1.1 Background

The reactor pressure vessel (RPV) of nuclear fission reactorsis closely monitored for material changes through microscopicand macroscopic testing and observation, andpredictions are supplemented with results from a rangeof computational tools. The focus of safety monitoringis the ability of the RPV's ability to withstand fractureat all normal operating temperatures, a property measuredby the Charpy test. This test measures the energyabsorbed during the fracture of identical specimens at arange of temperatures. Metals such as Alpha-Iron fromwhich an RPV is constructed show two distinct regimesof fracture; The first is the ductile regime, where materialstress is partially relieved by structural changes,and requiring the most energy to fracture. The second isbrittle fracture in which the material cannot easily relievestress and requires less energy to fail. The temperatureat which the material makes the distinct transition fromductile to brittle fracture (the DBTT) is important tosafety management and is itself strongly dependent onthe irradiation history of the material. Thus safety managementseeks to ensure that RPVs are never exposedto radiation such that the DBTT shifts to within normaloperating temperature margins. Fig.1 shows the effect ofradiation, annealing and re-irradiation on an RPV steel, demonstrating that whilst annealing (IARA) can largelyheal the deleterious effects of radiation (IAR), a permanentshift nevertheless occurs (IARA - Unirradiated).

The distinctive DBTT shift shown in Fig.1 can be interpretedin a simplified manner as resulting from twoprocesses. Firstly, radiation causes a high number densityof precipitates to form, and then annealing forcescoarsening of precipitates, lowering their number density.The effect on the Charpy energy is due to the precipitatesacting as obstacles to the movement of dislocationsacross the matrix, preventing the material from relievingstress [11, 7]. Thus a high number density of precipitatesresults in hardening, which leads generally to a more brittlefracture. The microscopic phenomena relating to theformation, and later coarsening, of precipitates are dealtwith in this study, although alternative non-hardeningembrittlement exists, in which the possibility of fracture at grain boundaries is increased by radiation-related phenomenasuch as void growth and impurity segregation[6].

1.2 Motivation and purpose

This work focuses on multi-scale modeling, the generalmethod employed by the safety management communityfor predicting macroscopic material behaviour from microscopicphenomena [15].

Multi-scale modeling relies upon useful and sufficientinformation being transferred from one stage to theother, (e.g. the results of a microscopic calculation beingused as inputs for a larger scale calculation).It hasbecome clear that there discontinuities currently exist inthe chain, so small features that may be significant at theupper end of the process are not identified and communicated.This work contributes to multi-scale modeling bydeveloping new computational methods that increase thescope of information accessible to certain stages, specificallyconcerning the formation and growth of impurityclusters and precipitates.

2 Cascade Simulation

The beginning of radiation induced precipitate nucleationis the cascade process, in which a fast neutronfrom a fission reaction interacts with the surroundingwall. The energy of the neutron is passed to a large numberof atoms through a growing chain of elastic and inelasticinteractions, giving many of them enough energyto leave their binding lattice sites and creating Frenkelpairs. Most of these recombine, but many will diffuseinto structures such as clusters of vacancies, interstitialatoms and impurity. atoms, , .

Molecular dynamics simulations (MD) of this cascade,using the Yang embedded atom method (EAM) potentialallows us to examine the effect of multiple impurityelements in a bcc iron cell during the cascade. The minorpart of this is testing 3 PKA energies in pure iron andbinary systems, with 10 repeats per composition. Themajor work is a 10KeV cascade in a volume of varyingelemental composition of side 14.3 nanometers (250,000atoms), lasting 5 Ps and evolved as an NVE ensemble.After cascading, the system is quenched and forceminimisedand counts are made of sites without atoms,and atoms without substitutional sites [1]. This gives agood measure of the damage produced by the cascade,and up to 15 repeats are made at every compositionalvalue. Data analysis reveals complex trends in synergisticeffects between elements, and one key point of interestis the difference in behaviour between an elements binaryeffect and its synergistic effect, see Fig.2

3 Parallel Tempering

Following the dissipation of a PKA's energy by cascadeinteraction, in addition to the generation of many Frenkelpair, a thermal spike occurs. In the fission reactor environment,this is of magnitude of thousands of degrees,and lasts the order of picoseconds [14]. Simultaneouslymodeling atomistic behaviour under the dissipation ofthe thermal spike remains difficult, since physical propertiesare uncertain and temperature gradients of 104K/nmare present. Current Kinetic Monte-Carlo methods ofanalysing clustering during this period consist of countingdefect number and type apx. 5 Ps after a cascade,and then modeling migration with a pre-calculated eventtable that relies on first principles calculations of binding and migration energy [16]. These current methodscannot explore phenomena outside the description of theevent table, which is restricted to known and likely reactions.We present a model which allows greater rangeof atomistic interactions, including those that are onlylikely to occur at very high-temperature. This will providegreater insight into the structure and composition ofclusters shortly after the dissipation of the thermal spike.In the simulation presented here, no prior calculation isnecessary, meaning that all local environment variablesare factored in to a diffusion event. Parallel Tempering(PT) works by simulating N exact replicas of a system atN different temperatures, where probability of swappingbetween neighbours increases as their potential energiesbecome similar, whilst preserving the detailed balancecriterion, guaranteeing reversibility [ 13, 2]:

Whilst PT could be used in multiple MD environments,it is particularly useful for cascade simulationssince the higher temperature replicas can be seen as representingthe possible events in the high temperature volumeof the cascade. Achieving a reliable heat distributionis not possible in conventional simulations, so it is anadvantage that PT can select stable configurations froma system undergoing high-temperature processes. Earlyresults of cascade simulation followed by parallel temperingindicates some aggregation of impurities during thispreviously unexplored period, Tab.l.

4 Optimisation of pair potentials

KLMC calculations require the calculation of the localconfiguration energy, typically employing empirical potentialfunctions to do so. The requirements of a KLMCare quite basic, needing only a static single-time energycalculation, which can be reasonably handled by a pairpotentials derived from chemical considerations [8]. Agenetic algorithm optimisation code forwards sets of parametersto a KLMC simulation, where cluster countingprogram returns a value of goodness to each set of parameters. The goodness of the parameters is improved byobserving biological rules of reproduction and mutation.

This method succeeded in optimising the pair potentialsfor a binary alloy against an EAM potential standard.In the future, this could be exchanged for an experimentalstandard since the detection efficiency of AtomProbe Tomography [ 4] has increased steadily to the pointwhere the chemical composition of clusters is clarified.The method established here would enable the experimentalatom map to be used as the standard to whichthe pair potentials could be optimised, thus linking onestage of the multi-scale modeling process to experiment

5 Distorted lattice monte-carlo

Exposure to radiation will have an embrittling effecton RPV materials, but this can be mostly recoveredby an annealing process, as seen in Figure 1. The annealingprocess allows for coarsening of precipitates, increasingtheir average size but decreasing their averagenumber density. During this process, the main agent fortransport of large impurity elements is vacancy diffusion.Since the jump of a vacancy is a rare yet energeticallywell understood event, the movement of a vacancy acrossa crystalline matrix can be simulated with a Kinetic LatticeMont-Carlo program. [5, 17] As the vacancy movesit drags impurity elements through the matrix which can,depending on the type of element, lead to clustering andlater precipitation.

Recent observations with one of the most accuratetechniques for determining chemical compositions, atomprobe tomography, reveals clustering to favour latticedefects for nucleation. This is of special importancewhen the defect is a dislocation, since other studies showthat the stress required for a dislocation to pass throughthe precipitate (CRSS) depends upon the relative location.To clarify this situation, a Kinetic distorted-LatticeMonte-Carlo (DLMC) program is introduced, in whichthe atomic positions are non-homogeneous, and representatomic positions near to a defect. The vacancy, andin turn the elements it drags, are sensitive to the differingatomic radii causes by the distorted lattice. In all realworldcrystalline materials, there are many dislocationsin every grain, thus the possibility that dislocations act asnucleation sites must be covered. Furthermore, contemporary MD simulations regard precipitates as homogenouslyformed across the lattice, and of regular shapeand distribution (see e.g. [9, 12]). The DLMC simulationsconducted here provide strong evidence that this isa misleading way to begin a simulation, since the impurityatoms tend to cluster in the highly stressed regionsinside the dislocation core, where they can have the mostsignificant effect on hardening, see Fig.3.

6 Conclusion

This work represents contributions to the multi-scalemodeling community in the study of fission reactor pressurevessels.

The effect of impurity elements such as found inJapanese RPVs on cascade damage processes is modeled,being the first such simulation to include a range of 5 impurityelements described by an Embedded Atom Modelpotential formulation.

Parallel Tempering is demonstrated for the first timeas a potential addition to the range of multi-scale modelingtechniques. Describing the behaviour of the RPVmatrix at times and spaces very close to the cascade,without the need for an event list, enables more realisticevolution of the system. Results show the tendencyof certain impurity elements to aggregate together veryshortly after the cascade, although the method of diffusionis unclear.

Linking computational Monte-Carlo techniques andmolecular dynamics methods is shown to enable morerealistic evolution, in this case by forcing diffusion of individualatoms (MC), and allowing the system to respond(MD). This allows the simulation to respond to two distincttime and length scales, giving new results about thenucleation and growth of clusters. A technique of optimisingpair potentials for use in a Kinetic Lattice Monte-Carlo expands the utility of such simulations, and pavesthe way for a new link between multi-scale modeling andexperiment.

Figure 1: Energy absorbed in fractureand reirradiated (IAR) and reirradiated(IARA) materials [4]

Figure 2: Binary vs. Synergistic effects

Table 1: PT effect on cascade damage of a simulationcell composition (%at.); Si 0.2, P 0.3, Mn 0.0, Ni 0.8,Cu 0. Data reflects the percentage increase of numberof 1NN and 2NN atoms containing an impurity solutefollowing PT relative to the number of 1NN and 2NNatoms containing an impurity solute before PT.

Table 2: Optimised pair potential parameters

Figurenn 3: Compressive stress cylinder effect on copperclustering. The dislocation core is at the top of the figure,demarked by a broken line of gray atoms in the zdirection. The simulation volume has been quartered inx and y directions and all iron atoms that are not partof the dislocation core have been erased.

本論文は原子炉圧力容器鋼の中性子照射による照射脆化の予測評価手法を対象として、その工学的意義を踏まえて、材料のミクロな変化機構に基づいた新たな評価手法をマルチスケールシミュレーションの立場から開発することを目的としている。

論文は7章で構成されており、第1章では原子炉圧力容器の照射脆化の概要を整理するとともに、中性子照射損傷に関する基本的に理解に基づいた照射脆化を引き起こすミクロな溶質原子クラスター形成と欠陥集合体に関する研究をレビューしている。またこれらの現象をマルチスケールなモデル化とその物理的ならびに数理的基盤の観点から整理し、最近の軽水炉の高経年化対策における中性子照射脆化評価を高度化する意義をまとめた上で、本研究の目的と構成について述べている。

第2章では、原子間ポテンシャルの適用性に関して、圧力容器鋼の照射脆化に寄与する6種類の元素(鉄、ニッケル、マンガン、シリコン、リン及び銅)を対象として電子スピンの効果を統一的な手法で考慮した最新ポテンシャルである原子挿入法に基づくYangポテンシャルを評価している。空孔メカニズムによる原子移動エネルギーを分子動力学法を用いて6つの元素すべてについて評価することによって、第一原理計算結果が存在する元素については、その値と比較しほぼ一致することを見出している。さらに鉄については、空孔移動エネルギー評価結果が実験事実とよく符合することを議論しており、Yangポテンシャルが中性子照射損傷と長期間にわたる溶質原子クラスターと欠陥集合体形成のマルチスケールシミュレーションに適した原子間ポテンシャルであるとまとめている。

第3章は、カスケード損傷に関する分子動力学シミュレーションを対象として、6種類の元素を含む系での中性子照射損傷に伴う直接的な溶質原子クラスター形成過程を評価している。各元素の含有量をパラメータとして多数回の分子動力学計算を行った結果に基づいて統計的な処理を行い、総残存格子欠陥量及び溶質原子クラスター形成に対する各元素の影響を解明することに成功している。

第4章は、中性子照射損傷に伴うカスケード冷却過程後の欠陥集合体と溶質原子クラスターの組成や構造に関するシミュレーション手法を議論している。広範囲の原子相互作用を評価しうるモデル開発の必要性を論じた上で、パラレルテンパリング法を用いた手法の物理的原理とモンテカルロ法へ適用する利点について物理的な検討を行い、カスケード損傷後の各種クラスター形成過程のシミュレーションコードを開発している。このコードを用いて、1次はじき出し原子の発生後の分子動力学計算が適用できる数ピコ秒後以降の長期間のミクロ組織発達に関する原理的実証に成功している。また、物理定数データベースを求めておく必要がない本手法の利点を活用することによって、効率的なマルチスケールシミュレーションが可能となることを定量的に提示している。

第5章は、モンテカルロ法に適用すべき原子間ポテンシャルの最適化を遺伝アルゴリズムを活用して行い、従来の古典的ポテンシャルを用いた溶質原子クラスター形成シミュレーションと比較して、論じている。

第6章は、新たな原子環境敏感型キネティックモンテカルロ法の開発を取り上げている。転位等のミクロな欠陥集合体の周囲に存在するひずみ場の影響を溶質原子クラスター形成過程のシミュレーションに組み入れる手法が原子環境敏感型キネティックモンテカルロ法であり、そのシミュレーションの原理をコード化するとともに、らせん転位のまわりの溶質原子クラスターの形成と転位の移動後のクラスター残存過程を鉄・リン合金及び鉄・銅合金について行うことに成功している。

第7章は、全体の総括であり、軽水炉圧力容器用低合金鋼の中性子照射脆化評価手法のための原子間ポテンシャル開発、分子動力学計算及びモンテカルロ法を含むマルチスケールシミュレーションの今後の課題を取りまとめている。

以上を要するに、本論文においては、原子レベルの理解に基づいてマクロな材料特性を予測評価するための、マルチスケールシミュレーションの課題を系統的に提示し、それを複数のオリジナルな手法によって結合することに成功しており、システム量子工学、特に原子炉材料学に寄与するところが極めて大きい。

よって、本論文は、博士(工学)の学位請求論文として合格であると認められる。