Materials Issues for Generation IV Systems: Status, Open Questions and Challenges

Materials Issues for Generation IV Systems

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Springer Springer Springer IOS Press IOS Press

Materials Issues for Generation IV Systems Status, Open Questions and Challenges

edited by

Véronique Ghetta CNRS, Laboratoire de Physique Subatomique et de Cosmologie (LPSC), Grenoble, France

Dominique Gorse CNRS, Laboratoire des Solides Irradiés (LSI), Palaiseau, France

Dominique Mazière CEA, DRI/DAE, Gif-sur-Yvette, France and

Vassilis Pontikis CEA, Laboratoire des Solides Irradiés (LSI), Palaiseau, France

Published in cooperation with NATO Public Diplomacy Division

Proceedings of the NATO Advanced Study Institute on Materials Issues for Generation IV Systems: Status, Open Questions and Challenges Cargèse, Corsica, France 24 September – 6 October 2007

A C.I.P. Catalogue record for this book is available from the Library of Congress.

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TABLE OF CONTENTS

Forword

ix

List of Authors

xi

The Energy Issue and the Possible Contribution of the Various Nuclear Energy Production Scenarios H. Nifenecker

1

Outlook on Generation IV Nuclear Systems and Related Materials R&D Challenges F. Carré, C. Renault, P. Anzieu, P. Brossard, and P. Yvon

25

Fundamentals of Neutronics: Reactivity Coefficients in Nuclear Reactors P. Reuss

49

Introduction to Thermodynamics G. Inden Kinetics of Phase Transformation in Multi-Component Systems G. Inden

73 113

Ab Initio Approaches to Designing Thermodynamic Properties of Materials A. Pasturel and N. Jakse

141

Correlation between Electronic Structure, Magnetism and Physical Properties of Fe-Cr alloys: Ab Initio Modeling I. Abrikosov, P. Olsson and A.V. Ponomareva

153

The Computational Modeling of Alloys: From Ab Initio Calculations and Thermodynamics to Heterogeneous Precipitation A. Caro

169

Numerical Modeling of Radiation Effects in Solids: Principal Features, Limitations and Perspectives P. Geysermans

187

v

vi

TABLE OF CONTENTS

A Multiscale Approach to Measuring and Modeling Cleavage Fracture Toughness in Structural Steels G. R. Odette, H. J. Rathbun, M. Hribernik, T. Yamamoto, M. He and P. Spätig Microstructures and Mechanical Properties of Irradiated Metals and Alloys S. J. Zinkle Multiscale Modeling of RPV Embrittlement N. Soneda Parametric Dislocation Dynamics and Boundary Element Modeling of Elastic Interaction between Dislocations and Precipitates A. Takahashi

203

227

245

263

Crystal Plasticity from Dislocation Dynamics V. V. Bulatov

275

Radiation-Induced Solute Segregation in Alloys A. J. Ardell

285

Research and Development of Oxide Dispersion Strengthened Ferritic Steels for Sodium Cooled Fast Breeder Reactor Fuels M. Inoue, T. Kaito and S. Ohtsuka

311

Some Aspects of the Structural, Mechanical and Electronic Properties of SiC P. Pirouz

327

Review: Oxidation of SiC/SiC Composites in Low Oxidising and High Temperature Environment C. Cabet

351

Fundamentals of Liquids J. -P. Hansen

367

Introduction to Interfaces and Diffusion P. Wynblatt

393

Some Aspects of Wetting at High Temperature D. Chatain and V. Ghetta

425

TABLE OF CONTENTS

Potentiometric Sensors for High Temperature Liquids J. Fouletier and V. Ghetta

vii

445

Influence of Liquid Sodium on Mechanical Properties of Steels, Refractory Alloys and Ceramics H. U. Borgstedt

461

Radiation Effects in Structural Materials for Fusion Power Plants: the Outcomes of the EU Fusion Program J. -L. Boutard, S. Dudarev and E. Diegele

481

Introduction to the Physics of Molten Salt Reactors 501 E. Merle-Lucotte, D. Heuer, M. Allibert, V. Ghetta, and C. Le Brun Physico-Chemical Properties of Molten Salts J. -C. Poignet and J. Fouletier

523

Combined Effect of Molten Fluoride Salt and Irradiation on Ni-based Alloys A. S. Bakai

537

Specific Features of Particule/Matter Interaction for Accelerator-Driven Sub-Critical Reactors S. Leray

559

Operation of High Power Liquid Metal Spallation Targets: a Challenge for the Structural Materials J. Henry, T. Auger and Y. Dai

575

Index

585

FOREWORD The NATO Advanced Study Institute, on MATerials issues for GENeration-IV systems, MATGEN-IV, was held from September 24 to October 6, 2007 at the “Institut d’Etudes Scientifiques de Cargèse” in Corsica, France. Principal objectives of the ASI were (i) the examination and critical comparison of the properties of existing structural materials, in view of their use for building the nuclear reactors of the 4th generation and (ii) the identification of the technical needs motivating the demand for new structural materials be developed. Prerequisite for reaching these objectives, is the close collaboration between academic researchers and engineers in nuclear science and engineering. Prior to and during the ASI, care was given to favor such contacts and to promote the sharing of available scientific and technical knowledge between participants, scientists or engineers, the cooperation of which would be decisive in designing and constructing safe and efficient Generation IV advanced systems. Introductory remarks by the Vice-Chairman of the French Atomic Energy Commission preceded the inaugural lecture focusing on energy issues related to mitigation of global warming via a drastic decrease of green house gas emissions during this century. The presentation of energy production scenarios, as proposed by the IIASA* and used by the WEC* and the IPCC*, has emphasized on nuclear intensive variants accounting for the sole economic and technological constraints. One particular scenario, consisting in the vast deployment of fast breeder reactors by 2050, has then been analyzed and various aspects of the fuel cycle have been considered with final conclusion that an intensive, worldwide R&D program on structural materials is crucially and urgently needed. Plenary lectures were about reactor physics, neutronics, the fundamentals of thermodynamics and few more specialized topics of “nuclear metallurgy”, such as wetting and techniques commonly used for the compositional control of coolants. Moreover, a detailed presentation of irradiation damage in materials, at different space and time scales, has focused on all aspects of ageing ranging from the micro-structural evolution to the associated macroscopic mechanical property change. Appropriately chosen

______ * IIASA: International Institute for Applied Systems Analysis ; WEC: World Energy Council; IPCC: Intergovernmental Panel on Climate Change

ix

x

FOREWORD

seminars have addressed different aspects of fracture and related experimental methods, the ductile to brittle transition (DBT), scaling approaches unifying experimental observations of fracture surfaces and environment driven changes of the mechanical properties. Additional lectures were devoted to the mechanisms of particle/matter interactions at different energy levels (from keV to GeV) and to a variety of projectile/target couples whereas others have summarized present knowledge on heavily irradiated materials in spallation-neutron sources* or in the future fusion reactors. Finally, special attention has been given to oxide-dispersion strengthened steels and SiC, both good candidates as cladding materials in Generation IV systems. Lecturers and participants, from over twenty countries, have contributed in interesting discussions and live exchanges at the occasions of scheduled events and informal meetings all along the eleven days the ASI has lasted. However, due to the lack of space, only 70% of the lectures are included in these proceedings. For the reader interested in other aspects of the ASI, program, presentation viewgraphs, pictures, …, information can be found on the web (http://www-matgen4.cea.fr). It is our hope that these proceedings provide the reader with a starting point while deeper understanding of specialized matters can be found in existing textbooks (e.g. neutronics, the fundamentals of liquids, …). The organizers thank warmly their colleagues who lectured in this ASI and/or, contributed to the proceedings. Moreover, our thanks go to the sponsoring institutions: the North-Atlantic Treaty Organization (NATO)/Scientific Affairs Division, the Commissariat à l’Energie Atomique (CEA), the Centre National de la Recherche Scientifique (CNRS), Electricité de France (EDF), The European Nuclear Education Network (ENEN), the association Ecoles en Sciences des Matériaux (ESM), the Institut National des Sciences et Techniques Nucléaires (CEA-INSTN) and the Groupement de Recherche Gestion des Déchets et Production d’Energie par des Options Nouvelles (GEDEPEON). V. Ghetta (CNRS-LPSC), D. Gorse (CEA/CNRS-LSI) D. Mazière (CEA-DRI), V. Pontikis (CEA/CNRS-LSI) Saclay, January 31st, 2008

______ *

MEGAPIE at Paul Scherrer Institute, Switzerland and SNS at Oak Ridge, USA

LIST OF AUTHORS

ABRIKOSOV I. A. Department of Physics, Chemistry and Biology, Linköping University, Sweden ALLIBERT M. LPSC, UJF Grenoble 1, CNRS/IN2P3, INPG, 53, avenue des Martyrs 38026 Grenoble Cedex, France ANZIEU P. French Atomic Energy Commission (CEA) Nuclear Energy Division Saclay Center, 91191 Gif sur Yvette Cedex, France ARDELL A. J. Department of Materials Science and Engineering, University of California at Los Angeles, Los Angeles, CA 90095-1595, USA AUGER T. CNRS, CECM-UPR 2801, 94407 Vitry-sur-Seine Cedex, France BAKAI A. S. Kharkiv Institute of Physics and Technology, 61108, Kharkiv, Ukraine BORGSTEDT H. U. FZK, Karlsruhe, Germany BOUTARD J.-L. EFDA-CSU Garching, Boltzmannstrasse 2, D-85748 Garching bei München, Germany BROSSARD P. French Atomic Energy Commission (CEA) Nuclear Energy Division Saclay Center, 91191 Gif sur Yvette Cedex, France BULATOV V. V. Lawrence Livermore National Laboratory, USA CABET C. Laboratoire d’Etude de la Corrosion Non Aqueuse, DEN/DANS/DPC/SCCME, CEA Saclay, 91191 Gif sur Yvette Cedex, France

xii

LIST OF AUTHORS

CARO A. Chemistry, Materials, and Life Sciences Directorate Lawrence Livermore National Laboratory, Livermore CA 94551, USA CARRÉ F. French Atomic Energy Commission (CEA) Nuclear Energy Division Saclay Center, 91191 Gif sur Yvette Cedex, France CHATAIN D. Centre de Recherche en Matière Condensée et Nanosciences CNRS, campus de Luminy, 13288 Marseille, France DAI Y. Spallation Source Division, Paul Scherrer Institut CH-5232 Villingen, Switzerland DIEGELE E. EFDA-CSU Garching, Boltzmannstrasse 2 D-85748 Garching bei München, Germany DUDAREV S. EURATOM/UKAEA Fusion Association, Culham Science Centre Oxfordshire OX14 3DB, UK FOULETIER J. Grenoble University, Laboratoire d’Electrochimie et de Physico-chimie des Matériaux et des Interfaces (LEPMI), E.N.S.E.E.G., BP 75, 38402 Saint Martin d’Hères Cedex, France GEYSERMANS P. CNRS – INSP UMR 7588, UPMC Université Paris 06, 140 rue de Lourmel, 75015 Paris, France GHETTA V. LPSC, UJF Grenoble 1, CNRS/IN2P3, INPG, 53, avenue des Martyrs 38026 Grenoble Cedex, France HANSEN J. -P. Department of Chemistry, University of Cambridge Lensfield Road, Cambridge CB2 1EW, UK HE M. Department of Mechanical Engineering, University California Santa Barbara, Santa Barbara, CA 93106, USA

LIST OF AUTHORS

HENRY J. CEA Saclay, DEN/DMN/SRMA 91191 Gif-sur-Yvette Cedex, France HEUER D. LPSC, UJF Grenoble 1, CNRS/IN2P3, INPG, 53, avenue des Martyrs 38026 Grenoble Cedex, France HRIBERNIK M. Department of Mechanical Engineering, University California Santa Barbara, Santa Barbara, CA 93106, USA INDEN G. Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf, Germany INOUE M. Advanced Nuclear System Research and Development Directorate Japan Atomic Energy Agency (JAEA), 4002 Narita-cho, Oarai-machi Higashi-ibaraki-gun, Ibaraki-ken, Zip code 311-1393, Japan JAKSE N. Sciences et Ingénierie des Matériaux et Procédés, INP Grenoble, UJF-CNRS 1130, rue de la Piscine, BP 75, 38402 Saint-Martin d’Hères Cedex, France KAITO T. Advanced Nuclear System Research and Development Directorate Japan Atomic Energy Agency (JAEA), 4002 Narita-cho, Oarai-machi Higashi-ibaraki-gun, Ibaraki-ken, Zip code 311-1393, Japan LERAY S. DAPNIA/SPhN, CEA/Saclay 91191 Gif-sur-Yvette Cedex, France LE BRUN C. LPSC, UJF Grenoble 1, CNRS/IN2P3, INPG, 53, avenue des Martyrs 38026 Grenoble Cedex, France MERLE-LUCOTTE E. LPSC, UJF Grenoble 1, CNRS/IN2P3, INPG, 53, avenue des Martyrs 38026 Grenoble Cedex, France NIFENECKER LPSC, UJF Grenoble 1, CNRS/IN2P3, INPG, 53, avenue des Martyrs 38026 Grenoble Cedex, France

xiii

xiv

LIST OF AUTHORS

ODETTE G. R. Department of Mechanical Engineering, University California Santa Barbara, Santa Barbara, CA 93106, USA OHTSUKA S. Advanced Nuclear System Research and Development Directorate Japan Atomic Energy Agency (JAEA), 4002 Narita-cho, Oarai-machi Higashi-ibaraki-gun, Ibaraki-ken, Zip code 311-1393, Japan OLSSON P. Department of Materials and Mechanics of Components, EDF R&D Les Renardières, Moret-sur-Loing, France PASTUREL A. Sciences et Ingénierie des Matériaux et Procédés, INP Grenoble, UJF-CNRS 1130, rue de la Piscine, BP 75 38402 Saint-Martin d’Hères Cedex, France PIROUZ P. Department of Materials Science and Engineering, Case Western Reserve University, Cleveland, OH 44106, USA POIGNET J.-C. Grenoble University, Laboratoire d’Electrochimie et de Physicochimie des Matériaux et des Interfaces (LEPMI), E.N.S.E.E.G., BP 75, 38402 Saint Martin d’Hères Cedex, France PONOMAREVA A. V. Moscow institute of Steel and Alloys Moscow, Russia RATHBUN H. J. Department of Mechanical Engineering, University California Santa Barbara, Santa Barbara, CA 93106, USA RENAULT C. French Atomic Energy Commission (CEA) Nuclear Energy Division Saclay Center, 91191 Gif sur Yvette Cedex, France REUSS P. Emeritus Professor Institut National des Sciences et Techniques Nucléaires, Point courrier 35, CEA/Saclay 91191 Gif sur Yvette Cedex - France

LIST OF AUTHORS

SONEDA N. Central Research Institute of Electric Power Industry, 2-11-1 Iwado-kita, Komae, Tokyo 201-8511, Japan SPÄTIG P. Fusion Technology-Materials, CRPP-EPFL Association EURATOM-Confédération Suisse, 5232 Villigen PSI, Switzerland TAKAHASHI A. Department of Mechanical Engineering, Faculty of Science and Technology Tokyo University of Science 2641, Yamazaki, Noda-shi, Chiba, 278-8510, Japan YAMAMOTO T. Department of Mechanical Engineering, University California Santa Barbara, Santa Barbara, CA 93106, USA YVON P. French Atomic Energy Commission (CEA) Nuclear Energy Division Saclay 91191 Gif sur Yvette Cedex - France WYNBLATT P. Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA ZINKLE S. J. Materials Science and Technology Division Oak Ridge National Laboratory, P.O. Box 2008 Oak Ridge, TN 37831-6138 USA

xv

THE ENERGY ISSUE AND THE POSSIBLE CONTRIBUTION OF THE VARIOUS NUCLEAR ENERGY PRODUCTION SCENARIOS H. NIFENECKER* Scientific consultant, LPSC, Université Joseph Fourier Grenoble 1, CNRS/IN2P3, INPG, 53, avenue des Martyrs, 38026 Grenoble Cedex, France Chairman of “Sauvons le Climat”

1. Introduction Since the beginning of the industrial era, less than two centuries ago, our society has relied heavily upon fossil fuels. It was, first, coal that provided ample energy for industry and transport, that allowed the generalization of electricity and even town gas obtained by reacting coal with water vapour. In the first half of the twentieth century oil took over coal as the most used fossil fuel. It was much easier to use and became intimately intertwined with the exponential development of the “automobile society”. It also started to displace coal as fuel in electric power plants. However following the 1973 oil price crisis the use of oil was restricted to transportation and petro-chemistry. Natural gas became more and more popular for electricity and heat production. In 2004 the World Total Primary Energy Supply (TPES) amounted to 11 Billion tons oil equivalent (toe) [1], of whom 34% was provided by oil, 25% by coal and 21% by gas. Thus fossil fuels provided 80% of our energy supply. It appears that the amount of oil and gas reserves discovered every year has fallen below their yearly consumption. It is predicted that the amount of extracted oil will start decreasing within the next 10 to 15 years (peak oil) and that of gas will behave similarly within 20 to 25 years. This means that the price of oil and gas will increase steadily until consumption decreases to the level of production. There might come a point where it will become cheaper to make oil and gas out of coal via chemical reactions like that of “Fischer Tropsch”. Reserves of coal are plentiful and should allow to pass this century without real energy shortage.

______ *

[email protected]

V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

1

2

H. NIFENECKER

It is clearer and clearer that the most difficult challenge in the energy sector is related to the mitigation of global warming via a drastic decrease of Green House Gas (GHG) emissions. 2. The Global Warming Challenge The main Green House Gases naturally present in the atmosphere are water vapour, carbon dioxide, methane and nitrous oxide. Water vapour has a very short cycling time so that its temperature dependent equilibrium concentration in the atmosphere is reached almost immediately and has a mere amplifying influence of the effect of other determinants. Green House Gases allow the average temperature of the earth to reach a gentle 15 d°C while it would be a chilling –18 d°C if they were not present in our atmosphere. In this respect they are useful and cannot be considered as pollutants. The historical records of temperature and GHG concentrations have been reconstructed from ice cores excavated from the Antarctic and the Greenland caps. A clear and positive correlation has been observed between temperatures and GHG concentrations with a quasi-periodical behaviour. The average temperature oscillated from minima close to 8 d°C lower than present to maximums 2 d°C higher. CO 2 concentrations oscillated from lows around 180 parts per million (ppm) to highs close to 280 ppm, methane between 300 parts per billion (ppb) and 700 ppb. The driving parameter behind those oscillations is the solar irradiation, which varies periodically according to the parameters of the earth orbit around the sun (Milankovitch oscillations). However, the magnitude of the change in solar irradiation is not sufficient to account for the amplitude of the temperature changes. Amplification factors are required. It is suspected that, starting from a glacial minimum, an increase of solar irradiation of the northern hemisphere leads to decrease of the surface of the sea ice in summer, which decreases the average earth albedo, which increases further the temperature. Increase of temperature of the ocean leads to CO2 release by degassing, which increases the temperature and the extent of boreal forests, here again accompanied by a decrease of the albedo etc. We can summarize the causality chain, which leads to the phasing out of a glacial era: ΔW→ΔT→ΔA→ΔT’→ΔA’→ΔT’’→ΔGHG→ΔT’’’→ΔGHG’→ΔT’’’’ where ΔW is the increase of the solar irradiation in the northern hemisphere, ΔT, ΔT’, ΔT’’, ΔT’’’, ΔT’’’’ the temperature increases, ΔA, ΔA’ decreases of the albedo, ΔGHG and ΔGHG’ the increase of the GHG concentrations. Because of the finite limits of the temperature and concentrations excursions one concludes that the series are converging.

ENERGY ISSUE AND SCENARIOS

3

The massive injection of GHG in the atmosphere due to billions tons of fossil fuels burning displays a new driving parameter of a new increasing series for temperatures and concentrations. Is it insured that such series are also converging? Nobody is sure, because of the infinitely more rapid present pace of change as compared to that observed previously. This is one of the reason, and, perhaps, the most frightening why massive injection of GHG in the atmosphere should be halted as soon as possible. 2.1. RECENT EVOLUTION OF GREEN HOUSE GAS EMISSIONS

Globally [2], emissions of the GHGs increased by about 70% (from 8 to 13 GtCeq*) from 1970 to 2004, with carbon dioxide CO2 being the largest source, having grown by about 80%. The largest growth in CO2 emissions has come from power generation and road transport. Methane (CH4) emissions rose by about 40% since 1970, with an 85% increase from the combustion and use of fossil fuels. Agriculture, however, remains the largest source of CH4 emissions. Nitrous oxide (N2O) emissions grew by about 50%, due mainly to increased use of fertilizer and the growth of agriculture. In the following I restrict myself to the case of CO2, which is the main responsible for global warming and is of crucial concern for the energy sector. 2.2. THE EFFORT TO MAKE

Out of the 7,4 GtC-eq injected into the atmosphere by the use of fossil fuels, about 3 seem to be absorbed by the biosphere and the Ocean. This is the level of emissions we should aimed at. Note that this amount supposedly absorbed by mother Nature might change according to the temperature, especially of the ocean, and to the concentrations of GHG in the atomsphere. In this respect future might reserve bad as well as good surprises. Present lines of thought point, unhappily, towards the bad (acidification of the ocean). Table 1 shows the evolution of the world population and GHG emissions between 2000 and 2004. It also shows what could be an acceptable future in 2050. It shows that, worldwide, one should decrease our individual emissions by a factor 3.5.

______

* Different gases have different warming powers and it is conventional to translate their warming power into that of CO2, and, more usually, into the mass of carbon included in the CO2. For example Methane has a warming power 20 times larger than that of CO2, thus 1 mole of methane (16 g) has a warming power equal to that of 20 moles of CO2 (880 g) and of 240 g Carbon equivalent (Ceq).

4

H. NIFENECKER TABLE 1. Observed and objective emissions from 2000 to 2050 [1]. 2000 2004 2050 Population Billions

6

6.4

9

Emissions CO2 GtCeq

6.4

7.3

3

Emissions/capita

1.06

1.15

0.33

Different countries should have different objectives, according to their present rate of emissions. A few examples are displayed on Table 2. The extreme effort would be that of US citizens who should decrease their emissions by more than 16! TABLE 2. 2004 CO2 emissions/capita in selected countries. 2004 CO2 emissions/capita USA

5.4

Germany

2.8

France

1.7

China

0.99

India

0.28

World Total

1.15

2.3. THE FACTORS TO CONTROL

Since the primary aim is the decrease of the amount of emission of CO2 per capita it is useful to isolate the main factors that influence it. This is done with the simple tautological equation:

QCO 2 GDP Energy Q = × × CO 2 N pop N pop GDP Energy

(1)

where QCO2 is the total amount of CO2 emitted by the energy sector, Npop the population, GDP the World Gross Domestic Product, Energy is the total world primary energy supply. The factor Energy/GDP is the “energy intensity” while the factor QCO2/Energy is the CO2 or Carbon intensity. It is rarely advocated that the first factor, the Gross Domestic Product per capita should decrease at the world level. The “energy intensity” has a clear tendency to decrease with increasing GDP, so that is seems reasonable to expect that, with persistent efforts, the energy consumed per capita might stay constant or increase only slightly. Therefore one expects that the total primary energy supply (TPES) might, in the best case, follow the population increase. It seems clear that the last factor will play a prominent role if one wishes to decrease strongly the amount of CO2 emitted worldwide.

ENERGY ISSUE AND SCENARIOS

5

2.4. THE IMPORTANCE OF ELECTRICITY

Before trying to see if such an effort has any chance of succeeding it is necessary to have a clear knowledge of where to put it the most efficiently. Table 3 shows that the electricity production sector is the main CO2 emitter. Since competitive technologies exist to produce electricity without CO2 emissions it seems natural to put the stress on it, in the first place. TABLE 3. Shares of CO2 emissions by sectors. Emitting sector Percent of total % Electricity 33 Refineries 5 Industry 25 Transportation 24 Other (non electric home heating etc.) 13

That electricity is, indeed an important key can be shown when one compares the CO2 emissions of several countries with similar degrees of development as function of their use of hydro or nuclear energies for producing their electricity.

Figure 1. Correlation between the Carbon intensity and the share of hydro and nuclear electricity in various countries.

This comparison is seen on Figure 1 and speaks for itself. The data corresponding to the figure are given in Table 4. The difference between Denmark with no use of hydro nor nuclear electricity and a carbon intensity of 2.57 and Sweden with 100% hydro+nuclear electricity and a carbon intensity of 1.09 is striking. Since year 2000 Denmark has considerably increased its share of wind electricity production, which reached 17% by 2006, while its carbon intensity decreased to 2.54 by year 2004. It is interesting to see that the curve of Figure 1 shows a favorable increase of slope for shares of hydro and nuclear electricity exceeding 50%. That this could be related to an increase use of electricity, especially for heating, is a possible explanation.

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H. NIFENECKER

TABLE 4. Data corresponding to Figure 1. The data on carbon intensities are from “Key Energy World Statistics 2002”and relate to year 2000. Country

Hydro + Nuclear (%)

tCO2/toe

Denmark

0.0

2.57

China

16.0

2.62

UK

23.0

2.28

US

26.8

2.46

Spain

28.0

2.28

Germany

30.0

2.45

Russia

33.8

2.45

Korea

37.0

2.24

Japan

38.9

2.20

Ukrain

45.0

2.16

Canada

71.2

2.10

Brasil

87.3

1.66

France

90.3

1.45

Norway

99.5

1.31

Sweden

100.0

1.09

2.4.1. The double face of electricity Aside from its specific applications, electricity can be used for heat production and for transportation, either collective or individual. It might also be used for hydrogen production via electrolysis. As far as CO2 emission is concerned, these use of electricity can be the best or the worse, depending on the nature of electricity production. A very good example is provided by heating. If electricity is produced by fossil plants with an efficiency of, say, 33%, 1 kWh heating via an electric furnace with efficiency 70% will require 4.3 kWh and, thus produce 0.39 kgCeq. If electricity is produced with renewables or nuclear energies, there will be no emission*. Using direct gas heating with an efficiency of 60% requires 1.66 kWh of gas and produces 0.09 kgCeq. It is clear that electric heating when electricity is produced with coal is catastrophic,

______ *

It is sometimes argued that one should count here the “life cycle” CO2 emissions which take into account, not only the CO2 emitted in the production process, but also that which is related to the mining and plant construction phases. We do not think that this method is correct since one should rather think in terms of marginal emissions, i.e. in terms of the additional emission due to the actual consumption. Construction and mining emissions are, then, allocated to the mining and building industries, not to the private consumer. Furtherrmore mining and construction emission are much larger for fossil electricity production than for the nuclear one.

ENERGY ISSUE AND SCENARIOS

7

4 times worse than direct gas use. Let us, now, assume that the electricity fossil production amounts to 10%. In most cases (especially in Europe) this production is concentrated during the winter months, say during 3 months. That means that during these winter months fossil electricity will represent almost 40% of the total. 70% of electric heating is supposed to take place during these three months. It follows that 1kWh electric heating would require 1.2 primary kWh of electricity produced by coal plants, i.e. produce 0.11 kg-Ceq. It follows that even a small fraction of fossil electricity leads to CO2 emissions due to electric heating similar to that of gas. Another example is given by electric cars. Consider a small diesel car consuming 4 liters of gasoline par 100 kms. It emits about 3 kg-Ceq for this distance. Taking efficiencies of 0.3 and 0.7 for the thermal and electric engine respectively, a consumption of 20 kWh of an electric car would allow the same service as that of the diesel car. If electricity is produced without fossils the electric car does not produce CO2. If the electricity is produced with coal the amount of CO2 produced will be 5.5 kg-Ceq, that is almost twice more than the diesel car. Thus, in order to minimize the CO2 emission, switching to electric cars is only efficient if electricity is produced without resorting to fossil fuels. 2.4.2. Learning from the past Before 1973, many countries and electricity operators thought that oil was the best choice for producing electricity since it was very cheap. After the oil price crisis of 1973, most of them decided to resort to other resources, mostly coal and (or) nuclear. As striking examples are Denmark who switched to an almost pure coal-based system and France which initiated a crash program of nuclear reactors. United States had already stopped building nuclear reactors due to the strong counteroffensive of the coal industry and to the opposition of environmentalists which gained impetus following the TMI accident in 1979. Germany who had a strong coal mining industry kept a large proportion of its electricity produced with coal but, also, started a strong program of reactor building. The Chernobyl catastrophe halted reactor construction almost in all industrialized countries with the exception of France, and Pacific States like Japan and Korea. These various choices produced a divergence between the electricity production structures of average OECD countries and countries like France, Sweden or Switzerland. This divergence is illustrated on Table 5 where the structures of electricity production of France and of the whole OECD ensemble are compared. Let us rewrite history and see what things would have looked like if all OECD countries had made the same choice as France did. Note that these countries have the technical knowledge and industrial strength to do so and

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TABLE 5. Comparison of the electricity mix of France and of the totality of OECD countries. Primary Energy source

share in electricity production in % for OECD countries

share in electricity production in % for France

Coal Oil Gas Nuclear and Renewables Total

43 6 21 31

5.5 1.1 3.5 89.9

100

100

TABLE 6. Comparison of actual OECD use of nuclear energy and CO2 emissions with those that would have been observed if OCDE countries would have adopted the same electricity mix as France.

Original OCDE Modified OCDE

Source of Primary Energy

Primary Energy Supply Mtoe

Total CO2 Primary Energy Emitted for Electricity Production MtCO2 Mtoe

CO2 emitted for Electricity Production MtCO2

Nuclear and Renewables

716

0

716

0

Total

5280

13311

2334

5284

Nuclear and Renewables

2097

0

2097

0

Total

5280

8922

2334

728

that the question of nuclear proliferation was not relevant in their case. Table 6 shows the result of this operation. While the total primary energy supply and the energy used for electricity production remain unchanged to 5280 Mtoe and 2334 Mtoe respectively, the CO2 emissions would have fallen by 33% from 13311 MtCO2 to 8922 Mtoe. Nuclear energy would have soared from 716 to 2097 Mtoe, i.e. almost a factor 3. The 33% reduction in CO2 emission is much larger than the 8% which have been set as a goal for industrialized countries by the Kyoto protocol. 3. Energy Scenarios for Imagining the Future In order for politicians and industrials to adjust their politics, futurologists imagine scenarios which try to foresee how the mixture of “given” evolutions and voluntary politics will shape our future. Scenarios are not “predicttions” but a way to understand how our decisions of today may influence our future.

ENERGY ISSUE AND SCENARIOS

9

3.1. THE IIASA SCENARIOS

I have chosen to give as examples the scenarios built by the Vienna International Institute of Applied Systems Analysis (IIASA) since they have been routinely used by the World Energy Council (WEC) and by the Intergovernmental Panel on Climate Change (IPCC). Three illustrative storylines, A2r, B1 and B2 are described in [3] and constitute the GGI Scenario Database, 2007. They are built at the regional level. They first make hypothesis on population evolutions. Storyline A2r assumes a continuous increase of the world population throughout this century up to more than 12 billions. Storyline B2 shows some kind of stabilization of the population to a little more than 10 billions in 2100. Finally storyline B1 goes through a maximum of 9 billions around 2050 and decreases to 7 billions in 2100. The second set of assumptions deals with the rates of increase of the GDPs. These are declined regionally. The fastest GDP growing storyline is B1 (low population) which raises to almost 350 trillion dollars in 2100 from 27 trillions in 2000 while the slowest growing is A2r raising only to 190 trillions dollars in 2100. Each storyline is then subdivided in different scenarios characterized by an assigned final concentration of CO2. The storyline A2r is subdivided into a baseline scenario and 5 additional scenarios with asymptotic CO2 concentrations between 670 and 1390 ppm. The baseline scenario exceeds 1450 ppm by 2100. These scenarios are clearly unacceptable, and would lead to catast-rophic consequences, not only climatic. The storyline B2 has 3 scenarios, the baseline which reaches more than 975 ppm by 2100, and scenarios with asymptotic CO2 concentrations of 670 and 480 ppm. The storyline B1 has a baseline scenario reaching 830 ppm of CO2 and 5 additional scenarios with asymptotic CO2 concentrations between 480 and 670 ppm. I believe that the storyline B2 is probably the most realistic as far as population projections are concerned. And I discuss it henceforth. Figure 2 shows the evolution of the CO2 equivalent GHG concentrations in the B2 scenarios. Note that, for the 480 ppm scenario, the concentration rises up to 600 ppm and, then decreases. This decrease is made possible by a very early decrease in the CO2 emissions which goes through its maximum of 12.7 GtCeq in 2020 (from 11 GtCeq in 2000) and, then decreases sharply to 9.4 GtCeq. in 2050 and 0.7 GtCeq. in 2100. The surprising decrease of the CO2 equivalent concentrations is related to a strong, probably optimistic, change of land-allocation which builds a CO2 drain well. Land use is a source of 1000 Mt CO2eq. (deforestation) in 2000 and becomes an absorbing drain well of 3500 Mt CO2eq. by 2100 (reforestation).

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Figure 2. Evolution of the GHG CO2 eq. concentrations for the three B2 scenarios.

We concentrate on the “optimum” 470 ppm B2 scenario. The total primary energy supply (TPES) increases from 9.5 Gtep in 2000 to 25 Gtep in 2100 with 19 Gtep in 2050. The regional evolutions of the TPES are shown on Table 7. By 2100 OECD countries will have a rather marginal contribution to the energy consumption. By 2050 the primary supply of non OECD Asia will exceed that of the whole OECD. TABLE 7. Regional evolutions of TPES. 2000 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 Asia

0.91

1.47

2.59

3.64

4.49

5.89

6.62

7.31

8.09

8.94

9.65

Form. USSR

0.63

0.94

1.56

2.12

2.90

3.76

4.67

5.50

6.32

6.91

7.43

OECD

2.87

3.18

3.79

4.19

4.39

4.38

4.32

4.21

4.07

4.02

3.99

LAM+Africa 0.61

0.75

1.03

1.35

1.76

2.08

2.17

2.32

2.41

2.50

2.53

Aside from the above mentioned changes of land use the key of the success of scenario B2-470 ppm to mitigate the increase of CO2 concentration is the increase of the electricity share and of that of fossil free electricity production methods. Figure 3 shows the spectacular increase of the shares of electricity and bio-fuels in final demand. Electricity essentially replaces coal and traditional biomass while modern biomass replaces oil. Figure 4 shows how the fossil electricity decreases from 62% to 10% with the share of nuclear electricity reaching around 60% in 2100.

ENERGY ISSUE AND SCENARIOS

11

Figure 3. Evolution of final energy demand structure The shares of bio fuels and district bio heat are added.

Figure 4. Evolution of the Electricity mix in scenario B2-470 ppm.

The decrease of hydroelectricity reflects an almost constant absolute production while the total electricity production increases by more than a factor 10 between 2000 and 2100. Wind and solar electricity reach a maximum share of 30% around 2050 but a maximum absolute production of more than 3 Gtoe by 2100. I believe that a 30% share is not realistic for intermittent energy sources as long as huge electricity storage is not available. In absolute value, nuclear energy is multiplied by more than 6 in 2050 and more than 35 by 2100. Such a surge would clearly require breeder reactors.

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3.2. THE IPCC SCENARIOS

The just presented scenarios of the IIASA belong to the family of scenarios which were built for the IPCC reports. The family counts a total of 40 different scenarios. It is to be feared that deciders will have a hard time finding their way in such a jungle. The complexity facing deciders is illustrated on Figure 5.

Figure 5. Emission profiles corresponding to the family of scenarios considered by IPCC.

3.3. SHARE OF NUCLEAR ENERGY IN “OFFICIAL” SCENARIOS

Most recent IPCC scenarios foresee a share of nuclear electricity of 18% in 2030, assuming a cost of CO2 less than 50 $/ton. The share of renewable electricity would be 35%. In 2030, the IIASA-470 ppm scenario (see Figure 4) foresaw a nuclear electricity share of 21% and a renewable electricity share of 28%, including hydroelectricity. Prediction of nuclear energy shares are highly political and reflect the state of public opinion. A few years ago it was ill behaved to even talk about it. As time passes and as the price of oil and gas increase it has become more admissible to mention nuclear energy. An example is given by the predictions of the World Energy Outlook of the International Energy Agency (IEA). WEO2004 predicted a decrease of nuclear production from 2975 TWh in 2020 to 2929 in 2030. This reflected “out of nuclear” energy policies dominance. The alternative WEO 2006 predicts a nuclear electricity production of 4106 TWh in 2030 with a share of 14% , in clear increase with the respect of the 8% in 2020 predicted by the WEO1998. It seems clear that scenario makers tend to apply auto censorship when it

ENERGY ISSUE AND SCENARIOS

comes to take fully in consideration nuclear power. I believe that important to determine how much nuclear energy could help dealing the CO2 emission question in absence of ideological opposition to it. ponsibilities of those who act this opposition will, then, be clearly luated.

13

it is with Reseva-

4. Nuclear Intensive Scenarios 4.1. REFERENCE IIASA-WEC 1998 SCENARIOS

A first approach of a nuclear intensive scenario with 3000 GWe nuclear power 30 years from time 0 and 9000 GWe 50 years from time 0 was given in 1999 [4]. A revised version is found in [5]. I will summarize the results presented there. The method was to use, as a reference, scenarios built by IIASA-WEC* in 1998 [6] and increase the share of nuclear power to the highest reasonably achievable level. The nomenclature of the 1998 IIASA scenarios are different from that used more recently in the frame of the IPCC studies. The philosophy of the 1998 scenarios was not essentially different from that of the 2007 ones. I believe that the conclusions of the studies of [4] and [5] are still essentially valid. In the IIASA reference study, storylines A correspond to a fast growth of the GDP/capita in all geographical regions of the world. It assumes a significant reduction of inequality between them. The growth is especially fast in former Soviet Union countries. Storyline C has a rather slow average GDP/capita growth but is, clearly, of the equalitarian type. For storylines A and B the energy intensities decrease as a consequence of the increase in the GDP/capita, as observed in the past. Storyline C assume a voluntary decrease of energy intensities, especially in the most developed countries. The main assumptions of the IIASA-WEC storylines are given on Table 8. The energy intensity is assumed to decrease markedly from 0.43 toe/k$ in 1990 to 0.245 for storyline A, 0.27 for storyline B and 0.19 for storyline C. The main differences between the scenarios are the energy mix producing the primary energy. In scenarios A1, A2 and A3 the relative shares of coal, oil and gas are different. In scenarios C1 and C2 the relative shares of renewable and nuclear energies are different. Note, also, the relative importance of gas in the energy mix of scenario C1. These features are displayed on Table 9. Whether the six scenarios are realistic must be evaluated with respect of the fuel reserves. This is done on Table 10.

______ *

WEC: World Energy Council

14

H. NIFENECKER TABLE 8. Assumptions of the IIASA-WEC storylines. GDP/capita k$ Population 2050 millions 362.42 494.6 148.12 394.67 141.06 838.58 924.25 1735.73 1984.17 750.55 2281.28 10055.43

North America Western Europe Pacific OECD Former Soviet Union Eastern Europe Latin America M. East N. Africa Africa Centrally planned Asia Other Pacific Asia South Asia World

1990 21.62 16.15 22.78 2.71 2.39 2.50 2.12 0.54 0.38 1.53 0.33 3.97

Scenarios B 45.84 37.06 45.80 7.48 7.83 7.07 4.03 1.03 3.36 7.86 1.33 7.24

A 54.47 45.88 58.68 14.09 16.27 8.33 5.64 1.57 6.99 12.21 2.00 10.10

C 38.79 32.95 42.80 7.14 7.97 7.39 4.09 1.19 5.40 10.20 1.75 7.46

TABLE 9. Primary energy mix for the IIASA-WEC scenarios. 1990 Coal Oil Nat. Gas Nuclear Hydro Biomass (comm) Biomass (nonc) Solar Others Total CO2(MtC)

B

2176 4136 3064 4040 1685 4499 450 2738 489 917 246 1122 849 860 0 432 17 1087 8976 19831 5932 9572

A1

A2

A3

C1

C2

3786 7901 4699 2904 993 1124 717 1858 852 24835 11619

7827 4781 5459 1092 1104 2207 747 420 1201 24840 14668

2241 4329 7913 2824 1062 2906 743 1636 1007 24661 9294

1504 2668 3919 521 1031 1481 822 1552 747 14246 5343

1472 2616 3344 1771 962 1357 824 1377 526 14250 5114

TABLE 10. Cumulative consumptions of fossil fuels for the IIASA-WEC scenarios compared to the 1990 reserves. Cumulative consumptions by scenarios Coal Gtep Oil Gtep Gas Gtep

A1 200 300 210

A2 275 260 211

A3 158 245 253

B 194 220 196

C1 125 180 181

C2 123 180 171

Reserves 1990 540 146 133

While it is true that reserves are subject to changes Table 10 shows that oil and gas will become rare, and, thus expansive. By 1999 this was still not clear and many economists foresaw a strong rise of the use of gas, especially for electricity production. It is, now, widely recognized that the Peak Oil will occur within the next 10 or 20 years and the Peak Gas 10

ENERGY ISSUE AND SCENARIOS

15

years later. It seems that coal will become, once more, the main fossil fuel: liquid and gas fuels can be synthesized from coal. This is why we have especially studied scenario A2 as an example of strong economic expansion. 4.2. NUCLEAR INTENSIVE ALTERNATIVES

The IIASA scenarios foresee a rather modest contribution of nuclear energy to the global energy mix. As said above, it seems that the main reason for this shyness is more related to “political correctness” than to economical or technological constraints. It is interesting to examine how much a deployment of nuclear energy limited only by these constraints might limit global warming and resource exhaustion. We, now, give the outline of such a treatment for the three scenarios A2, B and C2. We assume that, by 2030, (in our study, our time 0 was tyear 1999, a shift of 8 years should be made in 2007) the use of fossil fuels for electricity production will be drastically reduced. Table 11 displays the reduction factors used for several geographic aggregates. TABLE 11. Reduction factors used in the nuclear intensive scenarios in 2030 (column 2) and 2050 (columns 3 to 5). In the last column the share of Hydrogen and (or) electricity in the transportation sector is given. reduction factor for reduction factor reduction factor H2 (E) share for residual fossil use in electricity for residual coal use gas use production 2030 2050 (E) 2050 (N) 2050 (N) 2050 (H) OECD 0 0 0 0 0.8 former USSR 0.5 0 0 0.3 0.4 Eastern Europe 0.5 0 0 0 0.6 Latin America 0.5 0 0 0 0.6 Arab countries 0.3 0 0 0 0.3 South Sahara 1 0 0 0 0.3 China 0.3 0 0.3 0.3 0.3 India 0.3 0 0 0 0.2 Other Asia 0.3 0 0 0 0.3

The same reduction was applied in the three scenarios. Fossil fuels were assumed to be saved by resorting to nuclear power, although renewable energies could, equally well, be used provided they become reasonably competitive. The choice we have made of nuclear power provides a kind of existence theorem for a solution to curb Carbon Dioxide emissions, and, at the same time, tests the capabilities of nuclear power in terms of fuel availability, wastes produced and capital needs.

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Table 11 shows different types of reduction factors used for 2050. These reduction factors are used differently in three different scenarios with increasing use of nuclear power: •

The “electric” scenario, labelled “E”, assumes that fossil fuels are no longer used for electricity production,

•

The “nuclear” scenario, labelled “N”, assumes that coal and gas are no longer used in industry, homes or offices (in particular for heating), except, principally, in former USSR and China,

•

The “Hydrogen” scenario, labelled “H”, assumes that hydrogen and (or) electricity have largely replaced oil in the transportation sector. The share of hydrogen fuel is displayed in the last column of Table 11.

The performances for CO2 emissions of the different variant of the A2 and C2 scenarios are displayed on Figure 6. The Figure shows that politics based upon energy efficiency and use of nuclear electricity are operational to mitigate CO2 emissions. Even storylines which are rather energy proficient like the A one could be tamed with the use of nuclear power and CO2 emissions limited to less that one half of present emissions.

Figure 6. Influence of deployment of nuclear power on world CO2 emissions for different IIASA-WEC like scenarios.

The maximum extent of nuclear power would be around 14 Gtep, i.e. more than 20 times more than present. Will resources of Uranium allow such an intensive program? 4.3. DEPLOYMENT OF A WORLD BREEDER REACTOR FLEET

Light water reactors need approximately 200 tons of natural uranium each year to feed a 1 GWe reactor. A 7000 fleet would require 1.4 million tons annually. Estimates of uranium reserves depend upon the price one accepts to pay for it. For less than 130$/kg reserves are estimated around 4.7 millions tons, while for less than 260 $/kg they would reach 15 millions tons. Other non conventional reserves (phosphates, coal) might reach 25

ENERGY ISSUE AND SCENARIOS

17

additional millions tons. Finally one can note that the ocean contains 3 to 4 billions tons of uranium. Extraction of uranium from sea water has been technically demonstrated at a cost in the range of 500 to 1000 $/kg. Aside from the exploitation of oceanic uranium at a considerable economic and environmental cost, uranium reserves could not feed a fleet of 7000 standard reactors for more than a few decades. This is clearly not sustainable. There needs breeder reactors able to use natural uranium with an efficiency 100 times better than standard light water nuclear reactors. The breeding reaction consists in neutron capture by Uranium 238 to produce Plutonium 239, which is a fissile isotope as is Uranium 235. For the Uranium-Plutonium cycle, the breeding reaction reads: 238

239 239 239 β− β− U+n⎯ ⎯→ U ⎯⎯ → Np (period of 2 days) ⎯⎯ → Pu.

For the breeding reaction to be effective it is necessary that the number of neutrons emitted in the fission of the Pu239 nucleus should be large enough to ensure both a new fission and a capture by a U238 nucleus. In practice this condition is only realized for fast neutron spectra such as provided in fast neutron reactors. Another breeding reaction is based on the Thorium-Uranium cycle, which reads: 232

233 233 233 β− β− Th + n ⎯ ⎯→ Th ⎯⎯→ Pa (period of 27 days) ⎯⎯→ U

Unlike the Uranium-Plutonium case, the Thorium-Uranium cycle can work with thermal neutron spectra as well as with fast neutron spectra, although with difficulty. A rather large experience with fast neutron reactors have been collected in a number of countries like France, Japan and Russia. This is why we study more specifically how and if it is possible to deploy a fleet of 7000 fast breeders reactors. How fast such a park could be deployed will depend on the amount of Plutonium available from standard reactor fuel reprocessing, the initial inventory of the breeders, as well as on the doubling time of the breeder park. A relevant study has been made in [4] where the deployment of 3000 PWR reactors in 2030 and an additional 6000 breeders by 2050 was considered. The question addressed was whether such a deployment is compatible with Uranium reserves and doubling times of the breeders. The initial loads are assumed to be mixtures of the fertile element (U) with Plutonium taken from the spent fuels of PWR and BWR reactors. It is important to make sure that the amount of Plutonium available would be sufficient to supply all the breeding reactors by 2050. Experience with fast breeders shows [7] that a 1.2 GWe reactor requires an initial inventory of 5 tons of Pu. Such a reactor produces around 0.25 tons of Pu annually, corresponding to a doubling time of 20 years. However this value of the doubling time does not take into account the reprocessing stage. The longer the cooling time of the used fuel before reprocessing, the

18

H. NIFENECKER

longer the effective doubling time. As an example, if the residence time of the Plutonium in the reactor is 4 years, and the cooling time also 4 years, the Plutonium inventory is doubled, and so is the doubling time.

Figure 7. Size of installed nuclear power (in GWe) for the U-Pu cycle as a function of time. In a first stage a PWR park is developed which produces Plutonium used to start FR U-Pu breeder park.

The transition from a PWR(BWR) based system to a fast reactor one was studied. It was assumed that a strong PWR program starts in 2010, first breeders starting progressively in 2020*. By 2030 no new PWRs are connected to the grid, leaving the field to fast reactors. Figure 7 shows the evolution of the reactor park corresponding to a Plutonium production of 250 kg/GWe by the PWRs and 200 kg/GWe by the fast breeders. A cooling time of 1 year was assumed. The target of 9000 GWe by 2050 could then be reached. For longer cooling times it was found that the target cannot be reached. Cooling times as short as 1 year are probably not possible with standard aqueous reprocessing and would require pyro-chemical reprocessing. After 2050 the PWRs would be phased out progressively and the doubling time of the FR could be adjusted to the desirable evolution of the reactor park. In Figure 7, a 1.5% annual increase of the nuclear park was assumed. In our scenario the last PWR reactors will be phased out in 2070. At that time the total amount of natural Uranium used would reach 12 million tons, close to the presently estimated reserves. This means that the number and life time of the PWR park cannot be considered as an easily adjustable variable to achieve the strong increase of nuclear power between 2030 and 2050. This increase will be difficult to achieve and requires the early development of breeders, as well as the availability of as much as

______ *

Since this study was carried out in 1999 one should shift the time-table by 8 years in 2007

ENERGY ISSUE AND SCENARIOS

19

possible reprocessed Plutonium. The generalization of MOx incineration has to be weighed against this requirement. Similarly incinerating Plutonium in HTR reactors may be counterproductive if spent fuel reprocessing is not possible. Of course, accepting a lower value for the target in 2050 would make things easier. For example a target of 7000 GWe could be reached with a doubling time of 32.5 years. Another possibility would be to increase the share of more efficient Plutonium producing reactors such as the CANDUs. An alternative deployment scheme has, recently, been proposed by Merle et al. [8]. Aside from capitalizing the Plutonium produced in PWRs, it consists of using a limited number of Fast Reactors with Thorium blankets in order to produce the large quantities of 233U to be fed into fast neutrons molten salt converter reactors*. 450 1 GeV PWR like reactors and 300 FR suffice to feed a fleet of Molten Salt reactors reaching 3000 by 2050 and 5000 by 2100 with a consumption of 7 millions tons of natural Uranium. 4.4. FEASIBILITY AND COST

A multiplication of the number of reactors by a factor between 10 and 20 would require very important investments. Do they exceed the financial resources of the world energy sector? Such program has to be compared to the only alternative which is building a comparable fleet of coal power plants with CO2 capture and sequestration devices. For nuclear reactors, the investment cost amounts, approximately, to 2 billions euros per GeV. A program of 9000 reactors for 2050 would cost approximately 13500 billions Euros. During the 50 years when the deployment of this fleet would take place, the sales return of the electricity industry will be approximately 30000 billions Euros. Therefore the financial requirements for deploying a large fleet of nuclear reactors are within the capabilities of the power industry. The alternative to nuclear reactors are coal power plants. Their investment cost is only two thirds of that of nuclear ones, i.e. 1.3 G€/GWe. However, implementing the CO2 capture-sequestration scheme, an absolute necessity in view of the considerable CO2 emissions of coal plants, will increase the investment cost by at least 50%, which would suppress the relative advantage of coal versus nuclear plants. Furthermore the fuel and running costs of coal plants is much larger than that of nuclear reactors.

______ *

A converter reactor produces one fissile nucleus for each fissioning one. It is a breeder with infinite doubling time.

20

H. NIFENECKER

A serious problem for deployment of a large fleet of nuclear reactors will be the availability of skilled manpower and of specialized manufacturing equipment such as soldering of core containers. 4.5. THE QUESTION OF NUCLEAR WASTES

It is not possible, here, to treat the question of nuclear wastes with pertinence and completeness. Let us just stress that the use of breeder reactors reduces considerably the volume of highly radioactive wastes. For PWR or similar reactors the production of high level wastes amounts to 27.279 tons per GeV [5]. Of these, 26.047 correspond to Uranium, 266 kg to Plutonium and 20 kg to minor actinides. In a breeder fleet Uranium and Plutonium are resources and Minor Actinides may be reprocessed and incinerated. Thus the residual wastes are Fission products for a total of 946 kg. Among these only 63 are long-lived. Thus, for the same amount of energy produced wastes of breeders have a volume at least 30 times smaller than those of standard reactors. Furthermore, if Minor Actinides are reprocessed and incinerated the heat production after 300 years is almost a factor 100 less than that of UOx used fuels. 4.6. THE PROLIFERATION ISSUE

A multiplication of the number of reactors by an order of magnitude rises fears of uncontrollable proliferation of nuclear weapons. Evaluating whether such fears are justified requires examination of technical as well as political arguments. 4.6.1. Technical aspects of nuclear proliferation Any state that wants to acquire nuclear arms needs to have the technical skill necessary to make nuclear bombs and to master military grade nuclear matter production. In practice, only 235 U and 239 Pu can be used as nuclear explosive. 235 U allows simple (and low efficiency) designs of nuclear bombs. The critical mass of a simple (gun type) U235 bomb is in the range of 25 kg. Because of the spontaneous fissions of 240 Pu which tend to cause premature power excursion, 239 Pu bombs need a rather complex implosion device. But the higher neutron multiplicity of 239 Pu fission neutrons yields a smaller critical mass of around 11 kg. Military grade 235U is obtained by isotopic enrichment. The first used techniques of mass separation and gaseous diffusion are very electricity greedy. This makes detection and monitoring by IAEA inspectors rather easy, although the discovery of the importance of the Iraki program after the

ENERGY ISSUE AND SCENARIOS

21

first Gulf war, came as a surprise. The ultra centrifugation technique is much more discreet and difficult to detect, even for covert transport of equipment parts. The simplicity of conception of 235U bombs and the availability of ultracentrifuges makes this option the easiest for candidates to proliferation. 239 Pu are technically more difficult to make. Military grade 239Pu may be obtained in reactors with small fuel burn up. When reactors need to be halted for fuel extraction, monitoring by the IAEA inspector is easy since satellite observation can detect halted reactors and initiate inspection. CANDU reactors are equipped with continuous fuel discharge equipments operating while the reactor is working. It is easy to discharge weakly irradiated fuel, suited for military use. Chemical extraction of pure Plutonium may be simply carried out at the laboratory scale. Furthermore, CANDU and Graphite reactors may work with natural uranium. Most countries started their nuclear armament with such reactors. PWR or BWR reactors have never been used to obtain military plutonium because they have to be stopped for discharge. The fuel burn up is much higher than in the CANDU case (at least 35 Mega Watt days/ton versus only 5 Mega Watt days/ton), and, thus, the fraction of 240 Pu is so high that no one has ever made a bomb with plutonium extracted from used light water reactor fuel. Breeder reactors used fuels are planned to have very high burn ups (in the range of 200 Mega Watt days/ton). This will make the use of plutonium extracted from them difficult. Thus, provided the technical skill is present, any state has the possibility to produce 235 U and 239 Pu military grade material, independently of its nuclear electric power program. 4.6.2. Proliferation is a political question The non-proliferation treaty had the aim to limit the possession of the nuclear armament to those who had it already: USA, USRR, UK, France and China. Countries who did not have it pledged not to try to get it and got, in exchange, the right to have assistance in developing civil nuclear applications via the IAEA and by bilateral or multilateral agreements. Countries who had not signed the treaty were deprived of any help on nuclear matters from the IAEA as well as subscriber states. Selling strategic material to non-subscribers was forbidden to subscriber states. The main non-subscriber states were Israel, India, Pakistan, South-Africa (who signed later on). The stability of the treaty was insured by the cold war system, dominant states in the two camps keeping discipline among their allied. Recently, subscriber states tried to develop a nuclear armament: Iraq, Lybia, North Korea and, possibly, Iran. The non-proliferation treaty also

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implied that nuclear states should proceed to disarmament. Not much has been made in this direction. In general I believe that the motivation of states, who want to acquire a nuclear armament is that they want to get “sanctuarized“ by its possession. That there is some justification for that ambition can be seen if one asks what would have happened to Iraq if it had possessed the nuclear bomb. It is to be expected that states, who feel that they are threatened will, eventually, try to acquire, as discreetly and covertly as possible, nuclear capability. I believe that the present “unbalanced” situation where a few states have a monopoly over nuclear armament is basically unstable. The efforts made by “haves” to prevent “have nots” to become “haves” will, more and more, be interpreted as power policies. Already now, the multiplication of nuclear states present a risk of miscalculations and unwanted nuclear conflicts. The real danger, here, is a use of a nuclear arm without warning. I believe that a new treaty should be negotiated, treating all states on an equal footing. Since it is very doubtful that those in possession of nuclear armament will renounce it, one should accept that all states are untitled to it. In order to minimize the dangers of uncontrolled developments, all states should accept to submit their nuclear armament to international inspection. No armament apt to immediate use should be allowed. That is, airplanes in flight, submarines at sea carrying nuclear bombs should be banned. All missiles should be deprived of nuclear charge. This means that a minimum delay should be observed before use of nuclear armament could be made. Thus, clearly, nuclear arms could only be used as ultimate deterrent, and banned as surprise devices. Terrorist use of nuclear explosive devices is a really frightening prospect. Military grade 235U could be smuggled out or, even, provided to the terrorist groups by infiltrated militaries. 235U bombs are relatively easy to make, or could even be smuggled. I have no idea how to counter such threats except good secret services. It seems clear that an international and generalized inspection system of al military nuclear facilities might diminish the danger. 5. Proposal for Research and Development An important development of nuclear power requires very important R and D in three different but complementary fields: Breeder reactors, Fuel cycles including Thorium and Material and process studies. These lines of research imply specific work on the different types of reactors and their associated fuel cycle, namely.

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5.1. FOR THERMAL NEUTRON REACTORS •

for PWR reactors – Selective reprocessing: Extraction of Cs, Sr and M.A. This with the aim to reduce the cost and number of geological disposal facilities – Pyro chemical and electro metallurgical: Reprocessing techniques could be useful not only for standard reactor fuels but, also for that of advanced reactors as well as for molten salt fuels. – Th-Pu MOx fuel: In order to produce 233U. The use of 233U could be very useful for diminishing Plutonium and Minor Actinides production and limiting the technical possibility of proliferation.

•

for CANDU reactors – Use of Th-Pu and, then Th-U3 fuel: This will reduce the aptitude of CANDU reactors for proliferation and make use of their excellent neutronic properties. – Reprocessing of Th-U3 fuel: The Thorex or alternative processes should be evaluated at an industrial level. Fuel fabrication with high level gamma activity should be developed – Optimization of fuel regeneration: CANDU Th-U3 reactors are close to regeneration, giving the prospect of a very significant improvement of fuel use.

5.2. FOR FAST NEUTRONS REACTORS •

Sodium cooled: – Void coefficien: Present Sodium cooled breeders have a positive void coefficient which might be unacceptable for safety authorities. Geometries and, possibly, new fuels should be studied to prevent this unwanted property, – Core Recompaction: In case of partial or total fusion of the core present Sodium cooled breeders may lead to more reactive geometries. One should find geometries preventing such occurrence, – Th blanket: Study and experimental tests of a Thorium blanket in order to produce large quantities of 233U are recommended, – Reprocessing of Th blanket: in order to extract 233U.

•

Lead cooled reactors maybe a valuable alternative to Sodium cooled reactors since Lead do not react with air or water and since the vacuum coefficient of lead reactors is negative. The main problems that remain to be solved are:

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– Corrosion by hot lead under irradiation, – Pb-Bi alloys. Such alloys have a much lower melting temperature than lead. However a large production of the volatile polonium rises specific safety questions, as well as the availability of Bismuth. •

Molten salt cooled reactors. These are other alternatives to Sodium coolant with the nice property of being transparent. Chemical composition has to be optimized with regard to neutronic and corrosion properties as well as lowering the melting point.

•

Gas cooled reactors: – Reprocessing of refractory fuels, – Loss of coolant safety studies.

5.3. MOLTEN SALT REACTORS

Main R and D fields are based on neutron spectrum optimization, corrosionand fuel reprocessing. A demonstration reactor in the 10s thermal MW range is urgent in order to rise competences at a level comparable to that of other GEN IV reactors.

References [1] International Energy Agency (IEA): Key World Energy Statistics, http://www.iea.org. [2] Fourth Assessment Report of the Intergovernmental Panel on Climate Change (IPPC) 2007, http://www.ipcc.ch. [3] International Institute of Applied Systems Analysis (IIASA), http://www.iiasa.ac.at/webapps/ggi/GgiDb/dsd?Action=htmlpage&page=series. [4] Nifenecker, H., Heuer, D., Loiseaux, J. M., Meplan, O., Nuttin, A., David, S., and Martin, J. M., 2003, Scenarios with an intensive contribution of nuclear energy to the world energy supply, Int. J. of Global Energy Issues 19(1):63-77. [5] Nifenecker, H., Méplan, O., and David, S., 2003, Accelerator driven sub-critical reactors, IOP publishing, pp. 1-316. [6] IIASA-WEC, 1998, Global Energy Perspectives Report, http://www.iiasa.ac.at. [7] Bussac, J., and Reuss, P., 1985, Traité de Neutronique, Hermann. [8] Merle-Lucotte, E., Heuer, D., Allibert, M., Ghetta, V., Le Brun, C., Mathieu, L., Brissot, R., and Liatard, E., 2007a, Optimized transition from the reactors of second and third generations to the thorium molten salt reactor, Proceedings of the International Congress on Advances in Nuclear Power Plants (ICAPP), Nice, France (2007).

OUTLOOK ON GENERATION IV NUCLEAR SYSTEMS AND RELATED MATERIALS R&D CHALLENGES F. CARRÉ*, C. RENAULT, P. ANZIEU, P. BROSSARD AND P. YVON French Atomic Energy Commission (CEA) Nuclear Energy Division Saclay Center, 91191 Gif sur Yvette Cedex - France

Abstract:: Preoccupations of energy security and concerns about of the role that future reactors should play for minimizing long-lived radioactive waste led the French Government to formally approve a R&D strategy that gives first priority to developing a new generation of fast neutron nuclear systems and recycling technologies so as to assure a sustainable, proliferation resistant and environment friendly electricity production in the second half of the 21st century. This materializes though the goal of preparing the deployment of fast neutron reactors in the French generation fleet around 2040 that is being addressed by the CEA and industrial partners along two tracks: seeking for innovations to improve sodium cooled fast reactors current performances, and revisiting the potential merits of gas cooled fast reactors with safety features enhanced by the use of an innovative refractory fuel technology. This goal is closely linked to that of preparing the renewal of the spent fuel reprocessing plant of La Hague around 2040 through research and development on advanced recycling technologies capable of managing minor actinides so as to offer best nonproliferation features, as well as to minimize the long term burden of ultimate radioactive waste (heat load and radiotoxicity). The paper presents the R&D strategy to meet these goals, as it has been updated in 2006 to account for the presidential decision to launch design studies of a prototype of innovative sodium cooled fast reactor to be put in service by 2020. Furthermore, this strategy is devised according to the date of 2012 stipulated by the bill of June 2006 on the “management of radioactive materials and waste” as a major milestone to assess the potential of the considered fast reactor types for industrial deployment and transmutation of long-lived radioactive waste. The paper will also address the second objective of the French R&D programme on future nuclear energy systems that consists in supporting the strategy of industrial partners in the field of high temperature nuclear technologies for potential applications to the production of hydrogen and synthetic hydrogen fuels, or the delivery of process heat for other industrial applications. The prospects opened by the concrete implementation of the multilateral collaboration within the Generation International Forum and those opened by the perspectives of a strengthened cooperation on future nuclear fission systems within the 7th European R&D Framework Programme will also be covered, as well as the stakes of an efficient linkage between R&D programmes on the same nuclear system conducted at the national, European and broader international levels.

______ *

[email protected]

V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

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1. General Context of Nuclear Energy Development The rising awareness of a fast growing world primary energy demand at the beginning of the 21st century, together with considerations of energy security and concerns about greenhouse gas emissions led to acknowledge that all energy sources are to be used. Prospective studies, such as “Global Energy Perspectives” and “Energy to 2050 – Scenario for a sustainable future” respectively performed by the World Energy Council (1998) [1] and the International Energy Agency (2003) [2] show that, even with optimistic assumptions on the potential contribution of fossil and renewable energies, the use of nuclear energy will be necessary in countries where it can be developed safely and competitively (Figure 1).

Figure 1. IEA scenario of energy growth for a sustainable future from [2].

Furthermore, the safe operation of current power plants over the past 20 years, the increasing economic competitiveness of nuclear energy as fossil fuel prices escalate, as well as considerations of energy security pave the way for an active development of nuclear energy in Asia and a renaissance in the United States and Europe. This leads to anticipate an installed capacity of nuclear power of the order of 1000 to 1500 GWe by 2050, which is about four times the current installed capacity (370 GWe). Such a nuclear power capacity would require about 15 millions tons of natural uranium, if realized only with light water reactors which use less than 1% of the uranium (235U mainly) over a lifetime of 60 years. This amount, which is comparable to the estimated assured plus speculative reserves at a price below 130$/kg [3], incites to prepare the deployment by 2040 of fast neutron reactors with a closed fuel cycle that can burn more than 80% of natural uranium. Even if the situation around the middle of the century would not lead to a shortage of uranium because of additional reserves in phosphates or sea water, the rising cost of this resource that is already

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observed today (Figure 2), together with the accumulation of spent fuel, would drive the need to switch to fast neutron reactors with a proliferationresistant closed fuel cycle to achieve a more efficient use of uranium and minimize the ultimate long lived radioactive waste.

Figure 2. Recent evolution of Uranium market price (US$/lb).

Besides the goal of a sustained production electricity in the second half or the century, future prospects of cogeneration and needs of other energy products than electricity such as hydrogen, synthetic fuels and high temperature heat for industrial processes also trigger a renewed interest in high temperature reactors. Hydrogen is already needed to produce ammonia for fertilizers and to amend heavy crude oil, which will become an application of increasing importance in the next decades. Nuclear energy can also produce heat and/or steam to assist in the exploitation of tar sands or oil shells, and produce synthetic hydrocarbons as make-up fuels to gasoline from conventional oil resources. The rising perspective of the hydrogen economy and the subsequent recognition of the strategic nature of hydrogen technologies in the United States, Europe, Japan and other countries led to sign a multilateral agreement of International Partnership for Hydrogen Economy (IPHE) which paves the way for an important R&D programme on nuclear hydrogen production. In addition, continuing R&D on fuel and reactor technologies of 3rd generation light water reactors is of utmost importance to help optimizing these reactors in an evolving context over the entire century. Improving the conversion ratio of light water reactors that could permit to use up to 2% of the uranium energy content is of special interest to temporarily mitigate the consequences of rising natural uranium costs pending fast neutron reactors are deployed in the second half of the 21st century. In summary, research and development are essential to prepare the future of nuclear energy in three directions at least:

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•

Securing a sustainable electricity generation in the second half of the 21st century in a context of rising price of natural uranium, while assuring an adequate control of proliferation risks associated with the front and the back end of the fuel cycle (reprocessing and recycling);

•

Developing the production or cogeneration of other energy products than electricity (hydrogen, synthetic fuels, process heat for the industry…);

•

Innovating to adapt 3rd generation reactors to an evolving context over the 21st century.

2. The R&D Strategy in France on Future Nuclear Energy Systems Within the above context, prospective studies carried out by the CEA and industrial partners led to elaborate for France a R&D strategy on future nuclear energy systems for the medium and the longer terms (>2040), that aims at three complementary objectives: •

The development of fast neutron systems with a closed fuel cycle (sodium or gas cooled fast reactors) for a sustainable energy supply through breeding in the long term, and for managing actinides from light water reactors spent fuel in the medium term;

•

The development of key technologies for the nuclear production of hydrogen or the supply of high or very high temperature heat for the industry in close collaboration with industrial partners (high or very high temperature reactors and water splitting processes);

•

Innovations for light water reactor fuels, systems and high conversion cores to further optimize LWRs until 4th generation fast neutron systems are mature for industrial deployment around 2040.

The R&D priority is clearly put on fast neutron nuclear systems with a closed fuel cycle, the Sodium-cooled Fast Reactor (SFR) and the Gascooled Fast Reactor (GFR), owing to the general recognition of their capability to meet sustainability goals that include optimizing the use of natural uranium resources and minimizing long lived radioactive waste production (minor actinides) [4]. In addition, at a lower but significant level, R&D is conducted on high or Very High Temperature Reactors (V/HTR) in support to the AREVA project of multipurpose nuclear heat source ANTARES [5] presently viewed at 850°C for potential market needs from 2025 onwards. This R&D strategy proposed by the CEA and its industrial partners AREVA and EDF was formally approved by the French Government in March 2005. Since then, two major events came to sharpen the objectives

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and milestones of the French R&D programme on future nuclear energy systems: a presidential declaration in January 2006, and a bill promulgated on June 28, 2006 on the sustainable management of radioactive materials and waste. Both events set the objective to build and start in 2020 a prototype of 4th generation reactor intended to proceed with demonstrations of sustainable fuel cycle (“produce less waste and make a better use of uranium resources”). The decision of this prototype made it necessary to update the R&D strategy approved in 2005 through a second meeting of the French “Nuclear Atomic Energy Committee” that took place on December 20, 2006. This meeting that gathered representatives of the ministries of Research and Industry, CEA and industrial partners (EDF, AREVA) led to sharpen the R&D strategy on future nuclear energy systems along the following lines: •

Identify the sodium fast reactor (SFR) as reference fast neutron technology and select by 2012 innovative design features and technologies to be demonstrated in a 300-600 MWe around 2020;

•

Conduct in parallel a R&D programme on the gas fast reactor (GFR) with a major focus on key issues for the feasibility and performance of the concept (primarily a robust ceramic fuel and a safe management of depressurization accidents) with a view to preparing eventually a decision around 2012 for the construction of a 50-100 MWth experimental reactor abroad that could demonstrate around 2020 key principles and technologies of the GFR (the Engineering & Technology Demonstration Reactor (ETDR));

•

Proceed with the development in parallel of advanced fuel cycles that may allow recycling minor actinides and may replace current spent fuel treatment services around 2040 when fast reactors are deployed to eventually improve the ultimate waste form and the resistance to proliferation. Pre-industrial demonstrations will include pilot plants on the site of La Hague in France around 2015 to manufacture driver and experimental MOX fuels with minor actinides to be tested in the prototype of sodium cooled fast reactor.

Supporting AREVA’s industrial projects in the field of high temperature nuclear technologies for the production of hydrogen, synthetic hydrocarbon fuels, or the delivery of process heat for the industry remains an unchanged goal of the French R&D program on future nuclear energy systems. Besides, work on more prospective reactor types such as Supercritical water reactors (SCWR) and Molten salt reactor (MSR) or molten salt coolant is so far limited to assessing feasibility and performance issues and advancing special basic key technologies.

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An optimization of the global program is sought through common R&D pathways such as advanced closed fuel cycles for both fast neutron systems (SFR and GFR) and high temperature helium system technologies (materials, thermal insulation, purification…) between both gas cooled systems (VHTR and GFR). 3. Overview of R&D Programmes on Future Reactor, Fuel and Fuel Cycle Technologies The R&D programmes have been structured and funded in accordance to the strategic orientations and milestones presented above. This section develops the main challenges to be resolved by R&D and presents salient results obtained over the two past years. 3.1. SODIUM-COOLED FAST REACTOR (SFR)

The Sodium cooled Fast Reactor (SFR) is the reference technology for fast reactors. It may be considered for industrial deployment around 2040 since Europe, in cooperation with Japan, Russia and the United States, has acquired important expertise in this reactor type. However, innovations are needed for a Generation IV sodium cooled fast reactor to compete with Generation III LWRs in economics and safety. This will require systems’ simplification to reduce investment cost, enhanced safety with improved prevention and management of severe accidents, improved operability (fuel handling, maintenance and repair) to achieve high capacity factors, and advanced closed fuel cycles with multiple recycling of actinides offering appropriate resistance to proliferation and optimized waste forms. Given the maturity of the technology, the prototype reactor planned in France for 2020 will in the range of 300 to 600 MWe to demonstrate the innovations selected in 2012 to upgrade sodium cooled fast reactor performance and to open the way to a “first of a kind” commercial reactor. 3.1.1. High performance core with enhanced safety characteristics The optimized design of the SFR core combines a number of criteria potentially hard to reconcile: prevention and control of accidents (favourable reactivity coefficients), minimum recourse to breeding blankets (resistance to proliferation) and eventual incorporation of minor actinides (sustainability). The optimization of core design leads to consider dense and high thermal conductivity fuels such as carbide in place of oxide fuels, as well as high performance cladding materials as ferritic-martensitic, oxide-dispersed strengthened steels.

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Preliminary studies resulted in the definition of a 3600 MWt (~1500 MWe) reference core (Figure 3) with improved characteristics compared with those of the European Fast Reactor project (terminated in 1998). The core is self-breeding without radial fertile blankets, with improved safety parameters (reduced voiding effect). This result has been obtained while minimizing the inter-pin space which implies to make use of a cladding material with excellent swelling properties under irradiation. 10.25 9.87 9.48 9.09 8.70 8.32 7.93 7.54 7.15 6.77 6.38 5.99 5.60 5.22 4.83 4.44

Figure 3. SFR reference core design 3600 MWt (1500 MWe).

3.1.2. Innovative system and power conversion unit The motivation for investigating alternative power conversion systems to the steam turbine is twofold :simplification and/or increased compactness of the general system architecture (intermediate circuit) and elimination of water-sodium chemical interaction. The suppression of the intermediate system and direct coupling of the primary circuit with a Brayton power conversion unit using nitrogen or supercritical CO2 requires the development of specific sodium-nitrogen or sodium-CO2 heat exchangers. The use of nitrogen (or mixture of nitrogen with helium) implies increasing the temperature by 50 to 100°C in order to preserve an acceptable conversion efficiency. Issues associated to the supercritical CO2 option are (1) consequences of a possible ingress of gas in the primary system, (2) control of the chemical reaction with sodium and (3) design and qualification of specific turbines. Double-walled steam generators and the use of coolants reasonably compatible with sodium and water, such as liquid salts or metallic eutectic, are also investigated. Current investigations proceed along two tracks: a loop-type concept without intermediate circuit and gas power conversion system, and a pooltype concept with a simplified and compact design of the intermediate circuit with an alternative coolant.

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3.1.3. Overall system and component integration The main options investigated are integrated into point designs of innovative concepts to assess the consistency of all considered options and to support preliminary safety and performance assessments. Figures 4a and 4b below show typical lay-outs of innovative loop-type and pool-type designs.

Figure 4. a) Large pool-type concept (1500 MWe) with design optimizations (reduced size of the vessel), b) Modular loop-type concept (500 MWe) with gas conversion system (no intermediate circuit).

3.2. GAS-COOLED FAST REACTOR (GFR)

The helium cooled fast reactor is an innovative nuclear system with such attractive features as a chemically inert and optically transparent coolant, as well as a quasi-decoupling of the reactor physics from the state of the coolant. Other advantages of the GFR relate to its promise as a very/high temperature reactor (V/HTR) capable of producing hydrogen, synthetic fuels and process heat. On the downside, since gas is a poorer coolant than liquid metals, key aspects demonstrating the viability of the GFR include development of a refractory and dense fuel, and robust management of accidental transients, especially cooling accidents. Furthermore, the GFR has the potential to operate at high temperature (at least 850°C), which enables in principle the production of hydrogen or synthetic hydrocarbon fuels in a sustainable manner. The feasibility of the GFR is essentially linked to two demonstrations: the mastery (fabrication, thermo-mechanical behaviour) of a high fissile content refractory fuel, and the implementation of appropriate safety systems for the prevention and a robust mitigation of accidental scenarios (especially depressurization). Because there is no experience available on the GFR, a first step for demonstrating its feasibility is the operation of a

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50-100 MWth experimental reactor, the ETDR, to qualify its specific fuel, materials and operating principles. Ideally, R&D results expected by 201215 could support a decision to construct the ETDR, possibly as a European Joint Undertaking. The next step would be a prototype GFR that could come 10-15 years after. 3.2.1. A refractory fuel concept for the GFR The GFR fuel should comply with: •

an operating temperature of 1200°C in normal conditions and 1600°C in accidental conditions (to offset the gas poor efficiency as coolant);

•

a high fissile atom density and high thermal conductivity, thus triggering a renewed interest for carbide or nitride fuels;

•

a power density in the range of 100 MW/m3 as a trade-off between minimizing the plutonium content (lower boundary) and safety (slowdown of adiabatic heat-up).

Attempts to transpose attractive features of V/HTR fuel particles to fast neutron cores (fission product confinement, very high temperature resistance, thermal conductivity…) remained unsuccessful so far. Preliminary studies finally led to a reference fuel element consisting of fuel pellets arranged in cells within a ceramics clad plate (Figure 5).

Figure 5. Cellular fuel sub-assembly with composite cladding material (SiC-SiCf).

Each cell contains a fuel pellet composed of mixed uranium, plutonium and minor actinides. The cladding is made of composite silicon carbide reinforced with SiC fibres (SiC-SiCf) for an increased mechanical resistance. The sub-assembly is composed of a stack of such plates axially piled up in a triangular array and enclosed in a hexagonal wrapper. Significant progress has been made recently about the selection of constitutive materials(cladding, liner) to ensure leak-tightness to fission products and comply with requirements of thermo-mechanical integrity and adequate chemical compatibility between materials. An alternative design based on ceramics clad fuel pins is also investigated thoroughly.

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3.2.2. GFR design and safety studies The GFR reference design features a reactor of 2400 MWth unit power. Feasibility studies demonstrated the potential of high power concepts for increased margins on the fuel design features and for the optimization of both safety and economy. Indeed, larger core sizes afford to more easily accommodate lower fractions of actinide compound in the fuel assembly and do not emphasize threshold effects in the safety systems needed to safely manage accidental conditions. Initially envisioned with a direct cycle, the GFR reference design moved to the indirect-combined cycle option. The removal of residual power in case of accidental LOCA conditions is assured by gas circulating in forced convection for the short term (actuation of active systems) and in natural circulation for mid and long term. A guard containment is necessary to maintain the 5-10 bars pressure required for an efficient core cooling in natural circulation (coherent with the use of metallic guard containment). 3.2.3. Definition of the characteristics of GFR-type experimental reactor The ETDR is a small power experimental reactor (50 MWth) designed to be the pathway for direct qualification of GFR fuels (Figure 6), materials and other key specific technologies, as well as for testing GFR specific safety principles and associated systems.

Figure 6. ETDR system layout (50 MWt).

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Because of the very innovative character of the fuel type selected as a reference for the GFR, an evolutionary approach was advocated, involving two successive core configurations: a start-up core based on conventional pin sub-assembly with metallic cladding (ODS), then a demonstration core using the prototypic GFR fuel element (plate type (U,Pu)C-SiC subassembly). An intermediate stage is planned for preliminary testing of a few prototypic sub-assemblies inserted in the pin-type core. It was concluded that the power density for the demonstration core should be similar or slightly above that of the GFR. A 50 MWth reactor with a power density of 100 MW/m3 is currently retained as an acceptable trade-off between irradiation performances and economy. 3.3. R&D PROGRAMMES ON FUEL AND FUEL RECYCLING

The development and qualification of an appropriate and consistent set of technologies for spent fuel treatment and fuel re-fabrication is an essential component of a fast neutron system strategy. The head-end processes, aiming at dissolving the spent fuel refabricating fuel to be recycled are specific to the type of fuel (MOX fuel, carbide…). In return, the separation process is largely independent of the type of fuel (SFR or GFR) but it is rather determined by options selected for the recycling of plutonium and minor actinides: •

The COEX process, derived from PUREX, is able to manage uranium and plutonium together, but does not partition minor actinides which therefore are co-managed with fission products to constitute vitrified ultimate waste;

•

DIAMEX and SANEX processes which have been demonstrated in 2005 at laboratory scale to partition lanthanides and actinides in two steps can be combined with COEX to manage separately minor actinides for recycling eventually in breeding blankets or transmutation targets;

•

Based on above advances, the GANEX process has been developed to achieve a grouped extraction of actinides and enable a co-management of all actinides together (uranium, plutonium and minor actinides) so as to achieve a homogeneous recycling in core subassemblies with the intent to provide an improved resistance to proliferation.

Considering the time scale for industrial deployment of these various separation processes, it is important to keep all options open, as they could be deployed in sequence in accordance with international standards of non proliferation and with a goal of global optimization of the fuel cycle backend from the technical and economic viewpoints.

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As far as spent fuel treatment is concerned, the hydrometallurgy remains the reference technology. Pyrochemistry is an alternative option to be more deeply investigated, especially for metallic fuels. 3.3.1. Minor actinide recycling As stated above, two modes of minor actinide recycling strategies in fast neutrons reactors are being investigated (Figures 7a and 7b): homogeneous recycling in fuel subassemblies of the active core together with uranium and plutonium, and heterogeneous recycling in blankets at the core periphery.

Figure 7. a) Homogeneous recycling (Generation IV reference), b) Heterogeneous recycling minor actinides bearing blankets at the core periphery.

Important results have been obtained since 2005 on separation and fuel re-fabrication techniques. The potential of different processes for individual minor actinide partitioning (DIAMEX, SANEX) has been confirmed and a first test of oxalic co-conversion of actinides (uranium, plutonium, neptunium, americium) has been successfully realized. Exploratory experiments have also been conducted using the sol-gel technique, alternative to the oxalic technique, for the co-conversion of U-Pu (Figures 8a and 8b).

Figure 8. a) Microspheres of hydroxides U(VI)-Pu(IV) (85/15), b) Microspheres of hydroxides U(IV)-Pu(III) (85/15).

3.3.2. Fabrication of co-precipitated MOX fuel UPuO2 powders obtained by oxalic co-precipitation (COEX process) have been used for the fabrication of first samples of MOX fuel. Two tracks are being investigated:

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•

Fabrication from powder initially at the targeted plutonium content,

•

Fabrication from high plutonium content powder (+/– 45%) diluted with UO2 for adjustment to the aimed plutonium content.

Figures 9a, 9b and 9c compares the microstructure of MOX fuels obtained by co-precipitation and by more conventional fabrication processes (MIMAS, COCA).

Figure 9. Comparison of MOX fuel microstructures obtained by co-precipitation (first track) and more conventional fabrication processes (MIMA, COCA): a) UPuO2 MIMAS 11%Pu, b) (U, Pu)O2 6% Pu (COCA), c) (U, Pu)O2 27.5%Pu (co-precipitation) .

This shows the very good homogeneity of plutonium distribution in the co-precipitated MOX fuel. These results look promising in terms of thermomechanical behaviour and capability for fission production retention in the microstructure. The behaviour of MOX fuel manufactured from COEX separation will be tested under irradiation through a dedicated irradiation experiment in PHENIX beginning in 2008. 3.4. VERY HIGH TEMPERATURE REACTOR (V/HTR)

The HTR/VHTR (V/HTR) system corresponds to a thermal neutron heliumcooled reactor concept operated at high (850°C) or very high temperature (>950°C). AREVA currently develops the project ANTARES of multipurpose High Temperature Reactor (600 MWt, 850°C at core outlet) that delivers process heat through an intermediate heat exchanger (IHX). Electricity can be produced with a combined power conversion cycle that consists in a Brayton cycle (nitrogen/helium mixture 80/20) coupled to a classical Rankine cycle with steam superheat. This option takes the best benefit from a gas cycle in the high temperature range (850°C) and steam cycle below 550°C, with a potential for high efficiency: 47% or more. Essential R&D issues for V/HTR concepts are: •

The qualification of materials and components capable to sustain the service conditions of high temperature helium systems,

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•

The development of compact gas-gas heat exchangers with high thermal efficiency,

•

The mastering of the fabrication procedure of particle fuel.

Major R&D milestones to support a V/HTR prototype reactor around 2020 are a sufficient confirmation by 2012 of design features and key technologies to document a Preliminary Safety Analysis Report and a full qualification by 2015 to ground a decision to build the facility. 3.4.1. High temperature materials Typical V/HTR operating conditions require materials capable of sustaining service conditions in the temperature range of 350 to 1650°C: •

(~350-550°C) for the reactor pressure vessel and internals. Ferriticmartensitic steel 9Cr-1Mo (T91) that is selected as reference material has been qualified for thick wall welding. Further developments on T91 relate to additional qualification in areas such as welding properties, mechanical characteristics, emissivity, corrosion behaviour and irradiation effects;

•

(~650-950°C) for the materials of the primary circuit, heat exchangers and turbine. The candidate materials (Inconel 617 and Haynes 230) are being characterized and compared in terms of mechanical behaviour, ageing, weld-ability, corrosion with impure helium in order to select the reference material for the IHX in 2008. Besides, investigations are conducted on Fe-base and Ni-base ODS (Oxide Dispersed Strengthened) alloys, in order to develop high performance grades for very high temperature applications (up to 950°C);

•

(~950-1650°C) for core structural materials. The qualification of graphite grades for V/HTRs and that of composite ceramics (Cf/C, SiCf/SiC) for the VHTR and GFR core materials goes through the successive steps of screening tests and characterization, before and after irradiation.

3.4.2. Gas-gas heat exchangers A major technical issue is the development of the Intermediate Heat eXchanger (IHX) for an indirect power conversion into electricity and other energy products, and eventually also the recuperator for a direct cycle Brayton electricity generation (Figure 10). For a power conversion system that uses gas as working fluid, the challenge is to transfer several hundreds of megawatts from one gas to another (He/He or He/He+N2) while minimizing the temperature difference so as to optimize the power conversion efficiency. A heat transfer area of several 10,000s m2 is required while

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keeping outside dimensions compatible with an acceptable overall size for the reactor building. Among potential advanced technologies, plate concepts appear more promising because of their compactness. Their thermal performance is also better qualified owing to prior use in other industrial applications. Different plate concepts appear as good candidate technologies: Printed Circuit Heat Exchangers (PCHE), Plate Machined Heat Exchanger (PMHE), Plate Fin Heat Exchangers (PFHE), and Plate Stamped Heat Exchanger (PSHE). Each concept is being tested and assessed in terms of fabricability and mechanical resistance in normal and accidental conditions. These concepts are being currently investigated and compared at the scale of mock-ups in terms of thermal performances, manufacturing techniques and predicted lifetime, so as to select the most suitable technology by 2010.

Figure 10. AREVA’s ANTARES Project of multipurpose of high temperature heat source.

Molten salts are being assessed as another candidate coolant to transfer heat from the reactor to high temperature applications. 3.4.3. V/HTR TRISO fuel particle The V/HTR safety approach and subsequent design options are essentially based on the fact that fission products remain confined inside the fuel

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particle in normal operating conditions and accidental situations. In order to demonstrate adequate performances in this respect, it is necessary: •

To establish an industrial manufacturing process capable to produce fuel particles with very high quality (i.e. failure rate of particles below 10–5);

•

To qualify experimentally the behaviour of such particles in irradiation and temperature conditions relevant to HTR normal operation (1300°C) and accidental situations (1600°C) with a failure rate below 1.5 10–4.

These objectives are covered by a comprehensive program on the fabrication and characterization (before and after irradiation) of standard TRISO (UO2/SiC) particles. Uranium particle fuel is currently fabricated at CEA by the GAIA laboratory scale fabrication line (Sol-Gel and CVD processes) that was put in service in mid 2005 (Figure 11). The production has reached a sufficient quality today (diameter, sphericity) to prepare irradiation tests in the materials testing reactor OSIRIS that should begin in 2009.

Figure 11. TRISO fuel particles (natural uranium) fabricated by the GAIA facility.

A first batch of coated particles is being used for the fabrication of fuel « compacts » that will be loaded in the experimental reactor OSIRIS at the beginning of 2009. For the objective of very high temperature (>850°C), the R&D program on fuel is focused on innovative particle fuel forms that would allow an enhanced burn-up performance (20% FIMA) and a higher temperature of helium (up to 1000°C). UCO/SiC concept limiting CO production and internal gas pressure at high burn-up, or UO2/ZrC accommodating higher temperature in normal and accidental conditions, together with adequate techniques for the treatment of spent fuel of these types.

GEN. IV SYSTEMS AND MATERIALS CHALLENGES

41

4. Transition From Light W ater Reactors to Fast Reactors The path for Europe towards a closed fuel cycle depends on political, technical and financial contingencies. Indeed, such an evolution implies: •

Integration within the broader worldwide policy of safeguarding and proliferation resistance,

•

The renewal and addition of new plants for processing spent fuel and re-fabricating fuel to be recycled with advanced processes,

•

A funding process for the new investments required, especially in fuel cycle plants.

Figure 12 illustrates the transition from current reactors and a spent fuel treatment plant in France to Generation IV technologies around 2040. It shows how deployment of a progressive separation and recycling strategy allows the technical capabilities of fast neutron reactors and advanced recycling modes with a co-management of actinides to be implemented around 2040. This strategy is flexible enough to be adapted to the time line and type of fast neutron systems to be developed. Furthermore, it can preserve the possibility (if ever compatible with the technical and economic optimisation of the fuel cycle) of an integral recycling of actinides (Figure 13) capable of drastically reducing long term potential radiotoxicity and decay heat of the ultimate waste (Figure 14), thus eventually making the fuel cycle more resistant to proliferation.

Figure 12. Scenario of renewal of French nuclear plants and fuel cycle plants: a) 2040 – Deployment of Fast neutron systems (SFR or GFR), b) 2040 – Renewed spent fuel treatment plant at La Hague (grouped actinide extraction).

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Figure 13. Integral recycling of actinides (U-Pu + minor actinides) with Unat or Udep as make-up fuel.

Figure 14. Evolution of radiotoxicity of ultimate waste for direct disposal, recycling of U-Pu only, and integral recycling of actinides.

5. Major Stakes in International Collaboration The Generation IV International Forum that was launched by the UD-DOE in 2000 [6] and the initiative INPRO [7] under the auspices of the IAEA constitute today the most active multilateral framework of collaboration on 4th generation nuclear systems. In a context where the European contribution to the international R&D effort on future nuclear energy systems is not at the level that nuclear power currently represents for Europe, good prospects exist today to improve this unsatisfactory situation.

GEN. IV SYSTEMS AND MATERIALS CHALLENGES

43

5.1. STATUS OF 4TH GENERATION NUCLEAR SYSTEMS IN THE EUROPEAN R&D PROGRAMME

Diverging national visions of nuclear energy in Europe have limited work on future fission systems so far, although there is a visible and reasonably funded R&D programme on safety, waste management and radioprotection of current power reactors within the Euratom R&D framework programme. Strengthening the effort in Europe on future nuclear fission systems is in full agreement with Euratom formally joining the Generation IV Forum on May 11, 2006 and with the orientation of both Green Books issued by the European Commission in 2000 and 2006 for sustainable energy development in Europe [8, 9, 10 and 11]. Initiatives in this sense are all the more urgently needed in the 7th R&D Framework Programme (2007-2011). So far, the European R&D Framework Programme, together with national initiatives, has been successful in preserving competence in the nuclear field through a network of excellence, cooperative actions and by promoting the European dimension of education in nuclear engineering. Future nuclear energy systems will require all aspects of expertise involved in the design, technology development and safety demonstrations for light water reactors, nuclear fuels and fuel cycle processes including design and safety studies, fuel, materials and component technology, and fuel cycle and waste management processes and technology. Beyond this, the search for breakthroughs beyond LWR technologies, such as very high temperature and/or fast neutron systems with full actinide recycle, will require additional and non-nuclear-specific skills in materials science, very high temperature materials and components (composite C/C and ceramics), separation chemistry, and thermo-chemical and electrochemical processes for water splitting and hydrogen production. Other competence is required, either for prospective studies (uranium resources, hydrogen market) or assessments in economics, safety or proliferation resistance. Updating assessment methods may be effected nationally, through collaboration or through participation in INPRO and/or corresponding methodological groups of the Generation IV Forum (Economics modelling, Risk and safety, Proliferation resistance and Physical protection). Development of networks of experts in these fields in Europe may be considered. 5.2. RENEWAL AND EVOLUTION OF LARGE R&D FACILITIES IN EUROPE

Materials testing reactors and hot laboratories are essential R&D infrastructures to explore innovative research on fuels and fuel cycles, key technologies for 4th generation reactors:

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•

HFR, OSIRIS, LVR-15 and Jules Horowitz Reactor (JHR) in 2014…

•

ITU, ATALANTE, FZK, SCK, ACTINET core group facilities…

•

Bor-60, PHENIX.

Sizeable non-nuclear facilities are also needed to resume R&D on high temperature gas-cooled reactors, such as the particle fuel laboratory scale fabrication line, test benches for high temperature helium system technology (850, 950°C and above) and experimental loops in the 1 to 10 MW range for component mock-up tests. France is engaged in re-establishing such a basic R&D capability. Equipment costs for this effort on a European or a broader international basis are estimated to be about 100-300 M€. The cost for R&D equipment to resume the industrial development of sodiumcooled fast reactors is about the same. The decision to build a prototype of 4th generation fast neutron reactors in France by 2020 is important for Europe to remain credibly involved in R&D on fast neutron systems after PHENIX is shut-down in 2009. Other European countries, and possibly other international partners, may decide to develop experimental or prototype reactors of other Generation IV systems of specific interest, with invited external participation. Even though the costs of such facilities range from 0.5 to 2 B€, the prospect of such prototypes does not appear excessively ambitious in comparison with the number of experimental and prototype reactors operating in Europe in the 1980s: •

Sodium cooled fast reactors: – The experimental reactor KNK II (17 MWe) in Germany (1978-1991), – The prototypes PHENIX (250 MWe) in France (1973-2009), PFR (234 MWe) in the United Kingdom (1975-1994), BN600 in Russia (1980 Æ),

•

High temperature reactors (HTRs): – The experimental reactors Dragon (1964-1975) in the United Kingdom and AVR (13 MWe) in Germany (1966-1988), – The prototype THTR (300 MWe) in Germany (1983-1989).

5.3. TOWARDS A “SUSTAINABLE NUCLEAR FISSION TECHNOLOGY PLATFORM”

One of the major initiatives to integrate and strengthen R&D work on future nuclear energy systems in Europe is the proposal to organize fissionoriented R&D work in the European Union into a “Sustainable Nuclear Fission Technology Platform” (Figure 15). This is intended to address strategic R&D for European policy makers and industrial projects in the

GEN. IV SYSTEMS AND MATERIALS CHALLENGES

45

medium term such as LWR safe operation and life extension, fast neutron reactors with a closed fuel cycle, and high temperature reactors for cogeneration. Such an R&D organization on nuclear fission in Europe would help: •

directing R&D towards strategic goals,

•

identifying large experimental equipment for this research, so as to plan for investment or refurbishing consistent with the scale of Europe, and also

•

preparing decisions to realize and operate prototypes of 4th generation reactors within the framework of joint ventures.

It would also facilitate developing synergistic R&D with that for fusion systems already organized internationally. An appropriate level of R&D funding for 4th generation systems in Europe, should be comparable to the US and Japan in the same area, currently 300 MUS$/year exclusive of investments in experimental and prototype reactors. At stake are preserving the advance of the European nuclear industry in the world, preparing the deployment of sustainable fast neutron systems with advanced recycling modes in Europe by 2040, and promoting the development of key technologies for non-electricity applications of nuclear energy. Nuclear Fission Technology Platform: Operation LWR Safety & Economics

SRA and Platform

Innovative Materials and Fuels

V/HTR Process Heat, Electricity & H2

Simulations and Experiments: Reactor Design, Safety, Materials and Fuels Strategic Research Agenda; Platform Operation

TSO: Mirror Group

Training and R&D Infrastructures

Fast Systems With Closed Fuel Cycles Sustainability

EU High Level Group on Nuclear Safety & Security

Geological Disposal Technologies, design, safety assessment

Waste Management (CARD)

Figure 15. Goals and components of the “Sustainable Nuclear Fission” Technology Platform.

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6. Future Prospects Prospective studies carried out by the CEA and industrial partners led to elaborate for France a R&D strategy to support the development of nuclear reactors for the continuous and sufficient supply of energy over the 21st century. To prepare the deployment of fast neutron systems in the French reactor fleet around 2040, CEA and its industrial partners AREVA and EDF have promoted a dual approach including the search of innovations for the SFR and in parallel the evaluation of the GFR promising potentialities. Closely linked to R&D on reactor concepts, a consistent programme on fuel cycle technologies is also conducted with the objective to achieve significant progress on the nature of final waste and on proliferation resistance towards an optimized management and recycling of minor actinides. The overall R&D program is oriented towards the construction of a prototype of 4th generation reactor in 2020, subsequently to the selection of the main design options in 2012. The 4th generation nuclear energy systems also support the goal of generating other energy products than electricity such as hydrogen and synthetic hydrocarbon, or process heat for industry. In this respect, CEA supports AREVA’s ANTARES project of 600 MWth multipurpose high temperature nuclear heat source (850°C). These fast neutron and high temperature reactors both require breakthroughs beyond 2nd and 3rd generation light water reactors (hence recognition of the switch to a new generation). They pose real technological challenges for nuclear fuels, systems materials and technology, spent fuel treatment processes and non-conventional applications. Their application in recycling spent fuel for efficient use of the uranium and burning long-lived radioactive waste, leads to consideration of these 4th generation reactors as “nuclear systems” consisting of reactor, fuel and fuel cycle, optimised as a whole. Such technology challenges require cooperation among European research partners (National Laboratories, Universities and other research organizations) and industrial partners on corresponding R&D objectives. They also require development through international cooperation, to share the cost of innovation, experimental reactors and prototypes in Europe. Securing sustainable electricity generation in the second half of the century suggests the need for possible deployment of at least one type of fast neutron reactor in Europe around 2040. This in turn suggests work along two complementary lines of research: (1) innovation to develop a new generation of sodium cooled reactors, already a mature technology, and (2) diversifying risks and market opportunities by developing at least one other

GEN. IV SYSTEMS AND MATERIALS CHALLENGES

47

reactor type such as the gas cooled fast reactor with properties inverse to sodium cooled systems or lead cooled fast reactors. The French decision to build a prototype of sodium cooled fast reactor by 2020 as a successor to PHENIX, could permit a comparative evaluation of the merits of alternative fast reactor types to select a second technology and build an experimental test reactor in another European country. With a core outlet temperature of at least 850°C, the gas fast reactor concept could also represent a nuclear technology useful for high temperature applications. This stresses the need to fully integrate R&D work on future nuclear energy systems at the European level. This makes sense in view of the important share of nuclear electricity in Europe (32%) and of the fact that Euratom affiliated countries joined the Generation IV International Forum in 2006. Even though some European countries still maintain a nuclear moratorium, the preparation of the 7th European R&D Framework Program (2007-2011) offers prospects for strengthening work in this field. This could increase exchanges with the Generation IV International Forum, an essential condition to achieve balanced cooperation with major nuclear partners such as the United States and Japan, which spend about 300 MUS$/year each on future nuclear systems, and also with Russia and China. Last but not least, another goal for European stakeholders of nuclear research and industry, is to become sufficiently involved in international R&D on future nuclear energy systems to benefit from past experience in precursor reactors of Generation IV technologies (principally sodium fast and high temperature reactors), to keep current with advances in technologies such as sodium cooled fast reactors and fuel cycle processes, and ultimately to become involved in development of standards and commercial technologies strategic for Europe and international markets. Overcoming the current limitations of the European R&D Framework Program on Fission and maximizing European contributions to international R&D on advanced nuclear technologies, requires an integrated organization gathering research laboratories and industry. It is essential to identify and set strategic priorities on R&D needs and necessary competence, to define needs and elaborate plans for new large experimental facilities such as material testing reactors, hot laboratories, large experimental loops, and to take decisions for the construction of experimental or prototype reactors within the joint undertakings. Organizing R&D on nuclear fission in Europe along these lines would not only help direct R&D towards strategic goals and make Europe a major partner of international collaboration, but would also increase the European potential to profit from building and operating prototypes of 4th generation reactors. Such a strategy would offer the best prospects for European stakeholders in nuclear energy to preserve their current leadership.

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References [1] Global Energy Perspectives to 2050 and beyond, 1998, Report of the World Energy Council with scenario studies with the International Institute for Applied Systems Analysis, Cambridge University Press. [2] Energy to 2050, Scenario for a Sustainable Future, 2003, Report of the International Energy Agency. [3] Uranium 2005: Resources, Production and Demand, 2006, Joint report by the OECD/NEA and the IAEA Nuclear Energy Agency homepage http://www.nea.fr. [4] Fast Reactors R&D Strategy in France for a Sustainable Energy Supply and Reduction of Environmental Burdens, 2005, Carré, F., JAIF International Symposium, Tokyo, March 24. [5] ANTARES, 2006, Ready for the Combined Heat and Power Market, Gauthier, J. C., Lecomte, M., Carré, F., and Renault, C., HTR2006, Sandton, Republic of South Africa, October 2-4. [6] A Technology Roadmap for Generation IV Nuclear Energy Systems, Dec. 2002, GIF002-00, Issued by the U.S.DOE Nuclear Energy Research Advisory Committee and the Generation IV International Forum - http://gif.inel.gov/roadmap/. [7] IAEA International Project on Innovative Nuclear Reactor (INPRO), 2006, Final report of phase 1. [8] Towards a European Strategy of Energy Security, Nov. 2000, European Commission, COM (2000) 769. [9] Green paper on “A European Strategy for Sustainable, Competitive and Secure Energy”, March 2006, Commission of the European Community, COM (2006) 105. [10] An Energy Policy for Europe, 2007, Communication from the Commission to the European Council and the European Parliament. [11] The Role of Nuclear Power in Europe, Jan. 2007, World Energy Council.

FUNDAMENTALS OF NEUTRONICS: REACTIVITY COEFFICIENTS IN NUCLEAR REACTORS P. REUSS* Emeritus Professor Institut National des Sciences et Techniques Nucléaires, Point courrier 35, CEA/Saclay, 91191 GIF-SUR-YVETTE CEDEX, France

Abstract: The objective of this lecture is to discuss some aspects of importance for the safety performance of a nuclear fission system, related to the production of energy, the incineration of nuclear wastes and/or the fuel reprocessing. This presentation will be first devoted to the analysis of the basic principles of the fission-neutron chain reaction behaviour. The design parameters will then be introduced, and some exemplary systems described. Particularly, the aspects concerning, from the neutronics point of view, the choice of the materials (theme of this autumn school) will be emphasized: main options for the core design, reactivity effects, conditions for a working stability, neutron utilization. The coupling between the neutronics and the other branches of physics involved in the reactor design (thermal-hydraulics, irradiation of materials, safety...) will be recalled in the conclusions.

Foreword The present concerns of the public opinion which more and more emerge in our word have brought a new interest in nuclear fission energy, as this source appears to be safe, competitive and almost without any emission of carbon dioxide [3]. However, it also appears that the known reserves of uranium will be rapidly exhausted, with the present mode of utilization of uranium using merely the isotope 235 which represent only 0.72 % of the natural uranium. It is the reason why the specialists of numerous nuclear laboratories have studied, since one or two decades, new concepts of fission reactors (said “forth generation systems”), safer, more competitive, and above all more economical in natural resources [1]. The Generation 4 Forum compared the various proposals and selected the six most promising ones. The young scientists and engineers will now have to develop these projects: the objective of this autumn school is to contribute to the training they will need to carry out this important mission.

______ *

[email protected]

V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

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P. REUSS

The topics of this school concern the materials which will be needed for the forth generation reactors. The choices must be compatible with the criteria of the neutron physics (or neutronics): it is the reason why some basic knowledge of this physics, specially about the reactivity effect, is necessary. The present paper is an introduction to neutronics through analyses of safety coefficients of PWRs. 1. Fission Chain Reaction and Reactivity [5, 6, 7, 8, 9, 10, 17] The fission chain reaction in the nuclear reactors is based on the facts that fissions yields neutrons and that neutrons can induce new fissions, specially of fissile material: one natural nuclide, uranium 235, and some artificial nuclides as plutonium 239, plutonium 241, uranium 233. The neutrons are emitted by fission around 2 MeV (about 20 000 km/s). At this energy, the cross-sections are small. Then it is necessary to use a material quite rich in fissile atoms in order to reduce the number a sterile captures by other materials. This is the way of the “atomic bombs” and of the “fast (neutron) reactors”; these last ones are characterized by a good neutron balance which allows, for instance, some breeding. A second way takes advantage of the very large cross-sections of the fissile nuclides for the slow neutrons. Then, in addition to the fuel, another material is used, called “moderator”, to slow down the neutrons by successive scatterings until they come close to the thermal equilibrium at an energy of a few tenths of eV and a velocity of a few km/s. For these slow neutrons, it is possible to get a chain reaction with a much less enriched fuel, and even with natural uranium if one of the best moderators is used (heavy water, graphite, beryllium or beryllia [beryllium oxide]). Most of the world power reactors are such “thermal (neutron) reactors” and, among them, the water reactors (PWR and VVER, BWR) using water as moderator; because of a not negligible capture cross-section of hydrogen for the thermal neutrons, the fuel of these systems must then be enriched at least to a few percents in fissile nuclei. The equilibrium or imbalance of a chain reaction is characterized by the multiplication factor k defined as the mean ratio between the number a neutrons of a given generation and the number of neutrons at the preceding generation or, equivalently, the number a fissions at a given step and the number of fissions at the preceding step. Indeed, these numbers evolve like N(0) kn with the generation n: •

if k > 1, these numbers increase when time goes on: the system is overcritical;

•

if k < 1, these numbers decrease: the system is under-critical;

FUNDAMENTALS OF NEUTRONICS •

51

if k =1, these numbers remain constant: the system is critical: its power (proportional to the number of fissions per unit of time, or to the number of fissions per generation) is constant.

In its nominal state, a reactor must therefore be critical; to start it or to increase its power a transient (small) over-criticality is created; to reduce the power or to stop the system, a transient, resp. permanent, undercriticality is created. (In a nuclear weapon, a larger as possible overcriticality is looked for.) The multiplication factor k is the product π ν of the probability π that a neutron emitted by fission induces a new fission and the mean number ν of new neutrons that this fission will yield. This last number is a nuclear characteristic. To adjust k, π must therefore be adjusted: in practice, some movement of material allows to modify the respective probability of fission and (sterile) capture; generally, control rods are used: they are made with a highly capturing material (boron, cadmium, etc.): when inserted, the captures increase and k decreases, and conversely when extracted. Instead of k, the reactivity ρ is often used:

ρ = (k −1) k It allows to characterize the criticality by 0 instead of 1; as ρ is small in practice, it is generally expressed in pcm (p.c.m: pour cent-mille, i.e. 10–5). A simple modelling of the reactor, shows that its power varies like eωt or a linear combination of such functions [11, 12]. The dominant mode is characterized by a value of ω of the same sign as ρ , i.e. increasing if ρ > 0, decreasing if ρ < 0 and constant if ρ = 0. The other values of ω , always negative, correspond to transients due to the several groups of delayed neutrons (neutrons emitted a few seconds after the fission and a preliminary beta decay of a “precursor”; average delay: about 10 seconds). The key parameter for the kinetic behaviour is the proportion β of the delayed neutrons, for instance: • •

β = 680 pcm for uranium 235, β = 220 pcm for plutonium 239.

If the reactivity is less than this proportion, the kinetics remains slow and quite manageable because the delayed neutrons play the major role despite their small proportion (the life of the “prompt” neutrons is very short, because they travel rapidly through the matter: between 10−7 s and 10−4 s according to the type of the reactor). Note that a reactor using plutonium is a little more “nervous” than a reactor using uranium because β is smaller. If ρ exceeds β , the power increase very rapidly (prompt critical mode). Fortunately, in practice and for all the types of reactors, temperature feedbacks stop the explosion.

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2. Analysis of a Neutron Balance These orders of magnitude show that a precise analysis of the multiplication factor is necessary. It is the reason why it is useful to present the neutron balance obtained in practice by a computer code in a form allowing the physical analyses. The most famous and usual presentation of such a neutron balance is due to Enrich Fermi himself, who elaborated the basic neutron physics during the late 1930’s and early 1940’s. At that time, the formulae below were used to calculate the multiplication factor. Nowadays, a precise spectrum and space calculation of the neutron tux is performed by a computer code (for instance, in France, by the CEA code APOLLO2) and the factors are reconstituted a posterior for a physical analysis. First of all, Fermi distinguished the reactions occurring in the neutron multiplying medium and the neutrons escaping the reactor (leakage):

k = k∞ PNL where k∞ would be the multiplication factor without leakage, that is to say, if the reactor would be innate, and PNL the non-leakage probability (Figure 1).

Figure 1. Variation of the “effective” multiplication factor k of a reactor (here spherical) as a function of its size, showing the critical size (depending on the used materials).

For a bare homogeneous reactor, this last one can be written:

PNL =

1 1 + M 2 B2

where M 2 is the sixth of the mean square distance travelled by the neutrons and measured as the crow flies, and B2 a geometrical parameter characterizing both the dimensions and the form of the reactor; for instance: • •

B 2 = π 2 R 2 for a sphere of radius R, B 2 = π 2 H 2 + j 2 R 2 for a cylinder of height H and radius R ( j = 2.405: first zero of the Bessel function J0),

•

B 2 = π 2 a 2 + π 2 b 2 + π 2 c 2 for a parallelepiped of edges a, b and c.

FUNDAMENTALS OF NEUTRONICS

53

According to the neutron histories in a thermal neutron system*, Fermi broke down k∞ into four factors:

k∞ = ε p f η The fast fission factor ε takes into account the rare fissions of non-fissile materials as uranium 238 induced by the very fast neutrons (typically of energy greater than 1 MeV) which produce few extra neutrons beyond the main production by the fissions of fissile material induced by thermal neutrons. For instance, for a PWR, ε is about 1.07. Here, we shall not comment further this corrective term. The resonance escape probability or antitrap factor p [13-16] is the probability for a neutron to escape the numerous resonances, specially of uranium 238, which risk to capture it without fission during its slowing down. Order of magnitude: 0.75 for a PWR. The classical formula giving this factor was established by Fermi:

⎡ V f N f I eff ⎤ p = exp ⎢− ⎥ ⎢⎣ Vm N m (ξ σ s )m ⎥⎦ where f and m refer to the fuel and the moderator respectively, and where V are the volumes, N the atomic or molecular concentrations, Ieff the “effective integral” of the fuel capture cross section (see Figures 2 and 3), σ S the moderator scattering cross-section and ξ m the mean gain of lethargy† during a neutron scattering by the moderator. The effective integral depends on a parameter called “equivalent dilution cross-section” which characterize the reactions other than the resonant captures in the fuel, and which can approximately be written:

σd =

1 (1− C)b l N f 1− C + C b

where l is the diameter of the fuel element. The Bell factor b (about 1.1 for uranium 238) is a correction due to the transposition from a homogeneous geometry to a heterogeneous one. The Dancoff factor (see Figure 4) is the probability for a neutron emerging from a fuel element to cross the

______

* At that time, Fermi prepared his famous experiment CP1 (Chicago pile number one) which used natural uranium and was moderated by graphite, and which reached criticality on December 2th, 1942. † In neutronics, we often use, instead of the neutron kinetic energy E, its “lethargy” u defined by u = – ln(E/Ereference); Ereference is chosen conveniently, e.g. 10 MeV.

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moderator without collision and, consequently, to re-enter a fuel element and to risk to be captured in a resonance. The smaller the lattice pitch (indicated next the curves), the greater the Dancoff effect, i.e. the self shielding of the resonances.

Figure 2. Zoom on the uranium 238 capture cross-section showing the numerous resonances in the epithermal energy range of the neutrons.

Figure 3. Effective integral of uranium 238; this integral increases if the fuel temperature increases thanks to the so called “Doppler effect” (see paragraph 3).

Figure 4. Dancoff’s factor for a square lattice of cylindrical fuel elements (clads neglected).

FUNDAMENTALS OF NEUTRONICS

55

The thermal utilization factor f is the probability for a neutron which escaped the “traps” (uranium 238 resonances) and became thermal to be absorbed into the fuel and not elsewhere in the reactor lattice (moderator, clads...). Considering, for instance these three materials (indices f, m and c), we can write:

f=

V f Σ a, f Φ f V f Σ a, f Φ f + Vm Σ a,m Φ m +Vc Σ a,cΦc

where the Φ are the values of the fluxes averaged over the thermal spectrum and over each volume respectively, the Σ the macroscopic crosssections, and the V the volumes. For a PWR, the flux tilts are small (a few percents less in the fuel than in the moderator); f is about 0.92 (without boron in the water). The reproduction factor η is the mean number of neutrons emitted by fission (induced by a thermal neutron) for one (thermal) neutron absorbed into the fuel:

η=

(ν Σ f ) f Σ a, f

(The fuel volume and the averaged flux in the fuel cancel between the numerator and the denominator.) For a PWR using enriched uranium, η is about 1.8: the product k∞ of the four factors is therefore about 1.3; such a high value is required to compensate the leakage (a few percents of the neutrons) and, above all, the large decrease of k∞ which will appear during irradiation due to the burning up of the fissile materials and to the capture by the non-fissile by-products, specially the fission products. At the end of its irradiation, the value of k∞ of a PWR assembly is less than one! Of course, the reactor always runs at criticality: that is possible because it contains a mixture of fresh and irradiated fuels*. This Fermi’s presentation of the neutron balance is quite convenient for the thermal neutron reactors; even for the PWR, is which the spectrum is rather “hard” (not well thermalized), this breakdown into four factors is relevant, and it appears that the elementary formulae we recalled can

______ *

Generally, for the PWRs, there are 3 or 4 batches of fuel elements. At each end of cycle (1 cycle: between one and two years) the reactor is stopped, the most irradiated batch is removed from the core and replaced with fresh fuel, and all the fuel elements are repositioned in the core in order to optimize the multiplication factor and the power distribution. At the beginning of the cycle, the over-criticality is compensated thanks to boron acid in solution in the primary water, and possibly burnable poisons; the end of cycle is at the moment when there is (almost) no boron in the primary circuit.

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produce good orders of magnitude of the “reactivity coefficients”, that is to say the (small) variations of the multiplication factor due to (small) variations of the state of the reactor (temperatures, concentrations...) or of the cross-sections. However, such a presentation cannot apply for fast reactors. It is the reason why other breackdowns, similar to the one of Enrico Fermi but usable for any type of reactor, have been proposed, for instance by Roger Naudet and by Alain Satamarina. We will present here below the decomposition used by the author of this paper (similar to Santamarina’s one). We can go on distinguishing the reactions in the core and the leakage, consequently writing:

k = k∞ PNL with the same approximate expression of the non-leakage probability PNL. But we cannot consider that the neutrons follow two successive distinct steps, slowing down, then thermalization. Therefore, is is preferable to analyse the reaction rates not according to the neutron energy range but according to the type of material liable to absorb the neutron. Here we shall propose to distinguish four types of materials: 1. the fissile* (heavy) nuclei, 2. the fertile† (heavy) nuclei, 3. the non-fissile-non-fertile heavy nuclei‡, 4. the other (non-fissile) nuclei§. Using the following notations**: S number of neutrons emitted by fission (source), A number of neutron absorptions, P number of neutron productions (resulting from these absorptions), L number of neutron leakages, and the indices of the nuclide types (no index means “total”), we can write:

______ *

Fissile: which can be fissioned by a neutron of any energy, even (and generally easily) by slow neutrons; e.g. uranium 235, plutonium 239, etc. † Fertile: which gives a fissile nucleus by a neutron capture; e.g. thorium 232, uranium 238, plutonium 240, etc. ‡ E.g. uranium 236, plutonium 242, etc. § E.g. moderator, coolant, clads, etc. ** The numbers of reactions can refer to any unit of time, as the normalization do not appear in the definitions. Very often, we take in the codes S =1.

FUNDAMENTALS OF NEUTRONICS

k=

57

P = k∞ PNL S

with:

PNL =

A A+ L

k∞ =

A+ L P P =ζ S A A

The factor ζ =(A + L)/S takes into account the extra neutrons produced by (n,2n) and (n,3n) reactions: because of these reactions, the number A + L of neutrons disappearing is greater than the number S of neutrons emitted (in practice of a few hundreds pcm). We propose to split the remaining fraction P/A of the expression of k∞ into five factors, in order to get the six factor formula:

k∞ = ζ ε pq f η with the following definitions:

ε=

P1+2+3 P , η= 1, P1 A1

p=

A1 A 1+2

, q=

A 1+2 A 1+2+3

,

f =

A 1+2+3 A 1+2+3+4

Note: if we introduce

η 1 = P1 A1 (= η), η 2 = P2 A 2 , η 3= P3 A 3 , we can write:

ε = 1+

(1− p) q η2 + (1− q) η3 pq η1

The last five factors have a physical meaning similar to the one they had in the Fermi breakdown: ε expresses the fissions of the secondary heavy nuclides (types 2 and 3) and η expresses the reproduction by the fissile nuclides (type 1); the factors p and q take into account the loss of neutrons due to the absorptions by the heavy nuclides which generally are not fissioned, respectively the nuclides of types 2 and 3; the factor f takes into account the loss a neutrons in the captures not in the fuel (type 4 materials). The factor η is the reproduction factor of the main fissionable material, i.e. η 1 of the type 1 material. In the following tables some examples are given. Table 1 gives the Fermi four factors for PWR assemblies: a uranium oxide assembly (uranium 235 enrichment: 3.7%) and a MOX assembly (6.5% of plutonium and 93.5% of depleted uranium), both at the beginning of life without leakage (wl) and the first one at the beginning (0 GWd/t) and the end of irradiation (46 GWd/t) with critical leakage (cl).

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P. REUSS

TABLE 1. The infinite multiplication factor and the four factors of its breakdown according to the conventions of Fermi for some typical PWR cases (500 ppm of B, Xe at equilibrium).

Case

ε p f

η k∞

UOX-0 1.0698 0.7621 0.8937 1.7490 1.2743

UOX-0-cl 1.0766 0.7537 0.8936 1.7479 1.2674

UOX-46-cl 1.1045 0.6878 0.8965 1.2846 0.8749

MOX-0-wl 1.0888 0.6382 0.9498 1.7239 1.1379

Table 2 gives the six factor proposed breakdown for the same PWR cases (the columns for the MOX cases with critical leakage both at the beginning and the end of irradiation have been added). TABLE 2. The infinite multiplication factor and the six proposed factors of its breakdown for the typical PWR cases (500 ppm of boron, xenon at equilibrium). Case

ζ ε p q f

η k∞

UOX-0-wl

UOX-0-cl

UOX-46-cl

MOX-0-wl

MOX-0-cl

MOX-46-cl

1.0018 1.0679 0.6748 1.0000 0.8928 1.9770 1.2743

1.0014 1.0752 0.6669 1.0000 0.8937 1.9751 1.2674

1.0020 1.1023 0.5373 0.9673 0.7949 1.9173 0.8749

1.0020 1.0867 0.5931 0.9719 0.9546 1.8989 1.1379

1.0017 1.0905 0.5908 0.9720 0.9547 1.8998 1.1378

1.0019 1.1065 0.5320 0.9511 0.8553 1.9428 0.9321

The neutron leakage in the water reactors occurs above all for the fast neutrons: therefore the effects on ε, p and q, but no effect on the Fermi f and η factors. The irradiation induces a very big decrease of k∞ , particularly for the UOX case; at the end of life, the media are clearly under-critical. The change of nuclear fuel gives a decrease of p and q due to the great capture by the 240 and 242 isotopes of plutonium, respectively. Because of the much greater thermal absorption of the MOX, the Fermi f is better for this case; but the Fermi η ’s are similar. Table 3 gives the six factor proposed breakdown for five examples of helium cooled fast neutron media [2] (preliminary design studies of one of the Generation IV concepts: helium cooled fast reactor), calculated with critical leakage. They differ by the cladding (C: graphite; S: SiC; Z: zirconium), the percentage of plutonium in the MOX fuel (first and second digits), the volume percentage of the MOX in the compact (third and fourth digits), and the volume percentage of helium in the cell (last digits). Both the cases with carbon differ by the plutonium contents: a greater contents improves ε because plutonium 240 is more easily fissioned than uranium 238, but, above all, the improve appears on p because uranium 238 almost disappeared; there is also an improve on η because of the spectrum

FUNDAMENTALS OF NEUTRONICS

59

hardening; the first case is largely under-critical. The comparison of the third and the fourth cases show an improve of ε and η for the zirconium case because of the spectrum hardening; but the main effect is the decrease of f because of the great capture by zirconium. The last two cases differs only by the helium contents, i.e. the migration area which is twice greater for the last case: the infinite multiplication factor are almost the same, but the critical dimensions would be very different. TABLE 3. The infinite multiplication factor and the six factors of the proposed breakdown for examples of helium cooled fast neutron media.

Case

ζ ε p q f

η k∞

C-25-12-65 1.0001 1.0314 0.4420 0.9596 0.9701 1.7776 0.7545

C-99-12-65 1.0000 1.0530 0.8019 0.9651 0.9744 1.8580 1.4754

S-25-60-40 1.0006 1.1523 0.5362 0.9814 0.9852 1.9589 1.1709

Z-25-60-40 1.0012 1.1837 0.5295 0.9812 0.8789 2.0715 1.1210

Z-25-60-40 1.0012 1.1752 0.5294 0.9813 0.8701 2.0547 1.0928

3. Reactor Operation The reactor operation is characterized by two types of reactivity effects: •

actions of the operator to start the system, modify the power, or stop the chain reaction; any term of the neutron balance (fission, capture, leakage) can a priori be used for such changes of reactivity; most generally, the modification of the capture is chosen, often thanks to the movement of “control rods” (or bundles), sometimes by other means as the introduction or dilution of some boric acid in the primary circuit of the PWRs;

•

spontaneous variations of the multiplication factor due to the chain reaction and the production of heat. These spontaneous effects are themselves of two types:

•

the effects resulting from changes of the composition of the materials, above all the fuel, due to the consumption and the transmutation of the heavy nuclides, and to the accumulation of fission products; these variations of reactivity are slow: the time constants can vary between a few hours for some fission products as the xenon 135 to years for other fission products and for the heavy nuclides;

•

the effects resulting from changes of the temperatures of the materials; these last ones are almost instantaneous (for the fuel temperature which is directly linked to the power production, i.e. the fission rate) or

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P. REUSS

characterized by times constants of a few seconds (coolant and moderator temperatures whose variations need some heat transfer from the fuel element). In any case, the time constants of the power variations are of the order of a few tens of second, as governed by the delayed neutrons. Here we shall comment only these temperature effects. First of all, do note that it is compulsory to design the system in such a way that the temperature coefficient* α = ∂k/∂T (where k is the multiplication factor and T the considered temperature) is negative: indeed, then the system is stable: if there is, for instance, an increase of power, there will be an increase of the temperature, then a decrease a the multiplication factor (if α is negative), and consequently an under-critical situation, and therefore a decrease of the power. Conversely, if α was positive, any perturbation would be amplified. For example, Figure 5 shows the curves giving the convergence of the power towards its equilibrium level (k = 1) from initial states characterized by an initial power greater or smaller than this equilibrium power.

Figure 5. Evolution of the power towards the equilibrium level from some non-equilibrium initial states, less or greater than the equilibrium level, and characterized by the values at time zero (instantaneous negative temperature effect; one group of delayed neutrons).

Let us considerer the example of the PWR (the physical effects are similar for the other types of reactors). Three temperature effects can be distinguished: The Doppler effect is linked to the fuel temperature: because of the thermal agitation, the resonances, as “seen” by the neutrons in the

•

______ *

Or, at least, the coeffcient of the most rapid effect (as we shall see, it is the effect linked to the fuel temperature, i.e. the Doppler effect).

FUNDAMENTALS OF NEUTRONICS

61

laboratory (reactor) system, broaden and their peaks decrease, the integral remaining unchanged (Figure 6); but, because of the “self-shielding effect”, the associated absorption (that is to say the effective integral in the formula of the factor p) increases (see Figure 3). The main Doppler effect occurs on the uranium 238 captures and consequently decreases the factors p, and therefore k. So this effect, though not very great in absolute value ( ≅ –2 pcm/°C), is the main contribution to the reactor stability, and consequently safety in case of reactivity accident.

Figure 6. Broadening of the main uranium 238 resonance because of the Doppler effect. •

The thermal spectrum effect is linked to the temperature of the material thermalizing the neutrons, i.e. the moderator: if its temperature increases, the thermal agitation of its atoms increases, and so do the neutron spectrum which consequently becomes “harder”. Because several of the most important cross-sections do not vary exactly as the inverse of the velocity v of the neutrons* (Figure 7), some variations of the factors f and η occur, generally giving a small negative temperature coefficient for uranium 235 and a small positive temperature coefficient for plutonium 239. For the water reactors, this spectrum effect is almost negligible in comparison with the following third effect.

•

The moderator expansion effect, indeed, is preponderant for the moderator temperature coefficient of the PWRs, because water expands considerably with a temperature increase around the nominal condition (about 300°C): (1/ρ ) · ∂ ρ /∂T ≅ –250 pcm/°C, where ρ is the water density. This expansion modifies the factors p (stabilizing effect) and f (destabilizing effect). It can easily be calculated from the elementary formulae of these factors that:

______ *

The absorption (capture and fission) cross-sections for the slow neutrons generally follow the “ 1/v law ”.

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P. REUSS

1∂p 1 1 ∂ρ = ln ⋅ , p ∂T p ρ ∂T

1∂f 1 ∂ρ = − (1− f ) ⋅ f ∂T ρ ∂T

i.e. –72 and +20 pcm/°C, if we use the values of p and f given previously. Therefore the global water temperature coefficient is negative (which is required because of the safety criteria) if (neglecting the spectrum effect):

ln(1 p) > (1 − f ) This is verified for the PWR examples.

Figure 7. Variation of some important cross-sections in the thermal range [v σ /v0 σ 0 versus v/v0; v0 = 2 200 m/s and σ 0 = σ (v0)].

But, look out! If some boron is added in solution in the primary water in order to reduce the thermal utilization factor f and consequently the multiplication factor k, this condition may no more be fulfilled: as we shall explain further, in this case solid (and burnable) poisons, modifying η and no longer f, will be used. 4. Reactor Design: Choice of the Main Parameters We will now discuss some examples of design problems: we will take these examples in the well known generation 2 and 3 pressurized water reactors (for which a very great experience has been accumulated), in order to show how various problems are interwoven. It is clear that similar interweaving will occur in the generation 4 reactors design, though it is problably still too early to completely appreciate them. We chose three examples of problems: the first one concerns the very basic choices which were to be made at the very beginning of the PWR development. The second one appeared when the utilities tried to improve the fuel utilization thanks to longer burnups; the last one cropped up when great quantities of plutonium (got thanks to the reprocessing of the standard

FUNDAMENTALS OF NEUTRONICS

63

fuels) accumulated because of the lack a fast neutron breeder reactors to use it and when it appeared more judicious to use it in the PWRs rather than to wait hypothetical breeders. The three main parameters of a PWR lattice are: the radius of the fuel pins, the pitch of the lattice and the initial isotope 235 content of uranium in the fuel. The neutron balance is not very sensitive to the radius of the pins if we compare situations differing only by a homothety ratio, i.e. without change of the volume ratios: if the dimensions increase, the effective integral in the formula of the factor p slightly decreases (because of l, though the decrease of the Dancoff factor C partly compensates) and p increases; in the formula of the factor f, the flux ratio Φ f Φ m slightly decreases and f decreases. As all these effects are small, the choice of the fuel radius is made not by neutronics but by thermal-hydraulics criteria, particularly the central temperature which must remain below the fusion temperature of the uranium dioxide (with a security margin) for the power density which is looked for. Conversely, the neutron balance is very sensitive to the so-called “moderation ratio”, that is to say the ratio of the moderator volume (or mass) to the fuel volume (or mass). Indeed this ratio directly appears in the formulae of the factors p and f. If it increases, p improves and f deteriorates: therefore, their product (and consequently the multiplication factor k) goes through a maximum between a null moderation ratio (where p goes to zero) and an infinite moderation ratio (where f goes to zero), as shown schematically on Figure 8.

Figure 8. Optimum of the moderation ratio (the numerical values are orders of magnitude for PWRs).

It can be seen that an under-moderated configuration is chosen. There are two main reasons of this choice: 1) the first one is the compacity of the core: if it is smaller, the vessel and all the associated materials are smaller and less expensive; 2) the second reason is linked to the safety: in any incidental situation (leak of water, formation of bubbles or water expansion due to a temperature increase) the mass of water decreases whereas the

64

P. REUSS

mass of fuel does not change; if the configuration is under-moderated, the multiplication factor decreases, and so does the power. In the PWRs, the potential (i.e. without compensation) multiplication factor continuously decreases with irradiation. Therefore a value greater than 1 is required at the beginning of cycle, the excess of reactivity being compensated with poisons. The end of cycle is reached when the potential excess of reactivity (and this poisoning) go to zero*. The initial multiplication factor of course increases if the initial contents in uranium 235 of the fuel increases: it is consequently chosen in order to satisfy the lenght of cycle which is looked for. 5. Reactor Design: Problem of Using a Boron Solution in the Primary Circuits of the PWRs It can be seen on Figure 8 that the volume moderation ratio of the PWRs is about 2, whereas the optimum value is about 4. This loss of profit (some 7 000 pcm!) is accepted because, as we said, it allows to reduce the size of the core and, above all, because a reactivity compensation by boron acid B(OH)3 is used. This boron poisoning reduces the value of the thermal utilization factor f... but also induces a positive component on the moderator expansion coefficient: indeed, the boron in solution expands like water and consequently its concentration (and poisoning) decreases if the moderator temperature increases. The curve of Figure 8 corresponds to a situation without boron: it can easily be shown thanks to the elementary formulae, that if the boron reduces f, therefore k, it also moves the maximum of the curve (optimum of moderation) towards the left. As a negative moderator expansion coefficient is associated to a positive slope of the curve for the chosen moderation ratio, this last one must remain smaller than the optimum one in spite of the addition of boron. The designers of the PWRs evaluated the needs of reactivity compensation, deduced the maximum boron concentration to be used, and finally chose the moderation ratio in order to keep a negative moderator temperature coefficient with this maximum concentration. At that time, the maximum fuel burn-up was about 30 000 MWd/t: that correspond to a loss of reactivity of about 30 000 pcm, or with the usual

______ *

As the poisoning is made by boron in solution, the end of cycle occurs not for a zero concentration but a minimum small concentration for which the dilution becomes ineffective. Some flexibility is possible: a cycle can be stopped before the normal instant; it can also be extended (stretch-out) thanks to the negative moderator temperature coefficient by a small decrease of the primary water temperature.

FUNDAMENTALS OF NEUTRONICS

65

core management in three batches, to 10 000 pcm of reactivity excess at the beginning of cycle, i.e. about 1 000 ppm of boron (efficienty ≅ –10 pcm/ppm). Such a order of magnitude keeps an optimum moderation ratio slightly greater than the chosen moderation ratio, that is to say, the following criterium remains satisfied:

ln(1 p) > (1 − f ) A better knowledge of the effects of irradiation (particularly the interactions between the pellet and the clad) allowed, some decades after the first operation of the PWRs, to propose longer irradiations (in spite of the greater consumption of natural uranium and separative work units per unit of mass of fuel this appears to be interesting on a economy point of view). The consequence of greater cycles is a greater initial potential reactivity and greater needs of compensation. It appeared that sometimes this needs would exceed what is possible thanks to boron in solution, that is to say the limit associated to a negative moderator temperature coefficient. Then, as this limit is compulsory, what to do? The solution of this difficulty has been the use of burnable poisons. This poisons are: •

solid, in order not to expand and not to induce a positive temperature coefficient. Boron carbide in specific pins or gadolinium oxide mixed with the uranium oxide in some pins have been used as such burnable poisons;

•

efficient, in order to be useful for the reactivity compensation;

•

burnable, in order to disappear (or, at least, almost disappear) at the end of cycle, when there is no more need of reactivity compensation.

Figure 9. Evolution of the multiplication factor of a PWR without and with burnable poisons.

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P. REUSS

As an example, Figure 9 shows (schematically) what can be done thanks to the burnable poisons: of course, they cannot compensate exactly for the reactivity excess, but their contribution allows to reduce the use of boron and, consequently, to insure a negative moderator temperature coefficient. 6. Reactor Design: Problem of Recycling Plutonium in the PWRs The history of the plutonium use in France is interesting to recall to the young scientists preparing the next generation of reactors and particularly their breeding! [4] After irradiation, the standard fuel of the PWRs (uranium initially enriched to a little more than 3% of uranium 235) still contains about 1% of uranium 235 and about 1% of plutonium (Figure 10). Since the beginning of the developments of the pacific nuclear energy during the late 1940s, France had been a convinced supporter of the fast neutron breeder reactors: the official politics was to built a few thermal neutrons reactors (at that time natural uranium-graphite-gas (UNGG) reactors) in order to produce some plutonium, then to introduce this plutonium into breeders which will produce more and more plutonium for more and more fast neutron reactors. It is the reason why an important development was devoted to these fast reactors, and why a “close cycle” was chosen, that is to say a systematic reprocessing of the irradiated nuclear fuels and recycling of the energy-giving materials, particularly plutonium.

Figure 10. Typical evolution of the concentrations of the main heavy nuclides in a PWR fuel.

In reality, a very small number of fast reactors were built in France: after the experimental facility Rapsodie, the prototype Phénix (250 MWe, 1973, still running) and the industrial plant Superphénix (1 200 MWe, 1986, stopped in 1998 not for technical but for political reasons).

FUNDAMENTALS OF NEUTRONICS

67

Meanwhile, the UNGG program was stopped and an ambitious REP (“réacteur à eau sous pression”, i.e. PWR) program was carried out: 58 plants between 1977 and 1999. The “closed cycle” was confirmed and, after the plant which operated at Marcoule, new reprocessing plants were built at La Hague. It can be calculated that about 10 tons of plutonium per year are produced at La Hague*. Of course, the only running fast reactor Phénix cannot use such a huge amount of plutonium! It the reason why Électricité de France (EdF) decided to recycle plutonium into the water reactors: some PWRs use plutonium fuels since 1987. The recycling of plutonium in the thermal reactors are less interesting than in fast reactors, because the reproduction factor η is smaller: 2.1 (almost the same value than the uranium 235 one) instead of 2.4 in a fast spectrum, for plutonium 239; particularly, a plutonium loaded (as a uranium loaded) thermal reactor cannot be a breeder. Nevertheless, it was thought that it should be more judicious to use the available plutonium in such conditions rather than to wait hypothetical fast reactors†. In practice, the standard uranium fuels are replaced with so-called “MOX” fuels, that is to say mixed (natural or depleted) uranium and plutonium oxides, e.g. 6.5% of plutonium and 93.5% of depleted uranium instead of uranium enriched up to 3.7% in isotope 235. Remark that the plutonium content in the MOX fuel is greater than the uranium 235 content in the standard fuel, because the plutonium is fissile for about the two thirds only. When the recycling of plutonium was envisaged in the PWRs, the neutron physicists had to adapt and requalify the codes, thanks to experimental programs, and the designers had to calculate the core neutron distribution and management for such a new fuel. Indeed, the difference is great! If the standard and MOX fuels are very similar for the fast and epithermal neutrons (the amounts of uranium 238 and water remain the same) they are completely different for the slow neutrons: the MOX assembly is about 2 to 3 times more absorbing for such neutrons‡. This is the consequence: 1) of the greater concentration of plutonium than uranium 235; 2) of a greater microscopic absorption crosssection for plutonium 239 than uranium 235; 3) of the presence of resonances in (or near) the thermal range for all the main plutonium isotopes (Figures 11 and 12).

______ *

Nowadays, 1 150 tons of irradiated fuels are unloaded from the PWRs and 850 tons are reprocessed per year. † Note that, if plutonium is stored, the isotope 241 (fissile) disappears by radioactive decay with a half-life of 15 years; note also that about 70% of the initial plutonium remain in the irradiated plutonium fuels and could be recovered (presently, these fuels are not reprocessed). ‡ We compare the macroscopic mean absorption cross-sections.

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P. REUSS

Figure 11. Absorption cross-sections for slow neutrons of uranium 235 and the main isotopes of plutonium.

Figure 12. Comparison of thermal neutron spectra in the standard and MOX fuels (multigroup calculations by the code APOLLO [18]) Note the great decrease of the flux due to a large absorption in the MOX fuel and the selfshielding effect (flux depression) of the plutonium resonances.

The first consequence of a such a big absorption is a drastic reduction of the efficiency of the control rods, boron and poisons*. The core calculation which are made for the nominal and the main transient situations showed

______ *

These materials mainly absorb thermal neutrons; their efficiency is proportional to the ratio between the absorbent and the assembly macroscroscopic absorption cross-sections: therefore, the greater this last one, the less the efficiency.

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69

that a core completety loaded with MOX could not be controlled in some cases, and consequently could not be authorized by the safety authority. Conversely, a core half loaded with MOX and half with standard fuels could be managed. In order to keep some margins, EdF decided to restrict the proportion of MOX in the core (and consequently in each reload) to one third. Consequence of such a choice: the presence in the core of interfaces between very different zones with uranium 235 and plutonium fissile material respectively; for such an interface, a peak of power appears near the interface in the most absorbent zone, i.e. the MOX assembly (Figure 13).

Figure 13. Flux and power behaviour near a uranium-plutonium interface (scheme).

The flux is lower in the plutonium (more absorbent) zone, but must be a continuous function, therefore the indicated behaviour; the power is proportional to the product of the flux by the fission macroscopic crosssection, which is greater in the plutonium zone, therefore the indicated behaviour showing a power peak in the plutonium zone near the interface. Precise calculations show that such a peak cannot be accepted*. The only mean to reduce such a power peak is to reduce the plutonium content in the pins situated near the interface, and to increase the content in the internal pins in order to keep the average content. Calculations showed that at least three zones are necessary. We see that such a design leads to an even more expensive manufacturing of the MOX fuels†... The question arises: is the plutonium recycling in the PWRs still interesting on an economic point of view? The answer is a matter of controversy... In the facts, almost all the first generation plutonium produced in France from the standard PWR fuels is recycled once in most of the 900 MWe plants.

______ *

The maximum pin temperature, therefore the maximum pin power is limited for safety reasons. Consequently the smaller the form factor Pmaximum/Paverage, the greater the total power of the plant. A too high power peak therefore induces a too low efficiency of the plant. † The manufacturing of the MOX fuel is more expensive than the standard one: it has to be made in “ glove boxes ” because of the α -activity of plutonium, much greater than the uranium one.

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Such a controversy, during the 1990s, and the concerns of the opinion about the radioactive wastes let sometimes think that the main interest of the plutonium recycling was to reduce the amount of such a “waste”: from a energy-giving produce the statute of plutonium became more and more a waste to incinerate as much as possible! We even saw EdF develop some design studies (notably thanks to a greater moderation ratio) to increase the consumption of plutonium, and the Commissariat à l’Énergie Atomique (CEA) lunch a programm CAPRA, which means “greater consumption of plutonium in the fast reactors”: the main idea was to reduce as far as possible the plutonium breeding by removing the blankets and reducing the uranium 238 mass in the core (some of it must be kept in order to preserve a significant negative Doppler effect). Nowadays, after the Generation IV reflexions, and the awareness of the limited ressources of uranium, such considerations sound rather strange! 7. Conclusions At the very beginning of the neutron physics, the main concern was to get the critical condition, particularly when only natural uranium was available for such a purpose. Nowadays, we are able to enrich uranium or to substitute plutonium for uranium 235, and to get criticality is no more the main difficulty. Conversely, we see that the economic optimization (particularly the flatening of the power distribution and the extension of the burn-ups) and, above all, the intrinsic safety thanks to negative reactivity feedbacks have become the major concern of the reactor designers. The example of the boron in solution in the PWRs showed the technical constraints brought by such criteria. We could also have develop the example of the recycling of second generation plutonium in the PWRs, or many examples which can be find in the fast reactor design, such as the replacement of oxide fuel with nitride or carbide fuels, the change of fissile materials, the change of coolant (lead or helium instead of sodium), etc. The young generations of physicists and engineers will have a lot of studies to perform in order to elaborate the best choices! We must underline that even a deep knowledge of neutron physics will never be sufficient to get these good choices. Indeed, they also depends on all the physical characteristics of the cores, and therefore let arise numerous domains of physics. Particularly, all the problems of heat transfers, hydraulics, effects of irradiation on the materials... are strongly linked to neutronics. Only a good knowledge of all these aspects by all the designers (and discussions between the various specialists) can lead to sound solutions. Let us hope that this autumn school will be a step in that direction!

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In practice, the codes treating the various aspects of the core behaviour are to be linked and run in an iterative process until convergence. For instance, the neutron distribution depends on the temperature of the various materials through the Boltzmann equation and the cross-sections it contains... and these temperatures depends on the power sources through the equation of the thermal-hydraulics: therefore, for any core calculation, both the codes of neutronics and of thermal-hydraulics have to be coupled and iterated. Similarly, codes treating the whole system are necessary to take into account the counter-reactions of a part of the reactor on the other parts, for instance of the secondary circuit on the primary one. Finally, we would like to emphasize two conclusions which can be drawn from the examples relative to the PWRs that we developed and that will certainly remain pertinent for the Generation IV developments: The first conclusion is the importance to design the reactors with margins as large as possible. Indeed, new operational modes appears, which were not anticipated. If the margins are not large enough for these new modes, technological adaptations are generally possible, but they are time consuming in developments and expensive to realize. For example, we see that longer irradiations could no more be managed with boron in solution only, but necessitated the use burnable poisons – which necessitated code developments and costly experiments to qualify them. Similarly, the introduction of MOX fuel into the PWRs led to great technical and expensive difficulties, particularly the necessity to zone the MOX assemblies – which, again, necessitated many codes developments and experiments. The second conclusion is the “weight of history”. The decisions taken at a given time not only govern the next operations but often direct the technical choices for a very far future. A good example is the history of plutonium in France. The initial direction was to develop as rapidly as possible the fast neutron reactors. Immediate consequence: use plutonium, reprocess the irradiated fuel and build prototypes of fast reactors. Far and not anticipated consequence: go on with reprocessing, introduce the MOX fuel into the water reactors with all the difficulties which arose. It is the reason why it is so important to make, as far as possible at the first attempt, the good choices!

Ackownledgements The author would like to thank his colleagues who provides the APOLLO calculations [18, 19]: P. Bioux and L. Payen from EdF (Clamart) for the PWR cases, and A. Nicolas and L. Buiron from CEA (Saclay) for the GFR cases.

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References [1] Commissariat à l’énergie atomique, 2005, L’énergie Nucléaire du Futur: Quelles Recherches pour quels Objectifs?, e-den, monographie de la Direction de l’énergie nucléaire. [2] Commissariat à l’énergie atomique, 2006, Les Réacteurs Nucléaires à Caloporteur Gaz, e-den monographie de la Direction de l’énergie nucléaire. [3] Reuss, P., 2006, L’énergie Nucléaire, série: Que sais-je?, PUF. [4] Reuss, P., 2007, L’épopée de l’Energie Nucléaire, une Histoire Scientifique et Industrielle, série :Génie atomique, EDP Sciences. [5] Barjon, R., 1992, Physique des Réacteurs Nucléaires, Institut des sciences nucléaires, Grenoble. [6] Bussac, J., and Reuss, P., 1985, Traité de Neutronique, Physique et Calcul des Réacteurs Nucléaires, Enseignement des sciences, Hermann. [7] Reuss, P., 1998, La Neutronique, série: Que sais-je?, PUF. [8] Reuss, P., 2003, Précis de Neutronique, série: Génie atomique, EDP Sciences. The english translation will be published. [9] Reuss, P., 2004, Exercices de Neutronique, série: Génie atomique, EDP Sciences. The english translation will be published. [10] Tellier, H., 1898, Réactions Nucléaires Induites par les Neutrons, CEA/Institut national des sciences et techniques nucléaires, collection Enseignement. [11] Rozon, D., 1992, Introduction à la Cinétique des Réacteurs Nucléaires, éditions de l’École polytechnique de Montréal. [12] Tellier, H., 1994, Cinétique des Réacteurs Nucléaires, CEA/Institut national des sciences et techniques nucléaires, collection Enseignement. [13] Livolant, M., and Jeanpierre, F., 1974, Autoprotection des Résonances dans les Réacteurs Nucléaires, rapport CEA-R-4533. [14] Reuss, P., 1991, Théorie de l’Absorption Résonnante des Neutrons, note CEA-N-2679. [15] Coste, M. and Reuss, P., 2003, Development of computational models used in France for neutron resonance absorption in light water lattices, Prog. Nucl. Energy 12(3): 237282. [16] Coste, M., 2006, Modélisation du Phénomène d’Autoprotection dans le Code de Transport Multigroupe APOLLO-2, thèse CNAM (7 mars 2006) and rapport CEA-R6114. [17] Lewis, E. E., and Miller, W. F., 1994, Computational Methods of Neutron Transport, John Willey & Sons. [18] Hoffmann, A., Jeanpierre, F., Kavenoky, A., Livolant, M., and Lorain, H., 1972, APOLLO Code Multigroupe de Résolution de l’Equation du Transport pour les Neutrons Thermiques et Rapides, note CEA-N-1610. [19] Coste, M., Hébert, A., Sanchez, R., Stankovski, Z., Zmijarevic, I., 1999, APOLLO-2 Notice Théorique, rapport CEA/DMT/SERMA/LENR/RT/99-2719/A. [20] Fédon-Magnaud, C., 1994, Résolution de l’Equation de Transport dans le Code CRONOS, note CEA-N-2763.

INTRODUCTION TO THERMODYNAMICS G. INDEN* Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf, Germany

Abstract: At first, the fundamental concepts and their mathematical formulation are recalled: integral quantities like various forms of thermodynamic potentials, exchanged energy forms, partial molar quantities like chemical potentials, activities etc.. The emphasis will be on multi-component systems of solid, liquid and gaseous phases. The thermodynamic treatment of chemical reactions is presented and applied to oxidation reactions. The basis of Ellingham diagrams and their use is illustrated. The thermodynamic description of multi-component solution phases is presented as well as the construction of phase diagrams. The restricted equilibrium in the case of immobile substitutional elements is discussed (para-equilibrium). Keywords: Thermodynamics; potentials; phase equilibria; phase diagrams; para-equilibrium

1. Reminder of Thermodynamic Principles 1.1. STATE, VARIABLE, SYSTEM

The dynamic description of physical systems and processes is based on the concept of an exchange of quantities. The physical quantities are represented by variables, indicating that only their numerical value is of primary importance, not the way an exchange is achieved. A state is defined by the values taken by the variables. A system is then an ensemble of states. It is defined by: •

the number of degrees of freedom r

•

by defining (r +1) state variables

•

by defining a mathematical relation (often called fundamental relation) between the (r +1) state variables.

At this point it may be interesting to make the link with concepts in geometry. The state corresponds to a point in geometry. The state variables

______ *

Prof. Dr. Gerhard Inden, Im Weidengrund 19, D-40878 Ratingen, Germany; e-mail: [email protected]

V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

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correspond to the space variables (e.g. x, y, z), which are given by the coordinates relative to a coordinate system. A state is defined by the values of the state variables, just as the point is defined by the values of the coordinates. A system corresponds to a geometric object, like a sphere, representing an ensemble of points. A sphere is defined by: •

the dimension of space r (here r = 3)

•

by defining r +1 variables (here x,y,z and R)

•

by defining a mathematical relation between the variables (here x2 + y2 + z 2 = R2 )

The definition of a system given above is mathematical and very general and it is not certain that every system defined this way can be realized. However, it is sometimes helpful to have recourse to idealized systems in order to derive certain conclusions. The variables characterising a system may be divided into two groups: extensive variables Xi (e.g. volume V, entropy S, number of species Ni, magnetisation M,…) and intensive variables Yi (e.g. pressure P, temperature T, chemical potentials μ i , magnetic field H,… ). It is not trivial to identify the extensive or intensive character of a variable. The term extensive suggests that the variable scales with the size of a system. This is, of course, true for energy, volume, entropy, number of particles: taking e.g. two identical systems together to form a new system, these variables double their value. There are, however, many variables which cannot be classified on the basis of this heuristic scheme, such as the position in a field, the extension of a spring. Intensive variables do not depend on the size of the system. Yet, not every quantity that does not depend on the size is automatically intensive. Examples of this type are reduced quantities like molar volume, mole fractions etc.. For every system with degree of freedom r one may chose r variables, e.g. (X1,... Xi, Yi+1,…Yr), to define a coordinate system. At least one extensive variable must be within this set in order to define the size of the system. The (r +1)-th variable, Ψ , is then defined by the fundamental relation:

Ψ = Ψ ( X 1 ,... X i , Yi +1 ,...Yr ) . The function Ψ is called Massieu*- Gibbs*- Function (or simply GibbsFunction) of the system defined by the coordinate system.

______ *

François Jacques Dominique Massieu, 1832-1896, professor at Ecole des Mines de Paris; he discussed the mathematical properties of such functions.

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1.2. INTERNAL ENERGY

Axiom: Any realizable system of dimension r admits at least one set of (r+1) extensive variables such that the fundamental equation defines a Gibbs-Function, which is a first order homogeneous function of the r independent extensive variables. This first order homogeneous function is the energy E. Let us denote by Xi the r independent extensive variables. The energy may then be written as r

E ( X1,...X r ) = ∑Y j ⋅ X j

(1)

j=1

Since E is a first order homogeneous function of the independent extensive variables, it has to fulfil the following criterion:

E (λX1, λX 2 ,...λX r ) = ∑Y j ⋅ (λX j ) = λ∑Y j ⋅ X j = λ E (X1, X 2 ,...X r ) (2) Consequently, the Yi have to be of zero order in the extensive variables and thus cannot depend on the size of the system. The Yi are the energyconjugate intensive variables of the corresponding extensive ones, Xi. They are defined by

Yj =

∂E ∂X j

(3) X k ,k≠ j

1st law: Energy is a conserved quantity. Its value for a given system can change only by exchange with other systems. The change in energy caused by exchange of extensive quantities is given by r

∂E dE = ∑ j=1 ∂X j

r

⋅ dX j =∑Y j ⋅ dX j X k ,k≠ j

(4)

j=1

Systems in physical metallurgy are assumed to be at rest. In that case they do not exchange linear or angular momentum, and the energy is then called internal energy U. Common extensive variables are e.g. volume, *

Josiah Willard Gibbs, 1839-1903, professor of mathematical physics at Yale University; he introduced this fundamental concept in physics.

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entropy, number of particles of type i, Xj =V, S, Ni. The internal energy is given by

U (V,S,N i ) = ∑Y j ⋅ X j = −PV + TS + ∑ μi N i j

i

with pressure P, temperature T and chemical potential μi :

P =− def

∂U ∂V

∂U def ∂S V ,N i

T= S,N i

∂U def ∂N i V ,S ,N

μi =

j ≠i

as energy-conjugate intensive variables. For any change dXj of the extensive variables Xj one gets

dU = ∑Y j dX j = −PdV + TdS + ∑ μi dN i j

(5)

i

The various terms in Eq. (5) are called exchanged energy forms. The sign convention of the individual terms is such that if the fluxes of extensive quantity and of energy are oriented towards the same direction, the energy terms are counted positively. This is the case of entropy, particle number etc. However, in the case of mechanical work by volume change, a “flux of volume from the outside” ( dV > 0 ) means work supplied by the system to the outside. Thus the fluxes of energy and volume are oriented into opposite directions. Consequently, −P and not P is the conjugate variable of V. Equation (5) suggests to associate the term –PdV with work performed on or by the system. Indeed, considering a system with gas in a cylinder and a piston of area A, the pressure can be written as P = F/A and dV = A dx, so that PdV = F dx, i.e. equivalent to mechanical work. This, however, is not generally true, it only holds if the process of volume change is performed reversibly. The same holds for the term TdS representing the amount of heat exchange only in reversible processes. The difference between reversible and irreversible processes can be illustrated with the expansion of a gas. In Figure 1a a reversible realization is shown. The pressure from the outside is decreased by infinitesimal steps. The piston is then moved outwards by the gas, the pressure decreases until the level imposed from outside is reached. The process is reversible: it is sufficient to reset the external force from (F-ε) to F, the gas contracts until it reaches the original state. The irreversible realization is shown in Figure 1b. The volume expansion is performed by pulling out walls. The gas then has access to the increased volume without performing any work. This process cannot be reversed, since the probability for the event that a volume segment does not contain any gas, is zero. It is to be emphasized that in both cases the term -PdV appears and takes the same value for the given volume change. However, only in the first case this value represents work performed by the system. Since in the irreversible case the system does not

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exchange any energy form, the internal energy stays constant. From Eq. (5) it follows dU = −PdV + TdS = 0, TdS = PdV . In the irreversible process the quantity PdV is thus converted into T dS . Such conversion is also called entropy production.

Figure 1. a) Realization of a reversible expansion and compression of a gas. If the external force F is reduced by an amount ε, the gas expands the volume until the gas pressure equals the external pressure. The process can be reversed by resetting the external force to its original value F, b) Realization of an irreversible expansion of a gas. The volume increase is not performed by the gas. It is made available externally by removing the walls. The gas simply accesses the additional volume. This process cannot be reversed.

2nd law: Entropy cannot be annihilated: a reduction of entropy can only occur by exchange, while an increase can occur by exchange and/or production. Reversible process: process with no entropy production. Ideal gas: system which fulfils the equation PV = nRT (n=number of moles, R=8.3144 J/(K·mole)). From the PV = nRT equation it follows that

∂U(V,T,N) =0 ∂V

i.e. at constant T and N the internal energy of the ideal gas does not depend on volume. Consequently, if the reversible expansion of the gas is performed at constant temperature, the internal energy does not change. The energy loss due to the work − PdV performed by the gas is thus compensated by a heat flux TdS from the heat reservoir, which keeps temperature constant. 1.3. CONJUGATE VARIABLES, LEGENDRE TRANSFORMATION

The experimental conditions are often characterized by keeping certain variables constant. In order to be able to impose such constraints the corresponding variables must be independent. However, many practical cases require the control of intensive rather than extensive variables, e.g. temperature rather than entropy or pressure rather than volume. For such

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cases it is necessary to change from independent extensive variables to intensive ones. By definition, see Eq. (4), the intensive variables are the first order partial derivatives of the internal energy. The problem of changing from an extensive variable to its conjugate intensive variable is thus equivalent to expressing the functional relationship in terms of first order derivatives. This problem is solved by the Legendre transformation.

Figure 2. Legendre transformation of a function y(x) with only one variable.

A one dimensional illustration of the Legendre transformation is shown in Figure 2. Suppose a one dimensional functional relationship y(x) describing a curve in the (y,x)-plane. This functional relationship shall be described in terms of the first derivative ξ = dy dx which represents the slope of the tangent to the curve y(x). Every point of the curve is characterized by the slope and the intercept u(ξ ) of the tangent to the curve y(x). The variation of this intercept with the slope represents the property of the functional relationship in terms of the slope. The intercept is given by u(ξ ) = y (x ) − ξ x . In this one-dimensional example, the transformation is given by x → ξ = dy dx and y (x ) → u(ξ ) = y − ξ x . Introducing the symbolic description [y ]x for the Legendre transform of y with respect to x, we can write

[y ]x = y − ξ

x

(6)

The situation in the case of two variables is illustrated in Figure 3. Figure 3a shows the partial transformation with respect to the variable x. The construction is the same as in Figure 2, here of course performed within the section y = cst. The same procedure applies to the partial transformation with respect to y, the construction being performed within the section

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x = cst., Figure 3b and 3c shows the transformation with respect to both variables. The tangent plane is defined by the two slopes ξ1 and ξ 2 . Its intersection with the Z-axis defines the value of the Legendre transform:

[Z(x, y)]x,y = Ψ(ξ1,ξ 2 ) = Z (ξ1,ξ 2 ) − ξ1 x − ξ 2

y

(7)

The Legendre transformation is often written in a way as (6) which could suggest that the transformation consists of simply adding a product term of the two conjugate variables to the original function. Equation (7)

Figure 3. Legendre transformation of a function with two variables. a) Partial Legendre transformation with respect to x, b) Partial Legendre transformation with respect to y, c) Complete transformation with respect to both variables, d) Transformation of a cone: all tangent planes define the same intercept: the origin zero.

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specifies clearly that the transformation requires the substitution of the variables x and y by ξ1 and ξ 2 in the original function Z, which means that the relation ξ1 = ∂Z ∂x defining the intensive variable must be resolved with respect to its conjugate x. The cone is a geometrical example of a homogeneous function of first order in (x,y). In this case the tangent planes define a unique intercept, the origin zero (Figure 3d). 1.4. LEGENDRE TRANSFORMATION OF THERMODYNAMIC POTENTIALS

This last situation is indeed encountered in thermodynamics since the internal energy U is a homogeneous function of 1st order. It thus represents a hyper-cone in the space of the extensive variables. The Legendre transform with respect to all of the extensive variables thus leads to the identity zero. This is known as Gibbs-Duhem equation. It is, of course, possible to perform partial Lengendre transforms. These correspond to hyper-planes tangent to conic sections. These planes do not pass through the origin. When a partial Legendre transformation is performed, a new function is obtained which is still a homogeneous linear function of the remaining extensive variables. Let us denote the Legendre transform of a function

Φ(X1, X 2 ,... X r)

with respect to Xj symbolically by

[Φ]X

j

= Ψ(X1, X 2 ,...Y j ...X r) .

The Legendre transform is then given by

Ψ = Φ(X1, X 2 ,...Y j ,...X r )− X j Y j

(8)

In order for this transformation to be possible it is necessary that the equation

Y j = ∂Φ ∂X j

can be inverted such that one gets

X j = X j (Y j )

as a function of Yj, otherwise one cannot replace Xj in Φ as required in Eq. (8). Because of the following relations:

d Φ = ∑ Yk ⋅ dX k + Y j ⋅ dX j k≠ j

and

d (Y j ⋅ X j ) = X j ⋅ dY j + Y j ⋅ dX j

the differential of Ψ is obtained as

dΨ = d (Φ − Y j X j ) = ∑ Yk dX k − X j dY j . k≠ j

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The general case of a multiple Legendre transformation with respect to m variables, e.g. X1, X2, ... Xm, is obtained as

[Φ]X ,...X 1

m

m

= Ψ(Y1 ...Ym , X m +1 ...X r ) = Φ(Y1,...Ym , X m +1,...X r ) − ∑ X k Yk

(9)

k=1

and

dΨ =

r

∑Y j= m +1

m

j

dX j −∑ X k dYk

(10)

k=1

1.4.1. Enthalpy The Legendre transform of U with respect to V, is called the enthalpy:

H (P,S,N i ) = [U (V,S,N i )]V = U + PV = TS + ∑ μi N i

(11)

dH = VdP + TdS + ∑ μi dN i

(12)

def

i

with i

At constant pressure (!) enthalpy is again a state function since it is a homogeneous function of 1st order in the remaining extensive variables. At constant pressure and particle numbers, Eq. (12) leads to dH = TdS, i.e. the enthalpy change represents the reversible exchange of heat. 1.4.2. Free energy Free Energy is the Legendre transform of U with respect to entropy, i.e. with

F (V,T,N i ) ≡ [U (V,S,N i )]S = U − TS

dF = −PdV − SdT + ∑ μi N i i

1.4.3. Gibbs energy Gibbs energy is the Legendre transform of U with respect to both volume and entropy such that now pressure and temperature become independent variables.

G(P,T,N i ) ≡ [U ]V ,S = U + PV − TS = ∑ μi N i i

with

dG (P ,T , N i ) = VdP − SdT + ∑ μ i dN i i

(13)

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G. INDEN

At constant temperature and pressure Gibbs energy is again a state function. There are many more Legendre transforms if one considers different starting functions, e.g. entropy. 1.4.4. Gibbs-Duhem-Relation If we consider the Legendre transform of U with respect to all extensive variables, we do not end up with a new potential function, but with a relation between the intensive variables: n

[U ]V ,S,N = U + PV − TS − ∑ μi N i ≡ 0 1

(14)

i−1

and n

d[U ]V ,S,N = VdP − SdT − ∑ N i dμi = 0 i

(15)

i=1

At constant temperature and pressure Eq. (15) takes the familiar form: n

∑ N dμ i

i

= 0.

i=1

Thus, the Legendre transform with respect to all extensive variables does not lead to a new Gibbs function, but leads to a constant which can be set to zero. Therefore, it is sometimes called “zero potential ”. It is not possible to define the size of a system with intensive variables only. In order to define a system at least one extensive quantity must be among the variables characterizing the system. 1.5. PARTIAL QUANTITIES

1.5.1. Molar quantities In most cases of physical metallurgy the size of the system is not really important. It is then common practice to refer the extensive quantities to 1 mole of atoms. Taking Gibbs energy as an example, we may write :

G(T,P,N i ) = N with

G(T,P,N i ) = N G (T,P, x i ) N n

N = ∑ N i and xi = i =1

Ni N

Setting N = 1 mole of atoms, we obtain the molar Gibbs energy:

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83

n

Gm (P, T, x i ) = ∑ x i μi i=1

where the chemical potentials have to be given in energy per mole of atoms. 1.5.2. Partial molar quantities Let us suppose a homogeneous first order extensive quantity A that has only the particle numbers as extensive variables, e.g.

A = A(T , P , N1 ,...N k )

The first order homogeneity implies

A ( P, T , λ N1 ,..., λ N k ) ⇒

= λ ⋅ A ( P, T , N1 ,..., N k )

dA ( P, T , λ N1 ,..., λ N k ) = A ( P, T , N1 ,..., N k ) dλ

Consequently,

A(P,T,N1,...,N k ) = ∑ i

∂A(P,T, λN1,..., λN k ) ∂λN i P ,T , λN

=∑ i

⋅ j

, j≠ii

∂A(P,T, λN1,..., λN k ) ∂λN i P ,T , λN

j , j≠ i

dλN i dλ

⋅ Ni

This relation must hold for any value of λ. We may thus take λ = 1 and get

A(P,T,N1,...,N k ) = ∑ N i i

∂A(P,T,N1,...,N k ) ∂N i P,T ,N

= ∑ N i ⋅ Ai i

j , j≠ii

The partial quantity Ai is thus defined as

Ai =

∂A(P ,T , N 1 ,...N i ,...N k ) ∂N i P ,T ,N

(16) j , j ≠i

As a result, any homogeneous function of first order in the particle numbers can be set up by the weighted sum of the contributions of the components. The individual contributions Ai are called partial quantities. Since we have

Ai (P,T, λN1,..., λN k ) =

∂A(P,T, λN1,..., λN k ) ∂λN i P ,T , λN

= Ai (P,T,N1,...,N k )

j

, j≠i

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the Ai are indeed intensive quantities and they are constant along lines defined by (λN1,..., λN k ) within the space spanned by axes defining the particle numbers. That means: the whole variety of Ai values can be obtained from a section in this space. If we take

λ=

1 1 = ∑ Ni N i

the coordinates of this section are given by the mole fractions x i = N i N and the partial quantity is a function of the mole fractions, Ai= Ai ( P, T , x1 ,..., xk ) . 1.5.3. Geometrical interpretation of partial molar quantities The quantity A(P,T,N1,...,N k ) defines a hyper-cone in the space defined by the coordinate system (A,N1,...,N k ). For a fixed number of N, e.g. N = 1 mole of atoms, the quantity Am (P, T, x1,..., x k ) defines the values of A for 1 mole of substance. These values are within a vertical section defined by a hyper-plane parallel to the A-axis and passing through the points for 1 mole on every axis Ni, i.e. (1,0,...0), (0,1,...0),... (0,0,...,1). This is shown in Figure 4 for the case of a binary system. The partial molar quantity Ai (P, T, N1,..., N k ) defines the slope of A(P, T, N1,..., N k ) in the direction of the Ni-axis at a point of the hypercone, defined by the coordinates (N1,..., N k ). This direction being parallel to the Ni-axis, the coordinates Nj remain unchanged, in accordance with definition (16). It will now be shown that the partial molar quantity

A i (P , T , x10 ,..., x k0 )

is obtained as intersection of the tangent plane to the surface Am (P, T, x1,..., x k ) defined in the space (x1,..., x k ) at the point

A m (P , T , x10 ,..., x k0 )

We start from equation:

Ai =

∂A ∂N i

= P ,T , N j , j≠i

∂N ⋅ Am (P, T, x1,...x k ) ∂N i P ,T , N

Since

∂Am ∂N i and

= P ,T , N j , j≠i

= Am + N j

, j≠i

∂Am ∂N i

P ,T , N j , j≠i

k ∂Am (P, T, x1,..., x k ) ∂Am ∂x i ∂A ∂x j = +∑ m ∂N i ∂x i ∂N i j≠i ∂x j ∂N i

Ni N = N − N i = 1 (1 − x ) i ∂N i N2 N

∂

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∂

85

Nj

N =−Nj =− xj ∂N i N2 N

we get

∂Am ∂N i

= P ,T ,N j , j≠i

k 1⎡ ∂A ∂A ⎤ ⎢(1− x i ) m − ∑ x j m ⎥ ∂x i j≠i ∂x j ⎥⎦ N ⎢⎣

For the partial molar quantity Ai we thus obtain

Ai = Am + (1− x i )

k ∂Am k ∂A ∂A ∂A − ∑ x j m = Am + m − ∑ x j m ∂x i j≠i ∂x j ∂x i j=1 ∂x j

(17)

In the derivation of equation (17) the particle number Ni was varied while taking the other numbers Nj constant. Yet, all the mole fractions varied since the total number varied. In fact, Ai represents a property that is not related to the vertical section of 1 mole, since it is not possible to add atoms of a specific kind, say i, if the system is forced to remain within the section defined by Σ i x i = 1.

Figure 4. Binary system with components 1 and 2. The extensive property A(T,P,N1,N2) defines a cone because of its linear homogeneity. A vertical section, defined by the condition N1 + N2=1, intersects the cone leading to the curve A(T,P,x1,x2). At point (0.1,0.9) the partial quantity A2(T,P,0.1,0.9) represents the slope of the cone into the direction of axis N2. This slope is the same as that of the triangle in the (A,N2)-plane defined by the origin, the point (0,1) and the intersection of the tangent plane with the N2-axis.

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This becomes clear in Figure 4 where it is shown that Ai represents a slope in a direction that is not within the section. Yet it is possible to derive the value of Ai within the section. For this purpose we have to take account of the constraint Σ i x i = 1 and select one of the components as dependent. Let us take x1 as dependent variable and replace it by k

1− ∑ x j j =2

in the expression Am. This implies:

∂Am = 0. ∂x1

Furthermore, the remaining xi are all independent. The partial derivatives then become in fact derivatives and we get

Ai = Am + (1− x i )

k dAm k dA dA dA − ∑ x j m = Am + m − ∑ x j m dx i j= 2 dx j dx i j= 2 dx j j≠i

k

A1 = Am − ∑ x j j= 2

i ≠1 (18)

dAm dx j

This is in fact the equation of the intercept of the (hyper)-plane tangent to the (hyper)-surface Am at the coordinates (x 2 ,..., x k ) with the Am-axis at the pure component i. 1.5.4. Examples Partial molar Gibbs energy of a binary system (Figure 5): For a binary system we take e.g. the mole fraction of component B as independent variable.

Figure 5. Binary system A-B: graphical construction of the partial Gibbs energy G i according to equations (19). The partial molar Gibbs energy is the chemical potential μ i . The chemical potential of component i is obtained as intercept of the tangent to the molar Gibbs energy curve with the axis of molar Gibbs energy of component i.

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From equation (18) we get

dGm dGm dGm − xB = Gm + x A dx B dx B dx B dG μ A = GA = G m − x B m def dx B

μ B = GB = G m + def

(19)

Partial molar Gibbs energy of a ternary System (Figure 6): In this case we take the mole fractions xB and xC as independent composition variables and obtain from Eq. (18):

dGm dGm − xC dx B dxC dG dG μC = Gm + (1− xC ) m − x B m dxC dx B

μB = Gm + (1− x B )

μ A = Gm − x B

dGm dGm − xC dx B dxC

(20)

The graphical construction is shown in Figure 6 where the molar Gibbs function Am the Gibbs energy has been taken, namely Am = Gm .

Figure 6. Ternary system A-B-C: Graphical construction of the partial Gibbs energy Gi according to Eq. (20), here demonstrated for component B. Similar constructions apply for components A and C. In a ternary system the chemical potential of a component i is obtained as intercept of the tangent plane to the molar Gibbs energy curve with the axis of molar Gibbs energy of component I.

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1.6. EQUILIBRIUM

Systems can be coupled together to exchange quantities, provided the quantities are of the same type, e.g. extensive. It will turn out that this offers a way of defining the character of a variable which cannot be identified easily by simple arguments like a scaling with the size of the system. Let us consider two systems (1) and (2) characterized by the MassieuGibbs functions

(

)

v

Φ( 1 ) X i( 1 ) = ∑ Yi( 1 ) X i( 1 )

(

)

i =1 v

Φ ( 2 ) X i( 2 ) = ∑ Yi( 2 ) X i( 2 ) i =1

The variables

X i(ν ) and Yi (ν )

represent respectively extensive variables of the two systems ν = 1, 2 and the conjugate intensive variables. The potential of the combined system (1)+(2) is given by Φ = Φ(1) + Φ(2) . Equilibria between systems are established by a free exchange of coupling variables, while the remaining variables are kept constant. A free exchange of the extensive quantity X j between the two systems means two conditions:

(

)

(1) (2) (2) 1. X (1) = 0 or ordX j j + X j = X 0 = const. and thus dX 0 = d X j + dX j

(1)

= −dX (2) j 2. Φ = const . , i.e. no change of the potential of the combined system due to the exchange, and thus dΦ = d(Φ(1) + Φ(2) )= 0. Since the remaining X i(≠n )j are kept constant, one gets:

(2) (2) d Φ = Y j(1) dX (1) = (Y j(1) − Y j(2) ) dX (1) j + Y j dX j j =0

i.e. Y j(1) = Y j(2) At equilibrium, the intensive variables conjugate to the exchanging extensive quantities take the same value. The condition dΦ = 0 defines an extremum. It can be shown that in the case of “energy type” potentials like enthalpy H, free energy F or Gibbs energy G, the extremum is a minimum. In the case of “entropy-type” potentials the extremum is a maximum. 1.6.1. Reduction of dimension due to equilibrium: Every condition for equilibrium reduces the degree of freedom of the system by one. Taking the system Φ as the combination of two subsystems

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89

(1) and (2) with dimension r1 and r2 , the number of extensive variables is r1 + r2 . The Gibbs function of the combined system is r1

r2

j =1

j =1

Φ = Φ (1) + Φ ( 2 ) = ∑ Y j( 1 ) X (j1) + ∑ Y j(2 ) X (j2 ) Let us suppose an equilibrium by a free exchange of the variables

X k( 1 ) + X m( 2 ) = const . = X 0 this implies:

dX m(2 ) = −dX k(1) and Yk(1) = Ym(2 )

Consequently:

(

(1)

( 2)

d Φ = Yk − Yk r1

) dX

(1) k

r1

r2

j ≠k

= ∑ Y j( ) dX (j ) + ∑ Y j( ) X (j 1

1

j ≠k

1 = d Φ (( r1)−1)

r2

+ ∑ Y j dX j + ∑ Y j( 2) X (j 2) (1)

2

(1)

j≠k

2)

j ≠k

2 + d Φ (( r2)−1)

Due to the equilibrium condition each of the subsystems has a dimension reduced by 1. 1.6.2. Equilibrium with a reservoir Reservoir: A system with only one degree of freedom is called a reservoir of a given energy form dE = YdX if it can exchange energy only at a fixed value of the intensive variable Y. Let us denote this fixed value by Y0. Let us take system (2) as a reservoir. We then can write

(

)

r1

r1

j≠k

j≠k

d Φ = Yk(1) − Y0 dX k(1) + ∑ Y j(1) dX (j1) = ∑ Y j(1) dX (j1) This shows that for an equilibrium with a reservoir the dimension of the system is reduced by one: the Massieu-Gibbs function is of first order in the r1 − 1 extensive variables. Remark: it is important to realize that an equilibrium is only established if a free exchange of variables is occurring. This can be confirmed in experiments. If e.g. iron is exposed at high temperatures to a high carburizing atmosphere, graphite will be deposited on the surface. If this deposition occurs with the basal plane of graphite parallel to the iron surface, then the C-atoms are so strongly bound within the basal plane that no free exchange is possible. Consequently, equilibrium between graphite and iron is not established in that case.

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G. INDEN

2. Chemical Reactions 2.1. REACTIONS BETWEEN GASES

At a given temperature T and pressure P the Gibbs energy of an ideal gas with reference to the state at pressure P0 and n=1 mole, is given by

G(P, T, n ) − G(P0 , T, n ) =

P

P

P0

P0

∫ VdP = nRT ∫

⎛P⎞ dP = nRT ln⎜ ⎟ P ⎝ P0 ⎠

(21)

The chemical potential is obtained as

μ=

⎛ P ⎞⎤ ⎛P⎞ ∂G d ⎡ 0 = ⎢nGm + nRT ln⎜ ⎟⎥ = G 0 (P0,T,n = 1) + RT ln⎜ ⎟ ∂n dn ⎣ ⎝ P0 ⎠⎦P ,T ⎝ P0 ⎠

(22)

Let us consider a mixture of ideal gases with species A1, A2,… Ak, and mole numbers ni within a volume V. Each of the gases has to fulfil the equation Pj V=nj RT. Pj is called partial pressure of component j. It is defined as

Pj =

nj

∑n

P = xjP j

j

The system composed of the different gaseous species thus fulfils the relation

V ⋅ ∑ P j = V ⋅ P = RT ∑ n j = nRT

Let us consider a system with n1 moles of species A1, n2 moles of species A2, etc.. A reaction may be defined by the symbolic relation

n1 A1 + n 2 A2 + ... + n i Ai = n i +1 Ai +1 + n i + 2 Ai + 2 + ... + n r Ar r

or

∑ν

j

Aj = 0

(23)

j =1

with coefficients ν j = −n j for 1 ≤ j ≤ i and ν j = n j for i + 1 ≤ j ≤ r . Since the reaction proceeds with fixed values of ν j , we can define the degree of advancement ξ of the reaction in such a way that for every step dξ the amount of reactants i

∑n

j

dξ

j

dξ

j =1 r

transforms into the products

∑n j = i +1

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At constant temperature and pressure the change in Gibbs energy associated with the reaction is given by r ⎛ r ⎞ ΔG = ∑ μ j dn j = ⎜⎜ ∑ν j μ j ⎟⎟ ⋅ dξ j=1 ⎝ j=1 ⎠

(24)

The term in parenthesis is called Gibbs energy of reaction:

Δ rG =

def

r

∑ν

j

μj

j=1

According to (22) the chemical potentials are given by

⎛ Pj ⎞ ⎟ ⎝ P0 ⎠

μ j = G 0j (P0, T,1) + RT ln⎜

(25)

ΔG 0j defines the Gibbs energy of 1 mole of species j in its standard state, i.e. at pressure P j = P0 = 1 bar . The Gibbs energy of reaction thus becomes r ⎛r ν ⎞ r Δ rG = ∑ν j G 0j + RT ln⎜ Π P j j ⎟ = ∑ν j G 0j + RT ln(Q) (26) ⎝ j=1 ⎠ j=1 j=1

For Δ rG < 0 the reaction proceeds forward and for Δ rG > 0 the reaction proceeds backward. The first term on the right hand side of (26) is called standard Gibbs energy of reaction: r

Δ rG 0 = ∑ν j G 0j

(27)

j=1 r

ν

while Q = Π P j j is called reaction quotient. j=1

At equilibrium: Δ rG dξ = 0 . Here dξ represents an arbitrary positive or negative deviation from equilibrium. Consequently, Δ r G = 0 , or

Δ rG 0 = −RT ln(K )

(28)

The reaction quotient at equilibrium is called equilibrium constant K=Qequil. If the value of Δ rG 0 is given for a temperature T0 the value corresponding to a temperature T is obtained from: Δ rG 0 (P0 ,T ) = Δ rG 0 (P0 ,T0 ) − (T − T0 )Δ r S 0 (P0 ,T0 ) +

T

∫Δ c r

T0

T

P dT − T ∫ T0

Δ rc P

τ

dτ

(29)

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G. INDEN

Δ r S 0 = ∑ν j S 0j and Δ rc p = ∑ν j c (p j ) .

with

j

The specific heat capacity of species j is given by

c

=

( j) p

∂H ( j ) ∂T

P

Equation (29) shows that the Gibbs energy of reaction can be expected to be a linear function of temperature since for gases the term Δ r c P is small and virtually constant. The corresponding diagram versus temperature is called Ellingham diagram. Table 1 gives the thermodynamic data for gaseous substances in their standard states at T = 298 K. There from the standard Gibbs energies of reaction can be deduced for the following reactions:

(H 2 )g + 1 (O2 )g = (H 2 O )g 2

ΔrG0

= −228570 − ( T − 298 )(188.83 − 130.684 − 102.57 ) + ( −9.9 )( T − 298 )

(30)

= −228570 + 54.25 (T − 298 ) = −244736 + 54.25 T

J/mole H 2O

( CO ) g + ( O2 ) g = ( CO2 ) g 1

2

Δ rG 0 = −394360 + 137170 − (T − 298)(213.74 −197.67 −102.57) + (−6.7)(T − 298) = −257190 + 79.8 (T − 298) = −280970 + 79.8 T

(31)

J/mole CO2

Application: It is often required to maintain a very low partial pressure of oxygen in a gas atmosphere. This can be realized by taking an appropriate mixture of water and hydrogen. Suppose a partial pressure of 10–10 bar O2 is required at T = 2000K. Introducing the data of Eq. (30) into Eq. (28), one gets the equilibrium constant at T = 2000K:

⎛ PH O 2 − RT ⋅ ln⎜ ⎜P ⋅ P O2 ⎝ H2 PH 2O PH 2

⎞ ⎟ = −136236 ⇒ K = 3616 ⎟ ⎠ = 3.616 ⋅ 10 −2 .

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TABLE 1. Thermodynamic data of gaseous substances, The values correspond to T=298K, P=1bar and 1mole Gaseous species

(O2 ) g (H 2 ) g (H 2O) g (CO) g (CO2 ) g

ΔGi0 J/mole

S 0 J/mole·K

c P J/mole·K

0 0 –228 570 –137 170 –394 360

205.138 130.684 188.83 197.67 213.74

29.35 28.82 33.58 29.14 37.11

If the partial pressure of H2 is set to 1 bar, then

PH 2O = 0.036 bar . This partial pressure corresponds to the saturation pressure of water at 28°C. The required gas mixture can thus be obtained by bubbling hydrogen gas at 1bar through liquid water at 28°C. The partial pressure of oxygen can also be controlled using reaction (31). Suppose a pressure of 10–10 bar O2 is required at T = 1500K. The equilibrium constant is obtained from

⎛ P ⎞ CO2 ⎟ = −161270 ⇒ K = 4.132 10 5 −RT ⋅ ln⎜⎜ ⎟ ⎝ PCO ⋅ PO2 ⎠ PCO2 = 4.132 PCO

If the total pressure is 1 bar, i.e. PCO2 + PCO = 1 bar then

PCO2 = 0.805 bar

PCO = 0.195 bar

The required gas mixture can thus be obtained by taking the gases in the ratio 80 volume% CO2 and 20 volume% CO. 2.2. REACTION BETWEEN GASES AND SOLID PHASES

In the same way as shown for the reaction with gases, the standard Gibbs energies of reaction between solid and gaseous phases are obtained. However, for solid phases the pressure term in Eq. (21) is practically zero because of negligible variation of volume with pressure. Consequently Eq. (25) yields for solid phases:

μi = Gi0 (P0,T,1)

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G. INDEN

The missing pressure term for the solid phases implies that the reaction constant contains only the partial pressures of the gaseous species. This means: equilibrium according to Eq. (28) for the reaction

M

s

+ (O2 )g = MO2

s

is thus given by

⎛ 1 ⎞ ⎟⎟ = RT ln(PO2 ) Δ rG 0 = −RT ln⎜⎜ P ⎝ O2 ⎠

(32)

and named standard Gibbs energy of reaction. The Ellingham diagram represents the standard Gibbs energy of reaction as function of temperature. The essential features of the diagram shall first be discussed with schematic diagrams. In Figure 7a the case of oxidation of a solid metal to form a solid oxide is shown. The standard Gibbs energies of two reactions are represented. It is possible to include the right hand side of Eq. (32) into this diagram. This leads to lines with slope RT Ln PO2. In the range of negative energy values the negative slope induces pressures less than 1 bar. The equilibrium condition given by Eq. (32) is fulfilled at the intersection of these lines with the Δ rG 0 curves. The slope of the line gives the corresponding equilibrium partial pressure of O2. These equilibrium partial pressures can be labelled on a separate nomographic axis as shown in Figure 7a. In this figure the standard Gibbs energies of reaction are shown for two reactions:

2 M s + (O2 )g = 2 MO 2 A s + (O2 )g = 2 AO

(i)

s

(ii)

s

For a given temperature the equilibrium vapour pressure O2 can be read from the nomographic pressure axis by drawing a line from the origin 0 (T = 0K, Δ rG 0 = 0 ) to the corresponding point on the line Δ rG 0(T ) of the reaction. At the temperature Tequil of the intersection of the two lines Δ rG 0(T ) for (i) and (ii) the equilibrium for both reactions is fulfilled. The difference

Δ rG 0(iii) = Δ rG 0 (i) − Δ rG 0(ii) represents the standard Gibbs energy of reaction (iii):

(i)

− (ii) ⇔

2 M s + AO s = 2 A s + MO

s

(iii)

Since Δ rG (iii) > 0 at temperatures below T , the reaction proceeds backward, i.e. both 2 <M>S and S are stable relative to 2 S and <MO>S. Above Tequil it is the reverse. This means that if A were to be used as a reducing agent to reduce MO to form M and AO, then the reduction could only be effected at temperatures T > Tequil. 0

equil

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95

Figure 7. a) Schematic Ellingham diagram of oxidation reactions of solid phases to solid oxides, b) Schematic Ellingham diagram for an oxidation reactions between gases at different pressure ratios P(CO2)/P(CO).

The case of an oxidation reaction between gaseous species is illustrated in Figure 7b. The standard Gibbs energy of reaction corresponds to 1 bar partial pressure of the reacting gases. Maintaining PO2 = 1bar the ratio of partial pressures of CO2 and CO can be varied leading to an additive term 2 ⎞ ⎛ PCO RT ln⎜ 2 2 ⎟ ⎝ PCO ⎠

according to Eq. (26). This is shown in Figure 7b. The linear variation of the additive term with temperature allows drawing a nomographic scale for the pressure ratio.

Figure 8. Effect of a phase transition on the standard Gibbs energy of reaction: melting of metal A as reactant and melting of the metal oxide MO as reaction product.

96

G. INDEN

In Figure 8 the effect of a phase transformation on the standard Gibbs energy of reaction is shown. Taking the melting of a species I as an example, the entropy changes abruptly by the amount of entropy of melting,

ΔS Is →liq = ΔH Is →liq Tm leading to a change in slope of the corresponding standard Gibbs energy curve. Depending on whether species I belongs to the reactants or the reaction products the slope is increased or reduced. In Table 2 the standard Gibbs energies for some reactions are given. In the case of reaction 2 C

s

( )

( )

+ O2 = 2 CO g g

the solid is reacting with 1 mole of gas to produce 2 moles of gas. This leads to a substantial value of the entropy of reaction. In the case of reaction

C

s

+ (O2 ) g = (CO2 ) g

a solid is reacting with 1 mole of gas to form 1 mole of gas. Consequently, the entropy of reaction is very small. Figure 9 shows the Ellingham diagram for these and other reactions. There are three main uses of the Ellingham diagrams: •

estimation of the ease of reducing a metal oxide into the pure metal,

•

the determination of the partial pressure of oxygen in equilibrium with a metal oxide at a given temperature,

•

the determination of the CO/CO2 ratio and of the gas composition that will allow a reduction of the oxide into the pure metal. TABLE 2. Standard Gibbs energies of reaction between gases and solid phases

Reaction 2 < C > S + (O2 ) g = 2 (CO) g < C > S + (O2 ) g = (CO2 ) g < Mn > S + (O2 ) g = 2 < MnO > S 2 < Mg > S + (O2 ) g = 2 < MgO > S < Ti > S + (O2 ) g = < TiO2 > S 2 < Ni > S + (O2 ) g = 2 < NiO > S

Δ r G 0 J/mole O2 –223425–175.3 T –394133–0.837 T –769437+145.6 T –1207921+284 T–10.73 T lnT –897468+173.2 T –489109.6+197 T

The position of the line Δ rG 0 in the Ellingham diagram determines the stability of the oxide as function of temperature. Reactions closer to the top of the diagram define oxides which are less stable and easy to reduce, e.g. Ag2O. A given metal can reduce all the oxides with Δ rG 0 lines above in the Ellingham diagram. E.g. the line of 2Mg+O2=2MgO lies below the line of Ti+O2=TiO2. Consequently, Mg can reduce TiO2 to metallic Ti.

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97

Figure 9. Ellingham diagram of various metal and carbon oxides. The reactions are defined for 1 mole of O2. M=melting point, A=austenite/δ-ferrite transition of Fe.

The line for 2C+O2=2CO has a negative slope. It thus intersects many of the lines of other reactions. As soon as the carbon oxidation line goes below a metal-oxide line, carbon can reduce the metal oxide to pure metal. This makes carbon so important as a reducing agent. As an example, C can reduce Cr2O3 once the temperature exceeds approximately 1225°C. The nomographic axes at the right hand side of the Ellingham diagram allow reading the partial pressures of the atmosphere for an equilibrium between a metal and its oxide. If this value is exceeded the metal will be oxidized, if it is lower the oxide will be reduced. The axis giving the ratio of the partial pressures of CO and CO2 can be used to determine the minimum partial pressure of CO needed for reduction of the oxide. The more stable the oxide, the greater must be the proportion of CO.

98

G. INDEN

To illustrate the use of the Ellingham diagram the problem of reduction of MnO by solid carbon at 1200°C shall be taken as example. The oxidation reaction is

2 Mn s + (O2 )g = 2 MnO

(33)

s

Carbon in equilibrium with oxygen implies the following reactions:

C s + ( O2 ) g = ( CO2 ) g 2 ( CO ) g + ( O2 ) g = 2 ( CO2 ) g

(a) 2 C s + ( O2 ) g = 2 ( CO ) g (c)

(b)

C s + ( CO2 ) g = 2 ( CO ) g (d )

(34)

Figure 10 illustrates the procedure of determining the experimental conditions. At first the point corresponding to T=1200°C is fixed on the line of equilibrium (33). From the starting points at T=0K of the equilibrium lines defining the equilibria (34), lines have to be drawn to the point at 1200°C. These lines intersect the corresponding nomographic axes and define the equilibrium partial pressures.

Figure 10. Ellingham diagram of carbon and manganese oxides. The construction illustrates the condition for a reduction of MnO by a gaseous atmosphere of O2, CO and CO2.

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99

In the present instance, this construction leads to PCO = 5 10−2 , PCO2 = 2.2 10−6, PO2 = 2 10−20 (bar), PCO PCO2 = 2.310 4 Thus, at equilibrium the total pressure is practically 5·10–2 bar. If the total pressure is reduced, then spontaneous reduction of MnO will occur. 3. Thermodynamics of Alloys In the following the experimental conditions T=Const. and P=Const will be assumed. The thermodynamic description must therefore be based on Gibbs energy. As long as the size of the system does not have any effect, as in most cases of physical metallurgy, it is sufficient to consider molar quantities. Therefore, in the following only molar quantities will be described. In order to simplify the notation an index “m” will be omitted. The molar Gibbs energy of an alloy is then constructed as a sum of various contributions, starting with the contribution coming from the pure components, then the contribution due to the mixing process to form a random alloy, followed by further contributions due to the formation of special configurations like ordering or magnetic effects etc.. 3.1. REFERENCE FRAME, LATTICE STABILITY

Energies can only be defined as differences. Therefore it is necessary to define a reference frame. For a pure component i the enthalpy of the stable state, say α, at T = 273K and P = 1bar is usually taken as reference for Gibbs energy of the component:

Δ 0Giα (T ,1 bar ) = 0Giα (T ,1 bar ) − 0 H iα ( 273K ,1 bar ) The reference term is denoted 0

H iα (273K,1 bar) ≡ H iSER def

where SER stands for Stable Element Reference. Since a pressure of 1 bar is taken as the standard situation, the pressure will not be explicitly indicated in the following. If at some temperature T the state is different from the one at 273K, Gibbs energy is then described by

Δ 0Giβ (T ) = 0Giβ (T )− 0H iα (273K ) = 0Giβ (T ) − H iSER = 0Giβ (T )− 0Giα (T )+ 0Giα (T ) − H iSER = Δ 0Giα →β (T ) + Δ 0Giα (T )

The term Δ 0Giα →β represents the Gibbs energy of transformation from α to β. This term is called lattice stability. In the case of a compound θ with

100

G. INDEN

fixed (stoichiometric) composition, the molar Gibbs energy of formation from the pure elements is given by

Δ f0Gθm = 0Gθm − ∑ν i 0Giα where the ν i are the stoichiometric coefficients. The value of Δ f0G mθ then corresponds to 1 mole of formula unit, i.e. to a total amount of atoms n = Σν i . If the stoichiometric coefficients are ν i = xi its value corresponds to one mole of atoms. The Gibbs energy of formation representing an energy difference, the reference state does not enter explicitly. It can, of course, be introduced in order to use tabulated data which usually are given relative to a reference state REF (which may correspond to SER):

Δ f0Gθm =0Gθm − H REF − ∑ν i (0Giα − H iREF ) 3.2. PURE COMPONENTS

In thermodynamic databases the molar Gibbs energy of a pure component is written as a power series in T and P:

Gm − H REF = a + bT + cT lnT + dT 2 + ...+ eP + fTP + gP 2 + ... If the power series is truncated at the quadratic terms the coefficients can be related to the following quantities: •

specific heat capacity c P . Since

⎛ ∂G ⎞ H (T , P ) = G + TS = G + T ⎜ − ⎟ ⎝ ∂T ⎠ P the specific heat capacity is obtained as:

cP =

∂H ∂T

= P

∂G ∂G ∂2G ∂2G − − T 2 = −T 2 ∂T P ∂T P ∂T P ∂T P

Consequently: c P = −c − 2 d T •

molar entropy: From:

S=−

∂G ∂T P

it follows Sm = −b − c − c lnT − 2 d T − f P

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101

molar volume: From:

V=

∂G ∂P T

it follows Vm = e + f T + 2 g P •

molar enthalpy: H m − H REF = a − c T − d T 2 + e P + g P 2

•

molar free energy: Fm − H REF = a + b T + c T lnT + d T 2 − g P 2

•

molar internal energy: U m − H REF = a − c T − d T 2 − f T P − g P 2

3.3. SOLUTION PHASES

The Gibbs energy of solution phases is usually built up starting from the pure components as a so-called “mechanical mixture” which is just the amount of different elements. The Gibbs energy of the mechanical mixture is thus given by

ΔG

mech.− mixt. m

n

= ∑ x i 0ΔGi i= 1

The Gibbs energy of the solution phase is then set up in the form

ΔGm = ∑ x i 0ΔGi +

M

Gm

by various additive terms which shall describe the various forms of solution phases. 3.3.1. Ideal solution The simplest model for a solution is the random mixing of the atomic species without any particular interaction between unlike atoms. In this case only the contribution of random mixing is to be added to the Gibbs energy of the mechanical mixture. This contribution is given by the entropy of mixing: n n n ideal ideal 0 Sm = − R ∑ xi ln xi ΔGm = ∑ xi ΔGi + RT ∑ xi ln xi i =1 i =1 i =1

(35)

The ideal entropy term is not only important for the hypothetical model system, but also for real solutions. At high enough temperatures every system approaches states, which are close to random mixing since due to the factor T the entropy becomes dominant against other terms due to interactions.

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G. INDEN

We may calculate the partial molar entropy according to Eq. (18): n

Siideal = Smideal − R (ln x i + 1) + R∑ x i (ln x i + 1) = − R ln x i i= 1

Similarly, we obtain for the chemical potential:

μ i = Gi = G

n

ideal m

+ Gi + RT (ln x i + 1) − ∑ x j 0

j=1

(G 0

j

)

+ RT [ln x j + 1]

(36)

= 0Gi + RT ln x i 3.3.2. Regular solution The regular solution model tries to take into account a pair-wise interaction between unlike atoms leading to an enthalpy of mixing term. in the Gibbs energy expression. Defining by (1) ε AA ,ε

(1)

(1) and ε AB

BB

the interaction energies (at 0K) between the various nearest neighbour pairs, the energy contribution upon mixing derives from a counting of such pairs. Given the coordination number z1 of atoms in the nearest neighbour shell, the total number of pairs between N atoms is z1 N 2 . The probability for a pair to be of type i-j is given by pij = x i x j . The energy is thus:

U=

(

z1 N 2 (1) (1) (1) (1) x A ε AA + xB2 ε BB + x A xB ε AB + x A xB ε BA 2

)

Taking the pure components as the reference:

U − x A 0U A − xB 0U B = U − =

(

z1 N (1) (1) x A ε AA + xB ε BB 2

(

z1 N (1) (1) (1) − ε AA − ε BB 2 ε BA 2

)

)

Defining the energy per nearest neighbour bond A-B as (1) ε (1) = ε AB −

(1) (1) ε AA + ε BB

2

the Gibbs energy of the regular solution becomes n

G

regular m

n

n

n

= ∑ x i Gi + ∑ ∑ z1εij x i x j + RT ∑ x i ln x i 0

i= 1

i≠ j j =1

(1 )

i= 1

If the nearest neighbour interaction energy is positive, this means that A-B bonds are less stable than the average between the two like bonds. At low temperatures where the energy terms are dominating the mechanical mixture is thus more stable than the solution. This is called a phase separation tendency. If the nearest neighbour interaction energy is negative, this means

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103

that at low temperatures the most stable state is with a preference of unlike bonds. This leads to ordering. The regular solution can be generalized to higher order pair interactions. Defining the coordination number z k for the k-th neighbour shell, one can write: n n n ⎛ n (k ) ⎞ Gmregular = ∑ x i 0Gi + ∑ ∑⎜ ∑ zkεij ⎟ x i x j + RT ∑ x i ln x i (37) ⎠ i= 1 i≠ j j =1 ⎝ k i= 1

Defining the interaction coefficient Lij =

∑z

k

k

n

Gmregular = ∑ x i 0Gi + i= 1

n

∑

(ε

n

(k)

ij

)

+ ε ji( k ) one can write: n

∑ Lij x i x j + RT ∑ x i ln x i

i= j +1 j =1

i= 1

3.3.3. Chemical potentials of regular solutions The chemical potential μ i of a component i is the molar partial Gibbs energy. From Eq. (18) one gets for a ternary system:

μB = 0 GB + LAB x A (1− x B ) − LAC x A xC + LBC xC ((1− x B ) + RT ln x B μC = 0 GC − LAB x A x B + LAC x A (1− xC ) + LBC x B ((1− xC ) + RT ln xC μA = 0 GA + LAB x B (1− x A ) + LAC xC (1− x A ) − LBC x B xC + RT ln x A Similarly, for a binary system:

μB = 0 GB + LAB x A (1− x B ) + RT ln x B = 0GB + LAB x A2 + RT ln x B

μA = 0 GA + LAB x B (1− x A ) + RT ln x A = 0GA + LAB x B2 + RT ln x A 3.3.4. Activity of binary regular solutions The regular solution may be described in the same format as the ideal solution if one introduces a new variable, the activity a i , instead of xi by means of the following equation:

μ i = 0 μ i + RT ln a i

(38)

The activity thus plays the same role for the regular solution as does the mole fraction for the ideal solution. The molar Gibbs energy being given by the expression

ΔGm = Gm − ∑ x i 0Giref

= x A Δ 0GA + x B Δ 0GB + x A x B L + RT (x A ln x A + x B ln x B )

we obtain for the chemical potential:

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μB =Δ 0GB + L x A2 + RT ln x B According to the definition of activity (38) we get:

⎛ Δ 0GB + L x A2 ⎞ RT ln aB = RT⎜ ⎟ + RT ln x B RT ⎝ ⎠ ⎧ ⎛ Δ 0G + L x 2 ⎞⎫ B A = RT ln⎨exp⎜ ⎟⎬ + RT ln x B RT ⎠⎭ ⎩ ⎝ ⎛ Δ 0G + L x 2 ⎞ B A ⎟⎟ aB = x B exp⎜⎜ RT ⎝ ⎠

or

The reference state for the activity is the same as that for Δ0 GB , usually SER. Raoult’ian reference state: If the structural state of component B at a given temperature T is the same as that of the alloy, one may equally well adopt the state of the pure component at temperature T as a reference. In that case we have Δ0 GB = 0 .

⇒

aB

x B =1

=1

In the Raoult’ian reference system the activity of the pure component is 1. For the composition x B we get in the Raoult’ian reference system

⎛ L (1− x ) 2 ⎞ B ⎟⎟ aB = x B exp⎜⎜ RT ⎝ ⎠ In the case of an ideal solution, L = 0 , the activity is identical with the mole fraction. In activity vs composition plots this gives a line from x B = 0 to x B = 1 . For L ≠ 0 , the activity deviates from this line. For L>0 the deviation is towards higher activity values, see Figure 11b, for L<0 it is towards lower activity values, see Figure 12b. The limiting behaviour of the activity at xB = 0 and xB = 1 is obtained by taking for e y ≅ 1 + y at y = 0 :

⎛ L ⎞ x B → 0 : aB ≅ x B exp⎜ ⎟ Henry’s law* ⎝ RT ⎠ ⎛ L 2⎞ x B → 1 : aB ≅ x B ⎜1+ (1− x B ) ⎟ ≅ x B Raoult’s law* ⎝ RT ⎠

______ *

William Henry, 1775-1836. English physician and chemist who in 1803 proposed what is known as Henry’s law.

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Figure 11. Variation of activity with composition and temperature for a system with segregation tendency between the atoms A and B (L>0). At low temperatures a miscibility gap opens. a) Iso-activity lines of component B. The tie-lines in the miscibility gap are branches of iso-activity lines, b) Variation of activity vs composition. The activity is constant for compositions within the miscibility gap.

Figure 12. Variation of activity with composition and temperature for a system with ordering tendency between the atoms A and B (L<0). At low temperatures long range ordered phases become stable. The ordered phase in the case of the body centered cubic lattice is B2 : a) Isoactivity lines of component B, b) Variation of activity vs composition. The strong bulging downwards at low temperatures results from the ordering reaction.

3.3.5. General substitutional solution The description of a general solution is based on an extension of the concepts shown above in terms of polynomial developments. The starting *

François Marie Raoult, 1830-1901. French chemist and physician, professor at the University of Grenoble.

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point is the ideal solution, and any extra term is then subsumed in a socalled “excess term” E Gm : n

n

Gm = ∑ x i Gi + RT ∑ x i ln x i + E Gm 0

i =1

i=1

The excess term is written in form of a series expansion with the first term representing the regular solution: E

with

Gm = ∑ ∑ x i x j

[L + L 0

1

ij

ij

(x

]

− x j ) + 2Lij (x i − x j ) + ... 1

i

2

ν

Lij =ν Aij + ν Bij T+ ν Cij T 2 + ... The coefficients ν Lij are called interaction terms. If the expansion is truncated at the first power of ( x i − x j ) and no temperature dependence of the interaction parameters is assumed, then Gibbs energy describes the socalled sub-regular solution. 3.4. PHASE DIAGRAMS

Phase diagrams are two-dimensional representations of stability ranges of multiphase equilibria. In materials science these diagrams are usually at constant pressure of 1 bar. Temperature is either one of the axes or is also fixed. Consequently, phase equilibrium is given by the minimum of Gibbs energy. The numerical calculation of phase diagrams by computer programs like THERMO-CALC consists in finding this minimum starting from the given Gibbs energy functions of all phases. The connection between Gibbs energy and phase diagram is shown in Figure 13 for the case of the binary system Ag-Cu. This system is characterized by a segregation tendency within the face centered cubic phase (fcc). The stable phases are liquid and fcc. The Gibbs energy of both phases as function of composition is shown for three temperatures: above the eutectic* temperature TE, at T = TE and below TE. The single phase regions are those compositions where the Gibbs energy of one of the phases represents the

______

* Eutectic means either “good to make” or “well constructed” (greek: ευ = good, well; τεκτειν = to make, to build, to construct; architect = chief constructor). The term “eutectic” can be found in acient documents. It was used in the sense “good to make”, since eutectic systems exhibit lower melting points than the constituent pure components. The eutectic microstructure usually shows “well constructed” arrangements of the two solid phases formed simultaneously from the melt, leading to improved mechanical properties compared to the pure components. However, the observation of the particularly “well constructed” microstructures was possible only after the invention of incident light optical microscopy.

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lowest value. Within the two-phase regions the points of a common tangent are lower than the Gibbs energy values of either phase. These points on the common tangent represent the Gibbs energy value of a mixture of both phases

x ( s ) ΔGm( s ) + x (liq ) ΔGm( liq )

with x (α ) =volume fraction of phase α. The limits of the two-phase regions are obtained from the tangent points.

Figure 13. Phase equilibria in the binary system Ag-Cu: a) Phase diagram with eutectic point (liq)Æfcc+fcc, b) At 900°C two-phase equilibria exist between solid and liquid phases, c) Three phase equilibrium at the eutectic temperature, d) Below the eutectic temperature the common tangent defines a miscibility gap.

The common tangent construction is the graphical solution of the condition for equilibrium, i.e. equality of the chemical potentials of the exchanging species. It was shown in paragraph 1.5.3. that the chemical potentials are obtained from the intercepts of a tangent to the Gibbs energy

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curve. The common tangent thus defines equal potentials of the components for the two phases. Usually the eutectic situation is between phases of different crystallographic structures. In those cases the same construction is to be made with three different Gibbs energy curves. The situation for a three component system is schematically shown in Figure 14a. The two-phase equilibrium is obtained from a common tangent plane to the molar Gibbs energy surfaces. Figure 14b shows the calculated isothermal section of the ternary system Fe-Co-Cr at T = 1100°C. Three phases are stable at this temperature: the body centered cubic (bcc) phase ferrite (α), the fcc phase austenite (γ) and the sigma phase (σ). In the isothermal section the iso-activity lines for Co have also been plotted. The tie-lines are branches of iso-activity lines since they define compositions with constant chemical potential. The iso-activity lines are of particular importance for discussing diffusion in these systems. The gradient of chemical potential or equivalently of activity represents the driving force for diffusion.

Figure 14. a) Common tangent plane construction for a two-phase equilibrium in a ternary system for the determination of the tie-line, b) Calculated isothermal section of the ternary system Fe-Co-Cr. The dotted lines represent the Co-iso-activity curves.

In Figure 15a the calculated isothermal section of the ternary system FeMn-Ni at T = 600°C is shown. This system is characterized by a large domain of homogeneous austenite γ. This system can be described by a

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regular solution approach. Here again the iso-activity lines are shown. The activity values are also given. It is worth noticing that the activity values are very different from the mole fraction of the component. This is of course to be expected for a regular solution. In Figure 19 the Gibbs energy of mixing of the two binary limitrophe systems Fe-Ni and Mn-Ni is shown together with the corresponding ideal solution. The origin of the difference is vizualised with two cases: for Fe-Ni the case of aNi = 0.3 and for Mn-Ni the case aNi .= 0.5. In the case of the ideal solution these values are identical to the mole fraction of Ni. Taking the tangent at these compositions the intercept with the Ni-axis defines the chemical potential of the ideal solution. Taking from these intercepts the tangent to the regular solution curves gives the compositions corresponding to these activities. By his construction the origin of the differences becomes transparent.

Figure 15. a) Calculated isothermal section of the ternary system Fe-Mn-Ni. The difference between the activity values and the mole fractions comes from the regular solution, b) Gibbs energy of mixing of the limitrophe binary systems Fe-Ni and Mn-Ni. In order to show the regular solution contribution, the hypothetical ideal solution term is also shown.

3.5. RESTRICTED EQUILIBRIUM: PARAEQUILIBRIUM

In alloys with substitutional elements like Fe, Mn, Cr, etc. and interstitial elements like carbon, nitrogen, hydrogen, there exist temperature ranges where the substitutional elements are practically immobile and thus cannot take part in the equilibration process, while the interstital elements are sufficiently mobile to allow a free exchange. We thus have to consider restricted equilibria, the restriction being that the free exchange is restricted to certain elements. We will consider here the case of one interstitial element C. The amount of substitutional elements is:

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∑N j =1

j

= N 1,s

def

If the size of the system is given by the total number of atoms (1 mol), then of course dN1,s = d (N − N c ) = −dNC . However, the fractions of substitutional elements, the so-called u-fractions ui = N i N1, s (i=1,..,s) stay constant. We may write the Gibbs energy change at constant temperature and pressure as s +1 s ⎛ s ⎞ NdGm = ∑ μi dN i = ∑ μ j d(ui N1,s ) + μC dNC = ⎜⎜∑ μ j ui ⎟⎟ d (N1,s ) + μC dNC i= 1 j =1 ⎝ j =1 ⎠

or

⎛ s ⎞ N N dGm = ⎜⎜∑ μ j u j ⎟⎟d 1,s + μC d C = μM dx M + μC dxC N ⎝ j =1 ⎠ N

(39)

This shows that the multi-component system is reduced to a binary system M-C, where the properties of component M are given by the average chemical potential s

μM = ∑ u j μ j def

j =1

and its mole fraction xM = N1,S /N. Taking then the composition of M as the dependent quantity, Eq. (39) becomes: s ⎛ ⎞ dGm = ⎜⎜μC − ∑ u j μ j ⎟⎟ dxC ⎝ j =1 ⎠

The case of a ternary system A-B-C with two immobile components A, B and one mobile interstitial component C is illustrated schematically in Figure 16. The para-equilibrium phase boundaries and tie-line are shown by solid lines. The ortho-equilibrium phase boundaries and tie-lines are shown as broken lines. The case shown in Figure 16a is representative for austeniteferrite equilibria in alloys like Fe-Mn-C or Fe-Ni-C, while that in Figure 16b corresponds to an equilibrium with a carbide like cementite. Calculated phase diagrams for Fe-Mn-C and Fe-Ni-C are shown in Figure 17.

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Figure 16. Para-equilibrium in a ternary system with one interstitial component C. Paraequilibrium is determined by a common tangent construction to the Gibbs energy curves obtained by cutting the Gibbs energy surfaces by a vertical section for fixed u-fraction uB. The tangent defines the chemical potential μ C on the axis 0 GC and, in the plane defined by the axes 0 GA and 0 GB, the value μ A uA + μ B uB , a) Situation corresponding to ferriteaustenite transitions, b) Situation corresponding to the case of a carbide like cementite.

Figure 17. Calculated - para-equilibrium (shaded area) and ortho-equilibrium of Fe-MnC and Fe-Ni-C. In (a) cementite was suppressed in the calculation. In the case of Fe-Ni-C cementite was included in the ortho-equilibrium calculation.

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References The numerical calculations of thermodynamic equilibria were performed using the software THERMO-CALC together with the database TCFE5 [1] Thermodynamic data were taken from Tables in [2] and [3]. For further reading the references [3]-[6] are recommended. [1] Thermo-calc Software AG, Stockholm, Sweden; http://www.thermocalc.com [2] Barin, I., 1989, Thermochemical Data of Pure Substances, Part I and II, VCH Verlagsgesellschaft, Weinheim/Germany. [3] Atkins, P.W., 1986, Physical Chemistry, Oxford University Press. [4] Callen, H. B., 1985, Thermodynamics and an Introduction to Thermostatics, J. Wiley &Sons, New York. [5] Lupis, C. H. P., 1983, Chemical Thermodynamics, Elsevier Science Publishing Co., Amsterdam. [6] Gaskell, D. R., Introduction to the Thermodynamics of Materials, 2003, Taylor&Francis, New York.

KINETICS OF PHASE TRANSFORMATION IN MULTI-COMPONENT SYSTEMS G. INDEN* Max-Planck-Institut für Eisenforschung GmbH Düsseldorf, Germany

Abstract: The classical approaches of the kinetics of phase transformations are based on empirical descriptions. In multi-component systems the effects of large differences in the mobility of the elements as well as the strong coupling of the thermodynamic driving forces lead to short and long term effects that cannot be treated by these approaches. In order to arrive at a predictive modelling numerical simulations are required which take full account of the thermodynamics and of mobilities. It is shown that this fundamental approach is capable of predicting the temporary existence of metastable phases, the various reaction regimes of fast and slow reactions and the transition between them leading to a transformation stasis. An excellent agreement between simulation and experiment is obtained without fitting. Keywords: Phase transformation; kinetics; local equilibrium; para-equilibrium; precipitation stasis

1. Introduction The properties of materials are strongly dependent on the microstructure which is given by the distribution of precipitates within a matrix. The precipitation is usually obtained by means of a heat treatment. It is thus important to know which phases are possible candidates for precipitation during a given heat treatment. Since a heat treatment is performed for a limited amount of time, the resulting microstructure cannot be expected to correspond to full equilibrium. Due to limits in time scale of industrial production and treatments, meta-stable phases may be kinetically favoured within a certain time interval compared to stable phases. Then it becomes important to now about the competition between stable and meta-stable phases as well as about the stability of a given microstructure under long term service conditions.

______

* Prof. Dr. Gerhard Inden, Im Weidengrund 19, D-40878 Ratingen, Germany; e-mail: [email protected]

V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

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The microstructure also depends strongly on alloying additions. Here the question arises which element promotes or prevents the formation of precipitates. Technical materials are generally multi-component systems. This increases the complexity of microstructure formation to an extent that in the past drastic simplifications and empirical concepts had to be used in the treatment of phase transformations. The class of materials to be covered in this report are steels. For these materials thermodynamic and kinetic databases are available as well as the software for performing thermodynamic and kinetic calculations. Steels offer a large variety of different transformation regimes due to large differrences in the kinetic parameters between phases as well as between substitutional and interstitial elements. The regimes are controlled by different boundary conditions at the moving interfaces which are imposed by thermodynamics. 2. Inducement to New Approaches 2.1. KOLMOGOROFF-JOHNSON-MEHL-AVRAMI APPROACH

In view of the complexity of the matter the methods applied for treating microstructure formation were essentially based on empirical approaches. A very successful approach was provided by the so-called KolmogoroffJohnson-Mehl-Avrami (KJMA) formalism [2-4]. Here the case of isotropic spherical growth shall be recalled. Kolmogoroff starts with the assumption of a growth rate of a particle. In the case of isotropic spherical growth this takes the form r (t ) = g ⋅ t n . Assuming site saturated nucleation, i.e. all N nuclei are formed at t=0, the evolution of the volume fraction is given by 3 1 N 4π 4 π 3 3n Vk = N g t n ) = NV gt ∑ ( V k=1 3 3

(1)

Kolmogoroff considered a random distribution of particles and hard impingement. The fraction of untransformed matrix 1 − f can be defined as the probability of a point being not within any of the particles, i.e.

⎛ V ⎞⎛ V ⎞⎛ V ⎞ 1 − f = ⎜1 − 1 ⎟ ⎜1 − 2 ⎟ ⎜ 1 − 3 ⎟ ..... ⎝ V ⎠⎝ V ⎠⎝ V ⎠ N Vk ⎛ V << 1 ⇒ ln (1 − f ) = ∑ ln ⎜1 − k V ⎝ V k =1 Combining (1) and (2) yields

N Vk ⎞ ≈ − ∑ ⎟ ⎠ k =1 V

(2)

KINETICS OF PHASE TRANSFORMATION

⎛ 4 π 3 3n ⎞ f = 1− exp⎜−N V gt ⎟ ⎝ ⎠ 3

115

(3)

The KJMA equation is usually written as

f = 1− exp(−K t ν )

(4)

The exponent takes the value ν = 3 for constant growth rate (n = 1) and ν = 3 2 for parabolic growth rate (n = 1/2). The KJMA- equation, Eq. (4) assumes that the transformation continues until the matrix has transformed completely, i.e. f = 1 . If the reaction stops at an equilibrium fraction f equil less than unity, as e.g. in diffusion controlled transformations, the KJMA-equation must be modified. First, Eq. (3) must be multiplied by f equil in order to get the end of the reaction at the equilibrium fraction. Furthermore, the total volume accessible to the precipitate is not V, but V equil = f equil V . Consequently, in Eq. (3) the quantity NV must be replaced by

NVequil =

N V

(

equil

=

NV f equil

f = f equil 1 − exp ( − K tν ) with K = −N V

)

(5)

4π 3 g . 3 f equil

The applicability of this approach shall be illustrated with the austenite/ferrite transformation in Fe-0.51%C at T = 749°C. For this example experimental data of volume fraction of ferrite and of density of nuclei are available from a 3-D metallographic analysis which was performed by serial sectioning [5]. The result of the DICTRA simulation and of the KJMA equation is shown in 1a. In the DICTRA simulation only the density of nuclei enters. This defines the spherical cell size: NV = 290 nuclei/mm3 ⇒ r = 94μm. For the KJMA equation three parameters are needed: NV, the parabolic growth rate factor g, and the equilibrium fraction –7 1/2 f equil . The factor g is taken from the DICTRA simulation, g =8·10 m/s , the equilibrium fraction is obtained from a thermodynamic equilibrium calculation, f equil = 0.17 . Figure 1a shows a good agreement between the DICTRA simulation and the KJMA equation in this simple example. The same holds for the ternary case shown in Figure 1b where the effect of different densities of nuclei is shown.

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It is tempting to deduce from Figure 1b a faster kinetics in the case of higher nucleus density. This, however, is not correct. This misinterpretation is suggested due to the normalisation of volume precipitated. If instead of volume fraction the radius of the particles is plotted, it becomes evident that the growth rate is the same in both cases, only the time of impingement is decreased in the case of higher nucleus density, see Figure 2.

Figure 1. a) Comparison between experimental data, DICTRA simulation and KJMA equation of ferrite growth in a binary Fe-C alloy, b) Ternary Fe-Mn-C alloy: effect of different cell sizes, or equivalently two different nucleation densities: NV5 μm = 512 ⋅ NV40 μm .

These examples suggest that the KJMA approach might be generally applicable. This, however, is definitely not true. For the purpose of illustration, the example of the Fe-Mn-C system shown in Figure 1b shall be taken up. If the temperature is reduced by just 50K to T = 1000K, the evolution of the precipitation reaction shows a pronounced stasis shown in Figure 3a.

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Figure 2. Variation of particle radius vs time for two different cell sizes. The parabolic growth rate is the same in both cases. Calculation using DICTRA.

Figure 3. a) Ferrite precipitation in the Fe-Mn-C alloy of Figure 1b, but at a temperature of T=1000K, b) Ferrite precipitation in Fe-Si-C. The metallographic analysis was performed by serial sectioning [5].

This behaviour cannot be accounted for in the KJMA approach, and it is by no means singular. In Figure 3b it is shown that the addition of 1.15%Si to the Fe-0.51%C alloy discussed in Figure 1a yields an evolution which obviously shows a stasis. The volume fraction of the plateau is well below equil ≅ 0.26 was that of thermodynamic equilibrium. Of course, if the value f taken from experimental observation, the experimental data could be fitted with the KJMA equation, but this does not reproduce the long term behaviour. This shows clearly the impact of thermodynamic calculations on the interpretation of experimental results. In both instances two different growth regimes control the reaction. It will later be shown that these correspond on the one hand to a local equilibrium condition (LE), on the other to a local equilibrium with negligible

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partitioning of the substitutional elements (LENP). This cannot be treated nor being predicted by the KJMA approach. This clearly shows that there is the need for a more sophisticated approach with reliable predictive capabilities covering all these situations. 2.2. DIFFUSION IN MULTI-COMPONENT SYSTEMS

Diffusion is generally treated on the basis of Fick’s diffusion equations. The process of substitutional diffusion is usually by means of a vacancy mechanism. Taking the lattice fixed frame of reference and the component A as dependent quantity the fluxes are given for a binary system by:

J B = − DB

∂c B D ∂x =− B B ∂ξ Vm ∂ξ

∂c D ∂x D ∂x J A = −DA A = − A A = + A B ∂ξ Vm ∂ξ Vm ∂ξ

(6)

Here ξ represents the space coordinate, xi the mole fraction of component i, Vm is the molar volume. In a binary system we thus need to know two diffusion coefficients. Instead of the lattice fixed frame of reference one may introduce the socalled “number fixed” frame of reference. In that reference frame the two fluxes are coupled in opposite direction such that J A' + J B' = 0 . Instead of Eq. (6) one gets

J B' = −

D˜ ∂x B Vm ∂ξ

D˜ = x A DA + x B DB

(7)

and one may work with only one diffusion coefficient. The extension to multi-component systems is straight forward, with the number fixed reference frame defined by n

∑J k =1

' k

=0

In a ternary system one needs four diffusion coefficients. However, since the work of Gibbs we know that the true driving forces for diffusion are chemical potential gradients rather than concentration gradients. Therefore, the diffusion coefficients turn out to be strongly composition and temperature dependent since they contain all thermodynamic properties and correlations. It is not feasible to determine these complicated variations from experiments. The importance of this thermodynamic coupling becomes evident if one considers the phenomenon of “up-hill” diffusion found experimentally by

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Darken in a famous experiment with a diffusion couple. The experimental finding is shown in Figure 4a together with the result of DICTRA calculation. Both show an excellent agreement. If the concentration gradient was the driving force for diffusion of C, one would expect an error function profile levelling out the tiny difference in C-contents between the two halves of the diffusion couple. At the time step of the experiment, t = 1123200s, the Si profile did not change in time. It remains practically constant up to about t = 108s, see Figure 4b.

Figure 4. a) Up-hill diffusion of C. Comparison between experiment and DICTRA simulation, b) Calculated Si-profiles. Up to 108s there is practically no Si diffusion. Only at later stages the Si profile changes towards a constant equilibrium value.

Figure 5a shows the C-profiles for a series of time steps up to t = 108s calculated with DICTRA which works with the true driving forces. The corresponding C-activity profiles are shown in Figure 5b. They clearly show that the thermodynamic driving force, i.e. the gradient of chemical potential or equivalently of activity, drives C into the part which is Si-free. After 108s the activity gradient is almost constant and varies only according to the developing Si-profile. This phenomenon of up-hill diffusion also plays a role in precipitation reactions. As an example, the growth of Laves phase in a Fe-12%Cr-3%W0.1%C alloy shall be discussed. The composition of the Laves phase is close to Fe0.5W0.35Cr0.15. The growing phase thus absorbs a high amount of W and Cr. Figure 6 shows the resulting Cr-concentration profiles during the growth of Laves phase. Figure 6a representing mole fractions shows that Cr diffuses against its concentration gradient towards the Laves phase. This cannot be deduced from profiles in wt% shown in Figure 6b which seem to suggest that Cr diffuses away from the growing Laves phase. This is due to

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the take up of a high amount of W in the Laves phase and the resulting depletion of W around the growing phase. It is thus mandatory for any reliable treatment of diffusion controlled transformations to take the full thermodynamics into account in order to obtain the true driving forces for reaction.

Figure 5. a) Up-hill diffusion of C. Calculated C concentration profiles for various time steps during the C-diffusion regime, b) Calculated C activity profiles during the C-diffusion regime. After 108s the activity is almost constant. Further C-diffusion is coupled to Si diffusion.

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Figure 6. Cr concentration profiles calculated with DICTRA for various times steps during growth of Laves phase: a) Profile in mole fraction. The Laves phase absorbs Cr which diffuses towards the Laves phases against its composition gradient, b) Cr-profiles expressed in wt%. This representation cannot be used to discuss the mechanism of diffusion. The uphill diffusion of Cr does not become evident from those profiles.

3. Numerical Treatment of Diffusion Controlled Transformations The thermodynamics of multi-component systems is available with the aid of the so-called CAPHAD approach [6-8] which provides the software and databases for thermodynamic calculations of equilibria, phase diagrams and thermodynamic properties. The method is based on the concept of deriving, by numerical optimization techniques, the thermodynamic functions of a system from all available experimental data. The thermodynamic functions are written as polynomials of composition and temperature. The numerical values of the polynomial coefficients are the result of the optimization. Although this method aims primarily at calculating complex phase diagrams, i.e. focusing on equilibrium states, the thermodynamic functions also provide information for metastable states. Such information is needed for the treatment of phase transformations. In multi-component systems the condition of local equilibrium at a moving interface differs very much from the global equilibrium represented in a phase diagram. It is thus very important to get information about the thermodynamics of phases in their ranges of metastability. Meanwhile, a variety of thermodynamic databases and software has been developed covering large parts of the wide field of materials and phases: THERMO-CALC [9], CHEMSAGE and FACTSAGE [10-11], PANDAT and MTDATA [12]. In the present work the thermodynamic calculations were performed with the software THERMO-CALC.

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3.1. MOBILITY

Thermodynamics states that the flux of an extensive quantity is proportional to the thermodynamic driving force. This driving force is given by the gradient of the conjugate intensive quantity. In the case of entropy flow (heat flow) the driving force is given by the temperature gradient since temperature is the conjugate intensive quantity to entropy. In the case of particle flow the conjugate intensive variable to the particle number is the chemical potential. The particle flux is also proportional to the concentration of particles. The proportionality factor is called mobility M:

J i = −M i c i

∂μi ∂μ ∂c = −M i c i i i ∂ξ ∂c i ∂ξ

(8)

∂μi ∂ξ

(9)

Comparing (6) and (8) yields

Di = M i c i In the case of an ideal solution with

∂μ i RT = ∂ci ci

one gets Di = M i RT . Considering the ternary system A-B-C and taking component A as the dependent component the chemical potentials depend on both concentrations c B and cC : μ B = μ B (c B , cC ) and μ C = μ C (c B ,cC ) . Consequently,

⎡∂μ ∂c ∂μB ∂μ ∂c ⎤ = −M B c B ⋅ ⎢ B B + B C ⎥ ∂ξ ⎣ ∂c B ∂ξ ∂cC ∂ξ ⎦ ∂c ∂c = −DBB B − DBC C ∂ξ ∂ξ ∂c ∂c JC = −DCB B − DCC C ∂ξ ∂ξ J B = −M B c B

In order to describe diffusion in a ternary system four diffusion coefficients are needed. In an n-component system (n-1)2 coefficients are needed, forming the diffusivity matrix. This matrix is not symmetrical, and all the terms depend strongly on temperature and composition due to the thermodynamic coupling. In contrast to this, only (n-1) mobilities are needed. From those, the diffusivity matrix can be evaluated using the thermodynamic potentials.

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The mobility contains contributions related to the vacancy concentration and to the exchange rate of atoms with a vacancy. Both terms refer to thermally activated processes. Therefore, the mobility is given the form

Mk =

⎛ − ΔGk∗ ⎞ M k0 ⎟⎟ exp⎜⎜ RT RT ⎝ ⎠

where M k0 is related to a frequency factor and ΔGk∗ is a Gibbs energy barrier. Both terms depend on composition. In the mobility database these terms are represented by a Redlich-Kister polynomial very much the same way as it is done for the thermodynamic functions in the corresponding databases. For magnetic systems a magnetic factor has also to be introduced. A mobility database was developed based on assessments of diffusion experiments [13-15]. In the mean time, the data for a large number of elements and phases are available which are included in the database MOB2. 3.2. THE CONCEPT OF A SHARP INTERFACE

The way of treating diffusion in multi-component systems was outlined in [13] forming the basis of the DICTRA program. The treatment of the moving boundary problem in DICTRA is outlined in [16,17] Applications were presented in several review articles [1,18-21]. The concept of a sharp interface assumes that two phases, e.g. α and γ, are in direct contact and can exchange diffusing elements across the boundary. If this is a free exchange, i.e. that no extra driving force is needed for the elements to cross the interface, then a local equilibrium with respect to this exchange is established. This local equilibrium is defined by the equality of the chemical potentials of all diffusing elements. The numerical value taken by these chemical potentials is determined by the mass balance condition:

⎛ x γk x αk ⎞ Jk − Jk = v ⎜ γ − α ⎟ ⎝ Vm Vm ⎠ γ

α

(10)

The mass balance provides n equations. The unknowns are the (n-1) independent chemical potentials and the velocity v. The values of the chemical potentials determine the compositions at the interface within the two phases, i.e. xkα and x kγ . 3.2.1. Local equilibrium (LE) Local equilibrium at an α/γ interface means μ αk = μ γk k=1,2,..n-1. In a binary system the degree of freedom is zero. Consequently there exists only one

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solution for the equality of the chemical potentials, and this solution is given by the global equilibrium. The compositions at either side of the boundary are given by the corresponding tie-line, see Figure 7a.

Figure 7. a) LE boundary condition in a binary system, b) LE boundary condition in a ternary system.

The situation in higher order systems is different and may be illustrated with a ternary system as shown in Figure 7b. The system Fe-Ni-C was selected with C being a fast diffuser. Given an alloy composition in the twophase field α+γ, the operating tie-line is defined by the velocity of the interface obtained from the flux balance with respect to every diffusing species. To meet this condition, C as a fast diffuser needs only a small driving force compared to Ni. The first operating tie-line is thus very close to the intersection of the iso-activity line of the austenite matrix with the phase boundary. The reaction proceeds with this boundary condition until the driving force for C-diffusion has decreased to an extent that Ni diffusion becomes competitive. Then the operating tie-line shifts towards the global equilibrium tie-line. During the whole growth process both Ni and C must diffuse. This leads to a slow reaction controlled by the substitutional element diffusion. In alloys with interstitial elements like C two sublattices must be considered, one for the substitutional elements defining the volume, and the interstitial sublattice. Then it is useful to introduce in addition to the mole

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fractions so-called u-fractions which define the fraction of substitutional elements S:

ui =

Nj Ni xi = = ∑ N j N − N C 1 − xC

(11)

j ∈S

The operating tie-line shown in Figure 7b is not only valid for the alloy considered. It holds for a whole series of compositions falling on the isoactivity line which corresponds to the alloy. The difference in Ni content between the γ-matrix and the α-particle defines the amount of redistribution of Ni during the reaction. With decreasing Ni content the amount of Ni redistribution decreases. This is shown in Figure 8a for the alloys 1 to 3. When the Ni content reaches the line of Ni content in growing α, ferrite inherits the Ni content of austenite and no redistribution of Ni is required. This holds for the alloys 4 to 6 in Figure 8a. The reaction being only Cdiffusion controlled, this regime is called “local equilibrium no partiioning” (LENP). The transition line between LE and LENP is obtained by the intersection of a line of constant Ni content with the iso-activity line corresponding to the interface composition in γ (open circles in Figure 8a). The LENP reaction with no redistribution of substitutional elements is the result of local equilibrium with respect to all components. It is not a conesquence of limited mobility.

Figure 8. a) Regions of compositions where the precipitation reaction γÆα proceeds according the LE or LENP, b) Para-equilibrium phase diagram.

3.2.2. Constrained equilibrium (PARA-EQUILIBRIUM) Limited mobility comes into play when the reaction temperature is lowered. The substitutional elements become more and more immobile compared to

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the velocity of the interface. Then equilibrium cannot be established with respect to these elements, but it may still be valid for the fast diffusing interstitial components. This situation was discussed by Hultgren [22]. He assumed the constraint of immobility for the substitutional elements and called the corresponding constrained equilibrium “para-equilibrium”. Under this assumption a multi-component system Fe-X-C (X=Ni, Cr, V,…) is regarded as a pseudo-binary system M-C where all the substitutional elements (Fe, Ni, Cr,…) are replaced by an “average” component M. The conditions for para-equilibrium are:

μCα = μCγ

α α γ u Fe μ Fe + ∑ u αX μ αX = uFe μ γFe + ∑ u γX μ γX X

(12)

X

The first equation imposes the local equilibrium for C, the second imposes the constraint that this equilibrium has to be within the section of constant fraction of substitutional elements X. A reaction according to PARA cannot lead to the final global equilibrium state. Figure 8b shows the para-equilibrium phase diagram of Fe-Ni-C superimposed on the orthoequilibrium diagram. Since it is a pseudo-binary diagram the tie-lines are horizontal if the u-fraction axis is selected. The condition for PARA to occur is immobility of the substitutional elements, i.e. the flying distance of the interface per unit time must be larger than the average diffusion distance of the elements. This is expected to occur at very low temperatures. It may also occur at early stages of transformations with high interface velocities. 3.3. CASE STUDIES

The various reaction regimes shall be demonstrated with some case studies. Two Fe-Ni-C will be considered which correspond to the alloy numbers 3 and 4 in Figure 8a. The result is shown in Figure 9a and Figure 9b. The LE reaction is very slow, while the LENP reaction is extraordinary fast. Figure 10a shows the evolution of volume fraction including the PARA reaction. It is purely accidental that the final volume fraction of the PARA reaction turns out the same as that for LENP. The lever rule applied to the tie-lines of this alloy in Figure 8a and Figure 8b yields the same fraction of ferrite in this particular instance. Coming back to the example already discussed in Figure 3b, we can now run a DICTRA calculation using only the experimental value of the density of nuclei NV = 2200±400/mm3. It was found that nucleation took place at the cell vertices. It is thus reasonable to adopt spherical geometry.

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This implies a cell radius of r = 50μm. The result is shown in Figure 10b. The experimental data are very well reproduced without any fitting. This clearly confirms the existence and importance of LENP reactions as well as the impact on the long term behaviour of materials.

Figure 9. a) Ferrite growth according to LE, b) Ferrite growth according to LENP.

Figure 10. a) Growth kinetics of ferrite according to LE, LENP and PARA, b) Comparison between experiment and calculation. The experimentally observed stasis is well reproduced by the calculation. Equilibrium is reached only after 2.5 years.

3.4. EFFECT OF MICROSTRUCTURE

Information about the microstructure is required in order to obtain good agreement between calculation and experiment. It was already mentioned that the density of nuclei must be known in order to define the cell size. Figure 1b showed the effect of different cell sizes on the evolution of the fraction of phase formed. It is further important to make an appropriate

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choice of geometry. In the case of nucleation of individual particles spherical geometry with a nucleus at the centre is a good approach. In the case of thickening of a grain boundary layer on may select spherical linear geometry as long as the volume fraction remains small. Otherwise it is possible to assume the growth of a phase from the periphery of the spherical cell. The definition of the cell parameters is given in Figure 11. The definition has been chosen such that spherical cell radius and linear cell size are given by the half inter-particle distance. The correspondence between the two is R = 1.24 D in order to get the same volume per growing particle.

Figure 11. Definition of the cell parameters: - spherical geometry: cell radius R and particle radius, - linear geometry: half size of particle and half distance between particles D.

The numerical value for the cell size can be obtained from metallographical information such as the number of particles per unit volume, NV, or from the area fraction of particles, NA: Lin. geom..:

Spher. geom.:

D= R=3

13 1 2 NV 3 4 π NV

or

or

D=

12 1 2 NA

R=

3 4 π N A3/2

(13)

These parameters have a strong effect on the particle growth. This shall be illustrated for the case of ferrite precipitation in austenite. The alloy considered is Fe-1.73%Si-0.4%C at T = 725°C. The cell size has bee defined such that the same total volume is obtained in linear and spherical geometry. The parabolic growth rate varies considerably with the geometry, Figure 12a. The growth rate is smallest for the growth from periphery since the volume change associated with a change in thickness of the precipitate is largest in that case. The opposite holds for the growth of a spherical particle at the cell centre. The linear case falls in between the tow other extremes. Similar differences appear in the representation of volume fraction as function of time, Figure 12b.

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These results clearly show the strong sensitivity of the growth behaviour on the spatial arrangement. It is thus very important to define the geometrical parameters in accordance with the microstructure.

Figure 12. a) Variation of ferrite particle size with t for various geometries, b) Evolution of the volume fraction of ferrite for different geometries.

4. Applications 4.1. FERRITIC STEELS FOR HIGH TEMPERATURE APPLICATIONS

A way of increasing the efficiency of power plants could be a higher service temperature than presently used. Focusing on a target temperature of about 650°C, martenitic/ferritic 9-12%Cr steels are potential candidates. Austenitic stainless steels exhibit good mechanical properties at this temperature, but their physical properties like thermal expansion and thermal conductivity are inferior. The critical issue is to reach sufficient creep strength and corrosion resistance. In order to provide sufficient corrosion resistance, the 12%Cr variant shall be considered. The addition of Si can also contribute to a better resistance. The initial microstructure should be martensite with no δ-ferrite. A fine dispersion of fine dispersed particles is wanted in order to block the movement of subgrains (M23C6 carbides) and dislocations (MX carbonitrides). To achieve a sufficient creep strength over the whole life time period it is necessary to introduce further precipitates with a kinetics which is slow compared to the first carbides in order to compensate possible coarsening processes. For this purpose, Laves phase seems to be a good candidate.

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The starting point is thus given by the following requirements: •

No δ-ferrite at the austenitization temperature

•

No MX carbonitrides forming in the melt, otherwise this precipitates would be too coarse.

•

Carbides shall be precipitated during a tempering at 780°C

•

Laves phase should precipitate during long term service

In order to obtain the Laves phase, elements like W and Nb have to be considered.

Figure 13. Effect of W (a) and Nb (b) on the phase equilibria in Fe-12%Cr-0.1%C.

In Figure 13 the effect of W and Nb on the phase phase equilibria in Fe12%Cr-0.1%C is shown. In both cases there is a Laves phase field at lower temperatures, but the γ-phase field for austenitization is not available. Adding the austenite stabilising element Co changes the situation in the case of W, but not for Nb, see Figure 14. Noibium can thus be excluded as possible alloying element for the present purpose. This shows the utility of such equilibrium calculations, by means of which the candidate elements can be checked and excluded before designing experimental work. The kinetics of transformation shall be visualised by a two-step heat treatment after austenitisation: first a 2h annealing at 780°C to precipitate M23C6, followed by a treatment at 650°C representing the service conditions. In order to treat the simultaneous growth of M23C6 and Laves phase, two spherical cells have been introduced. They are coupled together with free exchange of particles, i.e. with the boundary condition of equal chemical potentials for all elements. The value of these potentials are defined by the condition that mass balance must be fulfilled. Particles leaving one cell have to go entirely into the other cell. The result of this simulation is shown in Figure 15.

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Figure 14. Effect of Co on the phase equilibria in Fe-12%Cr-0.1%C. Si and V have only a negligible effect on the phase equilibria.

The effect of Co on the transformation of the Laves phase is two-fold. Co is not taken up by the Laves phase and has thus to be transported in front of the growing Laves phase. This slows down the reaction as seen in Figure 16a In addition, it changes the diffusion of Cr from up-hill diffusion (Figure 6a) to normal diffusion along the gradient of mole fraction, Figure 16b.

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Figure 15. Two-step heat treatment: first annealing at 780°C 2h, followed by isothermal holding at 650°C corresponding to service condition. Most of the carbide M23C6 is formed at 780°C. At 650°C the precipitation continues at a slower rate. The Laves phase starts appearing only after about one year and approaches equilibrium after 10 years. For comparison, the situation corresponding to a single step annealing at 780°C is shown by broken lines.

Figure 16. Effect of Co on the growth of Laves phase in Fe-12%Cr-3%W-0.15%C.

4.2. COMPETITION BETWEEN STABLE AND METASTABLE PHASES

The precipitation of carbides in a ferritic steel Fe-12%Cr-0.1%C at 780°C shall be taken as an example. The stable phase equilibria can be taken form the phase diagram in Figure 17a. It shows that the alloy is located in the two-phase field (α+M23C6). If the starting state is ferrite, the precipitation of M23C6 is expected to occur. However, it cannot be excluded that the other

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carbides, M7C3 and even cementite Fe3C, may show up because of a fast kinetics. Of course, eventually, these metastable carbides will disappear. In order to examine this situation, three spherical cells with ferritic matrix were assumed, each with radius 5μm, with a carbide nucleus inserted at the centre of each cell. The result of the calculation is shown in Figure 17b. The formation of metastable carbides does indeed occur. It depends on the density of nuclei whether this intermediate existence holds for ours or for days during the annealing treatment.

Figure 17. a) Isothermal section of the phase diagram Fe-Cr-C, b) Simultaneous growth of stable and metastable carbides.

Figure 18. a) Cr-profile of M23C6 during growth in the ferritic matrix, b) Dissolution of M23C6 at T = 1100°C formed at T = 780°C after 104s (see Figure 18a). The particle is dissolved within t = 1590s.

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Another interesting aspect is the variation of the boundary condition with time. Because of negligible diffusion within the carbides the variation of the boundary condition is reflected by the composition profile. As an example the Cr- profile within M23C6 is shown in Figure 18a Such profiles are indeed found experimentally. It is interesting to analyse the dissolution of such a carbide particle by an austenitisation treatment, e.g. at 1100°C. At this temperature the ferritic matrix transforms directly into austenite. The carbide is then surrounded by an austenite matrix. The result of the corresponding simulation is shown in Figure 18b. The shrinking of the particle is accompanied by a smearing out of the Cr profile so that there is no Cr heterogeneity left when the particle is completely dissolved. 4.3. THE ROLE OF MICROSTRUCTURE

The effect of the microstructure on the precipitation reaction was already emphasized. It can be demonstrated by looking at recent experiments of ferrite growth in steel CK45 with composition Fe+0.49%C-0.18%Cr0.22%Cu-0.71%Mn-0.19%Ni-0.31%Si. The experiments were performed by Pariser [23]. The steel was austenitized at 1200°C and subsequently annealed at 725°C. The microstructure showed ferrite nucleated at the grain boundaries. At later stages the grain boundaries are fully covered by a layer of ferrite. The area fraction of ferrite nuclei was determined as NA = 77±21 mm–2. This corresponds to a spherical cell with radius R = 71μm or a linear cell with D = 57μm. The result of the calculation is shown in Figure 19a. It is evident that the linear geometry and the growth from the centre of a spherical cell cannot reproduce the observation. On the contrary, growth from the periphery which comes close to the microstructure observation gives a good agreement.

Figure 19. Growth of ferrite calculated for different geometries. Exp. data from [23].

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The microstructure observation also showed that ferrite on the grain boundary was growing into one grain, not into the other one. This situation is typical when nucleation takes place with a coherent interface to one grain and an incoherent interface to the other. This situation can be simulated by considering two spherical cells, but ferrite growing only in one cell. The cell boundary represents the immobile interface. C is then transported away by two ways: (a) in front of the growing interface into the second cell, (b) into first cell by short-circuit diffusion through ferrite. The result is shown in Figure 19b. Due to the high diffusivity of C in ferrite, there is only a small difference between the situation of two cells or just one cell. 4.4. LATERAL GROWTH OF GRAIN BOUNDARY CEMENTITE

The thickening kinetics of grain boundary pro-eutectoid ferrite was determined by Bradley and Aaronson [24] at temperatures between 725°C and 825°C. Some of the results are shown in Figure 20. The experiments show a parabolic growth kinetics as expected for diffusion controlled reactions. The DICTRA calculations were performed for both LENP and PARA conditions. For the early stages of thickening the linear

Figure 20. Thickening of grain boundary allotriomorphic ferrite.

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geometry was adopted with cell dimensions according to the microstructures. At the temperatures 725°C and 775°C the agreement is best for the PARA condition, although the difference between LENP and PARA is not large (Figure 20a,b). At T = 825°C it becomes evident that the reaction does occur under PARA conditions. At this temperature the LENP regime is not possible. This can be deduced from Figure 20d which shows that in the case of LENP the C-activity at the interface, aCα /γ , is less than the C-activity aCγ in the matrix. Thus, C diffuses towards the interface, a condition which is opposite to the growth of ferrite. Consequently, instead of LENP the LE condition applies which results in a very slow growth kinetics, shown by the large difference occurring between PARA and LE in this instance. 4.5. LENP- AND PARA-EQUILIBRIUM REACTIONS

In Figure 8a and Figure 8b the ranges of possible LENP and PARA reaction for the austenite/ferrite transformation were shown for the case of Fe-Ni-C alloys and T = 1000K. Recently Hutchinson et al. [25] performed experiments with very short heat treatment times and rapid quenching. The austenitization was performed at 1200°C followed by a heat treatment at 700°C. Two alloys shall be discussed: Fe-2.02%Ni-0.0885%C and Fe-2.76%C0.715%C. At 700°C both alloys may transform according to LENP and PARA. Figure 21 shows the reaction for the first alloy.

Figure 21. Austenite to ferrite transformation in Fe-2.02%Ni-0.0885%C at 700°C. The DICTRA calculation was performed assuming either PARA condition or a change of regime after 85s.

The calculations were performed assuming growth from the periphery of a spherical call. The cell radius was derived from the observed microstructure which showed the growth of ferrite from the grain boundary. At early stages of the reaction the experiments indicate a reaction according to para-equilibrium, although the difference to LENP is marginal. At later

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stages a transition from PARAto LENP occurs when the flying distance of the interface becomes comparable to the average diffusion distance per unit of time [26]. The experimental data of the volume fraction of ferrite are in accordance with the LENP reaction. The results of the second alloy are shown in Figure 22a. Also in this instance the initial transformation regime is obviously para-equilibrium. The difference between PARA and LENP is sufficiently large to justify this conclusion. At later stages, here after 66s, a transition to LENP is assumed in the DICTRA simulation which confirms the stasis observed experimentally. The starting point of the transition between PARA and LENP can be estimated as follows. At T = 700°C the diffusion coefficient of Ni in bcc Fe bcc −17 2 is DNi = 1.45 ⋅ 10 m / s . The average diffusion distance is then obtained as bcc x = 2DNi ⋅t . The change from PARA to LENP consists in building up a Ni spike at the interface such that the chemical potential of Ni becomes the same in both fcc and bcc phases. This spike is built up by diffusion in bcc, since the chemical potential of Ni in bcc is higher than in fcc for the same composition, as can be deduced from the slope of the tie-lines in Figure 8a and Figure 8b. It is reasonable to assume an average diffusion distance of about four bcc unit cells to assure that equilibrium can be locally established at the interface [25]. The velocity of the interface must then define an average flying distance per unit time which less or equal to this average diffusion distance. Taking x = 11 ⋅ 10 −10 m , the average time for diffusion over this distance is

t=

x2

≅ 0.042 s 2D The velocity of the interface should thus be lower than v≤

x ≅ 2.6 ⋅10−8 s t

In Figure 22b the interface velocity is shown as function of time. The interface velocity falls below the critical value after about 66s. Using this value for a change from PARA to LENP in the DICTRA simulation yields good agreement with the experimental observation as shown in Figure 22a. Recently, experiments of the austenite to ferrite transformation in a steel 16MnCr5 steel were performed by Pariser [23]. This steel is a multicomponent alloy with composition Fe-1.02%Cr-0.2%Cu-1-13%Mn-0.16%Si0.18%C. The comparison between the experimental results and DICTRA simulations are shown in Figure 23. In one set of experiments the austenitization was performed at 1200°C followed by annealing at 745°C, 700°C and 660°C, Figure 23a-c, a second austenitization was performed at 900°C, shown in Figure 23d.

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Figure 22. a) Austenite to ferrite transformation in Fe-2.76%Ni-0.715%C at 700°C. The DICTRA calculation was performed assuming para-equilibrium, LENP or a change from PARA to LENP after 66s, b) Velocity of interface during the para-equilibrium regime. At 66s the velocity of the interface is 2.6·10–8 m/s.

This lower austenitization temperature leads to a smaller grain size and thus also to a smaller spherical cell radius in the simulation. At T=745°C diffusion is too fast to allow a para-equilibrium reaction. The experimental data follow nicely the calculation according to LENP Figure 23a. At 700°C the reaction starts with the PARA reaction and switches at later stages to the LENP regime, Figure 23b. At 660°C the PARA reaction is obviously operative, Figure 23c. There is, of course, only a little difference between both PARA and LENP regimes. The experimental results after austenitization at 900°C, Figure 23d show more clearly the change of regime. The smaller cell size leads to an earlier soft impingement effect which reduces the velocity of transformation. Consequently, the change from PARA to LENP occurs at an earlier time. In conclusion, these results clearly show that para-equilibrium is a reaction regime that is occurring at low temperatures (in Fe-alloys essentially below 700°C) and at very early stages of the precipitation reaction when the velocity of the interface is so high that the flying distance per unit time is larger than the average diffusion distance of the substitutional elements. In that case the atoms cannot exchange freely across the interface, a condition which must be fulfilled in order to get a local equilibrium.

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Figure 23. Austenite to ferrite transformation in a steel 16MnCr5. The spherical cell sizes were determined from microstructure observations.

Acknowledgement The author is particularly grateful to Dr. G. Pariser for making available his PhD thesis on Modeling the Austenite to Ferrite Phase Transformation for Steel Development.

References [1] Schneider, A., and Inden, G., 2004, in: Continuum Scale Simulation of Engineering Materials, D. Raabe, F. Roters, F. Barlat, L-Q. Chen (Editors), Wiley-VCH, Weinheim. [2] Kolmogoroff, A., 1937, Akad. Nauk SSSR, Izv. Ser. Matem. 1:355. [3] Johnson, W., and Mehl, R., 1939, Trans AIME 135:416. [4] Avrami, M., 1939, J. Chem. Phys. 7:1103, 1940, 8:212, and 1941, 9:177.

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[5] Zureck, C., 1993, Thesis RWTH-Aachen/Germany 1993; H.P. Hougardy, G. Inden, U. Knoop, C. Zurek, Final Report of COSMOS Project. [6] Kaufman, L., and Berstein, H., 1970, Computer Calculation of Phase Diagrams, Academic Press, New York. [7] SGTE Case Book: Thermodynamics at Work, 1996, K. Hack (Ed.), The Institute of Materials, London. [8] Saunders, N., and Miodownik, A. P., 1998, CALPHAD, Pergamon Materials Series Vol. I, R.W. Cahn (Ed.), Elsevier Science, Oxford. [9] Sundman, B., Jansson, B., and Andersson, J. -O., 1985, CALPHAD 9:153, 190. [10] Erikson, G., and Hack, K., 1990, Met. Trans. B 21:1013-1023. [11] Bale, C.W., Chartrand, P., Degterov, S. A., Erikson, G., Hack, K., Mahfoud, B., Melancon, R. J., Pelton, A. D., and Petersen, S. , 2002, CALPHAD 26:189-228. [12] Davies, R. H., Dinsdale, A. T., Chart, T. G., Barry, T. I., and Rand, M., 1990, High Temp. Science 26:251-262 [13] Andersson, J. O., and Ågren, J., 1992, J. Applied Physics 72:1350-1355. [14] Jönsson, B., 1994, Z. Metallkunde 85:498-509. [15] Franke, P., and Inden, G., 1997, Z. Metallkunde 88:795-799. [16] Crusius, S., Inden, G., Knoop, U., Höglund, L., and Ågren, J., 1992, Z. Metallkunde 83:673-678. [17] Crusius, S., Höglund, L., Knoop, U., Inden, G., and Ågren, J., 1992, Z. Metallkunde 83:729. [18] Inden, G., and Neumann, P., 1996, Steel Research 67:401. [19] Borgenstamm, A., Engström, A., Höglund, L., and Ågren, J., 2000, J. Phase Equilibria 21:269-280. [20] Inden, G., 2002,CALPHAD and Alloy Thermodynamics, P. E. Turchi, A. Gonis, R. D. Shull eds., The Minerals, Metals and Materials Society (TMS), Warrendale/PA, 107. [21] Inden, G., 2003, Thermodynamics, Microstructures and Plasticity, A. Finel, D. Mazière, A. Olemskoï, M. Véron, P. Vorhees eds., Kluwer , pp. 135-153. [22] Hultgren, A., 1947, Trans. ASM 36:915-989. [23] Pariser, G. C., 2005, PhD-Thesis, RWTH-Aachen/Germany, Shaker Verlag GmbH Aachen (2006), ISSN: 0943-4631. [24] Bradley, J. R., and Aaronson, H. I., 1981, Metall. Trans. A 12:1729-1741. [25] Hutchinson, C., Fuchsmann, A., and Brechet, Y., 2004, Metall. Mater. Trans. A 35:1211. [26] Inden, G., 2007, Defect and Diffusion Forum 263:11-20.

AB INITIO APPROACHES TO DESIGNING THERMODYNAMIC PROPERTIES OF MATERIALS A. PASTUREL1, N. JAKSE Sciences et Ingénierie des Matériaux et Procédés, INP Grenoble, UJF-CNRS, 1130, rue de la Piscine, BP 75, 38402 Saint-Martin d’Hères Cedex, France

Abstract: This paper presents a brief overview of recent developments in the application of ab initio calculations to the study of bulk thermodynamic properties and phase equilibria in alloys. We also emphasize the links that presently exist between ab initio methodologies and the Calphad approach to obtain a valuable tool in the calculation of complex, multicomponent phase equilibria often found in industrial alloys. Keywords: Ab initio calculations; computational thermodynamics; phase diagrams; thermodynamic properties

1. Introduction One of the greatest challenges in computational materials science is the ability to design a new material, optimizing the composition and processing steps in order to achieve the properties required for specific technological properties. To achieve this goal, it is necessary to have a thorough understanding of how atomic features influence macroscopic behaviour. In particular, one must understand how alloy properties may change with composition, temperature and pressure. Computational thermodynamics approaches known as the CALPHAD methodology have been highly successful in studying such effects on alloy properties and in calculating the phase equilibria in complex, multi-component, industrial alloys. These methods rely on databases of free energies, obtained from an optimization process involving experimental thermodynamic data combined with observed phase diagrams. Once built the database, the computational thermodynamics programs perform minimization of the multi-component free energy functional of interest to predict phase equilibria. However these

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[email protected]

V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

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methods cannot predict the existence of phases that are not already in the database. Ab initio electronic structure theory of ‘real materials’ aims at understanding materials properties from an atomistic quantum-mechanical point of view, retaining the complexity and specificity of actual materials, without losing track of the basic laws of physics. In particular they produce properties such as energetics without reference to any experimental data. Moreover, statistical mechanics can use this information as input to investtigate the problem of phase stability as a function of composition, temperature and pressure. Then such methods appear to offer a way of predicting the thermodynamics of phases well suited to supplementing experimental thermodynamics databases. During the last decades, important theoretical and algorithmic advances in theory of condensed matter and in ab initio calculations have boosted the development of such approaches. With the continuing rapid increases in computational power, theses advances enable us to tackle problems of practical importance. As a result, ab initio modelling is starting to pay off in a dramatic way, bridging the gap between fundamental research and materials engineering. Such an approach to modelling alloy phase equilibria however raises some specific questions. Ab initio calculations are performed by solving the Schrödinger equation, able to describe the bonding between atoms using only the atomic number of the constituents as input. The challenge is to find this solution even for complex problems occurring in phase diagrams and to achieve at the same time the exceedingly high accuracy necessary to calculate phase equilibria. The complexity of actual materials occurs at different levels: (a) The number of constituents. Most industrial alloys are multicomponent with often more than ten constituents. In principle, ab initio calculations are only limited by the number of atoms in the unit cell but not by the number of constituents in the phase. However the increase of constituents is often related to the occurrence of some degree of disorder of the underlying crystalline structure. In this case, the number of possible configurations increases as CL, where C is the number of constituents and L the number of sites where constituents can occupy. The situation becomes more problematic in liquids where we have to find the configuration of minimum free energy within a continuum of possible liquid structures. (b) Another complexity is related to the physical contributions to the free energy of a phase. Configurational excitations are required to deal with the configurational disorder as discussed above. However other contributions like magnetic, vibrational contributions may also play a crucial role in the stabilization of a phase. As this time, a full accounting for all physical contributions and their couplings from ab initio calculations is not a straightforward matter.

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Predicting phase stability with absolute certainty is currently an intractable problem in materials science. When calculating phase equilibria, we often need to know phase boundaries to within a few degrees. Even for simple cases, ab initio calculations do not have this exceedingly high accuracy necessary to such determinations. The complexity of materials of interest combined with the need to achieve high accuracy defines the twofold role of ab initio calculations to designing thermodynamic properties: (a) Ab initio calculations combined with statistical mechanics must be used to unravel the main physical contributions to the free energy of a phase and to predict what will happen when composition, temperature and pressure are varying. Such calculations can be used to build the coarse-graining theories that have been so successful in other parts of science. (b) Ab initio calculations can be combined with the Calphad methodology to form a hybrid ab initio/computational-thermodynamics method. In recent years, the lattice stability data for pure elements and the formation energies of compounds from ab initio methods have been used to supplement or modify exiting thermodynamic databases. This approach is particularly useful when there is little or no information on the phases that take part in equilibria in the system under study. The coarse-graining theories may be also used to develop thermodynamic models whose form is constrained by theoretical considerations and whose parameters are physically reasonable. In section 2, we briefly present the density functional theory that is the foundation of modern electronic structure calculations. More particularly, we focus on current limitations of ab initio calculations, emphasizing that the accuracy of their predictions may depend critically on their underlying approximations. An excellent and complete summary of electronic structure theories can be found in the recent book by Richard Martin [1]. In section 3, we discuss the different links between ab initio calculations and statistical mechanics to compute not only zero-temperature energetics but also finitetemperature contributions to alloy free energies arising from configurational and vibrational excitations. In section 4, we review the possible links that may be established between ab initio calculations and the Calphad methodology. 2. Ab initio Calculations: an Overview The starting point of a quantum theory of materials is the Hamiltonian of the many-ion-many-electron system and the possibility to solve the Schrödinger equation (SE). The first step to reduce the complexity of such a system is the adiabatic approximation which allows one to study the

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electrons and the lattice dynamics independently. In this approximation, ions are assumed fixed in space while solving the SE for the many electron subsystem. But even with this approximation, the SE is rarely solvable analytically. For practical calculations in condensed matter, Hohenberg and Kohn [2], and Kohn and Sham [3] reformulated the SE, which involves all the 3N spatial coordinates of N interacting electrons, into density functional theory (DFT), a theory based on the electron density, a function of only three spatial coordinates. The system of interacting electrons is mapped in a unique way onto an effective non-interacting system with the same total density. Within this approach, the ground state energy functional may be written in the form:

EKS = Ts [n]+

∫ drV

ext

(r)n(r) + E Hartree[n]+E II + E xc [n]

(1)

Here Vext(r) is the external potential due to the ions and EII is the interaction between the ions. Ts is the kinetic energy of independent electrons and all of the many-body interactions are contained in Exc[n]. Basically, this term incorporates all the difficulties related to the original many-body problem and the task of finding a functional that embodies the required information seems just as hopeless as that of calculating the exact many-body wave function for hundreds electrons. What saves this approach is that very oversimplified approximate functionals like the localdensity approximation (LDA) work correctly at least for metallic systems. However it has some shortcomings mostly due to the tendency of overbinding, which cause too small lattice constants and too high cohesive energies. Recognising that LDA failures must in part be due to ignoring spatial variations in the density, functionals have been developed that include dependence on the gradient of the density. This approximation, known like the generalized-gradient approximation (GGA), improves predicted binding and dissociation energies. More important is that the correct magnetic ground state is predicted by GGA for ferromagnetic Fe and antiferromagnetic Cr and Mn [4]. However both approximations fail to describe correlated materials, like oxide materials and/or systems containing f-electron like Plutonium. A lot of work has been done recently to design new functionals able to describe these systems. Among them, the LDA+U scheme combines a DFT description of extended states with a Hartree-Fock like treatment of the Coulomb repulsion U between states in narrow d- and f-bands. Such an approach leads to a significant improvement in the ground state properties of oxides based on transition metals (like NiO) and actinides (UO2, PuO2). The dynamical mean-field theory (DMFT) may be considered as a many-body extension of DFT that attempts to capture the true manybody physics of the electron-electron interaction. It provides a meaningful description of the cohesive properties of high temperature phases of Pu,

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namely δ and ε phases. At this time, no unified “post-DFT” theory exists, and users will need to decide with care which improvement strategy is best suited for a property that cannot be accurately computed at the DFT level. A variety of mature DFT codes [5] are nowadays available to provide an accurate framework for calculating the relative energetics of competing structures in solids, liquids for a wide range of materials. However DFT calculations are actually limited to structures containing around 500 atoms, while ab initio Molecular Dynamics are practically limited to time scales less than 1ns. It is clear that for most cases in materials science, direct applications of DFT to the calculation of thermodynamic properties remain intractable. Effective strategies have nonetheless been developed to bridging the size and time-scale limitations imposed by DFT. In the next section, we present integrated approaches to the design of phase equilibria that combine accurate ab initio total-energy calculations and statisticalmechanical modelling. This combination of methods makes possible parameter-free predictions of the finite-temperature thermodynamic properties governing the phase stability of specific metallic materials like compounds and solid solutions. 3. Ab initio Thermodynamics 3.1. ORDERED SYSTEMS

In an ordered non-magnetic solid, thermal fluctuations may come from different contributions like electronic excitations and lattice vibrations and accordingly the free energy can be written as F=E0+Fele+Fvib where E0 is the absolute zero total energy. The electronic contribution can be obtained easily from the knowledge of the electronic density of states and the Fermi distribution [6]. As a rule this term becomes significant only at high temperatures (beyond ~1000K).The quantum treatment of lattice vibrations in the harmonic approximation provides a reliable description of thermal vibrations for low and moderately high temperatures. The thermodynamic properties of a harmonic system [6] are entirely determined by the phonon density of states, i.e. g(ν). ∞

Fvib (T ) = k B T ∫ ln(2sinh( 0

hv )) g (v)dv 2 k BT

(2)

where h is Plank’s constant and kB is Boltzman’s constant. To illustrate this approach we refer to a recent paper [7] in which the phase stability of TiSi2 in the equilibrium C54 and metastable C49 structures has been investigated using first-principles calculations. The surprising result from that work is

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that the energetic ground state of TiSi2 is not C54, as might be expect from the phase diagram. Instead, the calculated T = 0 energy of the C49 phase is lower than that of C54. It was shown that this apparent discrepancy between calculated T = 0 energetics and the experimental observations at finite temperature can be reconciled by appealing to the vibrational entropies of both phases: a significant difference in vibrational entropies between the two phases serves to reverse the T = 0 energetic tendency to stabilize C54 at modest temperatures. The harmonic approximation can be improved by making g(ν) volume dependent, which can be easily done by calculating, within the harmonic approximation, the phonon density of states for several volumes. This approximation is called the quasi-harmonic approximation. However for the purpose to modelling thermal expansion, it is more convenient to directly parametrize the volume-dependence of the free energy itself: F(T,V) = E0 ( V ) + Fele( T,V ) + Fvib( T,V ). This dependence has two sources: the change in entropy due to the change in the phonon frequencies and the elastic change due to the expansion of the lattice. Currently, there are two methods to provide the necessary input parameters from ab initio calculations: the so-called direct force method and the linear response theory [8]. Although that the linear response theory is considered to be more accurate and is particularly useful when a system is characterized by non-negligible long-range interatomic forces (ionic solids), the direct force method is easier to use since it can be implemented regardless of the ab initio codes. This method is also very appropriate for thermodynamic studies since it allows to determine the vibrational properties of structures more complex that what can be done with the linear response theory. Finally let us mention that while harmonic theory is known to be highly accurate for a wide class of materials, important cases exist where this approximation breaks down due to large anharmonic effects. Examples include the modelling of martensitic phase transformations and phase transitions in Pu [9]. In this case, direct application of moleculardynamics offers a general framework for modelling thermodynamic properties of anharmonic solids [10]. 3.2. DISORDERED SYSTEMS

Let us now consider compositionally disordered solids. To access configurational energetics on a large scale, one can renormalize away the explicit electronic coordinates of the total energy as written in (1) and formulate instead a model Hamiltonian with fewer and more transparent parameters, providing an efficient representation of the configurational dependence of the energy of an alloy. An example is the cluster expansion

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[11-13] of the configurational energy E(σ) in terms of effective atom-atom interactions J: ECE (σ ) = J 0 + ∑ J iσ i + ∑ J ij σ iσ j + sites

pairs

∑

triplets

J ijk σ iσ j σ k + ... (3)

here, the variable σi is the spin-like occupation variable to each site i of the parent lattice, which takes the value –1 or +1 depending on the type of atom occupying the site. A particular arrangement of spins of the parent lattice is called a configuration and can be represented by a vector σ containing the value of the occupation variable for each site in the parent lattice. The interaction energies or effective cluster interactions {J} represent the contribution of each group of atoms to ECE(σ). In practice, the cluster expansion is known to converge rapidly but it is necessary to know which effective cluster interactions (ECI) are most important for a given material. Having this information, we could determine their numerical values by fitting equation (3) to the energy of a set of configurations obtained through ab initio calculations. Once we know these interactions, we can then compute easily ECE(σ) for many other configurations and search for the lowest energy configuration at each composition of the alloys system. Furthermore, the model Hamiltonian can be coupled with various mean-field methods or Monte Carlo simulations to provide thermodynamic quantities, short-range order, phase diagram and so on. The cluster expansion thus presents an extremely concise and practical way to model the configurational dependence of the energy of an alloy but choosing exactly the set of ECIs which must be retained is not easy. For instance, certain distant pairs or three-body interactions may be more important than nearest-neighbour pairs and the set of “the most efficient” ECIs depends on the alloy. Essentially two different approaches to this problem are actually used. The first one, known as the hierarchical approach postulates that if a given cluster is included, all other clusters of same size and all subclusters should be included [14]. However this restriction becomes rapidly impractical unless the series converge after few terms. Van de Walle and Ceder [15] have shown how to make a hierarchy-based approach manageable by introducing cross-validation as a systematic criterion to assess the predictive power of the cluster expansion but adding specific rules to capture “physically meaningful” candidate clusters. An alternative approach developed by Zunger et al. [16] is to attempt to identify the leading cluster interactions independent of hierarchical constraints. The set of input structures is split into two parts, one for fitting ECIs, and one to check predictions made with these ECIs. The procedure is repeated for different choices of fitting or prediction sets and the average prediction error is the cross-validation score. Very recently, it was shown that the

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search of leading terms can be efficiently performed using a genetic algorithm [17]. Recently, the study of complex phases, where several inequivalent sites exist in the unit cell, has become a focus of theoretical investigations [18-20] based on the technological importance of such phases. More particularly, the site preference in typical Frank-Kasper σ and μ phases for different alloys has been modelled using DFT methods combined with the cluster expansion-cluster variation method. Predicted site occupancies agree well with available experimental data. It was also shown that unlike order-disorder transformations in simple crystal structures, the preferential occupancy of atomic species for particular sites can not be removed by annealing. Such results allowed us to examine also the validity of the compound energy model widely used in the Calphad methodology to modelling thermodynamic properties of such complex phases. Such a discussion is made in the next section. 4. Links Between Ab initio and CALPHAD 4.1. DATA BASES

In the previous section we have presented the most recent advances in ab initio calculations that allow accurate prediction of thermodynamic properties, such as formation energies of binary alloys. Accurate calculations of entropic contributions, such as configurational or vibrational entropies, while more difficult, are also possible. However, it is not possible computationally tractable, for the foreseeable future, to use ab initio calculations alone to determine the total free energy directly for multicomponent systems, at least not with an accuracy comparable to experimental phase diagram measurements. On the other hand the Calphad methodology has been very successful in predicting the phase equilibria of multi-component commercial alloys by computing thermodynamic functions for multi-component systems using thermodynamic data obtained in unary, binary and ternary systems. Therefore, ab initio methods may be used to supplement or modify existing databases necessary to the Calphad approach [21]. This combination is particularly desirable for phases of practical interest like metastable precipitate phases which are often absent from computational thermodynamic databases. Additionally, in cases where an experimental controversy exists, ab initio calculations can offer a pragmatic tool to solve it. Such an approach may be particularly efficient when the structural stability of intermetallic compounds, like for instance in the Fe-Zr system, is found to be different [22,23]. More particularly, it has been shown [24] that the Zr2Fe compound does not form at low temperatures, supporting experimental results of Stein et al. [23]. Let us also

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mention that the ab initio data have been used to extend the thermodynamic base Zircobase and the Zr-rich part of the Fe-Zr phase diagram was reassessed with this new database. Similar attempts were made to discuss the occurrence of ternary compounds in the Al-Li-Si system by combining calorimetric experiments and ab initio calculations [25]. The existence of phases, the necessary step to modeling phase equilibria, is a recurrent problem in materials science. If most unary and binary systems have been studied, the number of experiments falls drastically for ternary and high-order systems. The application of data mining techniques to databases coupling structural data and ab initio energetics appears to be a promising development towards the practical prediction of crystal structures [26,27]. However the actual limitations come from the construction of these databases since the real payoff will come with the inclusion of multicomponent systems. Pragmatic solutions will require also the development of efficient algorithms to search through the space of possible crystal structures even for complex systems. As discussed recently by Turchi et al. [28], the input of phase diagram information obtained from ab-initio calculations may be also particularly relevant. This procedure is well adapted to the study of short-range order effects in binary solid solutions. In this case, the simplest way is to convert the output ab initio thermodynamics, calculated with similar approaches to those discussed in section 3.2, to a Redlich - Kister/Bragg - Williams formalism with an acceptable level of accuracy. The advantage of this approach is that this conversion provides a robust and yet simple scheme to study multi-component alloys. 4.2. MODELS

The compound energy formalism (CEF) [29] is commonly used within the Calphad methodology to describe, in complex multi-component systems, substitutional and interstitial solutions, intermetallic phases with wide homogeneity range, order/disorder transformations. However the Gibbs energy of the Frank-Kasper (FK) phases is still poorly modeled despite the versatility of the Calphad approach. The situation is in fact much more complex than that of solid solutions with a simple lattice (e.g. bcc). Indeed most FK phases have multiple sublattices (five in the case of the μ and σ phases). The CEF allows thermodynamic descriptions considering as many distinct sublattices as there are crystallographic sites. Many parameters are then needed, 32 for the binary m phase, and they are generally poorly estimated due to the lack of experimental information on these phases, particularly site occupancy data as a function of composition and temperature. However, it is possible to calculate all these parameters via the

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possible ordered configurations directly by using ab initio calculations. Thus a composition-dependent Gibbs energy is obtained over the whole binary composition range. The temperature dependence is then provides by the Bragg-Williams-Gorsky approximation (BWGA) which implies random distributions of species in each crystallographic site, i.e. short-range order is ignored. Such an approximation can be easily checked by comparing the so obtained results with those given by the more sophisticated approach that combines the Cluster Expansion with the cluster variation method (CVM). Comparisons made for different alloys and different types [20, 30] of FK phases, i.e. σ and μ phases, emphasize that the CEF model yields as good an approximation of the temperature and composition dependencies of site occupancies as the CE-CVM calculations, at least in the range of temperatures and compositions of practical interest. This suggests that shortrange order, which was ignored in the CEF calculations, may not be essential for modeling FK phases. 5. Conclusion It is clear that substantial gaps still exist between the largest systems accessible to DFT calculations and the phase stability of multi-component alloys with accuracy comparable to experimental phase diagram measurements. While this is often described as a length and time-scale problem, this is much more a knowledge problem. Bridging the length scale from microscopic to macroscopic requires that one understands the relevant phenomenon on each scale. For simple cases, like ordered compounds and solid solutions based on a simple lattice, these gaps may be partly overcome by efficient coarse-graining theories. More particularly, we have shown how ab initio calculations can be used in conjunction with statistical mechanics to yield finite-temperature thermodynamic properties with a great accuracy. For more complex problems, like multi-component industrial alloys, we have discussed the possible routes to integrate ab initio calculation in the Calphad methodology. Ab initio calculations provide formation and structural energies even for hypothetical structures and then they seem well suited to augmenting the experimental databases that form the basis of the Calphad approach. Another promising route is to integrate directly ab initio calculations into the CEF model commonly used to model complex multi-component systems. We have shown that such integration is fruitful to describe site occupancies and thermodynamic properties of the FrankKasper phases. With improvements in the accuracy of ab initio methods and developments of new coarse-graining techniques, it is anticipated that ab initio calculations will find growing applications in predictive studies of phase stability for a wide range of materials.

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References [1] Martin, R.M., 2004, Electronic Structure Basic theory and Practical Methods, Cambridge University Press, Cambridge. [2] Hohenberg, P., and Kohn, W., Inhomogenous electron gas, Phys. Rev. 136:B864-871. [3] Kohn, W. and Sham, L.J., 1965, Self-consistent equations including exchange and correlations effects, Phys. Rev. 140:A1133-1138. [4] Hafner, J., 2000, Atomic-scale computational materials science, Acta Mater. 48:71-92. [5] http:// ElectronicsStructures.org/software.asp [6] Wallace, D.C., 1998, Thermodynamics of Crystals, Dover Publications Inc. Mineola, New York. [7] Colinet, C., Wolf, W., Podloucky, R., and Pasturel, A. 2005, Ab initio study of the structural stability ofTiSi2 compounds, Appl. Phys. Lett. 87:41910-41912. [8] van de Walle, A., and Ceder, G., 2002, The effect of lattice vibrations on substitutional alloy thermodynamics, Rev. of Modern Physics 74:11-38. [9] Robert, G., Pasturel, A., and Siberchicot, B., 2003, Calculated thermodynamic properties of plutonium metal, J. Phys. Condens. Matter 15:8377-8387. [10] Allen, M. P., and Tildesley, D. J., 1987, Computer Simulations of Liquids Clarendon Press, Oxford. [11] Ducastelle, F., 1991, Order and Phase Stability in Alloys, Elsevier Science, New York. [12] Zunger, A., 1994, Statics and Dynamics of Alloy Phase Transformations, eds P. Turchi and A. Gonis, Plenum New York, pp. 361-419. [13] de Fontaine, D., 1994, Cluster approach to order-disorder transformation in alloys, Solid State Phys. 47:33-176. [14] Zarkevitch, N. A., and Johnson, D. D., 2004, Reliable first-principles alloy thermodynamics via truncated cluster expansions, Phys. Rev. Lett. 92:255702-255705. [15] van der Walle, A., and Ceder, G., 2002, Automating first-principles phase diagram calculations, J. Phase equilibria 23:348-359. [16] Blum, V., and Zunger, A., 2004, Mixed-basis cluster expansion for thermodynamics of bcc alloys, Phys. Rev. B 70:155108-155124. [17] Hart, G., Blum, V., Walorski, M. J., and Zunger, A., 2005, Evolutionary approach for determining first-principles Hamiltonians, Nature Materials 4:391-394. [18] Berne, C., Sluiter, M., Kawazoe, Y., Hansen, T., and Pasturel, A., 2001, Site occupancy in the Re-W sigma phase, Phys. Rev. B 64:144103-144111. [19] Sluiter, M., Pasturel, A., and Kawazoe, Y., 2003, Site occupation in the Ni-Nb μ phase, Phys. Rev. B 67:174203-174213. [20] Dupin, N., Fries, S. G., Joubert, J. M., Sundman, B., Sluiter, M., Kawazoe, Y., and Pasturel, A., 2006, Using first-principles results to calculate finite-temperature thermodynamic properties of the Ni-Nb μ phase in the Bragg-Williams approximation. Phil. Mag. 86:1631-1641. [21] Wolverton, C., Yan, X. Y., Vijayaraghavan, R., and Ozolins, V., 2002, Acta Mater. 50:2187-2197. [22] Jiang, M., Oikawa, K., Ikeshoji, T., Wulff, L., and Ishida, K., 2001, Thermodynamic Calculations of Fe-Zr and Fe-Zr-C Systems, J. Phase Equilibria 22:406-417. [23] Stein, F., Sauthoff, G., and Palm, M., 2002, Experimental Determination of Intermetallic Phases, Phase Equilibria, and Invariant Reaction Temperatures in the FeZr System, J. Phase Equilibria 23:480-494. [24] Barberis, P., Dupin, N., Lemaignan, C., Pasturel, A., and Grange, J. M., 2005, Microstructure and phase control in Zr-Fe-Cr-Ni alloys: thermodynamic and kinetic aspects, J. of ASTM Int. 2:129-156.

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[25] Pisch, A., Jakse, N., Pasturel, A., Harvey, J. P., and Chartrand, P., 2007, Structural stability in the Al-Li-Si system, Appl. Phys. Lett. 90:251902-251904. [26] Curtarolo, S., Morgan, D., Persson, K., Rodgers, J., and Ceder, G., 2003, Predicting Crystal Structures with Data Mining of Quantum Calculations, Phys. Rev. Lett. 91:135503-135506. [27] Fischer, C., Tibbetts, K. J., Morgan, D., and Ceder, G., 2006, Predicting crystal structure by merging data mining with quantum mechanics, Nature Materials 5:641-646. [28] Turchi, P., Abrikosov, I., Burton, B., Fries, S., Grimvall, G., Kaufman, L., Korzhavyi, P., Manga, V., Ohno, M., Pisch, A., Scott, A., and Zhang, W., 2007, Interface between quantum-mechanical-based approaches, experiments, and Calphad methodology, Comp. Coupling of Phase Diagrams and Thermo. 31:4-27. [29] Ansara, I., Chart, T. G., Fernadez-Guillermet, A., Hayes, F. H., Kattner, U. R., Pettifor, D. G., Saunders, N., and Zeng, K., 1997, Thermodynamic Modelling of Selected Topologically Close-packed Intermetallic Compounds, Calphad 21:171-181. [30] Fries, S., and Sundman, B., 2002, Using Re-W σ-phase first-principles results in the Bragg-Williams approximation to calculate finite-temperature thermodynamic properties, Phys. Rev. B 66:12203-12206.

CORRELATION BETWEEN ELECTRONIC STRUCTURE, MAGNETISM AND PHYSICAL PROPERTIES OF FE-CR ALLOYS: AB INITIO MODELING I. A. ABRIKOSOV Department of Physics, Chemistry and Biology, Linköping University, Sweden P. OLSSON Department of Materials and Mechanics of Components, EDF R&D, Les Renardières, Moret-sur-Loing, France AND A. V. PONOMAREVA Moscow institute of Steel and Alloys, Moscow, Russia

Abstract: We review recent developments in the field of ab initio electronic structure theory and its application for studies of phase stability of alloy systems. Basic ideas behind state-ofthe-art techniques for first-principles theoretical simulations of the phase stabilities and properties of intermetallic compounds and alloys based on the density functional theory are outlined. We concentrate on methods that allow for an efficient treatment of disorder effects, and illustrate their predictive power for the case of Fe-Cr system. We show that in the ferromagnetic alloys there are peculiarities of the mixing enthalpy in the low-Cr region in the bcc phase. Thus the stability of the Cr containing steels stems from the negative mixing enthalpy at low concentrations of chromium. We explain the effect by the strong concentration dependence of the interatomic interactions in Fe-Cr system. Keywords: Phase stability; Electronic structure theory; Interatomic interactions; Intermetallic alloys; Fe-Cr system

1. Introduction Recent technological developments have led to increasing demands for materials with special properties. Accordingly, qualitatively new classes of materials have emerged with properties that are fascinating for scientists, as well as for engineers. Experimentally one can now alloy immiscible elements. It is possible to grow magnetic nanoparticles in diamagnetic or semiconductor matrices, as well as at surfaces, to prepare tiny nanowires, and to synthesize crystals with forbidden symmetries, the so-called quasicrystals.

V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

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To understand fundamental properties of materials is a challenging problem for the theory. Indeed, the dominating approach in studying materials has been experiments. However, at present the electronic structure theory allows one to obtain reliable results for the thermodynamic, mechanical, electrical and magnetic properties of metals, semiconductors or insulators without any adjustable parameters fitted to the experiment [1]. Of course, until recently this was mostly done for relatively simple systems (for example, a perfectly ordered compound or a completely random alloy). At the same time, we can see today that the first-principles simulations are extended towards more realistic materials, which are of direct importance for practical applications. In particular, the binary body centered cubic (bcc) Fe-Cr alloy is the base of many important industrial steels, especially in the nuclear industry. A moderate amount of Cr, up to 10%, has proven to be most beneficial when it comes to the ductile to brittle transition temperature as well as corrosion resistance and resistance to neutron radiation induced swelling. Though there is a miscibility gap for the alloy concentrations above 10 at.% Cr at room temperature for lower concentrations of Cr the alloy is stable with respect to this decomposition. Moreover, diffuse-neutron-scattering experiments show an anomaly associated with a change of the ordering tendency from the clustering in Cr-rich alloys towards the short-range order for low Cr Fe-Cr alloys. In this paper we show how the anomalous stability of these steels can be simulated and understood by means of ab initio theory. The physical and mechanical properties of materials depend on the chemical content and on the internal structure which is formed during their manufacturing and service. A straightforward way to describe phase equilibria and to construct phase diagrams is offered within the phenomenological thermodynamics approach. For alloy systems one considers most often temperature-composition phase diagrams at ambient pressure. Thus, one deals with a system in thermal and mechanical contact with a constant-temperature constant-pressure heat bath, whose equilibrium is described by the thermodynamic potential G = E + PV – TS = H – TS, where G is the Gibbs free energy, E is the energy of the system, H is the enthalpy, S denotes the entropy, and thermodynamic variables P, V, T represent pressure, volume, and temperature, respectively. During natural processes G decreases and general conditions for equilibrium are represented by its minimum. If phase α transforms into phase β in a pure material at equilibrium transformation temperature Tc, Gα = Gβ at Tc. For alloy solutions, the chemical terms for the free energy are included by adding the chemical potential

FE-CR ALLOYS: AB INITIO MODELING

⎛ ∂G ⎞

⎟⎟ μ I = ⎜⎜ ⎝ ∂nI ⎠ T , P , n

155

(1) J

≠ nI

for each component I, where nI is the number of moles of component I. At equilibrium, the Gibbs-Duhem relation holds:

SdT − VdP + ∑ nI μ I = 0

(2)

I

For example, in a binary system A1–xBx it immediately follows that at constant P and T the G curves as a function of an alloy component fraction x = xB = nB / [nA + nB] for the phases in equilibrium must share a common tangent. 2. First-Principles Approach In contrast (or perhaps more properly as a complement) to the thermodynamic approach, the first-principles simulations of the phase stability of alloys are based on the atomistic description of the problem. But because the number of particles is of the order of the Avogadro number, the theory must use a statistical approach. The main goal of statistical mechanics is to derive the properties of a macroscopic system from its microscopic properties by averaging over unobservable microscopic coordinates leaving only macroscopic coordinates, say volume, and macroscopic variables, say temperature. This is done by introducing an idea of a statistical ensemble which can be defined as a large number of replicas of a system, considered all at once, each of which represents a possible state that the real system might be in. For the phase stability simulations one most often uses the so-called canonical ensemble (CE), which is a statistical ensemble that corresponds to a system in the heat bath (equivalently, to an assembly of identical replicas of a system) subject of constrain by macroscopic parameters T, V, and number of particles N, and with the distribution function given by Boltzmann distribution. The thermodynamic potential which describes a system at constant temperature and volume is the Helmholtz free energy F = E - TS, and general conditions for equilibrium are represented by minimum of F. For the canonical ensemble the Helmholtz free energy is calculated as:

⎡ ⎛ E{R} ⎞⎤ ⎟⎟⎥ (3) F (T ,V , N ) = −k BT ln Z (T ,V , N ) = − k BT ln ⎢∑ exp⎜⎜ − ⎝ k BT ⎠⎦ ⎣ { R}

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where Z(T,V,N) is the partition function and kB is the Boltzmann constant. In Eq.(3) the sum runs over all possible states {R} of the system (for example, if we need to find the lowest energy configuration for A1–xBx alloy at fixed composition x and underlying crystal lattice, say, bcc, {R} represents all possible occupations of the bcc lattice by (1–x)N A and xN B atoms). Note that we have used the classical statistical mechanics for our description above, and it is definitely sufficient for most of the problems within the field (some interesting deviations may occur, e.g. while working with hydrogen). However, a determination of energies E{R} in Eq.(3), generally speaking, requires a solution of a complex many-body quantum mechanical problem for each particular atomic configuration {R} of the solution phase A1–xBx. The essence of the alloy theory is to avoid this impossible task, and modern approaches to this problem are discussed below. However, it is still necessary to carry out quantum mechanical calculations for selected configurations {R} in order to derive ab initio parameters for the models used in subsequent statistical mechanics simulations. A solution to the quantum mechanical problem is provided by the electronic structure theory, and an excellent summary of it has been recently given by Martin [1]. Here we would like to go briefly through some points of the theory which are relevant for simulations of alloy thermodynamics. Nowadays, the majority of calculations within the field are carried out in the framework of the density functional theory (DFT). Within the DFT one solves the so-called Kohn-Sham (KS) equations for a system of independent “electrons” (characterized by single-particle wave functions). The essence of the KS theory is that it is constructed in such a way that the density of the independent “electrons” (which we denote as n) obtained from a selfconsistent solution of the KS equations is exactly the same as the electron density in the original system of real interacting electrons. According to the first theorem of the DFT, the potential in which electrons move is a unique function of the electron density n, and therefore the latter determines all properties of the material. The second theorem ensures that the converged solution corresponds to the charge density in the (electronic) ground state. The total energy of a configuration {R} of the solution phase A1–xBx can now be calculated as a functional of the oneelectron density:

G G E{ R}[n] = Ts [n] + ∫ d 3 rVext (r ,{R})n(r ,{R}) G G n(r ,{R})n(r ',{R}) 3 3 + ∫∫ d rd r ' G G r −r ' + EII + Exc [n]

(4)

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Note that the G density n depends on the alloy configuration {R} and coordinate r (in contrast to many body wave function which depends on G the coordinates of all the electrons in the system). In Eq.(4) Vext ( r , {R}) is the external potential (e.g. due to nuclei), EII is the classical electrostatic interaction between the nuclei,

G G Ts [n] = ∑ ε one − el − ∫ d 3rVKS (r ,{R})n(r ,{R})

(5)

occ

is the kinetic energy of independent electrons with eigenstate energies εone-el moving in the effective potential VKS (rG,{R}) . These quantities are obtained by the self-consistent solution of the KS equations, together with the charge density n. The last term in Eq.(4) is the so-called exchange-correlation energy Exc[n], which incorporates the effects of many-body electron-electron interactions. Unfortunately, its exact expression is not known. The first approximation exchange-correlation functional used the uniform electron gas as a model system for the calculation of Exc[n], and was called the local density approximation (LDA). Several useful parameterizations were suggested in the literature (see Ref. [1], Appendix B.1). The local density approximation seems to be quite crude because real charge densities vary strongly, especially close to nuclei. Still, numerous applications of the LDA were very successful, and lead to a wide acceptance of the DFT in the physical and materials science community. At the same time, in many applications the accuracy of the LDA-DFT calculations is not satisfactory and therefore more accurate approximations are clearly needed. Unfortunately, there is no systematic way to improve approximations for the Exc[n]. But in the end, an approximation that accounts for the density gradients was found, the so-called generalized gradient approximation (GGA). In contrast to the LDA, which is exact for the uniform electron gas, there is no physical (model) system for which the GGA is exact. The different GGA expressions obey certain known exact properties, but all GGA functionals, which are available today, have basically been constructed by hand. In particular, in Fe different forms of the GGA functional led to somewhat different results for the magnetic contribution to the total energy [2]. GGA does fix some remarkable problems encountered within LDA. For example, it predicts the correct ground state for iron. Generally, it gives bulk properties, most importantly lattice parameters, in better agreement with experiment as compared to the LDA. However, it is somewhat less known that the reliability of the GGA for the estimation of the surface properties is still under discussion, and there are evidences that the intrinsic

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surface errors of current GGA functionals show up to much larger degree, as compared to the LDA, in calculations of surface energies [3] and vacancy formation energies [4]. Moreover, neither LDA nor GGA approximations are known to be suitable for calculating certain properties (for instance, band gaps in semiconductors), and for treating certain systems (the so-called correlated materials). A lot of work has been done recently in finding approximations suitable for treating these problems. One can mention the LDA+U scheme, the GW approximation, dynamical mean-field theory (DMFT), the exact exchange method, and hybrid functionals [1]. 3. Treatment of Solution Phases A serious problem with the application of the DFT formalism occurs if a configuration {R} of the solution phase A1–xBx does not have any translational periodicity, which is the most common situation in practice. The point is that in this case group theory, which is the corner stone of the modern electronic structure calculations, cannot be used directly. Below we outline modern approaches for solving the electronic structure problem and calculating the total energy, Eq.(4) and (5), for systems with substitutional disorder. One obvious way to deal with a disorderd system is to consider its fragment(s), to impose periodic boundary conditions, and to solve the KS problem for such “supercells”. A tremendous progress taken place in computer technologies, together with a development of efficient ab initio algorithms, makes it nowadays possible to carry out calculations for systems with up to a thousand of atoms. Within this so-called supercell method one can use the codes developed for the periodic solids, and minimize the total energy with respect to the electronic degrees of freedom, as well as with the positions of atoms within the supercell. In many cases this way of simulating the disorder is acceptable for energetic properties. However, it is important to remember that this model is based on the use of transla Department of Materials and Mechanics of Components, EDF R&D, Les Renardières, Moret-sur-Loing, France. In Figure 1 we show two supercells, which were used in Ref. [6] for the calculations of the segregation energy of a Cr impurity towards the (001) surface of bcc Fe. The use of a 2 × 2 × 7 supercell (left panel) led to the segregation energy of –0.043 eV, indicating weak Cr segregation towards the surface. On the contrary, calculations carried out for larger supercell (3 × 3 × 9 right panel) gave the segregation energy 0.090 eV, indicating that Cr prefers to be in the bulk in the dilute limit. This is so

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because in the former case the surface concentration of Cr is actually very high, 25 at%. In the latter case it is smaller, 11 at%. Because interatomic interactions in Fe-Cr system depend strongly on the concentration, as will be shown explicitly below, the use of the smaller supercell leads to a qualitative error of the calculated segregation energy [6].

Figure 1. Two supercells used in Ref. [6] for the calculations of the segregation energy of Cr impurity towards the (001) surface of bcc Fe. The left panel shows 2 × 2 × 7 supercell constructed at the bcc underlying crystal lattice with 1 Cr (here at the surface) and 27 Fe atoms. The right panel shows 3 × 3 × 9 supercell at the same lattice with 1 Cr (here at the surface) and 80 Fe atoms.

As a matter of fact, Zunger et al. [7] suggested a method for supercells construction using the intelligent design of the atomic distribution within the cell rather than ad hoc “flip coin” technique. In their so-called special quasirandom structures (SQS) the atomic correlation functions are required to be the same as in completely random alloys for the first few coordination shells. Of course, the long-range correlation functions in the SQS are nonzero. However, from the discussion presented in Ref. [5] it is obvious that in order to calculate correctly the total energy of a random structure using the supercell method one needs to satisfy two important conditions: (i) interactions between all the atoms for which correlation functions are nonzero should be negligibly small, and (ii) and vice versa short-range order parameters should be zero for all the coordination shells for which interactions are nonzero. Since the effective interactions are usually quite longranged, these conditions are never satisfied in the supercell calculations exactly. But in many cases the SQS technique gives a reasonable representation of a random alloy due to the fact that the strongest effective interactions are usually present at the nearest neighbor coordination shells.

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It is clear, that the quality of an SQS is improving with the number of correlation functions satisfying the second condition. It is also obvious what total energies calculated with supercells, where the correlation functions are not controlled, e.g. with ad hoc constructed supercells, may lead to unpredictable results even if one carries out an averaging over several (but still limited) number of different supercells. As an alternative to the supercell approach one can reconstruct threedimensional periodicity of the substitutionally disordered system (we need it in any case!) by mapping it onto a suitably chosen ordered lattice of “effective” atoms, which describe the original system on the average. The simplest way here is to apply the mean-filed approximation, which are common in all areas of physics. The basic principle of the method [8] is the following: one considers a few degrees of freedom concentrated in a limited region of the space where they are defined, and assumes that their interactions with all the others can be replaced by what is called a meanfield. Averaging out the considered degrees of freedom, one obtains a homogeneous medium which yields self-consistent relations determining these fields. The mean-field equations are non-linear, and can be difficult to solve, but this non-linearity is the root of the richness of the method. In terms of the electronic structure problem, one is ultimately interested in processes of electron scattering off the atoms in the system, the so-called multiple-scattering. The simplest mean field method, the coherent potential approximation (CPA), is constructed by placing effective scatterers at the sites of the original system. Scattering properties of these effective atoms have to be determined self-consistently from the condition that the scattering of electrons off the alloy components, embedded in the effective medium as impurities, vanishes on the average. The CPA is currently one of the most popular techniques to deal with substitutional disorder, and it is implemented in several codes [9-11]. Numerous applications of the CPA show its excellent efficiency and rather high accuracy in the description of electronic and thermodynamic properties of alloys. In particular, an implementation of the CPA within the exact muffin-tin orbital (EMTO) theory [11] was used with great success in studies of bcc Fe-Cr alloys [6,12,13]. In Figure 2 we show mixing enthalpies calculated for this system by means of EMTO-CPA and supercell methods. For the supercell calculations we employed the projector augmented wave (PAW) approach, as well as ultra-soft pseudopotential (USPP) approach as implemented in the Vienna ab initio simulation package (VASP) [14]. The calculations within the PAW were carried out for SQS with 128 atoms. For the USPP calculations supercells with 1 Cr per cell were constructed at each concentration. All supercell were fully optimized both respect to ionic positions and volume. Note that the large

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number of atoms is required in order to minimize the short range order parameters for the lower concentrations and was used consistently for all calculations.

Figure 2. Concentration dependences of the mixing enthalpy for disordered FM bcc Fe-Cr alloys from 0—100 at.% Cr, as predicted by the EMTO-CPA (squares) and PAW-SQS (diamonds), and USPP-SQS (circles) methods. Inset in (b) is a blow up of the Fe-rich region with 0-15 at.% Cr to better show the predicted change in sign of the mixing enthalpy in the FM alloys.

The mixing enthalpy H of an alloy with x parts Cr is defined as 0 H Fe1−xCrx = EFe1−x Crx (Ω 0Fe1−xCrx ) − (1 − x) EFe (Ω 0Fe ) − xECr (Ω Cr ) (6)

where all total energies E are calculated at corresponding equilibrium volumes Ω 0, and the two last terms represent the energies of the standard states for pure ferromagnetic bcc Fe and nonmagnetic bcc Cr, respectively. The use of the nonmagnetic phase of Cr is motivated by the fact that its Neel temperature is below temperatures of interest for Fe-Cr steels power applications. In Figure 2 one can clearly see that all the computational techniques predict similar behavior of the mixing enthalpy with well-defined minimum for Fe-rich alloys, if somewhat different in strength. The negative sign of H clearly indicates the ordering tendency in the system that stabilizes the alloys against the phase separation that occurs at higher Cr concentrations, and is in agreement with the anomalous stability of Fe-Cr observed on the iron side of the alloy phase diagram. As a matter of fact, this example is quite representative for the accuracy of the modern first-principles calculations. They allow one to predict reliable trends, but it is unlikely that one can predict numbers with metallurgical accuracy, that is with accuracy

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which allows to calculating phase diagrams fully ab initio with a precession of 1K for phase transition temperatures. It is also very difficult to define “the best” ab initio methodology. For instance, there are a lot of reasons to believe that the PAW method is more accurate than the USPP method, and it is highly recommended to using it in VASP simulations. But in the particular case of the Fe-Cr system it seems to underestimate the stability range of the alloys. Indeed, experiments show that a transition from a tendency towards ordering to a tendency towards clustering behavior in bcc Fe-Cr alloys occurs at around 10% Cr. This may indicate that in our simulations both, EMTO-CPA and USPP calculations reproduce the experimental trends better than PAW technique. Of course, one must remember that calculations presented in Figure 2 are carried out at zero temperature, while the experiments are done at finite temperatures, which can affect the comparison. Therefore the statement above should not be viewed as a proof that one technique is better than the other. Generally speaking, it is probably impossible at present to recommend any single electronic structure technique as the best among all the others. Despite the success of the CPA calculations for the Fe-Cr alloys, it is important to underline that it gives an approximate, mean-field description of the scattering properties of a system. Within the CPA one considers a solution phase A1–x Bx with only one type of disorder, the completely random substitutional disorder at each of the system sublattices. This means in particular that there must exist an underlying crystal lattice, and the sites of this lattice are occupied by the alloy components with probabilities equal to their sublattice fractions. Occupation of each site within a sublattice does not depend on the type of an atom which occupies neighboring sites, i.e. there is no short-range order in the alloy. Though size mismatch of alloy components can cause different interatomic distances to be different, i.e. socalled local relaxation effects, they are not taken into consideration within the CPA. Note however that schemes have been developed, which allow one to correct the CPA total energies for the contribution due to the local lattice relaxations [15, 16]. A way beyond the mean-field theory in total energy calculations for solution phases is associated with the development of so-called O(N) methods [1]. The point is that computational efforts for a conventional electronic structure code scale as N3 with respect to the number of atoms in a system (or a supercell). In O(N) methods the scaling is linear, which allows for simulations of systems with much larger number of atoms. Unfortunately, most of them can not deal with metallic systems at present. However, two methods based on a concept of local self-consistency within the multiple scattering theory, the locally self-consistent multiple scattering method (LSMS) [17] and the locally self-consistent Green’s function

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method (LSGF) [18] have been applied for metals with considerable success. In particular, the LSGF method [5,18] allows for self-consistent calculations of the electronic structure and total energy for solution phases, nonmetallic as well as metallic, with arbitrary distribution of atoms at the sites of the underlying crystal lattice. Within the LSGF method each atom is considered together with few shells of its nearest neighbors, the so-called local interaction zone (LIZ) [17]. The LIZ is embedded in an effective medium, and the multiple scattering problem is solved in terms of cluster Dyson equation for the LIZ. The individual LIZ are associated with every atom in the system under consideration (usually, SQS with much large number of atoms N than in original SQS calculations [7]). The effective medium represents on the average scattering properties of the system beyond the LIZ, and is most often constructed using the multicomponent generalization of the CPA. The accuracy of the method is determined by the size of the LIZ, and one can expect that the convergence with respect to the LIZ size will be achieved as soon as the central site becomes insensitive to the interchange of positions of two atoms from the same sublattice at the LIZ’s boundary, in contrast to a requirement of the complete insensitivity in conventional O(N) schemes. Applications of the LSGF method are discussed, e.g. in Refs. [5, 18]. 4. Interatomic Interactions in Solution Phases Though in principle Eqs. (4) and (5) allow one to determine the energies E{R} in Eq.(3) for each particular atomic configuration {R} of the solution phase A1–xBx, such a path is highly impractical, and perhaps also impossible at present. An alternative to the direct quantum mechanical calculations can be found within the so-called multiscale modeling. By the latter one understands a solution of the complete problem step-by-step employing theoretical methodologies, which are suitable for the particular length and/ or time scale followed by an appropriate coarse graining when proceeding towards the next (larger) scale. For the phase stability simulations, one starts with the solution of the quantum mechanical problem within the DFT for a relatively small system (~100 atoms). From these results one determines interactions between different atoms and uses them in statistical mechanics simulations (classical molecular dynamics or statistical mechanics, e.g. Monte-Carlo technique), which includes 103 to 105 particles. In doing so one can study, for example, melting or ordering in metals and alloys. For simulations of mechanical properties and microstructure (i.e. properties on the scale of microns) one often needs to substitute the

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atomistic description by continuum theories. The latter can be done, for example, in the framework of the phase-field model. For the problems related to the alloy phase stabilities it is most appropriate to consider atomic configurations {R} at the sites of an underlying crystal lattice. In this case the alloy energetics can be described by the Ising Hamiltonian:

⎡ ⎤ E{R} = V ( 0 ) + ∑ ⎢V (1) σ + ∑ V ( 2, s ) σ iσ j + ...⎥ s ⎣ ⎦

(7)

where i,j are lattice sites, the spin variables σi takes on values +1 or –1 depending on the type of atom occupying site i. The average products of the spin-variables, < σi σj >, are the multisite correlation functions which form the complete basis for the total energy expansion [8], V0 is the reference energy, which, in fact, is the total energy of a random equiatomic alloy, and V(d,s) are the effective cluster interactions, which correspond to clusters of nn nn nn the order d and type s. For instance, V ( 2,1) = VAA is the + VBB − 2VAB effective pair (d=2) interaction at the first coordination shell (s=1), which describes interactions between all the different types of pairs (AA, BB, and AB) of nearest neighbor (nn) atoms in the A1–xBx alloy. The on-site interaction V(1), which is the effective chemical potential, can be neglected in the canonical ensemble calculations (that is, if the number of particles in a simulation box is conserved). The coarse-graining procedure in going from quantum mechanical description, Eqs. (4) and (5), to classical description, Eq.(7), consists of the determination of parameters in Eq.(7), the effective cluster interactions V on the basis of first-principles DFT calculations. A number of different procedures exist for extracting many-body interactions for alloys out of electronic structure calculations [19]. They can be classified into two main categories, (1) methods based on the electronic structure of random alloys and (2) methods based on the calculation of the electronic structure of specific ordered configurations of alloys. Methods within the first category correspond to perturbation-like expansions from the disordered state either in direct configuration space or in reciprocal space. The most well-known technique representing this type of methods is the generalized perturbation method [8], recently updated by Ruban et al. [20] in order to account for the screened electrostatic interactions. The second group of methods, nowadays often referenced as the cluster expansion method [21], Eq.(7) is used as a fitting function to extrapolate between the total energies (or most often between the enthalpies) calculated for a set of ordered structures. Then the least square fit is used to extract the effective cluster interactions. The cluster expansion method can be relatively

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easy implemented, requires standard software for the DFT calculations, and takes care of all the terms for the total energy (the GPM considers only the dominant terms, that is the one-electron energy, the first term in the righthand side of Eq.(5), and the electrostatic interactions EII in Eq.(4), and accounts for the effect of local lattice relaxations (in contrast to the GPM). On the other hand, in the majority of applications the effective interactions are assumed to be concentration independent, and their volume dependence is not always treated correctly. Moreover, a treatment of effects associated with magnetic disorder in the paramagnetic phase of magnetic alloys is a real problem for the methods from the second group. As will be discuss below, their neglect may lead to erroneous conclusions [21]. 5. Anomalous Stability of Low Cr Alloys: Results and Discussion Let us now present and discuss the results obtained by the first-principles simulations for the Fe-Cr system. In Figure 2 one can clearly see that the mixing enthalpy of ferromagnetic bcc Fe-Cr alloys has a well defined minimum for low concentrations of Cr. Although the depth and width of the minimum differs for PAW-SQS, USPP, and EMTO-CPA calculations, the trend is clearly similar in both the supercell and the effective medium simulations. Thus, the low Cr alloys show anomalous stability, as compared to the alloys in this system with higher Cr concentration. Note that this effect is in agreement with experimental observations and experience gained in technological applications of Fe-Cr steels. However, to the best of our knowledge, the experimental data for the mixing enthalpies is available only at high temperatures. For instance, Ref. [22] gives it at T = 1600 K. But at this temperature the alloys are paramagnetic (the local magnetic moments exist, but they are disordered), and their mixing enthalpy is very well described by the CPA calculations for the magnetically disordered phase [12,13]. In this respect calculations carried out in Ref. [21] for the mixing enthalpy, as well as for all other properties, at high temperatures (above the magnetic transition temperature in Fe-Cr system, which varies with composition but has its maximum at about 1050 K) appears to be irrelevant, as they were done based on the effective cluster interactions determined for the ferromagnetic alloys, that is for the quailtatively different system. In order to explain the variation of the mixing enthalpy of the ferromagnetic bcc Fe-Cr alloys as a function of concentration we have studied effective cluster interactions in this system using the screened version of the GPM method [8,20]. Our results for the most important pair interaction at the first coordination shell, term V(2,1) in Eq.(7), are shown in Figure 3. One can clearly see that the interaction depends strongly on the concentration,

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and it changes sign at about 14 at% Cr. Note that the positive sign corresponds to the tendency towards ordering, while the negative sign indicates a tendency towards the clustering. Thus, in agreement with experiment, the calculations reproduce the change of the character of the interactions in Fe-Cr alloys at low Cr concentrations. Thus, mixing should decrease the total energy of dilute Fe-based alloys, while the clustering should be energetically preferable at higher Cr concentrations. From these simple arguments one can see that the mixing energy for bcc Fe-Cr must change its sign as a function of alloy composition, and therefore must pass through the minimum, making low-Cr steels particularly stable with respect to decomposition into more Fe-rich and more Cr-rich phases.

Figure 3. Effective pair interactions, V(2,1) in Eq.(7), for Fe-Cr alloys as a function of Cr concentration calculated by the generalized perturbation method. Dots connected with solid line show the interactions in the bulk. Circles show the interactions within the surface layer.

As a matter of fact, the behavior of the interatomic interactions in Fe-Cr alloys can be explained in its own turn from the band filling arguments. Indeed, based on the model calculations it was demonstrated earlier [8] that ordered phases ought to be expected for alloys of transition metals with half filled bands whereas phase separation should occur in alloys with almost filled or almost empty bands, with a crossover point in the region of electron concentrations of interest. Of course, the situation is complicated by the presence of the magnetism. But Hennion [23] has demonstrated that, indeed, the effective interatomic potential between the nearest neighbors varies with concentration in ferromagnetic bcc Fe-Cr alloys, and even changes sign at around 25 at.% Cr due to the filling of the alloy d-band. This prediction is in remarkably good agreement with our first-principles result, giving the simplicity of the model used in Ref. [23].

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Note that in Figure 3 we also show effective pair interactions between nearest neighbors in the surface layer at (001) surface of Fe-Cr alloys. Their concentration dependence is somewhat weaker as compared to the interaction in the bulk. This fact was used in Ref. [6] for the explanation of the strong dependence of the segregation energy in Fe-Cr alloys on the supercell size. Thus, the suitability of a particular supercell for the calculations of any property of a random alloy is indeed determined by the range of interatomic interactions relevant for this particular property. 6. Conclusions In summary, we have demonstrated that first-principles simulations can be carried out for real materials of technological importance. The results allow for the cautious optimism. In particular, while the mixing enthalpy for paramagnetic alloys is positive at all concentrations, in excellent agreement with experiment, our calculations show that ferromagnetic bcc Fe-Cr alloys are anomalously stable at low Cr concentrations. Therefore, magnetic effects must be taken into account in simulations of properties in this system. The stability of the ferromagnetic alloys is determined by the strong concentration dependence of interatomic interactions in this system, which therefore must also be taken into account in simulations. Acknowledgements IAA is grateful to Andrej Ruban for most useful discussions. We would like to thank the Swedish Research Council (VR), the Swedish Foundation for Strategic Research (SSF) and the Royal Swedish Academy of Sciences (KVA) for financial support. The authors gratefully acknowledge the support of the HPC Umeå.

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THE COMPUTATIONAL MODELING OF ALLOYS: FROM AB INITIO CALCULATIONS AND THERMODYNAMICS TO HETEROGENEOUS PRECIPITATION A. CARO Chemistry, Materials, and Life Sciences Directorate. Lawrence Livermore National Laboratory. Livermore CA 94551

Abstract: This paper gives an introductory view of some methods of computational thermodynamics to study equilibrium finite temperature properties of model systems. We present a methodology to obtain free energies from empirical potentials and applied it to the study of the phase diagram of FeCr. As an application, we use Metropolis Monte Carlo to analyze homogeneous and heterogeneous precipitation of the Cr rich solid solution α’. Keywords: Computational thermodynamics; free energy; FeCr; precipitation; Monte Carlo

1. Introduction This lecture addresses a particular methodology to study thermodynamic driving forces, one of the two terms (thermodynamics + kinetics) that determine evolution in non-equilibrium processes at the atomic level. It is written at an introductory level and focused into the applications to a particular system, FeCr alloys. Since most of the problems related to materials behavior under irradiation are non-equilibrium processes, the challenge for computational modeling is to capture the relevant aspects of this wide class of problems. As a lecture, this paper is self contained. However, for additional readings we list below a series of references where particular aspects are discussed in more detail. P. Hasen, 2005, for thermodynamic of alloys; N. Saunders and A. P. Miodownik, 1998, for a description of the CALPHAD methodology; and a series of papers by our group and collaborators, which includes E. Ogando Arregui, M. Caro, E. M. Lopasso, D. Farkas, P. E. A. Turchi, and D. Crowson, 2002-2007 which cover in detail the methodology discussed here. It can clearly bee seen that today’s belief on the power of the computational approach to physics and materials science has become a V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

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commonplace. The combination of models and computers constitutes a paradigm in modern science. It is important to highlight that quantitative computational materials science, i.e. a discipline aimed at predicting properties of real materials under real conditions, indubitably stands on two legs: quantum mechanics and thermodynamics. Unfortunately the quantum mechanics problem can never be solved exactly and therefore, we always make use of models that necessarily capture partial aspects of reality. But it is important to emphasize that models at all levels have ultimately to search for their physical content deep inside these two roots. This is the reason that justifies the multiscale approach to real problems, so well established in the research on nuclear materials. An un-exhaustive list of techniques within the multiscale approach would certainly contain the ab initio formalism for electronic structure, the empirical potentials for large scale molecular dynamics, MD, the kinetic Monte Carlo, k-MC, approach for long time atomic scale evolution, the phase field methodology for microstructure evolution, and the continuum mechanics for the macroscopic scale response, among others. Regarding thermodynamics, its importance cannot be overstated. Free energies determine equilibrium, i.e. where a system wants to go. Additionally, free energy differences determine the strength of the forces that drive systems towards equilibrium. Unfortunately free energies are difficult to measure both experimentally and numerically. As examples, note that experimentally, equilibrium can not always be reached in the time scale available at the laboratory if diffusion is too slow; numerically, free energies are given by integrals over the phase space accessible to the system, impossible to do except for some very particular cases (we come back to this issue below). This last fact is what drives a family of computational approaches to obtain these key functions through different and complementary approaches. At this point it is important to make a clear distinction between modeling and simulations. Setting up a model for a physical system, that is equivalent to say giving a recipe for the total energy, even the simplest one as for example the Lennard Jones, LJ, pair potential for interaction among noble gas atoms, already completely determines its equilibrium and non equilibrium thermodynamic behavior. However, solving the model, i.e. getting its properties, is a task of diverse complexity. Models can be classified into two big families: the ab initio models that are models with no free parameters, and the empirical models. The first class contains only a few examples that are so successful that dominate the scene. These are models that solve the unsolvable many body quantum mechanics equations by making judicious approximations that, luckily enough, do not necessitate

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any case-specific parameter. Among them, the Local Density approximation (LDA) and the Generalized Gradient Approximation (GGA) to the Density Functional Theory (DFT), deserve particular mention. They are an extraordinary achievement of modern condensed matter physics. When addressing a particular problem, these models can give highly accurate results, as is the case for many elements and combination of elements in the periodic table, or can fail, as in the case of Actinides or late transition metal oxides. If they fail for the case of interest, then we have no other alternative than moving to the next class of models, the empirical models. By their very definition, there are infinite many empirical models, from the complex LDA+U and LDA+DMFT to the simplest LJ. Models always capture partial aspects of reality, those aspects that we believe are important for the problem at hand. As said, models can have different degrees of complexity and solving them is a discipline of science by itself, namely computer simulations. Running a simulation means solving a model using computers. This task can be done exact or using approximations. For example, running a molecular dynamics (MD) simulation of a set of classical LJ particles gives exact results, because Newton equations are the actual equations governing the time evolution of the system. Perhaps MD cannot afford the time scale needed for some problems, like diffusion, but this is another story that does not invalidate the fact that the solution is exact. The complementary example is running a lattice k-MC simulation of a LJ alloy; it will give approximate results since, just to mention one, the vibrational entropy contribution to the free energy is missing is such a lattice approach.

Figure 1. Block scheme of the strategy followed in this work to obtain thermodynamic functions and empirical potentials for alloys. For details, see text.

Having defined the computational modeling and simulations landscape, let’s move now to the methodology we address in this lecture, namely, obtaining thermodynamic functions via empirical potentials so as we understand the driving forces acting in non-equilibrium processes. Figure 1

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shows a conceptual flow chart that helps us understanding the role of different components of the alloy modeling strategy. Other lectures in this Volume by A. Pasturel and I. Abrikosov describe the basics of the ab initio models and their practical ways to solve them; they also discuss the approximations used to go from ab initio energetics to finite temperature properties, and in particular to thermodynamic functions. G. Inden discussed the CALPHAD (CALculations of PHAse Diagrams) approach to thermodynamics, which is based essentially on experimental data. In summary, the two left-most blocks of the diagram in Figure 1 were covered by these lectures. We emphasize here, by way of the vertical arrow linking the ab initio to the Thermodynamics modules, and by the ‘~’ sign attached to it, that thermodynamic properties are always the result of approximations, when coming from ab initio energetics. There is not such a thing as ‘ab initio thermodynamics’. For example, sample size limitations preclude exploring samples large enough to effectively average over all possible local order configurations, as the thermodynamic limit would require; consequently, the energetics of a solid solution needs to be obtained by for example mean field approximations such as the Coherent Potential Approximation (CPA), or by the Cluster Expansion (CE) method. Similarly, vibrational entropy requires long MD runs at various temperatures, as we’ll see below, unaffordable by ab initio techniques that then need to recourse to the harmonic or quasi-harmonic approximations to it. In our work we follow the path indicated by the red arrows in Figure 1, namely, from the ab initio energetics for the alloy at 0 K, from experimental results, and from thermodynamics phase diagrams information extracted from the CALPHAD database, we first develop an empirical potential for the alloy and then, using MD in a ‘thermodynamic package’ that performs a series of calculations that we describe below, we calculate free energies and phase diagrams exact, i.e. with no approximations, as the horizontal black arrow in Figure 1 suggests. The approximation in this method resides, of course, in the description of reality through an empirical potential, as indicated by the ‘~’ sign over the red arrow. The success of this step is system-dependent: some alloys can quite accurately be described by empirical potentials, some others can not, and it is precisely here where the interest of this methodology stands: if a particular alloy can satisfactorily be described by an empirical potential, i.e. if the thermodynamic functions reasonably describe free energies and phase diagrams as they appeared in nature, in the CALPHAD database, or in the ab initio results, then we have a classical potential that can be used in large scale MD and MC computer simulations to study equilibrium and non equilibrium processes, which are at the core of radiation damage studies.

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2. Computational Thermodynamics (in one of its many flavors) The evolution of a physical system is determined by both i- thermodynamic driving forces and ii- kinetic effects. The scheme below (Figure 2) shows the basic theoretical steps that lead to this assertion. The first law of thermodynamics states that equilibrium for an isolated system is given by the maximum of entropy, S, as a function of the internal variables of the system, ζ. This is a postulate, i.e. it does not emanate from any other law; it is therefore the starting point of thermodynamics. When this postulate is reformulated for a system in contact with its surroundings, and can therefore exchange heat Q, work W, and particles N, it translates into saying that equilibrium of such a system is defined by the minimum of a new function, the Free Energy g.

Figure 2. Basic statement of thermodynamics defining equilibrium .

Having defined equilibrium, we introduce now a simplifying assumption that allows us to treat non-equilibrium systems on an analytic basis: the linear response theory (we go through this step just for the purpose of this explanation; computer simulations usually are not bound to it). Let’s assume that for small departures from its minimum, g is quadratic in its variables; then forces are proportional to displacements (equivalent to the Hook’s law in elasticity). Repeating once again the argument, we assume that currents (or fluxes, or velocities) are proportional to forces (equivalent to the Ohm’s law in electricity). Equation (1) summarizes the assumptions and introduces an important quantity in the theory of irreversible processes: the Onsager Phenomenological Matrix, L:

Jζ = ∑ L j,ζ ∇μ j j

μj =

∂g ∂N j

(1)

This equation is similar to the well known Fick’s first law, but it is much richer: the gradient of the chemical potential replaces the gradient of composition. It is in this term that all thermodynamics resides. The relation between Fick’s law and Onsaguer Phenomenological equations for the particular case of ideal solutions is discussed in the lecture by G. Inden in this volume. In what follows we do not make any assumption on the nature of the alloy and let μ carry all the complexity of the actual system under

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study. Important to highlight here is the fact that due to the matrix character of L in Eq. (1), there is a coupling between forces and fluxes of all species in the system, represented by the off-diagonal terms of L. Examples of this are the well know Soret effect (coupling of grad T with flux of solutes), Peltier (coupling of grad E -electric field- with flux of heat), Seeback (coupling of grad T with flux of electrons), etc. The chemical potential of each species, μj, is the derivative of the free energy with respect to the number of particles of species j. It carries all the information regarding how these particles interact with their surroundings. For heterogeneous precipitation studies, as we describe in this lecture, it carries also the information of strain fields produced by extended defects or external constraints. The Gibbs free energy g and the Helmholtz free energy F are given by

g = H − TS = E + PV − TS

F = E − TS

(2)

For computational thermodynamics, the vocabulary of statistical mechanics is more adequate since it expresses thermodynamic quantities in terms of the microscopic mechanical degrees of freedom; we introduce then the partition function Ω, with the integral in Eq. (3) running over the all degrees of freedom of the system:

F = −kT ln(Ω)

Ω=

1 h3

∫ exp(−H (p, x )/kT )d

3

x d3 p

(3)

V

From F, all thermodynamic quantities can be obtained; for example:

E = −T 2

∂(F /T) ∂T

S=−

∂F ∂T V

(4)

Entropy is a difficult quantity to measure; there is no ‘entropymeter’, contrary to a voltmeter, barometer, etc. Some quantities, like F and S, called thermal quantities, can not be measured as an average over an ensemble, as mechanical quantities can (E, V, T, P, …), but require calculating volume integrals in phase space, as in Eq. (3). This is the core of the difficulty to calculate entropies and free energies from computer simulations. The basic assumption that links statistical mechanics to molecular dynamics is the hypothesis of ergodicity that says that a time average equals an ensemble average: At = A (5) where

1 At = t − t0

t

∫ A(x (τ ))dτ t0

A =

∫ A(x )exp(−H (x)/kT )dx ∫ exp(−H (x)/kT )dx

V

V

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Unfortunately free energy, Eq. (3), is clearly not an ensemble average (i.e. it is not a ratio of two integrals as in Eq. (5), and can therefore not be obtained as an average over a MD run. Computational thermodynamics addresses the fact that F can not be obtained using averages of MD runs. Let’s then look in more detail at the integral defining the partition function Ω, Eq. (3). There are only a few cases where it can be evaluated analytically, namely, in cases where the Hamiltonian H(x) is quadratic in space q’s and/or momentum coordinates, p’s, because in such cases variables are not coupled and Gaussian integrals are just numbers. Such cases are the ideal gas (H=p2/2m) and the harmonic oscillator (H~p2+q2). For one particle in one dimension the ideal gas partition function can easily be seen to be 3/2 ⎛ 2πmkB T ⎞3 / 2 V V V 3 / 2⎛ 2m ⎞ 3 − β p 2 / 2m Ω= 3 ∫ d p e = 3 π ⎜ ⎟ = V⎜ ⎟ = 3 (6) ⎝ h2 ⎠ h h Λ ⎝ β ⎠ where Λ is called the De Broglie thermal wave length. For the harmonic oscillator (H=p2/2m+mω2q2/2) in turn, 3/2 ⎛ 2 ⎞ 3 / 2 ⎛ 1 1 ⎞3 ⎛ T ⎞ 3 1 3 / 2⎛ 2m ⎞ 3/2 Ω= 3π ⎜ ⎟ π ⎜ =⎜ ⎟ =⎜ ⎟ 2⎟ h ⎝ β ⎠ ⎝ β mω ⎠ ⎝ = β ω ⎠ ⎝ TE ⎠

(7)

where TE is the Einstein frequency. The generalization for binary mixtures of ideal gas particles or harmonic oscillators is straight forward, except for some configurational entropy considerations; a detailed treatment can be found in Ogando Arregui, Caro, and Caro, 2002-a. The reader can verify that from Eqs. (6) and (7), and using Eq. (4), the familiar expression for the energy of these systems is obtained. So far, the knowledge of the free energies for these two very simple systems seems not to help much in our quest of the free energy of more realistic Hamiltonians. However, a very helpful procedure, known as the ‘switching Hamiltonian technique’, provides the path to our goal. Let’s assume a system described by a Hamiltonian H that is a linear combination of two different Hamiltonians, U and W, U being the system we are interested on, and W being one of the systems described above, for which the free energy can be obtained analytically,

H( λ) = (1− λ)W + λU

Let’s now evaluate the derivative of the free energy of this Hamiltonian with respect to λ. Using Eq. (3) it is easy to show that

∂H

∂F kT ∂(Ω) =− = ∂λ Ω ∂λ

∫ ∂λ exp(−H (x, λ)/kT )dx = ∫ exp(−H (x, λ)/kT )dx

Ω

Ω

∂H ∂λ

(8)

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Equation (8) states that the derivative of a thermal quantity, F, with respect to an internal parameter of the Hamiltonian, λ, equals the ensemble average of the derivative of H with respect to that variable, the latter being a mechanical quantity. Remember that ensemble averages (mechanical quantities) can be obtained by MD, using the hypothesis of ergodicity, Eq. (5). Then our problem is solved because if we can calculate Eq. (8), then F can be obtained simply by integration over λ! λ=1 ∂H (λ ) F (U ) = F (W ) + ∫ dλ = F (W ) + ∫ dλ U − W ∂λ λ= 0 λ= 0 λ=1

(

) (9)

The last step that completes the methodology is the Gibbs-Duhem integral of the enthalpy that gives free energy at a temperature T given its value at temperature T0,

F F0 T E (τ ) + PV (τ ,P ) = −∫ dτ T T0 T0 τ2

(10)

A detailed description of this approach can be found in Ogando Arregui, Caro, and Caro, 2002-b. The methodology described above is exact, i.e. there is no approximations to the free energy of the Hamiltonian H. However, the computational cost of evaluating Eqs. (9) and (10) is too high to be used with ab initio Hamiltonians. For such Hamiltonians it is necessary to use approximations (typically harmonic or quasi-harmonic for the vibrational entropy, and cluster expansion for the energy; see lectures by A. Pasturel and I. Abrikosov in this volume). For illustration purposes, a view of the integral in Eq. (9) for the bcc phase of Fe as described by an Embedded Atom empirical potential is given in Figure 3, and a view of the enthalpy integration appearing in Eq. (10), for the bcc and fcc phases of the same potential is given in Figure 4.

Figure 3. Average values of vs λ for bcc Fe corresponding to the integrand of the switching Hamiltonian (cf. Eq. 9). Numerical accuracy is critical for the prediction of the free energy.

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The numerical accuracy of these quantities is crucial for the precision of the results, which then requires long runs to make the hypothesis in Eq. (5) valid. Finally, the free energies of the three relevant phases (bcc, fcc, and liquid) of the empirical potential for Fe are shown in Figure 5. The constraint of high numerical accuracy is evident from this figure that shows how similar all three functions are. Intersections of these lines signal the first order phase transitions among these phases. From these free energies and Eq. (4), we can get enthalpies and entropies to compare with metallurgical databases, like CALPHAD SSOL and validate in this way the empirical potential.

Figure 4. Enthalpy of Fe as a function of temperature. Heating and cooling runs are done consecutively. Overheating and undercooling due to size effects are clearly observed, which help obtaining enthalpy of solid and liquid phases beyond its equilibrium temperature ranges.

Figure 5. Free energies of bcc, fcc, and liquid phase of Fe as described by an Embedded Atom Model empirical potential.

Figure 6 shows enthalpy and entropy for the bcc and liquid Fe. This figure clearly shows one of the main difficulties in modeling Fe, both ab initio and empirically. The discrepancies with experimental data are notorious in both phases. The reasons for it are to be found in the contribution of magnetic excitations. Around the Curie temperature of Fe,

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TC = 1033 K, an anomaly is observed in the enthalpy as well as in the entropy of the CALPHAD bcc phase, which is, of course, absent in the empirical prediction. The anomaly is the consequence of the excitation of the magnetic degrees of freedom that, on approaching the ferroparamagnetic (FM-PM) second order phase transition from below, start to absorb energy, and consequently increase enthalpy and entropy. Above TC, the magnetic system is completely disordered and does not absorb energy any more. The empirical potential does not show the anomaly, as the dynamics of magnetic degrees of freedom are not modeled; only their contribution to the 0 K energetics is accounted for.

Figure 6. Enthalpy and entropy for the bcc and liquid phases of pure Fe as functions of temperature. The solid (dashed) lines refer to the CALPHAD (empirical potential) results. The enthalpy of the bcc phase at 298.15 K is taken as zero of energy. For a discussion on the discrepancies, see text.

This problem has a fundamental as well as a technical origin. For CALPHAD, the FM and PM phases are not treated as different phases. Both are identified as bcc Fe with thermodynamic functions that include the magnetic contribution and its T-dependence. For ab initio or empirical potentials, the bcc FM phase is different from the bcc PM phase. It can then be said that the broken lines in Figure 6 report enthalpy and entropy of FM bcc and liquid Fe and that other lines should be added to describe PM phases. A different empirical potential, based on a different ab initio calculation, would be needed to describe the PM phases. In what follows, then, we will

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discuss the empirical potential modeling of the bcc FM phase of Fe and FeCr alloys, i.e. the bcc phase well below ~ 1000K. To move from pure elements to binary alloys we use the standard formulation developed by the CALPHAD group, i.e. the free energies of a pure element and a solid solution are written as, Unary Phase:

gφ (T) = aφ + bφ T + c φ T 2 + dGφ T ln(T) Multi-component Phases:

gφ (x,T)= ref gφ (x,T)+ id gφmix (x,T)+ xsgφmix (x,T) where ref

and id

(11)

φ gφ (x,T) = (1− x)gφFe (T) + xgCu

gφmix (x,T) = kT ((1− x)ln(1− x) + x ln(x))

The excess enthalpy of mixing is written in terms of the Redlich-Kister polynomial expression, n

g (x,T) = x(1− x) ∑ Lp,φ (T)(1− 2x) p

xs φ mix

(12)

p= 0

The meaning of Eq. (11) is that a description of a multicomponent system can be constructed by describing the properties of the elements first, then the properties of the binaries, and finally the combination of binaries. For binaries, the free energy is written as excess quantities with respect to the linear interpolation between the two components. These excess quantities have two contributions, one from the entropy associated to mixing, and the other from the enthalpy and vibrational entropy associated to the mixing. The advantage of writing the thermodynamic functions for alloys in the same vocabulary as CALPHAD resides in the possibility to directly compare results from empirical models to those of experiments, i.e. to make possible a direct ‘dialog’ between quantities calculated in the computer and quantities measured and adjusted by the CALPHAD group. A description of Eqs. (11) and (12) can be found in the G. Inden lecture in this volume. We focus now the attention on the results we get for the FeCr alloy. As shown in the I. Abrikosov lecture, this system presents an anomaly in the heat of mixing, that shows a change in sign from negative for low Cr content, indicating tendency to order, to positive for large Cr content, indicating a tendency to segregate into an heterogeneous solution composed of α and α’, both bcc solid solutions, rich in Fe and in Cr respectively. The empirical potential for this alloy had to incorporate explicit composition

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dependence in its terms, reflected in high order Redlich-Kister polynomial expansion, up to fourth order. Explicitly, the Embedded Atom type potential we suggested for this alloy has the form, N

E = ∑[Fαi (∑ ραi,βj (rij )) + 1/2∑Vαi,βj (x,rij )] i

j≠i

(13)

j≠i

with

Vαi,βj (x,r) = h(x)v AB (r) where α and β refer to atom species, i and j to atom location, rij is the distance between atoms at sites i and j, r is the atomic density function, V is the pair potential function, and v is a pair potential function between sites occupied by atoms of different chemical identity. Here h(x) is a polynomial depending on local composition x, of the same order (4th) that needed to fit the heat of mixing results of P. Olsson et al. (see lecture by I. Abrikosov in this volume). The resulting representation of the heat of mixing, Δh, compared to the ab initio – CPA results and to the CALPHAD data, is shown in Figure 7.

Figure 7. Heat of formation versus composition for the FeCr bcc solid solution. Solid line is the CPA result for the PM phase and the fitted result using the phenomenological potential based on Eq. (13) (both coincide). Broken line is the CALPHAD data for the bcc phase. Note the change in sign of the CPA and the empirical potential curves at low Cr composition.

With the potential so developed, we apply the ‘thermodynamic package’ to evaluate the free energy of the FM bcc phase, and obtain a phase diagram for that phase. The free energy surface is shown in Figure 8; for details see the references. Black lines on the free energy surface are isotherms that, with the help of a ruler for example, can be used to determine the common tangent points, i.e. the solvus line. The phase diagram is shown in Figure 9 up to temperatures of 1000 K. Several differences are readily noticeable. Neither the miscibility gap nor the spinodal go to zero composition at 0 K on the Fe rich side of the diagram. This curious effect is due to the change in the sign of the heat of

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mixing of this alloy at about xCr = 0.08, as shown in Figure 7 which implies finite solubility at low temperatures. We can say that the ab initio results showing a negative heat of mixing (instead of positive as for a normal segregating system) at low Cr composition has as a consequence that the phase diagram is substantially different at low T - low xCr from the standard phase diagram for this system as represented by the CALPHAD solvus. Our methodology, described in Figure 1, gave us the path to link the 0 K result obtained from ab initio calculations to the finite T properties reported in Figure 9.

Figure 8. Free energy of a FeCr random solid solution in the ferromagnetic phase, as a function of temperature T and Fe composition xFe. Note the anomalous behavior at low T and high xFe related to the change in sign of the heat of formation (see text).

Figure 9. Part of the phase diagram of FeCr corresponding to the ferromagnetic bcc phase below 1000 K. Other phases have been suspended. Solid line is the CALPHAD solvus corresponding. Dashed line is the prediction of the solvus for the FM bcc phase. Comparison has to be made only for temperatures well below TC. Dot line is the location of the spinodal inside the miscibility gap.

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The non-zero solubility at 0 K seems a contradiction to the fact that the entropy at 0 K should be zero. In fact, it is not. The phase diagram at 0 K between xCr = 0 and the start of the solvus line, as shown in Figure 9, is incomplete. Because the formation energy is negative, we know that there must be one or more intermetallic (ordered) phases. Although this/these phases may not be relevant for applications due to its very small ordering energy, it is a subject of research now as a way to further understand this system. The implications of the results shown in Figure 9 are that Cr is much more soluble at low T that previously thought. Several experiments on irradiation and annealing of the binary FeCr suggested that the standard phase diagram might not be accurate at low Cr composition. The present result seems to be more in agreement with those experiments. For a discussion see Caro, Caro, Lopasso, and Crowson, 2006. We show the phase diagram below 1000 K. The fact that the critical temperature at which the miscibility gap closes is much higher for the ab initio–empirical potential description of FeCr than for CALPHAD as can be concluded from Figure 9 can be traced back to the heat of formation reported in Figure 7 which is already much larger than the value reported in CALPHAD database. This should not be considered as an error in the ab initio–empirical potential approach because, as we discussed already, the theoretical/numerical prediction corresponds to the ferro-magnetic phase, while the real gap develops across a temperature regime where the magnetic transition takes place. Several groups are working today in the extension of the ab initio and empirical approaches to include magnetic degrees of freedom into the models, so as to include their contribution explicitly. 3. Precipitation in FeCr Alloys Having obtained the thermodynamic behavior of an empirical potential for FeCr, and seen that this behavior is quite complex as to suggest large Cr solubility at low T, we move now to the applications of the model to precipitation processes. We show two examples, one on homogeneous and one on heterogeneous precipitation in FeCr. Since we have an empirical potential for this system that satisfactorily captures its complexity, we use it for MD simulations which would provide exact results for it time evolution, as discussed in the introduction. However, precipitation kinetics involves diffusion, with a time scale inaccessible to MD. We then need to use then stochastic methods. Ideally we are interested in solving a non equilibrium process by some method that would account for both terms in Eq. (1), namely, driving forces and kinetics. There exists no such method at present, mainly due to intrinsic

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complexity hidden in Eq. (1). To simplify the problem two strategies are currently used based on whether we focus the accuracy on the kinetic or on the thermodynamic aspects. Kinetic Monte Carlo approaches provide the time scale of the kinetics but at the price of simplifying the driving force landscape; just as an example among several others, the stress field induced by the presence of a grain boundary can not be accounted for. In this work we focus on the driving forces, i.e. on where the system wants to go, at the price of neglecting the kinetic effects. We emphasize that this approximation limits the predictions to true equilibrium only, while often in nature what counts are states of meta-stability. We come back to this point at the Conclusions. Metropolis Monte Carlo is the algorithm for the search of equilibrium at finite temperature. In a similar way as conjugate gradients or steepest descent are algorithms to find minima of energy, Metropolis MC is an algorithm to find minima of free energy. From a starting configuration, say a saturated solid solution, the algorithm will drive the system towards a minimum of free energy (a segregated alloy in our example) through a series of states where each one is obtained from the previous one by an ad hoc choice of fluctuations. In our examples below, these fluctuations involve both displacements and random switch of chemical identity, in what is known as the Semi-Grand Canonical Metropolis MC. We have developed a massively parallel code for this purpose, whose description is a meter of a future publication. We first study homogeneous precipitation of the α’ phase in a saturated FeCr solution at two compositions, xCr = 0.15 and 0.5, at T = 750 K, both compositions inside the miscibility gap as can be seen in Figure 9; xCr = 0.15 is just above the solubility limit, while xCr =0.5 is well within the spinodal. Figure 10 shows two dimensional, 2D, slice sequences from 3D MC simulations that represent steps towards equilibrium, starting from a solid solution and evolving towards a state with the sample separated into regions of α and α’. All three simulations correspond to perfect crystal cubic samples with ~2 million atoms and periodic boundary conditions. Frames a-d show homogeneous precipitation and coarsening of α’. The fact that Cr-Cr interaction is repulsive (see Caro et al. 2007) implies that there is a critical size for the nuclei to be stable; and the fact that the decomposition mechanism at this concentration is nucleation and growth implies that the concentration of the nuclei is always the saturation composition of α’, i.e. xCr ~ 0.99. As MC steps go on, the precipitates adopt a spherical shape, indicative of small anisotropy on the interface energy. Panels e-h focus on one particular precipitate, highlighting the fact that there is a critical size below which there are no stable precipitates, and the evolution to spherical shape.

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Figure 10. 2D slice sequences along three MC simulations starting from a saturated solid solution and evolving towards thermodynamic equilibrium. For details see text. Taken from Caro et al. 2007.

Panels i-l show evolution of the alloy via spinodal decomposition, i.e. continuous increase of composition differences from xCr = 0.5 to the terminal solution values xCr = 0.12 for α and xCr = 0.99 for α’. It is important to point out that these simulations, which include displacements and energy of local configurations, make no approximations in the driving force for segregation, in particular, the surface energy and eventual size mismatch of the precipitate is fully accounted for.

Figure 11. 2D slice of a nanocrystalline Fe sample with an average grain size of 5 nm. We observe several configurations of grain boundaries and triple junctions. Blue dots are Fe atoms sitting at boundaries, red dots are Cr atoms. Fe atoms at perfect bcc positions are not shown. For details see text.

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We now analyze the case of heterogeneous precipitation, i.e. precipitation in the presence of defects. We choose to explore this process in a polycrystalline sample with average grain size of 5 nm. In this way we can observe in a single run a multiplicity of configurations of grain boundaries, triple and higher order junctions, and eventually surfaces. Figure 11 shows a slice taken from a cubic sample; simulation parameters are xCr = 0.15 and T = 750 K. Blue dots are Fe atoms sitting at boundaries, red dots are Cr atoms. Fe atoms at perfect bcc positions are not shown. A surprising effect can readily be observed: excess Cr does not precipitate at boundaries, instead all α’ precipitates are in the interior of the grains, avoiding contact with the grain boundary network. We observe a similar behavior for free surfaces. These two examples, which are fully developed in the References, are used here to highlight the relevance of this thermodynamic approach to MD/MC simulations. Traditionally, k-MC is used at approximation levels that do not fully account for the free energy landscape. FeCr, as we have seen, is a complex material that shows a change in sign in its mixing energy, which implies a competition between ordering and segregation. Without explicitly considering this anomaly, as would be the case with a k-MC simulation that does not evaluate ‘on the fly’ the actual local free energy landscape for the moving atom, the effects of this change in sign would not be captured. Similarly, because FeCr is the basics binary for ferritic stainless steel, understanding heterogeneous precipitation for example at surfaces, is necessary to interpret corrosion resistance behavior. Heterogeneous precipitation is controlled by the properties of defects, such as surfaces, grain boundaries, dislocations, etc., where the crystalline periodicity is altered. Monte Carlo algorithms must explicitly consider these displacements to correctly account for the phenomenon. There is no k-MC today able to address both these issues (actual free energy landscape and displacements) due to the exceedingly large computer resources needed. By addressing equilibrium issues using Metropolis MC, as we do here, we explore the force terms in Eq. (1), as a step on the path to a full description of complex alloys in heterogeneous microstructures. 4. Summary In this lecture we presented a methodology to obtain free energies from empirical potentials and applied it to the study of the phase diagram of FeCr. Subsequently, we used Metropolis Monte Carlo to analyze homogeneous and heterogeneous precipitation of the Cr rich solid solution α’. These examples are part of our work in the area of steels for nuclear

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applications and can be found in several publications of our group cited as references. Acknowledgments This lecture describes work done in a period of several years by a group of people form different institutions: M. Caro, B. Sadigh, P. Turchi, P. Erhart, (LLNL, US); E. M. Lopasso (Bariloche, Argentina), D. Crowson (Virginia Tech, US), P. Klaver (Belfast-UK), S. Srivilliputhur (LANL, US), G. Bonny (SCK-CEN, Belgium) Work prepared by LLNL under Contract DE-AC52-07NA27344

References We give references to books describing the basics of thermodynamics of alloys, the CALPHAD methodology, and to papers published by our group where details of the methodology can be found as well as general references to the literature on the subject. Phase Transformations in Materials, Materials Science and Technology: A Comprehensive Treatment, Vol. 5, P. Haasen Editor, Wiley-VCH, 1996. CALPHAD (Calculation of Phase Diagrams): A Comprehensive Guide, Pergamon Materials Series, N. Saunders and A. P. Miodownik Ed., Pergamon 1998. Caro, A., Caro, M., Klaver, P., Sadigh, B., Lopasso, E.M., and Srinivasan, S.G., 2007, The Computational Modeling of Alloys at the Atomic Scale: From Ab initio and Thermodynamics to Radiation-Induced Heterogeneous Precipitation., Journal of Minerals, Metals and Materials (JOM) April 2007:52-57. Caro, A., Caro, M., Lopasso, E.M., and Crowson, D. 2006, Implications of ab initio energetics on the thermodynamics of Fe-Cr, Appl. Phys. Lett. 89:121902-04. Caro, A., Caro, M., Lopasso, E.M., Turchi, P.E.A., and Farkas, D. , 2006, Thermodynamics of Fe-Cu Alloys as Described by a Classic Potential, J. Nucl. Mater. 349:317-326. Caro, A., Crowson, D., and Caro, M., 2005, Classic many body potentials for concentrated alloys and the inversion of order in iron-chromium, Phys. Rev. Lett. 95(7):075702. Caro, A., Turchi, P.E.A., Caro, M., and Lopasso, E.M., 2005, Thermodynamics of an empirical potential description of Fe-Cu alloys, J. Nucl. Mater. 336:233-242. Lopasso, E., Caro, M., and Caro, A., 2003, Phase diagram of Fe-Cu as predicted by the Embedded Atom Model, Phys. Rev. B 68(21):214205. Ogando Arregui, E., Caro, M., and Caro, A., 2002, Reference systems for computational free energy calculations of binary solutions: role of the constrained center of mass motion, Computational Materials Science 25 (3):297-304. Ogando Arregui, E., Caro, M., and Caro, A., 2002, Numerical evaluation of the exact phase diagram of an empirical Hamiltonian: Embedded atom model for the Au-Ni system, Phys. Rev. B 66(5):054201.

NUMERICAL MODELING OF RADIATION EFFECTS IN SOLIDS: PRINCIPAL FEATURES, LIMITATIONS AND PERSPECTIVES P. GEYSERMANS* CNRS – ICMPE UMR 7182, 15 rue Georges Urbain, 94407 Vitry-sur-Seine cedex, France

Abstract: This paper focuses on atomistic computer modeling of radiation effects with special emphasis on the achievements of this approach as well as on its space and time scales inherent limitations. These last are the principal motivations for the development of a multiscale approach ideally consisting in a succession of models providing input information to the model acting at the immediate next time and space scales. Some methodologies used to bridge the gap between multiple space and time scales are considered together and reasons encountered preventing the approach from being predictive are examined. Moreover, we briefly comment methods improving modeling of radiation in solids. Keywords: Radiation effects; multiscale modelling; atomistic simulations; molecular dynamics; kinetic monte carlo; density functional theory

1. Introduction The non-linear and far from the thermodynamic equilibrium phenomena triggered in solids under irradiation result in the formation of historydependant microstructures controlling their physical properties (e.g. mechanical). These features prevent the development of predictive associated theoretical approaches and computer simulation becomes therefore the unique methodology available for modelling irradiation and for help deconvoluting experimental results. Phenomena arising in materials under irradiation involve a multiple hierarchy of spatial and temporal scales. This means that processes that occur at a certain scale govern the behavior of the system across usually several larger scales. The notion of multiscale modeling arises therefore quite naturally in the field of radiation effects in solids. In the general context of materials simulations, different levels of length and time scales can be distinguished. Different computational approaches have been developed to tackle with the relevant

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CNRS – INSP UMR 7588, UPMC Univ Paris 06, 140 rue de Lourmel, 75015 Paris.

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phenomena and the suitable accuracy at each level: (1) the atomic scale where the quantum electronic state of the electrons dictates the interaction among the atoms. Methods based on density functional theory and its different approximations can be applied to systems of few hundreds of atoms for the computation of static properties. At this microscopic scale, interactions between atoms can also be described by classical interatomic potentials, which empirically incorporate the electronic nature of bonding. Classical molecular statics, molecular dynamics and atomistic Monte Carlo simulations are not as accurate as their quantum mechanics founded counterparts but are able to examine atomic processes in systems of size up to 108 atoms on a timescale up to 10–8 second. (2) The mesoscopic scale where lattice defects such as dislocations, grain boundaries and other microstructural elements are the principal ingredients. Dynamics of latter replaces dynamics of atoms thus eliminating degrees of freedom inappropriate to the treated problem. Properties of systems of size of a few tens of microns can be investigated. (3) Finally, the macroscopic scale where the system is described as a continuum. In this context, finite elements methods are used to provide insight into the large-scale properties of materials (mechanical properties under various loading conditions). But the challenge in materials science is that materials usually exhibit phenomena on one scale, requiring a very accurate and computationally expensive description and other phenomena on another scale for which, a coarser description would be sufficient. An example is given by the driving force for dislocation climb which corresponds to the diffusion of vacancies through the crystal. An accurate accounting of the elemental vacancy diffusion mechanisms is therefore required to properly describe the plasticity in metals, e.g. the different dependence in the temperature of dislocation slip and climb. Diffusion is accurately studied in atomistic simulations and values of interest such as migration energy associated to different diffusion mechanisms in regions of local stress can be introduced in continuum dislocation dynamics simulations. In this case, atomistic approaches provide accurate quantities which can be introduced in higherlevel models of plasticity. This procedure thus consists in a hierarchy of computational approaches which are put together. Information obtained from the more detailed approach is input into the larger-scale simulation. This is an efficient procedure provided the different scales are weakly coupled. On the other side, full atomistic simulations via, for instance, molecular dynamics or statics, capture the local behavior but at the expense of limited sample size and attendant boundary effects. Since the latter can modify the true behavior, important efforts are made in developing concurrent methods attempting to link methods appropriate at each scale all together in

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a combined model. In other words, the system is spatially partitioned into domains characterized by phenomena occurring at different scales which are considered concurrently and communicate with each other through a mixing procedure. The development of predictive models of damage evolution and associated properties modifications requires understanding of the elementary processes involved in the production of radiation damage. Most damage caused by fast neutrons and ions results from the creation in elastic collisions of energetic primary knock-on atoms (PKA). They in turn produce cascade of atomic displacements. Space scale of a cascade produced by one PKA is about 10 nm and its time scale is of the order of 1 ps. At the end of this collisional phase, it remains a core of vacancies surrounded by self-interstitials [1]. Since molecular dynamics with phenomenological potentials can simulate the time evolution of a set of about 106 atoms, the elementary mechanisms of defect production in displacements cascades have been abundantly investigated by this technique [1,2,3]. In the following, through different examples, we present important contributions of numerical simulations in this field, but also related shortcomings. We show how smaller scale techniques like that including quantum mechanical descriptions of interactions are able to overcome the difficulties related to the classical approaches. On the other side, we emphasize how the atomic details are smoothed leading to a description where defects are included as objects, allowing to go beyond the problem of the limited size in molecular dynamics and to remarkably reproduce experimental results like those obtained from resistivity recovery after irradiation by annealing. Additionnaly, we present a promising work in which three space scales are concurrently treated in a multiscale simulation. A dislocation with segregated impurities is modeled by describing the core region at the atomic scale with a continuum medium surrounding it, with the aim to reproduce the long range deformation due to the dislocation and constituting therefore the boundary conditions of the simulation box. An energy minimization with quantum mechanics technique is applied to the core region, avoiding thus complexity and difficulties related to the use of empirical classical potentials as long as the number of induced chemical species is higher than two. A short overview of classical molecular dynamics in the next section will help us to understand next-coming examples with some indications about the applicability of quantum mechanical calculations. Next sections will be firstly devoted to the role played by point defects in irradiation processes and secondly to the relation existing between the irradiation processes and the resulting microstructure as well as the corresponding mechanical properties. We will discuss their

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strong points and their shortcomings and will present some directions in which innovative research should investigate. 2. Technical Prerequisites Molecular dynamics (MD) is a computer based simulation modeling the material as a set of N interacting classical point particles, the time evolution of which is studied by integrating their newtonian equation of motion

G d 2 ri G mi 2 = Fi dt

(1)

where mi is the atom mass. According to equation G (1), the second derivative of the atomic position is its acceleration. Fi is the force acting on it, due to the interaction with other atoms. Consequently, MD is a deterministic technique, in which all atoms will move, eventually bump into each other, similarly to what they do in a real material. The classical approximation, underlying equation (1) is justified as long as de Broglie thermal wavelength (Λ) is smaller than the nearest neighbor distance. It’s verified for light elements such as Li or Ar at the triple point and the ratio Λ/a (where a is the lattice parameter) decreases rapidly for heavier elements [4]. When the temperature is sufficiently low, quantum effects become important in any system. We here focus on systems and temperature range allowing for a classical description be employed. The computer calculates a trajectory in a 6N-dimension phase space, thus producing a set of configurations distributed according to the distribution functions. Physical quantities are represented by temporal averages over successive configurations. The measurement of a physical quantity is therefore simply the arithmetic average of the instantaneous values taken by the microscopic observable of interest during the run. It’s remarkable that equilibrium thermodynamic properties quickly converge even in systems of limited size (about 103 atoms) and for time scales less than the picosecond [5]. This ability of MD of reproducing equilibrium properties in systems tractable by computers is the ingredient of success of MD since the sixties. In its standard version (eq.1), MD produces constant energy trajectories, i.e. in microcanonical statistical ensemble. Extended methods allow for other statistical ensembles, like constant temperature and/or pressure (constant stress), in principal applicable only to spatially homogeneous system [6]. Since irradiation processes give rise too largely inhomogeneous affected regions, they are preferentially studied in constant energy simulations. But what are the limitations of molecular dynamics ? The main approximation rendering this technique sometimes inappropriate, is that forces are obtained

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as the gradientGof aG potential energy function, depending on the positions of the particles (r1 ,...rN ) , according to

G G G Fi = −∇V (r1 ,...rN )

(2)

which contains all the physics of the considered material. The of G problem G modeling is therefore that of finding a potential function V ( r1 ,...rN ) for the given material. In the past, the pairwise approximation has been used, consisting in writing the energy as the sum of pairwise terms. But the class of materials which can be realistically modeled using this kind of approach is in practice limited to rare gases. In these materials, cohesion is due to dispersive forces rising from the weak van der Waals like interactions. Considering systems of more practical interest, such as metals, many-body effects must be correctly accounted. High ratio between the cohesive energy and the melting temperature, low ratio between the vacancy formation energy and the cohesive energy (<1, but only 1 in pairwise if no relaxation) and high ratio between the elastic constants C12/C44 (≠1) are as much as typical properties of metals which are not reproduced by a pair potential. Since the mid 80’s, many-body effects are incorporated in phenomenological potential energy functions via the following equation N

V =∑

N

N

∑ϕ (rij ) +∑φ ( ρi ) , with ρi = ∑ψ (rij ).

i =1 j = i +1

i =1

(3)

j

This equation expresses the interaction between two atoms as a function of only their interatomic distance but also of their local environment, through the quantity ρi , representing a sum over the local electronic charge density ψ induced at site i from atoms at sites j. Of course, parameters appearing in these functions are to be determined by fitting on experimental values of quantities like cohesive energy, lattice parameter, and elastic constants. When dealing with metallic alloys, equation (3) is taken its two species version where indexes i and j run over all particles, and different fitted parameters apply to each kind of atomic pair. In these cases, the reliability and transferability of these potential models have to be examined through a series of validations, such as a dynamic stability test using MD simulations and predictions of optimal structural parameters, bulk modulus and binding energies of several compositions of the studied intermetallic compound. The problem is far from trivial since potentials are density dependent in the case of metals, and so, concentration dependent in the case of alloys. As mentioned above, it’s worth noticing that thermodynamical averages converge rapidly as a function of N. Mean squared displacements are observed to linearly depend on the system size and achieve 95 % of their

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infinite size value for a system size of around two thousand atoms in a simple metal. Nevertheless, periodic conditions usually used in these simulations avoiding presence of excessive surface on volume ratio, imply a cutoff frequency in the phonons spectrum which could be dramatically in the computation processes like fusion [5]. Quantum mechanical calculations offer an alternative way to evaluate the total energy of the system and could therefore go without developing semi-empirical classical potentials. All properties of materials could be investigated and understood from first-principles, without any parameters, by solving the Schrödinger equation for the electrons moving quantummechanically in the potential field of a set of atomic nuclei, which in the Born-Oppenheimer approximation are considered to be static (electrons movement is decoupled from nucleus one). In practice it is almost impossible to treat within a full quantum mechanical framework, since the large number of freedom degrees implies a too important number of variables. Ab initio techniques are therefore largely developed with the main aim of solving the problem of interacting electrons with the use of various approximation schemes. The most basic approximation is the Hartree-Fock theory in which the electrons are considered independent. Only a mean effect of their interaction is introduced in the calculations. But for 30 years, the density-functional theory has become the method of choice by reducing the complex multi-component many-body problem of interacting electrons and nuclei into a set of one particle coupled equations, by replacing the dependence on the external potential (nuclear degrees of freedom) by a dependence on the electron density distribution. It is therefore able to treat many problems with high accuracy and is computationally simple. Only the electron density of the ground state determine all the mean electronic and thermodynamical observables of the material (such as charge density and band structure of solids, total energies and many related quantities such as bonding distances, defect energies or migration barriers). Different implementations at different degrees of sophistication of density functional theory make the method very versatile (Plane waves, pseudopotentials acting on valence electrons, linearized methods for all-electrons calculations). But it is worth to stress that, because of the number and the variety of different approximation schemes, the validity of the method is in practice established by its ability to reproduce experimental results. The accuracy of the method is discussed in the framework of other alternative approaches, essentially depending on the specific problem one has to investigate.

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3. Irradiation and Point Defects Atomistic simulations are able evaluating the energy required to displace an atom from its site, as a function of the temperature and the PKA energy. This value can be then introduced in binary collision codes similar to TRIM. Such simulations give evidence of that with increasing PKA energy, the number of surviving free defects in the cascade is less than predicted by the NRT model. This behavior relates to the fact that for energetic PKA’s, the cascade volume is large, and few interstitial atoms are transported beyond the highly agitated cascade core. The recombination is then efficient and the surviving defect fraction is reduced. Nevertheless the question of the discrepancies between binary collision calculations and atomistic simulation remains not well understood and further interpretation is still expected. MD also succeeded by putting in evidence the channelling phenomenon. In the above mentioned aspects of irradiation effects studied by using MD, valuable successes of this approach are highlighted which should not hidden its severe limitations and others candidates for further investigation. Because of the size limitation of the technique, attention should be given to the possible artefact of interaction of the cascade region with itself across the periodic boundary conditions. This constrains the studied system since it can only include one cascade and low PKA energies, keeping the cascade zone well confined inside the simulated cell. Microcanonical statistical ensemble is usually chosen to run such simulations requiring the approximation of no heat extraction from the simulated cell. The temperature of the system therefore increases to a value related with the PKA energy, usually not exceeding 100-200K in the range of energies currently employed in MD simulations, and some papers report on the formation of a molten zone inside the cascade core, which subsequently recrystallizes during the cascade quenching stage [1]. Since thermal conductivity in metals is dominated by electronic effects, a right description of heat dissipation should be revisited in the context of a coupling between electrons and phonons. We have to notice the overwhelming lack of statistics in the published results, since generally, the number of observed cascades is smaller than four. But irradiation is a phenomenon completely far from equilibrium. In this case, for small perturbation, the system response is linear (e.g. without creating any defects) and MD computes averages that are expected to be convenient, like for temperature. But when the perturbation is not linear, like those producing point defects, MD calculations suffer from being not reproducible. Finally, studying intermetallic compounds which are still poorly investigated by MD, is of crucial importance for structural materials in reactor systems. Therefore, understanding of microstructural evolution of

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alloys, such as high chromium ferritic martensitic steels, under irradiation conditions constitutes a real challenge. To date many investigations have performed MD cascade simulations in pure iron using a variety of interatomic potentials. But cascade simulations in iron alloys have not yet been extensively studied because of the lack of reliable interatomic potentials for alloys [7,8]. Since during irradiation, atomic defects, i.e. vacancies and interstitial atoms, are produced in important quantities, their properties are of crucial importance in the prediction of temporal defect production and therefore microstructure evolution. In this context, electronic structure calculations advantageously complete molecular dynamics. In the example of α-iron, the configuration of the self-interstitial (SIA) is non typical, with respect to other body-centered cubic metals since the migration energy is unusually large [9]. Some discrepancies, such as stability and mobility of selfinterstitial atoms, exist between results from experiments and empirical potentials predicted values. More particularly, ab initio calculations provide the right relative stability of the <110> and <111> dumbbell configuration of self-interstitial in α-iron by incorporating spin effects [10-12]. In agreement with experiments the <110> dumbbell configuration is found to be the most stable. The large energy difference between the <111> and the <110> dumbbell configuration was not reproduced by other semi-empirical potentials, e.g. Akland [13], and even if this particular deficiency is more recently corrected, e.g. Mendelev [14], semi-empirical potentials still fail reproducing configurations not included into the fit generating potential parameters (Figure 1). A systematic investigation of the stability trends of self-interstitial defects in body-centered transition metals, using density functional theory, shows that in all the nonmagnetic body-centered metals, the most stable self-interstitial defect adopts the <111> configuration (Figure 2) [12]. The relative energies of self-interstitial atoms defect in the non-magnetic body-centered-cubic metals are different from the ferromagnetic α-iron. The calculation of local magnetic moment reveals the presence of a local anti-ferromagnetic moment in the associated to the presence of the interstitial defect α-iron. This means that the atomic configuration and the electronic structure are intimately connected. Contrary to what Willaime et al. claim, correct prediction of the relative stabilities of different dumbbell configurations of interstitial defect should not be so crucial for the predictive modeling of the behavior of irradiated iron based materials, since upon clustering, the relative stability of interstitials in the latter is inversed adopting <111> orientation. But during irradiation, a large amount of interstitial defect is produced, suggesting that the isolated configuration is not the significant one. Classical interaction potentials provide results for multiple interstitials in agreement with

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the experiment. This is the reason why atomistic simulations of defect production cascades presented in the previous paragraph cannot be rejected as unsuccessful modeling attempts. Nevertheless quantum mechanical calculations are interested in the sense that they point out the limitations of classical simulations, which do not take into account the magnetic component of α-iron.

Figure 1. Formation energies of several self-interstitials configurations in α-Fe for various configurations: comparision of quantum DFT results [10] and two empirical potentials (Akland [13] and Mendelev [14]).

Willaime et al. have extended their study by investigating the dynamics of self-interstitial atoms by calculating various migration jumps up to third nearest neighbor positions [15]. The activation energies are determined by the calculation of the migration-rotation pathways. Excellent agreement with the experimental values of migration energies of mono-, di-, and tri – interstitial and vacancy type defects is found. Experimentally, the population of vacancies, interstitials and their clusters can be followed only indirectly, by resistivity measurements. Results are generally difficult to interpret, so it appears very promising to inject ab initio calculated quantities into a Kinetic Monte Carlo (KMC) simulation [16] in order to simulate the kinetics of defects and reproduce the main features in experimental resistivity curves [15,17,18]. The different defects are considered as objects without any atomic details and their distribution evolution is followed. The time and length scales are those of macroscopic experiments (hours and micrometers). The key point in this multiscale modeling is that the atomistic properties of defects clusters entering in the Monte Carlo simulations are generated by electronic structure calculations.

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Figure 2. Formation energies of several self-interstitials configurations calculated for bcc transition metals of group 5B (top) and group 6B (bottom). Data are taken from [12], except for Fe [10].

Assuming that resistivity is proportional to the remaining number of interstitials and vacancies, resistivity curves can be obtained. The principle of the experiment consists in irradiating the material at low temperature and to raise the temperature gradually. The aim is to activate a sequence of thermally activated defect migrations and reactions. A sequence of recovery stages is observed consisting in changes of resistivity in the sample during isochronal annealing. The derivative of the resistivity with respect to the temperature, providing effective activation energy values, is in remarkable qualitative agreement with the experiment. Agreement is obtained in the peak positions and allows concluding that the different stages are qualitatively reproduced by Monte Carlo simulations. The largest discrepancy is found for stage III, with a resistivity peak found at a temperature about 57 K lower, which corresponds to a difference of only 0.1 eV in the value of the migration energy of vacancy type defects. The results obtained by atomistic simulations are poorly

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compatible with experimental results, since the migration and dissociation energies associated to different defects configuration are quantitatively badly reproduced by empirical potentials, used for α-iron. Notwithstanding these positive results, main shortcomings of KMC could be summarized as follows. Real time is not present in these simulations in the meanwhile that only qualitative results can be obtained. KMC consists in numerically solving a master equation. But the equivalence between analytical and numerical solutions of this equation is sometimes questioned. And finally how is incorporated the evolution of chemical potential ? 4. Irradiation and Mechanical Properties Dislocations and dislocation mobility are responsible of large plastic deformation in ductile materials at lower stresses than in perfect crystals. Understanding the structure and the mobility of dislocations appears therefore to be of crucial importance in building a predictive approach of the long term behavior of irradiated materials, especially if including interaction of these dislocations with impurities and point defects produced during irradiation processes. The energy associated with the dislocations is stored in the whole material. At the length scale of several lattice constants away from the dislocations, i.e., outside the core region of the dislocations, the strain and stress fields are described well by the continuum dislocation theory. Within the core region of the dislocations, the continuum theory does not apply, and atomistic models are used to describe the details of the dislocation cores. The hereafter depicted work by Aslanides et al. [19] attempts to link the two different scale modeling by introducing in the continuum theory the dislocation core radii evaluated in atomistic simulations. By introducing a perfect edge dislocation in an aluminum slab, with suitable boundary conditions, in order to enable shear deformation of the crystal and to avoid any tilt of it, and by applying an energy minimization procedure, Aslanides et al. find a good agreement between the simulated and experimental values of distance of dissociation. They show how the two descriptions, atomistic and continuous, can be joined to provide the excess energy of a system containing a constrained perfect edge dislocation in its center (Figure 3). Since elastic theory predicts important displacements when approaching the dislocation line, the excess energy is divided into two contributions. The core energy is defined in the region where the non elastic excess energy is dominant and where a discrete description of the material is needed while for larger values, continuous approach is still valid.

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Figure 3. Excess energy stored in a cylinder with axis the dislocation line and outer radius R. Excess energy in eV per unit length of dislocation, R in Å. The solid line corresponds to the non-elastic region where an atomistic treatment has to be applied, the dashed line corresponds to the elastic prediction. [112 ] projection of the core region of the perfect edge dislocation in aluminium is shown in insert. Each symbol stands for an atomic site. Dimensions in Å; calculated excess energies in meV/atom. (Black - grey - Open symbols: triangles ≈ 70 - 35 meV/at; diamonds ≈ 60 - 30 - 10 meV/at; circles ≈ 45 - 20 - 3 meV/at.) [19]. Figure courtesy of V. Pontikis.

The key point of this approach resides in that the chosen value of the dislocation core radii ensures the continuity of excess energy calculated either atomistically or elastically. This is a multiscale approach in the sense that a quantity provided by atomistic simulations is input into the continuum dislocation theory. It is worth mentioning that temperature or stress effects could be included by extending the above described work with molecular dynamics calculations. We could regret that an isotropic metal, i.e. aluminium is chosen in this work, without addressing the problem of anisotropic metals. The problem of the structure of dislocation core in aluminium with segregated impurities is initiated by Kaxiras et al. [20,21] by coupling quantum mechanical, classical atomistic and continuum mechanical simulations in a unified fashion. This could be very promising in tackle with systems where chemical interactions, such as impurities, in a small region of space, can affect the macroscopic properties of a material. The idea underlying this concurrent technique is that atomistic processes of interest often occur in a very small region of the material while the majority of atoms behave accordingly to well-established continuum theory. The small central region is described by a full quantum mechanical approach, while a classical semi-empirical formalism is applied to the surrounding region. These two descriptions are finally embedded in a

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finite elements continuously treated external region, constituting in that way a new kind of boundary conditions. This method is particularly suitable for system in which the central part contains many different atomic species, which is then fully described by quantum theory, while the surrounding region contains only one atom type. Consequently there is no need to develop reliable empirical potentials able to describe each species and their crossed interactions and only one is needed for the pure material. This renders the method particularly interesting in dealing with impurities, which can become an impossible task for atomistic simulations.

Figure 4. Dislocation core structures obtained from quasi continuum (QC) coupled with DFT approach [21](top). The black circles are Al atoms. The contours correspond to out-ofplane displacement in Å. The contours clearly indicate the splitting of the dislocation. The atoms within the black box are DFT atoms. The finite elements mesh serve as a guide to the eye to help visualize deformation. Dislocation core structure in the presence of a column of H impurities, same QC –DFT technique (bottom). The white circles are H atoms. The black lines are a guide to the eye, indicating atomic planes. Figure courtesy of E. Kaxiras.

This technique was presented by Kaxiras et al. [20,21] and applied to the study of an edge dislocation in aluminum, in the absence and presence of hydrogen impurities. It’s a central problem in the hydrogen embrittlement of metals, which is nowadays impossible to tackle with by using empirical potentials. It is worth noticing that this work suggests new ways to the understanding of this phenomenon. The presented results indicate that a density functional theory box containing less than hundred atoms is sufficient to reproduce the dislocation splitting behavior into two partials,

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with the correct dissociation distance (Figure 4). The calculated value is very close to the experimental one. Moreover, the authors investigate the effect of adding one column of hydrogen impurities at the dislocation core. This results in a spreading of the core, i.e. the dissociation distance is increased up to 13 Å. This is consistent with the fact that the hydrogen could lower the stacking fault energy. This was previously suggested by some density functional theory calculations. It is suggested that a wider dislocation core could give rise to enhanced dislocation mobility in the presence of hydrogen, itself being the cause of the hydrogen embrittlement phenomena. To reinforce this hypothesis, the electron density distribution at the dislocation core in absence and in presence of hydrogen has been studied (Figure 5).

Figure 5. Charge density distribution in dislocation core region in the absence (top) and in the presence of one (middle) and two H impurity atoms (bottom). The blue spheres are Al atoms, and the red spheres are H atoms. The gray isosurfaces illustrate the charge density distribution (0.28 electrons/ Å3). Electron density values range from 0 to 0.30 electrons / Å3 [21]. Figure courtesy of E. Kaxiras.

In presence of the impurity, negative charge accumulation occurs. The strong and covalent bonding between aluminum atoms across the slip plane is disrupted, while ionic bonding between aluminum and hydrogen atoms is developed. The ionic character of the bonding appears to be responsible of the wider dislocation core. This work marks the first evidence of significant modification of the atomic bonding at the dislocation core in aluminum and in this sense fills out the work by Alslanides et al. [19] previously presented. Nevertheless the benefits of Kaxiras’s approach in the field of impurity effects are achieved at the expense of an important computational cost because of the quantum mechanics treatment applied to the dislocation core region. Moreover the absence of time and temperature in the model

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could be blamed. Boundary conditions between different scale modeling in the system are delicate and should be further investigated. 5. Conclusion and Perspectives In this paper, a number of examples of materials simulation have been briefly described. They have been chosen to illustrate some current trends in multiscale computational modeling of radiation damages in materials. They constitute just a tiny and not necessarily representative sample of the widespread activities in this field. The latter rapidly progresses, as increasingly complex problems associated with real materials can be addressed. Powerful techniques like those based on DFT and MD simulations allow to tackle with more and more complex and large systems which would not be even considered just a decade ago. These include for example the quantum mechanical based study of extended defects such as dislocations with impurity segregated inside its core. The presented works are not only qualitative like KMC but also quantitative like when computing the core radii of a dislocation or simulating resistivity recovery experiment. These progress naturally owes a lot to the rapidly growing computing power at affordable costs. Even if right components in simulations have been identified to travel through the different scales of description (e.g. interstitials and vacancies point defects and related migration energies), we cannot deny that the encountered successes are often due to lucky empiric approaches. A lot of different techniques have been developed providing a better understanding of irradiation processes controlling the ageing of radiated materials. Nevertheless, it is admitted that an exact treatment of interactions in intermetallic alloys has still not been answered satisfactorily. Similarly, time and space are not simultaneously handled with in presented examples and considerable efforts are still required to converge to a multiscale modeling where time, space and temperature should be concurrently and exactly treated. In that sense, experiments are not yet replaced by simulations !

Acknowledgement The author acknowledges Dr V. Pontikis (V.P.) for fruitful discussions as well as for Figure 3. Dr Efthimios Kaxiras is sincerely acknowledged for Figures 4 and 5.

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References [1] Calder, A. F., and Bacon, D. J., 1993, A molecular dynamics study of displacement cascades in α-iron, J. Nucl. Mater. 207:25-45. [2] Zinkle, S. J., and Singh, B. N., 1993, Analysis of displacement damage and defect production under cascade damage conditions, J. Nucl. Mater. 199:173-191. [3] Bacon, D. J., Diaz de la Rubia, T., 1994, Molecular dynamics computer simulations of displacement cascades in metals, J. Nucl. Mater. 216:275-290. [4] Hansen, J.-P., 2008, Fundamentals of Liquids, this volume. [5] Rosato, V., Guillope, M., and Legrand, B., 1989, Thermodynamical and structural properties of f.c.c. transition metals using a simple tight-binding model, Phil. Mag. A 59:321-336. [6] Allen, M. P., and Tildesley, D. J., 1987, Computer Simulation of Liquids, Clarendon Press, Oxford. [7] Terentyev, D. A., Malerba, L., Characova, R., Nordlund, K., Olsson, P., Rieth, M., and Wallenius, J., Displacement cascades in Fe–Cr: A molecular dynamics study, 2006, J. Nucl. Mat. 349:119-132. [8] Justin, N., Nordlund, K., Wallenius, J., and Malerba, L., 2005, Nucl. Instr. and Meth. in Phys. Res. B 228:92. [9] Scultz, H., 1991, in: Atomic Defects in Metals, Landolt-Börnstein New Series, Group III, Vol 25, Springer-Verlag, Berlin, p.115. [10] Fu, C.C., Willaime, F., 2004, Ab initio study of helium in α-Fe: Dissolution, migration, and clustering with vacancies, Phys. Rev. Lett. 92:175503. [11] Willaime, F., Fu, C. C., Maranica, M. C., and Dalla Torre, J., 2005, Nucl. Instr. and Meth. in Phys. Res. B 228:92. [12] Nguyen-Manh, D., Horsfield, A. P., and Dudarev, S. L., 2006, Self-interstitial atom defects in bcc transition metals: Group-specific trends, Phys. Rev. B 73:020101(R). [13] Akland, G. J., Bacon, D. J., Calder, A. F., and Harry, T., 1997, Computer simulation of point defect properties in dilute Fe - Cu alloy using a many-body interatomic potential, Phil. Mag. A 75:713-732. [14]Mendelev, M. I., Han, S., Srolovitz, D. J., Akland, G. J, Sun, D.Y., and Asta, M., 2003, Development of new interatomic potentials appropriate for crystalline and liquid iron, Phil. Mag. A 83:3977. [15] Fu, C.C., Dalla Torre, J., Willaime, F., Bocquet, J. -L., and Barbu, A, 2005, Multiscale modelling of defect kinetics in irradiated iron, Nat. Mat. 4:68. [16] Lanore, J. M., 1974, Rad. Eff. 22:153. [17] Dalla Torre, J., Bocquet, J.-L., Doan, N. V., Adam, E., and Barbu, A., 2005, JERK, an event-based Kinetic Monte Carlo model to predict microstructure evolution of materials under irradiation, Phil. Mag. 85:549-558. [18] Dalla Torre, J., Fu, C. C., Willaime, F., Barbu, A., and Bocquet, J.-L., 2006, Resistivity recovery simulations of electron-irradiated iron: Kinetic Monte Carlo versus cluster dynamics, J. Nucl. Mat. 352:42-49. [19] Aslanides, A., and Pontikis, V., 1998, Atomistic study of dislocation cores in aluminium and copper, Comput. Mat. Sci. 10:401-405. [20] Choly, N., Lu, G., Kaxiras, W. E., and Kaxiras, E., 2005, Multiscale simulations in simple metals: A density-functional-based methodology Phys. Rev. B 71:094101. [21] Lu, G., Tadmor, E. B., and Kaxiras, E., 2006, From Electrons to Finite Elements: A Concurrent Multiscale Approach for Metals, Phys. Rev B 73:024108.

A MULTISCALE APPROACH TO MEASURING AND MODELING CLEAVAGE FRACTURE TOUGHNESS IN STRUCTURAL STEELS G. R. ODETTE*, H. J. RATHBUN, M. HRIBERNIK, T. YAMAMOTO AND M. HE Department of Mechanical Engineering, University California Santa Barbara Santa Barbara, CA 93106, USA P. SPÄTIG Fusion Technology-Materials, CRPP-EPFL, Association EURATOM-Confédération Suisse, 5232 Villigen PSI, Switzerland

Abstract: This paper overviews the physical processes that mediate cleavage fracture toughness from atomic to structural length scales. We first describe the toughnesstemperature master curve shift method to assess embrittlement following neutron irradiation of structural steels. Based upon a well-calibrated local approach of cleavage, we present an optimized adjustment procedure to transfer fracture toughness data from one specimen geometry to another one. The adjustment procedure is illustrated by two large databases: one for a reactor pressure vessel steel and one for an advanced high-chromium steel. The observed constant shape of the master-curve after irradiation is believed to result from a moderate temperature dependence of the critical local microcleavage stress. We briefly summarize the results of a recent experimental study of the temperature dependent initiation (KIc) and arrest toughness (Ka) of cleavage oriented (100)[010] and (100)[011] Fe single crystals that support this hypothesis. Keywords: ductile-brittle transition; ferritic steels; toughness-temperature master-curve; local approach; iron single crystal toughness

1. Introduction Since cracks are presumed to be present, defect tolerant structural integrity assessments are required to design and safely operate critical nuclear reactor components. Integrity assessments must properly relate pertinent characteristics of both the material and structure to the loads and displacements (or stresses and strains) that lead to fracture by crack propagation. The

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G. R. Odette, Phone: +1-805 893 3525. Fax: +1-805 893 3525. Email: [email protected].

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pertinent characteristics include the specific size and geometry of crackedbody and appropriate measures of the material’s constitutive properties and local measures of their corresponding fracture toughness. While generally not an issue at the start-of-life, the probability of fast fracture increases in service due to the growth of sub-critical cracks by fatigue and other mechanisms, as well as the degradation of fracture toughness by irradiation embrittlement. As illustrated schematically in Figure 1, fracture is inherently a multiscale phenomenon, mediated by processes from the atomic to component dimensions. In this paper we summarize progress on developing and applying a multi-scale understanding of the fracture of structural steels in the cleavage transition. We first address continuum macroscale aspects of cleavage fracture, in the framework of the master toughness-temperature curve (MC) concept. This is followed by a summary of the atomic and mesoscale processes leading to cleavage. We close by briefly outlining some outstanding issues and research needed to resolve them. Note the reader is directed to a recent paper that comprehensively discusses many of these same issues in much more detail (Odette et al. 2003). 2. Cracked body Loading Parameters The term cracked-body refers to both structures and test specimens. Length scales range from about 10–3 to >1 m. The crack loading parameter that relates the applied loads (P) and displacements (Δ) to the concentrated stress and strain fields at the crack tip of that ultimately cause cleavage. The size and geometry of the cracked region and plastic deformation play a key roles in determining the critical loads and displacements (Pc– Δc) at fracture. For very large, deeply cracked linear elastically loaded bodies, (see Figure 2) with plasticity confined to a small, embedded crack tip region, the pertinent parameter is the linear elastic stress intensity factor (SIF), KI (Anderson, 2005). The KI is determined by P, the crack length (a) and a known nondimensional geometric function (Anderson, 2005), f(a/W), as:

⎛ P ⎞ K I = f (a w) ⎜ ⎟ a ⎝ Bw⎠

(1)

Here, a is the crack length and W is the specimen width (see Figure 4 below for a description of specimen dimensions). The elastic region is limited to the initial linear portion of the P–Δ curve up to the point of plastic yielding at Py–Δy (see Figure 2). Elastic fracture occurs at P < Py when KI = KIc, where KIc is the material’s linear elastic fracture toughness. In this sense, the KI = KIc criteria is analogous to

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Figure 1. Schematic illustration of the multiscale nature of fracture.

Figure 2. Schematic illustration of a typical load-displacement fracture curve.

specifying the onset of plastic deformation in a tensile test as being when the applied stress (σ ) equals the material’s yield stress (σ ). The SIF is no longer applicable outside the linear elastic region and must be replaced by an elastic-plastic loading parameter KJ based on the evaluation of a corresponding J-integral, Jep (see Figure 2). In this case Jep is given by:

J ep =

K I2 η A pΔ + B E′ b

(2a)

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K J = E ′ J ep

(2b)

Here E’ is the plane strain elastic modulus, η (≈2) is a constant that primarily depends on geometry, ApΔ is the plastic work area under the P-Δ curve and B and b are the specimen thickness and uncracked ligament length, respectively. Again the applied loading at fracture, KJ = KJc, where KJc is the material’s elastic-plastic fracture toughness. Thus, integrity assessment requires measuring KJc with coupon tests and applying this measure of a material’s fracture resistance to structures by evaluating the corresponding Pc–Δc condition when the structures are loaded to KJ = KJc. As discussed below, this single parameter description of determining cleavage fracture conditions can be directly used only in very limited circumstances. A much more general approach is to use local microscale measures of fracture toughness to specify the crack-tip stress-strain fields that cause a material to cleave, that are independent of the cracked body geometry. These local measures of fracture toughness can be used to determine the Pc–Δc condition and corresponding measured KJm (for coupon tests) or effective KJe (for components). Note, our focus here is on simple deterministic measures of local fracture toughness, that are elaborations of the critical stress-critical distance criteria, first proposed by Ritchie et al. (1973). However, these same concepts can be incorporated in statistical local models of the probability of cleavage as a function of the crack loading (examples include, Beremin 1983, Wallin 1984, Petti and Dodds 2004a). A detailed review on the French development of local approaches was recently published by Pineau (2006). 3. Master Curve-Temperature Shift Method to Assess Embrittlement To begin, it is important to define “fracture toughness” since it has both different values and significance in its various manifestations (see Anderson 2005). Further, the effective toughness of a cracked body extrinsically depends on size and geometry. Indeed, fracture toughness can be considered an intrinsic material property only for very restricted and well-defined conditions. These conditions are seldom met in practice. In this paper, we define a material fracture toughness (KJc) for cleavage under plane strain, elastic-plastic small scale yielding (SSY) conditions with a T-stress = 0 (see below) and quasi-static Mode I loading of through-thickness edge fatigue cracks with a crack depth (a) to specimen width (W) ratio of a/W ≈ 0.5. However, even in this very restricted case, KJc itself depends on the specimen thickness, or the crack front length (B). Thus, we will further define a reference KJc(KJr) for a specified reference Br = 25.4 mm. These defined values will generally differ significantly from the KJm measured in

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coupon tests, and typically differ even more from KJe, pertinent to the fracture of a cracked structure. The fracture behavior of ferritic alloys, along with other body-centered cubic (bcc) alloys, including refractory vanadium, tungsten and molybdenum alloys, all share several key characteristics that must be properly understood and managed. •

As shown in Figure 3a, between lower and upper shelf regions, KJc increases with increasing test or service temperature. Fast fracture occurs due to unstable propagation of a macroscopic pre-existing crack by locally brittle cleavage or quasi-cleavage. We note that locally brittle fracture can also occur by intergranular processes; and that cleavage may follow a limited amount of ductile tearing.

•

There is a large inherent scatter in KJc (or KJr), thus toughness is more properly described by a distribution function defined by the mean (), median and statistical confidence intervals (Beremin 1983, Wallin 1984, Petti and Dodds 2004a).

•

A variety of in-service microstructural evolutions, driven by irradiation, often lead to significant hardening (Δσy) and degradation of KJc. Embrittlement is manifested as an increase in a temperature (To), or ΔTo, at a reference median value (Kor) of KJc, as well as reduction in the upper shelf ductile tearing toughness.

•

The effective and measured fracture toughness (KJe and KJm) depend on size, geometry and loading rate of the cracked body, as well as other extrinsic factors that must be accounted for in practice.

•

Most fracture tests on irradiated alloys will continue to be based on small to ultra-small specimens.

We have previously described a systematic and highly efficient approach to dealing with embrittlement, as well as all the other factors that control the KJm and KJe, known as the master curves–shifts (MC–ΔT) method (Odette et al. 1998, Odette and He 2000, Odette et al. 2002b, Odette and He 2002, Odette et al. 2003, Yamamoto et al. 2006, Spätig et al. 2007). While differing in detail and philosophy, the MC–ΔT method extends the American Society for Testing and Materials (ASTM) E1921-03 Master Curve Standard (2003) to evaluate median KJc_med(T) and corresponding curves at specified confidence intervals (e.g., KJr(T) at the 5% and 95% bounds) in the transition (ASTM E1921, 2003). Both ASTM E1921-03 standard and the MC–ΔT method assume there is a universal invariant toughness temperature curve shape, KJc_med(T-To), or small family of shapes, that is indexed on an absolute temperature (T) scale by a reference

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Figure 3. a) Schematic illustration of the fracture toughness-temperature behavior of bcc alloys, b) Illustration of the MC method for positioning Ko(T) curves (dashed lines) on an invariant Ko(T-To) scale (solid line).

temperature To at a median toughness, typically 100 MPa√m. This is illustrated in Figure 3b. The KJc_med(T-To) found in the ASTM E1921 standard is primarily for low alloy pressure vessel steels, but this shape also appears to be applicable to cleavage fracture in martensitic steels and, perhaps, vanadium alloys (Odette et al. 1998, Donahue et al. 2000, Odette et al. 2003). Note that the KJc_med(T-To) curve described in the ASTM E1921 is specifically for specimens with a 25.4 mm thickness and is given by: KJc_med = 30 + 70 exp(T-To))

(3)

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Extensive research on reactor pressure vessel (RPV) steels has shown that hierarchical approaches can be used to model both the fundamental mechanics/micromechanics of fracture and irradiation embrittlement (Odette et al. 2001, Jumel et al. 2002, Odette et al. 2003) as further discussed below. The enormous advantage of the MC approach is that the KJc(T) curves can be measured with a relatively small number of relatively small specimens; and that effects such as irradiation and loading rate variations can be treated in terms of ΔTo. Indeed, without such efficiencies, the testing required for integrity assessments of irradiated components would be prohibitive. However the MC method largely rests on an empirical foundation and is specifically aimed at evaluating fracture resistance under a restricted set of conditions that are almost never met in practice. Thus it is necessary to address at least the following issues. •

Is the MC shape indeed invariant for a wide range of alloys and conditions including after irradiation?

•

If the MC shape is invariant, what is the physical basis for such behavior?

•

How does toughness vary with the size and geometry of the cracked body and how can coupon tests be used to inform defect tolerant integrity assessments of components?

•

Are there experimental and model based methods to evaluate the ΔTo from simpler measurements that can limit the need for extensive fracture testing of irradiated alloys?

4. Size and Geometry Effect on Measured Toughness It is well known that measured values of fracture toughness often depend on the test specimen size and geometry. For example, materials science textbooks typically describe the increase in toughness associated with the transition from conditions of plane strain to plane stress with decreasing specimen thickness (B – see Figure 4) showing standard b specimen geometries). However, this is only one manifestation of size effects that are related to the stress and stress-state distributions in the local volume of material near the tip of a blunting crack. Under plane strain, the crack tip stresses are higher than the material’s flow stress due to triaxial constraint, with peak stresses ≥ 3 times the uniaxial yield stress, σy. In general, the crack tip stress fields depend on the true stress-strain constitutive properties of the material, σ(ε), the size and geometry of the cracked body and loading conditions. For limited conditions of deep cracks with a/W ≈ 0.5, (where a is the crack depth, W the specimen width and S the span for cracked-beam specimens)

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loaded in bending, and with a small plastic zone embedded in a larger elastic region, the stress fields (σij) have a self-similar, and are described by a fixed function (fssy) when distances from the crack tip (r) are normalized by J/(b σy), or the crack tip opening displacement, δ,

⎛r

⎞

σ ij = f ssy ⎜ ,θ ⎟ ⎝δ ⎠ where θ is the angle from the crack plane. Note, full self-similarity also requires that the so-called T-stresses to be negligible; the T-stresses that occur in actual specimens are non-singular (weakly depend on position) and oriented in the direction of the specimen width, W, with either positive (tensile) or negative (compressive) signs. Small-scale yielding (SSY) fields depend on the constitutive properties and KJ, but are independent of the size and geometry of the cracked body.

Figure 4. Three point bending specimen dimensions with self-similar geometries.

Size and geometry factors that reduce the amplitude of the crack tip stress fields below SSY levels are responsible for what are collectively known as constraint loss effects. There is a large literature on constraint loss effects based on finite element simulations of crack tip fields for various specimen configurations (Dodds et al. 1991, Dodds et al. 1993, Odette 1994, Nevalainen and Dodds 1995, Gao et al. 1998; Gao and Dodds 2001, Anderson 2005, Rathbun et al. 2006a, 2006b). Constraint loss increases measured fracture toughness, KJm, of body-centered cubic alloys in the cleavage transition over corresponding SSY values, KJc, (KJm/KJc > 1). Even under plane strain conditions, constraint loss is caused by large scale inplane plastic deformation, when the size of the plastic zone is no longer small compared to the uncracked ligament, b, or other key cracked body dimensions, leading to a size dependence of KJm, schematically illustrated in Figure 5. Further, non-singular compressive T-stresses, associated with shallow cracks or loading in tension, also reduce constraint and increase KJm (Dodds et al. 1991, Odette 1994 Gao et al. 2000, Gao and Dodds 2001, Odette and He 2002, Petti and Dodds 2004b, Anderson 2005).

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Figure 5. Schematic illustration of the concept of small and large scale yielding requiring larger KJ to produce the same stressed volume, V.

In addition to constraint loss effects, there are also statistical size effects related to the volume of material under high stress near the crack tip. At equal SSY J-loading, the highly stressed volume increases roughly with the specimen thickness, B (neglecting near side-surface effects). A larger volume increases the statistical probability that a sufficiently high stress field location will encounter a sufficiently weak, so-called, trigger particle that produces a propagating microcrack resulting in macroscopic cleavage. Trigger particles typically include large grain boundary carbides, or clusters of carbides, and other brittle inclusions. The trigger particles fracture under the high crack tip stress fields, enhanced by local strain incompatibilities. However, the critical event is continued propagation of the particle nucleated cleavage microcrack through the ferrite matrix. Based on a modified Griffith criterion, the stress required to produce a propagating microcrack varies as 1/√dt, where dt is the effective trigger particle size. As schematically illustrated in Figure 6, the probability (Pf) of triggering a large weakest link microcrack nucleation site increases with increasing stressed volume, hence, B. Pf also statistically varies from specimen to specimen at a given KJ. Thus, stressed volume and statistical effects are responsible for both the B-dependence and natural scatter in measured cleavage fracture toughness.

Figure 6. Illustration of the need for a larger KJ to produce the same stressed volume in a thinner specimen.

Within the framework of the ASTM E1921 standard, the statistical effects are accounted for by adjusting the measured toughness KJm (with a

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thickness B) to the reference toughness KJr, corresponding to a specimen thickness of Br=25.4 mm using the equation: 1/ 4

⎛ B⎞ K Jr = [K Jm − K min ] ⎜ ⎟ ⎝ Br ⎠

+ K min

(4)

The equation is based on an equivalent stressed volume resulting in equal Pf, that is 0 below a minimum Kmin. Prior to developing the database as described in the next section and published by Rathbun et al. (2006a), there was no direct basis to unambiguously discriminate constraint loss from statistical size effects. This was due to the fact that the vast majority of toughness data was for specimens that varied in dimension, but had self-similar geometries, like the ones shown in Figure 4 with a relatively small range of B/b ratios. Notably, models of the initial stages of constraint loss predict roughly the same size scaling as statistical size effects (b–1/4 versus B–1/4) in self-similar specimens (Rathbun et al. 1999). Thus, the effects of reductions in B attributed to statistical size effects could actually be associated with reductions in b originating from constraint loss effects. 5. Size/Geometry Adjustment Based Upon the Calibrated Local Approach A comprehensive, single variable-type experiment that provides a robust database for evaluating the individual and combined contributions of statistical and constraint loss mediated size effects was carried out (Rathbun et al. 2006a). The database was designed to be amenable to detailed analysis using physically based cleavage fracture models and to address the following questions (Rathbun et al. 2006b): •

What are the relative contributions of constraint loss and statistical mediated size effects on KJm?

•

Is the (B/Br)1/4–Kmin statistical size scaling procedure consistent with toughness variations over a large range of B for conditions of high constraint?

•

How do KJm and constraint loss effects vary as a function of deformation?

•

Is there an effective and practical approach to adjusting for the combination of constraint loss and statistical size effects and their interactions?

To answer these questions, a full matrix of a/W ≈ 0.5 three point bend specimens with B from 8 to 254 mm and b from 3.2 to 25.4 mm were fabricated from the 3/4 thickness position in a A533B steel plate section of

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the Shoreham reactor vessel. The b–B test matrix covered an unprecedented range of specimen size and geometry. Eight specimens were tested under a common set of conditions at -91 °C for each b–B combination. The test results are shown in Figure 7. Figure 7a shows the variation of the measured KJm with B for the 4 ligament dimensions (increasing b from left to right) illustrating the strong effect of the stressed volume, which scales with B. Figure 7b shows the corresponding KJm averaged for the various B as a function of b, illustrating the effect of increasing constraint loss with decreasing ligament length.

Figure 7. a) KJm versus B for various b, b) Average adjusted to B r=25.4 mm versus b=(W-a) showing considerable constraint loss for small b.

As described in detail in (Rathbun et al. 2006a, 2006b), a micromechanical local fracture model was developed to quantitatively discriminate between constraint loss and statistically stressed volume effects, and to provide a basis the adjust KJm data to other cracked body sizes and geometries. The analysis was based on a model that posits that cleavage occurs when a crack tip normal stress field contour (σ22), equal to a critical micro-cleavage stress (σ*), encompasses a region of material at the crack tip (V) equal to a critical volume (V*), as illustrated in Figure 8.

Figure 8. Schematic illustration of a critical stress σ*. encompassing the critical volume V*.

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In SSY the V(σ22) increases with KJ4. However, when deformation is more extensive in the large scale yielding (LSY) regime, V increases more slowly with KJ, and the condition for cleavage specified by σ22 = σ * and V = V* requires a larger KJm compared to the SSY KJc. This is schematically illustrated in Figure 9, showing the variation of the σ22 normal stress with the non-dimensional distance from the crack tip (x/δ, where δ is the crack tip opening displacement) on the crack plane. The stress fields remain high near the tip, but fall off more rapidly under LSY, thus reducing the V(σ22) at a specified KJ.

Figure 9. Schematic illustration of crack plane σ22 fields for SSY and large scale yielding (LSY).

As illustrated in Figure 10a, the [KJm/KJc] at σ*-V* can be determined by finite element method (FEM) calculations based on the materials σ(ε) and σ*. Three-dimensional FEM calculations were carried out to determine the average in plane stress area (σ22) as a function of KJ, where V = B, for all the specimen geometries in the b-B matrix. The model was first calibrated by fitting B = 25.4 mm compact tension test data over a range of temperatures to determine the local fracture properties σ* and A*.

Figure 10. a) Illustration of toughness scaling between a test specimen and SSY conditions, b) The fully adjusted KJr b-B database.

The calculated [KJc/KJm] factors at cleavage were then used to adjust the measured KJm to KJc for each b-B combination, accounting for constraint

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loss effects. Further, it was found that the statistical stressed volume size effect treatment in Eq. (4), provided a good statistical size adjustment of KJc to KJr. The results of the adjustment on the b-B database are shown in Figure 10b. Figure 11 show the application of the constraint loss-statistical size adjustment procedure to a much larger UCSB and literature database on the Shoreham plate for a very wide range of specimen sizes (Rathbun et al. 2006a, 2006b).

Figure 11. a) Measured toughness data, for the Shoreham RPV steel, b) Statistically stressed volume and constraint loss adjusted KJr data for the Shoreham RPV steel.

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Figure 11a shows that there is large scatter in the measured adjusted KJm without either statistical stressed volume or constraint loss adjustments and with a corresponding wide range of To. In contrast, Figure 11b shows that the statistical stressed volume and constraint loss adjusted data can be represented by a single master curve with To = -84°C. Figure 12 show the results on a similar exercise on the 8Cr normalized and tempered martensitic steel F82H, that yielded a single master curve with oT =-98±5°C (Odette et al. 2002b).

Figure 12. a) Measured KJm for the reference F82H steel, b) Statistically stressed volume and constraint loss adjusted KJr data for the reference F82H steel.

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The amount of deformation at cleavage initiation, and the corresponding loss of constraint, is represented by a M parameter given by

M = 0.5

b

δ

= σy E′

b K Jc2

(5)

The ASTM E19921 Standard proposes a value of M = 30 as a censoring limit for measuring supposedly valid KJc data. However, the results of the Rathbun et al. study clearly show M = 30 is far too permissive, and that constraint loss in three point bend tests begins at M = 150 to 200 and is large below about 75. Note, compact tension specimens have a positive Tstress that helps them maintain constraint to larger values of M; however, recent data on 8Cr normalized and tempered martensitic steel Eurofer97 suggests constraint loss begins below M ≈ 100 even for compact tension specimen tests (Spätig et al. to be published). In summary, the values of measured fracture toughness depend on the specimen size and geometry due to both statistical stressed volume and constraint loss effects even for deep cracks loaded in bending under plane strain conditions. In this case, calibrated micromechanical models can be used to adjust the KJm to other sizes and geometries subject to specific limitations. While not discussed in this paper, the σ*-V* adjustments do not work as well in cases with large negative (compressive) T-stresses associated with shallow cracks and/or tensile loading geometries, which result in large loss of constraint even at low deformation levels. In this case the overall amplitude of the crack tip stress fields are reduced, rather than just at increasing distances from the crack tip as was shown in Figure 9. Since, compressive T stresses are much more typical of component defects and loading, it will be important to extend the micromechanical models to treat such low constraint conditions. The volume averaged power weighted crack tip stresses used in statistical models may be a viable approach to treating low such constraint conditions (Gao and Dodds 2001, Petti and Dodds 2004a), but further discussion of this topic is beyond the scope of this paper. 6. Irradiation Embrittlement and To Shifts While the MC method provides a very efficient way to measure fracture toughness in the cleavage transition, it must be coupled with a systematic procedure to evaluate To shifts (ΔTo) due to irradiation embrittlement, and other factors, such as variations in the loading rate. Here we will focus on the ΔTo in the irradiation hardening dominated regime at around 300°C.

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Hardening, typically measured as increases in σy (Δσy) and the corresponding embrittlement, diminish rapidly above about 350°C and are negligible above 450°C. As discussed elsewhere (Odette et al. 2003, Yamamoto et al. 2006), non-hardening embrittlement mechanisms may emerge at higher irradiation temperatures. However, discussion of this issue is beyond the scope of this paper. It has been empirically observed that for low alloy reactor pressure vessel steels (Sokolov and Nanstad 1999, Odette and He 2000): ΔTo = CoΔσy

(6)

where Co ≈ 0.7°C/MPa. Odette and He have developed a SSY KJc(T) MC shape and ΔTo = Coσy model based on a σ* critical area (A* = V*/B) cleavage criteria. The in-plane stressed A(σ22, KJ) increases with KJ4, and cleavage occurs at A(σ∗, KJc) = A*. The stressed area, A, can be represented in a compact non-dimensional form (Ao) as Ao = log[A(σ22/σy)σy4/KJ4]

(7)

The A(σ22/σy) are computed using a finite element (FE) code, like ABAQUS, for a specified σ(ε), and fitted by a polynomial as Ao(σ22/σy) = C0 + C1(σ22/σy) + C2(σ22/σy)2

(8)

Thus, in the framework of this model, the microstructurally sensitive material properties that control the SSY KJc are σ(ε,T), A* and σ*, as ∗ 1/4 KJc ≈ σy [A*10Ao( σ σ y )]

(9)

Odette and He (2000) showed that at low To, characteristic of unirradiated alloys, the temperature dependence of σy(T) alone is sufficient to predict the shape of the KJc(T) curve. However, at high To, associated with large ΔTo due to irradiation hardening, the classical assumption that σ* is independent of both temperature and irradiation results in a strong layover in the predicted KJc(T) curve, and very large ΔTo that are not consistent with observation. It was shown that the model can be made consistent with observation by incorporating a mild temperature dependence of σ*(T). The resulting predictions for KJc(T) and ΔTo are shown in Figure 13a. Figure 13b shows the corresponding σy(T) and σ*(T) curves. The model σ*-A* model also predicts a Co = 0.7°C/MPa in agreement with the RPV steel data. The Co for irradiated tempered martensitic steels (TMS) are lower than for irradiated RPV, with estimates ranging from about 0.3 to 0.6 (Rensman et al. 2002, Odette et al. 2003, Yamamoto et al. 2006, Spätig at al. 2007, Lucon et al. 2007, Sokolov et al. 2007). It has been recently shown that this is due to the fact that while irradiation hardening has only a small effect on

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strain hardening in the RPV steel case, TMS experience large reductions in strain hardening, or even strain softening, as well as large Δσy (Odette et al. 2002a, Odette et al. 2007) after 300°C irradiations to several displacements per atom. A detailed finite analysis (Odette et al. 2007) showed that the ΔTo correlate with changes in the average strain hardened flow stress (Δσfl) between 0 and 10% strain, pertinent to crack tip process zone conditions, rather than Δσy, as ΔTo ≈ 0.7Δσfl

(10)

Figure 13. a) Predicted KJc(T) (dashed lines) versus MC shapes (solid lines) for the σ* - A* model, b) The σy(T) and σ*(T) curves used in the model.

Thus it is possible to estimate ΔTo with much simpler measurements of Δσfl. However, in cases where necking nearly coincides with yielding in irradiated TMS, estimating the true strain Δσfl is difficult. Notably, however, these authors also showed that hardness (H) also measures the same average Δσfl, in a way that can be quantified using a master H - Δσfl

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curve (He et al. 2007). Thus measurements of unirradiated and irradiated H, converted to Δσfl, can be used to estimate ΔTo. It is also noted, without further elaboration, that the Δσy and Δσfl can be modeled based on observed or irradiation induced changes in the alloy microstructure. 7. Underlying Micromechanisms of Cleavage and Significance of σ* The unresolved key to understanding the MC shape and irradiation induced ΔTo is the temperature dependence of σ*(T). As discussed elsewhere (Odette and He 2000, Odette et al. 2003), σ* is controlled by the propagation of dynamic micro-cracks formed at broken brittle trigger particles. It is important to emphasize that these atomistic scale mechanisms exert a powerful effect on continuum KJc, and reflect the critical underpinning of any multiscale model of cleavage fracture. Assuming a cleavage orientated penny-shaped particle crack with a effective diameter dt σ∗(T) ≈ Kµ(T)(π/2dt)1/2

(11)

The micro-arrest toughness of the alloy, Kµ(T), is the underlying property that controls the temperature dependence of σ∗(Τ). There are many potential trigger sites in a stressed volume, but only those with sizes, local stresses and orientations that can activate low toughness cleavage systems will produce propagating micro-cracks. Note the local stresses include the effects of strain incompatibilities between the matrix and trigger particle and stress fluctuations associated with crystal plasticity; hence, the local stresses are likely to be even larger than those produced by the blunting macro-crack itself. The preferred cleavage system in Fe are {100} planes and <100> or <110> directions. The Kµ can be expressed in terms of a modified Griffith fracture criteria that accounts for the combination of creation of new surfaces, a quasi brittle process contributing < 1 MPa m , and dislocation activity at or near the microcrack tip, contributing a few to several MPa m in the semibrittle clevage region. While the dislocation activity has been classically viewed as dislocation emission from the crack tip, and corresponding nanoblunting, other dissipative processes, such as crack interactions with pre-existing dislocations, may also be important. Thus cleavage in oriented Fe single crystals is semibrittle, with low toughness values (≤ ≈ 5 MPa m at low temperature); however, this single crystal toughness acts as a valve in controlling the much higher macroscopic toughness (>> than 20 MPa m ) in ferritic alloys. A number of models have been proposed for static (KIc) and a fewer number for arrest toughness (KIa as a surrogate for Kµ) in cleavage oriented Fe single crystals. A common feature in the models is that they predict an

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increase in toughness with increasing temperature up to a brittle to ductile transition temperature (TBDT), above which cleavage cannot occur. The TBDT increases with increasing loading rate reflecting the activated character of the underlying dislocation processes. However, previously there have been no reliable measurements of KIa(T) in unalloyed single crystal Fe, and only limited useful data on Fe-3Si and W (Qiao and Argon 2003, Gumbsch et al. 1998). We have recently carried out the first rigorous measurements of KIa and KIc over a wide range of loading rates and temperatures for cleavage oriented Fe-single crystals (Hribernik, 2006), and a series of publications are currently being prepared to provide a detailed description and analysis of these results. However, it is useful to summarize these finding, especially with respect to their implications the MC shape (Hribernik et al. to be published). •

The average KIa gradually increases from about 3.5 MP m at -196°C to about 9 MPa m at 0°C, and is similar in both the (100)[010] and (100)[011] orientations.

•

Initiation, propagation, arrest and re-initiation produce complex slip step structures, river patterns and etch pits on the fracture and side surfaces of the Fe crystals, indicating significant dislocation activity, that is associated with {110}<111> and {112}<111> slip systems. The dislocation activity is qualitatively consistent with the observation semibrittle cleavage fracture toughness.

•

Measurements of initiation toughness show that KIc increases with temperature and decreases with loading rates from < 1 to 2x104 MPa m /s, follow dynamics that are controlled by, and indeed are identical to, thermally activated dislocation dynamics that mediate the corresponding yield stress of Fe based alloys.

•

At low temperatures the fracture and yield dynamics are dominated by double-kink nucleation on screw dislocations. Both fracture and yield dynamics are governed by a stress-dependent activation energy with a maximum value of about 0.6 to 0.7 eV.

•

These activation parameters can be used to adjust the KIc(T) and KIa(T) to form a single master curve on a strain rate compensated temperature scale for any specified reference strain rate.

•

The temperature dependence of the unalloyed Fe single crystal KIa(T) is too strong to rationalize the shape of the macro KJc(T) MC and is inconsistent with measured KIa in Fe-Si (Qiao and Argon 2003).

•

It appears that the KIa(T) and KIc(T) curves are associated with the low strength of unalloyed Fe. Indeed, the shape of the KIa curves for Fe and Fe-3Si approximately scale with the inverse of the total dynamic yield

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stress (σyd(T)) as KIa(T) = Ca/σyd(T). Here, Ca is a normalization factor fit to the Fe single crystal KIa(-196°C) data. Likewise KIc(T) data for W scale as KIc = Ci/σy(T). These results suggest that KIa and KIc are controlled by the total strength of an alloy including both thermal and athermal components. •

The σ*-A* model can be used to calculate the KJc(T) curves for using KIa(T) mediated σ*(T) curve shapes based on the total alloy σyd(T). Assuming a minimum KIa for polycrystalline materials of about 2 - 4 (3.25 is used in the example shown here) MPa√m produces better agreement between the MC shapes and the model predictions MPa√m.

•

The KJc curves for an unirradiated alloy with a static σy = 475 MPa and Δσy from 100 to 300 MPa are shown in Fig. 14a. The predicted KJc(T) are too steep at low To but are in good agreement with the MC shape at higher To. Likewise the ΔTo versus Δσy results shown in Fig. 14b show a curvature that is not experimentally observed. Better results are obtained for a KIa shape that has a weaker dependence on the total σy and that is closer to the universal curve derived in the original model (Odette and He 2000).

•

Clearly the model is oversimplified. For example it assumes that the macroscopic KIa(T) is a surrogate for the micro Kµ(T), which may not be entirely valid. Nevertheless, given it’s simplicity, the revised KJc(T) model predicts reasonable trends and provides a good basis for further experiment and modeling studies.

•

These results suggest that a number, if not all, of the components of the total alloy σy contribute to the confinement of dislocations, and retardation of dislocation energy dissipation processes, near a crack tip, that results in decreased KIa. Thus KIa is not an intrinsic property of the ferrite lattice. Rather the KIa(T) curves are shifted to higher temperatures with increasing alloy strength. These shifts KIa are approximately sufficient to compensate for the reduced temperature dependence of σyd(T) with increasing temperature (and To) resulting in approximately constant MC shapes.

•

One important issue to explore is the relative importance of various contributions to the alloy σy on KIa. These contributions include the lattice resistance to slip, various dispersed dislocation obstacles, dislocations themselves and grain boundaries. For example, the effect of dislocations on KIa(T) would be expected to be somewhat different than on σy(T). Indeed, it has been shown that higher dislocation densities flatten the single crystal KIc(T) curve in W (Gumbsch et al 1998), and our preliminary data suggests the same trend in Fe single crystals.

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Figure 14. a) Predicted KJc curves (dashed lines) for increasing increments of irradiation hardening, Δσy. The MC are shown by the solid lines, b) The predicted ΔTo - Δσy relation. •

Likewise, the effects of solid solution strengthening are more complex than simply adding this contribution to the alloy’s athermal strength component. And variations of strain hardening with microstructure and temperature have not yet been treated.

•

The limitations of current understanding can be addressed systema-tically by carefully designed combinations of experimental and modeling studies

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8. Outstanding Issues and Concluding Remarks Tremendous progress has been made on developing a multiscale framework for understanding and modeling cleavage fracture in the transition. At the mesoscopic and continuum scales, local fracture models have provided a basis to account for size and geometry effects for standard specimen geometries. The continuum models can be linked to mesoscopic scale microstructural features, especially the distributions of trigger particle. Stronger links with dislocation and atomic scale processes of microcrack propagation-arrest have been established along with a fundamental database on Fe single crystal initiation and arrest toughness dynamics. However, while the framework for understanding and developing robust multiscale models are generally in place, many issues and details remain to be resolved. A partial and subjective list includes the following items: •

Size and geometry adjustments for the effective toughness of shallow cracks and defects in structures that often involve low constraint conditions with qualitatively large deviations from classical SSY fields.

•

Local effects of strain incompatibilities, secondary singularities and crystal plasticity on the stress fields in the process zone of a crack that result in the generation and propagation of micro-cracks.

•

More detailed treatments of the constitutive behavior of a material on KJc(T) including factors such as the temperature and strain rate dependence of strain hardening and solute strengthening.

•

Developing improved methods to calibrate KJc(T) models, including micro-micro mechanics approaches that are based on detailed characterization of the alloys microstructure, complementing, or even replacing, mechanical tests.

•

Rigorous dislocation dynamics and atomistic models of KIa and KIc that include the effect of alloy composition and microstucture, closely linked to a wide range of systematic experiments that characterize dislocation structures associated with cleavage and explore the combination other factors that also mediate cleavage toughness.

References Anderson, T. L., 2005, Fracture Mechanics Fundamental and Applications, 3rd Ed. CRC, Boca Raton. ASTM, 2003, “Standard Test Method for Determination of Reference Temperature, To , for Ferritic Steels in the Transition Range”. In: Annual Book of ASTM Standards: Metals

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Mechanical Testing; Elevated and Low Temperature Tests; Metallography. Vol. 03.01, American Society for Testing and Materials. Beremin, F. M., 1983, A local criterion for cleavage fracture of a nuclear pressure vessel steel, Met. Trans. A 14A:2277-2287. Dodds, Jr R. H., Anderson T. L., and Kirk, M. T., 1991, A framework to correlate a/W ratio effects on elastic-plastic fracture toughness (Jc), Int. J. fract. 48:1-22. Dodds Jr, R. H., Fong Shih, C., and Anderson, T. L., 1993, Continuum and micromechanics treatment of constraint in fracture, Int. J. fract. 64:101-133. Donahue, E. G., Odette, G. R., and Lucas, G. E., 2000, On the mechanisms and mechanics of fracture toughness of a V-4Cr-4Ti alloy, J. Nucl. Mater. 283–287:518-522. Gao X., Ruggieri C., and Dodds Jr R. H., 1998, Calibration of Weibull stress parameters using fracture toughness data, Int. J. Fract. 92:175-200. Gao, X., Dodds Jr, R. H., and Tregoning, R. L., 2000, Prediction of the To shift between specimens of different constraints using the T-stress based T-functions, Int. J. fract. 104 (3):L3-L8. Gao, X., and Dodds Jr, R. H., 2001, An engineering approach to assess constraint effects on cleavage fracture toughness, Eng. Fract. Mech. 68:263-283. Gumbsch, P., Riedle, J., Hartmaier, A., and Fischmeister, H. F., 1998, Controlling factors for the brittle-to ductile transition in tungsten single crystals, Science 282 (13):1293-1295. He, M. Y., Odette, G. R., Yamamoto, T., and Klingensmith, D., 2007, A universal relationship between indentation hardness and flow stress, J. Nucl. Mater. 367-370:556560. Hribernik, M., 2006, Cleavage Oriented Iron Single Crystal Fracture Toughness, PhD Thesis University of California Santa Barbara. Jumel, S., Domain, C., Ruste, J., Van Duysen, J. C., Becquart, C., Legris, A., Pareige, P; Barbu, A., Van Walle, E., Chaouadi, R., Hou, M., Odette, G R., Stoller, R E., and Wirth, B. D., 2002, Simulation of irradiation effects in reactor pressure vessel steels: the reactor for virtual experiments (REVE) Project. J. Test. Eval. 30(1):37-46. Lucon, E., Chaouadi, R., and Decréton, M., 2004, Mechanical properties of the European reference RAFM steel (EUROFER97) before and after irradiation at 300°C, J. Nucl. Mater. 329-333:1078-1082. Nevalainen, M., and Dodds Jr, R. H., 1995, Numerical investigation of 3-D constraint effects on brittle fracture in SE(B) and C(T) specimens, Int. J. fract. 74:131-161. Odette, G. R., 1994, On the ductile to brittle in martensitic stainless steels – Mechanisms, models and structural implications, J. Nucl. Mater. 212–215:45-51. Odette, G. R., Edsinger, K., Lucas, G. E., and Donahue, E., 1998, Developing fracture assessment methods for fusion reactor materials with small specimens, in: Small Specimen Test Techniques, ASTM STP 1329, W. R. Corwin, S. T. Rosinski, E. van Walle, Eds., American Society for Testing and Materials, 298. Odette, G. R., and He, M. Y., 2000, A cleavage toughness master curve model, J. Nucl. Mater. 283–287:120-127. Odette, G. R., Wirth, B. D., Bacon, D. J., and Ghoniem N. M., 2001, Multiphysics modeling of radiation-damaged materials: embrittlement of pressure-vessel steels. MRS Bulletin 176-181. Odette, G. R., and He, M. Y., 2002, Micromechanical modeling of master curve temperature shifts due to constraint loss, J. Nucl. Mater. 307–311:1624-1628. Odette, G. R., He, M. Y., Donahue, E. G., Spätig, P., and Yamamoto, T., 2002a, Modeling the multiscale mechanics of flow localization-ductility loss in irradiation damaged bcc alloys, J. Nucl. Mater. 307-311:171-178.

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Odette, G. R., Rathbun, H. J., Rensman, J. W., and van den Broek, F. P., 2002b, On the transition toughness of two RA martensitic steels in the irradiation hardening regime: a mechanism-based evalutation, J. Nucl. Mater. 307–311:1011-1015. Odette, G. R., Yamamoto Y., Rathbun H. J., He M. Y., Hribernik M. L., and Rensman J. W., 2003, Cleavage fracture and irradiation embrittlement of fusion reactor alloys: mechanisms, multiscale models, toughness measurements and implications to structural integrity assessment, J. Nucl. Mater. 323:313-340. Odette, G. R., He, M. Y., and Yamamoto, T., 2007, On the relation between irradiation induced changes in the master curve reference temperature shift and changes in strain hardened flow stress, J. Nucl. Mater. 367-370:561-567. Petti, J. P., and Dodds Jr R. H., 2004a, Coupling of the Weibull stress model and macroscale models to predict cleavage fracture, Eng. Fract. Mech. 71:2079-2103. Petti, J. P., and Dodds Jr R. H., 2004b, Constraint comparisons for common fracture specimens: C(T)s and SE(B)s, Eng. Fract. Mech. 71:2677-2683. Pineau, A., 2006, Development of the local approach to fracture over the past 25 years: theory and applications, Int. J. fract. 138:139-166. Qiao, Y., and Argon, A. S., 2003, Brittle-to-ductile fracture transition in Fe-3wt.%Si single crystals by thermal crack arrest, Mechanics of Materials 35(9):903-912. Rathbun H. J., Odette G. R., He M. Y., Lucas G. E., and Sheckherd J. W., 1999, Statistical versus constraint size effects on the transition regime toughness: a preliminary study of a pressure vessel steel. Fracture, fatigue and weld residual stress, PVP. 393. American Society of Mechanical Engineers: 17-22. Rathbun H. J., Odette G. R., He M. Y., and Yamamoto T., 2006a, Influence of statistical and constraint loss size effects on cleavage fracture toughness in the transition-A model based analysis, Eng. Fract. Mech. 73:2723-2747. Rathbun H. J., Odette G. R., Yamamoto T., and Lucas G. E., 2006b, Influence of statistical and constraint loss size effects on cleavage fracture toughness in the transition-A single variable experiment. Eng. Fract. Mech. 73:134-158. Rensman, J., van Hoepen, J., Bakker, J. B. M., den Boef, R., van den Broek, F. P., and van Essen, E. D. L., 2002, Tensile properties and transition behaviour of RAFM steel plate and welds irradiated up to 10 dpa at 300°C, J. Nucl. Mater. 307-311:245-249. Ritchie, R. O., Knott, J. F., and Rice, J. R., 1973, On the relationship between critical tensile stress and fracture toughness in mild steel, J. Mech. Phys. Sol. 21:395-410. Sokolov, M. A., and Nanstad, R. K., 1999, Comparison of irradiation induced shifts of KJc and Charpy impact toughness for reactor pressure vessel steels, in: Effects of Irradiation on Materials: 18th International Symposium, American Society for testing and Materials, Philadelphia, PA, 167-190. Sokolov, M. A., Tanigawa, H., Odette, G. R., Shiba, K., and Klueh, R. L., 2007, Fracture toughness and Charpy impact properties of several RAFMS before and after irradiation in HFIR, J. Nucl. Mater. 367–370:68-73. Spätig, P., Bonadé, R., Odette, G. R., Rensman, J. W., and Campitelli E. N., Plastic Flow Properties and Fracture Toughness Characterization in the Low Transition of Unirradiated and Irradiated Tempered Martensitic Steels, 2007, J. Nucl. Mater. 367– 370:527–538. Wallin, K., 1984, The scatter in KIc-results, Eng. Fract. Mech. 19:1085-1093. Yamamoto, T., Odette, G. R., Kishimoto, H., Rensman, J. W., and Miao, P., 2006, On the effects of irradiation and helium on the yield stress changes and hardening and nonhardening embrittlement of 8Cr tempered martensitic steels: Compilation and analysis of existing data, J. Nucl. Mater. 356:27-49.

MICROSTRUCTURES AND MECHANICAL PROPERTIES OF IRRADIATED METALS AND ALLOYS S. J. ZINKLE Materials Science and Technology Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6138 USA

Abstract: The effects of neutron irradiation on the microstructural evolution of metals and alloys are reviewed, with an emphasis on the roles of crystal structure, neutron dose and temperature. The corresponding effects of neutron irradiation on mechanical properties of metals and alloys are summarized, with particular attention on the phenomena of low temperature radiation hardening and embrittlement. The prospects of developing improved high-performance structural materials with high resistance to radiation-induced property degradation are briefly discussed. Keywords: Dislocation loops; displacement cascades; void swelling; radiation hardening; flow localization; fracture toughness embrittlement; structural materials; ferritic/martensitic steels

1. Introduction The importance of neutron irradiation on the properties of materials for nuclear energy applications was first highlighted by Wigner (1946) in the mid-1940s. Whereas the structural materials in current (2nd-generation) commercial light water reactor fission reactors are subjected to moderate temperatures (~290-330°C) and relatively low doses (<0.1 displacements per atom, dpa, for the reactor pressure vessel steel and 10 to 30 dpa for core internal structures), the anticipated temperature and dose requirements for structural materials in 4th-generation fission reactors are much more challenging, as summarized in Figure 1. 2. Fundamentals of Microstructural Changes Neutron irradiation can produce a wide variety of microstructural changes in metals and alloys, including formation of point defects such as selfinterstitial atoms (SIAs) and vacancies, defect clusters such as dislocation loops and stacking fault tetrahedra (SFTs), and cavities (voids and gas-filled V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

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bubbles). Numerous tools are available to qualitatively or quantitatively evaluate the radiation-induced microstructural changes. Positron annihilation spectroscopy is best suited for investigation of vacancies and small voids or bubbles. Neutron diffraction techniques such as small angle neutron scattering are very effective for evaluating small precipitates. Quantitative analyses of dislocation loops, cavities, and precipitates are usually performed using transmission electron microscopy (TEM) techniques.

Figure 1. Overview of projected operating temperatures and damage levels for structural materials in the six Gen-IV concepts compared to existing (Gen-II) fission reactors.

The amount of radiation damage is quantified by the international standard unit of displacements per atom, dpa (Norgett et al. 1975). A damage level of 1 dpa corresponds to metastable displacement of all atoms in the material. At elevated temperatures, thermally activated diffusion of the defects produced by irradiation leads to a large amount of recombination (typically 99 to 99.9% recovery of the initial displacement damage). It is important to recognize that vigorous transient atomic mixing occurs within the fast neutron-induced displacement cascades, such that the amount of atomic mixing is roughly two orders of magnitude higher than the dpa value (Zinkle and Singh 1993). Electrical resistivity studies (in conjunction with other property measurements, such as X-ray diffraction monitoring of lattice expansion and positron annihilation spectroscopy) were very useful in the early years of radiation damage studies to identify the characteristics of key point defects such as SIAs and vacancies. Five distinct recovery stages were observed in isochronal annealing studies on pure metals following irradiation at temperatures near absolute zero (where no point defect diffusion is possible). Three of the most important recovery stages are “Stage I” recovery due to SIA migration, Stage III recovery due to monovacancy

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migration, and Stage V recovery due to thermal dissolution of small vacancy clusters produced directly in displacement cascades (Zinkle and Singh 1993). 2.1. PRIMARY DAMAGE STATE IN FCC AND BCC METALS

The displacement cascade event is initiated by transfer of kinetic energy from an incident neutron to a primary knock-on atom in the host lattice (interaction time scale ~10–15 s). The peak transient atomic mixing occurs at times ~0.1 to 5 ps after the initial knock-on collision, and the residual displacement cascade structure is largely established within ~10 ps following the initiation of the displacement event. Molecular dynamics (MD) simulations of the primary damage state in displacement cascades (Bacon et al. 2000, Bacon et al. 2003) and experimental studies of defect accumulation (Singh and Zinkle 1993, Eldrup et al. 2002) have observed several fundamental differences in the behavior of face centered cubic (FCC) and body centered cubic (BCC) metals. Whereas the overall defect production efficiency (normalized to the dpa value) and amount of in-cascade clustering is similar in FCC and BCC metals, the average cluster sizes produced within displacement cascades are substantially larger in FCC metals compared to BCC metals for typical transition metals with comparable atomic mass. Figure 2 shows a comparison of the residual primary damage state in FCC copper and BCC iron (Zinkle 2005a). The degree of in-cascade clustering (and average cluster size) increases with increasing atomic mass. Irradiation temperature has a rather weak effect on the primary damage state of irradiated metals for temperatures above recovery stage I. 2.2. DOSE DEPENDENCE OF MICROSTRUCTURAL EVOLUTION BELOW RECOVERY STAGE V

At very low doses (below 0.0001 to 0.01 dpa), the rate of defect cluster accumulation is directly proportional to dose. For intermediate doses at temperatures where long-range point defect migration is possible, interacttions among defects created in different displacement cascades often leads to a regime where the defect cluster accumulation is sublinear, e.g., square root dependence on dose (Singh and Zinkle 1993). At damage levels above ~0.1 dpa, the defect cluster density typically reaches a constant value that depends on dose rate and temperature for irradiation at temperatures above recovery Stage V (higher concentrations for high dose rate and low irradiation temperature). The saturation value of the defect cluster density at low temperature increases with atomic mass of the bombarded metal.

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Figure 2. Comparison of the residual radiation defects in Cu and Fe for a 20 keV PKA displacement cascade, as determined from molecular dynamics simulation.

The dose dependence of the size and predominant geometry of defect clusters varies for different metals. For example, in pure copper the SFT is the predominant defect cluster for all doses up to at least 10 dpa and the average cluster size exhibits a constant value of ~2.5 nm (Zinkle and Singh 2000). This has been taken as evidence for highly efficient direct production of SFTs within neutron displacement cascades in copper. Conversely, other medium-mass FCC metals such as nickel or austenitic stainless steel exhibit a much more complex evolution of defect cluster size and density. For example, in pure nickel the predominant defect cluster changes from SFTs at low doses (<0.1 dpa) to dislocation loops at higher doses (Zinkle and Snead 1995, Yao et al. 2005). Whereas the SFT size under neutron irradiation conditions is typically constant in most metals, the dislocation loop size increases with increasing dose.

IRRADIATED METALS: MICROSTRUCTURES AND PROPERTIES 231 2.3. TEMPERATURE DEPENDENCE OF MICROSTRUCTURE CHANGES

The temperature dependence of the microstructure of irradiated metals and alloys can be divided into three broad categories: Below recovery Stage V, the cavity swelling regime, and very high temperatures. The high density of small defect clusters introduced during low temperature irradiation (below recovery Stage V, corresponding to homologous temperatures of 0.3 to 0.4 TM, where TM is the melting temperature) can produce pronounced radiation hardening and loss of tensile elongation (Zinkle and Ghoniem 2000). Fracture toughness embrittlement is also of concern, particularly in BCC metals. The dominant defect cluster visible by TEM typically changes from vacancy-type defects (SFTs or vacancy dislocation loops) below Stage V to a mixture of interstitial dislocation loops and cavities above Stage V. At temperatures above recovery Stage V, the increased vacancy supersaturation associated with release of vacancies from cascade-produced vacancy clusters can lead to substantial cavity formation. The cavity swelling regime extends up to temperatures where thermal evaporation of vacancies from cavities becomes very high (near ~0.6 TM). For austenitic stainless steel, the cavity swelling regime extends from 300 to 650°C (Zinkle et al. 1993). Due to the less pronounced in-cascade formation of large vacancy clusters in BCC metals compared to FCC metals (Figure 1), the lower temperature boundary of the void swelling regime in BCC metals typically extends to much lower homologous temperatures (~0.2 TM) compared to FCC (~0.3-0.35 TM) metals (Singh and Evans 1995). At very high temperatures, the increased thermal equilibrium vacancy concentration becomes comparable to the vacancy concentration produced by irradiation and the microstructure tends to approach that of thermally annealed material. Two important exceptions to this benign high temperature behavior can occur. Helium produced by (n, alpha) nuclear transmutation reactions in the host metal can lead to formation of helium-filled cavities that are very resistant to thermal annealing. Under applied tensile stress, these cavities tend to preferentially form at grain boundaries where they can cause pronounced loss of grain boundary strength via a phenolmenon known as high temperature helium embrittlement (Ullmaier 1984). A second potential concern involves radiation-induced segregation of one or more chemical species in an alloy to interfaces such as grain boundaries and/or radiation-induced modifications to the stability of second phases in alloys at elevated temperatures. This can produce grain boundary embrittlement such as what is observed in many irradiated superalloys at high temperatures (e.g., Boothby 1996).

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2.4. EFFECT OF ALLOYING ADDITIONS

Solute additions to pure metals generally increase the nucleation of point defect clusters such as dislocation loops at irradiation temperatures below recovery Stage V. Figure 3 compares the dose-dependent dislocation loop density in pure copper and the solid-solution Cu-5%Ni alloy following irradiation to ~1 dpa near room temperature (Zinkle et al. 1994). Qualitatively similar behavior has been observed in numerous other materials, such as pure iron versus ferritic steel. The most important role of alloying additions is to create second phases that can provide improved mechanical strength and radiation resistance. A fine dispersion of highly stable nanoscale precipitates can be very effective in providing resistance to radiation damage degradation such as void swelling by producing a high sink strength for recombination of the radiation defects (Mansur and Lee 1991). This approach was very effectively used to produce swelling-resistant Ti-modified austenitic steels for fast breeder reactor applications (Maziasz 1993, Tavassoli 2002, Ehrlich et al. 2004).

Figure 3. Effect of addition of 5%Ni on the dislocation loop density in irradiated copper.

2.5. FIVE KEY CATEGORIES OF RADIATION DAMAGE

The temperature-dependent microstructural changes induced by neutron irradiation give rise to five major categories of radiation damage phenolmena (Zinkle 2005a). At low temperatures (below recovery Stage V, <0.30.4 TM), radiation hardening and embrittlement is of concern for doses above ~0.1 dpa. At intermediate temperatures, three phenomena become technologically significant for doses above ~1 to 10 dpa: radiation-induced

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segregation and precipitation (0.3-0.6 TM), void swelling (0.3-0.6 TM), and radiation induced creep (0.2-0.5 TM). At very high temperatures (>0.5 TM), high temperature helium embrittlement due to helium cavity formation along grain boundaries may occur particularly for helium concentrations >100 appm. In the following section, some aspects of the radiation hardening and embrittlement category are discussed in more detail. 3. Deformation and Fracture in Irradiated Metals and Alloys There are two major practical consequences of high levels of radiation hardening (Zinkle and Ghoniem 2000). First, the radiation hardening is generally accompanied by a reduction in uniform elongation under tensile test conditions due to highly localized plastic flow. This may require the use of more stringent structural design rules if the uniform elongation falls below a critical value, e.g., 2%. In particular, secondary stresses such as thermal stress that are usually ignored in the allowable stress analysis of structures may need to be included in the allowable stress evaluation (Majumdar and Kalinin 2000). A second consequence of radiation hardening that is particularly important for BCC alloys is reduction in fracture toughness and a potential shift in the ductile-brittle transition temperature to values that are above the operating temperature (Odette et al. 2003). Operation of structural materials in the “lower shelf ” fracture toughness regime is usually not feasible based on safety considerations, and this could lead to premature shutdown of the reactor before the design operating lifetime is achieved. 3.1. DOSE DEPENDENCE OF RADIATION HARDENING BELOW STAGE V

As reviewed elsewhere (Zinkle 1999, Zinkle and Matsukawa 2004), analytic models predict radiation hardening that is proportional to (N*d)1/2 or d*N2/3, where d is the defect cluster diameter and N is the defect cluster density. The former equation is more appropriate for obstacles that have relatively strong resistance to dislocation propagation, whereas the latter equation is more relevant for obstacles with relatively weak barrier strengths. For irradiation temperatures below recovery Stage V, the yield strength and hardness increase at a rate proportional to (dose)n at low doses, where n varies between 0.3 and 0.5 depending on the material. Similar to the defect cluster evolution in FCC vs. BCC metals, the initial rate of radiation hardening in FCC metals is typically stronger than in BCC metals due to the influence of relatively large sessile defect clusters that are produced directly

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in displacement cascades in FCC metals. Hardening saturation occurs for doses above ~0.1 dpa and irradiation temperatures below recovery Stage V (Byun and Farrell 2004), which coincides with the saturation in defect cluster density as observed by TEM. If large amounts of solute transmutation occur as the result of neutron bombardment, additional hardening may occur at high doses due to precipitation of small gas cavities or solute dispersoids. 3.2. TEMPERATURE DEPENDENCE OF RADIATION HARDENING

In general, the temperature dependence of the strength of metals and alloys following irradiation is closely correlated with the density of defect clusters discussed in section 2.3. This is because the defect cluster density is typically strongly dependent on temperature near and above recovery Stage V, whereas other parameters such as cluster size vary more slowly with irradiation temperature. Figure 4 shows the dose and irradiation temperature dependence of the yield strength of V-4%Cr-4%Ti alloy (a representative BCC alloy). Note the decrease in radiation hardening above 0.3-0.4 TM.

Figure 4. Effect of dose and temperature on the yield strength of fission neutron irradiated V-4%Cr-4%Ti alloy, for tests performed at the irradiation temperature (Zinkle et al. 1998).

In addition to the typical strong irradiation temperature dependence near and above recovery Stage V, at high doses the strength of irradiated metals can exhibit a different test temperature dependence compared to unirradiated samples. At low doses, the radiation hardening increment (i.e., σirrσunirr) is usually constant at all test temperatures up to the irradiation temperature. At higher doses where the irradiated strength becomes large compared to the unirradiated strength, the radiation hardening increment can exhibit significant variations with test temperature. Figure 4 shows an

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example of the temperature-dependent strength of pure iron before and after neutron irradiation near 350 K, where the test temperature dependence was much stronger following irradiation. Similarly, the strength dependence on strain rate can change after irradiation. Therefore, it is important to obtain mechanical property measurements over a wide range of test temperatures and strain rates, not just at a single temperature and strain rate.

Figure 5. Effect of test temperature on the yield strength of single crystal iron irradiated to a dose of ~0.002 dpa at an irradiation temperature near 80°C (Diehl et al. 1968).

At low doses, the amount of radiation hardening generally decreases with increasing irradiation temperature due to decreased thermal stability of sessile defect clusters produced directly in displacement cascades. It is interesting to note that the amount of radiation hardening for neutron irradiation near 300°C in both 9%Cr tempered martensitic (Rensman 2005) and austenitic stainless steels is initially much less than that for irradiation near 80°C, whereas at doses above several dpa the hardening is higher for irradiation at 300°C. This may be attributable to the gradual nucleation and growth of high-strength defect cluster hardening centers such as small helium filled cavities at 300°C. 3.3. FUNDAMENTALS OF RADIATION HARDENING AND FLOW LOCALIZATION

3.3.1. Dislocation-obstacle interaction mechanisms Fundamental studies on the resistance to dislocation motion of radiationproduced defect clusters have determined that small dislocation loops and SFTs are not Orowan obstacles, i.e., the gliding dislocation can bypass the

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obstacle at intermediate stress without forming Orowan loops during deformation (Bement 1970, Zinkle 1999). Typical values of the obstacle strengths for dislocation loops and SFTs are 0.2 to 0.4 (normalized to the Orowan impenetrable obstacle value). Since these obstacles are not impenetrable, this raises the possibility that the obstacles might be annihilated by interaction with gliding dislocations. Molecular dynamics simulations (Rodney 2005, Osetsky et al. 2006, Lee et al. 2007) and in-situ TEM deformation studies (Matsukawa et al. 2006) have identified several detailed mechanisms whereby partial or complete annihilation of SFTs and other defect clusters may occur during interactions with gliding dislocations. 3.3.2. Proposed plastic instability mechanisms Although low uniform elongations are generally observed in FCC and BCC metals following irradiation below recovery Stage V, examination of the fracture surfaces of tensile specimens typically reveal high reductions in area (Zinkle et al. 1998, Byun and Farrell 2004). Therefore, the low uniform elongation is usually associated with highly localized ductile plastic deformation. There are numerous proposed mechanisms for localized flow in unirradiated (Gerold and Karnthaler 1989, Basinski et al. 1997) and irradiated (Wechsler 1972, Byun and Farrell 2004) metals. One common mechanism that can lead to localized flow in unirradiated and irradiated materials is the presence of a dominant hardening component of ordered precipitates or other dislocation obstacles that can be sheared by gliding dislocations. For the past forty years, the most common explanation for localized flow in irradiated metals has centered on “dislocation channeling”, where the defect clusters produced by irradiation are annihilated by the leading dislocations emitted from a dislocation source. Subsequent dislocations emitted from the same source would thereby experience a locally soft (obstacle-free) slip plane region, and the remainder of the matrix outside the localized dislocation channel regions would not participate in the deformation process. The presence of cleared dislocation channels following deformation of irradiated BCC, FCC and hexagonal close packed metals has been well established in hundreds of investigations (Wechsler 1972, Luft 1991, Victoria et al. 2000, Zinkle and Lucas 2003). However, it is not yet proven that dislocation channeling is the cause of loss of uniform elongation; it is only known that channeling is nearly universally present when flow localization occurs and that it is a plausible mechanism. Recently, an alternative explanation for flow localization based on exhaustion of strain hardening capacity has been proposed (Byun and Farrell 2004). The analysis by Byun and Farrell of experimental data suggest the

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existence of a constant plastic instability stress for unirradiated and irradiated metals when the data are evaluated in terms of true stress vs. true strain. Further research is needed to conclusively identify the responsible mechanism(s) of flow localization in irradiated metals. 3.3.3. Deformation mechanism maps A variety of different deformation mechanism maps have been developed to describe the behavior in irradiated metals. One type of deformation mode map is useful for delineating the regimes for cell structure formation and dislocation channeling as a function of irradiation dose and amount of deformation (Okada et al. 1989, Farrell et al. 2004). Dislocation channeling is favored at high doses and relatively low levels of deformation. Ashbytype deformation mechanism maps are useful for evaluating the effect of neutron irradiation in altering the dominant deformation mechanism between dislocation glide (of which dislocation channeling may be considered a special case) and other mechanisms such as twinning (Hashimoto et al. 2000, Zinkle and Lucas 2003). 3.4. FRACTURE TOUGHNESS OF IRRADIATED METALS

The hardening associated with irradiation below recovery Stage V generally results in a reduction in fracture toughness for both FCC and BCC metals. The quantitative reduction in fracture toughness for irradiated FCC alloys such as Type 316 austenitic stainless steel is relatively small even when the uniform elongation is reduced to values near zero (Rowcliffe et al. 1998a). Irradiation at temperatures below ~0.3 to 0.4 TM generally causes pronounced increases in the ductile to brittle transition temperature and a reduction in the upper shelf toughness of BCC alloys such as tempered 9-12%Cr ferritic/martensitic steels (Rowcliffe et al. 1998b, Odette et al. 2003). Radiation hardening can increase the flow stress above the critical value for ductile fracture and thereby induce brittle cleavage failure with resultant low fracture toughness. There are two general strategies to mitigate degradation in fracture toughness of irradiated BCC alloys. In the first approach, specific alloying modifications are used (in conjunction with thermomechanical treatment, if appropriate) to reduce the amount of radiation hardening for a given irradiation temperature and dose. This approach has proven particularly effective for improving the fracture toughness of irradiated pressure vessel steels in existing fission reactors (e.g., by reducing the Cu content in the steel). In the second approach, metallurgical changes are used to increase the critical stress for initiation of brittle cleavage fracture. A detailed

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discussion of fracture mechanics is given in the accompanying lecture by Spätig and Odette. 3.5. IRRADIATION CREEP

Application of a mechanical stress during irradiation causes plastic deformation beyond what is associated with normal thermal creep processes. Both transient and steady state creep mechanisms may occur, with transient accelerated creep behavior observed in austenitic stainless steels at neutron irradiation temperatures below 400°C (Grossbeck et al. 1999). The steady state irradiation creep deformation is directly proportional to the applied stress and the dose level. The steady state creep compliance coefficient for BCC steels (B~0.5×10–6/MPa-dpa) is about half that of FCC steels (Garner et al. 2000). Considering that deformation levels up to several percent can usually be accommodated in the engineering design of structural components, irradiation creep becomes of practical concern for structures exposed to damage levels above ~20 dpa and stresses above ~100 MPa (the plastic strain due to irradiation creep for Type 316 stainless steel at this dose and stress is ~0.2 to 0.5%, with the higher values relevant for transient creep that may occur at irradiation temperatures below 400°C). Irradiation creep is of concern for irradiation temperatures up to ~0.5 TM; at higher operating temperatures, thermal creep typically becomes the dominant contributor to plastic deformation. 3.6. HIGH TEMPERATURE HELIUM EMBRITTLEMENT

Helium produced by nuclear transmutation of solute atoms is highly insoluble in all metals. At high temperatures (>0.5 TM) and under applied tensile stress, the helium preferentially nucleates at grain boundaries in the form of cavities, which can lead to intergranular fracture at relatively low stresses due to the transformation of the helium bubbles to voids (Ullmaier 1984, Trinkaus 2003). The critical amount of helium necessary to induce grain boundary embrittlement decreases with increasing test temperature and decreasing strain rate, and is much higher for ferritic/martensitic steels compared to austenitic stainless steels (Schroeder and Ullmaier 1991). For tensile strain rates near 10–4/s and test temperatures near 550°C, helium embrittlement of grain boundaries is induced for helium concentrations near 100 appm in austenitic stainless steel whereas grain boundary embrittlement is not observed even with 1000 appm He in ferritic/martensitic steel (Yamamoto et al. 2002). Introduction of high densities of nanoscale precipitates that serve to trap migrating helium before it can reach grain

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boundaries has been found to be effective in mitigating helium embrittlement (Schroeder 1988). 4. Prospective Structural Alloys for Gen IV Reactors 4.1. BEHAVIOR OF CURRENTLY AVAILABLE ALLOYS

In addition to having good mechanical properties, well developed industrial fabrication capability and acceptably low cost, structural materials for future Gen IV fission reactors must have acceptably high resistance to several radiation damage threats including radiation hardening and fracture toughness embrittlement, void swelling, irradiation creep, and helium embrittlement (Zinkle and Ghoniem 2000). Figure 6 compares the void swelling behavior for several different types of steels following fission neutron irradiation.

Figure 6. Void swelling behavior of two austenitic stainless steels and several 3-12%Cr bainitic/ferritic/martensitic steels (Gelles 1996, Garner et al. 2000, Klueh and Harries 2001).

Utilization of high-density nanoscale TiC precipitation in the Timodified Type 316 stainless steel has a dramatic effect on improving the void swelling resistance of austenitic stainless steels up to doses in excess of 100 dpa. However, even better swelling resistance (up to doses of at least 200 dpa) is exhibited by the 3-12%Cr bainitic, tempered martensitic, or ferritic steels. The superior radiation resistance of 9-12%Cr ferritic/martensitic steels compared to austenitic stainless steels for irradiation creep and high temperature helium embrittlement has already been discussed in sections 3.5 and 3.6. Among currently available structural materials, tempered ferritic/martensitic steels containing ~9%Cr are leading candidates for Gen IV

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applications since they exhibit a favorable combination of good mechanical strength, ductility and toughness in the unirradiated condition (with extensive industrial fabrication expertise), along with good radiation resistance with respect to void swelling, irradiation creep, radiation-induced phase instabilities, and high temperature helium embrittlement. Alternative structural materials such as titanium alloys and nickel-base superalloys either have limited high temperature strength or poor resistance to intense neutron irradiation. Two key concerns with the use of currently available 9%Cr steels in fission reactor structural applications are susceptibility to low temperature radiation embrittlement and an inferior thermal creep strength compared to austenitic steels at elevated temperatures. Several 9%Cr steels developed over the past 15 years have significantly better resistance to low temperature radiation embrittlement compared to earlier 9Cr-1Mo and 12Cr steels, while simultaneously exhibiting improved high temperature thermal creep strengths compared to earlier generation 9%Cr steels (Zinkle 2005b). However, further improvement in resistance to low temperature radiation embrittlement and high temperature thermal creep strength is desirable in order to broaden the range of potential operating temperatures. 4.2. PROSPECTS FOR DEVELOPMENT OF IMPROVED ALLOYS

In general, the most effective strategy for a high-performance reactor structural material appears to center on utilization of BCC alloys (to take advantage of their typical good resistance to many forms of radiation damage) combined with the development of a fine-scale distribution of highly stable precipitates or nanoscale solute clusters (to provide additional radiation damage resistance, along with enhanced high temperature strength capability). Either an evolutionary approach (based on incremental improvements of traditional ingot metallurgy) or a higher-risk revolutionary approach (e.g., new nanoscale oxide dispersion strengthened steels) can be envisioned. In order to speed up the traditionally long and expensive process for developing new steels, modern materials science tools such as computational thermodynamics codes should be utilized to the maximum extent possible. The leading base alloy candidates include 3–9%Cr bainitic or ferritic/ martensitic steels (Klueh 2004, Klueh 2005, Klueh et al. 2005, Zinkle 2005b) and 9–14%Cr nanocomposited or oxide dispersion strengthened (ODS) ferritic/martensitic steels (Ukai et al. 1998, Ukai and Fujiwara 2002, Hoelzer et al. 2007). Engineered design of nanoscale features in these steels can simultaneously confer improved mechanical strength (particularly at high temperatures) and improved radiation resistance. Substantial improvements in the

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high temperature strength of 3-12%Cr steels have recently been achieved by introducing uniformly dispersed fine-scale precipitates within the matrix by carefully tailored minor alloying modifications and thermomechanical processing (Klueh et al. 2005, Klueh et al. 2007). There are two main options for nanocomposited or ODS steels, building upon the pioneering work by Ukai and coworkers (1998, 2002): 12-16%Cr ferritic ODS steel (better high temperature capability, but typically lower fracture toughness and anisotropic mechanical properties) and ~9%Cr ferritic/martensitic ODS steel (potential for more isotropic properties and higher fracture toughness after heat treatment, but lower ultimate high temperature capability). Recent work has found that the key strengthening features in high-performance ODS steels may not be true oxide particles, but instead nanoscale clusters consisting of a mixture of oxygen, Ti, Y and Fe (Miller et al. 2006). Acknowledgements This work was supported by the Office of Fusion Energy Sciences, U.S. Department of Energy.

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IRRADIATED METALS: MICROSTRUCTURES AND PROPERTIES 243 Miller, M. K., Russel, K. F., and Hoelzer, D. T., 2006, Characterization of precipitates in MA/ODS ferritic alloys, J. Nucl. Mater. 351:261-268. Norgett, M. J., Robinson, M. T., and Torrens, I. M., 1975, A proposed method of calculating displacement dose rates, Nucl. Eng. Des. 33:50-54. Odette, G. R., Yamamoto, T., and Rathbun, H. J., He, M. Y., Hribernik, M. L., and Rensman, J. W., 2003, Cleavage fracture and irradiation embrittlement of fusion reactor alloys: mechanisms, multiscale models, toughness measurements and implications to structural integrity assessment, J. Nucl. Mater. 323(2-3):313-340. Okada, A., Kanao, K., Yoshiie, T., and Kojima, S., 1989, Transition of deformation structures in Ni and Au by D-T neutron irradiation, Mater. Trans. JIM 30(4):265-272. Osetsky, Y. N., Rodney, D., and Bacon, D. J., 2006, Atomic-scale study of dislocation – stacking fault tetrahedron interactions. Part I–mechanisms, Phil. Mag. 86(16):22952313. Rodney, D., 2005, Atomic scale modeing of clear band formation in FCC metals, Nucl. Instr. Meth. Phys. Res. B 228:100-110. Rowcliffe, A. F., Zinkle, S. J., Stubbins, J. F., Edwards, D. J., and Alexander, D. J., 1998a, Austenitic Stainless Steels and High Strength Copper Alloys for Fusion Components, J. Nucl. Mater. 258-263:183-192. Rowcliffe, A. F., Robertson, J. P., Klueh, R. L., Shiba, K., Alexander, D. J., Grossbeck, M. L., and Jitsukawa, S., 1998b, Fracture toughness and tensile behavior of ferriticmartensitic steels irradiated at low temperatures, J. Nucl. Mater. 258-263:1275-1279. Schroeder, H., 1988, High temperature helium embrittlement in austenitic stainless steelsCorrelations between microstructure and mechanical properties, J. Nucl. Mater. 155157:1031-1037. Schroeder, H., and Ullmaier, H., 1991, Helium and hydrogen effects on the embrittlement of iron- and nickel-based alloys, J. Nucl. Mater. 179-181:118-124. Singh, B. N., and Evans J. H., 1995, Significant differences in defect accumulation behaviour between fcc and bcc crystals under cascade damage conditions, J. Nucl. Mater. 226:277-285. Singh, B. N., and Zinkle, S. J., 1993, Defect accumulation in pure fcc metals in the transient regime: a review, J. Nucl. Mater. 206:212-229. Tavassoli, A. A. F., 2002, Present limits and improvements of structural materials for fusion reactors – a review, J. Nucl. Mater. 302:73-88. Trinkaus, H., and Singh, B. N., 2003, Helium accumulation in metals during irradiation– Where do we stand? J. Nucl. Mater. 323(2-3):229-242. Ukai, S., Nishida, T., Okuda, T., and Yoshitake, T., 1998, R&D of oxide dispersion strengthened ferritic/martensitic steels for LMFBR, J. Nucl. Mater. 258-263:1745-1749. Ukai, S., and Fujiwara, M., 2002, Perspective of ODS alloys application in nuclear environments, J. Nucl. Mater. 307-311:749-757. Ullmaier, H., 1984, The influence of helium on the bulk properties of fusion reactor structural materials, Nucl. Fus. 24(8):1039-1083. Victoria, M., Baluc, N., Bailat, C., Dai, Y., Luppo, M. I., Schaüblin, R., and Singh, B. N., 2000, The microstructure and associated tensile properties of irradiated fcc and bcc metals J. Nucl. Mater. 276:114-122. Wechsler, M. S., 1972, Dislocation channeling in irradiated and quenched metals, The Inhomogeneity of Plastic Deformation, R. E. Reed-Hill. Metals Park, OH, Am. Society for Metals: 19-54. Wigner, E. P., 1946, Theoretical physics in the metallurgical laboratory of Chicago, J. Appl. Phys. 17:857-863.

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Yao, Z., Schäublin, R., and Victoria, M., 2003, Irradiation induced behavior of pure Ni single crystal irradiated with high energy protons, J. Nucl. Mater. 323(2-3):388-393. Yamamoto, N., Murase, Y., Nagakawa, J., and Shiba, K., 2002, Creep behavior of reduced activation martensitic steel F82H injected with a large amount of helium, J. Nucl. Mater. 307-311:217-221. Zinkle, S. J., 1999, Fundamental radiation effects parameters in metals and ceramics, Radiat. Eff. Defects Solid 148:447-477. Zinkle, S. J., 2005a, Fusion materials science: Overview of challenges and recent progress, Phys. Plasmas 12(5):058101. Zinkle, S. J., 2005b, Advanced Materials for Fusion Technology, Fusion Engineering and Design 74:31-40. Zinkle, S. J., and Ghoniem, N. M., 2000, Operating temperature windows for fusion reactor structural materials, Fusion Engineering and Design 51-52:55-71. Zinkle, S. J., Horsewell, A., Singh, B. N., and Sommer, W. F., 1994, Defect microstructure in copper alloys irradiated with 750 MeV protons, J. Nucl. Mater. 212-215:132-138. Zinkle, S. J., and Lucas, G. E., 2003, Deformation and fracture mechanisms in irradiated FCC and BCC metals. Fusion Materials Semiann. Progress Report for Period ending June 30, 2003, DOE/ER-0313/34, Oak Ridge National Lab: 101-125. Zinkle, S. J., Matsui, H., Smith, D. L., Rowcliffe, A. F., van Osch, E., Abe, K., and Kazakov, V. A., 1998, Research and development on vanadium alloys for fusion applications, J. Nucl. Mater. 258-263:205-214. Zinkle, S. J., and Matsukawa, Y., 2004, Observation and analysis of defect cluster production and interactions with dislocations, J. Nucl. Mater. 329-333:88-96. Zinkle, S. J., Maziasz, P. J., and Stoller, R. E., 1993, Dose dependence of the microstructural evolution in neutron-irradiated austenitic stainless steel, J. Nucl. Mater. 206:266-286. Zinkle, S. J., and Singh, B. N., 1993, Analysis of displacement damage and defect production under cascade damage conditions, J. Nucl. Mater. 199:173-191. Zinkle, S. J., and Singh, B. N., 2000, Microstructure of Cu-Ni alloys neutron irradiated at 210°C and 420°C to 14 dpa, J. Nucl. Mater. 283-287:306-312. Zinkle, S. J., and Snead, L. L., 1995, Microstructure of copper and nickel irradiated with fission neutrons near 230°C, J. Nucl. Mater. 225:123-131.

MULTISCALE MODELING OF RPV EMBRITTLEMENT N. SONEDA* Central Research Institute of Electric Power Industry, 2-11-1 Iwado-kita, Komae, Tokyo 201-8511, Japan

Abstract: The multiscale modeling of the neutron irradiation embrittlement of reactor pressure vessel (RPV) steels is described. In this approach, information from experiments and computer simulations that cover very wide ranges of time and length scales is accumulated, correlated and integrated to develop a model for the prediction of embrittlement. Some examples from experiments and simulations are presented to show how they provide information to understand the fundamental mechanism of embrittlement, and then we describe how all the information is integrated into a mathematical form. A practical output of this approach is the development of a new embrittlement correlation method for Japanese RPV steels. Keywords: Multiscale modeling; multiscale computer simulation; microstructural characterization; reactor pressure vessel; neutron irradiation embrittlement

1. Introduction Reactor pressure vessel (RPV) steels are required to have sufficient toughness to prevent their fracture under the loading conditions subjected to during operation. The steels used for the RPVs of light water reactors are low-alloy ferritic steels with body-centered cubic (bcc) crystal structure. In this class of materials, the fracture mode changes from brittle at lower temperatures to ductile at higher temperatures within a narrow temperature range called transition temperature. The exposure of RPV steels to fast neutron flux during operation causes a shift of the transition temperature region to higher temperatures, and this phenomenon is known as neutron irradiation embrittlement. For the structural integrity assessment of RPVs, it is very important to predict the amount of transition temperature shift with high accuracy to demonstrate that the RPV steels have sufficiently high fracture toughness against failure.

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* Naoki Soneda, Materials Science Research Laboratory, Central Research Institute of Electric Power Industry, 2-11-1 Iwado-kita, Komae, Tokyo 201-8511, Japan; e-mail: [email protected]

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The neutron irradiation embrittlement of RPV steels has been extensively studied since the 1960s. It is now well recognized that the embrittlement is primarily caused by the hardening of the materials due to the accumulation of radiation damage (for example, Odette and Lucas 1998). Copper atoms in RPV steels as an impurity element are known to cause large embrittlement, but even in low-Cu materials, an increase in transition temperature has been observed, particularly at high fluences. Although such radiation damage features are very small with a typical diameter of a few nanometers (Miller and Burke 1990, Auger et al. 1995, Carter et al. 2001), they act as obstacles to dislocation motion, resulting in the hardening of the materials and thus embrittlement. For the accurate prediction of embrittlement, an understanding of the details of the embrittlement mechanism and the quantitative modeling of the effects of metallurgical and environmental variables are necessary. In this paper, we will describe the microstructural characterization of RPV steels as well as multiscale computer simulations. Then we will describe the multiscale modeling of RPV embrittlement based on the understanding on the embrittlement mechanism. 2. General Consensus on RPV Embrittlement Mechanism The embrittlement of RPV steels is in general thought to be caused primarily by 1) the formation of Cu-enriched clusters, 2) the formation of matrix damage features, and 3) the segregation of P atoms at grain boundaries. A Cu-enriched cluster is a cluster of solute atoms with a special enrichment of Cu atoms that accumulate because of their low solubility in an Fe matrix. The experimental characterization of the clusters by 3D atom probe (3DAP) and small-angle neutron scattering show that Cu-enriched clusters are solute atom clusters with a size of 2-3 nm in diameter associated with other elements such Ni, Mn and Si (Carter et al. 2001). Matrix damage is another type of damage of matrix damage that occurs in the Fe matrix. Its nature has not yet been characterized in detail, but the candidates of matrix damage are point-defect clusters, such as vacancy clusters and/or selfinterstitial atom (SIA) clusters, and solute atoms – point-defect complexes. Embrittlement due to the formation of Cu-enriched clusters and matrix damage features is categorized as hardening embrittlement, because both Cu-enriched clusters and matrix damage impede dislocation motion to cause hardening. On the other hand, the segregation of P atoms at grain boundaries may weaken the strength of grain boundary bonding so that fracture occurs preferentially at grain boundaries. Thus, embrittlement due to P segregation is called nonhardening embrittlement. At present, however, although P segregation has been identified in RPV materials such as A533B

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Cl.1 and A508 Cl.3 steels used in many reactors in western countries, the evidence for grain boundary fracture is extremely limited, and the nonhardening embrittlement does not appear to be a major contributor to the embrittlement of such materials. The outline of the embrittlement mechanism is now understood, but its details are not necessarily clear, particularly in terms of developing a quantitative model. For example, the role of Cu in high-Cu content materials is relatively clear, but its role in solute atom cluster formation in low-Cu (i.e. <0.1wt.% Cu) material is not very clear. Also, the effects of environmental variables such as dose, dose rate and irradiation temperature have not yet been well modeled. The effect of the dose is very important because long term operation of reactors with lifetimes of over 60 years has recently been planned, and the end-of-life neutron fluence of RPV materials is expected to be much larger in these reactors. Regarding the dose rate, the difference between RPV embrittlement in boiling water reactors (BWRs) and pressurized water reactors (PWRs) is of interest. The dose rate is one or two orders of magnitude lower in BWRs than in PWRs, and thus, large embrittlement of BWR RPV materials was not expected because of the low dose. However, recent surveillance data show that the embrittlement in high-Cu BWR RPV steels is large, which is probably partly due to the effect of low dose rate irradiation. The irradiation temperature is another factor of interest because the coolant temperature of BWR plants is in general lower than that of PWR plants by more than 10oC, and an accurate model on the effect of the irradiation temperature is necessary. In the last ten years, a couple of projects have been conducted worldwide to develop mechanism-guided embrittlement correlation methods (Eason et al. 1998, ASTM 2002, Hiranuma et al. 2004, Debarberis et al. 2005). Clarification of the issues described above will contribute to the development of more accurate correlation methods by improving the embrittlement models used. 3. Microstructural Characterization of Surveillance Materials 3.1. MICROSTRUCTURAL CHARACTERIZATION OF PWR SURVEILLANCE MATERIALS

Among the issues raised in the previous section, we will discuss the effect of Cu in low-Cu steels, the effect of the dose rate in high-Cu steels and the nature of matrix damage based on the experimental characterization of nanometer-scale radiation damage features. We investigated such features formed in several different surveillance materials irradiated in commercial

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reactors with different chemical compositions using 3DAP tomography (Soneda et al. 2007). Figure 1b shows the result of the 3DAP measurement of a PRV surveillance material with a Cu content of 0.12 wt.% irradiated to a fluence of 4 × 1019n/cm2, E > 1 MeV. The Cu content of 0.12 wt.% is relatively low compared with the RPV materials used worldwide, but relatively high for Japanese RPVs. It is evident from the figure that Cu-enriched clusters are formed in the material, and the clusters are associated with Ni, Si and Mn. The cluster number density is 2.2 × 1023m–3 and the mean cluster size is 3.1 nm. Figure 1a shows the 3DAP result of a lower-Cu material with a Cu content of 0.06 wt.% irradiated to a fluence of 6 × 1019n/cm2, E > 1MeV. Although the Cu content in the clusters is considerably lower than that of Figure 1b with the decrease in bulk Cu content, one can clearly see that clusters containing Ni, Si and Mn solute atoms are formed. The number density of the clusters is 1.2 × 1023m–3, which is lower than that of Figure 1a in spite of the higher fluence of the material, but the mean cluster size is 3.4 nm which is slightly larger than that of Figure 1a. In Figure 1c, we show the result of 3DAP for a material with an even lower Cu content of 0.03 wt.% irradiated to a fluence of 6 × 1019n/cm2, E > 1MeV.

Figure 1. a) 3DAP result of Cu, P and Si atom distribution in 33 × 38 × 284 nm3 volume of RPV material with 0.06 wt.% Cu irradiated to 6x1019 n/cm2, b) 3DAP result of Cu and P atom distribution in 35 × 41 × 491 nm3 volume of RPV material with 0.12 wt.% Cu irradiated to 4 × 1019 n/cm2, c) 3DAP result of Cu, P and Si atom distribution in 41 × 49 × 264 nm3 volume of RPV material with 0.03 wt.% Cu irradiated to 6x1019n/cm2.

Because of the very low bulk Cu content, we can no longer observe any Cu atoms in the clusters but we can still identify the formation of soluteatom clusters that consist of Ni, Si and Mn atoms. The number density of clusters is 5.6 × 1022m–3, which is lower than that in Figure 1b and the mean cluster size is 3.1 nm. These results clearly demonstrate that solute atom

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clusters can be formed under irradiation irrespective of the presence of Cu atoms in the materials. However, the Cu content and number density of the clusters are strongly affected by the bulk Cu content. To identify the formation of matrix damage features, TEM observations were performed on the same surveillance materials. Figure 2 shows the TEM images of the materials with 0.12wt.% Cu.

Figure 2. Dark field TEM image of the material with: a) 0.12 and b) 0.06 wt.% Cu irradiated 4 × 1019 n/cm2.

Figure 3. Comparison between the recovery response of ΔVHN and ΔS. EP2 material with a Cu content of 0.06wt.% Cu was irradiated in the supplemental surveillance program using the commercial reactors in the U.S., a) Hardness, b) S-parameter.

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We observed the formation of SIA type dislocation loops, whose diameter and number density are 2-3 nm and of the order of 1021m–3, respectively. We did not identify the formation of dislocation loops with size less than 1nm due to the resolution limit of TEM, and therefore the loop number density may be larger. On the other hand, to investigate vacancy-type defects, which are another candidate for matrix damage, positron annihilation S-parameter measurement was performed combined with post-irradiation annea-ling (PIA) and hardness test (Ortner 2004). Figure 3 shows the recovery behavior of hardness increase (ΔHv) and S-parameter increase (ΔS) during PIA. Both ΔHv and ΔS increase after irradiation, indicating that vacancy-type defects are formed by irradiation. During the PIA, however, ΔHv starts to recover at 400°C, while ΔS starts to recover at 350°C which is different from that of ΔHv. This clearly indicates that the change in ΔS is unrelated to the change in ΔHv, and thus, vacancy-type defects are not matrix damage. One possible conclusion from these results is that the nature of matrix damage is dislocation loops or loops associated with solute atoms. 3.2. EFFECT OF DOSE RATE ON SOLUTE-ATOM CLUSTER FORMATION

Figure 4c shows a comparison of the transition temperature shifts of three materials from the same heat with a Cu content of 0.24wt.%, but irradiated at different dose rates of ~109, ~1010 and 7 × 1011 n/cm2-s, which correspond to irradiation on the inner wall of a BWR RPV, at an accelerated position in a BWR RPV, and in a material test reactor, respectively. The material irradiated at the lowest dose rate shows a much larger transition temperature shift than the other two materials, which have similar shifts in spite of the difference in dose rate. To understand the effect of the dose rate on the microstructural evolution, the materials designated as SP1 and SPT1 in Figure 4c with dose rates of 109 and 1010n/cm2-s were investigated using 3DAP, as shown in Figures 4a and 4b, respectively. At the lower dose rate, SP1, very distinct and welldeveloped clusters are formed with very high number density, and this microstructure is consistent with the large transition temperature shift of this material. On the other hand, at the higher dose rate, SPT1, solute atom clusters are formed with similar number density, but the sizes of the clusters are smaller than those of SP1, i.e., 3/4 of the diameter and 1/2 of the volume. This can be used to explain the difference in the transition temperature shifts. These results suggest that a dose rate effect does exist in highCu material, but that it affects the growth of clusters rather than their nucleation.

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Figure 4. a) 3DAP result of Cu and P atom distribution in 41 × 48 × 149 nm3 volume of SP1, b) 3DAP result of Cu and P atom distribution in 24 × 29 × 175 nm3 volume of SPT1, c) The comparison of the transition temperature shifts of the three materials irradiated at three different dose rates.

4. Computer Simulations on Microstructural Evolution 4.1. MULTISCALE COMPUTER SIMULATION

Computer simulation techniques are increasingly used to understand the mechanisms that explain experimental observations and to investigate the response of materials under conditions that cannot easily be studied in experiments. Multiscale computer simulation techniques, where information obtained at picosecond and nanometer scales is passed to higher-level simulations with longer time and larger length scales up of to tens of years and meters, are recognized to be very promising for the study of changes in materials. In this approach, different computer simulation techniques, such as first-principles calculations, molecular dynamics, the KMC method and dislocation dynamics, are appropriately used and combined to obtain the necessary information. In this chapter, the idea of the KMC method is briefly explained, and then its application to the understanding of microstructural changes is demonstrated.

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4.2. KINETIC MONTE CARLO METHOD

In metals irradiated by energetic particles (neutrons in our case), various phenomena such as point defect production by displacement cascades, diffusion of the defects, the clustering (or dissociation) or segregation of point defects and solute atoms, and the annihilation of point defects by recombination occur simultaneously to form microstructures. The KMC simulation is a technique for simulating all these events using models that describe all the possible events that may occur in the crystal (Soneda et al. 2003). Point defect production by displacement cascades is studied and modeled by performing molecular dynamics simulations over a wide range of primary knock-on atom (PKA) energies that the atoms receive during the collision with neutrons. Figure 5 shows one example of point defect production in a cascade of 50 keV PKA energy (Soneda et al. 2001). In such high PKA energy cascade simulations, the formation and distribution of point defects can occur in a variety of ways, and thus, we need to repeat this type of simulation many times to obtain sufficient statistics on the defect production. A cascade database for various PKA energies is also important for simulating the PKA energy spectrum. In our case, we performed cascade simulations from 100 eV to 50 keV and modeled the PKA energy spectrum of fast neutron irradiation using the cascade database. Displacement cascades in the KMC method are simulated by introducing point defects with an interval equivalent to an assumed dose rate. Since the cooling phase of displacement cascades normally finishes within about 10 ps, we use the point defect distribution 10ps after cascade initiation obtained using MD as the initial point defect distribution in KMC simulations.

Figure 5. One example of defect production by a displacement cascade with a 50 keV PKA energy. Circles are SIAs, and dots are vacancies. Defect distribution at 10 ps after cascade initiation is shown.

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The diffusion and thermal stability (binding energy) of point defects and their clusters can also be calculated using MD (Soneda et al. 1998, Soneda et al. 2001). Diffusion calculation can be performed by introducing a single point defect (cluster) in the computation box and monitoring the motion of the defect while keeping the crystal temperature, T, at a given temperature. The diffusion model used in the KMC simulations can be written as a thermally activated process whose mathematical expression is given as D = D0 exp(–Em/kT), where D0, Em and k are the pre-exponential factor, migration energy and Boltzmann’s constant, respectively. The values of D0 and Em can be obtained by fitting the expression to the calculation data obtained at several different values of T. The thermal stability of point defect clusters can also be modeled as a thermally activated process using a similar expression of v = v0 exp(–Ed/kT) , where v is the dissociation frequency and Ed is the dissociation energy, given by the sum of migration energy Em and binding energy Eb. In the KMC simulations, the occurrence frequencies of all the possible events are calculated at every Monte Carlo step, and then one event out of the possible events is selected with a probability proportional to the relative occurrence frequency of the event. After the occurrence of the selected event, such as defect production, a defect jump or defect dissociation, a new spontaneous event such as clustering or recombination occurring in the computation box is searched, the time is incremented by Δt, and then we return to the event selection stage. The time increment, Δt, due to the occurrence of one event is calculated as Δt = –log(R x P)/P, where P is the sum of all the event occurrence frequencies and R is a uniform random number between 0 and 1. A periodic boundary condition is normally applied to calculate the bulk behavior, but we can consider the effect of crystal grain size by limiting the number of repetitions of the periodic boundary condition to a finite number (Soneda et al. 2003). 4.3. EFFECT OF ENVIRONMENTAL VARIABLES ON DAMAGE ACCUMULATION

The effects of dose, dose rate and irradiation temperature on the defect accumulation in bcc-Fe are studied by the KMC method. Figure 6 shows the effects of dose and dose rate on the number densities of vacancy clusters and SIA clusters (Soneda et al. 2003). Stable vacancy cluster formation is observed at dose rates of 10–4 and 10–6dpa/s, but not at lower dose rates. Since vacancy clusters are not thermally stable at the irradiation temperature of 600 K because of their small binding energies, the vacancy clusters observed at high dose rates are apparent stable clusters, i.e., they are in an balanced state of dissociation and growth. On the other hand, SIA clusters

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are formed at all dose rates, and the number density increases with dose. The cluster number density decreases with decreasing dose rate, but remains the same level at the low dose rates of 10–8 and 10–10dpa/s.

Figure 6. The KMC results of defect production in bcc Fe. The irradiation temperature of 600 K, a PKA energy spectrum due to fast neutron irradiation and a 10 μm grain size are assumed.

The above results can be understood as follows. Point defects produced by a cascade diffuse in a computation box until they reach a final state, which is either annihilation at sinks such as dislocations or grain boundaries, annihilation by recombination or the formation of stable clusters. When the dose rate is high, point defects will interact with other point defects generated in other cascades before they reach the final state. This interaction, i.e., inter-cascade interaction, enhances point defect clustering, resulting in an increase in the number density of defect clusters. On the other hand, when the dose rate is low, point defects from one cascade reach a final state before the point defects from other cascades enter the computation box, and thus, we observe no dose rate effect at dose rates below a threshold. Since the dose rate of commercial RPV steels is lower than 10–10dpa/s, they are in

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a condition where no inter-cascade interaction occurs and no dose rate effect can be found in the dislocation loop density. However, in material test reactors, the dose rate can be as high as approximately 10–8dpa/s, and it may be necessary to consider the effect of the dose rate. The effect of irradiation temperature can also be studied using the KMC simulation (Soneda et al. 2004). Jones and Williams (1996) proposed a model that describes the irradiation temperature dependence of the embrittlement of low-Cu materials, ΔT = α × FT × (φt)1/2, where ΔT, α and φ t are the transition temperature shift, constant coefficient and dose, respectively, and FT = 1.869 – 4.57 × 10–3T (oC). This model was studied by the KMC simulation, as shown in Figure 7. The vertical axis shows the square root of the number density of the point defect clusters divided by the dose and the horizontal axis shows the irradiation temperature. The number densities of both vacancies and SIA clusters have a linear temperature dependence with a slope equivalent to that of FT, and thus, the origin of the form of the FT term can be understood from the temperature dependence of point defect cluster formation.

Figure 7. The effect of temperature on the defect production. Both SIAs and vacancies have similar temperature dependence.

Another example of the application of the KMC simulation is to investigate the dose rate effect on the Cu-enriched solute atom cluster formation in high-Cu RPV steels. As is apparent from the explanation above, our KMC simulation has been applied only for pure Fe, and it is very difficult to simulate the behavior of the solute atoms directly. However, since Cu atoms diffuse in the Fe matrix by a vacancy mechanism, counting the number of vacancy jumps in the computation box of the KMC simulation will give indirect information on the amount of diffusion of Cu atoms. Thus, we performed KMC simulations up to a dose of 0.01dpa at various dose rates and

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counted the number of vacancy jumps, as shown in Figure 8 (Soneda et al. 2003). As has been described in the beginning of this section, inter-cascade interaction increases at high dose rates resulting in the suppression of the number of vacancy jumps. At dose rates below the threshold dose rate, no inter-cascade interaction occurs, and the number of vacancy jumps does not change for different dose rates. Here, however, one should note that there are so-called thermal vacancies in all the metals, which minimize the entropy of the crystal. The number of thermal vacancies is negligible at high dose rates because the irradiation time required to reach a given dose is very short at high dose rates. However at very low dose rates, the irradiation time increases markedly, and the total number of vacancy jumps becomes larger than at high dose rates. In Figure 8, the number of thermal vacancy jumps is also plotted, and it is demonstrated that the number of thermal vacancy jumps at very low dose rates becomes comparable or even larger than that at high dose rates. This enhances the diffusion of solute atoms such as Cu, and eventually enhances their clustering. This mechanism supports the experimental results shown in Figure 4.

Figure 8. The contribution of irradiation vacancies and thermal vacancies to the total number of vacancy jumps.

4.4. INTERACTION BETWEEN SOLUTE ATOMS AND POINT DEFECTS

As we discussed in Section 3, the formation of Cu-enriched clusters is one of the primary causes of RPV embrittlement. Although the driving force of Cu clustering appears to be the very low solubility of Cu atoms in the Fe

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matrix (~0.05wt.%), it is also necessary to consider another possibility, i.e., the interactions of Cu atoms with point defects under irradiation conditions, because irradiation introduces a number of vacancies and SIAs. We performed MD simulations using the Ackland potential (Ackland et al. 1998) to investigate the stable configurations of Cu atoms and vacancies or SIAs. Vacancies have positive binding energies with Cu atoms, and the binding energy of a vacancy or a Cu atom increases with increasing complex size (Takahashi et al. 2002). Figure 9 shows the structure of stable Cu-vacancy complexes. Vacancies are located at the center and Cu atoms are located around the vacancies. In other words, vacancies form a nanovoid, and the inner surface of the void is covered with Cu atoms.

Figure 9. The structures of Cu – point defect clusters obtained by the kinetic lattice Monte Carlo (KLMC) and MD. Stable configuration is first searched by KLMC and then the atom positions are relaxed by MD to determine the energy.

On the other hand, a single Cu atom has a repulsive or a very small binding energy with an SIA depending on the configuration (Ackland et al. 1998, Marian et al. 2001). However, the situation drastically changes when two or more Cu atoms interact with an SIA or an SIA cluster (Takahashi et al. 2004). For example, two Cu atoms can replace two Fe atoms of an SIA to form a Cu-Cu dumbbell. The Ackland potential gives a binding energy of 0.5eV for this configuration, which is also confirmed by a first principles calculation using the VASP code (Kresse and Furthmuller 1996). This complex can be formed when a single SIA runs over the two Cu atoms located at the first- nearest neighbor distance. When we increase the number of Cu atoms and SIAs in a complex, Cu atoms form a precipitate at one position around the SIA cluster (loop) rather than be uniformly distributed around the loop. These results suggest that the formation of vacancies in an Fe matrix by irradiation enhances the clustering of Cu atoms in the solid solution, but it is also very likely that the Cu atoms and point defects of both vacancies and SIAs tend to form complexes, which can act as nuclei for the further clustering of Cu atoms and point defects. The former can be referred to as irradiation-enhanced clustering, and the latter can be referred to as irradiation-induced clustering. In high-Cu materials, irradiation-enhanced

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clustering dominates, but in materials with a relatively low Cu content below 0.1wt.%, irradiation-induced clustering should play a role. 5. Modeling of RPV Embrittlement On the basis of the above discussion together with our knowledge from the literature on the embrittlement mechanism over very wide range of time and length scales, we have developed a new embrittlement model for predicting the shifts of transition temperature due to neutron irradiation (Soneda et al. 2007). In conventional practice in Japan, U.S. and France, a simple equation is used to predict the transition temperature shift that was developed with the help of statistical analysis (JEA 1991, USNRC 1988, Brilland 1987). In our approach, however, we first write down our understanding of the mechanism of microstructural changes in the form of rate equations, and then the transition temperature shift is calculated using the predicted microstructural changes. The microstructural changes, namely, the formation of solute atom clusters and matrix damage features, due to irradiation are modeled using the following equations: ∂CSC = ξ3 ⋅ ( CCumat + ε1 ) ⋅ DCu + ε 2 ⋅ CMD + ξ8 ⋅ CCuavail ⋅ DCu ⋅ (1 + ξ 7 ⋅ CNi0 ) ∂t 2 ∂C ∂CMD = ξ 4 ⋅ Ft 2 ⋅ ξ5 + ξ6 ⋅ CNi0 ⋅ φ − SC ∂t ∂t

(

)

(

(

)

2

(1)

)

(2)

mat ∂CCu ∂C ′ ⋅ CSC = −vSC ⋅ SC − vSC ∂t ∂t

(3)

avail vSC = ξ 2 ⋅ ( CCu ⋅ DCu ) ⋅ tr

(4)

2

′ = ξ1 ⋅ CCuavail ⋅ DCu vSC

⎧0 avail CCu = ⎨ mat sol ⎩CCu − CCu

(5) mat sol C Cu ≤ C Cu mat sol C Cu > C Cu

thermal irrad thermal DCu = DCu + DCu = DCu + ω ⋅φ η

(6) (7)

where CSC and CMD are the number densities of solute atom clusters and matrix damage features, CCumat and CNi0 are the bulk chemical contents of Cu and Ni, Dcu is the Cu diffusivity, φ is the dose rate, t is the irradiation time and tr is the relaxation time, respectively. Equation (1) represents the time evolution of solute atom clusters. The first and second terms on the right-hand side of the equation are the irradiation enhanced term and irradiation induced term, respectively. The first

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term indicates that solute atom clustering occurs with matrix damage features as the nuclei. This process can occur without Cu atoms but is accelerated by their presence. On the other hand, the second term is zero if the bulk Cu content is lower than the solubility limit. Equation (2) represents the time evolution of matrix damage features. The formation of matrix damage features is affected by the irradiation temperature and also the bulk Ni content. The amount of matrix damage features used as the nuclei of solute atom clusters is subtracted in equation (2). Equation (3) models the depletion of the matrix Cu content due to the formation and growth of Cuenriched solute atom clusters. Note that the depletion of the matrix Cu reduces the formation rate of Cu-enriched solute atom clusters. Equation (7) models the diffusivity of Cu atoms, and the idea of contributions from both irradiation-induced vacancies and thermal vacancies is implemented. DCuthermal becomes relatively larger when the dose rate is very small. Mechanical property changes are correlated with the microstructural changes using the following equations: mat 0 ΔTSC = ξ16 ⋅ V f = ξ16 ⋅ ξ15 ⋅ f (CCu ,CSC )⋅ g(C Ni )+ h(φt ) ⋅ CSC mat f ( CCu , CSC ) = ξ11 ⋅

0 mat CCu − CCu + ξ12 CSC

(

0 g ( C Ni0 ) = 1 + ξ13 ⋅ ( CNi )

ξ14

)

ΔTMD = ξ17 ⋅ C MD ΔT =

(10) DSC ≈ DCu

(11) (12)

(ΔTSC ) + (ΔTMD ) 2

(9)

2

h(φt ) = ξ 9 ⋅ (1+ ξ10 ⋅ DSC ) ⋅ φt

(8)

2

(13)

where ΔTSC and ΔTMD are the contributions of solute atom clusters and matrix damage features, which are calculated using equations (8) and (12) as functions of CSC and CMD, respectively. In calculating the contribution of solute atom clusters, an empirical model, in which the transition temperature shift is proportional to the square root of the volume fraction of solute atom clusters, is used. The average volume per cluster, which is necessary for calculating the volume fraction, is modeled using equations (9) ~ (11), which take into account the effect of chemical composition and the growth of the clusters during irradiation. The Greek characters in the above equations, except for φ, are coefficients, and were optimized using the surveillance database of Japanese commercial reactors. Figure 10 shows a comparison between the calculated and measured transition temperature shifts. The standard deviation of the

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prediction error is smaller than that of the other correlation equations used in Japan and in the U.S. as shown in Figure 10. When we apply a plantspecific adjustment to the initial transition temperature, the standard deviation of the prediction error becomes much smaller and is as low as 6°C.

Figure 10. The comparison of predicted and measured transition temperature shifts. Plant specific adjustment is performed by offsetting the initial values.

6. Concluding Remarks The multiscale modeling of RPV embrittlement based on information obtained from experiments and computer simulations covering very wide ranges of time and length scales was demonstrated. A practical output of this approach is the development of a new embrittlement correlation method for Japanese RPV steels, and this method has been adopted in the Japan Electric Association Code (JEAC) 4201-2007. Thus, this study is a good example of how the understanding of a fundamental mechanism can be applied in a real-world engineering application. The key to the success of multiscale modeling is the close coupling of results obtained from experiments and simulations. Conventionally, simulations are very powerful and useful for understanding the results of experiments, and in some cases, can provide new information that cannot be obtained by experiments. However, another aspect of the relationship between experiments and simulations should be the design of new experiments based on information obtained from simulations to validate the predictions

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of simulations. Such an approach will provide essential information for further understanding of mechanisms.

References Ackland, G. J., Bacon, D. J., Calder, A. F., and Harry, T., 1998, Computer simulation of point defect properties in dilute Fe-Cu alloy using a many body interatomic potential, Phil. Mag. 75:713-732. ASTM E900-02, Standard Guide for Predicting Radiation-Induced Transition Temperature Shift for Reactor Vessel Materials, E706(IIF), Annual Book of ASTM Standards, Vol. 12.02, American Society for Testing and Materials, West Conshohocken, PA. Brillaud, C., Hedin, F., and Houssin, B., 1987, A Comparison between French Surveillance Program Results and Predictions of Irradiation Embrittlement, Effects of Radiation on Materials: 13th International Symposium, ASTM STP 956, pp. 420-447. Carter, R. G., Soneda, N., Dohi, K., Hyde, J. M., English, C. A., and Server, W. L., 2001, Microstructural characterization of irradiation-induced Cu-enriched clusters in reactr pressure vessel steels, J. Nucl. Mater. 298:211-224. Debarberis, L., Acosta, B., Sevini, F., Krykov, A., Gillemot, G., Valo, M., Kikolaev, A., and Brumovsky, M., 2005, Role of nickel in a semi-mechanistic analytical model for radiation embrittlement of model alloys, J. Nucl. Mater. 336:210-216. Eason, E. D., Wright, J. E., and Odette, G. R., 1998, Improved EMbrittlement Correlations for Reactor Pressure Vessel Steels, NUREG/CR-6551. Hiranuma, N., Soneda, N., Dohi, K., Ishino, S., Dohi, N., and Ohata, H., 2004, Mechanistic Modeling of Transition Temperature Shift of Japanese RPV Materials, Proc. of 30th MPA-Seminar in conjunction with the 9th German-Japanese Seminar, Stuttgart, October 6 and 7, p. 3.1. Japan Electric Association, 1991, Method of Surveillance Tests for Structural Materials of Nuclear Reactors, JEAC4201-1991. Jones, R. B., and Williams, T. J., 1996, The dependence of radiation hardening and embrittlement on radiation temperature, in: Effects of Radiation on Materials, 17th International Symposium, vol. 1270, D. S. Gelles, R. K. Nanstead., A. S. Kumar, E. A. Little eds., ASTM. Kresse, G., and Furthmuller, J., 1996, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54:11169. Marian, J., Wirth, B. D., Perlado, J. M., Odette, G. R., and Diaz de la Rubia, T., 2001, Dynamics of self-interstitial migration in Fe-Cu alloys, Phys. Rev. B 64:094303. Miller, M. K., Burke, M. G., 1990, Fine-Scale Microstructural Characterization of Pressure Vessel Steels and Related Materials Using APFIM, Effects of Radiation on Materials, Proc. 14th International Symposium (Volume II), ASTM STP 1046:107. Odette, G. R., and Lucas, G. E., 1998, Recent progress in understanding reactor pressure vessel steel embrittlement, Radiation Effects & Defects in Solids 114:189-231. Ortner, S., 2004, Microstructural Characterization of Reactor Pressure Vessel Steels: PostIrradiation Annealing Experiments, Joint EPRI-CRIEPI RPV Embrittlement Studies (1999-2004). EPRI, Palo Alto, CA: 2004, 1003531, and CRIEPI, Tokyo, Japan: Q980401. Soneda, N., and Diaz de la Rubia, T., 1998, Defect production, annealing kinetics and damage evolution in α-iron: an atomic-scale computer simulation, Phil. Mag. A 78:995-1019.

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Soneda, N., Diaz de la Rubia, T., and Ishino, S., 2001, Vacancy loop formation by ‘cascade collapse’ in α-Fe: a molecular dynamics study of 50 keV cascade, Phil. Mag. Lett. 81:649-659. Soneda, N., Ishino, S., Takahashi, A., and Dohi K., 2003, Modeling the microstructure evolution in bcc-Fe during irradiation using kinetic Monte Carlo computer simulation, J. Nucl. Mater. 323:169-180. Soneda, N., Dohi, K., Ishino, S., and Takahashi, A., 2004, Computer simulation of the effect of neutron irradiation on the microstructural evolution in bcc-Fe, CRIEPI T03076. Soneda, N., Dohi, K., Nomoto, A., Nishida K., and Ishino, S., 2007, Development of Neutron Irradiation Embrittlement Correlation of Reacvtor Pressure Vessel Materials of Light Water Reactors, CRIEPI Q06019. Takahashi, A., Soneda, N., Ishino S., and Yagawa, G., 2002, Structures and energies of vacancy-Cu clusters in a-Fe, Phys. Rev. B 67:024104. Takahashi, A., 2003, Multiscale simulation of materials strength, Ph.D. Thesis, The University of Tokyo. US Nuclear Safety Commission, 1988, Regulatory Guide 1.99 Revision 2, Radiation Embrittlement of Reactor Vessel Materials, Office of Nuclear Regulatory Research, U.S. Nuclear Regulatory Commission, Washington, DC.

PARAMETRIC DISLOCATION DYNAMICS AND BOUNDARY ELEMENT MODELING OF ELASTIC INTERACTION BETWEEN DISLOCATIONS AND PRECIPITATES A. TAKAHASHI* Department of Mechanical Engineering, Faculty of Science and Technology, Tokyo University of Science, 2641, Yamazaki, Noda-shi, Chiba, 278-8510, Japan

Abstract: This paper introduces a computational method for studies of the interaction between dislocations and precipitates. The method is based on 3-dimensional parametric dislocation dynamics and boundary and volume element method. The accuracy of the method is examined with the comparison with a Molecular Dynamics (MD) simulation result. The comparison shows that the method has a good accuracy and consistency with the MD simulations. The method is used to investigate the influences of precipitate geometry and the ratio of precipitate-to-matrix elastic shear modulus on the critical resolved shear stress (CSS). Finally an extension of the method to incorporate the dislocation core contribution to the dislocation-precipitate interaction is discussed. Keywords: Dislocation; precipitate; elastic interaction; dislocation dynamics; boundary element method

1. Introduction In reactor pressure vessel (RPV) steels in nuclear power plants, the formation of precipitates and the interaction with dislocations are known to be responsible for the degradation of the material strength and the embrittlement during high-energy neutron irradiations. On the other hand, precipitation strengthening is known to be one of very effective technique to design advanced alloys with superior strength characteristics, such as yield strength and hardness. In both situations, the basic mechanisms are based on blocking the motion of dislocations by precipitates. It is therefore essential that the interaction between dislocations and precipitates is accurately and efficiently

______ *

Akiyuki Takahashi, Department of Mechanical Engineering, Faculty of Science and Technology, Tokyo University of Science, 2641, Yamazaki, Noda-shi, Chiba, 278-8510, Japan; e-mail: [email protected]

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modeled in order to explore the basic physics of the embrittlement of the RPV steels and to enable rational alloy design. In the dislocation-precipitate interaction, there are four essential factors that control the physics and strength of the interaction: (1) mismatch between the elastic properties of the matrix and precipitate; (2) coherency strains between the precipitate and matrix; (3) misfit dislocations at incoherent precipitate-matrix interfaces; and (4) changes in the core structure of dislocations as they penetrate precipitates. In practice, the first two factors exist in many cases. Thus, we will focus here on the influence of the first two factors on the dislocation-precipitate interactions. Several theoretical studies on precipitation strengthening have been based on statistical assumptions for precipitate distributions and shapes, and on simplified dislocation geometry (Friedel 1964, Fleischer 1961, 1963, Nembach 1983). During a dynamic interaction process, dislocations can have a complicated shape, and frequently change their shapes. Moreover, precipitates are not generally spherical. Recently, Van der Giessen and Needleman developed a computer simulation technique to simulate dislocation dynamics in a finite volume with boundary conditions based on a superposition principle (van der Giessen and Needleman 1995). The method is a hybrid of Dislocation Dynamics (DD) and Finite Element Method (FEM). Shin (Shin et al. 2003, 2005) applied the idea to the simulations of the interaction between a dislocation and an impenetrable precipitate. Molecular Dynamics (MD) method can deal with the interaction between dislocations and precipitates, and can give us atomistic information of the interaction. However, the MD simulation generally require a large number of atoms, such as 1 million or 10 million atoms, to simulate the interaction accurately. Therefore, we need a 3-dimensional simulation technique that can deal with the elastic interaction between dislocations and precipitates with arbitrary shape, and that must have a possibility to simulate the collective behavior of dislocations in metals with randomly distributed precipitates. In this paper, a computational method for solving dislocation-precipitate interaction problems based on linear elasticity theory, which enables us to investigate the interaction between flexible dislocations and penetrable precipitates, is introduced. The dislocation-precipitate interaction problem is decomposed into two problems based on the superposition principle: (1) a dislocation problem with image stresses from interfaces between precipitates and the matrix, and (2) a precipitate problem with initial stresses. The first problem is solved by the parametric dislocation dynamics (PDD). To solve the second problem, boundary and volume integral equation for the problem is derived, and solved by the boundary element method (BEM). The benchmark test to show the accuracy of the method is presented. Then,

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the method is applied to investigations of the critical shear stress (CSS) of precipitates. Finally, an extension of the method to incorporate the dislocation core effect is discussed, and demonstrated by the simulation of a FrankRead source in copper. 2. Computational Method 2.1. ELASTICITY OF DISLOCATION-PRECIPITATE INTERACTION

Precipitates are described as a situation where a coherency strain Εij is prescribed in a precipitate Ω within a body D, and is zero in the matrix D- Ω, and the precipitate Ω has different elastic properties from the matrix D. Let us consider Np precipitates with coherency strains Εij and elastic constants C mijkl in an infinite matrix with elastic constants Cijkl. The superscript m denotes the mth precipitate. The infinite matrix D contains an arbitrary number of dislocations, and is subjected to an external stress σ 0ij at infinity. Following Mura, we describe the stress in D- Ω as (Mura 1982)

σ ij0 + σ ij = Cijkl (εkl0 − εkl ) in D − Ω

(1)

−1 where, ε ij0 = Cijkl σ kl0 , εij is the strain produced by dislocations and the exter0 nal stress σ ij. The domain Ω contains the total number of Np precipitates. Similarly, the stress in the mth inhomogeneous inclusion Ωm can also be defined as

m σ ij0 + σ ij = C ijkl (ε kl0 + ε klm − Eklm ) in Ω m

(2)

Figure 1. Application of the superposition principle to a dislocation-precipitate interaction problem: a) the original problem, b) the dislocation problem with image stresses, and c) the m correction problem with initial stresses Σ ij .

As shown in Figure 1, using the superposition principle, the interaction problem between dislocations and precipitates can be decomposed into two problems: (1) a dislocation problem in an infinite matrix, and (2) a correction problem, which is an elastic problem of precipitates.

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At first, the stress in the original problem is described by a summation of the stress in the dislocation problem and the stress in the correction problem. The stress σ~ij in the dislocation problem is defined by

σ˜ ij = Cijkl (εkl0 + ε˜kl ) in D

(3)

where, ε~ij is the strain produced by dislocations in D. The stress σˆ ij in the correction problem can then be defined by

σˆ ij = C ijkl εˆ kl

in D − Ω

m m m σˆ ij = C ijkl εˆ klm − C ijkl E klm + (C ijkl − C ijkl )(ε kl0 + ε~kl )⎫⎪

m ⎬ in Ω ⎪⎭

m = C ijkl εˆklm + Σ ijm

(

(4)

)

m m where, Σijm = −Cijkl ∈mkl + Cijkl − Cijkl εkl .

From above equations, the correction solution can be considered as the solution to a precipitate problem with an initial stress Σ ijm . 2.2. SOLUTIONS TO DECOMPOSED PROBLEMS

The dislocation problem (Eq. 3) is solved here by the PDD method. The detail of the PDD method can be found in elsewhere (Ghoniem et al. 2000). On the other hand, to solve the correction problem (Eq. 4), boundary and volume integral equations are derived as

cij ( P ) uˆ j ( P ) = ∫ (U ij ( P, Q ) tˆj ( Q ) − Tij ( P, Q ) uˆ j ( Q ) ) d Γ

for D − Ω

Γ

cij ( P ) uˆ mj ( P ) = ∫

Γm

(U ( P, Q ) tˆ ( Q ) − T ( P, Q ) uˆ ( Q ) ) d Γ ⎫⎪ m ij

m j

m ij

− ∫ m Σ mjk ( q ) U ijm, k ( P, q ) d Ω Ω

m j

(5)

m ⎬ for Ω ⎪ ⎭

In Eqs (5), cij is the shape factor, U ij and Tij are the kernel functions, u j and t j are respectively the displacements and tractions on the interface

Γ between the infinite body and precipitates. Assuming that the infinite body and precipitates are perfectly bonded, the displacements and the tractions on the interfaces must satisfy the continuity condition for displacements and the equilibrium condition for tractions. Using the conditions, we can solve Eqs. (5) and compute the displacements and tractions on the interfaces by the BEM. Once the displacements and tractions are obtained, the stress at any point can be easily computed by

PDD-BEM MODELING

σˆ ij ( p ) = ∫ ( Dijk ( p, Q ) tˆk ( Q ) − Sijk ( p, Q ) uˆk ( Q ) ) d Γ Γ

267

(6)

It is worth noting that there is a singularity in the stress calculation by the boundary element method when the calculation point is close to the boundary. Also the stress of the dislocation involves a strong singularity within a range of the dislocation core. In this work, these singularities are simply removed by introducing a cut-off radius 2b, which is typically used as a radius for the dislocation core. The stresses within the cut-off distance are kept to be a constant. In this work, the methodology of PDD and the boundary and volume integral equations for correction problems are mainly for the isotropic materials. However, the effects of elastic anisotropy are also of interest. The PDD method has already been extended to elastic anisotropic materials (Han et al. 2003), and the BEM has also been applied to elastic anisotropic problems (Vogel and Rizzo 1973). Therefore, applying these achievements in the PDD and BEM, the present method can also be easily and directly extended to the anisotropic materials. Additionally, the present method can provide the information on the effects of the strain rate and temperature on the dislocation-precipitate interactions. The PDD method can calculate the effect of the strain rate on the dislocation motions. The effect of the temperature on the dislocation motion can also be taken into account by changing the mobility and friction stress on the dislocations and including the cross slip of screw segments. 3. Numerical Accuracy and Convergence The accuracy of the present PDD and BEM method (PDD-BEM) for dislocation-precipitate interaction problems is shown in this section. In order to show it, the results of PDD-BEM simulations are compared with those of a MD simulation. Kohler (Kohler et al. 2005) performed MD simulations of the interaction between an edge dislocation and a copper precipitate in iron. In the work, they calculated the critical shear stress (CSS) for the edge dislocation to break away from the interaction with the copper precipitate (Kohler et al. 2005). In this work, we choose the results of the MD simulations to compare with those of the PDD-BEM simulations to show the accuracy and consistency with the MD results. In the simulations, the size of the simulation volume is 19.7 × 9.73 × 19.7 nm, and the x, y and z axes are along [111], [11 0] and [112 ] directions, respectively. A spherical copper precipitate with a diameter dCu and an edge dislocation with [111] Burgers vector on (1 1 0) slip plane are placed in the volume. The center of the copper precipitate is on the slip plane of the

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dislocation. Periodic boundary conditions are applied along [112 ] direction, which implies that the edge dislocation is infinitely long, and that the precipitate is one of a periodic array in the [112 ] direction. The CSS is evaluated by gradually increasing the external shear stress by 5 MPa each 1000 time steps. Their calculated shear modulus for the iron matrix and copper precipitate are μ1 = 69.76 GPa and μ2 = 21.842 GPa, respectively. Although the experimental values of the shear moduli are higher, we used the same elastic constants of Kohler et al. in the present simulations. This allows direct comparison between the elasticity results (PDD-BEM method) and the results of MD simulations, which are based on interatomic potentials. In order to show the convergence of the accuracy of the present method, we used different boundary and volume element mesh densities resulting in successively higher resolution, while the length of each parametric segment is set to be 1 nm in all simulations. Figure 2 shows the dislocation configurations during the interaction with a 3 nm diameter copper precipitate. In the Figure, the dislocation line near the precipitate is bent, meaning that the dislocation is likely to come into the precipitate. Then the dislocation moves to the middle of the precipitate, and forms again a straight shape. The configuration must be the most preferable in the sense of the elasticity. The external applied shear stress is continued to increase, and pushes the dislocation more. Then the dislocation comes to the other edge of the precipitate, and is bent by the pinning effect of the precipitate. Finally, the dislocation can successfully break away from the precipitate, when the external applied shear stress sufficiently increases to the CSS.

Figure 2. Dislocation configurations in the interaction with a precipitate with a diameter of 3nm: a) The dislocation feels an attractive force from the precipitate, b) An equilibrium configuration of the dislocation at the center of the precipitate, c) The precipitate gives a pinning effect to the dislocation, d) The dislocation broke away from the precipitate.

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Figure 3. Critical shear stresses of copper precipitates in iron calculated by the PDD-BEM method. The MD simulation results of Kolher et al. (2005) are also plotted in the Figure.

Figure 3 shows the comparison between the results of the PDD-BEM simulations and those of the MD simulations. In the Figure, the CSS calculated with a coarse mesh displays a large deviation from MD simulation results. However, the CSS converges gradually to MD results as the mesh resolution is increased. Note that the results for two fine meshes are almost the same indicating convergence, and that the results are in reasonable agreement with MD simulations. 4. Influence of Precipitate Geometric and Elastic Parameters on Strength We applied the PDD-BEM method to the investigation of the influence of elastic parameters on the CSS. The simulation volume is 50 × 50 × 50 nm, with the x, y and z axes taken along the [111], [1 1 0] and [112] directions, respectively. A spherical precipitate with a diameter of 5 nm is placed in the middle of the simulation volume. A straight edge dislocation with [111] Burgers vector on the (1 1 0) slip plane is also introduced, and is initially placed 10 nm away from the precipitate. Periodic boundary conditions are applied along the [112 ] direction to simulate the interaction of a onedimensional precipitate array with the dislocation. The elastic shear modulus of the matrix material μm is taken as 81.8 GPa, and of the precipitate shear modulus, μp, is changed in the range from 0.01 ≤ μm /μ p ≤ 5 in order to investigate the effect of the elastic modulus mismatch between

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the precipitate and the matrix on the CSS. To investigate the influence of the precipitate diameter and modulus on the strength, a set of simulations were performed for a range of precipitate-to-matrix shear modulus ratios, and for initial precipitate diameters of 5, 7.5, and 10 nm. The results of the simulations are shown in Figure 4. In the Figure, the CSS is normalized by the reference shear stress τ o = μm b/L , where L is the inter-precipitate spacing which corresponds to the length of the simulation volume in the [112] direction. It is shown that larger precipitates have a stronger strengthening effect, and that the CSS is dependent also on the relative elastic modulus mismatch. Softer precipitates ( μ p /μ m ≤ 1 ) still result in strengthening ( τ /τ 0 > 0 ). The maximum CSS of the softer precipitate can be found around 0.3-0.4τ0.

Figure 4. Critical shear stresses of precipitates with different elastic constants and diameter.

On the other hand, harder precipitates are still shearable up to μ p /μm ratios on the order of 3-4, where the Orowan looping mechanism starts, as shown in the Figure with hollow symbols. After the Orowan looping mechanism is initiated, the CSS seems to be saturated, and converges to a constant CSS. The maximum CSS of the harder precipitate can be found around 0.6-0.8τ0, which is twice bigger than that of the softer precipitate. The transition from the shearable precipitate to the Orowan regime depends on the precipitate size, as shown in the figure. The self-force on the dislocation surrounding the precipitate mainly depends on the diameter of the precipitate, and is large when the diameter of the precipitate is small. For small precipitates, the dislocation is likely to cut through them as the result

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of the strong self-force on the dislocation. Therefore, the precipitate with a small diameter must have a large elastic shear modulus to stop the dislocation from cutting it, and hence the onset of the Orowan mechanism must take place at larger μ p /μm values for a precipitate with a smaller diameter. 5. Dislocation Core Effects Banerjee (Banerjee et al. 2004) proposed an abinito and continuum hybrid method, which enables the simulation of the dislocation core structure based on the continuum elastic theory with a direct input of atomistic information The basic idea to deal with the dislocation core structure is the use of the Pierels-Nabarro model (Nabarro 1947). given by

p=

μ

π (1 −ν ) ∫

∞

−∞

1 du dx ' x '− x dx '

(7)

where, p is the lattice restoring stress, x is the position of an infinitesimal element of displacement in the dislocation core. In the model, the dislocation core structure is expressed by the balance between the lattice restoring stress and an elastic interaction stress between displacements in the dislocation core. Because it is very difficult to apply the model to dislocations with complicated shapes, the model is here discretized using a number of tiny dislocations. When the equation is discretized with N dislocations, the Burgers vector of the tiny dislocations is b/N to keep the total displacement of the dislocations.

pi =

μ

b π (1 − ν ) 2 N

N

∑x i≠ j

j

1 − xi

(8)

The discretization changes the elastic interaction term between displacements in the dislocation core to the elastic interaction between tiny dislocations. The elastic interaction between tiny dislocations can be easily solved by the PDD method, meaning that the model can now be implemented into the PDD-BEM. On the other hand, the lattice restoring stress can be calculated by

p=−

dγ s du

(9)

where, γs is the generalized stacking fault (GSF) energy. The GSF energy is evaluated using ab-initio calculations, and is directly input to the total stresses acting on dislocations in the PDD method.

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5.1. A DEMONSTRATION OF PDD WITH PN MODEL SIMULATION

To demonstrate the PDD simulation with the PN model, the nucleation of dislocation loops from a Frank-Read source in copper is simulated. In the simulation, a straight edge dislocation segment with two fixed ends is prepared as the Frank-Read source. The x, y and z axes are along [112 ], [1 1 0] and [111] directions, respectively. The slip system of the dislocation is (111)[1 1 0]. In the PDD simulation with the PN model, the straight edge dislocation is divided into a number of tiny dislocations. In copper, dislocations generally split into two partial dislocations to make extended dislocations. Therefore, the half of the tiny dislocations has the Burgers vector of leading partial dislocation, a0/6 [2 1 1], and the rest of the tiny dislocations has that of trailing partial dislocation, a0/6 [12 1 ]. Also, to find the appropriate number of tiny dislocations, the dislocation core structure of an infinitely long straight edge dislocation in copper is simulated using the PDD method with the PN model. The dislocation core structure is converged, when the number of tiny dislocations is more than ten. Therefore, in the simulation, we use ten tiny dislocations to describe the dislocation core structure. External shear stresses of τyz = 350MPa and τxz = 700MPa are applied to the dislocation. To investigate the dislocation core effects on the dislocation nucleation from the Frank-Read source, we also performed the PDD simulation without PN model, where the dislocation is described with only one dislocation line with a perfect Burgers vector, [1 1 0]. Figure 5 shows the results of the PDD simulations with and without the PN model. In Figure 5a and 5b, without the PN model, the Frank-Read source expands in a circular shape. On the other hand, as shown in Figure 5c and 5d, the result with the PN model is clearly different from the result without the PN model. The leading partial dislocation of the Frank-Read source expands in an inclined circular shape. The direction of the incline corresponds to the direction of the Burgers vector of the leading partial dislocation. Likewise, the trailing partial dislocation of the Frank-Read source also expands in an inclined circular shape, of which the direction is along the Burgers vector of the trailing partial dislocation. Moreover, the external shear stress component of τxz interacts with the screw component of the two partial dislocations, which expands the stacking fault between two partial dislocations very widely to form a twin structure. Obviously, it is impossible to simulate the dynamics and expansion of the dislocation core without the PN model. Therefore, the results suggest that the PDD simulation with the PN model can provide information on the dynamics of the dislocation core, and that the PN model must be used in the PDD-BEM simulation of the interaction between dislocations and precipitates to incorporate the effects of the dislocation core dynamics to the interaction.

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Figure 5. Dislocation nucleations from a Frank-Read source in copper: (a) and (b) shows the initial and expanded shapes of the Frank-Read source simulated by the PDD without the PN model, (c) and (d) are respectively the initial and expanded shape of the Frank-Read source simulated by the PDD with the PN model.

6. Conclusions We introduced a computational method for the dynamics of the most general dislocation-precipitate interaction in 3-D, which is based on a hybrid of the PDD and the BEM. The method shows excellent consistency with the results of MD simulations and convergence behavior of accuracy in terms of the mesh resolution. The method was then applied to the investigation of the CSS with different precipitate-to-matrix elastic modulus. The simulation found out that the maximum CSS of the harder precipitate is twice bigger than that of the softer precipitate. After the Orowan looping is initiated, the CSS was saturated and converged to a constant CSS. Finally, we presented an idea, which is the use of the PN model, to incorporate the

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dislocation core contribution to the present method. Also, we presented a demonstration of the PDD simulation with the PN model, which is the simulation of the dislocation nucleation from a Frank-Read source. Comparing the results with and without the PN model, the dynamics and shapes of the dislocation core show a clear difference, indicating that the effect of dislocation core must be accounted for in the PDD-BEM simulation, and that the PN model must be an promising approach to incorporate the dislocation core effects and very suitable to the PDD-BEM simulation.

References Banerjee, S., Ghoniem, N. M., and Kioussis, N., 2004, A Computational Method for Determination of the Core Structure of Arbitrary-shape 3D dislocation Loops, Proc. MMM-2, 23-25. Fleischer, R. L., 1961, Solution hardening, Acta Metall. 9: 996-1000. Fleischer, R. L., 1963, Substitutional solution hardening, Acta Metall. 11: 203-209. Friedel, J., 1964, Dislocations, Pergamon Press, Oxford. Ghoniem, N. M., Tong S. -H., and Sun, L. Z., 2000, Parametric dislocation dynamics: A thermodynamics-based approach to investigations of mesoscopic plastic deformations, Phys. Rev. B 61:(31-34) 913-927. Han X., Ghoniem, N. M., and Wang, Z., 2003, Parametric dislocation dynamics of anisotropic crystals, Phil. Mag. 83(31-34): 3705-3721. Kohler, C., Kizler, P., and Schmauder, S., 2005, Atomic simulation of precipitation hardening in α-iron: influence of precipitate shape and chemical composition, Modelling Simul. Mater. Sci. Eng. 13: 35-45. Mura, T., 1982, Micromechanics of Defects in Solids, Kluwer Academic, Dordrecht. Nabarro, F. R. N., 1947, Dislocations in a simple cubic lattice, Proc. Phys. Soc. 59: 256-272. Nembach, E., 1983, Precipitation hardening caused by a difference in shear modulus between particle and matrix, Phys. Stat. Sol. (a) 78: 571-581. Shin, C. S., Fivel, M. C., Verdier, M., and Oh, K. H., 2003, Dislocation-impenetrable precipitate interaction: a three-dimensional discrete dislocation dynamics analysis, Philos. Mag. 83(31-34): 3691-3704. Shin, C. S., Fivel, M., and Kim, W. W., 2005, Three-dimensional computation of the interaction between a straight dislocation line and a particle, Modelling Simul. Mater. Sci. Eng. 13: 1163-1173. Van der Giessen, E., and Needleman A., 1995, Discrete dislocation plasticity: a simple planar model, Modelling Simul. Mater. Sci. Eng. 3: 689-735. Vogel, S. M., and Rizzo, F. J., 1973, An integral equation formulation of three dimensional anisotropic elastostatic boundary value problems, J. Elast. 3: 203-216.

CRYSTAL PLASTICITY FROM DISLOCATION DYNAMICS V. V. BULATOV Lawrence Livermore National Laboratory

Abstract: This contribution overviews the premise and the promise of Dislocation Dynamics (DD) as a method for direct calculation of plasticity and strength of single crystals from the underlying motion of dislocation lines. First, a few aspects of dislocation physics more relevant for the DD method development will be discussed followed by a brief introduction of several key algorithmic and numerical aspects of the DD method. Then, specific examples of applications of the DD method will be given to illustrate the method’s ability to capture the essential physics of collective dislocation motion. The primary focus will be on BCC metals where several key results have recently emerged from DD simulations. Finally, the method’s applicability to simulations of irradiated metals and alloys will be discussed along with the associated method development issues. Keywords: Crystal plasticity; dislocation theory; dislocation dynamics; computer simulations; numerical algorithms; irradiated materials

1. Introduction Dislocations were introduced as a physical concept in 1934 in three papers published simultaneously and independently by Taylor, Polanyi and Orowan [1-3]. The need for dislocations as agents of crystal plasticity had arisen from the then unresolved gross discrepancy between the plastic strength of crystals that could be expected from sliding along crystal planes and the actually observed, much lower strength of single crystals. That dislocations actually exist was confirmed much later, in the mid-50s, with the development of electron microscopy [4]. At present, the key role of crystal dislocations in defining plastic strength of single- and polycrystalline materials is firmly accepted common knowledge established through the decades of extensive research on the topic. Dislocations affect numerous properties and behaviors observed in crystalline materials. In fact, with a few notable exceptions, the dislocation can be thought as inherent to crystals, even if it is commonly thought that dislocation formation energy can not be furnished by thermal activation alone. Dislocations grow within the material along with the growing crystal. Then, during the material lifetime, dislocations move around, multiply and interact with various other crystal defects. It is through these processes that many V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

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important properties of crystalline materials are determined, such as yield and flow strength, brittleness and ductility, creep and fatigue, indentation hardness and friction, functioning of electronic devices, etc. Additional complex behaviors occur under irradiation: radiation induced/enhanced creep, swelling, embritllement, etc. The method of Dislocation Dynamics is intended to establish a quantitative connection between the mentioned behaviors and the details of dislocation motion under appropriate conditions. 1.1. BASIC FACTS ABOUT CRYTAL DISLOCATIONS

Figure 1 illustrates several ways by which a dislocation can form in a crystal. First, a dislocation can be thought of as a boundary between the slipped and yet un-slipped areas of a crystal plane. The upper arrow on the figure points to shear on a crystal plane as one of the processes by which a dislocation is introduced. The reason why crystals shear through formation and motion of dislocations is because this is a path of least resistance: only or few atomic bonds are broken over a unit of dislocation motion whereas shearing a crystal uniformly across the shear plane would break at once all atomic bonds reaching across the shear plane. to shift relative to each other.

Figure 1. Three ways by which a dislocation can be inserted in a crystal. On top: by applying a shearing stress, a dislocation can be inserted from the left surface of the crystal. It then propagates to the right along the horizontal plane leaving a surface step behind. In the middle: by removing a half plane of atoms. At the bottom: by inserting an extra plane of atoms. The red circles surround the highly distorted atoms in the dislocation core.

Crystal dislocations are sometimes likened to a wrinkle in a rug: it takes much less effort to move a rug by a small distance by making and propagating a wrinkle along its whole length rather than by pulling on the whole rug. In a way, dislocation is a “topological wrinkle” in the crystal lattice whose propagation along crystal planes makes different parts of the crystal to shift relative to each other.

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The second way a dislocation can be inserted is by removing a halfplane of atoms from the bottom half of the crystal, as shown in the middle of Figure 1. The third way is by inserting an extra half-plane of atoms from the top of the crystals. Remarkably, while the number of atoms and the existence of absence of the surface steps make these three final configurations different, the character of atomic distortion in and around the dislocation center - the core - is the same. In fact, there is an infinite number of ways, such as various combinations of the three procedures illustrated in Figure 1, by which a dislocation can be inserted into the crystal lattice. However, once in the crystal, the properties of a given dislocation do not depend on the insertion mechanism. Dislocations are specific crystal defects with their own properties and characteristic behaviors. In real crystals, dislocations are indeed formed by all three ways just mentioned, i.e. by shearing along crystal planes and by condensation of vacancies or interstitials. To begin to appreciate the importance of dislocations in crystal behaviors, let us realize that, except for very few special materials and processes, dislocations are essentially unavoidable. In fact, crystals grow from supersaturated solutions by atoms attaching themselves to crystal steps spiraling a round a screw dislocation. Thus, dislocations grow together with the growing crystal. Likewise, in crystals obtained by solidification from the melt, dislocations are introduced due to unavoidable thermal gradients occurring in transition zone between the melt and the growing crystal. Subsequently, during the service life of a crystalline material, dislocations often multiply causing various changes in material properties. Properties affected by dislocations are numerous: yield and flow stress, ductility and brittleness, creep and fatigue, indentation hardness and friction, reliable functioning of electronic devices, etc. The magnitude of lattice distortion associated with a dislocation is given by the Burgers vector. Two things are important to understand or, at least remember, about the Burgers vector. First is that the Burgers vector is not arbitrary but must be equal to a repeat vector of the crystal lattice, usually one of the smallest repeat vectors, also called primitive vectors. The second is that the Burgers vector is conserved along a dislocation line. Both mentioned properties follow from the topological connectivity of the host crystal lattice. For any given dislocation line its Burgers vector can be revealed by a Burgers circuit test described in Figure 2. Conservation of the Burgers vector is illustrated in Figure 3. Lattice distortions associated with a dislocation are obviously largest in the dislocation but also reach far into the crystal. However, at distances of the order a several Burgers vectors from the core, distortions become small enough for a linear elasticity theory to apply. Continuum theory of dislocations based on linear elasticity is well developed and useful for understanding

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dislocation behaviors on large scales. Understanding behaviors related to dislocation core is less well established and demands theories resolving atomic and electronic degrees of freedom. The roles of the dislocation core and that of the long-ranging lattice distortions in defining dislocation properties are different. Crudely, one can assume that the dislocation’s energy and the mechanical forces acting on the dislocation are mostly due to the long-ranged elastic distortions whereas how the dislocation moves in response to the mechanical force is defined mostly by the atomic core.

Figure 2. Burgers circuit is an arbitrary sequence of steps each connecting two neighboring atoms so that, when taken in a perfect lattice, the end of the path (E point) coincides with the path’s beginning (S point). Burgers circuit 1 on the right does not enclose any dislocation and points E1 and S1 coincide. Circuits 2 and 3 enclose the same dislocation and reveal the same closure defect vector equal to the Burgers vector b.

Figure 3. Burgers vector is conserved along a closed loop (left) and when dislocations 1 and 2 merge together to form dislocation 3.

Since the dislocations were first invented in 1934, it has become clear that dislocation behavior is the single most important factor defining crystal strength. However, even if the connection between the motion of individual dislocations and macroscopic mechanical properties has been established in principle, it has been so far impossible to take advantage of this understanding in an ultimate sense, i.e. to compute crystal strength directly from the underlying motion of dislocation lines.

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2. The Method of Dislocation Dynamics The idea of the Dislocation Dynamics (DD) method is very simple: assuming that one knows how each individual dislocation moves in response to the forces it sees, let us assemble a large group of dislocations in a simulation volume, let the dislocations interact with the external stress and with each other and start moving dislocation over small time increments in response to their local forces. Then, the evolution of the whole dislocation ensemble should mimic what dislocations do en masse in the real material. The DD method is uniquely suited for addressing the key issue in the physics of crystal strength – what is the role of collective dislocation behaviors, owing to the interaction among the dislocations, in defining materials strength. More specifically, dislocation interactions are known to be responsible for a very important property of strong crystals, called strain hardening. The latter quality means that the material becomes stronger with increasing strain. Engineers actually count on this very important property in designing bridges or space ships. Possibly the simplest experiment on strain hardening is to take a aluminum paper clip and start bending it back and forth: one can feel how the clip becomes stiffer with each bending cycle until, eventually, the clip breaks. In principle, in DD crystal strength can be computed in a simulation that mimics the laboratory experiments, e.g. when the specimen is compressed along its vertical axis at a constant compression rate. The latter rate defines the vertical dimension of the specimen as a linear function of time. In a DD simulation, this current dimension is used to compute the average stress in the simulated specimen. This is done using the following feedback loop: on each time step the increment of planar area swept by moving dislocation on this step is obtained and used to compute the corresponding increment in the plastic (irreversible) strain produced by the dislocation motion. Then, this plastic strain is subtracted from the total strain (always known from the preset straining rate) to the average stress is computed using the Hooke’s elasticity law from the current elastic strain. Once the average stress is updated, the stress resulting from the interaction among dislocations also needs to be computed on each time step because this interaction depends on the dislocation positions. Strain hardening results from two dislocation behaviors – multiplication and intersections. Under most straining conditions, dislocations multiply under stress making their own subsequent motion increasingly more difficult as dislocations now need to cross more numerous other dislocations (dislocation forest) to continue to move. To deform plastically, higher external stress is required – this is the essence of strain hardening. Potentially, not only should DD be able to compute strain hardening, but it can also

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serve as an ideal virtual microscope that can reveal in real time how dislocations move, multiply and interact and exactly which of their behaviors define the hardening. These characteristics make DD the method of choice for studying the fundamental mechanisms of crystal plasticity and strength. The method development has been taking place since the 60s but gained new momentum in the early 90s when first 3D simulations were performed. Generally, there are two challenges that are currently addressed to make the DD method more useful. The first challenge is that of fidelity: the unit mechanisms of dislocation behavior are multiple and complex and require much effort to sort out this complexity [5]. The second challenge is that of computability: in order to deliver on its promise the DD method should allow simulations on length and times scales of real straining experiments. Only if and when both mentioned challenges are addressed, the DD method can be expected to develop its full potential. Project ParaDiS in the Lawrence Livermore National Laboratory, aims at making DD simulations more accurate and computationally efficient. ParaDiS stands for Parallel Dislocation Simulator and, as the name implies, attempts to use parallel computing to solve the computability challenge. Simultaneously, fidelity of unit mechanisms of dislocation behavior remains a topic of active research in the ParaDiS team. New developments in dislocation physics, mechanics and computational algorithm that went into the development of ParaDiS are described in [6,7]. In terms of the computational performance, the code progressed from about 0.02% of maximum simulated strain (in a cube-shaped simulation volume 5-10 microns on edge) in 2001, to 0.5% in 2003, to 3-5% in 2005. Likewise, the size of the dislocation ensemble that the code can comfortably handle increased from a few tens of thousands of dislocations segments in 2001 to tens of million in 2006. The improvements in the code performance resulted from more efficient numerical algorithms and from porting the code on the progressively larger parallel computers. Making the code run efficiently on parallel machines with hundreds, thousands or even 131 thousand processors has been and remains the key component of ParaDiS project. The second release of ParaDiS source code is now freely available. Those interested are advised to contact Tom Arsenlis at [email protected] and ask him to send a CD with the code distribution. Even before trying to obtain ParaDiS, we advise potential users to first download, install and run a toy version of ParaDiS called DDLab. The latter code is written in MatLab and can be downloaded from http://micro.stanford.edu. DDLab is very easy to use and can give one a good idea how ParaDiS works. The ParaDiS team uses DDLab to work out various issues in physics and

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numerical implementation of the DD method. Only after a new feature is found to work well in DDLab, it is then implemented in the parallel code. 3. Dislocation Dynamics in Action: Strain Hardening ParaDiS simulations were recently used to study strain hardening in BCC metal molybdenum. The function describing dislocation mobility was adjusted to represent dislocation motion at an elevated temperature of 450°K. A series of three simulations were performed for [001] and [011] orientations of the compression axis in the virtual BCC single crystal at a constant straining rate of 1 s–1. First reported in [8], the ParaDiS results agreed with an earlier experiment in predicting an enhanced hardening rate for the crystal deformed along [001] axis and a reduced hardening rate for deformation along [001] axis. At the same time, when analyzing the concomitant evolution of the dislocation network obtained in the same simulation, we observed rather unusual dislocation formations that termed multi-junctions. One of such multi-junctions is shown in Figure 4*

Figure 4. A schematic of a multi-junction (line 4) formed by merging together lines 1, 2 and 3. In BCC molybdenum, the four lines connecting at a multi-node have four different, but symmetry-equivalent Burgers vectors of the 1/2<111> type.

In BCC crystals, there are four symmetry-equivalent directions of the Burgers vector of 1 2 <111> dislocations. A multi-junction is where three dislocation lines with three different Burgers vectors merge together to form a forth line, a multi-junction. The Burgers vector of the multi-junction is different from the Burgers vectors of three parent dislocations and is parallel the forth symmetry equivalent <111> direction. As was discussed in [8], the multi-junctions form from the regular binary junctions when a third dislocation with an appropriate Burgers vector approaches the binary junction. The frequency of such encounters is low because the binary junctions

______ *

[8].

Multi-junctions were also identified in Transmission Electron Microscopy experiments

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are weak, holding the two lines zipped together for a while. Hence, the total length of the binary junctions is relatively small even in otherwise dense dislocation networks. However, once a collision of a third line with a binary junction takes place with an appropriate combination of the Burgers vectors, the resulting multi-junction can be very strong. Indeed, our data provide ample evidence that the multi-junctions are much stronger than the binary junctions. So strong in fact that, rather than unzipping under stress as the regular binary junctions do, the multi-junctions often hold the lines together forever, so that the only way a line can move under stress is by bowing out and forming a closed loop. Thus, the multi-junctions can form new sources of dislocation multiplication. Since it takes three different Burgers vectors to form a multi-junction, the latter are more likely to form under straining conditions when at least three of the four symmetry equivalent slip directions (Burgers vectors) are active. These are the so-called multi-slip conditions. Inspection of the simulated stress-strain curves revealed that for the [011] there was essentially no hardening observed. Notably, under [011] straining, only two of the four slip directions feel any external force. Two other slip directions see zero force and are not expected to be active making the formation of multijunctions in such conditions unlikely. This observation implicated that multi-junctions may play a role in strain hardening in BCC metals. To better understand this possible role, we performed another special simulation of straining along the high symmetry [001] where all four slip directions must be equally active and where we observed the multi-junctions to form in abundance. However, in this new simulation we doctored the initial dislocation configuration in such a way that, where it was previously equally populated by dislocations of all four Burgers vectors, we now converted the Burgers vector of type 3 into Burgers vector of type 1, approximately doubling the length of dislocations with Burgers vector 1. Likewise, Burgers vectors 4 were converted into Burgers vectors 2. Thus, the doctored configuration had the same initial idslocation density as before but only two of the four Burgers vectors and multi-junction formation was precluded. The resulting stress-strain curve showed no strain hardening under the same straining conditions where the full, “un-doctored”, system showed rapid hardening. Driven by a very large energy reduction, formation of multi-junctions is likely to occur in other high symmetry crystals under multi-slip conditions. In particular, in FCC crystals, the multi-junctions are predicted to form and to be just as strong as in BCC. Yet there should be important differences in this case. For example, the binary junctions in FCC metals must be relatively stronger than in BCC crystals and, therefore, the contrast in their strength with the multi-junctions is less dramatic. Then, there are more

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symmetry-equivalent orientations of the 1/2<110> Burgers vector in the FCC crystals – six – and it is impossible to find a straining orientation in which only two of the six slip direction can be active. In this sense, straining in FCC is always multi-slip even if some of the slip directions can be more active than others. One important question concerning the multi-junctions is whether still more complex and, most importantly, still stronger multi-line formations exist than the reported ternary multi-junctions? Analyses of network topologies suggest that nodes with more than four connections do occasionally form in the ParaDiS simulations. Such nodes are observed to be unstable, eventually dissociating into simpler 3- or 4-nodes. To add to this observation, I offer the following simple geometric conjecture: In the three-dimensional space, the maximum number n of different vectors b1, b2, …, bn that form only strictly negative dot products, e.g. b1.b2<0, b1.b3<0, b2.b3<0, …, is four. That is nmax=4. While I do not know if this conjecture can be proven as a theorem, it appears plausible. If so, then the ternary multi-junctions should be the strongest possible multi-line reactions that can take place in the dislocation networks simply because all four Burgers vectors labeled on Figure 4 are pair-wise attractive, i.e. they all have negative dot products. Our recent simulations further suggest that ternary multi-junctions play an important role in the formation of dislocation patterns, such as cellular structures often observed in cold worked metals. It appears that, under multi-slip straining the multi-junctions form a strong skeleton of the emerging dislocation microstructure and make for an essential mechanism of accumulation and storage of dislocation density. 4. Dislocation Dynamics in Irradiated Materials What would be the future developments required to make the method of Dislocation Dynamics useful for simulations of irradiated metals and alloys? The key issue is how to treat the complex interaction between dislocations and numerous other defects formed in the material under irradiation. This interaction is mutual and highly non-linear: not only the dislocation behavior is affected by the presence of irradiation defects but also the latter are strongly affected by the presence of dislocations. As an example of relevant interactions, the dislocations work as sinks that absorb a large fraction of interstitials introduced in the material by the continued irradiation. This depletion of the interstitial population promotes formation of vacancy clusters (sometimes filled with He atoms as in

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He-bubbles) that contribute to swelling. Consumption of point defects and clusters by the dislocations can lead to radiation-induced or radiationenhanced creep contributing to dimensional instability. On the other hand, glide mobility of dislocations embedded in the irradiated microstructure is impeded by dislocation collisions with the irradiation defects possibly leading to embrittlement and reduction in the ductile-to-brittle transitions temperature (DBTT). Then there are behaviors involving chemistry, such as irradiation enhanced corrosion and stress-corrosion cracking. In all of this, dislocation behaviors are important and yet poorly understood. While this lecture does not offer any solutions to the difficult issues mentioned above, it appears plausible that extension of DD to dealing with various effects of irradiation are possible and that reasonable progress can be achieved in a near future for some of the simpler materials, such as pure α-iron or its diluted alloys. In concentrated and multi-component alloys the picture is mind boggling because of the additional and very complex interactions between the dislocations and the alloy microstructure. In order to apply DD method to modeling the realistically complex alloys, retaining computational and physical tractability of dislocation dynamics will require approximations to treat at least some of these complex interactions. Acknowledgements The author wishes to express his gratitude to F. Willaime and the rest of the Laboratory of Physical Metallurgy at CEA Saclay, for hospitality and many enlightening discussions during the author’s sabbatical in the fall of 2007.

References [1] Taylor, G. I., 1934, Plastic deformation of crystals. Proceedings of the Royal Society of London 145:362-404. [2] Polanyi, M., 1934, Lattice distortion which originates plastic flow. Z. Phys. 89(9-10): 660-662. [3] Orowan, E., 1934, Plasticity of crystals. Z. Phys. 89(9-10):605-659. [4] Hirsch, P. B., Horne, R. W., and Whelan, M. J., 1956, Direct observations of the arrangement and motion of dislocations in aluminium. Phil. Mag., 1(7):677-684. [5] Cai, W., Bulatov, V. V., Chang, J., Li, J., and Yip, S., 2004, Dislocation core effects on mobility in: Dislocations in Solids, F. R. N. Nabarro and J. P. Hirth, North-Holland, 12:1-117. [6] Bulatov, V. V., and Cai. W., 2006, Line Dislocation Dynamics in: Computer Simulations of Dislocations, Oxford University Press, pp 196-240. [7] Arsenlis, A., Cai, W., Tang, M., et al., 2007, Enabling strain hardening simulations with dislocation dynamics, Model. Simul. Mat. Sci. Eng. 15:553-595. [8] Bulatov, V. V., Hsiung, L. L., Tang, M., et al., 2006, Dislocation multi-junctions and strain hardening, Nature 440:1174-1178.

RADIATION-INDUCED SOLUTE SEGREGATION IN ALLOYS A. J. ARDELL* Department of Materials Science and Engineering University of California at Los Angeles

Los Angeles, CA 90095-1595, USA

Abstract: The state of knowledge of radiation-induced solute segregation (RIS) in alloys is reviewed, with special emphasis on solute segregation in steels. Background information is summarized and some theoretical perspective is provided. Examples illustrating the factors affecting RIS are presented and some important outstanding issues that impact the current understanding of RIS are discussed.

Keywords: Radiation damage; solute segregation; precipitation, kinetic rate theory; Ni-Si alloys; austenitic and ferritic stainless steels

1. Introduction When crystalline solids are subjected to irradiation by energetic neutrons, ions or electrons their microstructures can experience a variety of radiationinduced damage phenomena. The process begins with the production of displacement-damage collision cascades, the sizes of which depend primarily on the mass of the bombarding species. The collisions produce vacancies and interstitial atoms in equal numbers, in concentrations far exceeding their equilibrium values. A large fraction of these excess point defects recombine, eliminating the original damage in the local environment in which the collision cascade was created. However, interstitials interact much more strongly with pre-existing internal sinks for point defects, such as dislocations and grain boundaries, primarily because of the large strain fields associated with them. Furthermore, the mobility of interstitials far exceeds that of vacancies, so they are generally eliminated more quickly than vacancies. The elimination of interstitials that survive recombination at preexisting sinks is therefore biased, leaving behind an excess concentration of less mobile vacancies, the concentrations of which significantly exceed

______ *

Alan J. Ardell, Department of Materials Science and Engineering, UCLA, Los Angeles, CA 900951595, USA; e-mail: [email protected]

V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

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their thermal equilibrium values. Though vacancies are far less mobile than interstitials, they are nevertheless mobile enough to contact each other and eventually condense into voids. In alloys bombarded by energetic particles preferential interactions can exist between the different atomic species, interstitials and vacancies. The origins of the interactions are related to factors such as size differences, electronic structure, etc., and can lead to attraction or repulsion between solute atoms and point defects. Fluxes of interstitials or vacancies towards pointdefect sinks therefore often produce concomitant preferential fluxes of atoms. Even in the absence of strong interactions (or binding) between point defects and solute atoms it is possible, for other reasons to be discussed, for atom fluxes to be induced by fluxes of point defects. In the simplest possible terms, the fluxes of atoms induced by fluxes of point defects, leads to radiation-induced segregation of solute (and solvent) atoms; this phenomenon is generally referred to as radiation-induced segregation (RIS). RIS will often produce a concentration of solute that slightly exceeds its limit of solubility. Under these conditions a new phase will nucleate and grow, leading to radiation-induced precipitation (RIP). In this paper the history and principal causes of RIS will be discussed. Different approaches to modeling RIS will be presented, along with selected experimental observations in conjunction with attempts to predict these observations quantitatively. The influence of experimental variables (e.g. radiation temperature, the addition of trace elements) and microstructural parameters (e.g. grain boundary character) will be highlighted, particularly in steels, which are candidate alloys for use in Generation IV nuclear reactors. Processes related to RIS, in particular thermal non-equilibrium solute segregation, and its implications regarding atom transport, will also be discussed. Finally, a few examples of the role of RIS on mechanical behavior will be presented. 2. Physical Nature of RIS 2.1. HISTORICAL PERSPECTIVE

RIS and RIP were discovered by Okamoto and Wiedersich (1974). They observed the precipitation of the intermetallic compound Ni3Si (with the L12, Cu3Au crystal structure) at the surfaces of radiation-induced voids in an austenitic stainless steel. Okamoto and Wiedersich postulated that the concentration of interstitial atoms would be preferentially populated by undersized Si atoms, the diffusion of which towards void surfaces (which are very strong point-defect sinks) would enable the precipiation of Ni3Si once the solubility limit of this phase was exceeded. Shortly after this

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discovery, Barbu and Ardell (1975) observed the precipitation of Ni3Si in a Ni-4 at% Si alloy at interstitial dislocation loops as a consequence of Ni+ion irradiation at 500°C. To place this observation in its proper context, the solubility of Si in Ni exceeds 10 at.% at temperatures above 600°C (Rastogi and Ardell 1969) and the lattice constant of Ni-Si solid solutions decreases with increasing Si content (Kamara et al. 1996), i.e. Si is an undersized solute atom. It is evident that an alloy containing 4% Si is highly undersaturated at 500°C. An example of the radiation-induced micro-structure observed by Barbu and Ardell (1975) is shown in Figure 1, where the segregation of Si to the interstitial loops and precipitation on them is evident. Janghorban and Ardell (1979) later observed the RIP of Ni3Si in undersaturated Ni-Si alloys at other kinds of point defect sinks, such as grain boundaries, coherent twin boundaries and free surfaces; examples are shown in Figure 2. These early findings have since been supplemented by many other observations of RIS and RIP in alloys containing undersized solutes (Rehn and

Figure 1. Bright field (BF) and dark field (DF) transmission electron micrographs of the same area (see arrowheads) in a Ni-6 at.% Si alloy irradiated at 500°C to a dose of 10 dpa using Ni+ ions (Barbu and Ardell, 1975). The precipitation of ordered Ni3Si is evident at the dislocation loops in the DF image, taken using a (100) superlattice reflection.

Figure 2. Dark-field images of 3 different manifestations of RIP in undersaturated NiSi alloys: a) discontinuous precipitation at a grain boundary, b) RIP at a coherent twin boundary, c) a continuous surface film of Ni3Si, showing the anti-phase domain boundary structure (Ardell and Janghorban 1983).

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Okamoto 1983). It is firmly established that there is preferential coupling or binding between undersized atoms and interstitials generated during irradiation. Rehn and Okamoto (1983) noted nearly 25 years ago that there had never been an observation of the depletion of an undersized solute at a point defect sink; I believe that this is still the case today. In general, oversized solute atoms tend to become depleted at point defect sinks. A notable exception to this behavior is the segregation of oversized Ge atoms to point-defect sinks, specifically interstitial dislocation loops, in undersaturated Ni-Ge alloys (Barbu 1979), leading to the formation of Ni3Ge, which is isostructural with Ni3Si. In this binary alloy system the solubility of Ge in Ni is similar to that of Si (Ardell 1994). Rehn and Okamoto (1983) have also observed the formation of surface films of Ni3Ge. An explanation for this apparent exception to the “size-effect” rule has been offered by Gupta and Lam (1979). They calculated the binding energy of mixed Ni-Ge dumbbell interstitial complexes and showed that it was large enough to enable the segregation of Ge by interstitial diffusion, in the same manner as strongly bound Ni-Si complexes. The preceding discussion and examples presented all have their explanations rooted in the coupling between atoms and point defects, especially undersized atoms with interstitials That such coupling is not necessary for the occurrence of RIS was first point out by Marwick (1978). Anthony (1969) proposed that solute segregation near the surfaces of quenched binary Al alloys could be explained by invoking the so-called Inverse Kirkendall Effect (IKE). The classic Kirkendall effect (see Shewmon 1963) involves fluxes of vacancies that compensate for differences between the fluxes of atoms during normal diffusion by the vacancy mechanism in crystals. Anthony realized that fluxes of quenched-in vacancies to free surfaces during annealing at lower temperatures could produce fluxes of atoms in the opposite direction, leading to depletion at the surfaces. Marwick realized that the same situation could obtain during irradiation. This type of behavior in a binary A-B alloys is illustrated schematically in Figure 3. In this case A atoms diffuse faster than B atoms, so the flux of vacancies towards the point defect sink induces a preferential flux of A atoms in the opposite direction. The sink becomes depleted in A atoms and since the concentration of A + B atoms must remain constant, the sink becomes enriched in B atoms. The relative importance of diffusion of these so-called point-defect/atom complexes and the IKE has remained an active area of research over the past several decades, and some aspects of that issue will be addressed later in this paper. To simplify the discussion, complexes involving vacancies and solute or solvent atoms will be referred to as V-S complexes. The interstitial counterpart will be called I-S complexes.

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Figure 3. Schematic diagram illustrating the fluxes of vacancies and atoms in the inverse Kirkendall effect.

2.2. MECHANISMS OF POINT-DEFECT ATOM DIFFUSION

Solute redistribution by either vacancy-solute or interstitial-solute complexes can occur provided that their dissociation energies are much larger than their migration energies. An example of a V-S complex in a dilute binary alloy is shown in Figure 4, adapted from the review article of Wiedersich and Lam (1983). The oversized solute atom (labeled 1) and its nearest neighbor atom (4) are both depicted as neighboring a vacant site for illustrative purposes; it is highly improbably that atoms 1 and 4 would both have nearest neighbor vacant sites in a real dilute alloy. The other atoms nearest the vacant sites are numbered differently because their jump freuencies into these sites will differ due to their own nearest-neighbor atom configurations. The jump frequencies of atoms 3 are assumed equal, as are those of atoms 5, consistent with the description of Howard and Lidiard (1965) and Howard and Manning (1967). Because the atoms labeled 2 and 4 are also nearest neighbors of the solute atom 1, they exchange places with their nearest-neighbor vacant sites with different jump frequencies than the other atoms. Diffusion in this model is expressed in terms of ratios of the calculated jump frequencies, which depend on the V-S binding energy.

Figure 4. The atom configurations around vacancies in a dilute substitutional alloys containing oversized solute atoms (1). The atoms are numbered differently to emphasize that their jump frequencies into the vacant lattice site are different and influenced by their local surroundings. The diagram is adapted from Wiedersich and Lam (1983).

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An example of an I-S complex and the manner in which the interstitial atom can jump from one lattice position to another in a dilute alloy is shown in Figure 5, also taken from the article by Wiedersich and Lam (1983). The lowest-energy configuration of an I-S complex is the so-called mixed dumbbell, in which an undersized substitutional solute atom and a solvent atom share a lattice site. In the example shown in Figure 5 the solute atom migrates by rotation of the mixed dumbbell. Other possible configurations for the migration of mixed dumbbells are discussed by Wiedersich and Lam (1983), and in more detail by Robrock (1983). Diffusion of undersized solute atoms in dilute face-centered cubic alloys, even in the simplest case of mixed dumbbells in 〈100〉 orientation, is quite complicated. Approximate solutions can be found in papers by Barbu (1980) and Johnson and Lam (1976). It is emphasized that the formation of mixed dumbbells is probable only when self-interstitials have been introduced into the crystal by irradiation; the formation of these complexes under thermal equilibrium conditions is nearly impossible. Diffusion of an undersized substitutional solute atom typically proceeds by the vacancy exchange mechanism. In irradiated alloys the energy of a mixed dumbbell is lower than that of a solvent dumbbell because the strain energy of this configuration is lowered by the smaller size of the undersized solute atom.

Figure 5. Illustration of the diffusion of a mixed dumbbell interstitial by rotation inside the unit cell, followed by exchange of the undersized solute atom with a neighboring solvent atom. The diagram is adapted from Wiedersich and Lam (1983).

3. Theories of RIS There are 2 fundamentally different classes of theories of RIS. The first class consists of kinetic rate theories (KRT) that involve solutions to a hierarchy of partial differential equations. These include concentration gradients of the various diffusing species, their diffusion coefficients and correlation factors, generation and recombination rates of point defects, assorted treatments of the solution thermodynamics and descriptions of the boundary conditions. There are 2 subclasses of KRTs. They differ in the way that diffusion is modeled. One class is typified by the theory of Wiedersich et al. (1979), which uses a continuum model of diffusion of which the random alloy model of Manning (1971) is representative. The other class, proposed

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by Granjean et al. (1994), uses a kinetic model of diffusion on a discrete lattice (Martin 1990) This model also takes the surface energies of the elements into account. The other class of theories is due to Faulkner et al. (1993), and will be referred to here as a non-equilibrium segregation (NES) theory. It attributes RIS, as well as other kind of non-equilibrium solute segregation, entirely to the diffusion of V-S and/or I-S complexes, and is built on theories of equilibrium segregation at grain boundaries. A positive binding energy with point defects always leads to enrichment of atoms, solute or solvent, at sinks, while a negative binding energy (i.e. repulsion between the atom and point defect) always produces depletion at sinks. Since diffusion in the context of Fick’s laws plays no role in the NES theory, there can never be a contribution to segregation from the IKE. There is yet another kind of RIS mentioned here for the sake of comleteness. It predicts destabilization of a homogeneous solid solution under irradiation, leading to radiation-induced homogeneous precipitation (Cauvin and Martin 1981). Recent Monte Carlo simulations by Krasnochtchekov et al. (2007) have verified some of the predictions of the theory Cauvin and Martin, but this topic will not be dealt with further. The theory of Wiedersich et al. (1976) has gained the most traction among the numerous investigators who have constructed models of RIS under a variety of different conditions. It is an extension of the original kinetic rate theory of Johnson and Lam (1976), which is valid for dilute solid solutions and utilizes models for diffusion of point-defect solute-atom complexes such as those depicted in Figures 4 and 5. In concentrated solid solutions, even those exceeding 1 at.% or so, such models break down because the probabilities that vacancies or interstitials have only one solute atom as a nearest neighbor are very small. In the case of oversized solute atoms, as depicted in Figure 4, the probability is large that another solute atom will replace any of the atoms labeled 2 to 4. The jump frequencies of the atoms neighboring a vacancy, as well as other parameters like the correlation factor, can then no longer be estimated with any degree of accuracy. Likewise, when the concentration of undersized solute atom increases the number of solute-solute dumbbells will increase, so calculations of diffusion involving the transport of mixed dumbbells become unrealistic. To deal with the formidable problem of formulating a theory of diffusion involving the numerous types of point-defect solute-atom complexes, Wiedersich et al. introduced the idea of partial diffusion coefficients related to 4 different diffusivity coefficients in a binary alloy defined by the equations

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d A,B / v ,i =

ν A,B / v ,i bv ,i Z v ,i

(1)

6Ω

where the notation dA,B/v,i represents partial diffusivity coefficients of A or B atoms, respectively, by interchange with vacancies (v) or interstitials (i); these are identical to diffusivity coefficients of vacancies or interstitials by interchange with A or B atoms. The quantities bv,i and Zv,i are the jump distances and coordination numbers, respectively, of vacancies and interstitials, which can differ in principle, and Ω is the average atomic volume in the solid. The important quantities in equations (1) are the νA,B/v,i, which represent the effective jump frequencies with which A or B atoms exchange places with vacancies or interstitials. Equations (1) are readily extended to multi-component alloys (Lam et al. 1982). The νA,B/v,i, have temperature dependencies of the Arrhenius type, i.e. ν A,B / v ,i = ν ( A,B / v ,i )0 exp ( − E( A,B / v ,i ) m /k B T

)

(2)

where the E(A,B/v,i)m are the activation energies for motion of A or B atoms by vacancies or interstitials, and the ν(A,B/v,I) are fundamental vibration frequencies, kB is Boltzmann’s constant and T is the absolute temperature. The activation energies can be very difficult to estimate because the point defects and solute atoms can be bound to, or repelled from the individual atoms, or groups of atoms, in the crystal. Furthermore, the energies will depend on the configuration of the solute-atom-point-defect complex and can depend on the composition of the alloy. At the same time, however, their estimation provides enormous flexibility in modeling the behavior of the solid solution under irradiation. The next step in the theory of Wiedersich et al. (1979) is to develop equations for the fluxes of the various components in the alloy system, including not only the atomic species, but also the vacancies and interstitials. This is done by first defining partial diffusion coefficients D

A, B / v,i

= d

f

A, B / v,i A, B / v,i

C

v,i

D

v,i / A, B

= d

f

A, B / v,i v,i / A, B

C

A, B

(3)

where the concentrations Cv,i and CA,B are expressed in numbers per unit volume and fA,B/v,i and fv,I/A,B are correlation coefficients for the jumps of A or B atoms into vacant or interstitial sites and the corresponding correlation coefficients for vacancy and interstitial jumps with A and B atoms, respecttively. Note that fA,B/v,i ≠ fv,I/A,B but dA,B/v,i = dv,i/A,B. Assuming for simplicity that the correlation coefficients are all equal to unity, the total diffusion coefficients for the atoms and point defects are given by the equations

SEGREGATION DURING IRRADIATION D

A, B

= D

A, Bv

+ D

D

A, Bi

v,i

= D

v,iA

293

+ D

v,iB

(4)

The diffusion equations for the various species under irradiation conditions are

∂C A,B = −∇J A, B ∂t

∂Cv,i = −∇J v, i + G − RCiCv ∂t

(5)

where G is the rate per unit volume of production of interstitials and vacancies, R is their rate per unit volume of recombination and the fluxes, JA,B,v,i are given by the equations J A, B = – D A, Bα∇C A, B + d A, Bv Ω C A, B ∇C v – d A, Bi Ω C A, B ∇C i

J v = – Dv ∇C v + d AvαΩ C v ∇C A + d BvαΩ C v ∇C B

(6)

J i = – Di ∇Ci – d AiαΩ Ci ∇C A + d BiαΩ Ci ∇C B ,

where α is the thermodynamic factor. The physical significance of Eqs. (6) is that under irradiation conditions the fluxes of all the atomic species depend on the concentration gradients of both vacancies and interstitials, as well as their own gradients, and that the fluxes of point defects depend on the gradients of the atomic species as well as their own gradients. These are the most important distinctions between diffusion under irradiation and diffusion under normal conditions. With respect to a fixed lattice plane the fluxes of A atoms, B atoms, vacancies and interstitials are not independent, and must satisfy the equation J

A

+ J

B

= – J + .J v

(7)

i

Wiedersich et al. (1979) showed that under steady-state irradiation conditions ΩC C d d ∇C

A

=

(

A

B

Bi

Ai

α d C D + d C D Bi

B

A

Ai

A

B

)

⎧⎪ d Av ⎨ ⎪⎩ d Bv

⎫⎪ ⎬ ∇Cv d ⎪ Bi ⎭

d –

Ai

(8)

which follows from the conditions JA = JB = 0 and Jv = Ji. In a coordinate system such as that in Figure 6a the gradient of Cv is always positive. According to Eq. (8) the sign of the gradient of CA will therefore depend on

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the sign of term in curly brackets, which from Eqs. (1) depends on the ratios of the νA,B/v,i. If the vibration frequencies are such that dAv/dBv > dAi/dBi, the gradients of A atoms and vacancies will have the same sign, as shown in Figure 6b. Physically, if A and B atom interstitials jump at approximately equal rates and the exchange of A atoms with vacancies is much faster than the exchange of B atoms with vacancies, then the diffusion of A atoms is faster than the diffusion of B atoms. RIS at the point defect sink is thus realized by the IKE, leading to the depletion of A atoms and enrichment of B atoms.

Figure 6. Schematic diagram illustrating a prediction of the theory of Wiedersich et al. (1979), Eq. (8), when dAv/dBv > dAi/dBi.

An example of the predictive capability of the theory of Wiedersich et al. (1979) is demonstrated in Figures 7 and 8 which shows data on the growth rate of Ni3Si surface films in a Ni 12.7 at.% Si alloy irradiated using 2 MeV He+ ions at a dose rate of 2.6 × 10–5 dpa/s. The theory assumes that the near-surface region is rapidly depleted of Si as strongly-bound Siinterstitial complexes migrate towards the surface and form an Ni3Si film, similar to that seen in Figure 2c, when the solubility limit is reached.

Figure 7. Plot of Ni 3Si surface film thickness vs. square root of the dose as a function of temperature in a Ni-12.7 at.% Si alloy.

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Vacancies play an important role in the theory through their affect on recombination, but their role in diffusion is ignored because of the strong interaction between Si and interstitials. A steady-state concentration profile is established in the bulk alloy behind the surface film, where the solute is depleted as the film thickens, and the flux of solute in this region is calculated. Using a simplified model of the diffusion geometry, which is justified by numerical solutions, Okamoto et al. (1982) predict that the thickness of the film should increase linearly with the square root of the dose. The parameters used in the model are summarized in Table 1.

Figure 8. Arrhenius plots of growth rates of the3 Ni Si films vs. reciprocal temperature.

The data in Figure 7 illustrate how well this dose-dependence of the film thickness is observed at all irradiation temperatures. The growth rate, given by the slopes of the curves in Figure 7, increases with increasing temperature up to about 565°C, then decreases such that the growth rate at 630°C is actually lower than that at 452°C. TABLE 1. Parameters used by Okamoto et al (1982) in fitting the theory of Wiedersich and Lam (1979) to the data on Ni3Si film growth during irradiation. Parameter Fundamental frequency factor for vacancies Fundamental frequency factor for I-S complexes Migration energy of vacancy Formation energy of vacancy Formation entropy of vacancy Lattice constant Recombination volume Sink annihilation probability for vacancies Sink annihilation probability for I-S complexes Migration energy of I-S complex Binding energy of I-S complex

Notation

ν 0ν ν 0 Bi

Emv Efv Sfv ao

Ω

Emi Ebi

Value 5 × 1013 s–1 5 × 1012 s–1 1.2 eV 1.5 eV 4.0 K 0.352 nm 3 150 a0 1 to 3 × 104 1 to 3 × 104 0.6 eV 0.9 eV

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redicts that the activation energy for growth at lower temperature should be Emv/4, where Emv is the migration energy of vacancies, but at higher temperatures the growth rate should decrease with an Arrhenius-type temperature dependence controlled by the formation energy of vacancies, Efv; specifically, at high temperatures the growth rate should vary as exp(Efv/2kT). The theory also predicts that at lower temperatures the growth rate should increase with decreasing dose rate. The predicted temperature and dose-rate dependencies are displayed in Figure 8. While the magnitudes of Emv and Efv are not known with complete certainty, the values implied by the results in Figure 8, Emv ≈ 1.2 eV and Efv ≈ 1.5 eV, are consistent with expectation. Additionally, the growth rates at dose rates of 2.6 × 10–5 dpa/s are roughly a factor of 2 higher than those at a dose rate of 3.1 × 10–4, consistent with what is predicted. The physical reasons for the observed temperature dependencies are manifestations of the dominant role played by recombination. If all the interstitials and vacancies generated during radiation recombined there would be no RIS, and other deleterious effects of radiation damage, such as void swelling, would also vanish. However, recombination is incomplete because interstitials are far more mobile than vacancies and are much more strongly attracted to sinks because of their large strain energies. In Ni-Si alloys the undersized Si atoms form stable I-S complexes during irradiation, and as stated the diffusion of these complexes towards sinks leads to the formation of Ni3Si when the solubility limit is reached. At lower irradiation temperatures, below the temperatures at which the thermal equilibrium concentration of vacancies is negligible, the concentration of Si-interstitial complexes in the bulk of the alloy depends on the concentration of vacancies produced by irradiation. Since the mobility of vacancies always increases with T, the concentration of vacancies available for recombination with interstitials decreases as T increases. Recombination therefore decreases in this temperature range and the growth rate of the Ni3Si films increases. The temperature dependence in this range of T is governed by the mobility of the vacancies, hence the dependence on Emv. At higher temperatures the concentration of vacancies is dominated by thermal equilibrium. Since Cv increases as T increases in this range of temperatures, recombination also increases as T increases, hence the growth rate of the film decreases. The effect of dose rate in the lower range of temperatures is also explained by the role of radiation-produced vacancies in recombination. Assuming that the concentration of Si-Interstitial complexes is unaffected by temperature, the smaller rate of point-defect production leads to smaller concentrations of vacancies at a given irradiation temperature. Recombination is therefore reduced and the growth rate of the films increases.

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4. RIS in Steels Because the theory of Wiedersich et al. (1976) has promising quantitative predictive capability it has been used, with various modifications, to examine RIS in steels, primarily in austenitic steels. Since different kinds of steel alloys are anticipated for use in Generation IV nuclear reactors, the remainder of this paper will focus on RIS in steels. The progress in understanding segregation of the main alloying elements in steels, as well as microstructural factors and alloying behavior that influence RIS, will be presented, and some remaining perplexing issues will be discussed. Before proceeding, it is worth mentioning that modeling the behavior of RIS in steels has involved a variety of modifications. These include: 1. Changing the boundary conditions used to solve Eqs. (4), after substitution of Eqs. (5); 2. Calculating the correlation factors for diffusion of the various species; 3. Calculating the magnitude of α, which depends on the thermodynamics of the solid solution; 4. Calculating the interaction energy between solute atoms and point defects, which strongly influence the vibration frequencies, hence the diffusivity coefficients in equations (1); 5. Allowing the parameters involved in factors 1 to 3 be concentration dependent; 6. Varying the details of point-defect generation under different types of irradiation conditions and hence the magnitude of G; 7. Evaluating th recombination of point defects, which depends strongly on microstructural features such as the dislocation density, grain size, grain character and even grain boundary motion. Typical RIS behavior at grain boundaries in austenitic stainless steels is shown in Figure 9, taken from the work of Norris et al. (1992). Their particular steel contained ~20% Cr, ~25.% Ni and~0.8% Nb (all wt.%) and all the specimens were irradiated by neutrons in an advanced gas-cooled reactor. The irradiated specimens were examined using analytical electron microscopy. In all steels Ni, which is undersized, is enriched at grain boundaries, while Fe, and especially Cr, are depleted. At higher temperatures the concentration profiles are much broader, reflecting the increasingly important role of recombination as temperature increases. Ferritic and ferritic/martensitic steels generally have more complicated microstructures than austenitic steels, so measuring RIS at defects is somewhat more difficult. Nevertheless, Muroga et al. (1989) have observed the depletion of Cr and enrichment of Ni at ferrite-ferrite grain boundaries in steel containing 10% Cr and 5% Ni (wt.%) irradiated using 1 MeV electrons at 500 °C to a dose of 3 dpa; this result is shown in Figure 10. It is often found that the concentration profiles produced by RIS at grain boundaries do not monotonically depend on distance from the boundary. This is most evident for the profiles of the Fe and Ni atoms in Figure 9 as

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well as the Ni and Cr profiles in Figure 10. It is not possible to reproduce this behavior using models that invoke only the IKE (Marwick 1978) and ignore the role of interstitials, as is evident in the modeling work of Marwick et al. (1984), Perks and Murphy (1987) and Allen and Was (1998). Nastar et al. (1998), using a modified version of the theory of Grandjean et al. (1994), had some success in reproducing this feature, but it is quite 0.50

0.45

0.45

0.40

Ni

0.40

0.35

348 ¡C

atom fraction

0.35

471 ¡C

0.30

0.30

0.25

0.25

0.20

0.20

0.15

0.60

0.65

0.55

0.60 0.55

0.50

Fe

0.45

0.50

0.40

0.45

0.35

0.40

0.25

0.25

0.20

Fe

0.20

Cr

0.15 0.10 -40

Ni

-30

-20

0.15 -10

0

10

distance

20

30

40

Cr

0.10 -160 -120 -80

-40

0

40

80

120

160

distance

Figure 9. Concentration profiles of Ni, Fe and Cr at grain boundaries in a stainless steel irradiated at 348°C to a dose of 1.4 dpa (left column) and 471°C to a dose of 4.7 dpa (right column). The dashed horizontal lines indicate the average concentrations. The atom fractions of Fe, Ni and Cr were normalized to unity, which is why the average concentrations in the left and right columns of figures differ slightly (Norris et al. 1992).

Figure 10. Concentration profiles of Ni and Cr at grain boundaries in a ferritic steel irradiated at 500°C to a dose of 3 dpa using 1 MeV electrons. Data of Muroga et al. (1989).

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apparent from the work Soisson (2006) that non-monotonic concentration profiles are readily reproduced using the theory of Wiedersich et al. with the appropriate choice of physical parameters and boundary conditions. This is evident in Figure 11, which shows the results of a Kinetic Monte Carlo (KMC) simulation of RIS in a ideal solid solution of a body centered cubic alloy containing 10% A, 90% B. The shaded band in the figure represents the scatter in the data on simulation and the solid curve represents the calculations using the theory of Wiedersich et al. The agreement is exceptionally good.

Figure 11. Results of the KMC simulations of Soisson (2006) for an A-B ideal solid solution irradiated at 500 K to a dose of 0.5 dpa using a dose rate of 10–3 dpa/s. The scatter in the simulations is represented by the shaded band and the solid curve shows the results of the calculations according to the theory of Wiedersich et al. (1979).

It is often observed that grain boundaries in alloys migrate under irradiation conditions. In these cases the segregation profiles are usually asymmetric, sometimes to a quite large extent. An example, also taken from the work of Norris et al. (1992), is shown in Figure 12. These profiles were obtained from the chemical analysis of curved boundaries pinned along their length by inclusions, probably carbide particles, in the steels investigated. Some data on RIS associated with radiation-induced grain boundary migration, observed directly in an electron-irradiated austenitic stainless steel, are shown in Figure 13, taken from the work of Sakaguchi et al. (2005). They developed a model based on the theory of Wiedersich et al. (1979) that includes the velocity of the grain boundary, which depends on the difference between the radiation-induced and equilibrium vacancy concentrations via a parameter αv,i, which they call the rearrangement factor. When αv,i = 0 RIS is predicted to occur without grain boundary migration, but values of αv,i as small as 10–8 produce measurable migration. The predictions of the model of Sakaguchi et al. are compared with experimental data in Figure 13. The agreement, while not perfect, is quite good. Sakaguchi et al. (2005) note that the physical significance of the parameter αv,i is not well understood. From the perspective of modeling the migration behavior of grain boundaries under irradiation, it seems that αv,i

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represents the behavior of a boundary subjected to unequal fluxes of point defects within the grains adjacent to the boundary. That these unequal fluxes induce unequal segregation profiles at the boundaries is probably not too surprising. But the origin of the unequal fluxes is not known.

Figure 12. Concentration profiles of Ni, Fe and Cr at moving grain boundaries in an irradiated stainless steel. The dashed horizontal lines indicate the average concentrations. The atom fractions of Fe, Ni and Cr were normalized to unity, as in Figure 9 (Norris et al. 1992).

Any factor that affects the efficiency of grain boundaries as sinks for point defects should influence RIS. The character of a grain boundary, specifically whether it is a low-angle boundary consisting of a wall of dislocations, or whether it is a special boundary, such as a coherent twin boundary or another special boundary with a small number of coincident lattice sites in the grains that it separates, has been shown experimentally to affect RIS (Duh et al. 2000, Watanabe et al. 2000). The first observations were made on proton irradiated and high-voltage electron irradiated austenitic stainless steels, respectively, irradiated at temperatures in the range 350 to 450°C. Both investigations showed that the extent of RIS, characterized by the depletion of Cr and enrichment of Ni, was significantly reduced at low-angle grain boundaries and special boundaries of the typeΣ 3 (coherent twin boundaries) and Σ 9. An example, taken from the work of Sakaguchi et al. (2000), is shown in Figure 14 for grain boundary orientations symmetric around a 〈110〉 axis of rotation.

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Figure 13. Concentration profiles at a grain boundary that migrated during electron irradiation of an austenitic stainless steel at 400°C to a dose of 14.4 dpa. The solid curves represent the predictions of the model of Sakaguchi et al. (2005).

Figure 14. The effect of grain boundary misorientation on RIS of Cr and Ni in an austenitic stainless steel irradiated to a dose of 3 dpa at 350°C. Data of Sakaguchi et al. (2004).

The qualitative explanation for the data seen in Figure 14 is clear. Lowangle and special boundaries are poor sinks for point defects, so the gradients of vacancies and interstitials in their vicinities become increasingly small as the ideal boundary orientations are approached. Consequently the fluxes of point defects decrease dramatically, hence preferential diffusion of solute by any mechanism is eliminated. Modeling of the dependence of RIS on grain boundary character requires the introduction of a parameter that captures the strength of the boundary as a point-defect sink. Models that do this successfully have been developed by Duh et al. (2001) and Sakaguchi et al. (2003). Both approaches are based on the KRT approach of Wiedersich et al. (1976) and reproduce the observed dependence of RIS on grain boundary misorientation with reasonable accuracy. The curves in Figure 14 show the results of the calculations of Sakaguchi et al. (2004). The agreement between the calculated dependencies of the concentrations

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of Cr and Ni is excellent. The agreement between the predictions of their theory and the data of Duh et al. (2000) is also excellent. The addition of oversized solute atoms to stainless steels has been shown to strongly affect RIS of the main alloying elements. The effect of deliberate and controlled additions of solute elements to an austenitic stainless steel appears to have been investigated for the first time by Kato et al. (1992), who added approximately 0.35 at.% Ti, Zr, Hf, V, Nb or Ta to type 316 stainless steel and examined RIS by analytical electron microscopy in specimens irradiated using 1 MeV electrons to doses up to 10.8 dpa at temperatures in the range 350 to 500°C. They found that the RIS of Cr and Ni at grain boundaries was essentially eliminated by the additions of Zr and Hf, though there was no clear correlation between atomic size and the extent to which RIS was reduced. The beneficial effect of Hf in reducing or eliminating RIS in austenitic steels has been confirmed. Fournier et al. (2003) irradiated type 316 stainless steel, doped with 0.3 at.% Hf or Pt, at 400°C to either 2.5 or 5 dpa using energetic protons. At 2.5 dpa the reduction in the RIS of Cr was significant for both Pt and Hf additions, as seen in Figure 15. It was found that the suppression of RIS of Cr, Ni and Fe by the addition of Hf was not as large at a dose of 5 dpa as it was at 2.5 dpa, but was still significant. Lu et al. (2006) showed that the addition of 1 at.% Hf to a ferritic/martensitic steel (9.2 Cr, 1.0 Mo, 0.7 Nb, 1.0 W, all wt.%) suppressed RIS of Cr and P in specimens irradiated at 300°C using 200 keV Ni+ ions to 0.3 dpa. Additions of Hf thus reduce or eliminate RIS in both austenitic and ferritic steels. The reason for the reduction of RIS in steels by the addition of oversized solute atoms is that vacancies form V-S complexes with the alloying elements, thereby reducing vacancy diffusion, hence maintaining a higher vacancy concentration and promoting recombination. The calculations of Stepanov et al. (2004) support this conjecture in that they show the effect of V-S binding energy on the enrichment of Ni at a grain boundary. However, there are no calculations of binding energies for specific atom species, so there is as yet no way to predict which sub-stitutional solute additions are expected to be most effective in reducing or eliminating RIS in steels. There is evidence that RIS in steels is reduced by the addition of interstitial alloying elements as well as oversized solute atoms. Kano et al. (1998) investigated the RIS of Cr, Ni and Si in 2 austenitic steels similar to types 304 and 316. The 316 alloys contained 0.004, 0.019 and 0.052% C and the 304 alloys contained 0.003, 0.032 and 0.095% N. Specimens were irradiated with 12 MeV Ni3+ ions at 300°C. The changes in concentration at the grain boundaries compared to their values in the bulk were small, but

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appear to have been systematic, especially regarding the concentrations of Ni and the impurity element Si, both of which segregate to grain boundaries during irradiation. The main results are shown in Figure 16 where it is evident that the influence of interstitial C and N is most pronounced on the RIS of Ni, with rather smaller effects on the Cr and Si concentrations.

Figure 15. The effect of the oversized solute atoms Hf and Pt on the segregation of Cr in an austenitic stainless steel. Data of Fournier et al. (2003).

Figure 16. The effect of alloying with interstitial C and N atoms on RIS in an austenitic steel. Data of Kano et al. (1998).

Kano et al. (1998) did not offer an explanation of their results, but noted that the nucleation of dislocation loops was enhanced by the interstitial additions. It is possible that the concentration of Ni and Si at grain boundaries was reduced by enrichment of these atoms at the dislocation loops. Unfortunately, there do not appear to have been any subsequent investigations of this effect, and more work must be done to both verify and, if necessary, explain these results.

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5. TNES and W-shaped Profiles Thermal non-equilibrium segregation (TNES) occurs when alloy solid solutions are cooled at relatively rapid rates from a high temperature. It was first observed in bicrystals of Pb doped with Sn, Ag, Au or In by Westbrook and Aust (1963) indirectly through measurements of microhardness and attributed to excess solute segregation over and above values expected from equilibrium segregation. TNES was investigated more systematically by Aust et al. (1968), again using microhardness measurements. Aust et al. proposed that the hardening or softening measured at grain boundaries was due to diffusion of V-S complexes either towards or away from the boundaries. The specimens of doped Zn, Pb and Sn were all air-cooled from temperatures in the range 200 to 300°C. Cooling rates were not measured, but would qualify as being “relatively rapid” in the context of this discussion. Anthony (1969) provided a robust explanation for these observations in terms of either the motion of V-S complexes or the IKE. TNES has been observed in both austenitic and ferritic steels. Faulkner (1996) has reviewed some of the earliest observations. A representative concentration profile for the TNES of Cr is shown in the top drawing of Figure 17, taken from the work of Goodwin et al. (1999). The steel is type 304, which was solution treated at 1050°C, cooled at ~20°C/s and examined by analytical electron microscopy. The data on the unirradiated specimen show that the concentration of Cr at grain boundaries is ~7% larger than in the bulk. Busby et al. (1999) observed similar behavior in proton-irradiated type 304 and 316 stainless steels cooled from high temperatures by forced air convection, though the magnitude of TNES was not quite as large as that found by Goodwin et al. Subsequent irradiation of specimens that have experienced the enrichment of Cr produces the other concentration profiles shown in Figure 17. At intermediate doses (1 dpa in Figure 17) the so-called “W-shaped profile” is observed, while at larger doses the profile becomes more comparable to those in steels in which TNES has not occurred (Figure 9). At present, there is no completely satisfactory explanation for the behavior seen in Figure 17. Nastar (2005) has reproduced a W-shaped profile using the theory of Grandjean et al. (1994), though quantitative agreement with the experimentally observed profile is not very good. She also shows that the pre-existing profile produced by TNES is not crucial in the formation of the W-shaped profile. Allen et al. (2007) suggest that the interaction of Cr with interstitial alloying elements B, C and N, or the ordering tendency of Ni and Cr, might be important in producing the Wshaped profile.

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Figure 17. Data on the Cr concentration profiles in a moderately rapidly cooled austenitic stainless steel. The uppermost plot illustrates TNES in an unirradiated specimen. The Wshaped profile at intermediate doses is shown in the middle plot and normal RIS is observed at higher doses. Data of Goodwin et al. (1999).

In some ways the most convincing explanation of all the behavior seen in Figure 17 is the one offered by Faulker (1996). It is based on a combination of Faulkner’s theory of TNES (1981) and the theory of Faulkner et al. (1993) on RIS. In describing this theory it is first important to point out that TNES cannot be explained by the IKE. Figure 18a illustrates what the expected Cr concentration profile during the action of a flux of vacancies to grain boundaries under intermediate cooling conditions. Since Cr is the fastest diffusing element in austenitic stainless steels (Rothman et al. 1980), a flux of vacancies towards the grain boundary would induce a Cr flux in the opposite direction, leading to depletion of Cr rather than enrichment. In the TNES theory of Faulkner, oversized Cr atoms have a positive binding energy with vacancies and a negative binding energy with interstitials. TNES would therefore lead to enrichment of Cr, since V-Cr complexes would diffuse towards grain boundaries during cooling, as indicated in Figure 18a. On the other hand, during irradiation the flux of interstitials towards grain boundaries exceeds that of vacancies, since interstitials are much more mobile. The theory of RIS of Faulkner et al. (1993), when applied to the case of Cr in steels, attributes the depletion of Cr at grain boundaries to the repulsion of Cr by interstitial complexes such as Ni-Ni dumbbells or perhaps mixed Ni-Fe dumbbells. The flux of interstitial complexes towards grain boundaries therefore produces a flux of

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Cr in the opposite direction, as illustrated schematically in Figure 18b. This explanation is also consistent with the RIS of undersized Ni atoms at grain boundaries. Allen et al. (2007) state that Ni and Fe are both depleted by TNES (slightly in the case of Ni) at grain boundaries, though they do not cite the source of their information. Faulkner does not address the segregation of Ni and Fe in his papers, so it is not known what kind of segregation behavior this approach would predict. A completely satisfactory and self-consistent explanation of TNES and RIS is not yet available.

Figure 18. Schematic illustration of the segregation tendencies of Cr. In (a) the flux of Cr from the IKE is opposite to that if Cr atoms are bound to vacancies, as postulated by Faulkner et al. (1993). In (b) The flux of Cr is opposite to that of interstitials according to the theory of Faulkner et al.

6. Known Effects on Mechanical Properties A major reason for understanding RIS in steels is to provide suitable information on its role in determining the mechanical behavior under irradiation conditions. The radiation environment is hostile in so many different ways that it is truly difficult to isolate the effect of RIS from others, such as the dislocation and void microstructures generated during irradiation. We can certainly expect the microstructures of irradiated alloys to be strengthened at lower temperatures, with concomitant decreases in ductility. We might also expect a reduction in creep strength at operating temperatures due to the large numbers of point defects generated by radiation damage. RIS can be expected to compromise mechanical properties that depend on the local chemistry of grain boundaries; intergranular fracture and the susceptibility to stress-corrosion cracking come immediately to mind. Despite the inferences and implications of RIS it is difficult to isolate its role. For example, Busby et al. (2002) attempted to do just that in an austenitic stainless steel subjected to proton irradiation and post-irradiation annealing experiments. However, they were not able to correlate the

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stresscorrosion cracking behavior with either RIS or other radiation-induced features of the microstructure. Lu et al. (2006) nave demonstrated that the RIS of phosphorus during neutron irradiation of a reactor pressure vessel steel has the deleterious effect of raising its ductile-to-brittle transition temperature (DBTT). The change in the DBTT with neutron fluence is shown in Figure 19, where it is seen that the DBTT increases by more than 250°C at high doses. The implication is that the room-temperature fracture toughness of the steel is significantly compromised by the RIS of phosphorus to grain boundaries. I am not aware of any other published investigations in which a correlation between mechanical behavior and RIS has been definitively established. Nevertheless, for the variety of reasons noted above RIS should be an important factor contributing to mechanical properties of alloys under irradiation.

Figure 19. Data of Lu et al. (2006) on the dependence of the change of the ductile-brittle transition temperature, ∆DBTT, with dose in a neutron irradiated pressure vessel steel. The solid curve is the behavior predicted by the empirical theory of the authors that includes the effect of phosphorus segregation to grain boundaries.

References Allen, T. R., and Was, G. S., 1998, Modeling radiation-induced segregation in austenitic FeCr-Ni alloys, Acta Mater. 46:3679-3691. Allen, T. R., Tan, L., Was, G. S., and Kenik, E. A., 2007, Thermal and radiation-induced segregation in model Ni-base alloys, J. Nucl. Mater. 361:174-183. Anthony, T. R., 1969, Solute segregation in vacancy gradients generated by sintering and temperature changes, Acta Metall. 17:603-609.

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Ardell, A. J., 1994, Measurement of solubility limits from data on precipitate coarsening, in: Experimental Methods of Phase Diagram Determination, J. E. Morral, R. S. Schiffman S. M. Merchant eds., TMS Warrendale, PA, pp. 57-66. Ardell, A. J., and Janghorban, K., 1983, Irradiation-Induced Phase Transformations in Binary Ni and Pd Base Alloys, in: Phase Transformation During Irradiation*, F. N. Nolfi Jr ed., Applied Sci., London. Aust, K. T., Hanneman, R. E., Niessen, P., and Westbrook, J. H., 1968, Solute induced hardening near grain boundaries in zone refined metals, Acta Metall. 16:291-302. Barbu, A., 1979, Précipitation induite et gonflement dans la solution solide sous-saturée Ni 6% at Ge irradiée aux electrons de 1 MeV, in: Comportement sous irradiation des materiaux métalliques et des coeurs des réacteurs rapides, J. P. Poirier, J. M. Dupouy eds., Commissariat à l’Energie Atomique, Gif-sur Yvette, pp. 65-73. Barbu, A., 1980, Expressions microscopiques des flux de solute et d’interstitiels dans des gradients de concentration de solute et d’interstitiels pour un crystal de structure cfc, Acta Metall. 28:499-506. Barbu, A., and Ardell, A. J., 1975, Irradiation-induced precipitation in Ni-Si alloys, Scripta Metall. 9:1233-1237. Busby, J. T., Was, G. S., Bruemmer, S. M., Edwards, D. J., and Kenik, E. A., 1999, Influence of initial grain boundary composition on the evolution of radiation-induced segregation profiles, MRS Symp Proc. 540, pp. 451-456. Cauvin, R., and Martin, G., 1981, Solid solutions under irradiation. I. A model for radiationinduced metastability, Phys Rev B 23:3322-3332. Duh, T. S., Kai, J. J., and Chen, F. R., 2000, Effects of grain boundary misorientation on solute segregation in thermally sensitized and proton-irradiated 304 stainless steel, J. Nucl. Mater. 283-287:198-204. Duh, T. S., Kai, J. J., Chen, F. R., and Wang, L. H., 2001, Numerical simulation modeling on the effects of grain boundary misorientation on radiation-induced solute segregation in 304 austenitic stainless steels, J. Nucl. Mater. 294:267-273. Faulkner. R. G., 1981, Non-equilibrium gain-boundary segregation in austenitic alloys, J. Mater. Sci. 16:373-383. Faulkner, R. G., 1996, Segregation to boundaries and interfaces in solids, Int. Mater. Rev. 41:198-208. Faulkner, R. G., Waite, N. C., Little, E. A., and Morgan, T. S., 1993, Radiation-induced grain boundary segregation in dilute alloys, Mater. Sci. Eng. A171:241-248. Fournier, L., Sencer, B. H., Was, G. S., Simonen, E. P., Bruemmer, S. M., 2003, The influence of oversized solute additions on radiation-induced changes and post-irradiation intergranular stress corrosion cracking behavior in high-purity 316 stainless steels, J. Nucl. Mater. 321:192-209. Goodwin, C. C., Faulkner, R. G., and Fisher, S. B., 1999, The modelling of combined radiation-induced and thermal non-equilibrium segregation of chromium in neutronirradiated austenitic stainless steels, in: Effects of radiation on materials, R. K. Nanstad, M. L. Hamilton, F. A. Farner, S. A. Kumar eds., 18th international symposium, ASTM STP 1325, pp. 634-646. Grandjean, Y., Bellon, P., and Martin, G., 1994, Kinetic model for equilibrium and nonequilibrium segregation in concentrated alloys under irradiation, Phys. Rev. B 50:4228-4231. Gupta, R. P., and Lam, N. Q., 1979, Defect-solute interactions and radiation-induced segregation in Ni(Si) and Ni(Ge) alloys, Scripta Metall. 13:1005-1010. Howard, R. E., and Lidiard, A.B., 1965, The annealing of vacancies in dilute alloys, Phil. Mag. 11:1179-1187.

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RESEARCH AND DEVELOPMENT OF OXIDE DISPERSION STRENGTHENED FERRITIC STEELS FOR SODIUM COOLED FAST BREEDER REACTOR FUELS M. INOUE*, T. KAITO, S. OHTSUKA Advanced Nuclear System Research and Development Directorate, Japan Atomic Energy Agency (JAEA) 4002 Narita-cho, Oarai-machi, Higashi-ibaraki-gun, Ibarakiken, Zip code 311-1393, JAPAN

Abstract: Oxide dispersion strengthened (ODS) ferritic steels is eligible for fuel pin cladding tubes to endure heavy displacement damages to 250 dpa at high temperatures up to 973 K in commercialized sodium cooled fast breeder reactor cores. Japan Atomic Energy Agency has been developing ODS steels since 1987. For ODS steels, powder metallurgy process including mechanical alloying and hot consolidation produces highly stable and fine oxide dispersoids in matrix, and provides excellent dispersion hardening. Although cold rolling process in tube manufacturing tends to cause anisotropic grain growth, alpha to gamma phase transformation for the 9Cr-ODS steel and recrystallization for the 12Cr-ODS steel can modify grain morphology and improve creep rupture strength under internal pressure. Mechanical properties of the tubes have been extensively tested in air and stagnant sodium environments to establish Material Strength Standard for fuel pin mechanical design. Also, hundreds of specimens have been irradiated in the experimental fast reactor JOYO to investtigate irradiation effect on dimensional stability and mechanical properties. Pressurized resistance welding technology has been applied to join a tube and end plugs. Both 9Cr- and 12Cr-ODS steel tubes were assembled into fuel pins, and have been irradiated since 2003 under the collaborative program in the BOR-60 in Russia.

1. Fast Reactor Cycle Development in Japan (Sagayama 2007) Last year (2006) was an epoch-making year for fast reactor cycle development in Japan. Fast reactor cycle technology was selected as one of the key technology of national importance by Council for Science and Technology Policy in March. Research and development (R&D) plan for fast breeder reactor cycle was issued by Ministry of Education, Culture, Science and Technology in November, and the plan targets to introduce a

______ *

Tel: +81-29-267-4141 ext.6821, Fax: +81-29-267-7173 E-mail: [email protected], [email protected], [email protected]

V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

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demonstration fast breeder rector (DFBR) by around 2025 and to commercialize fast reactor cycle before 2050. Feasibility Study (FS) Phase II (JFY 2001-2005) on Commercialized Fast Reactor Cycle Systems had been carried out by Japan Atomic Energy Agency (JAEA) and Japan Atomic Power Company, and final reports of FS were compiled in March. The reports concluded that a combination of “sodium cooled fast reactor (SFR) with mixed oxide (MOX) fuel”, “advanced aqueous reprocessing” and “simplified pelletizing fuel fabrication” is the most promising concept to commercialize fast reactor cycle. Fast reactor cycle technology development (FaCT) project succeeded to FS in October. In FaCT project, candidates of innovative technology for commercialized fast reactor cycle will be screened by 2010, and then, eligible technologies will be furthermore developed and demonstrated by 2025. 2. Introduction to ODS Steels Oxide dispersion strengthened (ODS) ferritic steels, which is one of the innovative technologies in FaCT project, is the most promising material for fuel pin cladding tubes in commercialized SFR cores. JAEA has been developing ODS steels for the tubes since 1987. In design studies for large and medium scale SFR cores (JAEA 2006, Kaito et al. 2005, 2007), discharge average burnup is targeted to be 150 GWd/t and peak burnup to be 250 GWd/t. The tubes need to endure heavy displacement damages to 250 dpa. In order to improve thermal efficiency of SFR plants, maximum outlet coolant temperature at reactor vessel is set to 823 K, and maximum (hot-spot) temperature of the tubes corresponds to be 973 K. Internal pressure increases with burnup due to noble gas accumulation, and approaches to be about 12 MPa in the highest burnup fuel pin after nine years service. Mechanical properties of ODS steels are targeted to be over 300 MPa at 973 K for ultimate tensile strength (UTS), over 1% for uniform elongation (UE) and 120 MPa at 973 K for 10,000 hr for creep rupture strength against internal pressures. Historically, modified type SUS316 austenitic stainless steel named as PNC316 has been applied for fuel pin cladding tube, spacer wire and hexagonal duct in driver fuel pins or subassemblies in the experimental fast reactor JOYO MK-II core and in the prototype fast breeder reactor MONJU (Tateishi et al. 1988). In conventional austenitic stainless steels such as PNC316, detrimental void swelling will emerge over 120 dpa. Since body centered cubic structure is more resistant against displacement damages than face centered cubic structure, ferritic steels are indispensable for the tubes. Precipitation hardened (PH) high strength ferritic/martensitic steel

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named as PNC-FMS has been developed since 1983 in JAEA (Shikakura et al. 1991). However, PH ferritic steels such as PNC-FMS will rapidly lose their strength over 923 K. In contrast, dispersion hardening by thermally stable oxide particles can be effective even over 923 K. This paper reviews our state-of-the-art technology for ODS steels in JAEA, and consists of following six technical paragraphs: “Alloy Design” concepts based on strengthening (hardening) mechanism in paragraph 3, “Manufacturing Process” which combines powder metallurgy and seamless precision tube production processes in paragraph 4, “Microstructure Control” to grain morphology during cold rolling process in paragraph 5, “Mechanical Property” with environmental effects in paragraph 6, “Irradiation Tests” in paragraph 7 and “Current Status and Project Schedule” in paragraph 8. 3. Alloy Design In JAEA two types of ODS steels have been selected as the candidates for the tubes. One is martensitic 9Cr-ODS steel for higher radiation resistance and the other is fully ferritic 12Cr-ODS steel for higher corrosion resistance (Shikakura et al. 1991). We have prioritized the 9Cr-ODS steel as the primary and the 12Cr-ODS steel as the secondary. Chemical composition of the 9Cr-ODS steel is Fe-0.13C-9Cr-2W-0.20Ti-0.35Y2O3 in mass%, and that of the 12Cr-ODS steels is Fe-0.03C-12Cr-2W-0.26Ti-0.23Y2O3 in mass%. Chemical compositions of the ODS steels are mainly designed to maximize high temperature strength by dispersion and solution hardening mechanism (Ukai et al. 2002, 2003, 2006, Otsuka et al. 2004, 2005a, 2005b, 2007). The number of alloying elements in the ODS steels are much less than PH ferritic ones as listed in table 1. In practice, carbon, chromium, tungsten, titanium, yttrium and oxygen are defined to be alloying elements. Silicon, manganese, phosphorus, sulfur, nickel, nitrogen and argon are defined as impurity elements. Carbon and chromium contents are defined to obtain alpha to gamma (martensitic) transformation for the 9Cr-ODS steel or fully ferritic for the 12Cr-ODS steel. In general, martensitic grains contain high density dislocations and large areas of packet and block boundary, which are useful to absorb point defects and provide radiation resistance. Experiences in PNC-FMS alloy designing recommends that chromium content in the 12CrODS steel should be less than 12% to prevent from embrittlement by chromium rich phase precipitation.

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TABLE 1. Alloying Elements for the 9Cr- and 12Cr-ODS steels in mass%. Ex.O = Total Oxygen content – 48/178 Yttrium content. The final heat treatments are: normalizing at 1,323 K for 1 hour + Tempering at 1,053-1,073 K for 1 hour (9Cr-ODS) and annealing at 1,423 K for ~1 hour (12Cr-ODS). 9Cr-ODS Target

C

Cr

W

Ti

Y2O3

Ex. O

0.13

9.0

2.0

0.20

0.35

0.07

Specification 0.11 - 0.15

8.6 - 9.4

1.8 - 2.2 0.20 - 0.35 0.33 - 0.37 0.04 - 0.10

12CR-ODS Target

0.03

12.0

2.0

0.26

0.23

0.07

Specification 0.01 - 0.05 11.6 - 12.4 1.8 - 2.2 0.24 - 0.28 0.20 - 0.26 0.05 - 0.09

Tungsten contents are optimized to balance solution hardening effect against embrittlement by laves phase precipitation. Titanium, yttrium and oxygen are optimized to maximize the high temperature strength with microstructure controllability. The ODS steels always contain “excess oxygen” which originates from contamination during manufacturing process (mechanical alloying and following degassing). The amount of oxygen from contamination and content in yttrium sesquioxide (yttria: Y2O3) are comparable in the final products. An appropriate amount of titanium, excess oxygen with yttria is essential for uniform and fine distribution to enhance dispersion hardening effect and, in particular for the 12Cr-ODS steel, to ensure recrystallization. It is noteworthy here that the 9Cr-ODS steel consists of duplex phases: one is martensite and the other is delta-ferrite. Phase diagrams by thermodynamical prediction indicate that the 9Cr-ODS steel should be fully martensitic. At present, the reason for the origin of delta ferrite is not clarified. 4. Manufacturing Process Powder metallurgy and seamless precision tube production processes are combined to manufacture the ODS steel tubes as summarized in Figure 1. Powder metallurgy process includes mechanical alloying, canning, degassing and hot-consolidation to produce mother tubes. The mother tubes are supplied to seamless precision tube production process which includes cold rolling, various heat treatments and inspections. Manufacturing process of the latest 9Cr-ODS steel tube production for a JOYO fuel pin irradiation test is described below in detail. Argon gas atomized pre-alloyed steel powders (10 kg) and yttria particulates are mechanically alloyed by use of an attrition type ball mill (Attritor) in high purity argon gas atmosphere. Mechanically alloyed powders are sieved, filled into mild steel cans, degassed at 673 K, and then hot-extruded at

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1,423 K into fully dense billets. The billets are sectioned, machined and drilled into mother tubes: 18 mm in outer diameter, 12 mm in inner diameter and 180 mm in length. The mother tubes are cold-rolled by 4 times with a reduction rate (area) of about 50% in each. The size of final products is 8.5 mm in diameter, 7.5 mm in inner diameter and 1,850 mm in length. In practice, one mother tube can be one cladding tube (~300 g).

Mild Steel Can Ferritic Steel Powder Mechanically Alloyed powder

Y2O2 Powder

Hot-Extrusion Mother Tubes (OD18mm 3 ID12mm 3 L180mm)

Mechanical Alloying (10kg Attritor)

4 times repeat

Cold-rolling (Rd = 50%)

Inspection Chemical composition Visual inspection Dimension Ultrasonic inspection Mechanical properties

Heat-Treatments

Cladding Tubes (OD8.5mm 3 ID7.5mm 3 L1,850mm)

Figure 1. Current manufacturing process for fuel pin cladding tubes of the ODS

steels.

Severe and repeated plastic deformation during high energy ball milling, yttria eventually dissolves into ferritic steel matrix. During hot-extrusion process, titanium, yttrium and oxygen precipitate into complex oxide particles and finely disperse in a few to several nanometer scales. In other words, powder metallurgy process controls oxide particle size distribution, subsequent dispersion hardening effect and microstructure controllability in the tube production process. The ODS steel tubes can be cold-rolled by Pilger type equipments which is the same as Zircalloy tube manufacturing process for water reactor fuel rods. 5. Microstructure Control In order to manufacture the tubes, four times or more cold rolling passes is necessary. Cold rolling process always makes grain morphology anisotropic (elongated to rolling direction); in extreme, it looks like “bamboo” as

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shown in Figure 2a. In case that grain morphology is very anisotropic in a tube, creep rupture strength under axial stress becomes much higher than that under hoop stress. At elevated temperatures grain boundary sliding easily occurs and the extent of the sliding increases with decreasing grain sizes. Therefore, the preferential sliding contributes to deform much more under hoop stress than under axial stress, and results in strength anisotropy. In addition, grain boundary sliding at lower temperature around 673 K tends to significantly degrade its ductility under hoop stress since cavities induced by grain boundary sliding will not be easily accommodated by diffusion assisted grain deformations. Grain morphology of the ODS steels should be equiaxed to suppress detrimental contribution of the sliding. In practice, to eliminate “bamboo” like grains and to obtain equiaxed ones, alpha to gamma transformation is available for the 9Cr-ODS steel and recrystallization is desirable for the 12Cr-ODS steel as shown in Figure 2b and 2c. 9Cr-ODS Steel Significant Grain Morphology Anisotropy without Microstructure Control

α / γ phase transformation (b)

200 µm

12Cr-ODS Steel

Recrystallization (c) (a)

1 µm 200 µm

Figure 2. Grain morphology anisotropy and improvements by alpha to gamma

transformation for the 9Cr-ODS steel and recrystallization for the 12Cr-ODS steel.

Hardness of the tubes is enhanced by cold rolling process due to work hardening. If appropriate heat treatment to soften, in other words; reduce the hardness, is not combined, it is impossible to continue further cold rolling. Grain morphology of the tubes should be controlled simultaneously with the hardness. For the 9Cr-ODS steel, alpha to gamma transformation is reversible when heated at 1,323 K far beyond Ac1 point. We can apply the transformation for softening and hardening methods by changing cooling rates after heating. Before cold rolling, slower cooling rate (furnace cooling) in

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the intermediate heat treatments can prevent from martensitic transformation and then soften as shown in Figure 3. In the final heat treatment, martensitic transformation is introduced by faster cooling rate (air cooling) to produce desired mechanical performance. During alpha to gamma transformation, elongated grains will be restructured to equiaxed grains and grain morphology anisotropy disappears. T (C) 1100 900 18000 ? /h

700

α +γ

γ

α

500

Cold-rolling (RD = 50%)

Hardness (Hv)

450

1st

2nd

3rd

4th

100

400

101

350

300

300

1st

2nd

Mother tube

10508C 3 30 min - FC (150 8C/h)

α + m

m 1 02

103

104

105

Time from 1000C (s)

Intermediate HT Softened ferrite

3rd

Intermediate Heat treatment

γ + m

Final HT Martensite

Final Heat treatment 1050C360 min - AC + 8008C360 min - AC Tempered Martensite

As cold-rolled

ferrite After heat treated

20µm

Figure 3. Microstructure control by alpha to gamma.

In practice, control of recrystallization for the 12Cr-ODS steel is much more difficult than that of alpha to gamma transformation for the 9Cr-ODS steel, because recrystallization is not reversible. Although recrystallization is favorable to soften before cold rolling, recrystallization must be induced in the final heat treatment, because recrystallization in intermediate heat treatments prevent from further recrystallization in the final heat treatment. To prevent from premature recrystallization and subsequent strength anisotropy in final products, one-step annealing is not enough and two-step annealing in the intermediate heat treatments is invented as shown in Figure 4 (Narita et al. 2004).

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Figure 4. Two-step annealing to suppress premature recrystallization for the 12Cr-

ODS steel.

6. Mechanical Properties Mechanical properties of the ODS steels are systematically examined, correlated and compiled into Material Strength Standards (MSSs) to apply to fuel pin mechanical design. In practice, the tubes will deform under internal pressure due to mechanical interaction with fuel pellets and noble gas accumulation with burnup. Therefore, tensile and creep rupture strength under hoop stress are major concern. To predict plastic strains, stress-strain and creep curves are correlated. In addition, both out-of-pile and in-pile tests characterize environmental effect of exposure to fuel and flowing sodium on mechanical properties and on dimensional stability such as void swelling and irradiation creep. Ring specimens wire-cut from the tubes are prepared to examine tensile properties under hoop stress. UTS and UE of the ODS steels are plotted in Figure 5 in comparison with PNC-FMS and PNC316. The ODS steels exhibit much higher strength than PNC-FMS over the entire temperature range. Although UE of PNC-FMS tends to decrease with increasing temperature, that of the ODS steels is adequately stable and increases over 773 K. Larger UE in the ODS steels than in PNC-FMS will arise from the retardation of localized recovery and continuing work-hardening due to pining the dislocation by dispersoids. Creep rupture strength under internal pressure for the ODS steels is plotted in Figure 6, and compared with PNC-FMS and PNC316. The creep rupture strength of both 9Cr- and 12Cr-ODS steels are comparable with each other and meets the target of 120 MPa for 10,000 hr at 973 K. This

319

1500

15 9Cr-ODS 12Cr-ODS

1000 PNC316

500

0 200

PNC-FMS

400

600

800

1000

Temperature (K)

1200

Uniform Elongation (%)

Ultimate Tensile Strength (MPa)

ODS FERRITIC STEELS 9Cr-ODS 12Cr-ODS 10 PNC316 PNC-FMS

5

0 200

400

600

800

1000

1200

Temperature (K)

Figure 5. Out-of-pile tensile properties tested by ring specimens for the 9Cr- and 12Cr-ODS steels.

Figure 6. Out-of-pile creep rupture strength under internal pressure for the 9Cr- and

12Cr-ODS steels.

strength level is far beyond that of PNC-FMS and superior to PNC316 in lower stress than 120 MPa and longer rupture time than 10,000 hr. Tensile tests after stagnant sodium immersion revealed that UTS of the 9Cr-ODS steel decreases with increasing the duration of the immersion. In contrast, UTS of the 12Cr-ODS steel showed little change after the immersion. Above 873 K, PNC-FMS clearly exhibits a strength reduction due to decarburization during the immersion. Even though decarburization is observed in the ODS steels, its effect on mechanical properties is not so significant (Yoshida and Kato 2004). This indicates that the oxide

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dispersoids in the ODS steels are chemically stable and then its strengthening effect remains unchanged ever after the long term immersion (Yamashita et al. 2004). Creep rupture properties have been tested in stagnant sodium immersion with internally pressurized tubular specimen, and, as shown in Figure 7, there is no sodium environmental effect on creep rupture strength at 973 K. He gas detector Hoop stress (MPa)

Internal creep rupture specimen (Pressurized tube)

He 60 mm gas

Na outlet

500

(*) Sodium flow rate : <0.001m/s

300

100 80 60

923K 973K in Air in Sodium

40 101

102

103

104

105

Time to rupture (h)

Na inlet into testing vessel

Figure 7. The effect of stagnant sodium environment on creep rupture strength 1200

10

1000

8

Uniform elongation (%)

Engineering stress (MPa)

under internal pressure for the 9Cr-ODS.

800 600 400

9Cr-ODS (Unirrad.) 9Cr-ODS (3dpa) 12Cr-ODS (Unirrad.) 12Cr-ODS (3dpa)

200 0

0

5

10

15

20

25

Strain (%)

30

9Cr-ODS 12Cr-ODS 12Cr-ODS (Unirrad.)

6 4 2 0 650

9Cr-ODS (Unirrad.) 3dpa

14dpa

700

15dpa

750

7dpa

800

850

Temperature (K)

Figure 8. Stress-Strain Curves and Uniform Elongation after JOYO Irradiation

for the 9Cr- and 12Cr-ODS Steels.

Irradiation effect on tensile properties for the ODS steels were investigated by use of the material specimen irradiation rig at temperatures between 670 and 807 K up to 15 dpa in JOYO MK-II core (Yoshita et al. 2004). As shown in Figure 8, the irradiation effect on tensile properties is not distinguishable. Based on both out-of-pile and in-pile tests, the MSS for JOYO fuel pin irradiation test was tentatively compiled in 2005. The longer term creep

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rupture tests with pressurized tube specimens in air and stagnant sodium will be applied to upgrade the MSS. 7. Irradiation Tests After reviewing the out-of-pile experiments, irradiation performance of the ODS steels are examined by the following three steps. The first step is material specimen irradiation tests to investigate the effect of radiation (displacement) damages on mechanical properties and dimensional stability. The second step is fuel pin irradiation tests to investigate the effect of exposure to fuel and flowing sodium in addition to radiation damages. Material specimens are usually irradiated in nearly stagnant sodium, and temperature fluctuation in fuel pin irradiations is much larger than in material specimen irradiations. The third step is subassembly irradiation tests to demonstrate high burnup capability and to assure its integrity with statistically enough number of fuel pins. The MSS will be upgraded by a series of irradiation tests to design ODS clad driver fuel pins in MONJU and DFBR fuels. To investigate the irradiation effect over 200 dpa by 2015, several lots of the ODS steels in material specimens have been irradiated by use of Core Material Irradiation Rigs (CMIRs) in JOYO. Also, in-pile creep rupture tests with pressurized tube specimens were carried out by use of Material Testing Rigs with Temperature Control (MARICOs) in JOYO. Fuel pins clad with both 9Cr- and 12Cr-ODS steels and loaded with the vibro-packed granulated mixed oxide (MOX) fuels and uranium metal getter particles have been irradiated in BOR-60 since 2003 under the collaborative program between JAEA and Research Institute of Atomic Reactors (RIAR) (Ukai et al. 2005). The objectives of the program are to irradiate ODS steel clad fuel pins up to 150 GWd/t, 75 dpa and 973 K and to evaluate their chemical compatibility with high burnup fuel by 2010. The first irradiation tests started in June 2003 and completed in May 2004 in which maximum cladding mid-wall temperatures were 943 and 993 K, and peak burnup of 50 GWd/t and peak neutron dose of 21 dpa were achieved without fuel pin failure. To attain the cumulative creep damage fraction (CDF) value similar to end of life condition, plenum length of the fuel pins was shortened for the first irradiation test. Two more subassemblies have been irradiated since August 2005; one targets 100 GWd/t and 50 dpa by JFY2008 and the other 150 GWd/t and 75 dpa by JFY2009. Six fuel pins clad with the 9Cr-ODS steel will be irradiated up to 180 GWd/t and 210 dpa by 2015 in JOYO MK-III core after 2008. The irradiation test condition and schedule in JOYO are summarized in Figure 9.

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CY 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 JOYO Reactor Operation

Material Specimens Irradiations

3-6

7-11

CMIR 6 40 dpa off-Line

CMIR 7 90 dpa off-Line

MARICO 2

30 dpa

17-21

22-26

27-31

CMIR CMIR CMIR CMIR 8 9 10 11 140 dpa 190 dpa 240 dpa 270 dpa off-Line

MARICO 3

50 dpa Pin (B12) 50 dpa 40 GWd/t

Fuel Pins Irradiations

12-16

MARICO 4

90 dpa Pin Pin (B12) (B12) 100 dpa 150 dpa 90 GWd/t 130 GWd/t

Subassembly Demonstrations

Pin (B12) 250 dpa 220 GWd/t

Bundle (CX) <Scheduled at beginning of FaCT project>

Figure 9. Schedule of JOYO irradiation tests.

For joining a tube to end-plugs in fuel pin assembling, JAEA has developed Pressurized Resistance Welding (PRW) technology, which is based on electrical resistance heating across the interface under sufficient continuous contact force (Seki et al. 2004). Instantaneous and forging process expels liquid weld and obtains metallurgical bonded weld with the original microstructure. The existing fusion welding methods such as TIG welding significantly degrades mechanical properties around the weld, because large size of blowholes and coarsened dispersoids will emerge. Electric current, voltage and contact force have been selected to weld reproducibly. For the welded specimens, tensile, burst and creep rupture tests were conducted to assure its integrity and quality. In addition, an ultrasonic inspection method has been developed to non-destructively examine the weld. 8. Current Status and Project Schedule 8.1. CURRENT STATUS

Since 1987, JAEA has been developing the ODS steels, and recently selected two promising candidates. One is martensitic 9Cr-ODS steel and the other is fully ferritic 12Cr-ODS steel. Alloying tungsten up to 2 mass% offers solution hardening with little detrimental laves compound precipitations. Titanium and Y2O3 addition under controlled oxygen contamination

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enables to uniformly disperse a few to several nanometer size complex oxides and enhances dispersion strengthening effect. Powder metallurgy and seamless precision tube production process are combined to manufacture the tubes. Powder metallurgy process controls oxide particle size distribution. Since anisotropic grain growth degrades high temperature strength against hoop stress, alpha to gamma phase transformation for the 9Cr-ODS steel and recrystallization for the 12Cr-ODS steel are applied to control grain morphology. Appropriately combining intermediate or final heat treatment temperatures with cooling rates can either soften or harden matrix in Pilger cold-rolling process. Mechanical properties of the tubes have been extensively tested in air and stagnant sodium environments to establish MSS for fuel pin mechanical design. Hundreds of specimens have been irradiated in JOYO to investigate the effect of radiation damages on mechanical properties and dimensional stability. Post-irradiation examinations for tensile specimens exposed up to 15 dpa revealed that no effect on UE is observed and that there are slight increases in proof strength and UTS of both 9Cr- and 12Cr-ODS steels. Both 9Cr- and 12Cr-ODS steel clad fuel pins with vibro-packed MOX fuel have been irradiated since 2003 under the collaborative program between JAEA and RIAR in BOR-60. PRW technology has been developed to join a tube to end-plugs. 8.2. PROJECT SCHEDULE

In FS Phase II, we prioritized that the 9Cr-ODS steel is the primary candidate and the 12Cr-ODS steel is the secondary, and scheduled long term R&D plan as shown in Figure 10 (Kaito et al. 2007). Manufacturing process for mass production to supply annually 10,000 tubes for future SFR fuels will be developed only for the 9Cr-ODS steel by 2010, where scale-up from laboratory to engineering in mechanical alloying and hot-consolidation processes will be the most important issues. MSS will be upgraded for both 9Cr- and 12Cr-ODS steels by a series of material specimen and fuel pin irradiation tests in BOR-60 and JOYO until 2015. Six fuel pins, which are clad with the 9Cr-ODS steel and loaded with annular MOX fuel pellets, will be irradiated to demonstrate high burnup capability in the JOYO after 2008; target dose is 210 dpa with peak burnup: 180 GWd/t by 2015. The upgraded MSS will be applied to design ODS clad driver fuel pins in MONJU and DFBR fuels.

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M. INOUE, T. KAITO AND S. OHTSUKA

2005

2010 Technology Development

- Technology development with large scale manufacturing - Quality assurance & control - Irradiation test (~150GWd/t, ~100dpa) (Fuel pins demonstration)

2015

2020

2025

Engineering Demonstration

- Engineering scale manufacturing with commercial technology - Irradiation test - (~180GW d/t, ~240dpa) - Fuel pin and subassembly demonstration

973 K,150 GWd/t, 75 dpa BOR-60 / Fuel pin irradiation

973 K,_250 GWd/t, 250 dpa

JOYO / Material & fuel pin irradiation

JOYO or MONJU / Subassembly irradiation <Evaluate and Select Innovative Technologies by 2010> - ODS clad fuel pin data at 150GWd/t and evaluation of high burnup capability <Establish Innovative Technologies by 2015> - ODS clad fuel pin data at 250GWd/t and fuel pin bundle irradiation at medium burnup - Fuel pin bundle/subassembly demonstration in MONJU

MONJU Driver Fuel

D-FBR

Figure 10. Project schedule.

Acknowledgments R&D of the ODS steels has been conducted with many staff of JAEA. We wish to acknowledge Professor Shigeharu UKAI of Hokkaido University, Dr. Masayuki FUJIWARA and Dr. Takanari OKUDA of Kobelco Research Institute, Inc. and Mr. Toshimi KOBAYASHI of Sumitomo Metal Technology, Inc. for their grateful contributions.

References Bottcher, J., Ukai, S., and Inoue, M., 2002, ODS steel clad MOX fuel-pin fabrication and irradiation performance in EBR-II, Nucl. Technol. 138(3): 238-245. Kaito, T., Ukai, S., Ohtsuka, S., and Narita, T., 2005, Paper No.169, GLOBAL2005, October 9-13, 2005, Tsukuba, Japan. Kaito, T., Ohtsuka, S., and Inoue, M., 2007, Paper No.005-175716, GLOBAL2007, September 9-13, 2007, Idaho, USA. Narita, T., Ukai, S., Kaito, T., Ohtsuka, S., and Kobayashi, T., 2004, Development of twostep softening heat treatment for manufacturing 12Cr-ODS ferritic steel tubes, J. Nucl. Sci. Technol. 41(10):1008-1012. Narita, T., Ukai, S., Kaito, T., Ohtsuka, S., and Fujiwara, M., 2005, Development of manufacturing process of PNC-FMS wrapper tube with SUS316 short joint, J. Nucl. Sci. Technol. 42(9):825-832.

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Ohtsuka S., Ukai S., Fujiwara, M., Kaito, T., and Narita, T., 2004, Improvement of 9Cr-ODS martensitic steel properties by controlling excess oxygen and titanium contents, J. Nucl. Mater. 329-333(1):372-376. Ohtsuka, S., Ukai, S., Sakasegawa, H., Fujiwara, M., Kaito, T., and Narita, T., 2005, Improvement of creep strength of 9Cr-ODS martensitic steel by controlling excess oxygen and titanium concentrations, Mater. Trans. 46(3):487-492. Ohtsuka, S., Ukai, S., Fujiwara, M., Kaito, T., and Narita, T., 2005, Nano-structure control in ODS martensitic steel by means of selecting titanium and oxygen contents, J. Phys. Chem. Solids. 66(2-4):571-575. Ohtsuka, S., Ukai, S., Sakasegawa, H., Fujiwara, M., Kaito, T., and Narita, T., 2007, Nanomesoscopic structural characterization of 9Cr-ODS martensitic steel for improving creep strength, J. Nucl. Mater. 367-370(1):160-165. Seki, M., Hirako, K., Kono, S., Kihara, Y., Kaito, T., and Ukai, S., 2004, Pressurized resistance welding technology development in 9Cr-ODS martensitic steels, J. Nucl. Mater. 329-333(2):1534-1538. Ukai, S., Mizuta, S., Fujiwara, M., Okuda, T., and Kobayashi, T., 2002, Development of 9CR-ODS martensitic steel claddings for fuel pin by means of ferrite to austenite phase transformation, J. Nucl. Sci. Technol. 39(7):778-788. Ukai, S., Hatakeyama, H., Mizuta, S., Fujiwara, M., and Okuda, T., 2002, Consolidation process study of 9Cr-ODS martensitic steels, J. Nucl. Mater. 307-311(1):758-762. Ukai, S., Kaito, T., Ohtsuka, S., Narita, T., Fujiwara, M., and Kobayashi, T., 2003, ISIJ International 43(12):2038-2045. Ukai, S., Kaito, T., Seki, M., Mayorshin, A. A., and Shishalov, O. V., 2005, Oxide dispersion strengthened (ODS) fuel pin fabrication for BOR-60 irradiation test, J. Nucl. Sci. Technol. 42(1):109-122. Ukai, S., Kaito, T., Ohtsuka, S., Narita, T., and Sakasegawa, H., 2006, ANS Annual Meeting, June 4-8, Reno, USA, Trans. Am. Nucl. Soc. 94:786-787. Yamashita, S., Akasaka, N., and Ohnuki, S., 2004, Nano-oxide particle stability of 9-12Cr grain morphology modified ODS steels under neutron irradiation, J. Nucl. Mater. 329333(1):377-381. Yoshitake, T., Abe, Y., Akasaka, N., Ohtsuka, S., Ukai, S., and Kimura, A., 2004, Ringtensile properties of irradiated oxide dispersion strengthened ferritic/martensitic steel claddings, J. Nucl. Mater 329-333(1):342-346. Yoshida, E., and Kato, S., 2004, Sodium compatibility of ODS steel at elevated temperature, J. Nucl. Mater. 329-333(2): 1393-1397.

SOME ASPECTS OF THE STRUCTURAL, MECHANICAL, AND ELECTRONIC PROPERTIES OF SIC P. PIROUZ Department of Materials Science and Engineering, Case Western Reserve University, Cleveland, OH 44106, U.S.A.

Abstract: Some of the most important challenges in nuclear reactors involve materials issues. These include the material employed in the reactor walls, the cladding material for the fissile nuclear fuel, and the material used for retaining the fission products. In addition to long term resistance to irradiation, the materials used in a nuclear reactor must have excellent mechanical and corrosion properties in order to tolerate the thermal shocks as well as the high temperature and severe corrosive environments that they will encounter during operation. One of the materials that has long been considered for this purpose is SiC, a material that is used both for its ceramic characteristics, as well as its wide bandgap semiconducting properties. In this paper, after reviewing the structure of various SiC polytypes, we discuss recent results on the mechanical and fracture properties of single polytype, single crystal 4H-SiC that has recently become available. We also discuss one of the many cases where lattice defects in the crystal affect the electronic properties of this high-temperature semiconductor.

1. Introduction In polycrystalline form, SiC has been used as a ceramic material for a long time. The strong covalent/ionic Si-C bond in SiC gives it high intrinsic strength even at elevated temperatures and makes it a superior structural material. For this reason, many types of composites employ SiC in the form of particulate, whisker or fiber reinforcement in a ductile matrix. More recently, following the pioneering work of Tairov and Tsvetkov [1], commercial growth of bulk single crystalline SiC has become available [2] and consequently electronic devices are being fabricated that utilize the many advantageous electronic properties of this wide bandgap semiconductor. In addition to its superior high-temperature mechanical strength, SiC has very good oxidation properties and excellent long term resistance to irradiation. As a result, SiC is considered for mechanical and electronic applications at high temperatures, in corrosive environments, as well as in nuclear reactors and outer space where radiation can quickly render other materials useless. This paper starts with a short description of the structure of SiC and the various variants (or polytypes) that this crystal can adopt. Following this, V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

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some recent results on the mechanical and fracture properties of SiC are presented. Finally, a specific problem with the use of SiC in a particular electronic device, the PiN diode, is discussed. Investigations of the latter problem have shed light on the energy levels of a few lattice defects in the bandgap of SiC. 2. Structure of SiC We discuss the structure of SiC as an assembly of Si-C tetrahedra which arise out of the tetrahedral bonding between silicon and carbon atoms [3,4]. A SiC4 (or Si4C) tetrahedron is shown in Figure 1a with the silicon atoms at the four corners of the tetrahedron and a carbon atom at its centroid (or vice versa). There are strong covalent-ionic bonds between the carbon and silicon atoms as shown by the thick lines within the tetrahedron. The moderate ionic contribution to the bonding is primarily because of the different sizes of carbon and silicon atoms [5]. We take one of the Si-C bonds - the vertical one in Figure 1a - to be parallel to the c-axis of the SiC, parallel to the [0001] direction of the crystal that is to be formed from an assembly of these tetrahedra. The tetrahedron face opposite the c-axis is the “basal plane”, parallel to the (0001) plane, and the three sides of the basal triangle in Figure 1a are parallel to the 1120 directions of the crystal.

Figure 1. a) A Si-C tetrahedron T with the silicon atoms (open circles) at the corners and the carbon atom (solid circle) at the centroid. The Si-C bonds are shown by bold dark lines and the vertical bond is taken to be parallel to the [0001] c-axis thus making the tetrahedron face normal to it parallel to the (0001) (or basal) plane. The sides of the base triangle are parallel to the 1120 directions, b) Rotation of the tetrahedron around the c-axis results in the “twinned” variant T’.

An equilateral tetrahedron has 3-fold symmetry about any of the lines joining its centroid to its corner atoms, including the c-axis shown in Figure 1a. Thus, if we rotate the tetrahedron about this axis by 60°, symmetry is broken and a new variant is produced (Figure 1b). We call this variant a “twinned tetrahedron” and denote it by T’, as distinct from the unrotated (“normal”) tetrahedron that is denoted by T.

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All SiC polytypes can be envisioned as assemblies of normal and twinned tetrahedra with the topological restriction that the various tetrahedra must be joined at their corners. In order to show the different assemblies of tetrahedra, it is more convenient to project the tetrahedra on a plane and deal with 2-D figures. In Figure 2, we have projected the normal and twinned tetrahedra along one of the 1120 direction, i.e. along one of the sides of the basal triangle. Each tetrahedron projects as an acentric triangle; note that the two parallel bonds in the triangles do not represent a double bond between carbon and silicon atoms but are two single C-Si bonds that overlap when the tetrahedron is projected along any of the 1120 directions.

Figure 2. Projection of the (a) normal and (b) twinned tetrahedra T and T’, respectively, along the ⎡⎣1120 ⎤⎦ direction. The two parallel bonds in (a) and (b) are not Si-C double bonds but result from the overlap of the front and back silicon atoms with the centroid carbon atom in the tetrahedron.

Just as there are three spatial positions A, B, C, when spheres are assembled in a close-packed arrangement, so are there three spatial positions for a tetrahedron in an assembly of tetrahedra. As shown in Figure 3, we call these T1, T2, and T3; in a similar way, there are three spatial positions for the twinned tetrahedra that we denote by T’1, T’2, and T’3. There are topological restrictions when the tetrahedra are to be joined at their corners (Figure 4). Thus, starting with the normal tetrahedron Ti (i = 1,2,3), a tetrahedron of the same variant (i.e. also normal) can sit on Ti if it follows the cyclic sequence, i.e. it can only be Ti+1 (Figure 4a). Correspondingly, starting with the twinned tetrahedron T’i, a tetrahedron of the same variant (i.e. twinned tetrahedron) can sit on it if it follows the anti-cyclic sequence, i.e. it can only be T’i-1 (Figure 4c).

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Figure 3. Three different spatial positions for the (normal) tetrahedron: T1, T2 and T3.

In this way, an assembly of same variant tetrahedra, with a stacking sequence …T1T2T3… (Figure 5) or …T’1T’3T’2…, can form. A tetrahedral assembly that consists of the same variant, i.e., all normal or all twinned, exhibits cubic symmetry and has a periodicity of 3 tetrahedra. In the Ramsdell notation [6], this is known as 3C (3 for the periodicity and “C” for cubic). In the Zhdanov notation [7], the number of consecutive tetrahedra of the same variant are shown; thus the arrangement in Figure 5a is denoted as n0. In most other semiconductors, this arrangement is known as the “sphalerite” or “zincblende” structure. It is also possible for tetrahedra belonging to different variants to stack on top of each other. Thus Ti must be followed by T’i-1, or equivalently, T’i must be followed by Ti+1 (Figure 4b and 4d). In this case, the tetrahedral assembly will be a mixture of normal and twinned tetrahedra and it exhibits hexagonal (H) or rhombohedral (R) symmetry. In the Ramsdell notation, a polytype consisting of i normal tetrahedra and j twinned tetrahedra will either be (i+j)R or (i+j)H. The hexagonal or rhombohedral symmetry of the polytype is determined by the parity of the number (i+j). Thus if (i+j) = even = 2m (m = integer), then the crystal will have hexagonal symmetry and the polytype will have a Ramsdell notation (i+j)H. On the other hand, if (i+j) = odd = 2m+1, then the polytype will be rhombohedral and its Ramsdell notation is (i+j)R. The simplest polytype having mixed tetrahedral variants is …T1T’3… shown in Figure 5b. This is 2H (in the Ramsdell notation) or 11 (in the

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331

Zhdanov notation) and often referred to as the “wurtzite” structure in other materials. In SiC, the 2H polytype is rather rare and whenever it occurs, the crystal transforms quickly to the cubic 3C-SiC phase [8,9].

Figure 4. Examples of allowed (a-d) and forbidden (e-h) stacking rules for tetrahedra. Starting with a normal Ti tetrahedron, either a normal Ti+1 or a twinned T’i-1 tetrahedron can sit on top of Ti. Similarly, starting with a twinned T’i tetrahedron, either a normal Ti-1 or a twinned T’i+1 tetrahedron can sit on top of T’i.

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Figure 5. (next page) a) The tetrahedral arrangement for the cubic 3C-SiC polytype …T1T2 T3… (n0 in the Zhdanov notation), b)The tetrahedral arrangement for the hexagonal 2H-SiC polytype …T1T’3… (11 in the Zhdanov notation), c) The tetrahedral arrangement for the hexagonal 4H-SiC polytype …T1 T2 T’1T’3… (22 in the Zhdanov notation), d) The tetrahedral arrangement for the hexagonal 6H-SiC polytype …T 1 2 T 3 T 2T’ 1T’ 3T’ … (33 in the Zhdanov notation).

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The tetrahedral arrangement of two other common polytypes, 4H-SiC and 6-SiC, are shown in Figures 5c and 5d, respectively. Note in Figure 1 (or Figure 2) that (0001) planes passing through silicon atoms alternate with (0001) planes passing through carbon atoms. Conventionally the (0001) silicon planes are denoted by Roman letters A, B, C and the (0001) carbon planes by Greek letters α, β, γ. Thus, in 3C-SiC (Figure 3a), the stacking of (0001) basal planes is …αAβBγC… and in 4HSiC it is …αAβBαAγC…. Also note in Figure 1 that the height of the tetrahedron with side a is given by:

h=

6 a 3

(assuming this is an equilateral tetrahedron). The (0001) carbon plane, passing through the centroid of the tetrahedron, is at a distance of h/3 from the bottom (0001) silicon plane and 2h/3 from the top (0001) silicon plane, i.e. the tetrahedral height is divided in a ratio 1:2 into a narrowly-spaced SiC (Aβ) and a widely-spaced C-Si double plane (αA). 3. Dislocations in SiC In most crystals the slip plane of dislocations is in between the most widelyspaced planes [10]. Thus, in considering the glide plane of tetrahedrallycoordinated semiconductors (including SiC), it is natural to assume that the slip plane will be between the more widely-spaced αA planes than between the narrowly-spaced Aβ planes This was actually first suggested to be the case by William Shockley [11] about fifty years ago because a dislocation on this slip plane - said to belong to the “shuffle set” [10] - would then cut only one silicon-carbon bond per tetrahedron. On the other hand, a dislocation on a slip plane passing in between the narrowly-spaced Aβ planes – said to belong to the “glide” set [10] – would cut three siliconcarbon bonds per tetrahedron and its formation would presumably be energetically more costly. In Figure 6 we show the motion of atoms by the passage of a partial dislocation on the (0001) slip plane for the two cases. As may be seen, in the case of the slip plane in between the widely spaced (0001) planes (Figure 6a), the motion of a partial dislocation with a Burgers vector:

b=

1 [1 1 00] 3

severely distorts one of the three Si-C bonds. On the other hand, when the slip plane is located in between the narrowly-spaced (0001) planes (Figure 6b), passage of the partial dislocation changes the normal tetrahedron to a twinned one without producing any distortion in the interatomic bonds. Thus,

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Figure 6. Atomic rearrangement after the motion of a partial dislocation belonging to the shuffle set (a), and the glide set (b).

from an energetic point of view, motion of partial dislocations in the “glide” set is presumably favored. We note that the primary slip plane in hexagonal and rhombohedral SiC is 1/3 < 1120 > (0001) and the secondary slip system is 1/3 < 1120 > { 11 00 }. In cubic 3C-SiC, as in all crystals with the zincblende structure, the primary slip system is 1/2 < 1 1 0 > {111} and the secondary slip system is 1/2 < 11 0 > {110}. The cleavage plane in hexagonal/rhombohedral SiC is {110}. As in all semiconductors, basal dislocations in SiC are dissociated into leading/trailing partial dislocations according to the following reactions: •

In hexagonal/rhombohedral SiC: 1 1 1 1120 → 10 10 + 01 10 3 3 3

•

In cubic SiC: 1 1 1 1 10 → 2 1 1 + 121 2 2 2

In each case, the leading/trailing partials bound a ribbon of stacking fault that, because of the low value of stacking fault energy, γ, in SiC, is often very wide. The value of stacking fault energy in 4H- and 6H-SiC has been experimentally measured to be ~15 and ~3 mJ/m2, respectively [12,13]. These values are in good accord with theoretical estimates [13]. Note that the presence of, say, an edge dislocation belonging to the shuffle set would produce a row of broken vertical Si-C bonds. A broken (dangling) bond in a semiconductor has a high energy and is not very stable. Sometimes - usually when the dangling bond is isolated - a reactive element (such as hydrogen) attaches itself to the broken bond and “compensates” it. At other times, when broken bonds occur in neighboring atoms along a row (such as in a surface step or along a dislocation line), they may join each

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other to form double bonds. The latter process can only take place if the energetic cost of this “reconstruction” is not high (less than the cost of a row of broken bonds). Reconstruction would be very difficult for dangling bonds which are vertically parallel to each other – as in dislocations belonging to the shuffle set - because bending of two bonds through a large angle (~90°) would be involved (see Figure 6a). In contrast, the three broken bonds in the case of partial dislocations belonging to the glide set make shallow angles relative to their neighboring bonds (Figure 6b) and they can easily form double bonds and reconstruct. In this way, a whole row of dangling bonds in the core of the dislocation line is eliminated and there would be appreciable savings in energy (Figure 7). This has indeed been confirmed by theoretical calculations that show that the cores of both Si(g) and C(g) partial dislocations in SiC are reconstructed [14,15]. Hence, we shall henceforth assume that the partial dislocations in SiC belong to the glide set and their cores are reconstructed.

Figure 7. The atomic structure of the two 30° partials, Si(g) and C(g), in SiC bounding a ribbon of stacking fault. Note the row of dangling bonds in the unreconstructed core (a), and elimination of all dangling bonds in the reconstructed core (b).

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The core of the partial (or perfect) non-screw dislocations in SiC consists of either carbon or silicon atoms. If such partial dislocations belong to the glide set (“g”), they are respectively denoted as C(g) and Si(g). For a shuffle set, the letter “s” is used and the corresponding partial dislocations are denoted by C(s) and Si(s) [16]. The cores of both the leading and trailing 90° or 30° partials in a dissociated 60° dislocation have the same atom species, i.e. they are both either Si(g) or C(g). On the other hand, the 30° partial components of a dissociated screw dislocation have different core natures, i.e., one is Si(g) whilst the other is C(g) (Figure 7). Since the motion of a dislocation (say, by kink formation and migration) involves breaking and re-forming interatomic bonds in its core, and the strengths of Si-Si and C-C bonds are very different, it is clear that the mobilities of the Si(g) C(g) partial dislocations will be very different; this has also been confirmed by theoretical calculations [14,15]. 4. Plastic Properties of SiC The plastic properties of a crystal in its ductile regime are controlled by the formation and motion of dislocations. Consequently, the investigation of the plastic behavior of a crystal is intimately related to a study of its dislocations. The following well-known phenomenological expression relates the yield stress τy of a semiconducting crystal to the activation enthalpy ΔHτ for dislocation glide in that crystal [17-20]:

⎛ ΔH τ ⎞ ⎟ ⎝ kBT ⎠

τ y (T , ε ) = Aε exp ⎜ 1

n

(1)

where A and kB are constants (Boltzmann constant in the latter case) and ε is the strain rate. This exponential decrease of the yield stress with temperature has been experimentally verified in most semiconductors and is shown schematically in Figure 8. If, instead of plotting τy versus T (as in Figure 8), logarithm of τy is plotted as a function of 1/T, according to Eq. (1) a straight line would result. Also, according to this equation, the slope of the straight line would be equal to the activation enthalpy ΔHτ for dislocation glide in the crystal. Unlike the yield stress, the fracture stress σF of a crystal – equal to the normal component of the applied stress σn – is weakly temperature dependent. This is also shown in Figure 8 where the weak temperature dependence of σF is indicated by the low slope of the σF(T) line. The two curves intersect at the brittle-to-ductile transition temperature TBDT indicating that at temperatures above TBDT deformation of the crystal takes place by plastic yielding and at temperatures below TBDT, the crystal cleaves by

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brittle fracture [21,22]. Also note that at a higher strain rate ε2 (> ε1 ), the yield stress curve τ y (T ) shifts to the right and its intersection with the σ F (T ) curve occurs at a higher temperature. This increase of the TBDT with strain rate is common to practically all crystals. The schematic diagram of Figure 8 is also useful in noting the effect of a hydrostatic pressure on TBDT: if the crystal is under a compressive pressure p, the σF(T) line rises by p because the hydrostatic pressure counteracts the tensile component σn by p [23]. As a result, the BDT temperature shifts to a lower temperature than the TBDT observed under ambient pressure (see Figure 8). This is what happens in indentation experiments on crystals where cracks emanating from the corners of the indent occur at temperatures much lower than the ambient BDT temperature of the indented crystal.

Figure 8. Schematic diagram showing the temperature dependence of the yield stress τy and fracture stress σF (=σn) at two different strain rates ε1 and ε2 . The intersection of the two curves τ y (T ) and σ F ( T ) occurs at the brittle-to-ductile transition temperature TBDT. As the strain rate increases from ε1 to ε2 the τ y ( T ) curve shifts to the right and TBDT shifts to a higher temperature. On the other hand, in the presence of a hydrostatic stress (e.g., during an indentation test), the σF(T) curve moves up along the stress axis and TBDT shifts to a lower temperature.

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Using the assumption that the fracture stress is temperature independent, the intersection of the τ y (T ) and σ F (T ) curves gives the following expression for the brittle-to-ductile transition temperature TBDT [21,22]:

TBDT (S, ε ) =

ΔH τ ⎛ Sσ ⎞ kB ln ⎜ 1Fn ⎟ ⎝ Aε ⎠

(2)

Note the appearance of parameter S (Schmid factor) in this equation that accounts for the relative orientation of the primary slip system in the crystal with respect to the tension/compression axis. Also, both equations (1) and (2) show that parameters such as the constants A, n and, most importantly, ΔHτ, the activation enthalpy for dislocation glide, are required in order to predict the temperature dependence of the yield stress τ y (ε,T ) , and the strain dependence of the brittle-to-ductile transition temperature TBDT (ε ). Since these parameters were not available for SiC, deformation experiments at various strain rates were performed on single crystals of 4H-SiC, cut and oriented for single glide [22,24]. In addition fracture experiments were conducted on pre-cracked samples cut from the same 4H-SiC ingot to obtain the brittle-to-ductile transition temperature TBDT and its variation with the strain rate ε [25]. In the next two sections we describe these experiments and discuss the results in section 8. 5. Yield Stress Measurements in 4H-SiC Dislocations in SiC have been investigated since the 1950s where chemical etch-pitting and X-rays were first employed for their study. The initial investtigations were mostly in polycrystalline SiC that were used for ceramics applications. The first unambiguous characterization of dislocations in SiC were performed by Amelinckx et al. using transmission electron microscopy (TEM) [26]. Maeda and his coworkers [12,27,28] later conducted a careful study of dislocations in single crystal 6H-SiC. More recently, Samant et al. [29-33] and Dement et al. [24,34-36] investigated the plastic properties of 6H- as well as 4H-SiC single crystals by compression tests over a wide range of temperatures and strain rates. These authors also characterized the deformation-induced dislocations by TEM and interpreted the mechanical properties of the tested crystals in terms of the behavior of dislocations [22,37-39]. More recently, Lancin and her coworkers [40,41] made direct measurements of partial dislocation velocity in SiC. In this section, we shall summarize some of the results of compression experiments on 6H- and 4H-SiC as described in our work [22,24,32,39]. These experiments reveal that like other semiconductors, the yield stress τ y of

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Figure 9. Plots of the logarithm of the yield stress τy of 4H-SiC versus reciprocal temperature at three different strain rates ε (from [23]).

4H-SiC decreases exponentially with increasing temperature T at a fixed strain rate ε . Similarly, at a fixed temperature, τy increases with an increasing strain rate. Figure 9 shows the dependence of τy on temperature at three different strain rates. The presentation is in the form of ln(τy) versus 1/T which, as mentioned before, should be a straight line with a slope equal to that of the activation energy E for dislocation glide in the crystal. The surprising outcome in this study was that in both 6H- and 4H-SiC, instead of one straight line, each lnτy(1/T ) plot consists of two straight lines with different slopes, intersecting at a critical temperature Tc. This indicates that the deformation behavior is different in the two temperature regimes TTc. For this reason, the microstructure of the samples deformed below and above Tc were investigated by TEM paying particular attention to the type and configuration of deformation-induced dislocations [39]. The results showed that in both 4H- and 6-H SiC, in the T >Tc regime, the dislocations participating in plastic yielding were invariably dissociated into leading/trailing partial pairs bounding a ribbon of stacking fault with a width equal to the equilibrium separation of partials. In contrast, at temperatures below Tc (i.e. in the T
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samples also revealed that all the samples tested below Tc contained microcracks whereas those tested above Tc had a large number of slip lines with typical features of a strained ductile sample. Thus it was suspected that the critical temperature Tc could correspond to the brittle-to-ductile transition temperature TBDT. 6. Fracture Experiments on 4H-SiC In order to verify the possibility that the critical temperature Tc corresponds to the brittle-to-ductile temperature TBDT, fracture tests were conducted on samples cut from the same ingot as the one used for the compression tests. The specific tests were 4-point deformation tests on pre-cracked samples to directly measure the BDT temperature of 4H-SiC [25]. The pre-cracks were in the form of Knoop indents and the geometry was such that fracture occurred effectively in mode I. The TBDT temperature is known to be very sharp in semiconductors (and many ceramics) and to occur within a few degrees Celsius (generally less than 5°C). This is unlike the transition in, say, f.c.c. metals where the TBDT temperature is generally soft and can extend up to 100°C. In addition, TBDT is known to be a sensitive function of the rate of application of the applied stress σ or, equivalently, the strain rate ε , and to increase with increasing strain rate. Figure 10 shows plots of the logarithm of the strain rate ln (ε ) versus the reciprocal of the BDT temperature 1/TBDT.

Figure 10. Plot of the logarithm of the strain rate ln (ε ) versus reciprocal temperature 1/TBDT for of 4H-SiC. The slope of the straight line is equal to that of the activation energy for dislocation glide in the crystal (from [25]).

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As expected from Eq. (2), this plot should be a straight line with a slope equal to the activation enthalpy ΔHτ for dislocation glide in the crystal. From the plot of Figure 10, one can see that at a strain rate of, e.g., 3.10–7s–1, 4H-SiC becomes ductile above 1000°C. Thus, any working object fabricated from single crystal 4H-SiC is liable to catastrophic failure (by brittle fracture) below this temperature. Further an increase of the strain rate by an order of magnitude raises the BDT temperature by about 100°C. Our fracture experiments for the measurement of TBDT also enabled us to testimate the fracture toughness of single crystal 4H-SiC to be 1.9 ± 0.3 MPa m [25]. 7. Brittleness versus Ductility in Tetrahedrally-Coordinated Single Crystals Based on the yield stress and TBDT measurements on 4H-SiC, together with the microstructural investigation by TEM of the samples deformed at TTC, a model for the transition from brittle fracture to ductile plasticity in semiconductors was proposed. In Figure 11, a schematic picture of a crack is shown under mode I loading. The TEM experiments showed that the deformation-induced dislocations in 4H-SiC crystals compressed at temperatures lower than Tc in Figure 9 consisted of leading partials each dragging a stacking fault. On the other hand, when the temperature was above Tc, the deformation-induced dislocations were invariably perfect leading/trailing pairs bounding a ribbon of stacking fault. Figure 11a illustrates the situation at TTc.At these higher temperatures, both partials are nucleated and the source can act as a FrankRead source and an avalanche of dislocations form from this source (and other sources) and the crack stops propagating. Thus, the onset of ductility in this model is the nucleation of the trailing partial (and thus perfect) dislocations at T≈TBDT. Quantifying this model leads to the following equation for the BDT temperature [22,37-39]:

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TBDT =

n ΔH eff

(

)

⎡ νo Ky − Ko ⎤ kB ln ⎢ ⎥ K ⎢⎣ ⎥⎦

(3)

where νo (≈10–13 s–1) is a frequency factor, Ko is the initial value of the stress intensity factor before the load is applied at a rate K until K ultimately reaches KIc and the crystal fails catastrophically, and Ky is the stress intensity factor at which yielding of the crystal occurs (i.e., dislocations first n nucleate); ΔH eff is the effective enthalpy for dislocation nucleation given by n ΔH eff = ΔH n − ατ yV ∗ with α as a stress concentration multiplier and V* as the activation volume.

Figure 11. Schematic diagram of a mode I crack opening under a tensile stress σ. In (a) at temperatures below TBDT, a leading partial dislocation has nucleated from a source at the crack tip without the accompanying trailing partial and has expanded to produce a faulted half-loop; this basically shuts off the source and no more leading partials can nucleate from this source. In (b) at temperatures above TBDT, both partials have nucleated and the source can act as a Frank-Read source and dislocation multiplication occurs.

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Equation (3) for the brittle-to-ductile transition temperature in tetrahedrally-coordinated materials is a more sophisticated form of Eq. (2) and gives the variation of TBDT with the strain rate ε (proportional to K ) and the yield stress (proportional to Ky). 8. Effect of Lattice Defects on Electronic Properties of SiC Among the variety of different silicon carbide (SiC) polytypes, the hexagonal 4H form is currently considered as the most appropriate for highpower applications due to its wide band gap and minor anisotropy of electron mobility. Continuing development of 4H-SiC technology has recently resulted in prototype devices with high-voltage blocking capabilities, approaching theoretical limits of the material. However, high standards of reliability and reproducibility immanent for commercial scale applications remain an urgent issue. Until recently, maturity of the SiC technology was mainly delayed by material growth related problems and occurrence of defects such as micropipes and threading unit screw dislocations. This can be a particularly severe problem because the epitaxially grown layers commonly inherit numerous extended structural defects from the substrate, mostly different kinds of dislocations. Though it is generally agreed that defects such as micropipes severely diminish the high-blocking capability of SiC devices, their density has been reduced to very low values (less than 50 cm–2) in the newer generation of SiC wafers and they are no longer considered to be a serious problem for SiC electronic devices. SiC-based applications are generally anticipated to operate reliably under harsh environment conditions. However, some experiments involving overstress of SiC material by high forward-current and electron-beam irradiation have indicated formation and motion of dislocations at room temperature. More recently, a phenomenon apparently affecting the longterm stability of SiC power device performance was reported [42,43]. A drift of the forward voltage drop observed during continuous long-term operation in PiN diodes was attributed to the decrease of the effective carrier lifetime in the material by the formation of a high density of stacking faults. At the moment, this degradation phenomenon is a top-priority issue to be resolved for commercial implementation of SiC devices in high power applications and, in spite of growing world-wide interest, understanding of the phenomenon is still limited (for a recent review see ref. [44]). What has become clear in the last few years, however, is that the SiC device degradation occurs because of a relatively little-discussed phenomenon in semiconductors known as “recombination-enhanced defect reaction” (REDR). Research on recombination-enhanced processes had a burst of activity in the seventies when degradation of lasers manufactured from

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GaAs and related compounds was a major problem (for an excellent review of this phenomenon up to 1996, see ref. [45]). The problem has now become topical again in SiC electronic devices. That REDR is the underlying cause of SiC PiN diode degradation is supported by the fact that degradation is not observed under blocking conditions or in unipolar devices, thus indicating that charge carrier recombination is a necessary condition for the forward voltage drift to occur. Since degradation is also interrelated with appearance of structural defects in the active region of device, this section briefly discusses the degradation of 4H-SC PiN diodes by recombination-enhanced glide of partial dislocations.

Figure 12. Spectrally-selective EL-OEM images in a degraded 4H-SiC PiN diode: a) In the wavelength range 450±20 nm only the SF’s can be observed, b) The bounding partial dislocations (one or both of which are mobile) and the bright spots of threading dislocations (fixed) are revealed in the 700±20 nm range.

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In our work on the optoelectronic properties of SiC, we employed the technique of optical mission microscopy (OEM) [46] to image radiative extended defects in SiC. Basically, OEM enables one to observe radiative defects in a semiconductor when the concentration of minority carriers is disturbed from its equilibrium value [47]. This is because radiative defects in the crystal luminesce when the minority carrier population is perturbed by applying a forward bias to the pn junction (resulting in electroluminescence) or, alternatively, when SiC is irradiated with a laser of the appropriate wavelength (resulting in photoluminescence). In [48], we used spectral filtering of the integrated EL radiation at selected wavelengths to determine the energy levels of radiative defects (partial dislocation an stacking faults) in the bandgap of SiC PiN diode. The intrinsic SFs in 4HSiC were visualized by imaging in a wavelength range of 450±20 nm with an emission peak corresponding to ~2.8 eV. Similarly, the partial dislocations that bound these SFs, became visible in the range 700±20 nm, indicating that there are traps on the partials with an energy level of 1.8 eV. These traps act as radiative recombination centers giving rise to luminescence of the line defects (see Figure 12). In addition, by sequential imaging of the partial dislocations in a forward-biased PiN diode, their in situ motion and the corresponding expansion of stacking faults was observed and measured in the active area of 4H-SiC PIN diodes [48]. Finally, by measuring the temperature dependence of the velocity v, a first estimate of the dislocation glide activation enthalpy under electrical stressing was made (optical stressing was later found to decrease the glide activation enthalpy by the same amount [49]). This activation energy turned out to be the remarkably low value of ~0.27 eV which, as a comparison with Figure 10 shows, is about 2.2 eV less than the activation enthalpy for dislocation glide in the absence of recombination events (Figure 13).

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Figure 13. Arrhenius plot of ln(v) versus 1/T in a forward-biased 4H-SiC PiN diode. The slope of this plot gives an activation energy of (0.27±0.02 eV) for dislocation glide in 4HSiC when recombination enhancement takes place. The inset shows that at high injection currents, temperature variation of dislocation velocity cannot be observed because of saturation.

We have suggested that this motion takes place by recombination enhancement at specific sites on the partials that increase the kink nucleation rate and/or the kink migration velocity [47,50]. More precisely, such changes occur because of a decrease in the activation energy of kink formation and/or kink migration by an amount equal to the recombination energy released at the trap sites. The specific sites for trapping of charge carriers were proposed to be a particular defect on the partial dislocation line known as an anti-phase defect (APD) [51] (sometimes known as a soliton [52]). An APD is a reconstruction defect that is associated with an isolated broken bond on a partial dislocation line that is otherwise reconstructed. The existence of a dangling bond at an APD can act as a potent trap site for charge carriers and make this defect a likely site for channeling the recombination energy to lowering the activation energy for kink pair nucleation [47]. Thus, one can assign the value of 2.2 eV to the energy level of APD in the bandgap of SiC; these defects act as nonradiative recombination centers and lead to the remarkable lowering of the glide activation energy by nearly an order of magnitude (from ~2.5 eV to ~0.27 eV). In brief, on the basis of experimental measurements, the following energy levels were proposed in 4H-SiC (Figure 14).

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Figure 14. Energy levels of defects induced by forward-biasing a PiN diode and determined by optical emission microscopy. The ~2.8 eV and ~1.8 eV levels corresponding to the stacking fault (SF) and partial dislocation (PD), respectively, are radiative while the ~2.2 eV level is non-radiative and is speculated to correspond to the anti-phase defect (APD).

An energy level of 2.8 eV was attributed to (sites on) intrinsic SFs that acts as a radiative recombination center. Two energy levels were attributed to specific sites on partial dislocations in 4H-SiC: a radiative site with an energy level of 1.8±0.2 eV and a nonradiative site (speculated to be an APD) at ~2.2 eV from the top of the valence band. Acknowledgement The author has been privileged to collaborate with his former graduate students and postdocs, Drs. Moonhi Hong, Xian-Jie Ning, Anand Samant, Jin-Wei Yang, and Ming Zhang and with Professor Jean Luc Demenet of the University of Poitiers, France, Professor Jan Linnros of the Royal Institute of Technology (KTH), Stockholm, Sweden, and Dr. Augustinas Galeckas of the University of Oslo, Norway. This work was supported by the Basic Energy Sciences Division of the Department of Energy (grant number FG0293ER45496) and the National Science Foundation (grant number DMR-0108303).

References [1] Tairov, Y. M., and Tsvetkov, V. F., 1978, Investigation of Growth Processes of Ingots of Silicon Carbide Single Crystals, J. Crystal Growth 43(2):209-212 [2] Cree, Inc., Durham, NC, U.S.A. [3] Pirouz, P., and Yang, J. W., 1993, Polytypic Transformations in SiC: The role of TEM, Ultramicroscopy 51:189-214. [4] Pirouz, P., 1997, Polytypic transformations in SiC, Solid State Phenomena 56:107-132.

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[5] Lambrecht, W., and Segall, B., 1991, Electronic structure and bonding at SiC/AlN and SiC/BP interfaces, Phys. Rev. B 43(9):7070-7085. [6] Ramsdell, L. S., 1947, Studies on silicon carbide, Am. Mineral. 32:64-82. [7] Zhdanov, G. S., 1945, The numerical symbol of close packing of spheres and its application to the theory of close packings, C. R. Acad. Sci. USSR 48:39. [8] Powell, J. A., and Will, H. A., 1972, Low-temperature solid-state phase transformations in 2H silicon carbide, J. Appl. Phys. 43(4):1400-1408. [9] Stan, M. A., Patton, M. O., Warner, J. D., Yang, J. W., and Pirouz, P., 1994, Growth of 2H-SiC on 6H-SiC by Pulsed Laser Ablation, Appl. Phys. Lett. 64(20):2667-2669. [10] Hirth, J. P., and Lothe, J., 1982, Theory of Dislocations, J. Wiley & Sons, Inc., New York. [11] Shockley, W., 1953, Dislocations and edge states in the diamond crystal structure, Phys. Rev. 91:228. [12] Maeda, K., Suzuki, K., Fujita, S., Ichihara, M., and Hyodo, S., 1988, Defects in plastically deformed 6H-SiC Single Crystals Studied by Transmission Electron Microscopy, Phil. Mag. A 57(4):573-592. [13] Hong, M. H., Samant, A. V., and Pirouz, P., 2000, Stacking fault energy of 6H-SiC and 4H-SiC single crystals, Phil. Mag. A 80(4):919-935. [14] Sitch, P. K., Jones, R., Öberg, S., and Heggie, M. I., 1995, Ab initio investigation of the dislocation structure and activation energy for dislocation motion in silicon carbide, Phys. Rev. B 52(7):4951-4955. [15] Blumenau, A. T., Fall, C. J., Jones, R., Öberg, S., Frauenheim, T., and Briddon, P. R., 2003, Structure and motion of basal dislocations in silicon carbide, Phys. Rev. B 68(17): 174108. [16] Alexander, H., Haasen, P., Labusch, R., and Schröter, W., 1979, Foreword to, J. Phys. (Paris) 40:Colloque C6. [17] Alexander, H., and Haasen, P., 1968, Dislocations and plastic flow in the diamond structure in: Solid State Physics, vol. 22, F. Seitz ed., D. Turnbull, and H. Ehrenreich, Academic Press: New York and London, pp. 27-158. [18] Alexander, H., 1986, Dislocations in covalent crystals, in: Dislocations in Solids, Vol. 7, ed. F. R. N. Nabarro, Elsevier Science Publishers B.V.: Amsterdam, pp. 114234. [19] George, A., and Rabier, J., 1987, Dislocations and plasticity in semiconductors. I - Dislocation structures and dynamics, Rev. Phys. Appl. 22:941-966. [20] Rabier, J., and George, A., 1987, Dislocations and plasticity in semiconductors. II. The relation between dislocation dynamics and plastic deformation, Rev. Phys. Appl. 22:1327-1351. [21] Pirouz, P., Zhang, M., Demenet, J. -L., and Hobgood, R. M., 2002, Transition from brittleness to ductility in SiC, J. Phys.: Condens. Matter 14(48):12929-12945. [22] Pirouz, P., Zhang, M., Demenet, J. -L., and Hobgood, H. M., 2003, Yield and fracture properties of the wide band-gap semiconductor 4H-SiC, J. Appl. Phys. 93(6):32793290. [23] Pirouz, P., Wang, S., Bayu-Aji, B., Zhang, M., and Démenet, J. -L., 2005, The effect of dislocation core structure on the plastic and fracture behaviour of GaAs and InP, Phys. Stat. Sol. C 2(6):1973-1986. [24] Demenet, J. -L., Hong, M. H., and Pirouz, P., 2000, Plastic behavior of 4H-SiC single crystals deformed at low strain rates, Scripta Mater. 43(9):865-870. [25] Zhang, M., Hobgood, H. M., Demenet, J. L., and Pirouz, P., 2003, Transition from Brittle Fracture to ductile behavior in 4H-SiC, J. Mater. Res. 18(5):1087-1095.

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[26] Amelinckx, S., Strumane, G., and Webb, W. W., 1960, Dislocations in silicon carbide, J. Appl. Phys. 31(8):1359-1370. [27] Fujita, S., Maeda, K., and Hyodo, S., 1986, Anisotropy of high-temperature hardness in 6H silicon carbide, J. Mat. Sci. Lett. 5:450-452. [28] Fujita, S., Maeda, K., and Hyodo, S., 1987, Dislocation glide motion in 6H SiC single crystals subjected to high-temperature deformation, Phil. Mag. A 55(2):203-215. [29] Samant, A. V., and Pirouz, P., 1998, Activation parameters for dislocation glide in αSiC, Int. J. Refractory Metals and Hard Materials 16(4-6):277-289. [30] Samant, A. V., Wei, X. L., and Pirouz, P., 1998, An optical and transmission electron microscopy study of deformation-induced defects in 6H-SiC, Phil. Mag. A 78(3):737746. [31] Samant, A. V., Zhou, W. L., and Pirouz, P., 1998, Effect of temperature and strain rate on the yield stress of monocrystalline 6H-SiC, Phys. Stat. Sol. A 166:155-169. [32] Samant, A. V., 1999, Effect of Test Temperature and Strain-Rate on the Critical Resolved Shear Stress of Monocrystalline α-SiC, Ph.D. Thesis, Case Western Reserve University. [33] Samant, A. V., Hong, M. H., and Pirouz, P., 2000, The relationship between activation parameters and dislocation glide in 4H-SiC single crystals, Phys. Stat. Sol. B 222(1):7593. [34] Demenet, J. -L., Rabier, J., Milhet, X., Hong, M. H., Pirouz, P., Stretton, I., and Cordier, P., 2002, Microstructures of 4H-SiC single crystals deformed under very high stresses, J. Phys.: Condens Matter 14(48):12961-12966. [35] Démenet, J. -L., Zhang, M., Hobgood, H. M., and Pirouz, P., 2003, La transition fragileductile de 4H-SiC monocristallin, J. Phys. IV France 106:53-62. [36] Demenet, J. L., Milhet, X., and Rabier, J., 2005, TEM observations of the coexistence of perfect and dissociated dislocations in SiC under high stress, Phys. Stat. Sol. C 2(6):1987-1991. [37] Pirouz, P., Samant, A. V., Hong, M. H., Moulin, A., and Kubin, L. P., 1999, On temperature-dependence of deformation mechanism and the brittle-ductile transition in semiconductors, J. Mater. Res. 14(7):2783-2793. [38] Pirouz, P., Kubin, L. P., Demenet, J. L., Hong, M. H., and Samant, A., 2000, Transition temperatures in plastic yielding and fracture of semiconductors, Symposium on multiscale phenomena in materials: Experiments and modeling, Materials Research Society: Pittsburgh, PA, Mat. Res. Soc. Symp. Proc. 578:205-216. [39] Pirouz, P., Demenet, J. L., and Hong, M. H., 2001, On transition temperatures in plasticity and fracture of semiconductors, Phil. Mag. A 81(5):1207-1227. [40] Idrissi, H., Regula, G., Lancin, M. , Douin, J., and Pichaud, B., 2005, Study of Shockley partial dislocation mobiltiy in highly N-doped 4H-SiC by cantilever bending, Phys. Stat. Sol. C 2(6):1998-2003. [41] Regula, G., Lancin, M., Idrissi, H., Pichaud, B., and Douin, J., 2005, Structural characterization of double stacking faults induced by cantilever bending in nitrogendoped 4H-SiC, Phil. Mag. Lett. 85(5):259-267. [42] Lendenmann, H., Dahlquist, F., Johansson, N., Söderholm, R., Nilsson, P. Å., Bergman, J. P., and Skytt, P., 2001, Long term operation of 4.5 kV PiN and 2.5 kV JBS diodes, Mater. Sci. Forum 353-356:727-730. [43] Bergman, J. P., Lendenmann, H., Nilsson, P. Å., Lindefelt, U., and Skytt, P., 2001, Crystal Defects as Source of Anomalous Forward Voltage Increase of 4H SiC Diodes, Mater. Sci. Forum 353-356:299-302. [44] Skowronski, M., and Ha, S., 2006, Degradation of hexagonal silicon-carbide-based bipolar devices, J. Appl. Phys. 99(1):011101-24.

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[45] Maeda, K., and Takeuchi, S., 1996, Enhancement of Dislocation Mobility in Semiconducting Crystals by Electronic Excitation, in: Dislocation in Solids Vol. 10 (Chapter 54), F. R. N. Nabarro and M. S. Duesbery eds., North-Holland Publishing Co.: Amsterdam, pp. 443-504. [46] Konstantinov, A. O., and Bleichner, H., 1997, Bright-line defect formation in silicon carbide injection diodes, Appl. Phys. Lett. 71(25):3700-3702. [47] Pirouz, P., Zhang, M., Galeckas, A., and Linnros, J., 2004, Microstructural Aspects and Mechanism of Degradation of 4H-SiC PiN Diodes under Forward Biasing, Mat. Res. Soc. Symp. Proc. 815:91-102. [48] Galeckas, A., Linnros, J., and Pirouz, P., 2002, Recombination-enhanced extension of stacking faults in SiC PiN diodes under forward biasing, Appl. Phys. Lett. 81(5):883885. [49] Galeckas, A., Linnros, J., and Pirouz, P., 2006, Recombination-Induced Stacking Faults: Evidence for a General Mechanism in Hexagonal SiC, Phys. Rev. Lett. 96(2):25502-4. [50] Galeckas, A., Hallén, A., Majdi, S., Linnros, J., and Pirouz, P., 2006, Combined photoluminescence-imaging and deep-level transient spectroscopy of recombination processes at stacking faults in 4H-SiC, Phys. Rev. B 74(23):233203. [51] Hirsch, P. B., 1980, The structure and electrical properties of dislocations in semiconductors, J. Microscopy 118(1):3-12. [52] Heggie, M., and Jones, R., 1983, Solitons and the electrical and mobility properties of dislocations in silicon, Phil. Mag. B 48(4):365-377.

REVIEW: OXIDATION OF SIC/SIC COMPOSITES IN LOW OXIDISING AND HIGH TEMPERATURE ENVIRONMENT C. CABET Laboratoire d’Etude de la Corrosion Non Aqueuse, DEN/DANS/DPC/SCCME, CEA Saclay.

Abstract: Due to their thermal, mechanical, and nuclear properties, silicon carbide fiberreinforced, silicon carbide matrix (SiCf/SiCm) composites are promising candidate materials for fuel assembly in gas-cooled fast reactors (GFR), but their long term oxidation resistance must be evaluated under GFR relevant environments: high temperature, impure helium, yearlong in-core exposure. Experience from other high temperature applications shows that in an inert atmosphere SiC fibers may undergo a thermal decomposition. On the other hand at high P(O2), SiC based elements form surface silica that provides an outstanding corrosion resistance. However, oxidative volatilization of silica occurs at intermediate P(O2) and SiC suffers from active oxidation. Finally, the carbon layer at the fiber/matrix interface is highly sensitive to oxidation. The overall SiCf/SiCm behaviour is complex because it may involve degradation of several components and coupled interactions as well. A comprehensive R&D program is required to design a candidate composite material and to determine suitable environmental conditions. Fundamental as well as technical knowledge would be the basis of models for life time prediction.

1. Introduction Based on their promising thermal, mechanical, and nuclear properties, silicon carbide fiber-reinforced, silicon carbide matrix (SiCf/SiCm) composites are candidate materials for fuel assembly in gas-cooled fast reactors (GFR). The SiCf/SiCm composites have been elaborated to withstand exposures at high temperatures in oxidizing atmospheres due to the properties of SiC: a high melting point (Td~2800°C) and an excellent oxidation resistance based on the formation of a silica protective layer (passive oxidation regime). However, their compatibility must be ascertained in the GFR specific operating conditions. Structural materials for the GFR fuel shall reach high temperatures, typically 1000°C in nominal service conditions and up to 1600°C in transient accidental situations, in an atmosphere of industrial-grade helium. Despite efficient purification systems, it is expected that the cooling gas helium will be polluted by low amounts of oxidizing impurities – oxygen, water vapor, carbon oxide – plus possibly other gasses, such as nitrogen, methane… Besides, in relation to V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

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the criterion of high fuel burn-ups (target: 5 to 10%FIMA), the fuel assembly shall dwell in-core for extended durations. There are no data available on the oxidation resistance of composites in the unique GFR environment but it was evidenced in other ambiances with low oxidizing potential and high temperature that thermal aging and active oxidation of some components may occur, which dramatically affect the composites properties. We first give a short overview on the structure of a SiCf/SiCm composite and we examine the thermal stability of its main elements under inert atmosphere or vacuum. Then we try to identify the high temperature oxidation properties of each component, especially at low oxidizing potentials. Next, we look at the aging and oxidation processes of the whole composite, its consequences on the mechanical behavior and the possible means to improve the global oxidation resistance. Finally we give some concluding remarks on the design of a SiCf/SiCm composite for GFR applications as well as on the gap between the available corpus of data and the R&D needs for qualifying such a structure in the reactor particular service conditions. 2. Structure of a SiCf/SiCm Composite and Thermal Stability A SiCf/SiCm composite consists of SiC-based continuous fibers embedded in a SiC matrix. Its attractive mechanical properties (tensile strength, elastic modulus, creep resistance) originate from the reinforcing fibers. Thermal instability of some fibers under inert atmosphere or in vacuum above a given temperature level is thus of main importance for the composite resistance. Composites also exhibit non-brittle mechanical behavior when the fibers are bonded to the matrix via a thin layer of a compliant material, generally pyrocarbon, referred to as the interphase. 2.1. SIC-BASED MATRIX

The matrix nature (composition, structure, microstructure) depends on its fabrication route. Generally speaking, Chemical Vapor Impregnation (CVI) and Polymer Impregnation and Pyrolysis (PIP) produce rather porous silicon carbide while Reactive Melt Infiltration (RMI) and Slurry Infiltration and High Pressure Sintering (SI-HPS) result in a denser structure although this last process, using sintering additives, leads to a SiC matrix which contains other phases (e.g. oxides). Besides, matrix may include an excess of carbon, so that the involved system is in fact SiC+C.

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SiC is stoichiometric with a non-congruent melting and the Si-C system does not have any other intermediate compound. It decomposes following a peritectic reaction: 2540°C SiC(s) ⎯⎯ ⎯ ⎯→ C(s) + Si(C)(l)

(1)

At high temperature (far below the melting point), SiC is in equilibrium with a gas phase which mainly includes silicon. Above the binary system SiC+C, the partial pressure of Si(g) is somewhat decreased (factor ~10) and the gas phase also contains SiC2(g). As the evaporation process is not congruent, keeping SiC (or SiC+C) at high temperature under an inert atmosphere induces the formation of free carbon that is to say a thermal degradation. Araki et al. (1998) also claimed that a recrystallisation process of the SiC grains occurs. Thermodynamics calculations can establish the equilibrium conditions in the gas phase. However due to kinetics factors (growth of a carbon layer), the effective partial pressure of Si(g) appears to be far lower than the equilibrium value (Chatillon et al. 1999). Therefore, SiC thermal decomposition only actually occurs at very high temperature, above 1600°C. 2.2. THREE GENERATIONS OF FIBERS AND THEIR THERMAL STABILITY

SiC-based fibers have a small diameter (10-20µm). They are generally manufactured by spinning of a polycarbosilane (possibly doped) precursor into fibers which are then cross-linked and pyrolysed. Fiber layout can be unidirectional UD or 1D, 2D with plies or woven fabrics, as well as 3D (generally with a smaller volume fraction along the z-axis). The fabrication routes and the fibers properties have constantly improved over the last 30 years resulting in the development of three generations of fibers (Bunsell and Piant 2006). Table 1 gives some commercial brand names of fibers. •

First generation fibers The first generation fibers, cured in oxygen, in fact belong to the Si-C-O system with 10-15wt% oxygen: they consist of SiC nanocrystals (mean size: 2nm) in a matrix of amorphous silicon oxycarbide and may also contain a small fraction of free carbon. Mainly due to the silicon oxycarbide, these fibers are thermodynamically unstable at high temperature (Bodet 1996): this phase decomposes beyond approx. 1200°C producing a highly porous structure of mixed SiC and C, with evolution of SiO(g) and CO(g). Several authors (Shimoo et al. 1996, Bodet et al. 1996) proposed models to account for the process mass balance, which globally formulates:

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(SiC, SiC2xO1-x) → w SiC(s) + x C(s) + y SiO(g) [ + z C(s) ]

(2)

The decomposition also promotes the growth of SiC crystals e.g. grain size increases from 1-2.5nm up to ~1µm in Nicalon CG fibers aged under vacuum for 3hrs at 1460°C (Labrugère et al. 1997). The decomposition rate is influenced by several parameters: pressure, kind of atmosphere (inert gas or vacuum) and especially temperature. The thermal decomposition yields to a significant loss in fiber strength, elastic modulus and creep resistance (Takeda et al. 1998, Bodet et al. 1996). •

Second generation fibers These fibers, cured by electron beaming, are nearly oxygen free (about 0.5wt.%), and consist of SiC nanocrystals (grain size 2-5nm) with a large excess of free carbon. They exhibit a much higher thermal stability than the Si-C-O fibers with only a slight degradation related to the reaction (Araki et al.1998): SiC1+x(s) + Otrace(g,s) → SiC(s) + CO(g)

(3)

with Otrace coming directly from the fibers or from pollution in the gas phase. Reaction (3) causes changes in the fiber microstructure, especially the growth of SiC grains (Chollon et al. 1997). •

Third generation or nearly stoichiometric fibers Called stoichiometric fibers, their bulk has a low excess of carbon and no residual oxygen although the surface can be covered by a thin carbon layer. Except the Hi-Nicalon S (from Nippon Carbon), they contain additives such as Al (Tyranno SA) or B (Sylmaric). They are characterized by a high density but a coarse microstructure and a relatively low fracture strain (Dong 2001, Bunsell 2006). They are stable up to their synthesis temperature, typically 1800°-2000°C and maintain good mechanical properties up to 1400°-1500°C although the SiC grains grow. At this temperature level, the thermal decomposition of SiC through reaction (1) can be significant. Therefore a Tyranno SA reinforced composite creep tested in vacuum for 84hrs under 150MPa in the range 1350°1700°C exhibits a radial decomposition of its fibers that affects the creep behavior (Sauder and Lamon 2007).

2.3. INTERPHASE

Fibers and matrix are bounded by a thin layer, typically 100nm or so, of a compliant material with a low shear strength (τi <10MPa). Generally

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speaking, the interphase has a leaf microstructure (with leaves parallel to the fiber surface) such as graphitic carbon. 3. Oxidation of SiCf/SiCm Components Basically all silicon-rich materials, silicon or silicon-based ceramics, oxidize according to the same mechanisms: either a passive oxidation with the formation of a protective silica surface layer, or at low P(O2) an oxidative volatilization. In a first section, the base line oxidation properties of massive SiC are reviewed, in particular at low oxidizing potentials. Then, the oxidation behaviors of the matrix, the fibers and the interphase are successively surveyed. 3.1. OXIDATION OF MASSIVE SIC

The oxidation behavior of massive SiC is well documented at high and low P(O2) and data are available on the effect of water vapor and carbon dioxide. This knowledge is the starting point in describing the oxidation properties of SiC-based components of the composites. 3.1.1. Passive oxidation In an oxidizing medium such as air, massive SiC produces at high temperature a thin, adherent, surface silica layer. The main oxidation reactions are: SiC(s) + 3/2 O2(g) = SiO2(s) + CO(g)

(4)

SiC(s) + 2 O2(g) = SiO2(s) + CO2(g)

(5)

The silica growth kinetics, in terms of mass gain or scale thickness (x), is said to be linear-parabolic according to the work by Deal and Grove (1965) on silicon, that is to say:

KP KL + = (t + τ ) x2 x

(6)

with KP and KL the parabolic and linear rate constants and τ a time constant that accounts for the initial scale thickness. After a while, the parabolic term becomes the main contribution. For single crystal α-SiC, the growth rate is limited by the transfer of oxygen through the scale. Two oxidation regimes prevail: at a temperature below about 1400°C, oxygen mainly permeates as a molecule (this process is independent of P(O2) and Ea ≅ 120kJ/mol) while the atomic diffusion

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increasingly takes part to the transfer at high temperature (KP is proportional to P(O2)n and Ea = 200-300kJ/mol) (Cawley 1993). Polycrystalline SiC essentially follows the above characteristics but the data published are widely spread with activation energies from 120 to 420kJ/mol (Narushima et al. 1989, Goto et al. 2002). On the one hand, the silica produced by oxidation is first amorphous, then depending on the environmental conditions the film may crystallise. For pure SiO2 above 1200°C, the stable phase is cristobalite (Rodriguez-Viejo et al. 1993). Crystallisation changes the oxidation kinetics in modifying the transfer of oxygen and CO. On the other hand, additives as well as impurities in the gas phase globally increase the oxidation rate (Tressler 1993) as they incorporate in the surface film to form silicates with lower viscosity than pure silica. The transport of oxidizing species is thus enhanced (Jacobson 1993, Costello and Tressler 1981). Besides, porosity also plays a role in the oxidation kinetics (Jacobson 1993). 3.1.2. Active oxidation Passive oxidation with the growth of a surface silica film explains the outstanding high temperature oxidation resistance of silicon-based ceramics. However in given atmosphere conditions, the surface silica is not stable anymore. For pure SiC below a critical oxygen partial pressure, oxidation does not follow reactions (4) and (5), but occurs through reaction (7): SiC(s) + O2(g) = SiO(g) + CO(g)

(7)

SiC is thus converted into gaseous products with a mass loss. There is a transition point between active and passive oxidation which has been theoretically and experimentally assessed. The corrosion rate (mass loss) as well as the transition point depends on many factors: the SiC purity (Vaughn and Maahs 1990), the temperature, the gas flow velocity and gas phase composition (Hinze and Graham 1976), the total pressure, the oxygen partial pressure (Opila and Jacobson 1995). 3.1.3. Other oxidizing gasses and role of the gas composition In GFR atmospheres, oxygen shall not be the only oxidizing species. Water vapour and possibly carbon oxide could take part to oxidation processes. •

Passive oxidation mode In the passive mode, oxidation by water vapour occurs through reactions (8) and (9): <1400° C SiC(s) + 2 H2O(g) ⎯T⎯ ⎯⎯→ SiO2(s) + CH4(g)

(8)

OXIDATION OF SIC/SIC COMPOSITES >1400° C SiC(s) + 3 H2O(g) ⎯T⎯ ⎯⎯→ SiO2(s) + CO2(g) + 3 H2(g)

357

(9)

The silica growth rate is enhanced by the presence of water vapour. The factor can be 10 to 20 when comparing oxidation in water vapour to oxidation in pure oxygen, and 2 to 3 when 10% H2O is added to oxygen (Tressler 1993). The oxidation rate increases due to the higher solubility of water than oxygen in silica and because H2O can break the Si-O links which enhances the diffusion of water molecules. The kinetics is linearparabolic (Narushima et al. 1990), but the linear contribution in Eq. (6) lasts then for longer. Moreover in H2O-rich atmospheres, Si-O-H(g) volatile species may form. Volatilization due to steam for instance follows reactions (10) and (11) with pressure dependences [P(H2O)] and [P(H2O)]2; therefore, the higher the moisture, the more important the mass loss (Jacobson et al. 1990). SiO2(s) + H2O(g) = SiO(OH)2(g)

(10)

SiO2(s) + 2 H2O(g) = Si(OH)4(g)

(11)

Unlike water vapour, the parabolic rate constant is lowered by a factor 20 or so if SiC is oxidized under CO2 (Opila and Nguyen 1998). •

Active oxidation mode Kim and Moorhead (1990) established that mixed H2O/H2 atmospheres can produce reducing conditions toward silica where SiC actively oxidizes through reactions (12) and (13) at 1300°-1400°C. SiC(s) + H2O(g) = SiO(g) + CH4(g) SiC(s) + 2 H2O(g) = SiO(g) + CO(g) + 2 H2(g)

(12) (13)

Besides in such gas mixtures, high P(H2) can reduce silica through reaction (14): SiO2(s) + H2(g) = SiO(g) + H2O(g)

(14)

Low oxygen partial pressures may also be reached under CO2/CO mixtures. Narushima and Goto. (Goto et al. 1993, Narushima et al. 1993) showed that active oxidation occurs for P(CO2)/P(CO) ratios in the range 10–4 to 10–1 and that the SiC mass loss exhibits a maximum for intermediate ratios. Finally, Balat et al. (1998) observed SiC oxidative volatilization under low partial pressures of pure CO2 as well.

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3.2. OXIDATION OF THE SIC MATRIX

Filipuzzi et al. (1992) verified that the behaviour of CVI β-SiC under pure oxygen in the range 900°-1500°C complies with the general characteristics described in Sec. 3.1.1 for passive oxidation. The surface layer grows following a parabolic law and silica is amorphous except after treatment at 1500°C. 3.3. OXIDATION OF SiC FIBERS

Oxidation behaviour of SiC fibers derives from the general features explained in Sec. 3.1, especially regarding an active to passive transition point. However most fibers contain exogenous materials: oxycarbide (first generation), free carbon (second generation) or additives (stoichiometric) that may basically influence the oxidation properties. Shimoo et al. (2000a) tested Nicalon CG fibers (based on the Si-C-O system) at 1500°C under various P(O2). At 100Pa, oxidation was active and the fibers lost mass following the same trend that for thermal decomposition but while the mass loss stopped in inert atmosphere (with a Δm of about 25%), it went on in low oxidizing atmosphere. There was a transition between active to passive oxidation in the range 250Pa-1kPa and above 1kPa the oxidation was passive with the growth of a surface silica layer and a mass gain. It is noteworthy that passive oxidation can notably decrease the fiber thermal decomposition as it forms a surface silica layer that prevents the gaseous products from evolving. Therefore, the fibers maintain rather good tensile properties (σr=1.5GPa vs 3.5GPa as-received) after passive oxidation, but their strength to rupture drops after an active oxidation. Passive oxidation of second generation of fibers follows a parabolic law with a radial silica growth (Shimoo et al. 2000b) and an active to passive transition occurs (e.g. at 10-25Pa for Hi-Nicalon fibers at 1500°C (Shimoo et al. 2002b). As for first generation of fibers, development of a surface silica film improves the high temperature behaviour (Shimoo et al. 2000a): when exposed to argon at 1500°C, Hi-Nicalon fibers did not keep any residual strength, but pre-oxidation at 800°-1500°C allowed to maintain a quite high tensile strength. Shimoo et al. (2002b) compared the experimental active to passive transition of some fibers at 1500°C and low P(O2). They measured the transition point for Hi-Nicalon S fibers (stoichiometric) at very low P(O2) (quite unexpectedly the Tyranno SA fibers did not undergo active oxidation in the investigated range i.e. above 1 Pa O2) and generally speaking there was a continuous improve of the resistance against oxidation from 1st generation to stoichiometric fibers.

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Moreover, Naslain et al. (2004) verified for Hi-Nicalon fibers between 900° and 1400°C that, as exemplified for massive SiC, the oxidation kinetics dramatically increases (factor 20 or so at 1000°C) when water vapour (20 kPa) is added to oxygen (20 kPa in Ar). 3.4. OXIDATION OF THE INTERPHASE

Carbon is basically sensitive to oxidation following equations (15) and (16) that transforms a C interphase into gaseous products with a corresponding mass loss. C(s) + O2(g) = CO2(g)

(15)

C(s) + 1/2 O2(g) = CO(g)

(16)

4. Aging and Oxidation of SiCf/SiCm Composites, Consequences and Improvement Processes Fibers and carbon interphase, components that play key roles in the composite mechanical properties, are sensitive to aging and oxidation, and as composites are essentially porous, oxidizing species can access to the innermost components and gaseous corrosion products can evolve as well. These physico-chemical processes may thus strongly impact the composite strength. Improving the lifetime of composites requires developing protecttion methods to tackle with the fibers thermal instability and the interphase preferential oxidation. 4.1. THERMAL AGING AND CONSEQUENCES ON THE COMPOSITE PROPERTIES

Araki et al. (1998) examined the thermal stability of UD-SiCf/C/CVI-SiC composites with three types of SiC fibers of commercial grade Nicalon CG (1st gen.), Hi-Nicalon (2nd gen.) and Hi-Nicalon S (stoichiometric) with respective oxygen concentration of 11.7%, 0.5% and 0.2 wt.% in vacuum at 1000° - 2000°C (heat treatment: 1hrs). They verified that the damaging physico-chemical processes described in Sec. 2.2 actually occur for the fibers within the composites: the Nicalon CG fibers deoxidized according to reaction (2) above about 1400°C and the SiO2xC1-x oxycarbide completely decomposed above 1700°C with a concurrent growth of the SiCf grains. This caused a drop in the RT flexural strength and fracture toughness of the SiCf(CG)/SiC composite. Labrugère et al. (1997) aged 2D Nicalon CG/C /CVI-SiC composites at 1100-1500°C under vacuum or argon and

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evidenced that side reactions may also occur between gaseous products of reaction (2), mainly SiO(g) and the carbon interphase, for instance: SiO(g) + 2 C(s) = SiC(s) + CO(g)

(17)

This interaction produced a crown of large SiC crystals around the fibers which is deleterious for its sliding properties. The authors were able to explain why different mechanical behaviours were observed according to the extent of the decomposition/reaction process: mild aging conditions result in a weakening of the fiber/matrix bonding due to interfacial decohesion, more severe thermal treatments yield to a partial consumption of the interphase with the formation of large SiC-crystals on the fiber surface, and for drastic aging treatments the interphase is totally removed and the fibers undergo a marked crystallisation; the composite becomes brittle. Consistent with reaction (3), Araki et al. (1998) observed a slight degradation of the RT bending properties of the SiCf(Hi-N)/SiC composite. Shibayama et al. (2000) also claimed that the carbon interphase may undergo oxidation by SiO(g) according to reaction (17) that sinters together the SiC fibers and the CVI-SiC matrix. On the other hand, the SiCf(S)/SiC composite with stoichiometric fibers, that are stable at high temperature, exhibited no significant change in its RT strength after the thermal treatments. The same team (Araki et al. 2004) also aged a Tyranno-SA/C/CVI-SiC composite for 1hr in vacuum and confirmed that it maintained a high strength up to aging temperatures of 1700°C. Beyond 1700°C, the post-treatment strength decreased with increasing temperature which was attributed to recrystallisation of the CVI-matrix (see Sec. 2.1) as well as to the degradation of the Tyranno-SA fibers (see Sec. 2.2). 4.2. OXIDATION AND CONSEQUENCES ON THE COMPOSITE PROPERTIES

This Sec. focuses on the corrosion behaviour in the passive domain of SiC which is mainly driven by the oxidation of the fiber/matrix interphase. Filipuzzi et al. (1994) used model 1D-SiC/PyC/SiC to exemplify the behaviour of a composite when exposed to environmental conditions (700°C
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the pore, and (iii) reaction of oxygen with the pore walls following reactions (4) and (5) leading to the growth of a silica layer on both the fibers and the matrix. In case the C interphase is thin enough, silica is able to seal the pore and the active oxidation of the interphase then stops. However a thicker layer of pyrocarbon forms a larger pore and reactions (15) and (16) proceed up to the interphase complete consumption. It was verified (Huger c1994, Kleykamp et al. 1995) that the above steps are actually valid for 2D composites (Nicalon fibers) as well. In oxygen at 1000°C or so, the SiCf/C/SiCm samples first experience a mass loss attributed to the pyrocarbon layer oxidation which produces a recession of the interphase, then a parabolic mass gain corresponding to the development of a silica layer. The recession rate of thick C interphase was said to be linear (reaction control) (Windish et al. 1997) or linear-parabolic (Springer et al. 1996, Eckel et al. 1995) at 800-1200°C under oxygen; reaction orders between 0.5 and 1.0 were measured with respect to O2 at 1100°C (Jones et al. 1995, Windisch et al. 1997). Because the role of the interphase is to bound the fibers to the matrix, its volatilization has a significant impact on the composite mechanical strength. Therefore, porous SiCf/SiC composites demonstrate a post-oxidation embrittlement. Huger et al. (1994) correlated TGA curves to in-situ measurements of the Young’s modulus for SiCf(CG)/PyC/CVI-SiC and concluded that the interphase recession induces a sharp decrease of E that eventually may recover if a silica layer bridges the gap between fiber and matrix. Moreover, oxidation induces an environmentally assisted cracking as widely discussed in the literature (Evans et al. 1996, Jones et al. 1995, Jones et al. 2000, Henager et al. 1996, Lewinsohn et al. 2000, Windisch et al. 1997). This phenomenon called “pest” by some authors (Evans et al. 1996) may occur at stresses above that needed to cause matrix cracks (typically 0100 MPa) at intermediate temperatures (500°-1000°C). When creep tested, the composites exhibit different creep laws depending on the oxygen partial pressure and temperature. Several mechanisms were proposed to account for the environmental influence. The first step is the transport of oxygen to the carbon interphase through porosity or microcracks produced by the applied stress. Then, in the IRM regime (Interphase Removal Mechanism) the creep behaviour results from the fiber creep coupled to the effect of the interphase removal due to reactions (15) and (16) which time dependently debonds fibers and matrix. When the interphase has been totally removed, the crack velocity shows a marked increase. Another regime called OEM (Oxidation Embrittlement Mechanism) results from the formation of a brittle surface layer by reactions (4), (5) or (17) that creates a surface crack on the fiber, degrading the fiber strength. Alternative mechanisms include (i) the development of a brittle silica scale by reactions (4) and (5) within

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the fiber-matrix pore that increases the interfacial bound strength and decreases the fiber debonding and pullout; (ii) the decrease in the fiber strength as a result of inner reactions. Several variables may be involved such as time, thickness of the carbon interphase, composition of the phase that forms on the fibers… Finally, a FRM regime (Fiber Relaxation Mechanism) is controlled by the mere fiber creep. 4.3. POSSIBLE MEANS TO IMPROVE THE OXIDATION RESISTANCE

As composites, especially their interphase, are sensitive to oxidation, several protective strategies were developed to improve the oxidation resistance of the interphase and to limit the transport of gasses (oxygen or oxidizing species as well as decomposition and corrosion products). •

Improving of the oxidation resistance of the interphase: BN and multilayer interphases As pyrocarbon appears to be highly sensitive to oxidation, it can be replaced by another compliant material; the most common is the hexagonal boron nitride or alternatively boron-doped carbon. B-based interphases form a boron oxide phase (solid or liquid up to about 1100°C) in oxidizing media according to reaction (18): 2 BN(s) + 3/2 O2(g) = B2O3(s,l) + N2(g)

(18)

B2O3 or mixed SiO2- B2O3 in situ forms a viscous glass around the fibers that can seal pores and cracks. Multilayer interphases of the type (X-Y)n with X = PyC or BN, Y = SiC and n = 2 to 10 have also been developed; they alternate a compliant material and a self-healing oxidation resistant material. Oxidation of multilayer interphases (PyC or BN -SiC)n involves a mix of reactions (4), (5), (15), (16), and/or (18) (Bessman et al. 2000). •

Outer or intrinsic oxidation barrier Different approaches have been followed to limit the oxygen transport within the composite. On the one hand, the outer surface can be covered by an environmental barrier coating: either a thick CVD-SiC scale or a multilayer scale – for instance made of a SiC bound-layer, a self-healing material (B4C or Si-C-B), and a thick out-layer (oxide). On the other hand, the composite matrix can be self-healing because it contains a material such as B4C or Si-C-B that forms a fusible oxide (see reaction (18)).

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5. Conclusion SiCf/SiC is a leading type of ceramic composite being evaluated for use as a structural material for fuel assembly in GFR. Its performances under GFRrelevant environments – high temperature (1000°C or so), impure helium, yearlong times – need to be established. It must also be demonstrated that the fuel assemblies retain their confining properties after a transitory accidental exposure up to 1600°C. Although SiCf/SiC composites have been considered for gas cooled blanket in a fusion reactor where they would have been exposed to impure helium (O2, H2, H2O…) at 800-1200°C (Springer et al. 1996, Shibayama et al. 2000, Henager et al. 1996, Perez and Ghoniem 1993, Lewinsohn et al. 2000), there is a huge gap between the needs to select and qualify a SiCf/SiC in the GFR unique environment and the available corpus of data on the behaviour of composites, components and whole assembly, at high temperature in impure helium. Experience from other high temperature processes and data from the open literature may help in designing the features of a primary candidate composite and allows to propose recommendations on the most suitable environments. On the one hand, because of the very long in service times, it is necessary to use the best state-of-the-art components and composites: stoichiometrics fibers are the most thermally stable (but only Hi-Nicalon S fibers are made of pure SiC; for other fibers, the acceptability of additives must be ascertained), a low porosity matrix limits the transport of oxidizing gasses to fibers and interphase, application of an Environmental Barrier Coating further reduces the gas access into the composite, an appropriate interphase with improved oxidation resistance – structure, composition, thickness – is necessary (use of small quantity of boron-containing phases must be contemplated in regards of the neutron exposure). On the other hand, the atmosphere within the GFR core shall be controlled. It is not possible to give precise criteria on the impurity contents but an acceptable P(O2) range must be thoroughly determined at all temperatures. When the oxygen supply is not sufficient, oxidative volatilization of SiC occurs at high temperature, which totally damages the composite; other processes related to the thermal decomposition happen in inert atmospheres as well. On the contrary, oxidation of the pyrolitic carbon interphase proceeds even at intermediate temperature and is directly related to the oxygen potential. Besides, water vapor as well as hydrogen or methane interferes with the silica growth and their partial pressure in the gas phase must be controlled. It is of major importance to note that most experiments referred to in this paper are very short term, typically few minutes to some hours.

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Besides, there is little field experience on the most advanced composite (multilayer interphase, nanostructured self-healing matrixes…). A significant R&D program is thus necessary to assess the performance of SiCf/SiC in GFR conditions and to validate their lifetime. Main axes of an experimental program requires the study of the long term oxidation behaviour of each component in relevant ranges of atmosphere, temperature and duration, then to assess the overall reistance of the composite. Moreover, coupled interactions between stress and environment as well as between neutron irradiation and chemistry have to be investigated. Fundamental knowledge on the degradation mechanisms is required for the development of lifetime predictive models.

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Narushima, T., Goto, T., Iguchi, Y., and Hirai, T., 1990, High-temperature oxidation of chemically vapor-deposited silicon carbide in wet oxygen at 1823 to 1923 K, J. Am. Ceram. Soc. 73:3580-3584. Narushima, T., Goto, T., Yokoyama, Y., Iguchi, Y., and Hirai, T., 1993, High-Temperature Oxidation of Chemically Vapor-Deposited Silicon Carbide in CO-CO2 Atmosphere, J. Am. Ceram. Soc. 76:2521-2524. Naslain, R., Gette, A., Rebillat, F., Le Gallet, S., Lamouroux, F., Filipuzzi, L., and Louchet, C., 2004, Oxidation mechanisms and kinetics of SiC-matrix composites and their constituent, J. Mater. Sci. 39:7303-7316. Opila, E. J., and Jacobson, N. S., 1995, SiO(g) Formation from SiC in Mixed OxidizingReducing Gases, Ox. Met. 44:527-544. Opila, E. J., and Nguyen, Q. N., 1998, Oxidation of Chemically-Vapor-Deposited Silicon Carbide in Carbon Dioxide, J. Am. Ceram. Soc. 81:1949-1952. Perez, F. J., and Ghoniem, N. M., 1993, Chemical compatibility of SiC composite structures with fusion reactor helium coolant at high temperatures, Fus. Eng. Des. 22:415-426. Rodriguez-Viejo, J., Sibieude, F., and Clavaguera-Mora, M. T., 1993, Study of the Oxidation of CVD β-SiC:Influence of Crystallisation, Duran, P., and Fernandez, J. F. (Eds), Third Euro-Ceramics, pp. 865-871. Sauder, C., and Lamon, J., 2007, Tensile Creep Behavior of SiC-Based Fibers With a Low Oxygen Content, J. Am. Ceram. Soc. 90:1146-1156. Shibayama, T., He, G. W., Takahashi, and H., Kohyama, A., 2000, Application of HRTEM to studies of electronic structure and atomic configuration of interface in SiC/SiC composites, J. Phys. IV 10:97-103. Shimoo, T., Morisada, Y., and Okamura, K., 2000a, Oxidation behavior of Si-C-O fibers (Nicalon) under oxygen partial pressures from 102 to 105 Pa at 1773 K, J. Am. Ceram. Soc. 83:3049-3056. Shimoo, T., Morisada, Y., and Okamura, K., 2002a, Oxidation behaviour of Si-M-C-O fibers under wide range of oxygen partial pressures, J. Mater. Sci. 37:4361-4398. Shimoo, T., Okamura, and K., Morisada, Y., 2002b, Active-to-passive oxidation transition for polycarbosilane-derived silicon carbide fibers heated in Ar-O2 gas mixtures, J. Mater. Sci. 37:1793-1800. Shimoo, T., Toyoda, F., and Okamura, K., 2000b, Oxidation kinetics of low-oxygen silicon carbide fiber, J. Mater. Sci. 35:3301-3306. Springer, G. D., Windish, C. F., and Jones, R. H., 1996, Stability of SiC/SiC composites in environments containing O2 and H2, J. Nucl. Mater. 233-237:1271-1274. Takeda, M., Urano, A., Sakamoto, J. -I., and Imai, Y., 1998, Microstructure and oxidative degradation behavior of silicon carbide fiber Hi-Nicalon type S, J. Nucl. Mater. 258263:1594-1599. Tressler, R. E., 1993, Theory and experiment in corrosion of advanced ceramics, in: Nickel, K.G. (Eds) Corrosion of Advanced Ceramics - Measurement and Modelling, the Netherlands, pp. 3-22. Vaughn, W. L., Maahs, H. G., 1990, Active-to-Passive Transition in the Oxidation of Silicon Carbide and Silicon Nitride in Air, J. Am. Ceram. Soc. 73:1540-1543. Windisch, C. F., Henager, C. H., Springer, G. D., and Jones, R. H., 1997, Oxidation of the carbon interface in Nicalon-fiber-reinforced silicon carbide composite, J. Am. Ceram. Soc. 80:569-574.

FUNDAMENTALS OF LIQUIDS J. -P. HANSEN Department of Chemistry, University of Cambridge Lensfield Road, Cambridge CB2 1EW, UK

Abstract: An informal introduction to Statistical Mechanics of liquids is presented, emphasizing the key concepts and theoretical methodologies for a molecular description of “simple” liquids. The objective is to explain routes to explicit calculations of microscopic structure, thermodynamic and interfacial properties, time-dependent correlation functions and their link with transport coefficients. Possible extensions of this methodology to liquid metals and to water are briefly discussed. Keywords: Liquid matter; statistical mechanics; microscopic structure; pair distribution function; correlation functions; equation of state; transport coefficients

1. Introduction Liquids play a key role in reactor design and operation, in their capacity of vectors of heat transfer (cooling and energy transport). It is hence desirable to develop a reliable predictive description of essential thermodynamic and transport properties of liquids, like the equation of state (e.o.s.), the specific heat, the phase diagram or the thermal conductivity, in terms of realistic molecular models. The objective of the present lectures is to provide a brief overview of liquid state theory which provides a link between the microscopic and macroscopic levels of description by using and extending the standard concepts and tools of classical Statistical Mechanics. The challenges of a molecular theory of liquids lie in the intrinsic microscopic disorder (compared to the long-range order of crystalline materials) and in the strong spatial and temporal correlations associated with the high density of liquids (compared to their vapour). These challenges can be met by appropriate approximation schemes, specifically adapted to dense, disordered systems [1], and by the systematic use of computer simulation techniques (or “computer experiments”) [2], like Monte Carlo (MC) and Molecular Dynamics (MD) methods, in close relation with theoretical developments. In subsequent sections we will try to give a flavour of liquid state modelling and theory, avoiding formal developments as much as possible while focussing on relevant applications. V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

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J. -P. HANSEN

For pedagogical reasons we will first focus on models of “simple” liquids, but the technologically more relevant polar liquids (including water) and liquid metals will be discussed in the last part of the lectures. 2. Molecular Description of Liquids For the sake of simplicity, we first consider a pure (one-component) liquid made up of N identical, spherical molecules occupying a volume V; the corresponding number density is ρ = N V . If σ is the molecular diameter, it is convenient to define the dimensionless (reduced) density ρ * = ρσ 3 ; in liquid or solid condensed matter ρ * ≈ 1, which expresses the fact that neighbouring molecules touch on average, i.e. the mean nearest-neighbour distance d ≈ ρ −1 3 ≈ σ . On the microscopic scale the system is characterized by the Hamiltonian H N , a function of the position ( r i ) and momentum ( pi ) vectors of the N molecules ( 1 ≤ i ≤ N ), which, in the absence of external forces, is the sum of kinetic ( K N ) and molecular interaction ( VN ) energies. The interaction energy between non-polarizable molecules is generally well approximated by a sum of N(N −1) 2 pair interactions:

N p2 ⎛ ⎞ H N ({ri },{pi }) = K N ({pi })+ VN ({ri }) = Σ i + Σ Σ v⎜ r − r ⎟ j⎠ i=1 2m i < j ⎝ i

(1)

where m is the mass of a molecule, and the pair potential between spherical molecules depends only on the distance

r = r −r ij i j between the centres of any pair of molecules (i, j ). The pair potential v (r) between spherical molecules can be determined by low density Quantum Chemistry calculations [3]; it comprises a stiff short-range repulsion (due to the Pauli exclusion principle) and a long-range dispersion attraction, which decays like 1 r 6 . A popular and useful interpolation is provided by the Lennard-Jones potential: 12 6 v(r ) = 4 e ⎡ ( σ r ) – ( σ r ) ⎤ (2) ⎣ ⎦ where the energy scale ε determines the depth of the potential minimum at

r = 21 6σ .

In liquids (except Helium) intermolecular quantum effects are negligible because

d ≈ σ >> Λ =

h 2π mk B T

FUNDAMENTALS OF LIQUIDS

369

the thermal de Broglie wavelength (where h and kB are Planck’s and Boltzmann’s constants, and T is the temperature). Hence, with the exception of valence electrons in metals, liquids can be described by classical Statistical Mechanics. A microstate of a classical system is entirely characterized by a point { ri ,p i } in 6N-dimensional phase space Γ. Macroscopic equilibrium properties of an N-particle system are identified with statistical averages of appropriate microscopic “dynamical” variables A(Γ) ≡ A({ ri , p i }) obtained by integrating A over all accessible phase space with the normalized Boltzmann weight:

PN (Γ) =

1 N!ΔΓQN

exp{−βH N ({ri ,pi })}

(3)

β = 1 kB T ; ΔΓ is a phase space element, conventionally taken equal to

h 3N ; N! accounts for the fact that permutations of the labels of identical particles lead back to the same microstate; the dimensionless normalization factor QN is the canonical partition function

QN =

1 exp {− β H N ( Γ )}d Γ N !ΔΓ ∫

(4)

where dΓ ≡ dr1 " drN dp1 " dpN Thus, statistical averages are defined by

A =

∫ A(Γ) P (Γ)dΓ N

(5)

In particular, the internal energy U of a liquid is given by:

U = H N = K N + VN where the mean kinetic energy K N tition theorem) [4]:

KN =

(6)

is easily calculated to be (equipar3N 2

kB T

(7)

This would be the total internal energy of an ideal gas of non-interacting molecules (VN = 0). In a liquid, intermolecular interactions are predominant, but the calculation of VN is much less straightforward. The Helmholtz free energy F = U − TS (where S is the entropy) is directly related to the partition function via the “bridge” relation [4]:

F = −k B T ln QN

(8)

Note that for a pure (one component) fluid, the internal and free energies are thermodynamic functions of the macroscopic variables N, V and T.

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J. -P. HANSEN

Another important dynamical variable is the microscopic stress tensor, defined by: α β N ⎛ pα p β N r r ⎞ ij ij Παo β = Σ ⎜ i i + 12 Σ 2 rij v′ rij ⎟ i =1 j ≠i r ⎝ m ⎠ ij

()

(9)

where α, β denote the Cartesian indices x, y or z and v ′(r) = dv (r) dr . The macroscopic pressure P is then given by the average of the diagonal elements of the stress tensor, while, due to the isotropy (rotational invariance) of liquids, the equilibrium average of the off-diagonal elements vanishes:

PV = Παα o Παo β = 0,

α ≠β

(10a)

(10b)

Equation (10a) provides the virial route to the e.o.s. 3. Microscopic Structure and Thermodynamics The spatial structure of liquids on the molecular scale is essentially determined by excluded volume effects due to the strong short-range repulsion between molecules when r < σ . This leads to short-range order, (as opposed to the long-range order in crystalline solids) characterized by the pair distribution function (p.d.f.) g ( r ) . Consider first a fluid of noninteracting molecules (an ideal gas), of number density ρ ; since molecules are uniformly distributed, the average number of molecules inside a spherical shell of radius r and width dr around a central molecule is simply:

n ( r ) dr = 4π ρ r 2 dr

(11a)

For interacting molecules, this must be corrected by a correlation factor, which defines the p.d.f.:

n ( r ) dr = 4π ρ g ( r ) r 2 dr

(11b)

At low density correlations are essentially due to pair collisions, so that

g (r ) ≈ exp {− β v(r )}, pointing to a “correlation hole” ( g ( r ) ≈ 0, r < σ ). At

liquid densities neighbouring molecules form coordination shells (radial layering) beyond the correlation hole, as illustrated in Figure 1. This damped oscillatory structure is characteristic of short range order, which dies out when r >> σ , i.e. g(r) → 1 as r → ∞ . The above heuristic definition of the p.d.f. may be formalized as follows. Define the microscopic density associated with a molecular configuration { ri } as

371

FUNDAMENTALS OF LIQUIDS N

ρ˜ (r) = Σ δ (r - ri )

(12a)

i=1

Its Fourier components are:

ρk =

N

∫ ρ˜ (r) eik⋅r dr = Σ eik⋅r

i

i=1

(12b)

Translational invariance (due to the homogeneity of a fluid in the absence of any external force field) implies that

ρ˜ (r ) = ρ = N V

(13)

Spatial correlations between molecules at points r and r ′ are characterized by the two-point correlation function ρˆ (r ) ρˆ (r ′) ; the p.d.f. is then defined by:

g(r, r ′) =

ρ˜ (r )ρ˜ (r ′) 1 − δ (r − r ′) ρ2 ρ

(14)

where the second term on the r.h.s. is subtracted to eliminate the trivial “self term” corresponding to i = j in the double sum arising in the first term if (12a) is substituted into (14). Due to translational invariance and isotropy, g (r, r ′) = g ( r - r ′ ).

Figure 1. Pair distribution function g(r ) (left hand curve) and structure factor of liquid Na near the triple point. The curves are obtained by MC simulations of the purely repulsive 4 potential v (r) = ε(σ r) , while the dots are X-ray scattering data from Greenfield et. al [27]; after Hansen and Schiff [28].

The static structure factor S(k ) is defined by:

S (k ) =

1 ρ k ρ -k N

(15)

In view of Eqs. (12a), (12b) and (14), it is clear that S (k ) is related to the Fourier transform (FT) of g(r), or the corresponding pair correlation function h (r) = g(r) −1 , namely:

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J. -P. HANSEN

S (k ) = 1 + ρ ∫ eik ⋅r h ( r ) dr = 1+ ρ hˆ (k )

(16)

The structure factor is in fact directly measurable by X-ray or neutron diffraction [1]; an example is shown in Figure 1. The main peak of S (k ), which occurs roughly at a wavenumber k ≈ 2π d is reminiscent of the first Bragg diffraction peak of crystalline solids, and characterizes the degree of short range order in a liquid. The amplitude of the main peak increases with density, and reaches a quasi-universal value of 2.85 at freezing for most non covalent liquids [5]. The functions g(r) and S(k ) provide a direct link with thermodynamics. Thus, combination of Eqs. (1) and (11b) leads to:

VN = 2πρ N

∞

∫ g ( r ) v ( r ) r dr 2

0

(17)

so that, remembering the result (7), the internal energy per molecule may be expressed as ∞ 1 1 3k T U= H N = B + 2πρ ∫ g ( r )v ( r ) r 2 dr 0 N N 2

(18)

Similarly Eqs (9) and (10a) lead to the following expression for the e.o.s.

PV 2πβρ = 1− Nk BT 3

∞

∫ v′ ( r ) g ( r ) r dr 3

0

(19)

U and P are first order derivatives of the free energy (with respect to temperature and volume). Second order derivatives, like the isothermal compressibility χ T , or the specific heat at constant volume CV , may be related to fluctuations of density or energy within a small, but macroscopic sub-volume of the total system, which fixes the chemical potential μ . The fluctuation of the number N of molecules inside the sub-volume is directly related to the isothermal compressibility [1,4]:

(ΔN )

2

N

=

(N −

N

N

)

2

= ρ k BTχ T

(20)

The fluctuation ΔN can be expressed as an integral of the microscopic density (12a):

ΔN =

∫ [ρ˜ (r) − ρ]dr V

(21)

Substituting Eq. (21) into Eq. (20) and remembering the definitions (14) and (16), one arrives at the compressibility equation:

373

FUNDAMENTALS OF LIQUIDS

S (k = 0) = 1+ ρ ∫ h(r )dr = ρk BTχ T

(22)

The long wavelength ( k → 0 ) limit of the structure factor thus measures the macroscopic response in volume or density to a change in pressure:

χT = − V1 ( ∂ V ∂ P )T More generally S(k ) measures the response of the system to an external force on the molecular scale (finite wavelength [1,6]). Consider a weak external potential acting on the molecules, which is spatially modulated with a characteristic wave vector k:

φ ( r ) = V1 δφk e – ik⋅r

(23)

The amplitude δφ k is assumed to be small compared to k B T ; the total potential energy ΦN of the N molecules in the external field is:

Φ N = ∫ φ ( r ) ρ ( r ) dr = V1 δφk ρ−k

(24)

V

and the total potential energy in the Boltzmann factor (3) is now VN + ΦN . When ΦN = 0 , the fluid is homogeneous and the local density ρ (r) = ρˆ (r ) reduces to the bulk density ρ (c.f. Eq. (13)). In the presence of an external field, translational invariance is broken so that ρ (r) is no longer constant, and the Fourier components ρ k are non-zero for finite k:

ρk =

∫ ρ exp {− β (V + Φ )} dr " dr ∫ exp {−β (V + Φ )} dr " dr k

N

N

N

1

N

1

N

(25)

N

By expanding the Boltzmann factors in numerator and denominator to linear order in βΦN , one obtains the linear response result:

ρ k = − Vβ δφk ρk ρ− k

0

+O

(( βδφ ) ) = −βρδφ 2

k

k

S ( k ) (26)

where the subscript 0 refers to a statistical average of the unperturbed (homogeneous) fluid (Φ N = 0) and S (k ) is the corresponding structure factor (16). The static linear susceptibility χ (k) is hence related to the structure factor of the unperturbed fluid by the static version of the fluctuation-dissipation theorem [1,6]:

χ (k) =

δ ρk ρS(k) =− δφ k k BT

(27)

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J. -P. HANSEN

which generalizes Eq. (22) to finite k, provided one identifies lim δφ with k→0

k

–PV. The formal relations derived so far are exact, but approximations are required to enable explicit calculations of the pair functions g(r)or S (k ), and the resulting thermodynamic properties derived from eqs. (18), (19) and (22). The key to approximate calculations of these quantities lies in van der Waals’ recognition (in 1873!) that the microscopic structure of most liquids (with the notable exception of highly polar liquids, like water) is determined by the excluded volume effects due to the strong short-range repulsion between molecules, while the long-range attractive interactions may be treated within perturbation theory. The pair potential v (r) is consequently divided into short-range and long-range parts:

v (r) = v 0 (r) + w(r)

(28)

and a hypothetical system of molecules interacting only via the short-range potential v 0 ( r) is called the reference system, henceforth designated by the subscript 0. The equilibrium properties of the reference system are generally easier to calculate; in particular, in the absence of attractive interactions between molecules, only a single fluid phase exists, i.e. there is no longer a liquid-vapour transition. The strongly repulsive potential v 0 ( r) can often be accurately represented by a model system of hard spheres, with an effective diameter σ0, i.e.:

v HS (r) = ∞,

r < σ0

v HS (r) = 0,

r > σ0

(29)

with a temperature-dependent σ0 roughly determined by v 0 (σ 0 ) = k B T . The hard sphere model is athermal, i.e. the reduced dimensionless thermodynamic properties are independent of temperature (since v HS (r) has no energy scale), and depends only on the reduced density ρ * , or equivalently on the packing fraction η = πρ * 6 . The e.o.s. of the hard sphere fluid is accurately represented by the semi-empirical Carnahan-Starling equation:

βP 1+ η + η 2 − η 3 = 3 ρ (1− η)

(30)

which agrees very well with MC or MD simulation data [1,7]. The corresponding Helmholtz free energy per molecule is obtained by thermodynamic integration:

βF N

= ln (η) −1 +

η(4 − 3η) 2 (1− η)

(31)

F (= −TS ) is purely entropic in the case of hard spheres (U = 0) ; the fluid crystallizes into an FCC crystal for η ≈ 0.5 ; at higher packing fraction the

FUNDAMENTALS OF LIQUIDS

375

entropy of the ordered solid is larger than that of the metastable fluid (jamming effect)! The structure factor and e.o.s. of the liquid of interest (i.e. including the intermolecular attraction w(r)) can be related to those of the reference system by a mean field argument (or “random phase approximation” RPA) [6]. Setting aside the short range repulsive forces, the local potential φ (r ) felt by a molecule at r is the sum of any external potential φext (r ) and of the mean potential energy due to the slowly varying long range interaction w(r − r′) with all other molecules situated at positions r′ in the fluid. If δρ (r′) denotes the deviation of the local density ρ (r′) from its mean ρ , then:

φ (r) = φext (r) + ∫ w(r - r′)δρ (r′)dr′

(32)

The second term is of mean-field type since it neglects correlations between molecules at r and r′ . Taking FT’s of both sides of Eq. (32) one arrives at:

φ k = δφ kext + wˆ (k )δ ρ k

(33)

According to the linear response result (27):

δ ρk = χ 0 (k )⎡⎣δφ kext + wˆ (k )δ ρk ⎤⎦

(34)

Where χ 0 (k ) is the susceptibility of the reference system (which has only short-range interactions). Solving (34) for δ ρ k , one arrives at:

δ ρk =

χ 0 (k) δφ kext ≡ χ (k)δφ kext ˆ 1− χ 0 (k)w(k)

(35)

which yields the (approximate) relation between the susceptibility χ (k) of the system of interest, and that of the reference system. The fluctuationdissipation relation (27) then leads to the corresponding RPA relation between the two static structure factors: ρwˆ (k ) 1 1 (36) = + kB T S (k ) S0 (k ) and to the following relationship between the compressibility of the two systems (cf. Eq. 22):

ρkB Tχ T =

ρkB Tχ T(0) 1+ ρ 2 wˆ (0)χ T(0)

Since ρ k BT χT = ( ∂β P ∂ρ )T , Eq. (37) can be integrated over e.o.s.: −1

βP βP0 ρ = + wˆ (0) ρ ρ 2

(37)

ρ to yield the (38)

376

J. -P. HANSEN

One further thermodynamic integration leads to the approximate relation for the free energy:

βF N

=

β F0 N

+

ρ 2

β wˆ ( 0 ) =

β F0 N

+

ρ 2

∫ β w ( r ) dr

(39)

The mean field result (Eq. 38) generalizes the classic van der Waals e.o.s. which uses a rough, free-volume approximation for the e.o.s. of the reference system. Correlations within the perturbation term in Eq. (39) may be approximately taken into account by adding an additional g0 (r) factor in the integrand. The result may also be derived within a systematic cumulant expansion of the free energy in powers of βW N , where W N is the sum of w(r) over all pairs of molecules [1]. Variants of this thermodynamic perturbation theory differ by the prescription for the effective diameter σ 0 used to map the reference system onto an underlying hard sphere fluid [1,8,9]. At high densities (close to freezing) thermodynamic perturbation theory predicts free energies and e.o.s. with an accuracy of a few percent (compared to computer simulation data for the same system). Thermodynamic perturbation theory tends to be less accurate at higher temperatures and lower densities, where fluctuations become increasingly important, particularly so in the vicinity of the critical point of liquid-vapour coexistence. As regards comparison with experimental data, thermodynamic perturbbation theory relies on the accuracy of the pair potential between molecules. Large deviations of the predictions of thermodynamic perturbation theory from experiment can signal the lack of convergence of the theory, or the inadequacies of the assumed pair potential, or both. The first source of discrepancy can be checked by a direct comparison between theory and computer simulations of the same model system. The pair distribution function (or the structure factor) of the reference system, required within thermodynamic perturbation theory, or of the full system, can be extracted from computer simulations of periodic systems of a few hundred or thousand molecules, or can be calculated approximately from fluid integral equations which combine the so-called Ornstein-Zernike equation (in Fourier space):

1 = 1− ρcˆ (k ) (40a) S (k ) with a closure relation between g(r) and c (r) (the FT of cˆ(k) ) for a given pair potential v (r): g(r) = G(g(r ), c (r); v (r)) (40b) Equation (40a) defines the direct correlation function (d.c.f.) c, while the approximate closure relation is generally derived from functional expansions or diagrammatic resummations [1]. Comparing Eqs. (40a) and (36),

FUNDAMENTALS OF LIQUIDS

377

one observes that with a non-interacting reference system (S 0 (k ) = 1), the RPA amounts to setting c (r) = −βw (r). In fact the RPA provides the correct long-range behaviour of c (r). For the hard sphere fluid the PercusYevick (PY) closure is derived by noting that w(r) = 0, r > σ 0 , namely:

g(r) = 0, r < σ 0 (exact)

c (r) = 0, r > σ 0 (approximate) (41)

Equation (41), together with Eq. (16) and Eq. (40a) provides a closed set of equations, which can in fact be solved analytically to determine c (r) for r < σ 0 , and hence g(r) and S (k ) [1]. The resulting g(r) agrees well with simulation data for η ≤ 0.4 , and is routinely used in thermodynamic perturbation calculations. The concepts and methods discussed in this sector are easily generalized to multi-component systems (liquid mixtures), and applications include the calculation of thermodynamic mixing properties, of phase diagrams of binary or ternary mixtures, of solubilities of impurities and of the n (n + 1) 2 partial structure factors characterizing an n-component mixture [10]. 4. Inhomogeneities and Interfaces As already noted in the previous section, when the molecules of a fluid are subjected to the potential φ (r) of an external field, translational invariance is broken, and the inhomogeneous fluid is characterized by a local density (or density profile) ρ (r ) = ρ˜ (r ) . Within the linear response to a weak external field, the Fourier components of ρ (r) are given by Eq. (26). We are interested here in strong inhomogeneities due either to confinement (e.g. a liquid near a solid substrate or inside a pore), or near the spontaneous interface between two phases, like the liquid-vapour interface. In the simplest cases of a liquid above a horizontal solid substrate, or of the liquidvapour interface under gravity, the density profile will depend only on the vertical coordinate z (ρ (r ) → ρ (z )). In the former case the external potential φ (z) which induces the vertical inhomogeneity is due to the interactions between the molecules in the solid substrate, and the fluid molecules, averaged over the distribution of substrate molecules, assumed to be homogeneous. If the interaction between substrate and fluid molecules is of the Lennard-Jones form (2), the resulting external potential φ (z ) ~ −1 z 3 beyond a few molecular diameters; this long-range substrate-fluid attraction leads to adsorption of fluid molecules and to wetting of a solid substrate by a liquid film in the presence of saturated vapour. Excluded volume effects between molecules in a liquid lead to layering of the molecules in the immediate vicinity of a solid substrate, signaled by an oscillatory density profile ρ(z).

378

J. -P. HANSEN

In the presence of an interface, the usual bulk thermodynamic relations must be modified to account for an additional extensive variable, the area A of the interface. Thus the variation of the Helmholtz free energy of a onecomponent system now reads:

dF = −SdT − PdV + μdN + γdA

(42)

where γ is the surface tension, i.e. the work required to increase the interfacial area by dA . Similarly the variation of the grand potential Ω = F − μN reads:

dΩ = −SdT − PdV − Ndμ + γdA

(43)

The surface tension is obtained as a thermodynamic derivative of F or Ω :

⎛∂ F ⎞ ⎛ ∂ Ω⎞ γ =⎜ =⎜ ⎟ ⎟ ⎝ ∂ A ⎠V ,T , N ⎝ ∂ A ⎠V ,T , μ

(44)

Since Ω is a homogeneous function of first order in the extensive variables V and A, integration of (43) yields:

γ=

1 (Ω + PV ) ≡ ω (s) A

(45)

i.e. γ appears as the surface excess grand potential per unit area (in the bulk Ω = −PV !). If the planar substrate is situated at z = 0 , the adsorption (i.e. the excess or deficit of molecules per unit area) is defined as: ∞

Γ = ∫ ⎡⎣ ρ ( z ) − ρb ⎤⎦ dz 0 where

(46)

ρ b = lim ρ(z) z →∞

is the bulk liquid density. Standard thermodynamic manipulations [1,6] lead to the key relation between adsorption and surface tension:

⎛ ∂γ ⎞ Γ = −⎜ ⎟ ⎝ ∂μ ⎠T

(47)

If several molecular species are present (mixture), one defines adsorptions

Γα = − ( ∂γ ∂μα )T′{μ } β ≠α

for each species α ; such partial adsorptions characterize the competition between different species to be adsorbed or depleted near a given substrate. The key surface property is the surface tension, which must obviously be related to the density profile ρ (z). These two quantities can be approximately calculated within density functional theory (DFT) [1,11]. While for a homogeneous fluid, the free energy is a function of the bulk density ρ b ,

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FUNDAMENTALS OF LIQUIDS

an inhomogeneous fluid may be characterized by a free energy which is a functional of the local density, F[ ρ (r)], for a fixed chemical potential. If φ (r) is the external potential, then the equilibrium density is that which minimizes the grand potential

Ω = F ⎡⎣ ρ ( r ) ⎤⎦ − ∫ {μ − φ ( r )}ρ ( r ) dr

i.e.

δF [ρ ] + φ (r ) = μ δρ (r)

(48)

where δ δρ (r ) is the functional derivative with respect to the local density. Note that the functional F[ ρ ] is an intrinsic property of the fluid, which is independent of the external potential φ (r ). For a homogeneous fluid, φ (r) = 0, ρ (r) = ρ and (48) reduces to the thermodynamic relation:

( ∂ F ∂ N )T ,V = μ The key problem of DFT is to construct a free energy functional F[ ρ ] for a given Hamiltonian (1). This is a difficult problem and approximations adapted to given physical situations must be made. For an ideal gas of noninteracting particles (V N = 0 ), the bulk free energy is trivially known,

{

}

Fid ( ρ ) = NkBT ln ( ρΛ 3 ) − 1

where Λ is the de Broglie thermal wavelength h generalization to an inhomogeneous ideal gas is:

{

2 π mk B T [4]. The

}

Fid [ ρ ] = k BT ∫ ρ ( r ) ln ( ρ ( r ) Λ 3 ) − 1 dr V

(49)

Substituting Eq. (49) into Eq. (48) one recovers the expected Boltzmann distribution of non-interacting particles in an external field:

ρ (r) = ρ b exp{−βφ (r)}

(50)

For a system of interacting particles, an unknown “excess” contribution must be added, i.e.

F [ρ] = Fid [ρ] + Fex [ρ]

(51)

In the case of a slowly varying spatial inhomogeneity, the local density varies over a wavelength ξ >> ξ 0 , where ξ0 is a typical correlation length in the bulk (a few molecular diameters σ ) i.e.

∇ρ ρ ≈ ξ1 << ξ1 0 The simplest assumption is to define a local free energy density at each point in the fluid, i.e. to assume that macroscopic thermodynamics applies

380

J. -P. HANSEN

locally, within volume elements ≈ ξ 3 . This leads to the “local density approximation” (LDA) for the free energy functional.

F [ ρ ] = ∫ f 0 ( ρ ( r ) ) dr

(52)

V

where f0 (ρ ) is the free energy per unit volume of the homogeneous fluid at density ρ = ρ(r ). Note that Fid in eq. (49) is precisely of this form. The variational principle (48) leads to the LDA Euler-Lagrange equation:

f0′ (ρ (r )) = μ − φ (r )

(53)

Taking the gradient of both sides of (53) and remembering that f 0′(ρ ) = [P0 (ρ ) + f 0 (ρ ) ρ ], one finds that (53) is equivalent to the macroscopic condition of mechanical equilibrium:

∇P(r) = −ρ (r)∇φ (r ) The first non-local correction to the LDA was pioneered as far back as 1893 by van der Waals. The intrinsic free energy functional may be formally expanded in powers of the inverse range 1 ξ of the inhomogeneity, i.e. of ∇ρ ρ . Use of symmetry consideration leads then to the “square gradient” approximation.

F [ρ] = ∫

{ f ( ρ (r )) + f ( ρ (r )) ∇ρ (r ) }dr 2

V

0

2

(55)

The coefficient f 2 (ρ (r )) can be determined by considering an inhomogeneity which varies slowly and is of small amplitude; a quadratic expansion in the deviation Δρ (r ) = ρ (r) − ρ 0 then leads to [1]:

f2 (ρ ) =

k BT 12

∫ c(r; ρ )r dr 2

(56)

where c(r;ρ) is the d.c.f. (defined in Eq. 40a) of the bulk fluid of density ρ . Within the RPA, c (r) = −βw(r) and f2 reduces to the constant:

f2 = −

π

m w ( r )r dr ≡ ∫ 3 2 ∞

0

4

(57)

where m is a positive, density-independent constant. The square gradient approximation (55) was first used by van der Waals to calculate the density profile ρ ( z ) of a planar liquid-vapour interface. In vanishing external field, the variational principle applied to the functional (55), with f2 approximated by (57) then leads to the Euler-Lagrange equation:

381

FUNDAMENTALS OF LIQUIDS

m

dW (ρ (z)) d 2 ρ(z) =− 2 dz dρ(z)

(58)

with W (ρ) = − f 0 (ρ ) + μρ = −ω (ρ) (grand potential per unit volume). The mechanical analogy is transparent, with m playing the role of mass, z the role of time, ρ (z) the role of position and W (ρ ) the role of potential energy in Newton’s equation of motion. The non-linear differential equation is subject to the boundary conditions

W ( ρ ( z ) ) → W ( ρb ) = − f 0 ( ρb ) + μρb = P as z → ±∞ where ρ b is the liquid bulk density ρ l (z → −∞)or the vapour bulk density ρg (z → +∞), and P is the bulk pressure. Thanks to the mechanical analogy, eq. (58) is easily integrated [1] to yield:

ρ ( z ) = 12 ( ρl + ρ g ) − 12 ( ρl − ρ g ) tanh

ξ = m / c (ρ − ρ

)

z

(59)

2ξ

l g where is the half-width of the interface, and c is the prefactor in a quartic parametrization of ω (ρ). Note that ξ diverges at the liquid-vapour critical point (where ρ l = ρ g = ρ c ), i.e. the interface vanishes. According to the thermodynamic definition (42), the surface tension γ is the extra free energy per unit area due to the presence of an interface. Accordingly, within the square gradient approximation for the liquid-vapour interface:

{

γ = ∫ dz f o ( ρ ( z ) ) + m2 ⎡⎣ ρ ′ ( z ) ⎤⎦ − fb +∞

−∞

2

}

(60)

which reduces (using Eq. 58) to [1]:

γ = m ∫−∞ [ρ ′(z)] dz +∞

2

(61)

The square-gradient result (Eq. 61) becomes exact in the slow modulation limit (ξ >> ξ 0 ) and follows from a more general, exact expression in that limit. Substitution of Eq. (59) in Eq. (61) shows that, as expected, γ vanishes at the liquid-gas critical point, as γ ~ (T c − T )3 2 ; the measured experimental exponent is somewhat lower than 32, and universal. For rapidly varying inhomogeneities, as in the case of a liquid near a solid, impenetrable substrate, molecular layering occurs, and the square gradient approximation becomes useless. An alternative approximation scheme which is better adapted to rapid modulation, generalizes the bulkmean-field expression (39), based on the separation (28) of the molecular interaction into short-range and long-range components. The corresponding free energy functional reads:

382

J. -P. HANSEN

F [ρ ] = F0 [ρ ] + 12 ∫ dr ∫ dr′ρ (r)w(r′ − r′)ρ (r′)

(62)

If the inhomogeneous reference system is a hard sphere fluid, a very accurate functional F0[ρ] is available, derived from purely geometric (excluded volume) considerations, the so-called “fundamental measure theory” of Rosenfeld [12]. The resulting functional has been successfully applied to the investigation of highly confined liquids and to wetting/drying phenolmena [11]. The surface tension γ is highly material and state-dependent. Near triplepoint conditions it is particularly high for liquid metals, as exemplified by the familiar droplet formation of liquid Hg on most substrates. As an application consider a spherical drop of liquid in equilibrium with an undersaturated vapour, at pressure P < Psat . The pressure inside the drop must be P + ΔP > P to balance the capillary force that tends to reduce the area of the drop. The excess pressure ΔP is related to the drop radius R by Laplace’s relation. A change in R will lead to a change in the volume 4 π R 3 3 and the area 4 πR 2 of the drop. The equilibrium radius R is obtained by expressing that the grand potential (43) must be at its minimum for fixed T and μ :

d Ω = − ( ΔP ) dV + γ dA = ⎡⎣ −ΔP × 4π R 2 + γ × 8π R ⎤⎦ dR = 0 (63) for an arbitrary variation dR. Laplace’s relation hence reads: 2γ

ΔP =

R

(64)

Chemical equilibrium between molecules inside the drop and in the vapour requires ⎛ ⎞ 2γ

μl ⎜ P + ⎝

R

, T ⎟ = μ g ( P, T ) ⎠

(65)

Laplace’s formula (64) also applies to vapour bubbles inside the liquid, with the roles of the 2 phases interchanged. The surface tension γ also controls the amplitude of capillary fluctuations of the interface between 2 phases, wetting and drying phenomena, and capillary condensation of a vapour within narrow pores [6]. 5. Molecular Dynamics and Transport Phenomena Since quantum effects are negligible in most liquids, the translational motion of spherical molecules, and the rotational motion of non-spherical polyatomic molecules, are governed by the classical equations of motion

383

FUNDAMENTALS OF LIQUIDS

derived from the Hamiltonian (1) (for spherical molecules), i.e. Newton’s equation: rij ij

m ri = Fi ( t ) = −∇iVN ({rj} ) = − Σ v′ ( rij ) r j( ≠i )

(66)

where a dot signifies a time derivative. A numerical solution of the N coupled non-linear second order differential equations for the position vectors ri will determine (together with pi = m ri ) the trajectory of the Nparticle system in phase space Γ . This is achieved in Molecular Dynamics (MD) simulations of systems of N particles in a cell of volume V, generally subjected to periodic boundary conditions to eliminate surface effects which are large for typical system sizes 10 2 < N < 10 7 [2]. The differential Eqs. (66) are replaced by finite difference equations with a time-step Δt which is typically of the order of 1fs, sufficiently short compared to characteristic timescales of molecular motion (typically of the order of a few ps). The most popular, time-reversible and stable finite difference algorithm for the numerical solution of (66) is the Verlet algorithm obtained by combining [1,2,13] the Taylor expansions of ri (t + Δt ) and ri (t − Δt ), namely: 2

(Δt ) F t + O Δt 4 ri (t + Δt) = −ri (t − Δt) + 2ri (t ) + ( ) i( ) m

(67)

Iteration of the Verlet algorithm, which conserves the total energy of the system to a high accuracy if Δt is chosen adequately, allows the phase space trajectory to be mapped out over time intervals up to a μs . The considerable amount of resulting microscopic information is then reduced to a manageable size by computing dynamical (time-dependent) correlation functions of the general form:

CAB (t ′, t ′′) = A(t ′)B(t ′′)

(68)

where A and B are dynamical variables, as defined in Section 2, and the statistical average is taken over initial conditions, i.e. with a Boltzmann weight (3) calculated for the positions and momenta of all particles at time t ′′ . For a system in equilibrium, time translation invariance implies that C AB (t ′, t ′′ ) = C AB (t = t ′ − t ′′ ) [1]. The physically most relevant correlation functions are autocorrelation functions (a.c.f.) where B = A * (complex conjugate of A). Just as static (or equal time, t = 0 ) correlation functions, like the pair distribution function or the structure factor, provide a link with macroscopic thermodynamics, so is the link with macroscopic transport coefficients provided by time-dependent correlation functions, within a dynamical generalization of linear response theory, as embodied in the fluctuation-dissipation theorem (27). A simple example is provided by the

384

J. -P. HANSEN

diffusion of a tagged (or tracer) particle through a fluid. The self diffusion coefficient D is defined by the Einstein relation:

r (t ) − r (0)

D = lim

2

(69)

6t

t→∞

now if v (t ) is the instantaneous velocity of the particle, then:

r (t ) − r (0) = r (t ) − r ( 0)

2

=

∫ v(t ′)dt ′ t

(70)

0

∫

t

0

t

v ( t ′ ) dt ′ ⋅ ∫ v ( t ′′ ) dt ′′ 0

t

t′

0

0

= 6 ∫ dt ′∫ dt ′′ Z (t ′ - t ′′ )

(71)

where Z (t ) is the velocity a.c.f. (v.a.f.)

Z (t ) = 13 v (t ) ⋅ v (0)

(72)

= v x (t )v x (0)

due to the isotropy of the fluid. According to the equipartition result (7), the initial value of the v.a.f. is:

Z (0) = 13 v 2 =

k BT m

(73)

Time translation invariance implies that any autocorrelation function is an even function of time; a Taylor expansion at short times thus contains only even powers in t: k T

B Z (t ) = m

{1 − Ω t 2 0

2

2

( )}

+ 0 t4

(74)

A straightforward calculation leads to: 2

F m Ω = v ( 0 ) ⋅ v ( 0 ) = 3k BT 3mk BT 2 0

= =

1 ∇VN 3mk BT

ρ

2

=

1 ∇ 2VN 3m

(75)

g ( r )∇ v ( r ) dr 3m ∫ 2

where F is the total force exerted on the diffusing particle by the fluid molecules, and v (r) is the interaction potential between the tagged particle

FUNDAMENTALS OF LIQUIDS

385

and the fluid molecules. If the tagged particle is one of the fluid molecules, 13 −1 Ω0 , called the Einstein frequency, is typically of the order of 10 sec . Substitution of Eq. (72) into Eq. (71) and into Eq. (70) leads to the Green-Kubo expression for D: ∞

D = ∫ Z ( t )dt

(76)

0

Computer simulations show that Z (t ) decays monotonically to zero at high temperatures, while close to triple point conditions, Z (t ) exhibits a negative region after the initial decay, for times of the order of a few picoseconds. Negative values of Z (t ) signal “backscattering” of the tagged particle, due to the cage formed by the shell of nearest neighbours. In most liquids D ≈ 10–5 cm2/s under triple point conditions. In the case where the tracer particle (e.g. an impurity or Brownian particle) is much heavier than the molecules of the liquid, a separation of timescales occurs naturally, and the deterministic equation of motion for the tracer particle may be replaced by a coarse-grained stochastic equation of motion, namely the Langevin equation [1,6].

M

d v ( t ) = −ξ Mv ( t ) + R ( t ) dt

(77)

where ξ is the friction coefficient measuring the macroscopic force exerted by the liquid on the Brownian particle, while R(t ) is the randomly fluctuating force arising from frequent collisions of the Brownian particle with the molecules. Since on the macroscopic scale the total force reduces to the friction force,

R(t ) = 0

(78)

The separation of time scales is reflected in the assumption of an infinitely short correlation time, i.e.

R ( t + s ) ⋅ R ( s ) = 2π R0δ ( t )

(79)

and in the assumption that R(t ) is uncorrelated with the velocity v at any earlier time:

R(t ) ⋅ v = 0, t > 0

(80)

Equation (77) may be integrated to yield the Brownian particle velocity at time t: t

M v ( t ) = M v ( 0 ) e −ξ t + e −ξ t ∫ eξ s R ( s )ds 0

(81)

386

J. -P. HANSEN

Squaring Eq. (81) and taking the statistical average one obtains, with the help of Eqs. (79) and (80):

v (t )

2

π Ro 1− e−2ξ t ] 2 [ M ξ

2

= v(0) e−2ξ t +

(82)

In the limit t → ∞ , the Brownian particle will be in equilibrium with the bath at temperature T, so that

v (∞)

2

= 3k BT M

Substitution in Eq. (82) leads to the Green-Kubo relation for the friction coefficient ξ

ξ=

πβR0 3M

=

β 3M

∫

∞ 0

R(t ) ⋅ R dt

(83)

The v.a.f. is easily derived by projecting Eq. (81) on the initial velocity v (0), taking a statistical average and exploiting Eq. (80): k T Z (t ) = 13 v(t ) ⋅ v (0) = B e−ξt M

(84)

Note that Eq. (84) is valid only for t > 0; Z (t ) is not time reversible, because of the macroscopic dissipation associated with the friction term in the Langevin Eq. (77). Substitution of Eq. (84) into Eq. (76) leads to the following relation between the diffusion constant and the friction coefficient:

D=

k BT ξM

(85)

Now the friction coefficient exerted by a liquid of shear viscosity η on a macroscopic sphere of diameter d dragged at constant velocity v through the fluid is given by Stokes’ law [14]:

ξ=

n πη d M

(86)

where n = 3 for stick and n = 2 for slip boundary conditions at the surface of the sphere. Combination of (85) and (86) leads to the Stokes-Einstein relation:

Dη =

k BT nπ d

(87)

FUNDAMENTALS OF LIQUIDS

387

It is a remarkable feature of this law that, although it is derived from purely macroscopic considerations, it provides a good empirical correlation of experimental data for the self-diffusion coefficient D of molecules and the viscosity η of simple liquids. Returning to the case where the tagged particle is identical to the bath molecules, the Langevin Eq. (77) no longer applies and memory effects (corresponding to non-locality in time) become important. A time-dependent generalization of linear response theory (embodied in Eq. (27) in the static case) allows macroscopic transport coefficients to be expressed in terms of time integrals of a.c.f.’s of appropriate microscopic dynamical variables [1,15], similar to Eq. (76) for the self-diffusion coefficient D. Thus, in a system containing charged particles (e.g. in ionic solutions) the frequency-dependent conductivity σ (ω ) in Ohm’s law, relating the electric current to an alternating applied electric field, is expressible in terms of the a.c.f. of the microscopic electric current: N

˜j(t ) = Σ qivi (t )

(88)

i =1

according to:

σ (ω ) =

1 3Vk BT

∫ ˜j(t )⋅ ˜j(0) e ω dt ∞

i t

0

(89)

Similarly, the shear viscosity η of a liquid is related to the a.c.f. of the offdiagonal elements of the stress tensor (9):

η=

1 VkB T

∫

∞ o

π oxy (t )π oxy (0) dt

(90)

Finally the thermal conductivity is given by the time integral of the a.c.f. of a properly defined microscopic heat current [1]. In practice the required a.c.f.’s are usually calculated from MD simulations. At high temperatures and moderate densities, transport coefficients may be calculated by approximate kinetic equations generalizing the familiar Boltzmann equation (valid for dilute gases), like the Enskog equation [16]. For most liquids, the measured self-diffusion coefficient D obeys an Arrhenius law:

D ≈ Do exp{−T To}

(91)

where k B To is an activation energy associated with the caging of a molecule by its neighbours. Combining (91) with (87), one finds that the liquid viscosity decreases with increasing temperature according to:

388

J. -P. HANSEN

η ≈ ηoT exp{To T}

(92)

6. Liquid Metals The key difference between metals and insulating materials is that the former are intrinsically two-component systems, made up of positive ions (e.g. Na+ or Al3+) and valence (or conduction) electrons. In “simple” metals, the latter interact weakly with the ions and behave, in the first approximation, as a quantum gas of free electrons. If ρ e is the number of valence electrons per unit volume, a commonly used dimensionless parameter characterizing the density is: 1

rs

(3 4 π ρe )3 = a0

(93)

where a0 is the Bohr radius. The Fermi energy ε F is the energy of the highest occupied energy level in the ground state of a gas of fermions: 2

εF =

= 2 ( 3π 2 ρe ) 3 2me

=

50.1 eV rs2

(94)

and the associated Fermi temperature is:

TF =

εF kB

=

58.2 × 10 4 K rs2

(95)

In liquid metals, under triple point conditions, 2 ≤ rs ≤ 5 , so that TF ≥ 2 × 10 4 K >> T ; in other words the valence electrons may be considered as being in the ground state. Since the ionic core radius rc is usually only a small fraction of the mean interionic spacing the ion cores occupy less than 10% of the volume of a liquid metal. In the “nearly-free electron” picture the conduction electrons are assumed to move more or less freely through the remaining volume (typically 90%) of the liquid, interacting only rarely with the ions, so that their mean free path is typically 10-100 times larger than the mean inter-ionic distance. The total Hamiltonian of a system of N i ions of valence z > 0 and N e = zN i electrons is:

H = H i + H e + Vei

(96)

where Vei is the ion-electron interaction term. All interactions are pair-wise Coulombic, except for the ion-electron coupling when r < rc . The combined

FUNDAMENTALS OF LIQUIDS

389

effects of the Coulombic interactions between valence and core electrons, of the attraction between the valence electrons and the ion nucleus, and of the Pauli exclusion principle, may be approximately described by a weak ionelectron pseudopotential [17,18], allowing the ion-electron interaction energy to be treated by perturbation theory, using the de-coupled mixture of interacting classical ions and of interacting quantum valence electrons as reference system [1,18]. To first order in the perturbation Vei , the total free energy reduces to

F (N i , N e ,V ) = Fi (ρ i , T ) + U e (ρ e , T = 0) + V0 (ρ i , ρ e )

(97)

where Fe (ρ e , T ) ≈ U e (ρ e , T = 0) since TF >> T , and V0 is a “volume” term, independent of the microscopic structure of the reference ionic and electronic systems, which is unaffected by the ion-electron coupling within first order theory. To second order in perturbation theory, the structure of the ionic fluid is modified by the linear response of the degenerate electron fluid to the ionic charge distribution, i.e. the ion-ion interactions are no longer purely Coulombic, but are screened by the polarization of the valence electrons. This leads to an effective atom representation, where the ions behave like neutral atoms, “dressed” by a valence electron cloud; the effective ion-ion interaction is now short-ranged, but depends on the valence electron density, i.e. on rs . The techniques (integral equations, thermodynamic perturbation theory) introduced in Section 3 can now be applied to the one-component system of effective atoms, provided the electronic contributions, and the volume term V0 are added to the calculated thermodynamic properties. The effective atom representation also leads to accurate predictions of the ionic structure, as measured by neutron diffraction experiments [19]. The liquid range of most metals is much larger than that of insulators, with a ratio of critical over triple point temperatures, Tc Tt ≈ 6 , compared to a ratio close to 2 for most other liquids. This appears to be a consequence of the fact that the liquid-vapour transition coincides with a metal-insulator transition, since the vapour phase of metals contains atoms or molecules and no longer nearly free valence electrons. The shear viscosity η of liquid metals is essentially determined by the ion dynamics, via. eq. (90), while the much more mobile valence electrons control the electrical and thermal conductivities σ and λ . The latter transport coefficients may be calculated from kinetic theory, taking into account the scattering of the electrons by the ions. The familiar Drude model provides a justification of the empirical Wiedemann-Franz law relating λ and σ : 2

λ π2 ⎛k ⎞ = L = ⎜ B ⎟ = 2.44 ×10−3WΩK −2 σT 3⎝e⎠

(98)

390

J. -P. HANSEN

A more complete kinetic theory calculation leads to Ziman’s formula for −1 the resistivity σ of a liquid metal [20]. 7. Water Water is the most common liquid under terrestial conditions, and of considerable technological importance, but it has anomalous behaviour in many respects, and its molecular description is complicated by the fact that it is both a highly polar and an associating liquid, with strong local directional ordering attributable to O-H––O hydrogen bonds. Some of its anomalous properties are the following: •

•

•

•

•

•

Water is one of the very few substances that contract on melting, by nearly 10%, i.e. Vliq − Vsol Vliq ≈ −0.09 for a given mass of water.

(

)

The density of liquid water increases between T = 0D C and T = 4D C before finally decreasing at atmospheric pressure.

The isothermal compressibility χ T of liquid water decreases between T = 0D C and T = 46D C before finally increasing with temperature. Liquid water has unusually high specific heats Cv and Cp (≈ 18 cal/mole K) D For T < 20 C , the shear viscosity η decreases with increasing pressure up to P ≈ 1500 bar, and increases only at higher pressures. X-ray diffraction experiments point to a low coordination number, ν ≈ 4.4 (close to the tetrahedral coordination in ice) compared to ν ≈ 10 for most non-associating liquids.

Liquid water also has a very high dielectric permittivity ( ε ≈ 78 at room temperature) compared to most other polar liquids. These anomalous properties can be traced back to the strong directional hydrogen bonds which favour local tetrahedral ordering of the molecules. Hydrogen bonds are long-lived at low temperatures (near freezing) but gradually break up as the temperature increases, in particular at vapourization, explaining the high values of the specific and latent heats. There exists a large number of intermolecular interaction models for water [21]. One of the simplest and most popular is the rigid SPC/E model [22]. It is of the interaction site form, i.e. interaction sites are associated with the O and two H nuclei in each molecule. A partial charge is associated with each site, viz q = −αe with the O site, and q = + αe 2 with each H site (α = 0.8476). The intramolecular OH bond length is 0.1nm, while the HOH angle is taken to be the tetrahedral angle θ = 109.5D ; the resulting

FUNDAMENTALS OF LIQUIDS

391

dipole moment is μ = 2.3 debye. The total interaction energy between 2 water molecules is then

ν (1, 2) = Σ Σ ν αβ (r1α − r2 β ) α β

(99)

where riα (1 ≤ i ≤ 2,1 ≤ α ≤ 3) is the position of site α associated with molecule i. The site-site potentials are of the form:

ν αβ (r) = ν 0 (r )δα ,0δβ ,0 +

qα qβ 4 πε0 r

(100)

where ν 0 (r) is a LJ potential (2) acting only between O atoms. The theoretical methods developed for simple liquids (presented in Section 3) are not easily adapted to water because of the strength of the Hydrogen bonds. Some success for thermodyamic properties can be achieved within theories specifically designed for strongly associating liquids, like SAFT [23,24], based on ideas by M. S. Wertheim [25]. These theories do not give access to the molecular pair structure. The latter, as well as dielectric properties, a.c.f.’s and transport coefficients can be obtained from MD or MC simulations based on potential models like the SPC/E potential (99-100). Such simulations lead to reasonable predictions of most experi-mentally measured properties of liquid water. Improvements are made by accounting for the electric polarizability of the water molecules, leading to many-body induction forces [26].

References [1] [2] [3] [4] [5] [6] [7] [8]

Hansen, J. P., and McDonald, I. R., 2006, Theory of Simple Liquids, 3d edition, Academic Press, London. Frenkel, D., and Smit, B., 2002, Understanding Molecular Simulation, 2d edition, Academic Press, San Diego. Maitland, G. C., Rigby, M., Smith, E. B., and Wakeham W.A., 1981, Intermolecular Forces, Clarendon Press, Oxford. See e.g. Chandler, D., 1987, Introduction to Modern Statistical Physics, Oxford University Press, Oxford. Hansen, J. P., and Verlet, L., 1969, Phase Transitions of the Lennard-Jones System, Phys. Rev. 184:151-161. Barrat, J. L., and Hansen , J. P., 2003, Basic Concepts for Simple and Complex Liquids, Cambridge University Press, Cambridge. Carnahan, N. F., and Starling K. E., 1969, Equation of State for Nonattracting Rigid Spheres, J. Chem. Phys. 51:635-636. Barker J. A., and Henderson, D., 1967, Perturbation Theory and Equation of State for Fluids. II. A Successful Theory of Liquids, J. Chem. Phys. 47:4714-21; 1976, What is “liquid”? Understanding the states of matter, Rev. Mod. Phys. 48:587-671.

392 [9] [10] [11] [12]

[13] [14] [15] [16] [17] [18] [19]

[20] [21] [22] [23] [24]

[25] [26] [27] [28]

J. -P. HANSEN Weeks, J. D., Chandler, D., and Andersen, H.C., 1971, Role of Repulsive Forces in Determining the Equilibrium Structure of Simple Liquids, J. Chem. Phys. 54:5237-47. Rowlinson, J. S., and Swinton, F., 1983, Liquids and Liquid Mixtures, 3d edition, Butterworths, London. Evans, R., 1991, in Fundamentals of Inhomogeneous Fluids, D. Henderson ed., Marcel Dekker, New York. Rosenfeld, Y., Levesque, D., and Weis, J. J., 1990, Free-energy model for the inhomogeneous hard-sphere fluid mixture: Triplet and higher-order direct correlation functions in dense fluids, J. Chem. Phys. 92:6818-32; Rosenfeld, Y., Schmidt, M., Löwen, H., and Tarazona, P., 1997, Fundamental-measure free-energy density functional for hard spheres: Dimensional crossover and freezing, Phys. Rev. E55: 4245-63. Verlet, L., 1967, Computer “Experiments” on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules, Phys. Rev. 159:98-103. Landau, L. D., and Lifshitz, E. M., 1987, Fluid Mechanics, 2d edition ButterworthHeinemann, Oxford. Kubo, R., 1966, The fluctuation-dissipation theorem, Rep. Prog. Phys.29:255-284. Résibois, P., and De Leener, M., 1977, Classical Kinetic Theory of Fluids, John Wiley, New York. Heine, V., 1970, The pseudopotential concept, Sol. State Phys. 24:1-36. Ashcroft, N. W., and Stroud, D., 1978, Sol. State Phys. 33:1. For a recent review of experimental data on liquid metals, see Scopigno, T., Ruocco, G., and Sette, F., 2005, Microscopic dynamics in liquid metals: The experimental point of view, Rev. Mod. Phys. 77:881-933. Ziman, J. M., 1961, A theory of the electrical properties of liquid metals. I: The monovalent metals, Phil. Mag. 6:1013-34. Guillot, B., 2002, A reappraisal of what we have learnt during three decades of computer simulations on water, J. Molec. Liquids 101:219-260. Berendsen, H. J. C., Grigera, J. R., and Straatsma T. P., 1987, The missing term in effective pair potentials, J. Phys. Chem. 91:6269-71. Jackson, G., Chapman W. G., and Gubbins, K. E., 1988, Phase equilibria of associating fluids. Spherical molecules with multiple bonding sites, Molec. Phys. 65:1-31. For a review, see Müller, E. A., and Gubbins, K. E., 2001, Molecular-Based Equations of State for Associating Fluids: A Review of SAFT and Related Approaches, Ind. Eng. Chem. Res. 40:2193-2211. Wertheim, M. S., 1984, Fluids with highly directional attractive forces. I. Statistical thermodynamics, J. Stat. Phys. 35:19-34. Sprik, M., and Klein, M. L., 1988, A polarizable model for water using distributed charge sites, J. Chem. Phys. 89:7556-60. Greenfield, A. J., Wellendorf, J., and Wiser, N., 1971, X-ray determination of the static structure factor of liquid Na and K, Phys. Rev. A 4(4):1607-1616. Hansen, J. P., and Schiff, D., 1973, Influence of interatomic repulsion on the structure of liquids at melting, Molec. Phys. 25:1281-1290.

INTRODUCTION TO INTERFACES AND DIFFUSION P. WYNBLATT* Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA

Abstract: This chapter covers selected fundamental topics in Materials Science, with the purpose of introducing certain concepts that will serve as a foundation for other chapters in this volume. The topics addressed include: interfacial Gibbsian thermodynamics, with special emphasis on adsorption (or interfacial segregation) phenomena; statistical thermodynamic treatments of interfacial segregation, with examples of the relationship between interfacial composition and thermodynamic properties of two-component materials, as well as the consequences of variations in interfacial composition on interfacial energy; anisotropy of interfacial energy and the effects of composition thereon; grain boundary structure and energetics; interfacial equilibrium in solid-vapor, solid-liquid and solid-solid systems; and finally a brief overview of diffusion in solid metals which introduces both the mathematical framework as well as atomistic aspects of diffusion mechanisms. Keywords: Interfacial thermodynamics; adsorption; segregation; interfacial energy and anisotropy; grain boundaries; interfacial vector equilibrium; diffusion in metals

1. Forward Interfaces are generally defined as those regions of a microstructure that separate two different phases. Interfaces can occur in either pure or multicomponent materials. For example, in pure materials, the region separating the pure liquid and its vapor may be referred to as the liquid-vapor interface, or more simply, as the liquid surface. But an interface may also separate the different solid phases of a given pure element, such as, for example the α and γ phases of pure iron, which coexist at a temperature of 912°C at atmospheric pressure (see Figure 1). Interfaces between two solid phases are referred to as solid-solid interfaces, and also as interphase interfaces or as interphase boundaries. Whereas phases in pure materials coexist only at one temperature (at a given pressure), phases in multi-component systems can coexist over a

______ *

Paul Wynblatt, MSE Dept. Carnegie Mellon University, Pittsburgh, PA 15213, USA; e-mail: [email protected]

V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

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range of temperature and composition, at fixed pressure (as shown for example in Figure 1 in the α + γ coexistence domain).

Figure 1. Partial Fe-C phase diagram.

In the examples cited above, the interfaces separate phases that differ either in structure (e.g. the α bcc and γ fcc phases of pure iron), or in composition (e.g. the α and β phases in the Cu-Ag phase diagram shown in Figure 2, which are both fcc), or in both structure and composition (e.g. the α and γ phases in Figure 1). This is the conventional definition of an interphase boundary, i.e., a region in the microstructure that separates phases which differ in structure and/or in composition. In this chapter, we shall deal with one other type of interface, namely, grain boundaries. This type of interface separates regions of the same phase that have identical structure and composition, but merely differ in crystallographic orientation. The topics to be covered in this chapter include the thermodynamics of interfaces, the phenomenon of interfacial segregation (or adsorption), grain boundaries and their anisotropy, the connection between segregation and wetting at grain boundaries, and a brief overview of diffusion in solids. 1100 1000 L

T, °C

900

β+L

α+L

800

α

β

779°C

700 α+β 600 500 400

0

20

40

60

80

wt % Cu

Figure 2. Ag-Cu phase diagram.

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2. Interfacial Thermodynamics In treating this subject, we will follow the approach developed by Gibbs (Gibbs 1961). However, Gibbs is not easy to read, and so it is convenient to use other sources which have "translated" Gibbs' work into a more modern thermodynamic language, such as the papers by Hirth (Hirth 1965, Hirth 1973) and by Mullins (Mullins 1963). Much of what is covered in this chapter on the topic of interfacial thermodynamics has been excerpted from a recent review of interfacial segregation (Wynblatt and Chatain 2006). 2.1. INTERFACIAL EXCESS PROPERTIES

The interfacial energy is denoted by γ (with units of [energy/(unit area)], typically mJ/m2), and is defined as the reversible work needed to create unit area of surface, at constant temperature, volume (or pressure), and chemical potentials. We begin by reviewing the thermodynamics of interfaces, using the liquid-vapor surface as an example, i.e., we consider a system composed of a liquid and a vapor phase at equilibrium, separated by an interface. The interface is generally not perfectly sharp, i.e. there is a transition region separating the two phases where the local density changes from that of the liquid to that of the vapor, as shown schematically in Figure 3. In order to avoid having to define the diffuseness of the interface, Gibbs defined a mathematical plane, known as the dividing surface, onto which he projected all the properties associated with the presence of the interface in the system. To do this for a given property, he subtracted the property obtained in a hypothetical system consisting of the two bulk phases, assumed to extend uniformly all the way to this mathematical plane (shown by dashed lines in Figure 3), from the desired property of the real system containing the interface (shown by the solid line in the figure).

Figure 3. Schematic of density variation across a liquid-vapor interface.

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The property obtained by this subtraction process is referred to as the interfacial excess property. As an example, we write the interfacial excess s internal energy of the system, E , as:

E S = E − E ′ − E ′′

(1)

where E is the real system internal energy, E' is the internal energy of phase ', and E" is the internal energy of phase ". This shows how the excess property is just the property of the real system, less the properties of the bulk phases (hypothetically extended to the dividing surface). It is important to note that the position of the dividing surface is arbitrary. It can be placed at any convenient location within the bounds of the physically diffuse region. Some consequences of this arbitrariness are addressed below. Every thermodynamic property of the system can be written in the same way as in Eq. (1); e.g. the number of moles of component i in the real s system is expressed as: ni = ni' + ni" + ni . The only exception is that the volume of the system is written: V = V' + V", i.e., there is no excess volume associated with the interface. The internal energy of any phase (say phase') is conveniently considered to be a function of the extensive variables entropy, S', volume, V' and number of moles, ni'. Then:

⎛ ∂E ′ ⎞ ⎛ ∂E ′ ⎞ dE ′ = ⎜ ⎟ dS ′ + ⎜ ⎟ dV ′ + ∑ ⎝ ∂S ′ ⎠V ′,n i′ ⎝ ∂V ′⎠ S ′,n i′ i

or

⎛ ∂E ′ ⎞ dn ′i ⎜ ⎟ ⎝ ∂n ′i ⎠ S ′,V ′,n i′ ≠i

dE ′ = TdS ′ − PdV ′ + ∑ μ′i dn ′i

(2)

i

where T is the temperature, P is the pressure, and μ i′ is the chemical potential of component i in the ' phase. This is the standard expression for the internal energy of a single uniform phase containing more than one component. Similarly, the internal energy of phase " is:

dE ′′ = TdS ′′ − PdV ′′ + ∑ μ′i′dn ′i′

(3)

i

For the surface (which does not have any excess volume) the excess internal energy is written:

dE s = TdS s + γdA + ∑ μisdn is

(4)

i

This shows that in the expression for the surface excess internal energy, the PdV term is replaced by a term consisting of the surface energy, γ, multiplied by a change in area, dA. Since, for a system in equilibrium, the

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chemical potential must be equal everywhere, we impose the condition: μi'= μi"= μis. Substituting for dE', dE" and dEs into Eq. (1), we obtain:

dE = T (dS′ + dS′′ + dS s )− P (dV ′ + dV ′′) + γdA + ∑ μi (dn′i + dn′i′ + dnis ) i

or

dE = TdS − PdV + γdA + ∑ μ idni

(5)

i

This is an important result. It shows that in the limit of a system where the contribution of the surface energy is small with respect to the bulk energy, the γdA term becomes negligible and Eq. (5) reduces to the standard thermodynamic expression for the internal energy. All other thermodynamic properties of a system with a surface can be derived from this expression. For example, the Gibbs free energy is given by: G = E + PV - TS, and its derivative may be written:

dG = dE + PdV + VdP − TdS − SdT = −SdT + VdP + γdA + ∑ μi dn i (6) i

and

dG s = −S sdT + γdA + ∑ μi dn is

(7)

i

2.2. GIBBS ADSORPTION EQUATION

We now return to Eq. (4) above, in which dE s is expressed only in terms of extensive independent variables (dS s, dA, dni s) while the intensive variables (T, γ, μi) are constant in the equilibrated system. Thus, we can integrate Eq. (4) to obtain:

E s = TS s + γA + ∑ μi n is

(8)

i

Re-differentiating this result yields:

dE s = TdS s + S sdT + γdA + Adγ + ∑ μi dn is + n isdμi i

But, by Eq. (4) dE s = TdS s + γdA +

∑ μ dn i

s i

i

⇒ S sdT + Adγ + ∑ n isdμi = 0 i

We now define the following new quantities:

ss ≡ S s / A , and Γi ≡ n is / A

(9)

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Quantities expressed per unit area are known specific interfacial excess quantities, e.g. ss is known as the specific interfacial excess entropy, and Γi [moles/(unit area)] is the specific interfacial excess number of moles of component i. However to avoid this cumbersome nomenclature, Γi is generally referred to simply as the adsorption of component i. With these definitions we can rewrite Eq. (9) as:

dγ = −ssdT − ∑ Γi dμi

(10)

i

This is the well-known Gibbs adsorption equation. It gives the variation of γ with changes in T and μI , where μi can easily be related to the composition of the bulk phases. Now that adsorption has been defined, it is useful to discuss briefly the relationship between interfacial segregation and adsorption. This issue has been discussed previously, for example, by Hondros and Seah (Hondros and Seah 1977). Historically, the term adsorption was first used to describe the presence of excess components at a surface that originated from the gas phase. In contrast, the term segregation was used to describe excess components at a surface that originated from the bulk of the material. In a system at equilibrium, consisting of a condensed phase in contact with a gas phase, the origin of the adsorbed or segregated component at the surface has no effect on the quantity adsorbed. For our purposes, therefore, these terms can be considered identical. However, it should be mentioned that the ways in which adsorption energy and segregation energy are defined, are not identical (Wynblatt and Chatain 2006). We now return to Eq. (10). At constant temperature, this equation simplifies to the Gibbs adsorption isotherm. In the case of a twocomponent system, we can write:

dγ = −Γ1 dμ1 − Γ2 dμ2

(11)

The Gibbs-Duhem equation for one of the two bulk phases, say phase ', can be expressed as:

n1′ dμ1 + n ′2 dμ2 = 0

(12)

which is valid either at constant T and P, or at constant T and V. Equation 12 can be used to eliminate one of the chemical potentials in Eq. (11). By convention, the chemical potential of the solvent (majority component, labeled 1) is eliminated, and the Gibbs isotherm is rewritten in terms of the solute (minority component, labeled 2). Thus, ⎞ dγ ⎛ n2′ = ⎜ Γ1 − Γ2 ⎟ dμ2 ⎝ n1′ ⎠

(13)

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Here, dγ dμ2 , is a measurable quantity, and therefore cannot depend on an arbitrary choice of the dividing surface. On the other hand, Γ1 and Γ2 do depend on the position of the dividing surface. Thus, the right hand side of Eq. (13): [Γ1n'2/n'1-Γ2], i.e., this particular combination of adsorption variables, must also be independent of the position of the dividing surface. In the case of interfaces between a condensed phase and a vapor phase, the difference in density across the interface makes it possible to select the position of the Gibbs dividing surface so as to make one of the adsorption terms vanish. In those cases, Eq. (13) can be simplified by choosing the position of the dividing surface so as to make Γ1 = 0. Under these conditions, Γ2 is no longer arbitrary (since we have now made a specific choice for the position of the dividing surface) and is written Γ2,1 to indicate that the choice Γ1 = 0 has been made: dγ dμ 2 = −Γ2,1

(14)

Γ2,1 is sometimes referred to as the relative adsorption (Defay et al. 1966). The value of Γ2,1 (or of any adsorption term) can be either positive or negative. From Eq. (14), it can be seen that if the adsorption of component 2 is positive (i.e. there is a positive excess of component 2 at the surface of the condensed phase) then dγ dμ 2 will be negative, and the surface energy of the solid will decrease as the chemical potential, and hence the bulk concentration of component 2, is increased. However, the simplification shown here for the case of interfaces between a condensed phase and a vapor cannot be applied to grain boundaries, or other interfaces between two condensed phases. Thus, in those cases, more complete versions of the Gibbs adsorption isotherm, such as Eqs. (11) or (13) must be used. The term dμ2 in Eqs. (13) or (14) can be transformed using the definition of chemical potential in terms of activity, a: μ2= μ2° + kT ln a2, where μ2° is the chemical potential in the standard state. In the case of ideal solutions, Raoult's law can be applied: a2 = x2, where x2 is the atom fraction (or mole fraction) of the solute; and in dilute solutions, Henry's law: a2 = kox2, where ko is a constant, can be applied. In either case, dμ2 = RT d(ln x2). Therefore, in these particular limits, the above relationships may be simplified as follows:

or

⎛ n′ ⎞ 1 dγ = ⎜ Γ1 2 − Γ2 ⎟ RT d ln x2 ⎝ n1′ ⎠

(15)

1 dγ = −Γ2,1 RT d ln x2

(16)

Thus, we can obtain the adsorption, Γ2,1 (which is related to the interfacial composition) from the dependence of the surface energy on bulk

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composition. In principle, we can also obtain the change in surface energy that results from the adsorption of solute. To do this, it is necessary to integrate the Gibbs isotherm, which requires some knowledge of the dependence of Γ2,1 on x2. Unfortunately, Gibbsian thermodynamics does not provide a suitable functional form for Γ2,1(x2). In the next section, we describe models that provide this needed functional relationship. 3. Statistical Thermodynamic Models of Segregation 3.1. USEFULNESS OF STATISTICAL MODELS

We have seen that the Gibbs adsorption isotherm relates the adsorption to the variation of interfacial energy with bulk composition (Γ2,1 = – dγ /dμ2). However, there are several features which detract from the usefulness of this relation. •

Neither γ, nor its variation with composition, are easy to measure in solids.

•

Γ2,1 is difficult to measure directly, because interfacial composition profiles can extend some distance from the interface, and one must determine the composition profiles of both components in the most general case.

•

Unless the relationship between Γ2,1 and μ2 is known, the Gibbs adsorption isotherm cannot be integrated.

For these reasons, alternative approaches have been sought to complement the Gibbsian formalism. These approaches have made use of the concepts of statistical thermodynamics, and go back to the work of Fowler and Guggenheim (Fowler and Guggenheim 1939), and Ono and Kondo (Ono and Kondo 1960). 3.2. THE MCLEAN AND RELATED MODELS

In the materials literature, the first treatment of interfacial segregation was put forward by McLean for the case of grain boundaries (GBs) (McLean 1957). The McLean formalism has also been used extensively to interpret segregation phenomena at other interfaces, such as liquid and solid surfaces and interphase boundaries. Like many other treatments, McLean's model used a monolayer representation of the interface. As was later pointed out by Defay et al. (Defay et al. 1966), models that employ a monolayer interface are not strictly consistent with Gibbsian adsorption, i.e. interfaces must be represented by a multilayer formalism in order to be compatible

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with the Gibbs formalism (see Figure 4). Nevertheless, we shall use the McLean model here for some illustrations.

Figure 4. a) Schematic of the monolayer model as applied to a two-component surface: lightcolored circles represent atoms of the segregating species, which is enriched in the topmost atom layer of the surface, and darker circles represent the other species, b) Plot of segregation profile at a (100) surface using either a monolayer or a multilayer model; the bulk atom fraction of the segregating species is taken to be x = 0.1.

In his model, McLean adopted a regular solution approximation, in which the free energy of the system (bulk plus interface) is assumed to consist of an entropy of mixing contribution, written in the ideal solution limit, and a non-zero energy of mixing contribution. In the regular solution approximation, the species in the interfacial and bulk regions are thus assumed to be randomly distributed, and the total entropy (in a binary system consisting of solvent and solute species) is written:

Sid = −R[n s x s ln x s + n s (1− x s )ln(1− x s ) + Nx ln x + N(1− x)ln(1− x)]

(17)

where xs and x are the atom fractions, and ns and N are the numbers of atoms, of the segregating component in the interface and the bulk, respectively. Whenever this ideal solution entropy approximation is used, the resulting expression for the composition of the interface adopts the following form: xs 1− x s

=

⎧⎪ ΔE ⎫⎪ x exp⎨− seg ⎬ 1− x ⎩⎪ RT ⎭⎪

(18)

where ΔEseg is the energy of segregation. For a substitutional solution, ΔEseg represents the internal energy change associated with the exchange of a segregant atom in the bulk with an atom of the other species located at the interface. It should be noted that xs is not identical with the Gibbsian Γ. The relation between these quantities is:

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P. WYNBLATT

Γ = ( x s − x) / σ

(19)

where σ is the area per mole at the (monolayer) interface. McLean assumed that the driving force for segregation was the complete elimination of the bulk elastic strain energy, ΔEel (defined as a positive quantity), associated with a misfitting solute atom in the matrix of the solvent, when a solute atom in the bulk is exchanged with a solvent atom in the surface, i.e.: ΔEseg = −ΔEel

(20)

Since only the solute can segregate to the interface in McLean's model, xs necessarily represents the atom fraction of solute at the interface. However, the segregating species is not always the solute, as it is possible for the solvent to segregate (a case that is sometimes referred to as desegregation, and which is equivalent in the Gibbsian description to a negative value of Γ for the solute). A more complete version of the McLean approach was subsequently developed, in which both chemical as well as solute elastic strain energy driving forces were considered (Wynblatt and Ku 1977, Wynblatt and Ku 1979). In this formalism, the energy of segregation takes on the following form:

[

]

ΔEseg = (γ B − γ A ) σ + 2ω z A ( x − x s ) + z v ( x − 1 ) − ΔEel 2

(21)

where γA and γB are the surface energies of the pure A and B components in the binary solution (component B taken to be the solute), and ω is the regular solution parameter for the AB solution. ω is defined as: ω = εAB – (εAA + εBB )/2, where εAB , εAA , and εBB (taken to be negative quantities) are the energies of bonds connecting A-B, A-A and B-B neighboring pairs of atoms, respectively. z A and z v are the numbers of in-plane bonds and half of the out-of-plane bonds of an atom in the surface plane, such that the total coordination number of an atom in the system is given by: z = z A + 2z v . To avoid possible confusion, species B is taken to be the solute, and the atom fractions x and xS are also taken to relate to the solute. In Eq. (21), the solute strain energy term can be expressed, for example, by a relation due to Friedel (Friedel 1954), as: ΔEel =

24 πK BG A rB (rA − rB )2 3KrB + 4GrA

(22)

where KB is the bulk modulus of the solute, GA is the shear modulus of the solvent, and rB and rA are the atomic radii of the pure solute and solvent atoms, respectively.

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Multi-layer models using a formalism similar to that of Eq. (21) have also been developed (e.g. (Wynblatt et al. 1998, Wynblatt and Chatain 2006)). One such model was used to compute the multi-layer segregation result illustrated in Figure 4. However, the details of the multi-layer model will be omitted here for the sake of simplicity. The model of Eq. (21) shows that there are three principal contributions responsible for segregation to surfaces or grain boundaries of alloys. There is a "chemical" driving force, which consists of two terms: one which depends on the difference between the surface energies of the pure components (i.e. a surface energy driving force) and a second one which depends on the regular solution constant (i.e. an interatomic interaction driving force). The latter vanishes in the case of ideal solutions (for which ω = 0). The third driving force is the one identified by McLean, namely an elastic strain energy contribution associated with the degree of misfit of the solute in a solid solution. This term also vanishes in an ideal solution. The extent of segregation depends not only on the magnitude of the three contributions mentioned above, but also on their signs. Negative contributions to the heat of segregation increase the value of xs in Eq. (18), whereas positive ones decrease xs. Since ΔEel (Eq. 22) is always positive, the solute strain energy contribution to the heat of segregation always favors solute segregation. In contrast, the two chemical terms can be either positive or negative for the solute. The surface energy of the pure solute can either be greater or smaller than that of the solvent, and a smaller surface energy will tend to promote segregation of that component. Similarly, a positive regular solution constant, ω, will add to the driving force for solute segregation. Thus, it is the sign of ΔEseg, arising from the sum of the three contributions, that will determine whether the solute or the solvent will tend to segregate to the interface. Furthermore, segregation will be strongest when all three contributions are negative. 3.3. EXPECTED TRENDS

An as example, we use the case of surface segregation. Figure 5 is a plot of ln(xs/1-xs) (i.e., the logarithm of the left hand side of Eq. (18)) versus 1/T, for various values of ΔEseg, as defined in Eq. (21). In the plots, the surface energies of the pure components, and the solute elastic strain energy have been kept constant, but the regular solution parameter, ω, has been changed from positive, to zero, to negative. Consideration of Eq. (21) shows that when ω = 0, ΔEseg becomes independent of surface composition. As a result, the line in Figure 5 corresponding to that condition is a straight line. For the other two cases, with finite ω, the dependence of ln(xs/1-xs) on 1/T is no longer a straight line, but varies with temperature through the

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dependence of xs on temperature. Furthermore, the figure shows that segregation is stronger for positive ω and weaker for negative ω. Positive ω corresponds to solid solutions with a tendency to cluster, in which the solute (and solvent) atoms prefer to be surrounded by atoms of their own species.

Figure 5. Dependence of surface segregation on temperature for solutions with a bulk composition of x = 0.01. The solute strain energy and surface energy driving forces are held constant, but the regular solution constant, reflecting interatomic interactions, is varied.

At suitable temperatures, this tendency will lead to phase separation into an A-rich phase and a B-rich phase, as in the case shown in Figure 2 for AgCu alloys. This incipient tendency to phase separation means that once some solute atoms are driven to the interface (by the surface energy and/or elastic energy driving forces) then it becomes energetically favorable for more solute atoms to be present there. Conversely, for cases where ω is negative, the solution will tend to order, i.e., atoms of one species will prefer to be surrounded by atoms of the other species. This tendency makes high concentrations of like atoms at an interface energetically less favorable, thereby decreasing the strength of segregation. Also, note that the value of ln(xs/1-xs), in the limit of infinite temperature (1/T = 0), is identical for all three cases. From Eq. (18), it can be seen that as 1/T tends to 0, ln(xs/1-xs) approaches ln(x/1-x); i.e., segregation tends to vanish in the limit of high temperatures. 3.4. INTEGRATION OF THE GIBBS ADSORPTION ISOTHERM

Recall that it was not possible to relate the Gibbsian adsorption, Γ, to the chemical potential, μ, within the Gibbsian framework of interfacial thermodynamics. This prevented integration of the Gibbs Adsorption

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405

Isotherm to obtain the change in interfacial energy resulting from segregation. However, now that we have used the statistical thermodynamic approach to obtain interfacial composition (as represented by xs) as a function of bulk composition (x) in Eq. (18), then in principle, we can use that information to obtain the requisite relation. We will show this here, for an ideal solution, where the integration can be performed analytically. (It should be noted that there are more general ways of computing the interfacial energy that can be applied to cases of non-ideal solutions – see for example ref (Wynblatt and Chatain 2006), but these are beyond the scope of this chapter). Consider surface segregation in an ideal solution. For that case (as discussed in connection with Figure 5) ΔEseg is a constant. Thus, let exp(-ΔEseg /RT) = k in Eq. (18), where k is a measure of interfacial enrichment at a given temperature, T. Then, from Eq. (18), we can write:

x s = k x /(1+ k x)

(23)

The variation in chemical potential of the solute in an ideal solution may be related to the solute bulk concentration by: dμ2 = RT d(ln x2). We have also seen in Eq. 14 that dγ dμ2 = −Γ2,1 , where component 2 refers to the solute. Thus:

dγ = −ΓRT d(ln x)

(24)

where Γ = ( x s − x) / σ by Eq. (19). Substituting for xs in Γ from Eq. (23), and inserting Γ into Eq. (24), we obtain

⎡ k 1⎤ dγ = −RT ⎢ − ⎥dx ⎣(1+ kx )σ σ ⎦

which can readily be integrated to give

Δγ = −

RT

σ

[ln(1+ kx) − x ]

(25)

This expression gives the change in surface energy, at constant temperature, which results from the increase in surface segregation that accompanies an increase in bulk concentration of solute in the solution. We can look, for example, at the change in surface energy that results from the segregation behavior for the case ω = 0, shown in Figure 5, where ΔEseg is constant (consistent with the assumption of a constant k). The results are displayed in Figure 6. The slopes in the curves shown are related to the adsorption, Γ = −dγ / dμ = −dγ / RTd ln( x) . As the bulk composition increases, at any temperature, so does the adsorption (by Eq. (18)), hence the gradual increase in absolute slope of the curves with increasing x. But note that the slope of the curves approaches an asymptotic value. This

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corresponds to saturation of the surface monolayer in solute. Note also that although the surface becomes saturated in solute, the surface energy continues to decline. This points to a common misconception. The surface energy is not proportional to the adsorption, rather it is the slope, dγ / dμ , that is proportional to adsorption.

Figure 6. Variation of the interfacial energy with bulk composition (log scale), at several temperatures, for values of the parameters used in the case ω = 0 of Figure 5.

4. Anisotropy of Interfacial Properties The interfacial properties of solids are anisotropic; i.e., the properties depend on the crystallographic orientation of the interface. This is most easily demonstrated for surfaces, where only two parameters are needed to describe the orientation. The orientation of any surface may be described by its normal. For the simplest case of cubic crystals, the normal of a plane with Miller indices (hkl) is the vector: hx + ky + lz, where x , y, and z are unit vectors along the three cartesian axes. The magnitude of this vector is: m = √(h2 + k2 + l2), thus the unit vector perpendicular to (hkl) is: (h/m)x + (k/m)y + (l/m)z. Since the coefficients of unit vectors are related by the equation (h/m)2 + (k/m)2 + (l/m)2 = 1, surface orientation is entirely described by 2 independent orientation variables or parameters. As an illustration of anisotropy, we compute the energies of two fcc surfaces by means of the nearest neighbor bond model, in which the energy of each atom is just proportional to the number of bonds it makes with adjacent atoms. Consider a (100) fcc plane, as illustrated in Figure 7a. Each atom in a perfect fcc crystal is connected to 12 neighbors, but an atom in a

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(100) surface plane is only connected to 8 atoms, and therefore has 4 dangling bonds. However, breaking 4 bonds/atom leads to the formation of two (100) surfaces. Thus, each atom in a (100) surface has 2 broken bonds, on average. The number of atoms per unit area on a (100) surface is 2/a2, where a is the lattice constant. Consequently, the excess energy of a (100) surface, per unit area, may be written:

γ (100 ) = 2φ

2 4φ = a2 a2

(26)

where φ is the energy required to break a bond (for a pure A-solid, φ = -εAA, in terms of previously defined bond energies).

Figure 7. Structure of (100) (a) and (111) (b) planes in an fcc crystal.

Let us now consider the (111) plane in Figure 7b. Each atom in a (111) surface plane is only connected to 9 atoms, and has 3 dangling bonds. Breaking 3 bonds/atom creates two (111) surfaces. The number of atoms per unit area is 4/(√3 a2). Consequently, the energy of a (111) surface, per unit area, is:

3 2

γ (111) = φ

4 3a

2

=2 3

φ a2

(27)

This illustration shows that the energies of those two surfaces differ, and that the energy of a (111) surface in fcc (~3.46 φ /a2) is lower than that of the (100) surface (4 φ /a2). The nearest neighbor bond model illustrated above can be used to compute the surface energy of any surface oriented along an (hkl) plane. For fcc surfaces terminated by planes other than (100) and (111), the computation is more complex, because there are several near-surface atom planes that possess broken bonds. The anisotropy of surface energy calculated by this approach (MacKenzie et al. 1962, Lee and Aaronson 1980a, Lee and Aaronson 1980b) is displayed in Figure 8, which shows that the minimum and maximum surface energies are expected for the (111) and (210) surface orientations, respectively. Although these results stem from an overly simplified model, measurements of surface energy anisotropy in a few fcc metals have confirmed that the (111) surface has the lowest energy

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and that the maximum surface energy occurs in the vicinity of the (210) orientation (Heyraud and Metois 1983, Chatain et al. 2004).

Figure 8. Standard stereographic triangle showing variation of relative surface energies (normalized to γ(111) = 1) with orientation in fcc metals.

If an isolated crystal is allowed to reach equilibrium, the shape adopted by the crystal is known as its equilibrium crystal shape (ECS). The ECS is directly related to the anisotropy of surface energy by means of a geometric construction known as the Wulff construction (Hirth 1965). While the details are beyond the scope of this chapter, it is worth mentioning that the ECS may display facets at orientations that correspond to local minima in the dependence of surface energy on orientation. As an example, Figure 9a shows a schematic of the ECS of pure Cu; it shows that the ECS consists mainly of curved surfaces with small flat facets at (100), (111), (110) and (311) orientations. All possible Cu surface orientations are present on the curved surfaces, and the maximum anisotropy is of the order of 2%.

Figure 9. Schematics of ECS of pure Cu (Chatain et al. 2004) (a), and ECS of Bi-saturated Cu (Chatain et al. 2005) (b).

Figure 9b shows a schematic the ECS of Cu saturated with Bi. Although the maximum solubility of Bi in Cu is only about 100 ppm, Bi segregates strongly to Cu surfaces to form a Bi monolayer. The effect of Bi

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adsorption is to change the ECS quite drastically. The ECS is entirely faceted (no curved surfaces), and the only stable surface orientations are represented by facets at (100), (111), (110) and (320). In addition, the maximum anisotropy has increased to 7% (i.e., by more than a factor of 3). 5. Grain Boundaries Most materials of technological interest are polycrystalline, i.e. they consist of many individual crystals (grains) which meet at grain boundaries (GBs). GBs play an important role in many properties of polycrystalline materials. As mentioned in the introduction, GBs are unlike interphase boundaries in that they separate regions of the same phase that have identical structure and composition, but merely differ in crystallographic orientation. A typical polycrystalline microstructure, in which the contrast between grains is visualized by their differences in orientation (as identified by electron backscattered diffraction (EBSD)) is shown in Figure 10.

Figure 10. Microstructure of a polycrystalline sample (tetragonal crystal structure) displaying EBSD orientation contrast, as described in the legend (Pang 2005).

The structure of a GB in a binary alloy is illustrated schematically in Figure 11. The figure shows that the atoms in the transition region between the two perfect crystals do not lie on the lattice points of either of the adjacent crystals, and that the extent of the disordered region amounts to a few atom diameters in thickness. In addition, this schematic shows that the solute atoms are present in excess at the GB, indicating GB segregation of

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solute. Although the segregating species at GBs is not invariably the solute, this is quite a common occurrence.

Figure 11. Schematic of the GB structure in a binary alloy, illustrating both the disordered region separating the perfect crystals on either side of the GB, as well as solute (type B atom) segregation.

Five parameters are required to describe the orientation of a general GB. These are often referred to as the GB character. In the most common description, 3 angles are used to quantify the rotation of the 3 crystallographic axes of one grain into the crystallographic axes of the adjacent grain across the GB (accounting for 3 of the 5 parameters), and a unit vector normal to the GB plane is used to quantify the inclination of the GB with respect to axes of one of the grains (accounting for the other 2 parameters). A less common, though more easily visualized method of accounting for the 5 parameters is the so-called interface-plane scheme (Wolf 1990). Here, character is defined by specifying the crystal planes on which each crystal terminates at the GB (each accounting for 2 parameters) and a twist angle about the GB normal (accounting for the 5th parameter), as shown schematically in Figure 12.

Figure 12. Description of GB character by the interface-plane scheme.

Thus, the orientation space of GBs, and hence the anisotropy of GB properties, is considerably more complex than that of surfaces (where only

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2 orientation parameters are needed). This makes it more difficult to discuss and to display the energy of GBs in terms of GB character. In order to gain some insights into GB energy, it is useful to simplify the GB orientation space by considering some simple limits. Figure 13 shows schematics of two such limiting GBs. The case displayed in Figure 13a is referred to as a symmetric tilt GB. Here, each of the two crystals terminating at the GB is tilted equally by a small angle (θ /2 ≤ ~7°) about an axis that lies in the GB plane. The schematic shows that the GB may be considered to consist of an array of edge dislocations lying in the GB plane, and running normal to the plane of the figure. The only variable that controls GB energy in this case is the tilt angle, θ. Figure 13b displays the second limiting case, which is known as a symmetric twist GB. In this case, each of the two crystals terminating at the GB is rotated equally by a small angle about an axis that lies perpendicular to the GB plane. The GB may be considered to consist of two perpendicular sets of screw dislocations, whose spacing is determined by the twist angle, θ, so that the only variable that controls GB energy in this case is also θ.

Figure 13. Schematics of (a) a symmetric tilt GB showing that the boundary can be described as an array of edge dislocations (⊥) whose spacing depends on the tilt misorientation angle θ, and (b) a symmetric twist GB showing that the boundary can be described as two perpendicular sets of screw dislocations (heavy lines), whose spacing also depends on the twist misorientation angle θ.

The energy of a symmetric tilt GB can be written as the energy of a planar array of edge dislocations:

γGB = Bθ [A – ln(θ)]

(28)

where A and B are constants. This is known as the Read-Shockley equation. As the misorientation angle increases, the spacing between dislocations in the GB becomes progressively smaller. At θ ~15°, the dislocation cores begin to overlap, and the Read-Shockley equation ceases to be physically

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realistic. A similar relation between energy and twist angle can be written for twist GBs. The energy of a GB in terms of the complete five parameter space of GB character can calculated if the character is described by means of the interface plane scheme. In this case, the energy of a GB, for a given pair of terminating planes, (hkl)1-(hkl)2, can be computed by an approach that is similar to the broken bond model introduced previously in Section 4 for surface energy (Wynblatt and Takashima 2001). The energy of GBs obtained for a given pair of terminating planes, plotted vs. twist angle, shows an essentially constant (or plateau) value with a few localized minima (exhibiting a Read-Shockley trend). This is illustrated in Figure 14a for a GB in an fcc material (Wolf 1990). In order to simplify the discussion, Figure 14b gives the results of computations of the GB energy for the plateau value only, plotted versus mean surface energy, defined as (γ(hkl)1 + γ(hkl)2)/2. As can be seen, the plateau value of GB energy varies with the type of terminating plane, and is linearly related (though not proportional) to the mean surface energy. Although the GB energy can approach zero, in the case of identical terminating planes as the twist angle approaches zero (see Eq. (28)), Figure 14b indicates that GB energy anisotropy at the plateau value is approximately twice as large as surface energy anisotropy. We shall see in the next section that this magnitude of the GB energy anisotropy has been confirmed experimentally.

Figure 14. a) Energy of a (111)-(511) GB in fcc as a function of twist angle. Note the existence of an energy plateau over most of the twist angle range, b) Computed GB plateau energies in fcc materials for all combinations of GBs constructed from 12 terminating planes distributed over orientation space, plotted vs. mean surface energies (Wynblatt and Takashima 2001).

We now comment on the nature of the GB energy minima displayed in Figure 14a. For the particular terminating planes illustrated in that figure, the low energy GBs that occur at twist angles of 0 and 60° correspond to

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asymmetric tilt GBs. As the name implies, these GBs are tilted by unequal angles away from the GB plane. Both symmetric and asymmetric tilt grain boundaries tend to be associated with energy minima. However, there are other "special GBs" with specific orientation relationships at which energy minima appear in plots such as Figure 14a. One important class of low energy boundaries can be identified by the so-called coincidence site lattice (CSL) model (Bollmann 1970). This model shows that there exist certain specific rotation angles of one crystal with respect to another where the two crystal lattices will coincide at some fraction of the lattice points. GBs that separate crystals with these special orientation relationships can also be associated with energy minima. The degree to which coincidence occurs is quantified by a parameter Σ, where 1/Σ defines the fraction of lattice points at which coincidences occur. A GB which lies between two crystals for which Σ=5 is referred to as a Σ5 GB. An example of such a boundary occurs when two fcc crystals terminated by {100} planes are rotated with respect to each other by a twist angle of 36.9°. A schematic of such a Σ5 GB is shown in Figure 15.

Figure 15. Schematic showing atomic arrangement of the two {100} GB terminating planes for a Σ5 boundary in an fcc material, where the crystals are rotated by 36.9° with respect to each other about an axis perpendicular to the GB. Light and dark circles represent atoms belonging to the two terminating planes, respectively. Lines indicate the CSL lattice, where coincidences occur.

Finally, a few words about segregation to GBs. As mentioned in Section 3.2., the McLean model was initially developed for the case of GB segregation, and Eq. (18) can therefore be applied to GBs. However, the interpretation of ΔEseg is more complex than in the case of surfaces; for example, the expression for ΔEseg given in Eq. (21) is not directly applicable to GBs. Appropriate models that take the 5 parameters of GB orientation

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into account have recently been developed (Wynblatt and Shi 2005, Wynblatt and Chatain 2006) but are beyond the scope of this chapter. 6. Interfacial Vector Equilibrium 6.1. EQUILIBRIUM OF A GB WITH A SURFACE

Figure 16. a) Schematic of GB-surface equilibrium at a GB groove, b) Atomic force microscope image of Cu surface showing GB grooves.

Although energies, such as interfacial energies, are scalar quantities, it is sometimes convenient to consider their equilibrium in vector terms. Below, we apply such an approach to the equilibrium of a GB with a surface. If the surface of a polycrystalline sample is annealed at a temperature where the intersection of GBs with the surface can equilibrate locally, then a groove will form along the line of intersection, as shown schematically in Fig. 16a, and for an anealed Cu surface in Fig. 16b. We denote the surface energy of the grain with a surface orientation (hkl)1 as γ(hkl)1, and similarly the second surface energy as γ(hkl)2. Then the GB energy may be written as:

γ GB = γ (hkl )1 cos(φ1 ) + γ (hkl ) 2 cos(φ 2 )

(29)

where φ1 and φ2 are partial dihedral angles. If the surface energy is weakly anisotropic (as in the case of Cu discussed above) then it is possible to approximate Eq. (29) as:

γ GB = γ S cos(φ )

(30)

where γs is the isotropic surface energy, and φ is the half dihedral angle at the root of the groove. Measurements of GB grooves in several pure fcc metals have shown that the ratio γGB/γs ranges from 0.25 to 0.5, indicating that the anisotropy of

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GB energy (ignoring special boundaries such as twin boundaries) ranges over about a factor of 2. However, it is useful to bear in mind that in multicomponent systems, segregation effects can intervene. Segregation at the two surfaces and at the GB will generally be different, as will the resulting changes in anisotropy of both GB and surface energies. 6.2. EQUILIBRIUM OF SECOND PHASES AT GBS AND GB WETTING

We shall first consider the equilibrium of a liquid phase at a GB, then say a few words about solid phases at GBs. Liquid phase-GB equilibrium can occur in a system located either in the α + L or the β + L domain of the a phase diagram such as Figure 2. If the temperature and composition of the material under consideration are such that the relative amount of liquid is small, then the microstructure could correspond to a polycrystalline solid with pockets of liquid phase present at the GBs, as shown schematically in Figure 17.

Figure 17. Section through a liquid drop in equilibrium at a GB.

The vector equilibrium of the previous section can be applied here also, to yield:

γ GB = γ SL1 cos(φ1 ) + γ SL 2 cos(φ 2 )

(31)

where γSL1 and γSL2 are the solid-liquid interfacial energies at the orientations in contact with the GB, and φ1 and φ2 are the partial dihedral angles. Again, if the solid-liquid interfacial energies are reasonably isotropic, then Eq. (31) can be simplified to:

γ GB = 2 γ SL cos(φ )

(32)

where φ is the partial dihedral angle. It is important to recognize that, since the equilibrium described in Figure 17 prevails in a two-component system, there can be segregation of one of the components at the GB, and that this segregation will affect the GB energy, and consequently the ultimate values of the dihedral angles. The behavior of liquids at interfaces is generally described under the title of wetting. Wetting can be "good" when the dihedral angles are small,

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or “ poor” when the dihedral angles are large. More details on the topic of wetting are provided in the chapter by Chatain and Ghetta in this volume (Chatain and Ghetta 2007). However, it is useful to introduce certain aspects of wetting here. Consider the region of a phase diagram where GB wetting by a liquid can occur (say the β + L domain in Figure 2 where a solid Cu-rich phase co-exists with a liquid Ag-rich liquid). As temperature is raised, the compositions of the solid and liquid phases become more and more alike. This similarity in composition gradually decreases the solidliquid interfacial energy, and a temperature is eventually reached where 2γSL= γGB; from Eq. (32), this implies that φ = 0. The temperature at which this condition is reached is known as the wetting temperature. A zero dihedral angle corresponds to a state where the liquid phase has spread to completely cover the GB, and is referred to as complete or perfect wetting of the GB. A micrograph showing the initial stages of complete GB wetting by a liquid is shown in Figure 18.

Figure 18. Fe-Cu alloy showing the Cu-rich phase wetting some of the GBs in the Fe-rich phase, quenched from ~1100°C, where the Cu-rich phase is liquid (Wynblatt and Takashima 2001).

Note that some GBs are wet, some are dry, and some are mixed (i.e., partly wet and partly dry). The fact that not all GBs are wet is an indication of the anisotropy of GB energy; i.e., if γSL is assumed to be constant at a given temperature, then relatively high energy GBs will encounter the condition 2γSL= γGB before relatively low energy GBs. At higher temperatures, where complete wetting occurs at all GBs, the liquid will form a continuous film through the microstructure. The presence of a continuous liquid film leads to a sharp deterioration of the mechanical properties of the solid, a phenomenon that is known as liquid metal embrittlement. One other wetting-related phenomenon may be useful to mention. Let us select a temperature above the wetting temperature, where complete wetting of the GBs would prevail in the β + L domain of Figure 2. Now

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consider a GB in an alloy with a composition within the β Cu-rich phase, at which no liquid is stable. Some segregation of Ag at the GB would be expected. If the Ag content of the alloy were to be increased gradually, so as to shift the composition towards the β/(β + L) phase boundary, one would expect a significant jump in the adsorption of the Ag to occur at the GB, somewhere before the liquid phase becomes thermodynamically stable. This jump in adsorption is known as a prewetting transition. As the Ag content of the alloy is increased further, the amount of Ag adsorbed at the GB would also continue to increase, until a bulk liquid film is formed as the alloy composition crosses into the two-phase β + L domain. Thus, prewetting is a phenomenon where a precursor of the liquid film will form at GBs before the liquid becomes thermodynamically stable (Luo 2007). Lastly, we address the case of solid phases at GBs. Here, the role of anisotropy of the solid-solid interface between the solid phases is generally more significant. Although complete wetting of GBs by a solid phase has been reported (Lopez et al. 2004), it does not seem to be as widespread as in the case of liquids at GBs. The equilibrium conditions for a solid phase at a GB has the same form as Eq. (31), except that the solid-liquid interfacial energies must be replaced by the energies appropriate for solid-solid interphase interfaces. 7. Diffusion in Metals – An Overview 7.1. INTRODUCTION

As an example, consider two alloys of different compositions, that both fall within a single phase domain at some specified temperature (e.g., two solids that fall within the γ domain of Figure 1, or the α domain of Figure 2). If these two solids are welded together and held at the specified temperature for some time, then material will flow between the solids so as to eliminate the presence of any chemical potential gradients. This material flow is referred to as diffusion, and the joined solids are referred to as a diffusion couple. In this simple example, the elimination of chemical potential gradients will also lead to elimination of the composition differences between the solids. It is possible to discuss diffusion in solids in two rather distinct manners. The first is a mathematical description of how the composition at any point in the diffusion couple will evolve with time, and the second is an atomistic view of how atoms of the various components present in the solids will move from one location to another. Below, we give an abbreviated description of both of these perspectives.

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7.2. MATHEMATICAL DESCRIPTION OF DIFFUSION

We will limit attention to cases where equilibrium is achieved when composition gradients vanish. A typical two-component diffusion couple is shown in Figure 19; it illustrates the variation of concentration, c, of one of the components, with distance along the couple, x, at three different times. c may be expressed in units of atoms per unit volume or mass per unit volume, and x in units of length.

Figure 19. Schematic of concentration profiles across a diffusion couple at different times, for two semi-infinite solids of uniform compositions c' and c'' at time t = 0.

The flux J (atoms/[unit area . time] or mass/[unit area . time]) of one of the components, at a given time, and any point along the profile, may be written as:

J = −D

∂c ∂x

(33)

where D is a constant known as the diffusivity or the diffusion coefficient (length2/time). Eq. (33) is referred to as Fick's First Law. The evolution of concentration with both time and distance along the couple is described by Fick's Second Law:

∂c ∂J ∂2c =− =D ∂t ∂x ∂x 2

(34)

where the concentration is expressed as a function of distance and time, c(x,t). For the example shown in Figure 19, it is possible to write the following initial conditions: IC:

c(x,0) = c' for x < 0, and c(x,0) = c'' for x > 0

(35)

For the case of semi-infinite solids, the solutions of the partial differential equation, Eq. (34), take on the following form: ⎛ x ⎞ c( x, t) = A + B erf ⎜ ⎟ ⎝ 4Dt ⎠

(36)

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where A and B are constants to be evaluated by applying appropriate initial (and/or boundary) conditions, and erf(z) represents the error function of the argument z. The error function is tabulated in various collections of mathematical tables. However, it is useful to mention that erf(0) = 0, erf(∞) = 1, and erf(-z) = - erf(z). Using the initial conditions of Eq. (35) in Eq. (36) one obtains A = (c'' + c')/2; B = (c'' – c')/2, giving the solution: c( x, t) − c′ =

⎛ x ⎞⎤ c′′ − c′ ⎡ ⎢1+ erf ⎜ ⎟⎥ 2 ⎢⎣ ⎝ 4Dt ⎠⎥⎦

(37)

There are, of course, many interesting diffusion geometries that differ from the simple example of diffusion in one-dimensional semi-infinite solids given above. Compilations of solutions for a variety of cases can be found in many textbooks (see for example (Crank 1975)). 7.3. ATOMIC MECHANISMS OF DIFFUSION IN ALLOYS

In order for diffusion to take place in crystalline solids, a mechanism must exist whereby atoms are able to move about and migrate through crystals. A great deal of indirect evidence has accumulated over the years to indicate that atom movements are associated with crystalline point defects, either vacancies or interstitials. In pure metals, atoms can move by exchanging positions with vacant sites, as shown schematically in Figure 20a, and atoms located in interstitial sites (which we will refer to as self-interstitials) can move from one interstitial site to another, as shown schematically in Figure 20b. However, the numbers of self-interstitials are much smaller than the numbers of vacancies, because of the high strain energy associated with

Figure 20. Schematic of diffusive jumps involving lattice defects: a) and b) Self diffusion by atom-vacancy exchange and self-interstitial jumps, respectively, c) and d) Solute diffusion by solute atom-vacancy exchange and interstitial solute jumps, respectively.

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self-interstitials. Thus, self-interstitials do not contribute much to the selfdiffusivity of pure metals. The figure also shows the schematics of atom movements in alloys. In substitutional alloys where the solute atoms occupy sites of the host metal lattice, atom migration by atom-vacancy exchange dominates, as it does in pure metals. In interstitial alloys, the solute atoms are much smaller than those of the solvent, and they tend to diffuse by moving from one interstitial site to another.

Figure 21. Schematics (a)-(c) of configurations and (d) of energy changes taking place during exchange of an atom with a vacancy.

In view of the importance of vacancies in diffusion, we provide here an expression for the fraction of vacant crystal sites, xV: ⎛ ΔG f xV = exp⎜ − V ⎜ RT ⎝

⎞ ⎟ ⎟ ⎠

(38)

where ΔGVf is the free energy of vacancy formation; i.e., the free energy change associated with the introduction of a single vacancy in an otherwise perfect crystal. Another important quantity in the context of diffusion is the rate at which atoms will jump into an adjacent vacant site. Figure 21 shows the stages in the exchange of an atom with a vacancy. As the atom advances towards the vacancy, the lattice is distorted, and this raises the energy of the system. Thus, an atom jumping into a vacant site must overcome an energy barrier ΔGVm , and the jump is likely to be thermally activated. The frequency at which the process will occur can be estimated as:

⎛ ΔGVm ⎞ ⎟ ⎝ RT ⎠

ω = ν exp⎜−

(39)

where ν is the vibration frequency of an atom in the crystal, and ΔGVm is known as the vacancy migration energy. One may view the frequency of atom-vacancy exchange as representing a product of the frequency of

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attempts that an atom makes at crossing the energy barrier, ν, and a Boltzmann factor, which gives the probability that a given attempt will be successful. The self-diffusivity, which is a measure of the rate at which atoms move around in a pure material, can be determined from experiments with diffusion couples, in which one side of the couple contains a certain concentration of a radio-active isotope of the pure material. The selfdiffusivity is an important parameter in determining the rate of such processes as sintering. This diffusivity is proportional to the product of the vacancy fraction (Eq. 38) and the frequency with which an atom exchanges with a vacancy (Eq. 39):

⎛ (ΔG f + ΔG m )⎞ V V ⎟ Dself ∝ xV ω = ν exp⎜⎜− ⎟ RT ⎝ ⎠ (40) ⎛ (ΔS f + ΔS m )⎞ ⎛ (ΔE f + ΔE m )⎞ V V V V ⎟ exp⎜− ⎟ = ν exp⎜⎜ ⎟ ⎜ ⎟ R RT ⎝ ⎠ ⎝ ⎠ Equation 40 provides a hint as to the origins the well-known empirical Arrhenius form for the temperature dependence of self-diffusivity, as well as for the diffusivity of various solutes in alloys: ⎛ Q ⎞ D = D0 exp⎜ − ⎟ ⎝ RT ⎠

(41)

where D0 is known as the pre-exponential factor and Q as the activation energy. Comparing Eqs. (40) and (41), one can conclude that in the case of self-diffusion Q represents the sum of the vacancy formation and migration energies. In the case of substitutional solute diffusion, the interpretation of Q is somewhat more complex. Figure 22 provides several instructive examples of diffusivity, and its temperature dependence. Figure 22a compares the self-diffusivity of Fe in the bcc and fcc structures. The higher self-diffusivity in bcc Fe is due to the relatively more open, or less closely packed, structure of bcc compared to fcc. This openness means that the exchange of an atom with a vacancy requires less lattice distortion, and hence a lower migration energy. Also, the diffusivities of C in both of the Fe structures are shown. Since C is present in Fe-C alloys in interstitial sites, no vacancies are needed to mediate the diffusion process. Hence, the activation energy in this case contains only the migration energy of the interstitial solute. As a result, the activation energy for diffusion of C is much smaller than that for Fe, with the result that the diffusivity is higher. C diffuses faster in bcc than in fcc Fe

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because of the previously identified greater openness of the bcc structure. The diffusivity of Mn, a substitutional solute in Fe, is slightly lower than that of Fe, but roughly comparable.

Figure 22. Arrhenius plots of diffusivity for various metals and alloys.

Figure 22b compares the diffusivities of Ag in crystalline Ag with Ag in Ag GBs. In general, the diffusivity is higher in GBs, because the local structural disorder renders the GB region less closely packed than the crystalline material. The self-diffusivity of Ag is higher than that of Cu. This is a reflection of the fact that Cu-Cu bonds are stronger than Ag-Ag bonds, as indicated for example by the lower melting point of Ag. Finally, the figure shows that Zn diffuses faster in Cu than Cu in Cu. This may be interpreted as the weakening of Cu-Cu bonds by the presence of Zn. 8. Summary This chapter has reviewed certain fundamental topics, which should provide a background for later chapters. Topics include interfacial thermodynamics, adsorption and its effects on interfacial energy, interfacial energy anisotropy, grain boundary structure and energy, interfacial vector equilibrium, and an overview of diffusion in metals. Acknowledgments The author wishes to acknowledge support of his research by the MRSEC Program of the National Science Foundation under Award No. DMR-0520425.

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References Bollmann W., 1970, Crystal Defects and Crystalline Interfaces, Springer-Verlag, New York, pp. 143-185. Chatain, D., Ghetta, V., and Wynblatt, P., 2004, Equilibrium shape of copper crystals grown on sapphire, Interface Sci. 12:7-18. Chatain, D., Wynblatt, P., and Rohrer, G. S., 2005, Equilibrium crystal shape of Bi-saturated Cu crystals at 1223 K, Acta Mater. 53:4057-4064. Chatain, D., and Ghetta, V., 2007, Wetting at high temperature, in THIS VOLUME Crank, J., 1975, The Mathematics of Diffusion, 2nd ed, Oxford University Press, Oxford. Defay, R., Prigogine, I., Bellmans, A., and Everett, D. H., 1966, Surface Tension and Adsorption, Wiley, New York, pp. 158-197. Fowler, R. H., and Guggenheim, E. A., 1939, Statistical Thermodynamics, Macmillan, New York, pp. 429-40. Friedel, J., 1954, Advan. Phys. 3:446-507. Gibbs, J. W., 1961, The Scientific Papers of J. Willard Gibbs, Dover, New York, vol. 1, pp. 219-331. Heyraud, J. C., and Metois, 1983, J. J., Equilibrium shape and temperature: Lead on graphite, Surface Sci. 128:334-342. Hirth, J. P., 1965, The kinetic and thermodynamic properties of surfaces, in: Energetics in Metallurgical Phenomena Vol II, W. M. Mueller ed., Gordon and Breach, New York, pp. 1-52. Hirth, J. P., 1973, Thermodynamics of surfaces, in: Structure and Properties of Metal Surfaces, Honda Memorial Series in Materials Science No.1, Maruzen Co. Ltd., Tokyo, pp. 10-33. Hondros, E. D., and Seah, M. P., 1977, The theory of grain boundary segregation in terms of surface adsorption analogues, Metall. Trans. 8A:1363-1371. Lee, Y. W., and Aaronson, H. I., 1980a, Surface concentration profile and surface energy in binary alloys, Surf. Sci. 95:227-244. Lee, Y. W., and Aaronson, H. I., 1980b, Anisotropy of coherent interphase boundary energy, Acta Metall., 28:539-548. Lopez, G. A., Mittemeijer, E. J., and Straumal, B. B., 2004, Grain boundary wetting by a solid phase; microstructural development in a Zn-5 wt% Al alloy, Acta Mater. 52:45374545. Luo, J., 2007, Stabilization of Nanoscale Quasi-Liquid Interfacial Films in Inorganic Materials: A Review and Critical Assessment, Crit. Rev. Sol. State Mater. Sci. 32: 67109. MacKenzie, J. K., Moore, A. J. W., and Nicholas, J. F., 1962, Bonds broken at atomically flat crystal surfaces, J. Phys. Chem. Solids 23:185-205. McLean D., 1957, Grain Boundaries in Metals, Oxford Press, London, pp. 116-122. Mullins, W. W., 1963, Solid surface morphologies governed by capillarity, in: Metal Surfaces: Structure, energetics and kinetics , W. D. Robertson and N. A. Gjostein, eds., ASM, Cleveland, pp. 17-66. Ono, S., and Kondo, S., 1960, Article in: Encyclopedia of Physics, vol. X, S. Flugge, ed., Springer, Berlin, pp. 134-277. Pang, Y., 2005, Relation between grain boundary segregation and plane orientation in Nbdoped TiO2, Ph.D. Thesis, Carnegie Mellon University, Pittsburgh. Wolf, D., 1990, Structure-energy correlation for grain boundaries in FCC metals. IV. Asymmetrical twist (general) boundaries, Acta Metall. Mater. 38:791-798.

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Wynblatt, P. and Chatain, D., 2006, Anisotropy of Segregation at grain boundaries and surfaces, Metall. Mater. Trans. 37A:2595-2620. Wynblatt, P., and Ku, R. C., 1977, Surface energy and solute strain energy effects in surface segregation, Surface Sci. 65:511-531. Wynblatt, P., and Ku, R. C., 1979, Surface segregation in alloys, in: Interfacial Segregation, W. C. Johnson and J. M. Blakely eds., ASM, Metals Park, OH, pp. 115-136. Wynblatt, P., and Shi, Z., 2005, Relation between grain boundary segregation and grain boundary character in fcc alloys, J. Mater. Sci. 40:2765-2773. Wynblatt, P. and Takashima, M., 2001, Correlation of grain boundary character with wetting behavior, Interface Sci., 2001, 9:265-273. Wynblatt, P., Saùl, A., and Chatain, D., 1998, The effects of prewetting and wetting transitions on the surface energy of liquid alloys, Acta Mater. 46:2337-47.

SOME ASPECTS OF WETTING AT HIGH TEMPERATURE D. CHATAIN* Centre de Recherche en Matière Condensée et Nanosciences, CNRS, campus de Luminy, 13288 Marseille, France V. GHETTA Laboratoire de Physique Subatomique et de Cosmologie, CNRS, 53 av. des Martyrs, 38026 Grenoble, France

Abstract: This chapter presents a comprehensive overview of wetting phenomena which take place at high temperatures. It relies in part on the chapter “Introduction to interfaces and diffusion”, elsewhere in this volume, as a foundation. The topics addressed include: definitions of intrinsic wetting and associated quantities, wetting anomalies due to diffusion and anisotropy of solid/fluid interfaces, wetting hysteresis on real surfaces, and orders of magnitude of the energies that determine wetting in metal/metal, metal/ceramic and glass systems. At the end of this chapter some examples of evolving contact between liquid and solid metal illustrate the complexity of wetting phenomena at high temperature. Keywords: Wetting thermodynamics; capillarity; wetting hysteresis; wetting and diffusion; metal/ceramic systems

1. Definitions 1.1. WETTING

Wetting characterizes the degree of spreading of a liquid on a solid, in equilibrium with a vapor phase. On a flat horizontal solid surface, a finite quantity of liquid adopts the shape of a sessile drop, as shown in Figure 1. Wetting depends on the structure of the solid, and the chemistry of the 3 coexisting solid-liquid-vapor phases. It is quantified by the contact angle, θY, between the liquid surface and the solid/liquid interface at the solidliquid-vapor triple line (see Figure 2). When the solid is in the shape of a small vertical tube, the contact angles inside and outside of the tube are identical and the phenomenon of capillary rise takes place if the tube has a

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* Dominique Chatain, CRMCN-CNRS, campus de Luminy, case 913, 13288 Marseille cedex 9, France; e-mail: [email protected]

V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

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sufficiently small inner diameter (see Figure 1). For a given solid-liquidvapor system, the intrinsic (local) contact angle does not depend on the macroscopic shape of the solid.

Figure 1. Examples of wetting and capillary rise of water and mercury on a same glass surface.

1.2. CONTACT ANGLE ON AN UNDEFORMABLE SOLID

The contact angle, θY, of a drop on a rigid solid is related to the interfacial energies by the Young equation:

cosθY =

γ SV − γ SL γ LV

(1)

where γIJ are the energies of the three interfaces IJ, and I and J are the phases that coexist at equilibrium (S = solid, L = liquid, V = vapor).

Figure 2. Young, or equilibrium, or intrinsic contact angle and interfacial energies.

This equation corresponds to the vector equilibrium of the energies of the 3 interfaces at the triple line, projected on the solid plane (see Figure 2). Contact angles can have any value between zero and 180°. Materials scientists working with inorganic materials at high temperature tend to separate the systems in which “good wetting” takes place, i.e. where θY < 90°, from those where “bad wetting” occurs (θY > 90°). This nomenclature is related to the ability of a liquid to spontaneously infiltrate a porous solid when θY < 90°. Physicists who handle organic systems at room temperature use the terms “wetting” and “non-wetting” when the contact angles are zero and positive, respectively. To avoid confusion, it is better to speak of complete or perfect

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wetting when the contact angle is zero. This phenomenon happens when a liquid film is stable on a surface (or at an interface or at a grain boundary), whatever its thickness as soon as the film acquires the characteristics of an autonomous bulk phase. When there is a gradient in any order parameter (e.g. composition, structure, etc..) in the film, this gradient must be associated with an adsorption phenomenon (cf. chapter “Introduction to interfaces and diffusion” in this volume). When the contact angle is 180°, wetting is “null”. In this case, it is the vapor phase that completely wets the solid in coexistence with the liquid. Such a phenomenon has never been observed, and the highest Young contact angles ever measured lie in the range 140-150°. However, apparent contact angles close to 180° can be obtained when the shape of the solid surface is specifically designed for reducing the actual contact between the liquid phase and the entire area of the solid surface. This phenomenon, recently named “lotus effect” will be addressed at the end of this paper. 1.3. CONTACT ANGLE ON A DEFORMABLE PHASE AND AT AN INTERFACE

Wetting on a deformable phase such as a liquid, sketched in Figure 3, is characterized by a dihedral angle, φ, which is related to the interfacial energies by the Newman relationship:

γ L1V γ γ = L 2V = L1L 2 sin α L 2 sin φ sin α V

(2)

The sketch of Figure 3 is available in the absence of buoyancy. The shapes of the interface between L1 and L2, and of the L1 surface confine the L1 drop in a minimum of areas of given energies. If the drop of the L1 phase is under gravity, there is an additional pressure due to its weight, and the surface of the L2 liquid around the particle, bends down in the vicinity of the triple line.

Figure 3. Wetting on a soft surface and dihedral angle.

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Here it must be emphasized that the apparent “contact angle” is not related to the interfacial energies by the Young equation (Eq. 1). A particle embedded at an internal interface has also a lens shape, and the wetting is characterized by the dihedral angle, φ, in the particle at the triple junction. A dihedral angle is also used to describe the equilibrium angle formed at the groove of a grain boundary merging at an interface (see Figure 4). The grain boundary can be replaced by an interface between two different solid phases. The dihedral angle, shown in Figure 4a, is related to the energy of the two interfaces of the particles, γi1 and γi2, and of the boundary, γ12, by the following Eq. (3):

γ i1 cos φ1 + γ i 2 cos φ2 = γ 12 with φ1 + φ2 = φ

(3)

Figure 4. Wetting and groove at internal interfaces: a) general case, b) classical sketches for a symmetrical grain boundary groove equilibrated under two different mechanisms of solid diffusion.

It is more usual to find the dihedral angle at a grain boundary defined by Eq. 4, with the restriction of a symmetrical shape of groove (where γi1 = γi2 = γi) (see Figure 4b).

φ

γ

cos = 12 2 2γ i

(4)

It should be emphasized that the definitions given in the above 3 sections are only valid if the bulk phases and the interfaces are in chemical and mechanical equilibrium.

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2. Surface and Interfacial Energies and Work of Adhesion 2.1. SURFACE AND INTERFACIAL ENERGIES

In this section, some general trends are reviewed. In order to understand wetting, it is important to understand the factors that determine the surface and interfacial energies in a multicomponent system. Technical details on these quantities are emphasized in the chapter “Introduction to interfaces and diffusion”. Surface and interfacial energies are positive quantities. They are excess energies referred to the bulk phases because surfaces and interfaces are defects of these bulk phases. They depend on the chemistry and the structure of the adjacent phases. Surfaces and interfaces have compositions and structures, which may differ from those of the adjacent bulk phases. However, they are not autonomous phases and cannot exist by themselves. If one of the bulk phases changes, the interface itself changes. The energy of a surface or an interface is the work (d W ) that must be expended to increase its area by one unit (dA):

γ=

dW dA

(5)

The USI units of interface and surface energies are J/m2 (or N/m). The magnitude of surface and interfacial energies depends on the type of materials. Some order of magnitude values of liquid surface energies are given below. These quantities are easy to measure. This is not the case for the surface energies of solids or of the energies of their interfaces with liquids. Some order of magnitude values of liquid surface energies are given in Tables 1 and 2. TABLE 1. Surface energy of organic liquids at room temperature (mJ/m2).

water fluorocarbons hydrocarbons polar organic compounds detergents in aqueous solution

72 8 - 15 18 - 30 22 - 50 24 - 40

TABLE 2. Surface energy of inorganic materials at their melting point (mJ/m2).

chlorides and fluorides glasses metals and alloys

50 - 300 200 - 400 350 - 1800

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In what follows, we will focus on high temperature materials. The surface energy of a pure liquid metal decreases slightly as temperature increases (by a few tenths of a mJ/m2 per Kelvin) (Keene 1993, Mills 2006). In multi-component systems, this trend is reversed because of adsorption phenomena (see chapter “Introduction to interfaces and diffusion”). The surface energy of a pure solid component is higher than that of its liquid by about 10-20% (Kumikov and Khokonov 1983). The energy of solid/liquid interfaces depends on the type of materials. In liquid metal/solid metal systems they are lower than the surface energy of the solid (Vinet et al. 2002). In liquid metal/ceramic systems, they are higher or have about the same magnitude as the surface energy of the solid (Saiz et al. 1999) depending on the oxygen partial pressure of the vapor phase in equilibrium with the condensed phases. Because crystalline solids are anisotropic in structure, the solid/fluid interfacial energies depend on the orientation of the solid interface. The magnitude of the interfacial energy anisotropy can vary from a few percent to a few tens of percents. It tends to be small for pure components close to their melting points and increases as temperature decreases, see for example (Heyraud and Metois 1983) and (Kitayama and Glaeser 2002). Adsorption at the surfaces of metals changes the anisotropy (Chatain et al. 2004, Chatain et al. 2005) but there are no systematic trends (Wynblatt and Chatain 2006). A solid/liquid interface behaves like a solid surface. The interfacial region, located between the two phases, contains the species of both phases at a particular composition and structure. Generally, the atoms of the liquid in the vicinity of the solid are ordered because they “feel” the periodic potential of the underlying solid (Kaplan 2006), and also because of the jump in density between the two phases (Geysermans et al. 2005). A solid/solid interface formed between two different phases (heterointerface) or between two grains of the same phase with different orienttations (grain boundary) also displays an anisotropic energy. It depends on 5 crystallographic parameters (Wynblatt and Shi 2005). The anisotropy of energy of grain boundaries is much more pronounced than that of solid/fluid interfaces. In multicomponent systems, anisotropic adsorption takes place at interfaces and decreases their energy (Wynblatt and Chatain 2006). The degree of coherency, of a solid/solid interface between the two adjacent crystals, is also an important parameter. When the lattices of the two adjacent phases are different, the interfacial energy is the lowest when there is no coherency (due to the absence of elastic strain of the lattices at the interface).

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2.2. WORK OF ADHESION

The work of adhesion is the work necessary to separate, in a reversible manner, two phases (P1 and P2) having an interface of unit area:

W = γ P1V + γ P 2V − γ P1P 2

(6)

After separation, the two newly created surfaces adopt their equilibrium adsorptions. Separation occurs at thermodynamic equilibrium, i.e. at constant chemical potentials of each species. There is no relationship between the compositions of the interface and of those of the two newly created surfaces. In the case of an interface between two (almost) pure components A and B, the work of adhesion cannot simply be calculated from the surface energies of the pure A and B phases which have been separated. This is because, at equilibrium, the lowest surface energy component (B) will tend to adsorb at the surface of phase A. Thus, the sum γA + γB – γAB does not correspond to the equilibrium state, and is referred to as the “spreading coefficient”. 3. The Equilibrium Shape of Interfaces in Wetting Experiments In the previous section, wetting on a rigid flat solid was defined by the classical Young equation. Whereas this configuration is simple and thus very popular, it often only describes a metastable equilibrium. At high temperature where diffusion processes are activated, the junction of 3 interfaces may achieve its equilibrium, by a reorganization of the matter of the 3 adjacent coexisting bulk phases. Thus, we think useful to address the consequent shape changes of their 3 interfaces. The chemical, structural and shape equilibration of the surfaces and interfaces can only be achieved by a reorganization of the local chemistry, structure and shape. In particular, the T-junction of 3 interfaces, as described above in Figures 3 and 4 for the case a drop on an immiscible liquid or in the case of grain boundary grooving, is generally not stable. Both Neumann and Young equations represent simplified cases of the general Herring equation of interfacial equilibrium, which takes into account the anisotropy of interfacial energies by including “torques terms” (Herring 1951), as described in more detail below. The anisotropy of interfaces involving crystalline solids must consider two features: the change of their energy with orientation (as described by the angles ψ1, ψ2), and their structure at the atomic scale. The interface with a solid can be facetted or rough at the atomic scale. On a crystal whose shape has been equilibrated, a faceted orientation appears as a flat surface

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between rounded surfaces of different orientations. Facets arise at orienttations where cusps occur in a plot of interfacial energy versus orientation (γ-plot). As an example, a two-dimensional (2D) polar γ-plot (Wulff construction, see (Hirth 1965)) of the surface energy of crystalline salts such as NaCl or LiF is shown in Figure 5a, together with the corresponding 2D equilibrium crystal shape. In Figure 5b, two simple examples of 2D equilibrium shapes show a crystal shape with all orientations stable and a crystal shape with missing orientations where the shape has singularities. Unstable surfaces with orientations that do not appear on this shape will decompose into the adjacent stable orientations, i.e. they will undergo microfaceting. In 3D, an unstable orientation will decompose into either two or three adjacent stable orientations (Figure 5c).

Figure 5. Equilibrium shapes and faceting: a) 2D gamma-plot and equilibrium shape of a crystal; b) Two 2D equilibrium shapes, the left one with all the orientations and the right one with missing orientations; c) 3D equilibrium shape of a fcc crystal with only 3 types of stable orientations ({111}, {100} and {110}). On the right, two AFM pictures show the microfaceting of unstable orientations between two stable facets or three stable facets.

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When the γ-plot of an anisotropic surface is such that the torque, (dγ/dψ), is not null, the complete equilibrium of three interfaces should be written as proposed by Herring (Herring 1951):

∑γ ij

ij

+∑ ij

dγ ij dψ

=0

(7)

When the interfaces are all isotropic, the second sum vanishes, and the Herring equation reduces to the Neumann equation. The second sum expresses the resistance of the solid/fluid interfaces to rotate from their initial orientations, under the pull of the liquid surface energy. When the torque of one faceted interface is large enough (i.e. when the cusp of the interfacial energy is deep enough), this interface remains flat. This is sketched in Figure 6. Large torques at the two solid/fluid interfaces allow both of these interfaces to remain flat and coplanar. This is the only case were the Young equation is strictly valid at high temperature.

Figure 6. a) Equilibrium at triple junction with one flat interface of energy γ1 because of its large torque; b) SEM image of a faceted ridge formed at the triple line of an ex-copper drop on a sapphire surface.

In the case of small torques, the species of the solid diffuse to achieve the local equilibrium shape required by Eq. (7). A ridge forms at the triple line (Saiz et al. 2000, Chatain and Carter 2004), as shown in Figure 6, for a copper droplet annealed at its melting point on a sapphire surface (Ghetta and Chatain 2002). This phenomenon is similar to the well-known phenolmenon of grain boundary grooving, which occurs where a grain boundary intercepts a surface or an interface (Mullins 1963). The dihedral angle formed at the groove may be used to determine the grain boundary energy from the solid/fluid interfacial energy. Ridging may invalidate the measurement of the wetting angle of small drops (Chatain and Galy 2006). However, it can be used to determine

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interfacial energies (Levi and Kaplan 2002), as in the case of grain boundary grooving. If the time of annealing is reasonable, ridging does not affect the contact angle for large drops (Saiz et al. 2000) in classical wetting experiments with non-reactive systems. 4. Wetting Experiments and Real Surfaces Actual solid surfaces have chemical and topological defects at different scales, and/or complex shapes. This generates wandering and/or sticking of the triple line, which makes the measurement of an equilibrium contact angle tricky. In what follows, the general equations, which drive capillary phenomena on real surfaces, are presented. They are used to explain the origin of wetting hysteresis, and of the scatter observed in contact angles measurements. 4.1. BASIS OF CAPILLARY PHENOMENA

The location of the triple line of a liquid on a solid surface is driven by the minimization of the total energy of the interfaces. This energy may be written as follows:

E = ∑ γ ij Aij

(8)

ij

where the Aij are the areas of the ij interfaces of energy γij. When the triple line is displaced by a small distance on the solid, the areas of the interfaces change; the area change of the solid/liquid interface is equal to the negative of the area change of the solid surface. It is possible to show that minimization of Eq. (8) reproduces the Young equation (Eq. 1). The minimum area of the liquid surface obeys the Laplace equation:

ΔP = γ LV (

1 1 + ) R1 R 2

(9)

where ΔP is the difference of pressure between the inside and outside of the liquid, and R1 and R2 are the principal radii of curvature of its surface. An analytical solution of this equation exists for the shape of the meniscus made by a liquid on a flat, infinitely large solid surface; this is because the inverse of one of the radii of curvature of the liquid surface tends to zero. Another simple solution of this equation arises when the two radii of curvature are identical: this is the case of a sessile drop which adopts the

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shape of a spherical cap when its size is below the capillary length, Lc, defined as follows:

Lc =

γ LV Δρ.g

(10)

where Δρ is the difference between the densities of the two fluids on each side of the surface, and g is the acceleration due to gravity. This characteristic length corresponds to a balance between the capillary pressure and the hydrostatic pressure. Numerical solutions of the Laplace equation are currently used for axisymmetrical sessile drops. This is because this configuration is often chosen for measuring the surface energy of liquids, knowing their density (see for example Keene (1993)). By minimizing Eq. (8) for a sessile drop in the shape of a spherical cap, it can be shown that the smaller the contact angle, the smaller the total energy of the interfaces. Thus, on a solid surface composed of different materials, a drop will adopt a position where the contact angle is the smallest. If the surface with smaller contact angle cannot accommodate the whole drop, the drop will acquire a complex shape. Once the axisymmetry of the sessile drop is lost, the search for the minimum of Eq. (8) requires the use of numerical calculations (e.g. by means of software such as Surface Evolver (Brakke 2003)). Figure 7 shows shapes of a liquid drop in contact with a well-wetted stripe surrounded by poorly wetted surfaces. These shapes have been calculated using Surface Evolver. For small volumes of liquid, the sessile drop shape is adopted, because the drop is accommodated on the stripe without touching its edges. For intermediate volumes of liquid, the triple line of the drop is stuck at the edge of the stripe, and develops feet which expand along the stripe. For large volumes of liquid, a bump forms in the middle of the liquid. Such a shape obeys the Laplace equation (Eq. 9) and optimizes the solid/fluid areas (Eq. 8). On these drops several contact angles can be measured, i.e. both intrinsic and apparent contact angles. On the stripe, the contact angle is the intrinsic contact angle of the liquid on the stripe material. Where the triple line is stuck at the edge of the stripe, the contact angle increases with increasing drop volume. Thus, the apparent contact angle is constrained. As soon as it reaches a value which corresponds to the intrinsic contact angle on the surrounding poorly wetted surfaces, the liquid expands onto this surface. In order to maintain a minimum surface energy, the drop spreads on one side of the stripe and the liquid drop adopts a T-like shape. One of the symmetry axes of the shape has been lost.

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Figure 7. Drops of increasing volumes (from a to c) confined on a well-wetted stripe calculated with Evolver. Top and profile views are displayed for the drops with parts of their triple line stuck on the edges of the stripe. a) the stripe hosts the volume of the whole sessile drop; b) the drop is confined between the edges of the stripe; c) a larger volume of liquid develops a bump before being able to spread laterally.

When the volume of the drop is decreased from this configuration, the shape does not change reversibly. The part of the liquid drop located outside the stripe recedes away from the poorly-wetted surfaces before relocating onto the edge of the stripe. This example shows that different apparent contact angles for the same volume of liquid can be observed, depending on whether a triple line advances or recedes on a heterogeneous surface. This is the basis of the wetting hysteresis phenomenon. 4.2. DISCONTINUOUS FLAT DEFECTS

The surface of an anisotropic polycrystal has regions of different surface energies, and the even surface of a single crystal is often spotted with chemical defects (inhomogeneities) of different sizes. Equation (8) can calculate the location of the triple line if all the different solid/fluid interface energies together with their areas and spatial distributions are accounted for. The energy-contact angle curve has a single minimum when the surface is perfectly homogeneous and flat. However, when there are small regions of different energy spotted on the surface, the energy-apparent contact angle curve will have several minima, the envelope of which is a curve with one minimum. The first and the last minima correspond to the minimum receding and the maximum advancing contact angle. Figure 8 illustrates these features, which are explained below.

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Figure 8. Sketch of the energy-apparent contact angle curve for a surface with randomly distributed defects. The equilibrium contact angle is at the lowest energy minimum. The wetting hysteresis ranges between the first and the last minima of the energy-apparent contact angle curve.

On advancing, the triple line sticks as it encounters regions which have higher contact angles, and spontaneously spreads on the regions which have lower contact angles. On a surface, which has randomly distributed defects, the triple line wanders about an average position. The apparent contact angle is deduced from the mean position of the triple line and the macroscopic shape of the liquid surface where it is not deformed by the presence of the defects. For a meniscus and a sessile drop, the average positions of the triple lines are a straight line and a circle, respectively. The maximum length of deformation of the triple line is the capillary length of the liquid, Lc. It is also the maximum distance of perturbation of the macroscopic shape of the liquid (Chatain et al. 2006). The capillary length is of the order of a few millimeters for almost all liquids. Defects much smaller than the capillary length cause local sticking of the triple line (Figure 9). Unsticking takes place when the local angle on the defect becomes the intrinsic local angle.

Figure 9. Discontinuous defects pinning a triple line.

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Several defects located close together cause cooperative sticking of the triple line, which raises the apparent contact angle on advancing, or lowers it on receding, in comparison to the angle on the clean surface (De Jonghe and Chatain 1995). The deformation of the triple line is damped when the defects are separated by less than the capillary length. The details depend on the magnitude of the contact angles on the clean and defect surfaces, as well as the shape and distribution of defects (De Gennes 1985). 4.3. WETTING ON A ROUGH SURFACE AND NEW EQUILIBRIUM CONTACT ANGLES

A real surface is rough. This is one of the main origins of scatter in contact angle data, which is often forgotten. A rough surface resembles a landscape of hills and valleys on which the contact angle is locally the intrinsic contact angle of the surface, θY. A local contact angle, θδ can be defined as a function of δ, the local tilt angle of the surface referred to the average plane of the solid surface. A rough surface has at least two types of local contact angles which arise because of the positive and negative slopes of roughness. Figure 10 shows a sketch of a saw-tooth roughness where the slopes are +δ and –δ (with δ > 0). The apparent contact angle can take on any value between the minimum and the maximum local angles, θ−δ and θ+δ, respectively. These extreme angles are the minimum receding and the maximum advancing angles. The maximum wetting hysteresis is equal to 2δ.

Figure 10. Wetting on a saw-tooth rough surface.

The contact angle which corresponds to the minimum of the total interfacial energy (Figure 8) is called the Wenzel contact angle. It takes into account the increase in the areas of the solid/fluid interface due to the roughness. If K is the ratio of the actual to the geometric solid/fluid interface, the Wenzel equilibrium contact angle, θW, is written as follows:

WETTING

cos θW = ( 1 + K )

γ SV − γ SL = ( 1 + K )cos θY γ LV

439

(11)

The lower limit of defect sizes that must be included in K is still not known. This can be an issue in the case of fractal roughness, where K tends towards infinity. A similar type of equilibrium contact angle can be defined for a solid with a randomly heterogeneous surface. For a surface consisting of two different solids, 1 and 2, with contact angles θY1 and θY2 and area fractions f and 1–f, respectively, the Cassie equilibrium contact angle, θC, is given by the following relation:

cos θC = f cos θY 1 + ( 1 − f )cos θY 2

(12)

The Wenzel and Cassie contact angles have values that lie within the wetting hysteresis range, i.e. between the maximum advancing and the minimum receding contact angles. They are difficult to measure because they correspond to local minima of the total interfacial energy, and are surrounded by high energy barriers that lead to trapping of the wetting system in metastable configurations (Eick and al. 1975). At the end of this section it must be emphasized that the measured contact angles are “apparent”. They depend only on the location of the triple line, which itself depends on the local pinning. When measurements are performed with a sessile drop, the measured angle is deduced from the macroscopic shape of the drop. Thus, it often differs from an equilibrium or a local contact angle. As an example, Figure 11 shows the effect of micron size heterogeneities on the shape of the triple line of a solidified tin droplet; it is pinned on silicon pillars, better wetted than the matrix surface of silica.

Figure 11. Secondary electron image of the triple line of a solidified droplet of tin attached to silicon heterogeneities organized on a silica surface. The edge of the silicon squares is 50 μm.

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4.4. CONTROLLING SOLID SURFACES FOR SPECIFIC WETTING BEHAVIOR

When the surface is very rough and the intrinsic or Young contact angles are either large or small, particular types of wetting behaviors can take place (Johnson and Dettré 1964), and can be used to control the location of liquids on surfaces. Composite wetting or super-hydrophobicity takes place when a liquid cannot enter narrow valleys in the surface for which the intrinsic contact angle is larger than 90°. In the limit of extremely steep and narrow asperities, the apparent contact angle is close to 180° (see Figure 12). This phenomenon has recently been renamed the “lotus effect”. On the other hand, capillary spreading is enhanced on such surfaces up to perfect apparent wetting when the intrinsic contact angle is lower than 90°. In the limit of extremely steep and narrow asperities the apparent contact angle is close to 0°. This can be called the “sponge effect”. Thus, by combining roughness and chemical heterogeneities natural materials are able to develop different wetting properties for their surfaces (see for example (Marmur 2004)). a

b liquid gas solid

Figure 12. The composite wetting phenomenon: a) Sketch of a drop with a composite interface on a lotus-like leaf surface; b) Image of almost non-wetting water droplets sitting on a water lily leave with a “hydrophobic” surface.

These phenomena prevent a proper measurement of the contact angles. However, they can be used to suppress contact between solid and liquid, or to control the location of a liquid. 4.5. EQUILIBRATION PHENOMENA AT LIQUID-SOLID INTERFACES

In order to approach the intrinsic contact angle, wetting experiments are performed on surfaces that have been cleaned and polished. They can be single-crystal, vitreous materials as SiO2 or C, or polycrystalline substrates. However, a careful preparation of the surface does not prevent the

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occurrence of several phenomena which take place at the interface. In what follow we detail some of them which may be of importance in the framework of nuclear plant design. Bubbles can form at the solid/liquid interfaces of sessile drops by trapping of the gas phase during the first step of melting of the material of the drop. Figure 13 shows the interface of a solidified drop with such features. The nucleation and growth of bubbles at the solid/liquid interface is fostered by residual oxide skins which are present on the substrate or around the metal piece before it melts. The gaseous cavities must achieve equilibrium with both the solid and the liquid phases. They are mainly located inside the initial drop base, as it can be seen on the picture of Figure 14.

Figure 13. Bubbles at the ex-solid/liquid interface of a Pb-Bi drop.

Figure 14. Tracks of gaseous cavity formed at the surface of a T91 steel sample annealed at 380°C in a Pb-Bi alloy for a month under hydrogen. The large light grey features are lumps of Pb-Bi which have remained trapped in the cavities during the emersion of the piece of steel.

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In wetting experiment these cavities artificially increase the volume of the drop and may perturb the contact angle measurement. On chemically complex substrates, the bubbles trapped at the solid/liquid interface can react with the solid substrate, and change its composition and its shape. The interface becomes heterogeneous and rough and a wetting contact angle is irrelevant to describe the solid/liquid/vapor system. As an example, Figure 14 shows tracks of gaseous cavity formed at the surface of a T91 steel sample annealed at 380°C in a Pb-Bi alloy for a month under hydrogen. Local crystallization of vitreous surfaces is another phenomenon which modifies the morphology and the wettability of a surface. Figure 15 shows an example of a SiO2 surface which has been in contact with a drop of Sn. The surface (ex-interface) shows different regions which correspond to different time of contact of the liquid metal with silica. It is again clear that the solid, originally flat and homogeneous, has become heterogeneous and rough.

Figure 15. A SiO2 surface which has been in contact with a drop of Sn. a) Two regions corresponding to two different time of contact of a sessile drop of liquid Sn at 700°C (I = surface II = short contact with liquid tin, III = longer contact with liquid tin); b) Detail of the morphology of the crystallized surfaces which have been in contact with tin.

Before ending this section, it must be emphasized that solid/liquid interfaces between complex materials for metallurgical applications are difficult to handle and may evolve depending on the experimental conditions. At high temperature, the thermodynamic equilibrium of small quantity of liquids, like drops, is rapidly reached. However circulating liquids, or nonisothermal system may behave differently. Even with almost equilibrated liquid and solid phases, the interfaces can slowly change composition by

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reacting with the impurities dissolved in the liquid (mainly oxygen but also sulfur and carbon), or with the species diffusing from the solid subsurface. The purpose of this paper is not to review the large variety of complex phenomena taking place at interface. Illustrations of such complex behaviors can be found in (Ford 1983). 5. Conclusions At first sight, wetting appears to be a simple geometric consequence of the equilibrium of 3 interfacial energies. However its measurement and control requires knowledge of the thermodynamics of the phases in equilibrium and of the shape of the surface at the microscopic scale.

References Brakke K.A., 2003, The Surface Evolver, http://www.susqu.edu/facstaff/b/brakke/evolver. Chatain D. and Carter W.C., 2004, Wetting dynamics: Spreading of metallic drops, Nature Mater. 13: 843-845. Chatain D. and Galy D., 2006, Epitaxy of Pb crystal on Cu, J. Mater. Sci. 41: 7769-7774. Chatain D., Ghetta, V. and Wynblatt P., 2004, Equilibrium shape of copper crystals grown on sapphire, Interface Sci. 12:7-18. Chatain D., Wynblatt P., and Rohrer G.S., 2005, Equilibrium crystal shape of Bi-saturated Cu crystals at 1223 K, Acta Mater. 53:4057-4064. Chatain D., Lewis D., Baland J.P., and Carter W.C., 2006, Numerical analysis of the shapes and energies of droplets on micropatterned substrates, Langmuir 22: 4237-4243. De Gennes P.G., 1985, Wetting: statics and dynamics, Rev. Mod. Phys. 57: 827 - 863. De Jonghe V. and Chatain D., 1995, Experimental study of the wetting hysteresis on surfaces with controlled geometrical and/or chemical defects, Acta Metall. et Mater. 43: 15051510. Eick J.D., Good R.J., and Neumann A.W., 1975, Thermodynamics of Contact Angles. II. Rough Solid Surfaces, J. Colloid Interf. Sci. 53: 235-248. Ford S.P., 1983, Stress corrosion cracking of iron-based alloys in auqeous environment, in: Treatrise on materials science and technology, C.L. Briant and S.K. Banerji, Academic Press, New-York, 25 pp. 235-274. Ghetta V. and Chatain D., 2002, Morphologies adopted by Al2O3 single-crystal surfaces in contact with Cu sessile droplets, J. Am. Ceram. Soc. 85: 961-964. Geysermans P., Elyeznasni N., and Russier V., 2005, Layered interfaces between immiscible liquids studied by densty-functional theory and molecular dynamics simumations, J. Chem. Phys. 123: 204711-204718. Herring C., 1951, Some theorems on the free energy of crystal surfaces, Phys. Rev. 82: 87-93. Heyraud J.C., and Metois J.J., 1983, Equilibrium shape and temperature: Lead on graphite, Surf. Sci. 128:334-342.

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Hirth J.P., 1973, Thermodynamics of surfaces, in: Structure and Properties of Metal Surfaces, Honda Memorial Series in Materials Science No. 1, Maruzen Co. Ltd., Tokyo, pp. 10-33. Johnson R.E. and Dettré R.H., 1964, Contact angle hysteresis, I: study of an idealized surface, Adv. Chem. Sci. 43: 112-135. Kaplan W.D. and Kaufmann Y., 2006, Structural order in liquids induced by interfaces with crystals, Annu. Rev. Mater. Res. 36:1-48. Keene B.J., 1993. Review of data for the surface tension of pure metals, Int. Mater. Rev. 38:157-192. Kitayama M. and Glaeser A.M., 2002, The Wulff shape of alumina: III, undoped alumina, J. Am. Ceram. Soc. 85: 611-622 Kumikov V.K. and Khokonov K.B., 1983, On the measurement of surface free-energy and surface-tension of solid metals. J. Appl. Phys., 54: 1346-1350. Mills K.C. and Su Y.C., 2006, Review of surface tension data for metallic elements and alloys: Part 1 - Pure metals, Int. Mater. Rev. 51: 329-351. Levi G. and Kaplan W.D., 2003, Aluminium-alumina interface morphology and thermodynamics from dewetting experiments, Acta Mater. 51: 2793–2802. Marmur A., 2004, The lotus effect: superhydrophobicity and metastability, Langmuir 20: 3517-3519. Mullins W.W., 1963, Solid surface morphologies governed by capillarity, in: Metal Surfaces: Structure, energetics and kinetics, W.D. Robertson and N.A. Gjostein, eds., ASM, Cleveland, p. 17-66. Saiz E., Cannon R.M., and Tomsia A.P., 1999, Energetics and atomic transport at liquid metal/Al2O3 interfaces, Acta Mater. 47: 4209-4220. Saiz E., Cannon R.M., and Tomsia A.P., 2000, Reactive spreading, adsorption, ridging and compound formation, Acta Mater. 48: 4449-4462. Vinet B., Magnusson L., Fredriksson H., and Desré P.J., 2002, Correlations between surface and interface energies with respect to crystal nucleation, J. Colloid Interf. Sci. 255: 363374. Wynblatt P. and Chatain D., 2006, Anisotropy of Segregation at grain boundaries and surfaces, Metall. Mater. Trans. 37A:2595-2620. Wynblatt P. and Shi Z., 2005, Relation between grain boundary segregation and grain boundary character in fcc alloys, J. Mater. Sci. 40:2765-2773.

POTENTIOMETRIC SENSORS FOR HIGH TEMPERATURE LIQUIDS J. FOULETIER* Grenoble University, Laboratoire d’Electrochimie et de Physico-chimie des Matériaux et des Interfaces (LEPMI), E.N.S.E.E.G., BP 75, 38402 Saint Martin d’Hères Cedex, France V. GHETTA Laboratoire de Physique Subatomique et de Cosmologie, CNRS, 53 av. des Martyrs, 38026 Grenoble Cedex, France

Abstract: The purpose of this chapter is to present a brief overview of the potentiometric sensors available for applications in high temperature liquids. The topics addressed include: the calculation rules of the e.m.f. of the electrochemical chain, with special emphasis on sources of errors, a discussion on the reference electrodes of the cells according to the temperature and the types of analyzed solutions. Finally, the state of the art of the potentiometric sensors available for monitoring of oxygen, hydrogen and carbon in molten metals and for studying oxoacidic properties of molten salts is presented. Keywords: Potentiometric sensors; oxygen; hydrogen; carbon; molten metals; molten salts; reference electrodes; sources of errors

1. Electrochemical Chains High temperature potentiometric sensors can be used for the measurement of the thermodynamic activity of ions (such as O2-), atoms (such as O or C) or molecules (such as O2 or H2) dissolved in molten salts or in molten metals. The choice of the constituents of the electrochemical chains has to fulfill different requirements according to their utilization, i.e., one-shot measurements (oxygen monitoring in molten steel) or long-term on-line monitoring.

______

* Jacques Fouletier, LEPMI, ENSEEG, BP 75, 38402 Saint Martin d’Hères cedex, France; e-mail: [email protected]

V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

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1.1. DEFINITIONS

Whatever the application, the entire electrochemical cell, which can be rather complex, is always a two-electrodes device involving one or several solid or liquid electrolytes, separated by membranes or liquid junctions, with two identical terminal metals. The sensor signal is the cell e.m.f. E which is the difference between the electrostatic potentials of the two terminal metals:

E = φ (+) − φ (−)

(1)

The poles (+) and (–) are chosen so that E is positive. According to the dedicated applications, all the constituents of the electrochemical chain indicated previously can be not necessary. First, the main characteristics and requirements of the individual cell constituents will be given. It should be recalled that an electrode is the interface between an ionic conductor and an electronic one. Moreover, the chemical species electrochemically reacting at this interface have also to be well-defined. Dealing with sensors, and owing to the nature of the interfacial exchange, four kinds of electrodes can be considered: •

1st kind electrode: metal/metal ion electrode (Mn+/M), corresponding to the equilibrium reaction: Mn+ + n e– = M.

•

2nd kind electrode: Ag/AgCl/Cl–, corresponding to the equilibrium reaction: AgCl + e– = Ag + Cl– in which AgCl is a sparingly soluble salt. This type of electrode is widely used as reference electrode.

•

3rd electrode: formation of a new phase (O2, M/β-alumina(Na+), corresponding to the equilibrium reaction: 2 Na+ + 2 e– + 1/2 O2 = Na2O

•

Gas electrode in which the overall reaction requires a three-phase boundary between an electrolyte, a metal and a gas:

•

1/2 O2(gas) + 2 e–(metal) = O2–(electrolyte).

For thermodynamic measurements, the electrochemical chain must include at least one reference electrode. As indicated previously, the most widely used reference electrodes are 2nd kind electrodes, such as Ag/AgCl/Cl–. The potential of this electrode obeys the theoretical Nernst law:

E = E° −

RT ln a − Cl F

(2)

Provided the chloride concentration is constant, the potential adopted by the reference electrode is constant. For high temperature applications, the electrolytes are either solids or fused salts that dissociate into ions. In the case of liquid electrolytes, a

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porous membrane separating both solutions ensures the electrical contact. There will be inevitably interdiffusion of ions through the membrane. Usually, the ion mobilities are different that will give rise to a potential difference between both electrolytes. Any measurement of the chain e.m.f. includes a contribution from this diffusion (or liquid junction) potential. The electrochemical chain can be a concentration cell, which has identical electrodes, and a membrane inserted between solutions differing only in concentration. Two cases are considered: •

the membrane is permeable to only one ion (solid electrolyte: in the example, yttria stabilized zirconia referred to as YSZ), such as:

(−)Pt / Fe /Pb(L ) / PbO − SiO2(L ) /YSZ(<< O2− >>) /PbO(L ) /Pb( L ) / Fe / Pt(+) In this case the e.m.f. can be calculated. •

the membrane is permeable to several charged particles (several ions or ions and electrons) or is a porous membrane (“liquid junction”), such as:

(−) Pt / Fe / Pb( L ) / PbO − SiO2( L ) / porousoxide / PbO( L ) / Pb( L ) / Fe / Pt (+) In this case, a flux of matter takes place through the porous membrane, inducing a source of error, which has to be estimated or minimized (see section 4.2). 1.2. EQUILIBRIUM CONDITIONS

At the interfaces between the different phases involved in the sensor, various situations can be considered: •

exchange of one particle j at the interface between two phases α and β. ~ The equilibrium condition is ΔG =0 , e.g.,

μ˜ αj = μ˜ βj

(3)

where

μ˜ αj = μαj + z j Fφ α

μ˜ βj = μ βj + z j Fφ β

μ˜ αj and μαj are respectively, the electrochemical potential of j in the α phase and the corresponding chemical potential. φα is the electrical potential within the α phase and zj has the sign of the charge of the particle j.

Consequently, at equilibrium, one obtains the Galvani potential between both phases that cannot be measured directly:

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φα − φ β = − •

(

1 μαj − μ βj z jF

)

(4)

exchange of more than one particle at the interface between two phases α and β, e.g., two ions or an ion and an electron. The resulting equilibrium condition is:

μ˜ αj = μ˜ βj and μ˜ αk = μ˜ kβ

(5)

It can be demonstrated that the interface is not in equilibrium unless one of the phases is saturated. •

junction between two ionic solids conducting by two different ions, e.g., interface between stabilized zirconia, O2– conductor, and β-alumina, sodium ion conductor (referred to as β). The equilibrium between both solids can be written: O2– + 2 Na+ = Na2O. The equilibrium condition is: β β μ~OYSZ +2μ~Na =μ Na O 2−

+

(6)

2

leading to the following Galvani potential:

φ β − φ YSZ = •

(

1 β μ − 2 μ β + − μYSZ O 2− Na 2F Na2O

)

(7)

gas electrode reaction (section 1.1). As an example, considering the oxygen electrode reaction 1/2 O2 + 2 e– = O2– on stabilized zirconia (O2, Pt/YSZ), the equilibrium condition is: 1 μO 2 + 2 μ˜ Pt− = μ˜ YSZ e O 2− 2

(8)

leading to the Galvani potential:

φ Pt − φ YSZ =

RT 1 ⎛ Pt ⎞⎟ ⎜ 2 μ − − μ YSZ ln PO 2 + 2− ⎠ ⎝ e O 4F 2F

(9)

1.3. E.M.F. CALCULATIONS RULES FOR AN IDEAL CHAIN

Let us consider the following electrochemical chain that can be proposed for the measurement of the oxide ion activity in a molten NaCl-KCl salt (cell of this type has been proposed, as an example, by Rouquette-Sanchez and Picard (2004)):

(−)Pt / Ag / AgCl /NaCl,KCl /Pyrex /NaCl,KCl,Na2O /YSZ /Pt,O2 (+)

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The cell can be considered as ideal, assuming that: – each solid electrolyte (Pyrex and YSZ) is conducting by only one ion (the minority carriers are neglected), i.e., Pyrex is considered as a pure sodium ion conductor, and YSZ as a pure O2– conductor, – the electronic conductivity of the electrolytes is negligible, – no current is passing through the cell, – all the interfaces are under equilibrium. The e.m.f. of the electrochemical chain can be calculated using the following rules: 1. Within each solid electrolyte (Pyrex and YSZ in the example), the electrochemical potential of the majority carrier is constant:

~ α ,YSZ =μ~ β ,YSZ μ˜ α ,Pyrex = μ˜ β ,Pyrex or μ + + O 2− O2− Na

Na

(10)

2. Each junction is characterized by an equilibrium involving only the majority carriers of the phases on contact, – same ionic carrier: ((Na2 O))/Pyrex

μ˜ α ,((+ Na2O )) = μ˜ α ,Pyrex + Na

Na

(11)

– different ionic carriers (YSZ//β-alumina), see equation (6). Figure 1 shows the variation of the electrical potential in the cell chosen as an example, in case of ideality.

Figure 1. Schematic presentation of the variation of the electrical potential in the cell.

The cell e.m.f. can be easily calculated in case of an ideal chain, using the preceding calculation rules: Pt Pt E = φ(+) − φ(−) =

RT RT ln PO2 − ln a((Na 2O )) + C 4F 2F

the constant C depends on the characteristics of the reference electrode.

(12)

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2. Electrolytes Two types of electrolytes are used in high temperature sensors, i.e., either solid electrolytes (single crystals, sintered ceramics or glasses) or molten salts. 2.1. SOLID ELECTROLYTES

The main requirement for a solid electrolyte is a pure ionic conductivity. However the nature of mobile species does not matter (e.g., sodium-ion conductors such as β-alumina have been successfully used in oxygen sensors) although the kinetics of the electrode reactions are generally more reversible for the sensing of the co-species, e.g., O2– conductors for sensing O2 or Cl- conductors for sensing Cl2 (Kleitz et al. 1991). The consequences of a non-zero electronic conductivity on the cell e.m.f. will be discussed in section 4.3. It should be pointed out that the total resistance of the ionic membrane is not an essential parameter, provided that the total impedance of the cell remains negligible compared to the input impedance of the millivoltmeter. The solid electrolytes must fulfill other criteria such as chemical stability, mainly in case of contact with a molten salt, mechanical stability and especially good heat shock strength and a thermal expansion coefficient compatible with those of the other constituents of the cell. Only a few solid electrolytes are used in potentiometric sensors: oxide ion conductors such as calcia or yttria stabilized zirconia or yttria doped thoria, fluoride ion conductors such as CaF2 or LaF3, and cationic conductors such as β-aluminas, Nasicon or silicate glasses (Kudo 1997). 2.2. MOLTEN SALTS

As indicated in another chapter of this book (Poignet and Fouletier), the molten salts exhibit interesting properties: – a large number of molten salts are available: chlorides, fluorides, carbonates, nitrates, etc. – wide range of temperature: 150°C to more than 1000°C – good stability – high electrical conductivity – high chemical and electrochemical reaction rates – wide electrolytic domain (redox and acid-base).

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However, a lot of difficulties have to be overcome: – corrosion towards containers, separators, and ionic membranes, especially at high temperatures, – handling not easy, – chemical interaction with water. 3. Reference Electrodes In high temperature electrochemistry, there is no generally accepted reference electrode. The reference systems are gas mixtures or 2nd kind electrodes. 3.1. SENSORS FOR MOLTEN METALS (LEAD, SODIUM)

The reference system can be a gas (pure or diluted in an inert gas), or a mixture of gases in contact with a noble metal or 2nd kind electrodes. The main reference electrodes currently used in sensors are air or pure oxygen, CO-CO2 or H2-H2O mixtures and 2nd kind electrodes, e.g., Pd-PdO, BiBi2O3, In-In2O3, Cr-Cr2O3, etc. (Kleitz et al. 1991). In this last case, the Gibbs free energies of formation of the compounds are generally not sufficiently accurate and a calibration of the sensor is generally necessary. The two main requirements concern the stability of the electrolyte in contact with the reference system and the good electrochemical reversibility, i.e., the low sensitivity to local fluxes of matter through the electrolyte membrane. 3.2. SENSORS FOR MOLTEN SALTS (CHLORIDES, FLUORIDES)

No universally accepted reference electrode is available although reference electrodes based on the Ag(I)/Ag(0) couple are undoubtedly the most common. Halogen electrode in halide melts are generally successful, but such electrodes are inferior in experimental convenience to those based on the Ag(I)/Ag(0) couple (Janz 1967, Winand 1972). The design of reliable reference electrodes in molten fluorides remains a major problem, due to the corrosive action on metal electrodes, and on glass or ceramics used as containers or diaphragms, and also because of the undetermined liquid junction potentials. A Ni(II)/Ni electrode contained in a boron nitride compartment gives good results over several days at 500°C (Jenkins et al. 1968). Single-crystal of LaF3 has also been proposed as ionic membrane between the reference compartment and the melt being investigated,as shown in Figure 2 (Bronstein and Manning 1972, Clayton et al. 1973).

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However, at higher temperatures, major difficulties are encountered and, until yet, no entirely satisfactory design has been proposed. A large number of authors use platinum or nickel wires as pseudo-reference electrodes, although the thermodynamic meaning is questionable: a constant potential within ±10 mV can be obtained over months (Mamantov 1969). Adhoum et al. (1995) developed a new concept of internal reference electrode based on the generation of an in-situ redox couple (Fe2+/Fe) by a galvanostatic anodic pulse (ca. 0.2 s) followed by open-circuit voltage relaxation. Such an electrode can be used to check periodically the potential of a reference or of a pseudo-reference.

Figure 2. Ni/Ni(II) reference electrode using a LaF3 single-crystal membrane for molten fluorides.

4. Sources of Errors All the constituent of the chain, i.e., the reference electrodes, the liquid junctions or porous membranes, the sensing membrane and the measuring electrode, can cause nonideal responses. 4.1. DEVIATION FROM IDEALITY ASSOCIATED WITH THE REFERENCE ELECTRODE

The main requirements concern the long-term stability of the electrode (no oxidation or reduction by the environments, no miscibility within the electrolyte, no partial reduction of the electrolyte inducing partial electronic conductivity) and a low sensitity to perturbations; to fulfill this condition, the corresponding electrode reaction must be reversible. If it is not the case, the stable e.m.f. of the sensor includes an overvoltage contribution, which constitutes a measurement error. From this point of view, Pd-PdO or CuCu2O can adjust to small disturbances more easily than the Ni-NiO mixture. The magnitude of the error due to the semipermeability flux through the

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electrolyte depends not only on the magnitude of the leak, but also on the buffer capacity at the electrode (Fouletier 1982/83). 4.2. DEVIATION FROM IDEALITY ASSOCIATED WITH LIQUID JUNCTIONS OR POROUS MEMBRANES

In molten salt electrochemistry, porous plugs or frits are used to prevent mixing of the contents of the various compartments in a manner analogous to aqueous bridges. These ionic junctions are responsible of concentration modifications, and especially of the reference salt and introduce a diffusion potential. When the solvent salt is the same in both compartments and the concentrations of solutes are low (less than 0.1 molal), the liquid junction potentials across the compartment separator is at most one or two mV and can be neglected. It should be pointed out that other sources of error could be ascribed to chemical attack of the membrane by molten salts or precipitation of insulating compounds within the porous plugs. 4.3. DEVIATION FROM IDEALITY ASSOCIATED WITH THE SENSING MEMBRANE

Two main types of sources of error can be considered: error due to a partial electronic conductivity of the membrane and interference phenomena at the interface between the membrane and the analyzed medium. Under high temperatures and noticeable reducing or oxidizing conditions, electronic species may be present in the membranes due to equilibria between the electrolyte and the gaseous phase. As shown in Figure 3 for three electrolytes, i.e., calcia stabilized zirconia (CSZ), yttria-doped thoria (YDT) and CaF2, at a given temperature, it is possible to determine an oxygen pressure range, referred to as the electrolytic domain, in which the ionic transport is higher than 0.99. As shown in the figure, stabilized zirconia is outside its electrolytic domain in case of oxygen monitoring in molten steel and, on the contrary to stabilized zirconia, yttria-doped thoria can be considered as a pure ionic conductor for monitoring of oxygen in molten sodium. The electronic conductivity of the electrolytes induces three mains sources of error (Fouletier and Siebert 1986): •

according to Wagner, the measured e.m.f. E is linked to the theoretical e.m.f. by the following equation:

E = t i E th where ti is the ionic transport number of the solid electrolyte,

(13)

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•

the electronic conductivity induces an oxygen semipermeability flux through the membrane inducing a change of the oxygen potential at one or both interfaces,

•

the oxygen semipermeability flux polarizes the electrodes deposited on the surfaces of the electrolyte.

Figure 3. Electrolytic domains of calcia stabilized zirconia and yttria doped thoria.

It has been shown that the resulting measurement error is negligible under normal working conditions when the electronic transport number is lower than 10–3. When an active species dissolved in a liquid (SOL) is in contact with a solid electrolyte, it can penetrate into the bulk by ionic exchange followed by diffusion:

iSE + j SE ⇔ j SE + iSE

(14)

where i is the majority carrier of the solid electrolyte (SE) and j an interfering ion. In these conditions, the electrochemical potential is not a constant within the membrane (see Eq. 10) and the interfering species induce a measurement error (Fabry and Siebert 1997). This source of error has been well studied in case of ion-selective electrodes. As an example, in the case of exchange of univalent cations, the cell e.m.f. obeys the NikolskyEisenman equation:

E = E° +

RT ln (ai + K ij a j ) F

(15)

where ai and aj are the activities of the ions i and j, Kij is the selectivity constant (or interfering coefficient).

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4.4. DEVIATION FROM IDEALITY ASSOCIATED WITH THE MEASURING ELECTRODE

The oxygen semipermeability flux can induce noticeable measurement error, especially at high temperatures and in case of a very low oxygen chemical potential in the molten metal (Fouletier and Siebert 1986). Moreover, the analyzed solutions are often complex and the cell e.m.f. can be not a thermodynamic voltage but a mixed potential. 5. Case Studies The main difficulties encountered in the potentiometric analysis of high temperature liquids concern the thermal shock resistance of the ionic membranes, the wide temperature range (200°C – 1600°C), the required time life, the stability domain of the electrolytes and the corrosion. Table 1 compares, oxygen concentration ranges and the times of operation of sensors used for metals analysis. TABLE 1. Comparison of the sensing conditions in molten metals.

Temperature (°C) Rate of temperature changes (K.min–1) CO (ppm) Time of operation Solid electrolyte material

Sodium

Copper

Steel

400

1150

1650

5

50

5000

0.5 - 50

1 - 1000

1 - 2000

10 000 h

100 h

5s

ThO2 - Y2O3 (cub.)

ZrO2 - CaO (cub.)

ZrO2 - MgO (cub.-mon.)

5.1. OXIDE ION ACTIVITY IN MOLTEN CHLORIDES

The effect of oxide ions introduced either by water vapor or oxygen contamination may be crucial for the electrochemical use of molten chlorides. The electrochemical sensor is based on stabilized zirconia or doped thoria (Ullmann and Teske 1991). The reference electrode is either a gas mixture (air or H2-H2O/Pt) or a metal-metal oxide mixture. The e.m.f. of the cell obeys the following Nernst equation:

ΔE = ΔE° −

( )

2.3 RT log O 2 − 2F

(16)

where (O2-) is the oxide ion molality. Figure 4 shows examples of determination of oxide solubilities in molten chlorides.

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Figure 4. Examples of the determination of solubilities of ZnO or Li2O in molten chlorides using a zirconia sensor (from Shenin-King 1994).

5.2. MONITORING OF OXYGEN, HYDROGEN AND CARBON IN MOLTEN METALS (SODIUM, LEAD AND LEAD-BISMUTH EUTECTIC)

The monitoring of oxygen levels in liquid sodium in fast reactor coolant circuits is necessary to control the corrosion processes. It has been demonstrated that the corrosion of metals and alloys increases with high oxygen concentration in sodium. An optimum amount of carbon in austenitic and ferritic steels used as structural materials of nuclear reactors is essential for keeping good mechanical properties during their life. Owing to the solubility of carbon in molten sodium, according to the temperature, carburization or decarburization can take place. Moreover, accidental ingress of oil from pumps or contamination from carbon dioxide in air will lead to a building up of carbon activity in sodium. Simultaneously, the reaction of rust with sodium gives rise to hydrogen release. A few set-ups have been proposed for oxygen, hydrogen and carbon monitoring in molten sodium. The main difficulties for the development of these devices concern the corrosive attack of sodium against ceramic materials, the high-radiation environment in primary coolant circuit, the stability of the electrolytes in highly reducing conditions, and the electrode reaction kinetics at low temperatures. The oxygen sensors are based on yttria-doped thoria tubes or crucibles (Taylor and Thompson 1983, Ullmann and Teske 1991, Jakes et al. 1984). Air/platinum electrode and In-In2O3 or Sn-SnO2 mixtures are used as reference electrode. The sensor temperature is in the range 350-500°C. Time lifes exceeding one year have been obtained. Hydrogen monitoring in molten sodium has been carried out with the following cell:

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Na(H) /Fe /CaH 2 − CaCl2 /Fe /Li,LiH The reference electrode was a Li-LiH coexistence electrode using argon as a protective gas, the hydride-ion solid electrolyte was a mixture of CaH2 (5 to 7.5%) and CaCl2, thin iron diffusion membranes were used between electrolytes and electrodes (Hobdell and Smith 1982, Gnanasekaran et al. 1982). Other types of sensors have been proposed for monitoring of hydrogen in aluminum (ca. 973 K), Cu (ca. 1423 K) or Zn (ca. 723 K) (Fukatsu et al. 1995, Iwahara et al. 2004, Fukatsu and Kurita 2005). Protonic conductors, i.e., Yb, Nd or Gd cerates, In doped zirconate (CaZr0.9In0.1O3-δ) were used as solid electrolyte. The monitoring of hydrogen in lead and leadbismuth eutectic using these types of protonic conductors is under progress. The sensors developed for carbon monitoring are concentration cells based on molten carbonates as electrolyte. The reference system is either cementite, Fe3C (Hobdell and Smith 1982) or graphite (Pillai and Mathews 1986). Thin permeable α-iron membranes as junctions between both electrodes and the molten electrolyte are used:

Fe3C or graphite / Fe / Na2CO3 − Li2CO3 / Fe /((C)) Na Assuming the following equilibrium at both electrodes;

CO32− + 4 e−

⇔ C + 3O2−

(17)

the e.m.f. of the cell obeys the Nernst law:

E=

RT a((C ))meas . ln 4F a<> ref .

(18)

Molten lead and lead-bismuth are proposed as spallation target and as a coolant in nuclear ADS reactors. The control of the corrosion of the structural and window materials requires the continuous monitoring of the oxygen and hydrogen potential in the liquid alloys. Experimental devices have been designed for the oxygen activity control and monitoring of molten lead and lead-bismuth eutectics at temperatures below 500°C (Ghetta et al. 2001, Ghetta et al. 2002, Ganesan et al. 2006). For e.m.f. measurements at temperatures below 600°C, the main sources of errors may be ascribed to non-equilibrium conditions at one or both electrodes of the cell. Intensive efforts are being made to improve the electrode reaction reversibility (reference and measuring electrodes). The main limitation of the Air, Pt/YSZ reference electrode is due to its irreversibility at low temperatures. The lower limit is assumed to be of ca. 450°C. Many substitutes for platinum were reported in literature. The electrode materials exhibiting the best performances are mixed conducting oxides with perovskite structure, e.g. La0.8Sr0.2CoO3. Co-existence

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electrodes such as Bi-Bi2O3, In-In2O3 have also been tested (Courouau 2004, Courouau et al. 2002, Konys et al. 2001). 6. Conclusion This brief overview shows that potentiometric sensors are powerful tools provided that the appropriate constituents of the cells are chosen taking into account the dedicated application. Moreover, a careful analysis of the sources of errors has to be carried out.

References Adhoum, N., Bouteillon, J., Dumas, D., and Poignet, J. C., 1995, Internal reference systems for electrochemical studies in molten sodium fluoride at 1025°C, J. Electroana.l Chem. 391:63-68. Bronstein, H. R., and Manning, D. L., 1972, Lanthanum trifluoride as a membrane in a reference electrode for use in certain molten fluorides, J. Electrochem. Soc. 119:125-128. Clayton, F. R., Mamantov, G., and Manning, D. L., 1973, Use of a lanthanum trifluoride membrane reference electrode in molten fluorides and fluoroborates [analytical methods for molten salt reactor streams], High-Temperature-Science 5:358-64. Courouau, J. L., 2004, Electrochemical oxygen sensors for on-line monitoring in leadbismuth alloys: status of development, J. Nuclear Mat. 335:254-259. Courouau, J. L., Deloffre, P., and Adriano, R., 2002, Oxygen control in lead-bismuth eutectic: First validation of electrochemical oxygen sensors in static conditions, J. Phys. IV France 12:141-153. Fabry, P., and Siebert, E., 1997, Electrochemical Sensors, in: The CRC Handbook of Solid State Chemistry, P. J. Gellings and H.J.M. Bouwmeester eds., CRC Press, London, pp. 329-369. Fouletier, J., 1982/83, Gas analysis with potentiometric sensors. A review, Sensors and Actuators 3:295-314. Fouletier, J., and Siebert, E., 1986, Potentiometric solid-state oxygen sensors, Ion-Selective Electrode Rev. 8:133-151. Fukatsu, N., Kurita, N., Yajima, T., Koide, K., and Ohashi, T., 1995, Proton conductors of oxide and their application to research into meal-hydrogen systems, J. Alloys and Compounds 231:706-712 Fukatsu, N., and Kurita, N., 2005, Hydrogen sensor based on oxide proton conductors and its application to metallurgical engineering, Ionics 11:54-65. Gnanasekaran, T., Ganesan, V., Periaswami, Mathews, C. K., and Borgsedt, H. U., 1990, Application of an electrochemical hydrogen meter for studying reactions in liquid sodium, J. Nuclear Mat. 171:198-202. Ganesan R., Gnanasekaran, T., and Srinivasa, R. S., 2006, Diffusivity, activity and solubility of oxygen in liquid lead and lead-bismuth eutectic alloy by electrochemical methods, J. Nuclear Mat. 349:133-149. Ghetta, V., Gamaoun, F., Fouletier, J., Hénault, M., and Lemoulec, A., 2001, Experimental setup for steel corrosion characterization in lead bath, J. Nuclear Mat. 296:295-300.

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Ghetta, V., Fouletier, J., Hénault, M., and Le Moulec, A., 2002, Control and monitoring of oxygen content in molten metals. Application to lead and lead-bismuth melts. J. Phys. IV France 12:123-140. Hobdell, M. R., and Smith, C. A., 1982, Electrochemical techniques for monitoring dissolved carbon, hydrogen and oxygen in liquid sodium, J. Nuclear Mat. 110:125-138. Iwahara, H., Asakura, Y., Katahira, K., and Tanaka, M., 2004, Prospect of hydrogen technology using proton-conducting ceramics, Solid State Ionics 168:299-310. Jakes, D., Kral, J., Burda, J., and Fresl, M., 1984, Development of electrochemical oxygen meter for liquid sodium, Solid State Ionics 13:165-173. Janz, G. J., 1967, Molten Salts Handbook, Academic Press, London, p. 501. Jenkins, H. W., Mamantov, G., and Manning, D. L., 1968, E.M.F. measurements on the nickel—nickel(II) couple in molten fluorides. J. Electroanal. Chem. 19:385-389. Kleitz, M., Siebert, E., Fabry, P., and Fouletier, J., 1991, Solid-State Electrochemical Sensors, in: Sensors, A comprehensive survey, W. Göpel, J. Hesse, J. N. Zemel eds.,VCH, Weinheim, pp. 341-428. Konys, J., Muscher, H., Voβ, Z., and Wedemeyer, O., 2001, Development of oxygen meters for the use in lead-bismuth, J. Nucl. Mat. 296:289-294. Kudo, T., 1997, Survey of types of solid electrolytes, in: The CRC Handbook of Solid State Chemistry, P. J. Gellings and H. J. M. Bouwmeester eds., CRC Press, London, pp. 195221. Mamantov, G., Molten Salts: Characterization and Analysis, Dekker, New York, 1969, p.537. Pillai, S. R., and Mathews, C. K., 1986, An electrochemical meter for measuring carbon activity in liquid sodium, J. Nuclear Mat. 137:107-114. Rouquette-Sanchez, S., and Picard, G. S., 2004, Chalcogenide chemistry in molten salts. I. Selenium (IV) acido-basic and redox properties in the LiCl-KCl eutectic melt at 450, 500, 550 and 600°C, J. Electroanal. Chem. 572:173-183. Shenin-King, J., 1994, Processus impliqués dans la chloruration d’oxydes métalliques en milieux chlorures fondues, PhD Thesis, Paris 6. Taylor, R. G., and Thompson, R., 1983, Testing and performance of electrolytic oxygen meters for use in liquid sodium, J. Nuclear Mat. 115:25-38. Ullmann, H., and Teske, K., 1991, Determination of oxygen activities in melts and solid materials by solid electrolyte cells, Sensors and Actuators B 4:417-423. Winand, R., 1972, Etat actuel des recherches visant à la mise au point d’une électrode utilisable dans les sels fondues, Electrochim. Acta 17:251-264.

INFLUENCE OF LIQUID SODIUM ON MECHANICAL PROPERTIES OF STEELS, REFRACTORY ALLOYS AND CERAMICS H. U. BORGSTEDT FZK, Karlsruhe, Germany

Abstract: The first part of this chapter deals with the corrosion processes that influence the material behavior. Corrosion causes mass losses and selective leaching of alloying metals. Intercrystalline effects can occur due to such corrosion reactions. Sodium and solid materials are able to exchange interstitial components as non-metallic elements the most important of which are carbon and nitrogen. Some alloys pick up oxygen from sodium even at very low concentrations of ≤5 μg/g in the liquid phase. Internal oxidation is the corrosion process in this case. Hardened or even brittle surface layers are produced. Creep-rupture and low-cycle fatigue tests in fast flowing sodium have shown that the time to rupture and also the number of cycles to failure of austenitic steels as type 304L or 316L(N) are larger in the liquid metal than in air. The tertiary creep region of these materials is, however, shortened by the influence of sodium, and short grain boundary effects occur.

1. Corrosion Mechanisms Corrosion of solid metals in pure liquid sodium at elevated temperature is based on the physical dissolution of their components in the liquid phase. The driving force of the dissolution process is the difference of the chemical activities of the components in the solid and liquid phases. The dissolution rate ΔS/Δt is ruled by the difference of the chemical activities in the solid phase, a(s), minus that in the liquid phase, a(l) (Epstein 1957, Weeks 1967). ΔS/Δt = ß (a(s) – a(l))

(1)

with ß as the corrosion constant. The activity at the interface is unity in the simplest case, if a pure metallic element is the solute and its concentration should be close to zero in the bulk of the liquid phase at the beginning of a corrosion reaction. The concentration of the solute grows in the liquid metal due to diffusion from the interface into the liquid volume. The difference of the chemical activities in Eq. (1) is dependent on the character of the liquid metal system that can be isothermal in stagnant

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sodium or non-isothermal in heat transfer systems with flowing liquid metal. 1.1. CORROSION IN STAGNANT LIQUID SODIUM

The saturation concentrations of solute metals in liquid ones depend strongly on temperature. Thus, the chemical activities for a given concentration vary with changes of the temperature. If we consider a system consisting of a metallic container filled with a liquid metal, the container metal is dissolved in the liquid metal up to the concentration, at which equilibrium is reached. The equilibrium concentration is first reached at the phase boundary of the container and the solvent. The dissolution proceeds with the diffusion of the solute into the bulk of the liquid metal. The process of dissolution is fast in the beginning and becomes slow due to the increasing concentration in the bulk. This fact decreases the diffusion rates by means of lowering the concentration differences and the gradients of the chemical activities. The corroding dissolution rate becomes zero, when the differences and gradients approach zero. Sodium corrosion of ceramic materials is not based on the physical dissolution of the solid phase in the liquid metal. The attack of liquid sodium is based on chemical reactions with the ceramics. Such processes are in many cases reactions with grain boundary materials that are often less stable against chemical attack of the liquid metal. Only ceramics that are stable on the basis of their thermodynamic data in temperature ranges of interest will be considered within this chapter. 1.2. CORROSION IN TURBULENT LIQUID SODIUM FLOW

The sodium corrosion in a fast turbulent liquid metal flow is also principally based on Eq. (1). The corrosion mechanism, however, based on the dissolution of a solid metal in the liquid one, may be different, if the liquid metal system is characterized by a temperature gradient. Such a system does not tend to saturate the liquid phase with the solute in the high-temperature region, since the solution becomes supersaturated in the cold part of the system, where the supersaturated solute metal crystallizes. The liquid metal reaches the heated zone again as a not saturated solution in respect to the solute. Thus, the corrosion process continues as long as the temperature gradient and the flow of the liquid are maintained constant. The liquid gets saturated at the boundary between container tubes and the liquid metal, while the bulk stream does not reach the state of saturation. Thus, the driving force for the dissolution of the solid metal exists for the operation period of a sodium loop. The situation may be complicated by the use of

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metallic alloys the components of which are dissolved to different saturation activities in the liquid metal. The activity at the interface is unity in the simplest case of a pure metallic solute, and under these conditions Eq. (1) becomes ΔS/Δt = ß (1 – a(l))

(2)

The concentration of the solute at unity chemical activity is the saturation concentration of this metal in liquid sodium at the given temperature. The chemical activity in the bulk stream of the liquid metal is orders of magnitude lower than unity, particularly in non-isothermal flowing systems. Its value can be neglected in comparison with the saturation concentration, when the temperature gradient is in the order of ≥100 K. Thus, the corrosion rate becomes proportional to the saturation concentration: ΔS/Δt = ß csat

(3)

The corrosion constant ß depends on the hydraulic parameters of the flow through the tubes of the system under consideration. The relation between ß and these parameters was also developed for liquid metal corrosion problems by Epstein (1957) and Weeks (1967): ß = Sh D dH–1

(4)

with Sh as the Sherwood number, D as the diffusion coefficient of the solute in the liquid metal and dH as the hydraulic diameter of the test tube. The Sherwood number is related to the flow velocity of the liquid metal via the Reynolds number: Sh = 0.0481 Re0.75 Sc0.42

(5)

Re = v dH ν–1

(6)

Sc = ν D–1

(7)

Re is the Reynolds number and Sc the Schmitt number with v as flow velocity and ν as viscosity of the liquid metal. With the knowledge of the corrosion constant ß and the saturation concentration for the given temperature (the chemical activity of the solute in equilibrium with the solid phase) one can calculate the corrosion rate for sodium passing tubes the hydraulic parameters of which are known. The corrosion constant depends on temperature through the viscosity ν and the diffusion coefficient D. The corrosion rates are also related to the temperature via the solubility that is a function of temperature. The calculation of the saturation concentration of nickel in sodium at 500 to 600°C from

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corrosion data in turbulent flow of the liquid metal has resulted in values that are in agreement with literature data (Borgstedt 1998). The importance of diffusion in the liquid phase can be understood by the assumption of a laminar boundary layer, which separates the turbulent bulk stream from the solid surface. A concentration gradient occurs within this boundary layer from saturation to nearly zero. The gradient is influenced by the flow velocity due to its thickness, that decreases with an increasing Reynolds number in the tubes. Since the temperature functions of the solubility of the components of alloys as steels are different, the dissolution rates of these components are also different. This fact is the reason for a selective leaching of alloying elements of steels due to corrosion in flowing sodium. Since the solubility of the alloy cannot be exactly related to that of one of the alloying elements, we have to consider an apparent solubility based on that of the mixture of solutes. Recommended solubility equations of steel components in liquid sodium are not available. However, comparable data can be taken from the data collection of (Borgstedt and Guminski 1996). Values for the temperatures 500 and 700°C are listed in Table 1. TABLE 1: Saturation concentrations of steel components in liquid sodium at 500 and 700°C Steel component Csat (mol%) at 500°C Csat (mol%) at 700°C Fe 3.5 10–5 2.7 10–4 –5 Ni 4.3 10 1.1 10–4 –7 Cr 2.2 10 5.6 10–5 Mo 9.9 10–7 4.0 10–6

Since the solubility of iron depends strongly on the oxygen concentration in sodium, the values in Table 1 are taken from measurements in a loop containing very pure liquid alkali metal, measured at 600 and 700°C; the value at 500°C is extrapolated from these measurements. The apparent solubility of the steel components in sodium containing impurities may be different from the data listed in Table 1. 1.3. SELECTIVE LEACHING OF ALLOY COMPONENTS

Sodium corrosion of austenitic stainless steels results in the formation of surface layers in which the contents of nickel, chromium and manganese are significantly reduced, while the contents of iron and molybdenum are increased. After reaching steady-state corrosion, the concentrations of austenite stabilizing components are decreased to such values, that the structure is changed to the ferritic one. The thickness of the layer remains more or

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less constant, since dissolution at the surface removes the ferrite, while at the interface to the austenitic matrix diffusion removes nickel, chromium and manganese, that ferrite becomes stable. The selective leaching of alloy components follows equation (1) in which the lower chemical activities of the minor constituents of stainless steels have to be used for calculations. The changes of compound concentrations in the surface area of a 16/13 Ni/Cr austenitic steel exposed 5000 h in sodium flowing with a velocity of 5 m/s at 700°C were significant: Nickel was reduced to 5%, chromium to 7%, manganese to nearly 0, while molybdenum and iron concentrations were increased (Borgstedt et al. 1977). The selective leaching of alloying components as Ni and Mn that stabilize the austenitic phase leads to changes of the structure in the surface region and surface near grain boundaries. Figure 1 shows the appearance of a thin ferritic surface layer and ferritic grain boundaries in stainless steel AISI 304 after the use as duct material in a loop that was operated for 80000 h in a cross sectional image (Vaidehi Ganesan et al. 2003). The appearance of the corroded surface of a specimen exposed for 10000 h at 700°C is shown in Figure 2. The surface grains are transferred into the ferritic structure, and grain boundaries have rounded shoulders. There are cavities visible at some triple boundaries that are deeper than one layer of grains. Bright points on the ferritic corrosion surface are FeMo particles the formation of which is favoured by the enrichment of Mo in the surface area.

Figure 1: Cross section of a tube that was operated 60000 h at 550°C + 20000 at 600 °C in the CREVONA loop of FZK showing ferrite layer and grain boundaries, micro cavities and local decohesion of the ferrite layer (Vaidehi Ganesan et al. 2003).

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H. U. BORGSTEDT

Figure 2: Scanning electron microscope image of the surface of a specimen of steel X10 NiCrMoTiB 1515 after 10000 hours sodium exposure at 700°C (from Borgstedt et al. 1977).

1.4. INFLUENCE OF IMPURITIES DISSOLVED IN LIQUID SODIUM

Corrosion studies of several classes of steels have indicated that the purity of liquid sodium influences the corrosion rates. The oxygen influence on corrosion rates of different alloys in liquid sodium was shown in a systematic study by Thorley and Tyzack (1967). The formation of compounds of the dissolving metals in the solution reduces their chemical activities and enhances the corrosion rates by increasing the difference between activities in Eq. (1). Such reactions occur in the presence of oxygen, since Na2O forms complex oxides with several metals.

Figure 3: Range of stability of sodium chromite in respect to temperature and oxygen concentration in sodium from Cavell and Nicholas (1980).

The oxygen effect on the sodium corrosion of austenitic stainless steels as Type AISI 316 was included in the equation that was developed by Thorley and Tyzack (1967) on the basis of the above mentioned studies. The dependency of the mass loss on the oxygen concentration is higher than linear.

LIQUID SODIUM ON MECHANICAL PROPERTIES

log S = 2.44 + 1.5 log cO – 18000/2.3RT

467

(8)

with mass loss S in mils/y, cO in ppm and T in K. They showed that this dependency on the oxygen concentration is only valid for stainless steels and not for nickel base alloys. The mechanism of the oxygen effect on iron base alloys is not fully understood. Probably the formation of double oxides Na2O.FeO is the way in that oxygen influences the corrosion reactions. Since the solubility of chromium in Na is still lower than that of iron (see Table 1) the dissolution of chromium seems to be stronger influenced by reactions with sodium oxide. The oxygen influence is due to the reaction of sodium oxide with chromium by which a double oxide is formed: 2 Na2O + Cr = NaCrO2 + 3 Na

(9)

This oxide, which was found at the steel-sodium interface, is not protecting the solid phase, since it is more or less transferred by the flowing sodium into the region of lower temperature. The stability of the sodium chromite in contact with liquid sodium depends on temperature and the oxygen concentration in the liquid metal. Figure 3 shows the stability range of Na2CrO3 after Cavell and Nicholas (1980), indicating the reduction of oxidized steel surfaces at temperatures above 600°C in sodium containing ≤ 5 10–4 mass% oxygen. Oxygen exchange between sodium and solid metals was also observed in corrosion tests with vanadium (Champeix et al. 1967, Whitlow 1969, Smith and Kassner 1970), and niobium alloys (Evans and Thorley 1958, Yaggee and Gilbert 1967). In this case, oxygen is not only enhancing the dissolution of the alloy components. These metals getter oxygen out of the alkali metal and form oxide layers on their surfaces or internal oxides (Borgstedt 1971). Internal oxidation causes the formation of brittle layers adjacent to the surface of the alloys. The embrittlement is most significant, if the alloys contain minor components, which form very stable oxides, as for instance in alloys of the type V-10Ti. Thus, such alloys are only more or less resistant against sodium corrosion, if the liquid metal is carefully purified in respect to its oxide content. Since the solubility of carbon in sodium is lower than that of the most metals, sources for dissolved carbon are carbon containing compounds or alloys in contact with sodium in which the chemical activity of carbon is relatively high. If such carbon sources are present, sodium acts carburizing on austenitic stainless steels. This process interferes with the dissolving corrosion. High carbon contents in the surface layers of the steels stabilize chromium against dissolution due to the formation of chromium carbides. The chemical activity and concentration of dissolved carbon in sodium is influenced by the cold trap temperature of a sodium system as was shown

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by Borgstedt and Frees (1990). Figure 4 exhibits the relations between the reciprocal cold trap temperature and the activity and concentration of carbon in the sodium system as measured by means of a gas analytical carbon meter (Asher et al. 1973). Sodium with very low carbon concentrations in which the chemical activity of carbon is lower than in austenitic steels leaches carbon out of the steels. This process may even decompose chromium carbides and thus enhance the dissolution of chromium out of the surface layers.

Figure 4: The chemical activity of carbon aC in sodium and its calculated concentration cC as function of the reciprocal cold trap temperature after Borgstedt and Frees (1990).

2. Corrosion Testing 2.1. DETERMINATION OF CORROSION RATES

2.1.1. Corrosion studies in stagnant and isothermal liquid metal Corrosion studies in stagnant and isothermal liquid metals are not representative for heat transfer systems. They can however be useful for studies of chemical and physicochemical reactions between sodium and solutions of impurities in the metal with materials. In order to have well defined

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parameters of sodium chemistry the filling procedure should be done with connections to a sodium loop or inside of a pure argon atmosphere. The material the reactions of which with sodium should be studied may be used as enclosing capsule that must be hermetically closed by welding or by fittings of vacuum tightness. If that material may be sensitive to reactions with the atmosphere, the capsule should be enclosed into a surrounding container in which it can be heated to the desired temperature. Reactions of Na2O dissolved in sodium with metals or metal oxides can be examined by introducing an electrochemical oxygen meter into the static devices. 2.1.2. Corrosion studies in flowing liquid Na and thermal gradients The simplest dynamic corrosion circuit uses a thermal gradient between a hot leg and a cold one to move the sodium through the system (Klueh and DeVan 1972). Such thermal convection loops are however characterized by low temperature differences and flow velocities. Corrosion tests in this type of loops are not representative for heat transfer systems that use sodium as coolant. Sodium loops for representative testing are equipped with sodium pumps, electrical heaters in the hot legs and coolers in the cold ones. In order to save energy they have heat exchangers between the hot and cold parts of the loop. They have therefore a figure of eight configuration in which one part is the hot leg containing the test sections while the other is the cold leg with a cold trap and the pump (Klueh and DeVan 1972). Electromagnetic pumps must be operated at temperatures below 400°C in order to avoid superheating of the electric coils. 2.2. DETERMINATION OF SODIUM INFLUENCE ON CREEP RUPTURE OF STEELS

In the early work in USA (Horton et al. 1967), the creep-rupture tests were often performed in vacuum or in air after long term sodium exposure at the same temperature. The creep specimens were pre-exposed in nonisothermal sodium loops with fast flowing liquid metal. The mechanical tests were subsequently made in conventional creep machines with vacuum chambers for uniaxial creep. This technique does not offer the possibility to study the influence of wetting sodium on the creep damage of the specimens. Thus, it does not produce representative results of the influence of sodium on the creep-rupture of stainless steel. For biaxial creep tests an apparatus for keeping pressurized tubes of ≈10 cm length at temperature with gauges for measurements of their diameters was constructed. The 304 steel tubes were pre-exposed in sodium at 650°C, while the creep tests were

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made at 550°C. The tubes were connected with the supply of pressurized Ar or He gas (Yaggee and Gilbert 1967). A method for testing mechanical properties as stress rupture behaviour of cladding materials in flowing sodium of high temperature was presented by Zebroski et al (1967). The stress was imposed onto the wall of steel tubes by means of internal pressure. However, the results of testing with this equipment were not reported. A similar technique was developed to insert pressurized fuel element cladding tubes of original dimensions connected to a pressure supply by means of capillaries into the high temperature sodium loop in the Austrian Centre of Sei-bersdorf (Matyas et al. 1967). 2.3. DETERMINATION OF SODIUM INFLUENCE ON FATIGUE AND CREEP-FATIGUE

Fatigue and creep-fatigue testing needed the development of particular testing chambers that could be passed by fast flowing sodium along the surfaces of specimens. These capsules were equipped with shrouds in order to lead the flowing sodium to the surfaces of specimens with high velocity of about 3 m/s. The cycling load was brought onto the specimens by servohydraulic testing machines. Devices for the measurement of strain had to be placed outside the sodium-leading capsule but in connection with the shoulders of the specimens. The whole equipment had to be kept on the testing temperature inside a safety chamber that could also follow the movement due to the strain of the specimens. Pre-corrosion in flowing sodium could also be performed with chains of these capsules connected to the hot leg of a sodium loop. 3. Corrosion of Reactor Materials The core materials are exposed to sodium of highest temperatures and flow velocities, and they are additionally irradiated by extensive fast neutrons. Since the fuel elements have to be exchanged after a service life of about two years, the lifetime of the core materials is only limited. The materials that are used under such severe conditions are stabilized or unstabilized austenitic steels. Martensitic 9 to 12% chromium steels are considered as duct materials, since they have sufficient strength and better swelling behaviour under neutron irradiation. The structural materials of the primary and secondary circuits are in contact with sodium at lower temperature levels and flow velocities. They get also less irradiation than the core materials. However, the lifetime of the structure components should be in the order of that of the reactor. Thus, the tubes and vessels have to withstand the conditions of the coolant flow for

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more than 30 years. The austenitic steels 304 and 316 are typical structural materials. Only for steam generators in the secondary systems ferritic steels are in use in order to minimise the risk of stress corrosion cracking from the waterside. The structural material of sodium loops operated for long periods of time was examined after dismantling of the circuits (Vaidehi Ganesan et al. 2003, Yoshida et al. 1995). 3.1. CORROSION OF IRON BASE ALLOYS

The iron base alloys that are of interest for sodium cooled reactors are austenitic and ferritic steels of sufficiently high strength at the temperatures of interest. Many studies in several countries were dedicated to get information of the corrosion resistance of such steels in flowing liquid sodium at temperatures up to 700°C (Thorley and Tyzack 1967). This high level of temperature was considered at the hot spot temperature of fuel element cladding tubes. These components are in the reactor core in the order of two years under full power conditions. The materials that are in use or under consideration for fuel elements of sodium cooled fast reactors are austenitic stainless steels as type AISI 316 or the German choice of 1.4970, a stabilized 15 15 CrNi steel with a boron content. The boron content of this steel was slightly decreased in the sodium corrosion tests at 700°C, however, there was no influence on the creep behavior of steel 1.4970 cladding tubes (Borgstedt et al. 1977). The sodium corrosion of austenitic stainless steel AISI 316 was studied in a wide range of parameters as temperature, flow velocity and oxygen concentration in sodium (Thorley and Tyzack 1967). Additional to mass losses, corrosion is also due to changes of chemical composition and structure in the surface near areas. The total effects after 2 years of operation at 700°C are mass losses from the surface of 15 μm followed by a ferritic layer of the same thickness and a region with ferritic grain boundaries of ≈20 μm. Stabilized austenitic material can be carburized and hardened behind the corrosion layers. Decarburization of 13% Cr steels in flowing sodium of 600 to 900°C that are not stabilized with alloying elements as Ti or Nb was also reported by Russian researchers (Agapowa et al. 1967). Already the early work of Zebroski et al. (1967) showed the decrease of sodium corrosion effects as weight losses and leaching of alloying components in a tube of constant dimensions over its length. This fact was called downstream effect and introduced into their corrosion equation by the geometric factor L/D (tube length of specimen position over hydraulic diameter):

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R = v0.884 cO1.156 exp ( 12.845 – 13300/T – 0.00676 L/D + 2.26/(t + 1) (10) with R = rate of metal removal in mg/dm2/month , V = sodium velocity in ft/sec, cO = concentration of oxygen in sodium in ppm, t = time in months, T = sample temperature in °C, L = distance from inlet to a given isothermal and constant velocity region, and D = hydraulic diameter. The downstream effect was related to hydraulic conditions in the constant conditions region that cause an increase in the thickness of the boundary layer with increasing L. This increased layer thickness was considered to be the reason for lower diffusion rates of dissolved atoms into the turbulent flow of sodium. Another effect that decreases the dissolution rates along the channel L is the increasing saturation of the liquid metal in respect to the solutes. Weeks and Isaacs (1972) formulated the corrosion rate at any L/D position in the hot leg of a sodium loop by a simple equation: R = R0 (1 – ε (L/D)2 )

(11)

The downstream effect on the sodium corrosion of stainless steel AISI 316 was later on confirmed as a function of the logarithm of the weight loss R over L/D by Suzuki and Mutoh (1986). They found that the effect is not dependent on the corrosion rate and surface composition of the specimens. Their conclusion is based on the absorption of upstream dissolved chromium, nickel and iron on active sites of the downstream specimens. Fuel element cladding materials as austenitic stainless steels are activated through neutron radiation. The activation products are isotopes of the same metal or even other metals as is shown in the Eqs (12)-(15) (Wozadlo et al. 1971). 50 59 54 58

Cr (n, γ) → 51Cr

τ = 27.8 d

(12)

60

τ = 5.3 a

(13)

54

τ = 314 d

(14)

58

τ = 71 d

(15)

Co (n, γ) → Co Fe (n, p) → Mn Ni (n, p) → Co

The corrosion and mass transfer behavior of the activation products was studied by Brehm et al. (1970), who found that mainly 54Mn was transferred in the direction of the heat flux. Due to this behavior of the manganese isotope the heat exchanger between the primary and secondary cooling system can be activated, a fact that may disturb maintenance work. 60Co, however, is kept back in the core region where activation has its origin. The isotope of chromium migrates with the bulk of these elements.

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Long term exposed structural material of a sodium loop (steel 304) was examined for Na corrosion effects after 80000 h loop operation. The tubes were cut from the test sections that have been operated 60000 h at 550°C and finally 20000 h at 600°C. Leaching rates of Cr and Ni have been obtained from the width of the depleted layer. The layers were found to be thicker than those that were calculated using the model of Thorley and Tyzack (1973). This may be due to the fact that the model was developed for steel 316 containing molybdenum, while steel 304 does not contain this element that decreases the leaching process (Vaidehi Ganesan et al. 2003). 3.2. CORROSION OF REFRACTORY ALLOYS

The sodium corrosion of nickel base alloys as Inconel 600 or Nimonic 80A was studied by Thorley and Tyzack (1967). They found weight loss rates at 650°C in fast flowing sodium for these alloys that were about three times compared to that of stainless steel AISI 316. While the corrosion of the nickel alloys was independent on the oxygen concentration in sodium, the rates were significantly enhanced by increased flow velocity of the liquid metal. The higher corrosion rates and the effect of flow velocity are understandable on the considerably higher solubility of nickel in liquid sodium compared to that of iron. The high nickel content of the two alloys was the reason for the fact, that no area of element depletion occurred in these corrosion tests. Sodium corrosion of nickel based alloys is additionally influenced by surrounding loop material the Ni content of which is considerably lower than that of the specimens. This is the case in stainless steel loops, in which mass transfer between nickel alloys and steel enhances the dissolution of Ni. Though metals as molybdenum, tungsten or rhenium are not usual reactor materials, they may play a role for particular applications. These metals are characterized by very low solubility in liquid sodium (Borgstedt and Guminski 1996). Thus, they are very well compatible with liquid sodium even at high temperatures and flow velocities. However, they are also sensitive to reactions with impurities in sodium, for instance the reactions of molybdenum with Na2O (Gnanasekaran et al. 1989). The formation of compounds as Na4MoO5 at 450°C needs however very high oxygen activities close to saturated solutions of Na2O in sodium. A lot of work has been done on the sodium corrosion of vanadium alloys, and the results of the early work were already presented on the Vienna conference 1967 (Champeix et al. 1967, Greenberg et al. 1967, Whitlow et al. 1969). The problems of internal oxidation of these alloys in liquid sodium purified by cold traps could not be solved; sufficient corrosion resistance could only be reached for vanadium alloys containing

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chromium and titanium as alloy components in sodium with oxygen contents below 1 10–4 mass% due to hot trapping with Zr (Borgstedt 1971). The oxygen distribution in alloys V-(1-3)Ti-15Cr after corrosion in sodium at 600°C was calculated from the analytical content and the hardness distribution. Oxygen diffuses fast in the alloy with the lowest Ti content, while diffusion rate decreases and embrittlement in the surface region increases with growing Ti content. 3.3. CORROSION OF SOME CERAMIC MATERIALS

A survey on the compatibility of liquid sodium with refractory ceramics at high temperatures was published by J.K. Fink et al. (1977). A number of refractory ceramic materials as alumina, magnesia, beryllia, thoria, zirconia, and spinel satisfy the thermodynamic stability criterion for sodium compatibility up to high temperatures. Static tests of bricks made of these ceramics have been performed at 890°C for one hour (Meacham 1975). The bricks made of ceramics containing only about 90 mass% Al2O3 were decomposed in these tests, while a brick of a material with ≥97.5 mass% Al2O3 retained its physical integrity, though surface cracks occurred. Generally, the compatibility of Al2O3 with sodium at high temperature depends on the purity of the ceramics and the character of impurities (Fink et al. 1977). But even at temperatures up to 890°C corundum of high purity was shown to be compatible with boiling sodium. The superior compatibility of advanced Al2O3 of 99.99% purity in sodium of 650°C was more recently shown in tests of 1000 h duration. Grain boundary attack was found to be the most often appearing corrosion phenomenon (Kano et al. 1995). Pure magnesia (99.9 mass% MgO) retained its physical integrity after reaction with sodium at 890°C for one hour (Meacham 1975), while impure material was completely destructed under these conditions. While a thoria rod was found to be compatible with sodium for 5 h at 850°C (Fink et al. 1977), the thoria - yttria ceramic used for oxygen meters was less resistant due to the fact that mixed oxides tend to suffer grain boundary attack. However, we have used such oxygen meters based on ceramics from different sources (United Nuclear, Westinghouse, Harwell and Rossendorf) for up to 10000 h at the operation temperature of 360 – 400°C, and we did never observe any traces of sodium attack. Failures were always caused by thermal shocks (Borgstedt 1995). Also some carbides were studied in respect to their compatibility with sodium (Kano et al. 1995). SiC did not suffer mass loss at 550°C even up to 4000 h, but showed already corrosion at 650°C after 1000 h. TiC suffered small weight losses at 550°C and large corrosion effects at 650°C. Among the nitrides, AlN was sufficiently stable up to 650°C, while Si3N4 withstood

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the attack only up to 550°C. BN was catastrophically corroded in the compatibility tests at both the temperatures (Kano et al. 1995). Ceramic materials have been considered for the use as core catcher materials in case of pipe ruptures and leakages in a sodium-cooled core. Basalt {anorthite NaAlSi3O8, albrite NaAlSi3O8, pyroxene ABSi2O6 [A = Mg, Fe, Ca or Na; B = Mg, Fe or Al] and olivine (Mg,Fe)2 SiO4} was one of the core catcher materials tested for this application with high-temperature or boiling sodium (Chasanov 1974). The sample tested for 6 h at 909°C fractured into several pieces, though the sodium remained bright and shiny. The pieces of the ceramic were transformed into crystalline compounds. Thus, basalt is not compatible with sodium under the conditions of a core catcher. The reactions of sodium with concrete were studied in large scale within a stainless steel pressure vessel (Cherdron and Schütz 1995). Two experiments with 0.6 m3 sodium and 1 m3 concrete were performed in an atmosphere of air. Since the reactions of sodium with the water and silicon contents of concrete are exothermal, the sodium was heated from its melting temperature to the temperature of the concrete with short time peaks at about 860°C. The results of the experiments indicated that the core catcher made of concrete should be surrounded by a steel liner. It is still open, if particular concrete types may form reaction layers that protect the main part of the structure against such violent reactions. Graphite that is in use in sodium circuits for trapping radioactive cesium (Lastov et al. 1995) was found to be compatible with sodium at a temperature of 850°C (Fink et al. 1977). The good compatibility should be due to the very low solubility of carbon in sodium. The trapping procedure is performed at relatively moderate sodium temperature at which the compatibility is excellent. 4. Sodium Effects on Mechanical Properties of Austenitic Stainless Steels 4.1. SODIUM EFFECTS ON THE CREEP-RUPTURE PROPERTIES OF AUSTENITIC STEELS

4.1.1. Creep –rupture tests with tangential stressed specimens Pressurized tubes have been exposed for up to 5000 h in the test sections at 700°C. However, the capillaries had partly blockades due to incorrect welding procedures. Thus, only a part of the specimens was tested under creep conditions (Borgstedt and Dietz 1974). The subsequent tests in the high temperature loop at KFK (Borgstedt et al. 1977) used therefore a

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different technique. The thin walled tubes (0.38 mm wall thickness) were closed with welded end plugs, one of which was connected with a capillary for the supply with pressurized gas. The gas pressure was controlled before closing the capillary by welding. The tangential stress that was applied to the steel 1.4970 at 700°C was in the range of 70 to 118 MPa, about 0.3 of the tangential stress for 0.2% strain in short time tests at the same temperature. Some of the specimens were exposed up to rupture of the material in the creep and corrosion tests at 600°C and 700°C. All specimens were widened by creep deformation that was measured after the exposure. Creep tests in flowing sodium at 650°C with austenitic (16 15CrNiMo) and ferritic steel (13CrMo) in size of tubes filled with Ar gas under pressure were also performed in Russia (Agapowa et al. 1967). 4.1.2. Creep–rupture tests with uniaxial stressed specimens Creep machines for uniaxial creep-rupture tests of steel 304L and 316L(N) were connected to sodium loops in the cooperation for the European Fast Reactor programme (Borgstedt and Huthmann 1991, Borgstedt et al. 1991). Base material and weldments of steel 304L were creep rupture tested in sodium of 550°C in two different loops, and the results were compared to reference tests in vacuum (Borgstedt and Huthmann 1991). The time to rupture was larger in sodium in both the facilities. However, the results in the CREVONA loop of KfK were somewhat below of those in the other loop. It could be showed that the sodium in CREVONA was slightly decarburising the specimens, while the loop of Interatom did not change the carbon content. Sodium did not influence the creep-rupture behaviour of welded specimens of this material. The steel 316L(N) was studied in sodium loops of five laboratories (see Figure 5), and the times to rupture in sodium and in air were close together up to exposure times of more than 20000 hours. The creep behaviour of this steel was not influenced by the sodium in the CREVONA loop, since the values were at the upper side of the scatter band of data. The creep curves are, however, somewhat changed by sodium environment. The tertiary creep range was shortened, and short intercrystalline cracks occurred in the surface zones of the specimens. The cracks in the surface zone were already formed early in the seconddary creep region. They arose from cavities that were corrosion phenomena and related to the selective leaching of alloy components. However, these surface cracks did not grow to fracture before the tertiary phase of the creep life. The shorter tertiary creep region was due to a sodium influence on the growth of the cracks.

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Figure 5: Creep-rupture data in Na at 550°C of steel 316L MOD for the European Fast Reactor programme compared to reference data in air (Borgstedt et al. 1991).

4.2. SODIUM EFFECTS ON THE FATIGUE AND CREEP-FATIGUE BEHAVIOUR OF AUSTENITIC STEELS

Low cycle fatigue (LCF) tests of steel 304 SS were performed in Germany and Japan (Borgstedt and Huthmann 1988, Maruyama et al. 1988). Steel 304 SS (German no.1.4948), base material and welded joints, and 316 L(N) were LCF tested in flowing sodium at 550°C. The hour glass specimens with minimum diameter of 8.8 mm and a gauge length of 21 mm were precorroded 3000 h at the test temperature inside the capsules. The strain range was kept within 0.6 to 1.5% in the tests. The same specimens and type of capsules were used for tests at Interatom in the AWN loop after a precorrosion of 500 h. Similar LCF tests were performed in flowing sodium by JAEA (Maruyama et al. 1988) with steel 304 SS at 500 and 600°C. The LCF life of steel 304 SS was longer in sodium compared to the reference tests in air in all these tests. A weak influence of the sodium purity was observed in respect to the carbon activity. The low carbon activity during the pre-corrosion and the LCF tests in the KFK loops resulted in a slight decarburization of the unstabilized steel 304 SS. The LCF life in the tests in this sodium loop was closer to the reference results in air than in the other loops. The duration of the pre-corrosion was not significant for the results of LCF tests. The LCF behaviour of steel 316 L(N) in sodium of 550°C was found to be comparable to that of steel 304. The number of cycles to failure was

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always larger than in the reference tests, and even pre-corrosion did not result in shorter fatigue life of the specimens (Mishra et al. 1996). 5. Conclusions Though the total corrosion of steel used for the fuel element claddings at 700°C exceeds the 15 μm mass loss in two years by the internal changes of the structure and composition, the corrosion effects are tolerable for the short living components. The experimental results are conservative due to the downstream effect, which decreases the corrosion rates in the long cooling channels. The corrosion of the material in the core influences the contamination of the sodium circuits with activated corrosion products. Corrosion should therefore be minimized. The sodium corrosion of structural materials up to 550°C was studied up to the order of 100000 h operation time of sodium loops. Mass loss and changes of composition and structure are not problematic for the lifetime of the pipes and vessels of a reactor. The creep-rupture and the fatigue behaviour of the steels favoured for the use in sodium-cooled reactors are not significantly influenced by the liquid metal. Refractory metals as Mo, W, Re and V are compatible with sodium of very high degree of purity. Ceramic materials that are stable against reactions with sodium are only compatible, if they are of high purity. Lesser degrees of purity cause intercrystalline corrosion.

References Agapova, N.P., Ioltukhovsky, A.G., and Romaneev, W.W., 1967, The behaviour of stainless steels in sodium at temperatures between 600 and 900°C, in: Alkali Metal Coolants, Internat. Atomic Energy Agency, Vienna, pp. 85-96. Asher, R.C., Bradshaw, L., Kirstein, T.B.A., Nixon, T.H., and Tolchard, A.C., 1973, Progress in the development of a carbon meter for sodium-cooled fast reactors, in: Proc. BNES Conf. on Liquid Alkali Metals, Nottingham, pp. 133-135. Borgstedt, H.U., 1971, Die reaktionen von V-Zr-Cr und V-Ti-Cr legierungen in flüssigen natrium mit geringen sauerstoffaktivitäten, Corrosion Sci. 11:89-105. Borgstedt, H.U., and Dietz, W., 1974, Corrosion and Mechanical Long Term Behaviour of Cladding Tubes in Sodium of 700°C, Report KFK 1932, Kernforschungszentrum Karlsruhe. Borgstedt, H.U., Frees, G., and Schneider, H., 1977, Corrosion and creep of pressurized stainless-steel tubes in liquid sodium at 873 and 973 K, Nuclear Technology 34:290-298. Borgstedt, H.U., and Frees, G., 1987, Low-cycle fatigue of stainless steel in liquid sodium environment, in: Low Cycle Fatigue and Elasto-Plastic Behaviour of Materials, K. T. Rie Ed., Elsevier, pp. 378-382.

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Borgstedt, H.U., and Huthmann, H., 1988, Low- and High-cycle fatigue behaviour of type 304 SS in liquid sodium, 4th Internat. Conf. on Liquid Metal Engineering and Technology , LIMET , Avignon, France, October 17-21 1988, Proc. Vol. 2, paper 509; Paris, Societé Francaise d’Energie Nuclaire, pp. 509, 1-7. Borgstedt, H.U., and Frees, G., 1990, Kohlenstoffaustausch zwischen sthal und natrium als korrosionserscheinung, Werkst. u. Korros. 41:1-6. Borgstedt, H.U., and Huthmann, H., 1991, Influence of sodium on the creep-rupture behaviour of type 304 stainless steel, J. Nucl. Mater. 183(3):127-136. Borgstedt, H.U., Debergh, M., Horton, C.A.P., Huthmann, H., and Wood, D.S., 1991, The influence of flowing liquid sodium on the creep-rupture behavior of SS AISI 316 L(N), Int. Conf. On Fast Reactors and Related Fuel Cycles FR’91, Atomic Energy Soc. of Japan, Tokyo 1991, Vol. IV P4.10, pp. 1-7. Borgstedt, H.U., unpublished work of FZK. Borgstedt, H.U., and Guminski, C., 1996, "Metals in Liquid Alkali Metals, Part I: Be to Os; Part II: Co to Bi", IUPAC Solubility Data Series, Vol. 63&64, Oxford University Press, Oxford. Borgstedt, H.U., 1998, The solubility of nickel in sodium calculated from corrosion tests in flowing liquid metal, Z. Metallkd. 89(3):199-202. Brehm, W.F., Kohmstedt, P.L., Kovacevich, E.A., and Shannon, D.W., 1970, Radioactive material transport in flowing sodium system, in: Corrosion by Liquid Metals, Draley, Joseph E. ed., Plenum Press, New York, 97-113. Cavell, I.W., and Nicholas, M.G., 1980, Some observations concerned with the formation of sodium chromite on AISI 316 exposed to oxygenated sodium, J. Nucl. Mater. 95(12):129-144. Champeix, L., Darras, R., and Sannier, J., 1967, Corrosion du vanadium et de ses alliages dans le sodium liquide, in: "Alkali Metal Coolants", Internat. Atomic Energy Agency, Vienna, pp. 45.-61 Chasanov, M.G., 1974, in: Chemical Engineering Div. Reactor Safety and Physical Property Studies Annual Report 1700, June 1974, Argonne National Laboratory Report ANL8120, p. 97. Cherdron, W., and Schütz, W., 1995, Large scale FAUNA experiments on the interaction of sodium, concrete and steel, in: Liquid Metal Systems, H.U. Borgstedt ed., Plenum Press, pp. 393-397. Epstein, L.F., 1957, Static and dynamic corrosion and mass transfer in liquid metal systems, in: Liquid metal technology, Chem. Engng. Symp. Ser. 53:20, Amer. Inst. of Chemical Engineers, New York, p. 67. see also: Epstein, L.F., and Weber, C.E., in: Report TID2501, Techn. Inform. Service, Oak Ridge, Tenn. 1950, 455, and in: Advances in Corrosion Science and Technology (1973) 3:1-66. Evans, J. W., and Thorley, A., 1958, UKAEA Report 1GR-TN/C1019, 1958. Fink, J.K., Heiberger, J.J., Kumar, R., and Blomquist, R.A., 1977, Intercations of refractories and reactor materials with sodium, Nucl. Technol. 35 (3):656-662. Gnanasekaran, T., Mahendran, K.H., Periaswami, G., Mathews, C.K., and Borgstedt, H.U., 1987, Stability of ternary oxygen compounds of molybdenum in liquid sodium, J. Nucl. Mater. 150(2):113-127. Greenberg, S., Ruther, W.E., and Levin, H.A., 1967, Corrosion of vanadium base alloys in sodium at 550 to 750°C, in: Alkali Metal Coolants, IAEA, Vienna, pp. 63-84. Horton, K.E., Andrews, R.C., Werner, R.C., and Atkins, D.F., 1967, Effects of sodium on the metallurgical and mechanical properties of candidate LMFBR alloys, in: Alkali Metal Coolants, IAEA, Vienna, pp. 247-260. Kano, S., Yoshida, E., Hirakawa, Y., Tachi Y., Haneda H., and Mitsuhashi, T., 1995, Sodium compatibility of ceramics, in: Liquid Metal Systems, H.U. Borgstedt ed., Plenum Press, 85-94.

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Klueh, R.L., and DeVan, J.H., 1972, Liquid metal test procedures, in: Handbook on Corrosion Testing and Evaluation, W.H. Ailor ed., J. Wiley and Sons, New York, 405433. Lastov, A.J., Konovalov, E.E., and Starkov, O.V., 1995, Physical chemistry of sodiumcaesium-graphite systems, in: Liquid Metal Systems, H.U. Borgstedt ed., Plenum Press, 1995, 291-295. Maruyama, R., Kato, S., Komine, R., Wada, Y., Hirano, M., Kano, S., and Nihei, I., 1988, Low-cycle fatigue properties of SUS 304 stainless steel in elevated temperature fluid sodium, 4th Internat. Conf. on Liquid Metal Engineering and Technology , LIMET , Avignon, France, October 17-21 1988, Proc.Vol. 2, paper 512; Paris, Societé Francaise d’Energie Nuclaire. Matyas, E., Rajakovics, G., and Schwarz, N., 1967, Problems of a sodium loop designed to operate at temperatures up to 800°,in: Alkali Metal Coolants, IAEA, Vienna, p 367-379. Meacham, S.A., 1975, Westinghouse Advanced Reactors Div. Report, WARD-D-0077. Mishra, M.P., Borgstedt, H.U., Valsan, M., Mannan, S.L., and Rodriguez, P., 1996, Effect of flowing sodium and prior exposure to sodium on LCF and creep-fatigue interaction behavior of type 316L(N) stainless steel, Trans. Indian Inst. Met. 49:497-503. Natesan, K., and Kassner, T.F., 1973, Thermodynamics of carbon in nickel, iron-nickel and iron, chromium-nickel alloys, Metall. Trans. 4(11):2557-2566. Smith, D.L., and Kassner, T.F., 1970, Application of thermodynamic and kinetic parameters of the V-O-Na system to the sodium corrosion of vanadium base alloys, in: Corrosion by Liquid Metals, Plenum Press, New York, 137-149. Smith, D.L., and Natesan, K., 1974, Influence of nonmetallic impurity elements on the compatibility of liquid lithium with potential CTR containment materials, Nucl. Technol. 22(3):392-404. Suzuki, T., and Mutoh, I., 1986, The downstream effect in the corrosion of metallic materials in a type 316 stainless steel sodium loop system, J. Nucl. Mater. 140:56. Thorley, A.W., and Tyzack, C., 1967, Corrosion behaviour of steels and nickel alloys in high-temperature sodium, in: “Alkali Metal Coolants”, Internat. Atomic Energy Agency, Vienna, pp. 97-118. Thorley, A.W., and Tyzack, C., 1973, Corrosion and mass transport of steel and nickel alloys in sodium, in: Liquid Alkali Metals, Brit. Nucl. Energy Soc., London, 257-270. Vaidehi Ganesan, Ganesan, V., and Borgstedt, H.U., 2003, Analysis of CREVONA sodium loop material, J. Nucl. Mater. 312:174-180. Weeks, J.R., Klamut, C.J., and Gurinski, D.H., 1967, Corrosion by the alkali metals, in: “Alkali Metal Coolants”, Internat. Atomic Energy Agency, Vienna, 3-24. Weeks, J.R., and Isaacs, H.S., 1972, A general model for the corrosion of steels in high velocity sodium, in: Chemical Aspects of Corrosion and Mass Transfer in Liquid Sodium, S.A. Jansson ed., Met. Soc. of AIME, New York 1972, p. 207-222. Whitlow, G.A., Hornack, R.J., Schrock, S.L., and Bishop, E.C., 1969, The effects of exposure to sodium on the metallurgical and mechanical properties of vanadium alloys, J. Less-Common Met. 18(4):357-371. Wozadlo, G.P., Boardman, C.E., and Weiss, M.L., 1971, USAEC Report GEAP-13671. Yaggee, F.L., and Gilbert, E.R., 1967, Effect of sodium exposure on the mechanical properties of potential fuel jacket alloys at 550°-700°C, in: Alkali Metal Coolants, IAEA, Vienna, pp. 215-230. Yoshida, et al., 1995, Post-corrosion and metallurgical analysis of sodium piping materials operated for 100,000 hours, in: Liquid Metal Systems, H.U. Borgstedt ed., Plenum Press, New York, p. 55-66. Zebroski, E.L., Young, R.S., Comprelli, F.A., and Dutina, D., 1967, Effects of mass transfer, and of changes in properties, on austenitic steels in flowing sodium, in: Alkali Metal Coolants, Internat. Atomic Energy Agency, Vienna, pp. 195-211.

RADIATION EFFECTS IN STRUCTURAL MATERIALS FOR FUSION POWER PLANTS: THE OUTCOMES OF THE EU FUSION PROGRAM J.-L. BOUTARD1 EFDA-CSU Garching, Boltzmannstrasse 2, D-85748 Garching bei München. S. DUDAREV EURATOM/UKAEA Fusion Association, Culham Science Centre, Oxfordshire OX14 3DB, UK. E. DIEGELE EFDA-CSU Garching, Boltzmannstrasse 2, D-85748 Garching bei München.

Abstract: The development of highly radiation resistant structural materials is a key issue for the development of fusion nuclear power plants. The selection criteria, main results and issues are presented for ferritic/martensitic and oxide dispersion strengthened steels. This paper also reviews the EU fusion modelling programme that aims at capturing the physics of radiation effects at the relevant scales for a EUROFER-type ferritic/martensitic steel.

1. Introduction Future thermo-nuclear fusion Nuclear Power Plants (NPPs) are expected to provide mankind with a sustainable energy source characterized by low release of green-house gases, and to contribute to the energy mix required to satisfy the growing demand of energy and to limit global warming. In this perspective, the decision of constructing the International Thermo-Nuclear Reactor (ITER) is an essential step towards accumulating the physical knowledge of thermo-nuclear plasmas for the safe and efficient operation of future fusion NPPs. The reliable operation of fusion NPPs also requires developing materials and components able to resist the high flux of the 14 MeV neutrons generated by the deuterium-tritium fusion reactions in plasma.

______ 1

[email protected]. T: 00.49.(0)89.3299.4318.

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In section 2 we outline the principles of fusion reactors. Section 3 describes design concepts together with operating conditions and structural materials selected for the Tritium-Breeding Blankets (TBBs) and the divertor of the future fusion reactors. Section 4 details the selection criteria for the range of candidate materials foreseen for fusion technology. Section 5 describes the main experimental results concerning the effect of radiation on the in-service properties of materials. In section 6 the main outcomes of the EU programme on modelling radiation effects are reviewed, followed by conclusions. 2. Thermo-Nuclear Fusion NPP In a fusion NPP the deuterium-tritium fusion reaction releases energy in the form of the kinetic energy of the products of the reaction:

D + T →4 He(3.56MeV ) + n(14.03MeV ) In order to have a significant effective cross section for this reaction, hot D-T plasmas are required with temperature ~100x106 K. In addition, the thermo-nuclear plasma in a fusion NPP has to meet the condition of selfsustained ignition, n.T.τE > 5 × 1021m–3 keV.s, where τE is the energy relaxation time in seconds, n the electronic density in m–3 and T the temperature in electron-volts. The magnetic configuration capable of meeting these conditions will be most probably of the Tokamak type [1], as in most of the present physics machines and ITER. Two in-vessel components are particularly important for the operation of a fusion NPP: the divertor and the Tritium-Breeding Blankets (TBBs). The divertor is essential for a self-sustained ignition, as it purifies the plasma by evacuating the alpha particles (i.e. helium produced in the fusion reaction) and impurities. It also extracts a significant part of the produced fusion energy. The role of TBBs is to extract the energy, to produce tritium for self-sufficient operation, and to contribute to the shielding of the vacuum vessel and of the superconducting magnets of the Tokomak (see Figure 1). The tritium breeding relies on the following nuclear reaction between an incident neutron and a 6Li nucleus:

n + 6 Li → T + 4 He + 4.78MeV For the tritium self-sufficiency every D-T fusion neutron has to be used to produce at least one tritium atom. Neutron multiplier materials such as Pb or Be are therefore introduced to compensate for the unavoidable loss of neutrons resulting for instance from transmutation reactions.

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Figure 1. A schematic view of Tokamak type fusion reactor [1].

3. Tritium-Breeding Blankets and Divertor Several design concepts are being envisaged for the TBBs [1]. In all the design concepts the structural materials close to the First Wall (FW) will be subjected to high doses of 14 MeV neutrons: ~30 to 80 dpa or even ~100 to 150 dpa for the first Demonstration (DEMO) or prototype reactors, respectively [2]. Simultaneously the 14 MeV neutrons will produce significant concentrations of He and H, with respective production rates of ~10 and ~ 45 appm/dpa in the components positioned near the FW. The FW itself will have also to be able to evacuate peak heat flux of the order of 0.5 MW/m2 . Figure 2 presents the dual-coolant TBB concept. The liquid Pb-Li eutectic is used as neutron multiplier (Pb) and as tritium breeding material (Li). It also contributes, with the helium gas, to the cooling of the TBB. The foreseen structural material is the reference Reduced Activation FerriticMartensitic (RAFM) steel EUROFER with an upper operation temperature limit of ~550°C. Oxide Dispersion Strengthened (ODS) EUROFER is foreseen to be used as a plate protected with W-tiles (not shown) facing the plasma, which will have to sustain operating temperatures of up to 700°C. SiCf/SiC composite inserts with low thermal and electrical conductivities are envisaged in the Pb-Li channels (i) to protect the structural steel components from exceeding the upper limit of the operational temperature range, and (ii) to mitigate the magneto-hydrodynamics effects by limiting the electrical current loops from the flowing liquid metal Pb-Li into the metallic structure, which are created by the Lorentz force.

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Figure 2. a) Schematic view of the dual-coolant concept of TBB with the selection of structural and functional materials, b) Operating temperature ranges, pressure and inlet & outlet temperatures of the He coolant and of the Pb-Li eutectic [1].

In addition to the 14 MeV neutron irradiation, the divertor will have to withstand a peak heat flux of ~10 MW/m2. Figure 3 gives elements of design, functional and structural materials for a He-cooled divertor design concept [3,4]. W is selected as protective material for its refractoriness high thermal conductivity and low sputtering yield under plasma loading. Walloys are envisaged as heat sink and structure materials. The ODS steels are foreseen as structural materials for the coldest parts of the divertor where temperature does not exceed ~700°C. The expected operational temperatures are relatively high but nevertheless compatible with thermal stability of the selected materials, at least in the absence of irradiation. 4. Selection Criteria The properties of any material are controlled by its crystalline structure, chemical composition and microstructure. The displacement cascades induced by the Primary Knocked-on Atoms (PKA) produced by the bombardment of the fast neutrons destroy locally the crystalline structure and generate defects, which partially recombine on the timescale between picoseconds and several seconds. The surviving point defects diffuse and agglomerate, especially vacancies with helium, which form a dense population of clusters. New microstructure is formed inducing hardening [5]. Swelling and voidage may also occur: due to the absorption of interstitials by the dislocation cores, the surplus of surviving vacancy clusters transforming

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Figure 3. He-cooled divertor concept. The radial Cassette (a) consists in a set of rows of actively cooled fingers (b). Functional and structural materials of the fingers are shown in (c). Operating calculated temperatures are given in (d) [3,4].

into stable cavity nuclei stabilised by the presence of gas [6]. The helium itself will strain the chemical bond of the material due to its high solution energy [7]. 4.1. EUROFER AND FERRITIC/MARTENSITIC STEELS

Ferritic and Ferritic/Martensitic (F/M) steels have been selected as structural materials for TBBs on the basis of the experience gained during the operation of various Fast Breeder Reactors from the 70’s to the present day. These steels, extensively irradiated in Fast Breeder Reactors, exhibit no swelling at doses as high as 150 dpa in the temperature range from 400 to 550°C [8]. In addition, conventional 9% Cr-1Mo steels showed very low embrittlement under irradiation within fast breeder reactors in the temperature range between 400 to 550°C [9]. For fusion reactors some specially

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tailored chemical compositions were developed to have Reduced Activation (RA) materials that can be recycled hands-on after ~100 years following the component end of life, as shown in Figure 4 [10]. For this purpose the carbide forming elements such as Mo or Nb are undesirable. In the lowactivation compositions they are replaced by W and Ta. Alloying elements or impurities such as Ni, Co, Cu, Ag or Al are also kept as low as possible to comply with the low activation requirements. The reduced activation 9% Cr-V-W-Ta ferritic/martensitic steels have mechanical and physical properties very similar to the classical modified 9% Cr-1Mo (9Cr-1MoVNb) ferritic/martensitic steel [11]. 4.2. ODS FERRITIC AND FERRITIC/MARTENSITIC STEELS

EUROFER has a maximum operating temperature limited at ~550°C by creep strength. The heat resistance of 9-12% Cr F/M steels with optimised chemical composition has been increased up to ~600°C [12,13]. The ODS

F/M steels retain their strength and creep resistance up to ~650°C and

Figure 4. Calculated decay of gamma surface dose rate in iron and ferritic/martensitic steels after irradiation up to 115 dpa equivalent to 12.5 MWyear/m2 in a first wall DEMO spectrum [10]. The radiologically undesired elements have been measured in the following ranges given in ppm mass for ten heats corresponding 11 tonnes and different product forms: Nb (2-7), Mo (10-32), Ni (70-280), Cu (15-220), Al (60-90), Ti (50-90) Si (400-700), Co (30-70). The Low Activation Requirements are: Nb < 0.01 ppm mass, Mo < 1, Ni < 10, Cu < 10, Al < 1, Ti < 200, Si < 400, Co < 10 [10].

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Figure 5. Creep Rupture Strength for EUROFER, ODS EUROFER, the 12YWT ORNL development alloy, and, the industrial grade PM 2000. PM2000 is tested in two metallurgical conditions: G6 with coarse grain (diameter in the mm range) and G4 with refined grains (diameter ~2 to 10 micrometers) [14,15]. Times to rupture are represented versus the LarsonMiller parameter P = T.(30 + log(t)) where the temperature T is in K and the time to failure in hours.

the ODS ferritic steels are capable of withstanding operating temperatures up to ~750°C as shown in Figure 5 [14,15]. ODS steels are produced by high-energy Mechanical Alloying (MA) followed by a consolidation treatment that could be either Hot Isostatic Pressing (HIP) or Hot Extrusion (HE) [16]. MA followed by HE at 1150°C results in a high number density ~1023 to 1024m–3 of nano-clusters highly enriched in Ti, Y and O with an average Guinier radius ~1 nm, as observed by Tomographic Atom Probe (TAP) [17-19]. These clusters are required for maintaining high creep strength of steels [14] and they remain stable after creep up to 800 and 850°C for 14,500 and 4,000 hours respectively [18]. High Resolution Transmission Electron Microscopy (HRTEM) observation of ODS EUROFER revealed crystalline Y2O3 particles with well-defined orientation relationship with the Fe-Cr matrix and mean diameter ~10 nm. [20]. Recent examination by TAP revealed that this material contains also a rather high density of oxygen and yttrium enriched clusters of mean diameter ~3 nm [21]. The high creep strength is achieved at the expense of fracture toughness [20, 22], although the actual values of fracture toughness remain acceptable. From a practical point of view the anisotropy of mechanical properties, the

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presence of inclusions and an optimised fabrication route are important issues that remain to be mastered to ensure the future use of ODS steels. As shown in Figure 6, both carbides and the process of the powder consolidation, HIP versus HE, are important for controlling the fracture toughness and the DBTT of the material [20,23]. Inclusions of carbide type, are also reported promoting cavitation and inducing brittle inter-granular creep failure in ODS ferritic steels [14]. In addition welding of ODS steels, is a critical issue: all conventional melting techniques (for example tungsten inert gas, electron beam or laser) will destroy the oxide dispersion resulting in a loss of strength in the molten zone. First trials on dissimilar diffusion welding, EUROFER/ODS-EUROFER have been successfully conducted [24].

Figure 6. a) Impact energy of the first generation and second generation of ODS EUROFER. In the first generation the impact energies are affected by heavy carbide precipitation, that disappeared in the 2nd generation by appropriate thermal-mechanical treatment [20], b) Impact energy of ODS EUROFER fabricated starting from the same MA powder as in (a) but consolidated using either HE or HIP. Clearly HE process results in better fracture toughness [23].

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5. Radiation Effects: Experimental Simulation 5.1. SIMULATING FUSION SPECIFIC RADIATION EFFECTS

In the absence of an intense prototypical neutron source such as the International Fusion Materials Irradiation Facility (IFMIF) [25], irradiations have been conducted using fission reactors with fast neutrons with typical energy of 2-3 MeV. The thermal recovery stages of the point defects after irradiation via fast fission or 14 MeV neutrons are the same [26]. Consequently any irradiation effect, controlled by super-saturation and diffusion of point defects, can be simulated using the fast spectra of fission reactors. Nevertheless due to the shift of the PKA spectrum to higher energy, replacement sequences will be generally larger under fusion spectrum and specific effects on phase stability are not to be excluded [27]. In the absence of an intense 14 MeV neutron source several methods or tricks have been used to simulate the accumulation of point defects and He under fusion reactor relevant conditions. Boron or nickel doped steels were irradiated in Materials Testing Reactors (MTRs) using 10B(n,α)7Li and 58Ni (nth,γ)Æ 59Ni(nfast,α) nuclear reactions respectively. Despite the significance of these experiments, the data remain questionable. Due to its low solubility, boron segregates very easily to grain-boundaries and hence He (and Li) is not produced homogeneously. Nickel strongly modifies the martensite transformation points and, consequently, the mechanical properties of the doped materials may change significantly in comparison with their initial metallurgical conditions. 54Fe doped F/M steels have been irradiated in MTR yielding a He/dpa ratio ~2 appm/dpa [28]. He and H can also be produced with ratio to dpa typical of the D-T fusion spectrum by spallation nuclear reactions [29,30]. Spallation reactions also produce a complete range of residues with the atomic mass ranging from the target nucleus down to H, and this can have non-negligible impact on microstructure and properties, especially for high radiation dose experiments. Finally, implantation of α-particles with energy ~20 to 100 MeV, produced by cyclotron facilities, can be uniformly carried out on thicknesses ~100 µm, with typical He/dpa ratio ~6,000 appm/dpa, allowing tensile testing and post mortem microstructure examination [31,32]. 5.2. FERRITIC/MARTENSITIC STEELS: EMBRITTLEMENT AND PHASE STABILITY

The glide motion of dislocations and the yield strength of ferritic and F/M steels are thermally activated at low temperatures [33]. Conversely the cleavage stress required to separate two crystalline planes is temperature

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independent. As a consequence, below a critical temperature called the Ductile Brittle Temperature Transition (DBTT), the critical stress for cleavage is less than the yield strength, resulting in the failure of the material via cleavage mechanisms with low energy. The increase of the yield strength at higher strain rate or due to radiation induced hardening, as well as a decrease of the cleavage stress due to the occurrence of embrittlement during operation will increase the DBTT. Embrittlement created by irradiation in fission reactors strongly depends on the irradiation temperature and chromium content as shown on Figure 7. After irradiation in Phénix above 380°C a large increase of the DBTT for the 17% Cr ferritic steels is observed as a result of intense α/α’ unmixing. For the 9% Cr-1Mo steels only fairly negligible α/α’ unmixing was detected and the radiation induced embrittlement is limited [9]. After irradiation in BOR60 at ~325°C, significant embrittlement was found for the conventional 9Cr1Mo steels and for EUROFER [34]. Contribution of α/α’unmixing cannot be excluded at this temperature since after irradiation in OSIRIS at 325°C up to 0.7 dpa the presence of noticeable α’ volume fraction was detected by Small Angle Neutron Scattering (SANS) in RAFM steels with Cr content above ~7-7.5%Cr [35].

Figure 7. Effect of the irradiation temperature on the radiation-induced shift of DBTT for a ferritic steel (17% Cr), conventional 12% Cr and 9Cr1Mo ferritic/martensitic steels and EUROFER. For irradiation temperature close to 320°C and above 380°C have been obtained in BOR60 [34, 36] and Phénix [9] respectively.

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5.3. FERRITIC/MARTENSITIC STEELS: HE INDUCED HARDENING AND EMBRITTLEMENT

As shown in Figure 8, the implantation of 23 MeV α-particles at 250°C results in significant hardening and nearly complete loss of ductility for He contents above ~2,500 appm He. Microstructure consists of a dense population of He clusters as shown in Figure 8c [32]. The magnitude of hardening that can be deduced using an Orowan model is in good quailtative agreement with measurements [32]. This hardening does not seem to be sufficient to promote the intergranular failure along the prior austenite grain boundaries since the same heat of modified 9Cr1Mo irradiated in a quenched martensite initial condition exhibits similar yield strength ~1,200 MPa after irradiation in OSIRIS at 320°C up to ~3 dpa [32], but failure surface obtained by tensile testing is transgranular and ductile. It is not the first time that metallurgy is facing an embrittlement issue due to possible segregation of impurities to planar defects. The solution has always been to try to get the impurities trapped within the grain. For steel-makers it was the case of oxygen and sulfur, which are now commonly trapped in alumina and manganese sulfides. For He the solution has to be found by selecting the most effective traps, carbides or dislocations for example, to

Figure 8. a) Tensile behaviour of modified 9% Cr-1Mo after uniform implantation of α-particles of 23 MeV up to 5,000 appm at 2500C carried out at the compact cyclotron of Forschungszentrum Jülich (FZJ) [31], b) Intergranular fracture surface after implantation at 250°C along the prior austenite grains observed by Scanning Electron Microscopy (SEM), c) TEM observation of He-V nano-clusters in samples implanted at 250°C up to 5,000 appm He [32], d) TEM observation of He-V bubbles along dislocations and interfaces in samples implanted at 550°C up to 5,000 appm He [32]. For 5,000 appm He the dose is ~0.8dpa.

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keep He within the grain. In this future work, determining the energetics of He in presence of these objects will be of prime importance. On this basis kinetics tools able to describe diffusion and clustering of He in the grains and grain-boundaries will guide the optimization of the microstructure. 5.4. ODS FERRITIC STEELS: GRAIN SIZE AND EMBRITTLEMENT

MA957 with the dense dispersion of nano-clusters revealed by TAP [19] was irradiated in BOR60 up to ~30 dpa, in two metallurgical conditions: (i) hot-extruded and stress-relieved at 920°C, which resulted in fine elongated grains of diameter ~0.5 micrometers and (ii) hot-extruded and recrystallised at 1,100°C resulting in grain diameters in the range 20-50 micrometers. Both materials exhibit similar radiation hardening, the sub-micronic grain material having yield strength ~500 MPa higher than the recrystallised batch. Nevertheless the fine-grained batch retains total elongation (7%) and reduction of area (41%), half the initial values, which indicate the initial fracture toughness is hardly affected, whereas the coarse-grained batch should suffer of a severe loss of fracture toughness due to total elongation and reduction of area close to zero after irradiation [36]. The microstructure of these samples is still to be characterised to obtain essential data about the stability of the nano-cluster dispersion and the α/α’ unmixing. Up to now most of the studies of the radiation stability of the oxide dispersion have been carried out with TEM and concerned oxide particles rather than the nano-clusters revealed by TAP (see for instance reference [37]). To our knowledge, only reference [38] based on TAP examination reported on the stability of the nano-clusters of an experimental ODS steel after 150 keV Fe ion irradiation up to 0.7 dpa at 300°C. In addition, it can be seen in Figure 5 that decreasing the grain diameter, decreases also the creep strength at high temperature, showing that a trade-off on the grain-size should be found between high fracture toughness and creep strength. 6. Radiation Effects: Physical Modelling The experimental data described above were obtained with a large variety of damage spectra. Using them for predicting kinetic evolution of materials in fusion reactors requires deep understanding of radiation damage physics, which is multi-scale in nature. Early in 2002, a group of EU experts concluded that a combination of computer based modelling tools developed by that time for pure metals and other simple systems [39] had sufficiently matured to warrant an attempt to extend these calculations to model alloys representative of the behaviour of steels [40]. It was decided to establish,

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within the EU Fusion Technology Materials Program, a modelling task focused on “the study of radiation effects in the EUROFER reduced activation F/M steel under fusion relevant conditions i.e. in the range of temperatures from room temperature to 550°C in the presence of high concentrations of irradiation induced impurities (e.g. H, He)”. A review summarising the main achievements of this program is given in Ref. [41]. 6.1. DISPLACEMENTS CASCADES IN α-Fe

The fundamental building block for the treatment of radiation effects in metals is the modeling of a displacement cascade. During the past 15 years, displacement cascades in α-Fe were simulated using Molecular Dynamics (MD) with various inter-atomic potentials. The primary stage of damage is characterised by the type of defects, the production efficiency, which relates the number of defects at the end of the cascade heat spike to those produced at the end of the collision phase, and by the fraction of clustered defects. The results of MD cascade simulations in α-Fe were critically reviewed [42] and reference therein for the potentials] with the objective of identifying the effect that dissimilar potentials have on the resulting defect structures. The production efficiency is of the order of 0.3 ± 0.1 and is independent of the potential used for the simulation, with only one exception. However the simulations made with these different potentials predict strongly varying fractions of clustered defects and do not explain what feature of the potential these differences can be attributed to. Since the distribution of defects produced in collision cascades forms the input for the study for long-term evolution of microstructure, these studies provide evidence that MD simulations of cascade evolution need to be performed using a more restricted class of potentials. 6.2. DFT CALCULATION OF POINT DEFECT ENERGETICS IN α-Fe

The empirical potentials mentioned above (references in [42]) predict similar energies for the <110> and <111> dumbbells, with the <111> configuration in most cases being the lowest energy configuration [43]. In addition they predict that a di-Self-Interstitial Atom (SIA) and larger SIA clusters form a set of parallel <111> dumbbells migrating rapidly along the Burgers vector with a low activation energy of the order of a few hundredths of an eV. These results do not agree with the measured migration energies of defects in α-Fe (0.35 eV for SIAs, and 0.42 eV for small clusters [44,45]) and those found in DFT calculations [46]. Calculations performed at CEA-Saclay [47,48] showed that the <110> dumbbell configuration has energy lower than the <111> defect

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configuration by 0.7 eV. The effect of magnetism on self-interstitial atom defect configurations in ferromagnetic iron was highlighted by ab initio calculations performed by Nguyen-Manh et al. of UKAEA-Culham [49], who showed that the pattern of formation energies of self-interstitials in α-Fe was different in comparison with all the other bcc transition metals. The <111> crowdion represents the minimum energy configuration in all the non-magnetic bcc transition metals, while in α-Fe it is the <110> dumbbell that has the lowest formation energy [46,49,50]. The “topology” of the migration pathway of the <110> SIA dumbbell is non-trivial, and migration proceeds via nearest-neighbour translation-rotation jumps [50] originally proposed by Johnson [51]. 6.3. EMPIRICAL POTENTIALS INCLUDING MAGNETISM FOR α-Fe

Discussion above amply showed the need for developing a semiempirical potential including magnetic effects. A many-body potential was developed by combining the Stoner model and the Ginzburg-Landau model of a second order phase transition. This model addresses the problem of environmental dependence of magnetic energy [52] within the embeddedatom type formalism, and reproduces the correct ordering of formation energies of interstitial defect configurations in α-Fe. A semi-empirical potential for α-Fe was also developed outside the EU fusion materials modelling programme [53], and the defect formation energies predicted using this potential are in good agreements with DFT data, too. For the extended defect configurations both potentials make somewhat different predictions, for example the predicted core energies of a 100 straight edge dislocation differ between potentials developed in [52] and [53] by approximately 20%. The non-degenerate structure of the core of a 111 screw dislocation is well reproduced [54] by the Mendelev potential [53]. At the same time the double-kink energy EDK ~ 1.7 eV predicted by the potential [53] is more than twice the experimentally measured value of 0.8 eV. For comparison, potential [52] predicts EDK~1.1 eV in better agreement with experimental observations. Potential [52] also has the capability of predictting local magnetic moments of atoms in large-scale defect structures, for example mesoscopic dislocation loops [52]. No study using either potential has been published so far, which treats magnetic excitations in iron or ferritic steels at elevated temperatures. Magnetic excitations are responsible for the α–γ phase transition occurring in α-Fe at approximately 912°C, just above the Curie temperature of the material [55]. Since magnon excitations do not affect the magnitude of local magnetic moments and only involve changing the mutual orientation of moments, the evaluation of the magnitude of a magnetic moment on an atom [52] represents an essential step in

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the development of a Magnetic Molecular Dynamics (MMD) approach to simulating inter-atomic interactions in iron and other magnetic alloys at elevated temperatures. 6.4. THE PHASE STABILITY OF THE Fe-Cr SYSTEM

At low temperatures the Fe-Cr system exhibits asymmetry of the mixing enthalpy curve, which is negative for Cr concentrations below 10%. The origin of this asymmetry is associated with magnetic ordering [56]. This was recently confirmed by a DFT study carried out by Olsson et al. [57]. On the microscopic level the origin of the asymmetry is probably related to magnetic frustration occurring in an alloy, where the two atomic species forming the material (Fe and Cr) and taken in pure form exhibit opposing tendencies towards ferro- and anti-ferromagnetic ordering [58]. Based on the two-band model potential developed by P. Olsson et al. [59] L. Malerba [60] showed that in the Fe-rich limit of the phase diagram the negative enthalpy of alloy formation is well reproduced. The formalism has also been applied to the study of thermal ageing of the alloy [59], and to the study of short-range ordering. Nevertheless the currently available numerical parameterisation of the potential does not seem to reproduce the closure of the miscibility gap in the high temperature region (>800 K). Another unresolved issue is that of the symmetry of the phase diagram. Within the conventional non-magnetic many-body potential formalism the phase diagram is predicted to be symmetric with respect to the transposition of Fe and Cr atoms, resulting in the occurrence of a negative enthalpy region at the Cr–rich end of the diagram. This question has been partially addressed by the cluster expansion formalism where a large database of energies found using DFT for a number of crystal structures is fitted to a generalized Ising-like model expansion [61]. In this formalism the asymmetry of the Fe-Cr phase diagram is now reproduced. This asymmetry can be also reproduced by a semi-empirical potential derived using an approach based on the Redlich-Kister expansion used in CALPHAD for approximating the heat of solution of an alloy [62]. 6.5. KINETICS OF HE AND POINT DEFECT ACCUMULATION

Almost all the microscopic processes associated with microstructure evolution in materials are driven by thermal activation. The rate of occurrence of an event (i.e. the frequency of this event per unit time) is given by the Arrhenius law ν = ν0exp(εa/kBT), where T is the absolute temperature and εa the activation energy, which is assumed to be significantly greater than kBT. The fact that this rate depends on εa and on T exponentially shows that it is

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necessary to know activation energies within an accuracy of ~0.1 eV. At present only DFT calculations approach this accuracy. Combined DFT and kinetic Monte-Carlo (kMC) [48] gave full description of the electrical resistivity recovery of electron-irradiated α-Fe [45]. He-desorption experiments [63] were modelled using both rate theory (RT) and kinetic Monte Carlo (KMC) [64]. Good agreement was obtained using ab- initio data [7] for defect energies describing various Fe–He configurations. These results indicated that only the vacancy formation and migration energies had to be defined to explain the experimental data, and that defect traps, typically associated with carbon atoms, are likely to be present in the α-Fe matrix. The SIA clusters were assumed to be immobile, in qualitative agreement with electron microscope observations. 6.6. MODELLING AT THE MESO-SCALE

In order to describe the behaviour of materials realistically, grain boundaries must also be included in the models. Effects of irradiation in the presence of grain boundaries were studied by MD simulation of nanocrystalline Fe. The simulations [65] show that grain boundaries play an important part since they absorb single and clustered interstitials, whereas the formation of small, void-like vacancy clusters is observed for some grain sizes. Furthermore, if the cascade size exceeds the size of a grain or if a cascade event develops across a grain boundary, then this may induce movement of grain boundaries. Another effect occurring at this scale is the glide of dislocations interacting with obstacles. The interaction of an edge dislocation in α-Fe with He in solution or in a cavity was compared, using MD, with the effect of a void or a dislocation loop [66]. Helium was found to affect the yield stress only for very large He concentrations (1 at.%). Both a 2 nm void and a 2nm sessile a0[100] loop were found to be strong obstacles, while a non-pressurised He-bubble was not, with the obstacle strength increasing as a function of the amount of He in a bubble. 6.7. VALIDATION EXPERIMENTS

Information on mechanical properties of the irradiated materials is mostly obtained using post-irradiation tests. An important question is how these properties differ from those of the materials during irradiation, i. e. during their actual use. To answer this question, preliminary tensile experiments were performed in-situ during neutron irradiation on Cu and Cu-alloys [67]. Comparison with post-irradiation results shows that in-situ specimens deform uniformly, with no yield drop or plastic instability. The level of maximum hardening and uniform elongation depends strongly on the level

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of pre-yield displacement damage, i.e. the amount of damage accumulated before plastic deformation. These results are presently being verified for the case of α-Fe and Fe-9%Cr. Validation experiments involving both mechanical testing and microstructure examination of Fe-Cr-C alloys of different compositions, neutron-irradiated in the BR2, are being carried out [68]. An extensive programme of model-oriented experiments has also been formulated [69] in connection with the expected launch of the JANNUS facility at the start of 2008. This joint facility operated by CEA, CNRS and University of Orsay consists of joint triple beam accelerators at CEA/Saclay and a dual-beam in-situ electron microscope at the Orsay University [70]. 7. Concluding Remarks Ferritic/martensitic steels are suffering at low temperature from hardening and embrittlement triggered by helium: inter-granular failure along the prior austenitic grains is promoted although no grain-boundary He-bubbles are observed using TEM. In addition contribution of the α/α’ unmixing is not to be excluded. Progresses have been achieved and are still to be made to master the initial microstructure, inclusion cleanness and joining techniques of ODS steels for higher heat resistance. The progresses made in the physically based modelling of radiation effects in EUROFER under reactor relevant conditions is expected to rationalise the existing experimental information and will provide a foundation for knowledge-based research for improving phase stability and microstructure of this reference steel for better radiation resistance. In the longer term the new modelling tools should be essential for (i) extrapolating the data obtained within the various spectra to the fusion one, (ii) optimising the qualification programs in IFMIF, the future neutron source with fusion relevant spectrum and (iii) reliably extrapolate the qualification data bases to the larger operating conditions of reactors. Finally, for a safe and economic operation of the future fusion NPPs, components made of structural materials with increased radiation resistance will have to be developed, tested and their integrity assessed. Here again physically based modelling should provide guidance to improve the realism of the constitutive laws and damage rules used in the present the present codes [71]. Acknowledgements We would like to express our gratitude to our colleagues, principal investigators and researchers of the EU Material Development programme for outstanding contribution and fruitful discussion.

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INTRODUCTION TO THE PHYSICS OF MOLTEN SALT REACTORS E. MERLE-LUCOTTE*, D. HEUER, M. ALLIBERT, V. GHETTA, C. LE BRUN LPSC, UJF Grenoble 1, CNRS/IN2P3, INPG, 53, avenue des Martyrs, 38026 Grenoble Cedex, France

Abstract: In the frame of a major re-evaluation of the molten salt reactor (MSR), we have developed a new concept called Thorium Molten Salt Reactor (TMSR), particularly well suited to fulfill the criteria chosen by the Generation IV forum. This reactor may be operated in simplified and safe conditions in the Th/233U fuel cycle with fluoride salts. Amongst all TMSR configurations, many studies have highlighted the configurations with no moderator in the core as simple and very promising. Since 233U does not exist on earth and is not being produced today, we aim at designing a critical MSR able to burn the Plutonium and the Minor Actinides produced in today’s reactors, and consequently to convert this Plutonium into 233U. Thus, the current fuel cycle can be closed thanks to TMSRs started with transuranic elements on a Thorium base, i.e. started in the Th/Pu fuel cycle, similarly to fast neutron reactors operated in the U/Pu fuel cycle. We analyze the characteristics of these reactor configurations, in terms of fissile matter inventory, salt reprocessing, waste production and burning, and finally deployment capabilities. Using a simple kinetic-point model, we analyze the reactor’s behaviour as the total reactivity margins are introduced in the core. We thus confirm, beyond the classical advantages of molten salt reactors, the satisfactory behaviour of the TMSR and the excellent level of deterministic safety which can be achieved in such reactors. We then illustrate how the reactor can be driven with no control rod, either by controlling the extracted power or by monitoring the operating temperature. Finally we stress the hardiness and the flexibility of this TMSR concept, allowing it to be adjustable without loosing its advantages in the event of any technological stumbling block. Keywords: Molten salt reactor; Thorium fuel cycle; TMSR; neutronic studies; safety analysis

1. Introduction The Generation-IV International Forum for the development of new nuclear energy systems has established a set of goals as research directions for nuclear systems: enhanced safety and reliability, reduced waste generation, effective use of uranium or thorium ores, resistance to proliferation, improved

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[email protected]

V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

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economic competitiveness. Molten Salt Reactors (MSRs) are one of the systems retained by Generation IV. MSRs are based on a liquid fuel, so that their technology is fundamentally different from the solid fuel technologies currently in use leading to specific advantages in terms of safety/reliability (Rosenthal et al. 1970). Furthermore, this type of reactor is particularly well adapted to the Thorium fuel cycle (Th-233U) which has the advantage of producing less minor actinides than the Uranium-Plutonium fuel cycle (238U-239Pu) (Lecarpentier 2001, Nuttin 2002, Nuttin et al. 2005). While breeding in the U-Pu cycle can be obtained only with a fast neutron spectrum, in the Th-233U fuel cycle it can, in principle, be obtained with a more or less moderated neutron spectrum. In a thermal neutron spectrum, poisoning due to the Fission Products (FP) being worse than in a fast neutron spectrum, the rate at which fuel reprocessing is performed can become a major issue. Because, in an MSR, the fuel is liquid, continuous extraction of the fission products is a possibility. Within the general TMSR concept, various reactor configurations can be obtained by varying the graphite moderator ratio, leading to a neutron spectrum that can range from very thermalized to fast. The performances of these configurations in terms of safety coefficients, breeding ratios, graphite lifetime and fissile inventories are briefly outlined here. We then focus on the TMSR without any moderator in the core, this configuration presenting the most interesting characteristics, among which the capability to use the TRU produced in Light Water cooled Reactors (LWRs) as fissile matter mixed with a thorium base. MSRs are thus described in this paper as energy producing critical systems as well as actinide burners. Our studies of these reactors are described from a physics standpoint: influence of the chemical reprocessing on the neutronic behaviour, burning capabilities, deployment capabilities, and finally a complete deterministic safety analysis. This work made use of the MCNP neutron transport code (Briesmeister 1997) coupled with an in-house materials evolution code REM (Nuttin 2002, Mathieu 2005). The former evaluates the neutron flux and the reaction rates in all the cells while the latter solves the Bateman equations for the evolution of the materials composition within the cells. These calculations take into account the input parameters (power released, criticality level, chemistry...), by continuously adjusting the neutron flux or the materials composition of the core. Our calculations rest on a precise description of the geometry and consider several hundreds of nuclei with their interactions and radioactive decay.

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2. Molten Salt Reactors and Thorium Fuel Cycle The Molten Salt Reactor (MSR) concept was developed in the early 1950’s at the Oak Ridge National Laboratory (ORNL). The point was to conceive reactors whose fuel would be liquid, serving both as fuel and as coolant. The primary advantage of this concept is to allow continuous adjustment of the fuel salt composition, and thus ensure its operation over time with no reactivity reserve. Moreover, it is possible to reach very high temperatures and very high power densities without large internal pressures, therefore without severe constraints for the structural elements. The first experimental MSR stemmed from an American military program concerning plane propulsion which led to the Aircraft Reactor Experiment (ARE) (Briant and Weinberg 1957) in 1954. This 2.5 MWth reactor ran satisfactorily during 100 hours. The continuation of this work led to the commissioning of the 8 MWth Molten Salt Reactor Experiment (MSRE) (Haubenreich and Engel 1970). This reactor, based on a lithium, beryllium and zirconium fluoride salt was operated with 30% 235U enriched uranium from 1965 to 1968, with 233 U from 1968 to 1969 and finally with plutonium mixed with 233U in 1969. The success of this reactor warranted the study of a thorium power breeder reactor coupled to a reprocessing unit for the on-line extraction of fission products: the Molten Salt Breeder Reactor (MSBR) (Bettis and Robertson 1970, Whatley et al. 1970). Even though the concept looked promising, the studies were stopped in 1976. Studies on this sort of reactor were resumed in the 1980’s in Japan with the Thorims-NES (Furukawa et al. 1990) then the FUJI-AMSB projects (Furukawa et al. 2005), as well as in France with studies on the MSBR by EDF† and the CEA‡. In the 1990’s, the concept was taken up again with a view to incinerating nuclear wastes in sub-critical reactors such as the TIER-1 project (Bowman 1998), proposed by C. Bowman to transmute the plutonium of pressurized water reactors, the CEA TASSE project (Berthou 2000), or the EDF AMSTER project (Lecarpentier 2001). The CNRS§ has been involved in this type of reactor since 1999 through complete studies aiming to re-evaluate the MSBR concept (Nuttin 2002, Nuttin et al. 2005). This revaluation led to at least two important results which questioned the MSBR concept. On one hand, the feedback coefficient was proved to be globally positive so that the reactor is intrinsically unstable, on the other hand, the reprocessing unit appears not

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Electricité de France – French Electricity Company Commissariat à l’Energie Atomique - French Atomic Energy Commission § Centre National de la Recherche Scientifique – French Scientific Research Center ‡

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only unrealistic from the operational standpoint, and probably too complex to ensure economic cost effectiveness. Taking this into account, we have undertaken a systematic study of MSRs to resolve these problems and to optimize the reactor in the framework of the deployment of a thorium based reactor fleet on a worldwide scale (Heuer et al. 2006, Merle-Lucotte et al. 2004, Merle-Lucotte et al. 2006). 3. General Thorium Molten Salt Reactor (TMSR) Concept The general Thorium Molten Salt Reactor (TMSR) concept is a 2500 MWth (1 GWe) graphite moderated reactor based on the 232Th/233U fuel cycle. The graphite matrix comprises a lattice of hexagonal elements with 15 cm sides. The density of this nuclear grade graphite is set to 1.86. The salt runs through the middle of each of the elements. For the systematic studies presented in this section, the salt considered in the simulations was a binary salt, LiF - (Heavy Nuclei)F4, whose (HN)F4 proportion is set at 22 mole% (eutectic point), corresponding to a melting temperature of 565°C. A graphite radial blanket containing a fertile salt (LiF 78 mole%-ThF4) surrounds the core to protect the external structures, so that approximately 80% of the escaping neutrons are stopped, and the breeding is improved. Concerning reprocessing (see paragraph 4.2 for details), we assume in our calculations that helium bubbling in the salt circuit is able to extract the gaseous fission products and the noble metals within 30 seconds. In this section, our calculations are based on a delayed reprocessing of the total salt volume over a 6 months period with complete extraction of the fission products and of the transuranic elements. The 233U produced in the blanket is assumed to be extracted within a 6 months period. The moderation ratio can be altered by changing the channel radius. This modifies the neutron spectrum of the core, placing it anywhere between a much thermalized neutron spectrum and a relatively fast spectrum. The core size is adjusted to keep the whole salt volume constant and equal to 20 m3. Figure 1 shows the influence of the channel radius on the neutronic behaviour. In order to evaluate the performance of a reactor configuration, we check a number of parameters: total reactivity feedback coefficient, breeding ratio, graphite life span and initial fissile inventory. A wide variety of neutronic behaviours is available by changing the moderation ratio. We define three types of configurations: thermal, epithermal and fast spectrum. Each one has advantages and drawbacks: a thermal spectrum leads to a low fissile inventory but slightly positive feedback coefficient due to the graphite, an epithermal spectrum like that of the MSBR requires efficient salt processing and leads to safety coefficients that are too limited while a fast spectrum implies a high breeding ratio and

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excellent safety coefficients but requires a large fissile inventory. Extensive studies have been carried out on these three fields of research to improve these configurations and to find relevant solutions (Mathieu 2005, Mathieu et al. 2005, 2006). In this paper, we focus on the configuration without any graphite moderator in the core, since it appears to be the most promising in terms of safety coefficients, reprocessing requirements, breeding and deployment capabilities.

Figure 1. Influence of the channel radius on the TMSR’s behaviour. Note: There is no graphite lifespan point for the 13.6 cm radius since there is no more graphite in the core.

4. The Non-Moderated Thorium Molten Salt Reactor 4.1. CORE DESCRIPTION

As shown in Figure 2, the core is now a single cylinder (1.25 m radius and 2.60 m height) where the nuclear reactions occur within the flowing salt. One third of the 20 m3 of fuel salt circulates outside of the neutron flux in external circuits flowing through the pumps, the gaseous extraction system,

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which removes the gaseous and non soluble fission products (see paragraph 4.2) and the heat exchangers before coming back into the core. A fertile blanket surrounds the core, as already mentioned in section 3.

Figure 2. To scale vertical representation of the TMSR, including pumps and intermediate heat exchangers (IHX).

4.2. SALT REPROCESSING

Beyond the on-line control and adjustment of the salt composition and properties (redox potential measurement, reactivity, temperature…). the salt reprocessing is split in two steps. First, an on-line gaseous extraction system with helium bubbling is dedicated to the removal of gaseous fission products which are strong neutron poisons. As shown by ORNL data, this bubbling also extracts at least some of the noble metals and non soluble fission products. In our simulations, we assume that this helium bubbling is able to extract the gaseous fission products and the noble metals within 30 seconds, a less efficient extraction having little effect on core behaviour: up to an extraction time of a few days, the breeding ratio remains nearly constant. To optimise the extraction efficiency which depends on the interaction between the liquid salt, the metallic clusters and the gas bubbles, scientific investigations and dedicated measurements are necessary. Studies are also needed to determine how to separate the fission products from the gas, store them and purify the gas. The second step in the reprocessing deals with the extraction of the other fission products, mainly lanthanides. A fraction of salt is periodically set aside to be reprocessed off-line (batch mode). The fissile matter (uranium) can be extracted quickly by fluorination and sent back into the core. The other actinides and lanthanides can be separated via several methods like electrolysis, reduction into metallic solvents, solid precipitation, or any other

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method developed in the frame of pyrochemistry reprocessing (Poignet and Fouletier 2007). Finally the actinides are sent back into the reactor core to be burnt, while the lanthanides are stored. The performance of the reactor, in terms of breeding and deployment capabilities, depends directly on the rate at which this off-line reprocessing is done (see next paragraph). 4.3. FUEL SALT COMPOSITION

The core contains a fluoride fuel salt, composed of LiF enriched in 7Li (99.999%) and heavy nuclei (HN) amongst which the fissile element, 233U or Pu. We have done parametric studies of this salt composition by varying the proportion of heavy nuclei in the fuel salt, and the reprocessing (MerleLucotte et al. 2007a). This has an influence on neutron energy moderation, actinide solubility, and the initial fuel inventory. For HN proportions ranging from 20 to 30 mole%, a binary salt LiF-(HN)F4 has been chosen whose melting point is around 570°C. For lower proportions of HN, the calculations have been done with a salt containing 80 mole% of LiF completed with BeF2 (other possible components like NaF or CaF2 for example are under study) to lower the eutectic point temperature and to allow operation at 630°C. The salt density ranges from 3.1 to 4.6 according to the HN proportion, with a dilatation coefficient of 10–3/°C (Ignatiev et al. 2005).

Figure 3. Amount of heavy nuclei reprocessed per day versus initial fissile (233U) inventory, for different heavy nuclei proportions in the fuel.

The results of these parametric studies are presented in Figure 3 where the reprocessing capacity required for a TMSR system is shown. The ordinate is the weight of heavy nuclei reprocessed per day while the abscissa corresponds to the initial fissile (233U) inventory. Heavy nuclei proportions in the MSR fuel are indicated (x mole% HN). For each proportion

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of heavy nuclei in the salt, ranging from 6 mole% to 27.5 mole%, we evaluate by simulation the breeding ratio of each reactor configuration as a function of the amount of heavy nuclei reprocessed per day. The black line separates breeding and non-breeding zones. In the breeding zone, the red and green lines indicate the reprocessing requirements which allow the generation of the 233U needed for a new TMSR reactor with the same inventory over 100 years and 50 years respectively. Under-breeder reactor configurations, which are located under the black line (Figure 3, bottom), will not be considered in the following since they do not allow sustainable reactor deployment. The 233U initial inventory ranges from 2400 kg for a HN proportion in the salt of 6 mole% to 6300 kg for a HN proportion of 27.5 mole%. This corresponds to a variation of the neutron spectrum from an epithermal to a fast spectrum, as shown in Figure 4.

Figure 4. Neutron spectrum of two TMSR configurations (6% and 27.5% of heavy nuclei in the salt) compared to the neutron spectrum in a Light Water cooled Reactor (LWR) and in a Fast Neutron Reactor (FNR).

We have chosen a medium configuration with 17.5 mole% of HN in the salt, that we will refer to as the “typical TMSR” in this paper. For the results presented below, we have also selected a reasonable reprocessing rate of 200 kg of HN per day, indicated by the orange horizontal line in Figure 3. As already stated, TMSRs are operated in the Thorium fuel cycle, using either 233U (233U-started TMSR) or Pu (transuranic-started TMSR or TRUstarted TMSR) as initial fissile matter. The initial load of the non-moderated TMSR can thus be a mixture of thorium and, for its fissile material, the transuranic elements (Pu, Np, Am and Cm) produced in the water moderated

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reactors fed at present with natural or slightly enriched uranium. Actually, to be more realistic, these TMSRs are started with the mix of 87.5% of Pu (238Pu 2.7%, 239Pu 45.9% , 240Pu 21.5%, 241Pu 10.7%, and 242Pu 6.7%), 6.3% of Np, 5.3% of Am and 0.9% of Cm, corresponding to the transuranic elements of an UOX fuel after one use in a standard LWR and five years of storage (de Saint Jean et al. 2000). For the typical TMSR configuration (with 17.5 mole % of HN in the salt), an amount of 7300 kg of fissile elements is needed initially, corresponding to 4.5 mole% of Plutonium. 4.4. HEAVY NUCLEI INVENTORY AND BURNING CAPABILITIES

Figure 5 illustrates the evolution of a typical fuel salt composition (17.5 mole% HN) all along the operation of this reactor, for the 233U-started (solid lines) and for the transuranic-started (dashed lines) TMSR. In terms of transuranic inventory, as shown in Figure 5, the TMSRs started with transuranic elements tend towards TMSRs directly started and operated with 233U after about forty years for a fuel salt with 17.5% of heavy nuclei, where more than 85% of the initial TRU inventories are burned.

Figure 5. Heavy nuclei inventory for the 233U-started TMSR (solid lines) and for the transuranic-started TMSR (dashed lines) with 17.5 mole% of heavy nuclei in the salt.

More generally, the assets of the Thorium fuel cycle are finally recovered for these transuranic-started TMSRs after 25 to 50 years for HN proportions ranging from 7 to 27.5 mole%, except for minor actinides where longer times are needed.

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4.5. DEPLOYMENT CAPABILITIES

The deployment capabilities of a fleet of 4th generation reactors are directly linked to the breeding capabilities of each reactor since their initial fissile matter (233U or Pu) does not exist on earth. The interesting variable is thus the operating time necessary to produce one initial fissile (233U) inventory, called the reactor doubling time. We have studied the amount of excess 233U produced in each TMSR configuration. Transuranic-started TMSRs allow the extraction of significantly larger amounts of 233U during their first 20 years of operation, thanks to the burning of TRUs which saves a part of the 233U produced in the core. The higher deployment capabilities allowed by the use of TRUs in the transuranic-started TMSR are visible on the reactor doubling times, displayed in Figure 6 (dashed line), where the configurations with HN proportions larger than 15% have the lower reactor doubling times, around 30-35 years, to be compared with the doubling times greater than 45-50 years reachable with the 233U-started TMSRs.

Figure 6. Reactor doubling time of 233U-started (solid lines) and transuranic-started TMSRs (dashed lines) for the different HN proportion configurations and for two off-line reprocessing schemes: 200 kg (full points) and 50 kg (empty points) of HN reprocessed per day.

The configuration with 17.5% of heavy nuclei in the salt appears particularly optimal in terms of deployment capabiliities, especially with the faster reprocessing scheme and for the transuranic-started TMSRs. With these results, we have simulated the deployment of a fleet of such reactors at a worldwide and at a European scale and verified that large scale deployment is possible both with regard to the availability of fissile matter and

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to the control of fissile matter stockpiles and radioactive wastes (MerleLucotte et al. 2006, 2007b). 5. Deterministic Safety Study of the Non Moderated TMSR 5.1. DETERMINISTIC SAFETY PARAMETERS OF THE TMSR

5.1.1. Fraction of delayed neutrons The fraction of delayed neutrons, β, is very important in reactor control, it is a determining safety parameter. The total number of fission product atoms giving rise to delayed neutron emissions will depend on the fissile composition of the reactor and, to a lower extent, on the type of neutron spectrum. The value of β at equilibrium for the TMSR considered here has been estimated by simulation to 450 pcm. TABLE 1. Abundances of seven delayed neutron precursors for two uranium isotopes. Group

1

2

Precursor

87

137

Half-Life

55.9 s

24.5 s

Br

I

3 88

4

5

93

139

16.4 s

5.85 s

Br

Rb

6

7

91

96

2.3 s

0.54 s

0.199 s

I

Br

Rb

Abundances 233

U (fast)

0.0788

0.1666

0.1153

0.1985

0.3522

0.0633

0.0253

233

U (thermal)

0.0787

0.1723

0.1355

0.1884

0.3435

0.0605

0.0211

235

U (fast)

0.0339

0.1458

0.0847

0.1665

0.4069

0.1278

0.0344

235

U (thermal)

0.0321

0.1616

0.0752

0.1815

0.3969

0.1257

0.0270

0.0742

0.1679

0.1209

0.1915

0.3533

0.0684

0.0240

Mean Value

We have considered in our study seven precursor families listed in Table 1 (d’Angelo 2002) for two fissile nuclei, 233U and 235U. Regarding the neutron spectrum of the TMSR, we take a value between those indicated for precursor abundances in a fast and in a thermal spectrum. We have considered two fissile nuclei since, in the TMSR, 90% of the fissions are due to 233U and 10% to 235U. The final values used for the safety evaluation are summarized at the bottom of Table 1. 5.1.2. Feedback Coefficient The feedback coefficient or temperature coefficient is the variation of the multiplication coefficient dk for a given variation dT in temperature of the whole or part of the core. This feedback coefficient has to be negative to ensure the intrinsic stability of the reactor. The practical evaluation of a feedback coefficient is done as follows. The multiplication coefficient k is first computed for the core with the matter compositions at equilibrium for a

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temperature of 900 K. It is then re-calculated using the same compositions but at a different reactor temperature. In practice, the modifications concern the temperature of the salt itself, the temperature of the graphite moderator (if any), together with the density of the salt because of its dilatation (dilatation coefficient of 10–3/°C in our case). Other temperature variations like those of the reflectors or the blanket are not considered since these materials have a very small contribution and heat up very slowly. The feedback coefficient dk/dT is traditionally broken down into three sub-coefficients related to the different components of the core presented above: dk ⎞ dk ⎞ dk ⎞ dk ⎞ = + + ⎟ ⎟ ⎟ ⎟ dT ⎠Total dT ⎠ Salt _ heating dT ⎠Salt _ dilatation dT ⎠ Moderator _ heating

(1)

The third sub-coefficient is negligible in our case since there is no moderator in core. The uncertainties indicated in this paper are a quadratic combination of the statistical and systematic uncertainties on the determination of the subcoefficients. The statistical errors are estimated precisely by simulation, the quantification of the systematic errors is more difficult. More precisely, concerning the systematic uncertainties on the contribution of salt heating, the cross-sections concerned are well known, inducing only negligible uncertainties. The uncertainties on the salt density and its dilatation lead to systematic errors less than 20% on the contribution of salt dilatation, the value used for the estimation of uncertainties in the next sections. The total feedback coefficient at equilibrium is displayed in Figure 7, together with its components, the contributions of the salt heating and salt density, as a function of the HN proportion in the salt. All these safety coefficients are significantly negative for all HN proportions, including the density coefficient which can be seen as a void coefficient. The total feedback coefficient is largely negative, ranging from –10 pcm/K to –5 pcm/K. Concerning TMSRs started with transuranic elements, the total feedback coefficients after one year of operation are also displayed in Figure 7 as a function of the HN proportion. They correspond to the initial safety behavior of the reactor, since the inventories after one year of operation are quite similar to the initial inventories, but we also take into account the effects of the fission products which are not present in the initial load. As shown in Figure 7, the safety coefficients are initially slightly smaller in the case of the transuranic-started TMSRs but they remain largely negative, being equal at equilibrium to the safety coefficients of the 233U-started TMSRs. We have applied the safety analysis presented in the next paragraph on the case of a typical TMSR configuration, i.e. with a heavy nuclei

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proportion of 17.5% at equilibrium. Our calculations lead to a total feedback coefficient equal to –7 (±1) pcm/K at equilibrium, the coefficient of the corresponding configuration started with transuranic elements ranging from –5 pcm/K at the beginning of its operation to –7 pcm/K at equilibrium.

233

Figure 7. Feedback Coefficients of the TMSR started both with U or with transuranic elements at equilibrium and of the TMSR started with transuranic elements after one year of operation, as a function of the HN proportion in the fuel salt.

5.1.3. Reactivity Margins in the TMSR Beyond the safety aspects, using a liquid fuel allows the adjustment of fertile and fissile matter without unloading the core, doing away with the need for any initial reactivity reserve, contrary to the case of a LWR where this reactivity reserve amounts to 10 000 pcm. Some reactivity margins may be introduced in the core of the TMSR involuntary through three possible perturbations: a direct insertion of reactivity, the loss of salt circulation, and finally the loss of the fertile blanket surrounding the core. The impact of different kinds of reactivity insertions has to be studied. If the adjustment of fissile and fertile matters in the MSR is an important advantage, it may indeed present potential specific dangers which are evaluated below. We first consider the unintentional introduction of 233U in core. The reactivity increase of the TMSR is equal to 13.6 ± 0.2 pcm per extra kilogram of 233U (Mathieu 2005). For an iso-breeder reactor, the core needs to be fed 2.6 kg/day in 232Th. If 233U is added, instead of 232Th, this would lead to an increase of the reactivity of 35 pcm/day. 233U is also directly produced in core as follows: 232

233 233 233 β β Th + n ⎯ ⎯→ Th ⎯ ⎯ → Pa ⎯ ⎯ → U The radioactive period of 233Pa being 27 days, it decays rather quickly in core - around 2.55% per day - giving 233U. This corresponds to (1) a

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disappearance of neutron capturing matter and (2) an appearance of fissile matter. If the reactor were to stop operating, the first effect would increase the reactivity by 33 ± 11 pcm/day, the second effect by 27 ± 1 pcm/day, giving a total reactivity margin of 60 ± 11 pcm/day. As detailed in paragraph 5.1.1, the fraction of delayed neutrons for the whole salt at equilibrium is equal to 450 pcm. As two thirds of the salt is in the core and one third outside in the heat exchangers during normal operation, 2/3 of the delayed neutrons (300 pcm) are emitted within the core. If the salt circulation is stopped, all the delayed decays will occur in the reactor, since all the fission products will stay in core. This will represent an addition of 150 pcm to the multiplication coefficient in some ten seconds. Two other problems have to be considered: the loss of coolant (mainly for reactors with solid fuel) and the loss of the fertile blanket. In our case, the draining of the core is equivalent not only to a loss of the coolant but also to a loss of fuel, leading to a decrease of the reactivity. Since the reactor is equipped with a draining system through stoppers made for example of solid salt and designed to melt at a given temperature, it was important to check that the salt evacuation will not worsen an accident, since it occurs mainly in abnormal situations. Finally, our reactor is surrounded by a thorium blanket so as to optimize the breeding ratio of the system. The structure of this blanket is made of ZrC, containing a salt composed of LiF (72.5%) – ThF4 (27.5%). This salt may be solid or liquid. The second solution simplifies the recovery of the fissile matter but the blanket could then be accidentally drained. Our simulations show that the extraction of the entire blanket leads to a decrease of reactivity. Because the ZrC structure of the blanket does not moderate neutrons, draining the blanket leads to increased neutron leakages and thus to a decrease of reactivity. Table 2 summarizes the reactivity margins, with their corresponding times of insertion. Table 2. Reactivity margins of a typical TMSR configuration. 233

Pa Decay

Salt Circulation

Core Draining

Blanket Draining

Reactivity Margins

Reactivity Margin 35 pcm

60 pcm

150 pcm

-

-

<500 pcm

Time of Insertion Some seconds

1 day

Some 10 s

-

-

> some seconds

Origin

233

U Addition

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5.2. SAFETY EVALUATION OF THE TMSR

Using a single kinetic-point evaluation presented in the next paragraph, we analyze how the safety parameters presented above impact the system when the total reactivity margins are simultaneously introduced in the TMSR. Deterministic safety implies the definition of a viability domain, corresponding to the range of acceptable core parameters. As internal pressure is very low in a TMSR, only phenomena following a temperature increase of the fuel salt could endanger the reactor. As a consequence, the viability region is limited by the fuel solidification temperature with Tmin = 800 K as lower limit, and by the salt dissociation temperature with Tmax = 1600 K as upper limit. 5.2.1. The kinetic-point model Transient simulations were carried out using a simple mathematical model which includes the following system of point reactor kinetics equations with seven groups of delayed neutrons (Table 3 gather the list of the parameters):

∂N ρ − β = N + ∑ λ iC i ∂t l i

(2)

∂C i βN = − λi C i ∂t l

(3)

∂T = ( P − Po ) /(C p d ) ∂t

(4)

TABLE 3. List of the parameters used in equations (2)-(4).

N : neutron flux in the core, t : time since the beginning of the transient, ρ : reactivity,

Ci : proportion of delayed neutron precursors of the ith-group, T : mean core temperature ( To = 900K),

3 : proportion of delayed neutrons (300 P : instantaneous power per cm , 3 Po : extracted power per cm (125 pcm in the TMSR), 3 l : the mean time between two fissions W/cm , constant), C p : specific heat (1.05 J/g/K), (8.46 μs here), λi : ln 2 /(t1 / 2 ) i where (t1 / 2 ) i is the d : density of the fuel salt (4.3 g/cm3). decay period of the delayed neutron precursors of the ith-group,

β

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This kinetic-point model implies: •

A uniform distribution of the fissions within the core (correct in our case)

•

No heat propagation

•

No follow-through of the precursors of the delayed neutrons

The goal of this section is to give an idea of the reactor’s global behavior during accidental transients due to reactivity insertion or heat evacuation loss. Further calculations taking in consideration heat propagation, a realistic distribution of the fissions and the propagation of the precursors would loosen the limits of the kinetic-point model. 5.2.2. Safety evaluation of the reference configuration From Table 2, we can conclude that the insertion of the total reactivity margins of the TMSR corresponds to the addition of less than 1000 pcm in some minutes (around 100 seconds). The insertion of this upper limit of 1000 pcm, in 0.001 to 100 seconds, is thus displayed on Figure 8 in terms of temperature, power and reactivity evolutions. The calculations have been done using a feedback coefficient equal to –5 pcm/K, close to the worst coefficient value of the typical configuration of the TMSR with a fuel salt containing 17.5 mole% of heavy nuclei. As expected, the final temperature reached at equilibrium is equal to

To +

1000 pcm = 1100 K. dK /dT

(5)

We can conclude from Figure 8 first that the prompt criticality is reached for an insertion time shorter than one second, since the maximal reactor reactivity is higher than the fraction of delayed neutrons (equal to 300 pcm). If we detail for example the reactivity insertion in 0.001 second (Figure 8, black curves), we first have to consider the evolution of the reactivity (Figure 8, top) since we modify it directly. In this case, the whole 1000 pcm reactivity is added before the reactor begins to react. The reactivity insertion then leads to an increase of the reactor power (Figure 8, middle) which reaches a maximum at 1 MW/cm3. Finally, the reactor temperature (Figure 8, bottom) begins to increase after some milliseconds. A significant temperature increase triggers the reactor feedback, and as a consequence reactivity and power decrease. The heat in excess has finally to be evacuated to recover normal operating conditions. However, even in the case of the prompt reactivity regime, a huge and dangerous increase of power and temperature could be avoided thanks to the reactor’s safety parameters, as seen for the insertion time of 0.1 second.

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In this case (Figure 7), the thermal feedback is fast enough to lower the reactivity back to zero before the end of the 1000 pcm reactivity insertion. A bounce can thus be observed, especially on the power evolution, when the remaining reactivity margins are inserted. A second feedback leads to the final reactivity and power decreases, and normal running conditions are reached again when the accumulated heat is evacuated. For insertion times greater or equal to one second, and a fortiori in a more realistic case of 100 seconds, the reactor feedback is fast enough to avoid the prompt reactivity regime. Thus the reactor succeeds in absorbing at least a 1000 pcm reactivity insertion in one second and behaves safely.

Figure 8. Evolution of the reactivity, the reactor’s power and the salt temperature following an insertion of reactivity of 1000 pcm in 0.001 to 100 seconds, the total feedback coefficient of the reactor being equal to –5 pcm/K.

Concerning the validity domain of our results, our simulations are reliable for the longer insertion times and are pessimistic for the shorter insertion times, since we consider a uniform power distribution within the core and we do not take into account the time to reach the density equilibrium (equilibrium considered as immediate). Larger local powers are indeed reached within the core, leading to a faster feedback to reactivity insertion than shown in Figure 8. For the insertion times between 0.01 and 10 seconds,

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things are more complex and precise calculations have to be done to confirm our results, mainly to take into account the follow-through of the precursors of the delayed neutrons. 5.2.3. Control of the reactor The TMSR can be driven by the extracted power, and thus by the energy demand through the secondary circuit, as illustrated in Figure 9. As the energy demand increases, the power extracted becomes larger than the power produced by fission in the core. As a consequence, the temperature of the fuel salt decreases, leading to a reactivity increase as shown in Figure 9 for the extracted powers greater than 100% (curves corresponding to an extracted power of 125% and 150% of the nominal extracted power). This logically brings on a larger production of power in the reactor. This produced power then temporarily exceeds the power demand, which restores the initial temperature and reactivity. The reactor finally returns to equilibrium at its nominal temperature, while producing the needed power.

Figure 9. Evolution of the reactivity, the reactor’s power and the salt temperature as a function of the extracted power (in percent of the nominal extracted power), for a total feedback coefficient of the TMSR equal to –5 pcm/K.

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As shown on the other curves of Figure 9 for extracted powers lower than 100% of the nominal reactor operation, the same goes in the case of a power demand decrease, this bringing on this time a rise of temperature, a drop of reactivity and thus a decrease of the produced power. The reactor’s reactivity can also be controlled by stabilizing its operating temperature. A decrease of temperature with a constant produced power reveals a drop of reactivity due to a lack of fissile matter in the core. Reversely, an increased operating temperature reveals a too large reactivity which has to be corrected by stopping the addition of fissile matter (in the case of an under-breeder reactor for example). 6. Conclusions In the frame of a major re-evaluation of the MSR concept, we have first introduced the innovative concept of Thorium Molten Salt Reactor or TMSR. We then focussed on the configurations without any graphite moderator in the core, since they appear to be really promising. More precisely, the non-moderated TMSR configurations with high proportions of heavy nuclei, leading to a rather fast neutron spectrum, present particularly interesting characteristics regarding their safety performance and their ability to be first loaded with transuranic elements produced by today’s water cooled reactors. Finally, their rather large initial fissile inventory does not inhibit their capability for a fast deployment thanks to their very good 233U breeding ratio during the burning of transuranic elements. Using a simple kinetic-point model, we have analyzed the reactor’s behaviour as the total reactivity margins are introduced in the core. For insertion times greater or equal to one second, and a fortiori in a more realistic case of 100 seconds, the reactor succeeds in absorbing the reactivity insertion and behaves safely. Beyond the classical advantages of molten salt reactors, our studies confirm the satisfactory behaviour of the TMSR and the excellent level of deterministic safety which can be achieved in such reactors. We have also illustrated how the reactor can be driven with no control rod, either by controlling the extracted power or by monitoring the operating temperature. Under these conditions the TMSR appears to be a very appealing concept; our calculations do not indicate any major reprocessing constraint, allowing batch mode reprocessing in the vicinity of the reactor. The main additional studies needed to demonstrate the scientific feasibility of the concept deal with the on-line control of the salt composition and of its chemical and physical properties. Such studies are in progress in the frame of the French concerted research program ‘Molten Salt Reactor’ (PCR-RSF).

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Finally we want to point out the hardiness and the flexibility of this TMSR concept, allowing it to be adjustable without loosing its advantages in the event of any technological stumbling block. Acknowledgments We are thankful to Elisabeth Huffer for her help during the translation of this paper.

References d’Angelo A., 2002, Overview of the Delayed Neutron Data Activities and Results Monitored by the NEA/WPEC Subgroup 6, Prog. Nucl. En. 41(1-4):5-38. de Saint Jean C., Delpech M., Tommasi J., Youinou G., Bourdot P., 2000, Scénarios CNE : réacteurs classiques, caractérisation à l’équilibre, CEA report DER/SPRC/LEDC/99-448. Berthou V., 2000, Le concept TASSE (Thorium ADS with simplified fuel cycle for long term energy production), Thèse de doctorat, Université d’Evry – Val d’Essonne. Bettis E.S. and Robertson R.C., 1970, The design and performance features of a single-fluid molten salt breeder reactor, Nucl. Appl. Technol. 8:190-207. Bowman C.D., 1998, Once-Through Thermal-Spectrum Accelerator-Driven System for LWR Waste Destruction Without Reprocessing: Tier-1 Description, ADNA/98-04. Briant R.C. and Weinberg A.M., 1957, Aircraft Nuclear Propulsion Reactor, Nuc. Sci. Eng. 2:795-853. Briesmeister J.F., 1997, MCNP4B-A General Monte Carlo N Particle Transport Code, Los Alamos Lab. report LA-12625-M. Furukawa K., Lecocq A., Kato Y., and Mitachi K., 1990, Thorium Molten-Salt Nuclear Energy Synergetics, J. Nucl. Sci. Technol. 27(12):1157-1178. Furukawa K., Numata H., Kato Y., Mitachi K., Yoshioka R., Furuhashi A., Sato Y., and Arakawa K., 2005, New Primary Energy Source by Thorium Molten-Salt Reactor Technology, Electrochemistry 73(8):552-563. Haubenreich P.N. and Engel J.R., 1970, Experience with the Molten Salt Reactor Experiment, Nucl. Appl. Technol. 8:107-117. Heuer D., Merle-Lucotte E., and Mathieu L., 2006, Concept de réacteurs à sels fondus en cycle Thorium sans modérateur, Revue Générale du Nucléaire 5:92-99. Ignatiev V. and Walle E., 2005, Density of Molten Salt Reactor Fuel Salts, Nureth Conference, Avignon, France. Lecarpentier D., 2001, Le concept AMSTER, aspects physiques et sûreté, Thèse de doctorat, Conservatoire National des Arts et Métiers, France. Mathieu L., 2005, Cycle Thorium et Réacteurs à Sel Fondu : Exploration du champ des Paramètres et des Contraintes définissant le Thorium Molten Salt Reactor, thèse de doctorat, Institut National Polytechnique de Grenoble, France. Mathieu L., Heuer D., Brissot R., Garzenne C., Le Brun C., Lecarpentier D., Liatard E., Loiseaux J.M., Méplan O., Merle-Lucotte E., Nuttin A., Wilson J., Garzenne C., and Lecarpentier D., 2005, Proposition for a Very Simple Thorium Molten Salt Reactor, Proceedings of the Global international conference, Tsukuba, Japan.

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Mathieu L., Heuer D., Brissot R., Le Brun C., Liatard E., Loiseaux J.M., Méplan O., MerleLucotte E., and Nuttin A., 2006, The Thorium Molten Salt Reactor: Moving on from the MSBR, Prog. Nucl; En. 48:664-679. Merle-Lucotte E., Mathieu L., Heuer D., Loiseaux J.-M., Billebaud A., Brissot R., David S., Garzenne C., Laulan O., Le Brun C., Lecarpentier D., Liatard E., Méplan O. Nuttin A., and Perdu F., 2004, Molten Salt Reactors and Possible Scenarios for Future Nuclear Power Deployment, Proceedings of the Physor 2004 Conference, The Physics of Fuel Cycles and Advanced Nuclear Systems: Global Developments, American Nuclear Society (Ed.), 1-12. Merle-Lucotte E., Heuer D., Le Brun C., and Loiseaux J.-M., 2006, Scenarios for a Worldwide Deployment of Nuclear Power, Int. J. Nucl. Gov. Econ. and Ecol., 1(2):168192. Merle-Lucotte E., Heuer D., Allibert M., Ghetta V., Le Brun C, Mathieu L., Brissot R., and Liatard E., 2007a, Optimized Transition from the Reactors of Second and Third Generations to the Thorium Molten Salt Reactor, Proceedings of the International Congress on Advances in Nuclear Power Plants (ICAPP), Nice, France (2007). Merle-Lucotte E., Heuer D., Allibert M., Ghetta V., Le Brun C., Mathieu L., Brissot R., and Liatard E., 2007b, The Thorium Molten Salt Reactor: Launching the Thorium Cycle while Closing the Current Fuel Cycle, Proceedings of the European Nuclear Conference (ENC), Brussels, Belgium (2007) Nuttin A., Heuer D., Billebaud A., Brissot R., Garzenne C., Le Brun C., Lecarpentier D., Liatard E., Loiseaux J.M., Méplan O., Merle-Lucotte E., and Wilson J., 2005, Potential of Thorium Molten Salt Reactors, Prog. Nucl. En. 46:77-99. Poignet J.-C. and Fouletier J.: Physico-Chemical Properties of Molten Salts, Proceedings of the NATO Advanced Study Institute on Materials for Generation-IV Nuclear Reactors, Cargèse, France (2007). Rosenthal M.W., Kasten P.R., and Briggs R.B., 1970, Molten Salt Reactors - History, Status, and Potential, Nucl. Appl; Technol. 8:107-117. Whatley M.E., McNeese L. E., Carter W.L., Ferris L. M., and Nicholson E.L., 1970, Engineering development of the MSBR fuel recycle, Nucl. Appl. Technol. 8:170-178.

PHYSICO-CHEMICAL PROPERTIES OF MOLTEN SALTS J.-C. POIGNET* AND J. FOULETIER Grenoble University, Laboratoire d’Electrochimie et de Physico-chimie des Matériaux et des Interfaces (LEPMI), E.N.S.E.E.G., BP 75, 38402 Saint Martin d’Hères Cedex, France

Abstract: The purpose of this chapter is to present an overview of the physico-chemical properties of high temperature molten salts. The topics addressed include: an overview of molten salts, i.e., their main characteristics, their advantages and drawbacks compared with aqueous electrolytes, and the industrial processes involving molten salts, the main physicochemical properties including the transport properties. Finally, the electrochemical properties of molten salts and transient methods for the study of electrode mechanisms are described. Keywords: Molten salts; physico-chemical and electrochemical properties; transport properties; molten salt composition; electrode mechanisms

1. Overview on Molten Salts 1.1. GENERAL PRESENTATION AND CHARACTERISTICS

Molten salts constitute a particular class of solvent, usually fully ionized, with cohesion resulting from coulombic forces. Their liquid state properties are similar to those of classical solvents, except for ionic concentration and therefore electrical conductivity, which are both much higher than those of aqueous or organic electrolytes. As an example, in Table 1 are compared the main physico-chemical properties of molten sodium chloride at 850°C (melting point of NaCl: 800°C) and an aqueous sodium chloride solution (10–1 mol.L–1), at 25°C. Because of the short interionic distances, the melts have tendencies to form ionic or uncharged complexes. Another specific feature of molten salts is the lack of inert molecular medium between the ions. The temperature range of melting (from –50°C up to more than 1000°C) of various molten salts is given in Table 2.

______

* Jean-Claude POIGNET, LEPMI, ENSEEG, BP 75, 38402 Saint Martin d’Hères cedex, France; e-mail: [email protected]

V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

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TABLE 1. Comparison of physico-chemical properties of molten NaCl and an aqueous solution of NaCl. Electrolytes

Molten NaCl at 850°C NaCl, 10–1 mol.L–1 at 25°C

Ions concentration (mol.L–1) 26 10–1

Density (g.cm–3) 1.53 1.002

Electrical conductivity Viscos (S.cm–1) ity (cP) 3.6 1.2 1.07x10–2 1.012

TABLE 2. Examples of molten salts from Bouteillon and Poignet (1996). Temperature range of melting

Examples of molten salts

– 50 to 150°C

HF-KF, ionic liquids (chlorides of alkylpyridinium, alkylimidazolium, and their mixtures with AlCl3)

150 to 250°C

Mixtures of alkaline halides and AlCl3

250 to 400°C

Mixtures of alkaline nitrates, nitrites, hydroxides

400 to 1000°C

Alkaline halides and their mixtures, alkaline sulfates, alkaline carbonates, and fluorides (cryolite)

>1000°C

Silicates, oxides and their mixtures (molten glass, slags)

This chapter is focused on molten halides, which are most suitable to applications in nuclear energy. 1.2. ADVANTAGES AND DRAWBACKS

An exceptional variety of molten salts are available, and according to their high mutual solubilities, the mixture composition can be adapted to fulfill the required properties. In the case of a binary mixture, the liquidus curve gives the lowest temperature of a liquid phase as a function of composition. The melting temperature of the eutectic composition is generally noticeably lower than that of the pure component (see, as an example, the LiCl-KCl diagram, Figure 1). In the case of ternary eutectics (such as the LiF-NaF-KF mixture, referred to as “FLiNaK”) the melting temperature is lower than the eutectic temperature of the three binary mixtures, see Figure 1. The ternary diagram LiF-ThF4-UF4 with indication of the composition range proposed for the molten salt reactor (see another chapter of this book: Merle-Lucotte et al.) is also schematized in Figure 1. Molten salts have good thermal stability with reasonably low vapour pressure, do not suffer radiolysis, are good solvents, exhibit a very large electrochemical stability domain (see section 2.2) so that it is possible to

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prepare or process most elements (from Li to F) in these media, and furthermore have a high electrical conductivity. Moreover, high reaction kinetics (either chemical or electrochemical reactions) are observed in fused salts.

Figure 1. Phase diagrams of binary and ternary mixtures: a) LiCl-KCl, b) LiF-NaF-KF, c) LiF-UF4-ThF4.

Apart from all their advantages, molten salts exhibit certain drawbacks ascribed to their chemical reactivity and the high temperature, i.e., specific materials have to be used for containers, vessels, membranes, etc.; due to their chemical interaction with water, molten salts must be carefully isolated from the open atmosphere; the high chemical and electrochemical kinetics entail corrosion problems if gradients are present (temperature, composition, etc.); molten salts may partially dissolve metals, among with alkali metals. 1.3. WORLD PRODUCTION

Many industrial plants and electrochemical devices (fuel cells, batteries, electrolysers, etc.) make use of halides (chlorides, fluorides), nitrates or car-

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bonates molten mixtures. In spite of engineering difficulties, metals are produced on a large scale by molten salt electrolysis. As an example, 20 million t./year of aluminium are produced in the world. It should be pointed out that the electrolysis of cryolite-alumina melts was patented in 1886. A continuous series of modifications have been incorporated in the cell designs. However, the fundamental production process remains unchanged. Other large-scale productions involve aluminium recycling (1 million t. per year), aluminium refining (50 000 t./year), magnesium (500 000 t./year), alkali metals (60 000 t./year), neodymium (1000 t./year), fluorine (1 000 t./ year), etc. Moreover, new processes are under industrial research and development, such as direct electrochemical reduction of solid oxides in molten CaCl2 and production of iron by molten salt electrolysis. Metals are also produced by pyrometallurgy in molten salts. One can quote the production of titanium and zirconium (reducing agent: Mg), of titanium (reducing agent: Na), of niobium and tantalum (reducing agent: Al), of magnesium alloys (reducing agent: Al-Si alloys), of potassium or rare earth metals (Nd, Dy, Gd) (reducing agent: calcium). 2. Properties of Molten Salts 2.1. MAIN PHYSICOCHEMICAL PROPERTIES OF MOLTEN SALTS

Physico-chemical properties of molten salts are reviewed in several books: Sundheim (1964), Janz (1967), Richardson (1974), Janz and Bansal (1982). Upon melting, the electrical conductivity rises drastically and, as shown by X-ray or neutron studies, some short-range order is observed, i.e., a cation is more likely to be found in the vicinity of an anion, and vice-versa. When melting, most salts expand by 10 - 20%, which indicates that the melts contain a large number of voids. Their mean size is between one and two interionic distances and they are distributed uniformly throughout the entire liquid volume. Owing to fluctuation phenomena, these holes form and disappear constantly. Since the forties, several models have been developed to explain the variation with temperature of either the conductivity or the viscosity of molten salts. The review paper of Moynihan (1971) can be referred to for detailed description of the mass transport models. Basically, in the hole theory (Furth 1941), the transport of an ion proceeds via an individual jump into a neighbouring void requiring the formation of a void left behind the migrating ion. In the transition state theory (Bockris 1952), the migration of the ion to a neighbouring vacancy or site in the liquid is impeded by a free energy barrier, leading to an Arrhenius variation of the conductivity with temperature. In the free volume theory (Cohen

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and Turnbull 1959), the voids within a liquid may be redistributed without energy change. The ion can migrate when the free volume exceeds some critical value roughly equal to the molecular diameter. Thus migration proceeds via cooperative movements of the neighbouring ions. Figure 2 illustrates the cooperative movements of the ions for a two-dimensional hard sphere system (Souquet 1994).

Figure 2. Illustration of the free volume motion. The ion displacement (from C to C’) is allowed when the neighbouring volume is higher than a critical value, from Souquet (1994).

The temperature dependence of the ionic conductivity λj of a particle j obeys the following equation, often referred to as VTF (Vogel-TamannFulcher) equation (the T–1/2 term is often omitted):

⎛

B ⎞ ⎟ ⎝ T − To ⎠

λ j = Aλ T −1/ 2 exp ⎜−

(1)

Adam and Gibbs (1965) consider that in a low-fluidity liquid, the cooperative rearrangement requires an activation free energy. It should be pointed out that, due to the large η (T) variation of viscosity of molten salts, none of these models can be applied in the whole temperature range. More recently, an ab initio molecular dynamic approach has been developed to examine the transport properties of simple binary fluoride mixtures (Salanne et al. 2006). A common characteristic of molten salt mixtures is that the equivalent conductances of the mixtures are less than those predicted on the basis of the additivity of the equivalent conductances of the pure components leading in some cases to a conductance minimum at some intermediate composition. This phenomenon can be ascribed to the formation of uncharged ion pairs, which cannot transport the current. For this reason, the NernstEinstein equation is not applicable. The viscosity and thermal conductivity differ little from those of aqueous solutions. However, their surface tensions are higher than those of aqueous solutions (see another chapter of this book: Chatain and Ghetta).

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Metals are soluble in varying degrees in molten salt. At sufficiently high temperatures, alkali metals and alkaline earth metals (with the exception of Mg) are miscible in all proportions. Such a solubility has to be taken into account in molten salt electrolytic processes: valence-unsaturated compounds, formed at the cathode by partial oxidation of the deposited metal by the molten salt, can be reoxidized at the anode, leading to a lowering of the current yield. Gases are slightly soluble in molten salts (the concentrations are about 100 times smaller in the salt than in the gas), the solubility can be important at high pressure (Henry’s law is obeyed) and the solubilities decrease as the size of the gas atoms increases. 2.2. ELECTROCHEMICAL PROPERTIES

Electrochemical data may be found in different books: Delimarskii (1961), Bard (1976), Tremillon (1993). There is no generally accepted reference electrode and no standard temperature for electromotive force series. The various reference electrodes proposed in molten chlorides and fluorides are described in another chapter of this book (Fouletier and Ghetta). The variation with temperature of the e.m.f. of various MFn/M couples in molten fluorides vs. F2/F- reference electrode is given in Figure 3. The e.m.f. range of actinides is sufficiently far away from that of lanthanides to envisage their separation by selective electrochemical reduction on solid or liquid cathodes or by using pyrometallurgical processes (see section 3.1).

Figure 3. E.m.f. series of MFn/M couples in fluoride melts.

Due to hygroscopicity or to the chemical interaction with water of most of molten salts, the monitoring of the free oxide contents is of great importance. The potentiometric sensor using stabilized zirconia electrolyte developed for the sensing of oxide ions in chloride melts is described in

PHYSICO-CHEMICAL PROPERTIES OF MOLTEN SALTS

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another chapter of this book (Fouletier and Ghetta). It should be pointed out that this sensor cannot be used in fluoride melts and electrochemical techniques have been proposed recently to monitor free oxide contents in molten fluorides (Massot et al. 2007). The oxoacidity concept (Lux acid-base reaction) is based on following equilibrium:

A + O2– = B

(2)

where “A” is an acid and “B” is its conjugated base. The acidity of the melt is defined by p(O2–) = – log [m(O2–)], where m(O2–) is the oxide ion molality. The higher p(O2–), the higher is the acidity of the molten salt. Other acidity scales have been proposed, i.e., p(Li2O), p(CO2) or p(H2O). Following the Pourbaix diagram concept, the potential-acidity diagrams for a given metal can be plotted in an E vs. p(O2–) (or E vs. p(Li2O)) diagram. As an illustration, the potential-acidity diagram of nickel is given in Figure 4. Three types of domains can be defined: •

the immunity domain, corresponding of the stability of the metal,

•

the corrosion domain, corresponding to an ion concentration in equilibrium with the metal higher than 10–6 mol.L–1,

•

the passivation domain, corresponding to the formation of an oxide.

Figure 4. Potential-acidity diagram of nickel.

The stability window of the molten salt has to be defined. Figure 5 illustrates the stability window of a fluoride melt: •

the basic limit (low p(Li2O)), corresponding to the precipitation of an oxide or a mixed oxide,

•

the acidic limit (high p(Li2O)), corresponding to the smallest concentration of oxide ions (arbitrary limit),

•

the reduction limit, corresponding to the reduction of a cation of the molten salt My+ (My+ + y e– → M: horizontal line in the diagram referred to as (Li)),

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J.-C. POIGNET AND J. FOULETIER

the oxidation limit, corresponding to O2 or F2 formation, according to the melt basicity: - low p(Li2O): oxidation of O2– (O2– → 1/2 O2 + 2 e–) corresponding to the equation (line referred to as (O2) in Figure 5):

E = A+

2.3 RT p ( Li2O ) 2F

- high p(Li2O): oxidation of F- (2 F- → F2 + 2 e-) corresponding to a horizontal line, referred to as (F2) in the Figure. Within the stability window, the lines that delineate the stability domains of the various oxocomplexes as functions of E and of p(O2-) (or p(Li2O)) can be drawn from the following table, given for Nd. TABLE 3. Qualitative valency vs. p(O2-) table for neodymium. Valency p(O2-) + III Nd2O3 NdF3 0 Nd

Figure 5. Potential-acidity diagrams of Nd (a) and U (b) in fluoride melts, at 1200 K, after Finne (2005).

Figure 5 gives the potential-acidity diagrams of neodymium and uranium taking into account the solid or gaseous oxo- and fluoro-complexes in LiF, at 1200K. 3. Electrochemistry in Molten Salts Electrowinning processes used for direct production of metals or oxides such as Hf, U, Pu, PuO2, etc., and electrorefining processes used for the production of high purity metals (U, Be, Pu, Th, Zr, Hf, etc.) will not be

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531

described in this chapter. This chapter is restricted to the description of two types of molten processes, i.e., molten salt reprocessing of nuclear wastes and non-stationary techniques used for the study of the electrode reaction mechanisms and of electroanalysis. 3.1. ELECTROCHEMICAL FUEL REPROCESSING

Different pyrometallurgical techniques for the separation and recycling of actinides contained in nuclear wastes are under development. They are based on direct electrolysis or reductive extraction using molten salts. The separation of actinides from lanthanides can be carried out by a displacement reaction between a liquid metal and a molten salt. The following reactions (3) and (4), when the metal is respectively M and La, give two examples:

MXm

+

m Li + i Cd =

(in fused salt)

LaCl3

(in molten Cd)

+ 3 Li

(in fused LiCl-KCl)

=

(in molten Cd)

MCdi +

m LiX

(3)

(molten intermetallic) (in fused salt)

LaCdi

+

3 LiCl

(4)

(molten intermetallic) (in fused LiCl-KCl)

Basically, the solute element M (or La in the form of LaCl3) is reduced by the reductant Li and transferred from the salt phase into the cadmium phase. Simultaneously, Li is transferred from the molten metallic phase into the fused salt phase. The efficiency of the liquid-liquid extraction is improved by mechanical stirring of molten mixture. It is estimated using the distribution coefficient of species M:

DM =

xM x MX n

( metal )

(5)

( molten salt )

xM and xMX n are the mole fractions. A similar expression is valid for Li :

DLi =

x Li xLiX

(metal )

(6)

( molten salt )

Putting the distribution coefficients of M and Li, after Lemort (1998) into the equilibrium constant of reaction (3), the distribution of species M between two immiscible phases corresponding at equilibrium reaction (3) obeys the following equation:

⎡γ m . γ ⎤ log DM = m log DLi + log K + log ⎢ Lim MX n ⎥ ⎣ γ LiX . γ Li ⎦

(7)

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J.-C. POIGNET AND J. FOULETIER

in which DM and DLi are the distribution coefficients of M and Li, respectively and the γis are the activity coefficients. The higher the DM value, the most efficient will be the transfer of M from the molten salt phase to the molten metal. The distribution coefficient depends on the compared redox properties of M and Li, and on the activities of M and Li in the molten salt and the molten metallic phase, respectively. For efficient liquid/liquid partitioning, the following options have to be considered: •

selection of the molten salt phase (chlorides, fluorides, etc.),

•

selection of the molten metal which is the solvent for the reducing agent,

•

selection of the reducing agent (or potential in the case of direct electrolysis of the molten salt),

•

influence of the phases compositions on the activity coefficients.

The distribution coefficient of various elements between Zn-Mg and LiCaF2 (4:1 molar) at 720°C are given in Figure 6 (Lemort 1998). The straight lines observed, with a slope of x/2 (where x is the cation valency in the molten salt) indicate that all the activity coefficients can be considered as constant.

Figure 6. Distribution of various elements between LiF-CaF2 (4:1 molar) and Zn-Mg at 720°C, after Lemort (1998).

3.2. ELECTROANALYSIS AND ELECTRODE REACTION MECHANISMS (BARD AND FAULKNER 1980, GREEF ET AL. 1985)

Electroanalytical techniques may be applied in stationary state or transient state conditions. Establishing well-reproducible hydrodynamic regimes in high temperature molten salts is not easy, therefore transient techniques,

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which necessitate the absence of convection in the electrolyte, are preferred. Among them, chronopotentiometry and linear sweep voltammetry will be addressed in this text. In chronopotentiometry, where a constant current is applied, the variations of the electrode potential are recorded as a function of time. In linear sweep voltammetry, where a potential varying linearly with time is applied, current-potential curves are recorded. Applying these perturbations causes variations of the concentration of the electroactive species in the diffusion layer, in the vicinity of the electrode-molten salt interface: the thickness of the diffusion layer, which increases with time, remains very small because the time scale of the experiment is kept short (inferior to a few seconds) in order to eliminate natural convection. Very few coulombs are passed, so that applying these methods do not create any significant modification in the bulk of the electrolytes. The determinations can therefore be repeated. Another advantage is that these methods necessitate merely the introduction of two or three electrodes with simple shapes (such as metallic wires, 1 mm in diameter), which can be easily replaced if necessary. If mass transport is purely diffusional (migration and convection are made negligible), the current density at the working electrode is proportional to the flux of the electroactive species, therefore to its diffusion coefficient D and to its concentration gradient at the interface:

i = n F J = −n F D (

∂C ) x=0 ∂x

(8)

Using this limiting condition yields the calculation of the distribution curves in the solution and at the electrode surface (Figure 7).

Figure 7. Concentration distribution of electroactive species (a) Chronopotentiometry, (b) Linear sweep voltammetry. The values on the curves of Figure 7a represent the time, expressed as t/τ, at which each distribution is calculated. The letters on the curves of Figure 7b represent the time at which each distribution is calculated (time increasing from a to f).

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J.-C. POIGNET AND J. FOULETIER

The electrode potential responds to the concentration of the electroactive species at the electrode surface. The resulting curves are given in Figures 8a and 8b, together with a few useful relations or criteria.

i τ nF π D = C 2

(9)

useful in chronopotentiometry to determine either the concentration C, or the diffusion coefficient D by measuring τ on the chronopotentiogram.

E p − E p/ 2 = - 0.77

RT nF

(10)

useful in linear sweep voltammetry to characterise reaction parameters such as rapidity of the reaction, number of electrons exchanged, by measurement on the voltammogram of the peak and half-peak potentials.

i p = 0.61(n F ) 3 / 2 C (

Dv 1 / 2 ) RT

(11)

useful, knowing the current peak ip, the number of electrons exchanged, the concentration C and the sweep rate v, to calculate the diffusion coefficient. Determination of C is also possible.

Figure 8. a) Chronopotentiogram, b) Linear sweep voltammogram.

Physico-chemical determinations offered by these techniques are illustrated by the following examples concerning recent studies of Am and Th in molten chlorides. The resulting curves are given in Figures 8a and 8b, together with few useful relations or criteria. 3.2.1. Electrochemistry of thorium in LiCl-KCl eutectic melts (after Cassayre et al. 2007) Main results, see Figure 9: – Th (IV) is the only stable valency in this melt, – Reduction into metallic thorium occurs via a simple step,

PHYSICO-CHEMICAL PROPERTIES OF MOLTEN SALTS

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– The diffusion coefficient of Th(IV) is determined, – Informations concerning potentials and activity coefficients are obtained.

Figure 9. Electrochemical study of thorium ions in LiCl-KCl eutectic melt containing 0.43 wt.% ThCl4 on W electrode (S = 0.31 cm2) (after Cassayre et al. 2007): a) Cyclic voltammogram, T = 773 K, scan rate: 100 mV.s–1 , b) Chronopotentiogram, T = 733 K.

3.2.2. Electrochemical behaviour of americium ions in LiCl–KCl eutectic melt (after Serp et al. 2006)

Figure 10. Electrochemical study of americium ions in LiCl-KCl eutectic melt (after Serp et al. 2006), [Am3+] = 3.65 × 10–5 mol.cm–3, working electrode: W (A = 0.2 cm2), T = 733 K, reference electrode: Ag/AgCl 1 wt.%, a) Cyclic voltammogram, scan rate 100 mV.s–1, b) Chronopotentiogram, I = –15 mA.

Main results, see Figure 10: – Am(III) and Am(II) are the two stable valencies in this melt, – Reduction of Am(III) to metallic Am occurs via a two steps, – The diffusion coefficients of Am(III) and Am(II) are determined, – Informations concerning potentials and activity coefficients are obtained.

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References Adam, G., and Gibbs, J. H., 1965, On the temperature dependence of cooperative relaxation properties in glass-forming liquids, J. Chem.Phys. 43(1):139-146. Bard, A. J., 1976, Encyclopedia of electrochemistry of the elements, Vol. X, Fused salt systems, Marcel Dekker Inc., Basel. Bard, A. J., and Faulkner, L. R., 1980, Electrochemical methods - fundamentals and applications, Wiley, N. Y. Bockris, J. O’M., Kitchener, J. A., Ignatowicz, S., and Tomlinson, J. W., 1952, Trans. Faraday Soc. 48:75-91. Bouteillon, J., and Poignet, J.-C., 1996, Constantes électrochimiques des sels fondus in: Techniques de l’Ingénieur, Constantes physico-chimiques, K 825. Cassayre, L., Serp, J., Soucek, P., Malmbeck, R., Rebizant, J., and Glatz, J.-P., 2007, Electrochemistry of thorium in LiCl-KCl eutectic melts, Electrochimica Acta 52:74327437. Cohen, M. H., and Turnbull, D., 1959, J. Chem. Phys. 31(5):1164-1169. Delimarskii, Iu. K., and Markov, B. F., 1961, Electrochemistry of fused salts, The Sigma Press Publ., Washington. Finne, J., 2005, Chimie des mélanges de sels fondus. Application à l’extraction réductrice d’actinides et de lanthanides par un metal liquide, PhD Thesis, Paris 6 University. Furth, R., 1941, Theory of the liquid state, the hole theory of the viscous flow of liquids, Proc. Cambridge Phil. Soc. 37:252-290. Janz, G. J., 1967, Molten salts handbook, Academic Press, London. Janz, G. J., and Bansal, N. P., 1982, Molten salt data: Diffusion coefficients in single and multi-component salt systems, J. Phys. Chem., Ref. Data, 11(3):505-693. Lemort, F., Piccinato, R., Boen, R., and Allibert, M., 1998, Modeling and estimating the distribution of trivalent actinides and lanthanides between molten fluorides and liquid metal, J. Alloys and Compounds 271-273:645-649. Massot, L., Cassayre, L., Chamelot, P., and Taxil, P., 2007, On the use of electrochemical techniques to monitor free oxide content in molten fluoride media, J. Electroanal. Chem. 606:17-23. Moynihan, C. T., 1971, Mass transport in fused salts, in Ionic interaction, from dilute solutions to fused salts, Petrucci S. Ed., Academic Press, London, pp. 261-384. Richardson, F. D., 1974, Physical Chemistry of Melts in Metallurgy, Vol. 1, Academic Press, London. Salanne, M., Simon, C., Turq, P., Heaton, R. J., and Madden, P. A., 2006, A first-principles description of liquid BeF2 and its mixture with LiF: 2. Network formation in LiF-BeF2, J. Phys. Chem. 110: 11461-11467. Serp, J., Chamelot, P., Fourcadot, S., Konings, R. J. M., Malmbeck, R., Pernel, C., Poignet, J.-C., Rebizant, J., and Glatz, J.-P., 2006, Electrochemical behaviour of americium ions in LiCl-KCl eutectic melt, Electrochimica Acta 51, 4024-4032. Souquet, J. L., 1994, Ionic transport in glassy and polymer electrolytes, in: Defects and Disorder in crystalline and amorphous solids, Catlow, C. R. A. ed., Kluwer Academic Publ., pp. 221-244. Greef, R., Peat, R., Peter, L. M., Pletcher, D., and Robinson, J., 1985, Instrumental Methods in Electrochemistry, Ellis-Horwood, Chichester. Sundheim, B. S., 1964, Fused salts, McGraw-Hill Book Company, London. Tremillon, B., 1993, Electrochimie analytique et réactions en solution, Tome 1, Masson, Paris.

COMBINED EFFECT OF MOLTEN FLUORIDE SALT AND IRRADIATION ON NI-BASED ALLOYS A.S. BAKAI* Kharkiv Institute of Physics and Technology, 61108, Kharkiv, Ukraine

Abstract: Results of investigations of combined effect of molten fluoride salt and electron irradiation on structure, mechanical properties and corrosion of Ni-based alloys are reviewed. Keywords: Molten salt reactor; Ni-based alloys; electron irradiation test facility; microstructure; corrosion; mechanical properties

1. Introduction Fluoride molten salts (MS) are considered as means of heat and/or fuel transport in Generation IV (GenIV) nuclear reactors. Fluid fuel in form of fissile and fertile materials dissolved in MS is promising fuel for both MS reactors and subcritical accelerator driven systems. MS also can serve as a coolant of high-temperature low pressure systems. It is commonly believed that Ni-Mo alloys (the Hastelloys) are proper structural materials for MS transportation pipes. The Hastelloy N was successfully used in the molten salt reactor experiment (Haubereich and Engel 1970) but the experiment duration time was in order of magnitude less than a reasonable time of GenIV reactor operation. Since that time corrosion tests of Ni–Mo alloys of different compositions were performed in loops without irradiation (DeVan 1969, Ignatiev et al. 2006). These tests were focused on investigation of role of different dopants on the corrosion rate. In part, it was revealed that small amount of Nb (0.5 ÷ 2wt%) depressses the corrosion of alloys in fluoride MS. Since corrosion is a result of transport and dissolution of the alloy components in MS, impact of irradiation on the corrosion is considerable

______ *

* Tel./fax: +380 57 3350854. e-mail : [email protected]

V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

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because, first, irradiation increases the chemical reaction rates within the salt-alloy interface and, second, irradiation accelerates the bulk diffusion due to generation of point defects. Therefore one can expect that corrosion of alloys under irradiation can dramatically differ from those without irradiation. It has to be noted that microscopic mechanisms of the structural and compositional changes of Ni–Mo alloys in the molten salts under and without irradiation were never studied in detail. Very scant information about precipitations and heterogeneities of these alloys is available. Experiments using reactor irradiation are rather expensive. Therefore the idea to simulate reactor irradiation by accelerated electrons or ions is in use for a long time. Evidently, the electron irradiation is an efficient tool to impact the corrosion process due to enhancement of the interface corrosion kinetics. But it is less efficient in simulation of the bulk effects of reactor irradiation. Therefore an important issue is corresponding data of the simulative tests under e-irradiation to the results of reactor irradiation. To solve this problem both reactor tests and computer simulations can be used. In this lecture I present a review of results of the corrosion tests of Ni– Mo alloys of two compositions (with and without Nb) under and without electron irradiation at T = 660°C. Partially the results obtained were published in the special issue* “Materials for molten salt reactors” (Guest eds. A.S. Bakai and F.A. Garner) and reviewed by Azhazha et al. (2005a). Results of extensive simulations determining quantitative irradiation characteristics of both 10 MeV electrons and MS reactors (MSR) with different neutron spectra are given, too. These results allow to judge about suitability of the simulation experiment data. 2. Methodology Specific features of the electron irradiation which are relevant to these investigations are the following: •

Since energetic thresholds of nuclear reactions induced by electrons and γ-quanta are ~10 MeV, use of the electron beams of 10 MeV energy does not induce considerable radioactivity in the irradiated materials. Therefore, hot cells are not needed in post-irradiation investigations.

•

The penetration depth of electrons with 10 MeV energy is of the order of several centimeters. On this length scale the electron energy decreases from 10 to ~1 MeV or less due to bremsstrahlung and another

______ *

Problems of Atomic Science and Technology. Ser.: Radiation Damage Physics and Radiation Materials Science, No. 4(87), 2005, available online at http://vant.kipt.kharkov.ua.

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inelastic energy-loss processes. Therefore, on one hand, the test ampoules can have reasonable sizes to mimic large scale experiments and, on the other hand, the irradiation conditions change strongly over a compareatively small length. Therefore we can use many specimens of submillimeter thickness within one ampoule to investigate the dependence of corrosion and radiation damage processes versus energy deposited by e– and γ fields. •

The electron beam can be used as an efficient heater of the irradiated target, to maintain a desired temperature if the electron beam power, target geometry and construction are chosen properly.

The Electron Irradiation Test Facility (Azhazha et al. 2005b) was designed to irradiate alloy specimens embedded into molten fluorides. The Facility and the target test bench design were chosen on the basis of the results of preparatory calculations and computer simulation. The e– and γ radiation fields, the depth profiles of deposited energy Edep in salt and samples, as well as the point defect generation in alloy have been calculated using radiation transport codes in the test bench geometry (Bakai et al. 2004, 2005a, 2007). A novel computer code RaT (Dyuldya et al. 2006) based on the CERN GEANT4 Monte Carlo toolkit (Agostinelli et al. 2003) was used for this simulation. The results of simulations allowed to optimize the target ampoule construction, as well as to evaluate quantitatively the dose dependencies of the specimen properties. The temperature fields in the target assembly were also calculated on the basis of the energy deposition simulation results (Bakai et al. 2005b). After ~10 MeV electron irradiation in EITF for 700 h, an extensive set of post-irradiated tests of alloys were carried out. Macroscopic mechanical properties are changing due to the influence of the molten fluoride salts and irradiation. These mechanical properties depend on the microstructure, precipitation in grains and grain boundaries, dislocation system and by compositional heterogeneities. All of them are changing due to irradiation. Along with the standard mechanical tests (Azhazha et al. 2005c), microindentations and nanoindentations of the materials in the vicinity of surface and in bulk were performed. Nanoindentation allows to measure the local value of Young modulus which is sensitive to local compositional and microstructural changes (Azhazha et al. 2005d). Compositional and phase changes in specimens are responsible for changes of mechanical properties and corrosion. The microstructure was characterized by high resolution TEM in a Philips CM30 microscope operating at 300 kV. For microchemical analysis, a 3D LEAP3000 (Imago Scientific Instruments) and 3DAP tomograph (APT, Cameca) were

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employed. The atomic absorption analysis was performed on a Pionicum SP-9 spectrophotometer. The IR spectroscopic investigations were made on a Specord 80M instrument, while X-ray microanalysis and visual control of the state of the sample surface was determined on a REM-101M scanning electron microscope-microanalyzer. Secondary ion mass spectrometry (SIMS) also was used to detect compositional changes in surface layers of tested samples (Azhazha et al., 2005e 2005f). The corrosion of alloys in the molten eutectic sodium fluoride-zirconium fluoride mixture without and under irradiation has been studied by cyclic voltammetry, X-ray analysis, scanning electron microscopy and metallography. Impact of small addition of La in the molten salt on the alloy corrosion was investigated, too. The corrosion rates of the samples exposed in salt at 660°C for different times and deposited energy doses were measured afterward in fresh molten salt by the voltammeteric method (Azahzha et al. 2005d, Bakai et al. 2005c). After irradiation by 10 MeV electrons a small amount of nuclear reactions happens, leading to formation of γ-active nuclei. Therefore reveal products of alloy corrosion can be revealed in molten salt by means of Roentgen spectroscopy. In spite of the fact that this method does not produce a proper quantitative compositional analysis it does provide both semiquantitative and qualitative information (Bakai et al. 2005d). For analysis of the experimental data theoretical models have to be developed to describe corrosion kinetics and phase transformation kinetics (Bakai et al. 2005e, 2005f). 3. Materials and Electron Irradiation Test Facility 3.1. TESTED MATERIALS

Purified ZrF4(51 mol%)–NaF(49 mol%) mixture which is molten at 660°C was used in experiments. To investigate impact of the lanthanide impurities on the alloy corrosion, La additions were used in one of them. Ni–Mo alloys showed a satisfactory corrosion resistance in contact with molten fluoride salts and ZrF4–LiF–BeF at temperatures 600 ÷ 800°C (DeVan 1969, Haubereich and Engel 1970, Novikov et al., 1990). It was revealed that comparatively small compositional changes of alloys can lead to considerable decrease or increase of the corrosion resistance. For example alloying with a few percent of Nb improves the corrosion resistance of the alloy. Taking these results as a starting point and background, two compositions the Hastelloy type (alloy A and alloy B, see Table 1) were designed from components of high purity (Azhazha et al. 2005g).

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EFFECT OF MOLTEN SALT AND IRRADIATION

Purification of the components allows to exclude impact of uncontrolled impurities. Composition of prototype alloys of Hastelloy family investigated earlier are also presented in Table 1. The alloys were homogenizied at 1100°C for 1 h and aged at 675°C for 50 h. One can see that alloy B differs from alloy A only by additions of Nb (0.5wt%) and Y (0.05wt%). Comparatively small amounts of these additions allows to judge the sensitivity of the alloy properties to dopants. TABLE 1. Compositions (wt%) of Hastelloy type Ni–Mo alloys for GenIV MSR applications.

Ni Mo Cr Fe Al Mn Nb Ti Si Y C 1

ORNL Hastelloy N standard modified

ІАЕ HN80MTY

SKODA MONICR

KIPT fabricated alloys alloy A alloy B

base 15 ÷ 17 6÷8 4÷6 — 0.5 — — 0.05 — 0.05

base 11 ÷ 12 5÷7 1.5 0.8 ÷ 1.2 0.5 — 0.5÷1 0.1 — 0.04

base 15.8 6.85 2.27 0.02 0.037 <0.01 0.026 0.13 — 0.014

78.15 11.70 6.70 1.50 0.83 0.50 — 0.47 0.15 — —

base 12 7 0.5 — 0.2 1 ÷ 21 ≤21 0.1 — 0.05

78.10 11.70 6.20 1.50 0.83 0.50 0.50 0.47 0.15 0.05 —

different Nb-to-Ti ratios are considered having the total (Nb + Ti) share less then 2 wt%.

3.2. ELECTRON IRRADIATION TEST FACILITY.

EITF has been constructed and built at Linac-10 electron linear accelerator in NSC-KIPT (see Figure 1). Schematically construction of the ampoule containing specimens in fluoride salt is shown in Figure 2. A container assembly consists of 16 ampoules made of the C–C composite. Ampoules hold specimens embedded in fluoride salt. The assembly was disposed in a water cooled chamber in Ar atmosphere. Electron beam of 10 MeV energy and up to 1 mA of average current (up to 1 kW of power) was in use. The electron beam was scanning over the chamber inlet window. The container assembly temperature was monitored with three thermocouples. The temperature is kept to be (660 ± 20)°C and controlled by the beam current.

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A. S. BAKAI

Figure 1. Construction of the EITF-KIPT container assembly and ampoules.

Figure 2. Top view of the filled ampoule loaded with Ni–Mo alloy specimens.

EFFECT OF MOLTEN SALT AND IRRADIATION

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4. Results 4.1. COMPUTER SIMULATIONS

4.1.1. EITF experiment simulation The specific requirement of the EITF experiment consisted in total utilization of the linac beam power and in obtainment of substantially different irradiation loads at successive MS–alloy interfaces concerning both energy deposition and point defects production rates. Various ampoule designs and arrangements of specimens were simulated to optimize the experimental setup (Bakai et al. 2004, 2005a) exactly reproduced by the RaT Monte Carlo code 3D computer models. Depth profiles of (e±,γ) fluxes, energy deposition and alloy dpa rates were built with 50 μm resolution over a beam penetration path. Preliminary optimization calculations were carried out for nominal EITF beam parameters, a monoenergetic 10 MeV, i = 1.25 μA/cm2 e-beam, and for ~2 · 1019 e–/cm2 projected fluence. It has been shown that at optimal setup the specific deposited energy at different MS-alloy interfaces varies from 5066 down to 64 eV/atom while the point defects concentration decreases from 2.12 10–3 to 4.42 10–6 dpa (Bakai et al. 2005a).

Figure 3. Depths dependencies of the deposited power (close markers) and the Ni–Mo alloy A dpa rate (open markers) in the vicinity of the loaded EITF target ampoule. The positions S1÷S6 enumerate the successive interfaces of alloy specimens with molten fluorides.

Monitored in course of the EITF experiment e-beam parameters differ from the nominal ones. The Linac-10 irradiation mode (Azhazha et al. 2005b) kept the average beam current i = 520 μA resulted in 1.625 μA/cm2 target front face current density; the total fluence over a 700 h long irradiation reached 2.56 1019 e–/cm2; the beam energy spectrum had a sharp peak at

544

A. S. BAKAI

9.6 MeV, and a tail spreading up to 12 MeV. These parameters were applied (Bakai et al. 2007) to get the revised EITF irradiation environment. Results of the EITF irradiation environment simulations using the monitored parameters of irradiation are shown in Figure 3. From front (S1) to the last (S6) MS–alloy interface the deposited energy decreases by 51 times (from 6192 down to 121 eV/atom), the absorbed dose falls from ≈10 GGy down to ≈0.2 GGy in both alloy and melt, and the primary damage decreases by 165 times from 3.31 · 10–3 dpa down to 2.01 10–5 dpa. The deposited energy and dpa profiles calculated for Ni–Mo alloys A and B agree within the limits of the Monte Carlo statistical error (≤0.5%). Therefore, the difference of alloys compositions (see Table 1) is too weak to affect the electron beam energy deposition and displacement of atoms. In the front specimens the irradiation effects are due to energy losses of primary beam electrons while in the last samples the contribution of seconddary bremsstrahlung γ-quanta dominates. The contribution of photonuclear reactions into the energy deposition and damage is found as marginal (<0.1% of totals) though the alloy and salt γ-activation mainly with the long-lived 57 Co and 95Zr radionuclides, respectively, was revealed by the γ-spectroscopy (Bakai et al. 2005d), and also observed in simulations. As a whole, the computer simulation results show that the strongly variable rates of major factor controlling the alloys corrosion, the energy deposition and primary damage, have been achieved in the EITF experiment. 4.1.2. Correlation with MSR irradiation environment conditions Obviously the irradiation environments of MS reactors are essentially differing from electron beam irradiation both in nature and intensity. In MSR Ni–Mo alloys are exposed to coupled (n,γ) irradiation, for alloys contacted with fissile MS fuel the impact of fission fragments on surface layers is significant (McCoy 1978). An issue of big importance is the investigation of applicability of the EITF test data for prognoses of alloys behavior in MSR. To simulate reactor radiation fields the “effective medium” model with homogeneously mixed MS fuel and graphite moderator was used (Bakai et al. 2007). The fuel enrichment was systematically adjusted to make a system critical. It ensures the correct ratio of correlated (n,γ) fluxes as well as the appropriate fission density. The MSR radiations source strength scales with its power, and was normalized to the realistic estimate of a total neutron flux φ0 = 1015 cm–2 ·s–1 in a core. Calculations have shown that the neutron and photon energy spectra in MSR are sensitive to the “moderator-to-fuel” ratio C/MS used as a core design parameter. As C/MS increases, the successive transition from the

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unmoderated (mainly epithermal) neutronics toward that of highly moderated thermal MSRs is observed (see Figure 4). Similar results are obtained using the MCNP code simulations of MSR cores (Merle-Lucotte et al. 2007).

Figure 4. Energy-weighted neutron (a) and photon (b) spectra (normalized to the total neutron flux φ0 = 1015 cm–2 · s–1) of critical MSR cores with different moderator-to-fuel (C/MS) ratios.

Effect of the alloy on the (n,γ) fluxes was neglected. The calculations of the Ni–Mo alloys engineering responses were performed within the kerma approximation (Abdou et al. 1975) for energy deposition (nuclear heating), and the NRT standard (Norgett et al. 1975) damage energy model for radiation damage of alloys. It has been found that the energy dependencies of these parameters of KIPT fabricated alloys A and B in average differ not more then by 0.6%, and small doping by Nb and Y is insignificant in respect to alloys nuclear responses. The code also calculated optional contributions of fission fragments using the simulation of depth profiles of energy deposition and damage produced in the vicinity of the MS–alloy interface by heavy ions from MS fuel. The simulation results depicted in Figure 5 show that for alloy–fuel interfaces fission fragments make major contributions to both energy deposition and damage of thin (~1 μm) surface layers. For interfaces with non-fissile salts (coolants), gamma heating and neutron damage are predominant. Generally, the EITF related energy deposition rates correspond to MSR irradiation environments with neutron fluxes φn ~ 1014 ÷ 1015 cm–2 · s–1 (see Figure 5a). One can conclude that the EITF is capable to simulate the energy deposition in Ni–Mo alloys embedded into the MS media for wide range of irradiation conditions they would experience in MSRs. The damage rates at EITF amount only to 10–2 ÷ 10–4 of those of MSR environments (see Figure 5b), and are comparable only with MSR related gamma damage that is marginal (≤0.01%) as compared with the neutron induced dpa rates. The EITF experiments can reproduce the alloy damage rates for

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low (~1010÷12 cm–2 · s–1) neutron fluxes more relevant to MSR vessel and piping.

Figure 5. The rates of the energy deposition (a) and the point defects (NRT dpa) production (b) in the Ni-Mo alloy A calculated for 1015 cm–2 · s–1 neutron irradiation in MSRs with different moderator-to-fuel (C/MS) ratios in comparison with the EITF electron beam deposited energy and damage rates at different MS–alloy interfaces S1 ÷ S6 (see Figure 3).

4.1.3. Temperature field Equation of the heat balance taking into account the integral deposited energy (see Section 4.1.1) and the heat irradiation from the surface is a base for estimation of the temperature balance of irradiated target in total. To calculate temperature field within an ampoule, the deposited energy distribution and heat transport equations were deduced and solved (Bakai et al. 2005b). The temperature distribution is an important characteristic of the irradiated target. In our experiments the goal was to make this distribution as homogeneous as possible. The calculations performed show

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that the temperature variance within the ampoule core is really small (660 ± 20)°C. 4.2. ALLOYS: STRUCTURE, COMPOSITION, PRECIPITATES

4.2.1. Metallography Metallography and scanning electron microscopy (SEM) was used to reveal macroscopic structural changes of the tested specimens in molten salt under irradiation. Unirradiated specimens of alloy A show no noticeable changes of the surface layer while some compositional changes up to 2 μm depth were revealed (Azhazha et al. 2005e, Bakai et al. 2005c). As the measurement of the corrosion rate of this alloy show (see Sec. 4.4.) that after 700 h of exposure it becomes rather low (~2 × 10–4 mm/yr). Thus the alloy A possesses an excellent corrosion resistance in fluoride MS without irradiation. It is believed that the alloy has the same corrosion rate with irradiation. Metallography of irradiated specimens in salt are shown in Figure 6 and Figure 7 (Azhazha et al. 2005a, Wanderka et al. 2005). It is seen that the surface layer of specimens of the alloy A irradiated in the molten salt for 700 h at T = 660°C (Figure 6) are corroded. The depth of intercrystalline corrosion essentially depends on the deposited energy. The depth is 25 ÷30 μm at Edep = 6192 eV/atom and 5 ÷10 μm at Edep = 121 eV/atom.

Figure 6. Structure of the alloy A after 700 hr exposure in the molten salt under irradiation at T = 660°C with the deposited energy Edep = 6192 eV/atom (a) and Edep = 121 eV/atom (b). Corrosion and precipitates within boundaries are larger at Edep = 6192 eV/atom.

No considerable intercrystalline corrosion of specimens of alloy B is visible at high and smaller values of the deposited energy (Figure 3) in spite the corrosion rate at Edep = 6192 eV/atom is larger than that at Edep = 121 eV/atom by two orders of magnitude. One can see that the mode of corrosion and the irradiation dose dependence of the alloy A differs considerably from that of the alloy B.

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The morphology, microstructure and compositions of the macroscopic precipitates indicate inclination of alloy to phase decomposition. The macroscopic precipitation is developed in both alloys after 700 h irradiation in molten salt at T = 660°C. The irradiated in molten salt alloy B shows much larger grain boundary precipitation as compare to that of alloy A (see Figure 8). Not only density but also sizes of precipitates in grain boundaries of alloy A are considerably less than in alloy B. In alloy A precipitates have mainly Ni–Ti–Si composition. The grain boundaries of alloy B are decorated by precipitates of 2 ÷ 7 μm in size and many of them include high content of Y (up to 30 at %).

Figure 7. Structure of the alloy B after 700 hr exposure in the molten salt under irradiation at T = 660°C with the deposited energy Edep = 6192 eV/at (a) and Edep = 121 eV/at (b). No considerable corrosion is seen.

Figure 8. a) Irradiated in molten salt alloy A. Macroscopic precipitates in grains and grain boundaries are seen. b) Irradiated in molten salt alloy B. Much more than in alloy A precipitates in grain boundaries are seen.

Precipitate formation in the unirradiated alloy A is similar to that in irradiated specimens but the X-ray phase analysis performed on a DRONUM diffractometer (CuKα radiation) reveals stronger reflections belonged to precipitates in the irradiated specimens (Azhazha et al. 2005c).

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4.2.2. Micro- and nano-structure of the tested specimens Since the alloy B shows more attractive corrosion behavior, its microstructure and compositional heterogeneities were investigated first (Wanderka et al. 2007a). Microstructure of alloy B was investigated by means of TEM, TEM/SAED, TEM/EDX and by means of 3D atom probe tomography (APT). The data obtained allow to connect the results of mechanical and corrosion tests with the structural properties. Figure 9a shows a bright field TEM image obtained from the non-irradiated alloy B. The grain boundaries are decorated by up to 2 µm large precipitates, indicated in Figure 9a by arrows, with varying compositions of the main elements Ni, Ti, Mo and Y; e.g. Mo19Ni26Ti44, Mo31Ni51Cr10 and Mo16Ni19Y31 as measured by EDX analysis in the TEM. Besides the large precipitates, the small spherical precipitates are formed in grains of the non-irradiated alloy, as it is seen in dark field image, Figure 9b.

Figure 9. a) Bright field TEM image, large Ni-Ti-Mo-Y precipitates have formed at grain boundaries; b) Dark field TEM image of homogeneously distributed small precipitates which have been formed in the interior of the grains. The [001] zone-axis electron-diffraction pattern (inset) exhibits L12 –type superlattice reflections.

These last small precipitates with a bright contrast and a mean diameter of 22 nm are homogeneously distributed throughout in the matrix without any obvious correlation with extended microstructural elements such as defect clusters, dislocations, or grain boundaries. The corresponding selectedarea electron-diffraction (SAED) pattern of a region containing matrix with these small precipitates is shown in the inset of Figure 9b. The [001] zoneaxis electron diffraction pattern exhibits superlattice reflections, which are generated by the small precipitates embedded in the matrix. The superlattice reflections correspond to the ordered L12 (γ') structure. Figure 10 displays a 3D reconstruction of the analyzed volume. Just the Al atoms are shown for clarity. The precipitates have ordered structure of (Ni3Al)-type with composition Ni+Fe+Mn)0.753(Al+Ti+Mo+Nb+Cr+Si)0.247.

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The chemical composition of the γ'-precipitates is measured by APT with nanometer spatial resolution. The rectangular parallelepiped analyzed by atom probe tomography (APT) has the dimensions 63×62×268 nm3 and contains ~2 × 107 atoms. Two phases appear in the investigated volume: the matrix phase depleted in Al and an Al-rich precipitate phase. One can see that the content of Nb within the matrix and L12 precipitates is almost the same. Therefore similar precipitates are expected to be formed in the alloy A also.

Figure 10. a) Distribution of Al atoms in non-irradiated alloy B after homogenizing at 1100°C for 1 h and aging at 675°C for 50 h. Investigated by 3D APT volume is 63×62×268 nm3. Precipitates of L-12 structure in the matrix are clearly seen, b) Concentration profiles of the main components obtained by the proxigram method. Precipitates possess composition (Ni+Fe+Mn)0.753(Al+Ti+Mo+Nb+Cr+Si)0.247.

In Figure 11 TEM bright-field images and selected-area electrondiffraction (SAED) patterns with [001] zone axis exhibiting diffuse SRO diffraction spots for the alloy B after electron irradiation to Edep ≈ 6 keV/at (B131) and Edep ≈ 100 eV/at (B139) are shown. The most dramatic changes of the microstructure appears at Edep ≈ 6 keV/at. The spherical L12 precipitates disappeared completely. This result is confirmed also by APT. From the other hand, the precipitates of Ni4Mo type were coarse graining in the course of the irradiation. At Edep ≈ 6 keV/at their mean size is ≈10 nm. For the specimen B139 deposited energy was much smaller, Edep ≈ 100 eV/atom. As it is seen in Figure 11, both γ ′ (L12) and smaller γ ′′precipitates of irregular shape (<1 1/2 0>fcc) exist in this specimen. One

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can conclude that the microstructure of the alloy B is rather sensitive to the electron irradiation of doses in the range Edep = 104 ÷ 102 eV/atom. In spite the diffuse diffraction spots are not observed at the <1 1/2 0>fcc positions in unirradiated specimens, thorough APT investigation of the composition heterogeneities of the unirradiated alloy B was performed looking for small clusters of Ni4Mo composition which have this type of the short range ordered (SRO). The result is shown in Figure 12.

Figure 11. TEM bright-field image and selected-area electron-diffraction patterns with [001] zone axis exhibiting diffuse SRO diffraction spots for the alloy B after electron irradiation to Edep ≈ 6 keV/at (B131) and ≈ 100 eV/at (B139). SAED pattern of the unirradiated alloy is shown in the upper left corner. The L12 ordered spherical γ ′ -precipitates and smaller γ ′′precipitates of irregular shape and the <1 1/2 0>fcc reflections are seen at Edep ≈ 100 eV/at.

Figure 12. APT image of the Mo enriched (with Mo content larger than 16 at%) clusters in volume 6×6×29 nm3.

It occurs that Ni–Mo clusters of Ni4Mo SRO (<1 1/2 0>fcc) exist in unirradiated alloy B. Composition of the clusters is given in Table 2. They have irregular shapes and diameters of 1÷3 nm. It is worth to note that Ni– Mo ratio within the clusters is 80Ni – 20Mo. Thus really Ni4Mo SRO

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cluster formation started in the initial alloy. Growth of these clusters along with dissolution of γ' precipitates occurs under irradiation. TABLE 2. Composition of clusters, (at. %). Ni

Mo

Al

Cr

Ti

67.18

17.6

2.37

8.4

0.72

4.3. MECHANICAL PROPERTIES

Mechanical properties of the tested alloys are tightly connected with their structural peculiarities. Impact of irradiation on the mechanical properties is an issue of a special interest because it reflects changes of microscopic and macroscopic structural elements of the alloys. Results of the mechanical tests of initial and exposed in molten salt without and under irradiation specimens of the alloys A and B are given in Table 3 (Azhazha et al. 2005c). One can see that the initial mechanical properties of both alloys are similar. The yield strength, σ0.2, and ultimate tensile stress, σB, are depending on the deposited energy, Edep. TABLE 3. Mechanical properties of irradiated and unirradiated alloys. Sample condition Edep (eV/atom) 6192 (sample #1) 121 (sample #3) 6192 (sample #2) 121 (sample #4) Initial non-irradiated Post-corrosion test (660°С, 700 h, non-irradiated) Edep (eV/atom) 6192 (sample #5) 121 (sample #7) 6192 (sample #6) 121 (sample #8) Initial non-irradiated

Ttest (°С) 20 660 20 600 20 660 20 660 20 600

σB(MPa)

σB (MPa)

δ %

626 848 276 362 900 397 1070 510

Alloy A 333 576 213 255 395 288 875 440

35 54 15 15 63 14 43 9

876 851 293 350 845 443

Alloy B 412 415 214 318 397 310

50 48 15 15 53 19

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This dependence is clearly seen from comparison of parameters σ0.2 and σB of irradiated to 6192 and 121 eV/at and unirradiated but also exposed in molten salt alloy A for 700 h at 660°C. The alloy A becomes softer due to irradiation. The plasticity of the irradiated alloy (15%) is considerably larger than that of the unirradiated specimen (9%) but not different from its plasticity in the initial state (14%). It is clearly seen that σ0.2 is the most sensitive to the irradiation feature. For example, at T = 20°C. σ0.2 of the alloy A at Edep = 121 eV/at is 1.8 times larger than that at 6192 eV/at. Both these values are considerably less than that of unirradiated alloy (875 MPa). Dependence of σ0.2 on Edep is less pronounced at 660°C. Unlike to alloy A, for alloy B the σ0.2 dependence on Edep is small at T = 20°C and appears to be considerable at T = 660°C. (Unfortunately the corrosion tests of this alloy without e-irradiation were not completed and impact of e-irradiation can not be separated from the annealing effect). Since σ0.2 depends of the initial structure of the specimen and σBВ in addition depends on the structure evolution in the course of deformation, one can conclude that microstructure including composition heterogeneities, precipitates and inner stresses, is sensitive to the irradiation. Results presented in Sect. 4.2 show that the microstructure of the alloy B is considerably changing under irradiation. 4.4. CORROSION RATES

The corrosion rates of the alloys A and B after exposure in ZrF4-NaF molten salt for 700 h at T = 660°C without and under irradiation are presented in Table 4. TABLE 4. Corrosion rates of alloys A and B. Sample

Corrosion current density ic, mA/cm2

Corrosion rate Km, g/m2–h

Kk, mm/yr

Alloy A

0.00002

0.0002

0.0002

Alloy A

0.018

0.20

0.19

Alloy A Alloy B Alloy B

0.0074 0.02 0.0004

0.081 0.22 0.0044

0.080 0.22 0.0043

Conditions

700 h of isothermal soaking in a fluoride melt at 650°С without irradiation 700 h of isothermal soaking in a fluoride melt at 660°С under electron irradiation Edep = 6192 eV/atom ––«»–– Edep = 121 eV/atom ––«»–– Edep = 6192 eV/atom ––«»–– Edep = 121 eV/atom

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It is seen that without irradiation the corrosion rate saturates on a rather low level ~2 × 10-4 mm/yr. Due to irradiation the corrosion rates considerably increase. The corrosion rates of both alloys are ~0.2 mm/yr at Edep = 6192 eV/at. At much smaller deposited energy, Edep = 121 eV/at, the corrosion rate of alloy A is twice smaller but alloy B has the corrosion rate smaller in two orders in magnitude. It is seen that Nb and Y additions not only depress the intergranular corrosion but also make the corrosion rate sensitive to the deposited energy. Small addition of La (10.8 wt%) in salt considerably changes the corrosion rate and nanohardness of the alloy A (Bakai et al. 2005c). In this case the nanoindentations allows determine the width of corroded layer very well. Investigation of the surface layer of the alloy A after exposure in molten salt with La impurities for 500 h at T = 600°C shows that the corrosion damaged the sample on depth 20 ÷ 25 μm. The nanohardness of the corroded layer of 20nm in thickness is poor. 5. Discussion The data reviewed show that •

irradiation considerably impacts microstructure, mechanical properties and the corrosion rates of Ni-Mo alloys;

•

corrosion resistivity is sensitive to small compositional changes. The alloys A and B have minor composition difference but their corrosion resistivity without and under irradiation are considerably different. The corrosion rate of alloy B is more sensitive to the deposited energy Edep;

•

impurities of La in the ZrF4-NaF molten salt dramatically enhance the corrosion of the alloy A even without irradiation.

Evidently, systematic investigations of corrosion of Ni-Mo alloys of different compositions in molten salts containing fluorides of actinides, lanthanides and other impurities expected to appear in blanket of MSR is severely needed. Since the role of the irradiation and dopants is so essential, it has to be investigated in much more detail. Theoretical models of the corrosion kinetics under irradiation have to be more developed. Since all of them include phenomenological coefficients like diffusion coefficients and chemical reaction rates, extensive empirical data are needed to identify them. A correlation of the nanohardness and yield stress of the irradiated specimens is revealed in our measurements. Similar correlation of microhardness and yield stress of irradiated alloys was demonstrated by Busby et al. (2005). Despite that a poor data base on this correlation is stored, it

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is believed that the nanoindentations can be used for semi-quantitative estimations of the macroscopic mechanical properties of irradiated materials. Direct reactor tests of the alloys are desirable to make clear availability of the test devices like EITF. 6. Conclusions •

EITF is an efficient device for corrosion tests of GenIV candidate construction materials.

•

EITF electron beam irradiation is capable to reproduce realistic irradiation conditions of GenIV molten salts reactor concerning the major factor affecting the Ni–Mo alloys corrosion, the energy deposition in surface layers of alloys.

•

Ni–Mo alloys A and B have acceptable corrosion resistance in molten salt at 660°C. After electron irradiation in EITF for 700 h the voltamperic data provide estimation of the corrosion rate to be ~0.1 mm/year.

•

The corrosion mode and resistance is rather sensitive to Nb and Y and presumably to other dopants. The alloy doped with Nb (0.5%) and Y (0.05%) does not show considerable intercrystalline corrosion, but its corrosion rate is sensitive to the deposited energy dose.

•

Compositional optimization is needed to minimize the corrosion rate.

•

Precipitates located in grain boundaries contain a lot of Y and apparently work as getters mitigating the intercrystalline corrosion attack.

•

Structure, composition and SRO of nano-scale precipitates in Ni–Mo alloys is sensitive to dopants and irradiation. This fact is promising for R&D of advanced materials for GenIV reactors.

Acknowledgments Collaboration and fruitful discussion with V.G. Baryakhtar, I.M. Neklyudov, V.M Azhazha, V.F. Zelenskii, A.M. Dovbnya, V.A. Gurin, A.A. Omel’chuk, S.V. Dyuldya, B.M. Shirokov, N. Wanderka, F.A Garner, and S.J. Zinkle are acknowledged very much. This research was partially supported by Science & Technology Center in Ukraine (STCU) within the framework of Project #294.

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Bakai, A. S., Tanatarov, L. V. Gonchar, V. Y., and Sergiyeva, G. G., 2005b, Calculation of temperature fields in ampoules under the radiation treatment by electrons with energy 10 MeV, Problems of Atomic Science and Technology. Ser.: Radiat. Damage Phys. and Radiat. Mater. Sci. No. 4(87):32–39. Bakai, A. S., Volkov, S. V., Azhazha, V. M., Andriiko, A. A., et al., 2005c, Corrosion processes in Zirconium Fluoride – Sodium Fluoride melts. Study of shortcut cell C/(Zr4– NaF)EUT/Hastelloy, Problems of Atomic Science and Technology. Ser.: Radiat. Damage Phys. and Radiat. Mater. Sci. No. 4(87):62–66. Bakai, A. S., Dovbnya, A. N., Zykov, A. I., Lavrinenko, S. D., et al., 2005d, Nuclide control of structural materials tested in Electron Irradiation Test Facility, Problems of Atomic Science and Technology. Ser.: Radiat. Damage Phys. and Radiat. Mater. Sci. No. 4(87):97–99. Bakai, A. S., Borisenko, A. A., Russel, K., 2005e, Amorphization kinetics under electron irradiation, Problems of Atomic Science and Technology. Ser.: Radiat. Damage Phys. and Radiat. Mater. Sci. No. 4(87):108–113 Bakai A. S., Tanatarov L. V., 2005f, Corrosion kinetics of an alloy immerged into a liquid metal melt containing oxygen, Problems of Atomic Science and Technology. Ser.: Radiat. Damage Phys. and Radiat. Mater. Sci. No. 4(87):114-119 Bakai, A. S., Bratchenko, M. I., and Dyuldya, S.V., 2007, to be published. Busby, J. T., Hash, M. C., and Was, G. S. 2005, The relationship between hardness and yield stress in irradiated austentic and ferritic steels, J. Nucl. Mater. 336:267–278. DeVan, J. H., 1969, Effect of alloying additions on corrosion behaviour of nickel-molybdenum alloys in fused fluoride mixtures, Report ORNL-TM-2021, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830. Dyuldya, S. V., Bratchenko, M. I., and Skorobogatov, M.A., 2006, Novel methods and software tools of statistic modeling in physics of radiation phenomena and technologies, in: Proc. of 17th Int. Conf. on Physics of Radiation Phenomena and Radiation Material Science. (ICPRP2006), NSC KIPT, Alushta, Crimea, Ukraine, pp. 30–31, in Russian. Haubereich, P. N., and Engel, J. R., 1970, Experience with molten salt reactor experiment, Nucl. Appl. Technol. 8:118–136. Ignatiev, V. V. Surenkov, A. I., Gnidoi, I. P., et al., 2006, Investigation of the corrosion resistance of nickel-based alloys in fluoride melts, Atomic Energy. 101:(4)730–738. McCoy, H. E., Jr., 1978, Status of materials development for molten salt reactors, Report ORNL/TM-5920, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830. Merle-Lucotte, E., Heuer, D., Allibert, M., Ghetta, V., Le Brun, C., 2007, Introduction to the Physics of Molten Salt Reactors, this book. Norgett, M. J., Robinson, M. T., and Torrens, I. M., 1975, A proposed method of calculating displacement dose rates, Nucl. Eng. and Design. 33:(1)50–54. Novikov, V. M., Ignatiev, V. V., Fedulov, V. I., Cherednikov, V. N., 1990, Molten Salt Nuclear Energy Systems — Perspective and Problems, Energoatomizdat, Moscow, in Russian. Wanderka, N., Bakai, A, Kropf, H., et al., 2005, unpublished. Wanderka, N., Bakai, A., Abromeit, C., Laheim, D., and Seidman, D.N., 2007a, Effects of 10 MeV electron irradiation at high temperature of a Ni–Mo-based Hastelloy, Ultramicroscopy. 107:786–790. Wanderka, N., 2007b, Atom probe and new results in material research, Int. Young Scientists School on Nuclear Physics and Energy, June 11–16, Alushta, Ukraine, unpublished.

SPECIFIC FEATURES OF PARTICULE/MATTER INTERACTION FOR ACCELERATOR-DRIVEN SUB-CRITICAL REACTORS S. LERAY* DAPNIA/SPhN, CEA/Saclay, 91191 Gif-sur-Yvette Cedex, France

Abstract: Accelerator-driven sub-critical reactors that are envisaged for the transmutation of nuclear waste use intense neutron fluxes produced through spallation reactions in a heavy metal target. The mechanism and specific characteristics of the high-energy nuclear reactions that occurs in ADS are described. The impact on target design, radioactivity production and material damage is discussed. Also, the general features of the simulation tools used to design ADS are presented and the precision with which each quantity important for applications can be predicted is given. Keywords: Spallation reactions; accelerator-driven systems

1. Introduction An accelerator-driven sub-critical reactor (ADS) is a system, in which an intense neutron flux produced through spallation reactions in a heavy metal target is used to drive aa sub-critical core. The spallation reactions are induced by a high-intensity proton beam of energy around 1 GeV. Minor actinides (MA) introduced in the reactor fuel can then be transmuted through capture and fission reactions. The goal being to finally fission the minor actinides, fast spectrum core, in which the ratio of fission to capture is more favorable, are generally preferred as transmuters. The transmutation of fission products, once considered, has been found to be not enough efficient and too expensive and is no longer foreseen. The main advantage of a sub-critical system compared to a critical one is the fact that fuels with a higher content of minor actinides can be used. Indeed, the introduction of MA into a reactor leads to the degrading of the safety parameters as the Doppler coefficient or the delayed neutron fraction. This limits the content

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e-mail: [email protected]

V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008

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of plutonium and minor actinides in a critical reactor but is less a problem if the system is sufficiently sub-critical. One of the objectives of GEN-IV nuclear systems is to minimize and manage their nuclear waste. A way to reduce nuclear waste is to transmute the minor actinides, after separation from the plutonium, which can be reused, and fission products that will go the storage. Two options are envisaged: either homogeneous re-cycling, in which a small amount of MA is transmuted in all the reactors of the park without degrading too much their efficiency and safety, or a double strata scheme, in which dedicated transmuters burn the waste arising from several commercial reactors. In the latter case, ADS are certainly promising since they could accommodate high contents of MA, as above explained, and therefore only a limited number of ADS would be needed. A possible deployment scenario has been for instance proposed by EDF, in which, in a park of fast reactors producing 60 GWe, ADS corresponding to 1.76 GWe would be needed to transmute the MA (Grouiller 2006 et al.). 2. Spallation Reactions 2.1. DEFINITION

The specificity of particle-matter interactions in ADS comes from the occurrence of spallation reactions in the neutron-producing target. A spallation reaction is defined as the interaction of a high-energy (above ~100 MeV) nucleon with a nucleus leading to the emission of light particles, mostly neutrons, and leaving a heavy residue. This process has actually been known for a long time, since it was first observed as particle cascades in cosmic ray interactions. Already with the first accelerators, reactions were found in which the target nucleus looses a lot of nucleons. Serber (Serber 1947) then proposed to describe spallation as a two step mechanism, description which is still used nowadays: the first stage can be viewed as a succession of individual nucleon-nucleon collisions which leads to the ejection of a few energetic particles, including some pions created in inelastic collisions, and to the deposition of part of the incident energy into the target nucleus. The excited remnant then decays by evaporating particles, mainly neutrons or by fission in the case of heavy nuclei. Some of the nucleons ejected during the spallation reactions have sufficient energy to induce further spallation reactions with neighbouring nuclei inside a thick target, leading to a multiplication of the emitted neutrons. Typically a 1 GeV proton on a lead target yields around 20 neutrons. It is therefore possible to produce with high-energy proton beams intense neutron fluxes that could serve for various applications.

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2.2. OTHER APPLICATIONS

Spallation neutron sources can be used for condensed matter and material structure studies. The fast neutrons produced by spallation are then moderated in a light or heavy water tank and reflectors are used to direct neutrons into beam tubes. The only continuous source is SINQ at PSI, Zurich, where the first Pb-Bi eutectic liquid target (Bauer 2001 et al.) has recently been successfully tested. New pulsed sources are under construction as SNS in USA, or JSNS in Japan. The European ESS project is still under discussion. Fundamental research in nuclear physics and astrophysics is demanding a new generation of high intensity radioactive ion beams (EURISOL project), with an emphasis on neutron-rich isotopes. High intensity proton or deuteron accelerators are foreseen to produce neutrons, directly or through a light element converter, that could induce fission reactions in an uranium block and then, after extraction and separation, provide neutron-rich fission fragment beams. Cosmic rays are composed mostly of protons and alpha particles with an energy spectrum presenting a maximum near 1 GeV/A. Cosmic rays bombardment of spacecrafts is responsible for severe problems of radiation damages, especially in computer electronics. Spallation reactions are also important to explain part of the elemental abundance distribution in galactic cosmic rays. 2.3. SPALLATION REACTION MODELLING

Simulation codes used for spallation source design generally consist of the coupling of a high-energy transport code, which handles the transport and interactions of the incoming proton and all the produced particles down to 20 (sometimes 150) MeV, and a low-energy neutron transport code utilizing evaluated nuclear-data files below. In the high-energy transport codes, the elementary cross-sections and characteristics of all produced particles are calculated by nuclear-physics models. To provide reliable predictions for quantities related to spallation reaction it is therefore necessary to have models describing correctly all the features of the reactions, validated on appropriate experimental data. During the last years, new high-quality experimental data have been collected, leading to a better understanding of the spallation reaction mechanism and allowing the testing of the currently used high-energy models (HINDAS 2005). This has resulted in the development of more reliable spallation models, as the INCL4 (Boudard et al. 2002) - ABLA (Junghans 1998 et al.) combination of intranuclear cascade and de-excitation models in Europe or CEM2k in USA (Mashnik

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et al. 2005), both now available in MCNPX (Hendricks et al. 2003). In this paper, we will try to give, when possible, the degree of reliability that can be expected from the model predictions. 3. Specificities of Spallation Reactions 3.1. NEUTRON PRODUCTION

In an elementary spallation reaction, high energy nucleons are ejected during the cascade stage. Their number is rather small, around 2 neutrons and 2 protons above 20 MeV per reaction in the case of Pb at 1 GeV, but they carry out the largest part (about 90%) of the energy and induce secondary reactions in a thick target. During the de-excitation of the excited remnant nucleus low energy (below 20 MeV) particles, mostly neutrons (around 15 in Pb) are emitted. The energy spectrum of neutrons produced in an elementary p+Pb reaction at 1 GeV is shown in Figure 1a. In an ADS, the high energy tail of the neutron spectrum can reach the fuel so it is important that simulations are done globally and do not assume that the spallation target can be decoupled from the rest of the ADS. Also, for radioprotection, these high energy neutrons should be properly taken into account. Although very few, the produced π0 should be considered since they decay into very high energy (70 MeV) photons. Because the reaction cross-section of a high energy proton is large (1.7 barns on Pb) a large fraction of the primary proton beam interacts in the first centimeters of the spallation target before slowing down. This is illustrated in Figure 1b (full line), which shows the respective numbers of reactions per incident proton as a function of the energy of the reaction for the different types of particles involved in the interaction of a proton with a thick target. The large number of low energy interactions is due to secondary particles and is dominated by neutron reactions because low energy protons have a large probability to stop without interacting. It can be noticed that pions also induce secondary reactions. All this explains that the maximum of the energy deposition in a spallation target is located close to the entrance of the beam, as it can be seen in Figure 2 in the case of the MEGAPIE target. A spallation target should be optimized to produce the maximum number of neutrons. Heavy elements, which lead to large numbers of evaporated neutrons, are chosen. Actually, there is very little difference between the different possible materials in terms of neutron production, as illustrated in Figure 3 from Letourneau et al. (2000), that shows the average neutron multiplicity per incident proton in different materials as long as the target thickness is sufficient to reach the plateau observed in Figure 3a.

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Figure 1. a) Energy spectra of the neutrons produced in the INC and de-excitation stages in the p+Pb reaction, b) Respective and total numbers of reactions per incident proton due to the interactions of the different particles involved in a thick Pb target bombarded by a 1 GeV beam.

Figure 2. Calculated local energy deposition along the beam axis in the MEGAPIE target. From Kirchner et al. (2003).

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In fact, the choice of the target material is mostly dictated by technological reasons. Generally, liquid targets (Hg, Pb, Pb-Bi eutectic) are preferred to solid ones because the necessity of a cooling system reduces the effective density of (Bauer 2004 et al.). It can be seen that solid W or Ta cooled by D2O produces significantly less neutrons than a liquid Hg target. Recently, the replacing of the standard SINQ target, made of Pb rods inside zircalloy cladding, by the liquid Pb-Bi eutectic MEGAPIE target has been found to increase the flux available to the neutron users at PSI by a factor 1.7 (Zanini et al. 2007). In addition, the liquid metal is its own coolant, the target does not suffer from radiation damage and volatile spallation products can be removed during operation. However, liquid targets have important constraints as the necessity of a container with, in particular, an entrance window for the beam, which is subjected to important radiation damage. Also the corrosion by the liquid metal of the structural materials is a matter of concern, as well as the highly activated circulating fluid. Solid targets (Ta, W, depleted U) have the advantage of simplicity (no container, spallation products confined) but have a limited power evacuation and are subjected to radiation damage. As regards the optimum beam energy, the relevant variable is the number of produced neutron per GeV in the beam. Several measurements (Letourneau et al. 2000) have shown that this number increases with increasing up to 1-2 GeV above which it tends to decrease. This is due to the opening of many meson production channels that carry out part of the proton energy without creating neutrons. Most of the available spallation models generally very well predicts total number of produced neutrons as well as their energy and angular distributions. 3.2. LIGHT CHARGED PARTICLE PRODUCTION

Reliable estimations of light charged particle production are important for spallation neutron sources: for instance, helium is a concern in structural materials, in particular the window separating the accelerator vacuum, because it can lead to swelling and embrittlement; tritium can escape from the liquid target and cause problems of radioprotection. The production rate of light charged particles increases with increasing incident energy up to a plateau around 1 GeV as can be seen in Figure 4 from Filgeset al. (2001). This means that production rates are much larger than in a conventional reactor. For instance, 2200 appm He per year are expected in the proton beam window of JSNS (Watanabe 1998) to be compared for instance to 250 expected in the first wall of a fusion reactor. DPAs due to the recoiling nuclei are also important (21 DPA/year are found in the same paper).

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Figure 3. Average neutron multiplicity per incident proton for different materials and different beam energies as a function of the target thickness in cm (a) and in atoms/cm2 (b). From Letourneau (2000), c), Comparison of the neutron current produced in a liquid Hg target and in solid W or Ta targets cooled with D2O as a function of the target depth. From Bauer et al. (2004).

Up to now, light composite particle production was rather poorly predicted by the different models implemented into MCNPX (Rapp et al. 2006) or by FLUKA (Pienkowski et al. 2006). Generally, in the models only emission during the evaporation stage is considered. In light composite particle energy spectra two different components are generally observed: a low-energy isotropic one coming from the evaporation stage, and a high energy tail more forward peaked. An attempt to separate the two components has been done recently by Herbach et al. (2006).

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Figure 4. Helium production cross-sections in Fe and Ta as a function of bombarding energy measured and calculated with different models. From Filges et al. (2001).

Figure 5 is taken from Herbach et al. (2006), and shows the relative contribution of this high-energy component, called pre-equilibrium in this paper, relative to the total yields of the different light composite particles, versus the target charge, at 1.2 GeV. It can be seen that, while 4He is produced predominantly by evaporation, for the other light composite particles the so-called pre-equilibrium contribution is far from being negligible, reaching 40% for tritium and even 60% for 3He on high Z targets. The level of this contribution in the production cross-sections of tritium and 3He shows the importance of being able to account for the high-energy tail with the models. This is the reason why the possibility to emit light composite particles through a mechanism of coalescence in phase space has been introduced in a new version of INCL4 (Boudard et al. 2004).

Figure 5. Contributions of the pre-equilibrium emission relative to the total yield of light composite particles. The data measured for reactions with 1.2 GeV protons are plotted as a function of the atomic target number ZT. From Herbach et al (2006).

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The ABLA model in his standard version only allows the evaporation of n, p and 4He. This means that tritium cannot be reliably predicted with INCL4-ABLA. This is a major deficiency as tritium appears, for instance, to be a major contributor to the radioactivity of the MEGAPIE target around 10 years after irradiation (Lemaire et al. 2007). In order to solve this problem, ABLA has been recently modified to include the evaporation of all types of light particles and emission of intermediate mass fragments (Ricciardi et al. 2007) with appropriate emission barriers. Figure 6 shows the result (solid line) obtained with the new versions INCL4.4 coupled to ABLA07, for total helium production cross-section in iron, important for material damage studies, as a function of proton incident energy. The comparison to the experimental data and to calculations with other models shows that obviously, the new models represent a progress compared to the previous one (dashed line) or to Bertini-Dresner (dotted line). Tritium production, not shown here, is now also reasonably predicted, the non-evaporative part coming from the INC stage being probably a little overestimated (Leray et al. 2007).

Figure 6. Helium production cross-sections in iron calculated with the new versions INCL4ABLA07 compared to data measured by different groups and to calculations with the standard INCL4-ABLA and Bertini-Dresner models. The dashed-dotted curve is the contribution from the cascade cluster emission. From Leray et al. (2007).

3.3. RESIDUE PRODUCTION

An important feature of spallation reaction is the large number of different residues that are produced. The example of proton on lead at 1 GeV, measured at GSI using the reverse kinematics technique (Wlazlo et al. 2000, Enqvist et al. 2001), is shown in Figure 7a.

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Figure 7. a) Nuclide production cross sections measured at GSI for the reaction p+Pb at 1 A GeV on a chart of the nuclides (Wlazlo et al. 2000, Enqvist et al. 2001), b) Number of nuclides per incident proton produced in a thick Pb target as a function of their mass for different bins in energy of the interaction.

The cross-sections of the thou-sand measured isotopes are represented on a nuclear chart, the empty squares indicating the stable nuclei. It can be seen that most of the residues are radioactive. Two main regions can be identified: one corresponding to evaporation residues beginning close to lead, which are mostly proton-rich nuclei, the other one due to fission residues,

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which represent about 10% of the total cross-section. In a thick target, a lot of residues are also produced by the secondary reactions. Actually, 3.5 isotopes per incident proton are created in average in a Pb target bombarded by a 1 GeV beam. The Figure 7b shows the mass distribution of the residues produced in the different interactions, divided into several bins in energy of the reaction, occurring in a thick lead target. It can be observed that the heaviest residues are created mostly in secondary low-energy (between 20 and 200 MeV) interactions while the fission products originate essentially from high-energy primary ones.

1017 Total Au195 Au193 Bi203 Bi205 Bi206 Bi207 Bi208 Bi210 H3 Os185 Hg194 Pb202 Pb203 Po210 T1204

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Figure 8. Main contributors to the activity calculated in the case of the liquid lead-bismuth MEGAPIE target. From Lemaire et al. (2007).

An important concern about spallation targets is the high activity generated by the numerous radioactive products. Figure 8 shows the total activity and the main contributors in the case of the MEGAPIE target (Lemaire et al. 2007). As expected from Figure 7b, the isotopes contributing the most to the activity are radioactive nuclides close to the target elements (Pb and Bi here). These nuclides are formed in spallation reactions in which only little excitation energy is deposited into the target nucleus, both because the energy of the interaction is low and because the highest crosssections are due to peripheral collisions. Generally, the models used to simulate the reaction are well established for small energy transfers. Therefore, their predictions for these isotopes are very similar and reliable in a wide range of incident energies as shown in Leray 2003 in comparisons to measurements (Gloris et al. 2001) of excitation functions.

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Figure 9. a) Comparison of INCL4-ABLA and Bertini-Dresner predictions with the production cross-sections as a function of the residue masses measured at GSI for the p+Pb system, b) Production rates of Xe isotopes from a thick Pb-Bi target, bombarded with 1.4 GeV protons, measured at Isolde, compared to different calculation. From Zanini et al. (2005).

Fission products are also important for radioprotection since some of them are gaseous elements, as Kr, Xe or I, which are released from the liquid metal during operation. Comparison of models to GSI elementary data have shown that Bertini-Dresner is not able to predict properly the fission residues (see Figure 9a) while INCL4-ABLA gives a very good agreement. The same is true in a thick target as shown in Figure 9b, in which the production rates of Xe from a Pb-Bi target irradiated by 1400

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MeV protons measured at Isolde are compared to different calculations. INCL4-ABLA, but also FLUKA (Fasso et al. 2005), gives a reasonable agreement with the data while Bertini-Dresner overestimates the production rates by nearly one order of magnitude. This indicates that Bertini-Dresner should never be used to calculate fission residues. Figure 9a also shows that a systematic misprediction of light evaporation residues is observed with INCL4-ABLA and other standard models (Boudard et al. 2002, Mashnik et al. 2005). This is even more pronounced in light systems as iron, not shown here, measured by Villagrasaet al. (2007) at GSI at different energies. This means that production rates of isotopes far from the target nucleus cannot be predicted with a good precision. This can be important for the assessment of some impurities in a given material. In addition, very light fragments with charge between 3 and 10, often called intermediate mass fragments, observed in a lot of experiments (Gloris et al. 2001, Herbach et al. 2006, Napolitani et al. 2004) are generally not predicted by the models (Leray 2003). Among those are for instance 7Be and 10Be, which are a concern for radioprotection. In the case of iron, a recent experiment (SPALADIN) has been performed at GSI, in which all the fragments and most of the particles emitted simultaneously in the reaction have been measured. This has allowed investigating in details the de-excitation mechanism and shown that intermediate mass fragments are probably formed in an asymmetric break-up mechanism (Le Gentil 2008), which is presently not taken into account in standard de-excitation models. The new version of ABLA is expected to solve this deficiency. 4. Conclusions Spallation reaction in ADS targets lead to specific radioprotection and material damage problems. The predicting capabilities of nuclear models implemented into widely used high-energy transport codes can now be assessed, thanks to their validation on elementary data. It can be concluded that neutron production, heavy spallation residues and total activity of the spallation target can be calculated reliably by most of the models. As regards volatile fission element production, INCL4-ABLA or FLUKA should be preferred to the default option of MCNPX. For gas production and intermediate mass fragments, none of the available models are satisfying. Recent developments not yet implemented into transport codes are expected to cure these deficiencies.

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References Bauer, G. S., Salvatores, M., and Heusener, G., 2001, MEGAPIE, A 1 MW pilot experiment for a liquid metal spallation target, J. Nucl. Mat. 296:17-35. Bauer, G. S., et al., 2004, The ESS Project Volume III Update, Technical Report, Chapter 4: target system, pp. 4-40, http://neutron.neutron-eu.net/n_ess/n_ess_documentation. Boudard, A., Cugnon, J., Leray, S., and Volant, C., 2002, Intranuclear cascade model for a comprehensive description of spallation reaction data, Phys. Rev. C 66:044615 (28 p). Enqvist, T. et al., 2001, Isotopic yields and kinetic energies of primary residues in 1 A GeV 208Pb+p reactions, Nucl. Phys. A686:481-524 Fasso, A., Ferrari, A., Ranft, J., and Sala, P. R., 2005, FLUKA, a multi-particle transport code, CERN-2005-10, INFN/TC_05/11. Filges, D. et al., 2001, Spallation neutron production and the current intra-nuclear cascade and transport codes, Eur. Phys. J. A11:467-490. Gloris, M. et al., 2001, Proton-induced production of residual radionuclides in lead at intermediate energies, Nucl. Instr. and Meth. A463:593-633. Grouiller, J. P. et al., 2006, GEDEPEON Workshop “Accelerator Driven System”, Aix-enProvence, France, May 18-19, 2006, http://www.gedeon.prd.fr. Hendricks, J. S. et al., 2003, MCNPX 2.5.d, report LA-UR-03-5916. Herbach, C. M. et al., 2006, Charged-particle evaporation and pre-equilibrium emission in 1.2 GeV proton-induced spallation reactions, Nucl. Phys. A765:426-463. HINDAS final report, 2005, EU contract FIKW-CT-2000-00031, J. P. Meulders, A. Koning and S. Leray ed. and references therein Junghans, A. R. et al., 1998, Projectile-fragment yields as a probe for the collective enhancement in the nuclear level density, Nucl. Phys. A629:635-655. Kirchner, T. et al., 2003, MEGAPIE Target Design and Dimensioning, proceedings of the 4th MEGAPIE Technical Review Meeting, March 18-19, 2003, Paris, France, report FZKA 6876. Le Gentil, E., 2008, Coincidence measurement of residues and light particles in the reaction 56Fe+p at 1 GeV per nucleon with SPALADIN, Phys. Rev. Lett., in press. Lemaire, S. et al., 2007, Simulation of helium and residue production in the Megapie target, contribution to Intern. Conf. on Nucl. Data for Science and Techn. (ND2007), Nice, France, April 22-29, 2007. Leray, S., 2003, HINDAS High-Energy Programme: Main conclusions and implications for spallation neutron sources, contribution to the Int. TRAMU Conference, Darmstadt, Germany, Sept. 1-5, 2003, http://www-wnt.gsi.de/tramu/. Leray, S. et al., 2007, Recent improvements of spallation models for better predictions of helium and tritium production, Int. Conf. on Accelerator Applications, AccApp’07, Pocatello, USA, July 30-August 2, 2007. Letourneau, A. et al., 2000, Neutron production in bombardments of thin and thick W, Hg, Pb targets by 0.4, 0.8, 1.2, 1.8 and 2.5 GeV protons, Nucl. Instrum. Methods B 170:299322. Mashnik, S. G., Gudima, K. K., Sierk, A. J., and Prael, R. E., 2005, Improved Intranuclear Cascade Models for the Codes CEM2k and LAQGSM, AIP Conf. Proceedings 769, 1188. Napolitani, P. et al., 2004, High-resolution velocity measurements on fully identified light nuclides produced in 56Fe+hydrogen and 56Fe+titanium systems, Phys. Rev. C 70: 054607 (21p). Pienkowski, L., Gawlikowicz, W., and Hilscher, D., 2006, FLUKA simulations for NESSI experiements, Report EURISOL DS/Task5/TN-06-05

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Rapp, B. et al., 2006, Benchmarking of the modelling tools within the EURISOL-DS project, Proceedings of International Workshop on Shielding Aspects of Accelerators, Targets and Irradiation Facilities (SATIF-8), Pohang, South Korea, May 22-24, 2006. Ricciardi, M. V. et al., 2007, Intermediate-mass-fragment production in spallation reactions, contribution to Intern. Conf. on Nucl. Data for Science and Techn. (ND2007), Nice, France, April 22-29, 2007. Serber, R., 1947, Nuclear Reactions at High Energies, Phys. Rev. 72:1114-1115. Villagrasa-Canton, C. et al., 2007, Spallation residues in the reaction 56Fe+p at 0.3A,0.5A,0.75A,1.0A, and 1.5A GeV, Phys. Rev. C 75:044603 (26p). Watanabe, N., 1998, Material Issues for Spallation Target by GeV Proton Irradiation, Symposium on Nuclear Data, JAERI, Japan. Wlazlo, W. et al., 2000, Cross Sections of Spallation Residues Produced in 1A GeV 208Pb on Proton Reactions, Phys. Rev. Lett. 84: 5736-5739. Zanini, L. et al., 2005, Volatile Elements Production Rates in a 1.4-GeV Proton-Irradiated Molten Lead-Bismuth Target, AIP Conference Proceedings 769, 1525. Zanini, L. et al., 2007, Int. Conf. on Accelerator Applications, AccApp’07, Pocatello, USA, July 30-August 2, 2007.

OPERATION OF HIGH POWER LIQUID METAL SPALLATION TARGETS: A CHALLENGE FOR STRUCTURAL MATERIALS J. HENRY * CEA Saclay, DEN/DMN/SRMA, 91191 Gif-sur-Yvette Cedex, France T. AUGER CNRS, CECM-UPR 2801, 94407 Vitry-sur-Seine Cedex, France Y. DAI Spallation Source Division, Paul Scherrer Institut, CH-5232 Villingen, Switzerland

Abstract: High power liquid metal spallation targets produce intense fluxes of neutrons which can be used for neutron research or to drive subcritical nuclear reactors (the so-called ADS: Accelerator Driven Systems) dedicated to the transmutation of radioactive wastes. In the present paper, we discuss the materials issues associated with such targets, whose components, in particular the proton beam window are subjected to severe operating conditions. We focus primarily on one example, MEGAPIE, a 1MW liquid Pb-Bi target which was recently successfully irradiated for several months, a world first and a key experiment for future development of ADS. Keywords: Accelerator Driven Systems (ADS), Spallation neutron sources, MEGAPIE experiment, 9Cr martensitic steel, irradiation embrittlement, liquid metal corrosion and embrittlement

1. Introduction Nuclear waste management is key issue for the future of nuclear energy and for the acceptance of this energy source by the general public. As detailed in (Leray 2008, these proceedings), one possible option would be to separate the Minor Actinides (MA) from the plutonium and “burn” them in a small number of dedicated subcritical Accelerator Driven Systems (ADS). However, several technical challenges lie ahead, such as the design of innovative fuels

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Jean Henry, CEA Saclay, DEN/DMN/SRMA, 91191 Gif-sur-Yvette Cedex, France

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with high MA contents (Wallenius et al. 2006). Moreover, an ADS requires a high power proton accelerator with a reliability which by far exceeds that of current accelerators (Burgazzi and Pierini 2007). Lastly, high power spallation neutron sources, which provide the necessary neutrons to maintain the chain reaction within the subcritical core, will have to be developed. In the following, we will concentrate on the latter point and more specifically on the materials issues related to the design and operation of these neutron sources. The emphasis will be put on a recent, successful first-of-a-kind experiment: the design and irradiation phase of the MEGAPIE spallation target. 2. The MEGAPIE Experiment 2.1. OBJECTIVES AND GENERAL DESCRIPTION

MEGAPIE (MEGAwatt Pilot Experiment) is a joint initiative of several research institutions worldwide to design, build and operate at the SINQ target location (Paul Scherrer Institut) a liquid metal spallation target of about 1 MW beam power (Auger et al. 2002, Groeschel et al. 2004). It is a key experiment for future ADS development. Figure 1 shows a sketch of the target and lists some of its main characteristics. The proton beam impinges through the beam entrance window on the spallation material, the liquid Lead-Bismuth Eutectic (LBE), from the bottom of the target. A main electro-magnetic pump forces an upward flow of hot LBE inside a flow guide tube and the heat deposited by the beam (650 kW) is removed by a 12-pin heat exchanger in the upper part of the target. The cooled LBE then flows down the outer annulus. An additional cooling of the beam window is performed by a jet of cold LBE pumped by a second pump through a pipe down to the window. A central rod is inserted in the LBE flow path. It contains a 20 kW heater in order to avoid LBE freezing during accelerator down times, as well as a neutron detector. 2.2. CHOICE OF T91 AS BEAM WINDOW MATERIAL

Figure 2 shows the bottom part of the target liquid metal container and the hemispherical beam window, which is the most critical target component as it is in contact with flowing liquid lead bismuth and subjected to thermal and mechanical loads as well as irradiation damage induced by high energy protons and spallation neutrons.

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Figure 1. Sketch of the MEGAPIE target.

Figure 2. Beam window and lower part of the liquid metal container.

This component, 1.5 mm in thickness, is made of martensitic 9Cr1MoVNb steel (commercial denomination: T91) in the normalized and tempered metallurgical conditions. The reasons for this choice, among other possible candidate materials (austenitic stainless steels, Ni based alloys, refractory metals and alloys etc.) are the following. This steel has excellent thermomechanical properties, since it displays high strength and very good resistance to failure from thermal loads due to its low thermal expansion

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coefficient and high thermal conductivity. In this respect, T91 is superior to 316 austenitic stainless steel which was envisaged as a back-up material for the window. For a given energy deposit, thermal stresses are twice as high in 316 than in T91. In addition, it has been demonstrated (Séran et al. 1994) that T91 has an excellent behaviour under irradiation at temperatures above about 400°C (moderate embrittlement, very low swelling, resistance to the high temperature helium embrittlement phenomenon). The main drawback of T91 is the large Ductile to Brittle Transition Temperature (DBTT) shift which occurs under spallation and fission neutron irradiation at low temperatures (below 400°C), which was evidenced by the results of the PIE performed on specimens irradiated in the SINQ target (Dai et al. 2005). The evaluation of the decrease in fracture toughness as a function of the proton fluence is a key element in the assessment of the window possible operating time, as discussed in the following. 2.3. IN-SERVICE CONDITIONS

The temperature distribution at the beam window was calculated taking into account the proton beam intensity profile, the total current and the main and by-pass LBE flows as described in (Dai et al. 2006). The maximum temperature in the beam window under normal conditions, with a 1.4 mA beam current, is 375°C at the outer surface and about 20°C lower at the inner surface cooled by the LBE. In the absence of beam, the target temperature is maintained at all times above 230°C. The mechanical load distribution, mostly due to thermal stresses, was calculated. Stresses are low compared to the yield point: the maximum value of the Von Mises stress is about 50 MPa, whereas the yield stress of unirradiated T91 at 300°C is about 400 MPa. In addition, frequent beam interrupts (so-called “beam-trips”) occur with a typical frequency of 1500 beam-trips per month and each beam-trip results in a load drop to about 20 MPa during a few seconds. The displacement damage and helium production due to the irradiation by high energy protons and neutrons were calculated using the FLUKA and MCNPX codes (Daiet al. 2006). The maximum dose in the window, expressed as a function of the proton charge (in Ampere hour, Ah), was calculated to be about 2.4 dpa/Ah corresponding to an accumulated helium content of 260 appmHe/Ah. 2.4. ASSESMENT OF FAILURE RISK

As mentioned above, the beam window is the most critical target component as it is subjected to multiple causes of damage acting synergistically:

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i) irradiation embrittlement in a spallation environment ii) corrosion/erosion by flowing Pb-Bi as well as possible Liquid Metal Embrittlement phenolmena iii) cyclic mechanical/thermal loadings due to the “beam trips”. In the following we will briefly discuss theses different damage causes in order to assess the possible service time of the window. 2.4.1. Irradiation embrittlement It is known that irradiation of tempered martensitic steels at temperatures below about 400°C induces a shift of the DBTT and a loss of fracture toughness. Until recently however, no data were available for T91 and other 9Cr martensitic steels irradiated in a spallation environment. Impact measurements performed on specimens irradiated in SINQ solid spallation targets (Dai et al. 2005, 2006) have shown that irradiation with high energy protons and neutrons results in a large DBTT shift, greater than that induced at similar temperatures and displacement damage levels by fission neutrons. Therefore, in order to evaluate the risk of sudden brittle fracture of the window, a Linear Elastic Fracture Mechanics (LEFM) analysis was carried out (Henryet and Lamagnère 2003).

Figure 3. a) semi-elliptical surface crack of depth a and length 2c, b) stress intensity factor for a large (c>>a) surface crack as a function of crack depth a in the window of thickness t = 1.5 mm (Henry and Lamagnère 2003).

The presence of a crack at the internal surface of the window was assumed and the stress intensity factor was calculated for mechanical loads corresponding to normal operating conditions. The stress intensity factor values as a function of crack depth are plotted in Figure 3. The stress intensity values were compared to fracture toughness measurements following T91 irradiation in a spallation spectrum. As shown in Figure 4, the calculated values for a large crack are one order of magnitude lower than the measured toughness at 250°C up to 8-9 dpa, which demonstrates that the presence of such a defect can not lead to sudden fracture under normal operating conditions and within this dose range.

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Figure 4. Fracture toughness versus dose for T91 irradiated in a spallation spectrum at LANSCE (Maloy et al. 2001) and SINQ (Jia and Dai 2006) tested either at Room temperature (RT) or irradiation temperature.

2.4.2. Liquid Metal corrosion and embrittlement The above analysis does not take into account possible effects due to the presence of flowing Pb-Bi in contact with the window. The oxygen solubility in LBE is low and varies, in the temperature range of interest for MEGAPIE, from 0.05 weight ppm at 250°C to 1.4 ppm at 400°C (Courouau and Robin 2004). At the start of operation, the oxygen content in the liquid metal is close to saturation, due to multiple sources of oxygen (oxygen content in the LBE ingots, release of oxygen from the targets metallic structures etc). The oxygen content is expected to slowly decrease at an unknown rate, as a result of the spallation induced production of hydrogen (44.6 10–3 mol H2 is produced per month in the LBE) and other reducing species, which might lead to dissolution of the steel during MEGAPIE operation. Therefore, tests in corrosion loops were carried out under flowing Pb-Bi and low oxygen concentrations (see Dai et al. 2006 for a review). The maximum dissolution rate was evaluated to be less than 100 μm for 9 months of operation. The corresponding reduction of the window thickness would not as such significantly weaken its mechanical resistance. However, dissolution of the oxide layer may lead to intergranular attack, as was for instance observed on T91 exposed for 4500 h to flowing Pb-Bi at 400°C and at an oxygen concentration lower than 10–3 wt ppm (Aiello et al. 2004). Intergranular attack provides defects which may propagate by cyclic loading in LBE leading to a strong reduction of the low cycle fatigue life (Vogt et al. 2007). However, fatigue crack formation and propagation is very unlikely under MEGAPIE normal operating conditions. The general

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trend is that the reduction in fatigue life and increase in fatigue crack growth in LBE compared to tests in air disappear at low stress/strain values (Veerlene et al. 2006, Vogt et al. 2007) which is the case for the MEGAPIE window. Moreover, if a small crack were to form, due to the very low stress intensity factor range values to which the crack would be submitted, its growth rate would be negligible (Auger et al. 2008). Therefore, there is a low probability that a crack of a few tenths of mm depth would form on the window inner surface. In addition, bending tests were carried out at 250°C in LBE on precracked T91 specimens irradiated in a spallation environment to a maximum dose of 9 dpa (Dai et al. 2006). The fracture toughness in LBE was evaluated to be about 40 MPam1/2 at 250°C for the specimen irradiated to 9 dpa. This value is significantly lower than the toughness measured in absence of LBE (about 100 MPam1/2). However the LEFM analysis described above still predicts that even if a deep surface crack were to form on the window inner surface, the risk of brittle fracture under normal operating conditions remains negligible up to a dose of about 9 dpa. 2.5. IRRADIATION PHASE AND PIE

The conclusion of the failure risk assessment was not proven wrong. Indeed, the MEGAPIE target was irradiated for more than 4 months in 2007, received an accumulated proton charge of 2.8 Ah (i.e. a maximum dose of 7 dpa in the window) and delivered a significantly increased neutron flux compared to previous solid SINQ targets. The outstanding success of this first-of-a-kind experiment is a big step forward for the development of ADS. The in-service performance of the structural materials will now be assessed by means of an extensive programme of post irradiation experiments (PIE) with the objective to get insight on: i) The changes of surface, microstructure, mechanical properties and chemical composition of the structural materials in the target induced by irradiation and LBE corrosion ii) The production, transportation and release of the spallation and corrosion products in the LBE, and the deposition of volatile elements in the whole target system. The PIE will include tests and examinations of the different parts of the target, in particular the beam window and the lower Liquid metal container made of T91. The PIE to be performed are: i) Non-destructive tests, for instance ultrasonic analysis of the beam window ii) Mechanical tests on specimens cut on selected locations, including hardness, tensile, bending, Small Punch Tests, some of which to be carried out in LBE iii) Surface and bulk microstructural/microchemical analyses, measurements of He and H

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contents in the specimens as well as chemical and radiochemical analyses of spallation and corrosion products in LBE. 3. A Specific Materials Issue for Pulsed Sources: Pitting Although they are not relevant for ADS applications which require continuous neutron sources, it should be mentioned that several projects of pulsed high power liquid metal spallation sources to be used for neutron research are underway worldwide. So far, one pulsed source has been built, the Spallation Neutron Source at ORNL (Mansur and Haines 2006) but operated only at low power. It has been found that cavitation erosion, or “pitting”, a damage cause specific to pulsed sources, has the potential to severely limit the service lifetime of such devices. This phenomenon is due to the intense energy input during a very short pulse, 1MW in less than 1 μs in the case of SNS, which induces an enormous rate of temperature rise. The liquid metal undergoes an instantaneous pressure increase as a result and a pressure wave propagates to the target walls. It is followed by a rarefaction which leads to the cavitation of the liquid metal in regions where the tensile strength of the liquid metal is exceeded. The subsequent collapse of the cavitation bubbles on target walls gives rise to high pressure microjets and produce localized erosion pits. Pitting damage was investigated using proton beam pulses or mechanical energy inputs in order to evaluate the decrease in fatigue lifetime (Futukawa et al. 2006) and the mean erosion depth as a function of the number of pulses (Haines et al. 2005). It was found that cavitation erosion may limit the service lifetime of high power pulsed sources to a few weeks only and therefore different ways of mitigating this phenomenon have been proposed such as covering the target walls with gas bubbles (Thomsen 2006). 4. Conclusion Structural materials issues are a prime concern for the development of high power spallation neutron sources. In particular, as discussed above, the proton beam window will be subjected to the combined embrittling effects of the irradiation by high energy protons and neutrons and the contact with a flowing liquid metal. The MEGAPIE experiment has demonstrated that a liquid lead-bismuth target with a T91 window can be successfully operated at 1MW power level for more than 4 months, which corresponds to a maximum irradiation dose of about 7 dpa in the window. The PIE to be performed now will provide a precious (at least unique in near future) opportunity for assessing the performance of structural materials and the

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distribution of spallation and corrosion products in a high power spallation target. In order to achieve significantly longer service life time than that of MEGAPIE, the present target configuration is not adequate. Possible improvements could include the control of oxygen content in order to avoid dissolution of the protective oxide layer and a higher window operating temperature in the range where irradiation embrittlement of 9Cr martensitic steels is less severe, i.e. above 400°C. The choice of a more advanced window material could be envisaged, such as nanostructured ferritic alloys which have the potential to exhibit less embrittlement than conventional ferritic/martensitic steels in fusion and spallation environments (Yamamoto et al. 2007). Finally, a “windowless” target concept has also been proposed (Cinotti et al. 2004), but its feasibility remains to be demonstrated.

References Aiello, A., Agostini, M., Benamati, G., Long, B., and Scaddozzo, G., 2004, Mechanical properties of martensitic steels after exposure to flowing liquid metals, J. Nucl. Mater. 335:217-221. Auger, T., Aphecetche, L., Cadiou, A., Dai, Y., Glasbrenner, H. , Gröschel, F., and Kirchner, T., 2002, MEGAPIE target design and LiSoR experiment - Status report, J. Phys. IV France 12:27-43. Auger, T., Hamouche, Z., Medina, L., and Gorse, D., 2008, Liquid Metal Embrittlement of T91 and 316L steels by heavy liquid metals: a fracture mechanics assessment, J. Nucl. Mater., accepted for publication. Burgazzi, L., and Pierini, P., 2007, Reliability studies of a high-power proton accelerator for accelerator-driven system applications for nuclear waste transmutation, Reliability Engineering & System Safety 92:449-463. Cinotti, L., Giraud, B., and Aït Abderrahim, H., 2004, The experimental accelerator driven system (XADS) designs in the EURATOM 5th framework programme, J. Nucl. Mater. 335:148-155. Courouau, J.-L., and Robin, J.-C., 2004, Chemistry control analysis of lead alloys systems to be used as nuclear coolant or spallation target, J. Nucl. Mater. 335:264-269. Dai, Y., and Marmy, P., 2005, Charpy impact tests on martensitic/ferritic steels after irradiation in SINQ target-3, J. Nucl. Mater. 343:247-252. Dai, Y., Henry, J., Auger, T., Vogt, J.-B., Almazouzi, A., Glasbrenner H., and Groeschel, F., 2006, Assessment of the lifetime of the beam window of MEGAPIE target liquid metal container, J. Nucl. Mater. 356:308-320. Futakawa, M., Naoe, T., Kogawa, H., Teshigawara M.,and Ikeda, Y., 2006, Effects of pitting damage on fatigue limit and lifetime in mercury target, J. Nucl. Mater. 356:168-177. Groeschel, F., Fazio, C., Knebel, J., Perret, C., Janett, A., Laffont, G., Cachon, L., Kirchner, T., Cadiou, A., Guertin, A., and Agostini, P., 2004, The MEGAPIE 1 MW target in support to ADS development: status of R&D and design, J. Nucl. Mater. 335:156-162.

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Haines, J. R., Riemer, B. W., Felde, D. K., Hunn, J. D., Pawel S. J., and Tsai, C. C., 2005, Summary of cavitation erosion investigations for the SNS mercury target, J. Nucl. Mater. 343:58-69. Henry, J., and Lamagnère, P., 2003, Risk of failure of the Pb-Bi container window: a tentative assessment, Proceedings of the MEGAPIE Technical review Meeting, FzK report FZKA 6876 p. 184 and CEA report DMN/SRMA/LA2M/NT/2003-2585. Jia, X., and Dai, Y., 2006, The change of fracture toughness of martensitic steels after irradiation in SINQ target-3, J. Nucl. Mater. 356:50-55. Leray, S., 2008, Specific features of particle/matter interaction for accelerator-driven subcritical reactors, these proceedings. Maloy, S. A., James, M. R., Willcutt, G., Sommer, W. F., Sokolov, M., Snead, L. L., Hamilton, M. L., and Garner, F., 2001, The mechanical properties of 316L/304L stainless steel, Alloy 718 and Mod 9Cr-1Mo after irradiation in a spallation environment, J. Nucl. Mater. 296:119-128. Mansur, L. K., and Haines, J. R., 2006, Status of the Spallation Neutron Source with focus on target material, J. Nucl. Mater. 356:1-15. Séran, J. L., Alamo, A., Maillard, A., Touron, H., Brachet, J. C., Dubuisson, P., and Rabouille, O., 1994, Pre- and post-irradiation mechanical properties of ferriticmartensitic steels for fusion applications: EM10 base metal and EM10/EM10 welds, J. Nucl. Mater. 215-215:588-593. Thomsen, K., Mitigation of cavitation effects by means of gas bubbles on a surface, 2006, J. Nucl. Mater. 356:321-324. Verleene, A., Vogt, J.-B., Serre, I., and Legris, A., 2006, Low cycle fatigue behaviour of T91 martensitic steel at 300 °C in air and in liquid lead bismuth eutectic, International Journal of Fatigue 28:843-851. Vogt, J.-B., Verleene, A., Serre, I., Balbaud-Célérier, F., Martinelli, L., and Terlain, A., 2007, Understanding the liquid metal assisted damage sources in the T91 martensitic steel for safer use of ADS, Eng. Fail. Anal. 14:1185-1193. Wallenius, J., Pillon, S., and Zaboudko, L., 2006, Fuels for accelerator-driven systems, Nucl. Inst. Meth. Phys. Res. A 562:625-629. Yamamoto, T., Odette, G. R., Miao, P., Hoelzer, D. T., Bentley, J., Hashimoto, N., Tanigawa, H., and Kurtz, R. J., 2007, The transport and fate of helium in nanostructured ferritic alloys at fusion relevant He/dpa ratios and dpa rates, J. Nucl. Mater. 367370:399-410.

SUBJECT INDEX

ab initio calculations ADS adsorption atomistic simulations austenitic stainless steels boundary element method

141 559, 575 393 187 285 263

capillarity carbon ceramics computational thermodynamics computer simulations correlation functions corrosion crystal plasticity

425 523 461 141, 169 245, 275 367 461, 537 275

density functional theory diffusion in metals dislocation dislocation dynamics dislocation loops dislocation theory dispersion hardening displacement cascades ductile-brittle transition

187 393 263 263, 275 227 275 311 227 203

elastic interaction electrochemical properties electrode mechanisms electron irradiation test facility electronic properties electronic structure theory energy issue equation of state Fe-Cr Fe-Cr system ferritic steels ferritic stainless steels ferritic/martensitic steels flow localization fracture toughness embrittlement free energy fusion power plant Generation IV GFR

263 523 523 537 327 153 1 367 169 153 203 285 227, 481 227 227 169 481 25 25

grain boundaries green house gas

393 1

hydrogen

523

interatomic interactions interfacial energy and anisotropy interfacial thermodynamics interfacial vector equilibrium intermetallic alloys iron single crystal toughness irradiated materials irradiation effect irradiation embrittlement

153 393 393 393 153 203 275 311 575

kinetic Monte Carlo kinetic rate theory kinetics

187 285 113

liquid matter liquid metal corrosion liquid metal embrittlement local approach local equilibrium

367 575 575 203 113

martensitic steel 9Cr 575 mechanical properties 327, 461, 537 Megapie experiment 575 metal/ceramic system 425 metal/metal system 425 microscopic structure 367 microstructural characterization 245 microstructure 537 molecular dynamics 187 molten metals 523 molten salts 523 molten salt composition 523 molten salt reactor 501, 537 molten sodium 461 Monte Carlo 169 multiscale computer simulation 245 multiscale modeling 187, 245 neutron irradiation embrittlement neutronic studies Ni-based alloys Ni-Si alloys numerical algorithms

245 501 537 285 275 585

586 ODS ferritic steels ODS ferritic steels oxidation of SiC oxygen pair distribution function para-equilibrium phase diagrams phase equilibria phase stability Phase transformation physico-chemical properties potentials potentiometric sensors precipitate precipitation precipitation stasis

SUBJECT INDEX 311 481 351 523 367 73, 113 73, 141 73 153 113 523 73 523 263 169, 285 113

R&D strategy radiation damage radiation effects radiation hardening reactor pressure vessel reference electrodes refractory alloys

25 285 187 227 245 523 461

safety analysis scenarios

501 1

segregation SFR SiC SiC structure SiC/SiC composites sodium solute segregation sources of errors spallation neutron sources spallation reactions statistical mechanics structural materials

393 25 327, 351 327 351 461 285 523 575 559 367 227, 481

thermodynamics thermodynamics properties thorium fuel cycle TMSR toughness-T master-curve transport coefficients transport properties

73 141 501 501 203 367 523

VHTR void swelling

25 227

wetting and diffusion wetting hysteresis wetting thermodynamics

425 425 425