.8: (a) Crystal structure of an FCC lattice (b) <100> orientation in the x, y and
z direction where the uniaxial deformation will be applied (Courtesy: P.M.
Anderson, [95]).
The equilibration step was conducted at a temperature 300 K (Ambient
temperature) with a pressure of 1 atm (0.01 MPa) at each simulation cell boundary
terminating after 12000 simulations at a time step of 0.002 ps.
The simulation cell was then subjected to uniaxial tensile deformation in the x-
direction under controlled temperature at a strain rate of 5 x 1010 1/s, while the
lateral boundaries were controlled using the NPT (constant Number of Atoms,
Pressure and Temperature ensembles) equation of motion at 25 MPa pressure.
The dump file that included the x, y, and z coordinates, the centrosymmetry values,
the potential energies, and forces for each atom were outputted by the program as
it runs for direct visualization by VMD.
The stress and strain values were also outputted into a separate file named
Fe.Deform for the extraction of the mechanical properties of the materials.
The Simulation was terminated after 20000 iterations at time step of 0.002 ps.
The simulation was then repeated for temperatures of 573 K, 673 K, and 773 K
(300 ºC, 400 ºC and 500 ºC respectively) for each Fe-Ni-Cr alloy (304, 308, 309,
316) evaluated.
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3.3.5 Output of LAMMPS Simulation
After successful running the in.file, the three out-put files obtained were:
Dump files (contained the atomic co-ordinates of the final structure after
simulation, and data on deformation for VMD visualization of deformation
processes, see Appendix VI)
Fe.deform (contained the stress component values as well as the corresponding
strain values to determine the bulk modulus, see Appendix VI).
Log.lammps file (contained thermo-physical data of temperature, pressure, volume
and total energy after a particular number of simulation steps, see Appendix VI).
Copies of the last two output files are shown in the Appendix II and the structures at various
time steps, the final structure after simulation and the first output file that contained the
atomic co-ordinates were visualized and animated with VMD code.
3.3.6 Visualization of Simulation Output by VMD and MATLAB
The atomic structures in the Dump file output file from the LAMMPS simulation were
viewed in VMD, following the steps;
VMD program was opened;
The Dump file containing the atomic coordinates of the Fe-Ni-Cr alloy was then
imported through the VMD Main window;
Adjustments and changes were made to the colour and display format (perspective)
of final structure
The final output was then saved in JPEG Picture format.
The algorithm for animation of the tensile deformation can be seen at Appendix III.
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The mechanical properties - ultimate tensile strength, yield strength, breaking point
strength, and Young’s Modulus of Fe-Ni-Cr alloys were extracted through MATLAB
import terminal. The steps were:
MATLAB program was opened;
At the command window the import button was then pressed;
The output files Fe.deform were then imported into MATLAB
The imported data was analysed and then plotted to give the required graphs
The graphs were then saved in JPEG (Picture format) form.
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CHAPTER FOUR: RESULTS AND DISCUSSIONS
The Chapter is divided into three main Sections. Section 4.1 consists of results obtained
from the irradiation assessment, Section 4.2, the evaluation of the mechanical integrity
while the Section 4.3 gives the general discussions of the research findings. Typical output
data for SS304 would be presented under the results section, and the other results be
provided in Appendix IV while the comparison results will be in tabular form and that will
be placed at Appendix V. Thermal neutron irradiation damage were compared with fast
neutron damage.
4.1 THERMAL AND FAST NEUTRON IRRADIATION DAMAGE IN SS304
4.1.1 Collision Cascade
The SRIM-TRIM output window of the collision events of the thermal spectrum compared
with the fast spectrum are shown in Fig. 4.1 (a) and (b) both illustrating at the Calculation
Parameters window the resulting calculations of Vacancies/ Ion, Longitudinal range or
depth of penetration, Percentage Energy Loss and then Sputtering Yield of the SS304.
Fig 4.1 (a)
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Fig 4.1(b)
Fig 4.1: Collision Cascade window for (a) thermal and (b) fast neutron irradiation damage in Fe-
Ni-Cr alloy SS304
4.1.2 Projected Ion Range Distribution
The output plots of the longitudinal range are shown in Fig. 4.2 illustrating the depth of
penetration of both the thermal and fast neutron into the Fe-Ni-Cr alloy SS304. The thermal
neutrons penetrated to 11.3 µm as compared with 32.3 µm in the fast spectrum. The
penetrating depth in the fast neutron spectrum was about three times greater than in the
thermal spectrum. Hence material for the fast spectrum design must be of larger thickness.
Nonetheless, comparing their depth with the thickness of 4.6 x105 µm indicates
insignificant penetration and possible minimal damage.
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Fig 4.2 (a)
Fig 4.2 (b)
Fig 4.2: Projected Range of (a) thermal and (b) fast neutrons in the Fe-Ni-Cr Alloy SS304
4.1.3 Lateral Ion Range Distribution
The average final y-z displacement of the ions assuming a perpendicular incidence of the
ion beam for both the thermal spectrum and fast spectrum are shown in Fig. 4.3 (a) and
(b) respectively. It can be seen that the ions had a wider lateral distribution (spreading)
in the thermal neutron spectrum than the fast neutron spectrum implying that increase in
incident ions energy decreases the lateral range and rather increase the projected range
or penetration power as shown in Fig 4.2(a) and (b). Hence the materials to be used in
the thermal spectrum should be wide enough to curtail any penetration in the y-z
direction.
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Fig 4.3 (a) Fig 4.3(b)
Fig 4.3: The lateral Range distribution of the (a) thermal and (b) fast neutron ions in the Fe-Ni-Cr
Alloy SS304
4.1.4 Ionization Energy Distribution
Figures 4.4 (a), (b), (c) and (d) show the energy neutrons lost to the Fe-Ni-Cr Alloy target
electrons in 2D ( Fig 4.4(a) and (b)) and 3D Fig 4.4(c) and (d)) views. At the very bottom
of the two plots are the tiny ionization contributions from the recoils (that blue line),
implying that the bulk of the energy dissipated to the Fe-Ni-Cr Alloy SS304 electrons was
from the Ions (red). The percentage of energy loss in Ionization in the simulation as shown
in Fig 4.1 (a) and (b) at “% Energy Loss” was 97.18 % (355 MeV – comprising of 94.37
% from Ions and 2.81 % from recoil atoms) in the thermal spectrum as compared with
99.33 % (1808 MeV – also comprising of 98.49 % from Ions and 0.45 % from recoil atoms)
of the fast spectrum. The above result implies that more heat would be produced in the
latter than the thermal spectrum. This means that should the SS304 be used for the fast
neutron spectrum design, irradiation induced mechanical deterioration through diffusion of
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point defects as a result of ionization would be massive and hence much attention would
be needed.
Fig 4.4 (a) Fig 4.4(b)
Fig 4.4 (c)
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Fig: 4.4 (d)
Fig 4.4: 2D view of the Ionization energy distribution of (a) thermal neutrons as compared with the
(b) fast neutrons (c) and (d) 3D view of the Ionization energy distribution in the Fe-Ni-Cr alloy
SS304
4.1.5 Phonons
From Fig 4.1 (a) and (b) in section 4.1.1, the “% Energy Loss” pane shows that 2.56 %
(0.975 MeV – comprising 0.01 % of the Ions and 2.55 % of the recoil atoms) of the
energy was loss in a form of phonons in the thermal spectrum as compared with only
0.06 % (1097 MeV) in the fast spectrum which was only from the recoil atoms. It could
also be seen in Fig. 4.5 (a) that due to the low energy of the incident ions, there was a
thorough interaction with the Target material and hence a higher percentage of the energy
was loss as a result of the vibration unlike in the case of the Fig 4.5 (b) where the ion
penetrated deeper with little interaction.
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Fig 4.5 (a) Fig 4.5(b)
Fig 4.5: Distribution of (a) thermal and (b) fast neutrons energy loss to Fe-Ni-Cr Alloy SS304
phonons
4.1.6 Ion’s Energy to Recoil Distributions
The energy absorbed by the component elements Fe, Ni, Cr, and C in the case of the Fe-
Ni-Cr alloy SS304 is shown in Fig 4.6 (a) and (b). It could be seen that the higher the
percentage of the component element, the higher the absorption.
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Fig 4.6 (c) Fig. 4.6 (d)
Figure 4.6: Distribution of energy absorbed by the SS304 Fe-Ni-Cr Alloys elements from (a) the
thermal and (b) fast neutrons spectrum
4.1.7 Collision Events
Since the kinetic energies in the cascades were very high, the material was driven locally
far outside its thermodynamic equilibrium. This led to the production of point defects such
as vacancies in the materials. Though the material was subjected to different energy levels
in the two proposed spectrum it was realized that the rate of vacancy production were the
same. Thus in the thermal neutron spectrum (Fig 4.7 (a) and (c)), the total target
displacement was averagely 337242/Ion and out of that, 325279 representing 96.5 %
remained as vacancies. Similarly, out of the total target displacement of 394519/Ion in the
fast neutron spectrum (Fig 4.7 (b) and (d)), 380528 (96.5 %) were left as vacancies. This
therefore implies that the Collision event may not be a good tool to use in assessing the
damage of a particular material used in different neutron spectrum.
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Fig 4.7 (a) Fig 4.7 (b)
Fig 4.7 (c)
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Fig 4.7 (d)
Figure 4.7: 2D view of the collision events of (a) thermal and (b) fast neutron spectrum,
(c) and (d) gives the 3D view of the collision events
4.1.8 Sputtering Target Atoms
Fig. 4.8(a) and (b) shows the integral sputtering for the thermal and fast neutron spectrum
respectively. This plot shows the number of elemental atoms reaching the target surface
that did not have enough energy to escape. It could be seen that the Fe components
sputtered much, followed by the Cr, then Mn and Ni. This therefore implies that sputtering
does not depend on the percentage of the elements. Also whiles in the thermal spectrum
about 0.1 atoms of the Fe were sputtered per Ion, only 0.05 atoms were sputtered per ion.
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Fig 4.8 (a) Fig 4.8 (b)
Figure 4.8: Distribution of the integral sputtering yield of Fe-Ni-Cr Alloy SS304 in (a) thermal and
(b) fast neutron spectrum
The output results obtained on the irradiation damage assessment of the other three Fe-Ni-
Cr alloys 308, 309 and 316 are displayed at Appendix IV and Comparison of the irradiation
damage of all Fe-Ni-Cr alloys under thermal and fast neutron spectrum of the SCWR are also shown
in Appendix V.
4.2 EVALUATION OF MECHANICAL DETERIORATION OF THE Fe-Ni-Cr
ALLOYS
4.2.1 Cohesion Energy of the Fe-Ni-Cr Potential File
The results of LAMMPS cohesive energy Ecohe, and equilibrium lattice constant, a
simulation for the FCC FeNiCr.eam.alloy potential are as shown in Table 4.1 compared
with their experimental values.
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Table 4.1: Summary of Equilibrium Lattice Constant and Cohesive Energy from the simulation
compared with theoretical values.
LAMMPS Expt. Data Δ/%
Equilibrium Lattice
Constant /Å
3.489 3.499 [94] 1.0
Cohesive Energy /
eV/atom
-4.117 -4.120 [94] 0.3
4.2.2. VMD output for Tensile Deformation
The detailed VMD output files for the tensile deformation are shown in Figure 4.9,
representing the snapshots of the simulation at 8 picosecond intervals.
t= 0 ps t= 8 ps t= 16 ps
t = 24 ps t = 32 ps t = 40
Fig: 4.9: VMD Snapshot showing Fe-Ni-Cr alloy model of size 10 Å x 10 Å x 10 Å (No. of atoms 4000)
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4.2.2 Stress-Strain Plots at Ambient and Supercritical Water Conditions
Figure 4.10 presents the stress- Strain curve to failure showing Young’s Modulus, UTS,
and Breaking Strength of SS304 alloy under ambient conditions (23 ºC, 1 atm). It shows
the Young’s Modulus, the Ultimate Tensile Strength, the Breaking Strength (Fracture) and
their corresponding strain values. The same approach was used in extracting the
mechanical properties of the other alloys under investigation at Ambient and Supercritical
Water Cooled Condition shown in Appendix VII and Figure 4.11 (a) through to Figure
4.11(d).
Fig 4.10: Stress- Strain curve to failure showing Young’s Modulus, UTS, and Breaking Strength
of SS 304 alloy under Ambient Conditions
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
0
2
4
6
8
10
12
Strain (%)
S
t
r
e
s
s
(
G
P
a
)
Corresponding Strain
Value of BS = 0.19
Corresponding Strain
Value of UTS = 0.07
Young's
Modulus
= Slope of
line AB
A
B
Breaking Strength
of SS304
at Ambient
conditions
UTS of SS304 at
Ambient conditions
Breaking
Strength
= 8.71
UTS
= 10.55
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Fig 4.11 (a) (SS304)
Fig 4.10 (b) (SS308)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
0
2
4
6
8
10
12
Strain %
S
t
r
e
s
s
/
G
P
a
SS304 at Ambient Condition
SS304 at 300ºC
SS304 at 400ºC
SS304 at 500ºC
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
0
2
4
6
8
10
12
Strain (%)
S
t
r
e
s
s
/
G
P
a
S308 at Ambient Condition
SS308 at 300ºC
SS308 at 400ºC
SS308 at 500ºC
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Fig 4.11 (c) (SS309)
Fig 4.11 (d) (SS316)
Figure 4.11: Stress-Strain plots for (a) SS304 (b) SS308 c) SS309 and d) SS316 alloys at
ambient condition and Supercritical Water Condition at strain rate of 5x1010 s-1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
0
2
4
6
8
10
12
Strain (%)
S
t
r
e
s
s
/
G
P
a
SS309 at Ambient Condition
SS309 at 300ºC
SS309 at 400ºC
SS309 at 500ºC
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
0
2
4
6
8
10
12
Strain (%)
S
t
r
e
s
s
/
G
P
a
SS316 at Ambient Condition
SS316 at 300ºC
SS316 at 400ºC
SS316 at 500ºC
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4.2.3 Mechanical Properties of Fe-Ni-Cr Alloys
The mechanical properties of the Fe-Ni-Cr alloys extracted from the Stress-Strain curves
of Fig. 4.11 are shown in Fig 4.12. Figures. 4.12(a), 4.12(b), 4.12(c) and 4.12(d)
respectively show the variation of Young’s Modulus, Yield Strength, Ultimate Tensile
Strength and the Breaking Strength with different temperature and pressure(treatment
conditions). It could be seen that all the four plots above show similar trend. The
mechanical properties decreased with increasing temperature and pressure. Though the
materials had similar mechanical strength at the Ambient condition, Fe-Ni-Cr alloy SS304
was found to be deformed least while SS309 was realized to be deformed the most.
Fig 4.12(a)
0 50 100 150 200 250 300 350 400 450 500
110
120
130
140
150
160
170
180
190
200
Temperature (°C)
Y
o
u
n
g
'
s
M
o
d
u
l
u
s
(
G
P
a
)
SS304
SS308
SS309
SS316
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Fig 4.12(b)
Fig 4.12(c)
0 50 100 150 200 250 300 350 400 450 500
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
Temperature (ºC )
Y
i
e
l
d
S
t
r
e
n
g
t
h
(
G
P
a
)
SS304
SS308
SS309
SS316
0 50 100 150 200 250 300 350 400 450 500
6
6.5
7
7.5
8
8.5
9
9.
10
10.5
11
Temperature (°C)
U
l
t
i
m
a
t
e
T
e
n
s
i
l
e
S
t
r
e
n
g
t
h
(
G
P
a
)
SS304
SS308
SS309
SS316
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Fig 4.12(d)
Fig 4.12: Variation of (a) Young’s Modulus (b) Yield Strength (c) Ultimate Tensile
Strength and (d) Breaking or Fracture Strength and of the alloys with respect to ambient
and SCW condition.
4.3 DISCUSSIONS
4.3.1 General Discussions
In general, neutron irradiation damage and mechanical deterioration of materials are two
of the most prevalent reactor structural material challenges in general. Hence in the process
of identifying suitable structural materials for the proposed Supercritical Water-Cooled
Reactor (SCWR), it was necessary to take candidate materials through these tests.
Based on available knowledge on the Fe-Ni-Cr alloys usage in LWR and fossil fueled
boilers and with the assumption that SCWR operation will be based on the two
0 50 100 150 200 250 300 350 400 450 500
5.5
6
6.5
7
7.5
8
8.5
9
Temperature (ºC)
B
r
e
a
k
i
n
g
/
F
r
a
c
t
u
r
e
S
t
r
r
e
n
g
t
h
(
G
P
a
)
SS304
SS308
SS309
SS316
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technologies [6], Fe-Ni-Cr alloys SS304, SS308, SS309 and SS316 were selected. These
materials were assessed for their susceptibility to neutron irradiation and mechanical
strength using the SRIM – TRIM code and LAMMPS code. Since the reactor core may
have a predominantly thermal or fast neutron spectrum depending upon the specific core
design [3, 6], the materials were assessed based on the thermal and fast neutron spectrum.
The neutron irradiation assessment in Section 4.1, Appendix III and Appendix V showed
that both thermal and the fast neutron caused defects in the alloys. It was realized that due
to the high neutron energy in the case of the fast neutron spectrum, it will experience high
penetration. This high penetration (infer from Figure 4.2) which led to high energy transfer
to the Fe-Ni-Cr target electrons (infer from Figure 4.4) which was dissipated later in a form
of heat energy. Also due to the high penetration power in the fast spectrum, the collision
cascade was located deeper under the surface of the target resulting in low sputtering yield
(infer from Fig 4.8). The closer a collision is to the surface, the higher the energy
transferred from the collision cascade to the surface or near surface atom and hence the
higher the sputtering. Therefore the fast neutron spectrum had low sputtering yield as
compared to the thermal and as such will lead to high heat energy dissipation.
The comparative studies of the irradiation damage of the Fe-Ni-Cr alloys under thermal
neutron spectrum of the SCWR in Appendix VII revealed that SS308 as well as SS304
were least damaged by the radiation.
The mechanical parameters extracted from the LAMMPS simulation, Figure 4.11 and
Figure 4.12, confirmed the results from the irradiation damage assessment since SS304 as
well as SS308 had slightly higher mechanical strength than the other two. Thus, as shown
in Figure 4.12, and in Appendix VII, the rate of decrease in the mechanical parameters
studied was slower in these two materials than the other two.
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4.3.2 Discussion on Neutron Irradiation Damage
4.3.2.1 Projected Range
The distance measured along the incident ion trajectory at which the highest concentration
of incident ions are as shown in Appendix V were found to be same, 11.3 µm, except SS309
which experienced 11.4 µm in thermal neutron spectrum. Also when the materials were
exposed to radiation in the fast spectrum SS308 recorded a depth of 32.4 µm while the rest
experienced penetration depth of 32.3 µm. Although there was a variation in the
penetration depth, there was invariably no significant difference in the values.
The results also revealed that the depth of penetration in the fast neutron spectrum would
be thrice that of the thermal and as such much attention should be given to the design of
the SCWR should the fast spectrum be considered.
The values seem insignificant since the structural materials are most often in order of
millimetres but continual exposure of these materials to be penetrated to this depth will
lead to the change in the elastic and plastic properties of the materials through the diffusion
of point defects.
4.3.2.2 Energy Loss to Ionization
Neutrons are uncharged particles, hence do not interact with electrons and therefore do not
directly cause excitation and ionization. They do, however, interact with atomic nuclei,
sometimes liberating charged particles or nuclear fragments that can directly cause
excitation and ionization.
The results revealed in the Figures 4.4 in Section 4.1.4 and Figures 4.16, 4.24, and 4.32 in
the Appendix IV and in Appendix V that the % Energy loss to ionization in the fast
spectrum was about 2% higher than in the thermal spectrum. The energy loss to ionization
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(both from Ions and Recoiling atoms) in the thermal spectrum was low in the SS309 with
a percentage loss of 97.14 % (354.5 MeV) followed by SS316 with a percentage of 97.14
% (354.6 MeV). However, SS308 and SS304 were found to have had high energy loss of
97.15 % (354.6 MeV) and 97.18 % (354.7 MeV) to ionization. And in the fast spectrum,
the least energy loss to ionization was recorded in SS316 and SS304; 99.33% (1807.9
MeV) energy loss was recorded.
Since the difference in the percentage of energy loss to ionization in the fast spectrum was
not different from one another, emphasis would be based on the thermal. That is, the least
energy loss was recorded in SS309 which shows how susceptible it is to mechanical
damage and that can even be testified with the high number of vacancies produced.
4.3.2.3 Fe-Ni-Cr Alloys’ Energy Loss to Phonons
The energy loss to the target phonons was realized to be higher in the thermal than the fast
neutron spectrum reactor design. Figures 4.5 in section 4.1.5 and Fig 4.17, 4.25, and 4.33
in the Appendix IV and in Appendix V. It was seen that in the thermal spectrum, as high
as 2.58% (9.43 MeV) was loss by the ions to phonons in SS316, SS309 and SS308 whereas
the SS304 recorded a loss of 2.56% (9.34 MeV) to Phonons. However in the fast spectrum
all the materials recorded 0.06% energy loss. This implies that there is high vibration in
the thermal spectrum than in the fast spectrum and also the SS304 will not experience
vibrations as high as the other materials in thermal spectrum and that could be seen from
the low number of vacancies produced.
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4.3.2.4 Energy Loss to Vacancies Production in Fe-Ni-Cr Alloys (Target
Damage Energy)
Target damage energy as a result of the vacancies produced in the materials could be seen
in Appendix V and was extracted from all the collision cascade diagrams (Fig 4.1, Fig 4.13,
Fig. 4.21, and Fig 4.29) at the “% Energy Loss” window. It could be seen that there is no
significant variation in the percentage loss in both spectrum.
The only difference was with the percentage recorded in the SS304 which was found to be
0.01 % less. This therefore attest to the lower number of vacancies.
It was revealed that the percentage of displaced atoms that were left as vacancies remained
same for a particular material for both spectrum. It was seen that 96.5% of the atoms that
were displaced in SS304 and SS308 were normally left as vacancies. Also 96.7% of SS316
atoms displaced were left as vacancies and 97 % in the case SS309. This implies that SS304
and SS308 would be least affected by point defects which later agglomerate to form voids
leading to swelling, hence SS309 was seen to be much prone that.
4.3.2.5 Energy to Recoil Cascade
In determining how much energy was transferred to the recoil cascade, all the energies
deposited by the recoils were added up as depicted in Fig. 4.1: (2.55 % + 0.26 % + 2.81
%) = 5.62 % and (0.73 % + 0.06 % + 0.60 %) = 1.39 % of the total energy loss to the
target (SS304) recoil cascade representing 20.4 MeV and 25.3 MeV for the thermal and
fast neutron spectrum respectively. Thus, 94.38 % (344.5 MeV) of the Ion’s energy was
deposited to the Fe-Ni-Cr alloy SS304 and only 5.62 % (20.4 MeV) was given up to the
recoil cascade in the thermal, whereas 98.60 % (1794.5 MeV) was deposited with 1.39
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% (25.3 MeV) given up for the recoil cascade in fast neutron spectrum of the SCWR
respectively.
Similar calculations for the other Fe-Ni-Cr alloys showed that recoils in SS308 deposited
6.10 % (22.3MeV) and 1.33 % (24.2 MeV) (see in Fig. 4.13), SS309 deposited 5.79 %
(20.9 MeV) and 1.35 % (24.6 MeV) (see in Fig. 4.21) and SS316 deposited 6.05 % (22.1
MeV) and 1.39 % (25.3 MeV) (see in Fig. 4.29) with each pair representing the thermal
and fast neutron spectrum respectively.
It can be seen that higher percentage of energy is given up for the recoil cascade in the
thermal neutron spectrum than in the fast spectrum leading to less energy deposition by
the ion in the thermal spectrum than the latter.
4.3.2.6 Sputtering Yield
When a collision cascade intersects the surface, sufficient energy can be transferred to a
surface atom to overcome its binding to the surface, so that it will be ejected from the solid
[96].
The sputtering yield was realized to have decreased from same materials being used in
thermal neutron spectrum to the fast neutron spectrum. The SS316 which recorded the
highest yield of 0.670 Atoms/Ion reduced to 0.150 Atoms/Ion in the fast spectrum. Also
the SS304 which had the least yield of 0.190 Atoms/Ion in the thermal spectrum and then
recorded 0.100 Atoms/Ion in the fast spectrum. It illustrates that sputtering yield is
inversely proportional to the energy of the incident ions and hence the lower the incident
ion’s energy, the higher the sputtering.
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However, comparing the sputtering yield results with the range of sputtering yield for
typical energies of 10-2 < Y< 102, it could be seen that sputtering would not be a serious
damage [97].
4.3.3. Discussion on Mechanical Deterioration
4.3.3.1 Cohesive Energy
The Equilibrium Lattice constant of 3.489 from the simulation in Appendix VI was seen to
be in agreement (only 1 percent difference) with the experimental Equilibrium Lattice
constants of 3.499 for the Potential file adapted [94]. This means that the potential file to
be adapted is good to use in the simulation since the results from literature or theory tallies
statistically with the LAMMPS figure. However the value of the 3.499 was lower than the
values used for the simulations as shown in Table 3.5 because the adapted potential was
designed for FCC Fe-10Ni-20Cr [94].
The cohesive energy of -4.117 in Appendix VI was also in agreement (only 0.3 percent
difference) with the experimental value of -4.120 [94].
The difference 1.0 implies that the potential file was good enough to be used in the
simulations.
4.3.3.2 Young’s Modulus
The comparison of the mechanical properties of the materials results in Appendix VII and
figure 4.12(a) show that Young’s Modulus, which gives the stiffness of the Fe-Ni-Cr alloys
decreased with increasing temperature. It was also realized that the Young’s modulus for
the ambient condition in Table 4.3 was 196 GPa for all the materials, which was within
95% confidence interval for the experimental values from literature in Table 3.1.
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The comparison also revealed in fig. 4.12(a) and in Appendix VII that all the materials had
a decrease in their Young’s Modulus values. The SS304 had the smallest change in its
elastic modulus from 196 GPa to 121 GPa (75 GPa) while SS309 and SS316 had greatest
change in the modulus from 196 GPa to 118 GPa (78 GPa). This therefore implies SS304
is stiffer than the others since it had high value even at the temperature of 500 ºC. [63 and
64] Hence a better material to consider in temperature ranging up to 500 ºC as in the
proposed SCWR.
4.3.3.3 Yield Strength
Figure 4.12 (b) and the comparison results in Appendix VII gives the yield strength of the
materials at 0.2 % offset. The results indicated that the yield strength which is the minimum
stress at which a plastic deformation occur in the materials decreased with an increasing
temperature.
The comparison results showed that SS309 though recorded the highest yield strength at
ambient condition, recorded lower yield than SS304 when it was further exposed to the
SCW condition. Also the highest change in the yield strength was in SS309 material from
9.97 GPa to 5. 30 GPa (4.67 GPa) whereas the lowest change was recorded in the SS304
from 9.88 GPa to 5.54 GPa.
4.3.3.4 Ultimate Tensile Strength
Figure 4.12(c) and in Appendix VII gives the trend of the Ultimate tensile strength (the
maximum stress the material can withstand before fracture) of the Fe-Ni-Cr alloys under
study. It was seen that the UTS values for the materials also decreased with increasing
temperature. However, the values in Appendix VII were not in agreement with the
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experimental values in Table 3.1 for the Ambient Conditions. And this might be that we
had perfect crystal arrangements with heterogeneous nucleation.
The Ultimate Tensile Strength of the materials were found to be decreasing in the same
trend as the Young’s Modulus for all the four materials considered. SS308 had the least
change of about 4.19 GPa in strength as the temperature increased while SS309 had a fast
decrease in UTS from the ambient to the SCW conditions with change of 4.42 GPa. But in
general the values registered were of the same order. Also the small change in UTS value
of the SS308 indicates that it would not fail as quickly as it is being exposed to extremely
harsh conditions.
4.3.3.5 Breaking or Fracture Strength
The results on the breaking strength, which is the strength at which the material fracture, is shown
in Figure 4.12 (c) in Appendix VII. It was realized that the Breaking Strength decreased when the
conditions in the reactor became harsh. Thus when the materials were subjected to ambient
conditions (pressure = 0.01MPa (1 atm) and temperature = 27 ºC) and the SCW condition
(pressure= 25 MPa and temperature range of 300 – 500 ºC), the breaking strength decreased
with an increase in temperature since the pressure was kept constant at the SCW condition.
Though the materials’ strength under all the conditions looked same, it was realized that
the SS304 had the highest value corresponding to rupture of the materials when the
temperature was increased to 500 ºC and that can be seen in Table 4.3 in Appendix VI.
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4.3.4. Discussions on Coupling Neutron Irradiation Damage and Mechanical
Deterioration
The research revealed in the two damage assessments that generally the Fe-Ni-Cr alloys
possess similar behavior under irradiation and stress. However, SS304, could be seen in
Appendix IV as stronger than the other three materials and hence was less damaged.
The results from the irradiation damage assessment was validated by the LAMMPS code
since materials like SS304 and SS308 after the irradiation, were seen to have had a very
strong. That is, 96.5 % of the atoms that were displaced in SS304 and SS308 were normally
left as vacancies. Also 96.7 % of SS316 atoms displaced were left as vacancies and 97 %
in the case SS309.
Also from Fig. 4.11 and 4.12, SS308 and S304 were seen throughout as possessing higher
strength. It was evidenced by the fact that there the percentage of atoms vacancies produced
from the number of displacements were low and that means that such materials may not
experience swelling.
Hence, the need to consider the two materials for in-core structural materials in the SCWR.
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CHAPTER FIVE: CONCLUSIONS AND RECOMMENDATIONS
5.1. CONCLUSIONS
The goal of the research was to examine neutron irradiation damage and mechanical
degradation of Fe-Cr-Ni alloys (SS304, SS308, SS309 and SS316) by high neutron dose
leading to 10 – 30 dpa and 100 – 150 dpa. Thermal and fast neutron bombardments at high
pressure and temperature conditions as would pertain in SCW conditions were assessed
using Binary Collision Approximation by SRIM-TRIM and Molecular Dynamics
simulations by LAMMPS code respectively along with VMD and MATLAB. The research
was successful and the conclusions drawn from the outcomes of the thesis were as follows:
The fast neutron irradiation damage assessment revealed very deep penetration. The
incidence ion penetration in the four Fe-Ni-Cr alloys ranged from 11.3 µm to 11.4
µm in the thermal neutron spectrum, the penetration ranged from 32.3 µm to 32.4
µm in the fast spectrum. The difference in penetration depth, for neutron spectrum
was significant and the depth of penetration of fast neutrons as compared to the
thermal neutrons was about three times that of the thermal neutron spectrum.
The SS304 required as high as 97.18 % (354.7 MeV) of the Ion’s energy for
Ionization whiles for SS309 and SS316 required 97.14 % (354.5 MeV) to be ionized
in the thermal spectrum. And in the fast spectrum, the least energy loss to ionization
was recorded in SS316 and SS304 with 99.33 % (1807.8 MeV) energy loss. There
was therefore a marginal difference in energy required to ionize the alloys.
The energy given off by the recoil atoms as phonons was higher in the thermal
neutron spectrum than the fast spectrum. But percentage of energy loss of 2.56 %
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96
recorded by SS304 which was lower than the others implies minimal vibration and
hence will lead to fewer atomic displacements.
The number of defects created by each projectile and energy used up in the
production of such defects showed that more energy was used up in the defects
creation in the thermal spectrum than in the fast. Also the SS304 and SS308 had
the least percentage of displacements left as vacancies.
The assessment also revealed that the SS316 had high sputtering yield in both
spectrum, and hence might experience pitting corrosion.
The evaluation of mechanical deterioration also revealed that Young’s, Ultimate
Tensile Strength and the Breaking/ Fracture Strength decreased with increasing
temperature.
The SS304 was found to have high mechanically strength than the other alloys. The
strength of the SS304 decreased less than the others when the conditions were
changed from the ambient to the supercritical water condition.
The ultimate tensile strengths, yield strength and breaking strengths were greater
than the experimental values.
Comparing the two methods of assessments, the alloys that performed well in the
BCA simulation, equally possessed high mechanical strength from the MD
simulation.
From the damage assessment and the evaluation process SS304 and SS308 were of
higher strength than the other alloys and hence could work with least damage under
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SCW condition and could be considered in the design of the SCWR internals and
pressure vessel.
5.2. RECOMMENDATIONS
Further research work on designing the TRIM.DAT input deck using the computer
program, Monte Carlo Neutron Program (MCNP code), which is the code widely used for
the transport of Neutron particles through matter should be embarked on. This will ensure
that uncertainties in the direction, energy and depth of particles are captured to improve the
accuracy of the simulation.
It is also recommended that students who go through the Radiation Damage and Corrosion
Models in Reactor Materials course be given demonstrations on special codes such as
SRIM-TRIM, ALICE, PHITS, MARS, FLUKA, LAMMPS, AMBER, CHARMM, VMD,
and ovito codes since these are the current and efficient codes for material analysis.
Additional research work should be done on the two Fe-Ni-Cr alloys, SS304 and 308 to
examine on hydrogen embrittlement, swelling and creep, and interactions with the
supercritical water environment; an extensive testing and evaluation program is required
to assess the corrosion effects that will be seen on the properties of these potential SCWR
materials.
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APPENDICES
APPENDIX I
TRIM.DAT Input Deck For SRIM-TRIM Simulation of Neutron Irradiation
Damage Assessment of Fe-Ni-Cr Alloys
Ûßßßßßßßßß TRIM with various Incident Ion Energies/Angles and Depths ßßßßßßßßßßßßßÛ
Û Top 10 lines are user comments, with line #8 describing experiment. Û
Û Line #8 will be written into all TRIM output files (various files: *.TXT). Û
Û Data Table line consist of: EventName (5 char) +8 numbers separated by spaces. Û
Û The Event Name consists of any 5 characters to identify that line.
Û
Û Cos(X) = 1 for normal incidence, and Cos(X) = -1 for backwards.
ßßßßßßßßßßßßßßßßßßßß Typical Data File is shown below ßßßßßßßßßßßßßßßßßßßßßßßßßßß
ÉÍ Neutron Ions from Uranium into Stainless Steel 0.46m thick (Energies 1GeV), Various Angles)
Event Atom Energy Depth Lateral-Position ----- Atom Direction ----
Name Numb (eV) X_(A) Y_(A) Z_(A) Cos(X) Cos(Y) Cos(Z)
A-1 92 1E9 7E5 3.5E4 0 1.00000 -.002000 -.100010
A-2 92 1E9 7E5 3.5E4 0 1.00000 -.022000 -.100020
A-3 92 1E9 7E5 3.5E4 0 1.00000 -.300000 -.100030
A-4 92 1E9 7E5 3.5E4 0 1.00000 -.000000 -.100040
A-5 92 1E9 7E5 3.5E4 0 1.00000 .000000 -.100050
A-6 92 1E9 7E5 3.5E4 0 1.00000 -.000000 -.100060
A-7 92 1E9 7E5 3.5E4 0 1.00000 -.000000 -.100070
A-8 92 1E9 7E5 3.5E4 0 1.00000 -.000000 -.100080
A-9 92 1E9 7E5 3.5E4 0 1.00000 -.000000 -.100090
A-10 92 1E9 7E5 3.5E4 0 1.00000 -.000000 -.100100
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
A-91 92 1E9 7E5 3.5E4 0 1.00000 -.000000 -.131000
A-92 92 1E9 7E5 3.5E4 0 1.00000 -.000000 -.132000
A-93 92 1E9 7E5 3.5E4 0 1.00000 -.000000 -.133000
A-94 92 1E9 7E5 3.5E4 0 1.00000 -.000000 -.134000
A-95 92 1E9 7E5 3.5E4 0 1.00000 -.000000 -.135000
A-96 92 1E9 7E5 3.5E4 0 1.00000 -.000000 -.136000
A-97 92 1E9 7E5 3.5E4 0 1.00000 -.000000 -.137000
A-98 92 1E9 7E5 3.5E4 0 1.00000 -.000000 -.138000
A-99 92 1E9 7E5 3.5E4 0 1.00000 -.000000 -.139000
A-100 92 1E9 7E5 3.5E4 0 1.00000 -.000000 -.140000
͢
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APPENDIX II
Input Files for Molecular Dynamics Simulation of Mechanical Damage Assessment
a) Potential File
The potential file FeNiCr.eam.alloy modified for the simulation was downloaded from
NIST Interatomic Potentials Repository Project (http://www.ctcms.nist.gov/potentials)
b) Input file for the Cohesive energy determination
# Determination of the cohesive energy and equilibrium lattice constant of the FeNiCr.eam.alloy potential with fcc
configuration Adapted from Mark Tschopp, 2010
#By Collins Nana Andoh (10443957)
# ---------- Initialize Simulation ---------------------
clear
units metal
dimension 3
boundary p p p
atom_style atomic
atom_modify map array
# ---------- Create Atoms ---------------------
lattice fcc 4
region box block 0 1 0 1 0 1 units lattice
create_box 1 box
lattice fcc 4 orient x 1 0 0 orient y 0 1 0 orient z 0 0 1
create_atoms 1 box
replicate 1 1 1
# ---------- Define Interatomic Potential ---------------------
pair_style eam/alloy
pair_coeff * * FeNiCr.eam.alloy.u3 Fe
neighbor 2.0 bin
neigh_modify delay 10 check yes
# ---------- Define Settings ---------------------
compute eng all pe/atom
compute eatoms all reduce sum c_eng
# ---------- Run Minimization ---------------------
reset_timestep 0.001
fix 1 all box/relax iso 0.0 vmax 0.001
thermo 10
thermo_style custom step pe lx ly lz press pxx pyy pzz c_eatoms
min_style cg
minimize 1e-25 1e-25 5000 100000
run 0
variable natoms equal "count(all)"
variable teng equal "c_eatoms"
variable teng equal "pe"
variable length equal "lx"
variable ecoh equal "v_teng/v_natoms"
print "Total energy (eV) = ${teng};"
print "Number of atoms = ${natoms};"
print "Lattice constant (Angstoms) = ${length};"
print "Cohesive energy (eV) = ${ecoh};"
print "All done!"
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c) A copy of the 16 Input Files used for this work
# This program is aimed at evaluating the mechanical #
# Integrity (Youngs modulus, Utimate tensile Strength, #
# Fracture point) of SS 304 treated under Ambient Temperature #
# condition #
# Adapted from materials developed by Mark A. Tschopp #
# (US ARL) and hosted at https://icme.hpc.msstate.edu #
# Designed By: Collins Nana Andoh 1044395 2015 #
####################################################################
# ---------- Initialize Simulation ---------------------
clear
units metal
dimension 3
boundary p p p
atom_style atomic
# ---------- Create Atoms ---------------------
lattice fcc 3.5918
region new_region block 0 10 0 10 0 10
create_box 1 new_region
lattice fcc 3.5918 orient x 1 0 0 orient y 0 1 0 orient z 0 0 1
create_atoms 1 region new_region
replicate 1 1 1
# ---------- Define Interatomic Potential ---------------------
pair_style eam/alloy
pair_coeff * * FeNiCr.eam.alloy.u3 Fe
neighbor 2.0 bin
neigh_modify delay 0 every 10 check yes
# ---------- Define Settings ---------------------
compute csym all centro/atom fcc
compute eng all pe/atom
# ---------- Equilibration---------------------
#reset timer
reset_timestep 0
#2 fs time step
timestep 0.002
#initial velocities
velocity all create 300 12345 mom yes rot no
#thermostat + barostat (1 degree= 273 K and 1 MPa= 10 bar
fix 1 all npt temp 473 473 2 iso 250 250 1 drag 1.0
# instrumentation and output
variable s1 equal "time"
variable s2 equal "lx"
variable s3 equal "ly"
variable s4 equal "lz"
variable s5 equal "vol"
variable s6 equal "press"
variable s7 equal "pe"
variable s8 equal "ke"
variable s9 equal "etotal"
variable s10 equal "temp"
fix writer all print 250 "${s1} ${s2} ${s3} ${s4} ${s5} ${s6} ${s7} ${s8} ${s9} ${s10}" #file Fe_eq.txt screen no
#thermo
thermo 500
thermo_style custom step time cpu cpuremain lx ly lz press pe temp
#dumping trajectory
dump 1 all atom 250 dump.eq.lammpstrj
#24 ps MD Simulation (assuming 2 fs time step)
run 12000
#clearing fixes and dumps
unfix 1
undump 1
#saving equilibrium length for strain calculation
variable tmp equal "lx"
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variable L0 equal ${tmp}
print "Initial Length, L0: ${L0}"
#------------------DEFORMATION-----------------------
#reset timer
reset_timestep 0
#2 fs time step
timestep 0.002
# thermostat + barostat
fix 1 all npt temp 300 300 1 y 0 0 1 z 0 0 1 drag 1.0
#nonequilibrium straining in x-direction at strain rate = 5e-3
variable srate1 equal 5e-3
fix 2 all deform 1 x erate ${srate1} units box remap x
#instrumentation and output for units metal, pressure is in #[bars] = 100 [kPa]= 1/10000 [GPa] => p2, p3, p4, are in GPa
variable strain equal "(lx - v_L0)/v_L0"
variable p1 equal "v_strain"
variable p2 equal "-pxx/10000"
variable p3 equal "-pyy/10000"
variable p4 equal "-pzz/10000"
fix writer all print 125 "${p1} ${p2} ${p3} ${p4}" file Fe.deform.txt screen no
#thermo
thermo 1000
thermo_style custom step cpuremain v_strain v_p2 v_p3 v_p4 press pe temp
#dumping standard atom trajectrories
dump 1 all atom 5000 dump.deform.lammpstrj
#dumping custom cfg files containing coords + ancillary variables
dump 2 all cfg 5000 dump.deform_*.cfg mass type xs ys zs c_csym c_eng fx fy fz
dump_modify 2 element Fe
#40 ps MD Simulation (assuming 2 fs time step)
run 20000
# clearing fixes and dumps
unfix 1
unfix 2
unfix writer
undump 1
undump 2
########################
print "All done"
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APPENDIX III
Algorithm for Animation of Tensile Deformation Using VMD
1. Open the VMD program
2. In the Main VMD window, click on file
3. In the file menu, select new molecules
4. In the Molecule file browser, browse for the dumb file (dump.deform.lammpstrj),
select file type(LAMMPS trajectory) and load
5. In the Main VMD window, click on the Graphics and select Representation
6. In the Graphical Representation dialog box, select Name, VDW and Opaque in the
Colouring Method, Drawing Method and Material slot and then click on the apply.
7. In the Main VMD window, click on the play bottom to give you the animation of
the deformation.
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APPENDIX IV
SRIM-TRIM Simulation Output Spectra for Neutron Irradiation Damage
Assessment
i) SS308
Fig 4.13(a)
Fig 4.13(b)
Fig. 4.13: Collision Cascade for (a) thermal and (b) fast neutron damage for SS308
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Fig 4.14 (a) Fig 4.14 (b)
Fig 4.14: Depth of penetration of (a) thermal and (b) fast neutrons in the SS308
Fig 15 (a) Fig 15 (b)
Fig 4.15: Lateral Range Distribution of (a) thermal and (b) fast neutrons in the SS308
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Fig 4.16(a) Fig 4.16(b)
Fig 16(c)
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Fig. 16(d)
Fig 4.16: 2D and 3D view of Ionization energy distribution of the Fe-Ni-Cr alloy SS308 in the
both thermal and fast neutron spectrum
Fig 4.17 (a) Fig 4.17 (b)
Fig 4.17: Distribution of Energy Loss as Phonons by the Fe-Ni-Cr Alloy SS308 in the (a)
thermal and (b) fast neutron spectrum
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Fig. 4.18(a) Fig. 4.18(b)
Fig 4.18: Plot of Energy absorbed by each element in the SS308 in the (a) thermal and (b) fast
neutron spectrum.
Fig 4.19(a) Fig 4.19(b)
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Fig 4.19(c)
Fig 4.19(d)
Fig 4.19: Collision events of SS308 in 2D and 3D view respectively in the thermal and
fast neutron spectrum
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Fig 4.20(a) Fig 4.20(b)
Fig 4.20: Plots of integral sputtering yield of SS308 in (a) thermal and (b) fast neutron
spectrum
ii) SS309
Fig. 4.21(a)
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Fig 4.21(b)
Fig. 4.21: Collision cascades for (a) thermal and (b) fast neutron irradiation damage in S309
Fig 4.22 (a) Fig 4.22 (b)
Fig 4.22: Depth of penetration of (a) thermal and (b) fast neutron in the SS309
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Fig 4.23(a) Fig 4.23(b)
Fig 4.23: Lateral Range Distribution of (a) thermal and (b) fast neutron in the SS309
Fig 4.24(a) Fig 4.24(b)
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Fig. 4.24(c)
Fig. 4.24(d)
Fig 4.24: 2D and 3D view of Ionization energy distribution of the Fe-Ni-Cr alloy SS309 in the
thermal and fast4neutron spectrum
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Fig 4.25(a) Fig 4.25(b)
Fig 4.25: The Distribution of Energy Loss as Phonons by the Fe-Ni-Cr Alloy SS309 in the (a)
thermal and (b) fast neutron irradiation
Fig. 4.26(a) Fig. 4.26(b)
Fig 4.26: Plot of energy absorbed by elements of the SS309 Fe-Ni-Cr Alloys in the (a) thermal
and (b) fast neutron spectrum.
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Fig 4.27(a) Fig 4.27(b)
Fig. 4.27(c)
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Fig 4.27(d)
Fig. 4.27: Collision events of SS309 in 2D and 3D view in the thermal and fast neutron
spectrum
Fig 4.28(a) Fig 4.28(b)
Fig 4.28: Plot of integral sputtering yield of SS309 in both spectrum
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iii) SS316
Fig. 4.29 (a)
Fig 4.29(b)
Fig. 29: Collision Cascade for (a) thermal and fast neutron irradiation damage in Fe-Ni-Cr
alloy SS316
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Fig 4.30 (a) Fig 4.30(b)
Fig 4.30: Projected Range Distribution of (a) thermal and (b) fast neutron in the Fe-Ni-Cr Alloy
SS316
Fig 4.31(a) Fig 4.31(b)
Fig 4.31: Lateral Range Distribution of (a) thermal and (b) fast neutrons in SS316
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Fig 4.32(a) Fig 4.32(b)
Fig 4.32(c)
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Fig 4.32(d)
Fig 4.32: 2D and 3D view of Ionization energy distribution of SS316 in the thermal and fast
neutron spectrum
Fig 4.33(a) Fig 4.33(b)
Fig 4.33: The Distribution of Energy Loss as Phonons in SS316 by (a) thermal and (b) fast
neutron irradiation
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Fig 4.34(a) Fig 4.34(b)
Fig 4.34: Plots of energy absorbed by elements in the SS309 in the (c) thermal and (d) fast
neutron spectrum.
Fig 4.35(a) Fig 4.35(b)
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Fig 4.35(c)
Fig 4.35(d)
Fig 4.35: Collision events of SS316 in 2D and 3D view in the thermal and fast neutron
spectrum
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Fig 4.36(a) Fig 4.36(b)
Fig 4.36: Plots of integral sputtering yield of SS316 in both spectrum
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APPENDIX V
Comparison of the Irradiation Damage of All Fe-Ni-Cr Alloys Both under Thermal and Fast Neutron Spectrum of the SCWR
DAMAGE TYPE
THERMAL NEUTRON SPECTRUM FAST NEUTRON SPECTRUM
SS304 SS308 SS309 SS316 SS304 SS308 SS309 SS316
Ion Projected Range in
Target (µm)
11.3 11.3 11.4 11.3 32.3 32.4 32.3 32.3
Target Displacement
(/Ion)
337242 339845 340557 340270 394519 392453 394733 395958
Replacement Collisions
(/Ion)
11963 11899 10244 11255 13991 13764 11841 13101
Target Vacancies
(/Ion) 325279 327946 330312 329015 380528 378688 382893 382856
Ion’s Energy to the
Target Electrons
(Ionization) (keV/Ion)
354684.3
(97.18%)
354607.7
(97.15%)
354573.1
(97.14%)
354586.6
(97.14%)
1807893.2
(99.33%)
1807965.8
(99.34%)
1807878.1
(99.34%)
1807847.4
(99.33%)
Ion’s Energy loss to the
Target Phonons
(keV/Ion)
9340.3
(2.56%)
9408.8
(2.58%)
9436.3
(2.58%)
9426.7
(2.58%)
10965.3
(0.60%)
10898.2
(0.60%)
10973.3
(0.60%)
11004.2
(0.60%)
Total Target Damage
Energy (keV/Ion)
983.13
(0.26%)
983.48
(0.27%)
990.59
(0.27%)
986.71
(0.27%)
1141.47
(0.06%)
1135.95
(0.06%)
1148.57
(0.06%)
1148.46
(0.06%)
Sputtering Yield
(Atoms/Ion)
0.190 0.380 0.440 0.670 0.100 0.020 0.030 0.150
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APPENDIX VI
Output Files of Molecular Dynamics Simulation of Mechanical Damage Assessment
a) Cohesive energy and equilibrium lattice parameter simulation output file
LAMMPS (30 Sep 2014-ICMS)
WARNING: OMP_NUM_THREADS environment is not set. (../comm.cpp:88)
using 1 OpenMP thread(s) per MPI task
# Determination of the cohesive energy and equilibrium lattice constants of the FeNiCr.eam.alloy potential wit fcc
configuration Adapted from Mark Tschopp, 2010
#By Collins Nana Andoh(10443957)
# ---------- Initialize Simulation ---------------------
clear
WARNING: OMP_NUM_THREADS environment is not set. (../comm.cpp:88)
using 1 OpenMP thread(s) per MPI task
units metal
dimension 3
boundary p p p
atom_style atomic
atom_modify map array
# ---------- Create Atoms ---------------------
lattice fcc 4
Lattice spacing in x,y,z = 4 4 4
region box block 0 1 0 1 0 1 units lattice
create_box 1 box
Created orthogonal box = (0 0 0) to (4 4 4)
1 by 1 by 1 MPI processor grid
lattice fcc 4 orient x 1 0 0 orient y 0 1 0 orient z 0 0 1
Lattice spacing in x,y,z = 4 4 4
create_atoms 1 box
Created 4 atoms
replicate 1 1 1
orthogonal box = (0 0 0) to (4 4 4)
1 by 1 by 1 MPI processor grid
4 atoms
# ---------- Define Interatomic Potential ---------------------
pair_style eam/alloy
pair_coeff * * FeNiCr.eam.alloy.u3 Fe
neighbor 2.0 bin
neigh_modify delay 10 check yes
# ---------- Define Settings ---------------------
compute eng all pe/atom
compute eatoms all reduce sum c_eng
# ---------- Run Minimization ---------------------
reset_timestep 0.001
fix 1 all box/relax iso 0.0 vmax 0.001
thermo 10
thermo_style custom step pe lx ly lz press pxx pyy pzz c_eatoms
min_style cg
minimize 1e-25 1e-25 5000 100000
WARNING: Resetting reneighboring criteria during minimization (../min.cpp:168)
Memory usage per processor = 3.4108 Mbytes
Step PotEng Lx Ly Lz Press Pxx Pyy Pzz eatoms
0 -14.730235 4 4 4 -94068.38 -94068.38 -94068.38 -94068.38 -14.730235
10 -14.844402 3.96 3.96 3.96 -99515.996 -99515.996 -99515.996 -99515.996 -14.844402
20 -14.966592 3.92 3.92 3.92 -111956.11 -111956.11 -111956.11 -111956.11 -14.966592
30 -15.106693 3.88 3.88 3.88 -136808.6 -136808.6 -136808.6 -136808.6 -15.106693
40 -15.280204 3.84 3.84 3.84 -175730.14 -175730.14 -175730.14 -175730.14 -15.280204
50 -15.492881 3.8 3.8 3.8 -210627.33 -210627.33 -210627.33 -210627.33 -15.492881
60 -15.725423 3.76 3.76 3.76 -218502.3 -218502.3 -218502.3 -218502.3 -15.725423
70 -15.945101 3.72 3.72 3.72 -196772.77 -196772.77 -196772.77 -196772.77 -15.945101
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80 -16.128136 3.68 3.68 3.68 -158787.1 -158787.1 -158787.1 -158787.1 -16.128136
90 -16.267326 3.64 3.64 3.64 -118869.93 -118869.93 -118869.93 -118869.93 -16.267326
100 -16.365806 3.6 3.6 3.6 -82373.473 -82373.473 -82373.473 -82373.473 -16.365806
110 -16.429061 3.56 3.56 3.56 -49883.162 -49883.162 -49883.162 -49883.162 -16.429061
120 -16.461675 3.52 3.52 3.52 -19192.249 -19192.249 -19192.249 -19192.249 -16.461675
130 -16.466526 3.4986965 3.4986965 3.4986965 1.3519338e-009 1.3509121e-009 1.3550405e-009 1.3498488e-009 -
16.466526
Loop time of 0.0636024 on 1 procs for 130 steps with 4 atoms
73.7% CPU use with 1 MPI tasks x 1 OpenMP threads
Minimization stats:
Stopping criterion = energy tolerance
Energy initial, next-to-last, final =
-14.730235479 -16.4665256808 -16.4665256808
Force two-norm initial, final = 11.2729 1.25995e-013
Force max component initial, final = 11.2729 1.23948e-013
Final line search alpha, max atom move = 1 1.23948e-013
Iterations, force evaluations = 130 135
MPI task timings breakdown:
Section | min time | avg time | max time |%varavg| %total
---------------------------------------------------------------
Pair | 0.013032 | 0.013032 | 0.013032 | 0.0 | 20.49
Neigh | 0 | 0 | 0 | 0.0 | 0.00
Comm | 0.0016911 | 0.0016911 | 0.0016911 | 0.0 | 2.66
Output | 0.0054446 | 0.0054446 | 0.0054446 | 0.0 | 8.56
Modify | 0 | 0 | 0 | 0.0 | 0.00
Other | | 0.04344 | | | 68.29
Nlocal: 4 ave 4 max 4 min
Histogram: 1 0 0 0 0 0 0 0 0 0
Nghost: 662 ave 662 max 662 min
Histogram: 1 0 0 0 0 0 0 0 0 0
Neighs: 496 ave 496 max 496 min
Histogram: 1 0 0 0 0 0 0 0 0 0
Total # of neighbors = 496
Ave neighs/atom = 124
Neighbor list builds = 0
Dangerous builds = 0
run 0
Memory usage per processor = 2.42293 Mbytes
Step PotEng Lx Ly Lz Press Pxx Pyy Pzz eatoms
130 -16.466526 3.4986965 3.4986965 3.4986965 1.3519338e-009 1.3509121e-009 1.3550405e-009 1.3498488e-009 -
16.466526
Loop time of 1.57894e-006 on 1 procs for 0 steps with 4 atoms
0.0% CPU use with 1 MPI tasks x 1 OpenMP threads
MPI task timings breakdown:
Section | min time | avg time | max time |%varavg| %total
---------------------------------------------------------------
Pair | 0 | 0 | 0 | 0.0 | 0.00
Neigh | 0 | 0 | 0 | 0.0 | 0.00
Comm | 0 | 0 | 0 | 0.0 | 0.00
Output | 0 | 0 | 0 | 0.0 | 0.00
Modify | 0 | 0 | 0 | 0.0 | 0.00
Other | | 1.579e-006 | | |100.00
Nlocal: 4 ave 4 max 4 min
Histogram: 1 0 0 0 0 0 0 0 0 0
Nghost: 1094 ave 1094 max 1094 min
Histogram: 1 0 0 0 0 0 0 0 0 0
Neighs: 736 ave 736 max 736 min
Histogram: 1 0 0 0 0 0 0 0 0 0
Total # of neighbors = 736
Ave neighs/atom = 184
Neighbor list builds = 0
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Dangerous builds = 0
variable natoms equal "count(all)"
variable teng equal "c_eatoms"
variable teng equal "pe"
variable length equal "lx"
variable ecoh equal "v_teng/v_natoms"
print "Total energy (eV) = ${teng};"
Total energy (eV) = -16.4665256808311;
print "Number of atoms = ${natoms};"
Number of atoms = 4;
print "Lattice constant (Angstoms) = ${length};"
Lattice constant (Angstoms) = 3.49869654884664;
print "Cohesive energy (eV) = ${ecoh};"
Cohesive energy (eV) = -4.11663142020778;
print "All done!"
All done!
b) A copy of the 16 Output Files from the Mechanical Damage Assessment
LAMMPS (30 Sep 2014-ICMS)
WARNING: OMP_NUM_THREADS environment is not set. (../comm.cpp:88)
using 1 OpenMP thread(s) per MPI task
# This program is aimed at evaluating the mechanical #
# integrity of (Youngs modulus, Utimate tensile Strength, #
# Fracture point)SS 308 treated under Ambient Temperature #
# condition #
# Adapted from materials developed by Mark A. Tschopp #
# (US ARL) and hosted at https://icme.hpc.msstate.ed #
# Designed By:
# Collins Nana Andoh #
# (10443957) #
# JULY 2015 #
###############################################################
# ---------- Initialize Simulation ---------------------
clear
WARNING: OMP_NUM_THREADS environment is not set. (../comm.cpp:88)
using 1 OpenMP thread(s) per MPI task
units metal
dimension 3
boundary p p p
atom_style atomic
# ---------- Create Atoms ---------------------
lattice fcc 3.5918
Lattice spacing in x,y,z = 3.5918 3.5918 3.5918
region new_region block 0 10 0 10 0 10
create_box 1 new_region
Created orthogonal box = (0 0 0) to (35.918 35.918 35.918)
1 by 1 by 1 MPI processor grid
lattice fcc 3.5918 orient x 1 0 0 orient y 0 1 0 orient z 0 0 1
Lattice spacing in x,y,z = 3.5918 3.5918 3.5918
create_atoms 1 region new_region
Created 4000 atoms
replicate 1 1 1
orthogonal box = (0 0 0) to (35.918 35.918 35.918)
1 by 1 by 1 MPI processor grid
4000 atoms
# ---------- Define Interatomic Potential ---------------------
pair_style eam/alloy
pair_coeff * * FeNiCr.eam.alloy.u3 Fe
neighbor 2.0 bin
neigh_modify delay 0 every 10 check yes
# ---------- Define Settings ---------------------
compute csym all centro/atom fcc
compute eng all pe/atom
# ---------- Equilibration---------------------
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#reset timer
reset_timestep 0
#2 fs time step
timestep 0.002
#initial velocities
velocity all create 300 12345 mom yes rot no
#thermostat + barostat (1 degree= 273 K and 1 MPa= 10 bar
fix 1 all npt temp 300 300 1 iso 0 0 1 drag 1.0
# instrumentation and output
variable s1 equal "time"
variable s2 equal "lx"
variable s3 equal "ly"
variable s4 equal "lz"
variable s5 equal "vol"
variable s6 equal "press"
variable s7 equal "pe"
variable s8 equal "ke"
variable s9 equal "etotal"
variable s10 equal "temp"
fix writer all print 250 "${s1} ${s2} ${s3} ${s4} ${s5} ${s6} ${s7} ${s8} ${s9} ${s10}" #file Fe_eq.txt screen no
# thermo
thermo 500
thermo_style custom step time cpu cpuremain lx ly lz press pe temp
#dumping trajectory
dump 1 all atom 250 dump.eq.lammpstrj
#24 ps MD Simulation (assuming 2 fs time step)
run 12000
Memory usage per processor = 3.83823 Mbytes
Step Time CPU CPULeft Lx Ly Lz Press PotEng Temp
0 0 0 0 35.918 35.918 35.918 -71838.912 -16381.467 300
500 1 18.584189 427.43641 35.034852 35.034852 35.034852 450.24634 -16387.164 161.01856
1000 2 36.363724 400.00099 35.040522 35.040522 35.040522 143.52865 -16381.732 173.20264
1500 3 57.872914 405.11042 35.047976 35.047976 35.047976 -740.27458 -16376.154 185.43939
2000 4 78.080024 390.40013 35.054911 35.054911 35.054911 -920.00382 -16372.052 200.44801
2500 5 96.593284 367.05449 35.053863 35.053863 35.053863 -510.4307 -16366.336 212.00241
3000 6 115.86609 347.59828 35.057679 35.057679 35.057679 -325.06191 -16360.184 222.06993
3500 7 133.09892 323.24025 35.055293 35.055293 35.055293 620.05444 -16355.15 233.37309
4000 8 147.4487 294.89741 35.058801 35.058801 35.058801 662.6319 -16349.486 242.27498
4500 9 163.50968 272.51614 35.060156 35.060156 35.060156 695.84364 -16347.193 256.23858
5000 10 180.13933 252.19506 35.073473 35.073473 35.073473 -396.87126 -16341.825 262.64355
5500 11 191.12924 225.88002 35.079207 35.079207 35.079207 -541.74259 -16338.201 270.64093
6000 12 197.72709 197.7271 35.079115 35.079115 35.079115 -55.798557 -16333.099 273.92674
6500 13 211.83854 179.248 35.068296 35.068296 35.068296 1470.9688 -16331.589 282.26106
7000 14 229.76504 164.11789 35.081638 35.081638 35.081638 392.05289 -16327.514 283.7951
7500 15 250.78482 150.47089 35.0801 35.0801 35.0801 369.02822 -16328.686 293.61679
8000 16 268.44122 134.22061 35.084015 35.084015 35.084015 48.125551 -16326.448 295.17463
8500 17 285.57829 117.59106 35.082331 35.082331 35.082331 231.76867 -16323.385 293.54886
9000 18 302.41941 100.80647 35.087561 35.087561 35.087561 -121.07472 -16322.314 294.26339
9500 19 319.33058 84.034366 35.092638 35.092638 35.092638 -1139.8359 -16325.895 302.6849
10000 20 336.19715 67.23943 35.082182 35.082182 35.082182 763.536 -16323.02 297.69095
10500 21 353.11178 50.44454 35.078052 35.078052 35.078052 873.88557 -16327.591 306.60518
11000 22 369.90504 33.627731 35.085554 35.085554 35.085554 365.48526 -16325.715 302.87981
11500 23 386.7937 16.817118 35.086145 35.086145 35.086145 -255.94453 -16323.474 298.43594
12000 24 403.63164 0 35.085311 35.085311 35.085311 -31.053508 -16324.877 301.05577
Loop time of 403.632 on 1 procs for 12000 steps with 4000 atoms
97.3% CPU use with 1 MPI tasks x 1 OpenMP threads
Performance: 5.137 ns/day 4.672 hours/ns 29.730 timesteps/s
MPI task timings breakdown:
Section | min time | avg time | max time |%varavg| %total
---------------------------------------------------------------
Pair | 391.82 | 391.82 | 391.82 | -1.$ | 97.07
Neigh | 0.049093 | 0.049093 | 0.049093 | -1.$ | 0.01
Comm | 1.6115 | 1.6115 | 1.6115 | 0.0 | 0.40
Output | 0.93485 | 0.93485 | 0.93485 | 0.0 | 0.23
Modify | 8.6272 | 8.6272 | 8.6272 | 0.0 | 2.14
Other | | 0.5938 | | | 0.15
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Nlocal: 4000 ave 4000 max 4000 min
Histogram: 1 0 0 0 0 0 0 0 0 0
Nghost: 8195 ave 8195 max 8195 min
Histogram: 1 0 0 0 0 0 0 0 0 0
Neighs: 347899 ave 347899 max 347899 min
Histogram: 1 0 0 0 0 0 0 0 0 0
FullNghs: 0 ave 0 max 0 min
Histogram: 1 0 0 0 0 0 0 0 0 0
Total # of neighbors = 347899
Ave neighs/atom = 86.9748
Neighbor list builds = 1
Dangerous builds = 0
#clearing fixes and dumps
unfix 1
undump 1
#saving equilibrium length for strain calculation
variable tmp equal "lx"
variable L0 equal ${tmp}
variable L0 equal 35.0853114166038
print "Initial Length, L0: ${L0}"
Initial Length, L0: 35.0853114166038
#------------------DEFORMATION-----------------------
#reset timer
reset_timestep 0
#2 fs time step
timestep 0.002
# thermostat + barostat
fix 1 all npt temp 300 300 1 y 0 0 1 z 0 0 1 drag 1.0
#nonequilibrium straining in x-direction at strain rate = 1x 10^10 / s = 1x10^2 / ps in units metal
#variable srate equal 1.0e10
variable srate1 equal 5e-3
fix 2 all deform 1 x erate ${srate1} units box remap x
fix 2 all deform 1 x erate 0.005 units box remap x
#instrumentation and output for units metal, pressure is in #[bars] = 100 [kPa]= 1/10000 [GPa] => p2, p3, p4, are in GPa
variable strain equal "(lx - v_L0)/v_L0"
variable p1 equal "v_strain"
variable p2 equal "-pxx/10000"
variable p3 equal "-pyy/10000"
variable p4 equal "-pzz/10000"
fix writer all print 125 "${p1} ${p2} ${p3} ${p4}" file Fe.deform.txt screen no
#thermo
thermo 1000
thermo_style custom step cpuremain v_strain v_p2 v_p3 v_p4 press pe temp
#dumping standard atom trajectrories
dump 1 all atom 5000 dump.deform.lammpstrj
#dumping custom cfg files containing coords + ancillary variables
dump 2 all cfg 5000 dump.deform_*.cfg mass type xs ys zs c_csym c_eng fx fy fz
dump_modify 2 element Fe
#40 ps MD Simulation (assuming 2 fs time step)
run 20000
Memory usage per processor = 5.6434 Mbytes
Step CPULeft strain p2 p3 p4 Press PotEng Temp
0 0 0 -0.0076650229 -0.029189782 0.046170857 -31.053508 -16324.877 301.05577
1000 642.87472 0.01 1.2081609 0.111532 -0.027848267 -4306.1487 -16321.834 300.05982
2000 603.66976 0.02 2.8906892 0.019841745 0.058671507 -9897.3414 -16316.581 301.21928
3000 559.83536 0.03 5.0431753 0.0087265977 0.048737926 -17002.133 -16305.377 299.81327
4000 522.55349 0.04 7.1591005 -0.02937889 -0.002314307 -23758.024 -16287.932 298.94816
5000 487.08493 0.05 9.228635 -0.077490074 0.0048635651 -30520.028 -16260.929 294.19783
6000 452.56928 0.06 10.396876 0.15961449 0.11953291 -35586.744 -16230.937 297.02056
7000 419.60397 0.07 9.886064 -0.051682975 -0.09163781 -32475.811 -16201.752 303.20434
8000 386.75543 0.08 9.2446009 -0.083955391 -0.16194994 -29995.652 -16173.548 303.05972
9000 356.54987 0.09 8.7976293 0.07863788 0.066336597 -29808.679 -16145.52 299.1521
10000 322.27974 0.1 8.7883277 0.071221379 0.14441091 -30013.2 -16120.017 295.91005
11000 288.39697 0.11 8.7161044 0.12374144 0.057888762 -29659.115 -16093.755 293.76039
12000 254.94453 0.12 8.4664141 0.11590022 0.053137264 -28784.839 -16072.727 302.16151
13000 222.10061 0.13 8.0312734 0.15479095 -0.00030987981 -27285.848 -16045.735 298.09456
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14000 189.48254 0.14 7.7593529 -0.067589459 0.023748449 -25718.373 -16021.048 296.6428
15000 157.24164 0.15 7.9620057 0.01935174 0.050888236 -26774.152 -15999.543 298.13657
16000 125.28295 0.16 7.9896281 -0.048697529 -0.2138625 -25756.893 -15981.117 305.28058
17000 93.654386 0.17 8.2825068 -0.14928122 -0.14198715 -26637.461 -15957.539 304.10016
18000 62.183767 0.18 8.4965527 -0.13383076 -0.20882948 -27179.641 -15925.861 289.97497
19000 30.970483 0.19 8.7027568 0.038033547 0.10172549 -29475.053 -15906.179 300.92811
20000 0 0.2 -4.1421569 -0.021888035 -0.25205478 14720.332 -16246.264 491.39709
Loop time of 618.338 on 1 procs for 20000 steps with 4000 atoms
99.1% CPU use with 1 MPI tasks x 1 OpenMP threads
Performance: 5.589 ns/day 4.294 hours/ns 32.345 timesteps/s
MPI task timings breakdown:
Section | min time | avg time | max time |%varavg| %total
---------------------------------------------------------------
Pair | 596.11 | 596.11 | 596.11 | 0.0 | 96.41
Neigh | 0.89945 | 0.89945 | 0.89945 | 0.0 | 0.15
Comm | 2.4723 | 2.4723 | 2.4723 | 0.0 | 0.40
Output | 0.70232 | 0.70232 | 0.70232 | 0.0 | 0.11
Modify | 17.229 | 17.229 | 17.229 | 0.0 | 2.79
Other | | 0.9249 | | | 0.15
Nlocal: 4000 ave 4000 max 4000 min
Histogram: 1 0 0 0 0 0 0 0 0 0
Nghost: 7625 ave 7625 max 7625 min
Histogram: 1 0 0 0 0 0 0 0 0 0
Neighs: 326047 ave 326047 max 326047 min
Histogram: 1 0 0 0 0 0 0 0 0 0
FullNghs: 652353 ave 652353 max 652353 min
Histogram: 1 0 0 0 0 0 0 0 0 0
Total # of neighbors = 652353
Ave neighs/atom = 163.088
Neighbor list builds = 29
Dangerous builds = 0
# clearing fixes and dumps
unfix 1
unfix 2
unfix writer
undump 1
undump 2
########################
print "All done"
All done
c) A copy of The Fe.deform file for SS304 under Ambient condition
# Fix print output for fix writer
0.00123999999999994 0.116564402264013 0.0464473137802112 0.00483803562523464
0.00248999999999981 0.210664407775859 00858898567310265 -0.0709971455330863
0.00373999999999987 0.421341288008089 0.00823323158588431 0.0164441171243528
0.00498999999999994 0.688853553139099 0.0761472295910508 -0.0350464974244329
0.0062399999999998 0.800749381530345 -0.0809253302455162 0.0287497894857198
0.00748999999999987 0.910864019197009 0.11111911577158 -0.0260790839192865
0.00873999999999994 1.05752203264959 0.0033984323810364 0.0787816266495031
0.0099899999999998 1.20817284939227 0.111533102518238 -0.0278485424353967
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
0.16499 8.12468386732701 -0.122944433536806 -0.0629838339958507
0.16624 8.11121348645094 -0.0384931493606099 0.027073254269727
0.16749 8.22193204181988 0.0132259402227129 0.016886999119873
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0.16874 8.18549920458727 -0.0954506289353622 -0.034732462635077
0.16999 8.28257759850691 -0.149282498303458 -0.141988368546309
0.17124 8.16345362543331 -0.0517945740057542 0.212603558695045
0.17249 8.28532678775858 0.0973838026767817 -0.0556655732552118
0.17374 8.33910632558225 0.0596424427678164 0.127755425697117
0.17499 8.35843778172521 0.0476820982142403 0.0354455163312715
0.17624 8.51631398059371 -0.00437531520986944 -0.0202301738650157
0.17749 8.46420804508895 0.00466010646728243 -0.0613997507025128
0.17874 8.37424278825909 0.0605217673410995 0.123093211465606
0.17999 8.49662466993194 -0.133831893335373 -0.208831248122214
0.18124 8.527957214023 0.030841747126963 0.0865655139947751
0.18249 8.52155398857506 0.165568812827459 0.158139344473259
0.18374 8.59531778441364 -0.0311383408288015 -0.0757330534870769
0.18499 8.60876451235429 -0.0665652530762197 -0.0622507067245766
0.18624 8.6965311777411 0.0923774166672753 0.00485308358310499
0.18749 8.61297499331869 0.02197279695611 0.149731469725766
0.18874 8.82854342641751 -0.0310480018312249 -0.155761526898982
0.18999 8.70282992130393 0.0380338668337344 0.101726343363716
0.19124 8.72246972603824 0.10215769205209 0.129522289471954
0.19249 8.70457054346254 -0.104732146381689 -0.208038310710638
0.19374 8.50732843319348 0.118066390027674 -0.106694703945573
0.19499 7.84998792874669 0.954151930411956 -0.217671611946641
0.19624 2.65161285144032 5.13773233925456 -2.72628959691421
0.19749 -2.52303837908483 2.15217169535593 -1.82216890834444
0.19874 -4.45373027963355 -0.372475631916491 0.25491075346315
0.19999 -4.14219144687075 -0.0218882176094841 -0.252056883048852
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APPENDIX VII
Mechanical Properties of the Fe-Ni-Cr Alloys under ambient temperature and supercritical water conditions
SPECIMEN
TESTING
CONDITION
YOUNGS
MODULUS (E) (GPa)
ULTIMATE TENSILE STRENGTH
(UTS)(GPa) BREAKING STRENGTH(GPa)
304 308 309 316 SS304 SS308 SS309 SS316 SS304 SS308 SS309 SS316
27 ºC
196 196 196 196
10.55
(0.07%)
10.53
(0.06%)
10.49
(0.06%)
10.46
(0.06%)
8.71
(0.19%)
8.72
(0.20%)
8.67
(0.193%)
8.84
(0.19%)
300 ºC
156 157 154 155
7.76
(0.07%)
7.72
(0.09%)
7.77
(0.07%)
7.66
(0.07%)
6.7
(0.16%)
6.57
(0.16%)
6.50
(0.154%)
6.56
(0.15%)
400 ºC
139 139 136 137
7.10
(0.08%)
6.96
(0.09%)
6.84
(0.07%)
6.82
(0.07%)
6.25
(0.15%)
6.01
(0.15%)
6.05
(0.148%)
6.13
(0.15%)
500 ºC
121 119 118 119
6.26
(0.11%)
6.34
(0.14%)
6.07
(0.10%)
6.15
(0.08%)
5.80
(0.15%)
5.75
(0.15%)
5.72
(0.143%)
5.77
(0.14%)
Note: The values in the bracket are the corresponding strain values of UTS and Breaking Strength.
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Yield Strength of the Fe-Ni-Cr Alloys under ambient temperature and
supercritical water condition
SPECIMEN TESTING
CONDITION
YIELD STRENGTH (GPa)
SS304 SS308 SS309 SS316
AMBIENT
CONDITIONS
T=27 ºC
P=0.01 MPa (1atm)
9.88
(0.07%)
9.90
(0.07%)
9.97
(0.07%)
9.77
(0.07%)
SCW Condition
T = 300 ºC
P = 25 MPa
7.56
(0.07%)
7.38
(0.07%)
7.46
(0.06%)
7.33
(0.07%)
SCW Condition
T = 400 ºC
P = 25 MPa
6.39
(0.07)
6.20
(0.06%)
6.17
(0.05%)
6.12
(0.06%)
SCW Condition
T = 500 ºC
P = 25 MPa
5.54
(0.06%)
5.10
(0.06%)
5.30
(0.05%)
5.34
(0.06%)
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