Doctoral Thesis in Physics

Radiation-Induced Damage Processes in Nuclear Reactor EnvironmentsELIN TOIJER

Stockholm, Sweden 2021

kth royal institute of technology

Radiation-Induced Damage Processes in Nuclear Reactor EnvironmentsELIN TOIJER

Doctoral Thesis in PhysicsKTH Royal Institute of TechnologyStockholm, Sweden 2021

Academic Dissertation which, with due permission of the KTH Royal Institute of Technology, is submitted for public defence for the Degree of Doctor of Physics on Friday the 26th of March,at 4:00 p.m. in F3, Lindstedsvägen 26, Stockholm.

© Elin Toijer ISBN 978-91-7873-823-6TRITA-SCI-FOU 2021;04 Printed by: Universitetsservice US-AB, Sweden 2021

iii

Abstract

It is essential that the materials used in nuclear reactor applications maintain theirstructural integrity. However, consequences of the strong radiation field in the re-actor core include radiolysis of the water surrounding the materials, as well as di-rect defect generation in the material itself. This may result in the combined cor-rosive attacks from the environment, and the internal weakening due to processesin the bulk. The goal of this thesis is to extend the current knowledge of the manyradiation-induced processes in the materials used in nuclear reactor applications,and to assess how such processes in turn can influence the materials’ stress toler-ance. Radiation-induced surface effects have here been evaluated from experiments,whereas bulk processes are investigated in a theoretical perspective. This providesa broad picture of the many phenomena which lead to material degradation in anirradiated environment.

In the first part of this thesis, the reaction mechanisms of H2O2, which is an im-portant product of water radiolysis, and oxide surfaces of interest in nuclear reactorapplications, have been evaluated. As a first step, the impact of Br– , Cl– and HCO3

–

on the reaction mechanisms between ZrO2 and H2O2 are determined. This providesan increased understanding for how competing reactions between the anions andH2O2 for the oxide surface can influence the kinetics of the system. As a secondstep, reaction mechanisms and reaction rate constants between H2O2 and the 304Lsteel in solution have been determined. Oxidative dissolution of steel componentsare here assessed, and the reaction paths of H2O2 towards the steel are discussed.

In the second part of this work, the objective has been to evaluate how direct effectsof radiation can influence the stress tolerance of the material. Radiation-inducedsegregation of chemical components in the model material fcc Ni have been quanti-fied, and focus has been on the driving forces behind this segregation. Based on thesegregation trends, the impact of solute enrichment on grain boundaries have beenassessed. In this context, focus has been not only to quantify the effects of the solutesegregation, but also to evaluate how common modeling techniques can influencethe results. Particularly, it is here shown that the applied model can have a deci-sive impact on observed trends. Moreover, it is shown that the common assumptionthat all solutes that are driven to sinks such as grain boundaries will end up in thegrain boundary center, can also give very misleading results. The concentration-dependent effect of Si, the slightly weakening effect of P, and the particular effectsof Cr are all quantified using state of the art atomistic modeling. As a final step, theimpact of magnetic disorder on thermal expansion and defect formation energieshave here been calculated in Ni. The model used in this part of the work very accu-rately predict the thermal expansion of the material, and shows that this is a majorcomponent in the temperature dependence of the defect formation enthalpies.

v

Sammanfattning

En stor del av de inre komponenterna i dagens kärnreaktorer tillverkas av olikatyper av stål och nickellegeringar. Valet av dessa material utgår från deras starkamotståndskraft mot den aggressiva miljön i reaktorn. Dock kan inga material gåopåverkade av den högintensiva strålningen, som dels skapar defekter i materialetsatomstruktur, och dels växelverkar med vattnet som omger materialet och på såvis skapar en oxiderande miljö som i sin tur kan leda till korrosion. Målet meddenna avhandling har varit att undersöka båda dessa effekter, för att på så vis utökakunskapen om hur materialens livslängd påverkas av den aggressiva miljön i reak-torn. Strålningsinducerade ytprocesser har här undersökts med hjälp av experiment,och processer inuti materialet har undersökts ur ett teoretiskt perspektiv. På så visförmedlas en bred bild av de olika fenomen som kan leda till försprödning och förtidigt åldrande av de material som används i dagens kärnkraftverk.

I avhandlingens första del undersöks hur växelverkan mellan joniserande strålningoch vatten (så kallad radiolys) kan påverka materialets yta. I denna växelverkanskapas bland mycket annat H2O2, och här undersöks hur denna molekyl interagerarmed det oxidlager som finns på metallernas ytor. Fokus är på att utvärdera reaktion-smekanismer och bestämma hastighetskonstanter i reaktionen mellan de två, samtatt bestämma hur dessa mekanismer påverkas av de olika komponenterna i lösnin-gen.

I avhandlingens andra del behandlas strålningseffekter inuti materialet. De defektersom genereras i materialet under bestrålning kommer att diffundera, tillsammansmed de kemiska komponenterna, vilket leder till segregation i materialet. Segrega-tionstrender har här kvantifierats i nickel, och påverkan på materialets spänningstå-lighet undersökts. Fokus ligger även på att utvärdera hur olika modeller och begyn-nelsevillkor kan påverka resultaten. De metoder som använts här har visat att borstärker korngränser, påverkan från kisel beror på dess koncentration, fosfor försva-gar något, krom har ett speciellt beteende och svavel försvagar korngränsen bety-dligt.

vii

Acknowledgments

My time as a PhD student has perhaps not been the most easy. Working in boththe chemistry and the physics departments meant I had to handle two very differ-ent ways of thinking, and two very different ways of doing research. This dualityhas indeed been difficult, however, it had one huge advantage. I met twice as manypeople, which meant I made twice as many friends. So if this journey was not easy,it just means that it is all the more valuable. To all the people who took part in myprocess, I am genuinely grateful for your company. I would like to mention a fewpeople who made an extra impact on my time, but I am well aware that this list isfar from complete.

Of course I have to start with my two supervisors, Pär and Mats. You both havefundamentally different ways of leading your students, but this is perhaps the bestcombination if a person really wants to grow. You have both taught me so much,probably more than I can even grasp myself. And I am so grateful (and probably abit nostalgic at the moment), for having worked side by side with both of you.

Luca, I cannot thank you enough for all the things you have done to help me. Youare a true inspiration for me, and I am so happy that you have become my friend. IfI know anything about radiation-induced segregation today, it is all because you sogenerously share your knowledge.

Thank you Malmö-Pär for not ignoring the e-mails from a seemingly crazy person.Thank you for inviting me to come, and thank you for the numerous times you tookyour time to help me. Without you, I would still be starring at my energy plots,wondering how the heck to make this stupid EEA model work.

Thank you Densie, Hongjie and Karl for not letting me get eaten by a cougar. Andthank you for a night I will never forget!

Thank you Torbjörn for always being available for a pleasant conversation. I willmiss having you as my colleague, perhaps most of all for your "never ending sup-port".

Thank you Ghada, whose name I unfortunately never learned to pronounce. In caseI never will, I am happy to know I can always call you my friend.

Thank you Björn for all the things you have done to help me. But most of all, thankyou for the memories we have created together.

Thank you Denise, Yulia, and Débora for all the good times we spent. Meetinganyone of you in the corridor was always the highlight of my working day.

And last but not least, to my family, thank you so much for your support.

viii

ix

List of publications

Included publications

I E. Toijer and M. Jonsson, Anion effects on the catalytic decompositionof H2O2 on ZrO2(s) in aqueous systems. ChemistrySelect, 5(43), 13754-13760, (2020)

II E. Toijer and M. Jonsson H2O2 and γ-radiation induced corrosion of 304Lstainless steel in aqueous systems. Radiation Physics and Chemistry, 159,159-165, (2019).

III E. Toijer, L, Messina, C. Domain, J. Vidal, C.S. Becquart, and P. Ols-son Solute-point defect interactions, coupled diffusion, and radiation-induced segregation in fcc nickel. Physical Review Materials, 5(1), 013602.(2021)

IV E. Toijer, P. A. T. Olsson and P. Olsson, Ab initio modelling of intergran-ular fracture of nickel containing phosphorus: Interfacial excess proper-ties. Submitted to Nuclear Materials and Energy (2021)

V E. Toijer, P. A. T. Olsson and P. Olsson, Ab initio investigation of effectsof solute segregation on intergranular fracture in nickel: Importance offracture path and structural modification, Manuscript in preparation

VI E. Toijer and P. Olsson, The impact of magnetic disorder on defect for-mation energies in fcc Nickel, Manuscript in preparation

Author’s contribution

I have performed all experimental work in papers I and II. In paper III, Iperformed a major part of the calculations. In papers IV-V, I developed acomputational code to perform the EEA analysis in addition to a computa-tional code to obtain the appropriate initial states for the DFT-calculations.In those two papers, I also performed all calculations, and all data analysis.In paper VI, I developed a computational code to continuously perform thespin flips throughout the DFT-MD calculations. I have had an active part inall data processing and have been the lead writer for all manuscripts.

xi

Abbreviations and nomenclature

BWR Boiling water reactorcX Defect concentration (vacancy or interstitial)CPG Chemical potential gradientCZM Cohesive zone modelDFT Density functional theoryDLM Disordered local momentEEA Excess-energy assessmentFcc face centered cubic structureGy Gray (J/kg−1)GB Grain boundaryIASCC Irradiation-assisted stress corrosion crackingIGSCC Intergranular stress-corrosion crackingk Reaction rate constantLWR Light water reactornn nearest-neighborNPP Nuclear power plantM Molar (mol/dm3)MD Molecular dynamicsOH• Hydroxyl radicalOH•(ads) Surface bound hydroxyl radicalPD Point-defectPDC Partial diffusion coefficientPWR Pressurized water reactorRGS Rigid grain shiftRIS Radiation-induced segregationSA Surface areaSA/V Surface area to solution volume ratioSIA Self-interstitial atomSCC Stress corrosion crackingSCMF Self-consistent mean field theoryTris Tris(hydroxymethyl)aminomethaneUBER Universal binding energy relationWOS Work of separation

xiii

Contents

Abstract iii

Sammanfattning v

Acknowledgments vii

List of publications ix

Abbreviations and nomenclature xi

1 Introduction 1

2 Radiation-induced processes at solid-liquid interfaces 52.1 Water radiolysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Chemistry of the heterogeneous system . . . . . . . . . . . . . 62.3 Chemical kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Radiation-induced segregation and its effect on fracture properties 93.1 Electronic-structure calculations . . . . . . . . . . . . . . . . . . 10

3.1.1 Density functional theory and molecular dynamics . . 123.2 Point-defect properties and solute transport . . . . . . . . . . . 133.3 Radiation-induced segregation . . . . . . . . . . . . . . . . . . 153.4 Modeling of cleavage fracture from first principles . . . . . . . 17

3.4.1 Excess energy assessment (EEA) of a fracturing region 183.5 Temperature dependence of defect properties . . . . . . . . . . 19

3.5.1 Disordered local moment molecular dynamics . . . . . 203.6 Calculation details . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.6.1 Development of computational codes to extract excessresponse and magnetic properties . . . . . . . . . . . . 21

4 Results I: Surface Processes 234.1 Anion effects on the catalytic decomposition of H2O2 on an

oxide surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.1.1 The homogeneous system . . . . . . . . . . . . . . . . . 244.1.2 The heterogeneous system . . . . . . . . . . . . . . . . . 25

4.2 H2O2 and γ-induced corrosion of austenitic alloys . . . . . . . 274.2.1 Reactivity of H2O2 towards a 304L oxide surface . . . . 284.2.2 Oxidative dissolution of steel components . . . . . . . 304.2.3 Discussion concerning the reaction mechanisms between

H2O2 and the 304L steel . . . . . . . . . . . . . . . . . . 31

xiv

5 Results II: Bulk and fracture properties 335.1 Systematic trends of defect stabilities in fcc materials . . . . . 335.2 Radiation-induced segregation in fcc Ni . . . . . . . . . . . . . 34

5.2.1 Dominant diffusion mechanisms in steady-state condi-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.2.2 Partial diffusion coefficients in the irradiated systems . 365.2.3 Radiation-induced segregation . . . . . . . . . . . . . . 38

5.3 Segregation-induced changes in fracture properties in Ni . . . 405.3.1 Impact of ab initio methodology: Rigid approach versus

excess energy assessment . . . . . . . . . . . . . . . . . 41Decohesion of bulk Ni and the clean Ni GB . . . . . . . 41Impact of P on the cohesive properties of Ni . . . . . . 42

5.3.2 Effect of Cr on the integrity of a Ni GB: importance offracture path and structural modification . . . . . . . . 46

5.3.3 Quantitative comparison of the impact of different ele-ments on Ni cohesive properties . . . . . . . . . . . . . 49

5.4 Defect formation energies in paramagnetic Ni . . . . . . . . . 50

6 Conclusions and outlook 53

xv

List of Figures

1.1 Factors contributing to irradiation-induced stress corrosion crack-ing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3.1 Illustration of the two main defect mediated diffusion paths inthe fcc crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Illustration of the cohesive zone . . . . . . . . . . . . . . . . . 17

4.1 Concentration of CH2O as function of dose for aqueous solu-tions containing 1 M Br– , Cl– and ClO4

– respectively, and incombination with 0.1 M HCO3

– . . . . . . . . . . . . . . . . . . 244.2 Concentration of CH2O as a function of time in presence of 2 g

of ZrO2 in combination with 1 M Br– , Cl– and ClO4– respec-

tively, and in combination with 0.1 M HCO3– . . . . . . . . . . 26

4.3 Combined impact of Br– and HCO3– on H2O2 decomposition

in the heterogeneous system. . . . . . . . . . . . . . . . . . . . 274.4 The natural logarithm of [H2O2]t/[H2O2]0 as a function of time

for different surface area to solution volume ratios . . . . . . . 284.5 Pseudo-first order rate constants versus steel surface area to

solution volume ratio (SA/V) . . . . . . . . . . . . . . . . . . . 29

5.1 Systematic defect stability trends in fcc material. Results arepresented in reference to the corresponding 〈1 0 0〉 dumbbell. 34

5.2 Ratios of vacancy and interstitial solute tracer diffusion coeffi-cients in steady-state conditions . . . . . . . . . . . . . . . . . . 36

5.3 Partial diffusion coefficient (PDC) ratios as functions of tem-perature.(a) Vacancy mechanism (b) Interstitial mechanism . . 37

5.4 Radiation-induced segregation tendencies of impurities in Ni 385.5 Illustration of the Σ5 GB used in calculations of the current work 405.6 Impact of ab initio stress-test model on Ni bulk/GB properties.

(a) Energy-separation (b) Traction-separation. . . . . . . . . . . 425.7 Impact of P on the stress-response of a Ni GB based on the

EEA approach. (a) Low surface coverage (θi,θs=1) (b) highersurface coverage (θi=2-4) . . . . . . . . . . . . . . . . . . . . . . 44

5.8 Impact of P on the stress-response of a Ni GB based on the RGSapproach. (a) Low surface coverage (θi,θs=1) (b) higher surfacecoverage (θi=2-4) . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.9 Energy versus separation for the Ni GB containing 4 Cr atoms(a) Energy per atom in the system (b) Total energy per surfacearea in the system (bulk+fracture plane) . . . . . . . . . . . . . 47

5.10 (a) Excess energy and (b) traction-separation behaviour of theGB containing Cr. . . . . . . . . . . . . . . . . . . . . . . . . . . 48

xvi

5.11 Traction-separation behaviour of the Ni GB including four in-terstitials of the respective elements the different elements . . 49

5.12 Fcc Ni lattice parameter as function of temperature from DLM-MD simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.13 Vacancy and SIA formation enthalpies in the static ferromag-netic and DLM-MD paramagnetic states as function of latticeparameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

1

Chapter 1

Introduction

Nuclear power is an important source of electricity, currently generating ap-proximately 10% of the world consumption [1]. The technology is based on acontrolled nuclear fission process in which the fuel, commonly consisting ofuranium oxide (UO2), is bombarded with neutrons. U-235 nuclei that absorbneutrons will split into two or more daughter nuclei, so called fission prod-ucts. In the process a large amount of energy is released, together with a fewneutrons that are used to sustain the chain reaction. Among the most com-mon types of nuclear reactors are the so called Light Water Reactors (LWRs).The fuel in such reactors is placed into rods, which in turn are grouped intoassemblies, and submerged into water in the reactor pressure vessel. The roleof the water is to both lower the kinetic energy of the emitted neutrons, andtransfer energy from the fission process to power turbines that generate elec-tricity. Common types of LWRs include boiling water reactors (BWRs) andpressurized water reactors (PWRs). The main difference between the twois that in BWRs, the water is heated to the point of boiling under about 75bar pressure, whereas in PWRs, the water is subjected to considerably higherpressure (150-160 bar), and is thus prevented from boiling.

As the uranium atoms are fragmented in the fission process, a number ofhighly radioactive elements are produced. The concentration of fissile mate-rial will thus decrease with time. When the concentration of UO2 is approx-imately 95%, the fuel is replaced. The spent nuclear fuel (SNF) is extremelyhazardous, and must be handled properly. In Sweden, a model to disposethe SNF has been developed by the Swedish Nuclear Fuel and Waste Man-agement Company (SKB) [2]. The model includes multiple barriers to protectthe SNF, and in this way assures the integrity of the repository until its toxi-city has reached background levels after a few hundred thousand years.

The steps the fuel goes through, from mining and refining, to operation andfinal repository, is known as the nuclear fuel cycle. It is fundamental thatsafety in operations can be guaranteed in all steps of this cycle. To date themajor technological factor, which can limit the efficiency and viability of nu-clear power generation, is represented by material degradation during ser-vice [3]. To decrease the probability of failure, the choice of material is care-fully evaluated for each and every component in the system. For internalstructural materials, the appropriate choice are materials with a high struc-tural integrity, and which form a stable, adherent oxide film that can provideprotection from the aggressive environment in the reactor core. Austeniticstainless steels and Ni-alloys have excellent high-temperature mechanical

2 Chapter 1. Introduction

properties, and are for this reason the common choice for many structuralcomponents [4–6]. The fuel cladding, on the other hand, are commonly madefrom zirconium alloys. This material has good corrosion resistance, resis-tance to irradiation damage, and importantly a high transparency to thermalneutrons.

Although structural materials used in today’s nuclear power plants (NPPs)have excellent corrosion resistance and stress tolerance, their integrity can-not always be sustained. The premature failure due to synergistic effectsof stresses, an aggressive water environment, and a susceptible material, isknown as stress corrosion cracking (SCC), and is a common problem in thiscontext. SCC is characterized by localized attacks involving transgranular(TG) or intergranular (IG) cracks, with the rest of the material remaining vir-tually unaffected. The three constituents of SCC; tensile stress, an aggressiveenvironment, and a susceptible material, are fundamental for the process tooccur. In a nuclear reactor environment, the radiation field has importanteffects of the latter two. The process has in this case been termed Irradiation-Assisted Stress Corrosion Cracking (IASCC). This can have a decisive impacton the life-time of nuclear reactor core components. The phenomenon hasbeen found to induce premature ageing and degradation of in iron- andnickel-based alloys in LWR environments [5, 7, 8]. The factors contributingto irradiation-assisted SCC (IASCC), are summarized Fig. 1.1.

H2O2

HO(ads)

HO(ads)

OH-H2O2

Defect ProductionVacancies/Interstitals

Changes in water chemistry

H2O2Grain boundary Segregation

Formation of defectclusters and loopsDiffusion

towards sinks

H+

H+

H+

H+

H2O2

H2O2

H

H

H

H

H2

H2

H2

e-(aq)

e-(aq)

e-(aq)

FIGURE 1.1: Factors contributing to irradiation-induced stress corro-sion cracking.

Due to the many factors at play, a quantitative description of this degradationprocess is far from straight forward. A better understanding of radiation-induced damage processes would enable a safe prolongation of the lifetimeof NPPs, and avoid costly shutdowns. In the current work, a number of fac-tors that can contribute to the premature degradation of structural compo-nents in current light-water reactors have been evaluated. The phenomenonhas been tackled from two aspects, namely:

Chapter 1. Introduction 3

• Impact of water chemistry: Processes at solid-liquid interfaces in com-bination with reaction mechanisms of important oxidants have been in-vestigated. The focus in this section has been on reaction mechanismsbetween the products of water radiolysis and the outer surfaces of thematerials used in reactor applications. Particularly, the reactions of bothZrO2 and stainless steel surfaces are investigated. Chemistry in both thehomogeneous (aqueous) and heterogeneous (containing a solid-liquidinterface) systems are evaluated to extract the reactions taking place atthe solid interface. The goal in this section is to quantify the reactionmechanisms between H2O2 and these oxides, in order to increase theunderstanding regarding how the materials are affected by the waterradiolysis.

• Changes in material properties: In this part, the focus is on radiation-induced changes in material stress tolerance as a consequence of redis-tribution of solutes. Segregation tendencies and diffusion mechanismshave here been evaluated in a model material. Based on the segregationtendencies, the impact of the solutes that display enrichment at sinks onthe stress-tolerance of a grain boundary has been investigated. Empha-sis in this section is on the models used to assess this stress response,and how they can influence the results. This section provides a thor-ough assessment of the common pitfalls and sources of error associatedwith performing stress tests in a first-principles framework.

The objective of this work is to improve the current understanding of the ef-fects of irradiation on the structural integrity of austenitic- and nickel-basedalloys used in nuclear reactor environments. The many processes have beeninvestigated in both an experimental and a theoretical framework in orderto provide a broad picture of the many phenomenon that contribute to thematerial degradation. The hope is that tackling the problem from multipleaspects can provide a broader picture of radiation-induced damage phenom-ena.

5

Chapter 2

Radiation-induced processes atsolid-liquid interfaces

For the austenitic and Ni-based alloys used in reactor applications, the maincorrosion resistance is provided by a passive oxide film at the outer sur-face [9]. In LWR environments, the oxide formed on austenitic alloys con-sists of a duplex structure. The inner oxide is primarily composed of Fe-Cr-Ni spinels, (Fe3O4 ,FeCr2O4, NiCr2O4, NiFe2O4) and the outer one is mainlycomposed of Fe spinels (Fe3O4), with some Ni [9, 10]. If the oxide film isruptured or damaged, the bulk material is directly exposed to the environ-ment. The repassivation will in this case start immediately, and the rate ofthis process will have a significant impact on the material damage and crackpropagation. If the process is quick, the corrosion attack can lead to only asmall crack propagation. A slow process, on the other hand, will lead to theblunting of the crack tip [9]. For this reason, maximum crack growth occursat intermediate repassivation rates, when the crack can grow without beingblunted. The stability of the oxide films is highly controlled by the chemistryin the system. The focus of the current work is to understand how radiationcan play a role in this process. In the following sections relevant theory andbackground is presented.

2.1 Water radiolysis

In LWRs, the water is in close contact with the nuclear fuel assemblies. Thefission process leads to the constant emission of neutrons, in combinationwith ionizing radiation from the daughter nuclei and actinides. Consequencesof this involve ionization, excitation, and decomposition of the water moleculesin a process known as radiolysis. A number of oxidizing and reducing speciesare formed in primary and secondary reactions: OH•, e−aq, H2, H2O2, H•,H3O+, O2, and O2

− (the dot notation in this case signifies a radical). Theamount of species produced or consumed (n) per unit absorbed energy isgiven by their chemical yield, or G-value:

G(n) =dn

dEabs

[moleV

](2.1)

Many of the species formed in water radiolysis are highly reactive, and ashort time after the energy transfer event only a minor part remains. H2O2,on the other hand, is relatively stable, and can accumulate in the system.

6 Chapter 2. Radiation-induced processes at solid-liquid interfaces

H2O2 is for this reason one of the more abundant products of water radiol-ysis at ∼300 oC [9, 11, 12]. As a consequence, although the hydroxyl radical,OH•, is by far the most potent oxidant, the impact of H2O2 in surface corro-sion processes in NPPs may very well exceed that of the hydroxyl radical byorders of magnitude [13].

2.2 Chemistry of the heterogeneous system

The impact of the individual products of water radiolysis in IASCC is dif-ficult to assess. The general consensus is however that the effect of waterradiolysis on IASCC is not due to any specific species, but to their combinedeffect as given by the corrosion potential [7, 8, 14–16]. H2O2 is considered amajor factor in increasing the corrosion potential under irradiated conditionsat reactor operating temperatures [7,10], and can for this reason play a crucialrole in the corrosion of the oxide surfaces in NPP environments.

There are two main reaction paths of H2O2 in the interaction with an ox-ide surface: (a) catalytic decomposition of H2O2 and (b) redox reactions. Inthe case of catalytic decomposition, the reaction mechanisms are given byreactions R1-R4.

H2O2 + 2 M −−→ 2 OH•−M (R1)OH•−M + H2O2 −−→ H2O + M + HO•2 (R2)

OH•−M −−→ OH− + M+ (R3)HO•2 + HO•2 −−→ H2O2 + O2 (R4)

where OH• – M signifies that the radical is adsorbed to the surface. If thematerial is in it highest oxidation state, reaction R3 cannot follow from thismechanism. However, if this is not the case, the possibility of surface oxida-tion by the adsorbed OH• should be considered. This mechanism has pre-viously been detected on UO2 [17]. The oxidation of the metal surface mayresult in dissolution of ions in solution. If the metal ions are soluble in multi-ple oxidation states, they may further react with H2O2 in solution accordingto the the Haber-Weiss mechanisms, given by reactions R5-R6.

H2O2 + Mn(aq) −−→ HO2

• + H+ + Mn−1(aq) (R5)

H2O2 + Mn−1(aq) −−→ OH• + OH− + Mn

(aq) (R6)

where Mn and M(n – 1) are the oxidized and reduced forms of the metal. Inthe case where M(n – 1) consists of ferrous iron (Fe2+), reaction R6 is known asthe Fenton reaction. The reaction can lead to the consumption of H2O2 andgeneration of OH• without the impact of the oxide surface. The process ishowever highly controlled by the metal concentration in the solution.

2.3. Chemical kinetics 7

2.3 Chemical kinetics

Whereas thermodynamics tells us in which direction processes in a systemshould occur spontaneously, chemical reaction rates can provide a quanti-tative description of the corresponding reaction characteristics and mech-anisms. In a homogeneous system, the rate of a chemical reaction can belinked to the concentration of each reactant by the rate law:

v =dcdt

= k ∏i

cmii (2.2)

where k is the reaction rate constant, ci is the concentration of reactant i, andmi is the partial order for each reactant. A heterogeneous system, on theother hand, contains both solid and liquid phases. The number of reactionmechanisms will in this case increase, and important steps include:

• Solution reactions

• Diffusion to the surface

• Adsorption to surface

• Surface Reactions

• Desorption of products to solution

It should be noted that both catalytic and non-catalytic processes can takeplace simultaneously in certain systems. In case of corrosion, a resultingdissolution of oxide components may induce further reactions in the liquidphase. Thus unlike in homogeneous solutions, where the reaction rate isdependent on the concentration in solution, the heterogeneous system in-cludes a number of possible rate-determining steps. In reactions that occurat a solid-liquid interface, the reaction is often limited by the surface area ofcontact, and the rate equation is given by:

− d[Solute]dt

= k×[

SAV

](2.3)

where SA/V is the surface area to solution volume ratio. If the solid materialis in excess, the kinetics of the system will display pseudo-first-order kinetics,and the corresponding rate constant can be obtained from plotting the loga-rithm of the solute concentration versus time. By repeating the procedure fordifferent SA/V ratios, the second order rate constants can be obtained. Dueto their high surface-to-volume ratio, powder suspensions are beneficial inevaluating reaction orders of a heterogeneous system.

2.4 Experimental details

The reaction mechanism between H2O2 and common oxides in nuclear reac-tor applications (ZrO2 in Paper I, and 304L steel in Paper II) have here beeninvestigated. This has been done in an experimental framework. A detaileddescription of the experimental procedures is presented in the corresponding

8 Chapter 2. Radiation-induced processes at solid-liquid interfaces

papers. In this section, general considerations are given.

Concentrations of H2O2 in solution were monitored using the Ghormley tri-iodide method [18], in which H2O2 reacts with iodine according to reactionsR7-R8. The product, I–

3 , can be detected spectroscopically at λ= 350 nm.

H2O2 + 2 H+ + 2 I− −−→ 2 H2O + I2 (R7)

I2 + I− −−→ I3− (R8)

The formation of OH• during decomposition of H2O2 can, due to the highreactivity of the radical, not be directly measured. The production can how-ever be quantified by the use of radical scavengers. A scavenger will reactwith the OH•, and in turn form a more stable product, which can be detectedin a straight forward manner. In Paper I, tris(hydroxymethyl)aminomethane(tris) was used, whereas in Paper II, methanol was used for this propose.The former provides the possibility for pH control in the system, which wasof importance in Paper I. However, since tris is an amine, which often formcomplexes with metal ions, methanol was considered a more appropriatescavenger in Paper II. In the reaction between OH• and either scavengerformaldehyde, CH2O, is formed and the production of the radical can thusbe estimated by monitoring the concentration of formaldehyde in solution[19, 20]. In aqueous solution, the rate constant for the reaction between trisand OH• is 1.1·109 M−1s−1 [21], whereas the reaction between methanol andOH• has a rate constant of 9.7·108 M−1s−1 [22]. In homogeneous solution,the yield of formaldehyde from tris is ∼35% [20, 23], whereas the yield formethanol is ∼70% [23].

The concentration of formaldehyde in solution was in this work measuredby a modified version of the Hantzsch method [24], introduced by Nash [25].Samples of formaldehyde were mixed with acetoacetanilide and ammoniumacetate, and the mixture was left to react at 40 oC for 15 minutes. In the reac-tion a pyridyl derivate is formed, and the concentration of the latter could bemeasured spectroscopically at λ= 368 nm.

Tracer elemental analysis to determine release of steel components in so-lution, was performed using inductively coupled plasma optics emissionspectroscopy (ICP-OES) on a ThermoFisher iCAP 6000 series instrument. γ-irradiation was performed by 137Cs, using a MDS Nordion 1000 Elite γ-cell,with a dose rate of 0.13 Gy s−1. All experiments have been performed inroom temperature.

9

Chapter 3

Radiation-induced segregation andits effect on fracture properties

In presence of a strong radiation field, the chemical components that makeup a metal or an alloy may start to diffuse in the solid matrix. Consequencesof this involve enrichment or depletion of the elements at sinks such as grainboundaries (GBs). Particularly, in austenitic steel, irradiation has been shownto result in GB enrichment of Ni, P, S, and Si, and depletion of Cr and Fe[26–30]. Similar tendencies have been observed for Ni-based alloys [31–35].Changes in the GB composition may significantly impact its cohesive prop-erties, and risk compromising the structural integrity of the whole material.This has been shown in many Ni-based and austenitic alloys used in reactorapplications [5, 36]. Particularly, Cr depletion at GBs is considered a key fac-tor in intergranular SCC (IGSCC) in austenitic stainless alloys [7, 9, 37, 38].

The driving forces behind the segregation are at the atomic scale related tothe presence of point defects (PDs), e.g. vacancies and interstitials. At hightemperatures or in a radiation field, the PDs are often mobile, and in the caseof strong solute interactions, diffusion of the latter will follow. The efficiencyof solute diffusion is for this reason strongly related to the defect concentra-tion in the material, which in equilibrium conditions is given by [39]:

ceqx = exp

(− Gf

xkBT

)= exp

(−Hf

x − TSfx

kBT

)(3.1)

where x corresponds to a vacancy or interstitial, Gfx is the formation free en-

ergy, Hfx the formation enthalpy, Sf

x, is the formation entropy, T the absolutetemperature, and kB the Boltzmann constant. Since Gf

Int � GfVac, the defect

population in equilibrium conditions is completely dominated by vacancies.In irradiated environments, however, significant amounts of energy are con-stantly transferred to the material. If the energy gained from such a transferevent is sufficient to displace an atom from its lattice site, the latter may travelthrough the material, and finally come to rest as an interstitial. In this casea vacancy remains in the original position, and the two form what is knownas a Frenkel pair. Under neutron irradiation conditions, such as in near-corecomponents in a reactor, atomic displacement cascades are caused by the en-ergy transfer from the fast neutrons many hundreds of point defects are cre-ated in a very short time frame. The concentration of PDs during irradiation

10 Chapter 3. Radiation-induced segregation and its effect on fracture properties

is given by the combined thermal and radiation effects:

cx = ceqx + cirr

x (3.2)

Due to such effects, defect concentrations can be significantly higher in irra-diated compared to equilibrium conditions. This is of particular importancefor the interstitials, which have high defect formation energies and are thusonly present at very low concentration at equilibrium. The interstitial con-centration an for this reason generally be increased by many orders of mag-nitude in a nuclear reactor environment.

Segregation of material components may significantly reduce a material’sstress tolerance. This will however depend on the properties of the enrichingsolutes. Thus, in order to determine if radiation makes a material prone tofailure under tensile stress, two factors should be considered: if the radiationcauses changes in the material’s microstructure, and how such changes affectits response to external stresses. In this work, both aspects have been evalu-ated, and the current chapter provides a detailed description of the relevanttheory. Since electronic structure calculations provide the very basis of themodels used in the current work, the chapter starts with a general descrip-tion of the corresponding theory. This will provide the reader with a base andrelevant references for understanding the models used in the current work,which are presented in the following sections.

3.1 Electronic-structure calculations

Many material properties e.g. PD-solute interactions and GB cohesive ener-gies, can be derived from its ground state electronic structure. This structurecan be quantified by solving the many-body Schrödinger equation, in theadiabatic approximation [40] given by:

HΨ(x1, x2, . . . , xn) = EΨ(x1, x2, . . . , xn) (3.3)

whereH = T + Vext + Vee (3.4)

T is the kinetic energy, Vext is the external potential, which corresponds tothe interaction of the electrons and the atomic nuclei, and Vee is the electron-electron interaction potential.

The Schrödinger equation describes the evolution of a quantum mechanicalsystem subjected to external forces, and is extremely powerful in this sense.However, problems emerge due to the many-body interactions which mustbe taken into account in systems of interest in condensed matter physics.Since the wavefunction will depend on the individual position of every sin-gle electron in the system, the complexity grows exponentially, and solutionscannot be obtained for systems containing even a handful of atoms. How-ever, in 1964 Hohenberg and Kohn showed that the ground state electrondensity, n(r), uniquely defines the external potential to within a constant [41].Additionally, they showed that it is possible to define a universal functional,

3.1. Electronic-structure calculations 11

E[n(r)], for which the ground state energy is minimized by the ground stateelectron density. Thus, if the functional E[n(r)] is known, the exact groundstate energy and density follow.

The power in the theorems proved by Hohenberg and Kohn lie in the real-ization that any property of a system of interacting particles can be describedas a functional of the system’s ground state electron density. The challengehowever remains to find the correct functional forms. Kohn and Sham pre-sented an elegant yet simple solution to this problem, in which the originalmany-body problem was replaced by an auxiliary independent-particle sys-tem [42]. In the case of the non-interacting system, the solution to the originalSchrödinger equation can be divided into contributions of the individual or-bitals accordingly: (

− h2

2m∇2 + veff

)ϕi(r) = εi ϕi(r) (3.5)

where veff is an effective potential, given by:

veff = Vext + VH + VXC (3.6)

where Vext is the external potential due to electron-nuclei interactions, VHis the Hartree potential, and VXC is the exchange-correlation potential. TheHartree potential describes the classical Coulomb interaction energy of theelectron density, and is given by:

VH = e2∫ n(r′)|r− r′|dr′ (3.7)

where e is the electron charge. Although both the external potential andthe Hartree potential can be calculated, ambiguity remains for the exchange-correlation. This potential incorporates all the errors in using a non-interactingkinetic energy and a classical electron-electron interaction. This functionalcannot be explicitly calculated, for which reason the accuracy of the calcu-lations is limited by this component. Common approaches to approximatethis functional include the local density approximation (LDA) [43], and thegeneralized gradient approximation (GGA) [44], given by:

ELDAXC [n] =

∫n(r)εXC(n(r))dr (3.8)

EGGAXC [n] =

∫n(r)εXC(n(r),∇n(r))dr (3.9)

where εXC is the energy of the uniform electron gas with the same local den-sity n(r). In the LDA it is assumed that the exchange-correlation energy de-pends on the local electron density in every point, whereas in the GGA, it isassumed that the same energy depends not only on the density, but also onthe gradient of the density. The theory and accuracy behind this and otherexchange-correlation approximations have been discussed in great detail inmany works, and will for this reason not be repeated here. The curious reader

12 Chapter 3. Radiation-induced segregation and its effect on fracture properties

is referred to e.g. [45].

Once all components of Eq. 3.6 are known, the total electron density canbe calculated as the sum of squares of all individual orbitals:

n(r) =N

∑i=1|ϕi(r)|2 (3.10)

However, veff can not be determined without knowledge of the electron den-sity. This problem calls for an iterative solution, and herein lies the basisof density functional theory (DFT). In this framework, the problem of solv-ing the many-body Schrödinger equation is divided into an ionic and anelectronic loop, and the ground state is searched based on the criterion thatboth individual loops must reach self-consistency. The separation betweenthe two loops is based on the adiabatic (Born-Oppenheimer) approximationwhich states that the two move on very different time scales [40]. Numericalalgorithms can thus be applied to find the ground state energy and config-urations. The term ab initio, meaning first-principles, is often used in thiscontext. This signifies that atomic and electronic properties are calculated bydirectly solving the Hamiltonian of the Schrödinger equation of the many-body system, and thus no quantities derived from experiments are used.

3.1.1 Density functional theory-molecular dynamics (DFT-MD)

The goal of performing ab initio calculations is to find the ground state energyof a static atomic configuration. It can however be of interest to evaluate howthe system evolves in time, and this can be simulated within the frameworkof molecular dynamics (MD). The trajectories of atoms are in this contextcommonly assessed based on Newtonian mechanics:

MIRI = −∂E∂RI

= FI [{RJ}] (3.11)

where MI and RI are the mass and the position of the ion. In large systems,this equation can be solved by numerical simulations by the use of a discretetime step. Thus, at each step, the nuclei move according to the forces that arecurrently acting on them due to the other nuclei in the system:

RI(t + ∆t) = 2RI(t) + RI(t− ∆t) +(∆t)2

MIFI [{RJ(t)}] (3.12)

vI(t + ∆t) = vI(t) +∆t

2MI[FI(t) + FI(t + ∆t)] (3.13)

where F and v are the corresponding force and velocity. The accuracy of thecalculations depends to a large extent on the potential used in calculations.Since DFT provides a precise description of energies in a system, the twomethods can be unified to extend the range of both. This approach is basedon the Hellmann-Feynman theorem, which states that if the electron densityhas been determined, the forces in the system can be calculated by classical

3.2. Point-defect properties and solute transport 13

electrostatics according to [46]:

FI(t) = −∂E0

∂RI= −

⟨ψ

∣∣∣∣dHdR

∣∣∣∣ψ

⟩(3.14)

Thus, from the ionic configuration R(t) at time t, the total energy can beused to determine the forces in the DFT framework. This information canthereafter be used to advance the atomic positions in the system, resultingin a new atomic configuration, on which the forces can again be calculated.This provides an elegant solution to estimate the time evolution of an atomicconfiguration.

3.2 Point-defect properties and solute transport

As discussed in the introduction of this chapter, the segregation of chemi-cal components can be attributed to coupling between point-defects and theelements that make up the alloy. Particularly, vacancy-coupled diffusion isrelated to the preferential exchange with the defect and a certain element. Ifthe preference is directed towards a solute, the probability of solute-vacancydissociation will be very low, and diffusion will occur as one coupled speciesin a process commonly termed vacancy drag [47]. If on the other hand thepreference is towards the surrounding elements, the defect is unlikely toexchange with the solute. This in turn may manifest itself in such speciesmoving in the opposite direction compared to the vacancy flux (inverse Kirk-endall mechanism). The mechanisms behind interstitial-mediated migration,on the other hand, will vary depending on if the solute preferentially takes onan interstitial or substitutional configuration in the crystal lattice. Interstitialsolutes can often migrate independently of other defects, and this processdoes not necessarily require radiation-inducement. Substitutional soluteshowever, can take on a split interstitial, or dumbbell, configuration whenthey encounter an self-interstitial defect. The efficiency of this diffusion de-pends on both the correlation between the solute and the dumbbell, and thesurrounding energy landscape. The two main diffusion paths, vacancy- anddumbbell mediated, are illustrated for the fcc crystal in Fig 3.1.

FIGURE 3.1: Illustration of the two main defect mediated diffusionpaths in the fcc crystal (a) Vacancy (b) 〈1 0 0〉 Dumbbell. The bluesquare marks the position of the vacancy and the red circle the posi-tion of the solute

14 Chapter 3. Radiation-induced segregation and its effect on fracture properties

The relative diffusion rates of the chemical components in the material willdetermine the segregation tendencies. Solvent (A) and solute (B) diffusioncoefficients induced by the two mechanisms (X) respectively, can be calcu-lated in a first-principles framework accordingly [48]:

D∗A = ga20 f0cXω0 = ga2

0 f0cXν0 exp(− Em

AkBT

)(3.15)

D∗B = ga20 fBcXωXp1nn = ga2

0 fBcXνB exp

(−Em

B − EbXB

kBT

)(3.16)

where g is a geometrical factor, a0 is the lattice constant, f0 and fB are the self-and solute correlation factors, related to the probability for the solute atomto make an immediate reverse jump back to its previous position, ω0,X arethe jump frequencies, ν0,B are the corresponding attempt frequencies, p1,nn isthe probability of having a solute-defect pair at a first nearest neighbor (1nn)distance, and Em is the migration barrier, and EBX is a solute-defect bindingenergy.

Since the relative diffusion rates determine the segregation of material com-ponents, an important part of this thesis has been to asses solute migrationrates in a first-principles framework. In this context, migration barriers andbinding energies were calculated. The former quantity describes the energyof a transition state between two locally stable configurations, and thus rep-resents the energy barrier the species must surmount in order to migrate. Thelatter describes the strength of the solute-defect interaction and shows if theinteraction is repulsive or binding. In the supercell approach, the bindingenergy can be calculated accordingly:

Eb(A1, A2, . . . , An) = ERef + E(A1 + A2 + · · ·+ An)− ∑i=1,...,n

E(Ai) (3.17)

where ERef is the energy of the supercell without any defects, E(Ai) is theenergy of a supercell with an isolated defect Ai and E(A1 + A2 + · · ·+ An)is the energy of the supercell containing all Ai interacting defects.

The diffusion efficiency is strongly correlated with the presence of point-defects. In irradiated conditions, the concentrations depend on factors suchas formation rate, sink strength, and vacancy-interstitial recombination rates.A proper consideration of all such effects is beyond of the scope of this thesis.In equilibrium conditions, however, defect concentrations can be estimatedby Eq. 3.1. In a first-principles framework, the concentrations can thus becalculated by the corresponding formation energies:

Ef,solvent = EDef −nDef

nRefERef (3.18)

where EDef/Ref and nDef/Ref are the energies and number of atoms in the de-fected and reference systems. If on the other hand the defect consists of animpurity atom, either in substitutional or interstitial configuration, the defect

3.3. Radiation-induced segregation 15

formation enthalpy is given by:

Ef,solute(nA + pB) = E(nA + pB)− nE(A)− pEimp(B) (3.19)

where n is the number of solvent atoms A, p is the number of solute atoms,B, and Eimp is the energy of the impurity in its reference configuration.

3.3 Radiation-induced segregation (RIS)

Since the radiation damage event is ultimately stochastic, the generated de-fects will not be evenly distributed in the material. As a consequence, localchemical potential gradients (CPGs) will arise, and fluxes of PDs will followto counteract the corresponding forces. The flux of a species i induced by aCPG, ∇µ, acting on a species is given by j:

Ji = −∑j

Lij∇µj

kBT(3.20)

where Lij are the transport coefficients, which dictate the kinetic responseof the system subjected to thermodynamic forces. Eq. 3.20 can be seen asan alternative to the more common Fick’s equation to describe solute flux.Based on the transport coefficients, the solute tracer diffusion coefficient canbe calculated accordingly:

D∗B =LBB

cB(3.21)

where cB is the solute concentration.

The combined vacancy and interstitial flux-coupling effects on radiation-inducedsegregation in a binary alloy is under steady-conditions given by [49–51]:

∇cB

cB= −α

∇cV

cV(3.22)

where cV is the vacancy concentration and

α =LAILAV

φ(LAIDB + LBIDA)

(LBV

LAV− LBI

LAI

)(3.23)

where the subscripts A, B, V, I represent solvent atoms, solutes, vacancies,and interstitial defects respectively, Lij are the transport coefficients (or co-efficients of the Onsager matrix), DA and DB are the intrinsic diffusion co-efficients of solvent and solute atoms, and φ is the thermodynamic factor,describing the change of chemical potential of one species with respect to aconcentration change of another.

As shown in Eq. 3.22, the direction of the solute flux is governed by thesign of α, which in turn is given by:(

LBV

LAV− LBI

LAI

)(3.24)

16 Chapter 3. Radiation-induced segregation and its effect on fracture properties

Consequently, the direction of the solute flux is dictated by the relative mag-nitude of the two ratios in 3.24, one which is related to vacancy- and one tointerstitial coupled diffusion. If one of the two has a value of 1, this describesan uncorrelated flux between the solute and that defect. The RIS tendenciesof the system will in this case be fully described by the opposite mechanism.The two ratios, termed partial diffusion coefficient (PDC) ratios, can thus beused to evaluate the respective contributions by the two mechanisms.

As discussed above, in the case of vacancy-mediated diffusion, the competi-tion between the solute and the bulk species can result in three distinct cases.In the case of preferential vacancy-solute exchange and positive flux coupling(LBV > 0), solutes migrate in the same direction as vacancies (vacancy drag),and solute enrichment at sinks occurs. This is indicated by a negative va-cancy PDC ratio since LAV is always negative. When vacancy drag does nottake place, enrichment can still take place, as in this case, both solvent and so-lute atoms diffuse against the vacancy flux (inverse Kirkendall mechanism),thus away from the sink. If the solute is slower than the solvent (preferentialsolvent-vacancy exchange), the solute will effectively be enriched at sinks.This is indicated by 0 < PDCvac < 1. In the case, instead, of preferentialsolute-vacancy exchange, solutes will diffuse away faster than solvent atoms,and depletion occurs (PDCvac > 1). For interstitial-mediated diffusion, theflux of solutes cannot be in the opposite direction to that of interstitials, so thePDCSIA is always positive. If solute transport is faster than solvent transport,this results in enrichment at sinks, and in this case the PDCSIA is greater than1. The PDCSIA is smaller than 1 in the opposite case.

The intrinsic driving mechanisms and the overall solute RIS tendencies canbe evaluated by computing the full matrix of transport coefficients [39]. Tocalculate these coefficients, the relative rates of the various atom-defect ex-changes are required. The original model to calculate such exchange rateswas developed by Lidiard and Le Claire [48, 52, 53]. In this model, five fre-quencies for atomic-vacancy exchange were determined for the fcc system,and based on the relative frequencies of the solute and solvent respectively,migration rates could be assessed. This five-frequency model was furtherdeveloped by Allnatt into an expression which could be used to calculatethe transport coefficients for vacancy-mediated diffusion [54]. Complicationshowever emerge since the relative diffusion rates are strongly dependent onthe local atomic environment. Since the five-frequency model neglect anysolute-defect interaction beyond the first nearest neighbor, and is valid onlyfor vacancy-mediated diffusion, it is not sufficient to capture the full com-plexity of the migration process. Additionally, the common five-frequencymodel accounts for only the vacancy mediated solute diffusion. The gen-eral approach can however be further expanded by means of self-consistentmean-field (SCMF) theory [55]. In the current work, the latter approach hasbeen used to calculate the transport coefficients in a binary alloy, and basedon model in Eqs. 3.22-3.23 the steady-state solution of Eq. 3.20 could be ob-tained, in turn describing the evolution of the irradiated material.

3.4. Modeling of cleavage fracture from first principles 17

After the segregation tendencies the material had been quantified, the im-pact of the enriching solutes on the structural integrity of the material can beassessed. The relevant theory behind the models used in the current work toestimate the latter effect, is presented in section 3.4.

3.4 Modeling of cleavage fracture from first principles

Basic approaches for theoretical description of crack propagation often in-volve the construction a cohesive zone model (CZM) [56,57]. In such models,the goal is to describe the stress response of the zone ahead of a preexistingcrack tip, since this is the part of the material that is susceptible to further de-cohesion. This region can be seen as an extension of the crack, characterizedby two cohesive surfaces, in turn held together by a cohesive traction. Thecrack will propagate in this region when the external forces are larger thanthe cohesive traction. Thus, by describing the energies and emerging forcesassociated with pulling apart the two halves, traction-separation curves cor-responding to the experimental ones can be obtained.

Cohesive zone

FIGURE 3.2: Illustration of the cohesive zone

Fracture is ultimately caused by the breaking of atomic bonds, and this pro-cess can be captured by means of ab initio methods. However, computationalresources limit the number of atoms which can be incorporated in such mod-els, and direct simulation of fracture from first-principles is not possible ata larger scale for the time being. On the other hand, common large-scalemodels can not resolve the strong stress gradients at the tip of a crack. Fora proper evaluation of the fracture process, it would thus be beneficial toevaluate the process of bond breaking in a first-principles framework, andcoarse-grain this information to construct analytical cohesive zone models ofarbitrary thickness. A first principles approach which can be used to generateatomistically informed interlayer potentials valid also in larger scale models,was first proposed by van der Ven and Ceder [58]. Their model not only pro-vides a sound basis for a multi-scale description of the fracture response ofa brittle material, but solves many of the problems associated with simple abinitio models (see paper IV). The relevant theory is presented below.

18 Chapter 3. Radiation-induced segregation and its effect on fracture properties

3.4.1 Excess energy assessment (EEA) of a fracturing region

The stress-response of a fracturing system can be assessed from first prin-ciples by straining the configuration in a number of steps, and fitting thecorresponding energies to a model curve. The stress-strain curve, or traction-separation behaviour, is given by the first derivative of the energy-separationcurve. In this method, known as the ab initio stress test, atomic relaxationsshould be included for a proper description of surface energies and workof fracture [58–61]. However, Nguyen and Ortiz have shown that in the abinitio stress test, both the peak stress and critical separation are strongly de-pendent on the number of atomic layers that are allowed to relax during thecalculations [62]. This apparent contradiction leads to the question if there isa possible way to allow for atomic relaxations, all the while results are keptindependent of the number of free layers in the system. In this context itshould be noted that the main reason that the obtained results depend on thenumber of free layers, is that in the ab initio calculations, the energy responsedoes not correspond only to that of the decohering region, but also to that ofthe surrounding bulk. However, since the fracture will occur at the interface(e.g. a GB), the bulk response is not the main interest in this context. This re-alization provides the basis of the excess energy assessment (EEA) method,first introduced by van der Ven and Ceder [58]. The goal of this approach isto describe the interface as an excess property in an otherwise perfect bulk,and to extract this response from the ab initio energy-separation curve.

Since the energy-separation curves obtained from first principles representthe combined response of the interface and the underlying bulk, the totalenergy can be separated into the individual contributions [62]:

Φtot(δ1, δ2) = NφBulk(δ1) + φInterface(δ2) (3.25)

where φBulk is the energy of an individual plane potential, N is the number ofatomic layers in the model that are allowed to relax, and the individual elon-gations, δ1 and δ2, correspond to the interplanar stretch associated with thesurrounding bulk and the decohering interface, respectively. The objective ofthe EEA method is to parametrize φInterface(δ2), and in this way characterizethe stress-response of the decohering region. This must be done by adher-ing to the requirement of mechanical equilibrium of stresses at the CZ/bulkboundary, and the condition that the total elongation consists of all individ-ual elongations:

δtot = Nδ1 + δ2 (3.26)

An appropriate ansatz of the potential form is helpful in this context. In thecurrent work, an extension of the universal binding energy relation (UBER),the xUBER, was used [63–65]:

φ(δ) = Cδ2

[1−

(mmax

∑m=0

αm

(δ

λ

)m)

exp(− δ

λ

)](3.27)

where δ is the separation and C is the stiffness. C, am, and λ are parame-ters to be fitted in this model. A thorough description of how this is done,

3.5. Temperature dependence of defect properties 19

is provided in paper IV, and will not be repeated here. The main take homemessage is twofold: Firstly, the EEA method allows for assessing how thestress response of a material is affected by defects such as grain boundaries,and how the presence of impurities will affect this behaviour. Secondly, andmost importantly, results obtained in the EEA framework are independenton the number of atomic planes that are included during calculations, andthe method thus enables unambiguous evaluation of the interplanar poten-tial [58, 65, 66]. The results obtained in this framework will for this reasonprovide a solid basis for further multi-scale modeling and analysis.

It should be noted that the use of CZM embedded in an elastic mediumis more accurate for materials undergoing brittle fracture, without signifi-cant atomic rearrangement [65]. The approximation is here considered valid,since alloys that are ductile in an inert environment, during SCC commonlyfail in a brittle manner [67].

3.5 Temperature dependence of defect properties

Electronic structure calculations have proven very successful in representingground state magnetic properties of 3d transition metals [68]. It should how-ever be noted that the calculations are generally performed at 0 K. At tem-peratures above the magnetic transition temperature, many systems behavevery differently compared to at the lower temperatures where the magneticmoments are more organized. Transposing properties to finite temperaturesmay for this reason result in both quantitative and qualitative errors [69]. Par-ticularly, since the defect concentration displays an exponential dependenceon Gf (Eq. 3.1), small errors in this quantity may result in the calculated defectconcentration being wrong by orders of magnitude. Generally, the problemsassociated with transposing the Gibbs free energy at 0-K to the paramagneticregion can be attributed to the adiabatic decoupling:

GPM = Gel + Gvib + Gmagn (3.28)

where the total free energy is separated into the corresponding electronic, vi-brational, and magnetic contributions. At 0 K, this decoupling is certainlyvalid, since the excitations have very different time scales. However, mag-netic excitations in the high temperature paramagnetic region are very dif-ferent from those at low temperatures. Additionally, in the high temperatureparamagnetic state, the impact of atomic motions become significant. Forthis reason, the three contributions cannot be separated in this temperatureregime, but they must all be handled simultaneously [70]. A correct repre-sentation of magnetic disorder in high temperature systems is not an easytask. Regarding the 3d transition metals, general consensus states that, al-though itinerant electrons determine their magnetization, the magnetic mo-ments are relatively strongly bounded to their respective sites [70]. There-fore, all electrons can be ascribed a local moments even in the paramagneticregion. The local moments are parallel to the net magnetization density atthe corresponding site, but become disordered above the magnetic transitiontemperature. There are a number of methods to handle the paramagnetic

20 Chapter 3. Radiation-induced segregation and its effect on fracture properties

state available in the literature, see e.g. [70] for a review. In the current work,a DFT-MD approach based on a disordered local moment model has beenused to describe this region. The relevant theory and approach are describedbelow.

3.5.1 Disordered local moment molecular dynamics (DLM-MD)

The static disordered local moment (DLM) model, was first introduced byHubbard [71–73] and Hasegawa [74, 75] to describe the paramagnetic phase.The DLM model is based on the assumption that, in the paramagnetic re-gion, an equal amount of atoms with spin up and spin down will exist andbe collinear, but the magnetic moments are fully disordered. For systemsabove the magnetic transition temperature, it is valid to neglect the impact ofnoncollinear orientations of local magnetic moments [69]. The DLM picturefurther assumes that the system of electrons does not cover its phase spaceuniformly, but the magnetic state of the system will get stuck in a configu-ration for certain time periods, denoted spin-flip times tSF. The spin stateof each configuration is completely random, with the caveat that the sum ofall spins is equal to zero. The energy and electronic structure of such a con-figuration can be calculated within the conventional alloy theory using thecoherent potential approximation, and based on the results a static approx-imation of the system is obtained [69]. The approach incorporates the in-terplay between magnetic and configurational degrees of freedom; however,since both charge and spin fields are dynamically fluctuating in both spaceand time, such interactions cannot be incorporated in the model. Althoughthe formalism is greatly simplified, the DLM model captures important partsof the correlations, and it may be sufficient to describe the paramagnetic stateof many simple systems [76]. However, a simultaneous treatment of latticevibrations with the magnetic excitations is hardly feasible within the DLMmodel, and it is very limited in treating systems with point-defects or grainboundaries [70].

The static picture of the DLM model becomes less accurate as temperaturesincrease and the impact of lattice vibrations become significant. For an im-proved description of the paramagnetic phase, Steneteg et al. suggested amethod for simultaneous modeling of magnetic and vibrational finite-temperatureeffects [77]. The approach is based on the picture presented in the DLMmethod, that the magnetic subsystem gets stuck for short periods of time atpoints in the phase space corresponding to certain magnetic configurations.However, calculations in the present work are performed within the frame-work of ab initio MD, which allows for the simultaneous treatment of latticevibrations. In this approach, the magnetic state of the system is randomizedperiodically at a time step given by tSF during the MD simulation. This isin line with the assumption of the DLM picture, that the system gets stuckfor periods of time, and that after such a period the system moves rapidly toanother point in phase space. This improved model provides a descriptionof the influence of the dynamically disordered magnetic state on the latticedynamics, and importantly the time evolution of such characteristics.

3.6. Calculation details 21

3.6 Calculation details

Detailed descriptions of the calculation procedures are provided in the re-spective papers. In this section, general considerations are presented.

All first-principles DFT calculations were performed using the Vienna Ab ini-tio Simulation Package (VASP) [78, 79], with pseudopotentials from the VASPlibrary generated with the projector augmented wave (PAW) method usingthe Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional [44]. Mi-gration barriers were calculated by the nudge elastic band (NEB) method[80, 81] implemented in VASP with the climbing-image algorithm [82] to ob-tain the saddle-point energy.

Solute-PD binding and migration energies were evaluated in the DFT frame-work. Based on the system properties, the transport coefficients could be ob-tained by means of the self-consistent mean-field (SCMF) theory [55], imple-mented in the KINECLUE code [83]. Calculations were performed on binaryNi-X systems in the dilute limit with a solute-to-solvent ratio of 10−4.

3.6.1 Development of computational codes to extract excess response andmagnetic properties

Previous studies in the literature have implemented the model proposed byvan der Ven and Ceder [58] and extracted the excess parameters associatedwith the decohering region (section 3.4.1). However, to the best of the au-thor’s knowledge, no computational code is available to do this straight-forwardly from first-principles calculations. An important part of the cur-rent work has for this reason been to develop such a code, which could beused in connection with VASP to extract the excess properties of a system. Toperform the calculations in accordance with the model which has here beentermed EEA, two computational codes have thus been developed: one whichallowed for simple generation of all the required input (POSCAR) files for VASPcalculations, and one which allowed for the simple extraction of the excessenergy and traction response. The necessary considerations are discussedbelow.

The calculations in the current work were based on a Σ5 [100](012) GB (il-lustrated in Fig. 5.5) and bulk supercell, both generated using the GB studiosoftware [84]. The length of the axis normal to the GB plane was first opti-mized in all systems (with and without the presence of solutes) to find thereference length from which the stress-response could be evaluated. As pe-riodic boundary conditions are implemented in VASP, a number of atomicplanes were locked at their equilibrium distance at all calculations followingthe initial relaxation, to assure that only one of the GBs underwent separa-tion. Thereafter, the free planes were strained in controlled (∼ 0.1 Å) stepsup till the point of fracture and the energy associated with each locally sta-ble configuration was evaluated. Two initial configurations were considered:one in which the free planes were homogeneously stretched, and one with a

22 Chapter 3. Radiation-induced segregation and its effect on fracture properties

vacuum in the fracture plane. This assured that the minimum energy config-uration was used at all points in the following EEA analysis. A code has beendeveloped in the current work which allows the user to easily define whichplanes should be allowed to undergo relaxation, and which locks all otherplanes at the equilibrium distance. The code outputs the required POSCARfiles with both initial configurations (uniform elongation and vacuum in cen-ter) at any strain. This allowed to easily perform all the required DFT calcu-lations based on any initial supercell.

As a next step, a computer code was developed which took the VASP gener-ated energy-separation data, and extracted the excess energy in accordancewith the theory presented in 3.4.1. If the energy-curves obtained in the firststep are smooth and the data points are sufficiently close, this code can beused straight forwardly and the energy- and traction-separation curves caneasily obtained. In this context, it should be noted that it is essential to prop-erly define initial configuration and fracture planes for a correct descriptionof the fracture process. Particularly in the case where impurities were addedto the system, since initial position and fracture paths must be defined bythe user, and if this is not done in accordance with the minimum energy, theanalysis can give misleading results.

A computer code was also developed by the author to facilitate the simula-tion of the paramagnetic region in accordance with the DLM-MD approach.This code set up a system with randomly oriented collinear local moments,after which spin-polarized MD calculations could be performed. The codewas used to continuously perform the spin randomization throughout theDFT-MD calculations. After each randomization, the DFT-MD relaxationwas allowed to proceed for a short period before the magnetic moments wererandomized again. The lattice configurations and velocities remained un-changed between relaxations. System energies were evaluated based on theaverages of the energy trajectories. However, as these fluctuated stronglyduring the simulation, the ensemble averages depended strongly on the cut-off points for averaging. In order to reduce the scatter, a set of ensembleaverages were generated with sliding starting points in time, and then theaverage of those ensemble averages were used to represent the actual aver-age.

23

Chapter 4

Results I: Surface Processes

The goal of the current chapter is to describe the kinetics and reaction mech-anisms for the reactions between H2O2 and oxide surfaces of importance innuclear reactor environments. In section 4.1, the impact of anions in solutionon H2O2 reaction mechanism has been evaluated, in order to provide a gen-eral understanding of how competing reactions can affect the system, andto better understand the reaction mechanisms. In section 4.2, the reactionmechanisms between H2O2 and a steel commonly used in reactor applica-tions, have been quantified.

4.1 Anion effects on the catalytic decomposition of H2O2 onan oxide surface

In systems of practical importance for nuclear applications, where water ispresent other elements are present as well (deliberately or not). These speciescan significantly alter the chemistry compared to the pure system in whichreaction mechanisms are generally identified. To provide an increased un-derstanding of the mechanisms and kinetics in such complex environments,a beneficial approach is to first look at a model system. ZrO2 has its metalcation in its highest oxidation state, and can thus only react with H2O2 viacatalytic decomposition according to reactions R1-R4.

The impact of Br– , and Cl– respectively, and in combination with HCO3– ,

on the mechanism and kinetics of the reaction between H2O2 and the ZrO2surface has been evaluated. Since ZrO2 is present on the outer surface of thefuel cladding, this system is more relevant for final repository conditions, butcan provide an increased understanding of both direct anion-H2O2 reactions,and competing reactions at the oxide surface. In the current work, reactionmechanisms were evaluated by addition of high concentrations of the anions(∼1 M) to solution. The increased background charge in solution will dimin-ish the impact of otherwise repulsive interactions. Particularly, since powdersuspensions were used in the current work, a significant ionic strength maylead to agglomeration of oxide particles. To account for such effects, refer-ence experiments were performed with ClO–

4 in solution, rather than ZrO2only. ClO–

4 is inert in the system, but gives the same ionic strength effects asthe potentially more reactive anions.

A challenge when evaluating reaction mechanisms in a heterogeneous sys-tem, is to quantify the relative contributions of solutes reacting in solution,

24 Chapter 4. Results I: Surface Processes

versus reactions on, or with, the oxide surface. I can for this reason be helpfulto first quantify the impact of the anions in a homogeneous aqueous solution.Of particular interest in this work was to assess the reaction mechanisms ofthe OH•, since these radical is formed in the decomposition of H2O2. Ra-diation chemical yield and reaction mechanisms of OH• are well-known inthe homogeneous aqueous system. γ-irradiation experiments were for thisreason first performed on this system, in order to evaluate the impact of theanions. In this way important information could be acquired, which couldbe further used in the next step when the impact of the anions in the hetero-geneous system was evaluated. To account for possible pH effects, all exper-iments in this work have been performed at both pH 7.5 and 9. It shouldbe noted that this pH range is mainly of relevance for final repository condi-tions [2].

4.1.1 The homogeneous system

In the current work, N2O-saturated aqueous solutions were γ-irradiated, andthe formation of CH2O, formed in the reaction between OH• and tris in so-lution, was monitored. Results in Paper I show that at pH 7.5, Cl– has verylittle influence, whereas the presence of Br– gave a significant increase inCH2O yield. The impact of ClO4

– was also evaluated in reference to the sys-tem containing tris only, and the minor differences between the two confirmsthe initial assumption that ClO4

– is inert in the system. At the higher pHof 9, the anions displayed a stronger influence on the CH2O yield. This isillustrated in Fig. 4.1.

0 200 400 600 800Dose (Gy)

0.00

0.05

0.10

0.15

0.20

[CH 2

O] (m

M)

BrBr / HCO3ClCl / HCO3ClO4ClO4 / HCO3HCO3

FIGURE 4.1: Concentration of CH2O as function of dose for aque-ous solutions containing 1 M Br– , Cl– and ClO4

– respectively, andin combination with 0.1 M HCO3

– . Dose rate=0.13 Gys−1, T=295 K,pH=9

4.1. Anion effects on the catalytic decomposition of H2O2 on an oxide surface 25

Fig. 4.1 shows that unlike the results at pH 7.5, at pH 9 the presence of Cl–

increases the CH2O yield compared to the reference system. However, inPaper I it was concluded that this effect can not be attributed to changes inthe radiation chemistry. Br– , on the other hand, was shown to significantlyincrease the yield of formaldehyde at both neutral and alkaline pH. As canbe seen in Fig. 4.1, the impact is even more important when Br– is combinedwith HCO3

– . The combined effect of the two anions will be further discussedbelow. However, as concluded in paper I, the impact of HCO3

– cannot beattributed to changes in the aqueous radiation chemistry, whereas Br– canreact with OH• accordingly [85]:

Br− + OH• −−→ BrOH•− (R9)

BrOH•− −−→ Br• + OH− (R10)

Br• + Br− ←−→ Br•−2 (R11)

Since the current system contains 1 M of Br– , with a reaction rate constantof reaction R9 of 1.2·109 M−1s−1 [85], and 20 mM of tris, which reacts withOH• with a rate constant of 1.1·109 M−1s−1 [21], the OH• is effectively scav-enged by Br– and converted into Br• –

2 . Thus, the high CH2O yield showsthat formaldehyde is formed in the reaction between tris and Br• –

2 , likely fol-lowing one-electron oxidation of the former. Additionally, since the presenceof bromide does not affect the overall concentration of radicals in the sys-tem, it was concluded that this reaction has a higher selectivity in producingformaldehyde compared to the reaction between OH• and tris. This resultis of great interest in the context of this analytic method, since it is generallybelieved that formaldehyde is only formed upon hydrogen abstraction fromtris.

4.1.2 The heterogeneous system

Once the effect of the reaction mechanisms between OH• and the anions hadbeen quantified, the impact of the anions in the more complex heterogeneoussystem could be assessed. The kinetics of H2O2 consumption together withCH2O production was for this reason monitored in presence of the consid-ered anions in a ZrO2 oxide suspension. General trends in CH2O yield werein line with observations in the homogeneous system, with a significant in-crease in the presence of Br– , and very little impact of Cl– and ClO4

– . In-terestingly however, in the presence of the oxide surface, the combined effectof Br– and HCO3

– differed significantly in the homogeneous and heteroge-neous systems. The CH2O concentration as a function of time in the hetero-geneous system is illustrated in Fig 4.2.

26 Chapter 4. Results I: Surface Processes

0 5 10 15 20 25 30 35Time (103 s)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

[CH 2

O] (m

M)

BrBr / HCO3ClCl / HCO3ClO4ClO4 / HCO3

FIGURE 4.2: Concentration of CH2O as a function of time in presenceof 2 g of ZrO2 in combination with 1 M Br– , Cl– and ClO4

– respec-tively, and in combination with 0.1 M HCO3

– . T=295 K, pH=9

As can be seen in Fig 4.2, when Br– is the only anion present, the CH2O yieldis significantly increased compared to all other systems. This is an indica-tion that, in accordance with the homogeneous system, Br– efficiently reactswith OH• and via Br• –

2 forms CH2O, although the radical is surface boundin this system. Indeed, based on the high rate constant (Br– + OH•: k ∼ 109

M−1s−1 [85]), the anion is very likely to react with OH• in the current system.Regarding the impact of HCO3

– , general trends are also in line with previ-ous observations. However, in clear contrast to the homogeneous system,the combination of HCO3

– and Br– significantly decreases the CH2O yield.The reason behind this trend is not straight forward to quantify. It is in thiscontext however important to note that the reactions are monitored by thepresence of a probe (tris), which has the sole purpose of interfering with thecatalytic decomposition of the H2O2. Indeed, by scavenging of the hydroxylradicals formed in reaction R2, these will no longer be available to contributeto further decomposition of H2O2 according to reaction R4. A high scaveng-ing capacity of the probe will of course maximize the yield of CH2O, butconsequences may involve an impact on the H2O2 reaction rate. The effect ofthe OH• scavenging becomes increasingly important when both HCO3

– andBr• –

2 are present in the system. The two anions can react accordingly:

Br•−2 + HCO3− ←−→ 2 Br− + CO3

•− + H+ (R12)

CO3• – can in turn react with H2O2, thus providing an alternative reaction

path for the molecule. This reaction will however not result in the formationof CH2O. The effective scavenging capacity of tris will decrease the proba-bility of the hydroxyl radical to react with H2O2, and the alternative reactionpath of the molecule, provided by CO3

• – becomes increasingly important.

4.2. H2O2 and γ-induced corrosion of austenitic alloys 27

In paper I it was concluded that the reaction path provided by CO3• – is the

likely explanation of the decrease in CH2O yield by the combination of thetwo anions.

Reaction mechanisms were also evaluated by monitoring the concentrationof H2O2 in solution. This concentration was seen to decrease very rapidly inthe initial state of the experiment. After the initial decrease, the molecule wasslowly consumed over a few hours. The effectiveness of the initial step wasseen as an indication that H2O2 strongly and quickly adsorped to ZrO2 in thecurrent system. The second part of the reaction was thus used as a basis forfurther evaluation of the impact of the corresponding anions. Results in Pa-per I showed that this process was significantly faster in presence of Br– , andat the higher pH in the presence of Cl– . Since it has been established in theprevious discussion that Br– effectively removes OH•ads from the surface, butdoes not react with H2O2 in itself, it is concluded that the effective decom-position can be attributed to an increased amount of available surface sites,when the Br– removes the surface bound hydroxyl radicals. This also impliesthat the corresponding reaction with Cl– , forming ClOH• – , only efficientlyremoves OH• from the surface at higher pH. The reaction mechanisms dis-cussed in the current sections, are summarized in Fig. 4.3.

2

HO(ads)

tris

HO(ads)

Br--Br2

H2O2

ZrO

CH2O

H2O2 HOO

tris

CH2O HCO3-

CO3

FIGURE 4.3: Combined impact of Br– and HCO3– on H2O2 decom-

position in the heterogeneous system.

4.2 H2O2 and γ-induced corrosion of austenitic alloys

The focus of the current section is to evaluate the reactivity of H2O2 towardsthe 304L alloy, commonly used in nuclear reactor applications. Unlike thepreviously discussed ZrO2 system, catalytic decomposition is in this case notthe only reaction path of H2O2, but the molecule can also oxidize the steel.Consequences may involve oxidative dissolution of steel components, fol-lowed by Haber-Weiss type reactions in solution. The various mechanismshave here been discussed based on reaction rate constants, and dissolutionof steel components.

28 Chapter 4. Results I: Surface Processes

4.2.1 Reactivity of H2O2 towards a 304L oxide surface

The reactivity of H2O2 in presence of the 304L alloy has been evaluated bymonitoring the consumption of H2O2 as a function of time and available ox-ide surface in aqueous suspension of the metal powder. From the rate equa-tion (Eq. 2.3), the time dependence of the H2O2 consumption could be de-rived:

ln([H2O2]t[H2O2]0

)= k2t×

[SAV

](4.1)

where SA/V stands for steel surface area to solution volume ratio. The re-sults from the kinetic study are presented in Fig. 4.4.

0 5 10 15 20Time (Days)

1.0

0.5

0.0

0.5

1.0

1.5

ln([H

2O2]

[H2O

2]0

)

2 103 m 1

4 103 m 1

8 103 m 1

16 103 m 1

FIGURE 4.4: The natural logarithm of [H2O2]t/[H2O2]0 as a functionof time for different surface area to solution volume ratios

As can be seen in Fig. 4.4, all considered oxide concentrations give linear cor-relations. This is an indication that pseudo-first order kinetics can be applied.In this case the relation between the rate constants is given by Eq. 4.2

k∗ = k2

[SAV

](4.2)

where k∗ is the pseudo first order reaction rate constant. The pseudo-firstorder rate constants as a function of SA/V is presented in Fig. 4.5.

4.2. H2O2 and γ-induced corrosion of austenitic alloys 29

0 2 4 6 8 10 12 14 16Surface Area/Volume (103 m 1)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5k

*(1

06

s1)

FIGURE 4.5: Pseudo-first order rate constants versus steel surfacearea to solution volume ratio (SA/V)

Based on the curve fit of data in Fig. 4.5, the pseudo-first order rate constantwas estimated to

k∗ = (6± 2)× 10−7(s−1) + (1.8± 0.2) · 10−10 ×[

SAV

] (m s−1

)(4.3)

Although the rate of H2O2 consumption increased with increasing SA/V,a significant intercept is seen in the current work. This is a indication ofH2O2 consumption in solution, possibly due to Haber-Weiss type reactionsresulting from oxidative dissolution of steel components. The linear con-tribution of k∗, on the other hand, increases with steel surface to solutionvolume ratio. This is likely a contribution of direct interactions betweenH2O2 and the oxide surface. The second order rate constant for this pro-cess, k2 = (1.8± 0.2)× 10−10 m s−1, is very low. Additionally, as discussedin section 4.1, the addition of H2O2 in the ZrO2 system was directly followedby a decrease in solution concentration. It was concluded that this was due toa strong adsorption of H2O2 to the oxide surface. Such effects were howevernot seen in the current system. Thus, the results in the current work showthat both the reactivity and adsorption of H2O2 on the steel is considerablylower compared to other systems of relevance in nuclear technology.

The yield of CH2O was measured during the consumption of H2O2 in thesteel system. After addition of 5 mM H2O2, the formaldehyde concentrationconverged to approximately 0.6 mM. This is a considerable yield comparedto what was found in the ZrO2 system, ∼ 0.05-0.1 mM. However, since bothcatalytic decomposition and Haber-Weiss type reactions have OH• as a com-mon intermediate [17], the detection of hydroxyl radicals cannot be used to

30 Chapter 4. Results I: Surface Processes

distinguish between mechanisms in the current system. It should howeverbe noted, that Fenton-like reactions in solution can lead to a close to 100%yield of formaldehyde, which is the case for homogeneous γ-radiolysis [23].Thus, the high yield in the current system can be seen as an indication thatsuch reactions are of importance.

4.2.2 Oxidative dissolution of steel components

In the current section, oxidative dissolution of steel components in solution isinvestigated, both by H2O2 addition, and by γ-irradiation, in which case OH•

is the predominant oxidant. Experiments have been performed in aqueoussolutions at room temperature, in which case the 304L steel surface consistsof a duplex oxide structure, generally with a layer of iron and nickel spinelsoutside a layer of oxidized chromium [86, 87]. The steel component in solu-tion can thus give an estimate of which part of the surface oxide is affectedby the various oxidants.

To determine the impact of H2O2, concentrations of Fe, Cr, and Ni in solutionwere measured after the steel had been submerged in 5 mM H2O2 solutionfor ten days. Results are presented in Table 4.1, where the correspondingsamples subjected to pure Milli-Q water, are included for reference.

TABLE 4.1: Concentration (M) of steel components in solution af-ter exposure of steel powder to aqueous solutions containing 5 mMH2O2 for 10 days. Results from the corresponding treatment wherethe steel was subjected to Milli-Q water are included for reference.Values not presented were below the detection limits (Fe: 9.0·10−8 M,Cr: 9.6·10−8, Ni: 8.5·10−8 M)

SA/V (m−1) Cr Fe NiH2O2 Ref H2O2 Ref H2O2 Ref

2000 2·10−6 - 9·10−8 - - -4000 4·10−6 - - - 5·10−7 2·10−7

16000 1·10−5 8·10−7 9 ·10−7 5·10−7 3·10−6 1·10−6

Results in Table 4.1 show that, although concentrations of Cr, Fe and Ni insolution are generally low, they are all higher after exposure to H2O2 com-pared to pure Milli-Q water. Of particular interest is the high Cr dissolutionin the H2O2 compared to the reference system. Since Cr oxides provide theprimary protection of the metal against an aggressive environment, resultsin Table 4.1 indicate that H2O2 can in fact impact the stability of this oxide.The results are in line with previous studies in literature, that conclude thatoxidizing conditions in general, and H2O2 in particular, can impair the struc-tural integrity of the protective Cr oxides on steel used in nuclear reactorapplications [88–90].

The system was also γ-irradiated at a dose rate of 0.13 Gys−1 for 24, 60, and 84hours respectively, and concentrations of steel components in solution weremeasured. Results are presented in Table 4.2.

4.2. H2O2 and γ-induced corrosion of austenitic alloys 31

TABLE 4.2: Concentration [M] of steel components in solution after16000 m−1 of 304L steel powder was subjected to γ-irradiations fordifferent time periods. Values not presented were below the detectionlimits (Fe: 9.0·10−8 M, Cr: 9.6·10−8, Ni: 8.5·10−8 M)

Time [h] Dose [Gy] Cr Fe Ni24 1.1·10−4 2·10−7 2·10−7 1·10−6

60 2.8·10−4 2·10−7 9·10−8 3·10−6

84 3.9·10−4 - - 4·10−6

In the case of γ-irradiation, mainly dissolved Ni was found in solution, whereasthe dissolution of Cr was comparable to background levels. The fact that Nidissolution is seen in this case, is an indication that the outer oxide layer hasbeen affected. Thus, results in this case can be seen as an indication that OH•

has a high impact on the outer layer. This can be explained by the high re-activity of the radical, which prevents it from reaching the inner oxide. OH•

is thus less likely to impact the corrosion resistance of the material comparedto H2O2.

4.2.3 Discussion concerning the reaction mechanisms between H2O2 andthe 304L steel

The results presented here, in combination with previous findings in the liter-ature, indicate that the H2O2 in solution affects the stability of the protectiveCr oxide. These results are of importance since Cr depletion is a significantfactor in IGSCC of stainless steel and Ni based alloys in nuclear reactor en-vironments [37]. However, the exact mechanisms driving the H2O2-induceddissolution of steel components are difficult to assess. Catalytic decomposi-tion is generally considered the main reaction path of H2O2 towards oxidesurfaces. Thus, although the release of steel components in solution indicatethat Haber-Weiss type reactions are at play in the current system, it is con-sidered likely that catalytic decomposition is the dominant reaction path ofH2O2 also towards steel surfaces. Since the general consensus in the litera-ture is that H2O2 is of importance in radiation induced corrosion of steel innuclear reactor environments, it is not unlikely that catalytic decompositionof H2O2 can in fact impact the stability of the oxide surface. It has been shownthat the surface-bound OH• formed in reaction R3 is actively involved in theoxidation of UO2 [17]. This is also a possibility in the steel system, although itshould be noted that since the radical cannot further contribute to the decom-position of H2O2, this this mechanism should be considered a redox reaction.However, in copper systems containing H2O2, O2 formed in reaction R4, hasbeen shown to be the main cause of corrosion of the metal [91]. If such mech-anisms are at play also in the steel system, the main reaction path of H2O2will, in accordance with many similar oxide surfaces, be catalytic decomposi-tion, all the while explaining the observed effect of the molecule on corrosionof stainless steel in nuclear reactor applications.

33

Chapter 5

Results II: Bulk and fractureproperties

In the current chapter, mechanisms behind radiation-induced segregation infcc materials are evaluated. This is followed by the investigation of how theenriching solutes can impact the cohesive properties of grain boundaries inNi. Finally, the point defect energies in the paramagnetic region of the samematerial are assessed.

5.1 Systematic trends of defect stabilities in fcc materials

As established in previous chapters, radiation-induced segregation is me-diated by the diffusion of point defects. While vacancies are always well-defined in a configurational sense, self-interstitials may come in many vari-ations. The relative stability of the configurations will ultimately determinethe possible migration paths in the material, and can therefore provide im-portant information regarding its evolution during irradiation. Due to thecomplex chemistry of austenitic alloys, building a correct description of therelative stability and mobility of defects in these materials is not an easytask. Simplified models can however provide important information to as-sess such properties in the more complex systems. In the search for system-atic trends, the relative stabilities of self interstitial defect in fcc metals ofgroups 10 and 11, in addition to fcc Fe, have here been evaluated. The de-fects considered are the 〈1 0 0〉, 〈1 1 0〉, and 〈1 1 1〉 dumbbells, together withthe octahedral and tetrahedral interstitials. Results are presented in Fig. 5.1,where the respective 〈1 0 0〉 dumbbell configurations were used as reference.

34 Chapter 5. Results II: Bulk and fracture properties

<100> <110> <111> OctahedralTetrahedral0.5

0.0

0.5

1.0

1.5

2.0

Rela

tive

form

atio

n en

ergy

(eV)

AgAuCuFeNiPdPt

FIGURE 5.1: Systematic defect stability trends in fcc material. Resultsare presented in reference to the corresponding 〈1 0 0〉 dumbbell.

Results in Fig. 5.1, show of clear tends in the considered fcc systems. The〈1 0 0〉 dumbbell and the octahedral are more stable compared to the otherconfigurations. In the case of both fcc Fe and Ni, which are important con-stituents of the steel of interest, the 〈1 0 0〉 dumbbell is most stable. Since thetrends in 5.1 for the pure metals are very clear, it is reasonable to assume theyhold also in the corresponding alloys of interest in the current work. For thisreason, the 〈1 0 0〉 dumbbell is assumed to be the dominant interstitial con-figuration in both irradiated Ni alloys and austenitic steels.

5.2 Radiation-induced segregation (RIS) in fcc Ni

The goal of the current section is to quantify the thermodynamic drivingforces and flux-coupling tendencies driving the RIS in nuclear reactor com-ponents. Many of the structural materials used in reactor applications areeither Ni-based or austenitic steels, and therefore fcc Ni was chosen to modelthe materials. This choice assured that the correct lattice structure was usedin calculations, and accounted for the fact that Ni is an important constituentin both materials. The focus of this section is to evaluate the response ofFe, Cr, Si, P, Ti, and Mn in the irradiated system. The choice of the Fe andCr was based on the fact that these in combination with Ni make up themain part of common alloys used for reactor application. Si and P are ofinterest since the former has been shown to have many beneficial effects inthe alloys [33, 34, 92–96], whereas the latter is know for its embrittling effectin both austenitic and ferritic materials [97–101]. Mn and Ti were includedsince they are common ingredients in the alloys, but previous studies regard-ing their impact in the material are scarce in the literature.

5.2. Radiation-induced segregation in fcc Ni 35

RIS tendencies have here been evaluated based on thermodynamic and ki-netic information between defects and solutes. Interaction energies, migra-tion barriers, and attempt frequencies were evaluated in the DFT framework.A detailed description of this methodology is provided in Paper III. Here it issimply noted that all considered solutes displayed a preference for the sub-stitutional compared to the interstitial configuration. If forced into the inter-stitial configuration, all solutes with the notable exception of P, were morestable in the 〈1 0 0〉 mixed dumbbell configuration. These results underlinethe conclusion of section 5.1, that the 〈1 0 0〉 dumbbell is more stable in fccsystems. P, however, was shown to be considerably more stable in an octa-hedral configuration. Based on the thermodynamic and kinetic interactionsand the relevant migration barriers, transport coefficients could be calculatedin a self-consisted mean field framework [55], implemented in the KINECLUEcode [83]. A number of material properties could be assessed from the re-sults, the main interest of the current work being the dominant diffusionmechanisms, the flux-coupling tendencies, and the overall radiation-inducedsegregation tendencies in the system. These properties are discussed in thefollowing sections.

5.2.1 Dominant diffusion mechanisms in steady-state conditions

The ratio between the vacancy- and interstitial mediated diffusion coeffi-cients can be used to assess the relative impact of each mechanism on theoverall solute diffusion. The diffusion coefficients, defined in Eq. 3.16, canin dilute conditions be calculated from the transport coefficients according toEq. 3.21. The diffusion coefficients will however depend on the PD concen-trations, which in turn depend on factors such as radiation flux, temperature,sink strength regimes etc. In order to keep the results generic, the transportcoefficients in the current work were calculated by assuming cI = cV = 1.However, to estimate the dominant diffusion mechanisms, a better approxi-mation is required. In steady state conditions, it has been shown that the ratiocV/cI can be approximated by DI/DV [102]. At low solute concentrations,the latter ratio is given by LI I/LVV . Under the assumption that self-diffusioncoefficients were not influenced by the presence of solutes in the system, theratio between the solute-vacancy and solute-interstitial diffusion coefficientscould thus be calculated. The results are presented in Fig. 5.2, in which theestimated relative defect concentrations are included in the lower inset. Asshown in Paper III, Ti and Mn do not migrate by the interstitial mechanism,and the two are omitted in the calculations. As can be seen in Fig. 5.2, allsolutes in the current study, with the exception of Cr, preferentially migratesvia the vacancy mechanism.

36 Chapter 5. Results II: Bulk and fracture properties

200 400 600 800 1000 1200 1400

T [K]

10 31

10 24

10 17

10 10

10 3

104

DB

,Int

DB

,Vac

* *

200 400 600 800 1000 1200 1400T (K)

106101110161021

cV cI

Cr

Fe

P

Si

II

II

I

FIGURE 5.2: Ratios of vacancy and interstitial solute tracer diffusioncoefficients in steady-state conditions. Interstitial diffusion is negli-gible for Ti and Mn, and the solutes are not included. The lower in-set displays the estimated vacancy-to-interstitial concentration ratioin thermal conditions. A solute-to-solvent ratio of 10−4 was used incalculations.

5.2.2 Partial diffusion coefficients (PDCs) in the irradiated systems

As discussed in section 3.3, the respective impact of the interstitial and va-cancy mechanisms on solute segregation can be assessed based on the cor-responding PDC ratios. Vacancy mediated diffusion gives three distinct re-gions: solute enrichment due to vacancy drag, solute enrichment due to sol-vent faster diffusion away from sinks with respect to the solute (inverse Kirk-endall), and solute depletion. Interstitials however, can not give rise to fluxesin the opposite direction to that of the defects. This mechanism will thus giveenrichment in the case of coupled migration, or depletion otherwise. ThePDC ratios for the considered solutes are presented in Fig. 5.3 as functions oftemperature.

5.2. Radiation-induced segregation in fcc Ni 37

200 400 600 800 1000 1200 1400

T (K)

106

104

102

100

0

100

102

104

Va

can

cy P

DC

Ra

tio

Cr

Fe

P

Si

Ti

Mn

Solute Enrichmentby drag

Solute Enrichment

Solute Depletion

(a)

-

-

-

-

200 400 600 800 1000 1200 1400

T (K)

10 10

10 7

10 4

10 1

102

Du

mb

be

ll P

DC

Ra

tio

Cr

Fe

Si

Solute Enrichment

Solute Depletion

(b)

-

-

-

-

FIGURE 5.3: Partial diffusion coefficient (PDC) ratios as functions oftemperature.(a) Vacancy mechanism (b) Interstitial mechanism. SinceP, Ti, and Mn migration via the dumbbell mechanism is negligible,these are omitted in (b)

Results in Fig. 5.3 show that Si and P are subjected to vacancy drag at alltemperatures, resulting in a strong enrichment of the two solutes due to thismechanism. Fig. 5.3(b) also indicates enrichment of Si due to the dumbbellmechanism. As discussed above, P is not stable in a dumbbell configuration,and the corresponding PDC ratio has therefore not been evaluated for this

38 Chapter 5. Results II: Bulk and fracture properties

element. Both Mn and Fe display depletion due to the vacancy mechanism.Results in Paper III show that the two are repulsed by vacancies. Based onFig. 5.3(a), it can be concluded that this interaction will lead to a preferen-tial exchange with the solvent, and the backward movement of the soluteswhich in turn results in depletion at sinks. Neither Mn nor Fe is susceptibleto interstitial-mediated migration (paper III), for which reason Mn is omittedin Fig. 5.3(b), and Fe displays a strong depletion due to this mechanism.

Interestingly, both Cr and Ti display a switchover between vacancy drag andno vacancy drag at 320 and 400 K, respectively. In the case of Ti, this mani-fests in a clear jump between enrichment and depletion. This effect is likelydue to the combination of a weak Ti-vacancy binding and repulsion, whenthe two are at first nearest neighbor (nn) and second nn distance, respec-tively (Paper III). A similar crossover between enrichment and depletion hasbeen found in bcc Fe, where the corresponding binding and repulsion at firstand second nn distance was also demonstrated [103]. It should however benoted that below 320 K, thermally-activated transport is very low in the ma-terial. It can for this reason be assumed that Ti enrichment at sinks is veryunlikely even in the lower temperature region. Similarly to Ti, Cr displaysa crossover between drag and no drag. However, vacancy-induced enrich-ment of this solute will occur even at temperatures up to∼1000 K. In the caseof interstitial-mediated diffusion however, the Cr and Ti behave very differ-ently. Fig. 5.3(b) indicates a strong enrichment of Cr due to this mechanism,whereas it was shown in Paper III that Ti migration by dumbbells can beconsidered negligible in this case.

5.2.3 Radiation-induced segregation

The solute-transport tendencies analyzed in previous sections will in steadystate irradiation conditions result in diffusion towards or away from sinks.The overall behaviour is defined by the radiation-induced segregation trends.In Fig. 5.4 the RIS tendencies in the dilute binary systems are presented.

5.2. Radiation-induced segregation in fcc Ni 39

200 400 600 800 1000 1200 1400

T (K)

� 3.0

� 2.5

� 2.0

� 1.5

� 1.0

� 0.5

0.0

0.5

1.0

RIS

Fa

cto

r (

/cB)

Cr

Fe

P

Si

Ti

Mn

Solute Depletion

Solute Enrichment

FIGURE 5.4: Radiation-induced segregation tendencies of impuritiesin Ni. Positive values indicate enrichment at sinks.

As can be seen in Fig. 5.4, results in this work display Cr, Si and P enrich-ment at sinks in the irradiated system, whereas Fe and Mn are depleted. Tion the other hand, displays a cross-over between enrichment and depletionat 320 K. In austenitic alloys, general experimental trends in the literatureshow Ni, Si, and P enrichment, and Fe, Cr, and Mn depletion at grain bound-aries [28, 29, 31, 104–109]. In Ni alloys on the other hand, Cr is observed toenrich and Fe to deplete at sinks [32,108,110]. Studies regarding the segrega-tion trends of Ti are scarce, although solute depletion following irradiationof binary Ni alloys has been observed [35].

In fcc Ni, interstitial mediated diffusion was shown in the current work notto be relevant for neither Fe, Ti nor Mn. The depletion of the solutes ob-served in Fig. 5.4 can for this reason be attributed to the vacancy-relatedinverse Kirkendall mechanism. In the case of Ti, the general trends in fcc Niare well in line with observations in bcc Fe, where the dumbbell diffusion isnegligible, and enrichment/depletion crossover has been found at approxi-mately 700 K [49]. In the case of fcc Ni, however this crossover lays at 320K, well below reactor operating temperature, whereas in the case of Fe thecrossover lays above these temperatures. The results can for this reason notbe directly transposed into austenitic alloys used in current NPPs. However,Ti has previously been shown to prevent swelling of austenitic materials inreactor applications [111–114], and to have a stabilizing effect on voids inNi-based model alloys [115]. These observations, in combination with theinefficient vacancy-mediated Ti transport in spite of a low solute-vacancymigration barrier (Paper III), can be seen as an indication that a preferentialexchange with Ti will result in trapping of vacancies in the material. A more

40 Chapter 5. Results II: Bulk and fracture properties

thorough investigation of such effects is however beyond the scope of thiswork.

As shown in Fig. 5.3, both the vacancy and the dumbbell mechanisms giverise to enrichment of Si. However, based on the magnitude of the PDC ratios,vacancy drag is considered the dominant mechanism up to temperatures ofapproximately 1200 K. Vacancy-mediated diffusion has also been shown tobe dominant in bcc Fe [49]. Since the same trends are observed in both alloys,it is considered likely that the experimentally observed enrichment in fcc Feand Ni alloys is mainly due to to the vacancy mechanism. Results in the cur-rent work also show that P diffusion is dominated by vacancy drag. This ishowever in contrast to bcc Fe, where the diffusion is dominated by dumbbellmigration [49]. However, P has a strong tendency for vacancy migration inbcc Fe as well [49]. One can thus suspect an important contribution of thevacancy mechanism to the P segregation observed in austenitic alloys.

In Fig. 5.4, it is shown that Cr is predicted to enrich at sinks during irradia-tion. However, general trends in Ni-based and austenitic alloys indicate thatthe Cr is depleted at sinks during irradiation. Thus, Cr is the only solute inthe current work where the observed trends are not in line with experimentalobservations. However, the calculations in the current work are performedin the dilute limit, and Cr is one of the main ingredients (∼20%) in austeniticand Ni-based alloys. Thus, perhaps not unexpectedly, the dilute binary alloyis not a proper representation for a solute in its highly concentrated environ-ment. With the exception of this solute, the findings in the current sectionare to a great extent in line with experimental observations. Additionally,the conclusion that the RIS behaviour is dominated by vacancy mediated-diffusion, corresponds very well with experimental studies in Fe–Cr–Ni al-loys [116–118]. The results in the current section shed light on the intrinsic ki-netic behavior of several important solutes in Ni alloys and austenitic steels.As the general segregation trends in the dilute fcc Ni alloys have now beenquantified, the question that remains is how the enriching solutes will impactthe cohesive properties of the material. This will be investigated in section5.3.

5.3 Segregation-induced changes in fracture properties in Ni

In section 5.2 it was shown that Cr, Si, and P will enrich at sinks in fcc Ni dur-ing irradiation. GBs are generally more susceptible to cracking, and soluteenrichment in such regions can significantly decrease their cohesive proper-ties. Experimental data regarding the impact of solute enrichment on cohe-sive properties in Ni are scarce in the literature and in the case of P, theoreticalstudies have given contradictory results [95, 96, 119–122]. Since the EEA ap-proach is generally considered more thorough compared to other methodsavailable in the literature, the impact of Cr, Si, and P on the cohesive proper-ties on a Ni GB has here been evaluated in this framework. The GB structureused in the current work is illustrated in Fig. 5.5, together with the consid-ered impurity configurations. A number of parameters, e.g. fracture pathand minimum energy configuration, are here discussed in significantly more

5.3. Segregation-induced changes in fracture properties in Ni 41

detail compared to previous studies. Additionally, common simplificationswhen evaluating a system’s stress-response in a first-principles frameworkare here assessed. The goal is to in this way not only provide a full under-standing of the impact of the solutes in question, but also to evaluate howgeneral model and method assumptions can impact the observed trends.

1

0

0

2

3

FIGURE 5.5: Illustration of the Σ5 [100](012) GB used in calculationsof the current work. The available interstitial solute configurationsare marked with 0, and the substitutional solute configurations con-sidered are marked by 1-3. Position 0 is empty in the case of the cleanGB (without solutes). The GB plane is marked with a dashed verticalline.

5.3.1 Impact of ab initio methodology: Rigid approach versus excess en-ergy assessment

For a simplified model of the fracture process, the ab initio stress test is of-ten performed with the approximation of rigid fracture. In this approach,known as the rigid grain shift (RGS), the distance between two rigid grains isincreased in discrete steps at a predefined fracture plane. The energy associ-ated with each configuration is thereafter evaluated in a DFT framework. Theenergy-separation data can simply be fitted to a model curve, φtot(δtot), andthe stress-response is given by its first derivative with respect to elongation.In the EEA approach (section 3.4.1), on the other hand, the energy given fromthe first-principles calculations represents the combined response of the frac-turing region and the underlying bulk. Thus, the energy associated with thedecohering region, φ(δ2), must be extracted and this can be associated withreal difficulties. Although the RGS approach is considerably more straightforward, this method does not necessarily give a correct representation ofsurface energies and the work of separation (WOS) [58–60]. Thus, the impactof this approach merits further investigation. In the current section, the stressresponse of the Ni GB has been assessed based on both the RGS and the EEAmethods. The impact of P in the system is thereafter evaluated in detail inboth frameworks. This will not only increase the current understanding ofhow enrichment of the solute can impact the system, but also highlight issuesassociated with common approximations and methodological choices.

Decohesion of bulk Ni and the clean Ni GB

In order to evaluate the impact of solute enrichment on GB properties, thestress response of the clean (impurity free) GB and bulk Ni were first quan-tified. The energy-separation response of the fracturing region obtained by

42 Chapter 5. Results II: Bulk and fracture properties

the RGS and the EEA methods are compared in Fig. 5.6(a). In Fig 5.6(b), thecorresponding traction-separation responses are presented.

0 1 2 3 4 (Å)

0

1

2

3

4

5 (J

m2 )

Bulk: EEABulk: RGSGB: EEAGB: RGS

(a)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 (Å)

0

5

10

15

20

25

30

(GPa

)

Bulk: EEABulk: RGSGB: EEAGB: RGS

(b)

FIGURE 5.6: Impact of ab initio stress-test model on Ni bulk/GB prop-erties. (a) Energy-separation (b) Traction-separation.

With the exception of the GB WOS, the RGS and the EEA methods only giveminor differences in the clean systems. The fact that the GB WOS obtained in

5.3. Segregation-induced changes in fracture properties in Ni 43

the RGS is higher than that from EEA (3.5 and 3.8 Jm−2, respectively) under-lines that importance of including atomic relaxations for a correct descriptionof the free surface energy. However, since the two methods display compa-rable results regarding the system’s stress-response, the rigid model can beconsidered as an adequate approximation in this case.

Impact of P on the cohesive properties of Ni

Once the stress-response of the clean system has been quantified, the im-pact of solutes could be assessed. In section 5.2.3, P was shown to mainlydiffuse by the vacancy mechanism in the current system. However, paperIII establishes that if a migrating dumbbell comes sufficiently close to P in asubstitutional configuration, the solute will be kicked out into an octahedralconfiguration, as one of the two Ni atoms takes its place in the crystal lattice.Due to the high defect generation rate in nuclear reactor environments, P isthus likely found also in the interstitial configuration. For this reason, theeffect of the element in both the substitutional and the interstitial configura-tion has been considered. In paper IV, the stability of the solute at availableGB sites is evaluated. Results show that P is stable in both interstitial andsubstitutional GB configuration, and both have been considered in the cur-rent work. In the case of the interstitial configuration, concentration effectswere evaluated by considering GBs containing one up to four interstitials,whereas the the impact of the solutes in the substitutional configuration wasevaluated based on one solute atom in the system. The number of impuritiesincluded in the calculations is within this context commonly termed surfacecoverage, θx, where the x signifies i or s, representing the interstitial or sub-stitutional configurations.

The impact of P on the stress response on the Ni GB was evaluated in boththe EEA and the RGS frameworks, and results are presented in Figs. 5.7 and5.8, respectively. The results have been separated into low and high surfacecoverages for clarity.

44 Chapter 5. Results II: Bulk and fracture properties

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

2 (Å)

0

5

10

15

20

25

30

(GPa)

Bulk

Clean GB

GB: s = 1

GB: i = 1

(a)

EEA

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

2 (Å)

0

5

10

15

20

25

30

(GPa)

Bulk

Clean GBGB: i = 2

GB: i = 3

GB: i = 4

(b)

EEA

FIGURE 5.7: Impact of P on the stress-response of a Ni GB based onthe EEA approach. (a) Low surface coverage (θi,θs=1) (b) higher sur-face coverage (θi=2-4). Bulk Ni and the clean GB are included forreference.

5.3. Segregation-induced changes in fracture properties in Ni 45

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

tot (Å)

0

5

10

15

20

25

30

(GPa)

Bulk

Clean GB

GB: s = 1

GB: i = 1

(a)

RGS

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

tot (Å)

0

5

10

15

20

25

30

(GPa)

Bulk

Clean GBGB: i = 2

GB: i = 3

GB: i = 4

(b)

RGS

FIGURE 5.8: Impact of P on the stress-response of a Ni GB based onthe RGS approach. (a) Low surface coverage (θi,θs=1) (b) higher sur-face coverage (θi=2-4). Bulk Ni and the clean GB are included forreference.

As can be seen in Figs. 5.7 and 5.8, the two approaches give significant dif-ferences in stress-response when P is present in the GB. When evaluating thestress-response in the RGS framework, the solute is shown to give a signifi-cant strengthening of the material, with a maximum increase in peak stress

46 Chapter 5. Results II: Bulk and fracture properties

of up to 14%, a value in line with that of the clean bulk. However, usingthe EEA approach, the solute has an important weakening effect at low con-centrations (θi,θs=1), whereas the impact diminishes as the concentration in-creases. Thus, although the stress-response of the clean systems could beaccurately represented in the rigid approach, the impact of the choice ofmethod is more considerable in the presence of P. In this context it is inter-esting to note that previous studies in the literature suggesting P to havea considerable strengthening effect in Ni, have been performed in the RGSframework [120, 122]. Since P is well known for weakening other metal ma-terials (e.g Fe [123–126] and common austenitic and ferritic alloys [97–101]),it is considered unlikely that the RGS model provides reliable results in thiscase. Particularly, the EEA model is considerably more thorough and is thusmore likely to give a correct physical representation. These results underlinethe importance of a proper description of the fracture processes when per-forming the ab initio stress test. The results also suggest that P has a weaken-ing effect in Ni, although less considerable compared to its effects in the bccand fcc Fe alloys.

5.3.2 Effect of Cr on the integrity of a Ni GB: importance of fracture pathand structural modification

The impact of Cr on Ni GBs has previously been assessed from first-principles,and general trends suggest the solute to have a strengthening effect [127–130]. In the previous studies, however, it has been assumed that if an inter-stitial Cr atom is present in the system, it will take on the position in the GBcenter (position 0 in Fig. 5.5). As a consequence, the solute will remain onthe outer surface as the structure fractures. However, in section 5.2.3 it wasshown that Cr migrates preferentially in a mixed-dumbbell configuration infcc Ni. Thus, the diffusion of Cr atoms in the system is strongly correlatedwith the transport of Ni atoms. It can for this reason not be assumed that themigrating Cr will come to rest in the GB center. Indeed, in Paper V it wasshown that Cr is significantly more stable in the substitutional bulk configu-ration compared to in the GB center. These results underline the differencebetween thermodynamics and kinetics. According to results in section 5.2,the Cr atoms will couple kinetically with the dumbbells and enrich in GBs.However, the GB center is a thermodynamically unfavorable position for thesolute, compared to the substitutional bulk configuration. The question thusremains if there exists a configuration with Cr in the GB vicinity, which ismore energetically favorable compared to the configuration with the solutein its center. To find the answer to this question, the relative energies of alarge number of GBs, with the Cr atoms in different configurations, wereevaluated. The possibility that the Ni atoms in this case take on the intersti-tial configuration in the GB center, while being replaced by Cr in the crystallattice, calls for a change in terminology. The Cr surface coverage will forthis reason henceforth be referred to as additional (and not interstitial), anddenoted θa. When evaluating the most favorable configurations, two fac-tors have been considered: which configuration gave the lowest GB energy,and which configuration gave the lowest WOS. The former aspect describeswhere the Cr atoms are likely to be found in the unstrained conditions, which

5.3. Segregation-induced changes in fracture properties in Ni 47

corresponds to δ = 0 in the energy-separation curves. The second factor rep-resents the energy required to fracture the system, regardless of initial config-uration. Calculations in the current work showed the configurations of maininterest were those where the Cr atoms replaced the Ni atoms on the GBsurface (position 1-3 in Fig. 5.5), while simultaneously pushing the Ni atomsinto the GB center. In Figure 5.9, the energy-curves of the θa=4 configurationsare presented, in combination with the common configuration with Cr in theGB center (position 0). In the figure, the energy curves are presented both interms of energy per atom in the system, and by total energy per surface area(the combined bulk and fracture response according to Eq. 3.25). From theformer, the more favorable GB (δ = 0) configuration can be extracted. Thelatter allows for the extraction of the WOS, and a more clear comparison withthe clean GB and the bulk Ni.

48 Chapter 5. Results II: Bulk and fracture properties

1 0 1 2 3tot (Å)

5.59

5.58

5.57

5.56

5.55

5.54

5.53

5.52

5.51

(eV/

atom

)

GB + Cr: Position 0GB + Cr: Position 1GB + Cr: Position 2GB + Cr: Position 3

(a)

1 0 1 2 3tot (Å)

0

1

2

3

4

(Jm

2 )

BulkClean GBGB + Cr: Position 0GB + Cr: Position 1GB + Cr: Position 2GB + Cr: Position 3

(b)

FIGURE 5.9: Energy versus separation for the Ni GB containing 4 Cratoms (a) Energy per atom in the system (b) Total energy per surfacearea in the system (bulk+fracture plane). In Fig. (b), bulk Ni and theclean GB are included for reference. Cr atom(s) in positions 1-3 (Fig.5.5), implicates that the Ni atoms have been pushed into the center ofthe GB (position 0).

As can be seen in Fig. 5.5, although the different configurations only resultin minor differences in GB energies, a considerable impact of WOS is seen.In particular, the difference between the commonly used configuration (Cr

5.3. Segregation-induced changes in fracture properties in Ni 49

in position 0) and the minimum energy configuration, results in a 0.4 Jm−2

difference. This effect is of particular interest since, based on the compari-son with the clean GB curve in 5.9(b), this results in a difference between astrengthening and a weakening effect of the solute. Thus, the initial configu-ration can have a decisive impact on the observed trends. To the best of theauthor’s knowledge, such effects have not previously been considered whenevaluating the stress-response of the current system.

The stress-response of the Cr enriched Ni GB has here been evaluated inthe EEA framework based on the minimum energy configuration found inFig. 5.9. The results are presented in Fig. 5.10. As can be seen in the fig-ure, Cr has a considerable weakening effect on the Ni GB. These results arein striking contrast to results previously reported for the system in the liter-ature [127–130]. Thus, when evaluating the stress-response of a system in afirst-principles framework, the importance of a proper consideration of theminimum energy configuration can not be overstated.

50 Chapter 5. Results II: Bulk and fracture properties

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.52 (Å)

0

1

2

3

4 (J

m2 )

BulkClean GBGB+Cr: s=1GB+Cr: a=1GB+Cr: a=4

(a)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.52 (Å)

0

5

10

15

20

25

30

(GPa

)

BulkClean GBGB+Cr: s=1GB+Cr: a=1GB+Cr: a=4

(b)

FIGURE 5.10: (a) Excess energy and (b) traction-separation behaviourof the GB containing Cr. The bulk Ni and clean Ni GB behaviour areincluded for reference.

5.3. Segregation-induced changes in fracture properties in Ni 51

5.3.3 Quantitative comparison of the impact of different elements on Nicohesive properties

Cr, Si, and P were all shown in section 5.1 to enrich at Ni sinks during irra-diation. In the current section, a quantitative comparison between the im-pact of the solutes on the cohesive properties of the considered GB is given.The results are presented for the GB containing four solutes in Fig. 5.11. Inthe figure, the stress responses of the pure bulk, the clean GB, and the GBcontaining four self-interstitial Ni atoms (SIAs) are included for reference.To benchmark the results, the impact of S, which is notoriously harmful inNi [95, 96, 120, 129, 131–137], and B, which is known to enhance cohesion inthe material [95,119,120,127–129,137] are also included. All calculations havebeen performed within the EEA framework.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.52 (Å)

0

5

10

15

20

25

30

(GPa

)

BulkClean GBGB + Interstitial NiGB + BGB + CrGB + PGB + SiGB + S

FIGURE 5.11: Traction-separation behaviour of the Ni GB includingfour interstitials of the respective elements the different elements. Theclean GB and bulk material are included for reference

Fig. 5.11 shows that a considerable weakening of the system in the presenceof S, whereas B gives a significant strengthening. These results can be seen asa validation of the approach used in the current work. Both P and Si displayonly minor effects on the cohesive properties of the GB at this surface cover-age. However, the impact of the two solutes were shown in Papers IV andV to be dependent on both solute concentration and configuration. The ori-gin of these effects remain for a future study. Perhaps the more unexpectedresults in Fig. 5.11, is the considerable weakening induced by the presenceof Cr. However, as shown in the same figure, the impact of the solute isvery closely related to that of the interstitial Ni. In this context, it should beremembered that following the energy minimization in section 5.3.2, the Cratoms are not present in the GB center, but in a substitutional configuration

52 Chapter 5. Results II: Bulk and fracture properties

in an inner layer. The GB is in this system instead enriched with Ni. Thusthe GB center of the GB+Cr and GB+SIAs systems are the same. This is aclear indication that, although the results in section 5.3.2 indicate that Cr hada considerable weakening effect of the GB, this effect can in fact be attributedto the presence of Ni SIAs in the GB center. Consequently, the direct effectsof Cr are in this case small.

The origin of the weakening of the GB enriched with SIAs can be under-stood based on the fact that the interstitials will want to arrange themselvesinto the original crystal lattice. Thus, when the structure is strained, boththe SIAs and the atoms of the outer surface of the two grains immediatelystart to rearrange. Indeed, in the ab initio calculations in the current work,the GB structure was in this case shown to be strongly depended on the to-tal elongation. It is interesting to speculate what could be the consequencesof such effects in an irradiated environment. The sinks in such systems areconstantly enriched in SIAs, and a continuous change in the GB structurecan therefore be expected. Thus, it is not unlikely that starting from a perfectGB structure, the enrichment of solutes causes disturbances and significantweakening until the point where the combination of strain and restructuringresults in the emergence of a new and stable GB structure. Such effects couldpossibly give rise to oscillating strengthening/weakening of the GBs in theirradiated system. These effects would indeed be interesting to study in afuture work.

5.4 Defect formation energies in paramagnetic Ni

Based on the previous discussion, it is clear that radiation-induced segrega-tion is highly correlated with the presence of point defects in the system. ThePD concentration is given by Eq. 3.1, where the defect formation enthalpycan be calculated according to Eq. 3.18. In transposing the results to finitetemperatures, it is commonly assumed that the defect formation free energyis constant with temperature or obeys simple Arrhenius relation that extrap-olates from 0 K to finite temperatures. However, this has been shown not tobe true for the vacancy formation energy [138, 139]. In the current section,the impact of magnetic disorder on the defect formation enthalpy in Ni hasbeen evaluated based on the disordered local moment molecular dynamics(DLM-MD) framework, described in section 3.5.1.

To ensure that calculations were performed at the equilibrium lattice param-eter at each temperature, simulations to evaluate the thermal expansion werefirst performed. This was done by applying the DLM-MD model using themicro-canonical (NVE) framework, and monitoring the external pressure inthe supercell for a large number of different volumes. For each consideredtemperature in the range 800-1600 K, a lattice parameter was found whichcorresponded to zero average external pressure as the simulation progressed.This value was considered to be the appropriate choice, and was used inthe further calculations. The thermal expansion of the lattice constant in theparamagnetic phase is shown in Fig. 5.12 together with experimental data

5.4. Defect formation energies in paramagnetic Ni 53

from [140, 141]. As can be seen in this figure, the model predicts lattice pa-rameters and the thermal expansion effect in very good agreement with ex-periments. The only significant deviation appears towards the highest tem-peratures where the model fails to follow the accelerated expansion seen inexperiments.

0 500 1000 1500 2000Temperature (K)

3,52

3,54

3,56

3,58

3,6

Latti

ce p

aram

eter

(Å

)

Hwang (1972)Jesse (1934)DLM-MD simulation

FM fcc Ni PM fcc Ni

FIGURE 5.12: The lattice parameter as function of temperature for fccNi. Two sets of experiments [140, 141] are presented along with theDLM-MD simulation results. Dotted lines demarcate the range of themagnetic phases.

Based on the determined lattice parameters, simulations to evaluate mate-rial properties were performed in the canonical ensemble (NVT) frameworkto provide temperature control. The temperature dependence of the ensem-ble average energies were extracted based on a sliding average method de-scribed in section 3.6. Based on the results, the defect formation enthalpiesas functions of temperature were calculated according to Eq. 3.18. In orderto better compare effects of temperature and lattice expansion, the resultsare compared to statically relaxed 0-K ferromagnetic formation enthalpies inFig. 5.13 as function of lattice parameter. It can there be clearly seen thatthe lattice expansion effect is a significant component to the formation en-thalpy. For vacancies, the magnetic excitation and vibration effects are seento be small in comparison to the lattice expansion effect. This is in line withthe reasoning of Gong et al. [138]. For the SIA, the expansion effect is signif-icantly larger but does at the same time not dominate in the same way. It is

54 Chapter 5. Results II: Bulk and fracture properties

clear that inclusion of magnetic and vibration effects are of importance for aproper description of SIA properties in paramagnetic fcc Ni.

3,48 3,5 3,52 3,54 3,56 3,58 3,6

Lattice parameter (Å)

0

1

2

3

4

5

6F

orm

atio

n en

thal

py (

ev)

Hvf (FM)

Hif (FM)

Hvf (PM)

Hif (PM)

FIGURE 5.13: Vacancy and SIA formation enthalpies in the static fer-romagnetic and DLM-MD paramagnetic states as function of latticeparameter

The full free formation energy calculations require appropriate models forthe formation entropy as well. In the literature there are formation entropymodels availabel for both vacancy and SIA, but not in the paramagnetic stateand direct application of these models lead to surprising results. In orderto progress further, additional development and analysis will be interestingfuture work that can pave the way for more accurate modelling of defectproperties in magnetically disordered systems.

55

Chapter 6

Conclusions and outlook

The primary objective of this work has been to provide an increased under-standing for the radiation-induced mechanisms driving the premature age-ing of structural materials used in light-water reactor applications. The ra-diation field in the reactor core may affect the internal components in twoways: By the generation of defects in the crystal structure, and by creatingan oxidizing environment. The latter effect is a consequence of the ioniza-tion, excitation and decomposition of the water molecules which surroundthe materials. To provide a comprehensive picture of the many processeswhich may lead to material degradation, radiation-induced effects have herebeen evaluated from both perspectives. Surface-processes have been studiedthrough experimental methods, and bulk effects have been investigated in amodeling framework. The goal is to in this way provide a broad picture ofthe many processes in an irradiated system.

In the first part of this work, the reaction mechanisms between the prod-ucts of water radiolysis, and two common materials in reactor applications(ZrO2 and 304L steel) have been investigated. In the ZrO2 system, the fo-cus has been to quantify the impact of anions (Br– , Cl– , and HCO3

– ) on thegeneral reaction mechanism for the reaction between H2O2 and the oxide.Results in this section show that the surface-bound OH•, formed in the de-composition of H2O2, is effectively scavenged by Br– transformed into Br2

• – .This reaction efficiently removed OH•(ads) from the surface. Additionally, theresults presented here show that the anions in solution can have an impor-tant influence on the experimental techniques commonly used in this con-text. Particularly, it is shown that tris, which is commonly used to quantifythe production of OH•, can react via one-electron oxidation with Br2

• – in areaction yielding formaldehyde. This is contrast to the previous assumptionthat formaldehyde is only formed from tris upon hydrogen abstraction. Al-though this result is not directly applicable to nuclear environments, it is ofgreat interest in the context of this experimental technique.

As a next step, the reactivity of H2O2 towards the 304L alloy has been in-vestigated. In this section, reaction rate constants were determined, and dis-solution of steel components in solution during the reaction was assessed.Possible reaction mechanisms between H2O2 and the steel are here discussed,with a focus on how the impact the stability of the steel surface. The possibil-ity that O2 formed in catalytic decomposition of H2O2 contributes to surfaceoxidation of the steel, opens the door for interesting future work.

56 Chapter 6. Conclusions and outlook

In the second part of this work, the direct effect of radiation on the bulk ma-terial properties were evaluated using the state-of the-art a first-principlesmethods and associated phenomenological theories. Radiation-induced seg-regation trends of Cr, Fe, Si, P, Ti, and Mn were evaluated in dilute fcc Nialloys, here used as a model material. Results show that Cr, P, and Si areenriched at sinks in the irradiated alloy, whereas Mn and Fe display deple-tion in such regions. Ti, on the other hand, displays enrichment below 320 K(well below reactor operating temperatures), and depletion for higher tem-peratures. Based on these segregation trends, the impact of Cr, P and Si onthe cohesive properties of a grain boundary (GB) in the material was inves-tigated. An important objective in this section was to evaluate the commonmodels used to asses the impact of the solutes on the material’s stress re-sponse. Results showed that the stress-test used in first-principles calcula-tions could have a decisive impact on the results. Moreover, it was shownthat the common assumption that interstitial Cr segregates to the center ofa GB, is not only wrong but this assumption gives decisively erroneous re-sults regarding the impact of the solute in this context. Instead it has herebeen shown that the solute will remain at the GB surface, and consequentlyself-interstitial Ni will migrate to the GB center. This configuration will givean important weakening to the structure, in clear contrast to previous stud-ies were Cr has been shown to promote strengthening of the same structure.However, the results in the current work further show that this weakening isin fact not due to the presence of Cr, but to the enrichment of self-interstitialNi atoms (SIAs) in the GB center. Moreover, it was found that as the GB wasenriched with SIAs as the structure was strained, the atoms rearranged in or-der to fit the interstitials into the original lattice structure. As a consequenceof this, the GB structure varied with strain. Since sinks in irradiated envi-ronments are constantly enriched with SIAs, these enrichment/restructuringeffects could possibly result in an interesting oscillatory behaviour betweenweakening through SIA enrichment, and strengthening through the emer-gence of a possible more stable structure as the GB is reconstructed. Sucheffects remain an interesting topic for future studies.

As a final step, the impact of magnetic disorder on defect formation ener-gies have here been evaluated in the DLM-MD picture of paramagnetism.Thermal expansion has been modeled and formation energies have been cal-culated for the common point defects, the vacancy and the self-interstitial, infcc Ni. The method is validated on experimental thermal expansion data andthen used to calculate the defect formation energies. The effect of the thermalexpansion is seen as a major component of the formation enthalpy for bothvacancy and SIA but magnetic disordering and vibration dynamics is also ofclear significance for the SIA. The difference between vacancy and SIA 0 Kformation enthalpies is usually in the order of a factor three, while here theabsolute value of the SIA formation energy is significantly lower than thatin the ferromagnetic phase. A interesting continuation of this work wouldbe to use this method to calculate defect concentrations in the high temper-ature region, which is generally considered very difficult in a first-principlesframework.

Chapter 6. Conclusions and outlook 57

In this thesis, a number of important mechanisms that contribute to radiation-induced material degradation have been quantified. A comprehensive pic-ture of the many factors that play a role in this context is presented by the useof both experimental and theoretical methods. This work will for this reasonprovide an important contribution to the ongoing effort to understand thematerial degradation in nuclear reactor environments.

59

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