Light Water Reactor Materials, volume I Fundamentals [1, 2 ed.] 9780894484612

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LIGHT WATER REACTOR MATERIALS Volume I: Fundamentals Donald R. Olander Arthur T. Motta

Arthur T. Motta received his B.S. and M.Sc. degrees from the Federal University of Rio de Janeiro, Brazil and a Ph.D. in Nuclear Engineering from the University of California, Berkeley working under Don Olander’s supervision. He joined Penn State in 1992 and is now Chair of the Nuclear Engineering Program. He has taught nuclear materials and has performed research in the broad field of materials degradation in the nuclear reactor environment for the last 25 years. He is a Fellow of the American Nuclear Society, and received the ANS Mishima Award for outstanding contributions in research and development work on nuclear fuel and materials as well as the ASTM Kroll Medal for impactful contributions to zirconium metallurgy. Donald R. Olander received his A.B. in chemistry and a B.S. in chemical engineering at Columbia University and his doctorate in chemical engineering at MIT. He joined the Chemical Engineering Department at the University of California, Berkeley in 1958. He moved to the then newly created Nuclear Engineering Department in 1961 where he taught and performed research in nuclear materials until 2008. He held the James Fife Chair in Engineering from 2003 to 2008. In 1976 he published the predecessor of this book, Fundamental Aspects of Nuclear Reactor Fuel Elements, which has been widely used in teaching nuclear materials. He has over 200 publications. He is a Fellow of the American Nuclear Society and a member of National Academy of Engineering, among many other honors. American Nuclear Society LaGrange Park, Illinois www.ans.org © 2017 American Nuclear Society All rights reserved. Printed in the United States of America ISBN-13: 978-0-894-48461-2 Figure 3.1, copyright 1986 by John Wiley & Sons, Inc. Figure 8.2, copyright 1995 by John Wiley & Sons, Inc. Figure 12.10, copyright 1981 by the American Physical Society The copyrights in the images included herein by Westinghouse Electric Company are owned by Westinghouse. Reprinted with permission. Library of Congress Cataloging-in-Publication Data Names: Motta, A. T. (Arthur T.) | Olander, Donald R. | Wirth, Brian, 1970Title: Light water reactor materials / Arthur T. Motta, Donald R. Olander. Description: La Grange Park, Illinois : American Nuclear Society, 2017- | Volume 1, chapter 15, by Brian Wirth. | Includes bibliographical references and index. Contents: volume 1. Fundamentals Identifiers: LCCN 2017013403 | ISBN 9780894484612 (hardcover v.1) Subjects: LCSH: Light water reactors–Design and construction–Problems, exercises, etc. | Light water reactors–Materials–Effect of radiation on–Problems, exercises, etc. | Nuclear reactors–Design and construction–Problems, exercises, etc. | Nuclear reactors–Materials–Effect of radiation on–Problems, exercises, etc. | Materials–Effect of radiation on–Problems, exercises, etc. Classification: LCC TK9203.L45 M68 2017 | DDC 621.48/33–dc23 LC record available at https://lccn.loc.gov/ 2017013403

1

Chapter Introduction 1.1 Introduction This book is intended as a textbook for the study of materials behavior in light water nuclear reactors. The target audience is upper-division undergraduate students and beginning graduate students in nuclear engineering. No specific knowledge of either nuclear engineering or materials science is assumed, since the basic concepts necessary to study materials behavior in the reactor environment are introduced in the book. Covered are the basic principles and practical problems related to materials used in a light water reactor, the primary coolant circuit, and the pressure vessel. The book is organized into two volumes: The first includes basic topics such as thermodynamics of solids, crystallography, dislocations, phase transformations, mechanical behavior, aqueous corrosion, and radiation effects; the second emphasizes applications, such as stress-corrosion cracking, dimensional instability, and materials degradation in the reactor environment.

1.2 What is a Light Water Reactor? Nuclear reactors have been built for numerous purposes: electric power generation, research and teaching, materials testing, production of

2 Light Water Reactor Materials

medical and industrial radioisotopes, weapons production, propulsion, and industrial and residential heating. Of these many variants, the power reactor is by far the most significant, in terms of number of units, nuclear fuel consumption, radioactive waste production, personnel employed, and total monetary investment. As such, the focus of this book is on the materials used in the most common of these power reactors, the light water reactors. Many power reactor types have been designed and developed, but most have had only brief commercial lives [1]. The light water reactor (LWR) has survived as the major provider (85%) of the nuclear-generated electricity in the world.1 Figure 1.1 illustrates the parts of what is called the nuclear steam supply, in this case for a pressurized water reactor. These range from the containment, which is the iconic symbol of a nuclear power plant, to the fuel pellet, the system’s smallest component. The nuclear steam supply includes all components needed to produce high-pressure steam. Figure 1.2 illustrates the most common types of light water reactors: the pressurized water reactor (PWR; Fig. 1.2a) and the boiling water reactor (BWR; Fig. 1.2b). In the PWR, steam is created in the steam generator, which is the link between the primary and secondary circuits, whereas in the BWR the Containment (40 m diam.)

300 Fuel pellets (8 mm diam. 1 cm height)

Reactor pressure vessel (5 m diam. 20 m high)

256 Fuel elements (1 cm diam. 4 m long)

Reactor core (4 m diam. 4 m high)

240 Fuel assemblies (20 × 20 cm, 4 m high)

FIGURE 1.1: Components of a PWR 1000 MWe nuclear steam supply. Typical numbers are given, although specific quantities may vary depending on type of reactor and fuel vendor. 1

Most of the remainder is generated by the heavy-water-moderated CANDU (Canadian DeuteriumUranium) reactor, which is not specifically covered in this book.

Introduction 3

steam is created in the core. The term balance-of-plant applies to the nonnuclear part of the reactor and covers the secondary circuit and turbine for the PWR and the turbine for the BWR. These components generate electric power from the steam’s energy. The reactor pressure vessel (RPV) contains the reactor core and the water coolant that receives the heat generated. Table 1.1 lists the characteristics of the RPVs of the two light water reactor types. The overall dimensions of the BWR vessel are significantly larger than those of the PWR, mainly because in the BWR, steam is generated directly in the core, and in the PWR, steam is generated in the steam generator, which is external to the core. Conversely, because the BWR vessel operates at less than half the pressure of the PWR, its pressure vessel wall is considerably thinner than that of a PWR. The reactor core contains fuel assemblies, support structures, and control components. Figure 1.3a shows a cutaway of the internal structure of a BWR, with a single fuel assembly in place, whereas Figure 1.3b shows the same view in an actual reactor during refueling. The blue-glow Cerenkov radiation generated by the radioactive decay of the fission products

Inside diameter (cm) Wall thickness (cm) Material Height (m) Water pressure (MPa) Outlet water temperature (K)

PWR BWR 450 600 25 15 ∗ Low-alloy steel Low-alloy steel∗ 13 24 15 7 598–602 575

TABLE 1.1: Typical characteristics of the reactor pressure vessel [2] ∗

SA508 is a typical alloy used for pressure vessel material. Its composition (%) is: Mn 0.5–1.0, Ni 0.5–1.0, Cr 0.25–0.45, Mo 0.55–0.7, Si 0.15–0.4, C < 0.27, V < 0.05, P(max) 0.025, S(max) 0.025

4 Light Water Reactor Materials (a)

Steamline

Walls made of concrete and steel 3–5 feet thick (1–1.5 meters) 4

Containment cooling system

3 Steam generator Reactor Control vessel rods

Turbine generator Conden- Heater ser Condensate pumps

Coolant loop

2 Core 1

Feed pumps Demineralizer

Reactor coolant pumps

Pressurizer Containment Emergency water structure supply systems

(b) Walls made of concrete and steel 3–5 feet thick (1–1.5 meters)

Containment cooling system

Steamline Reactor vessel

1, 2

Turbine generator Separators & dryers Feedwater

3

4

Heater Condenser

Core Feed pumps

Control rods Recirculation pumps Containment structure

Condensate pumps

Demineralizer

Emergency water supply systems

FIGURE 1.2: The principal components of two light water reactors: (a) a pressurized water reactor (PWR) and (b) a boiling water reactor (BWR).2 2

http://www.nrc.gov/reactors

Introduction 5

can be clearly seen in the fuel assembly being taken out of the core. In the BWR the core barrel surrounds the array of fuel assemblies. The waterfilled space between the core barrel and the RPV is approximately 50 cm wide. Its function is to reduce the energy of neutrons escaping the core in order to minimize radiation damage to the pressure-vessel wall. In addition to the RPV, Figure 1.2 shows other components inside the containment structure, principally piping leading to and from the reactor, the large feedwater pump, and the steam generator (in PWRs). In the PWR the water is heated in the reactor core (1) by fission, and is carried (2) to the steam generator (3) which creates steam in the secondary circuit which is then directed to the turbine (4) generating electricity. The primary circuit refers to the closed loop in which water circulates between the RPV and external components, such as the steam generator (PWR) or the turbine (BWR). In the PWR, the primary circuit includes the nuclear steam supply; in the BWR, the turbine and condenser are parts of both the primary circuit and the balance-of-plant. Fuel-handling device Pressure vessel

Duct Core barrel

Fuel assembly

Bottom support plate

FIGURE 1.3: (a) Schematic of a section of the inside of a BWR pressure vessel. (b) A picture of the BWR core during refueling. (Courtesy of AREVA.)

6 Light Water Reactor Materials

The low-alloy steel RPV contains the high-pressure water that flows up through the core and removes fission heat. The top hemisphere of the pressure vessel (called the upper head) is bolted to the cylindrical side wall to permit opening of the vessel for refueling and maintenance. There are three types of penetrations of the reactor pressure vessel: • inlet and outlet nozzles that connect the body of the pressure vessel to large pipes (~50 cm diameter). These pipes carry the hot coolant (liquid water for PWRs or saturated steam for BWRs) from the RPV to the other components in the primary circuit, and eventually return (relatively) cold water to the RPV. • control rods and safety rod drives, both of which contain neutron absorbers — the former are slowly removed during reactor operation to make up for the loss of U-235; the safety rods are used to shut down the reactor quickly in an accident situation. • instrumentation for measuring the in-core neutron flux and temperature. The balance-of-plant includes the steam turbine, the electric generator, and a condenser to convert the exhaust steam to liquid water. The balance-of-plant is similar to that connected to a fossil fuel unit using coal, natural gas, or oil for its steam supply. This book deals with the materials aspects of the components of the core and the primary circuit, but excludes the balance-of-plant and the concrete building that houses all components. The materials composing the nuclear steam supply are limited by the need to withstand a very hostile environment. The core, where nuclear energy is converted to thermal energy, is characterized by high stresses, high temperatures, extreme radiation fields, and aggressive chemistry. The materials used must satisfy stringent criteria regarding their mechanical, physical, chemical, and nuclear properties. This chapter describes the core features common to both LWR versions, then outlines the materials used in their construction and operation. The current challenges and economic forces that drive nuclear materials research and development are also described.

Introduction 7

1.3 Fuel Rods and Fuel Assemblies Energy from fission is converted into thermal energy in the fuel element or fuel rod, a collection of which is made into fuel assemblies. This energy is then transferred as heat to the water coolant that flows through the core past the fuel rods. Fuel rods are constructed (in fuel-fabrication factories, separate from the reactor site) by inserting a stack of cylindrical fuel pellets measuring approximately 1 cm diameter by 1 cm high (Fig. 1.4a) into a 4-m length of metal tubing called cladding (Fig. 1.4b). The fuel rods are then gathered into fuel assemblies (Fig. 1.4c). The cross-sectional (a)

(c)

(b)

Top end cap

(d)

Spacer grids

Spring

Plenum

Cladding

Nuclear fuel pellet Cladding Fuel rod Guide tube

Instrument tube

Fuel pellet

Fuel-cladding gap

Bottom end cap

FIGURE 1.4: The components of a fuel assembly: (a) the UO2 fuel pellet (Courtesy of AREVA); (b) fuel rods (Courtesy of AREVA); (c) schematic of a 15 × 15 PWR fuel assembly; (d) schematic cut out of a fuel rod.

8 Light Water Reactor Materials

schematic (Fig. 1.4d) shows the inside of the fuel rod. Note that a region on top of the fuel (the plenum) is left empty to accommodate released fission gases and fuel swelling/creep. A spring is placed on the pellet stack for mechanical stability, and caps are welded on the ends of the tube. The schematic PWR fuel assembly in Figure 1.4 contains fifteen fuel elements along each of its four sides and is referred to as a 15 × 15 assembly. The overall assembly shape is square, as is the pattern of the rods forming the bundle. The square dimension of the assembly is 20 to 25 cm, and its height is nearly 5 m. The cladding tube performs the following functions: • holds the fuel-pellet stack in a mechanically stable, coolable geometry; • retains fission products released from the fuel; and • prevents coolant water from contacting the UO2 pellets. The fuel pellets are made of high-purity uranium dioxide (UO2) enriched to 3 to 5% in U-235. A small fraction of the nuclear fuel used in the world’s reactors is mixed-oxide fuel (MOX) that is a blend of PuO2 in UO2 [(U,Pu)O2]. This makes it possible to utilize for power production the plutonium generated by irradiation of UO2 or recovered from nuclear weapons. Cladding is made of the zirconium alloy Zircaloy (Zr + Sn + some Fe,Cr,Ni), or more modern niobium-containing zirconium alloys, as discussed in more detail in Chapter 17.3 The choice of zirconium alloys is based mainly on the low thermal neutron-absorption cross section of zirconium. The mechanical, chemical, and physical properties are also optimized [4]. For the purposes of handling during transportation, loading/unloading, and storage, and for maintaining mechanical stability and proper spacing in the rapidly flowing coolant, fuel elements are packaged into the units known as fuel assemblies (Fig. 1.3c). The specific fuel assemblies used in PWR and BWR are shown in Figures 1.5a-d and 1.5e-f. 3

Chapters 16 through 29 are to be found in Light Water Reactor Materials, Volume II: Applications.

Introduction 9 (a)

(b)

ZIRLO® thimbles Reduced rod boy Alloy 718 top grid

(c)

Low-cobalt Westinghouse integral nozzle (WIN)

Optimized ZIRLO cladding

ZIRLOG grids

Advanced fuel rods for higher burnups and longer cycles: • Variable-pitch plenum spring • ZrB2 integral burnable absorber • Enriched annular axial blankets

Alloy 718 bottom grid Long solid-end plugs on the fuel rods

Thick thimble tubes to improve structural stability and resistance to incomplete rod insertion

(d)

Comprehensive debris mitigation: • Standard debris filter bottom nozzle (SDFBN) • Robust protective grid (RPG) • ZrO2 protective coating

FIGURE 1.5a-d: LWR Fuel Assemblies: (a) Schematic of a PWR fuel assembly, not to scale (Courtesy of Westinghouse Electric Company); (b) Full 17 × 17 PWR fuel assembly showing hold down springs, top nozzle, and guide tubes and grid spacers (Courtesy of Westinghouse Electric Company); (c) PWR control rod spider (Courtesy of AREVA); (d) Bottom nozzle showing control rod guide tubes and debris filter (Courtesy of AREVA).

The distance between the centers of adjacent fuel rods when loaded into the assembly is called the pitch. The pitch is kept as small as possible to minimize the size of the assembly, and hence of the reactor core and the RPV. The minimum pitch is dictated by the maximum attainable coolant flow rate through the core. In both cases, coolant water flows upward in the fuel assembly, but the ductless PWR version permits horizontal flow (or crossflow), which mixes coolant between adjacent assemblies.

10 Light Water Reactor Materials

Figure 1.5a shows a schematic PWR fuel assembly with the individual components, while the actual 17 × 17 assembly is shown in Figure 1.5b. The very large aspect ratio of the fuel assembly is clearly demonstrated in this figure. These fuel assemblies require control rods to shut down the fission reaction. Different control rod designs are used in the two reactors. The “spider”-type assembly, used in the PWR and shown in Figure 1.5c, consists of control rods that are filled with a neutron-absorbing material, such as silver-indium-cadmium alloys or gadolinium, and is inserted from the top onto the guide tubes. As the U-235 in the fuel is gradually consumed (or “burned”), the control rods may be withdrawn to maintain nuclear criticality. The alternative for controlling reactivity, which is used in most recent core designs, with greater core reactivity at the beginning of life, is to use dissolved boron in the reactor coolant or to coat the pellets with boron, which then acts as a “burnable poison.” The neutron absorption cross section of boron-10 is very high, and since the boron turns into non-absorbing Li as it absorbs neutrons, the neutron poison slowly decreases with time so as to keep reactivity constant. At the bottom of the fuel assembly is a steel lower end plate, or bottom nozzle (see Fig. 1.5d). This plate is penetrated by holes into which the control rod guide tubes are inserted. Above the debris filter, there are two sets of holes, each with a distinct function: The set of small holes receives the lower ends of the fuel rods; the set of larger holes serves as an entry for cooling water into the core region. The tops of the fuel rods are set into indentations in an upper end plate. The upper and lower end plates are connected by long rods firmly fixed with bolts on both. The BWR fuel assembly (Figs. 1.5e–f) is housed in a square box called a duct, whereas the fuel assemblies of PWRs (Figs. 1.5a–d) are open. Shown in Figure 1.5e is the square duct into which the BWR fuel bundle fits. The walls of the duct are ~3 mm-thick Zircaloy. By varying the

Introduction 11 (e)

(f)

2 1

5

3 4

15 8 14 16

6 7

13 9

11

10

12

FIGURE 1.5e-f: LWR fuel assemblies: (e) BWR fuel assembly showing fuel ducts; (f) BWR fuel assembly showing fuel pins (Courtesy of Westinghouse Electric Company).

diameter of the coolant entry orifices in the bottom end plate, the flow to each channel can be adjusted to compensate for the core radial power distribution so as to maintain a uniform steam quality (i.e., vapor fraction) at the exit of all fuel assemblies. In the BWR, the control rods take the form of cruciform metal control blades containing a neutron absorber (boron carbide). These blades fit between ducts and are moved through the bottom head of the pressure vessel, as will be shown below since the upper part of the BWR is occupied by the steam separators. For reactivity control, the burnable poison is gadolinia (Gd2O3), which is mixed with the uranium dioxide fuel pellets and which serves the same function as the dissolved boron in PWRs, as Gd is also a strong neutron absorber.

12 Light Water Reactor Materials

1.4 Fuel Rod Stabilization—Spacer Grids Cladding tubes are quite flexible, especially in 4-m lengths. If  held horizontally, a fuel rod would bend of its own weight. The stack of 1-cm-long fuel pellets inside the cladding does not provide rigidity to the fuel element. Bolting together the upper and lower end plates suffices only to hold the ends of the fuel rods. An additional design element is needed to prevent rod vibration (and possible damage to the rods) caused by the flow of high-velocity cooling water during reactor operation or during transportation to the reactor site. To stabilize the fuel rod bundle, spacer grids, a critical component of both the BWR and the PWR fuel assemblies, are placed at regular intervals along the assembly. (See Fig. 1.5a, and for more detail, 1.6a–1.6c). The principal function of this component is to hold the fuel rods and guide tubes in place. In addition, mixing vanes are added to improve cooling by increasing turbulence (Fig. 1.6b). The vertical spacing of the grids along the fuel bundle is chosen to minimize vibration of the rods produced by the flowing coolant. Typically, seven grid spacers are placed along the assembly. They are made either of Zircaloy or of the high-nickel alloy Inconel. The inserted PWR grid spacer is shown in Figure 1.6c. BWR grid spacers are similar to PWR designs but without the control-rod guide tubes. The spaces between the grid spacers are dubbed the grid spans. In both LWR types, the vertical sides of the square holes through which the fuel rods pass are formed into shapes resembling a leaf (spring clip), an “I” beam, or a dimple. These are the points of contact with the fuel rod, and the force that the contacts exert on the fuel rods must be set with great care. Too large a force causes cladding deformation, usually in the form of bowing of the rods. Too small a force allows the fuel rods to rattle about in their holes and can lead to failure by grid-to-rod fretting.

Introduction 13 (a)

Control rod guide tube I-Beam rod holder

(b)

Spot welds

(c)

Dimple rod holder Spring clip rod holder

FIGURE 1.6: (a) Grid spacers showing control rod guide tube; (b) Grid spacers showing mixing vanes (Courtesy of AREVA); (c) Grid spacer inserted into fuel assembly (Courtesy of AREVA).

1.5 Neutron-Absorber Devices 1.5.1 PWR control rods As mentioned above, in addition to fuel rods, LWR cores need control elements for regulation of the neutron population and burnable poisons to account for the decrease in nuclear fuel reactivity with time. In PWRs, movable control rods containing pellets of boron carbide, B4C (for scramming the reactor), or Ag-In-Cd (for fine tuning the neutron flux) fit into guide tubes that pass through and are welded to the spacer grid as shown in Figure 1.6c. Figure 1.5c shows the “spider” cluster of control rods passing through the spacer grids in a PWR. The guide tubes also hold gadolinia/ urania burnable poison rods, which are slowly withdrawn as fission proceeds. In addition, the guide tubes hold neutron sources for reactor startup.

14 Light Water Reactor Materials

1.5.2 BWR control blades Grid spacers for BWRs do not have control rods inside the fuel assembly. Instead, the burnable poison (boron nitride) is contained in pellets in small rods enclosed in a cruciform-shaped Zircaloy sheath. An example of a BWR control rod blade is shown in Figure 1.7; the neutron poison in this case is hafnium and boron carbide. These control blades are inserted upward between the ducts (Figs. 1.5e–f) and are moved up and down through penetrations in the bottom head of the RPV as needed to control nuclear reactivity.

CR 82M-1 Hafnium B4C powder

1.6 Coolant: H2O Besides the UO2 fuel, stainless steel reac- FIGURE 1.7: BWR control blade. tor internals, and the zirconium alloy (Courtesy of Westinghouse Electric Company.) cladding structures, the other material in the core and the primary circuit is ordinary water, the eponymous component of the LWR. Its high heat capacity and low viscosity make water a very efficient medium for collecting the heat emanating from the fuel rods and transporting it out of the RPV. The hydrogen in H2O permits the cooling function to be combined with the nuclear function of moderating (slowing down) neutrons. However, the neutron absorption cross section of hydrogen, while not large, is high enough to require enrichment of the fuel to 235U contents of 3% and up to 5%.4 4

As compared to the CANDU reactor, which can use natural uranium (0.7% uranium-235) because it uses heavy water, in which deuterium is much less absorbent of neutrons than hydrogen.

Introduction 15

Water has the advantage over other coolants in that, when turned into steam, it can directly drive a turbine-generator, just as in coal- or gas-fired plants. The major disadvantages of water in a nuclear system are its high vapor pressure and its chemical reactivity to the metals in the primary circuit. The efficiency of converting thermal energy to electrical energy increases with the temperature of the hot source, in this case the coolant leaving the core. However, a high exit temperature must be accompanied by a high system pressure (15 MPa for a PWR, 7 MPa for a BWR). The structures in the primary circuit must be sized to withstand the resulting stresses. In addition to purely mechanical constraints on material performance, these stresses can enhance certain corrosion processes (see Ch. 14 and 22). Corrosion of metal structures is a persistent problem in LWRs: Cladding corrosion (Ch. 22) can limit the length of time that the fuel can remain in the reactor; stress-assisted cracking/corrosion of stainless steel in BWRs (Ch. 25) has required expensive mitigation procedures or even complete replacement of primary circuit piping, and other corrosion mechanisms afflict the components of the steam generators of PWRs and necessitate plugging of leaking tubes or even complete replacement of the entire assembly unit. Coolant water also contains additives to help the metals in the primary circuit resist corrosion. In BWRs, hydrogen (as H2) is added to the water to remove oxidizing free radicals that are produced by radiolysis in the reactor. No solid additives are allowed because the water is vaporized to steam, and any solids in solution would concentrate in the liquid. In PWRs, solids in solution are added to the coolant for various purposes. Boric acid (H3BO4) is added for neutron reactivity control; LiOH is added to neutralize the acidity of the boric acid. Zinc is added to reduce radioactivity uptake in the primary-circuit piping (Ch. 21).

16 Light Water Reactor Materials

1.7 Water Reactors 1.7.1 PWR As the name suggests, the pressurized water reactor contains liquid water throughout its primary circuit.5 Figure 1.8 shows a complete PWR pressure vessel with all the inlets, outlets, and penetrations. The heated water exits the reactor pressure vessel at ~330oC, which is safely below the 345oC saturation temperature at the operating pressure of 15 MPa. After leaving the core the hot water passes through U-shaped tubes in the steam generator (Fig. 1.2) before returning to the RPV at ~290oC. The return water flows downward along the inner wall to the lower hemisphere of the RPV (called the lower head). From here, the flow is directed upward through the bottom end plate past the fuel rods and exits at the outlet plenum via a large (~50 cm diameter) pipe, where the primary circuit begins. Control rod drives (motors that move control rods up and down) are connected to pressure-tight penetrations in the upper head. The control rod drives are attached to the hub of the spoke-like control rod cluster mechanism shown in the lower-left fuel assembly cross section in the bottom of Figure 1.8. The partially filled circles at the ends of the spokes represent control rods that fit into the guide tubes of the PWR assembly in Figure 1.5d. Not all fuel assemblies contain control rods; only two of the four assemblies sketched in the bottom left of Figure 1.8 are provided with them. The other two assemblies contain only fuel rods. The core cross section on the right at the bottom of Figure 1.8 shows how the enrichments in the initial loading of fuel can be distributed to create a uniform power distribution. This is the realm of reactor physics and fuel management. 5

In some PWRs, significant localized boiling takes place in the upper core regions, which increases the void fraction. The result is a decrease in neutron moderation, as well as a consequent reduction in the thermal neutron flux and fission rate.

Introduction 17 Rod travel housing Instrumentation ports

Control rod drive mechanism Upper support plate

Thermal sleeve Lifting lug

Internals support ledge

Closure head assembly Hold-down spring

Core barrel

Control rod guide tube

Support column

Control rod drive shaft

Upper core plate

Inlet nozzle

Outlet nozzle Baffle radial support

Control rod cluster (withdrawn)

Baffle Core support columns Instrumentation thimble guides

Access port Reactor vessel

Radial support Lower core plate

Core support

Fuel assembly with rod cluster control

Enrichments Rod cluster control element Fuel rod Fuel assembly without

LOW

MED

HIGH

rod cluster control

FIGURE 1.8: (Top) Schematic of a pressurized water reactor; (bottom left) cross section of four adjacent fuel assemblies; (bottom right) cross section of a typical loading of fuel assemblies containing various levels of enrichment in the core.

18 Light Water Reactor Materials

At any one time there are three levels of burnup (and corresponding isotopic enrichment of the fuel) in the core; although in a given fuel assembly all fuel rods have the same enrichment. The enrichments correspond to fuel added in each of the three refueling outages. Every 18 to 24 months, one-third of the core is removed and replaced with fresh fuel. Fuel in its third cycle is the most highly burned and has the lowest neutron reactivity. Fresh fuel is the most reactive, so to flatten the radial power distribution in the core, it is placed mostly along the core periphery as shown in the bottom of Figure 1.8. The downside of this fuel management strategy is the enhanced embrittlement of the RPV wall by fast neutrons leaking from the outer portion of the core (see Ch. 26).

1.7.2 BWR The principal difference between the two types of LWRs is the conversion of liquid water to steam occurring in the core of a BWR instead of in an external steam generator, as used with PWRs. The reactor in a BWR plant is an integral part of the power cycle. As indicated in Figure 1.2, the steam leaving the reactor pressure vessel flows directly to the turbine-generator. The low-pressure exhaust steam from the turbine passes through a condenser to complete conversion to the liquid state. Following the condenser, the low-pressure liquid coolant is pumped through units that remove impurities and add corrosion inhibitors. Finally, the chemically prepared (or “polished”) coolant is repressurized by the main coolant pump to a system pressure of 7 MPa and fed into the RPV. Figure 1.9 shows a schematic of a boiling water reactor. In the BWR, the coolant enters the core as a subcooled liquid at 280oC. This feedwater is sucked into the lower head of the RPV by a jet pump (described below), then flows upward through the fuel assemblies stacked in the configuration in the lower right. In the lower third of the core, the coolant is heated to the saturation temperature (288oC at 7.1 MPa). In the middle third, the coolant rises as a foam, which is converted to wet steam (i.e., vapor with

Introduction 19 Head spray cooling nozzle Reactor vessel head Vessel head stud Dryer assembly lifting lugs

Vessel head nut

Steam dryer assembly Steam outlet nozzle Steam dryer and shroud head alignment and guide rods Steam separator and standpipe assembly

Shroud head lifting lugs Dryer seal skirt

Feedwater inlet

Core spray supply header

Feedwater sparger

Shroud head

Vessel wall

Shroud head hold-down bolts Shroud head floor support legs Jet pump inlet elbow and nozzle assembly Jet pump riser brace arm

Top guide Shroud head alignment pins Core spray sparger Fuel assembly

In-core flux monitor assembly Jet pump mixer

Control rod Fuel support

Recirculating water inlet nozzle

Core shroud Core plate

Jet pump inlet riser

Core differential pressure tap & liquid control inlet nozzle Recirculating water outlet nozzle Control rod guide tube

Jet pump diffuser

Vessel support skirt Vessel support ring girder In-core flux monitor housing Control rod drive housing

Control rod drive housing support structure

FIGURE 1.9: Schematic of a Boiling Water Reactor in cross section.

entrained droplets) in the upper third of the core. Because high-velocity wet steam erodes metal surfaces, the separator unit above the core outlet permits gravity removal of the water droplets from the rising steam. Dryers just before the exit pipe convert any remaining liquid to steam.

20 Light Water Reactor Materials

The outlet steam is not superheated (temperature greater than the saturation temperature) as it is in some fossil power plants. The device resembling nozzles on the RPV in Figure 1.9 is part of an auxiliary recirculation circuit that serves two purposes: First, via jet pumps, it delivers the main feed flow from the inlet pipe to the lower head; second, it captures the slowly-settling droplets from the steam separator and efficiently mixes them with the main coolant flow. As shown in the sketch, the recirculation pump is placed outside the RPV. The flow from this loop is injected into the funnel-like opening to the downcomer pipe situated between the core periphery and the inner wall of the RPV. The high flow rate of water in this device acts as a pump that efficiently returns all liquid above the core to the core entrance at the lower head. The drives for the cruciform-shaped control blades enter the RPV through the bottom head. Control blades containing neutron-absorbing materials are distributed among the fuel assemblies. The control blades are moved into and out of the core from the bottom of the reactor pressure vessel. The large number of control blades means that each blade controls a small amount of nuclear reactivity. This feature avoids large local power changes due to movement of a single blade, which could produce damaging thermal stresses in the nearby fuel rods. Reducing power around the failed rod decreases the loss of fission products and minimizes further degradation of the failed element. The fuel assembly shown in Figure 1.5e is a 9 × 9 BWR design. Although individual BWR rods are larger than those of PWRs, the lateral dimension of the BWR assembly is smaller because there are fewer rods in each row. For neutron moderation, a few of the rods at the center of the BWR assembly are filled with liquid water instead of fuel. In some designs, the center of the fuel assembly contains a square water-filled channel. The purpose of these “water rods” is to increase the hydrogen density and thereby provide more neutron moderation than is available in the steam filling the spaces between rods.

Introduction 21

1.8 Reactor Pressure Vessel Internals and Ex-Core Materials In addition to fuel and control materials, the reactor pressure vessel (RPV) contains numerous other components. The structural alloys used in these components include: • the reactor pressure vessel is fabricated from ferritic steel (“low-alloy steel”) lined internally with stainless steel; and • reactor internals use stainless steel and nickel-based alloys for:  mechanical support for the reactor core,  structures to direct the flow of coolant,  shielding for the RPV: core barrel and core shroud,  instrumentation guide tubes, and  steam separator (BWR); steam generator (PWR). Table 1.2 summarizes the major components of a 1000-MWe nuclear plant, the materials of which they are made, their approximate mass, and the principal materials-related issues encountered during operation. The last of these contains only the current issues; problems that appeared in the past but have been solved (e.g., irradiation densification of fuel) are not included in the list, although some are discussed in this book. Many of the materials problems in ex-core components listed in Table 1.2 have actually been encountered, while others are potential problems that nevertheless need to be considered in safety analyses. In the latter category are the possible consequences of irradiation hardening and embrittlement of the reactor pressure vessel, the chief one being brittle fracture during a postulated pressurized thermal shock. Although no RPV failure has occurred, preventative measures have resulted in power reduction in some PWR reactors believed to be prone to such an accident. Only pressure-vessel failure and massive fracture of large-diameter pipes are reactor-safety issues. These and accidents that can cause fuel damage such as the reactivity-initiated accident (RIA) are design basis accidents. Another example is fracture of large-diameter pipes leading to

Component

Material∗

Mass (Mt) Materials Issues

Fuel

UO2, (U,Pu)O2

100

Zircaloy: 1.7 Sn; 0.5 (Fe, Ni, Cr); 0.1 O; bal Zr; or modern alloys such as ZIRLO and M5 Neutron absorbers Ag-Cd-In (PWR) B4C (BWR); Gd2O3 (both) Reactor pressure vessel Low-alloy steel 2 Cr; 1 Mo; bal Fe Steam generator (PWR) Inconel 600: 25 Cr; 15 Fe; bal. Ni and 690 Reactor internals Stainless Steel: 18 Cr; 8 Ni; bal. Fe; Inconel Ex-core components, Stainless steel primary piping Valves, pumps Stainless steel; stellite (high-cobalt steel) Special components Alloy 718 Ni 52.5, Cr 19.0, Fe bal, Mo 3.0, Mn 0.35 Cladding, grid spacers

25

~1

Fission-gas release; fission product swelling; thermal conductivity and burnup (Ch. 20) Waterside corrosion and hydriding (Ch. 22 and 23); embrittlement (Ch. 26) growth (Ch. 27); pelletcladding interaction (Ch. 23); fretting None except in severe accident

350

Radiation embrittlement (PWR only) (Ch. 26) Tube plugging, cracking, and denting; leakage from the primary coolant to the secondary loop Stress-corrosion cracking (Ch. 25); fatigue

–

Stress-corrosion cracking (BWR) (Ch. 25)

–

Cobalt dissolution => activation in core => deposition in primary circuit (Ch. 21) Creep (Ch. 27); stress-corrosion cracking (Ch. 25)

TABLE 1.2. Materials in a 1000-Mwe nuclear steam supply ∗

The number next to each element is the weight percent in the alloy.

Introduction 23

a loss-of-coolant accident (LOCA). The other materials problems listed in Table 1.2 affect reactor operation and include issues that: • exceed the technical specifications that govern the limits of operating the reactor (e.g., radioactivity in the primary circuit coolant due to fission products released from a “leaker,” i.e., a failed fuel element); • require de-rating of the component (e.g., plugged tubes in the steam generator of a PWR); • force an unscheduled outage (e.g., a fuel element cladding failure large enough to release fuel to the coolant); • impede regular maintenance because of high radiation levels in the primary circuit (e.g., excessive deposition of activated cobalt-60 in the primary circuit [Ch. 21]); or • make it necessary to show that degradation of plant materials is not excessive before plant life extension can be granted.

1.9 Capacity Factor The single most important measure of the efficiency of a power plant is its capacity factor. This quantity is defined as the net quantity of electricity generated in one year divided by the maximum possible amount of continuous operation at the rated power. When nuclear plants are used as baseline power in the United States, this is equivalent to the availability factor (fraction of time the plant is available for electricity production when called upon to produce).6 The capacity factor is generally less than 100% because of shutdowns due to the following: • Scheduled outages for refueling and regular maintenance (typically for 20 to 30 days every 18 to 24 months—a capacity factor loss of ~5%) 6

That is, when nuclear power is needed it is usually available; this is in contrast to intermittent sources as wind and solar power, which generally have much lower availability and thus require greater installed capacities.

24 Light Water Reactor Materials

Forced outages due to equipment or fuel-element failures • Operation at reduced power due to:  load-following (changing power because of reduced or increased demand on the electrical grid, or because other sources, such as renewables, need to be used);  power maneuvering at a slow rate (to avoid damage to fuel elements);  avoid further degradation of damaged fuel, accommodate a de-rated steam generator or reduce the rate of accumulation of radiation damage by the reactor pressure vessel; or  seasonal variations in the temperature of the river water used for reactor cooling. Plant downtime is mostly caused by refueling. Over a five-year period in the 1990s, this represented a 29-day/year scheduled refueling outage, although this number is lower now. Numerous other problems in the nuclear steam supply such as shutdowns or de-rating due to steam generator problems together contribute a 6% (23 days/year) capacity-factor loss. Although fuel-rod failures (Ch. 23) are not even in the top 50% of reactorrelated causes of forced outages, their minimization is of continuing concern because problems with this component can cause operational difficulties that are not immediately reflected in the capacity factor and also because the degradation mechanisms may become more severe at high burnup. Although only one additional nuclear power plant has come online in the last twenty years (Browns Ferry in 2010), the percentage of nuclear power in the total electricity mix in the United States steadily increased in the 1980s and 1990s, and stabilized in the 2000s, even as the total electric-energy generated has increased. Most of the growth in the nuclear contribution is attributable to the increase in capacity factors from ~57% in 1986 to more than 90% in 2002, the equivalent of building twenty-two new nuclear power plants over that period. This was achieved by increasing the time between scheduled outages from 12 to 18 or 24 months •

Introduction 25

and decreasing the number of forced outages. Additional capacity factor increases were provided by power uprates. The capacity factors have more or less held steady since then.

1.10 Current Challenges As of the writing of this book, ~100 commercial nuclear power plants operated in the United States, providing a total of ~100,000 MWe or slightly less than 20% of the nation’s electricity. Over the past decade, operation of these power plants has continually improved, due in large part to the resolution of various materials problems. However, new problems, nearly all associated with materials performance, have arisen as nuclear plant operators seek to extract as much energy from the fuel as possible. The principal methods are: • Extended burnup: Increasing the average fuel discharge burnup to 60 to 70 MWd/kgU delivers a proportional decrease in the volume of radioactive waste from spent fuel. By using more highly enriched fuel and consuming a larger fraction of the 235U, less fuel is required to generate the same energy, thereby reducing fuel-fabrication costs. However, high burnup fuel rods experience increased radiation damage, corrosion, and more extensive hydriding of the cladding. Greater fission gas release leads to higher fuel-rod pressures, and swelling of the fuel increases the stress in the cladding. Note that to date the extended burnup has been achieved with no loss in capacity factor, which means fuel failures have decreased even as fuel duty increased. • Power uprates: At constant coolant flow rate, increasing the power in a PWR results in a higher outlet coolant temperature. This increases the efficiency of energy conversion and produces more electricity for a given thermal power. The higher fuel temperatures promote undesirable coolant boiling on the cladding surface, with corresponding increases in thermally activated processes that lead

26 Light Water Reactor Materials

to fuel degradation, including fission-gas release, corrosion, and hydriding. • Longer refueling cycle: Increasing the time between refueling outages from 12 months to 18 to 24 months results directly in an increased capacity factor. All of these factors increase the severity of fuel duty, such that fuel reliability needs to be constantly improved. Figure 1.10 shows the most 3.6% 25.1%

(a)

11% 5%

0.1% 0.4% 54.8%

24%

(b)

32%

9% 4% 31%

CRUD/Corrosion

Debris

Fabrication

Handling

PCI-SCC

Unknown

Grid-to-rod Fretting

FIGURE 1.10: Fuel leak causes in reactors worldwide from 1994 to 2006: (a) in PWRs; (b) in BWRs. (Redrawn from International Atomic Energy Agency).

Introduction 27

common causes of fuel failure worldwide in the period 1994 to 2006 for both PWRs and BWRs. Grid-to-rod fretting dominates the failures in PWRs, which has led fuel vendors to pay close attention to the design of the fuel assembly and fuel-coolant interactions. For BWRs CRUD/ corrosion and debris represent a large fraction of the fuel failure causes (see discussion in Ch. 23) and the design of debris filters is a current concern. Electricity deregulation has driven nuclear utilities to seek stringent cost-reduction measures in order to remain competitive. In spite of the increased demands on the fuel, cladding, and structures, the fuelperformance requirements continue to increase. Sixty percent of the original current reactor licenses in the United States were scheduled to expire before 2020. Nearly all of the operators of currently running reactors have applied for renewal of their operating licenses. Most plants in the United States have applied for life extensions of 20 years. The feasibility of continued efficient, economic, and safe operation of light water reactors depends crucially on safe aging of nuclear materials. Two of the thorniest issues are: • certifying that the RPV has not been excessively embrittled after exposure to fast neutrons during its normal life, and • assuring that steam generator tube cracking (in PWRs) and corrosion and cracking of reactor internals (in BWRs) will not accelerate during the extended plant life.

1.11 Layout of the Book To cover the materials issues that arise in light water reactors, the book is split into two volumes: the first, Fundamentals, (Ch. 1 through 15) deals with the fundamentals of materials, thermodynamics, atomic transport in solids, crystallography, defects, phase transformations, corrosion,

28 Light Water Reactor Materials

mechanical behavior, and radiation damage; the second, Applications, (Ch. 16 through 29) applies these fundamentals to the degradation processes that occur in service. The main focus is on the nuclear fuel and cladding, but the pressure vessel, reactor internals, and the other parts of the nuclear supply system are also covered. Understanding the behavior of materials in nuclear power plants is a complex endeavor. If, in the early 1960s, reactor materials experts had been asked to predict the problems that would later occur with UO2 fuel, Zircaloy cladding, pressure vessel steels, stainless steels, and other nuclear materials, they would have been hard put even to conceive of many of the problems that did arise, let alone predict their outcome. This is especially true for complicated failure mechanisms such as irradiation-assisted stress-corrosion cracking (which involves the synergistic interaction of mechanical, environmental, and materials factors) or the complex longterm evolution of damage in pressure-vessel steels. In applying predictive mechanistic models, the complexity of real physical phenomena and the limitations of current understanding must not be forgotten. This is what we strive for in this book.

References 1. Knief, R. A. Nuclear Engineering. Washington and London: Taylor and Francis, 1992. 2. Holmes-Siedle, A., and Adams, L. Handbook of Radiation Effects. Oxford: Oxford University Press, 1993. 3. Bailly, H., Menessier, D., and Prunier, C. The Nuclear Fuel of Pressurized Water Reactors and Fast Reactors. Paris: Lavoisier Publishing, 1999. 4. Lemaignan, C., and Motta, A. T. “Zirconium Alloys in Nuclear Applications,” Materials Science and Technology A Comprehensive Treatment. New York: Wiley VCH, 1994, 1–51.

2

Chapter Chemical Thermodynamics 2.1 Applications of Thermodynamics to Light Water Reactor Materials It is generally recognized that material failure limits both the maximum burnup attainable by light water reactor (LWR) fuel and the lifetime of structural alloys in and around the core. The processes that affect the behavior of these materials are both equilibrium and nonequilibrium; processes exhibit both thermodynamic and kinetic aspects. The nonequilibrium kinetic features are dominant in cladding corrosion by the coolant and the production of defects in metals by irradiation. Overall, however, thermodynamic properties either completely dictate the nature of the response of the materials to reactor conditions or control the driving force for the kinetic steps. Thermodynamics plays a particularly important role in the following: • pressure-temperature-volume properties (equations of state): water, fuel, fission gas • vapor pressures: fuel, coolant, fission products • thermal properties: specific heat capacity, coefficient of thermal expansion

30 Light Water Reactor Materials •

phase diagrams: single component and binary • chemical and physical state of fission products in fuel • hydride formation in cladding and nitride formation in steel • corrosion of metals and alloys • response of the O/U ratio in nonstoichiometric uranium dioxide to the oxygen pressure of the environment • thermal stability of point defects in solids • corrosion of cladding by steam This chapter is devoted to a brief review of chemical thermodynamics that is intended to serve as the basis of applications encountered in subsequent chapters.

2.2 Thermodynamics 2.2.1 Basic properties Thermodynamic properties depend only on the state or condition of the system but not on the process or the path by which the particular state was achieved. Five fundamental thermodynamic properties cannot be derived from other thermodynamic properties. These are, with their common symbols: T  = temperature p  =  hydrostatic pressure V =  volume U =  internal energy S   =  entropy In addition, there are two derived thermodynamic properties that are combinations of the primitive properties: H = U + pV = enthalpy

(2.1a)

Chemical Thermodynamics 31

G = H − TS = Gibbs energy

(2.1b)

These particular combinations of fundamental properties define new properties that describe commonly encountered processes. For example, when pressure and temperature are fixed, thermodynamic equilibrium is achieved when the Gibbs energy of a closed system is at a minimum. With the same constraints, the heat absorbed or evolved in a chemical reaction is the change in enthalpy. The basic thermodynamic properties can be classified as intensive or extensive. An intensive property is independent of the quantity of substance. Temperature and pressure, for example, are intensive properties. All of the others are extensive but can be made intensive by dividing by the quantity of the substance. Taking the number of moles, n, as the measure of quantity, the intensive counterparts of V and G are v = V/n and g = G/n. For these quantities, lower-case designations denote specific or molar values of the property. The molar volume v is the reciprocal of the molar density. Other thermodynamic properties are defined as partial derivatives of one of the properties with respect to temperature or pressure. The heat capacities (2.2) C v =  ∂u C p =  ∂ h  ∂T v  ∂T p represent the increases in internal energy and enthalpy, respectively, per degree of temperature increase. They are written as partial derivatives because of the constraints indicated by the subscripts on the derivatives. For Cv, the increase in temperature is required to occur at a fixed volume. For Cp, on the other hand, the pressure is maintained constant during the increase in temperature.

32 Light Water Reactor Materials

The coefficient of thermal expansion α and the coefficient of compressibility β involve the fractional changes in volume as temperature or pressure is increased: α=

1  ∂v  v  ∂T  p

β=−

1  ∂v  . v  ∂ pT

(2.3)

Since the specific volume (or reciprocal of molar density) of the α substance depends on both temperature and pressure, α and β are defined as partial derivatives to indicate the property that is held constant during the increase of the other property. Both α and β are positive numbers, which accounts for the negative sign in the definition of β. Four of the basic properties have absolute values: T, p, v, and s. Assignment of zero entropy to crystalline solids at 0 K is a consequence of the 3rd law of thermodynamics. The energy-like properties u, h, and g are relative. Any one of them may be given an arbitrary value at a standard state (e.g., specified pressure and temperature), which serves as a reference. To a good approximation, the internal energy and the enthalpy of both gases and condensed phases (liquids and solids) are nearly independent of pressure (or volume). This simplification permits the specific heats in Equation (2.2) to be integrated: T

u(T) − u(To ) = ∫ Cv (T ′) dT ′ ≅ Cv (T − To ) To

(2.4)

T

h(T) − h(To ) = ∫ C p (T ′) dT ′ ≅ C p (T − To )

(2.5)

To

where To is an arbitrary reference temperature (usually 298 K) and T ′ is the variable of integration (any letter would do). The second equalities in Equations (2.4) and (2.5) follow from the commonly employed approximation of temperature-independent specific heats.

Chemical Thermodynamics 33

Other important properties of pure (i.e., one-component) substances are the enthalpy changes that accompany phase changes. For the solid– liquid transition, the molar enthalpy of melting, or fusion, is ∆h M = h L − h S

(2.6)

where hL and hS are the molar enthalpies of the liquid and solid phases, respectively. Conversion of a liquid to its vapor is characterized by the enthalpy of vaporization: ∆hvap = hg − hL

.

(2.7)

These molar enthalpy changes are measured by the heat absorbed as the phase with the lower enthalpy is converted to the phase with the higher enthalpy. Consequently, ∆hM and ∆hvap are positive quantities. Relations similar to Equations (2.6) and (2.7) can be written for all types of phase transitions. Of particular importance are the transformations of crystalline solids from one type of crystal structure to another type.

2.2.2 Heat and work The concepts of heat and work are fundamentally different from the properties of a material. Heat, in particular, is often confused with the thermodynamic properties temperature and internal energy. It is neither. To say that a body (or system) contains a certain quantity of heat is incorrect; the body or system possesses internal energy. Heat appears as this energy crosses the system’s boundary by conduction, convection, or radiation. Work is a term for forms of energy transfer that have in common that they are not heat but are in principle completely interconvertible among themselves. The most common form of work is that produced by a force F acting over a distance ∆X, which represents displacement of the system

34 Light Water Reactor Materials

boundary. This action involves a quantity of work given by W = F∆X. If we multiply ∆X by A, the area over which the force acts, and divide F by A, the work equation becomes W = (F/A)(A∆X). Since F/A defines pressure p, and since the product A∆X is the volume change ∆V, the work involved can also be written as W = p∆V. This form of mechanical work done on (or by) the system is called “pV” work. Another common form of mechanical work is shaft work, by which a system exchanges work with its surroundings by means of rotational motion rather than expansion or contraction, as in the pV form. For example, shaft work is performed as high-pressure steam spins a turbine in an electric power plant. A third form of work is electrical work, which is best exemplified by a battery that runs a motor by means of the electrical current generated by a chemical reaction. These nonpV forms of work are known collectively as external work, denoted by Wext.

2.2.3 Thermodynamics laws The first law of thermodynamics is an empirical observation, never refuted, that the change in the internal energy of a closed system resulting from addition of heat and performance of work is given by ∆U = Q − W

(2.8)

where ∆U = U(final) − U(initial) = change in system internal energy;  Q = heat added to the system; and  W = work done by the system. Equation (2.8) applies to a closed system, which is a region of space whose boundaries enclose the substance characterized by the properties such as T and p (heat and work are not properties of a system). The material outside of the system boundaries is called the surroundings.

Chemical Thermodynamics 35

The law of energy conservation states that the sum of the energy changes of the system (∆U) and surroundings (∆Usurr) is a constant: ∆U + ∆Usurr = 0

.

(2.9)

For simplicity, potential and kinetic energies have been neglected. In the microscopic view of thermodynamics, entropy characterizes the state of disorder of a system. Consequently, entropy changes are closely related to heat, but are not associated with work. The connection of heat and entropy is embodied in the following statement of the second law:  Q (2.10) ∆S ≥ ∫ T where ∆S and the integral represent changes from an initial state to a final state. The equality in Equation (2.10) applies if the process is reversible (i.e., one that can be made to go backward without any change in the system or the surroundings). The total entropy change of the system (∆S) and the surroundings (∆Ssurr) is ∆S + ∆Ssurr ≥ 0 (2.11) where the equality again applies to reversible processes. Contrary to the energy analog given by Equation (2.9), entropy is not conserved in irreversible processes. In common with enthalpy changes, entropy changes are associated with phase changes of a pure substance. For melting, ∆sM = sL − sS

(2.12)

where sL and sS are the molar entropies of the liquid and solid phases, respectively. Conversion of a liquid to its vapor is characterized by the entropy of vaporization:

36 Light Water Reactor Materials ∆svap = sg − sL

.

(2.13)

Both ∆sM and ∆svap are positive quantities. The first law (Eq. [2.8]), written in differential form for a unit quantity of substance, is du = δQ − δW. For reversible changes, the second law (Eq. [2.10]) is δQ = Tds and, if only expansion/contraction work is permitted, δW = pdv. The combined first and second laws take the form du = Tds − pdv .

(2.14)

Equation (2.14) also applies to irreversible processes because it involves only properties of the system. For the same reason, it is equally valid if the system performs external (non-pV) work. Analogous differential forms of the combined first and second laws can be derived for the other two energy-like properties. For example, the molar enthalpy function is defined by: h = u + pv. The differential of this property is dh = du + pdv + vdp, which, when combined with Equation (2.14), gives dh = Tds + vdp . (2.15a) C dT At constant pressure, Equation (2.15a) gives ds = dh = p , or in T T integral form, T C p (T ′ ) s(T ) − s(To ) = ∫ dT ′ (2.15b) To T ′ where To is a reference temperature. For the Gibbs function, g = h − Ts, the differential is dg = dh − Tds − sdT. Substituting Equation (2.15a) gives dg = –sdT + vdp .

(2.16a)

Chemical Thermodynamics 37

At constant temperature, the above equation gives dg = vdp, or in integral form, p

g ( p) − g ( po ) = ∫ v ′( p ′) dp ′ po

(2.16b)

where po is a reference pressure, usually 1 atm ≅ 0.1 megaPascal (MPa).

2.3 Equations of State Equations of state (EOS) provide mathematical relationships between p, v, and T of a single-phase, single-component system (or a solution of fixed composition). Rather than total properties, molar properties are more conveniently used here.

2.3.1 Gases The EOS of gases are frequently represented by the ideal-gas law: pv = Rg T .

(2.17)

The gas constant, Rg , is the product of Avogadro’s number NAv and Boltzmann’s constant kB: Rg = NAv kB = 6.023 × 1023 (molecules/mole) × 1.3804 × 10−23 (J/molecule.K)  = 8.314 J/mole.K. (2.18) As is obvious from its genesis in two fundamental constants, the gas constant is not restricted to gases. Nonetheless, it is so often used with gases that its value in units compatible with Equation (2.17) is needed. Using the unit identities: J = N − m = Pa − m3, Equation (2.18) converts to

38 Light Water Reactor Materials

Pa − m 3 atm cm 3 × 9.87 × 10 − 6 × 10 6 3 mole − K Pa m atm − cm 3 .  = 82.1 mole − K

R g = 8.314

(2.19)

Nonideal behavior in gases [1, p. 51] arises from attractive and repulsive forces between colliding atoms or molecules. A frequently used modification of Equation (2.17) is the Van der Waals equation, (p + a/v2)(v − b) = RgT ,

(2.20)

in which a and b are unique to each gas. The constant a accounts for attraction between atoms or molecules. If p + a/v2 is thought of as an equivalent pressure in Equation (2.17), because it is larger than p, a reduction in v is required. This is what attraction should do: pull the gas together. The constant b represents the repulsive force; if v − b is considered to be an equivalent volume, an increase of p is needed to satisfy the right-hand side of Equation (2.17). Intermolecular repulsion requires a greater pressure to hold the gas. Example #1: Nonideality of rare gases The table below gives the Van der Waals’ constants for the rare gases: Gas He Xe

a, cm6 . atm/mole2 b, cm3/mole 24 3.5 × 104 6 4.25 × 10 51

What are the deviations from ideality at T = 600 K and v = 2.5 × 103 cm3/mole? The pressure of an ideal gas at this temperature and specific volume is 19.7 atm. The pressures calculated from Equation (2.20) and the errors incurred using the ideal gas law are:   He: p = 19.9 atm (0.9% error)   Xe: p = 19.4 atm (1.4% error)

Chemical Thermodynamics 39

The Van der Waals’ constants are to be used with caution. At lower temperatures and higher pressures than in the above example, a more accurate EOS, such as the hard-sphere EOS (Ch. 19), is needed.

2.3.2 Condensed phases Another way of describing the EOS is 1  ∂v  dv 1  ∂ v  = +   dp . dT v v  ∂T  v  ∂ p p T

(2.21)

For ideal gases, this gives dv dT dp v = T − p .

(2.22)

For condensed phases (liquids or solids), on the other hand, an EOS can be represented by substituting the coefficients of expansion (α) and compressibility (β) from Equation (2.3) into Equation (2.21): dv v = αdT − βdp .

(2.23)

Although α and β vary with temperature and pressure, the dependence is slight enough that Equation (2.23) can be integrated: v (T , p)  (2.24) ln  = α(T − To ) − β( p − po ) . v (To , po )  The subscript o signifies a reference state where the molar volume is known. Typical values of α are 10−5 K −1, so that a 100 K temperature change increases the molar volume by 0.1%.

2.4 Criteria of Thermodynamic Equilibrium Thermodynamic equilibrium in a closed system contains three subequilibrium requirements. For thermal equilibrium there can be no temperature

40 Light Water Reactor Materials

gradients in the system; mechanical equilibrium implies a constant pressure throughout the system. Chemical equilibrium, which fixes the composition of the system, is not as obvious. The combined first and second laws of thermodynamics, expressed by Equation (2.14), allows only for work due to expansion or contraction (pV work). If the system performs external work, such as shaft work or work derived from a chemical reaction, δWext must be subtracted from the right side of Equation (2.14). δWext also appears in the right sides of Equations (2.15a) and (2.16b). With the constraints of constant T and p, Equation (2.16a) reduces to dGT,p = − δWext . (2.25) A useful definition of a system at equilibrium is a system that cannot perform useful (nonexpansion) work. Applying this notion to Equation (2.25), the criterion of equilibrium for closed systems constrained by fixed T and p is dGT,p = 0 (2.26) or, at equilibrium, the Gibbs energy of a constant T-p closed system is at a minimum. This criterion of equilibrium is useful chiefly for systems with more than one phase or more than one component. For a single-phase, single-component system, fixing any two properties determines the others. For heterogeneous (multiphase) systems or homogeneous (single-phase) systems containing chemically reacting constituents, Equation (2.26) provides the essential starting point for determining the state of equilibrium.

2.5 Single-Component Phase Equilibria A pure substance can exist in one, two, or three phases, as long as the phase rule (Sec. 2.6) is satisfied. Important two-phase mixtures and the

Chemical Thermodynamics 41

process by which one phase is transformed to the other include: liquid–gas (vaporization), solid–gas (sublimation), solid–liquid (melting, or fusion), and solid I–solid II (allotropy). At fixed pressure and temperature, the condition of equilibrium between two phases that satisfies Equation (2.26) is gI = gII ,

(2.27a)

from which gII – gI = ∆gtr = ∆htr – T∆str = 0, or ∆ str =

htr . T

(2.27b)

Subscript tr signifies transition between phases.

2.5.1 Liquid-vapor equilibria An important application of Equations (2.27a) and (2.27b) is to liquid– gas transformation (vaporization), where I = L(liquid) and II = g(gas). At equilibrium, gL = gg. For a small move from equilibrium to another equilibrium state, dgg = dgL, or from Equation (2.16a), −sg dT + vg dp = −sL dT + vL dp, and dp s g − s L = dT v g − v L

∆s vap =

∆v vap

∆hvap =

T ∆v vap

∆hvap ≅

Tv g

≅

p∆hvap . RgT 2

In this equation, ∆svap has been replaced by ∆hvap/T according to Equation (2.27b), and vL has been neglected compared to vg , which has been approximated by Equation (2.17). Since the two phases are at equilibrium, p and T are connected. The link can be considered either as psat, the saturation pressure or vapor pressure at temperature T, or Tsat, the saturation temperature as a function of p. The former representation is most commonly used. ∆hvap can be constructed

42 Light Water Reactor Materials

by applying Equation (2.5) to the vapor and the liquid and subtracting, yielding ∆hvap(T) = ∆hvap(To) + (Cpg − Cpl)(T − To)

.

This second term on the right side of this equation is important for water, and at 350 K reduces ∆hvap by ~11%. However, for the refractory metals and ceramics with which we deal in this book, the first term in the above equation dominates the second term. ∆hvap is essentially independent of temperature and, as a consequence, dp/dT can be integrated to yield p  ∆ hvap  1 1  (2.28) = − ln  sat o   psat R g  To T  o is the vapor pressure at the reference temperature T . where psat o A formula of the same mathematical form but with ∆hsub in place of ∆hvap applies to vaporization from the solid phase. The saturation pressure is now called the sublimation pressure.

∆h p 1 1 ln  satM  = sub  −   psat  R g  TM T 

(2.29)

The reference temperature has been chosen as the melting temperature, TM, where the vapor/sublimation pressure is psatM . The heats of sublimation, vaporization, and melting are related by ∆hsub = ∆hvap + ∆hM

.

(2.30)

Figure 2.1 shows the saturation pressures of selected metals. Sublimation pressures are shown as solid lines and vaporization pressures as dashed lines. In each region, the lines are straight because both ∆hvap and ∆hsub are very nearly temperature-independent. The slopes of the lines are

Chemical Thermodynamics 43 −∆hsub/Rg

for the solid–gas transition and −∆hvap/Rg for the liquid–gas phase change. The changes in slope at the melting temperature for Ca, Zn, and Cd in Figure 2.1 are due to the replacement of ∆hsub by ∆hvap. According to Equation (2.30), the difference between these two enthalpy changes is the heat of fusion. The changes in slope are barely discernible because the heats of fusion are only ~4% to 5% of the heats of sublimation for all metals.

2.5.2 Solid-liquid-vapor equilibria In addition to the vapor pressure formula of Equation (2.29), the phase relations of a pure substance include the solid–liquid and solid–vapor equilibria. These three equilibria can be shown graphically on a phase

0

–2

log psat (atm)

Mn Ca

–4

Zn Cd

Sn –6 Ti Cr –8

–10

Pb W Pt

–12 0.4

0.8

1.2

1.6

2.0

2.4

103/T (K–1)

FiGure 2.1: Vapor pressures of selected metals: solid lines—sublimation; dashed lines—vaporization. Pressure is in atm and temperature is in Kelvin.

44 Light Water Reactor Materials

diagram. The most familiar graphical representation of the phase relationships of a pure substance is the p-T diagram shown in Figure 2.2. The three two-phase equilibrium lines intersect at the triple point. At pressures below the triple-point pressure, the line separating solid and vapor gives the sublimation pressure/temperature. At pressures above the triple point, the near-vertical line gives the melting temperature as a function of pressure and the line further out defines the vapor pressure as a function of temperature (Eq. [2.28]). At the triple point, all three phases coexist at equilibrium. In the areas labeled “solid,” “liquid,” and “vapor/gas,” a single phase is stable over a range of temperature and pressure. The melting line in Figure 2.2 is derived as follows: The Gibbs energies of the solid and liquid phases are related to the corresponding enthalpies and entropies at the melting temperature TM, where gL = gS, or

Melting

LIQUID

SOLID

Critical point ion

izat

Pressure

or Vap

Triple point o ati

n

blim

Su

VAPOR/GAS

Temperature

FiGure 2.2: Generic single-component phase diagram.

Chemical Thermodynamics 45

gL − gS = ∆gM = ∆hM − TM∆sM = 0 .

(2.31a)

If the pressure is increased by dp, the melting temperature changes by dTM. Since the new state is still in equilibrium, the change in the Gibbs energy of melting, d(∆gM), must be zero. Obtaining the difference in the liquid and solid Gibbs energy changes from the fundamental differential of Equation (2.16) yields d(∆gM) = − ∆sMdTM + ∆vMdp = 0 . Eliminating ∆sM between Equations (2.31a) and (2.31b) yields dTM TM ∆v M = ∆hM dp

(2.31b)

(2.32)

where ∆hM is the enthalpy of fusion and ∆vM = vL − vs is the volume change on melting. The slope of the melting line in Figure 2.2 represents the effect of pressure on the melting temperature. For most materials, ∆vM is positive; the solid is denser than the liquid. Water is the notable exception to this general rule because the liquid is denser than the solid near the melting point. A related aspect of the melting process is the variation of ∆gM with temperature at constant pressure. Under these conditions, the solid and liquid are no longer in equilibrium, so ∆gM ≠ 0. In Equation (2.31b), setting dp = 0 for the constant-pressure requirement and replacing TM by the variable temperature T yields ∆ hM  ∂∆ g M  − = −∆ = s  ∂T  M TM p

where Equation (2.31a) has been used to eliminate ∆sM. Assuming ∆hM to be independent of temperature, this equation can be integrated to give  T . ∆g M (T ) = ∆hM 1 − (2.33)  TM 

46 Light Water Reactor Materials

If T > TM, ∆gM is negative, implying that the liquid has a lower Gibbs energy than the solid, and hence, in a single-component system, is the stable phase. Conversely, if T < TM, only the solid exists at equilibrium. This sharp demarcation of phase stability breaks down in multicomponent systems; a component can exist in a liquid solution at temperatures well below its melting point when pure.

2.6 The Phase Rule Analysis of systems other than pure substances requires an understanding of the notions of phase, components, and degrees of freedom. Components are distinct chemical constituents whose quantities can be independently varied. The relative amounts of the components are designated by compositions, for which mole fraction is a common measure. Phases are regions of a system in which all properties are uniform and are separated physically from other regions in the same system by a boundary, or interface. The number of system variables (e.g., properties, composition) that can be independently specified without changing the phase(s) are called degrees of freedom. The numbers of components (C ), phases (P ), and degrees of freedom (F ) are related by the Gibbs Phase Rule: F=C+2−P .

(2.34)

Temperature and pressure are included in F. For single-component systems (C = 1), Equation (2.34) reduces to F = 3 − P. This relation can be understood using Figure 2.2. The areas labeled “solid,” “liquid,” and “vapor/gas” permit both p and T to be varied. This corresponds to P = 1 and F = 3 − 1 = 2. Or, these single-phase regions possess two degrees of freedom. Two phases are present for p − T combinations that fall on the sublimation, vaporization, and melting lines in Figure 2.2.

Chemical Thermodynamics 47

These correspond to P = 2 and F = 1. The single degree of freedom can be either p or T; specification of one fixes the other according to Equations (2.31a) or (2.31b). When three phases coexist at equilibrium, F = 0. This means that the three-phase mixture occurs at a unique combination of p and T (the triple point). Application of the phase rule to multicomponent systems is not as straightforward as it is for single-component systems. For example, take the problem of identifying the number of components in a gas containing H2, O2, and H2O. At a low temperature and in the absence of an ignition source, the hydrogen does not burn and the mixture is a true three-component, single-phase system. Temperature, pressure, and two mole fractions can be independently specified, which in Equation (2.34) corresponds to C = 3 and F = 3 + 2 − 1 = 4. At high temperatures, on the other hand, the chemical reaction 2H2(g) + O2(g) = 2H2O(g) provides a relation between the mole fractions of the three molecular components. This restraint effectively reduces the number of components from three to two. The two components are the elements H and O, irrespective of their molecular forms. The sole composition variable is the H/O element mole ratio. With C = 2, the phase rule gives F = 3, which corresponds to the variables p, T, and H/O needed to fix the equilibrium composition of the molecular constituents. If a phase does not change composition in a process, a binary system can be treated as a pseudo single-component substance. Thus, analysis of air flowing through an orifice need not consider N2 and O2 as distinct components as long as properly averaged properties are used.

2.7 Solution Thermodynamics The objective of this section is to understand the thermodynamics of single-phase, two-component (binary) systems. These include mixtures

48 Light Water Reactor Materials

of ideal gases and the simplest models of nonideal binary solid or liquid solutions. The terms “mixture” and “solution” are nearly, but not quite, synonymous. A solution unequivocally refers to a phase containing two or more constituents. The term is applied to condensed phases, but not to gases. Gaseous “solutions” are called mixtures. Thus, air is a mixture, not a solution, of O2 and N2. These semantic distinctions between mixtures and solutions are usually clear from the context in which the words are used. Whether a single-phase mixture or a solution, the composition is denoted by the mole fractions of the constituents present: x i = ni /n where n = ∑ ni .

(2.35)

n is the total moles of a phase, and ni is the number of moles of constituent i. By definition, the sum of the mole fractions is equal to unity. The other common measure of composition is the molar concentration (ci) in units of moles per unit volume. To a good approximation, gas mixtures and many liquid and solid solutions can be considered to be ideal. In an ideal mixture, there are effectively no intermolecular interactions between the constituents. Pure solids and liquids must exhibit strong intermolecular attractions simply to exist as condensed phases. A binary solution of A and B is ideal if the average of the A–A and B–B intermolecular forces is equal to the strength of the A–B interaction. With some exceptions, the properties of an ideal solution are molefraction-weighted averages of the properties of the pure components. If Q denotes an extrinsic property (i.e., for n moles of mixture or solution), and qA and qB are the properties per mole of the two pure components, the mixture property is Q

=

n A q A + n B qB or q = Q/n = x A q A + x B qB .

(2.36)

Chemical Thermodynamics 49

q is the intensive value of the property, or the value of the property per mole of mixture or solution. Equation (2.36) applies to the volume V, the internal energy U, the enthalpy H, and the heat capacities Cp and CV. The entropy, however, contains an additional term that arises from the increased randomness afforded by mixing. When two pure substances are mixed at constant temperature and pressure, the entropy change, or the entropy of mixing, is given by

(

)

∆smix = −Rg xAlnxA + xBlnxB

(2.37)

where Rg = 8.314 J/mole.K is the gas constant. Since the definitions of the Gibbs energy includes the entropy, the right side of Equation (2.37) appears in g analogous to Equation (2.36): G = n A g A + n B g B − R g T(n A lnx A + n B lnx B ) . (2.38a) Nonideal behavior is a common characteristic of condensed-phase solutions. Departure from ideality in a binary solution occurs when the mean of the A–A and B–B bond strengths differs from the A–B bond strength. Departure from ideal behavior can be positive or negative. Quantitative treatment of nonideality in a binary system is achieved by modification of Equation (2.36). The entropy of mixing, Equation (2.37), is unchanged and is handled just as for ideal solutions.

2.7.1 Partial molar properties Thermodynamic equilibrium analyses are based on minimization of the Gibbs energy of an isothermal, isobaric closed system. The nonideal equivalent of Equation (2.36) is expressed in terms of this property. The total Gibbs energy of a binary solution at a specified temperature and pressure and its value per mole of solution are given by G = n A g A + n B g B or g = x A g A + x B g B

(2.38b)

50 Light Water Reactor Materials

where g A and g B are the partial molar Gibbs energies of constituents A and B in the solution. g A depends on the nature of constituent B with which it shares the solution. In addition to depending on temperature (and to a lesser extent, on pressure), g A depends on the solution composition. The same characteristics apply to g B . These two partial molar quantities include the entropy of mixing. Formulas analogous to Equation (2.38b) hold for other thermodynamic properties (h, s). It is often necessary to express partial molar properties in terms of the molar property (of the solution) and its variation with composition. Consider the molar volume, which is easy to measure as the composition is changed. The inverted relations are [1, p. 193] vA = v + x B

dv dx A

vB = v + x A

dv . dx B

(2.38c)

2.7.2 The chemical potential The partial molar Gibbs energy of a component in solution is called the chemical potential of the component, or µi = g i . With this replacement, Equation (2.38b) becomes G = n A µ A + n B µ B or g = x A µ A + x B µ B .

(2.39)

The physical meaning of the chemical potential is best appreciated by recognizing that the total Gibbs energy of a binary solution depends on the number of moles of A and B as well as on temperature and pressure. Taking the differential of G(T,p,nA,nB) holding T and p constant gives ∂G ∂G dG =   dn A +   dn  ∂n A T,p,n  ∂n B T,p,n B B

or

dG = µ A dn A + µ B dn B .

A

(2.40a)

Chemical Thermodynamics 51

The partial derivatives in the first form of dG define the chemical potentials in the second form: µA =

 ∂G   ∂n A  T,p,n

µB = B

 ∂G  .  ∂n B  T,p,n

(2.40b)

A

This equation shows that µA represents the change in the Gibbs energy of the system when a small quantity of A is added while the amount of B is held constant. Equation (2.40a) can be “integrated” in a physical sense by simultaneously adding the pure components to a vessel at rates proportional to their concentrations in the final solution. This procedure maintains all concentrations constant during the process, so that the integral of Equation (2.40a) is identical to the first equality in Equation (2.39). This procedure demonstrates that µA and µB in Equation (2.40a) are identical to those in Equation (2.39) as expressed by Equation (2.40b).

2.7.3 The Gibbs-Duhem relation Another important relation involving the chemical potentials can be derived from the total differential of Equation (2.39), dG = µAdnA + nAdµA + µBdnB + nBdµB . Eliminating dG using Equation (2.40a) and dividing by n to convert mole numbers to mole fractions yields xA dµA + xB dµB = 0 .

(2.41)

This relation is known as the Gibbs-Duhem equation. As will be seen below, it is very important in analyzing nonideal solutions.

2.7.4 Standard-state Gibbs energy The basic data for solving chemical equilibria are the Gibbs energies of the compounds involved in the reaction. The pressure is specified as

52 Light Water Reactor Materials

1 atm, which defines the standard state. With this restriction and specified temperature T: g io = Gibbs energy of pure constituent i (compound) o    g i = identical to the standard Gibbs energy of formation of the compound, ∆Gif g io (or ∆Gif )  = zero if i is an element in its stable state (e.g., O2) f ∆Gi is related to measurable properties of the substance; namely, its enthalpy and entropy of formation by f

f

∆Gi = ∆H i − T ∆Si

f

(2.42a)

.

The enthalpy of formation of a constituent at a reference temperature To and 1 atm pressure is ∆H i

f

o (To ) = hio (To ) − ∑ whreactant (To ) . element

w is the balancing number of an element in the formation reaction. o Since there is no absolute value of enthalpy, all hreactant (To ) are set equal element to zero. At temperature T,   (T ) = ∆H i (To ) + ∫ C pi (T′ ) − ∑ w C p reactant (T′ ) dT′ element  To  T

∆H i

f

f

(2.42b)

where Cp is the specific heat. The analog of Equation (2.42b) for the entropy ∆Sif (T ) is C p reactant (T ′)   ( ) C T ′ element (To ) + ∫  pi  dT ′ (2.42c) − ∑w  T′ To  T ′ T

∆Sif

where

(T ) = ∆Sif

∆Sif (To ) = Sio (To ) −

o ∑ wSelements reactant (To )

.

(2.42d)

Chemical Thermodynamics 53

Example #2a: Standard enthalpy of formation of methane at 1000 K The formation reaction: C(s) + 2H2(g) = CH4(g). The enthalpy of formation of methane at 298 K: −75 kJ/mole. The heat capacity of methane: C pCH4 = 20.6 + 0.051 × T J/mole.K. In Equation (2.42b), the first integral between 298 K and 1000 K is 37.5 kJ/mole. The average heat capacity of graphite over the same temperature range is ∼12 J/mole.K, so the contribution to the sum is 8 kJ/mole. The average heat capacity of H2 gas over the same range is ∼15 J/mole.K, so the integral of CpH2 is 21 J/mole.K. In Equation (2.42b), ∆H i

f

(1000) = −75 + [ 37.5 − 8 − 2 × 21] = −88 kJ /mole .

Example #2b: Standard Gibbs energy of formation of methane at 1000 K o (300) = 186 s o ; (300) = 6 s o ; (300) = 131 Data are in J/mole.K: s CH C H2 4 f From Equation (2.42d): ∆S CH 4 (300) = 186 − 6 − 2 × 131 = −82; for CH4, Cp = 20.6 + 0.051 T, and the first integral in Equation (2.42c) is 60.5 J/mole.K; for C, Cp = 1.05 + 0.022 T, and the 2nd integral in Eq. (2.42c) is 18.5 J/mole.K; for H2, the 3rd integral in Equation (2.42c) is 35 J/mole.K. The second term on the right-hand side of Equation (2.42c) = 60.5 −18.5 f − 2 × 35 = −28 J/mole.K. From Equation (2.42c): ∆SCH (1000) = −82 − 28 4 = −110 J/mole.K = 0.11 kJ/mole.K. f Applying Equation (2.42a): ∆GCH (1000) = −88 − 1000(−0.11) = 22 kJ /mole . 4 Instead of the cumbersome calculations in Examples #2a and #2b, the website of the National Institute of Standards and Technology (NIST,

54 Light Water Reactor Materials

http://kinetics.nist.gov/janaf/) provides tabulations of ∆Gif (T ) for most important compounds.

2.7.5 Activity and activity coefficient Although the thermodynamic behavior of a constituent in solution is ultimately tied to its chemical potential, a connection between this property and the concentration of the constituent is needed. This relationship is made via the activity of a solution constituent that is related to the chemical potential by o µi = g i + RgT lnai . (2.43) o The activity ai tends to unity for pure i and µi reduces to g i , the molar Gibbs energy of pure i in its standard state. When component i becomes infinitely dilute in the solution, ai → 0 and its logarithm approaches −∞. This is also the limit of the chemical potential of i at infinite dilution. This inconvenient behavior of the chemical potential at zero concentration is avoided by using the activity coefficient in practical thermodynamic calculations. For real solutions, the activity coefficient of constituent i is the ratio of its activity and mole fraction γi = ai/xi . (2.44a)

Combining the above two equations gives o

µi = g i + RgT lnγi + RgT lnxi

.

(2.44b)

The first term on the right side represents the property of pure component i; the second term accounts for nonideality; and the third term is incorporated into the entropy of mixing of Equation (2.37). A useful connection between the activity coefficients of constituents in a solution is obtained by eliminating ai between Equations (2.43) and (2.44a) and substituting the resulting equation into the Gibbs-Duhem

Chemical Thermodynamics 55

equation, Equation (2.41). For two-component (A-B) solutions, this procedure yields xAdlnγA + xBdlnγB = 0 .

(2.45)

The activity coefficient is an important characterization of nonideal behavior in condensed solution phases. The significance of Equation (2.45) is that measurement of the activity coefficient of one constituent as a function of composition determines the activity coefficient of the other constituents by integration (of Eq [2.45]).

2.7.6 excess properties An alternative to partial molar properties as a means of characterizing nonideality in solutions is the concept of excess properties. A partial molar property and the corresponding excess property are not independent quantities [1, p. 194]. Instead of Equation (2.38b), the Gibbs energy of a binary solution can be expressed by g = g o + g ex − T ∆smix = x A g oA + x B g oB + h ex − Ts ex − T ∆smix

(2.46)

where g oA and g oB are the Gibbs energies of the pure constituents in the standard state (po = 1 atm) at the specified temperature. These terms and T∆smix represent the Gibbs energy of the solution if it were ideal. The nonideal features are contained in the g ex term, which has been broken into enthalpy and entropy contributions according to the definition of the Gibbs energy in Equation (2.1b). The motivation of this last step arises from the possibility of attaching physical meaning to hex and sex. Formulas analogous to Equation (2.46) apply to other thermodynamic properties. For example, the volume and enthalpy of a binary solution are given by v = xAvA + xBvB + vex and h = xAhA + xBhB + hex . (2.47)

=

56 Light Water Reactor Materials

The excess property that governs nonideality in solutions is the excess Gibbs energy, g ex. This is determined from Equation (2.39), wherein the chemical potentials are expressed by Equation (2.43), giving g = xA( g oA + Rg T lnaA)+xB ( g oB + Rg T lnaB )= xA g oA + xB g oB +Rg T ( xA lnaA+xB lnaB )

where aA and aB are the activities of A and B in the solution. Activities can be expressed in terms of activity coefficients by use of Equation (2.44a), so the above equation is g = (xA  g oA + xB g oB) + RgT(xAlnxA + xBlnxB) + RgT(xAlnγA + xBlnγB) . (2.48a) The middle term is the ideal mixing entropy, so the last term is the excess Gibbs energy: g ex = RgT(xAlnγA + xBlnγB) . (2.48b) Since gex = hex − Tsex, the right side of Equation (2.48b) includes both hex and sex. The excess volume and enthalpy are directly measurable, but a direct measure of gex requires measurement of activity coefficients, which are usually obtained from measurements of the partial pressures of A and B over the solution. Example #3: Excess enthalpy and excess volume When 50 cm3 of 20oC water is mixed with 50 cm3 of sulfuric acid at the same temperature in an insulated vessel, the temperature rises to 123oC, and after cooling, the final volume is 90 cm3. These changes are direct measures of vex and hex. The moles of each component are: nw = 50 cm3 × 1 g/cm3 ÷ 18 g/mole = 2.78 moles water nA = 50 cm3 × 1.83 g/cm3 ÷ 98 g/mole = 0.93 moles acid nw = 2.78/3.71 = 0.75; xA = 0.25 The molar volumes of the pure components are: v w = 18 g /mole ÷ 1 g /cm 3 = 18 cm 3/mole v A = le v A = 98 g /mole ÷ 1.83 g /cm 3 = 53.6 cm 3/mole

Chemical Thermodynamics 57

For an ideal solution, vid = xwvw + xAvA = 0.75 × 18 + 0.25 × 53.6 = 26.9 cm3/mole. The measured volume per mole: v = 90/3.71 = 24.3 cm3/mole. From Equation (2.44a): ∴ vex = 24.3 − 26.9 = − 2.6 cm3/mole Take the standard states of the two liquids to be at room temperature, so hw = hA = 0 and from Equation (2.47), h = hex. By cooling, heat is removed from the solution, which means that its enthalpy has decreased, so the final enthalpy of the solution is h = −C p ∆T = − (x w C pw + x AC pA ) ∆T where — Cp is the average heat capacity of the two liquid mixture. Cpw = 4.19 J/g.K × 18 g/mole = 75.4 J/mole.K; CpA = 1.38 J/g.K × 98 g/mole = 135.2 J/mole.K ex

∴ h = −(0.75 × 75.4 + 0.25 × 135.2)(123 − 20) = −9300 J/mole

In this example, water and sulfuric acid interact strongly and produce highly negative deviations from ideality.

2.7.7 regular solutions For ideal solutions, both hex and sex are zero. The behavior of a fair number of nonideal condensed solution phases can be adequately represented by the regular solution model. In this model, the molecules mix randomly as they do in ideal solutions, so that sex = 0 and the excess Gibbs energy reduces to the excess enthalpy. The analytical formulation of hex in terms of composition is restricted by the limiting behavior as the solution approaches pure A and pure B. In these limits, hex must be zero at xA = 0 and at xB = 0. The simplest function that obeys these restraints is the symmetric expression hex = ΩxAxB

(2.49)

58 Light Water Reactor Materials

where Ω is a temperature-independent property of the A–B binary pair called the interaction energy. The form of Equation (2.49) is supported by molecular modeling, which suggests that Ω is equal to the difference between the energy of attraction (bond energy) of the A–B pair and the mean of the bond energies of the A–A and B–B interactions. For regular solutions (i.e., solutions that obey Eq. [2.49]), the activity coefficients can be shown to be ([1, Sec. 7.7.2]) R g T lnγ A = Ωx B2

and

R g T lnγ B = Ωx A2 .

(2.50)

These activity coefficients satisfy the Gibbs-Duhem equation, Equation (2.41).

2.7.8 Mixtures of ideal gases Contrary to condensed-phase solutions, gas mixtures are generally nearly ideal (Sec. 2.3.1). Consider the following isothermal processes: Process No. 1: Two pure ideal gases (xA moles A and xB moles B) at the same p are combined at constant pressure. The mixture occupies a volume V equal to the sum of the initial volumes. Each constituent obeys the ideal gas law, but its pressure is termed the partial pressure. Denoting these by pA and pB, the ideal gas law applies to each constituent: pAV = nARgT    pBV = nBRgT . (2.51) The total pressure of the mixture (as measured by a gauge) is the sum of the partial pressures: p A + pB = p , (2.52) and n = nA + nB is the total moles of gas. The mixture also obeys the ideal gas law (Sec. 2.3.1), as can be seen by

Chemical Thermodynamics 59

adding Equations (2.51) and using Equation (2.52) to eliminate the partial pressures: pV = nRgT . Dividing each of Equations (2.51) by the above equation relates the partial pressures to the mole fractions: pA n A pB n B = = xA = =x . (2.53) p n p n B Equations (2.52) and (2.53) are known as Dalton’s law. The mixing rules for ideal gases for the volume, internal energy, and enthalpy follow the generic form of Equation (2.39). However, in addition, the Gibbs energy of the mixture must also include the entropy of mixing. In particular, the entropy change of process No. 1 (above) is ∆smix of Equation (2.37). An important related question is: How are the chemical potentials of a mixture of xA moles of A and xB moles of B at temperature T and 1 atm related to the partial pressures in the mixture? To answer this question, we consider isothermal process No. 2 divided into two steps: Process No. 2: 2a. The pure gases are reduced from 1 atm pressure to their partial pressures in the mixture pA and pB. 2b. The pure gases are mixed. Since the total pressure p = 1 atm and the mole numbers xA and xB are fixed, the partial pressures in the mixture are given by pA = xA p and pB = xB p . (2.54a) To calculate the Gibbs energy change of step 2a, we use the equation obtained by combining Equation (2.16a) with the ideal gas law. For constant temperature, this gives RgT dg i = vi = (2.54b) dpi pi

60 Light Water Reactor Materials

where i = A or B. Integrating from initial pressures p oA and p Bo to partial pressures pA and pB, g A = g oA + R g T ln ( p A /p oA )

g B = g oB + R g T ln ( p B /p Bo ) . (2.54c)

g oA and g oB are the molar free energies of A and B at the initial pressures (and the specified temperature). The initial pressures are chosen as 1 atm, and the condition is known as the standard state of the pure gas. The Gibbs energy at the termination of step 2a is g = x A g A + x B g B = x A( g oA + R gT ln p A ) + x B ( g oB + R gT ln p B ) . In step 2b, there is no change in either the entropy or the Gibbs energy (see [1, Fig. 7.3]). From Equation (2.39) and the above equation, g = x A µ A + g B µ B = x A ( g oA + R g T ln p A ) + x B ( g oB + R g T ln p B ) from which µ A = g oA + R g T ln p A

µ B = g oB + R g T ln p B

.

(2.55)

Equation (2.55) relates the partial pressures to the chemical potentials. The partial pressures pA and pB must be in units of atmospheres.

2.8 Two-Phase Equilibria An important application of the thermodynamics of solutions summarized in the preceding section is to the analysis of equilibrium of multiple components distributed between multiple phases. The most common combinations are a gas phase and one or two condensed phases. The latter includes liquid–solid, liquid–liquid, and solid–solid pairs. The two coexisting phases are denoted by I and II, but the number of components is restricted to two, labeled A and B. The total Gibbs energy of this two-phase mixture is G = GI + GII. A change in the state of the system at constant temperature

Chemical Thermodynamics 61

and pressure is provoked by moving dnAI moles of component A from phase I to phase II. If the system is at equilibrium, this movement does not change the system’s Gibbs energy, and Equation (2.26) results in dG = dGI + dGII = 0. The Gibbs energy changes of each phase are related to the chemical potentials according to Equation (2.40a), which yields µA(I)dnA(I) + µB(I)dnB(I) + µA(II)dnA(II) + µB(II)dnB(II) = 0

where nA(I) ….nB(II) are the numbers of moles of each constituent in each phase and µA(I)….µB(II) are their chemical potentials. Conserving component A, dnA(II) = −dnA(I), and because component B is not exchanged, dnB(I) = dnB(II) = 0. Applying these constraints to the above equation yields µA(I) = µA(II)

.

(2.56)

A similar equation applies to component B. Equation (2.56) is the multicomponent generalization of the equilibrium condition for two coexisting phases of a pure substance; namely, gI = gII, where g is the molar Gibbs energy. If phase I is a gas and phase II a condensed phase, the equilibrium criterion of Equation (2.56) becomes µA(G) = µA(L). Using Equation (2.55) for µA(G) and the combination of Equations (2.43) and (2.44a) for µA(L), the equilibrium condition becomes pA  g oA(g ) − g oA(L)  . = exp  − γ Ax A R g T   For pure A, µA(L) = g oA(L) and µA(g) = g oA(g) + RgT lnpsat,A, so the right side of the above equation is the saturation pressure of pure liquid A, and the above equation becomes p A = γ A x A psat,A .

(2.57)

This equation forms the basis for all analyses of phase equilibria in multicomponent systems involving a gas phase. A formula similar to Equation (2.57)

62 Light Water Reactor Materials

applies to component B, and to all other components if the system contains more than two constituents. It is equally valid for a solid solution of A and B. The dependence of pA on composition is in general nonlinear because the activity coefficient of A in the condensed phase, γA, is a function of composition if the solution is nonideal and not infinitely dilute in A.

2.8.1 raoult’s and Henry’s laws If the solution is ideal, γA = 1 for all xA, and Equation (2.57) reduces to Raoult’s law: pA = xA psat,A . (2.58) Component B also obeys Raoult’s law because Equation (2.45) shows that if γA = 1, then γB = 1 as well. In nonideal solutions, γA → constant ≠ 1 as the solution becomes dilute in A. Equation (2.57) reduces to Henry’s law: pA = kHAxA (2.59) where the Henry’s law constant, kHA, is the product of the compositionindependent activity coefficient of A in solution and the saturation pressure of pure A. In the concentration range where A follows Henry’s law, component B must obey Raoult’s law. This is a consequence of the Gibbs-Duhem equation, Equation (2.41). Figure 2.3 shows typical examples of nonideal solution behavior. The curves represent Equation (2.57) for positive and negative deviations from ideality (i.e., γA > 1 or γA < 1). The limiting cases of Raoult’s and Henry’s laws are shown as dashed lines.

2.8.2 Binary phase diagrams Equilibrium between condensed phases in two-component systems is displayed as a binary phase diagram.1 Most common phase diagrams 1

See Chapter 10 for a more extensive treatment of this topic.

Chemical Thermodynamics 63

Henry’s law

psatA

r

vio

lb

ea

n

-id

No

t’s ul

w

t’s No la nw ide al be ha vio r

a eh

la

R ao ul

Partial pressure of A (pA)

psatA

ao

R

Henry’s

0

law

0 0

XA (a) Positive deviation

1

0

XA (b) Negative deviation

1

FiGure 2.3: Equilibrium pressures of component A over an A–B solution. Temperature is fixed.

illustrate regions of a single- or two-phase stability in plots with temperature as the ordinate and composition as the abcissa. The total pressure is constant (usually 1 atm). The gas phase is usually ignored in this representation because the effect of pressure on the phase diagram is small. In two-phase regions, Equation (2.56) and the analogous equation for component B serve to determine the compositions of the two phases. The chemical potentials of A and B in each phase are expressed in terms of composition by eliminating the activity between Equations (2.43) and (2.44a). Melting in an ideal system is illustrated with Phase I (a solid) and phase II (a liquid). All four activity coefficients are unity (ideal behavior is assumed). With these restrictions, the conditions of equilibrium become gAL + Rg T lnxAL = gAS + Rg T lnxAS (2.60a) gBL + Rg T lnxBL = gBS + Rg T lnxBS .

(2.60b)

The differences gAL − gAS and gBL − gBS are the Gibbs energy changes on melting of the pure constituent. These are related to the melting properties

64 Light Water Reactor Materials

of A and B by Equation (2.31a). Since xAL + xBL = 1 and xAS + xBS = 1, the above equations contain two unknowns. Solving yields x BL =

with

 1 − exp (α )  1 − exp (α ) and x BS = exp (β)  (2.61) exp (β) − exp (α ) exp (β) − exp (α ) 

α=

1 − T  ∆h MA  TMA  R g T

β=

and

1 − T  ∆h MB . (2.62)  TMB  R g T

Figure 2.4 shows the phase diagram for an ideal binary system calculated from Equation (2.61) using the melting properties of metals U and Zr. In the left panel of Figure 2.4, the upper line (representing T versus xB(L)) is called the liquidus. All points lying above this line are completely liquid. Similarly, all points below the lower curve (the solidus, or T − xB(S)) are completely solid. In the region bounded by the solidus and the liquidus, two phases coexist. In the right panel of Figure 2.4, horizontal and vertical lines are superimposed on the phase diagram to illustrate important characteristics of the melting process. If the solid solution with a composition xB = 0.4 is heated, 2200

2000

liquid (L)

1800

Temperature (K)

Temperature (K)

2200

S+L

1600 solid solution (S)

1400 1200

2000 1800

E C B

1600

P

A

D

1400 1200

0

0.2

0.4

0.6

0.8

Mole fraction Zr (component B), x

1

0

0.2 0.4 0.6 0.8 Mole fraction Zr (component B), x

1

FiGure 2.4: Phase diagram of the U-Zr binary system with ideal behavior in both liquid and solid.

Chemical Thermodynamics 65

for example, the intersection of the vertical line with the solidus (at point A) shows that the first liquid appears at 1630 K and has a composition xBL = 0.21 (at point B). As the temperature is increased to 1700 K, the system lies at point P. Here a liquid phase with composition xBL = 0.31(point C) and a solid phase with xBS = 0.49 (point D) coexist. The fraction of the mixture present as liquid at point P is obtained from the mole balance known as the lever rule: x −x Fraction liquid at point P = PD = BS B = 0.49 − 0.4 = 0.50 . (2.63) CD x BS − x BL 0.49 − 0.31 Upon heating from point P, the last solid disappears at T = 1790 K (point E). Melting of this binary system at this particular overall composition is spread over a 160-K temperature range. In the above example, both solid and liquid phases were assumed to behave ideally. However, the majority of condensed-phase systems exhibit nonideal behavior. In a limiting version of such a system, the condensed phases exhibit positive deviations from ideality, so that there is negligible solubility of A in B or B in A. That is, molecules of B and A repel each other so strongly that significant solutions of one into the other are not possible. On the other hand, A-rich and B-rich liquid solutions exist and may even exhibit negative deviations from ideality (attraction between A and B on the molecular level). For pure solid2 A in equilibrium with an A-rich liquid, Equation (2.56) becomes g AS = g AL + R g T ln(γ AL x AL ) . (2.64) Solving for xAL, the A-rich portion of the phase diagram is expressed by x AL = 2

exp(−α ) γ AL

(2.65)

The solid cannot be pure A else xBS = 0 and Equation (2.60b) is invalid. However, Equation (2.65) is a close approximation to xAL.

66 Light Water Reactor Materials

where α is the temperature-dependent function given by Equation (2.62). To complete this portion of the phase diagram, γAL must be known as a function of temperature and composition (e.g., by Eq. [2.49] if the liquid obeys regular solution theory). An analogous treatment applies to the portion of the phase diagram in which pure solid B coexists with a B-rich liquid. The equilibrium condition is given by Equation (2.64) with A replaced by B, and the compositiontemperature equation is exp(−β) x BL = γ BL (2.66) with β given by Equation (2.62). Again, γBL must be known as a function of temperature and composition. The gold-silicon binary system shown in Figure 2.5 is representative of systems that exhibit this type of phase behavior. With A = Au, the Au-rich liquidus in the figure is a plot of Equation (2.65) and the Si-rich liquidus 1800

Temperature (°C)

1400

Liquid

Au-rich liquidus

1000

Si-rich liquidus Si + L

Au + L

600 Eutectic

Au + Si

200 0 Au

0.2

0.4 0.6 Mole fraction silicon

FiGure 2.5: The gold-silicon phase diagram.

0.8

1.0 Si

Chemical Thermodynamics 67

Temperature

c L g

α+L

f

b

L+β

α

a

Pure A

o g′ d

β n

m f′

α+β

Mole fraction B

Pure B

FiGure 2.6: Generic eutectic phase diagram.

represents Equation (2.66). The two liquidus curves intersect at a point called the eutectic (Greek for “lowest melting”). At this point, three phases coexist: the two pure solids and the liquid of the eutectic composition. At lower temperatures, only the two pure solids are present. When A and B are mutually soluble in each other, the phase diagram takes on the general appearance of the one shown in Figure 2.6. At this juncture, it is constructive to examine how binary phase diagrams relate to the phase rule (Sec. 2.6). For a two-component system, Equation (2.34) permits F = 4 − P degrees of freedom. The diagrams deal only with condensed phases and are minimally affected by total pressure. Ignoring the total pressure reduces the number of degrees of freedom by one, thereby allowing 3 − P properties to be independently varied. In the single phase (P = 1) portions of the phase diagram, two degrees of freedom are permitted. These are the temperature T and the composition, represented by the mole fraction of one of the constituents, say xB. Singlephase regions appear as shaded areas in the phase diagram. In two-phase regions (P = 2), only one system property can be specified. Fixing the temperature, for example, determines the compositions of the two coexisting condensed phases. These temperature-composition

68 Light Water Reactor Materials

relationships appear in the phase diagram as lines (or curves) called phase boundaries. A three-phase system (P = 3) has no degrees of freedom and is represented by a point on the phase diagram. Examples of these points are labeled “eutectic” in Figure 2.5 and “d” in Figure 2.6. The distinction between overall compositions and the compositions of individual phases is essential to understanding phase diagrams. For single-phase zones, the two are identical. When two phases coexist, the overall composition is the mole-weighted average of the compositions of the two phases (i.e., the lever rule, Eq. [2.63]). When A and B are mutually soluble, the phase diagram typically looks like Figure 2.6. Compared with Figure 2.5, two single-phase regions, labeled α and β, have been added to the generic eutectic phase diagram. The pure A component melts at point c and pure B melts at temperature o. The α phase retains the crystal structure of pure A, but some component B is dissolved in it, usually by substituting for A atoms in the lattice. The curves ab and bc represent the terminal solubility of B in A. Compositions along these curves are the maximum concentrations of B that the α phase can sustain. Starting from point f on ab, addition of B moves the system to point f ′, which is attained by precipitation of solid β at the composition along the terminal solubility line between points n and m. If B is added at a temperature above point d, a line gg ′ similar to ff ′ ends with precipitation of liquid L at a composition g ′ on the liquidus d. The iron-uranium phase diagram of Figure 2.7 exhibits two eutectic points located between high-melting entities called intermetallic compounds. The vertical lines at uranium atomic percentages of 33 and 86 represent Fe2U and FeU6, respectively. These are true compounds, with fixed and invariant Fe/U ratios, that are crystallographically distinct from the pure metals and have definite melting points. They form the boundaries of eutectic features in the phase diagram. Except for the presence of three allotropes of pure iron (the αFe, γFe, and δFe phases), the portion of the diagram between Fe and Fe2U is the same as in

Chemical Thermodynamics 69 1600 1538°C

1500 L+ δ Fe ∼1394°C L

1300

1228°C L+γFe

1100

1080°C

L + Fe2U

A

17.2 γFe+Fe2U ∼912°C

1000 900 800

L + Fe2U

Fe2U

725°C

600 0 Fe

10

20

C

L+Fe2U

αFe+ Fe2U

700

1135°C B

L+ γU

1200

L+FeU6

66 Fe2U+FeU6

70 60 50 40 30 Atomic percent uranium

FeU6

Temperature (°C)

1400

80

γU 778°C

FeU6+γU

βU 665°C αU

FeU6+βU FeU6+αU

90

100 U

FiGure 2.7: The iron-uranium phase diagram.

Figure 2.5. The zone between FeU6 and pure U is complex, owing to the presence of the three phases of uranium (αU, βU, and γU) and the limited solubility of Fe in these phases. The αU, βU, and γU phases indicated on the extreme right of the diagram are analogs of the β region in Figure 2.6. Except in these three single-phase solids and the liquid region, a point in Figure 2.7 indicates the coexistence of two phases. As illustrated in Figure 2.6, the coexisting phases lie at the intersections of an imaginary horizontal line at the temperature and between the boundaries of adjacent single phases. The lever rule (Eq. [2.63]) gives the relative proportions of the two phases.

2.9 Chemical Equilibrium Chemical reactions entail exchange of atoms between molecules. A system is designated as homogeneous if all constituents involved are contained

70 Light Water Reactor Materials

in a single phase, or heterogeneous if the constituents are in two or more phases. Typically, a homogeneous system may be a gas, a solid, or a liquid. A common heterogeneous system is metal oxidation, which involves a gas or a liquid (containing oxygen) and two solid phases (the metal and its oxide). Equilibrium analysis of chemical reactions provides an equation relating the mole fractions of all constituents. When supplemented with specified ratios of the elements involved, the composition of the equilibrium system is fixed. These generalities can be made more specific by considering the generic reaction between reactant constituents A and B to form product constituents C and D: aA + bB = cC + dD .

(2.67)

The coefficients of the reactant and product constituents (a, b, c, and d) are the stoichiometric coefficients or balancing numbers that conserve elements on the two sides of the reaction. The equal sign indicates that the four constituents are present at equilibrium. At equilibrium, there is no distinction between reactants and products; Equation (2.67) could just as well have been written with C and D on the left and A and B on the right. As in any system at constant temperature and pressure, thermodynamic equilibrium is attained when the Gibbs energy of the system containing all constituents involved is a minimum, or dG = 0. Extending the second equality in Equation (2.40a) to include constituents C and D yields µ A dn A + µ B dn B + µ C dnC + µ D dn D = 0

.

The changes in the mole numbers, dnA, … dnD, are related to each other by the balancing numbers in Equation (2.67). For example, for every a moles of A consumed, b moles of B disappear, and c and d moles of C and D, respectively, are produced. These stoichiometric restraints are equivalent to: dn B = b/a dn A ;dnC = − c/a dn A ; dn D = − d/a dn A, and the equilibrium condition reduces to

Chemical Thermodynamics 71

aµ A + bµ B = cµ C + dµ D .

(2.68) The equation applies to an equilibrium in a single phase or in multiple phases.

2.9.1 Law of mass action For a homogeneous gas-phase reaction, the chemical potentials are expressed in terms of the partial pressures by Equation (2.55), and Equation (2.68) becomes pCc p Dd  ∆G o   cg Co + dg oD − ag oA − bg oB  = K p = a b = exp  − exp   − R T  (2.69a) RgT pA pB  g where Kp is the equilibrium constant in terms of partial pressures. It depends on both temperature and total pressure. The effect of the latter variable can be made explicit by replacing the partial pressures using Dalton’s law Equation (2.53). With this substitution, the equilibrium constant becomes Kp = Kpm where m=c+d−a−b (2.70) x Cc x Dd (2.71) K= a b . xAxB K is the equilibrium constant in terms of gas-phase mole fractions, and is usually the preferred method for expressing equilibrium in a mixture. Equations (2.69a) and (2.71) relating compositions to Kp and K are sometimes called the law of mass action. Equation (2.67) is applicable to reactions involving both solids and gases. For example, the equilibrium constant for the reaction aA(g) + bB(s) = AaBb(g) is

Kp =

p Aa B b , p Aa

(2.72a) (2.72b)

72 Light Water Reactor Materials

provided that B is a pure solid, for which the activity is unity. The subscript AaBb denotes the molecule with this formula. An important class of reactions involves solution species as well as gases and pure phases. A type is Mz+(sol’n) + ½zH2(g) = M(s) + zH +(sol’n)

(2.73a)

where z is the valence of the metal ion. “sol’n” denotes an ion in an aqueous solution for which the chemical potential is given by a modification of Equation (2.44b): o (2.73b) µ i = µ i + R g T ln[i ] . µ oi is the chemical potential of species i in its standard state, which is 1 M

aqueous solution of i. [i] denotes the concentration of species i in moles per liter (molarity, symbol M). Equilibrium is defined by the general formula, Equation (2.68), applied to reaction (2.73a). Consider that • for the solution species the chemical potential in the form of Equation (2.44b) with xi replaced by [i], • for the gas as applied to Equation (2.72a), and • for pure solid components in which there is no composition dependence, the result is o

∆µ = µ oM + 2µ oH + − µ oM z + −

1 o zg 2 H2

 [H+ ]z  . (2.73c)  [M z+ ] pHz /22 

= − R g T ln 

The ratio in parentheses is the equilibrium constant for reaction (2.73a). For a reaction with equilibrium constant K, the aqueous analog of Equation (2.69a) is o (2.69b) ∆µ = − R g T lnK .

Chemical Thermodynamics 73

2.9.2 Standard Gibbs energy of reaction ∆G

o

in Equation (2.69a) is the standard Gibbs energy of reaction (2.67). It is expressed in terms of the Gibbs energies of formation ∆Gif (see Eq. [2.42a]). For reaction (2.67), o

f

f

f

f

∆G = c ∆GC + d ∆G D − a∆G A − b∆G B

,

(2.74a)

the superscript o indicates that it is to be evaluated at 1 atm pressure. The effect of pressure is contained in the integer m in Equation (2.70). Equation (2.71) applies to reactions in which the reactants and products are in any phase, not just the gas phase. If one or more of the components of the reaction are liquid or solid, their stoichiometric coefficients are not included in m of Equation (2.70). One of the most convenient sources of ∆Gif is the compilation known as the JANAF (stands for Joint Army-Navy-Air Force) tables, which have been maintained by NIST. Knowledge of ∆Go permits Kp to be obtained from Equation (2.69a). Armed with the equilibrium constant Kp, Equation (2.71) determines the composition of the equilibrium mixture. The closed system initially contains specified moles of constituents A, B, C, and D. In achieving equilibrium, the initial mole numbers change to new values. Let ξ be the number of moles of A reacted in achieving equilibrium. The changes in the number of moles of B, C, and D are related to a ξ by the stoichiometric coefficients of reaction (2.67). Table 2.1 gives the initial and final (equilibrium) mole numbers. In the last column, nT0 is the sum of the initial moles of the four constituents, and m is the combination of the stoichiometric coefficients shown in Equation (2.70). The mole fractions in the equilibrium system are obtained by dividing the moles of A, B, C, and D in the last row by the total moles in this row.

74 Light Water Reactor Materials Moles Initial Equilibrium

A

B

C

D

Total

n A0

n B0

nC0

n D0

nT0

n A0 − aξ

n B0 − bξ

nC0 − cξ

n D0 − d ξ

nT0 − mξ

TABLe 2.1: Initial and equilibrium mole numbers in gas-phase reaction aA + bB = cC + dD

Substituting these mole fractions into the law of mass action given by Equation (2.71) yields

(nC0 + cξ)c (n D0 + d ξ)d K= . (n A0 − aξ)a (n B0 − bξ)b (nT0 + mξ)m

(2.74b)

This equation is solved for ξ (in general, a numerical solution is required), and from this result, the mole fractions at equilibrium are calculated. Example #4: Combustion of 1 mole of methane by 2 moles of oxygen at 2000 K. The reaction is CH4(g) + 2O2(g) = CO2(g) + 2H2O(g) .

(2.75)

The letter g in parentheses following each constituent indicates it is a gas. Application of the phase rule: Two degrees of freedom are taken by fixing temperature and total pressure, so the remaining degrees of freedom are (Eq. [2.34]) F = C − P = 3 (elements C, O and H) − 1(gas phase only) = 2. One degree of freedom is consumed by the equilibrium of reaction (2.75).

Chemical Thermodynamics 75

The last degree of freedom is eliminated by specification of the 2:1 ratio of O2:CH4 in the initial charge. From the NIST tabulation, the formation reactions and their Gibbsenergies (in kJ/mole) at 2000 K are as follow: f

H2(g) + ½O2(g) = H2O(g)

∆GH O = −136

C(s) + O2(g) = CO2

∆GCO = −395

C(s) + 2H2(g) = CH4I

∆GCH = −131

(a)

2

f

(b)

2

f

(c)

4

Algebraically, reaction (2.75) = 2(a) + (b) − (c), so the standard Gibbs energy change of the reaction is f

f

f

∆G = ∆GCO 2 + 2 ∆GH2O − ∆GCH 4 o

= −395 + 2(−136) − (−131) = − 798 kJ/mole

. According to Equations (2.69a) and (2.70), the equilibrium constant is Kp = K = 7.5 × 1020. To solve for the composition of the equilibrium gas, a table similar to Table 2.1 is constructed with A = CH4, B = O2, C = CO2, and D = H2O. Second, Equation (2.74b) is specialized for this reaction. In the latter method, the input parameters are: a (CH4) = 1; b (O2) = 2; c (CO2) = 1; d (H2O) = 2; m = 0 and n A0 = 1; n B0 = 2; nC0 = 0; n D0 = 0 . Substituting these values into Equation (2.74b) yields 3

 ξ  7.4 × 1020 =   .  1 − ξ In this case, an analytical solution is possible, yielding 1 − ξ = 2.1 × 10−7. The corresponding mole fractions at equilibrium are xA = 4 × 10−8; xB = 8 × 10−8; xC = 0.333; xD = 0.667. The reaction goes nearly to completion (i.e., essentially all reactants are consumed).

76 Light Water Reactor Materials

2.9.3 Stability diagrams The class of heterogeneous reactions in which an element reacts with a diatomic gas to form a compound is both of practical importance and amenable to simple thermodynamic analysis. Reactions in this category include oxidation, nitriding, and hydriding of metals and halogenation of the electronic material silicon. The simplicity of the thermodynamics stems from the immiscibility of the reactants and products. Consider oxidation of a metal (M) to form a dioxide (MO2): M(s) + O2(g) = MO2(s) .

(2.76)

The letter “s” in parentheses indicates a solid phase; in this case, there are two solid phases because the metal and its oxide are essentially insoluble in each other. Since M and MO2 are practically pure substances, their chemical potentials are equal to their molar Gibbs energies. The chemical potential of oxygen gas is dependent on its partial pressure, and is given by Equation (2.55). With a = b = c = 1 and d = 0, the general criterion, Equation (2.68), becomes o g Mo + g Oo 2 + R g T ln p O 2 = g MO . 2

(2.77)

Rearranging this equation into more convenient forms gives f

∆G MO

ln p O 2 or where

=

R g T ln p O 2

RgT

f

∆H MO

2

=

f

f

∆S MO

2

−

RgT f

2

(2.78a)

Rg f

= ∆G MO = ∆H MO − T ∆S MO

2

(2.78b)

o o = go ∆G MO = g MO − g Oo − g M MO 2 2 2 2

(2.79)

f

2

2

is the standard Gibbs energy of formation of the oxide MO2. By definition, the standard Gibbs energies of formation of the elements are zero.

Chemical Thermodynamics 77

lnpO2

MO2 stable

Slope = M stable

∆H fMO2 Rg

RgT lnpO2

MO2 stable Slope = –∆SfMO2 M stable Intercept =∆H fMO2

1/T

T

(a)

(b)

FiGure 2.8: Stability diagrams for the M + MO2 couple.

The standard state is the substance at a total pressure of 1 atm and the f f and ∆ SMO specified temperature. ∆ H MO are the enthalpy and entropy of 2 2 formation (reaction [2.72a]). The former is the heat released when one mole of metal is oxidized. Equations (2.78a) and (2.78b) are plotted in Figure 2.8. These plots are called stability diagrams because the lines separate regions in which only one of the two phases is present. The line represents the pO2 −T combinations where both the metal and its oxide coexist. The oxide-metal stability diagram is a solid-phase analog of the p−T phase diagram of a single substance such as water, where lines separate regions of solid, liquid, and vapor phases (see Fig. 2.2). Example #5: Oxygen pressure over Ni/NiO Powdered nickel metal is contacted with a flowing mixture of CO2 and CO at 1 atm total pressure in a furnace at 2000 K. The quantity of metal is limited, but because of continual flow, the quantity of the gas mixture is unlimited. Therefore, the oxygen pressure established in the gas phase is imposed on the metal and determines whether or not it oxidizes. At what CO2/CO ratio do both Ni and NiO coexist? The O2 pressure in the gas is fixed by the equilibrium: 2CO(g) + O2 (g) = CO2(g). At 2000 K, the equilibrium constant is Kp = 4.4 × 105.

78 Light Water Reactor Materials 0

Temperature (°C) 2000 2500 1000 1500

500

2927

100 0 –100 PbO2

–200 Fe3O4

–300 –400 DGfMaOb (kJ/mole)

MgO

Fe2O3

B

B –500

Na2O

SiO2 CaO Al2O3 Eu2O3

MnO

B

–600

Yb2O3 Sm O 2 3

–700

B2O3

TiO2

EuO

–800 –900

B

MgO

CeO2

–1000 –1100

B

Pr2O3

Al2O3

M2O3, M = Tb, Dy, Ho, Er,Tn, Lu,Sc, Y

CaO Eu2O3 N2O3, N = La,Co, Pr, Nd, Sm, Gd, Yb

–1200 –1300 0

400

800 1200 1600 2000 2400 2800 3200 Temperature (K)

FiGure 2.9: Ellingham diagram for metal/metal oxide combinations. B is the boiling point of the metal.

The law of mass action for this equilibrium reaction is ( pO2 ) gas = ( pCO2 /pCO )2 /K p .

Chemical Thermodynamics 79

The nickel/nickel oxide equilibrium reaction is: 2Ni + O2 = 2NiO, for f 3 which ∆GNiO = −46 kJ/mole at 2000 K. According to Eq (2.71a), for coexisting Ni and NiO in the solid phase, the oxygen pressure is f ( pO2 )solid = exp( ∆GNiO /R g T ) = 6.3 × 10 −2 atm .

The mixed solid and the mixed gas are in equilibrium when (pO2)gas = (pO2)solid. From the above equations, this condition yields the required ratio of CO2 to CO in the gas: p CO 2 /p CO = K P ( p O 2 )solid = (4.4 × 10 5 )(6.3 × 10 −2 ) = 166 .

2.9.4 ellingham diagrams A common method of expressing the relative stabilities of metal oxides, chlorides, and fluorides is the Ellingham diagram. As shown in Figure 2.9, the temperature dependence of the standard Gibbs energies of formation of the oxides is plotted against T. The ordinate expresses ∆GMf 2 O3 per mole of O2, for the generic metal M with oxide MaOb reaction: 2a 2 M + O2 = MaOb b b The reason for the per-mole-O2 units is to permit easy assessment of the relative stabilities of the oxides. For example, application of Figure 2.9 to Al2O3 and TiO2 at 800oC yields 4 2 G Alf 2 O 3 = −975kJ /mole O 2 Al + O 2 = Al 2 O 3 3 3 O = −700 kJ /mole O G TiO Ti + O 2 = TiO 2 2 2 Subtracting the Ti reaction from the Al reaction gives 4 2 Al + TiO 2 = Al 2 O 3 + Ti 3 3 f f ∆G o = ∆G Al O − ∆G TiO = −275 kJ /mole . 2 3 2 3

The free energy of formation is per mole of O2, not per mole of NiO.

80 Light Water Reactor Materials

The negative standard Gibbs energy change indicates that the reaction favors the right side and, given sufficient aluminum, all TiO2 is reduced to the metal. Used this way, the Ellingham reactions are analogous to halfcell reactions in electrochemistry. The Gibbs energy of formation of MaOb can be written as f

f

f

∆G M O = ∆H M O − T ∆S M O a b a b a b

.

Because ∆H Mf a Ob and ∆SMf a Ob are very nearly temperature-independent, f f ∆G M O versus T is a straight line with a slope equal to −∆S M O and an a b a b f intercept at 0 K of ∆H Ma Ob given by 2 o 2a s M a Ob − s oM − sOo2 b b 2 2a H Mf a Ob = h Mo a Ob − h Mo − hOo2 b b where the lowercase letters apply to a property per mole of the constituent and the superscript o indicates the standard state of 1 atm. Because the oxide and the metal are solids, their entropies are small compared f with the entropy of O2. This means that ∆S Mf aOb ≅ − SOo2 , so that ∆G MO is 2 approximately f f ∆G M O ≅ ∆H M O + TSOo . a b a b 2 S Mf aOb

=

This is why the slopes of the lines in the Ellingham diagram are positive and nearly the same for all metals. The intercepts at 0 K (not shown in Figure 2.9) are all negative (because ∆H Mf a Ob is negative) and vary greatly with the metal. The sum of the enthalpies of the metal and oxygen is greater than that of the oxide, and most metals burn with considerable release of heat. The noble metals

Chemical Thermodynamics 81

Au, Ag, Pd, etc., are not shown in Figure 2.9. For these metals, ∆H Mf aOb is positive. The letters in Figure 2.9 indicate the three phase changes discussed in Sections 2.4.1 and 2.4.2: sublimation, vaporization, and melting. At the boiling point of the metal (B on the diagram), the slope of the line increases. The slopes of the lines with the metal liquid and vapor are slopeliquid

 2 so − f = −∆S M O = − a b  b M a Ob

slope vapor = −∆S Mo aOb

=−

2a o s M(L) − s Oo 2  b 

 2 s o − 2a s o − s o  ,  b M aOb b M(g) O 2 

so that the change in slope at the boiling point is the difference in the molar entropies of the vapor and liquid metal, or s oM(g) − s oM(L) = ∆s vap,M . Because ∆gvap,M = 0 at equilibrium, ∆svap,M = ∆hvap,M/Tvap,M. Example #6: Slope of the boiling line in Figure 2.9 For M = Al, the enthalpy of vaporization is 295 kJ/mole, and the boiling temperature (at 1 atm) is 2740 K, so ∆svap,Al = 0.11 kJ/mole.K. The slope of the Al(L) line in Figure 2.9 is ∼0.20 kJ/mole.K, so the slope of the Al(vap) should be 0.31 kJ/mole.K. The slope of the Al(vap) line from Figure 2.9 is ∼0.35 kJ/mole.K, which is close enough. There is a similar change in slope at the melting point of the metal. However, the enthalpy of melting divided by the melting temperature is too small for the expected change in slope to be visible in Figure 2.9.

2.10 Aqueous Electrochemistry and Ionic Reactions The chemical reactions between aqueous ions and metals are called electrochemical reactions. Such reactions are the source of some of the most important materials problems in the nuclear industry. Corrosion of the

82 Light Water Reactor Materials

metal components in the core and the primary circuit of light water reactors are prime examples of practical electrochemistry (see Ch. 15). The elements that, as ions, take part in electrochemical reactions often have more than one valence state, or oxidation state (z), that are stable in, or in contact with, water. Iron, for example, commonly occurs in the elemental (Fe0), ferrous (Fe2 +), and ferric (Fe3+) oxidation states. Hydrogen occurs as the diatomic molecule H2 dissolved in water or as the hydrogen ion H +. The forms of oxygen include OH − and the dissolved gas O2. Both hydrogen and oxygen appear in H2O2, which often takes part in electrochemical reactions. For an element to participate in an electrochemical reaction, it must change valence state in the reaction. For example, the overall reaction Mz + + ½zH2 = M + zH +

(2.80)

involves reduction of Mz + dissolved in water to the metal M. Accompanying the reduction of one element is the oxidation of another. In this case, hydrogen in its elemental state as H2 (either in solution or in the gas phase) is oxidized to the 1+ state in solution. Positive ions in solution are called cations, whereas negative ions, called anions, must accompany the positive ions to maintain electrical neutrality of the solution. Anions are not explicitly included in Equation (2.80). The reason is that most anions exhibit only one charge state in aqueous solution. If an ionic constituent is not capable of changing oxidation state, it cannot participate in the electrochemical reaction. Typical anions include 2− 3− Cl −, NO3− , SO 4 and PO 4 . Many (but not all) electropositive elements are immune from electrochemical effects because they exhibit only one stable oxidation state. Sodium, for example, is always present in solution as Na +; reduction to the element in water is not possible. The phases occupied by the constituents in the above overall reaction are not indicated, but are usually evident: the metal M is a solid; H2 is a gas; both cations are in solution.

Chemical Thermodynamics 83

Reaction (2.80) can be viewed as the transfer of z electrons from the hydrogen molecule to the metal ion. As a result, M decreases in valence from z + to 0 and is said to be reduced. Hydrogen increases its valence from 0 to 1+, or is oxidized. In overall electrochemical reactions, the exchange of the electrons is not physically manifest because it occurs in an intimate mixture of reactants and products. However, in a device known as an electrochemical cell, participants in a reaction are physically separated in a manner that makes it possible for the electron transfer process to be observed, utilized for doing work, or measured to provide information useful in understanding processes such as corrosion. This is accomplished by separating the overall reaction into half-cell reactions, in which the oxidation and reduction portions are shown explicitly. For example, the half-cell reactions corresponding to reaction (2.80) are Mz+ + ze = M

(2.81a)

2H + + 2e = H2 (2.81b) where e denotes an electron. By convention, the reduced species is on the right side of the equal sign. The overall reaction is (2.81a) minus ½ z of (2.81b), which gives (2.80). The rule for writing half-cell reactions is: electrons on the left.

2.10.1 Faraday constant Because electrochemistry deals with movement of electrons between species, a fundamental constant called the Faraday constant is frequently encountered. The Faraday constant F is the product of Avogadro’s number and the electronic charge: F = eNAv = (1.602 × 10−19)(6.02 × 1023) = 96,500 Coulombs/mole electrons. A Coulomb is a Joule per Volt, so F = 96.5

kJ . Volt − moleelectrons

84 Light Water Reactor Materials

2.10.2 The electric potential Figure 2.10 illustrates an electrochemical cell in which the half-cell reactions take place in individual compartments. Each half-cell contains a metal electrode. The left half-cell contains the metal M and the specified aqueous concentration of Mz +. In the right half-cell, the inert metal (platinum) electrode permits reaction (2.81b) to proceed efficiently. The concentration of H + and H2 pressure in the gas saturating the solution are both specified. The bridge separating the two compartments permits anions (associated with Mz + and H +) to move in order to maintain electrical neutrality. The two metal electrodes are connected by wires to one of two devices: a voltmeter or a battery. Shown in the figure is a meter that measures the voltage, or electric potential, between the two electrodes. In this mode, no current flows, and the system is at electrical equilibrium, but not chemical equilibrium. If, instead of the voltmeter, a battery is placed in the line, the applied voltage causes a current to flow as reactions (2.81a) and (2.81b) proceed in the two half-cells; the metal is either dissolved or electroplated, depending on the direction of current flow. The half-cell in which oxidation (increase in valence) occurs is the anode, and the one supporting reduction is the cathode. From a thermodynamic point of view, the utility of the electrochemical cell is that its electric potential is proportional to the difference in the Gibbs energies of the mixtures in the two electrodes; that is, ΦM,H is a direct measure of the equilibrium constant of the overall cell reaction (2.80). This connection is established by considering a nonequilibrium cell and equating the electrical work (cell potential times the charge transferred, z(eNAv) ΦMH) to the maximum possible work in a process at constant temperature and pressure (equal to the decrease of the Gibbs energy, Eq. [2.25]). The result is ∆G = −zF ΦMH . (2.82a)

Chemical Thermodynamics 85 ΦM,H

H2

Electrode, metal M Mz+

H+

Bridge

Inert metal electrode

FiGure 2.10a: An aqueous electrochemical cell with a metal half-cell and a hydrogen half-cell. ∆G is the difference in the Gibbs energy between the product and reactant

sides of the reaction. The cell in Figure 2.10a functions as an equilibrium cell since no current flows and no change occurs with time. However, this does not mean that the ion concentrations in the two electrodes are those that would be found if the two half-cell solutions were part of the same solution, which is equivalent to short-circuiting the cell in Figure 2.10a. In this case, the concentrations of Mz + and H + adjust until the equilibrium criterion of Equation (2.68) is satisfied for overall reaction (2.80). In the configuration of Figure 2.10a, on the other hand, the ion concentrations can be arbitrarily fixed and the potential ΦM,H reflects the imbalance of the chemical potentials in the two half-cells. Instead of the equilibrium condition of Equation (2.68), ∆G is nonzero and is given by z ∆G = µ ( M ) + z µ ( H + ) − µ ( M z+ ) − µ ( H 2 ) . (2.83) 2 The chemical potential of H2 depends on its partial pressure according to Equation (2.55):

86 Light Water Reactor Materials o

µ H = g H + R gT ln pH 2 2 2

.

(2.84a)

The chemical potential of H2 dissolved in water is the same as that of the gas (see Eq. [2.56]). • The chemical potential of pure metal M is equal to its molar Gibbs energy. • The chemical potentials of ions in solution are related to the concentrations by µ(H +) = µ

(H+) + RgT ln[H+] µ(Mz +) = µo(Mz +) + RgT ln[Mz +] . (2.84b)

o

The concentration of constituent I in moles per liter of solution (molarity, denoted by M) is [i]. The standard-state chemical potential for ion I in a 1-M solution is µo(i). Nonideality due to interaction of the ions with the surrounding water molecules is contained in the chemical potential in this standard state. All that is required is that the ion–water interaction be independent of [i], which is acceptable as long as the solution is not too concentrated in constituent i. Substituting Equations (2.84a) and (2.84b) into Equation (2.83) and eliminating ∆G using Equation (2.82a) gives Φ MH = −

∆µ o

zF

−

R g T  [H+ ]z  ln zF  [M Z+ ] pHz /22 

(2.84c)

where ∆µo = µo(M) − µo(M z +) + zµo(H +) − 1/ 2zg Ho 2 . The standard chemical potentials are converted to electric potentials of the half-cells: o o 1 o o o z o µ (M) − µ (M +) = −zF φM  ,  g H − µ (H +) = −F φH . 2 2

Chemical Thermodynamics 87 o

and φMo are the standard electrode potentials of the two half-cell reactions. All half-cell potentials are with respect to the standard hydrogen electrode (SHE), which is a H +/H2 half-cell containing [H +] = 1 M and pH2 = 1 atm. The standard electrode potential of the SHE is defined as φHo = 0. Combining the above equations gives

φH

∆µ

o o

= φH

− φMo .

zF Inserting this equation into Equation (2.84c) yields Φ MH = φMo − φHo −

R g T  [H + ]z  ln . zF  [M z+ ] pHz /22 

(2.84d)

2.10.3 The standard electrode potential Cell potentials for overall reactions such as that of reaction (2.80) are split into two half-cell potentials that characterize reactions (2.81a) and (2.81b) such that ΦMH = φM − φH. For Mz + + ze = M, RgT  1  ln zF  [M z+ ]

(2.85a)

R g T  pH 2  ln . 2F  [M+ ]2 

(2.85b)

φM − φMo −

whereas for 2H + + 2e = H2, φH − φHo −

Figure 2.10b shows the sequence of electrochemical cells that permit proceeding from standard electrode potentials to potentials of the overall cell reaction. The voltmeter reads the difference between the half-cell potentials.

88 Light Water Reactor Materials 0

(1) [H+] = 1M pH2 = 1 atm

SHE

φH

(2) [H+] pH2

SHE

φo

M

(3)

[MZ+] = 1M M

SHE

φM

(4) [MZ+] M

SHE

φMH

(5) [MZ+] M

[H+] pH2

FiGure 2.10b: Electrochemical cells.

In No. 1, a standard hydrogen electrode (SHE) is coupled to a half-cell containing H + at 1 M and hydrogen gas at 1 atm. Since the latter two conditions define the SHE, the potential of this overall cell is zero. No. 2 contains the same components as No. 1 except that the H + concentration and pH2 in the left half-cell are not unity. The potential of this H +/H2 overall cell is given by Equation (2.85b), again with φHo = 0. No. 3 couples a SHE and a half-cell containing the metal M and a 1-M concentration of the metal ion. The measured voltage is φMo , the standard electrode potential. No. 4 contains the same constituents as cell No. 3 except that the metal ion concentration is no longer specified as 1 M. The potential of this cell is given by Equation (2.85a). No. 5 mates the hydrogen half-cell of No. 2 and the metal half-cell of No. 4 to give the overall potential of the electrochemical cell in Figure 2.10b. The potential produced by this cell is

Φ MH = φM − φH = φMo −

R g T  [H + ]z  ln . zF [ M ]z+ pHz /22 

(2.86)

At this point, one might ask: Of what use are electrochemical potentials? The most important information is obtained by short-circuiting cell No. 5, which is equivalent to setting ΦMH = 0 in the above equation. This yields

Chemical Thermodynamics 89

[ M Z+ ] pHz /22  zF φMo  = exp  − . [ H + ]z  R g T 

(2.87a)

The right side is the equilibrium constant of reaction (2.80). The standard electrode potentials thus provide important equilibrium information for aqueous reactions. For a general reaction written as lhs = rhs, the law of mass action is [rhs] K= [lhs] where [rhs] denotes the product of the concentrations of solution species on the right side of the reaction. The same applies to [lhs]. The generalization of Equation (2.87a) is zF o (φox ,rhs − φoxo ,lhs ) . ln K = − (2.87b) RgT The subscripts ox,rhs and ox,lhs denote the oxidized species on the two sides of the reaction. The basic database for aqueous electrochemistry consists of the standard electrode potentials. Since most aqueous systems are at room temperature (with the notable exception of the coolant in light water reactors), a single table at 25oC suffices to accommodate the entire database. An abridged table of standard electrode potentials is given in Table 2.2. The last two columns of the top set of half-cell reactions refer to the conditions of the aqueous solution. φo for the hydrogen reaction (No. 3 in the table) is zero because this is the reference half-cell (the SHE). As shown in the examples below, the standard electrode potential for a half-cell reaction not found in Table 2.2 can often be derived from reactions that appear in the table. Standard electrode potentials are temperature-independent.

90 Light Water Reactor Materials

Half-cell reaction

φo, Volts

Involving H2 and O2 1 ½O2 + 2H + + 2e = H2O 1.23 − 1a ½O2 + H2O + 2e = 2OH  0.40 + 1.77 2 H2O2 + 2H  + 2e = 2H2O + + 2e = H 2H  0 3 2 − 2H O + 2e = H + 2OH  3a −0.83 2 2 Numbers 1a and 3a are obtained from numbers 1 and 3 using H2O = H+ + OH−. Involving metals 4 Au3 + + 3e = Au 1.50 2− + 2e = Cu Cu   0.34 5 2+ Ni   + 2e = Ni 6 −0.25 2+   + 2e = Fe Fe 7 −0.44 + Na  + e = Na 8 −2.75 Involving only ions 9 Fe3 + + e = Fe2 + 0.77 2+ + + 4 0.34 10 UO 2 + 4H + 2e = U   + 2H2O 11 PuO 22+ + 4H+ + 2e = Pu4 + + 2H2O 2.04 2+ 3+ 12 Pu   + e = Pu   0.98 + 2+ 13 Cu + e = Cu 0.16 Involving solid oxides or hydroxides 14 15 15a 16 17

2+ UO 2 + 2e = UO2(s) Fe2O3(s) + 6H + + 2e = 2Fe2 + + 3H2O FeO(s) + 2H + + 2e = Fe(s) + H2O Cu2O(s) + H2O + 2e = 2Cu(s) + 2OH − Cu(OH)2(s) + 2e = Cu(s) + 2OH −

TABLe 2.2: Standard electrode potentials.

−0.43

0.68 −0.03 −0.36 −0.22

Chemical Thermodynamics 91

Example #7a: The half-cell reaction Fe +3 + 3e = Fe This half-cell reaction is the sum of reactions 7 and 9 in Table 2.2, so φ o (Fe 3+/Fe) = φ7o + φ9o = −0.44 + 0.77 = 0.33 V

.

Example #7b: Standard electrode potential for No. 15 in Table 2.2 This reaction can be decomposed into the following: (7) Fe2 + + 2e = Fe (a) 2Fe + 3/2O2 = Fe2O3 (1) 1/2O2 + 2H+ + 2e = H2O No. 15 is obtained by algebraically combining the three reactions: (15) = −(a) − 2(7) + 3(1) . However, the electrode potential of half-cell reaction No. 15 cannot be determined simply by combining standard electrode potentials as in Example #7a because reaction (a) does not involve either aqueous ions or electrons. Instead, we only have the Gibbs energy of formation of Fe2O3 o at 25oC, ∆µ a . In order to combine the three component equations, φ1o o o and φ7o must first be converted to ∆µ1 and ∆µ 7 using the analog of Equation (2.82a), o o ∆µ i = −z F  φi , (2.82b) o

o

and the ∆µ i values combined to determine ∆µ15  : o

o

o

o

∆µ15 = −∆µ a − 2 ∆µ 7 + 3 ∆µ1

.

From Table 2.2, φio = 1.23 V, so

o

∆µ1 = −2 × 96.5 × (1.23) = −237 kJ/mole H2Ol o

and

φ7 = − 0.44 V

,

;

92 Light Water Reactor Materials o

∆µ 7 = −2 × 96.5 × (−0.44) = 85 kJ/mole Fe

so

from Figure 2.9, ∆µo (or ∆Go) of Fe2O3 is −500 kJ per mole of O2. So for reaction (a) as written, o 3 ∆µ a = (−500) = −750 kJ/mole Fe2O3 . 2  Combining o

∆µ15 = − (−750) − 2(85) + 3(−237) + = −132 kJ/mole

and using Equation (2.82b), the standard half-cell potential is − (−132) φ15o = = 0.68 V . (2 × 96.5)

2.10.4 The Nernst equation Half-cell reactions such as Equations (2.85a) and (2.85b) are written with electrons on the left, and in the corresponding potential equations, the reduced species is in the numerator and the oxidized portion in the denominator. For oxid + ne = red, φ N = φo −

R g T  red  o 2.3 R g T  red  =φ − ln log . nF  oxid . nF  oxid .

(2.88)

This general form of the half-cell potential is called the Nernst equation. At 25oC, 2.3RgT/F = 0.059 V. It is important to remember that the Nernst equation applies to half-cell reactions such as (2.81a) and (2.81b), not to overall reactions of which (2.80) is an example. The potential φN in Equation (2.88) is termed the Nernst potential of the half-cell. It is measured in a full cell relative to a SHE. n is the coefficient of e (the electron charge) in the half-cell reaction taken from Table 2.2. It need not be the same as the charge on the ion. For example, n = 2 for half-cell reactions 1 − 3a even though the charge of the moving ion is +1 or −1.

Chemical Thermodynamics 93

Example #8: Nernst potential of half-cell 10 in Table 2.2 For the following concentrations: [H +] = 0.01 M, [UO 22+ ] [U 4+ ] = 0.1 M. The Nernst equation for this half-cell is φ10N = φ10o −

=

2.0 M,

0.059  [U 4+ ]  log  2  [UO 22+ ][H + ]4 

= 0.34 −

0.059  0.1  log = 0.14V . 2  0.014 × 2.0

The Nernst equation, expressed in its general form by Equation (2.88), has three very important applications: (i) it gives the voltage (or potential) of an electrochemical cell consisting of two arbitrary half-cells; (ii) it provides a means of converting the data in Table 2.2 to the equilibrium constant of ionic reactions in a single solution; and (iii) it offers a systematic method for assessing the tendency for metals to corrode in water. The last of these applications are deferred until Chapter 14. The first two applications of Equation (2.88) are closely related and can be illustrated by the reaction by which tetravalent plutonium is reduced to the trivalent state by addition of a ferrous ion: Fe2 + + Pu4 + = Fe3 + + Pu3 + .

(2.89)

This reaction can be considered as either an overall reaction for an electrochemical cell consisting of electrodes with the half-cell reactions: Fe3 + + e = Fe2 + Pu4 + + e = Pu3 +

(2.90)

or as an equilibrium reaction with all four ions in the same solution. These two interpretations are shown schematically in Figure 2.11. As components of an electrochemical cell, the Nernst potentials of the Fe and Pu half-cells are, using Equation (2.88),

94 Light Water Reactor Materials

 [Pu 3+ ]  [Pu 4+ ]

N = φ O − 0.059log φPu  Pu

(2.91)

 [Fe 2+ ] φFeN = φFeO − 0.059log  3+  .  [Fe ]

When the Fe and Pu electrodes are joined (top of Fig. 2.11a), the potential of this full electrochemical cell is  [Fe 2+ ][Pu 4+ ] .  [Fe 3+ ][Pu 4+ ]

N = ( φ O − φ O ) − 0.059log φ = φFeN − φPu  Fe Pu

(2.92)

This equation has two interpretations. If φ represents an applied voltage, the ratios [Pu4+]/[Pu3+] and [Fe3+]/[Fe2+] adjust so that Equation (2.92) is satisfied. If the concentration ratios are fixed, φ adjusts according to Equation (2.92), and the combination forms a battery. Example #9: Pu4+/Fe2 + battery The Pu half-cell contains equal concentrations of trivalent and tetravalent plutonium, and the iron half-cell consists of a 5:1 concentration ratio of Fe2 + to Fe3 +.

Battery voltage Fe2+,

φ

Fe3+ Ne

rns

φFe

tp

ote

Pu3+, Pu4+

φPu

tia

ten

l

o tp

rns

nti

Ne

al

SHE

Fe2+, Fe3+ Pu3+, Pu4+

(a) Separate half-cells

(b) Solution

FiGure 2.11: Two ways of interpreting the plutonium-iron reaction in solution.

Chemical Thermodynamics 95

The electrode reactions are numbers 9 and 12 in Table 2.2. From N N Equation (2.91), the Nernst potentials are φPu = 0.98 V and φFe = 0.73 V. The overall cell potential is φ = 0.73 − (0.98) = −0.25 V.

2.10.5 ionic equilibria An important feature of the standard electrode potentials in Table 2.2 is their utility in calculating equilibrium constants for ionic reactions in aqueous solutions. The electrochemistry of ions contained in a single solution (Fig. 2.11b) is equivalent to short-circuiting the connection between the Fe and Pu half-cells (setting φ = 0 in Fig. 2.11a is equivalent to mixing the two together as in Fig. 2.11b). The condition for equilibrium in a solution containing the constituents of half-cells A and B is the equality of their Nernst potentials: N

N

φA = φB

. Two examples of utilizing this equation are given below.

(2.93)

Example #10: The Pu4 +/Fe2 + reaction The solution is made by mixing equal volumes of 1M FeSO4 and 0.5 M Pu(SO4)2. Find the concentration ratio Pu4+/Pu3+ at equilibrium. Element conservation for the two metals is [Fe 2+ ] + [Fe 3+ ] = 0.5 [Pu 4+ ] + [Pu 3+ ] = 0.25 . The total concentrations of the two elements are one-half of their initial values because two equal volumes are mixed, which doubles the volume of solution. The condition of charge neutrality requires that the total cation charges in the solution equal the total anion charges. This gives 2[Fe 2+ ] + 3[Fe 3+ ] + 3[Pu 3+ ] + 4[Pu 4+ ] = 2[SO 24− ] = 2[0.5 + 0.5] = 2 .

96 Light Water Reactor Materials 2−

The two 0.5 numbers in the brackets account for 1 M SO 4 concentrations in each of the solutions prior to mixing. With φ = 0 in Equation (2.92), the law of mass action for reaction (2.89) is K= o

[Fe 3+ ][Pu 3+ ] ( φPuo − φFeo ) / 0.059 = 10 . [Fe 2+ ][Pu 4+ ]

(2.94)

o

Using the φFe and φPu values for half-cell reactions 9 and 12 in Equation (2.94) gives K = 3600. To solve the above four equations for the four unknowns, it is convenient to simplify the notation using [Fe 2+ ] = x

[Fe 3+ ]= y

[Pu 3+ ] = u

[Pu 4+ ] = v ,

and the element conservation and charge equilibrium equations above become x + y = 0.5 (a) u + v = 0.25  (b) (c) 2x + 3y + 3u + 4v = 2 (d)  yu/xv = K The first step is to substitute (a) and (b) into (c), which yields u = 0.5 − x (e). (0.5 − x )u Substituting (a) and (b) into (d) yields K= , which, when u is (0.25 − u) x eliminated using (e), gives a quadratic equation for x: (K − 1)x2 − (0.25K −1)x − 0.25 = 0 . For K >> 1, this reduces to 4Kx2 − Kx −1 = 0, or x = 1/8[1 + 1 + 16 /k ] ≅ 0.25 + 1/K . Using this result in (e) gives u = 0.25 − 1/K and from (b), v = 1/K. The Pu4 +/Pu3 + ratio at equilibrium is v/u = 1/(0.25K − 1) ≅ 4/K = 4/3600 = 1.1 × 10−3.

Chemical Thermodynamics 97

Thus, addition of a ferrous ion solution to a solution of tetravalent plutonium very effectively reduces 99.9% of the latter to Pu3 +. Example #11: Water dissociation: H2O = H + + OH − The half-cell reactions from numbers 1 and 2 in Table 2.2 and their Nernst equations are  1 (1) 2H + + 1/ 2 O2 + 2e = H2O  φH = 1.23  − 1  / 2  ×  0.059 log   [H + ]2 p  O2  [OH − ]2  (2) 1/ 2 O2 + H2O + 2e = 2OH −  φOH = 0.40 − 1/ 2 × 0.059log   p O 2  At equilibrium, φH = φOH, which yields [H + ][OH − ] = K w = 10 −14 .

2.10.6 Solubility product Solubility product is an equilibrium constant for the dissolution of a solid compound into ions in water. For a metal hydroxide, for example, the reaction is M(OH)z(s) = Mz + + zOH − , which can be divided into half-cell reactions and the associated Nernst equations:     Mz + + ze = M N o 1 z+ φM = φM − × 0.059log (1/[M ]) z M(OH)z(s) + ze = M + zOH − 1 o z N φM( OH ) = φM( OH ) − × 0.059log[OH − ] z z z o

Setting φMN = φMN( OH ) z

z+

yields [M ][OH − ]z = 10

o

− z ( φM − φM OH )/0.059 ( )z

.

98 Light Water Reactor Materials

Example #12: Solubility of Ni(OH)2 in water For the above Nernst equations, φNio = −0.23V φNio (OH)2 = −0.66V. o

o

Solubility product = KSP = [Ni 2+ ][OH − ]2 = 10 − 2( φNi − φNi(OH)2 )/0.059 = 2.6 × 10 −15. (i) In neutral water, [Ni 2+ ] = 2.6 × 10 −15 /(10 −7 ) 2 = 0.26 M (ii) In an acid whose pH is 5, KSP is unchanged, but [OH − ] = K W /[H + ] = 10 − 14/10 − 5 = 10 −9 M [Ni 2 + ] = 2.6 × 10 −15/(10 − 9 ) 2 = 26 M . This high concentration of nickel in the acid can never be achieved. Either all of the Ni(OH)2 will be dissolved, or other solid solubility reactions will limit the amount of Ni in solution (e.g., NiO + H2O = Ni2+ + 2OH−).

2.10.7 electrolysis of water A classic application of the electrochemical cell is dissociation (electrolysis) of water, for which the overall reaction is H2O = H2 + 1/ 2 O2 .

(2.95a)

The cell (Fig. 2.12) consists of two platinum electrodes immersed in salt water. The cathodic reaction (reduction) is number 3 in Table 2.2: 2H + + 2e → H2 .

(2.95b)

The negative connection of the battery supplies the electrons for this reaction. At the anode, oxygen gas is produced by the reverse of number 1 in Table 2.2: H2O → 2H + + 1/ 2 O2 + 2e . (2.95c) The positive pole of the battery consumes the electrons produced by this reaction.

Chemical Thermodynamics 99 Battery Anode +

Cathode –

Oxygen

Diaphragm

Hydrogen

Hydrogen bubbles

Electrolyte

Oxygen bubbles

Solution

FiGure 2.12: Cell for electrolysis of water.

The H + converted to H2 by Equation (2.95b) at the cathode must be supplied by the H + produced by the anode reaction (2.95c). Unless a very large potential is applied to affect the autoionization of water, the electrolysis of pure water proceeds very slowly because the electrical conductivity of water is low. A water-soluble electrolyte (typically Na2SO4) is added to increase the electrical conductivity of water. The electrolyte disassociates into cations and anions; the anions move toward the anode and neutralize the buildup of positively charged H + there; similarly, the cations move toward the cathode and neutralize the buildup of negatively charged OH −. This allows the continued flow of electricity. For the cathodic reaction, the Nernst equation is pH 2  N = 0.0 − 0.059 log  φcath .  [H + ]2  2

100 Light Water Reactor Materials

Similarly, for the anodic reaction, N = 1.23 − φanod

 0.059  1 log  + 2 . 2  [H ] p O 2 

The battery voltage is the difference between φcath and φanod: N − φ N = 1.23 + φ = φanod cath

0.059 log (pH 2 p O 2 ) . 2

(2.96)

For the two gases leaving the anode and cathode at 1-atm pressure, this equation gives φ = 1.23 V. Water does not decompose at potentials less than this value.

2.11 Computational Thermochemistry Computational thermochemistry deals with the analysis of chemical equilibria using the extensive databases, software, and computer power that have become available in the last 20 years. The objectives include calculation of thermodynamic functions (e.g., equilibrium constants of chemical reactions, oxygen potentials of complex solids, and especially, phase diagrams). The field was initiated in 1958 by White et al. [2] and further developed in the 1970s by Eriksson [3]. Chemical equilibrium means a unique combination of constituents and phases that yield the lowest total Gibbs energy of a multicomponent, multiphase system at a fixed temperature and total pressure. The computational effort arises from the minimization of the system’s Gibbs energy, which is the condition of thermodynamic equilibrium (Eq. [2.26]). The numerical approach has been applied to materials of various kinds, such as slags, ceramics, alloys, and semiconductors. Recently, considerable effort has been devoted to using computational methods to analyze nuclear materials, with emphasis on fuel, cladding, and fission-product behavior in both normal and off-normal operating conditions in LWRs.

Chemical Thermodynamics 101

There are several ways of dealing with complex thermochemical problems. One is the classic approach described in Section 2.9, in which the input information consists of the Gibbs energies of formation of the chemical constituents4 involved, from which the standard Gibbs energy changes of the reactions (∆G o) are formed (e.g., Eq. [2.73a]). ∆G o provides the equilibrium constant that relates the concentrations of the constituents in the reaction (as in Eq. [2.69a]). This is called the law of mass action method. The problem with this approach is the complexity of the algebra if more than one or two reactions are simultaneously taking place. The equilibrium equations (i.e., laws of mass action) are nearly always highly nonlinear, and special techniques are required for their solution. While a single reaction such as methane combustion may be reducible to an analytic solution, as in Example #3, treatment of two simultaneous reactions almost always involves a numerical solution. Treating, say, thirty simultaneous reactions by this method is out of the question. Computer codes are designed to deal with complex thermochemical systems without the difficulties associated with the law-of-massaction method. Starting from Ericksson’s SOLGASMIX [4], FactSage, an extensive thermochemical database and calculational modules have been developed [5, 6]. These computational systems can handle ∼100 constituents (molecular or atomic), 20 elements, and as many as 10 solution phases. In addition to a single gas phase, the condensed phases can be solid or liquid solutions (with variable composition) or solids or liquids of fixed composition (i.e., stoichiometric). Figure 2.13 summarizes the notation describing phases, constituents, and elements. The right-hand sketch depicts phases: 1 = gas phase, and 4

Constituents are the chemical compounds present in the phases of an equilibrium system; components are the chemical units for the phase rule; elements are the true chemical elements in the system. (See footnote 2.)

102 Light Water Reactor Materials

q = solution phase; phase is designated by index p. The middle sketch shows the constituents (molecular or elemental) that are contained in phase p. Each constituent is denoted by an integer I, and there are mp constituents in phase p. The moles of constituent I are labeled nip, and the total moles of constituents in phase p are NP. The left-hand diagram in Figure 2.13 breaks the constituents into elements, with bjp specifying the number of moles of element j in phase p. The bjp are not input numbers because the initial mixture in general will not have the same number of phases and compositions in the phases as the equilibrium mixture. However, summing the quantity of element j in all phases, Bj is a specified quantity. The collection of all phases is termed the system. The number of phases and the number of components are constrained by the phase rule (Sec. 2.6); with temperature and pressure specified, the remaining degrees of freedom are F = ∑ B j − ∑ p − ∑ rxns .

(2.97)

j Phases p 1 Constituents Elements

j

ajip = Moles of element j in species i in phase p a1ip a2ip bjp = Σ ajipnip = i

Moles of element j in phase p

i

nip = Moles of constituents in phase p

1 Gas phase

2 p

q solution phases

n1p 1+q nip nmpp

Σp = 1+q = Phases in system

Σ i = mp = Constituents in phase p Σnip = Np = Moles of i constituents in phase p

Bj = Σ bjp = p

Moles of element j in system

FiGure 2.13: Notation for describing a multicomponent, multiphase system.

Chemical Thermodynamics 103

The last sum represents all of the equilibrium reactions in the system. The degrees of freedom that remain are consumed by specification of the initial material charged to the system.

2.11.1 Gibbs energy minimization Many computer thermochemistry codes use the method of Lagrange multipliers applied in a method known as Gibbs energy minimization. This technique potentially simplifies the problem and makes use of one of many readily available equation solvers. This method starts from the total Gibbs energy of the system. All phases and all constituents are contained in the following single equation for this property: m1

m

n il   q +1 p   o + ∑∑ n ip [g io + k BT ln a pi ] (2.98) G = ∑ n i 1 g i + k BT ln     N  1  p =2 i =1 i =1 where the first term on the right side corresponds to the gas phase (p = 1) and the second term corresponds to the solution phases and where kB is Boltzmann’s constant, T is the temperature, and g io is the standard Gibbs energy of pure constituent i at temperature T and 1 atm given by Equation (2.42a). The other symbols in Equation (2.98) are explained in Figure 2.13. In the second term on the right side, the inner sum adds constituents in a phase, and the outer sum adds the solution phases in the system. Example #13: Hydrocarbon combustion by Gibbs energy minimization Consider oxidation of methane (CH4) and ethylene (C2H4) at p = 1 atm, T = 2000 K. The reactions, CH4 + 2O2 = CO2 + 2H2O

(2.99a)

C2H4 + 3O2 = 2CO2 + 2H2O ,

(2.99b)

104 Light Water Reactor Materials

take place in one phase (gas) and contain five constituents (n1 = C2H4; n2 = CH4; n3 = O2; n4 = CO2; and n5 = H2O) and three elements (BH = H; BC = C; and BO = O). For reactions (2.99a) and (2.99b), the degrees of freedom from Equation (2.97) are F = 3 − 1 − 2 = 0. Or, there is a unique solution to this equilibrium. Since there is only one phase, the notation is simplified by dropping the second subscript: ni1 → ni and N1 → N . The relations between constituent numbers and element numbers are 4n1 + 4 n2 + 2n5 = BH  2n1 + n2 + n4 = BC n5 + 2n4 + 2n3 = BO .

(2.100a)

These three equations serve as constraints on the solution. From Equation (2.100a), the total moles are 5

1 1 N = ∑ n i = B H + BO . 4 2 i =1 For only a gas phase, Equation (2.98) becomes G RgT

(2.101)

51

 g io  ni   . = ∑ ni + ln  R g T  N   i =1

(2.98a)

The values of ni at the minimum of G, which is the condition of equilibrium (see Eq. [2.26]), is determined by the method of Lagrange multipliers (one for each element): 1 ∂G RT ∂ni

+ λH

∂ BH ∂ni

+ λC

∂ BC ∂ni

+ λO

∂ BO ∂ni

= 0, i = , ....5

.

(2.102)

Equations (2.100a) and (2.102) provide eight equations for determining eight unknowns (n1, …n5, λH, λC and λO). The partial derivatives of BH, BC,

Chemical Thermodynamics 105

and BO are obtained from Equation (2.100a) and the ∂G/∂ni terms from Equation (2.98a). With these conditions, Equation (2.102) becomes g 1o  n1  + 4 λ + 2 λ = 0 + ln H C  N RgT

g 2o  n2  + 4λ + λ = 0 + ln H C  N RgT

g 3o  n 3  + 2λ = 0 + ln O  N RgT

g 4o  n 4  + 2λ = 0 + ln O  N RgT

g 5o  n5  + 2λ + 1 λ = 0 + ln H  N RgT 2 O

. (2.103a)

To simplify the notation, these equations are redefined using g io  ni  , i = 1,... 5 , + ln [i ] =  N RgT so that

[1] + 4 λ H + 2λ C = 0

[2] + 4 λ H + λ C = 0

[3] + 2λ O = 0

[4] + λ C + 2λ O = 0

[5] + 2λ H + λ O = 0

(2.103b)

First, the Lagrange multipliers are eliminated in terms of the [i]: λH = −½[5] + ¼[3]  λC = [2] − [1]  λO = −½[3] . (2.104) Next, mole numbers are converted to mole fractions (xi = ni/N), so that Equation (2.100a) becomes 4x1 + 4x2 + 2x5 = βH  2x1 + x2 + x4 = βC x5 + 2x4 + 2x3 = βO (2.100b) βH =

where

H

/ BO

BH 1/ 4 B H + 1/ 2 BO βO =

βC =

BO 1/ 4 B H + 1/ 2 BO

BC 1/ 4 B H + 1/ 2 BO (2.105)

106 Light Water Reactor Materials

The Lagrange multipliers are eliminated by substituting Equations (2.104) into Equations (2.103b), a step that results in only two equations: −[1] + 2[2] + [3] − 2[5] = 0 and −[1] + [2] − [3] + [4] = 0

or, using the definition of [i], − g 1o + 2 g 2o + g 3o − 2 g 5o  x 22 x 3  = 0 + ln 2 RgT  x 5 x1  − g 1o + g 2o − g 3o + g 4o

RgT

+ ln

 x2x4  = 0 .  x1 x 3 

(2.106a)

Numerical Example Specify (arbitrarily) BH = 3, BC = 1, and BO = 2, so that from Equation (2.105), βH = 1.71; βC = 0.57; and βO = 1.14

.

o

The g i at 1000 K can be obtained from the NIST tables5 at 1000 K (in kJ/mole): g 1o (C 2H 4 ) = 119 g 2o (CH 4 ) = 19 g 3o (O 2 ) = 0 g 4o (CO 2 ) = −396 g 5o (H 2 O) = −193 so that Equations (2.106a) become x 22 x 3 −16 = 1.2 × 10 2 x 5 x1

x2 x4 26 = 1.2 × 10 . x1 x 3

(2.106b)

There are now five equations ([2.100a] and [2.106b]) for the five mole o fractions. Fortunately, because of the magnitudes of the g i , they are easily o solved. The g i are identical to the Gibbs energies of formation of the constituents (Sec. 2.7.4), so the more negative the value, the more stable the compound. This rule-of-thumb suggests that CH4 is much more stable

5

http://kinetics.nist.gov/janaf/

Chemical Thermodynamics 107

than C2H4, or x1 0. Different combinations of hkl with the atom positions in the unit cell give different values of F. A value of F = 0 corresponds to extinction of the diffracted beam. In those cases, diffraction from the particular (hkl) plane is said to be not allowed for that crystal structure. An example of extinction is the 100 reflection in the bcc structure. Referring to Figure 3.5, when constructive interference occurs between the top and bottom planes, the middle plane, lying halfway between the two, represents half the path difference between the X-rays scattered from the top and bottom planes and thus exactly cancels the constructive interference of the (100) planes. Thus, no diffracted beam from (100) is possible in the bcc; this is a forbidden reflection. Example #7: Structure factor in the bcc crystal The F-factor can also be calculated by considering the bcc unit cell for 1 1 1 which the atom coordinates are (0,0,0) and 2 , 2 , 2 . Inserting these positions for hkl = 100 in Equation (3.10a) gives

152 Light Water Reactor Materials

  1 1 1   F = f exp 2 πi (1.0 + 0.0 + 0.0) + exp 2 πi 1. + 0. + 0.      2 2 2   (3.10b)  F = f [1 + exp( πi )] = 0 ,

(3.10c)

meaning the diffracted intensity of 100 is zero. This is a characteristic of the bcc structure, and a set of these characteristics can help identify the structure when examined with X-ray diffraction. For the bcc structure, the only allowed reflections are those for which the sum h + k + l is even, while for fcc, h, k, l need to be all even or all odd. Because of this, the allowed reflections for bcc are (in order of increasing interplanar lattice spacing): 110, 200, 211, 220, 310, 222, etc. For fcc they are 111, 200, 220, 311, 222, 400, etc. For the simple cubic lattice, all reflections are allowed. The rules for hcp are more complex but follow the same principles. The observation of systematic presences or absences of these reflections in an X-ray pattern can help identify the structure (Prob. 3.13). In addition to crystal structure parameters, peak intensities and shapes are functions of many instrumental parameters (number of photons, source and detector slit size, energy spread of incoming radiation, distances between detector and sample, electronic noise, etc.). For a given peak in a particular crystal structure, the diffracted intensity depends on the following: • the atomic density of the corresponding planes; • their multiplicity (how many planes of the same type exist in the crystal structure); • the atomic scattering factor of the atom in question (see Eq. [3.10]); • the structure factor (how does the scattering from the atoms in the crystal structure combine to produce a scattered signal? Eq. [3.10]); • thermal scattering factors. A good review of these variables is presented in Reference [6]. The X-ray spectra can also be affected by characteristics specific to the sample, such as crystallographic texture, the presence of strain, and

Crystallography 153

grain-size effects. The last two cause peak broadening (i.e., the full width at half maximum of each peak increases), and peak shift. The first can result from a sample that exhibits preferential orientation as a result of fabrication processing, as discussed in more detail in Chapter 17 for the case of Zr. The information gathered in such experiments is available as part of several databases. A spectrum such as Figure 3.21 is compared with the theoretical values of the intensities for a given crystal structure. These structure determination calculations are involved and make use of sophisticated analysis methods such as Rietveld refinement [7].

3.8.2 Electron diffraction The wavelengths of high-energy electrons are small enough that they can interact with lattice planes. Two types of electron microscopes are in common use: a transmission electron microscope (TEM) and a scanning electron microscope (SEM). The TEM focuses a high-energy (∼100 to 400 keV) electron beam on a thin sample (typically less than 200 nm). The beam is partly transmitted through the sample and partly diffracted. Images can be formed from diffraction contrast by placing an aperture around the transmitted beam after focusing (bright-field image) or around one of the focused diffracted beams (dark-field image). In a bright-field image, objects that diffract strongly appear dark, and in a dark-field image they appear light. All transmitted and diffracted beams are simultaneously focused on the back focal plane. The image of these focused beams constitutes an electron diffraction pattern. An example of such a pattern from a sample of austenite (fcc Fe alloy) is shown in Figure 3.22a. The interplanar spacing can be calculated from the diffraction pattern 3.22a from the formula Rd = λL (3.11) where R is the measured distance in the micrograph, λ is the wavelength,

154 Light Water Reactor Materials

L is the camera length, and d is the d-spacing for the plane in question. The angles between planes can be measured directly from the figure. For example, the angle between the 220 and 220 planes is 45 degrees. Each spot in Figure 3.22a corresponds to the diffracted electron waves from one set of planes. Because the electron diffraction angle is very shallow (θ small), it is possible to simultaneously excite diffraction from many beams. This pattern of electron diffraction reveals the crystal symmetry better than in X-ray diffraction. In addition, because the electron beam can be focused on a region as small as a few nanometers, the spatial resolution of transmission electron microscopy diffraction is much higher than that of X-ray diffraction. Although beyond the scope of this book, much more detailed information can be gleaned from electron diffraction patterns than on the spot patterns obtained from thin regions. Convergent beam diffraction can yield more complex information from the dynamic interaction of electrons with the atoms in crystal structure (Fig. 3.22b). When the TEM sample is thicker, Kikuchi patterns can be observed (Fig. 3.22c) [2, 5], which serve as effective roadmaps for tilting the sample between different zone axes (crystal orientations relative to the beam). – –2 –2 0

0 –2 0

–2 0 0 –

– –2 2 0

2 –2 0

200

020

(a)

220

(b)

(c)

FIGURE 3.22: Examples of electron diffraction patterns: (a) Electron spot pattern taken using a parallel beam. The large spot in the center is the transmitted beam, and the smaller spots are diffracted from the various crystal planes in the structure. (b) Convergent beam diffraction pattern of pyrochlore, showing the interference pattern from higher-order Laue zones [8]. (c) A Kikuchi pattern, used for orientation imaging (electron backscatter diffraction) and also for proper tilting in the microscope.

Crystallography 155 Forward Scattering Electron Image

50µm 001

111 101

FIGURE 3.23: Electron backscatter diffraction orientation imaging. The electron beam is scanned across the sample, and grains are indexed according to the backscattered Kikuchi lines generated and colored according to their position in the orientation triangle.

However, in general the crystal lattice parameter determinations from TEM are less precise than those obtained with X-ray diffraction because of the larger volume sampled by the latter, and because of the greater precision in measuring angles and distances. The technique of electron backscattered diffraction allows diffraction information to be obtained with the SEM. The same Kikuchi patterns seen in TEM can be obtained in backscatter mode allowing the orientation of individual grains to be identified from the diffraction patterns, so that a much more extensive survey can be performed of the orientation distribution of grains in a sample than would be possible with the TEM. This allows the identification of grain-to-grain misorientations and strain partitioning between grains. Figure 3.23 shows a micrograph of different grains in a sample, all of which are oriented differently, as illustrated by the differently oriented cubes. A color can be associated with the grain orientation within the unit triangle in a standard stereographic

156 Light Water Reactor Materials

projection to give an overall view of the grain orientations in the sample. The color of the grains indicates that not many grains are oriented with the 001 plane parallel to the surface, and most grain orientations are close to 101 and 111.

References 1. C. Kittel, Introduction to Solid State Physics, 6th ed (Hoboken, New Jersey: Wiley, 1986). 2. J. Edington, Practical Electron Microscopy in Materials Science (1976). 3. T. Hahn, International Tables for Crystallography, Volume A: Space Group Symmetry, 5th ed. (Berlin, New York: Springer-Verlag, 2002). 4. F. Laves, “Factors governing structure of intermetallic phases,” Advances in X-Ray Analysis 6 (1963): 43–61. 5. G. Thomas and M. J. Goringe, Transmission Electron Microscopy of Materials (Hoboken, New Jersey: Wiley, 1979). 6. B. Cullity, Elements of X-ray Diffraction (Reading, Massachusetts: AddisonWesley, 1978). 7. R. Young, The Rietveld Method (Oxford: Oxford University Press, 1993). 8. A. R. Landa-Cánovas et al., “Electron microscopy study of the decomposition products at 1300ºC from jamesonite mineral FePb4Sb6S14,” Solid State Ionics 63–65 (1993): 301–306.

Problems 3.1 The crystal structure of α-uranium is shown in the sketch below. (a) What is the complete designation of this crystal structure? (b) One criterion for the suitability of a nuclear fuel is the uranium atom density. Calculate this parameter and the theoretical atom density for the three nuclear fuels: α-uranium, UO2, and UC. The lattice constants for UO2 and UC are 0.547 nm and 0.4961 nm, respectively.

Crystallography 157

Unit cell

c

a

b

Crystal Axis Vectors Direction a [100] b [010] c [001]

Length (Å) 2.852 5.865 4.945

3.2 For a tetragonal lattice where a = b = 3c, find the indices of the direction perpendicular to the (011) plane. 3.3 If atoms are considered as contacting hard spheres, show that: (a) the bcc lattice has a packing fraction of 0.68 (b) the fcc and hcp lattices have a packing fraction of 0.74 (c) the hcp structure has c/a = 1.633 3.4 Using the hard sphere premise, if a cube has side ao, what is the atomic diameter in bcc and fcc? 3.5 What low-index crystal planes in the tetragonal systems have the hkl direction perpendicular to the hkl plane? 3.6 Given a bcc structure formed by the packing of hard spheres, what is the largest sphere that can be introduced into the lattice without distortion, and where is it located? Such simple geometrical arguments often predict the position of interstitial atoms. 3.7 For the cubic system, verify Equation (3.7).

158 Light Water Reactor Materials

3.8 For the two-dimensional case, show that the choice of primitive unit cell is not unique but that all choices have the same area (volume for 3D case) (Fig. 3.2). 3.9 A cubic crystal lattice parameter is ao = 0.36 nm. Write an expression for a lattice vector from an atom at the origin to another atom that is two cells away along the a-axis, four cells along the b-axis, and, six cells along the c-direction. What is the length of the vector, and what angles does it make with the three crystal axes? Repeat for a monoclinic crystal where a = 0.2 nm, b = 0.36 nm, and c = 0.24 nm. 3.10 In a simple tetragonal crystal, the unit cell dimensions are a = b = 0.18 nm, and c = 0.24 nm. Find the spacing between adjacent (111) planes and adjacent (523) planes in the crystal. For the same crystal structure, find the distance between adjacent atoms along the [111] direction and along the [523] direction. 3.11 The conventional unit cell of NaCl (common salt) is shown in Figure 3.15. What is its crystal structure? How many nearest-neighbor atoms do the Cl atoms have, and of what type are they? 3.12 What are the Miller indices of the cubic and tetragonal planes below?

3.13 A material is studied by X-ray diffraction using the Debye-Scherrer method, with Cu Kα radiation (λ = 0.15418 nm). The measured Bragg angles for the first five lines are 10.83, 15.39, 18.99, 22.07, and 24.84 degrees, respectively. It is known that the lattice is simple cubic. (a) What are the Miller indices of the planes corresponding to those lines? (b) What is the lattice parameter? (c) Which lines would be missing if the lattice was fcc? What if bcc?

4

Chapter Point Defects 4.1 Classification of Point Defects Contrary to the perfect lattices discussed in the preceding chapter, all real crystals contain defects, which can be differentiated according to their dimension. Zero-dimensional defects are discussed in this chapter, line defects (dislocations) in Chapter 7, and area defects in Chapter 8. As discussed in Chapter 5, zero-dimensional defects are the agents of solid state diffusion. The zero-dimensional defect is called a point defect, implying that it involves only one atom site surrounded by an otherwise perfect lattice. However, the presence of a point defect may affect the properties of its nearest neighbors and, by elastic interactions, a sizable region of the lattice around the defect. In that sense, all defects are three-dimensional. Two types of point defects are intrinsic to elemental solids, meaning that they form spontaneously in the lattice without any external intervention. These two are the vacancy (V) and the self-interstitial atom (SIA), shown schematically in a two-dimensional (2-D) representation in the upper panels of Figure 4.1. The vacancy is simply an atom missing from a lattice site, which would otherwise be occupied in a perfect lattice. The self-interstitial is an atom lodged in a position between normal lattice

160 Light Water Reactor Materials Vacancy

Substitutional impurity atom (e.g.,Sn in Zr)

Self interstitial

Interstitial impurity atom (small, H, or C)

FIGURE 4.1: Point defects in an elemental crystal with examples of elemental impurities.

atoms; that is in an interstice. The qualification “self” indicates that the interstitial atom is of the same type as the normal lattice atoms. The lower two panels in Figure 4.1 show the two basic ways that a foreign (or impurity) atom exists in the crystal lattice of a host crystal. Large impurity atoms, usually of the same type as the host atoms (e.g.,  both metals, such as nickel in iron), replace the host atoms on regular lattice positions. These are called substitutional impurities. The structure of the lattice is not disturbed, although nearby host atoms may be slightly displaced from their normal lattice sites. Small atoms (compared to the vacancy size) that are also chemically dissimilar from the host atoms can occupy interstitial positions without appreciably distorting the surrounding host crystal. They are termed interstitial impurities. Typical examples are carbon in iron and hydrogen in zirconium. Aside from their identities relative to the host atoms, the SIA and the impurity interstitial differ in the way that they reside in the lattice. As shown in the two left-hand diagrams of Figure 4.2, in the bcc lattice, the selfinterstitial displaces a host atom from its normal lattice position, creating a dumbbell-shaped pair. This configuration is also called a split interstitial. The orientation of the dumbbell (or of the atomic configuration associated

Point Defects 161

Octahedral

Tetrahedral

FIGURE 4.2: (Left) interstitials in the bcc lattice. SIA; (right) impurity interstitial.

with the extra atom, in general) and the distance between the two atoms are determined by the condition that the potential energy of the lattice is a minimum. The small interstitial impurity atoms, on the other hand, occupy definite sites without significant distortion of the host lattice. However, even the small displacements of the neighboring host atoms require energy, which limits the solubility of the interstitial impurity. These sites are named after the shape of the polyhedron formed by joining the host atoms surrounding the interstitial. The examples shown on the right side of Figure 4.2 for the bcc lattice are the octahedral and tetrahedral sites. These two interstitial sites offer the most space for the impurity atom to reside in, but which site

162 Light Water Reactor Materials

is occupied is a sensitive function of the interaction energy between the impurity atom and the host atom.

4.2 Equilibrium Concentrations of Point Defects in Elemental Crystals In Section 2.4 of Chapter 2, it was shown that the condition of chemical equilibrium is the minimization of the Gibbs energy at constant temperature and pressure. The system in this case consists of atoms on regular lattice sites and the intrinsic point defects (e.g., vacancy, SIA) randomly distributed on sublattice sites. Even though there is only one element involved, thermodynamically the system consists of two components, the regular atoms and the empty sublattice sites in the case of a vacancy or atoms on interstitial-sublattice sites in the case of the SIA. In thermodynamic analyses, the two types of point defects are treated as independent entities. Normally both point defects do not exist simultaneously, as one type is more stable than the other. The reason that point defects form spontaneously lies in the components of the Gibbs energy of formation.1 For single point defects, denoted by j ( j = V for a vacancy or j = I for an SIA), the Gibbs energy contains two terms: g j = h j − Ts j (4.1) where hj and gj are differences in energy and sj is the difference in entropy for the processes shown in Figures 4.3 and 4.4. As illustrated in Figure 4.3, vacancy formation can be regarded as the movement of an atom from the interior of the crystal to its surface. When an SIA is formed (Fig. 4.4), the process is reversed as an atom is moved from the surface to the inside of the solid. 1

In this book, thermodynamic and transport properties are expressed in units of eV/atom. The literature often uses J/mole or kJ/mole. The relation between the two is 96.5 kJ/mole per eV. If kJ/mole is used, Boltzmann’s constant, kB, is replaced by the gas constant, Rg = NAv . kB.

Point Defects 163 Vacancy Removal

Atom on surface Vacancy Formation

Vacancy

FIGURE 4.3: Vacancy formation. Interstitial formation

Atom on surface Interstitial removal

Split interstitial

FIGURE 4.4: SIA formation.

Enthalpy The enthalpy change accompanying the creation of the point defects, hj, is from Equation (2.1): h j = E j − σ∆v j ≈ E j

(4.2)

where the formation energy Ej, the dominant contribution, is the energy required to create a mole of point defects from the perfect lattice. Ej is positive for both vacancy and SIA creation, but is much larger for the latter than the former. ∆vj is the volume of a mole of vacant lattice sites, which is nearly the same as the molar volume,2 2

If the above atomic units are used, v is replaced by the atomic volume Ω = v/NAv, where NAv is Avogadro’s number.

164 Light Water Reactor Materials

M

∆v j = ρ

(4.3)

where M is the atomic weight of the element and ρ is its density. The σ∆v term in hj represents the work involved in the change in system volume as a surface atom is moved between the interior site and the surface (see Fig. 4.3 and 4.4). σ is the hydrostatic stress (N/m2 or J/m3), which is positive in tension. Entropy The two parts of the entropy involved in the formation of point defects are expressed by the equation s = s vib + smix . (4.4) The svib component results from the difference between the vibrational motion of the atoms around the point defect from that of atoms in the perfect lattice. This term is [1, Sec. 6.3] s vib = k B ln(υ / υ′) α

(4.5)

where υ is the vibration frequency of atoms in the perfect crystal, which could be either the Einstein or Debye frequencies [1, Sec. 2.1], υ′ is the vibration frequency of the atoms immediately surrounding the point defect, and kB is Boltzmann’s constant.3 Around a vacancy, the nearestneighbor atoms relax into the hole, and in so doing, vibrate at a lower frequency. For this case, υ/υ′ > 1 and svib > 0. The more tightly packed atoms surrounding the SIA vibrate at a higher frequency than those in the perfect lattice, so υ/υ′ < 1 and svib < 0. The major component of the entropy change accompanying point defect formation is smix, the entropy of mixing (also called the configurational entropy). This term is a positive quantity that contributes negatively 3

Instead of kB, the gas constant Rg can be used in Equation (4.5) if the units of s are changed from J/molecule.K to J/mole.K.

Point Defects 165

to the system free energy, and thus promotes the formation of the point defects. This is due to mixing NV vacancies with N atoms. In this case, the expression for smix is N   N   NV   NV  smix = − k B  ln ln +  N + N V   N + N V   N + N V   N + N V   (4.6) = − k B [(1 − CV )ln(1 − C V ) + CV ln C V ] . The reason for this behavior is that smix is a measure of randomness, and the introduction of point defects into a perfect crystal reduces the system’s state of order. The entropy of mixing is responsible for the spontaneous existence of point defects at equilibrium; the magnitude of the positive energy of point defect formation, governs the concentration of these species at thermal equilibrium, as discussed in the following section.

4.2.1 Vacancy formation Figure 4.3 shows the creation of a vacancy by moving an interior atom to the surface of the solid, as described by the equilibrium reaction null ↔ V

+

Asurf

(4.7)

where null denotes the perfect crystal, V is a single vacancy, and Asurf is an atom on the surface of the solid. The equilibrium is maintained because of the equality of the formation and removal rates depicted in Figure 4.3. The components of the Gibbs energy equation, gV = hV − TsV ∼ EV, are considered as property differences between the right and left sides of Equation (4.7). The energy to create a mole of vacancies (EV) is positive, because the process involves breaking interatomic bonds.

166 Light Water Reactor Materials

The Gibbs energy of a solid containing NV moles of vacancies and N moles of atoms is GV = G(0) + N V (EV − σ V ∆v − Ts vib ) − T (N + N V ) smix       (4.8) where G(0) is the Gibbs energy of N moles of the perfect lattice (no vacancies). Even though the middle term in parentheses is always positive, the smix term is negative, effectively assuring that the minimum value of GV cannot occur without vacancies. The equilibrium vacancy concentration is obtained from Eq. (4.8) using the general criterion for equilibrium in a system at constant temperature and pressure, namely, that GV be a minimum. Since N >> NV, this criterion yields  dGV  = (E − σ∆v − Ts ) + k T = 0 . (4.9) V V vib B  dN V  N NV/N is approximately the site fraction of vacancies, denoted by CV. This concentration unit is the solid-state analog of the mole fraction unit that appears in the equilibrium equations in gases and liquids. The above equation is rewritten as N s   −E   σ∆v  CV = V = exp  vib  ⋅ exp  V  ⋅ exp   N  k BT   k BT   k BT   − EV   s vib  exp (4.10) ⋅  k T  .  k BT  B svib is unknown for most elements and is small for those elements for which it has been measured. The term involving the stress can be estimated using the following values: σ = 100 MPa; ∆v = 2.4 × 10−5m3/mole; Rg = 8.314 J/K-mole; and T = 1200 K. This combination yields σ∆v/RgT = 0.25, so the exponential of this factor in Equation (4.10) is ~0.8. For most applications, Equation (4.10) reduces to  −E  CV = exp  V  .  k BT  ≈ exp 

Point Defects 167

Example #1: The vacancies in copper EV = 1 eV/atom (96.5 kJ/mole). At 1300 K (which is just below the melting point) what fraction of the lattice sites are empty? Using these values in Eq. (4.10) and noting that kB is 8.62 × 10−5 eV/K, the site fraction of vacancies is CV = 10−4. Is value is too small to influence properties such as the density, but even smaller values of CV are critical in determining transport properties such as self-diffusion (see Ch. 5).

4.2.2 Self-interstitials The process analogous to that in Figure 4.3 for forming SIAs in elemental crystals is illustrated in Figure 4.4. The equilibrium concentration (as a site fraction) of SIAs is given by s   −E   −E   σ∆v  CV = exp  vib  ⋅ exp  I  ⋅ exp   ≈ exp  I  .  k BT   k BT   k BT   k BT 

(4.11)

For copper, the interstitial formation energy is EI ∼ 3.1 eV/atom (about 300 kJ/mole), which is about three times larger than the energy required to create a vacancy in this metal. As a result, the equilibrium concentration of interstitials is very much smaller than that of vacancies (by 8 orders of magnitude at 1300 K). This is true of all elements, so thermally generated interstitials can usually be neglected in most applications. However, in the presence of high-energy radiation, the two types of point defects are created at equal rates, and interstitials cannot be ignored. Although the vibration entropy and stress terms are usually omitted, for the few applications where they are necessary, understanding how they differ for the two types of point defects is useful.

4.2.3 Point-defect formation as a chemical reaction An approximate but extremely simple method for treating point defect equilibria regards the process as a pseudochemical reaction and directly utilizing the well-known theory of chemical-reaction equilibrium.

168 Light Water Reactor Materials

The “reaction” that produces vacancies is given by Equation (4.1). The law of mass action expresses the equilibrium constant for this reaction by activity of vacancies × activity of surface atoms (4.12) . activity of atoms in perfect lattice Because the concentrations of atoms in the undisturbed lattice and on the surface are much greater than that of the point defect, they are essentially unaffected by point defect formation. Consequently, their activities can be taken to be unity. The activity of the vacancies, however, is equal to their site fraction CV. The other feature of chemical-reaction theory that is adapted for the vacancy formation process is the relation between the equilibrium constant and the Gibbs energy change of the reaction (see Eq. [2.69] of Ch. 2): KV

=

−g CV = exp  V  .  k BT 

(4.13)

The Gibbs energy change for forming a mole of vacancies, gV, is equal to the sum of the terms in parentheses in Equation (4.8). A similar application of chemical equilibrium theory applies to self-interstitials as well.

4.2.4 Ordered compounds Ordered compounds consist of (at least) two sublattices, so that six types of point defects can exist: vacancies in the A sub-lattice, vacancies in the B sub-lattice, A-atom interstitials, B-atom interstitials and the so-called antisite defects, which contain A atoms in the B sublattice, and B atoms in the A sublattice. Although these are all possible, some will be energetically favored over the others and, because the dependence is exponential, only a few of the six may be stable.

4.3 Point Defects in Ionic Crystals Binary ionic solids consist of a nonmetal and a metal arranged on a crystal structure bonded by Coulomb forces (Ch. 3) and as a result have more complicated defect structures [1]. A subset of ionic solids of particular importance in

Point Defects 169

nuclear technology are oxides, the most important of which is uranium dioxide. Because of the multiplicity of valence states of uranium and the relative ease with which they can form, this oxide is one of the most difficult to understand. It has been the subject of more research than any other metal oxide. Characterization of a binary ionic crystal requires specification of: • the metal and the nonmetal; • their normal valences (oxidation states); • the crystal structure; • the types of point defects that spontaneously form. Because of the importance of ceramics such as UO2, PuO2, ThO2, and ZrO2 in light water reactors (either as fuel or corrosion product), the above list is focused as follows: 2 • Oxygen is the nonmetal; its only stable state is O −. • The metal is tetravalent in its most stable state, so the formula is MO2. • The crystal structure is fluorite (Fig. 3.15), consisting of a simplecubic anion (O2−) sublattice intermingled with a face-centered cubic cation (M4+) sublattice. Oxygen and metal interstitials have their own sublattices, so the solid with point defects has four sublattices.

4.3.1 Types of point defects Point defects in ionic solids can be either structural or electrical. As in elemental solids, a structural defect is either a missing ion (vacancy) or an ion in an irregular (interstitial) position. Because the cations and anions carry electrical charges, vacancy and interstitial formation are not independent processes. The middle two drawings of Figure 4.5 show the structural defects in the fluorite structure of MO2. The principal defect is the anion-Frenkel defect, which is created by displacement of an oxygen ion from its normal lattice position. The resulting structure consists of a vacant site on the anion sublattice and an O2− ion in an interstitial position.4

4

The actual anion-Frenkel defect in UO2 is more complex than that shown in Figure 4.5 (see Sec. 16.5).

170 Light Water Reactor Materials

This combination obeys electrical neutrality. The Schottky defect consists of two anion vacancies and one cation vacancy (for electrical neutrality). Schottky and Frenkel defects are created independently. In a particular crystal, one is dominant, while the other is either absent or a minor contributor. In UO2, for example, the major defect is anion-Frenkel, but Schottky defect formation occurs to a lesser extent. The defects in Figure 4.5 are termed intrinsic because they occur without the assistance of an impurity cation. Extrinsic point defects are generated due to the presence of aliovalent (alio = different) cations such as Q5+ or Q3+ that replace M4+ on the cation sublattice. A foreign 4+ cation causes no electrical disturbance in the MO2 lattice. The bottom diagram of Figure 4.5 is a purely electrical point defect that occurs only with cations that exhibit oxidation states other than the stable 4+ state. This is termed disproportionation; it amounts to loss

Perfect lattice

Schottky defect

5+ I

3+

Anion-Frenkel defect

Cation disproportionation Regular anion I

Anion interstitial Anion vacancy Regular cation Cation vacancy

FIGURE 4.5: Intrinsic point defects in an MO2-type ionic crystal.

Point Defects 171

or gain of an electron from M4+. It occurs only when the cation has valence states that are close in energy (such as U4+ and U5+ and Ce3+ and Ce4+).

4.3.2 Kroger-Vink notation The accepted method for characterizing defects in ionic solids is the one due to Kroger and Vink [2]. In this method, the defect’s charge is identified by the deviation from the absolute charge of the ions in the perfect lattice. In the oxide MO2, for example, M4+ and O2− in their normal lattice sites would be considered uncharged and identified as MM and OO. The normal letter denotes the species, and the subscript is the sublattice on which the species resides. Thus, in MO2, MM stands for cation M4+ on the cation sublattice. MI and OI designate occupation of the appropriate interstitial sites by a metal ion and/or an oxygen ion. As shown below, symbols for the charges of the various species (relative to the normal charges of 4+ for cations and 2− for anions) need to be added to the symbols. In place of + and − signs, deviant charges are denoted by apostrophes (′) for negative states and dots (•) for positive states. Thus, the anion vacancy is designated by VO , where V stands for vacancy, the subscript O indicates that the vacancy occupies a regular anion sublattice site, and the two dots show that the vacant site is doubly positively charged with respect to the anion that would normally occupy the site (O2−). The point defect symbols in MO2 are summarized here: VO =   doubly positively charged anion vacancy. VM″″ = vacancy on the cation sublattice. This species carries four negative  charges relative to the regular cation. This charge state is designated  by the quadruple apostrophe. M IM =  cation on a cation interstitial site designated by subscript IM. The superscript dots indicate the number of positive charges attached to the cation, relative to the empty interstitial site. ••

••

••••

172 Light Water Reactor Materials

O IO″ =     oxygen ion located on an anion interstitial site (IO). It carries two negative charges. M M′ = trivalent metal ion (i.e., M3+) on a normal cation sublattice site  (M). Compared to the tetravalent ion M4+, the trivalent ion carries one negative charge.

M •M  =  M5+ on a cation sublattice site (M). It is positively charged. Q •M = pentavalent impurity cation (Q5+) on a regular cation site. It has one extra positive charge compared to M4+ on a regular cation sublattice site. Q M′ =   trivalent impurity cation (Q3+) on a regular cation site. It is negatively charged. The concentration unit for component i is either site fraction (Ci) or moles/unit volume ([i]).

4.3.3 Site fractions Using MO2 as an example, defect site fractions are connected to Kroger-Vink notation (square brackets). For the ions in the normal lattice sites, these are: Cation: [M M ] CM  = . (4.14) [M M ]∗ Anion: [OO ] CO   = . (4.15) [OO ]∗ The starred concentrations are: [OO]∗  = concentration of sites on the anion (oxygen) sublattice. [MM]∗ = concentration of sites on the cation sublattice. Anion vacancy: [V ] (4.16) CVO = O ∗ [OO ] ••

Point Defects 173

Anion interstitial:

[O IO″ ] [I O ]∗

(4.17)

CVM =

[VM″″] [M M ]∗

(4.18)

C IM =

[M IM ] [I M ]∗

(4.19)

C IO = Cation vacancy:

Cation interstitial: ••••

Aliovalent cation:

[M M′ ] [M ] or C M M = M ∗ ∗ [M M ] [M M ]

(4.20)

[Q M′ ] [Q M] or C Q M = ∗ [M M ]∗ [M M ]

(4.21)

•

C M ′M =

•

Aliovalent impurity: •

C Q ′M =

•

CO in Equation (4.15) is a macroscopic measure of the gross oxygen concentration in the metal oxide (in terms of site fraction). On the other hand, the site fractions CVO , etc., are measures of the concentrations of atomic-size point defects. For the fluorite structure of MO2

[OO ]∗

=

∗

2 [M M ] .

(4.22)

The concentration of cation sites is  [M M ]

∗

=

ρ ox

M ox

=

ρ ox

M M + 32

(4.23)

where ρox is the mass density of the oxide (g/cm3) and Mox is its molecular weight (g/mole).

174 Light Water Reactor Materials

[IM]∗ = concentration of interstitial sites for cations. [IO]∗ = concentration of interstitial sites for anions.

4.3.4 Site-filling, electrical neutrality, and equilibrium The point-defect volumetric concentration [i] defined in the preceding section is employed in three types of equations: 1. Site-filling: These equations identify the species that occupy anion and cation sublattice sites. In the MO2 crystal structure, these are cation sites (4.24) [M M ] + [VM″″] = [M M ]∗ , which says that a cation sublattice site must contain either a metal ion MM or a vacancy VM″″. The metal ion may need to be divided into the various valence states and any impurity cations. Similarly, the anion site-filling condition is

[OO ] + [VO ] = [OO ]∗ .

(4.25)

••

2. Electric neutrality, or charge balance: This equation satisfies the requirement that the oxide be electrically neutral. Using MO2 as an example, but without multiple cation valences or cation impurities, 2[VO ] + 4 [M I ••

••••

] = 4[VM″″ ] + 2[O I″ ]

.

(4.26)

3. Equilibrium equations: These describe the thermodynamics among point defects or between point defects and the external gaseous environment (Sec. 4.3.7).

4.3.5 Anion-Frenkel defects Frenkel defects can form from either cations or anions. Anion-Frenkel defects are produced by moving an anion from a regular site on the anion

Point Defects 175

sublattice to a site on the anion interstitial sublattice. This process is expressed by (4.27) OO =  VO + O I′′ . ••

The law of mass action for this reaction is expressed in terms of site fractions rather than volumetric concentrations: −E s (4.28) K FO = CVO C IO = exp  FO  exp  FO   k B   k BT  where the subscript FO denotes Frenkel defects on the anion (oxygen) sublattice. The σ∆v term that appears in Equation (4.10) for elemental crystals has been neglected in Equation (4.28). If anion-Frenkel defects are the only point defects in the crystal, electrical neutrality is (4.29) [VO ] = [O I′′ ] ••

or, in terms of site fractions (from Eq. [4.16] and [4.17]), CVO [OO ]

∗

=   C IO [I O ]

∗

 .

(4.30)

With this relation, Equation (4.28) permits determination of both point-defect site fractions: C IO = 2 K FO

CVO = K FO / 2 ,

(4.31)

which assumes that 2[IO]∗ = [OO]∗ (for MO2).

4.3.6 Schottky defects Schottky defects consist of vacant sites on both the anion and cation sublattices (see Fig. 4.5 for the Schottky defect in MO2). The reaction producing a Schottky pair is M M + 2OO = 2VO + VM″″ + (M M + 2OO )surf ••

(4.32)

176 Light Water Reactor Materials

where VM″″ and VO are vacancies on the cation and anion sublattices, respectively. (MM + 2OO)surf represents bulk ions that have moved to the surface, leaving behind the vacancies. The law of mass action for Equation (4.32) is ••

 sS   − Es  . 2 C K S = CVO VM = exp   exp  kB k BT 

(4.33)

KS is the Schottky equilibrium constant expressed in terms of the entropy sS and energy ES of formation of a Schottky triplet. In the absence of impurity cations and competing defect processes, charge neutrality is expressed by 2 (VO ) = 4 (VM″″ ) ••

or, dividing by Equation (4.22), CVO =   CVM .

(4.34)

(4.35)

Combining Equations (4.33) and (4.34) yields the individual vacancy site fractions: (4.36) CVO = CVM = 3 K S . If the oxide contains aliovalent impurity ions (i.e., ions of different valence from the principal cation), their concentrations are included in Equation (4.22), but the solution method would be similar to that employed in the anion-Frenkel analysis of Section 4.3.5. If other defect formation processes are operative, the electrical neutrality condition of Equation (4.22) must include all charged species. For example, if anion-Frenkel defects were dominant, CVO would be determined by Equation (4.31), not Equation (4.36).

4.3.7 Gas-phase/defect equilibrium Finally, a connection between point-defect concentrations and the external environment is needed. For oxides such as MO2, this is supplied by

Point Defects 177

the equilibrium between the oxygen partial pressure in the gas phase and point defects in the solid. A possible reaction of this type is OO = 1 O 2 ( g ) + VO + 2e ′ . (4.37) 2 In the reverse of this reaction, an oxygen atom (½ O2) enters a vacant anion sublattice site (VO ) , at the same time extracting two electrons from the solid (2e′). The result is an O2− ion on a previously vacant anion sublattice site (OO). The equilibrium condition for the reaction is expressed by ••

••

(VO )(e ′) 2 po2 . K= (OO ) ••

(4.38)

Gas-solid equilibria other than reaction (4.37) may connect the oxygen partial pressure with point-defect concentrations. The case of UO2 outlined in Chapter 16 provides a more realistic example.

References 1. P. Kofstad, Nonstoichiometry, Diffusion and Electrical Conductivity in Binary Metal Oxides (Hoboken, New Jersey: Wiley InterScience, 1972). 2. F. Kroger and H. Vink, Relations Between the Concentrations of Imperfections in Crystalline Solids, Solid State Physics 3 (1956): 310.

Problems 4.1 A substance with the alumina-type (M2X3) crystal structure forms Schottky defects with an equilibrium constant KS.  This material is doped with a cation impurity with a valence of +2. The fraction of the cation sites containing the impurity ion is xDM. Derive the equation for the equilibrium fraction of vacancies on the anion sublattice in the doped crystal. Solve this equation for sS = 4Rg, ηx = 4 eV,

178 Light Water Reactor Materials

4.2

   4.3

4.4

T = 1800 K, and xDM = 10−2. Compare this result to the anion vacancy fraction in the undoped crystal and explain the difference. (a) Anion-Frenkel and Schottky defects exist in comparable amounts in an M2X type crystal. The anion X is divalent, and the cation M is univalent. The defect equilibrium constants are KFX and KS, respectively. The number of anion interstitial sites is equal to the number of regular anion sites. What fraction of the anion and cation lattice sites are vacant? (b) Repeat part (a) for an MX-type crystal. An M2X-type crystal forms Schottky defects with equilibrium constant KS. The crystal is doped with a cation site fraction xDM of an impurity of the DX type. The common anion has a valence of −2. Derive the equation giving the cation vacancy fraction xVM. The energy stored in graphite as displacements (i.e., vacancyinterstitial pairs, or Frenkel pairs) produced by low-temperature irradiation can be released if the temperature is raised beyond a critical point. The phenomenon is called “Wigner release” after the physicist who first predicted its existence. The phenomenon was responsible for the overheating of an early British reactor at Windscale, United Kingdom.  The magnitude of the stored energy, Q J/g, for irradiation of a graphite specimen to a particular fast-neutron fluence is measured in the following experiment. Ten grams of irradiated graphite are placed in a furnace and held at 200°C, and the temperature is monitored as a function of time. The upper curve on the graph shows the temperature history. The sample is then cooled to room temperature and the anneal repeated. The lower curve shows the time-temperature behavior of the second anneal. There is no stored energy to be released in the second anneal, so the lower curve represents simple heat transfer between the sample and the furnace. The rate of heat addition is found to be represented by the function

Point Defects 179

Temperature (°C)

300 0.44(200 − T) J/min, where T is in °C. The temperature at which First anneal 250 the two curves diverge is 128°C and the maximum temperature 200 achieved in the first anneal is Second 270°C. The area between the two anneal curves is 4200°C-min. The heat 150 capacity of graphite is 1.26 J/g-°C. (a) Determine the value of the 100 stored energy per gram of graphite. Hint: The rate of 50 release of stored energy is written as a temperature0 0 40 80 120 160 dependent function q(T), Time (min) which can be used in devel- FIGURE 4.P1: Time vs. temperature plot oping equations from which of graphite placed in a furnace held at Q can be determined. 200°C. Sample previously irradiated at (b) If the stored energy is 55°C (upper curve); same sample after due to recombination of annealing (bottom curve). (After G.J. Dienes radiation-produced Frenkel and G.H. Vineyard, Radiation Effects in Solids, pairs, what was the atomic Wiley-Interscience, Inc., New York, 1957 p. 100.) fraction of these pairs prior to annealing? The formation energies of vacancies and interstitials in graphite are 500 and 1340 kJ/ moles of defect, respectively. 4.5 In addition to single vacancies, a small concentration of di vacancies (denoted by V2) is present naturally in elemental solids. The equilibrium concentration of V2 is maintained by formation from single vacancies (V) and thermal decomposition of the divacancies.  The divacancy has a negative formation energy with respect to the single vacancies of which it is composed. The entropy of formation of the divacancy from single vacancies is sV2.

180 Light Water Reactor Materials

FIGURE 4.P2: Vacancy and divacancy.

Using the “chemical reaction” equilibrium approach: (a) What is the “chemical reaction” that describes formation of the divacancies from single vacancies? (b) Write the law of mass action that relates the equilibrium constant KV2 for the reaction of part (a) to the site-fraction concentrations xV of single vacancies and xV2 of divacancies. (c) Express the equilibrium constant KV2 in terms of εV2 and sV2. (d) Is sV2 positive or negative? Give reasons. (e) Does xV2 increase or decrease as the temperature is increased? Show why with appropriate equations. 4.6 An ionic crystal of the MX2 type forms Schottky defects, for which the energy of formation is 200 kJ/mole. By how much should the temperature be reduced from 2000 K order to reduce the concentration of anion vacancies by a factor of 10? 4.7 The vacancy formation enthalpy of a metal is 1.5 eV. The material is under a pressure of 850 MPa, and the lattice parameter of this simple cubic metal is 0.286 nm. Make a plot of the equilibrium vacancy concentration versus the inverse of absolute temperature (K −1) (a) for the case where no pressure is applied (b) for the case where the pressure above is applied. How much does the vacancy concentration change at 1500 K when pressure is applied?

5

Chapter Diffusion in Solids 5.1 General Features of Diffusion Solid-state diffusion is based on the ensemble of mechanisms by which atoms and point defects migrate through a crystal lattice. Diffusion is often the step that limits the rate of processes such as phase transformations or microstructure evolution in a solid. In this chapter, both the microscopic and macroscopic aspects of diffusion are reviewed. The former is useful in understanding the physics of migration and the origin of the material property called the diffusivity or the diffusion coefficient that quantifies the process. The latter is essential to calculating the concentration distribution of the diffusing species, the diffusion rate, and the effect of irradiation on these properties. The flow of solute atoms is called a flux, although strictly speaking, it is a current. In either terminology, it represents the net number of atoms that pass a plane of unit area per unit time. The flux of solute atoms is driven by a chemical potential gradient, generically referred to as a driving force. The most common driving force is a nonuniform distribution of solute atoms, otherwise known as a concentration gradient. Other forces can result in movement of solute atoms relative to the host crystal, including a temperature gradient and an electric-field gradient.

182 Light Water Reactor Materials

We concentrate primarily on diffusion generated by a concentration gradient, referred to as ordinary or molecular diffusion. Diffusion can occur in one, two, or three dimensions. The most common is three-dimensional diffusion, in which the diffusing species moves relative to the host lattice in any of the three directions. However, in anisotropic crystals, the diffusivities in different crystal directions may not be equal. For example, in the hcp crystal structure (Fig. 3.8), the rate of atomic movement in the c direction is different from that in the a directions. Two-dimensional diffusion occurs on the surfaces of solids or along internal surfaces that separate the grains of polycrystalline solids. The latter is termed grain boundary diffusion. Movement of a trapped foreign atom along a dislocation line is an example of one-dimensional diffusion. In addition, macroscopic diffusion may appear to take place in fewer dimensions than that of the physical medium. For example, diffusion in a sphere with solute uniformly spread on the surface is a one-dimensional (radial) problem even though the sphere is a threedimensional body.

5.1.1 Diffusion in nuclear processes Solid-state diffusion controls the rate of many important chemical and physical processes that take place in a nuclear fuel rod, a few of which are summarized in Table 5.1. In the fast-neutron and fission-fragment fields inside a reactor core, many diffusion processes are accelerated. Atom mobility enhancement results from the point defects created in copious quantities by collisions of the energetic particles with the host atoms. In addition, the motion of point defects proper is responsible for their agglomeration into extended defects, such as vacancies into voids and self-interstitials into disks (loops, see Ch. 7). Consequently, it is important to keep track of both the migrating species and the extended defects that result from the irradiation and diffusion processes.

Diffusion in Solids 183

Process Corrosion of cladding: • By water (normal operation) • By steam (severe accident) Hydrogen redistribution in cladding Hydrogen pickup into cladding Fission gas release from fuel or bubble formation in fuel Sintering and creep of fuel

Diffusing species

Host solid

O2−, e− O H

ZrO2 Zr/Zr suboxides Zr

H Xe, Kr

ZrO2 UO2

U 4+

UO2

TABLE 5.1: Solid-state diffusion processes in nuclear materials.

Extended defects can profoundly affect the mechanical and dimensional properties of the structural metals and ceramics in which they form. Self-interstitial diffusion combined with the preferred orientation (texture) of Zircaloy produces the phenomena of preferred-direction growth (in the absence of stress) and irradiation creep (with stress present).

5.1.2 Types of diffusion coefficients Diffusion processes are characterized by the identity of the moving species and path through the crystal (the mechanism) by which it moves. Below are several typical examples. Self-diffusion characterizes the movement of a species in the absence of a concentration gradient. It refers to the random motion of atoms or point defects in a solid. This is the type of diffusivity deduced from the atomic mechanisms discussed in Section 5.4. The measurement of a diffusion coefficient requires that a migrating species be identified. Thus, when similar atoms move, their motion is not detectable, so that strictly speaking, there is no direct method for measuring the self-diffusion coefficient.

184 Light Water Reactor Materials

However, if it is known, for example, that defect motion occurs by a vacancy mechanism, the motion of the vacancy can be measured, providing an indirect measure of the self-diffusion coefficient. Tracer diffusion describes the mixing of different isotopes of the same element. It is the principal means of measuring self-diffusion coefficients. This diffusivity is subject to the same conditions as the self-diffusion coefficient except that some of the atoms are radioactive isotopes of the host element or ion. Measurement of the tracer diffusivity requires a gradient (and hence a flux) of the tracer, but there is no gradient or flux of the combined radioactive and nonradioactive element or ion. Self-diffusivity and tracer-diffusivity are essentially equal. Mutual diffusion and chemical diffusion are terms describing the interchange of species A and B by diffusion. In binary solids, this diffusion coefficient, DAB, is related to the self-diffusion coefficients DA and DB by Darken’s equation (App. A): ∂ lna A  D AB = ( x A DB + x B DB )  (5.1)  ∂ lnx A  where aA and xA are the activity and mole fraction of component A, respectively. From the thermodynamics of the solution, aA is a function of temperature and composition. In an ideal solution, aA = xA. The selfdiffusivities DA and DB are functions of composition and temperature. In the second parenthetical term on the right, A can be replaced by B because the activities of the two species are coupled (Prob. 5.3). In addition to binary alloys, Equation (5.1) applies, for example, to the interdiffusion of U and Pu in (U,Pu)O2 in a concentration gradient of the two cations. The diffusivities defined above suffice for binary liquid systems, because the medium has no structure. In crystalline solids, on the other hand, the atoms are located on specific sites that possess distinctive geometrical properties (see Ch. 3). In addition, as discussed in Chapter 4, atoms can be

Diffusion in Solids 185

missing from some sites (vacancies) or occupy positions that are not regular sites (self-interstitial atoms, or SIAs). These are termed point defects. As shown in section 5.4 the point defects are the principal agents of solid state diffusion. Diffusion requires the presence of point defects, of which vacancies are the most common. These are characterized by a diffusion coefficient DV that is different from the atom diffusivity (see Eq. [5.40]). An example in an irradiated metal is diffusion of vacancies to voids (Ch. 19).1

5.2 Macroscopic Description of Diffusion Just as thermodynamics can be described either from a macroscopic, classical viewpoint or in a microscopic, statistical setting, so can the process of diffusion be viewed in either of these two ways. The macroscopic laws of diffusion are combinations of a species conservation equation with specification of the flux of the solute relative to the host substance.

5.2.1 Species conservation Conservation of a species whose concentration is c moles per unit volume is shown in Figure 5.1.2 The diagram shows a volume element of unit area and thickness dz. The flux of diffusing species, J, is the number of atoms (or moles) crossing the unit plane per unit time. The plane is fixed relative to the 1

dz J

Unit area J+

dJ dz dz

c(unit area x dz) atoms

FigurE 5.1: Species conservation in a differential volume.

Chapters 16 through 29 are to be found in Light Water Reactor Materials, Volume II: Applications. To simplify the notation, the subscripts denoting species in an AB solid are omitted. Thus, DAB is reduced to D, flux J means JA or JB, and concentration c can be either cA or cB.

2

186 Light Water Reactor Materials

crystal lattice. There may also be a source or sink of the species, S, inside the volume element. The statement of species conservation is as follows: The time rate of change of moles of the species in the volume element equals the net influx of species plus the creation of the species in the volume element. In mathematical terms, this word statement is ∂J (cdz ) = J − J + dz + Sdz  ∂z  ∂t ∂

or

∂c

=−

∂t

∂J ∂z

+S

.

(5.2)

This conservation statement applies no matter what force is driving the flux J. Among the common forces are gradients of (i) the solute concentration, (ii) the temperature, and (iii) the electric field. These lead, respectively, to fluxes describing ordinary molecular diffusion, thermal diffusion, and ionic transport. The electric field and ionic transport apply only to ceramics.

5.2.2 Fick’s laws of diffusion When a concentration gradient drives J, the flux is given by Fick’s first law: ∂c J = −D . (5.3) ∂z This equation follows the universal observation that matter moves from regions of high concentration to regions of low concentration, hence the minus sign. In principle, the flux J and the concentration gradient are measurable quantities, so Equation (5.3) effectively defines the diffusion

Diffusion in Solids 187

coefficient D. This definition of D is motivated by the fact that for a small enough perturbation from equilibrium (i.e., for a sufficiently small concentration gradient), the flux is proportional to this driving force and D is independent of the concentration gradient. This approximation is valid for a wide range of gradient values. The units of D are length squared per unit time (usually cm2/s). Clearly, J, c, and z must be in consistent units. Substituting Equation (5.3) into Equation (5.2) gives Fick’s second law, D ∂c + S . ∂t ∂ z  ∂ z

∂c

=

∂

(5.4)

This equation is also called the diffusion equation, by analogy to its heat transport counterpart, the heat conduction equation. In the common case of an isothermal system and D independent of solute concentration (and hence of z), and no source term, the diffusion equation simplifies to 2 ∂ c ∂c (5.5) =D 2 . ∂z ∂t Analogous equations for cylindrical and spherical geometry involving only one direction are shown here: Cylindrical geometry: 1 ∂  ∂c  ∂c (5.6) =D r ∂r  r ∂r  ∂t Spherical geometry: 1 ∂  r 2 ∂c  ∂c (5.7) =D 2 r ∂r  ∂r  ∂t

5.3 Useful Analytical Solutions There are a number of analytical solutions to the time-dependent, one-spatial-dimension diffusion equation. Compendiums of such solutions

188 Light Water Reactor Materials

are provided in books by Carslaw and Jaeger [1] and Crank [2]. If the concentration is nonuniform in the directions transverse to z or r, additional second derivative terms are required on the right sides of Equations (5.5) to (5.7). However, mathematical solutions of multidirectional diffusion equations are considerably more complicated than those involving only one spatial dimension. Diffusion problems that are not amenable to closed-form solution can be solved by numerical techniques for which numerous computer codes are available.

5.3.1 Surface-source and diffusion-couple methods Two common methods of measuring diffusivities in solids that are amenable to analytical solutions are depicted in Figure 5.2. Sketch (a) shows the so-called surface-source method, in which a layer of pure solute (A) is deposited on a thick block of pure B into which A diffuses. Source of diffusing species

Surface

Interface

Initially solute-free solid

Solid containing concentration co of diffusing species

C/Co

Same solid initially solute-free

C/Co

1

1

0.8

0.8

0.6

0.6 t

0.4

0.4

t

0.2

0.2

0

0 0

1

2

3

(a)

4

5

6

z

–2

–1.5

–1

–0.5

0.0

0.5

1

1.5

(b)

FigurE 5.2: Methods for the measurement of diffusion coefficients in solids.

2

z

Diffusion in Solids 189

The layer is assumed to be an inexhaustible source in which component B is insoluble. In Figure 5.2b, the source of diffusing species is a block of pure solute A pressed firmly against a block of solvent B containing an initial concentration co of A. This is termed a diffusion couple. In either method, the diffusing species can either be a different element from that which comprises the block on the right or an isotope of this element. The source of the diffusing solute need not be a layer of solid on the surface or another block containing the solute. It could equally well be a liquid or gas containing the solute that dissolves in the adjacent solid and provides the concentration co that drives the diffusion process. The experiment is initiated by raising the temperature to a value at which the diffusivity is large enough to permit sufficient atomic mobility of the solute into the initially solute-free zone for accurate measurement of the concentration to be made in a reasonable time. The diffusion equation is given by Equation (5.5) without the volumetric source term: 2 ∂ c ∂c =D 2 . (5.8) ∂t ∂z z extends from the surface or the interface into the initially solute-free blocks to the right in Figure 5.2. The initial conditions for the two versions are identical: c(z,0) = 0 for z > 0 . (5.9) The boundary conditions at z = 0 are Source method:

c(0, t) = c o

(5.10)

co 2

(5.11)

Couple method: c (0, t ) =

190 Light Water Reactor Materials

In the surface-source method, the surface layer produces an equilibrium concentration co in the adjacent block of B. This concentration remains constant until the layer A is completely depleted. Because of symmetry in the diffusion-couple technique, as soon as interdiffusion starts, the concentration at the interface immediately becomes co/2 in both solids, and this value is retained throughout the diffusion anneal. If the annealing time is sufficiently long, the diffusing solute reaches the far face of the initially pure-B block. However, for shorter times, this medium appears to be infinite in extent, and the boundary condition, c(∞, t) = 0 , (5.12) applies to both versions of the experimental method. Equations (5.8) to (5.12) are solved analytically by the similarity transform method given in Appendix B. The resulting solutions are Source method:  z  (5.13) = c erfc(ξ) 0 ≤ z ≤ ∞ c (z , t ) = c o erfc   2 Dt  o Couple method: c  z  co (5.14) c (z , t ) = o erfc  = erfc( ξ) −∞≤z ≤∞ 2  2 Dt 2 The function erfc(ξ) of the dimensionless argument ξ is called the complementary error function, defined as erfc(ξ) = 1−erf(ξ) where erf(ξ) is the tabulated error function: erf (ξ) =

ξ

exp(−ξ′ 2 ) d ξ′ ∫ 0 π

2

.

(5.15)

The complementary error function erfc(ξ) is unity at ξ = 0 and decreases rapidly with increasing ξ. In particular, at ξ = 1, erfc(ξ) ~ 0.15. That is, at the combinations of time and distance at which ξ = 1, the concentration is at 15% of the initial concentration. This is a useful rule of thumb, in that significant

Diffusion in Solids 191

diffusion can be said to happen when ξ = 1. Because the solution is selfsimilar, the shape of the curve is the same at short time/short distance or long time/long distance combinations that yield the same value of ξ. The evolution of the solute concentration distributions according to Equations (5.13) and (5.14) are shown in the bottom of Figure 5.2. Example #1: Time for appreciable diffusion A solute has a diffusion coefficient of 10−7 cm2/s in a solid. If an infinite source of the solute is available at the interface, how long would it take for significant diffusion to occur at 1 micron, 1 cm, and 10 cm from the interface? x2 Taking the rule of thumb of ξ = 1, we find that t = so that signifi4D cant diffusion happens at the times below for the distances listed: x (cm) 10−4 1 10

t (s) 0.025 2.5 × 106 (~1 month) 2.5 × 108 (~8 years)

The concentration at 1 micron and 0.025 second is the same as that at 10 cm and 8 years. The normalized profile at those two times is identical. One method of determining D from such experiments is to slice thin layers of the initially solute-free blocks and measure the concentration of solute in each layer. Alternatively, the solids can be cut to obtain a cross section perpendicular to the surface or interface. The solute concentration profile is measured by one of a number of methods. For example, an energetic beam of highly collimated electrons or ions excites the solute species, and radiation from decay of the excited atoms is recorded by a suitable detector. The solute concentration profiles so obtained are fitted numerically to Equations (5.13) or (5.14) to obtain the best-fitting value of D.

192 Light Water Reactor Materials

The arguments of the complementary error functions in Equations (5.13) and (5.14) can be interpreted as the ratio of the variable depth z to a characteristic diffusion depth, zdiff = 2 Dt , or ξ = z/zdiff . The nature of the complementary error function is such that the penetration of solute is effectively limited to depths z ≤ ~2zdiff (where erfc = 0.0047). This condition serves to restrict the anneal time in the diffusion experiments when the thickness of the blocks of solid is a finite value L ≥ zdiff. The methods described above are useful as long as the diffusivity is not dependent on composition (which is true if Fick’ s first law is valid), in which case the graphical methods described in Section 5.6 are employed. Example #2: Diffusion penetration depth If the solid blocks used in a diffusion couple experiment are 5-mm-thick slabs and the diffusion coefficient of the solute is 10−10 cm2/s, what is the maximum annealing time for which the boundary condition of Equation (5.12) is valid? For the solute concentration to be essentially zero at the back face of the slab z = L, the time must be such that L > 2 z diff = 4 Dt max . Setting L = 0.5 cm and solving for the time gives tmax = 1.6 × 108 s, which is >5 years. The solute penetration depth for a more realistic experimental time of, say, 2 months (5 × 106 s) is z diff = 2 10 −10 × 5 × 10 6 = 0.045cm or less than 0.5 mm. Sophisticated sampling methods are required to accurately measure a concentration distribution over such a small distance.

5.3.2 instantaneous-source method Instead of the inexhaustible surface source that led to the solution given by Equation (5.13), a common experimental technique involves depositing a

Diffusion in Solids 193

thin layer of the diffusing species on the surface. This limited quantity is assumed to be immediately dissolved in a very thin layer of the large block. Equations (5.8), (5.9), and (5.12) apply to this so-called instantaneoussource situation, but Equation (5.10) is not valid at the surface. In its place, the quantity of diffusing species per cm2 initially deposited on the surface (M) and subsequently completely diffused into the bulk is independent of time and must be conserved: ∞

M = ∫ c (z,t)dz .

(5.16)

0

A solution of Equation (5.8) that satisfies the initial condition Equation (5.9) and the boundary condition Equation (5.12) is c = bt−1/2exp(–z2/4Dt). b is a constant that is determined by substituting this solution into Equation (5.16), which yields b = M/(πD)−1/2. The final solution for the concentration distribution is c=

 z2  M exp  −  . πDt  4Dt 

(5.17)

5.3.3 Diffusion in finite geometries In this section, the geometries are a slab, a cylinder, and a sphere. The diffusion equation must be solved by the method of separation of variables and the concentration distribution expressed as an infinite series instead of the closed form represented by the complementary error function for a thick slab. The solutions for one-dimensional spherical, cylindrical, and slab geometries are given by Carslaw and Jaeger [1]. The equations solved are Equations (5.5), (5.6), and (5.7) without the source term. In slab geometry, the origin of z is the midplane, which permits the slab problem to be compared directly to the other two geometries. The initial conditions are

194 Light Water Reactor Materials

c(z,0)   =  0; 0 < z < L and c(r,0) =  0; 0 < r < R .

(5.18)

The boundary conditions at the surface are c(L,t) = c o and c(R,t) = c o

(5.19a)

where L represents the half-thickness of the slab and R the radius of a cylinder or sphere. Because of symmetry, the boundary condition at the origin is  ∂c  = 0 or  ∂c  = 0 (5.19b)  ∂r  r = 0  ∂z  z =0 Figure 5.3 graphs the solutions of Equations (5.5), (5.6), and (5.7) subject to the boundary conditions. Shown here is the volume-averaged concentration c– (relative to co) expressed in terms of the square root of the dimensionless time τ: Dt Dt (5.19c) τ = 2 or 2 . L R The analytical solution for the infinite medium can be regarded as a “short-time” approximation for the slab results shown later in Figure 5.3. Example #3: Accuracy of the short-term solution A 5-mm-thick slab is exposed to a source of solute on its surface for 107 seconds. The diffusion coefficient is 10−9 cm2/s. Compare the relative average concentration obtained from the analytical solution with that read from Figure 5.3. The dimensionless time is Dt/L2 = 10−9 × 107/(0.5)2 = 0.04. The average concentration in the slab is M/L, where M, the total quantity of solute in the solid, is given by Equation (5.11): c– co

=

M/L 2 co π

Dt L2

2 0.04 = 0.23

= π

Diffusion in Solids 195 1.0 0.9

Sphere (radius R)

0.8

Cylinder (infinite length, radius R)

0.7 0.6 – c 0.5 co 0.4

Slab (infinite width and length, thickness 2L)

0.3 0.2 0.1 0 0

0.5

1.0 Dt/L2 or

1.5

Dt/R2

FigurE 5.3: Average solute concentration in a slab, cylinder, and sphere with zero initial concentration and a fixed concentration co at the surface. (Redrawn with permission of Oxford University Press, USA.)

For comparison, on the curve for the slab in Figure 5.3 for Dt/L2 = 0.04 = 0.2, c–/c o = 0.23. Although the two methods agree here, deviations begin at longer times.

5.3.4 Fission-gas release in postirradiation annealing The classical model of the release of the fission gases xenon and krypton is treated in Chapter 20. In this section, attention is restricted to the mathematical aspects of the theory as an example of the use of Fick’s second law (the diffusion equation) in spherical coordinates. Briefly, the model assumes that the rate-limiting step is diffusion of the fission gases in the grains of uranium dioxide to the grain boundaries. The grains are modeled as spheres of radius Rgr. Upon reaching the periphery of the grains, the fission gas is assumed to be immediately released.

196 Light Water Reactor Materials

In a postirradiation anneal experiment designed to measure the diffusivity of the fission gases, the UO2 fuel specimen is irradiated in a neutron flux at a temperature sufficiently low that none of the fission gas is released. The irradiation produces a uniform concentration of Xe and Kr in the spherical grains representing the microstructure of the solid. The specimen is removed from the reactor and annealed at a high temperature. Fission gas that escapes is trapped and measured by its radioactivity. This information yields the fraction of the fission gas released as a function of time, which is the quantity predicted by diffusion theory. Assuming constant diffusivity, application of Equation (5.7) to this process gives ∂c fg 1 ∂  2 ∂c fg  (5.20) = D fg 2 r ∂r r ∂r  ∂t with the initial condition c fg (r ,0) = c fgo

(5.21)

and the boundary conditions  ∂c fg   ∂r  = 0 r= 0

and

c fg ( R gr, t ) = 0

(5.22)

where Rgr is the grain radius. The last of these boundary conditions arises from the assumption that the gas escapes without further resistance upon reaching the grain surface. The above analysis is referred to as the Booth model. The “short-time” solution is developed by the Laplace transform method in Appendix C. The result for the average concentration in the sphere or, alternatively, the fraction of the initial amount released is c fg 6 (5.23) f = 1− c = 1− τ + 3τ π fgo

Diffusion in Solids 197

where τ = Dfg t/R gr2 is a dimensionless time. Equation (5.23) can be directly compared to the upper curve in Figure 5.3, which is the exact solution. The comparison is direct because of the difference in the boundary and initial conditions used in the two methods; in Figure 5.3, the initial condition is cfg = 0 and the boundary condition at the periphery is cfg = cfgo. In the present solution, the initial and boundary conditions are reversed. Hence, the ordinate from Figure 5.3 is to be compared to the fraction release from Equation (5.23). Example #4: Fission-gas release Calculate the fission gas release from Equation 5.23 when the dimensionless time is approximately 0.1. For τ = 0.1, Equation (5.23) gives f = 0.23. From the upper curve of Figure 5.3 at τ = 0.1 = 0.32, read f = 1 − c fg/c fgo = 1 − 0.78 = 0.22. Analytical solutions such as Equation (5.23) are useful for approximations to realistic diffusion situations. However, more complex boundary conditions, a time-dependent source term, or concentration-dependent diffusivity need to be treated numerically (see Ch. 15).

5.4 Atomic Mechanisms of Diffusion in Solids Although the previous sections provide a quantitative description of diffusion in solids, nothing is said about the mechanism. The mechanism (Fig. 5.4) was originally thought to be either (a) a direct exchange where an atom switches places with a nearest-neighbor or (b) a “ring” mechanism in which atoms execute coordinated jumps that produce the same result. A distinction between self-diffusion of a host crystal and impurity diffusion in the host crystal is an important generalization. For most

198 Light Water Reactor Materials

(a)

(b)

(c)

FigurE 5.4: The (a) direct exchange, (b) ring, and (c) vacancy diffusion mechanisms. (Redrawn from The Minerals, Metals & Materials Society, 1997.)

systems, self-diffusion is dominated by vacancies, since creating selfinterstitials is usually a high-energy process. However, impurities can be substitutional or interstitial, and both types of diffusion typically play a significant role in atomic transport. The operative mechanism (c) was demonstrated by Kirkendall in 1947 [4, p. 132], which reported the results of the following experiment. In Figure 5.5, a layer of Cu was electroplated around a bar of brass (70Cu30Zn alloy) on which molybdenum wires were laid down to serve as markers. The wires are inert, as Mo neither dissolves in nor diffuses in the Cu/Zn alloy at the temperature of the experiment. This sample was then annealed at 1058 K for various times. The annealing results in interdiffusion of Cu and Zn. If diffusion involved an exchange or ring mechanism, the flux of Cu and Zn atoms across the tungsten wire boundaries would balance, and the distance between the tungsten wires would remain constant. Instead, it was observed that the brass core shrunk; that is, the molybdenum wires got closer together as the annealing time increased. The interpretation was that the operative diffusion mechanism was (c) in Figure 5.4, not (a) or (b). Because Zn diffuses outward faster than Cu diffuses inward, there is a net flow of Zn atoms out of the brass region, causing it to shrink. Simultaneously, there is a flux of vacancies in the

Diffusion in Solids 199 Molybdenum wire

Brass

Pure Cu

FigurE 5.5: Schematic setup of Kirkendall’s experiment [18].

opposite direction, which should have cancelled brass shrinkage due to loss of Zn. That this did not occur is due to the rapid re-establishment of the equilibrium vacancy concentration by removal of vacant lattice sites by filling with a metal atom. Had (a) or (b) been the driving mechanism, exchange of Cu and Zn atoms would not have caused movement of the Mo marker wires. This was the first demonstration that atomic diffusion in metals is mediated by point defects. A vacancy flux caused by a chemical concentration gradient is called the Kirkendall effect. In Chapter 24, we will see how the inverse Kirkendall effect (chemical species flux resulting from a point-defect concentration gradient) can cause irradiation-induced segregation. In general, diffusion of atomic species in metals depends on the presence of point defects that allow atomic motion to take place at the lowest energy. The movement of diffusing species (e.g., point defects, substitutional atoms, radioactively tagged host atoms, interstitial impurities) relative to the host crystal ultimately depends upon the interatomic potential with which a solute atom interacts with surrounding host atoms and the analogous potential function between host atoms. These interatomic potentials are responsible for sites in the crystal lattice in which the potential energy of the diffusing species is a minimum. The migration process consists of movement of an atom between neighboring equilibrium sites, which requires passing through a potential energy barrier known as the migration energy of diffusion, Em.

200 Light Water Reactor Materials

The diffusion coefficient depends on: • the lattice structure of the host crystal; • the equilibrium location of the diffusing species; • the path followed by the diffusing species through the lattice from one equilibrium site to another (the diffusion path will always be the one of least resistance, specifically the route that demands the lowest energy increase to effect the change in position); and • the temperature, which dictates the vibrational energy of all species in the system (the higher the temperature, the more likely the diffusing species is to acquire the necessary energy increment for migration).

5.4.1 impurity motion between equilibrium sites To illustrate the impurity-diffusion process, the elementary case of a small interstitial impurity atom migrating through a simple cubic lattice is shown in Figure 5.6. The equilibrium site of the impurity atom is the center of the cube. At this position, the distance between the impurity atom and the nearest host atoms is the largest possible, which usually minimizes the repulsive potential interactions. Because of the thermal energy shared by all atoms of the system, the impurity atom vibrates in three orthogonal [100] directions. The energy of each vibration is distributed according to a Boltzmann Barrier ao rb req

Impurity atom

x y z

FigurE 5.6: Idealized jump of impurity atom in crystal with simple-cubic structure.

Diffusion in Solids 201

distribution, and very few of the vibrations possess sufficient energy to move to the adjacent equilibrium site. The migration energy is the difference between the potential energy of the impurity atom at the midplane of its trajectory (the barrier) and the potential energy in the equilibrium site. The potential energy variation as the impurity atom moves along the jump direction is shown in Figure 5.7. The dashed curves represent the components of the interatomic potential between the impurity atom and each of the host atoms in the diagram of Figure 5.6. The energy in the equilibrium site, U(req), is the sum of the potential energies of the pairwise interactions between the impurity atom and all of the neighboring host atoms (to a first approximation, equal to the number of nearest-neighbors times the pairwise potential shown in Fig. 5.7). As the impurity atom moves along its path, its distance from the four atoms in the barrier plane is reduced from req to rb, where the total energy is positive (repulsive). The sum of the interaction energies at this location is Ub, which is higher than Ueq by the barrier energy Em = Ub − Ueq. Only a very small fraction of the impurity atom vibrations

Host-impurity interatomic potential (φ)

+

0

Repulsive

r rb

req

Total

– c tra At

e tiv

FigurE 5.7: Potential energy variation as the impurity atom moves along its migration path.

202 Light Water Reactor Materials

in the equilibrium site have the requisite activation energy Em to permit a jump. Because of the energy barrier, the diffusing species spends most of the time vibrating in equilibrium sites and only occasionally moves to an adjacent site. This movement is called a jump, the length of which is denoted by λ. For the diffusion mechanism illustrated in Figure 5.6, the jump distance is one lattice parameter, ao. For most realistic atomic mechanisms of diffusion, λ is smaller than a lattice parameter. A parameter critical to quantitative description of the diffusion process on an atomic scale is the vibration frequency of the impurity atom in its equilibrium site, denoted by υ. In order to estimate υ, the potential energy curve U(r) at the equilibrium site in Figure 5.7 is expanded in a two-term Taylor series about the equilibrium separation distance: 1  d 2U  U (r ) = U (req ) +  2  (r − req )2 + … . 2  dr  r

(5.24a)

eq

The absence of the first derivative reflects the minimum of U at this location. The force on the vibrating impurity atom is the derivative of the potential energy: F (r ) = −

dU dr

 d 2U  (r − req ) .  dr 2  r

= −

(5.24b)

eq

This equation is formally identical to the restoring force on an extended spring, with the second derivative of U at req serving as the spring constant. The linear restoring force leads to simple harmonic motion with an oscillation frequency 1/2

  1  1  d 2U   υ= 2π m  dr 2  r  eq   where m is the mass of the impurity atom.

(5.25)

Diffusion in Solids 203

A reasonably accurate estimate of υ can be obtained by approximating the potential energy function as a sinusoid of amplitude Em and wavelength λ:  π(r − req)  U (r ) = U (req ) + Em sin 2  .  λ 

(5.26)

Squaring the sine function has no other purpose than assuring positive U at all values of r. The second derivative of this function at req is  d 2U  2π 2 Em , = (5.27)  dr 2  λ2 r eq

which, when substituted into Equation (5.25), yields υ=

1/2 1  Em  . 2 mλ 2 

(5.28)

Example #5a: Vibration frequency of the diffusing atom in its equilibrium position In order to estimate υ, we take the mass of the diffusing atom to be that of hydrogen, a common impurity in many metals (m = 10−3 kg/mole). In Figure 5.7, the jump distance is a lattice constant, or λ ≈ 3 × 10−10 m. There is no simple estimate of the barrier height Em, theoretical knowledge of which would require a complete molecular model of the jump process (Ch. 15). Hence, we assume a typical value of 100 kJ/mole.3 Using these values in Equation (5.28) yields the impurity atom vibration frequency: υ=

3

1/2 10 5 J /mole  1  ≈ 2 × 1013 s −1 2  10 −3 kg /mole × (3 × 10 −10 )2 m 2 

Em is commonly expressed in electron volts per atom (eV/atom). The conversion factor is 96.5 kJ/mole per eV/atom.

204 Light Water Reactor Materials

or 2 × 1013 Hz. This computation makes use of the relationship for SI units J/kg = (m/s)2. The resulting vibration frequency is typical of atoms in a solid. The frequency at which a diffusing atom succeeds in moving from an equilibrium site to an adjacent one is much less than its vibration frequency in the equilibrium site. The vast majority of the vibrations do not possess sufficient energy to overcome the energy barrier Em. Assuming that the energy of the vibrating atoms follows a Boltzmann distribution the fraction with this energy is the factor exp(−Em /kBT). kB is Boltzmann’s constant, and T is in Kelvins. The jump frequency w is therefore −E w = υ exp  m  . (5.29)  k BT  Example #5b: Estimate the jump frequency of an impurity atom at 1000 K with a barrier energy of 100 kJ/mole = 1.04 eV/atom −1.04  = 2 × 10 8 s −1 w = 2 × 1013 exp  −5  8.62 × 10 × 1000 or ten in a million vibrations result in a diffusional jump. What distance will the atom cover within a second if the jump distance is 0.3 nm? In a random walk the average distance between the starting and end points is given by d N or 0.3 2 × 10 8 = 4.24 microns. The preceding explanation of the origin of Equation (5.29) is simple and leads to the approximately correct result. The rigorous derivation of this equation, however, is based on statistical mechanics and is considerably more complex than the “derivation” presented here (see [3, Sec. 7.5]).

5.4.2 The Einstein equation One of the most important equations in diffusion theory is attributed to Einstein. This equation connects the macroscopic property D, the diffusion coefficient, with the microscopic properties w, the jump frequency, and λ,

Diffusion in Solids 205

Impurity atom Host atom

J–

z J+

λ

FigurE 5.8: One-dimensional diffusion.

the jump distance. The original Einstein equation in three dimensions is presented in [3, Sec. 7.3] and [4,  Sec. 2.3]. Figure 5.8 is the basis of a simplified version in one dimension. In this case, the diffusing atoms occupy interstitial planes at an areal density of n atoms per unit area, decreasing from left to right in the diagram. The jump distance λ is equal to a lattice constant, so the areal density n is related to the volumetric concentration c (atoms per unit volume) by c = n/λ. The rate of jumping from the left plane of impurity atoms to the right plane is J+ = n(z)w, where z is distance along the jump direction. In the opposite direction, the flux is J− = n(z + λ)w. The net flux is J = J+ − J− = [n(z) − n(z + λ )]w = λ[c(z) − c(z + λ )]w .

(5.30)

 ∂c  Expanding c(z + λ) in a one-term Taylor series, c(z + λ ) = c(z) +  ∂ z  λ, converts the flux equation to ∂c J = − (λ 2 w ) . (5.31) ∂z

206 Light Water Reactor Materials

Comparison of this equation with the macroscopic definition of the diffusion coefficient given by Equation (5.3) shows that for this one-dimensional problem, (5.32) D1D = λ 2 w where w is the frequency with which an impurity atom jumps from an equilibrium site to a particular adjacent equilibrium site. The frequency with which the diffusing atom jumps to any available adjacent site is called the total jump frequency, and is denoted by Γ. In Figure 5.8, only jumps to the left or the right are allowed, so Γ = 2w, and the above equation becomes 1 2 λ Γ . (5.33) 2 In three dimensions, 1/2 in the above equation is replaced by 1/6 and Γ = Zw, where Z is the number of sites that are within a jump length of the diffusing atom: D1D =

D=

1 2 1 2 1  − Em  , λ Γ = λ Zw = λ 2 Z υ exp  k BT  6 6 6

(5.34)

which is the Einstein formula. The value of Z and the relation of λ to the lattice parameter depend on the crystal structure and the diffusion mechanism. Example #6: The Einstein formula in the simple-cubic lattice In Figure 5.6, λ = ao, the lattice parameter, and Z = 6, representing the six faces through which the impurity atom can jump into the six nearest equivalent jump sites. Using these values in Equation (5.34) and expressing w by Equation (5.29) gives  −E  (5.35) D = ao2 υ exp  m  .  k BT  Example #7: The diffusion coefficient of an impurity atom in the bcc lattice Figure 5.9 shows a (100) plane of the bcc lattice (Fig. 3.5). Also indicated above and below the (100) plane are two body-centered atoms

Diffusion in Solids 207

FigurE 5.9: Impurity-atom diffusion in the bcc lattice.

in adjacent unit cells. The equilibrium site for the impurity atom is the octahedral site (the center of the (100) plane; see also Fig. 4.2). The centers of the edges of the (100) plane are also octahedral interstitial sites. Allowable impurity atom (cube) jumps are indicated by the four arrows. The jump distance is λ = 0.5ao, and the jump frequency multiple is Z = 4. According to Equation (5.34), the diffusivity of the impurity atom is

2 −E −E 1 a 1 D =  o  4 υ exp  m  = ao2 υ exp  m   k BT  6  k BT  6 2

(5.36)

5.4.3 The vacancy mechanism In nearly all metals, self-diffusion of host atoms and migration of substitutional impurity atoms occur by the vacancy mechanism (see also Sec. 5.4). An atom (host or impurity) on a regular lattice site is surrounded by sites, nearly all of which are occupied by host atoms. The atom cannot exchange places with an adjacent host atom because the energy barrier is too high. It can move only if one of its nearest-neighbor sites is vacant. Figure 5.10 shows this mechanism operating in the fcc lattice. The atom jump shown

x

Jump into vacant lattice site

Vacancy x

FigurE 5.10: fcc structure: (a) atom jump into vacant lattice site; (b) vacancy jump into occupied lattice site.

208 Light Water Reactor Materials

in Figure 5.10a is equivalent to a vacancy jump in the opposite direction, which is depicted in Figure 5.10b. These two diagrams are two ways of portraying the same process. However, the end result depends on whether the diffusivity is considered to be that of the atom or the vacancy. The barrier that controls the activation energy Em is the shaded plane in the figure. In this location, the distance between the moving atom and the four corner atoms of the shaded plane is rb = 3/8 ao, which is less than the separation of nearest neighbors in the equilibrium site, req = ao / 2 . The energy barrier is due to the difference in these distances, as in Figure 5.6. We first consider the vacancy in Figure 5.10b as the migrating species. In the fcc lattice, the jump distance is ao / 2 , and the number of allowable jump directions is β = 12. Using Equation (5.31) for the jump frequency in the Einstein formula of Equation (5.34), the vacancy diffusivity in the fcc crystal is −E DV = ao2 υ exp  m  . (5.37)  k BT  To analyze the atom migration process illustrated in Figure 5.10a, the following difference between the movement of the two species must be recognized. For a vacancy jump, the neighboring site must contain an atom. This probability is 1 − CVeq, which is essentially unity because the vacancy fraction CVeq > CVeq and the self-diffusivity becomes D = DV CV . (5.40a) Although the interstitial variant of Equation (5.38) is negligible because C Ieq is very small, irradiation can generate substantial self-interstitial concentrations. Under irradiation the self-diffusivity driven by interstitial motion D = DSIA C I

(5.40b)

can be larger than the vacancy-mediated D of Equation (5.40a). The location of impurity atoms can be substitutional or interstitial, so diffusion of impurity atoms can be due to either a vacancy or an interstitial mechanism.

5.4.4 general diffusion formula Independent of the mechanism responsible for the species’ mobility, diffusion coefficients are often expressed as  −Q  D = Do exp   .  k BT 

(5.41)

210 Light Water Reactor Materials

The pre-exponential factor Do includes the jump distance and jump frequency, and the activation energy Q accounts for the barrier height Em and other energy requirements for atom movement, such as Ef in Equation (5.39).

5.5 Diffusion in Ionic Crystals In a binary solid such as MX, vacancies (V) and interstitials (I) are present on both the cation (M) and anion (X) sublattices. The two types of diffusion coefficients required to characterize atomic motion are: • Diffusivities of the point defects, DVX, DIX, DVM and DIM. • Self-diffusion coefficients of the atomic species, DX and DM. These are closely linked to the point-defect diffusivities by the mechanism of atom (ion) movement. For example, if the cations move by a vacancy mechanism, there must be a cation vacancy next to a cation to enable jumping of the latter into a new position. Diffusion in ionic crystals differs from that in metals and elements in numerous ways: • Ionic solids consist of at least two oppositely charged components: cations (+) and anions (−) (see Sec. 4.3). • Cations and anions generally exhibit very large differences in diffusivities, occasionally as large as seven orders of magnitude. • The defect that predominates (Schottky or Frenkel) controls the magnitudes of the diffusivities via C eq in Equation (5.38). The effect of substitutional doping of ionic solids with cations of different valence from the host cation has a profound effect on the diffusivities. • The requirement of local electrical neutrality affects the movement of the ions. On the other hand, there is considerable commonality between diffusion in metals and in ionic compounds: • The diffusion coefficients obey the form of the Einstein equation, Equation (5.34).

Diffusion in Solids 211 • • •

The diffusion mechanisms commonly found in metals and elements (vacancy and interstitial mechanisms) also predominate in ionic solids. The various types of diffusion coefficients discussed in the preceding section (tracer, self, mutual) also characterize ionic solids. As in intermetallic compounds, motion of the ions is restricted to the sublattice of the same charge. For an ionic solid with M = cation and X = anion, the self-diffusion coefficients are:

Vacancy mechanism: DM = DVM CVM and D X = DVX CVX

(5.40c)

Interstitial mechanism: DM = DIM C IM and D X = DIX C IX .

(5.40d)

As in metals, the self-diffusion coefficients depend linearly on the site fractions of the point defect responsible for atom motion. These are the equilibrium values in a thermal environment or the much greater concentrations sustained by irradiation. The diffusivities of the defects proper (DVM, DVX , etc.), are independent of their concentrations. Not all four mechanisms are operative in a particular ionic solid. The mechanism that dominates for the cations may be different from that controlling anion motion, leading to different diffusion coefficients for each atomic species. In NaCl-type crystals (MX), the cation diffusivity is substantially larger than the anion diffusivity, or DM >> D X . In fluorite lattices MX2, the opposite is true. Example #8: Cation diffusivity in MX Figure 5.11 shows a sketch of one-quarter of a full unit cell of the NaCl structure with a cation midway in its jump from site a along a face diagonal of the fcc sublattice into a nearest-neighbor cation vacancy at site b.

212 Light Water Reactor Materials

The jump distance is one-half of the face diagonal of the full unit cell, or λ = ao / 2 . Since each cation vacancy in the fcc sublattice has 12 nearest-neighbor cations, the jump-frequency multiple is Z = 12. Substituting these factors into Equation (5.34) and using Equation (5.29), the cation vacancy diffusion coefficient is DVM = ao2 υ exp 

− EmVM 

 k BT 

(5.42)

where ao ~ 0.35 nm is the lattice constant of the full NaCl unit cell (i.e., the height of the two cubes in Fig. 5.11, and υ ~1013 s−1 is the vibration frequency of the cations). At the midway position, the (110) plane ABCD is the barrier. Here, the Coulomb repulsion of the moving cation by the four corner cations is greater than the attraction of the two edge-centered anions. This net Coulomb repulsion is the source of the barrier energy EmVM , taken here to be 90 kJ/mole. The cation vacancy diffusivity at 1000 K from Equation (5.44) is DVM = 500 nm2/s (i.e., 5 × 10−12 cm2/s). According to Equation 5.40c, the diffusion coefficient of the cations is obtained by multiplying DVM by the vacancy fraction, CVM. For highpurity crystals, intrinsic point defects dominate and CVM = K S , where KS is the equilibrium constant for Schottky defects (method of Sec. 4.3.6 for MX). – + A The activation energy for diffusion is the sum of the migration energy of the cation B + – vacancy, EmVM (assumed here to be 90 kJ/ b – a mole or 0.93 eV/atom), and one-half of + – a the energy of formation of the Schottky defects, which in NaCl is ES ~ 220 kJ/ – + D mole (2.28 eV/atom). Neglecting the – C + entropy term in Equation (4.21), at 1000 K, eq KS = 3.2 × 10−12 and CVM ~ 1.8 × 10−6. From FigurE 5.11: Cation diffusivity Equation (5.42), DM = 10−17 cm2/s. in MX. o

Diffusion in Solids 213

Ionic solids are rarely so pure that intrinsic point defects control the point-defect concentrations. If a divalent impurity cation is present, the cation vacancy fraction in MX is given by 1 CVM = (C D + C D2 + 4 K S ) (5.43) 2 where CD is the site fraction of the divalent cation impurity. Even at very low divalent cation impurity concentrations, this impurity exerts a significant effect on the point-defect concentration. For example, for CD as low as 2 × 10−6, Equation (5.43) gives CVM = 3 × 10−6, a 70% increase over CVM in high-purity material. For impurity concentrations greater than a few parts per million, Equation (5.43) reduces to the extrinsic limit, CVM ~ CD. Excepting ultra-highpurity material, the cation diffusivity given by Equation (5.40a) in MX is controlled, via CVM, by the concentration of divalent impurity ions. The importance of diffusion in ionic crystals such as ceramics lies in the understanding of this process in nuclear fuels (see Refs. [6] through [15]).

5.6 Diffusivity Measurement Methods The basic transport properties in MX are the self-diffusivities of the two ions, DM and DX. Considerable effort has been expended in devising experimental techniques for their measurement.

5.6.1 Surface-tracer method This method requires that the metal component M has a radioactive isotope, typically an alpha-particle emitter. A thin layer of tracer-enriched MX is deposited on a surface of a single crystal of the same material but of normal isotopic composition. Figure 5.12 shows how the system operates. Alpha particles emitted at energy Eo lose energy according to the Bragg formula [5, Eq. (6)]. Fortunately, dE/dz is relatively independent of particle energy at these high energies and so can be considered to be a constant in the analysis.

214 Light Water Reactor Materials Thin layer of MX with tracer Mtr

α-particle detector output

Counts

MX single crystal

ctr (z)

α Eo

N(E)

E=(dE/dz)z α energy

z

dz

0

0

dE

Eo

FigurE 5.12: Measurement of cation self-diffusivity in MX using the method of alpha-particle detection.

The advantage of the method lies in the ability to collect alpha-energy spectra at different times during an experiment without disturbing the specimen. The thin layer of MX that is deposited on the single crystal provides Mtr moles of the tracer isotope per cm2 of surface. When heated to a fixed temperature and held for a time t, the tracer self-diffuses into the crystal to produce the volumetric concentration distribution ctr(z) = Ctr(z)/Ω shown in the figure (Ω is the volume of a cation). This is a classic instantaneous plane-source problem for which the solution is (Eq. [5.17]) Mtr z2  (5.44) c tr = exp −  4 Dtr t πDtr t where Dtr is the tracer self-diffusion coefficient of M. After a diffusion time t, the slice of the single crystal between z and z + dz emits λtrctr(z,t) dz alpha particles of energy Eo per second and per cm2, where λtr is the decay constant of the tracer. λtr is small enough that the loss of tracer by radioactive decay over the time interval of the experiment is negligible, which leads to ∞

∫0 ctr (z,t) dz = Mtr

.

(5.45)

Diffusion in Solids 215

Particles directed toward the surface of the crystal leave the surface at energy dE E=  z (5.46)  dz o where (dE/dz)o is the energy-loss rate at the birth energy. This energy-loss rate is essentially constant over the entire path length z of the alpha particles in the crystal. At time t, these particles generate N(E,t)dE counts in the energy range between E and E + dE. The alpha-emission rate and the count rate are related by N ( E , t )dE = Bλ tr c tr (z , t )dz .

(5.47)

B is a collection of instrumental constants and the fraction of alpha particles that are emitted in the solid angle subtended by the detector. Integrating Equation (5.47) and using Equation (5.45) gives Eo

B=

∫0 N(E,t)dE λ tr M tr

(5.48) .

In Equation 5.47, B is replaced by Equation 5.48, ctr(z,t) by Equation 5.44, and dE/dz by Equation 5.46. The result is  −z 2  exp   4 Dtr t  πDtr t

N ( E , t )  dE  .  dz  o ∫ N (E , t) dE

=E o

(5.49)

0

At a specified diffusion time t, the right side of this equation is a measurable function of E, or, using Equation (5.46), a function of z. This function is fitted to the left side of Equation (5.49) by choosing Dtr. This procedure

216 Light Water Reactor Materials

is repeated for several diffusion times, and the resulting best-fit Dtr values should be the same (within experimental error).

5.6.2 Diffusion-couple (Boltzmann-Matano) method The setup for determining the mutual diffusion coefficient of metals in an alloy or cations in ionic crystals is sketched in Figure 5.13. It is the analog of the method applied to elemental solids in Section 5.3.1. t=0

BX

(A,B)X

Fixed plane t>0

O

CA

CA

0

–∞

z

+∞

0.2 Fixed plane 0.1

AL

Matano plane (z = 0)

Tangent

CA

AR C A'R 0 –10

0 z, µm

10

FigurE 5.13: Matano’s method of determining a chemical diffusion coefficient [4, p. 134].

Diffusion in Solids 217

In the top drawing, a block of the mixed-cation crystal (A,B)X is tightly pressed against a block of BX to form a diffusion couple. The blocks are sufficiently thick to be treated as semi-infinite media for z < 0 and z > 0. After time t, the A-fraction profile CA(z,t) is shown in the middle drawing. In this example, the site fraction of A, CA, is the same as the mole fraction xA. Both the mutual diffusion coefficient DAB and the specific volume v = C A v A + C B v B are functions of composition (CA). v A and v B are the partial molar volumes of the two components (if the binary solid is ideal, the partial molar volumes are equal to the molar volumes of the pure species). The difference between the molar volumes of A and B is the origin of the marker movement in Kirkendall’s experiment (Sec. 5.4), irrespective of the mechanism of diffusion. The evolution of the A concentration from the initial state to the distribution shown in the middle sketch is governed by the diffusion equation: ∂C A ∂C ∂  D AB A  . = (5.50) ∂t ∂z  ∂z  The flux of component A (in parentheses) is defined relative to the volume-average velocity (Eq. [A.8] in App. A). The initial conditions are C A = 0 for z > 0 and C A = C Ao for z < 0 .

(5.51)

The boundary conditions for infinitely thick blocks are C A = C Ao for z = − ∞ and C A = 0 for z = ∞ ∂C A ∂z

=0

at z = ± ∞ .

(5.52)

218 Light Water Reactor Materials

A variable change suitable for this problem is u = z , which gives t ∂C A

=−

∂t

u dC A ∂ and 2t du ∂z

1 d . t du

=

(5.53)

For constant diffusivity, applying Equation (5.53) to Equation (5.50) yields u dC A  d 2C A (5.54) − = D AB , 2 2 du du which is integrated from CA = 0, where (dCA/du) = 0, to any location in the distribution CA dC A 1 . − ∫ udC A = D AB (5.55) 20 du Replacing u using u equation:

=

z/ t and dCA/du

=

tdCA/dz yields Matano’s

CA

1  dz  D AB (C A ) = −   ∫ z dC A′ . 2t  dC A  0

(5.56)

This equation permits the chemical diffusivity to be obtained graphically from the experimental CA(z) distribution at any time t. In order to use Equation (5.56), the dependent and independent variables are reversed, and the distribution in the bottom diagram of Figure 5.13 is treated as z(CA). The contents of the parentheses in Equation (5.56) is the diffusional flux of A (i.e., it is the flux relative to a plane through which the mixture velocity is zero). This plane, called the Matano plane, is located at the origin z = 0. The Matano plane moves with respect to the fixed plane in Figure 5.13.

Diffusion in Solids 219

The z = 0 location is obtained by noting that at z = −∞, CA = C oA, and (dCA/dz) = 0. This gradient condition in Equation (5.56) requires C oA

∫0 z dC A = 0

.

(5.57)

Equation (5.57) is the area under the z(CA) curve. The Matano plane is placed so that the area under the z(CA) curve to the left of the plane equals the area under the curve to the right of the plane. In terms of the notation in the bottom plot of Figure 5.13, this is AL = AR + AR′ . In order to determine D AB at an arbitrary point (say, point C in Fig. 5.13), the bottom plot is turned 90 degrees so that the curve appears as the z(CA) distribution. In this orientation, the integral in Equation (5.56) is the area AR′ and the derivative is the tangent to the curve at point C. Accurate methods of determining these quantities are discussed in [4, p. 35]. Example #9: Chemical diffusivity by the Matano method Find the chemical diffusion coefficient at point C in Figure 5.13 (CA = 0.05, z = 2.7 µm). The anneal time is 5 hours. The results are as follow: • The slope of the tangent line is: dz/dCA ~ −80 µm • An approximation to A′R is a rectangle with point C as one corner and a triangle with the hypotenuse connecting point C with 7.4 µm on the z axis: 0.05

A′R =

∫ z dC A = 0.32 µm 0

The result is 1 D AB t = (80(0.32)(10 − 4 )2) = 1.3 × 10 −7 cm 2 . 2

For t = 5 × 3600 = 1.8 × 104 s, D AB = 7 × 10 −12 cm 2/s .

220 Light Water Reactor Materials

5.7 Diffusion in a Thermal Gradient (Soret Effect) The Soret effect refers to the transport of atoms driven by a temperature gradient. Unlike ordinary diffusion in a concentration gradient, thermal diffusion is not a random-walk process; the affected atoms flow along the temperature gradient, either up or down; that is, they preferentially move in a particular direction. Diffusion under a temperature gradient acts on impurity atoms in a host crystal, one important example being hydrogen in Zircaloy (Ch. 23). The Soret effect is characterized by a property of the species in the solid called the heat of transport, which is denoted by Q∗ in units of kJ/mole. Thermal diffusion is accounted for by an additive term to the concentration gradient in the flux equation dc Q ∗c dT (5.58) Ji = − D i − D i2 . dz k BT dz A simple method for measuring Q∗ is available: A rod of the host solid initially containing a uniform concentration of an impurity species is heated to temperature TL at one end and maintained at a lower temperature To at the other end until the equilibrium distribution is reached. The rod is then cut into slices and the concentration of the impurity species ci measured as a function of z. Because the impurity species cannot leave the rod at either end, it merely redistributes in the rod under the influence of the temperature gradient. After a time sufficient for steady state to be achieved, the flux is zero everywhere. Setting Ji = 0 in Equation (5.58) and eliminating dz results in Q∗ 1 d ln c i = TQ d   where TQ = ; (5.59) kB T integrating yields

Diffusion in Solids 221

c i = Be TQ /T .

(5.60)

Plotting lnc vs 1/T yields a straight line whose slope is Q∗/kB. The slope, and hence Q∗, can be either positive or negative.

Appendix A: Darken’s Equation Darken’s equation [16] relates the self-diffusion coefficients of the two solutes (DA, DB) to the mutual diffusion coefficient, also known as the interdiffusion coefficient, D AB. Before dealing with the diffusion processes, several relationships between two solute components (A and B) in a solution (solid or liquid) are summarized. These are the principal quantities: cA, cB = concentrations, moles A,B/vol; xA, xB  = mole fractions, moles A,B/total moles; v–A , v–B = partial molar volumes, volume A,B/mole A,B; and µA, µB = chemical potentials of A,B. The total molar volume is v = x A v A + x B v B , volume /mole  .

(A.1)

Relation between mole fraction and concentration: x A = c Av xB = cBv .

(A.2)

Dividing Equation (A.1) by v and using Equation (A.2): (A.3)

cA v A + cB v B = 1 . The chemical potentials are o

µ A = g A + k BT

ln(a A ) and

o

µ B = g B + k BT

ln(aB ) .

(A.4)

222 Light Water Reactor Materials

g oA , g oB = Gibbs energies of pure A,B aA, aB = activities of A and B (see Sec. 2.6.5 in Ch. 2)  kB    = Boltzmann’s constant, 8.65 × 10−5 eV/atom  T   = temperature, K Dividing the Gibbs-Duhem equation (Eq. [2.41]) by v and using Equation (A.2) gives c A ∇µ A + c B ∇µ B = 0 . (A.5) The fluxes of the components relative to a fixed plane in the medium are the products of their velocities (relative to the same plane) and their concentrations: (A.6) J A = c Au A J B = c BuB . JA and JB are in opposite directions, as are the component velocities uA and uB. Each component has an intrinsic mobility MA, MB that is independent of the diffusive flux of the other component. The mobility is the ratio of a velocity to a force. In the present case, the force is the negative of the chemical potential gradient: (A.7) u A = − M A ∇µ A u B = − M B ∇µ B . The mutual diffusion coefficient of A and B is defined in terms of the volume-average velocity: uV = J A v A + J B v B . (A.8) This equation can be understood as follows. The product of the flux JA and the partial molar volume v–A is the velocity of A transported across a unit area of the fixed plane. Similarly, J B v–A is the velocity of B moving in the opposite direction. The sum in Equation (A.8) is the net volume per unit area moved across the fixed plane or the volumeaverage velocity uV.

Diffusion in Solids 223

The mutual diffusion coefficient of A and B is defined in terms of the flux of A through to a plane moving with volume-average velocity: J AuV = c A (u A − uV ) ≡ − D AB ∇c A .

(A.9)

Eliminating uv in Equation (A.9) using Equation (A.8), and using Equations (A.3) and the first of Equations (A.2) changes the first equality of Equation (A.9) to J Auv = ( J Ac B − J B c A )vB . (A.10) uA and uB are now eliminated between Equations (A.6) and (A.7) and the resulting equations for JA and JB are substituted into Equation (A.10). Then ∇µB is eliminated by means of Equation (A.5). The result is J Auv

 AB ∇c A = − c A v B (c B M A + c A M B )∇µ A = − D

.

(A.11)

Combining Equations (A.1) and (A.2), yields cA xA

1 =

.

xA v A + xB v B

Setting xB = 1 − xA in the above, solving for cA, and taking the derivative with respect to xA yields ∇ ln c A

=

∇ ln x A

vB

.

xA v A + xB v B

(A.12)

In Equation (A.12), ∇cA/cA = ∇lncA and ∇xA/xA = ∇lnxA. Multiplying and dividing Equation (A.11) by ∇lnxA, then using Equation (A.12) to eliminate by ∇lncA/∇lnxA results in D AB =

∂µA ∂ ln x A

(x B M A + x A M B ) ,

(A.13)

224 Light Water Reactor Materials

where ∇ has been replaced by ∂. The mobility of species and their self-diffusion coefficients (DA and DB) are related by the Nernst-Einstein equation: D D MA = A MB = B . (A.14) k BT k BT Substituting Equation (A.14) into Equation (A.13) and eliminating chemical potentials using Equation (A.4) yields Darken’s equation.

Appendix B: Similarity Transformation Solution to Equations (5.8) to (5.12) The objective is to convert Equation (5.5) into an ordinary differential equation that can be solved analytically. To this end, the following dimensionless variables are introduced: c (source) (B.1a) co c (B.1b) θ= (couple)  co   2 z ξ= (B.2) 2 Dt Equation (B.2) is called the similarity transformation. The new variable ξ is a function of both z and t. In terms of this variable, the partial derivatives in Equation (5.8) are: θ=

∂c ∂ ξ ∂c = = ∂t ∂t ∂ ξ

ξ ∂c z  −1  ∂c ; =− 3/2 2 D  2t  ∂ ξ 2t ∂ ξ

∂c ∂ξ ∂c = = ∂ z ∂ z ∂ξ

1 ∂c ; 2 Dt ∂ξ

Diffusion in Solids 225 ∂  ∂c  ∂ξ ∂  = = 2 ∂ z  ∂ z  ∂ z ∂ξ  ∂z 2 ∂2 c

and     

1 ∂c  1 ∂2 c . = − Dt ∂ξ 4 Dt ∂ξ 2

Using these transformations in Equation (5.8) converts the partial differential equation to an ordinary differential equation: d 2θ dθ + ξ =0 . 2 dξ dξ2 The initial and boundary conditions reduce to θ = 1 at ξ = 0 (Eq.[5.10] and [5.11]) θ=0

ξ=∞

at

(B.3)

(B.4)

(Eq. [5.12])

(B.5)

Solution of Equation (B.3) subject to Equations (B.4) and (B.5) is accomplished by introducing a new dependent variable U = dθ/dξ, which reduces Equation (B.3) to a first-order differential equation: dU + 2 ξU = 0 . dξ The first integral of this equation is U=

dθ = Aexp( − ξ 2 ) . dξ

Integrating again, θ=

ξ

A ∫ exp(− ξ ′ 2 ) d ξ ′ + G . 0

Applying the boundary conditions given by Equations (B.4) and (B.5) yields the integration constants: G = 1 and A = −

{∫

∞ 0

}

exp(−ξ 2 ) d ξ

−1

=−

2 . π

226 Light Water Reactor Materials

The dimensionless form of the solution is θ = 1−

2 π

ξ

∫0 exp (− ξ′ ) d ξ′ = 1 − erf (ξ) = erfc(ξ) 2

,

(B.6)

which, when returned to dimensional forms using Equations (B.1) and (B.2), yields Equations (5.13) and (5.14).

Appendix C: Laplace Transform Solution to Equations (5.20) to (5.22) The first steps are to introduce dimensionless variables and a new independent variable that converts the diffusion equation from spherical to slab geometry: 2

η = r/R gr ; τ = D fg t/R gr ;

u = ηc fg /c ofg .

(C.1)

Subscript gr means “grain,” and fg signifies “fission-gas.” Substituting Equation (C.1) into Equations (5.20) to (5.22) yields ∂u ∂ 2 u = ∂τ ∂η2

(C.2)

u(η,0) = η

(C.3)

u(0 ,τ) = 0 ; u(1,τ) = 0

(C.4)

Taking the Laplace transform of Equation (C.2) using Equation (C.3)4, d 2u = pu − η d η2

4

(C.5)

The Laplace transform of the time derivative of a function is the transform variable (p) times the transform of the function ~u minus the initial spatial distribution.

Diffusion in Solids 227

where p is the transform variable. The transformed boundary conditions of Equation (C.4) become u(  0) = u(  1) = 0 .

(C.6)

The solution of the transformed equation is u = −

exp( pη ) − exp(− pη ) η . + p[exp( p ) − exp(− p )] p

(C.7)

Before inverting the solution, the average concentration is obtained: c fg c ofg

=

a

4π ∫ r 2

1

1 c fg (r,t) dr 3 = ∫0 ηud η . c ofg

(C.8) 0 4 3 3 πa Taking the Laplace transform of Equation (C.8) and using Equation (C.7) yields 1 cfg 1 ( p − 1) + ( p + 1)exp(−2 p ) . = 3 η u d η = −3  p c ofg ∫0 p 2 [1 − exp(−2 p )]

(C.9)

Short times are equivalent to large values of p, so the terms with exp(−2 p ) can be neglected in Equation (C.9), giving cfg c ofg

=

p −1 1 3 1 −3 = − 3/2 p p p p2

3 + 2 p

.

This transform of the average concentration can be inverted from the table of inverse Laplace transforms to yield c fg 6 = 1− o cfg π

τ + 3τ

from which Equation (5.23) is obtained.

,

(C.10)

228 Light Water Reactor Materials

References 1. H. Carslaw and J. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford: Oxford University Press, 1959). 2. J. C. Crank, The Mathematics of Diffusion (Oxford: Oxford University Press, 1964). 3. D. R. Olander, Fundamental Aspects of Nuclear Reactor Fuel Elements, TID-26711-P1, National Technical Information Service (1976). 4. P. Shewmon, Diffusion in Solids, 2nd ed. (Hoboken, New Jersey: Wiley, 1991). 5. W. Breitung, “Oxygen self- and chemical diffusion coefficients in UO2 ± x,” J Nucl Mater 74 (1978): 10. 6. H. Matzke, “Atomic mechanisms of mass transport in ceramic nuclear fuel materials,” J Chem Soc Faraday Trans 86 (1990): 1243. 7. G. Murch and C. Catlow, “Oxygen diffusion in UO2, ThO2 & PuO2,” J Chem Soc Faraday Trans 2 83 (1987): 1157. 8. G. Murch and R. Thorn, “The mechanism of oxygen diffusion in nearstoichiometric uranium dioxide,” J Nucl Mater 71 (1977): 219. 9. G. Murch, “Oxygen diffusion in UO2—an overview,” Diffusion and Defect Data 32 (1983): 4. 10. H. Matzke, “Diffusion in ceramic oxide systems,” Adv in Ceramics 17 (1986): 1. 11. H. Matzke, “Diffusion processes in nuclear fuels,” J Less-Common Metals 121 (1986): 537. 12. H. Matzke, “On uranium self-diffusion in UO2 and UO2 + x,” J Nucl Mater 30 (1969): 26. 13. D. Glasser-Leme and H. Matzke, “Interdiffusion & chemical diffusion in the UO2-(U,Pu)O2 system,” J Nucl Mater 106 (1982): 211. 14. M. Stan and P. Cristea, “Defects & oxygen diffusion in PuO2 − x,” J Nucl Mater 344 (2005): 213. 15. F. Schmitz and R. Lindner, “Diffusion of heavy elements in nuclear fuel,” J Nucl Mater 17 (1965): 259.

Diffusion in Solids 229

16. L. Darken, “Diffusion, mobility and their interrelation through free energy in binary metallic systems,” Trans Amer Inst Mining Metall Engin 175 (1948): 184. 17. A. R. Cooper and J. H. Heasley, “Interdiffusion and Chemical Diffusion in the UO2 – (U,Pu)O2 System,” J Amer Ceramic Soc 49 (1966): 280. 18. A. D. Smigelskas and E. O. Kirkendall, “Zinc Diffusion in Alpha Brass,” Trans. AIME 171 (1947): 130–142. 19. H. Nakajima, Journal of Metals 49, no 6 (1997): 15–19.

Problems 5.1 Using the Schottky equilibrium constant for MX, derive Equation (5.36). d ln a A 5.2 Using Equations (2.42) and (2.43), prove that in Equation (5.1) d ln x A d ln aB can be replaced by . d ln x B   5.3 How long is required for a sphere of material A of radius Ro = 10 µm to dissolve in an infinite medium of B? The solubility of the sphere material in the matrix is [A]o = 10−2 moles/cm3. The diffusivity of A in B is DAB = 10−9 cm2/s. The density of solid A is 0.1 moles/cm3.  Even though the sphere is shrinking, assume steady-state diffusion. 5.4 The diffusion of element A into element B is described by Equation (5.13). At the temperature the experiment is conducted of 400 K, the diffusion coefficient of A into B is equal to 10−11 cm2/s. If Co = 1 (such as would occur when there is an inexhaustible supply of A at x = 0): (a) After 2 days of running the experiment, at what distances into the sample would the concentrations be equal to 0.5 and to 0.1? (b) For a distance of 1 cm, at what time would you expect to see a concentration of 0.1?

230 Light Water Reactor Materials

5.5

5.6

5.7

5.8

(c) How would the answers to the above questions change if the absolute temperature was twice the value of the initial temperature? Assume the activation energy for migration of A into B is 1 eV, and that all other terms of Einstein’s diffusion coefficient formula are independent of temperature. Given that you are able to perform a diffusion experiment during a period of 30 days and are able to analyze a diffusion layer with a thickness of the order of 10−3 cm, what is a rough estimate of the smallest diffusion coefficient that could be studied? If the sample is Cu, what is the lowest temperature at which the experiment could be made? If the temperature is increased to 950°C, how does this affect the time needed to achieve the same result? (Given that for Cu, Do = 0.6 cm2s−1; Q (activation energy) = 1.85 ev/atom.) It is desired to create a 0.2-mm carburized layer in a steel component by diffusing carbon into it through exposure to a carburizing gas at high temperature for a time t. The diffusion coefficient of C into fcc Fe at 1000 C is 4 × 10−11 m2s−1. The surface concentration of carbon is 0.5. (a) Use the Einstein formula to find the activation energy. Assume υ = 1013 Hz and a jump distance of 2.04 × 10−10 m. (b) Use the solution for the concentration profile established in a diffusion couple, to derive the annealing time necessary to obtain a 1 at % concentration of C over the desired thickness. Using Appendix A, show how the volume-averaged velocity uV can be obtained from the curve of cA versus z. Consider that the A-B solution is ideal. (a) At 800 K, how long is required for a vacancy to undergo 100 diffusive jumps? (see Ex. #6). The vacancy migration energy is 1.0 eV. (b) In the same time interval, how many jumps does a self-interstitial atom (SIA) make? The SIA migration energy is 0.7 eV.

Diffusion in Solids 231

5.9 The figure below shows a recombination event in two dimensions.

Atom on regular site Recombination: Interstitial atom Vacancy

ao

The vacancy-interstitial recombination rate is RR = α DI cI cV recombinations/unit area.s. The vacancy diffusion coefficient is very small compared to DI for interstitials. cI and cV are the areal concentrations (i.e., number per unit area). (a) What is the Einstein relation in 2-D? (b) What is the numerical value of α in the recombination rate equation? 5.10 In a solid-state diffusion experiment to measure diffusion coefficients, a layer of radioactive gold atoms (denoted Au∗) is deposited onto a block of nonradioactive fcc gold as shown in the figure below. Au*

fcc Au

x

(a) Explain briefly how the experiment would be conducted to determine the diffusion coefficient. (b) During the experiment, the activation energy (migration plus formation) for diffusion in this material is determined to be Q  = 1.67 eV.

232 Light Water Reactor Materials

Given that diffusion occurs by a vacancy mechanism, calculate the diffusion coefficient in Au at 1070 K. The lattice parameter for fcc gold is ao = 0.408 nm, and the vibration frequency is 5 × 1012 Hz. (c) Using the diffusion coefficient above, calculate the concentration of radioactive gold at x = 20 microns after one week. How long would one have to wait to obtain a concentration of 50% radioactive gold at that distance? Consider that the concentration of radioactive gold at x = 0 is 100% throughout the experiment.

6

Chapter Elasticity 6.1 Introduction The scope of this book is limited to the reactor pressure vessel and the associated piping (together called the primary circuit) and the fuel rods in the core. Because other texts address similar topics [1–3], only a limited portion of the theory of elasticity will be addressed in this text. When a force, or load, is applied to a solid, the body changes shape and perhaps size. These changes are called deformations. The objective of stress analysis is to quantitatively relate loads and deformations. Elasticity theory deals with deformations sufficiently small to be reversible; that is, the body returns to its original size and shape when the load is removed. Within this range, the distances between atoms in the solid, or, alternatively, the atom–atom bonds, are stretched or compressed, but are not broken. In order to remain in the elastic range, the fractional changes in interatomic distances (on a microscopic scale) or in the body’s gross dimensions (on a macroscopic scale) must be less than ~0.2%. When loads or deformations exceed the elastic range, the changes in the shape of the body are not recovered when the load is removed. This type of irreversible deformation is termed plastic deformation. On a microscopic level, atomic bonds are broken and new bonds are formed between different

234 Light Water Reactor Materials

atoms than in the original configuration. Plastic behavior of solids is discussed in Chapters 7 and 11. The maximum extent of plastic deformation is breakage or fracture of the material. Instead of load (or force) and deformation, elasticity theory utilizes the related quantities stress and strain. Stresses are forces per unit area acting on internal planes in the body, and strains are fractional deformations of the body.

6.1.1 Stresses and strains Figure 6.1 shows a rod of cross-sectional area A acted upon by force F. All planes perpendicular to the rod’s axis experience the same force. The stress on plane a–a′ is σn = F/A . (6.1) The stress is called normal (subscript n) if it acts in the direction perpendicular to the stressed plane. Determining the sign of the stress requires reference to a set of orthogonal coordinate axes such as that shown in Figure 6.2 (x and y). The normal stress is positive if it acts in the +x direction. The load F in Figure 6.1 generates a positive normal stress on plane a–a′. This stress tends to pull atomic planes apart and is termed tensile. If the force F in Figure 6.1 were reversed, the normal stress on a–a′ would also flip 180 degrees and the atoms in the solid would be squeezed together. Such a stress is called compressive. By convention, tensile stresses are positive and compressive stresses are negative. In Figure 6.2, the force F is applied on the top plane in the y direction. The stress is generated parallel to the surface rather than perpendicular to the surface. Such a stress is termed a shear stress (subscript s): σs = F/A

.

(6.2)

If the applied load (force) is not purely normal or purely shear, arbitrarily oriented planes in the body will experience both normal and shear stress components.

Elasticity 235 F Deformed

u σ

Original

F

a' Lo

a

v

x

σ wo

v

F

FIGURE 6.1: Normal stress.

Lo

y

F

FIGURE 6.2: Shear stress.

Stresses are generated in structures in a number of ways, which include: Externally applied loads (membrane stresses) • Mechanical loads represented by the force F in Figure 6.1. This type of loading is the basis, for example, of the uniaxial tensile test used to measure many mechanical properties of metals. • Different pressures of a fluid (gas or liquid) on opposite faces of a structure. An example of this type of loading is the high pressure of water or steam in the primary circuit of a light water reactor (LWR). • Reaction forces due to connection of a particular structure to its supports and to other components of a complex mechanical system. For example, the lower grid plate of an LWR supports the weight of hundreds of fuel assemblies. In this case, the stresses are induced by gravity. Thermal stresses • Stresses are generated when thermal expansion of a heated body is restrained. This topic is discussed in detail in Section 6.6.

236 Light Water Reactor Materials

Residual stresses This type of stress arises from two principal sources: • Fabrication of a component: processes such as cold-working (reduction in cross-sectional area by passing between dies) introduce internal stresses in the finished piece. These stresses remain during operation unless the piece is annealed at a high temperature prior to use. • Welding of two components: Welding involves melting of a metal and introduces large thermal stresses in the adjacent metal that does not melt (called the “heat-affected zone”). These stresses persist in the cooled piece and are supplemented by additional stresses arising from the solidification and cooling of the weld.

6.1.2 Notation for stresses In general, any point within a solid subjected to one or more of the loads just enumerated may possess as many as six components of stress. Three are normal, and three are shear. The simple designations σn for normal stresses and σs for shear stresses need refinement in order to cover complex stress patterns. Stress components are denoted with respect to the orthogonal coordinate system by which they are described (x, y, z for Cartesian; r, θ, z for cylindrical; r, θ, ϕ for spherical). The stress components are labeled σij, where i is the plane on which the stress component acts and j is its direction. The stress labeled σ in Figure 6.1 is properly denoted by σxx: it acts on the y–z plane, which is called the x plane after the direction of its normal; the stress component acts in the same direction, so the second subscript is also x. All normal stresses bear the generic designation σii, which is often shortened to σi, with the understanding that the stress component acts on the i plane in the i direction. In situations where the type of stress component is obvious, subscripts are often entirely dispensed with. Because

Elasticity 237

space is three-dimensional, there are at most three nonzero normal components of the stress state at a point. Proper designation of shear stress components cannot be reduced to a single subscript because i and j in σij are always different. The shear stress in Figure 6.2 should be written as σxz because the stress acts on the x plane and in the z direction. A moment is the product of a stress and its distance from an axis. Equilibrium of the moments in a solid require that σij = σji so that there are at most three nonzero shear components of the general state of stress.

6.1.3 Displacements and strains In what follows, definitions and derivations are given for two-dimensional Cartesian coordinates (i.e., x and y). This is done to minimize the complexity of the theory. Extensions to three dimensions or to other coordinate systems are stated without proof. Displacements are changes in the position of a point in a body between the unstressed and stressed states. Strain is a fractional displacement. In common with stresses, displacements and strains come in two varieties, normal and shear. Figure 6.1 shows the normal displacements u and v in the x and y directions, respectively, of a body subjected to a normal force acting uniformly on the upper and lower horizontal surfaces. The original dimensions of the piece are Lo and wo. The solid rectangle represents the stress-free solid, and the dashed rectangle is its shape following application of the axial force. The displacement u (positive) corresponds to the outward movement of the horizontal surfaces, and the displacement v (negative) represents the shrinkage of the body’s sides. The strains in the x and y directions are defined as fractional displacements: ε x = u/Lo (6.3)

238 Light Water Reactor Materials

.

ε y = v/w o

(6.4)

The point (or in this case, the plane) selected to follow the displacement need not be an outer surface; displacements and strains of interior points are defined in the same manner. Figure 6.2 shows a shear displacement v resulting from a force F acting on the side of the body at a distance Lo from the fixed bottom. The shear strain is defined as the ratio of these two lengths: ε xy = v/Lo

.

(6.5)

The two-digit subscript notation has been used—shear movement occurs on the plane normal to the x axis and in the y direction. A more general definition of shear strain is the angular distortion of both edges of an initially rectangular section (see below). Generalization of strain definition Equations (6.4) and (6.5) need to be generalized for stress analyses. Figure 6.3 shows how this is done. The solid rectangles are the original shapes, and the dashed figures represent the deformed shapes. Normal strain Figure 6.3a shows a deformation in the x direction. The original length of the body OB is taken to be a differential element dx. Upon application of a normal stress, the displacement of the bottom surface is OA = ux. The displacement of the upper surface is BC = ux + (∂ux/∂x)dx, assuming that the body is continuous. The strain in the x direction, εx, is the change in length, (OB + BC − OA) − OB = (∂ux/∂x)dx, divided by the initial length OB = dx. Expressing the lengths by their equivalents in terms of displacement (ux) and normal strain (εx) yields εx=

∂u x ∂x

.

(6.6a)

Elasticity 239 C x

A

uy

B y

B

dx

dx

α

D ux

A

β O

O

C dy

(a) Normal

(b) Shear

FIGURE 6.3: Definitions of strains.

The analogous equations for the y and z-directions are ε y=

∂u y ∂y

and

εz=

∂u z ∂z

.

(6.6b)

The analogous formulas for the normal strains in cylindrical coordinates with angular symmetry (i.e., ∂/∂θ = 0) are ∂u ur , and εz = z . (6.7) r ∂r ∂z For spherical geometry with spherical symmetry (i.e., ∂/∂ϕ = 0, ∂/∂θ = 0), εr =

∂u r

,

εθ =

∂u r

ur . (6.8) r ∂r In these equations, ur is the radial displacement, ϕ is the polar angle, and θ is the azimuthal angle. εr=

εθ = εϕ=

240 Light Water Reactor Materials

Shear strain Generalization of Equation (6.5) is accomplished with the aid of Figure 6.3b. The shear strain on the x–y (or y–x) plane is defined as the sum of the angles (in radians) of the deformed figure (dashed) relative to the stress-free shape (solid square); that is, εxy = εyx ≡ α + β. Similar definitions apply to shear strains on the x–z and y–z planes. What remains is to connect the angles α and β to the displacements u and v. For simplicity, the lower left corners of the original and deformed figures in Figure 6.3b are superimposed at point O. Also, the corners B and D of the deformed figure are taken to lie on the sides of the original shape. With these simplifications, AB = (∂uy/∂x)dx and tanα ≅ α = AB/dx = ∂uy/∂x and CD = (∂ux/∂y)dy and tan β ≅ β = CD/dy = ∂ux/∂y, ε xy =   ε yx =

∂u x ∂y

+

∂u y ∂x

.

(6.9)

Whereas normal strain is the fractional deformation parallel to a chosen direction, shear strain consists of fractional deformations perpendicular to particular directions; that is, the tangents of the angles α and β in Figure 6.3b are components of εxy. The shear-strain formulas for cylinders and spheres are not given here because all applications in this book involve only normal stresses in these geometries.

6.2 Equilibrium Conditions The so-called equilibrium conditions of elasticity theory are consequences of Newton’s third law: if a body is to remain stationary, the sum of the forces acting on it must be zero. This condition applies to all volume elements in a stressed body, and Figure 6.4 provides relations between stress components. Figure 6.4 shows the basis for deriving the x-direction force balance for a two-dimensional Cartesian body. The balance of x-direction forces is

Elasticity 241

net x force =  σxx 

+

dx − σxx  dy +  σyx   ∂x

∂σxx

∂σ xx

or

∂x

+

∂σ yx ∂y

=0

+

∂σyx ∂y

 dy − σyx  dx = 0 (6.10)

.

Note that the normal stresses σii (where i = x, y, z, r, θ, z) can also be written as σi. Both expressions are used interchangeably in this chapter. Note also that Equation (6.10) is a y ∂σ force balance expressed in terms of dy σ + x ∂y stresses. ∂σ σ dx Comparable equilibrium condiσ + ∂x tions apply to the y- and z-direcFIGURE 6.4: x-direction stresses on a volume tions, and the extension to three element dx in length and dy in height. dimensions is straightforward [1, Appendix]. In axisymmetric cylindrical coordinates, the radial equilibrium condition is yx

yx

xx

xx

xx

∂σ rr

+

∂r

σ rr − σ θθ

+

∂σ rz ∂z

r

=0

and for the z direction it is 1 ∂(rσ rz ) ∂σ zz =0 . + r ∂r ∂z

,

(6.11a)

(6.11b)

In spherical coordinates with spherical symmetry, the radial equilibrium condition is ∂σ rr ∂r

and, by symmetry

+2

σ rr − σ θθ

r

σ θθ = σ ϕϕ

=0

.

,

(6.12a) (6.12b)

242 Light Water Reactor Materials

6.3 Stress-Strain Relations The final set of equations that forms the basis of elasticity theory relates stresses and strains. Contrary to the strain-displacement relations (Eq. [6.6] to [6.8]) and the equilibrium conditions (Eq. [6.10] to [6.12]), the connection between stresses and strains involves material properties called elastic constants.

6.3.1 Elastic constants Figure 6.1 represents the simplest loading configuration because it produces only one component of stress, the normal component σx. The axial strain defined by Equation (6.4) is related to σx by ε x = σ x/E (6.13) where E is a material property called Young’s modulus or the modulus of elasticity. For steels, E ~ 2 × 105 MPa; E for aluminum is about one-third that for steel. The nuclear fuel UO2 has approximately the same Young’s modulus as steel, but this correspondence has little to do with its mechanical performance in a reactor environment, as will be seen in subsequent chapters. There is a stress limit (and consequently a strain limit) for the applicability of Equation (6.13), which is called the yield stress (see Ch. 11). At this point, the strain is 0.2%. For steel, the proportionality of stress and strain implied in Equation 6.13 fails at approximately 500 MPa. These conditions define the elastic limit of the material. As suggested in Figure 6.1, a positive displacement (or strain) in the x direction produces a negative displacement (or strain) in the y direction. In an isotropic solid, the other transverse direction (z) experiences the same strain as does the y direction, or εz = εy. The ratio of the magnitudes of the lateral strains to the axial strain in the uniaxial tensile situation of

Elasticity 243

Figure 6.1 defines Poisson’s ratio, υ: υ = −ε y /ε x = −ε z /ε x .

(6.14)

Like E, υ is a material property, which, for most materials is ~1/3. It is not surprising that the deformed body in Figure 6.1 shrinks in transverse dimensions as it elongates axially, as otherwise, a volume change would occur. However, the transverse shrinkage does not quite compensate for the axial elongation, and the solid does change volume as it deforms elastically.

6.3.2 Young’s modulus On a microscopic scale, macroscopic elastic strains are caused by the stretching or contraction of interatomic bonds. Figure 6.5 depicts the microscopic response to a tensile stress applied to a crystal structure. σn

a′ a c′

d′ zo

z

c

d b

Unstressed

b′

σn

FIGURE 6.5: Movement of atoms as a result of application of a tensile stress.

The origin of Young’s modulus lies in the increase of the distance between the adjacent atoms a and b from ab in the unstressed condition to the a′b′ with the applied stress. This stretching of the a–b bond

Potential energy (U )

244 Light Water Reactor Materials

0 Uo

z zo

FIGURE 6.6: Potential energy between atoms a and b in Figure 6.5.

is accompanied by an increase in the potential energy between the two atoms as shown in Figure 6.6.1 In the unstressed state, the atom separation ab lies close to the minimum on the potential curve (zo). Increasing the separation distance from ab to a′b′ is accompanied by an increase in the potential energy (U) between the two atoms. Around its minimum, the potential energy curve can be approximated by a parabola: U = Uo

+

1  d 2U  (z − z o ) 2 . 2  dz 2  0

(6.15)

The force between the two atoms is the gradient of the potential energy: F=

dU  d 2U  = (z − z o ) . dz  dz 2  z

(6.16)

o

Macroscopically, the force is equal to the stress σn times the projected area of an atom, π(zo/2)2, where zo/2 is the atomic radius. F The normal stress is  σ n = π(z o / 2) 2 1

(d 2U/dz 2 )z o (z − z o ) = . (6.17) π(z o / 2) 2

The curve in Figure 6.6 represents the interaction energy between a pair of atoms; however, interactions in metals, for example, are rarely exclusively pairwise.

Elasticity 245

The normal strain is ε n =

z − zo . zo

(6.18)

Combining the above two equations yields the expression for Young’s modulus in atomic terms: E=

σn εn

=

4  d 2U  . πz o   d z 2  z

(6.19)

o

Knowing the potential energy curve of Figure 6.6 is required to calculate the second derivative in Equation (6.19). Such interatomic potentials can be accurately estimated by a type of microscopic-level computation called molecular dynamics (Ch. 15). a′ θ a 60°

o

c′

c

Unstressed

Stressed

FIGURE 6.7: Diagram for calculating Poisson’s ratio in the fcc structure. The solid quadrilateral is the left drawing of Figure 6.5 and the dashed one is the right drawing.

6.3.3 Poisson’s ratio Poisson’s ratio can be derived by the same detailed analysis used for the Young’s modulus in the preceding section. However, the following simple method gives a close estimate.

246 Light Water Reactor Materials

First, the two diamond-shaped figures in Figure 6.5 are superimposed, as shown in Figure 6.7. The normal strain in the z direction, εn, and the normal strains in the transverse (y or z) directions, εt, are εn =

aa ′ and oa

εt = −

cc ′ oc

(6.20a)

where “o” is the midpoint between atoms a and b in Figure 6.5. The deformation consists of moving point a to a′, a positive displacement, and point c to c′, a negative displacement. The minus sign in the above equation accounts for the reduction of the segment oc to oc′. Now oc is the height of the equilateral triangle abc. Since all sides of the quadrilateral adbc are equal to twice the atomic radius (see Fig. 6.5), oc oa 3 1 = sin60 = = cos60 = . (6.20b) ac 2 ac 2 Second, considering the angle θ in Figure 6.7 and noting that from Figure 6.5, a′c′= ac = atomic diameter, sin θ = oc′ = oc − cc′ = sin60 − oc cc′ = sin 60(1 + εt) a′ c′ ac ac oc oa′ oa − aa′ oa aa′ = = cos60(1 + ε n ) . = cos60 + a′ c′ ac ac oa Squaring and adding the above two equations yields cos θ =

1 = sin 2 60(1 + ε t ) 2 + cos 2 60(1 + ε n ) 2 . Since the strains are always σFR, the 4-stage sequence of Figure  7.22 is repeated. The result is continuous production of loops that expand at speeds determined by obstacles that hinder their motion.

7.12 Impediments to Dislocation Motion Hindrance of dislocation movement is divided into two classes, source hardening and friction hardening.7 Source hardening is caused by impurity atoms, precipitate particles, or other microstructural defects that are attached to the dislocation in its unstressed state. These entities “lock” dislocations such that initiation of motion or operation of Frank-Read 7

The term hardening refers to the increase in yield stress (the stress required to initiate specimen-wide dislocation movement in the fifth largest slip system [see Ch. 11]) and to the continual increase in stress required to maintain dislocation motion as plastic strain increases.

322 Light Water Reactor Materials

sources requires a stress greater than that given by Equation (7.21). Friction hardening refers to the effect of obstacles of various types that impede the motion of mobile dislocations. The obstacles include other dislocations, foreign bodies ranging in size from impurity atoms to macroscopic precipitate particles, and in the case of metals irradiated by fast neutrons or fast ions, clusters of point defects. Friction hardening is further subdivided into long-range and short-range effects. Long-range friction hardening implies that the mobile dislocations do not directly contact the obstacle but feel its retarding force at a distance (Prob. 7.11). In short-range friction hardening, the mobile dislocations encounter obstacles lying in their slip planes.

7.12.1 long-range friction hardening The conventional picture of this type of impediment to dislocation motion is the interaction of two parallel edge dislocations on different slip planes, as depicted in the right diagram of Figure 7.16a. A critical value of the applied shear stress is necessary to force mobile dislocation 2 past immobile dislocation 1. From Equations (7.15) and (7.16), the retarding force exerted on 2 by 1 is Γbsin(4 θ) (7.22a) Fx = 4y where Γ is the collection of elastic constants given by Equation (7.8), and y is the separation of the two slip planes. The maximum back force occurs at θ = π/8, or b (7.22b) . ( Fx ) max = 4y The forward force due to the applied shear stress is given by Equation (7.10), which in the present notation is ( Fx ) app = σs b .

Dislocations 323

When (Fx)app just exceeds ( Fx ) max, dislocation 1 slips past dislocation 2 and goes on its way. With the help of Equation (7.8), the critical value of σs is

(σ scrit )friction = 4Γy = 8 π(1Gb− υ)y .

(7.23)

The separation of the slip planes (y) is related to the dislocation density. From the discussion in Section 7.4, the separation of parallel dislocations in the unstressed solid should be of the order of the unit cell radius R given by Equation (7.4). In a realistic collection of dislocations in a solid that is subject to an applied stress, two complications would need to be taken into account: 1. The force on a particular mobile dislocation in a random forest of dislocations is considerably more difficult to calculate than the above analysis for two parallel dislocations. 2. The angles identifying the direction of allowed motion of the mobile dislocation have to account for its slip system interacting with the force field of number 1 above. That is, a critical resolved shear stress analogous to Equation (7.1) or Equation (7.1a) would need to be calculated. (Prob. 7.11 is a simple example of two dislocations impeding a mobile one.)

7.12.2 Dislocation climb Even if the applied shear stress is less than the critical value given by Equation (7.23), another mechanism is available for the mobile dislocation to pass an immobile one. By increasing the separation of the slip planes (y in Figure 7.16a), the critical shear stress is reduced until it equals the applied shear stress. The process by which the mobile dislocation moves perpendicular to its slip plane is climb (see Sec. 7.3.4 and Fig. 7.11 in particular). With reference to the right diagram of Figure 7.16a, the normal stress σxx

324 Light Water Reactor Materials

on dislocation 2 due to the presence of dislocation 1 is compressive (if σxx is negative). The effect of this stress is to reduce the vacancy concentration at the core of dislocation 2 to a value below that in the bulk solid. This concentration difference generates a flux of vacancies to the dislocation, which responds by climbing upward, as shown in Figure 7.11. The key to establishing the reduction of the vacancy concentration at the dislocation core is the stress effect on the equilibrium vacancy concentration. This effect is analyzed in Section 4.2.1 and leads to Equation (4.10). In order to utilize this equation as a boundary condition on the vacancy diffusion equation, two modifications are needed. First, Fick’s laws of diffusion must be cast in terms of the volumetric concentration of the diffusing species. Equation (4.10) is converted to units of atoms/cm3 by dividing both sides by the atomic volume Ω. Second, the stress in Equation (4.10) is σxx in Equation (7.6b). With these modifications and vV, the specific volume of a mole of vacancies, the equilibrium vacancy concentration at the dislocation core is σ v c V (rd ) = c Veq exp  xx V   k BT 

(7.24)

where rd is the radius of the dislocation core and kB is Boltzmann’s constant. s Ef 1 c Veq = exp  V  exp  − V   k B   k BT  Ω

(7.25)

is the equilibrium concentration in the stress-free solid. The geometry in which a vacancy flux toward the dislocation (JV) is induced by the above concentrations is shown in Figure 7.23. The radius of the unit cell R is given by Equation (7.4) in terms of the dislocation density. The diffusion equation in the cylindrical annulus rd ≤ r ≤ R is 1 d  dc V  r =0 . r dr  dr 

Dislocations 325 Unit cell of dislocation Vacancy flux

^

Dislocation core

FIGURE 7.23: Vacancy diffusion in the unit cell of a dislocation.

The boundary conditions are given by Equation (7.24) at the dislocation core (radius rd) and by Equation (7.25) at the periphery of the unit cell (radius R). The solution is c Veq − c V ln( R/r ) . = eq c V − c V (rd ) ln( R/rd )

The flux of vacancies per unit length of dislocation is dc 2π J V = 2 πrd DV  V  = DV (c Veq − c V (rd )) .  dr  r ln( R/rd ) 4 The first term on the right side depends only on the geometry of the unit cell, and is written as 2π (7.26) ZV = , ln( R/rd ) so that JV becomes J V = Z V DV (c Veq − c V (rd )) . (7.27) For typical values R ~ 10−5 cm and rd ~ 3 × 10−8 cm. ZV ~ 1 is a value that will be used throughout for numerical work. cV(rd) is expressed by Equation (7.24) in which the exponential term is represented by its one-term Taylor-series expansion. Equation (7.27) reduces to σ v Z D σ xx v V J V = Z V DV c Veq xx V = V (7.28) k BT k BT Ω where the self-diffusion coefficient (D) has been substituted according to

326 Light Water Reactor Materials

Equation (5.28) and site fraction and volumetric concentration are related by: CVeq = c Veq Ω . The connection between the vacancy flux and the climb velocity can be deduced with the aid of Figure 7.11. Let δt be the time required to remove one row of atoms from the bottom of the dislocation’s half-plane. The empty volume created (per unit length of dislocation) is vcδt b, where vc is the climb velocity and b, the Burgers vector, is the approximate width of the half-plane of atoms. The volume supplied by the vacancy flux during this time is JVΩδt. From this volume balance, vc = JVΩ/b. Substituting Equation (7.28) for JV gives the climb velocity: vc =

Z V D σ xx Ω V . b k BT

(7.29)

Example # 5: Dislocation climb Two parallel edge dislocations on parallel slip planes initially 5 Burgers vectors apart are present in a specimen of iron. The applied shear stress pushing mobile dislocation 2 against immobile dislocation 1 in Figure 7.16 is one-half of the critical value given by Equation (7.23). Considering two temperatures, 500 K and 1000 K, how long will it take for dislocation 2 to climb until the critical stress is reduced to the actual stress? In Equation (7.29), substituting dy/dt for vc and Equation (7.6b) for σxx (in which r = y/sinθ) yields dy Z V ΩDΩ sin 2 θ(1 + 2cos 2 θ) . = dt bkB T y

(7.30)

Γ is given by Equation (7.8).

In order to integrate the above equation, θ is expressed in terms of y by equating the back force on dislocation 2 due to dislocation 1 given by Equation (7.22a) to the forward force σsb exerted by the applied shear stress. σs is equal

Dislocations 327

to one-half of the initial critical shear stress given by Equation (7.23), or σs = γ/8yo, where yo is the initial separation of the slip planes. This yields θ=

1 −1  1 y  1 −1  1  sin   = sin Y . 4 2   2 yo  4

(7.31)

Equation (7.30) is made dimensionless using y = yoY and t = t∗τ, where the characteristic time for climb is given by bk BTy o2 . t = Z V ΩDΓ ∗

(7.32)

For dislocation 2 to overcome the back force of dislocation 1, it must climb to double the separation of the slip planes, or to Y = 2. This separation of the slip planes requires a dimensionless climb time given by τf =

∫

2

YdY

1 sin 2 θ (1 + 2cos 2 θ)

.

Using Equation (7.31) to express θ in terms of Y, the integral is τf = 18.3. The time for dislocation 2 to escape dislocation 1 is tf = 18.3t∗. In order to evaluate t∗ from Equation (7.32), the following properties of iron are used: G = 73 GPa (room temperature value) b = 0.3 nm υ = 0.33 These values yield Γ = 5.2 N/m (using Eq. [7.8]). ZV = 1 yo = 5b = 1.5 nm −6 3 Ω = 6.6 × 10 m /mole  kB = 8.62 × 10−5 eV/K D = 1.9 × 10−4exp(−0.765/T) m2/s

328 Light Water Reactor Materials

At 1000 K, Equation (7.32) gives a characteristic climb time of t = 2.7 ms, or the time for the dislocation to climb to the point that the blocking dislocation is circumvented is 50 ms. At this temperature, climb is so rapid that friction hardening is negligible. At 500 K, on the other hand, the decrease in the self-diffusion coefficient is so pronounced that the comparable escape time is 2.5 millennia. At that temperature the only way that a blocked mobile dislocation can pass by an immobile one is to increase the applied shear stress until it exceeds the critical value. ∗

7.12.3 Jogs So far, only the interaction forces between parallel dislocations have been considered.8 However, even the simplified representation of dislocations in Figure 7.13 shows that moving dislocations are sure to encounter dislocations at an angle other than 0 degrees. When this occurs, the two dislocations stop at some distance apart or physically touch each other. If the stress is sufficiently high, one dislocation can cut through the other and both can continue on their way, albeit with more difficulty than before intersection. There are a large number of combinations of interacting dislocations. • Each combination requires a different stress for the crossed dislocations to disengage. • Each combination leaves one or both disfigured by a jog, or kink, in the initially straight or continuously curved line. • Upon disengagement from each other, one or both require a larger stress to continue moving than prior to the intersection. • The angle of approach of the two dislocations can be anywhere between 0 and 90 degrees. 8

Section 7.9 without applied stress; Section 7.10 with applied stress for screw dislocations; earlier in this section with applied stress for edge dislocations.

Dislocations 329

In order to provide a rudimentary understanding of dislocation intersection, six combinations grouped into two sets are shown in Figure 7.24. The interacting dislocations are at right angles to each other. In the two sets of interactions shown in Figure 7.24, the three vertical dislocations, which are the same in both sets, are assumed to be immobile. In the set on the left, a horizontal edge dislocation has cut through the three vertical ones. In the right set, the moving dislocation is a pure screw. Moving edge dislocation (left diagram in Fig. 7.24) 1(edge). The intersection of moving edge dislocation be with vertical edge dislocation b1 produces a jog in the latter, but not in the former. The length of the jog in b1 is of the order of an atomic spacing.9 It is treated as a small segment of a dislocation that maintains the Burgers vector of b1. Since the jogged section is perpendicular to b1, it must be of edge character (see Fig. 7.2b). Moreover, because the Burgers vector b1 lies in the slip plane of the segment, the jog does not interfere with the glide motion of the vertical edge dislocation that contains it. z y be

b1 b2

Moving edge dislocation

x

Moving screw dislocation

S b1 b2

S b3

S

bs

S σzx b3

S σzy

S

FIGURE 7.24: Intersection of three vertical dislocations by: (left) a moving edge dislocation, and (right) a moving screw dislocation. 9

It may appear odd that a jog one atomic diameter in length has the properties of a line defect. In fact, it does not [6, p. 131]. However, enough of the properties of a true line defect are at least partially retained by the nanometer-length line that treating it as a dislocation segment is useful.

330 Light Water Reactor Materials

2(edge). In the next case, the Burgers vectors b2 and be are parallel. The jogs that appear on both dislocations are parallel to the Burgers vectors, and consequently are short segments of screw dislocations. The shear stress responsible for moving b2 and be is parallel to the jogged segments, which move along the main dislocations in a manner similar to the motion of the wave in a garden hose when one end is given a sharp shake. Neither jog impedes the motion of the main dislocation. 3(screw). When the moving edge dislocation be cuts through the vertical screw dislocation b3, the jogged sections are perpendicular to the Burgers vectors of the original dislocations. Hence, the segments are of edge character. An edge dislocation can slip only in a plane containing both the line and its Burgers vector, which is true for the jogs in both b3 and be. However, for slip, the jog must move in a direction perpendicular to itself. This is true of the jog in be, which therefore does not hinder motion of the moving edge dislocation. The extra half-plane (or more precisely, half-row) edge jog in b3 lies in the plane perpendicular to the main screw dislocation. Depending on the orientation of the vertical shear stress component (i.e., σzx or σzy), the main screw dislocation b3 is capable of glide in a direction either perpendicular to be or parallel to it. In the former case, the edge-like jog would need to move parallel to itself, which it normally cannot do. In the latter case, movement of the jog in b3 is by climb (because the short “half-plane” of the edge jog is perpendicular to b3). Such movement necessitates a higher stress than that for glide and, in addition, requires absorption or emission of point defects. The consequences of the dislocation intersections in the three cases analyzed above illustrate the following general rules: 1. Jogs in pure edge dislocations do not impede glide motion of the main dislocation. However, the slip plane of the jogged section may not be as energetically favored for motion of the jog as that of the main dislocation.

Dislocations 331

2. Jogs in pure screw dislocations hinder movement of the parent dislocation, which requires a higher applied stress to drag along the recalcitrant jog. Moving screw dislocation (right diagram in Fig. 7.24) The shear stress σzy is required to move the bS screw dislocation decorated with a number of edge jogs in the x direction. The applied shear stress acts in the y direction in the plane perpendicular to the z axis. The edge jogs glide freely up and down the main screw dislocation. Some jogs are positive, and others are negative. Jogs of the opposite sign annihilate each other until only one kind remains. Since edge dislocations of the same sign repel one another, the jogs assume equidistant positions along the screw dislocation. At low shear stress, the mobile segments of the screw dislocation between the jogs are pinned by the jogs, and can only bow out in the manner shown in Figure 7.19. If the applied shear stress is sufficiently large, the array of jogs moves by climb along with the main screw dislocation. At temperatures low enough that the point defects are immobile, this process leaves a trail of point defects in the wake of each climbing jog. Figure 7.25 shows a screw dislocation moving in the x direction emitting vacancies. x

n

tio oca

isl

wd

y z

re Sc Edge jogs

L

Vacancies left by climbing jogs

FIGURE 7.25: Mechanism of movement of a jogged screw dislocation.

332 Light Water Reactor Materials

The stress required to initiate the climb process shown in Figure 7.25 is obtained by equating the energy to create a vacancy to the work done by the interjog segment of the screw dislocation as it moves one Burgers vector in its glide direction. The energy required to create a vacancy beneath a jog is assumed to be the same as that for a normal lattice vacancy (Sec. 4.2), even though the surroundings of the vacancy are quite different in the two cases. A normal vacancy resides in an otherwise unperturbed lattice site, whereas the jog vacancy sits in the highly disrupted core of an edge dislocation. In addition, the energy of vacancy formation is replaced by EVf ,jog = 0.2Gb3 . The right side of this equation is the energy added to that of the dislocation by the presence of the jog; Gb2 is the energy per unit length of a dislocation (Eq. [7.9]), and if the jog is regarded as a dislocation of length b, its energy is Gb3. The factor of 0.2 in the above equation arises from the much-reduced range of the stress field due to the jog compared to that around an ordinary dislocation. The above evaluation of the formation energy of a jog vacancy is given without physical justification in Friedel [8, p. 105] as well as in Hirth [9, p. 597]. However, there is no physical reason why the energy of the jog can be taken to be the energy to form a vacancy. Nonetheless, the approximation is remarkably accurate; for copper, G = 46 Gpa and b = 0.26 nm, for which the above equation gives EVf ,jog = 90 kJ/mole. The normal vacancy formation energy for copper is EVf = 100 kJ/mole. Creation of a vacancy beneath the jog is accompanied by glide of the screw dislocation at a distance b, the Burgers vector. On a per-jog basis, the work required is the force per unit length of dislocation given by

Dislocations 333

Equation (7.12), in which the shear stress is (σ crit s )jog . Multiplying by L, the interjog spacing (Fig. 7.24), the work performed by the stress is W = Lb 2 (σ scrit ) jog . Equating the work to the energy expended in producing the vacancy, the critical stress to move the screw dislocation by forcing the jogs to climb is Gb . (σ crit (7.33) s ) jog ≅ 0.2 L

7.12.4 Obstacles Metals used in nuclear reactors contain impediments to dislocation motion that are introduced into the solid by a variety of means, including: 1. Expressly during fabrication: Intermetallic precipitates in Zircaloy are in this class. 2. Incomplete purification during fabrication: Commercial aluminum is stronger than high-purity aluminum. 3. During operation: The voids, bubbles, and defect clusters produced in most metals by irradiation by high-energy neutrons or ions are the most notable examples of this class of obstacles. Irrespective of how they are introduced into the metal, obstacles have several features in common. First, they can be approximated by spheres or points for the purpose of analysis. Second, they impede or stop the motion of dislocations. Third, as a consequence of the second feature, their effect is to harden the metal and at the same time render it more brittle. Obstacles range in size from individual impurity atoms, to small clusters of point defects 1 to 2 nm in size found in metals irradiated by fast neutrons at a low temperature, to large precipitate particles. These obstacles

334 Light Water Reactor Materials R

2R Unit cells Unit area of slip plane

L L

FIGURE 7.26: The intersections of volume-distributed spherical objects with a plane.

have in common a stress field around them that either attracts or repels the dislocation. In the latter case, the applied stress must be large enough to overcome the repulsive force; in the former case, the applied stress must be able to tear the dislocation away from its attractive trap. Two mechanisms are available to dislocations to overcome an obstacle. 1. If the obstacle is small enough, the dislocation can cut through it. 2. The dislocation can bow around an array of large obstacles. The cutting mechanism is not discussed here; brief discussions of this process can be found in other sources [3, Sec. 18.5; 5, Sec. 10.6(a)]. The stress required to push a dislocation through an array of impenetrable spherical obstacles depends on their center-to-center spacing in the slip plane, L. This in turn is a function of the obstacle’s radius R and its number density in the solid, N. These two quantities can be related to their separation distance L with the aid of the sketch on the left of Figure 7.26. A sphere whose center lies within the square volume 2R  ×  1  ×  1 will touch the slip plane. The number of spheres within this volume is 2RN, which is also the number of objects intersecting the unit plane. The right sketch of Figure 7.26 changes the random positions of the circular intersections of the obstacles and the slip plane to the centers of unit cells. The unit cell is a square of side L with an object’s intersection

Dislocations 335

in its center. The area of this unit cell, L2, divided into the unit area of the plane is also the number of intersections of the objects with unit slip plane. Equating these two expressions and solving for L yields L = (2RN )−1/2  .

(7.34)

The average radius of the intersections shown as shaded circles in Figure 7.26 is πR/4. Figure 7.27 shows how a straight dislocation pushed to the right by the applied shear stress interacts with the array of obstacle intersections on its slip plane. Not only the intersections of the volume-distributed obstacles placed on a square grid, but their radii are all accorded the average value of πR/4. The dislocation interacts with the obstacles at their peripheries, whose separation distance l is less than the center-to-center distance of Equation (7.34) by twice the average radius: l = L − πR/ 2 = (2 RN ) −1/2 − πR/ 2 .

(7.35)

The sequence of events by which the dislocation passes through the obstacle array on its slip plane can be broken into four stages. The straight

l

σs

1

2

3

4

FIGURE 7.27: A dislocation moving through an array of obstacles in its glide plane.

336 Light Water Reactor Materials

dislocation approaches the obstacle array (number 1 in Fig. 7.27) and bows out between them (number 2). If the applied stress is too small to bow the line into a semicircle, the dislocation line is stuck in this configuration with the sections bowed to a radius of curvature given by Equation (7.22a). At an applied stress sufficiently large to bow the dislocation between obstacles into a semicircle (number 3), the shape becomes unstable and expands in the same manner as the Frank-Read source shown in Figure 7.22. When the portions of adjacent expanding bows touch, the dislocation line is reformed and proceeds on its way, leaving behind a circular dislocation loop of the type shown in Figure 7.5 around each obstacle (number 4). The stress required to reach the instability of stage 3 is given by Equation (7.20) with R = l/2:

(σ crit s ) obstacle = 2Gb/l

(7.36)

where l is given by Equation (7.35). As successive dislocation lines pass through the obstacle array, the residual loops accumulate around the obstacles and effectively increase their size. Because of this growth in obstacle size, their separation decreases below the initial value l and, according to Equation (7.36), the stress to pass the barrier increases. This effect contributes to the macroscopic phenomenon of work-hardening.

7.12.5 locking and unlocking The addition of impurity atoms, whether they are substitutional or interstitial, hardens metals by hindering the start of edge dislocation motion (they have no effect on screw dislocations). The impurities tend to accumulate around edge dislocations because at this location the system’s energy is reduced. The solid above the slip plane (i.e., in the section containing the extra half-plane of atoms) is in compression while the solid below the slip plane is in tension. A substitutional impurity atom that is larger than the

Dislocations 337

host atoms generates a compressive stress field in the immediate vicinity. Consequently, it has a tendency to accumulate on the tensile side of the slip plane, because the compressive stress is relieved by the dislocation’s tensile stress and vice versa. For the same reason, substitutional impurity atoms smaller than the host atoms tend to migrate to the compressive side of the slip plane. Interstitial impurity atoms are small compared to the host atoms (e.g., carbon, nitrogen, boron), yet they always create a compressive stress field around them because they are not in a regular lattice site. Consequently, interstitial impurities tend to collect just below the end of the edge dislocation’s half-plane (i.e., in the open space in the middle diagram of Fig. 7.2). In order for edge dislocations to accumulate impurity atoms, high temperatures and/or long times are needed. These can occur during fabrication as the molten metal cools and solidifies, the principal example being carbon in steel. Alternatively, high temperatures during operation of the component or very long times at low temperatures (called aging) can produce the same result. Analysis of this phenomenon is based on interstitial impurities, although similar results apply to substitutional alloying atoms. The first step is to determine the interaction energy between the interstitial atom and the edge dislocation. The interstitial atom is regarded as an oversized rigid sphere inserted into a hole in the solid of radius Ro, approximately equal to the cube root of the volume of an interstitial site in the host crystal structure. Insertion of an interstitial atom into this hole increases the radius of the hole to (1 + ε)Ro, ε being 4, grains with u < uA shrink and disappear. Grains with u > uB shrink until they reach uB, and grains in the range uA < u < uB grow until reaching uB. Thereafter u remains constant at uB (see Fig. 8.11). Since u is the grain size relative to the critical grain size, and since the latter increases with time, so would the actual grain size. This indefinite increase in R with time is physically impossible. Hillert’s conclusion is that the only physically acceptable value of ε is 4, for which Equation (8.6) reduces to du (2 − u) 2 =− . (8.8) dτ 2u The function on the right side has its maximum at u = 2, at which point du/dτ = 0. At all other values of u, du/dτ is negative. Grains of relative size u > 2 cannot exist because they shrink until u = 2. The conclusion is that there are no grains in the distribution with radii greater than twice the critical radius, or the maximum value of u in the distribution is 2.

364 Light Water Reactor Materials 1 A

0

B ε = 4.2

–1

du2/dt

ε = 3.8

–2 –3 –4 –5 0

1

2

3

4

u

FIGURE 8.11: Plots of Equation (8.6).

The time rate of change of the critical radius is obtained from the last of Equation (8.7) with ε = 4: 2 dRcrit = 2 M γ gb , dt

(8.9)

so that Rcrit increases as t1/2, and dR2crit /dt is constant.

8.2.2 Grain size distribution Determination of the distribution of grain sizes, such as in Figure 8.1, begins with the continuity equation in grain-size space. This equation is similar in meaning to: (i) the slowing-down equation for neutrons in energy space; (ii) the coalescence of bubbles in a solid in bubble-size space; and (iii) the mass continuity equation for fluids in real space. Figure 8.12 shows how the variables in the grain-size continuity equation

Grains and Grain Boundaries 365

relate to those of the common du ∂ du ϕ + ϕ du u (x) ∂u dτ dτ fluid mass continuity equation. du ϕ The function ϕ(u,τ) denotes the du number of grains in the size ϕ dτ range u to u+du at “time” τ. The grain-size continuity FIGURE 8.12: Diagram for deriving the equation equates the time rate of continuity equation in grain-size space. change of the number of grains The quantities in bold parentheses next to the parameters are the equivalents for mass in a differential size range due to continuity in a fluid. the net input of grains of size u by growth into this differential element. This is expressed by the equation ∂ϕ ∂τ

+

∂  du  ϕ =0 ∂u  d τ 

.

(8.10)

The boundary conditions for the continuity equation are: ϕ  = 0 at u  = 0 and u  = 2. Instead of an initial condition, the solution incorporates the requirement that the physical size of the system of grains given by 4 2 3 π R ϕ (u, τ)du 3 ∫0 gr be independent of time (or τ). With du/dτ the function of u given by Equation (8.6), Hillert obtains a solution for ϕ by rather unorthodox means. To eliminate a constant of integration, the expression for ϕ is divided by the total number of grains, given by 2

N(τ) = ∫ ϕ(u, τ)du , 0

which yields the desired probability distribution of grain sizes, P = ϕ/N, or P(u)du = probability that the size of a grain lies between u and u + du:

366 Light Water Reactor Materials

24ue 3 6  P(u) = exp  − 5  2 − u (2 − u)

(8.11)

where e is the base of the Naperian logarithm. Equation (8.11) is plotted in Figure 8.13. There is no explicit dependence of P on τ, which means that the distribution is self-preserving. If P were plotted as a function of grain radius Rgr, with time the distribution would shift to larger grain sizes but preserve its shape. The shift in time is a consequence of the definition of u given by Equation (8.7), Rgr  = uRcrit, with time dependence of Rcrit given by Equation (8.9). The average grain radius is obtained from u=

R gr Rcrit

2

=

∫0 uϕ(u, τ)du = 9

8

(8.12)

.

Other models of grain size distribution have been proposed with even better fits to the data than Hillert’s theory.

p(u)

1

– u 0

0

1

2

u

FIGURE 8.13: Hillert’s grain size distribution probability function.

Grains and Grain Boundaries 367

8.2.3 Average grain size If the mobility M of the grain boundary and the energy per unit area γgb of the grain boundary are known, integration of Equation (8.9) provides the increase of the critical grain radius Rcrit with time. Multiplication of Rcrit by 8/9 according to Equation (8.12) yields the mean grain radius. However, R is the mean radius in three dimensions, and is experimentally inaccessible. What are available are photomicrographs such as those on the right of Figure 8.1. These are images of a section cut through the polycrystalline solid, and as such represent the grain structure in two dimensions. What is termed “grain size” in all applications is the average diameter of the grains seen in the photomicrographs, designated as d. Hillert gives the relationship of the 3-D and 2-D grain sizes as (8.13) R gr = 0.89  d  .  2 Experimentally, the grain diameter is identified with the “mean intercept length,” which is obtained as follows. On a photomicrograph showing the grain structure, a straight line is drawn (Fig. 8.14). The line need not pass through the center of the photomicrograph; any line will suffice. The length of the line on the photomicrograph (L) is measured and then divided by the number of length = L intersections of the line with grain boundaries (n). For example, using the scale on the lower left in Figure 8.14, the length of the line is 535 µm. There are six intersections of the line with grain boundaries, yielding a grain diameter of 90 µm. The general 100 µm formula for this method is FIGURE 8.14: Photomicrograph with d (µm) = L(µm)/n . arbitrary line drawn on it.

368 Light Water Reactor Materials

Higher accuracy is obtained by drawing many lines on the photomicrograph and for each, determining d from the above formula. The average value is the best estimate of the 2-D grain diameter.

8.2.4 Grain growth Eliminating Rcrit from Equation (8.9) using Equation (8.12), replacing R gr by d using Equation (8.13), and integrating yields (8.14) d 2 − d o2 =  4 2 32 M γ gb  t = k gg t ,  0.89 81  which is the classical parabolic grain growth law. do is the initial mean grain diameter. The quantity in parentheses is a function of temperature only and is called the grain growth constant (kgg, µm2/s). The particular form of kgg in Equation (8.14) applies only to Hillert’s model. Preferred orientation and the collection of impurities at grain boundaries usually force deviations from the parabolic form of Equation (8.14). Under irradiation also grain growth follows different kinetics.

8.3 Grain Boundary Mobility The grain boundary mobility M in Equation (8.14) is intimately related to the curvature of the grain boundary. As suggested by Figure 8.10, a lattice atom (or, in the case of ceramics, an ion) on the convex side of the grain boundary has slightly fewer neighbors to bind to than one on the concave side. Consequently, the potential energy of atoms on the convex side of the boundary is slightly greater than those on the concave side. The variation of the potential energy of an atom/ion as it moves between the two sides of the grain boundary is illustrated in Figure 8.15. The extra potential energy of atoms/ions at the boundary in the convex side of the grain is denoted by ∆E, which is much smaller than the activation energy Q for atom motion across the barrier. The frequency with which

Grains and Grain Boundaries 369 Convex grain

Concave grain

Energy of atom

an atom jumps over the barrier in a particular direction is the vibration frequency in the equilibrium site, υ, times a Boltzmann factor (Eq. Q [5.29]). Because of the difference in the barrier heights, the barriercrossing frequency is slightly greater ∆E for convex-to-concave jumps than for those in the reverse direction. The net FIGURE 8.15: Potential energy of an atom flux of atoms from the convex side to on two sides of a curved grain boundary. the concave side, Jnet, equals the difference in the jump frequencies multiplied by the number of atoms per unit area in the grains abutting the boundary, 2 which approximately is 1/ao, ao being the lattice constant. Finally, the velocity of the grain boundary, which is in the opposite direction to the atom flux, is the 3 product of the flux and the atomic volume Ω, the latter approximated by ao. The result is 1  Q  − υ exp  − Q + ∆E   υ gb = ao3 J net = ao3 2 υ exp −  k T    k BT  ao   B ∆E Q exp  −  . ≅ υao (8.15) k BT  k BT  The last form arises from approximating the second exponential term by a Taylor series expansion, owing to ∆E > 1), the correction factor is kF = 1 − P 2/3 . k ox

(9.42)

Example #5: Fuel thermal conductivity reduction due to porosity Compare the thermal conductivity reduction factors using the following parameters: • for as-fabricated fuel, P = 0.05 • the pores are filled with 65% helium and 35% xenon – kpore ≅ 8 × 10−4 W/cm.K = 8 × 10−2 W/m.K. • UO2 thermal conductivity is kox ≅ 0.03 W/cm.K = 3 W/m.K, so ξ ≅ 36. Substituting these values into Equation (9.41) gives kF/kox = 0.87. That is, a 13% reduction in effective thermal conductivity relative to that of the solid UO2. This is close to the maximum reduction for this porosity that would be obtained if no heat flowed through the pore tube, which, from Equation (9.42), kF/kox = 0.86.

9.5.2 Thermal conductivity variation with temperature The integration of Equation (9.5) was performed assuming the thermal conductivity to be independent of the radial position r. In fact, the thermal conductivity of UO2 varies significantly with temperature, which in turn varies markedly with radial position.

Thermal Performance 413

Rather than a constant, a typical temperature-dependent thermal conductivity is (Sec. 16.7.2) k ox = 1 (W/m.K) (9.43) A + BT where A = 3.8 + 200 × FIMA cm.K /W = 0.038 + 2 × FIMA (m.K /W) and B = 0.0217 cm /W = 2.17 × 10 −4 m /W. Equation (9.43), a well-known empirical fit of thermal conductivity data, is often referred to as the Halden equation after the Norwegian laboratory where these data were generated. Neglecting porosity (kF ~ kox), the temperatures at the fuel centerline and fuel surface are related by (see Prob. 9.2) 1 ln  A + BTo  = LHR . (9.44) B  A + BTS  4 π Numerous correlations of kox have been prepared by Baron [5], which, in addition to temperature and porosity dependences, include the effects of plutonium and gadolinium (Fig. 9.7). 0.07

Hervé-Baron_95 Météore Harding-Martin Philipponeau MATPRO NFIR2 NFIR1 part2 NFIR1 part1

0.06 kox (W/cm-K)

0.05 0.04 0.03 0.02 0.01 0 0

500

1000

1500

2000

2500

Temperature (K)

FIgurE 9.7: Comparison of thermal conductivity of 95%-dense UO2 with correlations (after [5]).

414 Light Water Reactor Materials

To account for the dependence of thermal conductivity on temperature, stoichiometry, and plutonium content, the heat conduction equation would need to be solved with kox as a function of these variables using one of the correlations in Figure 9.7. This is normally done by numerical methods. Example #6: Account for the temperature effect on kF according to Equation (9.44) with A and B values following Equation (9.43) Assuming TS = 714 K, and using Equation (9.44), exp  

B × LHR   0.0217 × 200 = 1.41 = exp 4π  4π  

A + BTS = 3.8 + (0.0217)(714) = 19.3; To = [(19.3)(1.41) − 3.8] (0.0217) = 1081 K = 807°C . This is different from the To calculated in Example #1. This occurs because the thermal conductivity of UO2 increases with temperature and the constant kF = 0.03 W/cm.K = 3 W/m.K used in Example #1 is only a very rough average. In addition to variations in fuel thermal conductivity, the thermal conductivity of the cladding is degraded by the ZrO2 layer formed by waterside corrosion (Ch. 22) and also by CRUD deposits (Ch. 20). The slight upturn in kox above 2000 K is due to electronic conduction (Sec. 16.7.3), the main heat transport mechanism in metals. As in all ceramics, the Gibbs energy of free-electron formation in UO2 ( E elf ) is large, which accounts for the very-high-temperature onset of this contribution (i.e., exp[− E elf /k BT ]).

9.6 Thermal Margins and Operating Limits A nuclear reactor is different from conventional power plants in one important way: The central safety tenet in nuclear power plant operation is avoidance of fuel damage and attendant release of fission products.

Thermal Performance 415

Thus, thermal limits are prescribed both for normal operation and for accident situations, with the overall goal of avoiding fuel damage. We discuss operational limits in this section and accident limits in the next section. Comprehensive reviews of these limits can be found in [5] and in Chapter 28. The operational limits provide an envelope of conditions under which fuel failure should not occur. These are LHR limits (related to departure from nucleate boiling), centerline fuel temperature limit (to avoid fuel melting), and limits on pellet-cladding mechanical interaction (PCMI) to avoid cladding failure. To avoid fuel damage, the operational constraints are imposed on maximum fuel centerline temperature. Because the neutron flux and coolant temperature vary axially and radially through the core, so do the fuel-rod temperatures.

9.6.1 Critical heat flux As the outer surface temperature of a heated fuel rod immersed in a constant temperature liquid increases, the mode of heat transfer changes. The boiling curve can be determined from an experiment in pool boiling, in which the outer rod-surface temperature is gradually increased and the heat flux to the liquid is measured. Figure 9.8 shows a plot of the heat flux versus the temperature difference between the rod and the bulk liquid. Various forms of heat transfer can be discerned in the plot. 1) In the single-phase mode of heat transfer, a flux q is driven by the temperature difference between the cladding outer surface, TCO, and the bulk coolant at Tsat: (9.45) q = h(TCO − Tsat ) . The heat-transfer coefficient h that is most commonly employed for single-phase liquid is given by the Dittus-Boelter equation: hd eq /k cool = 0.023Re 0.8 Pr 0.4

(9.46)

416 Light Water Reactor Materials

where deq is the equivalent diameter of the flow channel, kcool is the thermal conductivity of the coolant, and Re and Pr are the Reynolds and Prandtl numbers, respectively.4 One of the main concerns for fuel rods is that the linear heat flux becomes so high that dryout occurs. Figure 9.8 shows the rise of the outer cladding temperature TCO as the heat flux q from the fuel to the coolant increases. At point B, the onset of nucleate boiling provides greater mixing and heat transfer to the coolant. Single phase

Nucleate boiling

Transition boiling

Stable film boiling

Forced convection 106 q (W/m2)

C C'

B

D

104 5

50 Tco – Tsat (°C)

FIgurE 9.8: Schematic of boiling curve in an LWR. Copyright 2008 from Nuclear Systems: Thermal Hydraulic Fundamentals by Neil Todreas and Mujid S. Kazimi. (Redrawn with permission of Taylor and Francis Group, LLC, a division of Informa plc.)

Nucleate boiling begins at point B (although the transition is not as distinct as shown in the figure). A typical nucleate-boiling empirical correlation applicable to PWR conditions is The Reynolds number (ratio of inertial forces to viscous forces) is defined as (ρv D/µu) where ρ is the density, v the velocity, D the diameter, and µ the viscosity) and the Prandtl number (ratio of moment diffusivity to thermal diffusivity) is defined as (cp µ/k) where cp is the heat capacity, µ the viscosity, and k the thermal conductivity. 4

Thermal Performance 417

q(W /m 2 ) = 6(TCO − Tsat )4 .

(9.47)

2) When a second phase is present, the mechanism of heat transfer from the cladding outer surface to the bulk coolant is more complex than the mechanism that drives the heat flux in single-phase water. As the temperature of the cladding outer diameter increases beyond point B, so too does the bubble concentration in the fluid near the wall. At a critical point (C), the bubbles coalesce and a continuous film of steam is formed. Point C is known as the critical heat flux (CHF). In PWRs, this point is identified as the departure from nucleate boiling (DNB). At this point, nucleate boiling turns into the very-much-less-efficient film boiling. 3) Beyond point C in Figure 9.8, the rod is blanketed by steam and the heat flux is severely decreased as the heat transfer coefficient from cladding to steam is much lower than that of cladding to water. The abcissa of Figure 9.8 is TCO − Tcool = (TCO − Tsat) + (Tsat − Tcool); this is because the boiling curves depend upon the difference between the wall temperature and the saturation temperature, not the bulk coolant temperature. The two temperature-driving forces differ by (Tsat − Tcool). If the system pressure is fixed, so is Tsat, but Tcool ranges from 553 K to 593 K (280oC to 320oC). In a PWR, Tsat = 615 K (342oC). A typical cladding OD temperature is TCO ~ 633 K (360oC). 4) Above point C, the bubble nucleation rate becomes high enough that a continuous vapor film forms at the surface. In this region the heat transfer becomes very unstable and the temperature of the wall can suddenly change to one in the film-boiling regime. The departure from nucleate boiling ratio (DNBR) is the ratio of the heat flux that causes dryout (CHF) to the actual heat flux. The DNBR scales with the incidence of fuel damage when these margins are obtained by calculations [6]. In an operating nuclear power plant, it is necessary to demonstrate, by calculation and statistical analysis, that the minimum DNBR for the hottest channel (Fig. 9.9) is larger than 1.15 to 1.3, or a margin of 15% to 30%.

418 Light Water Reactor Materials Critic

al hea

t flux

(calcu

lated)

Hottest channel

Minimum DNBR

Maximum heat flux Heat flux Average channel heat flux Core average heat flux

Bottom of core

Axial distance up the core

Top of core

FIgurE 9.9: Departure from nucleate boiling limits (adapted from [12]).

As shown in Figure 9.9, it is determined by identifying in the hottest channel the location where the heat flux most closely approached the critical heat flux. By this definition, the DNBR is a useful measure of how close a nuclear power plant gets to a dangerous situation during normal operation. When the critical heat flux is reached, the wall temperature increases precipitously, typically above 1100 K, and cladding failure is certain. To avoid reaching the critical heat flux, thermal margins are established. When that happens the liquid can only contact the cladding when the temperature is cooled down to point D, also called the Leidenfrost or rewetting temperature. The mechanism above is prevalent in PWRs, in which a majority of the liquid phase is present where bubbles are generated. In the case of a BWR, in which a significant amount of vapor already exists in the flow (higher quality), an annular flow (liquid film close to the wall with a high density of bubbles) may form. A dryout condition can then occur as a combination of droplet entrainment and evaporation takes place, at which point the flow

Thermal Performance 419

becomes vapor with entrained droplets. The heat transfer then decreases abruptly, and a critical heat flux condition is again reached, this time by a dryout mechanism. A more complete description of these phenomena can be found in [7].

9.6.2 Pellet-cladding mechanical interaction Early reports of fuel failures by pellet-cladding mechanical interaction (PCMI) first appeared in the 1960s. If the linear heat rate is suddenly increased, the pellet-cladding gap closes and the fuel mechanically loads the cladding. If the resulting tensile stress in the cladding is high enough, failure can occur. This frequently occurs in fuel rods adjacent to a recently withdrawn control rod, which causes a sudden LHR increase. The PCMI mechanism is exacerbated when the fuel is cracked or when there is missing chip on the fuel surface, which tends to localize the deformation in the cladding. With longer exposure, the cladding becomes increasingly brittle, so that failure occurs at a smaller imposed strain. Chapter 23 discusses fuel failures under these conditions. Localized PCMI can be avoided by eliminating “hourglassing” of the fuel pellet during transients by proper chamfering during fabrication. PCMI is most severe during high power ramp rates because the fuel does not have time to relieve internal stresses by creep. Pellet-cladding mechanical interaction can cause cracks to propagate from the interior of the fuel pellet, assisted by the presence of fission gases (iodine-assisted stress corrosion cracking, discussed in Ch. 26). Hard cladding results in failure during PCMI. Cladding with greater PCMI resistance than standard Zircaloy has also been investigated. Softer materials are more resistant to crack propagation. Duplex cladding consisting of co-extruded Zircaloy-2 and an inner (barrier) layer of (softer) Zr [8] enhances PCMI resistance [7]. Essentially, all BWR cladding uses some form of Zr-barrier cladding.

420 Light Water Reactor Materials

9.7 Limits for Accident Conditions In light water reactors, the accidents with the greatest potential for fuel damage are those (i) producing excessive power (as in a reactivityinitiated accident [RIA]), or (ii) having too little coolant, as in a loss-ofcoolant accident (LOCA). Both of these severe accidents are treated in detail in Chapter 28. We present a short introduction here.

9.7.1 loss-of-coolant accident (lOCa) The LOCA is a design-basis accident in light water reactors. The postulated LOCA is initiated by a guillotine break in the primary piping, allowing coolant discharge into the containment building. Although the control rods shut down the reactor, power continues to be produced by the decay heat of the fission products, which amounts to approximately 7% of the full-power thermal output of the reactor at the time of shutdown. For a 1000-MWe reactor, the decay of the fission products generates ~210 MW of heat. In addition, thermal energy is stored in the fuel from the temperature gradients necessary to drive heat flow during normal operation. PCMI and cladding overheating by the combination of decay heat and stored energy can be avoided by water from the Emergency Core-Cooling System (ECCS). The maximum allowable cladding temperature of 1205oC was determined by following the severity of the accident as the temperature of the cladding increases. At this temperature, and in the presence of steam, cladding corrosion becomes a runaway reaction. If the temperature is kept below the 1205oC limit, it is possible for the cladding to survive a LOCA. Thus, it is necessary using fuel-performance models to show with reasonable certainty that the cladding temperature remains below the limit for postulated LOCA accidents. As the fuel temperature increases during a transient, fission gas release accelerates with a corresponding increase of the internal rod pressure.

Thermal Performance 421

Simultaneously, the Zircaloy yield stress decreases (see Ch. 11), and the cladding yields, balloons, and may rupture (Ch. 28). Ballooning reduces the area for coolant flow in the fuel channel, which further enhances local overheating of the fuel rod. However, blockage and cladding rupture followed by release of fission products is not the worst potential outcome of a LOCA. If the cladding remains at a high temperature just prior to injection of cold water by the ECCS, embrittlement can cause the cladding to shatter. Ring compression tests were conducted to assess the acceleration of cladding corrosion from a high-temperature excursion during a LOCA and the accompanying reduction of post-quench ductility [9]. These results established that if the extent of oxidation from time-at-temperature remained below a certain level, cladding ductility would not be unduly compromised. High-temperature oxidation results in formation of a ZrO2 scale as well as increased oxygen absorption by the cladding. Consequently, a limit is established in terms of the equivalent cladding reacted (ECR), which is the fraction of the cladding that would be consumed if all the oxygen absorbed were used to form ZrO2. It was found that if the ECR is kept below 17%, the cladding retains enough ductility to remain intact during the LOCA.

9.7.2 reactivity-initiated accident (rIa) Although RIAs are improbable (none has ever occurred in a commercial LWR), it is necessary to demonstrate (by calculation) that whatever the extent of fuel damage, a coolable geometry is maintained. RIAs are discussed in detail in Chapter 28. Operating limits were established by tests in which reactivity was suddenly inserted by the ejection (PWR) or drop (BWR) of a control rod. The ensuing increase in fission rate leads to an expansion of the fuel against the cladding (PCMI), which then can rupture. If the failure is “severe,” a loss of coolable geometry may result.

422 Light Water Reactor Materials

The critical parameter is the total energy deposition during the transient. MacDonald and coworkers showed [10] that if the deposited enthalpy in the fuel was less than 180 cal/g (752 kJ/kg), no fuel failure occurred, and if less than 280 cal/g (1170 kJ/kg), no fuel dispersal occurred. It must then be demonstrated that reactivity insertions resulting in energy depositions higher than those values are extremely unlikely. Recent results have indicated that the above limits, which were derived for fresh fuel, may be lower for high-burnup fuel [11].

9.7.3 Stored energy One of the consequences of low fuel thermal conductivity is a large temperature gradient needed to drive the heat from the fuel to the coolant. An upshot is significant thermal energy stored in the fuel. This is one reason why cooling must continue following a loss-of-coolant accident. Otherwise, the temperature redistributes across the entire rod until the fuel and cladding temperatures equalize. This could cause the cladding temperature to rise above acceptable limits. The stored energy per unit length of fuel rod is RF

( LHR ) R F2 E stored = (ρC P ) F ∫ 2 πr (TF − TS ) dr = (ρC P ) F 8k F 0

(9.48)

for a parabolic temperature distribution (Eq. [9.10] and [9.11]). The coolant needs enough heat-transfer capability to remove at least the stored thermal energy, in addition to the heat generated by fission-product decay, before the fuel melts or the cladding ruptures. Example #7: Cladding temperature increase from decay heat A section of fuel rod is operating at LHR = 200 W/cm when the reactor is scrammed and the coolant flow is drastically reduced. What hcool (cladding outer diameter-to-coolant heat transfer coefficient) is required to

Thermal Performance 423

prevent an increase in the mean cladding temperature to 1205oC? For fuel-rod properties, see Table 9.1. Neglect decay heat and heat from cladding oxidation. Assume a constant fuel thermal conductivity of 0.03 W/cm-K. Ignore the thermal resistance of the gap. The fuel is 1 cm in diameter, and the cladding is 0.1 cm thick. The transient that follows is governed by ∂TF 1 ∂  r ∂TF  = kF (9.49) r ∂r  ∂r  ∂t where kF = kF /(rCP)F is the fuel thermal diffusivity. The initial condition is 2 2 2k TF (r ,0) = LHR 1 − r 2 + F  = 530 1.6 − r 2  . (9.50) 4 πk F  R F R F h  RF  The steady-state hcool = 2.5 W/cm2.K and kC/δC = 1.7 W/cm2.K, so from Equation (9.21), h = 1.0 W/cm2.K.  ∂TF  = 0  ∂r  r =0

and

TF ( R F , t ) = TC (t )

(9.51)

The average cladding temperature obeys: δ C (ρ C P ) C

dTC dt

 ∂TF   − h (T − T ) .  ∂r r =R F cool C cool

= −k F 

The initial mean cladding temperature is obtained from LHR = h o (T ) − (T ) ( ) 2 πRCO cool C 0 cool 0

(9.52)

(9.53)

where the quantities on the right side are pre-scram values. An additional difficulty is how to calculate Tcool for t > 0. This involves the entire fuel rod below the elevation in question. If this can be prescribed as a function of time, the value of hcool to keep TC < 1200°C can be calculated.

424 Light Water Reactor Materials

References  1. R. Yang, O. Ozer, and H. Rosenbaum, “Current challenges and expectations of high performance fuel for the millenium,” in Proceedings of Light Water Reactor Fuel Performance Meeting, April 10–13, 2000 (Park City, Utah: 2000).  2. R. O. Meyer, “Fuel Behavior Under Abnormal Conditions,” NUREG/ KM-0004. U.S. Nuclear Regulatory Commission, 2013.  3. M. M. El-Wakil, Nuclear Heat Transport (La Grange Park, Illinois: American Nuclear Society, 1978).  4. D. R. Olander, Fundamental Aspects of Nuclear Reactor Fuel Elements, TID-26711-P1., US Technical Information Center (1976).  5. D. Baron, “Fuel thermal conductivity: a review of modeling available for UO2, (U,Gd)O2 and MOX,” in Proceedings of Seminar on Thermal Performance of High Burn-up LWR Fuel 99 (Cadarache, France).  6. IAEA, “Safety Margins of Operating Reactors,” International Atomic Energy Agency, IAEA-TECDOC-1332, 2003.  7. N. E., Todreas and M. S. Kazimi, Nuclear Systems I, Thermal-Hydraulic Fundamentals (New York, London: Taylor and Francis, 1990).  8. J. S. Armijo, L. Coffin, and H. Rosenbaun, “Development of zirconium– barrier fuel cladding,” STP 1245, 11th ASTM International Symposium on Zr in the Nuclear Industry 3 (1995): 18.  9. H. M. Chung and T. F. Kassner, “Embrittlement Criteria for Zircaloy Fuel Cladding Applicable to Accident Situations in Light-Water Reactors: Summary Report,” NUREG/CR-1344 (Nuclear Regulatory Commission: 1980). 10. P. E. MacDonald, et al., “Assessment of light-water-reactor fuel damage during reactivity-initiated accident,” Nuclear Safety 21 (1980): 582–602. 11. R. Meyer, R. K. McCardell, and H. H. Scott, “A regulatory assessment of test data for reactivity accidents,” in Proceedings of the ANS International

Thermal Performance 425

Topical Meeting on Light Water Reactor Fuel Performance (Portland, Oregon, 1997) 729–744. 12. M. M. El-Wakil, Nuclear Heat Transport (La Grange Park, Illinois: American Nuclear Society, 1978).

Problems 9.1 Verify that the burnup when expressed in MWd/kgU is equal to 877 × FIMA. 9.2 The temperature dependence of the thermal conductivity of UO2 is given by Equation (9.35) with C = 0. From this, derive Equation (9.13). The rod linear power is LHR. 9.3 In calculating the temperature distribution in the cladding, the tube curvature is usually neglected and the geometry is approximated as a slab. Below is the corresponding analysis in cylindrical geometry. (a) Derive the equation for the temperature difference between the inside and outside surfaces for cladding of thickness δc and inner cladding radius RcI operating at linear power LHR. The cladding thermal conductivity is kc. (b) Show mathematically how the result of (a) reduces to that for the slab for the case of thin cladding (δc x2, single-phase alpha and single-phase beta, respectively, are formed. We should note that for the two coexisting phases at compositions x1 and x2, the chemical potentials of A and B atoms in the alpha and beta phases are the same. If x corresponds to the content of B in a binary solution, the chemical potentials are dg 1′ dg 2′ = µ β ; and µα = µβ . (10.8) = µαB ; B B dx x 2 B dx x1 (See Ch. 2.) That means that there is no driving force for a B atom in either phase to move to the other (since its chemical potential is identical in the two phases), which means the system is at equilibrium. The net result is that at a given temperature, the equilibrium phases are those that would be predicted by the lower envelope3 of the possible Gibbs energy curves, if it is considered that such envelope contains the common tangents such as g1–g2 shown in Figure 10.2. The above derivation allows us to relate the Gibbs energy variation with composition to the Gibbs energy-composition phase diagram. In Figure 10.2, which was sketched for a fixed temperature, the single-phase alpha stability domain ranges from 0 to x1, the two-phase domain from x1 to x2, and single-phase beta from x2 to 1. As illustrated in Figure 10.3, plots of Gibbs energy versus composition similar to that shown in Figure 10.2 at various temperatures can be used to construct a standard phase diagram, of which Figure 10.3 is an example. Temperature T1 Points x1 and x2 in Figure 10.2 translate to a horizontal line in Figure 10.3 extending between the two single-phase domains and indicating a composition region where at equilibrium, a mixture of two phases is observed. The twophase domain extends from a to b, which corresponds to x1 to x2 in Figure 10.2. 3

The lower envelope means the ensemble of the lowest lines of the Gibbs energy curves in the singlephase regions and the common tangents in the two-phase regions.

436 Light Water Reactor Materials

Temperature T2 As the temperature increases above T1, the Gibbs energy of the liquid decreases relative to the alpha and beta phases. At T2 the three curves on the g versus composition plot exhibit a single common tangent (upper left diagram in Fig. 10.4) at compositions c, d, and e in Figures 10.3 and 10.4. At this temperature the three phases are in equilibrium at the compositions determined by the common tangent, which means by Equation (10.8) that at this temperature the components in the three phases at those compositions have the same chemical potential. The lowest temperature and composition (point d) at which the liquid is stable is called the eutectic point. The eutectic transformation reaction may be written as cooling  →α+β . Liquid ← (10.9)  heating Note that the compositions of α and β at the eutectic transformation temperature (Eq. [10.9]) are c and e respectively; thus, when cooling, the liquid decomposes into two solids enriched in A and B respectively. T T5 T4

m j

g

f

T3

Liq

h

L+b

i

a+L c

T2 α

T1

l

k

a

d

a+b

e

b z b y

Pure A

Molar fraction of B

Pure B

FIGURE 10.3: Schematic phase diagram of a eutectic A–B system.

Phase Transformations in Solids 437

c T2

f d

e

T3

g h

i

m

j

T4

k

l

T5

FIGURE 10.4: Schematic Gibbs energy curves versus composition for the corresponding temperatures shown in Figure 10.3. The dashed curve is alpha, gray is liquid, and black is beta.

Temperature T3 As the temperature increases further, the Gibbs energy curve for the liquid falls below the common tangent of α and β so that there are now two common tangents, f−g (between α and liquid) and h−i (between liquid and β). In Figure 10.3, two two-phase regions (f−g and h−i) plus three single-phase domains (0−f, g−h, and i−1) appear. In addition to the single-phase α and β regions at the edges (below f and above i, respectively), there is now a liquid phase between points g and h. In between these phases, there are two regions of two-phase equilibria, α + L and β + L (f−g and h−i). Temperature T4 Above temperature T4, the Gibbs energy curve of the liquid becomes lower than that of the α phase throughout the entire compositional range, and the f−g two-phase zone disappears from the phase diagram. This temperature then corresponds to the melting temperature of the alpha phase

438 Light Water Reactor Materials

containing 100% A (point j in the two figures). The Liq + β two-phase zone persists, but is smaller. There is also a single-phase β region. Temperature T5 At this temperature the last remaining solid (β phase at 100% B) melts, so this is the lowest temperature at which the system consists of only a single liquid phase. This temperature is the melting point of single-phase B in the β crystal structure (point m in Fig. 10.3 and 10.4). By performing careful measurements of phase equilibria at a series of temperatures, it is possible to construct a phase diagram such as shown by the lines in Figure 10.3. Incidentally, the phase decomposition discussed above would occur when cooling from point z to point y in Figure 10.3, going from single-phase beta at z to a mixture of beta and alpha at y. Order of phase transformations Finally, phase transformations can be classified by their order. A phase transformation is first-order if during the phase transformation the change in Gibbs energy is zero but its first derivative is discontinuous at the transformation temperature:  ∂g  ≠0 . ∆  (10.10)  ∂T  T ,P t

In such a transformation, energy needs to be supplied or removed. For example, the phase transformation illustrated in Figure 10.1 is first-order because as freezing/melting occurs the Gibbs energy of the liquid or solid is continuous across the transformation (the curves match), but the first derivative of the Gibbs energy is not continuous (in the illustration, the first derivative of the liquid is higher than that of Sol1 at the transformation temperature). This implies that first-order phase transitions have a latent heat of transformation; i.e., energy either needs to be supplied or is released at constant temperature for one phase to transform into another. Thus, the phases coexist once the transformation is underway. For example, when a bucket of ice melts into water, both ice and water coexist in the bucket at 0°C until enough heat is supplied that all of the ice transforms to water.

Phase Transformations in Solids 439

For a second-order phase transformation, the discontinuity occurs in the second derivative of the Gibbs energy with temperature. As a result, in a secondorder phase transition, one phase changes into the other continuously, and the phases do not coexist during the transformation. This describes some magnetic transitions, but most phase transitions in nature are first-order.

10.2.3 Examples of important phase diagrams In analyzing the materials for the core of a nuclear reactor, phase diagrams are particularly useful, notably those dealing with the alloys used for cladding and structural materials and those related to the ceramic fuel material. Figures 10.5a to 10.5i show calculated binary phase diagrams for the main alloying elements used in zirconium alloys for nuclear fuel cladding, Fe-C (relevant for steels) and U-O (the component of fuel). The region of greater interest in the Zr-M phase diagrams for our purposes lies in the Zr-rich part of the diagram, as the concentrations of alloying elements are generally low. The exceptions are the H and O phase diagrams in which Zr oxides and hydrides are depicted. Zirconium hydrides and oxides can form in the fuel cladding during reactor operation. The phase diagrams in Figure 10.5 show various features that illustrate different phase equilibria. • The phase diagram of Zr-Sn (Fig. 10.5a) shows a large difference in melting temperature between the two constituents, so as the Sn content increases, so does the stable liquid phase domain. Various line compounds (exact stoichiometry) appear, such as Zr4Sn and Zr5Sn3, but are not observed in Zircaloy or ZIRLO™, likely because of the slow kinetics of their formation. • The Zr-Nb phase diagram (Fig. 10.5b) shows complete solid solubility in the beta phase at temperatures from about 1000°C to 1500°C. This means that body-centered cubic (bcc) Zr transforms smoothly into bcc Nb as the Nb content increases without any phase separation. The beta phase also shows a region of spinodal

440 Light Water Reactor Materials

•

•

•

•

decomposition (see Sec. 10.4.4), and more relevant to nuclear applications, the alpha Zr phase can take up to a significant amount of Nb in solid solution. The Zr-O phase diagram (Fig. 10.5c) shows an equilibrium between Zr(O) solid solution and ZrO2. Such an equilibrium is partially observed during waterside corrosion of zirconium alloys, since an oxygen-rich region is normally observed ahead of the advancing ZrO2 oxide layer. Because the oxide layer is constantly advancing and consumes the oxygen-rich layer as it advances, phase equilibria is constantly being re-established just ahead of the oxide front. The Zr-H phase diagram (Fig. 10.5d) shows the presence of zirconium hydrides as soon as the solid solution limit is exceeded, which is in fact seen in reactor. The Zr-Cr (Fig. 10.5e), Zr-Ni (Fig. 10.5f), and Zr-Fe (Fig. 10.5g) phase diagrams show very low solid solubility for all these elements in alpha Zr. Thus, as discussed in Chapter 17, these alloying elements are mostly found in intermetallic precipitates. Because the transformation temperature for Zr2Ni and ZrCr2 is lower than that for Fe-containing compounds, iron tends to appear as a substitutional solute in the Ni and Cr sublattices of these compounds. This is a simple example in which the presence of a third element affects the behavior of a binary system, illustrating that ternary phase equilibria (ZrFeCr and ZrFeNi) differ from binary equilibria (ZrFe). The Fe-C phase diagram (Fig. 10.5h) provides the basis for the development of the various steels used in the industry. It shows the alpha iron phase (bcc Fe), also called ferrite, as the stable low temperature phase, and gamma Fe (face-centered cubic [fcc] Fe) also called austenite, as the stable high temperature phase. (For clarity, beta Fe concerns a magnetic transformation in the alpha range). Stainless steel consists of Fe alloyed with Ni and

Phase Transformations in Solids 441

Cr, which causes the austenite phase to be stable at low temperature. The resistance of austenitic steel to corrosion is much higher than that of ferrite, as can be easily discerned in one’s kitchen by examining the behavior of carbon steel knives, which show rust in contact with water, and stainless steel flatware, which does not. The reactor pressure vessel is made of ferritic steel, while the reactor internals are of stainless steel. More discussion on this phase diagram is presented in Section 10.5. Finally, the U-O phase diagram illustrates the various phases of metallic uranium, including the low-temperature face-centered orthorhombic phase (see Ch. 3) and shows the small range of stability of UO2 in the substoichiometric range, allowing the use of UO2−x as fuel (Ch. 16).

•

2500

L

1988 ºC

2000

1592 ºC

1500

17% 19.1% 1327 ºC

β Zr

1142 ºC

863 ºC

α Zr

500

ZrSn2

1000

Zr5Sn3

982 ºC 7.3%

Zr4Sn

(a)

Temperature (ºC)

1855 ºC

232 ºC

0

20

40

60

80

100

Sn (at.%)

FIGURE 10.5a: Binary phase diagrams of importance to the nuclear industry: (a) Zr-Sn. (The data in all Fig 10.5 phase diagrams was graciously prepared by G. Lindwall and Z. K. Liu and were calculated using Thermocalc [J. O. Andersson, T. Helander, L. H. Hoglund, P. F. Shi and B. Sundman, “THERMO-CALC & DICTRA, computational tools for materials science,” CALPHAD, 26 (2002); 273–312.])

442 Light Water Reactor Materials 2469 ºC

2500 L

(b)

Temperature (ºC)

2000

1855 ºC 1740 ºC 21.7%

1500

β Zr + β Nb

60.6%, 988 ºC

1000

863 ºC 620 ºC 0.6%

500

91%

18.5%

α Zr

0

20

40

60

80

100

Nb (at.%)

L+G

2710 ºC

L

2500

2377 ºC 25%, 2130 ºC

10% 10.5%

19.5%

β Zr

1500

62%

1525 ºC

α Zr

63.6% 1205 ºC

1000 α ZrO2-x

Temperature (ºC)

(c)  

1855 ºC

γ ZrO2-x

40%

β ZrO2-x

2000

863 ºC

500

0

10

20

30

40

50

60

70

80

O (at.%)

FIGURE 10.5b–c: Binary phase diagrams of importance to the nuclear industry, continued: (b) Zr-Nb; (c) Zr-O.

Phase Transformations in Solids 443 1000 G

900

863 ºC

800

β Zr

(d)

Temperature (ºC)

δ ZrH2

700 ε ZrH2

600

α Zr

550 ºC 5.93%

500

56.7%

37.5%

400 300 200 100 0

10

20

30

40

50

60

70

80

H (at.%)

2000

1863 o C

1855 ºC

1800

L

γ ZrCr2

66.7%, 1673 ºC 82%

(e)

Temperature (ºC)

1600 1400 β Zr 8%

1332 ºC

β ZrCr2

22%

1532 ºC Cr α ZrCr2

1200 1000

1592 ºC

863 ºC 836 ºC

1.65%

800

α Zr

600 400 200 0

20

  FIGURE 10.5d–e: (d) Zr-H; (e) Zr-Cr.

40

60 Cr (at.%)

80

100

444 Light Water Reactor Materials 1855 ºC

1800 1600

1440 ºC

1455 ºC

1260 ºC

b Zr

1200

1120 ºC Ni

1000

960 ºC 24%

845 ºC

Ni5Zr

NiZr2

400

Ni10Zr7

a Zr

600

Ni10Zr7

800

NiZr

(f)

Temperature (ºC)

1400

Ni7Zr2 Ni3Zr

200 0

20

40

60

80

100

Ni (at.%)

2000 1855 ºC

1800

55.1%, 1673 ºC

1600

1538 ºC

1400 β Zr

1200

ZrFe2

γ Fe

24%

1000

974 ºC

928 ºC

863 ºC

885 ºC 775 ºC

800

4%

600

α Zr

400

Zr3Fe

(g)

Temperature (ºC)

90.2%

200 0

20

40

60

80

100

Fe (at.%)   FIGURE 10.5f–g: Binary phase diagrams of importance to the nuclear industry, continued (f) Zr-Ni; (g) Zr-Fe.

Phase Transformations in Solids 445 1800 1538 ºC

L

1600 1495 ºC

δFe

(h)

Temperature (ºC)

1400

1394 ºC

1200

1154 ºC 8.99%

γ Fe

17.28%

1000

γ Fe + Fe3C

912 ºC

800

738 ºC

3.09%

600

α Fe α Fe + Fe3C

400 200

0

5

10 C (at.%)

15

20

4500 4000

G L

3500

(i)

Temperature (ºC)

2852 ºC

3000 2425 ºC

2500 2000

UO2±x

1870 ºC

1500 1132 ºC

1000

γ -U

776 ºC

500

β -U

669 ºC

668 ºC

α -U

UO3

0 0

20

FIGURE 10.5h–i: (h) Fe-C; and (i) U-O.

40 60 80 O (at. %) U4O9 U3O8

100

446 Light Water Reactor Materials

The Zr-M binary phase diagrams determine much of the microstructure of the fuel cladding as discussed in Chapter 17. For example, the Zr-Cr phase diagram predicts that at a temperature of 300°C and Cr content of a few percent, the stable phases are α-Zr and ZrCr2, which are in fact observed in Zircaloy, but with Fe substituting in the Cr sublattice, Zr(Cr,Fe)2. The Zr-O phase diagram shows that only one oxide of zirconium (ZrO2) is stable at temperatures of interest, but many suboxide metastable phases are often observed during cladding waterside corrosion. The Fe-C phase diagram is the basis for understanding the composition of steels. These alloys, however, contain more than one metal, so ternary and higher-order phase diagrams would be needed to describe phase equilibria. These are, however, beyond the scope of this book.

10.3 Nucleation of a Second Phase The first part of this chapter dealt with the thermodynamic driving force for phase transformation, which governs the direction of a phase transformation. Once this is established, it is necessary to address how the transformation takes place, and its rate. When no change in composition is involved, the transformation occurs in a displacive manner, in which the atoms move a small distance (within the unit cell) in coordinated fashion to create a new phase. One example of this is the phase transformations in ZrO2 from cubic to tetragonal to monoclinic (Fig. 3.14). It is also possible to observe interface-reaction controlled transformations, such as the β → α transformation in Zr. Also, recrystallization is a transformation of this type since its kinetics are similar even though there is no change in crystal structure. More commonly, phase transformations involve compositional changes and large-scale transfer of matter; these are called replacive transformations, as one type of atom needs to be replaced by another through longrange atomic transport. The first step in a replacive transformation is the

Phase Transformations in Solids 447

creation of a nucleus (nucleation) of a new phase (α) from an unstable matrix phase (β). In homogeneous nucleation, the new phase appears without the aid of inhomogeneities or defects in the matrix. In heterogeneous nucleation, defects such as grain boundaries, dislocations, or free surfaces aid the nucleation of the new phase.

10.3.1 Homogeneous nucleation In replacive phase transformations, the creation of a new phase entails an energy cost associated with the formation of an interface between the nucleus of the new phase and the matrix in which it is embedded. The required energy has to be balanced by reduction in Gibbs energy associated with the formation of the new (more stable) phase. From this idea follows the theory of homogeneous nucleation. Consider a β phase that has been cooled below the equilibrium temperature for the simultaneous presence of the α and β phases (e.g., below Tm in Fig. 10.1). Consider the nucleation of a spherical alpha phase particle from the beta matrix. The question of interest is: What is the minimum cluster size that yields a stable α nucleus from a β-phase matrix? In this case, nα moles of α are formed from the original β phase as determined by 4 n α = 3 πrα3ρ α

(10.11)

where ρα is the molar density of α and rα is the radius of the nucleus. Let ∆gαβ = gβ − gα be the difference in Gibbs energies per mole of β and α. This is a positive quantity, since the β phase is subcooled. The change in Gibbs energy upon formation of a single spherical nucleus of α phase is ∆ g hom = ∆ g vol + ∆ g surf + ∆ g strain (10.12) where ∆g vol is the Gibbs energy change upon the transformation for the volume considered, ∆g surf is the interfacial energy that needs to be supplied

448 Light Water Reactor Materials

to create a new phase, and ∆g strain is the strain energy in the particle and in the matrix associated with a density change of the second phase with respect to the parent. This last term is often important and may determine the rate of nucleation but is not easily evaluated. If the strain term is for the moment ignored, the Gibbs energy of nucleation is ∆g hom ≈ ∆g vol + ∆g surf = − n α ∆ g αβ + γ αβ

(4 πr 2 ) α

(10.13)

where γαβ is the interfacial energy of the α−β surface and ∆gαβ is the Gibbs energy difference between alpha and beta. Eliminating nα between Equations (10.11) and (10.12) yields 4 3 πr ρ ∆g + 4 πrα2 γ αβ . (10.14) 3 α α αβ Equation (10.14) states that as a new particle is nucleated, the Gibbs energy initially increases as the surface energy term dominates. As the radius increases further, the volume term eventually overtakes the surface term and the energy becomes negative, which means that it is then favorable for precipitation to occur. Figure 10.6 shows a plot of Equation (10.14). The solid light gray curve is the surface energy term, while the solid medium gray curve is the volumetric decrease in Gibbs energy after the transformation, and the black solid curve is the sum of the two. When the derivative of the total Gibbs energy shown in the black curve becomes zero, any further increases will cause a decrease in overall Gibbs energy, and therefore, the growth of the particle can occur unimpeded. The point where the derivative is zero is called the critical radius for homogeneous nucleation of alpha phase, rcritα . This can be calculated formally by noting that ∆g hom = −

∆g hom

4 πr ′2 γ αβ

 rα  =−  r′

3

 rα  +  r′

2

(10.15)

Phase Transformations in Solids 449

r′ =

where

3 γ αβ ρ α ∆g αβ

(10.16)

∆ghom

in Equation (10.15) is positive for small values of rα and becomes negative as rα increases. When further increases in rα cause a decrease in Gibbs energy, the nucleus of the α phase becomes stable. Setting the derivative of Equation (10.15) to zero gives 2γ rαcrit 2 (10.17) = or rαcrit = αβ . r′ 3 ∆g αβ If nucleation produces an α sphere phase with a radius rα < rαcrit , the nucleus dissolves back into the β phase. If rα > rαcrit , the nucleus is stable and continued attachment of atoms from the β phase (growth) further lowers the Gibbs energy. One could question how the critical radius is

Gibbs energy

Dgsurf

r acrit

Dgtotal

Dgαβ

r

√

FIGURE 10.6: Schematic plot of the Gibbs energies upon the formation of a spherical nucleus from solid solution. The dashed line indicates the effect of greater undercooling.

450 Light Water Reactor Materials

ever formed, since there will always be a radius below which growth is unfavorable. The thought is that random thermal fluctuations are constantly creating nuclei, and when a fluctuation creates a nucleus above the critical size, its growth is thermodynamically favored. The nucleation rate (number of particles/second per unit volume) is proportional to the exponential of the critical free energy difference dN α  −∆g crit  ∝ exp dt  k BT 

(10.18)

where ∆gcrit is proportional to the critical radius. As ∆gcrit decreases, the nucleation rate increases. A high nucleation rate occurs when precipitates are small (by forming coherent precipitates or by heterogeneous nucleation4; see following section) or by large negative ∆gαβ. As shown in Figure 10.1 for the liquid (β)–solid (α) transformation, the Gibbs energy difference between the β and α phases, ∆gαβ, is zero at TM and increases with undercooling. From Equation (10.17), it can be seen that the critical radius is inversely proportional to the Gibbs energy difference. The effect of increased undercooling is shown in the dashed plot of Figure 10.6 in which the Gibbs energy change was increased by two-thirds relative to the initial calculation. It is clear that the critical radius decreases in proportion. Determination of a numerical value of the critical radius requires knowledge of the interfacial energy γαβ and the Gibbs energy difference between the β and α phases for the extent of subcooling. If the curves in Figure 10.1 can be approximated as straight lines, it can be shown that ∆ g αβ =

4

L M ∆T TM

(10.19)

Coherent precipitates have an orientation relationship with the matrix, such that atom positions match across the interface. Both coherent precipitation and heterogeneous precipitation can increase the nucleation rate.

Phase Transformations in Solids 451

where LM is the latent heat of melting, TM is the melting temperature, and ∆T (i.e., TM − T) is the undercooling. Substituting Equation (10.19) into Equation (10.17), the critical radius for solidification is 2 γ αβTM 1 . (10.20) L M ∆T For copper TM = 1365 K, γαβ = 0.18 J/m2, and LM = 1.9 × 109 J/m3, the critical radius as a function of the undercooling given by Equation (10.20) is plotted in Figure 10.7. Figure 10.7 shows that for ∆T = 10°C, the critical radius for solidification is 2.5 nm. rαcrit =

10.3.2 Heterogeneous nucleation Because real crystals have defects and impurities, homogeneous nucleation hardly ever occurs. Imperfections dramatically reduce the interfacial energy in Equation (10.14) that must be supplied, thereby providing an 3.0

Critical radius (nm)

2.5 2.0 1.5 1.0 0.5 0.0 0

50

100 Undercooling (K)

150

200

FIGURE 10.7: Critical radius of nucleus for precipitation of solid Cu from the liquid at 1365 K.

452 Light Water Reactor Materials

easier path for phase transformation. For example, when boiling water in a pot, if one looks closely, bubble nucleation occurs at scratches in the bottom of the pan. By analogy, when casting metals, the mold surface can provide nucleation sites for the solid to form from the liquid phase. Such processes are called heterogeneous nucleation. Decreasing the surface energy also reduces the critical radius and facilitates precipitation. This is shown by the dashed curve in Figure 10.8, in which the area needed for surface formation (and thereby the surface energy) was reduced by one-half. The effect of heterogeneous nucleation on the phase transformation can be modeled by a spherical cap of the new phase (α) forming on a surface (S), as shown in Figure 10.9. The upper portion represents the single-phase β phase and γαS, γβS, and γαβ denote interfacial energies at the three interfaces, surface–α, surface–β, and α–β.

Gibbs energy

Dgsurf

r′min

r

rmin Dgtotal

Dgαβ

FIGURE 10.8: Gibbs energy versus nucleus radius showing the effect of lower surface energy (dashed line) on the total Gibbs energy.

Phase Transformations in Solids 453 γαβ β γβs

θ

γαs

α

S

FIGURE 10.9: Heterogeneous nucleation at a surface (after [2], as cited by [3]).

The Gibbs energy change associated with forming a nucleus of the α phase from the β matrix on the surface S is given by ∆ g het = −∆ g αβ n α + γ αβ Aαβ + γ αS AαS − γ βS AαS (10.21) where the Aij are the areas of the i−j interfaces. The first term corresponds to the decrease of the Gibbs energy (per mole) in transforming nα moles from β to α; the second and third terms are the energies of the interfaces between the α and β phases and between α and the surface S. The last term is a reduction in the Gibbs energy resulting from the elimination of the β−S interface. Equilibrium of forces at the edge of the nucleus in Figure 10.9 means that γ βS = γ αS + γ αβ cos θ . (10.22) Substituting Equation (10.22) into Equation (10.21) yields ∆ g het = −∆ g αβ n α + γ αβ Aαβ − ( γ αβ cos θ) AS .

(10.23)

From the geometry in Figure 10.9, the number of moles of B in the spherical cap is ρ α πr 3 (2 + cos θ)(1 − cos 2 θ) nα = 3 where ρα is the molar density of the α phase. When θ = π/2, nα reduces to half the value of Equation (10.11) (half a sphere). Given that the area of the cap is Aαβ = 2πr2(1 − cos θ) and the area of the wall surface–beta interface is

454 Light Water Reactor Materials

AβS = πrα2 sin 2 θ where rα is the radius of the spherical cap of the α phase, the result is πr 3 ∆g αβ (2 + cos θ)(1 − cos 2 θ) ∆g het = ρ α + γ αβ πr 2 [2(1 − cos θ) 3 (10.24) + sin 2 θ cos θ] or

 

∆g het =  ρ α

with

 4 πr 3 ∆g αβ + 4 πr 2 γ αβ  f (θ) = ∆g hom f (θ)  3

(10.25)

(2 − 3cos θ + cos 3 θ) . 4

(10.26)

f ( θ) =

The bracketed quantity is the Gibbs energy change for homogeneous nucleation (Eq. [10.14]). Because f (θ) is between 0 and 1, heterogeneous nucleation diminishes the Gibbs energy barrier compared to homogeneous nucleation. That is, according to this model, ∆g het < ∆g hom. As can be seen from Figure 10.10 f (θ) → 1 when θ → π. That is, a sphere of α immersed in a β matrix without touching the surface S is equivalent 1 0.8

f (q)

0.6 0.4 0.2 0 0

45

90

135

q

FIGURE 10.10: The function f (θ) versus θ.

180

Phase Transformations in Solids 455

to homogeneous nucleation. When θ → 0, cos θ → 1, and f (θ) → 0, or the α phase “wets” the surface S. In this case, only a very small amount of energy is needed for nucleation, related to the edges of the disk, which are neglected above, and the α phase spontaneously replaces the β phase adjacent to surface S. Differentiation of Equation (10.25) with respect to rα (by Eq. [10.22], θ is independent of rα ) and setting the result equal to zero gives the critical radius: 2γ (10.27) (rαcrit )het = (rαcrit )hom = ∆g αβ . αβ That is, the radius of the spherical cap of α is the same as the radius of the entire sphere of α in homogeneous nucleation. The Gibbs energy change for the formation of a critical-radius second phase is 3 het = 16 πγ (10.28) ∆g crit f ( θ) . 3( ∆g αβ ) 2 Other surfaces or defects can also serve as nucleation sites. In particular, grain boundaries and especially triple points (Fig. 8.5) provide additional reduction of the Gibbs energy required for phase nucleation. Also, (rαcrit ) het with the additional nucleation sites is lower than that given by Equation (10.20). The practical implication is that heterogeneous nucleation is easier than homogeneous nucleation because the nucleation sites aid in the transformation. For example, salt water boils at a lower temperature than pure water because the salt acts as nucleation sites for formation of the steam phase.5 Another example is the seeding of clouds by throwing solid particles from airplanes into rain clouds, which causes the water vapor to condense and produce rain. 5

Note that this is a kinetic phenomenon; thermodynamically, a higher temperature is required to reach 1 atm vapor pressure in salt water than in pure water.

456 Light Water Reactor Materials

10.4 Phase-Transformation Kinetics In the previous sections, the reaction direction of a solid–solid phase transformation was discussed. The question can now be posed, as to how fast a given transformation occurs, that is, the phase-transformation kinetics, or the rate of the process. This is discussed in this section.

10.4.1 Homogeneous nucleation and growth in a single-component system Consider a single-component, initially β single-β-phase system in the absence of surfaces, crystal defects, or impuri1 ties. When the temperature falls below the phase-transformation temperature, homogeneous nucleation occurs. Once α formed, nuclei of the α phase continue 1 to grow by absorbing atoms from the 1 β matrix, as illustrated in Figure 10.11. FIGURE 10.11: Spheres of α phase It is of interest to know how fast the in an initial unit volume of β phase. beta-to-alpha transformation occurs, that is, how fast the alpha volume fraction increases. The key aspect of the kinetics of phase transformation by homogeneous nucleation and subsequent growth of the nuclei is the rate of increase of the fraction of a unit volume of the initial β phase that has been converted to an α phase. Figure 10.9 shows the two-phase mixture, but the following analysis does not account for attachment of the growing spheres. In this context, Vα(t) is the volume fraction of the α-phase spheres at time t since the start of nucleation. The two restrictions are: • The rate of α nucleation per unit volume of remaining β is condN stant at . The volume fraction of β phase at time t′ is 1 − Vα(t ′), dt dN at which time the rate of nucleation of α is [1 − Vα(t′)] . dt

Phase Transformations in Solids 457 •

Following nucleation, the growth rate drα/dt (i.e., the rate of increase of the radii of the α spheres) is constant at rα so at time t, the radius of a nucleus formed at time t′ is rα × (t − t′).

Consequently, the incremental volume of α at time t due to nucleation in the time period dt′ is the product of the number of nuclei created in dt′ and their volume at time t: 4π  ′ dVα = [rα (t − t ′)]3 × [1 − Vα (t ′)]Ndt (10.29) 3 integrating t′ from 0 to t and Vα(t′) from 0 to Vα(t), π  4 . (10.30) Vα (t ) = 1 − exp  − rα3 Nt  3  This is the Johnson-Mehl-Avrami-Kolmorogov equation. No explicit assumption has been made concerning the growth mechanism, except for the constancy of the nucleation rate and the growth rate of the α phase. Equation (10.30) 1.00

0.80

0.60 Vα

1 nm/s 0.2 nm/s

0.40 0.1 nm/s 0.05 nm/s

0.20

0.00 0

100

200

300

400

500

Time (s)

FIGURE 10.12: Fraction of phase transformed during homogeneous nucleation for various interfacial velocities rα in nm/s. The nucleation rate is 1018 nuclei/cm3.s.

458 Light Water Reactor Materials

predicts a transformation rate that is fast at the beginning but gradually slows down, showing a characteristic S shape in the curves in Figure 10.12. Both the growth rate rα and the nucleation rate N are strongly temperaturedependent. As the temperature decreases: • The nucleation rate increases as the undercooling increases since the thermodynamic driving force is greater, as shown by the double arrows in Figure 10.1. drα • decreases because of the slower atom diffusion rate at lower dt temperatures, resulting in a decreased rate of atom transfer from β to the α spheres at the interface. Other assumptions about the nucleation and growth rates lead to a more general formulation than Equation (10.30): Vα (t ) = 1 − exp (− k t n )

(10.31)

with n a constant depending on the nucleation and growth mechanism and k a factor that depends strongly on the temperature [4].

10.4.2 Interfacial reactions in two-component systems Since the system contains two components, growth of the α phase depends on the sequential processes of atomic migration by diffusion of component B in the β phase: (10.32) B(β) bulk → B(β) int followed by transfer of B atoms to the α phase at the α–β interface: B(β) int → B(α ) int . (10.33) The solute (B) concentration profile in the β phase assumes different shapes depending on which of the above processes is faster. In the case of fast atomic diffusion compared to the interface reaction kinetics, Equation (10.33), the flux of B is sustained without a significant solute-B gradient in the β matrix.

Phase Transformations in Solids 459

If the interfacial reaction is fast, a solute-depleted region appears in the β phase near the advancing interface. This diffusion limit is analyzed in the next section.

10.4.3 Diffusion-controlled growth in a binary solid-state system The question can now be posed as to what the equivalent calculation of a phase transformation kinetics would be in the case of a twocomponent system for a solid–solid transformation. That is, when two different atoms are present, how does the growth of one phase from another occur? We consider here a simple one-dimensional case, as the process is quite complex. Consider the phase diagram of a two-component, A and B system shown in Figure 10.13a. A solid solution of the β phase, initially at point z with mole fraction of B given by x βy , is cooled rapidly to point y in the α + β two-phase region. Because single-phase β at point y is thermodynamically unstable, the second phase nucleates and grows. The new solid

Temperature

Liq α + Liq

β + Liq z

α

β

α+β

y

X eq α

Mole fraction B

X y X eq β β

FIGURE 10.13a: Binary phase diagram showing cooling from z to y and separation into immiscible phases α and β.

460 Light Water Reactor Materials 1

Mole fraction B

X eq β Xy β

α

L

J

β

X eq α

0 0

z

Z

FIGURE 10.13b: Growth of an α particle into a β phase in a container of half-width Z.

has both the structure and composition of the α phase, which is leaner in element B than the β phase. For simplicity, the β → α conversion is considered for the case of a planar interface of alpha growing into beta, rather than the most likely physical case of spherical particles. Initially, the nucleation stage produces a thin layer of the α phase, which thereafter grows by diffusion of component B from the β phase. After some time, Figure 10.13b shows an α phase of half-thickness L in a β matrix that occupies the remainder of the container of dimension Z. x βeq and x αeq are the equilibrium concentrations of component B in the β and α phases, respectively, that prevail at the interface between the two phases. There is a B-composition gradient in the β phase but none in α because it is formed at the equilibrium (phase-diagram) composition. The physical case being considered is the precipitation reaction needed to effect the reaction (10.34) β (x βy ) → β (x βeq ) + α (x αeq ) , that is, the transformation from single-phase beta at composition x βy to alpha + beta at the equilibrium compositions. Because both alpha and beta phases are solids, their molar densities are considered to be approximately the same, and are designated by ρ. This approximation means that there is no volume change as β is converted to α at the interface.

Phase Transformations in Solids 461

The flux of component B (J in moles/cm2.s) in the β phase is described by Fick’s first law: ∂x J = −ρD AB β (10.35) ∂z and Fick’s second law: ∂ xβ ∂2 xβ (10.36) . = D AB ∂t ∂z 2 In these equations, DAB is the mutual diffusion coefficient of the A–B solid solution and t is the time since the original β phase was cooled to the temperature at point y. Movement of the interface is controlled by a balance on component B: dL dL ρx αeq + J ( L) = ρx βeq . (10.37) dt dt Because the α phase is growing, dL/dt is positive, so relative to the moving interface, the first term on the left is the convective flux of component B leaving the interface in the −z direction. The flux J(L) is positive in the +z direction (Fig. 10.13b), so it represents removal of component B from the interface. The term on the right in Equation (10.37) is the input of B to the interface from the matrix, which is effectively moving to the left relative to the interface. Rearranging Equation (10.37) and using Equation (10.35): dL J ( L)/ρ = dt x βeq − x αeq

D AB  ∂ x β   ∂ z  z =L

= eq x β − x αeq

(10.38)

Equations (10.36) and (10.38) are to be solved simultaneously with the initial conditions: x β (z ,0) = x βy L(0) = 0 (10.39)

462 Light Water Reactor Materials

and the boundary conditions: x β ( L , t ) = x βeq

∂ x β    =0 .  ∂ z z = Z

(10.40)

The equations are made dimensionless in the following instances. D AB t z η= τ= Independent variables: (10.41) Z Z2 x β − x βy θ = eq W= L Dependent variables: (10.42) y Z xβ − xβ With the above, Equations (10.36) and (10.38) become ∂θ ∂ 2 θ = ∂τ ∂η 2

dW dτ where

 ∂θ  ∂η η=W

= −g 

x βeq − x βy g = eq eq . xβ − xα

(10.43) (10.44)

(10.45)

The initial conditions become θ( η,0) = 0 W (0) = 0 , and the boundary conditions become

(10.46)

 ∂θ  ∂η = 0 . η=1

(10.47)

θ(W , t ) = 1

Before showing typical results, the equilibrium limit is determined. As time becomes very large, the α/β two-phase system attains equilibrium when the thickness of the α phase reaches L∞. This limit is determined by conservation of component B: The quantity of B at the starting

Phase Transformations in Solids 463

condition represented by point y in Figure 10.13a is Zx βy (the molar density ρ is omitted because it appears in all terms in this balance equation and so cancels out). In the final state, the concentrations of the components in both phases are the equilibrium values, so the quantity of component B is L ∞ x eqα + ( Z − L ∞ ) x βeq . Equating the initial and final quantities of component B and solving for L∞ yields L∞ Z

x βeq − x βy = eq =g x β − x αeq

(10.48)

where g is given by Equation (10.45). The dimensionless equations are solved using an explicit time-step, finite-difference scheme. Figure 10.14 shows the B mole fraction profiles for four dimensionless times and for τ = ∞. In the final equilibrium state, the α half-thickness occupies 36% of the container volume, and the β phase takes up the remaining 64%. Figure 10.15 shows the decreasing rate of approach of the α-phase half-thickness as a function of dimensionless time. The final equilibrium is attained at τ ∼ 0.9.

10.4.4 Transformation-time-temperature (TTT) diagrams A convenient way to illustrate the fraction transformed in terms of the time elapsed at a certain temperature is given by transformation-time-temperature (TTT) diagrams. These are obtained from successive isothermal curves of the fraction transformed after time t. The transformation process is depicted in Figure 10.16, in which the curves showing the degree of 1% transformation and 99% of the transformation affected are shown as a function of temperature. At the transformation temperature TT , the driving force for the transformation is zero. As the temperature decreases further, the driving force for the transformation increases, which decreases the time to transformation (temperature T1).

464 Light Water Reactor Materials t

0.003

0.023

0.013

0.2

∞

1 x eq β x y 0.75

Cβ

β

0.5

x eq α 0.25

0 0

0.25 0.5 0.75 Distance from midplane of a phase, z/Z

1

FIGURE 10.14: Kinetics of phase separation according to Equation (10.34).

0.4 τ=∞

L/Z

0.3

0.2

0.1

0 0

0.25

0.5 t

0.75

1

FIGURE 10.15: Length of alpha phase as a function of dimensionless time.

Phase Transformations in Solids 465 TT

Nucleation limited

T1 T2

Diffusion limited T3 99% 1%

In t

100% T3

T1

Fraction transformed

T2

0%

In t

FIGURE 10.16: Schematic temperature time transformation diagrams for three different temperatures (adapted from [4]).

Further increases cause the time to transformation to be reduced even more (temperature T2). Below T2 the atomic mobility decreases, at which point the nuclei that are present cannot grow fast, and thus, the time to transformation again increases (temperature T3). If the TTT diagram is known, a cooling rate can be approximately designed to achieve a certain degree of transformation. The kinetics of fraction transformed are shown

466 Light Water Reactor Materials

in the lower part of Figure 10.16 and should follow the S-shaped curve described in Equation (10.30). Spinodal decomposition In the phase transformations described above, there is a nucleation barrier, and phase formation occurs by a nucleation and growth process. This mechanism is valid when the second derivative of the Gibbs energy with respect to composition is positive. When the second derivative of the Gibbs energy is negative, phase formation occurs by spinodal decomposition: d2g (10.49) r and ri > r . Integrating and solving for the maxishrinkage when it is smaller than mum radius rmax > 2r , 6 Ω 2 γc f D 3 − r3 = (10.63) rmax t . o k BT dri dt

Lifshitz and Wagner, using a more sophisticated theory, were able to derive the evolution of the mean particle size as [7, 8] Ω 2 γc f D 8 3 3 r (10.64) r − o= t . 9 k BT Equation (10.64) shows that precipitates will coarsen according to cubic kinetics.

Phase Transformations in Solids 471

10.5 Diffusionless Phase Transformations: Martensitic Transformation At the other extreme from diffusion-controlled transformations are diffusionless phase transformations. In contrast with the diffusion-controlled (or replacive) transformations, diffusionless transformations (also called displacive) require no long-range transport of chemical species and thus can occur relatively quickly. Such transformations are also called martensitic transformations after A. Martens, who described them in steels. These transformations are technologically important and include the austenite-to-martensite transformation in steels and the monoclinic-to-tetragonal transformation in ZrO2 layers formed during zirconium alloy corrosion. Consider an Fe 0.4%C alloy held in the single-phase austenite (γ-Fe) field (say, at 1000°C in Fig. 10.5h) that is suddenly quenched (cooled rapidly by immersion in a liquid oil, for example). Although the stable phase equilibrium would be a combination of iron and Fe3C carbide (called cementite), the phase observed upon rapid cooling is called martensite. Martensite is a supersaturated solid solution of carbon in an α-Fe (bccFe or ferrite) matrix, which the carbon interstitials actually distort into a body-centered tetragonal structure. The high rate of cooling involved in quenching does not allow enough time for the atoms to come out of solution, which would permit decomposition into ferrite and cementite (Fe3C). Thus, martensite is a metastable phase and, by definition, it does not appear in the Fe-C equilibrium phase diagram. Because carbon is an austenite-stabilizing element, some austenite is retained upon quenching when the concentration of carbon in solid solution is high. The effect of martensite formation with its high carbon supersaturation is to increase the hardness of the material and to make it more brittle. The degree of hardening depends on the amount of carbon. As a result, tempering heat treatments are normally conducted to allow the carbon to partially come out of solution and form cementite. As this occurs, the material

472 Light Water Reactor Materials

becomes less hard and more ductile because the carbon gradually forms carbides and approaches phase equilibrium. By varying the heat treatment, different degrees of transformation and thus different levels of hardening and ductility can be achieved. The martensitic transformation has been the object of much still-ongoing study. It is beyond the scope of this text to analyze the detailed transformation mechanisms, which have been discussed elsewhere [3, 9]. It is clear, however, that this transformation occurs without change in chemical composition and by small atomic displacements within the unit cell. These atomic motions occur in a coordinated fashion so that the ensemble of atoms moves together in lock step. Because these transformations involve regimented atomic rearrangements, they have also been called military transformations, in contrast to the relatively disordered thermally activated nucleation and growth processes (also called civilian transformations). Finally, because no diffusion occurs, such atomic movements have to take place by athermal processes such as twinning or dislocation motion (see Ch. 11).

10.6 Order-Disorder Transformations Phase transformations in alloys can further be classified by whether they are clustering or ordering, that is, whether it is more favorable to form A–B bonds than A–A and B–B bonds. In many metal alloys, ordering is not important—in other words, the bonding energy between different pairs of atoms differ little from each other. In other compounds, such as

(a)

(b)

(c)

FIGURE 10.18: (a) w < 0, ordering; (b) w ≈ 0, disordering; and (c) w > 0, clustering.

Phase Transformations in Solids 473

the intermetallic compounds seen as precipitates in Zr alloys, the order is so important that they only form in near-stoichiometric compositions to maximize the ability to form unlike bonds. For nuclear materials, when subjected to irradiation, some of the atom pairs may be mixed by the energetic particles, which would cause these compounds to be less stable and either amorphize or dissolve, as discussed in Chapter 24. The tendency to form unlike or like bonds is measured by the ordering energy w: ϕ +ϕ w = ϕ AB − AA BB (10.65) 2 where ϕij is the bond energy of the ij pair. When w > 0, clustering is favored, and when w < 0, ordering is favored. For example in Figure 10.18(a) the lattice is completely ordered as AB bonds are favored, while in Figure 10.18(b) the atoms are indifferent as to which bonds are formed, and in Figure 10.18(c) the A–A and B–B bonds are favored over A–B bonds, so the atoms phase-separate by clustering. An ordered compound has at least two sublattices, each occupied by different chemical species. The degree of order in such materials can be measured by the long-range order parameter S and the short-range order parameter σ. The former is given by S=

Paa − x a 1− xa

(10.66)

where Paa is the probability of finding an atom of type “a” on an “a” sublattice site, and xa is the mole fraction of a in the alloy. In a perfectly ordered compound, the probability of finding an “a” atom in an “a” site is 100%, so Paa = 1 and S = 1. In a perfectly disordered compound, the probability of finding an “a” atom at an “a” site is purely random (i.e., the atoms are evenly mixed in the lattice), and thus, Paa = xa, so that S = 0.

474 Light Water Reactor Materials

The short-range order parameter is given by N ab − N abdis σ = ord N ab − N abdis

(10.67)

where Nab is the number of a-b pairs in the compound, N abdis is the number of a–b pairs in the perfectly disordered compound, and N abord is the number of a–b pairs in the perfectly ordered compound, resulting again in an order parameter of σ = 1 for a perfectly ordered material and σ = 0 for perfectly disordered material. Both parameters describe the state of chemical order of the material, but in general, the long-range order parameter is more easily accessible experimentally. The classical means of detecting long-range order is by the presence of superlattice reflections (see below), which are extinguished when the long-range order parameter goes to zero. Figure 10.19 shows a unit cell of Cu3Au. When the compound is fully ordered, only Cu atoms are found on Cu sites, and only Au atoms are found on gold sites. When the compound is fully disordered, the probability of finding atoms in particular lattice positions is given by the compound’s stoichiometry. In Figure 10.19, the crystal structure changes symmetry from simple cubic with a basis of 4 (3 Cu and 1 Au) to fcc with an average

FIGURE 10.19: The structure of Cu3Au. The (a) ordered structure is a simple cubic arrangement, while the (b) disordered structure constitutes an fcc lattice. The Au atoms (light-colored) occupy a simple-cubic lattice on the corners with the Cu atoms (dark-colored) on each face.

Phase Transformations in Solids 475

atom in each site. The particular atomic arrangement of the fcc lattice causes diffraction from some sets of planes to interfere destructively, so some reflections are not allowed (see Ch. 3). For example, when the (100) planes are at the exact Bragg condition for reinforcement from successive planes, there is a plane in between with the same atom density and that, by virtue of being midway between successive 100 planes, interferes destructively and causes a diffraction extinction [10]. This extinction does not occur in the simple cubic structure, and thus, as the material goes from disordered to ordered, many new (superlattice) reflections appear that can be used to monitor the degree of order. We should note that following the evolution of the parameter S from superlattice reflections is only possible in materials that change symmetry upon disordering as do the Cu3Au (simple cubic to fcc) or CsCl structures (simple cubic to bcc). To detect the changes in short-range order is a more complicated task. Furthermore, it is possible to have almost no long-range order but still a great deal of short-range order as described in Example #1. Example #1: Relationship between S and σ For a two-sublattice ordered crystal, the short-range order parameter corresponding to a long-range order parameter is calculated below. The number of a–b atom pairs in a crystal per sublattice site is given by N ab = Paa Pbb Z + Pba Pab Z (10.68) where Z is the number of nearest neighbors and the Pxy is the probability of finding atom x on sublattice y. From Equation (10.55), these are Paa = x a + S(1 − x a ); Pba = 1 − Paa Pbb = x b + S(1 − x b ); Pab = 1 − Pbb .

(10.69)

476 Light Water Reactor Materials

Substituting Equation (10.69) into Equation (10.68) and rearranging yields (2 x a − 1) 2  2 N ab = 2 Z x a (1 − x a )S + S + x a (1 − x a ) . (10.70)   2 For two sublattices with xa = 0.5, this reduces to 1 1 N ab = 2 Z  S 2 +  . (10.71) 4 4 The extreme cases are when S = 1 and S = 0. According to the formula above, for S = 1 (perfect order), the number of pairs per lattice site is Z, as expected. For S = 0, the number of a–b pairs per lattice site is Z/2, also as expected since in the case of random distribution of atoms in the sublattices in an equiatomic compound, there is an equal chance of finding either a or b atoms, and thus, the number of pairs is reduced by half. Returning to Equation (10.71), we can now write N abord and N abdis. The number of a–b pairs is given below for extreme cases. For S = 1: N abord = Z (10.72) For S = 0:

N abdis = 2 Zx a (1 − x a )

(10.73)

Substituting Equations (10.71), (10.72), and (10.73) into Equation (10.67), the short-range order parameter for a two-sublattice compound is Z[2 x a (1 − x a )S 2 + (2 x a − 1) 2 S + 2 x a (1 − x a )] − 2[ x a (1 − x a )]Z . σ= 1 − 2[ x a (1 − x a )]Z (10.74) Equation (10.74) gives the lower bound of the degree of short-range order that would be expected from a given degree of long-range order. This shows that order can be characterized by two different but related parameters. Figure 10.20 shows the relationship between short-range and long-range order in this model.

Phase Transformations in Solids 477

Short-range order parameter (s)

1

0.8

0.6

0.4

0.2

0 0

0.2 0.4 0.6 0.8 Long-range order parameter (S)

1

FIGURE 10.20: Relationship between the long-range and short-range order parameters.

10.7 Amorphization Although long-range order is absent in a liquid, there is some degree of short-range order. Normally, a liquid is stable at temperatures where the entropic contribution to the Gibbs energy increases the stability of structures with high degrees of disorder. Although the Gibbs energy of an ordered crystalline solid is intrinsically lower than that of an amorphous solid (no topological order) or of a chemically disordered solid (no chemical long-range order), metastable disordered or amorphous solids exist when the stable crystalline configuration is kinetically inaccessible. One common example is the formation of an amorphous solid upon very fast cooling from the melt. If the cooling rate is fast enough, short-range order may be established during cooling, with no topological order, which creates an amorphous solid. This is why amorphous solids are often referred to by the contradictory term “frozen liquid.”

478 Light Water Reactor Materials

Quantitatively, the degree of order of crystalline solids can be described by the radial distribution function, g(r). This function describes the probability of finding another atom at a given distance from a particular atom. Figure 10.21 shows an example of g(r) for a crystal and for a liquid. In the latter, as in an amorphous solid, although a well-defined relationship to their first nearest neighbors exists (as shown by the large first peak), this correlation decreases with distance such that there is only a wide peak in the second nearest-neighbor distance, decreasing to a random probability thereafter. In the crystal, although the peak heights decrease with distance because of the above factors, the correlation distance (distance over which atom positions are correlated) is much greater than in the amorphous solid. In an ideal crystal at 0 K, the probability of finding an atom at one of the lattice positions is unity. In a real crystal, lattice imperfections, such as point defects, dislocations, and grain boundaries, destroy the correlation that exists between lattice positions in a perfect crystal. As a 4.0 3.0 g(r) 2.0 1.0

0

1

2

3

r

4

5

6

1.5 g(r) 1.0 0.5

0

1

2

3

4

r

FIGURE 10.21: The X-ray diffraction radial distribution functions g(r) plotted versus r for a solid (top) and a liquid (bottom).

Phase Transformations in Solids 479

result, the interatomic distances vary, and these variations increase with distance such that the interatomic correlations decrease with distance. Besides fast quenching, a metastable amorphous phase can form during other nonequilibrium processes. Irradiation can also make crystalline phases amorphous and cause chemically ordered phases to become disordered, as discussed in Chapter 24. Amorphization comes as a response of the materials to reduce long-range order while maintaining short-range order. Because of the volume expansion and loss of stability associated with the transformation, amorphization should be avoided, normally by operating the material at a temperature that is high enough to dynamically anneal the material during irradiation.

References  1. P. J. Spencer, “A Brief History of CALPHAD,” Calphad—Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008): 1–8.  2. M. Z. Volmer, Elektrochem 35 (1929): 55.  3. J. W. Christian, The Theory of Transformations in Metals and Alloys, Part I: Equilibrium and General Kinetic Theory, 2nd ed. (Oxford: Pergamon Press, 1975).  4. D. A. Porter and K. E. Easterling, Phase Transformations in Metal and Alloys, 2nd ed. (London: Chapman and Hall, 1993).  5. J. Verhoeven, Fundamentals of Physical Metallurgy (Hoboken, New Jersey: Wiley, 1975).  6. G. W. Greenwood, “Particle Coarsening,” in The Mechanism of Phase Transformations in Crystalline Solids, Monograph 33 (London: Institute of Metals, 1969): 103–110.  7. J. M. Lifshitz and V. V. Slyozov, “The kinetics of precipitation from supersaturated solid solutions,” Journal of Physical Chemistry of Solids 19 (1961): 35.  8. C. Wagner, “Theorie de Alterung von Niederschlägen durch Umlösen,” Zeitshrift fur Elektrochem 65 (1961): 581.

480 Light Water Reactor Materials

 9. H. K. D. H. Bhadeshia, “Martensitic Transformation: Crystallography and Nucleation,” in Encyclopedia of Materials Science: Science and Technology, ed. K. Buschow, R.W. Cahn, M. C. Flemings, B. Iischner, E. J. Kramer, and S. Mahajan (Elsevier Science, 2001): 5203–5206. 10. B. D. Cullity, Elements of X-ray Diffraction (Reading, Massachusetts: Addison-Wesley, 1978).

Problems 10.1 Describe qualitatively what would occur to a melt of Zr-10%Cr as it is slowly cooled from 1750°C down to 700°C, in quasi-equilibrium. Describe what phases would appear, their compositions, and how the volume fractions and compositions of these phases would change as the temperature is steadily decreased. Sketch with illustrations the important points in the process, paying special attention to the regions where phase transformations take place. 10.2 Using the Zr-Cr phase diagram shown in the Figure 10.5e, answer the questions: (a) For a solid containing 22 atom %Cr, and at equilibrium at T = 1330°C, identify the phases present, their relative amounts (how much of each phase is present), and their composition (%Cr atom for each). (b) Find the maximum solubility of Cr (atom %Cr) in the beta phase of Zr and the temperature at which it occurs. At 1100°C, what occurs when the solubility limit of Cr in the beta phase is reached? (c) Construct Gibbs energy curves for T = 400°C, 1100°C, and 1650°C that are consistent with the Zr-Cr phase diagram given in the notes. Indicate the phases associated with each curve in each case. (d) For a 60 at %Cr, 40 atom %Zr mixture at 1100°C at equilibrium, what phases are present, in what composition, and in which proportions?

Phase Transformations in Solids 481

(e) Calculate the maximum possible equilibrium vacancy concentration in pure beta Zr, if the vacancy formation energy is 1.6 eV. 10.3 For the 2D figure below, where the different color circles indicate different types of atoms, (a) calculate the long-range S and short-range order parameter σ. Use Z = 4 and consider only the atoms that have all four nearest neighbors.

(b) how many atoms would have to be moved for perfect order (both S and σ) to be established? 10.4 During cooling from high temperature, a second phase “beta” precipitates from a solid solution in the “alpha” phase. (a) If the precipitate is spherical radius r, the surface energy of the alpha– beta interface is 1 J/m2, and the change in Gibbs energy from beta to alpha is 23 kJ/mol, calculate the critical radius at the transformation temperature, if the material density is 7 g/cm3 (0.7 kg/m3). What would likely occur to the critical radius if the temperature were further decreased below the precipitation temperature? Why? (b) Assume now that the precipitate is plate-like with a thickness t and that the surface energy of the plate surface is xΩ, where x is a number between 0 and 1. Estimate the plate dimensions as a function of x. Assume that the two in-plane directions are equivalent and that elastic forces play no role in the transformation.

11

Chapter Mechanical Behavior of Materials 11.1 Introduction In service, materials are subjected to mechanical loads of various intensities, types, and duration. The response of the material to these applied loads is termed mechanical behavior, and it is one of the most important factors in materials design. The most important questions to be answered are: How and when does the material undergo permanent deformation under load? When does the material fracture or otherwise fail? These apparently simple questions are addressed in the field of mechanical behavior of materials. The answers depend on whether the load is applied quickly or over a period of time, whether the material is pulled in one or in more than one direction at once (uniaxial or multiaxial loading), whether the load is cyclic, and whether there are preexisting flaws on the material. The ability of the material to resist deformation and failure under such conditions is measured by various mechanical properties, such as strength, ductility, and toughness. These macroscopic material properties and their link to the underlying material microstructures are discussed in this chapter.

484 Light Water Reactor Materials

11.2 Mechanical Testing Quantitative knowledge of material behavior comes from various mechanical tests from which material properties can be determined. To instruct the discussion, some common mechanical tests and the mechanical properties they determine are described in the following. (a)

Io P

P Ao

(b)

I P

P A

(c) P

P

FIGURE 11.1: Schematic of specimen deformation during a uniaxial tensile test.

11.2.1 Uniaxial tensile test A specimen of uniform cross section (usually cylindrical) is uniaxially loaded and its deformation measured as a function of the applied load (Fig. 11.1). A specimen of initial gauge length lo and cross-sectional area Ao is subjected to an increasing tensile load P, applied along its axis, and the resulting specimen deformation is measured. Upon application of the load, the specimen gauge length increases to l and the cross-sectional area is reduced to A. The engineering stress is defined as P (11.1) σ eng = , Ao but since during deformation the specimen cross-sectional area is reduced

Mechanical Behavior of Materials 485

from Ao to A, the true stress is σ true =

P . A

(11.2)

Similarly, the engineering strain e is l −l e= o , (11.3) lo while the true strain ε is the integral of the increments of strain along the specimen length: l dl  l . ε = ∫ = ln (11.4)   l l o l o

The true strain in tension is always somewhat larger than the engineering strain. For plastic deformation, volume is conserved: Al = Aol o .

(11.5)

From Equations (11.5) and (11.4), A

ε=

 − dA  = ln  Ao  . ∫ A   A A

(11.6)

o

The strain in the loading direction implies a strain of εt = υε in the two transverse directions, where υ is Poisson’s ratio.1 If the material is isotropic, volume conservation implies 2εt + ε = 0, which yields εt = −0.5ε. If the applied stress is plotted against the specimen strain, we obtain the stress-strain curve, a schematic example of which is shown in Figure 11.2. From zero up to the yield stress, the material deforms elastically 1

Poisson’s ratio is the ratio of the strain normal to the load to the strain along the load direction.

486 Light Water Reactor Materials True-stress/true-strain

σ

Work-hardening Engineering-stress/engineering-strain

σUTS Flow stress

σy Hooke’s law Elastic energy

εU

εf

ε

FIGURE 11.2: Schematic stress-strain behavior typical of an austenitic steel.

(i.e., reversibly) such that the strain is linearly proportional to the stress according to Hooke’s law: σ = Eε (11.7) where E is Young’s modulus or the elastic modulus. The region of validity of Hooke’s law is the elastic region. This region is characterized by small reversible deformation; that is, once the load is removed, the strain disappears. In the elastic region, the engineering stress/engineering strain curve is coincident with the true stress/true strain curve. Typical values of Young’s modulus and Poisson’s ratio are shown in Table 11.1. Note that both Young’s modulus and the yield stress decrease with temperature, as the material softens. Above the yield stress, material deformation starts to deviate from Hooke’s law as a result of plastic, or irreversible, deformation. In order to establish a definite onset of plastic deformation, a minimum deviation from Hooke’s law (normally 0.2%) is taken to be the point at which plastic deformation begins. The stress at which this occurs is called the

Mechanical Behavior of Materials 487

Material Aluminum Gold Iron Nickel Tungsten HT-9 Stainless Steel Alloy 690 T91 Zirconium Zirconium

E (GPa) 70 78 211 199 411 160 190–205 211 206 99 125

υ (Poisson’s ratio)

0.35 0.44 0.29 0.31 0.28 0.33 0.26–0.29 0.289 0.33 0.37–0.40

TABLE 11.1: Elastic moduli and Poisson’s ratios of different materials at room temperature.

yield stress, σy, beyond which the material deforms plastically, and is an important property of engineered materials. Note that some alloys, notably bcc materials, exhibit more complex yield behavior wherein the stress reaches an upper yield point before dropping down to a lower one [1]. The stress required for further deformation beyond the yield point may continue to increase because of work hardening (also called strain hardening), often described by an equation of the form σ = K ε np

(11.8)

where σ is the stress above σy, εp is the plastic strain, n is the workhardening exponent, and K is a constant, called the strength coefficient. Once yielding occurs, the stress required to continue plastic deformation is called the flow stress. At the yield point, the flow stress is equal to the

488 Light Water Reactor Materials

yield stress, but it is higher than the yield stress in the work-hardening region (Fig. 11.2). During tensile deformation, the material deformation occurring in the elastic region and in the beginning of the work-hardening region is uniform; that is, all the material within the gauge section participates equally in the deformation process. At a strain corresponding to the ultimate tensile strength (σUTS), diffuse necking sets in, causing deformation to become nonuniform. A small variation in the cross-sectional area in the gauge section of the material can cause a slightly larger stress at that location, which causes larger deformation, in turn further diminishing the area. As a consequence, once necking starts, the specimen loses load-carrying capacity, since the cross-sectional area (A) in the necked region becomes progressively smaller and, thus, no additional load (P) is needed to cause further deformation. The stress at which necking occurs is called the ultimate tensile strength of the material, or σUTS. This is the maximum loadbearing capacity of the material. The strain at which necking occurs is the uniform strain, εU  and is one measure of the ductility of the material (i.e., how much the material can deform before failing). As deformation proceeds, the neck becomes progressively thinner until the material fractures at strain εf . The percent elongation to failure (%EL) represents another measure of material ductility. Another quantity that can be obtained from Figure 11.2 is the toughness of the material or the energy absorbed during fracture. This is the area under the curve minus the shaded region on the right, which represents the elastic energy stored in the material and which is recovered upon fracture. The more energy the material can absorb before fracturing, the higher its toughness.

11.2.2 Biaxial testing: tube burst The uniaxial tensile test described above is experimentally convenient, and its theoretical interpretation is straightforward because only the normal stress σxx is active (stress in the direction of the rod axis). However, the stress in

Mechanical Behavior of Materials 489

nuclear fuel cladding and other core materials r is often biaxial or triaxial. For example, the stress-state of cladding subject to fission-gas θ z pressure closely resembles that in a long, thinwalled cylindrical tube closed at both ends and pressurized by a gas. This state of stress is biaxial, with components in the axial and hoop (aziFIGURE 11.3: Geometry muthal) directions. for a tube burst test. The response of tubing to biaxial stress is studied by a tube-burst test (illustrated in Fig.11.3), for which the following equations apply: pr (11.9) σθ = δ

pr 2δ σr ≅ 0 .

σz =

(11.10) (11.11)

In Equations [11.9] to [11.11], p is the internal pressure, r is the tube mean radius, and δ is the tube-wall thickness. For the tube to be considered thinwalled, the ratio of wall thickness to the radius should be smaller than five.

11.2.3 The von Mises equivalent stress To analyze plastic deformation and the yield condition in multiaxial stress configurations, the concept of an equivalent stress is employed. By definition, a uniaxially loaded specimen yields when the stress reaches σy. Under multiaxial loading, a yield criterion was developed by von Mises [1]. This criterion is derived here for the case of the tube-burst test based on an equivalent stress. Von Mises’ general yield criterion is based on the difference between the total energy density under a multiaxial state of stress ( Eel ) and the energy density resulting in the material when subjected to the mean of

490 Light Water Reactor Materials

the three principal stresses (Eel ). When this difference reaches a critical value Eel∗ , (11.12) Eel − Eel > Eel∗ the material will undergo plastic deformation. The mean of the three principal stresses is given by 1 σ m = (σ 1 + σ 2 + σ 3 ) . (11.13) 3 The elastic strain energy density Eel as a result of applied stresses is given by Equation (6.31): 1 υ 1 Eel = (σ12 + σ 22 + σ 32 ) − (σ1σ 2 + σ1σ 3 + σ 2 σ 3 ) + (σ122 + σ132 + σ 223 ) 2E E 2G (11.14) which, for a solid under hydrostatic stress (no shear stresses), simplifies to 1 υ Eel = (σ12 + σ 22 + σ 32 ) − (σ1σ 2 + σ1σ 3 + σ 2 σ 3 ) . (11.15) 2E E Substituting σ for σ1, σ2, and σ3 in Equation (11.15) yields the value of the homogeneous (average) elastic energy density Eel =

1 − 2υ (σ + σ + σ ) 2 . 6E 1 2 3

(11.16)

Subtracting Equation (11.16) from Equation (11.15), we obtain 1+ υ  2 2 2 Eel − Eel =   [(σ 1 − σ 2 ) + (σ 2 − σ 3 ) + (σ 1 − σ 3 ) ] .  6E 

(11.17)

By setting Equation (11.17) equal to the elastic energy density when there is only one applied stress, the critical equivalent stress σeq can be obtained. The elastic energy density difference for an uniaxial stress σeq that

Mechanical Behavior of Materials 491

is “equivalent” to the three-component hydrostatic stress is determined by setting σ1 = σeq, σ2 = σ3 = 0 in Equation (11.15):  1+ υ 2  1+ υ σ eq =  2 [(σ1 − σ 2 )2 + (σ 2 − σ 3 )2 + (σ1 − σ 3 )2 ] , (11.18)    6E   6E  from which we obtain the von Mises equivalent stress 1 σ eq = [(σ1 − σ 2 )2 + (σ 2 − σ 3 )2 + (σ1 − σ 3 )2 ]1/2 . 2

(11.19)

Equation (11.19) is an important and widely used relation that predicts the onset of plastic deformation (yielding) in an engineering component subjected to multiaxial stresses. The equivalent stress for the uniaxial tension case can be computed by noting that σ1 = σ x ; σ 2 = σ y = 0; and σ 3 = σ z = 0, which yields σ eq = σ x . Note that at the yield stress σ eq = σ y , again showing that Equation (11.19) can predict yielding in multiaxial stress states if the yield stress is known. For the tube-burst case, substituting Equations (11.9), (11.10), and (11.11) into Equation (11.19) yields 3 (11.20) σ eq = σ . 2 θ This equation permits plastic properties determined by the uniaxial tensile test (Sec. 11.2.1) to be applied to the response of a pressurized tube. For example, if the yield stress measured in the uniaxial test is σy, a pressurized tube should yield at a hoop stress of 2 δ 2 σ . (11.21) or a pressure of p yield = (σ θ ) yield = σ y 3R y 3 Since the equivalent stress derived above defines yielding in a multiaxial stress state, Equation (11.19) defines a yield locus, which is a surface in three-dimensional space defining the combinations of principal stresses

492 Light Water Reactor Materials

which cause the material to yield. The yield locus for two-dimensional plane stress (σ3 = 0) is analyzed in Example #1. Example #1: Yield locus for plane stress To determine the locus of points at which a specimen yields, Equation (11.19) for plane stress with σeq = σy is written as 1

(11.22)

2 = [( x − y ) 2 + x 2 + y 2 ] 2 where

x=

σ1 σy

y=

σ2 σy

.

(11.23)

Solving Equation (11.22) for y, 1 (11.24) y = [x ± 4 − 3 x 2 ] . 2 Equation (11.24) is plotted in Figure 11.4. For any and all combinations of the two stress components σ1 and σ2, which fall inside the tilted ellipse, the material behaves elastically and does not yield (as described in Ch. 6). All conditions outside the oval cause the material to yield. The yield locus shown on the left in Figure 11.4 is symmetric with respect to the deformation direction; i.e., it is the same in tension or compression as well as along the two orthogonal in-plane directions (e.g., x–y, θ–z). This has an important implication that is valid for most metals: the yield stress in tension is the same as in compression as long as there is no change in the deformation mechanism. Example #2: Application of plane stress yield locus to a pressurized tube Let σ1 = σθ and σ2 = σz; for the pressurized tube, Equations (11.9) and (11.10) are σ1 = 2σ2, or, in terms of Equation (11.23), y = 1/2x. When this relation is substituted into Equation (11.22), the solution is x2 = 4/3 or x = 2/√3, which is the same as Equation (11.21).

Mechanical Behavior of Materials 493 σ1

1.5 1.0

σy (1, 2)

σeq > σy A

0.5 s2/sy

Slip

–σy (1, 2) –σ1

σeq < σy

0.0

σy (1, 2) σ1

–0.5 Twinning σeq > σy

–1.0 –1.5 –1.5

–σy (1, 2) –1.0

–0.5

0.0 s1/sy

0.5

1.0

1.5 –σ2

FIGURE 11.4: (Left) Yield locus for plane stress. (Right) Yield loci for slip and twinning in a hexagonal close-packed sheet. (Redrawn from Backofen, Deformation Processing, 1st Ed. Pearson Education, Inc., New York, New York.)

To locate this point on Figure 11.4, x = 2/√3 = 1.155 and y = 1/√3 = 0.577. This is point A in Figure 11.4. The yield locus shown on the left in Figure 11.4 is symmetric with respect to the deformation direction; in other words, it is the same in tension or compression as well as along the two orthogonal in-plane directions. However, not all materials deform isotropically; zirconium, for example, has different yield stresses along different deformation directions. Figure 11.4 (right) shows the yield locus for textured hexagonal close-packed (hcp) sheet material. The locus is more elongated than that in Figure 11.4 (left), which means it does not obey Equation (11.22). The yield locus depends on the mechanisms of deformation, which could reduce the yield locus (i.e., achieve plastic deformation at a lower stress than the von Mises criterion). For example, Figure 11.4 (right) also shows a line that, under compressive stress, the alternative plastic deformation mode of twinning is favored over slip because it occurs at lower strains.

11.2.4 Hardness testing Hardness testing is a comparatively easy way to obtain information about the deformation behavior of a material. In this test, an indenter—typically

494 Light Water Reactor Materials

in the form of a ball or a pyramid—is applied with a certain force to a material. The indenter is much harder than the material tested, so that, to a good approximation, only the latter deforms plastically. For a given force, the indenter penetrates a certain depth into the material, which is measured microscopically by the area of indentation. As the indenter penetrates, the material flows plastically under it. The stress required for such deformation to occur is the flow stress (see text after Eq. [11.8]). The elastic region near the indent constrains the deformation so that a much higher stress than the yield stress is needed to cause plastic flow. For a spherical indenter, the relationship from which σy can be estimated is 4P p = 2 ≅ aσ y (11.25) πd where a = 2.5 to 3.0 depending on the material, p is the pressure on the indenter needed for penetration, P is the applied load, and d is the diameter of the indentation [3]. There are several types of hardness tests, depending on the indenter shape and measurement of indent area or depth. Typical examples include Brinell, Meyer, Vickers, and Rockwell tests. It is also possible to perform hardness testing on a microscale by using an indenter that is smaller than the dimension of one grain.

11.2.5 Impact testing Impact tests can be used to measure the fracture resistance of the material. Fracture energy is the area under the stress-strain curve up to fracture (Fig. 11.2). It is a measure of the ability of the material to deform plastically before fracture in the absence of a pre-existing crack. However, for failure analysis of nuclear components, it is often required to assume the presence of a crack, and show that it does not propagate. The resistance of the material to crack propagation is given by the fracture toughness of the material, which is rigorously measured using specially

Mechanical Behavior of Materials 495

designed specimens into which a fatigue precrack is induced. However, the impact test energy can be empirically related to the fracture toughness, as exemplified below.

h2

h1

FIGURE 11.5: Schematic geometry of a Charpy v-notch impact test. Note that in an actual Charpy test, the specimen would be positioned horizontally (coming out of the page in the figure); it is shown vertically for clarity.

The Charpy impact test is commonly used in the nuclear industry, for example, for assessing the degree of embrittlement suffered by the pressure vessel after exposure to neutron irradiation. In the Charpy test, a large hammer swung from a pendulum is released from an initial height h1 toward a specimen into which a v-notch groove has been machined (Fig. 11.5). The high speed of the hammer and the presence of the notch cause the material to be deformed at a high strain rate and in a triaxial state of stress, both of which favor fracture. The kinetic energy of the hammer is sufficient to break the sample. Subsequent to specimen fracture, the hammer continues along its arc, rising to a height h2 < h1. The energy absorbed by the specimen during fracture is given by E abs = mg (h1 − h2 ) (11.26)

496 Light Water Reactor Materials Eabs (J)

Upper-shelf energy Brittle

CVN = 41 J

Ductile

Lower-shelf energy

(DBTT)

Test temperature

FIGURE 11.6: Schematic representation of energy absorbed during a Charpy test as a function of testing temperature. The energy level of 41 J corresponds to the DBTT of steel.

where m is the mass of the hammer and g is the gravitational constant. A brittle material (such as glass) absorbs little energy as it breaks, while a ductile sample absorbs more energy before fracturing. Every material is brittle at a low enough temperature, so it is of interest to determine at which temperature the transition to brittle behavior occurs. To determine this, the test is repeated at several temperatures, and Eabs is plotted versus temperature. For some materials, in particular ferritic steels, the energy absorbed shows an abrupt change at a temperature called the ductile-tobrittle transition temperature (DBTT). Figure 11.6 shows schematically the energy absorbed during a Charpy impact test as a function of test temperature. For materials showing the ductile-to-brittle transition, the absorbed energy is high at an elevated temperature, undergoes a decrease within a relatively narrow interval as the temperature is decreased, then remains low thereafter. The energies absorbed at high and low temperatures are called the upper-shelf energy and lower-shelf energy, respectively. The midpoint of the transition from the upper-shelf to the lower-shelf energy (the Charpy v-notch energy [CVN]) occurs at the DBTT. For pressure-vessel steels, the absorbed energy at the transition is 41 J.

Mechanical Behavior of Materials 497

The analyses of the fracture surfaces from testing at low and high temperatures reveal differences in fracture morphology. At low temperature, the fracture surface is flat and shiny, indicating brittle failure by cleavage, while at high temperature, a fraction of the cross-sectional area (the fracture surface) is dull, indicating partial ductile fracture. The degree of ductile failure can be quantitatively evaluated by the fraction of cross-sectional area that shows a ductile appearance. For low-to-moderate strength steels, the Charpy test is a good indicator of the material’s fracture toughness. In fact, the fracture toughness often exhibits a marked decrease in a similar temperature interval as the absorbed energy [3]. Empirical relationships can be obtained between the Charpy v-notch energy and the fracture toughness (KIC, Sec. 11.5.3). Some of these relationships are in Table 4.2 of reference [4].

11.3 Microstructural Aspects of Deformation Having reviewed some of the tests used to determine mechanical behavior, we now turn to the underlying microstructural processes responsible for such behavior.

11.3.1 Critical resolved shear stress and the yield stress Plastic deformation in real crystals is principally achieved by the movement of dislocations, which requires the activation of multiple slip systems (combination of dislocations and slip planes). As discussed in Section 7.3, this occurs when the resolved shear stress exceeds the critical value for slip. For a single crystal, the onset of slip coincides with the yield stress. However, real materials are polycrystalline and contain grains of different orientations with different susceptibilities for slip. As a result, a uniaxial applied stress deforms some grains more than others during the deformation process. Because the deformation of a given crystallite (grain) is constrained by neighboring grains, grain-to-grain incompatibilities arise that need to be accommodated by further plastic deformation.

498 Light Water Reactor Materials

A simple two-dimensional illustration of this concept is shown in Figure 11.7. Consider two neighboring grains (a), the crystal orientations of which are oriented differently with respect to the loading axis (arrows) so that they shear in different directions. In such a thought experiment, if the grains were free, the slip system in the left grain deforms the grain to the left, while a different orientation of the same slip system in the grain on the right causes deformation in a different direction (b). In order to maintain the grain boundary intact, which is always the case in ductile materials, another slip system in the other direction is needed (c), so that the deformation in both grains is nil, thus keeping the grains together. This illustrates that several slip systems are needed to accomplish deformation in a polycrystalline solid. In the case of a three-dimensional solid, the above process is repeated in three orthogonal directions, which means that six independent slip

(a)

(b) Slip plane

Slip directions

(c)

FIGURE 11.7: Schematic of the role of various slip systems in ensuring deformation compatibility: (a) initial; (b) grain separation in the absence of grain boundary adhesion; and (c) excitation of compensating slip systems to prevent grain boundary separation.

Mechanical Behavior of Materials 499

systems are needed for grain-to-grain compatibility during plastic deformation. However, the requirement of constant volume means that one of the slip systems is not independent, so only five independent slip systems are needed to permit deformation in a polycrystalline solid. For a given direction of the load P = Aσ, the resolved shear stress σRSS acting in a particular slip direction inclined by an angle λ with respect to the loading axis and contained in a plane with a normal inclined by φ from the loading axis is (see Eq. [7.1]) P (11.27) cos λ cos φ = σ RSS , A σ σ RSS =

m

(11.28)

where the factor 1/m = cos λ cos φ is the Schmid factor for the slip system, and m is a constant. When σRSS is large enough to activate the slip system, it is equal to the critical resolved shear stress σCRSS for that slip system (Sec. 7.3.1). In a single crystal, the equation σy = mσ CRSS

(11.29)

describes the yield condition. Note that while σy is the axial stress at yielding, σCRSS is the critical shear stress acting on the slip plane and in the slip direction (which by necessity lies on the slip plane). In a polycrystal, each of the five independent slip systems needs to be operational to account for deformation incompatibilities (i.e., the angles of slip planes between adjacent grains, as in Fig. 11.7). For a given applied stress σ, each slip system has a different Schmid factor and consequently a different critical resolved shear stress. For all five independent slip systems to be operative, the product of the applied normal stress σ and the Schmid factor needs to be higher than the critical resolved shear stress. Only under this condition can plastic deformation begin; as a consequence this condition defines the yield stress, σy.

500 Light Water Reactor Materials

Because several grains are involved in ensuring deformation compatibility, it is not possible to directly relate Schmid factors and yield stress. The Schmid factor that activates a particular slip system depends on many factors, including the availability of slip systems, the ease of switching from one slip system to another, the texture of the material, and the availability of other deformation mechanisms such as twinning (see Ch. 172). A Schmid factor calculated by averaging the individual Schmid factors of the slip systems has been found to approximately describe yield in polycrystalline materials, but the most accurate value of the polycrystalline yield stress comes from the rigorous derivation of the factor m in Equation (11.29), which is called the Taylor factor, arrived at by selecting the combination of slip system deformation that causes the least amount of shear strain (see [3]). Detailed calculations show that in face-centered cubic (fcc) materials, the Taylor factor is 3.0 to 3.1, and it is a little lower (2.75) for the body centered cubic (bcc) structure [5].

11.3.2 Strengthening mechanisms Several mechanisms exist for increasing the yield strength of a given material, all of which involve changing the microstructure so as to hinder dislocation motion by creating obstacles that dislocations must bend around or cut through, or altering the dislocation to a shape that impedes its motion. As shown in Equation (7.36), the critical stress to induce motion of dislocations through a field of hard obstacles by a looping process is the Orowan Stress 2Gb σc = α (11.30) l where G is the shear modulus, b is the modulus of the Burgers vector, and l is the mean spacing between the obstacles. Depending on the degree of uniformity of the obstacle array, 0.8 < α < 1.0. For a uniformly distributed array of obstacles, α = 1 [6]. 2

Chapters 16 through 29 are to be found in Light Water Reactor Materials, Volume II: Applications.

Mechanical Behavior of Materials 501

The various hardening mechanisms refer to the different ways in which obstacles can be introduced into the material, as listed below. Increase in dislocation density Once yielding starts, dislocation tangling generates mobile dislocations by processes analogous to the Frank-Read sources (Sec. 7.11.2). A large fraction of these dislocations become entangled and sessile (immovable), thereby creating a long-range internal stress field that hinders the motion of the mobile dislocations, causing the stress required for further deformation to increase according to Equation (7.36). This is one of the principal causes of the work-hardening behavior discussed in Section 11.2.1. Like work-hardening (Fig. 11.2), cold-working increases the density of tangled and sessile dislocations. Grain boundary strengthening A similar hindrance to the motion of mobile dislocations is caused by grain refinement. As the grain size becomes smaller, the distance between grain boundaries decreases, and since these also impede dislocation motion, the stress required for deformation correspondingly increases. Solid solution hardening This is due to the addition of foreign atoms in solid solution in the material matrix. In solid-solution hardening, dislocations are pinned to impurities in the crystal. If the impurity atom size is substantially different from that of the matrix atoms, the strain-energy interaction between the solute atoms and the dislocation “anchors” the latter. Precipitation hardening In precipitation hardening, the insoluble impurity elements precipitate, and these precipitate particles become obstacles to dislocation glide. Dislocations either cut through small precipitate particles or bend around large particles (see Fig. 7.20), and hardening is approximated by Equation (11.30).

502 Light Water Reactor Materials

Thus, a high density of small particles with small spacing results in high strength. In either case, the critical resolved shear stress is increased. A review of strengthening mechanisms is available in [5] and [6].

11.4 Creep Deformation The tests described in Section 11.2 determine the instantaneous material response, that is, the response obtained when the structure of the material does not appreciably change during the test, as is the case at temperatures less than one-third of the melting temperature. At higher temperature, diffusion of atoms results in time-dependent deformation that occurs if the load is applied over a long period of time. One simple example is the deformation of a spring. If the spring is stretched in the elastic regime (σ < σy) and quickly unloaded, it returns to its original shape. However, if the load is applied over a long period, the full strain may not be recovered upon unloading; i.e., the spring has plastically deformed. This slow (timedependent) plastic deformation at a stress lower than the yield stress is called creep. Whether creep resistance or mechanical strength limits performance depends on the temperature/stress combination the material is subjected to during service [7]. At a fixed temperature, when the applied stress is smaller than that required for yielding, slow-strain-rate deformation can occur by creep. Similarly, for a fixed stress, yielding occurs at a high temperature, while creep is restricted to lower temperatures.

11.4.1 Phenomenological description of creep Creep under a constant load can be divided into the three stages shown in Figure 11.8. Stage I: The initially high strain rate decreases with deformation (or time). This stage is akin to strain hardening (Fig. 11.2), in which creep

Mechanical Behavior of Materials 503 ε

× I

II

III

t

FIGURE 11.8: Schematic description of creep strain versus time at fixed temperature. The X indicates fracture.

deformation changes the material structure and the strain rate (dε/dt ) decreases with time and strain. Stage II: When thermal deformation recovery and deformation hardening match, secondary creep, which is characterized by a constant strain rate, begins. This stage may last the greater part of the material lifetime. Stage III: This is tertiary creep; the strain rate increases and rapidly leads to failure (X in Fig. 11.8).

11.4.2 Creep lifetime Creep life is usually dominated by stage II, for which the steady-state strain rate is described empirically by the Dorn equation, which is a power law equation of the form  Q  k BT 

ε II = Aσ n exp −

(11.31)

where A and n are material constants, σ is the applied stress, and Q is the activation energy for creep. For small grain sizes, A may be a function of

504 Light Water Reactor Materials

grain size, especially at high temperatures, where grain boundary sliding becomes more prevalent. Often, stage II dominates the time to failure, and in this case the strain at fracture is given by (11.32) ε f = ε II t f where tf is the creep life of the material.

11.4.3 Larson-Miller plots Larson-Miller plots are a convenient method of predicting and comparing the creep strengths of engineering alloys. Taking the logarithm of both sides in Equation (11.31), we obtain Q + ln( Aσ n ) . ln ε II = − (11.33) k BT If stage II dominates, then inserting Equation (11.32) into the above yields  εf  Q ln( Aσ n ) − ln   = .  tf  k BT

(11.34)

From Equation (11.34), we define the Larson-Miller parameter LM by T [ln t f + C ] =

Q = LM . kB

(11.35)

Here, Q/kB = LM (units of Kelvins) is a material constant, and the terms on the left side of Equation (11.35) are the parameters of the creep test (temperature and time-to-failure). The LM parameter is a good indicator of a material’s creep resistance. Figure 11.9 shows LM plots for various steels [8, 9]. In this case, Equation (11.35) is slightly modified to T (ln t + 30) × 10 −3 = LM .

(11.36)

Note that Equation (11.36) is only approximately satisfied (a straight line would be expected to result if it did).

Mechanical Behavior of Materials 505 500 400

Stress (MPa)

300

200

F-82 F-82H HT9 9 Cr-2WVT a (HT1) 9 Cr-2WVT a (HT2)

100 90 80 70 60 50 24

26 28 30 Larsen-miller parameter P=T(log t + 30) ¥ 10–3, (K, h)

FIGURE 11.9: Larson-Miller plots for various ferritic-martensitic steels considered for advanced reactors. (Redrawn from Journal of Nuclear Materials, 371/1-3, R. L. Klueh, A. T. Nelson, “Ferritic/martensitic steels for next-generation reactors,” with permission from Elsevier.)

Example #3: Use the LM plot to determine the maximum allowable stress for HT-9 when operated at 700 K, so that failure does not occur before 24 × 365 hours (~1 year). The LM parameter under these conditions is LM = 700 (ln[8760] + 30) × −3 10 = 27.4, so, from Figure 11.9, the maximum allowed stress is ~140 MPa. Conversely, it is also possible to estimate the creep life under a given applied stress and temperature. For example, for a stress of 100 MPa at 700 K, alloy F-82 shows an LM parameter of 29, which translates to a creep life of 10.5 years. At 200 MPa for the same conditions, the LM is approximately 26, and the creep life would be approximately 52 days.

506 Light Water Reactor Materials

11.4.4 Creep mechanisms Contrary to low-temperature deformation, creep is thermally activated, relying on several mechanisms, such as dislocation climb and glide or diffusional flow of atoms and point defects. These mechanisms are briefly reviewed here. Thermally-enhanced glide Computer simulations have shown that a dislocation in a crystal is constantly “vibrating,” or changing shape because of the motion of atoms of which it consists. As the critical shear stress is approached, atom thermal motion provides the additional push that allows it to glide. This mechanism is also active at low temperatures, so it is not limited to creep deformation. Thermally-induced dislocation glide (climb and glide) An edge dislocation can be stopped in its slip plane by an obstacle that exerts a back force greater than the forward force from the applied stress. Glide can continue on an alternate slip plane by diffusional absorption of one type of point defect. This allows the dislocation to climb to another plane where the retarding force exerted by the obstacle is smaller and can be overcome by the forward force, thus permitting glide to continue. Figure 11.10 illustrates this mechanism.

FIGURE 11.10: Thermally enhanced climb-and-glide creep.

Mechanical Behavior of Materials 507

Thermally activated recovery As material deforms, dislocations multiply, most becoming entangled and sessile, forming long-range stresses that serve as barriers to dislocation motion. At elevated temperatures, a dislocation climb process enables such entangled dislocations to reorganize into lower-energy configurations. This recovery process decreases the back stress to mobile dislocations, permitting glide and consequent deformation to continue. Stress-induced diffusional flow σ (a) Grain difference (Nabarro-Herring creep)3 boundary When a stress is applied, the equilibrium concentration of vacancies in the solid adjacent to the grain boundaries is altered (see Ch. 4). Tensile stress augments the vacancy concentration relative to that in the unstressed solid, and a compressive stress reduces it. σ In Figure 11.11a, there is no tensile σ (b) stress on the vertical grain boundaries due to the applied stress indicated, so the vacancy concentration in the adjacent solid is not affected. On the tilted boundaries, however, there is a component of tensile stress; hence, the nearby vacancy concentration is increased. Therefore, a flux of atoms σ Before creep After creep occurs as shown by the arrows and the grain deforms in the direction of the FIGURE 11.11: (a) Nabarro-Herring applied load. creep; (b) Coble creep. 3

See also Chapter 8.

508 Light Water Reactor Materials

Coble creep In the same manner as in Nabarro-Herring creep, Coble creep occurs because of a vacancy gradient induced by the mechanical loading of the specimen (Fig. 11.11b). However, mass transport takes place in the grain boundaries rather than in the lattice adjacent to them. This mechanism also causes an overall grain extension and deformation in the direction of loading. Thermally-induced grain boundary sliding At high temperatures where vacancies can diffuse effectively, grain boundaries inclined to the stress can slide since the vacancies can move material to accommodate grain-to-grain incompatibilities (Fig. 11.7) that would otherwise occur. The process obeys Equation (11.31), where the parameter A increases with decreasing grain size.

11.4.5 Deformation mechanism maps Ashby and coworkers [10] have displayed the various deformation mechanisms on a single map (Fig. 11.12). In the region σ > σy, plastic deformation occurs by general yielding, which corresponds to dislocation glide. At higher stresses, the theoretical shear strength of the material is reached. At σ < σy, plasticity can occur by several creep mechanisms, each activated at different temperatures. Creep under irradiation is discussed in Chapter 27.

11.5 Material Fracture Increasing material deformation eventually leads to fracture, which is discussed briefly in this section. Most fractures of structural metals occur due to one or more of the following processes: Overload fracture Plastic deformation develops throughout the load-bearing cross section of the engineering component. Eventually, the plastic deformation drives a

Mechanical Behavior of Materials 509 1.E+00 Theoretical shear strength

Normalized shear stress (s/G)

1.E-01 Dislocation glide

1.E-02

Dislocation core diffusion

1.E-03

Power law (dislocation climb and glide) creep Lattice diffusion

1.E-04 1.E-05 1.E-06

Diffusional creep by bulk diffusion (NabarroHerring creep)

Diffusional creep by grain bondary diffusion (Coble creep)

1.E-07 1.E-08 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T/Tm

FIGURE 11.12: Schematic creep deformation mechanism map for pure nickel with a grain size of 1 µm [10]. Grain boundary sliding is insignificant. The stress is normalized to the shear modulus (σ/G) and the temperature to the melting temperature.

damage accumulation process within the metal, forming voids that subsequently grow and link as further plastic deformation occurs. The result is ductile fracture at stresses that exceed the yield stress. The component fractured through this process shows evidence of gross plastic deformation, usually in the form of “necking” of the cross section at the fracture surface. Failure due to crack growth This type of failure can occur when a crack is present and the external stress is sufficient to cause the crack-tip stress intensity to become critical at the fracture toughness of the material. Once the fracture toughness is exceeded, in most cases the crack propagates in an unstable manner, causing abrupt fracture. Importantly, the applied stress is less than the yield stress of the material in such failures. The mechanism of crack growth

510 Light Water Reactor Materials

may involve either ductile fracture or brittle fracture, but it is confined to the crack-tip zone, it occurs under elevated local tensile stresses, and it often takes place at high deformation rates within a small crack-tip plastic zone. The result is fracture that looks brittle to the naked eye. Failure due to subcritical crack growth This may occur when an existing small crack slowly propagates at a stress intensity smaller than the fracture toughness, as a result of external factors, such as stress-corrosion cracking, hydrogen-assisted cracking, or possibly load cycling (metal fatigue failure). Both stress-corrosion cracking and hydrogen-assisted cracking are often intergranular, resulting from grain boundary decohesion, which are weak spots for crack growth. Up to 90% of all in-service failures occur as slow, subcritical, stable crack growth until the crack reaches a length when it becomes critical at the fracture toughness of the material. Since the applied load is less than that of the yield stress, these time-dependent, “brittle-like” failures are almost always unexpected and can cause serious problems. The three failure mechanisms are briefly reviewed in the next sections.

11.5.1 Overload failure by diffuse necking: the Considère criterion As the material starts to deform plastically under tensile load, the strain is uniform (see Fig. 11.2) until the UTS, where diffuse necking starts. Once a small deviation from uniform strain appears in the rod, such as shown in Figure 11.1c, load-carrying capacity is lost; additional strain decreases the neck diameter, causing the stress in the necked section to increase, which causes further deformation and so on. However, with deformation comes work-hardening according to Equation (11.8), which raises the stress required to maintain plastic deformation. The onset of diffuse necking that occurs when the material deforms faster than it can strain-harden is identified with the Considère criterion [11].

Mechanical Behavior of Materials 511

Necking initiates near the maximum load: P = σA => dP = Adσ + σdA ,

(11.37)

so when dP = 0, dA = − dσ = −dε ∴ dσ = σ . (11.38) σ A dε Consider further that when a rod such as in Figure 11.1 is subjected to a load P, initial length l, and initial cross-sectional area A, the material is deformed plastically to a length l + dl, which results in a diminution of the cross-sectional area to A′. This causes the material to strain-harden. If the material deforms according to Equation (11.8), σ = Kε

n

.

The strain here is understood to be plastic strain. The change in strain hardening (dσ) caused by incremental strain dε is dσ n −1 nσ = Knε = . (11.39) ε dε Setting Equation (11.39) equal to (11.38) gives the following criterion for plastic instability due to necking: (11.40) εU = n where εU is the uniform true strain at the onset of necking (from Fig. 11.2). This equation, the Considère criterion [11], states that at the onset of a necking instability, the strain is numerically equal to the strain-hardening exponent n. For an engineering component loaded in uniaxial tension, the uniform strain also identifies the maximum load that components can sustain before failure occurs.

11.5.2 Ductile fracture Once necking starts, fracture of the material occurs by damage accumulation. Due to the presence of hard second-phase particles or inclusions

512 Light Water Reactor Materials

(e.g., intermetallic precipitates, carbides, sulfides, or hydrides), deformation incompatibilities of different grains develop within the material as it plastically deforms. At small plastic strains, these deformation incompatibilities are accounted for by other slip systems that allow particle-to-matrix interfaces to remain intact (Sec. 11.3.1). As deformation accommodation by activation of multiple slip systems becomes increasingly difficult, dislocation pileups and interactions and damage (such as voids or pores) accumulate near second-phase particles, interfaces, and triple points. As shown schematically in Figure 11.13b, larger strains nucleate voids at the particle-matrix interfaces, or they may cause the particles to crack, creating internal voids. Upon further straining of the rod, the voids grow and eventually coalesce as in Figure 11.13c. To predict failure (Fig. 11.13d), a void initiation criterion, a void growth law, and a void coalescence condition are needed. Many such models have been developed [12] but are beyond the scope of this book. As a result of the formation of voids before fracture (b) and their coalescence (c), the fracture surface will normally exhibit ductile dimples, as shown in Figure 11.14.

11.5.3 Fracture due to crack growth The presence of a crack and its propagation can cause early deformation and fracture (at stresses below the yield stress), representing an alternate

(a)

(b)

(c)

(d)

FIGURE 11.13: Schematic representation of void formation and coalescence in ductile fracture.

Mechanical Behavior of Materials 513

FIGURE 11.14: Micrograph showing the fracture surface of a steel containing spherical inclusions that initiate “primary” voids that are linked by small secondary voids during the ductile fracture process. (Reprinted from Materials Science and Engineering: A, 366/2, D. Chae, D. A. Koss, “Damage accumulation and failure of HSLA-100 steel,” Copyright 2004, with permission from Elsevier.) [13]

path to failure. The analysis of crack propagation is reviewed in this section. Griffith fracture theory When a crack is introduced in a stressed plate, two counteracting phenomena occur: Elastic energy is released, and surface energy is increased by the presence of the crack. If the former (which promotes growth) is larger than the latter (which impedes growth), crack propagation continues.

514 Light Water Reactor Materials

δ

a

FIGURE 11.15: An elliptical crack in a stressed plate.

If an elliptical crack of length a is introduced into a plate of thickness δ (Fig. 11.15), the total surface energy created is E surf = 2aδγ surf

(11.41)

where γsurf is the energy per unit crack surface area. The elastic energy released by the crack is estimated [14] as E el =

πσ 2 a 2 δ

4E where σ is the applied stress, and E is the elastic modulus. Thus, the overall change in energy is πσ 2 a 2 δ

(11.42)

+ 2aδγ surf . (11.43) 4E Differentiating with respect to a and setting the result equal to zero, the equilibrium condition is ∆E crack = −

Mechanical Behavior of Materials 515 πσ 2 a

= 2 γ surf , 2E from which the stress for unstable crack propagation is E γ surf σ=2 . πa Equation (11.45) can be rearranged as

(σ πa)

= crit

2 E γ surf

= K IC

.

(11.44)

(11.45)

(11.46)

Whereas the left-hand side of Equation (11.46) contains parameters specific to the test (crack size and applied stress), the right-hand side contains only material properties. The right-hand side can then be identified as a parameter that causes failure under a specific set of conditions, called the fracture toughness of the material K IC. The fracture toughness is measured in the laboratory with carefully prepared cracks in test specimens. This condition for unstable crack propagation based on these two energetic processes was derived by Griffith in 1921 [14]. However, except in very brittle materials, this theory underpredicts measured values of KIC (i.e., predicts brittle failure when it does not actually occur). This is because in materials that exhibit some ductility, it is necessary to induce plastic deformation prior to fracture. This occurs in a region ahead of the crack called the plastic zone. The Griffith derivation can be corrected for the creation of the plastic zone by

(σ πa)

= crit

2 E ( γ surf

+ γ pz ) = K IC

(11.47)

where γpz is the energy per unit crack area to create a fracture surface that requires plastic work within the plastic zone at the crack tip. Equation (11.39) indicates that the higher the elastic modulus and the surface and (especially) the plastic work energies, the greater the fracture toughness.

516 Light Water Reactor Materials

The stress intensity factor is defined by4 K I = Yσ a (11.48) where Y is a constant usually having values between 1 and 2, depending on the geometry of the crack. The crack is assumed to be a “mode I” crack oriented normal to the applied stress σ (see Fig. 11.15). Formulas for Y for many different geometries are available [15]. If the maximum allowable crack length is substituted into Equation (11.48), the design stress σ is obtained if KIC is known.

11.5.4 Subcritical crack growth In an inert environment with K I < K IC , cracks are indefinitely stable in the absence of creep. However, in an aggressive environment, failure can occur by subcritical crack growth if K IC > K I > K ISCC (11.49) where KISCC is the stress intensity required for crack growth due to stresscorrosion cracking as discussed in Chapter 25. Alternatively, under cyclic loading with KI < KIC, there may be sufficient localized deformation occurring within the crack-tip plastic zone to advance the crack. If ∆K is the stress intensity range (Kmax − Kmin), then fatigue crack growth will occur if ∆K > ∆Kth, where ∆Kth is the threshold stress intensity factor for fatigue crack growth.

References 1. 2. 3. 4

D. R. Olander, Fundamental Aspects of Nuclear Reactor Fuel Elements, TID-26711-P1, National Technical Information Service (1976). W. A. Backofen, Deformation Processing (Reading, Massachusetts: Addison-Wesley Publishing Co., 1972). T. H. Courtney, Mechanical Behavior of Materials (New York: McGrawHill, 1990).

The stress intensity factor (KI) is to fracture toughness as stress is to the yield stress (σγ).

Mechanical Behavior of Materials 517

 4. J. T. A. Roberts, Structural Materials for Nuclear Power Systems (New York: Plenum Press, 1981).  5. A. S. Argon, Strengthening Mechanisms in Crystal Plasticity (Oxford: Oxford University Press, 2008).  6. L. M. Brown and R. K. Ham, “Dislocation-Particle Interactions,” in Strengthening Methods in Crystals, ed. A. Kelly, and R. B. Nicholson (London: Applied Science, 1971).  7. H. J. Frost and M. F. Ashby, Deformation-Mechanism Maps, The Plasticity and Creep of Metals and Ceramics (Oxford, New York: Pergamon Press, 1982).  8. R. L. Klueh and D. R. Harries, High Chromium Ferritic and Martensitic Steels for Nuclear Applications (West Conshohocken, Pennsylvania: ASTM, 2001).  9. R. L. Klueh and A. T. Nelson, “Ferritic/martensitic steels for nextgeneration reactors,” J Nucl Mater 371 (2007): 37–52.  10. M. F. Ashby, Acta Metallurgica 20 (1972): 887.  11. A. Considère, Ann. Ponts et Chaussees 9 (1885): 574–775.  12. A. Gurson, “Continuum theory of ductile rupture by void nucleation and growth: Part I – Yield criteria and flow rules for porous ductile media,” J Eng Mater Technol 99 (1977): 2–15.  13. D. Chae and D. A. Koss, “Damage accumulation and failure of HSLA100 steel,” Materials Science and Engineering A366 (2004): 299–309.  14. A. A. Griffith, “The phenomena of rupture and flow in solids,” Philosophical Transactions of the Royal Society of London, A 221 (1921): 163–198.  15. R. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3e. (Hoboken, New Jersey: Wiley, 1989).

Problems 11.1 A metal rod of initial length l0 and cross-sectional area A0 is tested uniaxially by fixing one end of the rod to a stationary point and loading the other with a spring of force constant K. The change

518 Light Water Reactor Materials

in length of the rod, l − l0, is related to the change in length of the spring, ∆x, by ∆x = ∆x Y − a(l − l 0 ) 2 + b(l − l 0 ) where the first term on the right is the spring extension at the yield point of the rod and a and b are empirical constants. (a) Express the true stress σx in terms of the true (logarithmic) strain εx and the parameters K, A0, l0, a, b, and ∆xY. Do not assume a functional relation between these two variables (e.g., Eq. [7.3]). Neglect elastic strain. (b) Use the criterion for plastic instability to determine the true strain at the UTS. 11.2 The true stress–true strain curve in the work-hardening region can be written in dimensionless form: σ − σY = σY

A(ε − εY )n

where the subscript Y denotes the yield point, A is a dimensionless constant, and n is the work-hardening coefficient. Define X = (σUTS − σY)/σY, where UTS denotes the ultimate tensile stress. (a) What is the equation relating X to A and n? (b) Solve this equation for A = 23 and n = 0.7. Show that X increases with n at constant A. 11.3 For a particular metal, the true stress and the true strain obey the following plastic constitutive law: σx = σY exp(εx − εY). Prove that this material has no work-hardening region. 11.4 A closed cylindrical tube is pressurized to exceed yielding by 20%. The constitutive equation for the material of the tube in the plastic region is: σx − σY = C εxn (elastic strain at the yield point is negligible). (a) What are the stresses in the tube wall in terms of the uniaxial yield stress of the material?

Mechanical Behavior of Materials 519

(b) The plastic “strength” of the material is defined by: S = (σ∗ – σY)/ε∗, where σ∗ and ε∗ are the equivalent stress and strain, respectively. Determine S. (c) What are the radial, azimuthal, and axial plastic strain components in the tube wall? 11.5 A spherical shell of radius R and wall thickness δ is pressurized internally to a pressure p. (a) What stress components are nonzero, and what are their values? (Use a force balance as in Sec. 11.5 for the pressurized cylinder.) (b) If the uniaxial yield stress is σY, at what internal pressure will the shell yield? 11.6 The creep law of a particular steel is found in uniaxial tests to obey ε = 161σ 6.4 exp( −3 × 10 4/T )

where the strain rate is in hour−1, the stress is in MPa, and the temperature in Kelvin. A component fabricated from this steel operates at 330°C. The principal normal stress components at a particular location are σx = 380 MPa, σy = 220 MPa, and σz = 180 MPa. How long is required for the creep strain (in any direction) to reach 2%? 11.7 A weight of mass m is suspended from a closed thin-wall tube of radius R and wall thickness δ with internal pressure p. The yield strength of the tube material is σY. Allowing for a 25% safety factor, what is the maximum gas pressure in the tube? 11.8 A tensile test specimen is initially of length l0 and cross-sectional area A0. The specimen is subjected to a varying load P and the deformation ∆l is measured at each. The data are correlated by the empirical equation: P = c + b∆l − a( ∆l ) 2 where a, b, and c are positive constants. (a) What are the “engineering” yield and ultimate stresses of this material?

520 Light Water Reactor Materials

(b) Show how the true stress–true strain curve can be deduced from the above equation. Neglect elastic deformations and assume all deformation to be plastic. 11.9 When tested uniaxially at a particular temperature, the creep rupture behavior of a batch of metal tubing obeys the equation: σ x = C (t R ) −1/m , where C and m are material properties. (a) When this tubing is sealed at both ends and pressurized at constant pressure p, what is the rupture time? The tube diameter is D and its wall thickness is δ. (b) If the internal pressure varies linearly with time (p = kt), what is the time to failure? 11.10 A metal is tested in pure shear (σxy nonzero but all the other stress components are zero). If the yield strength in a uniaxial tensile test is σyuni, what is the yield strength in the shear test σy shear? 11.11 A rod just fits between two rigid plates when its temperature is To. The rod is heated quickly and uniformly to a temperature T, but the restraining plates do not move. (a) What is the initial axial stress in the rod when the temperature reaches T? Hint: Equation (5.14c) can be viewed as the total strain equal to the sum of the elastic and thermal strains, or εtot = εel + εth. (b) With time, the stress computed in part (a) is reduced because of creep (this is called creep relaxation). Since the thermal strain does not change with time, the total strain rate is the sum of the time rate of change of the elastic strain plus the creep strain rate, or ε tot = ε el + ε cr. At the final temperature T, the creep rate is ε cr = Bσ n, where the constant B includes the temperature effect. Derive the equation, giving the variation of the axial stress in the rod with time. Hint: In both parts (a) and (b), keep in mind that the total length of the rod remains constant.

12

Chapter Radiation Damage 12.1 Introduction While in service, reactor materials are exposed to intense fast neutron and gamma fluxes originating from the fission reactions in the fuel. The interaction of these energetic particles with the metallic structures and ceramic fuels can displace the atoms from their stable positions in the crystalline lattice, thereby creating lattice defects. The nature and density of these defects are at the root of the microstructual changes suffered by materials during irradiation. The mechanism of defect creation is determined by the radiation damage process, which is the subject of this chapter. After the Second World War, civilian applications of nuclear power were at the forefront of technological development. In that context, it was recognized that radiation damage to materials used in nuclear reactors would play a large role in determining their suitability for service. Wigner predicted in 1946 that radiation damage would degrade material properties [1], so the term Wigner disease was coined to encompass the deleterious changes in material properties when exposed to irradiation by energetic particles. Materials of interest for nuclear applications, such as metallic uranium, graphite, uranium dioxide, steels, and zirconium alloys were first developed in military programs, aimed at nuclear weapons or nuclear

522 Light Water Reactor Materials

propulsion. Information on the properties of these materials were shared in the international conference series on the Peaceful Uses of Nuclear Energy. In parallel with these technological applications of nuclear power, physicists recognized that energetic particles could produce point defects in these solids. This inspired efforts to understand the properties of these defects (migration, formation energies, morphology) using techniques such as resistivity changes and positron annihilation spectroscopy. Understanding the physical mechanisms of radiation degradation of materials is a continuing activity, not only in nuclear power applications but in many other fields, most notably the semiconductor processing industry [2, 3]. This chapter reviews the basics of radiation damage, following various previous works on the subject. See, for example, the proceedings of the Illinois Summer School, published in the Journal of Nuclear Materials, especially the article by Mark Robinson [4] as well as R. E. Stoller’s “1.11 – Primary Radiation Damage Formation” [5].

12.2 Particle-Solid Interactions It is useful to divide the effects of an external flux of energetic particles in a metal into two components: (i) creation of primary knock-on atoms (PKAs; these are the first atoms that absorb momentum from a collision with an energetic particle) and (ii) creation of transmuted atoms through nuclear reactions. Both of these processes result from the interaction of the particles with the atoms in the solid, as illustrated in Figure 12.1.1 Here φi (Ei) represents the flux of particle (i = neutron, gamma,..), which varies with particle energy Ei, while the macroscopic cross section Σκ(Ei, E) is a measure of the probability of interaction (the macroscopic 1

Note that in insulator materials it is possible to have other types of damage associated with charge deposition. One example is the creation of fission tracks in materials whereby an energetic ion leaves behind a trace of its passage (see [6] Fig. 15.6).

Radiation Damage 523 Primary knock-on atom energy distribution NPKA(E) φi (Ei ) i

Σκ (Ei , E)

= Neutron, Gamma Ions,…

Transmutation atoms Cκ

κ = Elastic scattering, absorption, inelastic scattering, …

FIGURE 12.1: The interaction of particles with atoms in the solid.

cross section of the various particle-solid reactions κ, transferring energy E to the atoms in the solid). The product of the interaction of the flux of energetic particles (represented by φ) and the atoms in the solid (represented by the macroscopic cross section Σ) is the creation of a distribution of NPKA(E) self-atom recoils called PKAs and a concentration of transmutation atoms Ck, where k represents the atomic species created. Among the many processes that can cause atomic displacements in solids under the flux of energetic particles, the largest contributor to displacement damage of structural components in reactor cores is the fast-neutron flux φn(E n) interacting by elastic scattering (Σs) with the atoms in the solid, creating a distribution of PKAs, which in turn displace other atoms.

12.2.1 Cross section The probability of occurrence of a particular reaction between the atoms in the solid and the incident particle flux is represented by a cross section. The concept of the microscopic cross section σ is illustrated in Figure 12.2. The microscopic cross section attributes an apparent size to the atoms in the solid, which is proportional to the measured reaction rates, such that for a given particle flux spectrum into a solid of a given atomic density, the apparent particle size increases with the probability of reaction. Figure 12.2 shows two types of atoms present in the solid. Atom A has a greater reaction rate for reaction 1 (fission of U-235) than does atom B,

524 Light Water Reactor Materials A

A B B

B

B B

B A

B

B

B

A

B B

B A

B

A

B

B B

B B Reaction 1

Reaction 2

FIGURE 12.2: Schematic representation of the microscopic cross section concept. If A atoms are U-235 and B atoms are B-10, then Reaction 1 would be nuclear fission, for which the U-235 atoms have a large cross section and the B atoms do not, while Reaction 2 could be the (n,α) reaction, for which the cross section of U-235 is much smaller than that of B-10.

and thus, A atoms appear very large (high cross section) when reaction 1 is considered. In contrast, for reaction 2, atom B has a much larger crosssection, and consequently, these atoms appear bigger when reaction 2 is considered. If, for example, we let atom A (light atoms) be uranium-235 and atom B (dark atoms) be boron-10, and Reaction 1 be nuclear fission and Reaction 2 be the (n,α) reaction (absorption of neutron and emission of an alpha particle), in a thermal neutron flux the microscopic cross sections would qualitatively appear as shown in Figure 12.2. The unit of the microscopic cross section σ is the barn (10−24 cm2). The macroscopic cross section Σ (cm−1) is the product of the microscopic cross section σ(Ei) (cm2/atom) and the atomic density c (cm−3). When multiplied by a particle flux (φ particle.s−1.cm−2), the reaction rate φΣ (reactions. s−1.cm−3) is obtained. In a solid containing c target atoms per unit volume through which a single particle of energy Ei passes, the differential probability dP that this

Radiation Damage 525

particle will interact with one of the target atoms in an element of small thickness dx is given by: dP = cσ ( E i )dx = Σdx . (12.1) Equation (12.1) is valid for any atomic reaction (absorption, fission, scattering, etc.). If σ(Ei) is an elastic scattering cross section, then the result of such an interaction is the transfer of energy E from the energetic particle to the struck atom, which then becomes an energetic recoil atom, the PKA. The PKA energy distribution NPKA(E) is the primary means of characterizing the damage caused by irradiation. Fast neutrons passing through a solid can scatter elastically or inelastically, and thermal neutrons can induce nuclear reactions, all of which deposit energy onto the material. The first result of a neutron-atom scattering interaction is the formation of the PKA followed by production of secondary recoils from the PKA, according to the following: En ⇒ E ⇒ T Neutron ⇒ PKA ⇒ Secondary Recoils The problem of calculating the displacement damage can be divided into (i) finding the energy spectrum of the PKAs created by interaction of the neutrons with the atoms of the solid and (ii) calculating the damage that a PKA of a given energy can produce. We address the second issue in the next few sections, and later in the chapter we consider the first part.

12.3 Primary Knock-On Atom Energy-Loss Mechanisms To find the number of displacements caused by a PKA of a given energy, it is first necessary to compute the energy transferred by the energetic PKA to the neighboring atoms. The total rate of energy loss of a PKA of energy E moving through a solid can be separated into three components:

526 Light Water Reactor Materials

dE dx

= TOTAL

dE dx

+ E

dE dx

+ N

dE dx

(12.2) In

where the terms on the right-hand side of Equation (12.2) refer to electronic energy loss (E), nuclear-elastic scattering (N), and nuclear-inelastic scattering (nuclear reactions) (In). Because the typical PKA energies in recoil spectra generated during reactor irradiation are considerably lower than the energies required for nuclear-inelastic scattering, the first two processes dominate energy loss in most materials of interest. In metals, because the electrons are shared by all atoms in the lattice, the collisions with electrons are of little permanent consequence to the solid; the energy of high-speed electrons is degraded into heat. In insulators or materials used in the electronics industry, however, electronic damage can be significant [2]. Nuclear-elastic scattering can cause permanent damage to the crystal in the form of atomic displacements. The partition between these two forms of energy loss determines the amount of radiation damage to metals and the subsequent radiation effects observed. We now consider these processes in turn.

12.3.1 Binary elastic collision dynamics The process of elastic transfer of energy is akin to that seen in hard sphere collisions, similar to what occurs when billiard balls strike each other. v1F

M1

M1 v10

M2 v20 = 0

ϕ1 ϕ2

M2

v2F

FIGURE 12.3: Elastic scattering process in the laboratory frame of reference.

Radiation Damage 527 M1 M1

M2 u10

+

u1F

θ +

u20 u2F

M2

FIGURE 12.4: Elastic scattering in the center-of-mass frame of reference.

We derive here the energy transfer between two hard spheres that collide. Figure 12.3 shows the energy transfer between two colliding hard spheres considered in the laboratory frame of reference. An atom of mass M1 and velocity v10 strikes a stationary atom of mass M2. After the collision, the scattering angles are ϕ1 and ϕ2, and the velocities are v1F and v2F. In the center of mass system, the above collision is shown as in Figure 12.4. The center of mass has the total mass of the system (M1+M2) so that from conservation of momentum from one frame of reference to another, the center of mass velocity vCM is M1 v CM = v . (12.3) ( M1 + M 2 ) 10 Using conservation of kinetic energy and momentum before and after the collision, the following relations apply in the center of mass system: M1u10 = M 2u 20

(12.4)

M1u1F = M 2u 2 F 1 1 1 1 M1u102 + M 2u 202 = M1u12F + M 2u 22F 2 2 2 2

(12.5) (12.6)

where u10 = v10 − v CM and u 20 = − v CM are the initial velocities and u1F and u2F are the final velocities in the center of mass frame.

528 Light Water Reactor Materials

Substituting Equations (12.4) and (12.5) into (12.6), we obtain uF = u0 for both particles, and thus, u2F = −vCM. The initial and final velocities in the center-of-mass frames of reference are vectorially related, as shown in Figure 12.5. Using the law of cosines (see Fig. 12.5), 2 + u 2 − 2v u cos θ v 22F = v CM 2F CM 2 F

.

(12.7)

But because v CM = u 2 F , 2 (1 − cos θ) . v 22F = 2v CM (12.8) Multiplying both sides by M2/2, and substituting for vCM using Equation (12.3), we obtain

M2 2 M 2  M 1 2 2  v (1 − cos θ) E 2F = v 2F = 2  2 2  M1 + M 2  10 1  4 M1 M 2 ( E )(1 − cos θ) . =   2 ( M1 + M 2 ) 2  10

(12.9)

Equation (12.9) describes the energy E2F transferred to an atom of mass M2 by an energetic particle energy E10 and mass M1 as a function of the scattering angle in the center of mass system. We define the energy transfer parameter 4 M1 M 2 ΛM M = (12.10) . 1 2 ( M1 + M 2 ) 2 1 Noting that the initial kinetic energy of particle 1 is E10 = M1v102 , 2 1 E 2F = ΛE10 (1 − cos θ), or if E 2F = T and E10 = E 2 then

T=

1 ΛE (1 − cos θ) 2

(12.11)

Radiation Damage 529

where we use Λ as a shorthand for ΛM1M2. The maximum amount of energy transferred occurs for a head-on collision in which θ = π and Tmax = ΛE . (12.12)

u1F

v1F

φ1 φ2

v2F

vCM

θ

u2F

The energy transferred changes with the mass ratio of the energetic particle and the FIGURE 12.5: Relationship between velocities in the center struck atom. For atoms of equal mass, Λ = 1; of mass and laboratory frames of that is, in a head-on elastic collision, the incom- reference. v = v + u 2F CM 2F ing atom transfers all of its energy to the struck atom. As the masses become increasingly different, the maximum possible transferred energy Λ decreases, and as a consequence, Tmax also decreases. Example #1: Energy transfer from neutrons to Fe-C: What is the maximum energy transfer from 1 MeV neutrons to atoms in steel? In steel (Fe-C) under 1-MeV neutron irradiation, in an elastic collision the maximum energy transfer from the neutron to the carbon atom is En–C = 0.28 En (280 keV) but only En–Fe = 0.07 En (70 keV) to the Fe atom. In turn the 280-keV C primary knock-on atom can transfer up to 100% of its energy to another C atom but only TC–Fe = 0.58 EC , and if the maximum EC is En–C max (280 keV), then TC–Fe is 162 keV to an Fe atom. Similarly, the 70-keV Fe primary knock-on atom can transfer up to 100% of its energy to another Fe recoil atom but can only transfer an energy TFe–C = 0.58 En–Fe (40.6 keV) to a C atom.

12.3.2 The limit between nuclear and electronic stopping Because the electron density in the material is far greater than the atom density, as long as the energetic particles can transfer energy to the electrons, they preferentially do so. For the electrons to be able to accept the energy from the moving particle, they need to receive energy equal to or greater than their ionization energy in the atom. For metals, since the average kinetic

530 Light Water Reactor Materials

energy of the conduction electrons is approximately 3/5 of the Fermi energy εFermi, the ionization energy (I) is approximately equal to 2/5 εFermi. A simple way to estimate the energy transferred to an electron in a collision between a particle of mass M and energy E is to use Equation (12.12): 4 me E . (12.13) M Setting Temax = I (ionization energy), we obtain the following condition for PKA energy to be transferred to electrons: M (12.14) E > E∗ = I . 4 me The value of 5 eV is a good approximation of the Fermi energy in metals; thus I ∼ 2 eV; if we take me = 1/1840 amu ≅ 1/2000 amu, and then we obtain a convenient formula for the cutoff energy: Temax = Λ M−e E ≅

E ∗ ≅ 1000 × M[eV] = M[keV] .

(12.15)

According to the approximation in equation (12.15), the cutoff energy for electronic energy loss E ∗ is numerically equal to the PKA atomic mass M, expressed in units of keV. For example, according to the model, a 2656Fe atom traveling through a solid in which the ionization energy is 2 eV starts losing energy to nuclei only below ∼56 keV. Clearly, electronic energy loss continues to occur to some extent below ∗ E  , in parallel with nuclear stopping, but it is difficult to evaluate its exact contribution. It is possible, however, to estimate this energy partitioning by Monte Carlo computer simulation. The computer programs TRIM (TRansport of Ions in Matter) and SRIM (Stopping and Range of Ions in Matter) [7] are Monte Carlo programs that use specially developed interatomic potentials to calculate the energy deposition of energetic atoms in solids. Both electronic and nuclear stopping are treated simultaneously to achieve a more realistic energy partition (see Sec. 12.9.3). In the following section, we show an analytic approximation.

Radiation Damage 531

12.3.3 Electronic stopping: energy transfer rate The energy transfer from a PKA of energy E, mass M to the electrons in the solid is now considered. The transfer is calculated differently depending on the ion energy (and consequently effective charge). Although energy transferred to electrons causes no permanent damage in metals, it is important to calculate the partition between electronic and nuclear stopping, as the latter produces permanent damage. For high-energy ions (in which the ion velocity v1 > 3 vEL, velocity of electrons), as the atom travels through the solid, some of its electrons are stripped off, causing the atom to become a charged particle of effective charge Zeff  . Because the effective charge changes with atom energy, it changes dynamically as the atom loses energy while traveling through the solid. If Zeff   is known, then the Bethe formula [19] gives an approximate expression for the electronic energy loss of the moving ion: dE dx

2 πne Z eff2 e 4 ( M/me )  4 E  ln  =  E EL ( M/me ) I 

(12.16)

where ne is the electron density, e is the electron charge, and I is the average ionization energy in the metal. The value of the effective charge was computed by Bohr as 1/2 Z 1/3   2 E  Z eff = 2 (12.17) e  M where  is Planck’s constant divided by 2π, and Z is the atomic number of the energetic particle. For low-energy ions, the energy transfer to an electron in an ion-electron collision is 1 Te = mo (v eF2 − v e2 ) (12.18) 2 where veF and ve are the final and initial electron velocities, and mo is the rest mass of the electron.

532 Light Water Reactor Materials

It can be shown that

and thus

Te = 2mo

Te ≅ 2mo vv e ,

(12.19)

2 E 2(3ε Fermi /5) =A E . M1 mo

(12.20)

where A is a constant. The energy transfer rate (called the stopping power) is given by dE collisions energy lost distance travelled (σ e J e )Te = × ÷ = (12.21) dx e PKA. s collision PKA. s v where σe is the cross section for atom-electron scattering, Je is the flux of electrons seen by the moving PKA and is equal to J e = ne (v e + v ) ≅ n e v e for low-energy PKAs, but since below the Fermi level all states are occupied, the “excitable” fraction of electrons is ne = Te /ε Fermi N e , where Ne is the total electron density in the material. Since the average electron initial kinetic energy is 3/5 εFermi, dE dx

= e

(σ e J e )Te v

= σe

 Te N v  (2mo vv e ) = 2.4σ N T . (12.22) e e e  ε F e e v

But Te = A E , and because, for a given material, the cross section for electron-ion collisions varies slowly with energy, and the other terms are constant, in the electron-stopping region, the energy loss rate by the PKA is proportional to the square root of the PKA energy: dE =K E . dx e According to detailed calculations by Lindhard [27], eV  K = 0.3c Z 2/3 ,   nm 

(12.23)

(12.24)

Radiation Damage 533

where c is the atomic density (nm−3) and Z is the atomic number. In summary, in the electron-stopping region, the energy loss rate by the PKA is approximately proportional to the square root of the PKA energy.

12.3.4 Nuclear stopping When the energetic particle (PKA) slows down enough that energy transfer to the electrons becomes difficult, it increasingly loses energy by collisions with the nuclei in the solid. For elastic collisions, a PKA with energy E strikes an atom in the solid, and transfers energy T, leaving it with energy (E-T ). The moving atom can transfer to the atom in the solid any energy between 0 and ΛE. The probability that a particle with energy between E and E+dE transfers an energy between T and T+dT to an atom in the solid is given by the differential energy-transfer cross section σ(E,T ). The differential cross section σ(E,T ) and the total cross section σ(E) are related by ΛE

σ(E) =

∫0 σ(E ,T )dT

.

(12.25)

The average energy loss dE in dx is given by integrating Equation (12.25) over all possible values of the transferred energy T: Tm

dE = ∫ cσ ( E , T )TdTdx ,

(12.26)

0

where c is the atom concentration, and the nuclear stopping power is given by ΛE

dE = cσ ( E , T )T dT . (12.27) dx N ∫0 The detailed process of atomic scattering and the energy loss for a single collision can be derived more exactly than in Section 12.3.1. Using conservation of energy and momentum, a relationship between the scattering angle in the center of mass system and the impact parameter is derived in the following section.

534 Light Water Reactor Materials

12.3.5 Ion-atom scattering; general binary collision dynamics A collision between two particles that have an interaction potential V(r) and which collide with an impact parameter p is shown in Figure 12.6. It is desired to find the orbit of two particles in an elastic collision and to relate the interaction potential to the differential cross section σ(E, θ). u20 r2sinα

r2

vCM r1 M1

r&1 α

S& 1

CM

M2 Impact parameter p

r1sinα

u10

FIGURE 12.6: Geometry for derivation of elastic collision between an energetic ion and a stationary atom, interacting by a potential V(r).

In the system considered, a particle mass M1 is moving initially with kinetic energy E toward an initially stationary particle mass M2. The center of mass (CM) is located on the line joining the two masses at a distance M2 (12.28) r and r = r1 + r2 M1 + M 2 from the mass M1. The initial velocity of particle 1 in the center-of-mass system, u10, is decomposed into two perpendicular components r1 and S1,    such that u10 = Sɺ1 + rɺ1 . The line between the particles makes an angle α with the initial direction of the particles in the CM system. Only the initial kinetic energy in the CM system is convertible to potential energy, and this is written 1 2 E CM = ( M1 + M 2 )v CM , (12.29) 2 and using Equation (12.3): M1 M1  M1v102  E CM = E= (12.30) M1 + M 2 M1 + M 2  2  r1 =

Radiation Damage 535

M2

r2

ψ CM

r1

r1

Impact parameter p

θ S1

M1

FIGURE 12.7: Geometry during the collision in center-of-mass coordinates.

where E is the initial kinetic energy of particle 1 in the laboratory frame. Now using conservation of energy and angular momentum during the collision, it is possible to derive a relationship between the scattering angle in the center of mass θ and the impact parameter p. The trajectory of the particles as they interact and are deflected by angle θ are shown in Figure 12.7. Conservation of energy As the two energetic particles approach each other, they convert kinetic energy into potential energy, V(r), so that at the distance of closest approach, the kinetic energy is minimized. Conservation of energy for the system is 1 1 1 E CM = V (r ) + M1u102 + M 2u 202 = V (r ) + M1 (r12 + S12 ) 2 2 2 1 + M 2 (r22 + S22 ) . 2 The tangential speed S is equal to r ψ , so Equation (12.31) is

(12.31)

1 1 E CM = V (r ) + M1 (r12 + r12 ψ⋅ 2 ) + M 2 (r22 + r22 ψ⋅ 2 ) , 2 2

(12.32)

536 Light Water Reactor Materials

and using the definition of the energy of the center of mass, 1 MM E CM = V (r ) +  1 2  (r2 + r 2 ψ⋅ 2 ) 2  M1 + M 2  dr d ψ   d ψ dt 

r2 = 

along with

yields

2

(12.33)

2

 dr  ψ⋅ 2 =  dψ

2 1  M1 M 2    dr  2 ⋅ 2 E CM = V (r ) + +r  ψ . 2  M1 + M 2   d ψ  

(12.34)

(12.35)

Conservation of angular momentum

The angular momentum of a mass M about an axis is r × ( Mu ) = rMu sin α = sin α = rMSɺ, where r is the distance between the particle and the axis, u is the velocity, and α is the angle between r and u. Then, the total angular momentum in the CM system is equal to L = r1 M1u10 sin α + r2 M 2 v CM sin α . (12.36) The tangential velocity S is equal to r ψ⋅ ; thus, L = L1 + L2 = M1S1r1 + M 2 S2r2 = M1r12 ψ⋅ + M 2r22 ψ⋅ .

(12.37)

Using the definition of CM velocity in Equation (12.36) and equating to Equation (12.37) M1 M 2 MM v10 (r1 sin α + r2 sin α) = 1 2 v10 p M1 + M 2 M1 + M 2 = M 1r12 ψ⋅ + M 2 r22 ψ⋅

L=

(12.38)

where p is the impact parameter. Using Equations (12.28) in Equation (12.38), we obtain an equation for ψ⋅ ψ⋅ = v 10 p/r 2

,

(12.39)

Radiation Damage 537

which can be eliminated in Equation (12.38) 2 2 1  M1 M 2    dr  2  v10 p E CM = V (r ) + . +r  2  M1 + M 2   d ψ    r2 

(12.40)

Using Equation (12.30) to eliminate v10, dψ p 1 = 2 . (12.41) dr r  V (r ) p 2 1/2 1− −  E CM r 2  When the particles are at their closest, r = ro and ψ = π/2, and when r → ∞, ψ → θ/2. Integrating Equation (12.41) between these limits: π /2

π

θ

∫/2 d ψ = 2 − 2

(12.42)

,

θ

produces the Classical Scattering Integral ∞

θ= π−2

∫ro

pdr p2 r 2 1 − V (r ) − 2   E CM r 

1/2

.

(12.43)

Equation (12.43) relates the impact parameter p to the scattering angle in the center of mass θ. At the distance of closest approach, dr/dψ = 0 and 1−

V (ro ) p 2 − =0 . E CM ro2

(12.44)

Equation (12.44) can be solved for the distance of closest approach as a function of impact parameter. In a head-on collision, p = 0 and Equation (12.44) reduces to V(ro)  = ECM, which, using Equation (12.27), yields the distance of closest approach in a head-on collision.

538 Light Water Reactor Materials dΩ = 2πd(cos θ) dp

M1

d

p

M2

θ

FIGURE 12.8: The geometry of the differential cross section showing its relation to the impact parameter.

Relation of impact parameter to differential cross section Figure 12.8 shows that if the impact parameter is in the range p to p + dp, the annular ring of area 2πpdp is the area associated with the annular cone 2πσ(E, θ)d(cos θ), and thus, 1 dp 2 . (12.45) 2 d (cos θ) If the potential V(r) is known, then Equation (12.43) can be integrated to give the final orbit of the particle after the scattering event (i.e., θ as a function of p). Then Equation (12.45) determines the cross section. This process can be performed analytically for a few simple potentials, but in most cases numerical integration is necessary. One of the analytical cases is the well-known Rutherford cross section, which is derived in Example #2. σ ( E , θ) =

Example #2: Rutherford cross section Substituting the unscreened Coulomb potential Z Z e2 V (r ) = 1 2 r into Equation (12.43) gives θ

=

π

2 2

(12.46)

1

−P

∫0 (1 − αu − p 2u 2)1/2 du

(12.47)

Radiation Damage 539

r where u = o ; r

α=

C ; C = Z1 Z 2e 2 ; and ro E CM

The solution of Equation (12.47) is   α + 2P 2  θ π = − sin −1  + sin −1   2 2 2 2   α + 4P  The dimensionless form of Equation (12.44) is

α

P=



 α 2 + 4P 2 

p . (12.48) ro

.

P 2 = 1− α . Using Equation (12.50), Equation (12.49) reduces to 2sin(θ/ 2) α= . 1 + sin(θ/ 2) Substituting Equations (12.48) and (12.51) into (12.50) yields p2 =

C 2  1 + cos θ . 4 E co2  1 − cos θ

(12.49)

(12.50)

(12.51)

(12.52)

The impact parameter p → 0 as θ → π (head-on collision) and as θ → 0 p → ∞ (complete miss). Substituting Equation (12.52) into Equation (12.45) results in the Rutherford cross section: Z12 Z 22e 4  M1 + M 2  2 1 σ ( E , θ) = , (12.53) 16 E 2  M 2  sin 4 (θ/ 2) which can be converted into an energy transfer cross section by noting that 2 πd (cos θ)  4π σ(E ,T ) =  σ ( E , θ) = (12.54) σ ( E , θ) Λ  dT  where λ is the energy transfer parameter. Combining Equations (12.53) and (12.54) yields  M1  1 . σ ( E , T ) = πZ 12 Z 22e 4 (12.55)  M 2  ET 2

540 Light Water Reactor Materials

The Rutherford cross section decreases with increasing PKA energy. Note also that small scattering angles (which correspond to small values of the transferred energy T) are strongly favored. The Rutherford cross section is based on an unrealistic unscreened potential, which causes the energetic particle to interact with the nuclei in the material as if the electrons did not exist. In reality, particle-atom Coulomb interactions are screened by the electrons, and the degree of screening decreases with increasing particle energy. So far, we have examined the stopping process from the point of view of the energetic particle as it deposits energy in the solid. While the electronic energy loss does not lead to permanent accumulation of damage, the nuclear energy loss can displace atoms from their equilibrium lattice positions. This process by which the deposited energy results in permanent atomic displacements is described in the next section.

12.4 The Displacement Process 12.4.1 Threshold displacement energy As a result of scattering by energetic particles, atoms in the solid are displaced from their equilibrium lattice positions, creating a vacant lattice site and a self-interstitial atom (SIA). This vacancy-interstitial pair is called a Frenkel pair (FP). Normally, in metals, the struck atom is not the one that eventually ends as a SIA; instead, the struck atom starts a replacement collision sequence (RCS) along one of the close-packed crystallographic directions (Fig. 12.9). If the initial energy of the struck atom is too low, the collision sequence propagates along the close-packed direction but then returns each atom to its original location; if the energy is high enough, the RCS returns only partway to the original struck atom location. At the end of the chain of displaced atoms, a SIA is formed and a vacant lattice site remains at the site of the original collision of the PKA. In between,

Radiation Damage 541 Vsr –

e

[110]

d

Energetic particle

“Explosion”

RCS

Focuson

FIGURE 12.9: Schematic representation of the atomic displacements involved in the creation of a permanent Frenkel pair [3].

each atom displaces its neighbor in a “domino” effect. The end result is the creation of a Frenkel pair. In the recovered part of the collision sequence (called a focuson; right side of Fig. 12.9), only energy is transferred along the line of close-packed atoms, but no atom is permanently displaced. The minimum energy required to sufficiently move the atoms so that they do not return to their initial sites is termed the displacement energy Ed. The magnitude of Ed depends on the crystallographic direction of the displacement chain. The minimum chain length to create a permanent Frenkel pair is that which avoids athermal recombination (i.e., an atomic jump that occurs without requiring thermal energy). The cross-shaped enclosure in Figure 12.9 shows schematically one such recombination volume. A vacancy created anywhere inside this volume spontaneously recombines with the interstitial shown at its center. Figure 12.9 illustrates that the displacement energy depends on the crystalline orientation. This has been demonstrated, both experimentally and computationally. Figure 12.10 shows the measured variation of Ed with crystalline orientation in pure Cu. The displacement energy is lowest along

542 Light Water Reactor Materials

the low-index crystallographic E (eV) 80 directions such as [110] and [100] 60 (∼20 eV in this case) but highest 40 along [111] (∼80 eV). 20 Because of the intrinsic uncer0 [100] tainties of displacement measurements and calculations, it is customary to use an average dis[110] placement energy. For metals, the average displacement energy is on FIGURE 12.10: Measured displacement energies the order of 20 to 40 eV. Compila- for various RCS directions in Cu [8]. (Redrawn from Wayne E. King, K. L. Merkle, and M. Meshii, tions of displacement energies for Physical Review B, 23, 6319–6334, 1981. American various materials can be found in Physical Society.) [9], but it has been proposed that 40 eV be used whenever more precise knowledge does not exist [10]. For intermetallic compounds and ceramics, in which sublattices and chemical ordering exist, the displacement energies can be much higher and can vary widely depending on the atomic species or sublattice considered [11; 8]. For example, the average displacement energies for the two atom types in UO2 are quite different, Ed ∼25 eV for O2− and Ed ∼ 40 eV for U4+. In addition, point defects of one type can be unstable, decaying to the other type of defect. This is also true in solid solutions where displacements of matrix and solute atoms occur at comparable rates, but if their formation energies differ widely, one type of interstitial converts to the other type, which causes the interstitial population to be enriched in one or the other type of atom. d

[111]

12.4.2 Displacement cascade and the final damage structure The simple picture of isolated atomic displacements described earlier does not hold at much higher PKA energies. As the PKA energy is divided between many atoms, a displacement cascade is formed. This is a region

Radiation Damage 543

where the dissipation of PKA energy causes many atoms to be displaced from their lattice positions. In fact, in such a case, so many atoms in a small region participate in the dissipation of energy that the very notion of crystalline lattice becomes difficult to define. In the displacement cascade region, a “thermal spike” can be defined as the region in which for a brief few picoseconds the average kinetic energy of the atoms corresponds to a “temperature” above the melting temperature. Figure 12.11 shows the results of a computer simulation at the end of a displacement cascade (see also Ch. 15). Extensive clustering of similar point defects occurs in the cascade core, as well as recombination of interstitials and vacancies, thereby reducing the total damage. The actual defect distribution left in the debris of the cascade should be considered. In the damage calculations later in this chapter, we divide the PKA energy equally among all atoms in the cascade, counting all energy transfers above Ed as permanent atomic displacements. In reality, because (a)

(b)

MD cascade simulations in iron at 100K: peak damage

Y Y 10 keV 100 keV

5 keV X Z

X

Z 5 keV – 0.26 ps 10 keV – 0.63 ps 100 keV – 0.70 ps

FIGURE 12.11: Defect configuration after (a) a 50-keV displacement cascade in Fe, interstitials in green, vacancies in red, (b) cascade substructures for PKA of different energies in Fe with 5 keV blue, 10 KeV green, and 100 keV red. (Courtesy R. Stoller, Oak Ridge National Laboratory.)

544 Light Water Reactor Materials

of the close proximity in which these defects are created, they interact with each other, creating defect clusters and/or restoring the undamaged lattice. In the latter case the final number of interstitials and vacancies is smaller than the total number of atoms displaced in the cascade. Since the subsequent microstructural evolution under irradiation depends on the actual number of defects and defect clusters created, it is useful to consider in detail the physical processes occurring in the cascade as it is created and cools. The typical energy given to a PKA (on the order of keVs to tens of keVs) is far in excess of thermal energies (∼10−2 eV). Because the PKA interacts with the surrounding atoms via a screened Coulomb potential, this energy is quickly shared with other atoms, which in turn displace other atoms, until the energy per atom is smaller than the displacement energy. This process, whereby atoms in close proximity to the PKA receive large amounts of energy during a short time, is what originates the displacement cascade. As shown in Table 12.1, the initial energetic particle–target atom energy transfer occurs in approximately 10−15 s (10−3 ps). During the following several picoseconds, the PKA shares its energy with other nearby atoms through successive collisions. Once a few tens of atoms participate in the cascade, a “temperature” calculated based on their average kinetic energy is far in excess of the melting temperature. Molecular-dynamics simulations (Ch. 15) show that during the first few picoseconds, the atomic configuration in the thermal spike associated with the displacement cascade is similar to that of a molten drop of the material [9]. However, the thermal spike lasts only a few picoseconds, since the solid surrounding the cascade represents a very large thermal reservoir that quickly dissipates the thermal energy, in effect quenching the cascade. Because of the very large difference between the PKA energies and thermal energies, the region in the material affected by the displacement cascade explores nonequilibrium crystal structures and defect configurations that are otherwise

Duration (ps) 10−3 10−3– 0.2

Event Transfer of energy from energetic particle Slowing down of PKA, generation of a displacement cascade and thermal spike

Result Primary knockon atom (PKA) Recoil atoms and lattice vacancies; formation of subcascades

0.2–3.0

Thermal spike cooldown

Stable interstitials (SIA); interstitial clusters; atomic mixing; athermal defect recombination

3–10

Cascade cooling to bulk solid temperature Thermal migration of defects and interaction with sinks

Depleted zone in cascade core

> 10

Microstructure evolution (segregation, precipitation, second phase dissolution, dislocation loop formation, etc.), leading to radiation effects (swelling, creep, embrittlement, hardening, growth, etc.)

Parameters σ(En,E) = cross section for particle energy En to transfer of energy E Ed = displacement energy υNRT = number of displaced atoms T = energy transferred to recoils dpa = displacements per atom rpa = replacements per atom υ(T) = number of stable point defects fj = clustering fraction for cluster size j rpa = replacements per atom arc-dpa = athermal recombination corrected dpa Vacancy loop collapse probability Tirr = irradiation temperature E im = migration energy for point defect i; i = SIA and vacancy Kji = strength of sink j for defect i FMD = Fraction of defects free for long-range migration

TaBlE 12.1: Chronology of events during the slowing down of an energetic PKA and its associated displacement cascade (adapted from [3]).

546 Light Water Reactor Materials Displacements within the cascade

I + V => null Recombine with opposite defect => Annealing

V + V => 2V; I + I => 2I React with similar defect Clustering

V, I Remain as free Point defects

FIGURE 12.12: Processes following defect creation in a displacement cascade.

inaccessible. This can cause the creation of nonequilibrium defect structures and phases that are not present in purely thermal conditions. During the “ballistic” or displacive part of cascade development, υ(T ) lattice atoms are displaced from their sites. υ corresponds approximately to the value calculated from the Norgett-Robinson-Torrens model (Sec. 12.5). As illustrated in Figure 12.12, three possible fates await the defects created by irradiation: annihilation (either by recombination or absorption at a defect sink), clustering (whereby they combine with like defects to form higher-order clusters), or remaining as individual point defects. As discussed in Chapter 13, the defect structure created and defect mobility at the irradiation temperature of interest determine the microstructural changes occurring under irradiation. In order to quantitatively evaluate radiation damage to solids, the number of displacements per PKA as a function of its energy E is calculated in the next section.

12.5 Displacements per PKA The final damage state results from the PKA interaction with its neighbors, causing many collisions and atomic displacements followed by intracascade clustering and recombination. The spatial distributions of the interstitial and vacancy cluster defects are not homogeneous. Because the newly created interstitials are energetic atoms, they are expelled from the center of the cascade via replacement collision sequences (Fig. 12.9), and a vacancyrich core forms, along with an interstitial-rich outer shell. This physical

Radiation Damage 547 Step

0

1

2

Average energy per atom

E

E/2

E/4

# of additional atoms in cascade

1

2

4

3

............

8

............

nF

Total

E 2n F

2Ed

2nF

E/2Ed

FIGURE 12.13: Derivation of the Kinchin-Pease formula.

separation between interstitials and vacancies enhances defect clustering and minimizes recombination. When the cascade cools, the final damage state contains a substantial number of point defects and point defect clusters. After this stage, the long-term interactions of the radiation damage with the remainder of the solid determine the evolution of the microstructure (Ch. 13).

12.5.1 The Kinchin-Pease (K-P) model The total number of displacements for a PKA energy E can be estimated by the following derivation attributed to Kinchin and Pease [12], and illustrated in Figure 12.13. If the initial PKA energy E > 2Ed, the PKA causes further atomic displacements in the solid. After the first collision, the number of displaced atoms is 2, and their average energy T is E/2. After the second collision, four atoms are now involved, and T = E/4. In general, after n collisions, 2n atoms are involved and T = E/2 n . The condition for the displacement cascade to end is that T ≤ 2 E d ; if the energy is more than Ed and less than 2Ed, the PKA can transfer enough energy to displace an atom, but the displaced atom is replaced by the PKA, so no new defects are produced. Thus, the condition

548 Light Water Reactor Materials

E , (12.56) 2nF where nF is the final number of collisions, determines the end of the cascade. But after nF steps, υ = 2nF, and the number of displacements is E υKP ( E ) = . (12.57) 2E d This is the Kinchin-Pease formula [7]. The arguments leading to Equation (12.14) and adopted in the K-P model include a sharp cutoff at an energy E∗, such that above E∗ all energy loss is electronic and below E∗ all energy loss is by elastic collisions with atoms. Thus, in the Kinchin-Pease model the displacements caused by a PKA energy E are given by 2E d =

E∗ 2E d E υ( E ) = 2Ed υ( E ) = 1 υ( E ) = 0 υ( E ) =

if E > E ∗ if 2 E d < E < E ∗ if E d < E < 2 E d if E < E d .

The displacements predicted by this model are shown in Figure 12.14 (solid lines) where the displacements per PKA against PKA energy are plotted in a log-log scale. Because the transition from electronic stopping to nuclear stopping occurs at different energies for different materials, the transition occurs at a higher energy for U than for Cu and C.

12.5.2 lindhard-Scharff-Schiott (lSS) energy partition model In contrast with the sharp partition used by the K-P model, the LSS model uses a more realistic potential (Thomas-Fermi) than the hardsphere approximation to predict the partition between electronic and nuclear stopping. This model provides a smooth transition from the

Radiation Damage 549 E*/2Ed

Displacements per PKA

10000 1000 100 10

Lindhard C Lindhard Cu Lindhard U KP C KP Cu KP U

Ed 2Ed

1 0.1 –5 10

10

–4

10

–3

10

–2

10

–1

0

10

PKA Energy (MeV)

FIGURE 12.14: Displacements in the Kinchin-Pease (solid lines) and the LSS (dashed lines) models in carbon, copper, and uranium.

electronic-loss–dominated regime to the nuclear-stopping regime [27]. Assuming that a fraction of the nuclear-stopping part of the deposited energy is converted into displacements, the number of displacements given by the LSS model, υLSS, is E υ LSS ( E ) = ζ ( E , Z ) (12.58) 2E d ζ ( E , Z ) = damage efficiency =

1 1 + 0.88 Z 1/6

(3.4 ε1/6 + 0.4 ε 3/4 + ε)

(12.59)

where Z is the atomic number, ε = E/ 2 Z 2e 2 /a ; and a = screening radius = 2a B / ( Z 12/3 + Z 22/3 )1/2 with aB the Bohr radius. In the case that the PKA is the same as the struck atom, the screening radius is a = aB/Z1/3. The damage efficiency ζ(E,Z) originates from a numerical solution of the energy partition between electronic and nuclear stopping. This solution is plotted in Figure 12.15 and approximated by Equation (12.57). Because this model predicts a smooth transition rather than a sharp cutoff, there are

550 Light Water Reactor Materials

Damage efficiency (z)

1.0 0.8 U Au

0.6

Nb Cu AI

0.4

C Be

0.2 0 101

102

103 104 PKA energy (E) (eV)

105

106

FIGURE 12.15: Damage efficiency as a function of PKA energy for various elements. The dashed line shows the nuclear-stopping limit [11].

less displacements below E∗ and more displacements above it. This is shown in the dashed and dotted lines in Figure 12.14 for each element. As seen in Figure 12.15, the damage efficiency decreases slowly with PKA energy, and decreases more steeply for light elements than for heavy elements. According to Equation (12.14), the electronic-stopping cutoff (dashed line in figure) occurs at different energy levels for different elements. For U, the damage efficiency is higher than 0.7 at the cutoff, while for Be, it is lower than 0.5.

12.5.3 The Norgett-Robinson-Torrens (NRT) Model The above derivations do not take into account the fact that real collisions are not between hard-spheres, as assumed in the Kinchin-Pease model; as specified by Lindhard, instead of the displacement process exhibiting a sharp cutoff at E∗, the partition of nuclear and electronic stopping varies smoothly with energy. Consideration of these two factors was given in deriving the Norgett-Robinson-Torrens formula [13] for the number of displacements caused by a PKA energy E, υNRT.

Radiation Damage 551

The nuclear and electronic partition is evaluated by using the concept of damage energy Tdam, which is the fraction of the recoil energy that can cause displacement damage. This is given by Tdam = where

E [1 + k L g (ε)]

(12.60)

3/2 1/2 32  me  [ 1 + ( M 2 /M1 )] Z12/3 Z 1/2 2 kL = 2/3 2/3 3/4 3π  M 2  ( Z1 + Z 2 )

(12.61)

where M1 and Z1 indicate the mass and atomic number of the incident atom. The same quantities with the subscript 2 stands for the target atom. If the target and projectile are the same atom, then kL and where with

ε=

 = 0.1337 Z 1/6

Z  M

1/2

,

(12.62)

g(ε) = ε + 0.40244 ε 3/4 + 3.4008ε1/6

(12.63)

( M1 + M 2 ) 2 E E a = 2 2 a12 . Z1 Z 2e 2 12 M 2 Ze

(12.64)

In equation (12.65) e is the electron charge and the screening length a12 is given by 1/3 aB 9π2   a12 = (12.65) 1/2  128  2 Z 1/3 where aB is the Bohr radius. Using Equation (12.60) for calculating Tdam, the displacements caused by the PKA energy E are υ NRT ( E ) ≅ 0.8

Tdam 2E d

(12.66)

552 Light Water Reactor Materials

with Tdam given by Equation (12.62). This formulation is used in codes to calculate the displacements per atom for a given neutron spectrum. The K-P and NRT formulas have various limitations, for example, in taking no account of collisions between atoms of different masses (for calculations of displacements in polyatomic solids, see [14]) or of the variation of displacement energy with crystalline orientation. Appropriate accounting needs to be taken of electronic excitations, and there are approximations in the method above. However, the greatest value of the NRT model and of other standard damage parameters (see Sec. 12.7) is not that it gives exact numbers of displacements but, rather, that it provides a unified and consistent physically based damage parameter that can be used to compare different types of irradiations. In the following section, the models above are used to calculate the damage produced by neutron irradiation.

12.6 Displacements per Atom Caused by Neutron Irradiation In this section, the atomic displacement rate caused by a given neutron flux incident on the material of interest is calculated. The displacement rate is in units of displacements per atom (dpa) per second, which is simply the volumetric displacement rate (displacements per second per unit volume) divided by the atom density. In the reactor, neutrons and gamma rays bombard the nuclear fuel and reactor components. To evaluate the displacement rate k (dpa/s) in a neutron flux spectrum φ(En), the displacement cross section σd(En) is needed. The displacement crosssection is ΛE n

σ d (En ) =

∫ σ s (En , E)υ(E)dE Ed

(12.67)

Radiation Damage 553

where λ is the energy transfer parameter between neutrons and atoms in the solid and En is the neutron energy. Once this is known, the displacement rate k is given by ∞

k=

∫ Ed

σ d ( E n )φ( E n ) dE n

(12.68)

.

Λ

The lower limit of the integral represents the lowest neutron energy capable of transferring Ed to the atoms in the solid. Assuming Ed 0.255MeV



.

562 Light Water Reactor Materials

To calculate the damage due to the Compton electrons, we need the differential cross section in terms of the electron energy, not the energy of the scattered gamma ray. These two differential cross sections are related by σ s ( E γ , E e )dE e = σ s ( E γ , E γ′ )dE γ′

or

σ s ( E γ , E e ) = σ s ( E γ , E γ′ )

dE γ′ dE e

= σ s (E γ , E γ − Ee )

.

(12.86)

The last equality results from taking the differential of Equation (12.83) and using the same equation to eliminate E γ′ . Therefore, σ s ( E γ , E e ) is obtained by replacing E γ′ by Eγ – Ee in Equation (12.85) and on the left side of Equation (12.84) and eliminating θ between these two equations. From the analog of the neutron case, the displacement cross section for a photon of energy Eγ can now be calculated for the electron produced by Compton scattering (Eq. [12.84]) and weighting the probability of a Compton electron with the Klein-Nishina formula (Eq. [12.85]) converted to electron energy by Equation (12.72): Eγ

σ d (E γ ) =

∫ Ed /Λ

σ s (E γ , Ee ) σ s (E γ )

ΛE e

dE e

∫ σ s (Ee , E)υ(E)dE

(12.87)

Ed

where from Equation (12.10) with M1 = me and M2 = M, the mass number of the atoms in the solid, Λ = 2.2 × 10−3/M2. Sufficient energy transfer to cause a displacement in iron, for example, requires Ee ≥ 0.65 MeV, assuming a displacement threshold Ed = 25 eV. The total Compton scattering cross section for a gamma ray of energy Eγ is Eγ

σ s (E γ ) =

∫ me

c 2 /2

σ s ( E γ , E e ) dE e

.

(12.88)

Radiation Damage 563 Displacement cross section (barn)

100

10

1 Compton Photoelectric

0.1

Pair production TOTAL

0.01

0.001 0

4

8 12 Gamma energy (MeV)

16

FIGURE 12.19: Gamma displacement cross section as a function of gamma energy in iron [20].

Electron–atom scattering can be assumed to be isotropic in the centerof-mass system, so the differential energy-transfer cross section is the electron analog of Equation (12.70) for neutrons: σ s (Ee ) σ s (Ee , E) = (12.89) ΛE e where σs(Ee) is the total scattering cross section for electrons and lattice atoms. To cause one or more displacements, the electron must interact with the bare nuclei of the atoms. The interaction for this process is purely Coulombic, so the cross section is given by the Rutherford formula, Equation (12.53). Figure 12.19 shows the displacement cross sections calculated for Fe for all three sources of energetic electrons. It is clear that the dominant component is Compton scattering. Finally, the dpa rate due to a specified photon spectrum φ(Eγ) is, by analogy to the neutron displacement damage in Equation (12.68) neutrons, given by: ∞

kγ

=

∫ E γ min

σ d ( E γ )ϕ( E γ ) dE γ

(12.90)

564 Light Water Reactor Materials

where Eγ min is the photon energy that will just deliver energy Ee/Λ to the struck electron, which in turn is just capable of imparting Ed to a lattice atom. Eliminating E γ′ between Equations (12.83) and (12.84), Eγ min is E γ min 1 = B ( 1 + 2/B + 1) me c 2 2 where

B=

Ed

Λ

me c 2

.

(12.91) (12.92)

If B were very large (which it is not), the result would be Eγ min  = Ed/Λ. For iron, B = 1.28, and Equation (12.77) gives Eγ min = 0.85 MeV. Inside a reactor core, and in most cases for the pressure vessel as well, the displacement rate caused by the gamma flux arising from both fission products and activation products decaying to their stable species is much smaller than the neutron elastic collision displacement rate (typically < 1%). In a few cases, however, especially when a large water gap exists between the core and the pressure vessel to thermalize the fast neutrons leaking from the core, kγ can be of the same order of magnitude as kn as in Equation (12.72). One well-known case relates to the accelerated embrittlement of pressure vessel materials observed in the High Flux Isotope Reactor (HFIR) at Oak Ridge National Laboratory in the early 1990s [21]. The measured levels of embrittlement at the pressure vessel were found to be higher than expected for the levels of damage calculated by the NRT model (which considers only neutron damage). It was then realized that because of the large water gap, the gamma displacements were comparable to the neutron displacements. Taking the gamma displacements from Equation (12.90) into account removed the apparent discrepancy. Gamma-induced displacement damage is also important in situations where not only the total amount of damage but also the damage structure (in particular, the freely migrating

Radiation Damage 565

defect fraction) is important in evaluating the effects of irradiation. This will be discussed in more detail in the following section and in the next chapter.

12.8.2 Thermal neutron reactions Thermal neutron reactions can also cause damage to reactor components. Having absorbed a thermal neutron, a nucleus undergoes radioactive decay, typically emitting an energetic gamma or alpha particle. The emitted particle can cause damage. By conservation of momentum, the decaying atom recoils in the opposite direction, often with an energy sufficient to cause further displacements. The specific reaction depends on the atoms present, but several reactions are possible, including (n,α), (n,γ), (n,n′), (n, p), etc. One example, for gamma decay in Fe, is shown below. Example #5: Displacements due to neutron activation of iron Iron can be activated by absorbing a thermal neutron according to 56 Fe + n → 57 Fe + γ (7MeV) 26 26

(12.93)

By conservation of momentum Pγ = E γ /c = PFe (c is the velocity of light) and the energy of the recoil nucleus of mass M is E Fe = PFe2 / 2 M , which is 460 eV. This is enough to cause nine displacements according to Equation (12.57) if Ed = 25 eV. For a thermal neutron flux of 1013 n.cm−2.s−1 and a thermal cross section for the (n,γ) reaction in iron equal to 3 barns, the displacement rate would be 2.7 × 10−8 dpa/s, which is quite a bit lower than the displacement rate by elastic collisions with fast neutrons. Note that the displacements caused by the 7-MeV photon would also have to be considered here.

12.8.3 Inelastic scattering Inelastic scattering involves forming a compound nucleus by absorption of a fast neutron by the nucleus of an atom and subsequent emission of

566 Light Water Reactor Materials

a neutron of lower energy. This reaction is called inelastic scattering or (n, n′). It only occurs above a certain threshold energy of the incident neutron, which for typical metals is above 1 MeV. Because the fission spectrum contains a considerable fraction of neutrons of energy above 1 MeV, the (n, n′) reaction is a significant damage mechanism. The threshold neutron energy for inelastic scattering for an atom mass M is M +1 E nth = E (12.94) M ex where Eex is the excitation energy, or the energy level above the ground state that is populated by the collision with the incident fast neutron. The threshold energy being greater than the excitation energy is a consequence of momentum conservation. The analog of Equation (12.9) for inelastic scattering is  1 E nth  E nth 1/2  1 E = ΛE n 1 − 1 −  cos θ . 2  2 En  En  

(12.95)

The maximum energy transferred corresponds to cos θ = −1 and the minimum to cos θ = 1, so the analog of Equation (12.11) is E max,min =

  1 1  α 1− ± 1− α  2   2α 

ΛE nth

(12.96)

where α = E nth /E ex . Since inelastic scattering involves the formation of a compound nucleus, the emitted neutron is isotropic in the CM. The analog of Equation (12.70) is σ in n (En , E) =

σ nin ( E n )

E nth ΛE n 1 − En

,

(12.97)

Radiation Damage 567

and the displacement cross section for inelastic scattering is σ ind ( E n ) =

σ in n (En )

E th ΛE n 1 − n En

E max

∫ υ(E) dE

.

(12.98)

E min

Example #6: Displacements from inelastic scattering of neutrons from iron The inelastic scattering–induced displacements become important above 1-MeV neutron energy. Reference [22] calculates the inelastic displacement cross section at approximately 500 barns and at 2200 barns at 10 MeV. Assuming a monoenergetic flux of 1013 n.cm−2.s−1 at these energies would give displacement rates of 5 × 10−9 dpa/s and 2.2 × 10−8 dpa/s, respectively. Both are much smaller than in reactor displacement rates caused by fast neutrons.

12.9 Charged-Particle Irradiation For the sake of completeness, charged-particle irradiation (electrons and ions) generated by accelerators is discussed in this section. The motivations for using charged-particle irradiation in experimental radiation damage studies are as follow: 1. The ion irradiation displacement rates are orders of magnitude higher than those achievable under neutron irradiation, so equivalent doses (in dpa) are attained in hours instead of years. 2. The effects of experimental variables such as temperature and dose rate can be explored with greater ease. 3. Ion- or electron-irradiated samples are less radioactive than samples irradiated with neutrons, making them easier to handle. The disadvantage of accelerator-produced particle irradiation is that the damage is produced within, at the most a few tens of microns of the surface. The free surface is a sink for point defects and has to be taken into account.

568 Light Water Reactor Materials

The microstructures obtained from such experiments do not necessarily have a one-to-one correspondence with the results of neutron irradiation, nor should such a close correspondence be expected, as the irradiations are quite different; in particular, ion irradiation has a much higher dose rate, and often has to be run at higher temperatures to obtain equivalent microstructures [23, 24]. Table 12.2 shows the differences between the different types of irradiation for typical values of irradiation parameters in reactors, accelerators, and electron microscopes.

12.9.1 Electron irradiation Energetic electrons include Compton electrons, beta particles from nuclear disintegrations, and electrons produced by an accelerator. These electrons create atomic displacements by direct scattering from atomic nuclei. Electron irradiation differs from neutron and ion irradiation in two ways: (i) even for electron energies in the MeV range, the energy of the PKA created by an electron-atom collision is not much greater than the displacement energy, making displacement cascades rare, and (ii) the electron velocities are high enough to require relativistic treatment of collision dynamics. The maximum energy transferred to a nucleus struck by an electron is given by 1 E E max =  e 2 + 2 ΛE e (12.99) 2  me c  instead of the classical result Emax = ΛEe used in the upper limit of the integral in Equation (12.87). The minimum electron energy to cause displacements is arrived at by setting Emax= Ed in Equation (12.99) and rearranging 2 E /Λ E e min = me c 2 −1 + 1 + d 2  . me c  

(12.100)

Fast Neutron 5 × 1013 Typical flux −2 −1 (LWR core) (particle⋅cm ⋅s ) Displacement rate (dpa/s) 10−7 Irradiation time to 1 dpa ~4 months Temperature Reactor temperature Macroscopic gradients of Small gradient; damage follows neutron flux attenuation Sample volume irradiated Bulk (homogeneous over large volumes, whole components irradiated) Microscopic spatial Inhomogeneous (dense distribution of damage cascades present) Freely migrating defect 1−5 production (% of NRT value)

Ion 1011−1012 (Ion accelerator) 10−4−10−5 4 to 40 hours adjustable Sharp nuclear-stopping peak at the end of ion range

Electron 5 × 1019 (High voltage electron microscope) 10−2–10−3 2 to 20 minutes adjustable Homogeneous in thin foil, but sharp lateral gradients (beam spot)

Thin foil (100 nm thick, beam is Near surface (on the order ~1 µm2) of 1–10 µm for typical 2 irradiations, laterally ~1 cm ) Inhomogeneous (variable Homogeneous (mostly isolated density cascades) Frenkel pairs) ~10−50 ~100

TaBlE 12.2: Comparison of neutron, ion, and electron irradiation in metals Note: For ion and electron irradiation, typical fluxes are given for ion accelerators and for high-energy electron microscopes when the beam is condensed.

570 Light Water Reactor Materials

If the second term in the square root is small compared to unity (that is, if the energy is small compared to relativistic energies), a Taylor series expansion gives the classical result Eemin = Ed/Λ.

12.9.2 Ion irradiation The principal difference between irradiation with ions and with neutrons is that the former interacts with atoms with a cross section on the order of 10−16 cm2, whereas typical neutron–nucleus cross sections are on the order of 10−24 cm2. The result is that ions lose their energy much faster than neutrons do as they transverse a solid. This difference causes both a higher displacement rate due to ion interactions and a spatial gradient of damage as the displacement cross section increases sharply with decreasing ion energy. Since irradiation effects depend on the balance between irradiation damage and thermal annealing, increasing the displacement rate affects phenomena such as, for example, the peak void swelling temperature. In Chapter 24,4 we show that this temperature is higher under ion irradiation than under neutron irradiation. Ion irradiation illustrates all the interactions of radiation with solids that were presented earlier in this chapter. In particular, the regions of electron stopping and nuclear stopping are illustrated in Figure 12.20a, which is a transmission electron microscopy (TEM) micrograph of a cross section of MgAl2O4 (spinel) irradiated with 2-MeV Al+ ions at 650ºC to a level of 14.1 dpa. The contrast observed is the formation of dislocation loops that are more numerous in the nuclear-stopping region, as seen from a TRIM simulation. The energetic particles penetrate the solid from the left. At the beginning, they interact mostly with electrons, causing very few displacements. After a distance of approximately 0.5 to 1 µm, (corresponding to a decrease of ion [PKA] energy down to the limit between nuclear and electronic 4

Chapters 16 through 29 are to be found in Light Water Reactor Materials, Volume II: Applications.

Radiation Damage 571 (a)

2 MeV Al+

0

1.0 1.5 Depth (mm)

2.0

Ion ranges Ion range = 1.38 um Skewness = 1.5251 Straggle = 1562 A Kurtosis = 7.6788 30000 25000 20000 15000 10000

Layer 1

Atoms/(cm3) / atoms/(cm2)

(b)

0.5

0A

(c)

5000 Target depth

0 2 um

Energy to recoils Kinchin-Pease damage calc.

eV/ion/angstrom

35 30 25 20 15 Layer 1

10

0A

5 Target depth

0 2 um

FIGURE 12.20: (a) TEM bright field image showing the variation of displacement damage with depth during 2-MeV Al+ ion irradiation of MgAl2O4 (spinel) [25]. (b) Plot of implanted ion distribution. (c) Displacement damage as a function of distance. Both simulations performed using SRIM [7]. The displacement energy was 40 eV for all atoms, and the K-P approximation was used.

572 Light Water Reactor Materials

regimes), the ion starts interacting with nuclei. This is seen in the center of Figure 12.20c, where the displacement rate increases abruptly, thereby generating a high defect concentration. At the end of the displacement region, the ion comes to rest as a neutral atom (Fig. 12.20b).

12.9.3 TRIM and SRIM codes The TRIM (Transport of Ions in Matter) and SRIM (Stopping and Range of Ions in Matter) codes calculate the interactions of energetic ions with the atoms in solids [2]. Since their inception, these codes have become the standard method of calculating dpa and range of ions in matter due to the ease of use and sound physical basis. The Monte Carlo program in these codes runs a simulation of many histories, during which the probabilities of nuclear and electronic collisions are weighted according to their probabilities, as discussed below. The program does not consider nuclear reactions, the crystal structure of the material (material is disordered [or noncrystalline] with a specified density), or the accumulation of damage (each ion sees virgin material). The program does explicitly account for polyatomic targets and atomic sputtering, and it allows for up to three layers of material with up to four elements in each. The importance of atomic sputtering can also be directly evaluated. No thermal spikes are considered. The inputs are the incident atom’s energy and angle of incidence; target thickness, density and composition; displacement energy for each atom-type in the solid; and the surface energy. The user may choose a detailed calculation with full damage cascades or a K-P-based approximation, which takes the primary recoils and calculates the number of displacements using Equation (12.57). Recently, Stoller and others have shown that for recent versions of SRIM, it is more accurate to use the latter option [26]. The program relies on the development of a universal interatomic potential that depends on ion energy and the masses and atomic numbers of the atoms in question. The potential is of the form

Radiation Damage 573

Z1 Z 2e 2  r  Φ (12.101)  a r where Z1 and Z2 are the atomic numbers of the energetic ion and the target atoms, e is the electron charge, r is the interatomic distance, and a is a screening length that depends on the atomic numbers of the two atoms by the semi-empirical formula: 0.8854 a (12.102) a = 0.23 Bohr Z1 + Z 20.23 V (r ) =

where aBohr is the Bohr radius (the radius of the hydrogen atom, 0.53 Å). Φ is the “universal” screening function determined by fitting of the calculated interatomic potentials of 521 randomly selected element combinations given by 4   r r  Φ = ∑ Ai exp − Bi (12.103)  . a    a   i =1 The remarkable feature of Equation (12.103) is that it depends only on the atomic numbers Z1 and Z2. It is very accurate for high-energy collisions but does not accurately give the number of displacements for collisions less than a few tens of electron volts. This empirical interatomic potential shown in Equation (12.103) has been validated by extensive comparison to experiment. It has been implemented into a code that partitions the energy loss between electrons and nuclei. The procedure is to substitute Equation (12.87) into Equation (12.42) to find a relation between impact parameter and angle θ. At the beginning of each step, the code determines a “free flight” distance in the material equal to the target interatomic spacing at low energies and by a complicated function of the ion energy and scattering-atom density at high ion energies. At the end of the free-flight distance, the ion undergoes a nuclear collision with an atom at a randomly selected impact parameter. The analog of Equation (12.51) for the universal potential determines the

574 Light Water Reactor Materials

energy transferred to the PKA and the scattering angle. If the ion energy after the collision, E i′ = E i − E , or the transferred energy E are lower than a specified displacement energy Ed, the calculation is terminated. If not, the program recalculates the free-flight distance for the two branches (moving ion and struck atom), and the process continues until the energies of all atoms involved fall below Ed. The code keeps track of energy loss, number of displacements, ion range, and various other parameters. The program has been validated by comparison to experiments and is available for free downloading at www.srim.org. Example #7: Calculating dpa doses from SRIM Considering the plot shown in Figure 12.20b, what is the maximum total displacement rate (dpa/s) observed in the spinel sample if the Al+ ion flux is 100 µA/cm2? The density of spinel is 3.6 g/cm3. In the example given, atoms of Mg, Al, and O are displaced; we consider the sum of all displacements in this calculation. The plot in Figure 12.20b shows a maximum value of approximately 0.2 displacements/ion.Å (2 displacements/ion.nm). The ion flux can be obtained by noting that 100 µA/cm2 is 10−3 Coulomb/ s.cm2, which for single-charged Al is equivalent to 10−3/1.6 × 10−19 (ion/ Coulomb), or a flux of 6.25 × 1013 Al+/cm2.s. We now note that for a 1-cm2 irradiated area a 1-Angstrom (0.1 nm) slice (a volume of 10−8 cm3) has a mass of 3.6 × 10−8 g. One mole of MgAl2O4 contains 6.023 × 1023 formula units and weighs 142 g. This slice thus contains 1.52 × 1014 formula units. Since the calculation does not follow the detailed cascades, it is not possible to partition the displacements among the different atoms. Assuming an equal probability of displacement for each atom, we have 1.06 × 1015 atoms in the slice. The value is then 0.2 displacements/ion.Å, which is equivalent to 0.2/1.06 × 1015 = 1.88 × 10−16 [displacements per atom/(ion/cm2.s)], which, when multiplied by the ion flux, gives a peak damage rate of 0.0118 dpa/s.

Radiation Damage 575

References  1. E. P. Wigner, “Theoretical physics in the Metallurgical Laboratory of Chicago,” J Appl Phys 17 (1946): 857–863.  2. C. Claeys and E. Simoen, Radiation Effects in Advanced Semiconductor Materials and Devices (Berlin: Springer, 2002).  3. A. Holmes-Siedle and L. Adams, Handbook of Radiation Effects (Oxford: Oxford University Press, 1993).  4. M. T. Robinson, “Basic physics of radiation damage production,” J Nucl Mater 216 (1994): 1–28.  5. R. E. Stoller, “1.11 – Primary Radiation Damage Formation,” Comprehensive Nuclear Materials, ed. R. J. M. Konings (Oxford: Elsevier, 2012) 293–332.  6. D. R. Olander, Fundamental Aspects of Nuclear Reactor Fuel Elements, TID-26711-P1, National Technical Information Service (1976)  7. J. Ziegler, J. P. Biersack, and U. Littmark, The Stopping and Range of Ions in Matter (New York: Pergamon Press, 1985).  8. W. E. King, K. L. Merkle, and M. Meshii, “Determination of the threshold-energy surface for copper using in-situ electrical-resistivity measurements in the high-voltage electron microscope,” Physical Review B 23 (1981): 6319–6334.  9. P. Jung, “Atomic displacement functions of cubic metals,” J Nucl Mater 117 (1981): 70–77. 10. ASTM, “Standard Practice for Neutron Radiation Damage Simulation by Charged-Particle Irradiation,” E521–96, American Society for Testing and Materials Standard Practice (1996). 11. S. J. Zinkle and C. Kinoshita, “Defect production in ceramics,” J Nucl Mater 251 (1997): 200–217. 12. G. H. Kinchin and R. S. Pease, “The displacement of atoms in solids,” Reports on Progress in Physics 18 (1955): 1–51. 13. M. J. Norgett, M. T. Robinson, and I. M. Torrens, “A proposed method for calculating displacement dose rates,” Nuclear Engineering and Design 33 (1975): 50–54.

576 Light Water Reactor Materials

14. D. M. Parkin and C. A. Coulter, “Total and net displacement functions for polyatomic materials,” J Nucl Mater 101 (1981): 261–276. 15. “Evaluated Nuclear Data File VI (ENDF-VI),” Brookhaven National Laboratory (2006). 16. L. R. Greenwood and R. K. Smither, “SPECTER: Neutron Damage Calculations for Materials Irradiations,” ANL/FPP/TM-197, 1985, Argonne National Laboratory. 17. K. Nordlund et al., “Primary Radiation Damage in Materials,” OECD NEA (2014). 18. L. E. Rehn and R. C. Birtcher, “Experimental studies of free defect generation during irradiation—implication for reactor environments,” J Nucl Mater 205 (1993): 31. 19. R. D. Evans, The Atomic Nucleus (New York: McGraw-Hill, 1955). 20. J. Kwon and A. T. Motta, “Gamma displacement cross-sections in various materials,” Annals of Nuclear Energy 27 (2000): 1627–1642. 21. I. Remec et al., “Effects of gamma-induced displacements on HFIR pressure vessel materials,” J Nucl Mater 217 (1994): 258–268. 22. D. G. Doran, Irradiation Effects 2 (1970): 249. 23. L. K. Mansur, “Theory of transitions in dose dependence of radiation effects in structural alloys,” J Nucl Mater 206 (1993): 306–323. 24. L. K. Mansur, “Void swelling in metals and alloys under irradiation— assessment of theory,” Nuclear Technology-Fusion 40 (1978): 5. 25. S. J. Zinkle, “Dislocation loop formation in ion irradiated spinel and alumina,” in 15th ASTM International Symposium on Radiation Effects on Materials (Nashville, Tennessee, 1992), STP 1125, 749–763. 26. R. E. Stoller et. al., “On the use of SRIM for computing radiation damage exposure,” Nuclear Instruments and Methods in Physics Research, Section B (Beam Interactions with Materials and Atoms) 310 (2013): 75–80. 27. J. Lindhard, M. Scharff and H.E. Schiott, “Range concepts and heavy ion ranges,” Mat Fys Medd Dan Vid Selsk 33 (14) (1963): 1–42.

Radiation Damage 577

Problems 12.1 Two atoms of the same kind interact with an energy transfer crosssection given by C σ(E ,T ) = ET where C is a constant. What is the probability of scattering with a center-of-mass angle greater than 90 degrees? 12.2 The (n, α) reaction in 5928 Ni releases a prompt α-particle of Eα = 4.8 MeV. (a) Using momentum conservation, calculate the recoil energy of the 5726 Fe product nucleus. (b) Calculate the number of displaced atoms caused by the recoil according to the Kinchin-Pease and NRT models. Assume Ed = 25 eV. (c) Using the K-P model, calculate the displacement rate (displacements/s.cm3) due to this mechanism, in a thermal flux of 1013 n/cm2s, if the density of iron is 7.8 g/cm3 and the thermal absorption cross section for Fe is 2.5 barns. (d) Compare this displacement rate to the displacement rate caused by fast neutrons for a fast flux of 1013 n/ cm2s, ( E n = 1 MeV ), also using K-P. Assume elastic isotropic scattering and a scattering cross section of 3 barns. 12.3 For a monoenergetic fast neutron flux of energy 0.5 MeV, calculate the number of displacements per atom in iron after a fast neutron fluence of 1022 n.cm−2. Compare this calculation with that due to the displacements for a flux of 10-MeV neutrons. The displacement crosssection for the 10-MeV neutrons is 3000 barns. 12.4 An iron primary knock-on atom (PKA) is created with an energy of 100 KeV. According to the Kinchin-Pease model of displacement calculation and the Lindhard electronic-stopping formula (Eq. [12.23]), how far does the PKA travel before starting to interact with the nuclei in the solid?

578 Light Water Reactor Materials

12.5 The figure below shows a portion of a fission fragment track in UO2. At one point, the track changes direction, indicating that the fragment has undergone a nuclear collision with an atom. (This is best seen by looking at a grazing incidence at the track on the page.)

Assuming this is a 10042 Mo fragment with a birth energy of 100 MeV, which has traveled 2 µm from where the fission took place to the place where it collides with the atom: (a) What is the effective charge of the fragment at birth? (b) Before the collision, the fragment loses energy by electronic excitation according to the Bethe formula. Calculate the energy at the point of the collision, assuming the mean excitation energy in the Bethe formula I = 8.8 Z (eV). (c) If the scattering angle on the photograph is 5 degrees, calculate the energy transferred to the struck atom in the case of (1) an oxygen atom and (2) a uranium atom.

Radiation Damage 579

12.6 Calculate the average iron PKA energy in a fission neutron spectrum: φ( E n ) = A exp(− E n )sinh(2 E n )1/2

where En is the neutron energy in MeV. How does this value compare with the approximation of calculating the average PKA energy due to collision with the neutron of average energy? Assume isotropic, elastic scattering and an energy-independent scattering cross section. 12.7 It is desired to evaluate the number of displacements per atom suffered by a steel core shroud and pressure vessel subjected to a neutron flux in a boiling water reactor (BWR). For the purposes of this calculation, assume the material is 100 at% Fe and that the displacement energy is 25 eV. Use the neutron flux provided below, divided into 47 energy groups. Consider that the total iron scattering crosssection is constant and equal to 3 barns. Assume isotropic scattering (the probability of generating a PKA of energy E is independent of E for the energy range considered). For Zr scattering, use the equation σZr (barn) = 10 − 3En(MeV) for En < 2 MeV

and σZr = 4 barn for En < 2 MeV. The methodology to be utilized is the following: (a) Each of these neutron energy groups will generate a distribution of energetic recoils of varying energy. Characterize each group by the total group flux at the average neutron energy in the group. Calculate the average recoil energy for each group. (b) Using the K-P formula, find the displacement cross section for each energy group. (c) Multiply by the neutron flux per group to find the number of displacements per group, and sum them over all energy groups to find the total number of displacements NRT for the material under consideration.

580 Light Water Reactor Materials

En (MeV) Neutron Neutron lower bound Flux BWR Flux PresCore sure Vessel of group Shroud Surface (n.cm−2.s−1) 1.00E−11 7.61E+11 2.87E+08 1.00E−07 1.07E+11 4.53E+07 4.14E−07 3.50E+10 1.69E+07 8.76E−07 3.66E+10 1.84E+07 1.86E−06 5.03E+10 2.65E+07 5.04E−06 3.85E+10 2.13E+07 1.07E−05 6.46E+10 3.69E+07 3.73E−05 5.16E+10 3.05E+07 1.01E−04 3.83E+10 2.30E+07 2.15E−04 3.76E+10 2.26E+07 4.54E−04 6.18E+10 3.90E+07 1.59E−03 3.96E+10 2.66E+07 3.36E−03 4.05E+10 2.72E+07 7.10E−03 3.62E+10 2.29E+07 1.50E−02 1.74E+10 1.77E+07 2.19E−02 6.19E+09 5.61E+06 2.42E−02 7.97E+09 7.14E+06 2.61E−02 1.04E+10 6.36E+06 3.18E−02 1.32E+10 9.11E+06 4.09E−02 3.19E+10 2.55E+07 6.74E−02 3.93E+10 3.46E+07 1.11E−01 4.86E+10 4.43E+07 1.83E−01 5.46E+10 4.67E+07 2.97E−01 3.42E+10 3.69E+07

En (MeV) Neutron Neutron lower bound Flux BWR Flux PresCore sure Vessel of group Shroud Surface (n.cm−2.s−1) 3.69E-01 4.45E+10 4.26E+07 4.98E-01 3.82E+10 3.43E+07 6.08E-01 4.70E+10 4.27E+07 7.43E-01 2.25E+10 1.73E+07 8.21E-01 4.11E+10 3.37E+07 1.00E+00 6.97E+10 5.58E+07 1.35E+00 5.03E+10 3.58E+07 1.65E+00 3.72E+10 2.62E+07 1.92E+00 3.77E+10 2.62E+07 2.23E+00 1.54E+10 1.03E+07 2.35E+00 3.06E+09 2.05E+06 2.37E+00 1.29E+10 8.38E+06 2.47E+00 2.62E+10 1.77E+07 2.73E+00 2.34E+10 1.68E+07 3.01E+00 3.23E+10 2.55E+07 3.68E+00 4.34E+10 4.12E+07 4.97E+00 2.15E+10 2.95E+07 6.07E+00 1.31E+10 2.49E+07 7.41E+00 4.69E+09 1.13E+07 8.61E+00 2.35E+09 7.40E+06 1.00E+01 1.07E+09 4.15E+06 1.22E+01 2.22E+08 1.28E+06 1.42E+01 5.05E+07 3.10E+05 1.73E+01

Radiation Damage 581

12.8 It is desired to estimate the importance of considering polyatomic processes on displacement calculations in Problem 12.7, considering K-P and full damage cascade calculations. To do this, use the TRIM code and compare it to the values calculated using the NRT model. (a) Choose two neutron groups and determine the appropriate maximum PKA energies for these groups, E1 and E2. Divide the energy interval between 0 and E1, and 0 and E2, and derive a set of PKA energies to run TRIM for the two groups. (b) Run TRIM (using full damage cascades) for the set of PKA energies for the two pure elements using the atomic densities of these elements in ZrO2. Obtain from TRIM in each case (1) the number of oxygen displacements and (2) the number of Zr displacements created by the set of PKA energies E, for all intervals between 0 and the group energy—either E1 or E2. (c) Use the displacement values calculated to calculate a weighted displacement rate in the compound. Compare this value with the value obtained in Problem 12.7 for the neutron energy groups in question. (d) Run TRIM for the compound material (2/3 O, 1/3 Zr). Compare the oxygen, zirconium, and total displacement rates obtained with those from part (c). What are the differences? 12.9 It is an experimental observation that during electron irradiation of a material below a given electron energy, no displacements are possible, as predicted by Equation (12.100). Occasionally, it is possible to observe damage even below Ed as a result of secondary displacements through light element impurities. If bcc Fe has a displacement energy of 40 eV and contains carbon, calculate: (a) The minimum electron energy to cause displacements in pure Fe. (b) The minimum electron energy considering secondary displacements of the type electron => C => Fe. What is the maximum Fe recoil energy obtainable from 400-keV electrons through a secondary C displacement mechanism?

582 Light Water Reactor Materials

(c) What purity Fe would be necessary to ensure that secondary displacements are limited to 1/100 of the primary displacements at an electron energy of 900 keV? The displacement cross section for 900-keV electrons is 30 barns in Fe and 20 barns in C. 12.10 Given the following two-region neutron flux incident on a Zr component, calculate the total displacement rate from fast neutron collisions assuming that the number of displacements per recoil is given by the Kinchin-Pease model. Φ (n.cm–2.s–1)

1013

5 ×1012

I II 0.5 MeV

1.0 MeV

En

The displacement energy for Zr is 33 eV. See Figure 12.17 for the scattering cross section of neutrons in Zr; use the average energy for each region. 12.11 Redo the calculation in Example #3, but take into account the variation of the scattering cross section between 0 and 3 MeV, which you can approximate as a linear function. 12.12 Calculate the distance of closest approach for two particles that meet head-on, and whose interaction is governed by the unscreened Coulomb potential (Eq. [12.46]). 12.13 A 40-eV atom of mass M1 strikes a lattice atom of mass M2 = 2M1. (a) What is the probability that the lattice atom is displaced? (b) If the lattice atom is displaced, what happens to the other atom? Assume hard-sphere scattering and a displacement energy of 25 eV.

13

Chapter Microstructure Evolution Under Irradiation 13.1 Introduction As shown in Chapter 12, irradiation of materials by energetic particles creates a population of point defects and defect clusters issuing from the debris of the displacement cascades. After they are created, stable point defects can migrate thermally, and eventually react—that is, be absorbed, annihilated, or cluster—with other point defects, or with fixed defect sinks that can absorb defects and defect clusters, such as dislocations, grain boundaries, bubbles, and voids. During irradiation, because of the high rates of defect generation relative to the concentration of point defects that can be sustained in the lattice under thermal conditions, a large fraction of the defects created by displacement damage are constantly annihilated at the sinks. Since the defects must migrate through the material to arrive at the sinks, one of the main effects of irradiation is to create persistent and significant defect fluxes that permeate

584 Light Water Reactor Materials

the material while it is subjected to the energetic particle flux. One speaks of a “vacancy wind” or an “interstitial wind,” because it is the motion of the defects and the persistence of these high defect fluxes and their interaction with the material microstructure, rather than higher concentrations of the point defects themselves, that most strongly impact the microstructure and, by extension, the macroscopic properties of the material. The interactions of the defects and the defect fluxes with the material microstructure are at the origin of the irradiation-induced processes of microstructural evolution that cause macroscopic irradiation effects, which affect material performance, for example, by changing mechanical properties, material dimensions, or causing phase transformations. In ceramic nuclear fuel, point-defect interactions with fission-gas bubbles strongly affect fuel behavior (Ch. 201). In metals, microstructure evolution is the basis for changes in mechanical properties (Ch. 26), increases in material susceptibility to corrosion and stress corrosion cracking (Ch. 25), and for dimensional instability (Ch. 19 and 27). The relevant physical processes and the rate-theory methodology that permit quantitative description of microstructure evolution under irradiation are presented in this chapter.

13.2 Microstructure Evolution Under Irradiation Figure 13.1 illustrates the processes occurring during exposure of a solid to neutron irradiation that can lead to an evolution of the microstructure. In the following discussion the numbers in parentheses refer to those in Figure 13.1. In the upper left corner, a displacement cascade creates a number of defects and defect clusters (1). The large number of displacements in the cascade can cause new phases to appear where the cascade hits (e.g., precipitation, disordering, amorphization), sometimes 1

Chapters 16 through 29 are to be found in Light Water Reactor Materials, Volume II: Applications.

Microstructure Evolution Under Irradiation 585 Steady-state concentration of interstitials and vacancies

En – T

En

Absorption at dislocation loops

4

Displacement cascade

1

I Absorption at dislocation lines

Vacancy rich core

8

V

Interstitial clustering; loop formation

Loop unfaulting Defect absorption at gbs

5

7

11

Recombination Irradiation-Induced segregation Irradiation induced precipitation

Grain boundary bubble formation

Absorption at incoherent precipitate

13 12

10

14

Grain boundary solute enrichment

9

7

Solute-defect trapping

Cascade recovery

Absorption at voids

Void formation

Vacancy clustering

6 3 Vacancy cluster collapse; loop formation

2

Defect-assisted dislocation climb

Enhanced recombination

15

Precipitate dissolution

16

FIGURE 13.1: Point-defect processes in irradiated materials.

requiring the overlap of two or more cascades (2). The vacancy-rich core in the cascade may collapse into a dislocation loop (3) or remain as a depleted zone (1). After intracascade clustering and defect interaction has produced the final defect configuration (Ch. 12; see also [1]), the remaining isolated defects and defect clusters can then migrate to other defect clusters, dislocation loops, voids, and to preexisting lattice defects, such as the as-fabricated dislocation network, grain boundaries, and fissiongas bubbles in ceramic nuclear fuel. Voids are nearly spherical clusters of vacancies (Ch. 19), and loops are dislocations formed by a planar cluster of either vacancies or interstitials (see Ch. 7). The interaction of the point defects with the sink structure creates a steady-state concentration

586 Light Water Reactor Materials

of defects that is higher than the equilibrium concentration found in the absence of irradiation (4). Vacancies and interstitials react with each other by recombination (5), whereby both defects disappear and the perfect crystal structure is restored locally. The recombination reaction may be enhanced by trapping of defects at solutes (6). Point defects may also interact with defects of the same type, forming defect clusters (7). Defect clustering and absorption in the material alters the original microstructure, creating voids and loops. The sink density and strength of the material are continually modified during irradiation. As shown in Figure 13.1, interstitials and vacancies may also be absorbed at extended sinks, such as the original network dislocations or dislocation loops, whether vacancy or interstitial loops, (8), voids (9), incoherent precipitates (10), and grain boundaries (11). Interaction of point defects with solute atoms (12) may cause solute enrichment or depletion in the vicinity of grain boundaries (13), creating local solute supersaturations, which can lead to the precipitation of new phases (14). At the same time, cascade atomic mixing drives atoms of preexisting precipitates back into solid solution in the matrix (15). Absorption of the defects by dislocations can cause dislocation climb (16), which is one of the mechanisms of irradiation creep. Figure 13.1 includes a mix of “athermal” processes (i.e., rates independent of temperature) such as (2) and (15) and thermally activated processes such as (5), (7), and (8). Microstructural evolution is seen to be a mixture of processes that: (i) depend on long-range thermal migration, such as defect-assisted dislocation climb; (ii) depend on short-range atomic rearrangements; and (iii) are completely athermal. Consequently, the effects of irradiation on the microstructure depend on the balance between the formation of radiation damage and its thermal annealing. The potential energy stored in the material during irradiation is converted to thermal energy in defect-defect and defect-sink reactions.

Microstructure Evolution Under Irradiation 587

The stored energy is then released as lattice thermal vibrations (or phonons). The recombination of interstitials and vacancies is the most obvious example, releasing approximately 5 eV. In this manner reactions such as divacancy formation and defect absorption at dislocations also convert potential energy to thermal energy. The energies of the nuclear particles (neutrons, accelerated ions, fission fragments, beta particles, and gamma rays) passing through the material are so high compared with the thermal energy of solids that processes that would not occur under purely thermal driving forces are enabled in the presence of energetic particle irradiation. This includes irradiationinduced precipitate dissolution and amorphization. Finally, all of the above processes shown in Figure 13.1 occur in parallel, so they compete and interact with each other. For example, the absorption of point defects by sinks causes the sinks to grow, thereby increasing sink strength and reducing point-defect recombination. Mean-Field Reaction Rate theory (hereafter simply rate theory) is an effective tool that was designed for treating the complex phenomena described above and is the subject of the remainder of this chapter. In this model, the microstructure is homogenized so that the kinetic equations become spatially independent.2 In some situations hybrid schemes can also be used, in which some spatial dependence on an outside sink such as a free surface can be preserved (see Eq. [13.82] and [13.83] and Prob. 13.3). Also, although in the simple rate theory formulation, an isotropic solid is assumed, for many phenomena (such as irradiation growth; see Ch. 27), the crystallographic distribution of damage—the one-dimensional or two-dimensional migration of defects and annealing— is often crucial. 2

In spite of this approximation, the spatial correlation of damage can be important. For example, even if defects are completely free after a cascade cools, the probability that a vacancy interacts with the interstitials formed in the same cascade is considerably higher than with other interstitials, so that the recombination is correlated. In the following sections we will bring in spatial inhomogeneities as needed.

588 Light Water Reactor Materials

13.3 Rate Theory of Defect Evolution Under Irradiation The complex interactions of point defects with the microstructure shown schematically in Figure 13.1 are difficult to model explicitly, especially in the vicinity of the extended defects, where point-defect gradients exist. In the rate-theory formulation [2, 3, 4, 5], the bulk defect concentrations, reaction rates, and sink strengths are averaged over the material, in effect “smearing them out” so that the whole solid is homogeneous. The homogenization approach used in this chapter requires solving the spatially-dependent differential equations for the reactions of mobile point defects with other mobile defects and fixed sinks in an effective unit cell and applying the volume-averaged rate over the whole solid. Reaction rates of point defects with other defects and with extended sinks are calculated based on an idealized local geometry relevant to these interactions. These locally calculated defect-sink reaction rates are combined with the sink densities and the defect generation rate to obtain equations describing the microstructure changes in terms of these rate processes. These equations are then used to calculate the average (bulk) concentrations of defects and the consequent microstructural evolution for different irradiation conditions.

13.3.1 Basic rate theory assumption In the classical rate theory model (see also Sec. 20.5.4), the point-defect reactions are modeled as first-order chemical reactions, such that the rate of reaction between species j and species i per matrix atom is reaction rate (s −1 ) = K ij C i C j

(13.1)

where Ci and Cj are the concentrations (site fractions) of i and j atomic species, defined as the ratio of the number of a particular atomic species to the number of sites for that species in the matrix of the material. The reaction rate between atomic species i and j is characterized by a rate

Microstructure Evolution Under Irradiation 589

constant Kij (s−1), which is independent of the defect concentration and generation rate, but could be dependent on temperature, defect geometry, migration path, etc. For defect-sink interactions, the reaction rate is written in terms of the sink strength for the absorption of atomic species j by sink S, denoted by k2j,S (cm−2) reaction rate = k 2j ,S D j C j (s −1 ) (13.2) where Dj is the diffusion coefficient for species j. The square indicates that the rate constants for defect absorption into sinks are always positive. Defect-defect reactions In defect-defect reactions, whether clustering or recombination, the nature of the defect is significantly altered. For example, when two vacancies react to form a divacancy, or point defects are completely eliminated when recombination occurs. Some of the more important defect-defect reactions are written in chemical-reaction form. Vacancy–self interstitial atom (SIA) recombination: K

IV V + I  → null

(13.3)

Divacancy formation: V + V  VV  → V2

K

(13.4)

K

(13.5)

K

(13.6)

Vacancy clustering: V −V2 V + V2           → V3

Di-SIA formation: II I + I  → I2

SIA clustering: K

I−I 2 I + I 2  →I3

(13.7)

590 Light Water Reactor Materials

Defect–fixed sink reactions Alternatively, point defects disappear by absorption into a fixed sink, which is the generic term for larger microstructure features, including spherical defects such as bubbles, pores, and voids (cavities in general); line defects, of which the only example is a dislocation; and planar defects, including grain boundaries and free surfaces. When point defects are absorbed into these larger features, the point defects disappear but the sink retains its character, although slightly modified. For example, beyond a very small radius change, the absorption of a vacancy into a 100-vacancy void does not appreciably alter the void. These reactions can be illustrated as k2

j ,S j + sink        →"modified" sink . (13.8) In this class of reactions, point defect j vanishes and the sink is slightly modified. Examples are: • Arrival of an interstitial at a free surface; the interstitial disappears as a separate entity and adds to a preexisting ledge of atoms or creates an adatom on the surface. • Fission-gas atom absorption by a bubble causes a slight increase in bubble volume and gas pressure in the cavity. • Pickup of an interstitial atom by a void; the void shrinks by one atomic volume. The rate of a point defect–sink reaction per matrix atom is expressed by Equation (13.2): rate per matrix atom = k 2j ,S D j C j

where Dj is the diffusion coefficient of atomic species j and k2j,S is the sink strength of the extended-sink S for defect j (cm−2). The concentrations in this derivation (Cj) are expressed in site fraction (number per atom) but can also be volumetric concentrations (cj) in units of atomic species per unit volume. The two units are related by: C j = c jΩ (13.9)

Microstructure Evolution Under Irradiation 591

where Ω is the atomic volume of the matrix atoms. Most of the radiation damage literature utilizes site-fraction units in point-defect reaction rate expressions, and this convention is followed in this book.

13.4 Reaction Rate Between Point Defects In this section, the individual rates of defect reactions between point defects in the material are calculated for divacancy formation and interstitial-vacancy recombination. Reaction rates between point defects are calculated by identifying the set of lattice sites around one defect, which, if occupied by the other defect, assures reaction. The number of such sites is termed a combinatorial number, generically designated as Z. The combinatorial number Z increases as the energy released in the defect-defect reaction increases. For example, for divacancy formation, the number of assured-reaction sites is equal to the number of nearest neighbors multiplied by the number of second-nearest neighbors from where a jump can occur (as in the fcc divacancy formation derivation below). In contrast, for interstitial-vacancy recombination, the strain fields extend further, and therefore, the defect-defect reaction can occur at larger distances, which entails a larger interaction volume and a higher Z. In general, the rate of reaction is equal to the probability of finding a defect at one jump distance from one or more combinatorial sites times the probability that a reaction jump occurs.

13.4.1 Divacancy formation The initial step of vacancy-cluster formation is the production of a divacancy from two single vacancies, as expressed by the reaction in equation (13.4). This reaction proceeds preferentially from left to right because the formation of a divacancy from two isolated single vacancies reduces the Gibbs energy of the system by the difference between twice the formation energy of a vacancy and the formation energy of a divacancy (Fig. 13.2).

592 Light Water Reactor Materials

E = 2 E Vf

E’= E f2V

FIGURE 13.2: The formation of a divacancy. Vacancies are depicted by a square. The total Gibbs energy of the system decreases with divacancy formation.

The probability per unit time that a vacancy in one of the secondnearest-neighbor positions to a particular vacancy jumps into a nearestneighbor position (thus forming a divacancy) is PVV . The rate of divacancy formation per atom is PVV CV (13.10) where PVV is the product of the number of favorable configurations for a jump, the vacancy site fraction CV, and the jump frequency PVV

= βz 2 CV w

(13.11)

where β is the number of nearest neighbors to the particular vacancy, and z2 is the number of lattice positions around the ring of nearest neighbor from which a second vacancy can jump into a nearest-neighbor site. Then, z2 CV vacancies are available around a particular target nearest-neighbor site to a vacancy to perform a jump that creates a divacancy. For all vacancies in a mole of matrix atoms, the rate of divacancy formation per matrix atom is then βz 2 CV2 w . (13.12) From Chapter 5, the jump frequency w is − E mV   w = υ exp .  k BT  (13.13)

Microstructure Evolution Under Irradiation 593

From the rate-theory formalism expressed in Equation (13.1), the reactionrate constant is KVV = combinatorial number × jump probability = βz 2 w = ZVV w (13.14) or, with Equation (5.25), KVV

=

6 Z VV DV βλ 2

(13.15)

where λ is the vacancy jump distance (on the order of a lattice constant). Example #1: Combinatorial number for divacancy formation rate in the fcc lattice Figure 13.3 shows the relevant geometrical configuration needed to calculate the combinatorial number for divacancy formation for the face-centered cubic (fcc) lattice. There are twelve nearest-neighbor sites in the fcc lattice, and thus, β = 12. For each of these nearest-neighbor sites, there are also twelve nearest-neighbor sites. Of these, one is the original vacancy, and five are shared with that vacancy, so jumps from these sites are eliminated. The remaining seven are sites from which a jump into the nearest-neighbor site would create a divacancy from two separate vacancies, so z2 = 7, and the combinatorial number ZVV for divacancy formation in the fcc lattice is 84. Since both vacancies are equally mobile, the number is actually twice as high, or 168. Similar calculations can be performed for other crystal lattice structures, such as body-centered cubic (bcc), simple cubic (sc), and hexagonal close-packed (hcp). For anisotropic structures, migration energies are directionally dependent, so it is necessary to identify the correct migration direction and migration paths, but the same principles apply.

594 Light Water Reactor Materials ao

Vacancy Nearest neighbor to the vacancy Other lattice sites

FIGURE 13.3: Divacancy formation in the fcc structure [6].

13.4.2 Point-defect recombination The rate constant for vacancy-interstitial recombination is calculated in a similar fashion to the formation of divacancies from single vacancies. Because vacancy-interstitial recombination results in a perfect lattice (defects are annihilated), the amount of energy released is considerably higher than that from divacancy formation, so the number of assured recombination sites is higher than the geometrical size of the defects. As seen in Chapter 12, if an interstitial enters a region near the vacancy called the recombination volume, mutual annihilation occurs without need for thermal motion. Because the energy released upon recombination is quite high (~5 eV), the region around the vacancy where athermal recombination can occur can comprise many hundreds of atoms. By analogy with the divacancy derivation, Section 13.4.1: Recombination rate = number of favorable configurations × jump probability = PIV w

(13.16)

Microstructure Evolution Under Irradiation 595

The number of favorable configurations is computed in a manner similar to that for divacancy formation but in a less straightforward way because of the complex geometry. In this case, the critical rate-determining step is the jump from the outside to the inside of the recombination volume. Thus, z1 is the number of sites within the recombination volume but next to its outer surface and z2 is the number of sites just outside the recombination volume from which a jump could occur to each of the z1 sites. The number of favorable configurations for a recombination jump is then PIV = CV z 1z 2C I = CV Z IV C I .

(13.17)

The jump frequency is the sum for the two defects, w = wI + wV. In metals, the interstitial migration energy is normally much lower than the vacancy migration energy, and so, wI >> wV and w ∼ wI. Since the threedimensional (3-D) configuration of sites z1 and z2 is complicated and crystal-structure dependent, we give a two-dimensional (2-D) example below. Example #2: Interstitial-vacancy recombination number The recombination rate constant is calculated for the schematic 2-D recombination volume configuration shown below. The migration energies and interatomic potentials are such that any interstitial entering the dotted line would athermally recombine with the vacancy in the center (the definition of recombination volume). Only the interstitial is mobile. In Figure 13.4, one could imagine the interstitial replacing any one of the atom locations right outside the recombination volume. One interstitial is shown as a dumbbell on the upper left.. Figure 13.4 shows three different types of recombination sites (labeled 1, 2, and 3); there are four of each, so that is a total of twelve. As seen in the figure, the number of next-nearest-neighbor sites outside the recombination volume is different for each recombination site. For type 1 sites, there are four; for type 2 sites, there are two; and for type 3 sites there is only one.

596 Light Water Reactor Materials

1 2 3

FIGURE 13.4: Two-dimensional configuration of the athermal recombination volume around a vacancy (dashed line), showing the three different types of sites. A dumbbell interstitial (outlined) is shown about to enter the recombination volume.

The combinatorial number is thus 4 (4 + 2 + 1) = 28 possible jumps leading to recombination, and thus, KIV = 28w. Clearly, for a 3-D case, the total possible number of jumps would be much higher, leading to a higher value of the combinatorial number. In the general case, the total number of favorable configurations is given by PIV = C I CV ∑ z 2 j (13.18) all j

where j = all sites z1, which are nearest neighbors to the defect. Thus, the recombination reaction rate is C I CV w I ∑ z 2 j = Z IV C I CV w I ,

(13.19)

all j

and the recombination rate constant is KIV = ZIV wI where ZIV is the combinatorial number (also called the recombination number). The values of the summation Z IV = ∑ z 2 j over all possible configurations can j

range from 50 to 500. Typical recombination numbers could be in the 100s, but each crystal structure has its own recombination volume, which

Microstructure Evolution Under Irradiation 597

tends to extend along close-packed directions. The recombination rate constant is Z (D + D ) Z D K IV = IV I 2 V ≅ IV 2 I . (13.20) λ λ λ

is the diffusive jump distance. The final approximation assumes DI >> DV. The difference in the formulation above from that in Equation (13.14) stems from the fact that not all sites at the edge of the recombination volume are equivalent, as is the case for divacancy recombination.

13.5 Point-Defect Reactions with Extended Sinks An important feature of point-defect behavior in both metals and actinide oxides under irradiation is their removal from the matrix by diffusion and annihilation at defect sinks in the microstructure. Many common sinks are spherical, such as voids and helium bubbles in a metal (Ch. 19) and fission-gas bubbles in oxide fuel. In contrast, dislocations in metals are line sinks for both vacancies and interstitials, and grain boundaries are surface sinks for point defects and for diffusing rare-gas atoms. The determination of the vacancy and interstitial absorption rates into these extended sinks is an important step in determining the material behavior under irradiation. To a first approximation, the voids, bubbles, and dislocations are “perfect” sinks for the point defects. In a perfect sink, all defects stick completely to the sink and never leave; i.e., the sink acts as a black hole. In the analysis, the point-defect concentration at the surface of these sinks is zero, so the reaction rate is given by Equation (13.2), where Cj is the volume-average site fraction of point defect j in the solid, and the rate of point-defect absorption is Rate of defect j absorption into sink S (s −1 ) = DC j k S2, j . (13.21)

598 Light Water Reactor Materials

The sink strengths k2S,j (units of cm−2) are derived for spherical and line sinks. D is the diffusivity of the point defect. The rates are calculated by using the geometric shape of the sink and assuming that point-defect diffusion limits the defect ingress rate into the sink. The sinks are considered to be inexhaustible; i.e., absorption of defects does not change the sink strength. Strictly speaking, this is not the case, since the strength of extended sinks can change during irradiation. For example, an incoherent precipitate with a larger volume than the matrix has a volume mismatch with the matrix, and is a sink for vacancies, since vacancy absorption helps eliminate strain energy. However, as vacancies are absorbed, the mismatch decreases, until the precipitate ceases to be a net vacancy sink. These effects are neglected in the following derivations.

13.5.1 Sink strengths: isolated sinks Dislocation sink Point defects are absorbed at dislocations because there is a net elimination of lattice strain energy in this process. Defect absorption into a dislocation line eliminates the defect and causes a small jog to appear in the dislocation line. The additional energy on the dislocation line from the jog is much smaller than the vacancy formation energy EVf so that there is a net decrease in energy upon defect absorption. Multiple absorptions of one type of point defect cause the dislocation line to climb (see Ch. 7). Absorption of point defects by dislocations is modeled in rate theory as follows: 1. The dislocation distribution is represented by the dislocation density ρd. An even distribution of dislocation lines constituted of only one type of dislocation is assumed. In reality, the distribution of dislocations is quite anisotropic and inhomogeneous, and many different types of dislocation exist.

Microstructure Evolution Under Irradiation 599

2. There is a region next to the dislocation line, the dislocation core (Sec. 7.5), into which defects enter and are irrevocably captured by the line. 3. The effect of the dislocation line on the concentration of point defects in the matrix diminishes with distance from the core such that at the dislocation unit-cell radius (Eq. [7.4]), the concentration of defects is equal to the average bulk concentration. 4. Within the unit-cell radius of the dislocation Rcell (Fig. 13.5), the defects undergo random walk, with no influence of the dislocation strain field. 5. The rate of absorption of defects is limited by diffusion to the dislocation core. Although dislocations normally form a chaotic tangle in a solid, mathematical analysis requires a highly simplified geometry. To this end, the dislocation population is represented as an array of parallel lines on a square pattern, such as shown in Figure 13.5. Each infinitely long line occupies a square parallelepiped, and only this geometry needs to be analyzed. To further simplify the problem, the cell is taken as an equivalent cylinder of the same area as the square, as shown at the top of the drawing. Because the line and its unit cell are infinite in length, the diffusion equation to be solved is D d  dC ′ = −k r (13.22) r dr  d r  where C ′ is the spatially varying defect site fraction, r is the radial distance from the dislocation line, and k is the net point-defect production rate (dpa/s). The production rate is assumed to be constant over Rd < r < Rcell . The radius of the dislocation core is Rd, and it represents the distance from

600 Light Water Reactor Materials Unit cell, radius Rcell Dislocation line radius Rd

FIGURE 13.5: Representation of dislocation lines in a solid.

the dislocation, which, if penetrated by a point defect, assures capture of the defect by the line. It is less than 1 nm. The cell radius Rcell is fixed by the dislocation density ρd, in units of length of line per unit volume of solid, or cm−2. The reciprocal, (ρd)−1, is the area associated with each dislocation or: 2 = ρ −1 πR cell d

.

(13.23)

So, each dislocation has a “claim” on all defects inside “its” cylinder. The dislocation core, radius Rd, acts like a perfect sink. This is represented in Figure 13.6. In the figure, C is the average atom fraction of point defects in the lattice. Since the problem is symmetrical in the θ direction z direction, the defect concentration depends only on r. Assuming that the defect generation rate in the cylinder is negligible during the diffusion time of the defect from Rcell to Rd, the balance equation in the region associated with the dislocation is 1 d  dC ′ =0 . r (13.24) r dr  dr 

C’(r)/C

Microstructure Evolution Under Irradiation 601

C

1

Rd

Rcell

r

FIGURE 13.6: Geometry for the derivation of the reaction rate for defect-dislocation line interaction.

This equation is to be solved with the following boundary conditions: (i ) C ′( R d ) = 0 (ii ) C ′( Rcell ) = C

(13.25)

Integrating Equation (13.24) and applying the boundary conditions yields C ′(r ) ln(r/R d ) = . (13.26) C ln( Rcell /R d ) Figure 13.7 shows a typical concentration profile around a dislocation according to Equation (13.26). The defect flux per unit length of dislocation line is then given by   dC ′  j dis = 2 π R d  D  dr   .  Rd  

(13.27)

602 Light Water Reactor Materials 1.0

0.8

Rcell /Rd = 100

C'(r)/C

0.6

0.4

0.2

0.0 0

20

40

60

80

100

r/Rd

FIGURE 13.7: Point-defect concentration distribution around a dislocation line.

Substituting Equation (13.26) into Equation (13.27), we obtain j dis = 2 π

DC ≅ zDC . ln( Rcell /R d )

(13.28)

Since there are ρd cm of dislocation line per cm3, we obtain J dis = j dis ρ d = zρ d DC . Thus, the vacancy flux into all dislocations is J Vdis = jVdis ρ d = z V ρ d DV CV .

(13.29)

(13.30)

This equation has units of s−1, as C is given in atom fraction. For reasonable values of the dislocation core radius and dislocation density, 2π zV = ≈1 . ln( Rcell /R d )

Microstructure Evolution Under Irradiation 603

For interstitials, the flux to dislocations is J Idis = j Idis ρ d = z I ρ d D I C I .

(13.31)

The assumption of random walk of defects inside the dislocation cylinder is less valid for interstitials, which interact more strongly with the dislocation than vacancies do. As evaluated quantitatively in the example below, the interaction factor is zI ~1.02. More recent work has suggested that zI is as high as 1.2 [7,8]. Example #3: Interstitial bias factor of dislocations To estimate the interaction between dislocations and interstitials, we assume that the compressive stress field in a solid near an interstitial interacts with the tensile stress field at the underside of edge dislocations (see Sec. 7.5.2). The initial hole is of atomic size, in other words, Ro ≈ Ω 1/3

(13.32)

where Ω is the atomic volume. The radius is expanded to a radius (1 + ε) Ro when the interstitial is inserted: 4 ∆V = π [(1 + ε) 3 R o3 − R o3 ] ≈ 4 πR o3 ε . (13.33) 3 Lattice expansion performs work against the hydrostatic component of the dislocation stress field σh, which leads to an interaction energy, EI

= −σ h ∆V

.

(13.34)

From Equations (7.6) and (7.8), the hydrostatic stress generated by an edge dislocation is σh =

1 sin θ Gb . (σ rr + σ θθ + σ zz ) = −  1 + υ 3 3 π  1 − υ r

(13.35)

604 Light Water Reactor Materials

Combining Equations (13.34) and (13.35), we obtain EI

=

4  1 + υ sin θ . GbεRo3 3  1 − υ r

(13.36)

According to Equation (13.36), when the interstitial is in the compressive stress field of the dislocations (near the extra half-sheet of atoms), the dislocations repel the interstitial, while when in the tensile stress field of the dislocation (π < θ < 2π), the interaction is attractive. The interstitial migration in this stress field is biased by this interaction. To obtain the biased flux, the interstitial diffusion equation is solved accounting for the force exerted by the dislocation. To simplify the calculation, we set sinθ = −1 to make the force cylindrically symmetric, and combining all other properties in Equation (13.36) into a single constant B so that B EI = − . (13.37) r So that the force exerted by the dislocation on the interstitial is FI

=−

dE I dr

B =− 2 r

.

(13.38)

This force creates an inward drift velocity (the migration is no longer random) of v drift = MFI , where M is the interstitial mobility. From [6, p. 238], M = D I /k BT . The drift term modifies Fick’s law (written this time in terms of volumetric concentration [cm−3]) as dc I dc I DI + c I v drift = − D I + cI F , dr dr k BT I which is introduced into the conservation equation as 1d (rj ) = 0 . r dr I jI

= − DI

(13.39)

(13.40)

Microstructure Evolution Under Irradiation 605

Substituting Equation (13.39) into Equation (13.40), d  dc I  B  c I 1 dc I  r − − =0 . dr  dr  k BT  r 2 r dr 

(13.41)

This equation is solved with the same boundary conditions as Equation (13.24). However, now the interstitial bias factor is given by −1

Rcell  B/k BT  dr  z I = 2 π  ∫ exp  − . r  r   Rd 

(13.42)

In the limit where B/k BT > R d , the integral in Equation (13.42) is approximated as Rcell

∫ Rd

B/k T R B/k T exp  − B  dr ≅ ln  cell  − B ,  r  r  Rd  Rd

(13.43)

−1

so that

R B/k T z I = 2 π ln  cell  − B    R d  R d  2π 1 + B/R d k BT  . ≅ ln( Rcell /R d )  ln( Rcell /R d ) 

(13.44)

The second term in brackets is the bias for interstitials absorbed into dislocations: z I − z V B/R d k BT . (13.45) = zV ln( Rcell /R d ) We can evaluate the magnitude of this term. For Cu at 573 K, with

606 Light Water Reactor Materials

G = 4 × 1010 Pa, b = 1.5 × 10 −10 m, Ro = 2.34 × 10 −10 m, Rd = 5 × 10 −9 m, Rcell = 1/ πρ d = 5.6 × 10 −7 m, ε = 0.02, υ = 0.34, we obtain (zI − zV)/zV = 0.022 . Because of this difference, whatever its magnitude, dislocations are biased sinks for interstitials. From the above, by inspection, the dislocation sink strengths for vacancies and interstitials are then

and

k I2,dis = z I ρ d

(13.46)

kV2 ,dis = z V ρ d .

(13.47)

Although the above analysis was derived for straight dislocations, it is also used for dislocation loops. In this case, ρd is used, with no consideration of geometry, so that circular dislocations are effectively “straightened out.” Spherical sinks Similar arguments for point-defect absorption can be made in the case of spherical cavities, such as voids or bubbles uniformly spaced on a cubic lattice (Fig. 13.8). Spherical sinks in solids occur in numerous radiation

Cavity sink, radius Rc Unit sphere, radius Rcell

FIGURE 13.8: Spherical cavity sinks uniformly distributed in a solid. The cubes are unit cells for each sphere.

C'(r)/C

Microstructure Evolution Under Irradiation 607

C

1

Rcell

RC

r

FIGURE 13.9: Geometry for calculation of defect absorption at voids.

environments, including fission-gas bubbles in oxide nuclear fuels, as well as voids and helium bubbles in metals generated by fast-neutron irradiation or by ion irradiation in an accelerator. Figure 13.9 shows the geometry used for this calculation. In a solid containing NC cavities per unit volume, the volume associated with each void is equal to 1/NC. All defects created within that volume are absorbed into the void. The cell radius is 1 3

3 Rcell =  4 πN C 

.

(13.48)

The diffusion of defects to the cavity at the center controls the defect absorption rate at the cavity. If defect generation during diffusion is neglected, the spatially dependent defect concentration (site fraction) C′ depends only on r, and the appropriate equation (see also [6], Ch. 13) is 1 d  2 dC ′ =0 r (13.49) r 2 dr  dr 

608 Light Water Reactor Materials

with the following boundary conditions (i) C ′ = C at r = Rcell (ii) C = 0 at r = RC .

(13.50) (13.51)

The solution of equation (13.49) with boundary conditions (13.50) and (13.51) is R R 1 1 C ′(r ) = cell C −  −  C . (13.52) ( Rcell − RC )  RC r  The rate of absorption of defect j per cavity is dC ′ j Cj = 4 πRC2 D j = 4 πR C D j C . dr r = RC

(13.53)

Since there are NC cavities per unit volume, the rate of vacancy and interstitial absorption into cavities per matrix atom is J VC = 4 πRC DV CV N C J IC = 4 πRC D I C I N C .

(13.54)

This is the same as Equation (13.70) of [6], and the cavity sink strength is k C2 = 4 πRC N C .

(13.55)

The same technique can be applied to derive grain boundary and free surface sink strengths (Sec. 13.7).

13.5.2 Sink strengths for dislocations and voids: accurate calculation The sink strength derivations in the two previous sections are normally used in the literature. The derivations assume that migration is isotropic, which has been shown not to be the case [9]. In addition, Hayns and

Microstructure Evolution Under Irradiation 609

others have provided various corrections to the sink strengths [10]. For the sake of illustration, we show one such correction, namely, to the assumption that each sink can be treated in isolation, such that there is no interaction between sinks and that the production rate of the point defects can be neglected. This assumption is removed in the more accurate sink-strength derivations below, which do not specify k ~ 0. Dislocation sink As above the dislocation geometry is simplified to that of Figure 13.5 so that instead of Equation (13.23), the point-defect balance is D d  dC ′ = −k r r dr  d r 

(13.56)

where D is the diffusivity of the mobile point defect, r is the radial distance from the dislocation line, C′ is the spatially dependent defect concentration, and k is the net production rate (per atom or dpa/s) of the point defect considered. The production rate is assumed to be constant over Rd < r < Rcell. As above the cell radius is fixed by Equation (13.23). The boundary conditions for Equation (13.56) are C ′( R d ) = 0  d C ′  d r  R

and

=0

(13.57)

cell

The solution for the concentration distribution around the line is 2    r  1  r 2 − R d2    k Rcell ln − . 2   2 D    R d  2  Rcell 

C ′(r ) = 

(13.58)

610 Light Water Reactor Materials

This solution is recast in terms of the following dimensionless quantities: η=

r Rd

Θ=

C′ [kR d2 / 2 D]

Θ ( η) = β 2 ln( η) −

yielding

β=

Rcell Rd

(13.59)

1 2 ( η − 1) 2

(13.60)

The area-averaged concentration is β

Rcell

C=

1

∫ 2πr C ′( r ) dr

2 − R2 ) π( R cell d R d

or

2

Θ= 2 β −1

∫ ηΘ(η) d η

.

1

(13.61) Substituting Equation (13.60) into Equation (13.61) gives β2  2  Θ= 2 β ln β − β − 1 

3 1 1 2 3 +1− ≈ β ln β −  ,  2 4 4β  4 

(13.62)

which is valid for β >>1. In most practical situations β > 100, so that the approximation is valid. Figure 13.10 plots the point-defect distributions around the dislocation line (Eq. [13.60]) (solid curves) and the average concentration given by Eq. (13.62) (dashed lines). Curves for β values correspond (by Eq. [13.59]) to Rd = 0.6 nm and Rcell from Equation (13.23) for −2 −2 10 8 ρd = 10 cm and ρd = 10 cm . Both of these values are typical of commercial metals and ceramics. Replacing Θ with C using Equation (13.59) yields the connection between the net point-defect production rate and its mean concentration:  kR d2  2  3 C =   β ln β −  . (13.63) 4  2D  

Microstructure Evolution Under Irradiation 611 h = r/Rd 20

0

40

60

80

100

7 avg 6 β = 1000 5 β = 100 avg

Q/b2

4

3

2

1

0 0

200

400

600

800

1000

h = r/Rd

FIGURE 13.10: Point-defect concentrations around dislocation lines. Volume-averaged concentrations are shown as dashed lines.

The rate at which the dislocation line captures point defects is  dC ′ J = D   dr 

Rd

D  kR d2   dΘ D  kR d2  2 = = (β − 1) , (13.64) R d  2 D   d η  η=1 R d  2 D 

or, from Equation (13.63), in terms of C, Equation (13.64) is D Rd

β2 − 1

D 1 C . Rd ln β − 3 ln β − 3 4 4 The point-defect absorption rates are 2πRdJ/Ω per unit length of dislocation line and 2πRdJρd/Ω per cm3 of solid. Combining the latter rate with J=

β2

1

C≈

612 Light Water Reactor Materials

the flux J from the above equation yields the dislocation sink strength defined as 2 πρ d 2 πρ d 2 = = zρ d [s −1 ] . = k disl (13.65) 3  Rcell  3 ln β − 4 ln  R  − 4 d The conventional derivation of the line sink strength, as given by Equations (13.46) and (13.47) (and [6, Eq. (13.93)]), is missing the factor of ¾ in the denominator of Equation (13.65). The error incurred is 11% for 3 2 β = 10 and 19% for β = 10 . That is, as Rcell decreases, the dislocation lines are closer to each other, and the error of treating them in isolation becomes more significant. For example, for Rd = 1 nm, a dislocation density of 108 cm−2 implies β ~ 560 while 1010 cm−2 implies β ~ 56. Typical dislocation densities in engineering materials are on the order of 5 × 109 cm−2. Spherical sinks The cubic unit cell for a cavity is approximated as a sphere of the same volume with the cavity at its center, as shown in the previous section. The sink strength of these cavities is governed by the diffusion equation of the mobile atomic-size species: D d  2 dC ′ = −k r r 2 dr  d r 

(13.66) where the cell radius Rcell is determined by Equation (13.48). Boundary conditions are  dC ′ C ′( RC ) = 0 and    dr  R

=0 cell

.

(13.67)

Microstructure Evolution Under Irradiation 613

As with the line sink, appropriate dimensionless parameters are R r C′ , Θ= 2 , and β = cell . η= (13.68) RC RC (kRC /3 D) Written in terms of these dimensionless parameters, the solution to Equations (13.66) and (13.67) is Θ = β3

η−1 η

−

1 2 ( η − 1) . 2

(13.69)

The volume-averaged concentration is β

Rcell

C=

3 3 − R3 ) 4 π( Rcell C

∫ 4 πr 2C ′(r ) dr

or

3

Θ= 3 β −1

Rc

∫ η2Θ(η) d η

.

1

(13.70) Substituting Equation (13.69) into Equation (13.70) yields Θ = β3 +

1 3  3 5 1 3 1  3  9 − 3 β − β − ≈β β− . 2 β −15 2 10  5

(13.71)

The second form retains the two highest-order terms in β. Converting from Θ to C by means of Equation (13.68) gives  kRC2  2  9 C = β β− .  3 D   5

(13.72)

The rate of absorption by a cavity is  d C ′ D kR 2  dΘ jC = 4 πRC2 D   = 4 πRC2  C    RC  3 D   d η η=1  dr  R d

= 4 πR C2

 kRC2 

D (β 3 − 1) . RC  3D 

(13.73)

614 Light Water Reactor Materials

Combining the above two equations and multiplying by NC cavities per cm3 converts the rate from a per-cavity basis to a per-matrix atom basis and yields the following cavity point-defect absorption rate: β3 − 1 J C = 4 πRC N C DC . (13.74) 9   2 β β−  5 3 For β >> 1, the cavity sink strength for both vacancy and interstitial is k C2 = 4 πRC N C

β

(13.75) 9 . 5 If β >> 9/5, Equation (13.75) reduces to the conventional result given by Equation (13.55): k C2 = 4 πRC N C . β−

Using NC ~ 4 × 1013 cm−3 and RC ~ 2 nm is typical of fission-gas bubbles in oxide fuel, so β ~ 150. The correction factor in Equation (13.74) is only ~ 1.01. However, for voids in a void lattice (Sec. 19.6.1) (β ~ 6), the correction factor is a non-negligible 1.45. Figure 13.11 plots the dimensionless concentration against the dimensionless distance from the cavity surface. For large β = Rcell/RC  , the common assumption C = C ′(Rcell) is perfectly adequate. This is true for most 5

160 140

4

120

3 β = 150

80 60 40

Q/b2

Q/b2

100

β=6

2 1

20 0

0 0

20

40

60 80 100 120 140 h = r/Rc

1

2

3 4 h = r/Rc

5

6

FIGURE 13.11: Point-defect site fractions around cavities for two values of β. Volumeaveraged values are shown as dashed lines.

rz

on

e

Microstructure Evolution Under Irradiation 615

Ou

te

Outer zone

Inner zone CV

C Inner zone

CI 0 Outer zone

0

z

FIGURE 13.12: Partitioning of grain for calculation of grain boundary sink strength.

applications, such as bubbles in fuel and voids in metals. For small β (right graph, Fig. 13.12), the above assumption is not acceptable, although such cases are uncommon.

13.5.3 Grain boundary sink strength Planar sinks such as free surfaces and grain boundaries can substantially affect microstructure evolution under irradiation. Unlike dislocations, grain boundaries are neutral sinks, i.e., the same physical interaction for both types of point defects. Having determined the steady-state pointdefect concentrations, the sink strength of grain boundaries, k gb2 can be calculated. As before, steady state and no recombination are assumed. As shown in Figure 13.12, the grain is divided into two zones. In the inner zone, Equations (13.98) and (13.99) apply, giving the following point-defect concentrations: k k CV = 2 and C I = 2 . (13.76) k I DI k V DV In the outer zone, bordering the grain boundary, the same simplifications apply, but terms accounting for concentration gradients from CV to CVeq and C I to 0 are added to the simplified point-defect balance

616 Light Water Reactor Materials

equations. For vacancies, a simple balance of point-defect production and annihilation gives d 2C DV 2V + k = kV2 DV CV , (13.77) dz and there is a similar equation for interstitials. The solution of Equation (13.77) is k CV′ (z ) = A exp(− kV2 z ) + B exp( kV2 z ) + 2 . (13.78) k V DV The following boundary conditions apply: CV must be finite as z →∞, so B = 0. According to Equation (13.76), the last term is the bulk site fraction CV . At the grain boundary C ′V (0) = 0, so A = − CV . Inserting the above into Equation (13.77) gives the solution CV′ (z ) = 1 − exp(− k V2 z ) . CV

(13.79)

The flux of vacancies to the grain boundary is J Vgb = (4 πR gr2 ) DV

dCV′ = (4 πR gr2 ) DV k V2 C V . dz z =0

(13.80)

Rgr is the equivalent radius of the grain. The vacancy sink term in Equation (13.76) k gb2 DV represents the rate of vacancy absorption by the grain boundaries per matrix atom (s−1). From Equation (13.79), the concentration gradient at the grain boundary is k2VCV , so J Vgb 3 kV2 DV CV 2 k gb DV CV = 4 3 = R gr πR 3 gr

Microstructure Evolution Under Irradiation 617

kV2 , cm −2

k gb2 /kV2

1/ kV2 , µm

1011 1010 109

0.02 0.06 0.20

0.03 0.10 0.31

TaBlE 13.1: Characteristics of the grain boundary sink strength for a 10-µmdiameter grain containing other sinks.

from which the grain boundary sink strength is k gb2 =

3 kV2 . R gr

(13.81)

Table 13.1 lists two features of the grain boundary sink strength for typical values of the strength of the extended sinks (dislocations plus cavities, Eq. 13.46, 13.47, and 13.55). The second column gives the ratio of the strengths of the grain boundary and intragranular sinks. For large kV2 the ratio is less than 10%, which means that the grain boundary sink can safely be neglected in the pointdefect analysis. However, when the strength of the intragranular sinks is 109 cm−2 or less, the ratio is > 20%. For such relatively clean crystals, the grain boundary sink term k gb2 DV CV and k gb2 D I C I should be added to the right sides of the point-defect balances. Also, the grain boundary sink effect becomes larger the smaller the grain size. This approach has been used to develop radiation-tolerant materials based on nanograins in which the extended grain boundary sink is at a short distance from its creation. There is no sharp separation between the inner and outer regions of the grain as shown in Figure 13.12. However, according to Equation (13.79), the concentration in the outer zone reaches (1 − 1/e) of the inner value at a distance 1/ kV2 from the grain boundary. These characteristic lengths are listed in the last column of Table 13.1. They are to be compared to the

618 Light Water Reactor Materials

grain radius, which is 5 µm for the table. The characteristic lengths do not exceed 0.3 µm, which in most cases is sufficiently smaller than the grain radius that the two-zone approximation is valid.

13.6 Point-Defect Balances The point defects created during irradiation are the agents of microstructural evolution. Their migration through the lattice and subsequent annihilation at sinks, clustering, or recombination is the cause of the irradiation effects observed macroscopically. The microstructure evolution has often been described by a theory in which the point-defect balances are calculated from a balance of defect creation, interaction with defect clusters, and annihilation at various extended sinks; the latter as homogenized as in the derivations shown in the previous sections. This so-called rate theory was initially developed in the 1950s to describe the annealing of defects in irradiated materials [2, 11, 12] and further developed in the late 1960s and early 1970s [3, 4, 5] to explain void growth in irradiated metals. The theory is essentially a collection of point-defect balance equations and their solutions. A brief summary of the theory is given in this section, and other summaries are also available in the literature [13, 14].

13.6.1 Point-defect balance equations According to rate theory the overall change in the concentration of the defect species results from the balance between defect production, reaction, and diffusion: Rate of Change = Generation rate − Diffusion − Recombination − Annihilation at sinks − Clustering In principle, one balance equation would have to be written for each size of defect cluster. Indeed, the production of damage in displacement

Microstructure Evolution Under Irradiation 619

cascades results not only in the formation of Frenkel pairs, but also of defect clusters of different sizes, depending on cascade energy, density, etc. Clusters also evolve from individual point-defect reactions as shown previously in this chapter. Thus a reaction of a vacancy with a divacancy would result in a trivacancy, the reaction of a vacancy with a di-interstitial in a single interstitial, and so on. Depending on which defect clusters are mobile, a very large number of reaction rates and balance equations would result from this more precise formulation of the problem. Such an approach is indeed now used in clusters dynamic calculations (see Ch. 15), which take advantage of the increased computational power available today. In this chapter we only model the single point-defect balances, as this llustrates many of the basic physical concepts that more detailed formulations would not show as easily. The rate equations for single vacancies and interstitials are3 ∂C V ∂t

and

= DV ∇ 2C V + k − K IV C I CV − k V2 DV C V

∂C I ∂t

= D I ∇ 2 C I + k − K IV C I CV − k I2 D I C I

(13.82)

(13.83)

where DI and DV are the defect diffusion coefficients, k is the defect generation rate in units of point defects per second per matrix atom (dpa/s), CI and CV are the point-defect site fractions, and k j2 is the total sink strength for defect j (units of length−2). Equations (13.82) and (13.83) are solved with the appropriate boundary and initial conditions. If only spatially averaged concentrations are 3

In addition, each cluster size requires a separate balance equation; the larger cluster sizes are neglected in this simple derivation. Cluster dynamics treatments take into account explicitly the full range of defect clusters (see Ch. 15).

620 Light Water Reactor Materials

needed (e.g., far from grain boundaries or free surfaces), Equations (13.82) and (13.83) are simplified to

and

dCV = k − K IV C I CV − k V2 DV C V dt

(13.84)

dC I = k − K IV C I CV − k I2 D I C I . dt

(13.85)

These forms of the balance equations are particularly useful for identifying the various stages of microstructure evolution. The last terms in Equations (13.84) and (13.85) reflect the absorption of point defects in all the sinks in the material. For example, assuming that the only sinks are cavities and dislocations, according to Equations (13.46), (13.47), and (13.55), the total sink strengths are k I2 = 4 πRC N C + z I ρ d (13.86) and

kV2 = 4 πRC N C + z V ρ d .

(13.87)

Note that the only difference between these two sink strengths is zI and zV, which differ by as little as ~2% (Ex. #3).

13.6.2 Steady-state solutions of point-defect balances Given the previous formulation, the next step is to solve the defect balance equations for particular cases. The quantities of interest are the average defect concentrations as functions of time, including t = ∞ (steady state), and the time to reach steady state. In the following, we assume that one defect (interstitial) diffuses fast, while the other (vacancy) is slow (a reasonable simplification in metals). The resulting coupled nonlinear partial-differential equations have no analytical solutions, and must be solved numerically. Because the coefficients in the equations often differ

Microstructure Evolution Under Irradiation 621

by several orders of magnitude, a robust equation solver is needed, and often the equations have to be nondimensionalized to make the numerical calculation stable. The steady-state versions of Equations (13.84) and (13.85) are

and

0 = k − K IV C I CV − kV2 DV CV

(13.88)

0 = k − K IV C I CV − k I2 D I C I .

(13.89)

Equations (13.88) and (13.89) show that the defect production k is balanced by a combination of annihilation at all the extended sinks in the material and annihilation by reaction with other small distributed defects as represented by recombination.4 In reality, rather than just recombination of single point defects, it is the sum total of defect clusters formed during irradiation that soon become the principal sinks for the newly formed mobile point defects. The crucial point is that because these small defect clusters are formed homogeneously throughout the material, the point defects created do not have to undergo long-range migration to be annihilated. There are, however, materials with very fine-grained microstructure (e.g., nanograined materials or oxide dispersion strengthened steels) in which the defects do not have to migrate far to be annihilated. The distinction is then between (i) a sink-dominated regime in which most newly created defects are annihilated through long-range diffusion to extended sinks, which are little modified by the defect absorption and (ii) a defect cluster or recombination-dominated regime in which defects are annihilated in smaller clusters, 4

We should note that, historically, an incorrect simplification of recombination-dominated regimes at low temperatures versus sink-dominated regimes at higher temperatures has been assumed. Because the material quickly develops an irradiated microstructure that increases sink density these assumptions are not necessarily correct.

622 Light Water Reactor Materials

which are homogeneously distributed through the material and which are either eliminated (as is the case with the recombination of Frenkel pairs) or significantly modified after point-defect absorption. In the formulation above, the recombination term serves as a substitute for the effect of the full defect cluster distribution. Manipulating Equation (13.89) yields CI

k =

K IV C IV + k I2 D I

(13.90)

.

Inserting Equation (13.90) into Equation (13.88) yields CV2 +

k I2 kk I2 D I =0 . DC − K IV I V K IV kV2 DV

(13.91)

The solution of (13.91) is k 2D CV = − I I 2 K IV

 kk I2 D I  k I2  2  K k 2 D +  4K   IV  IV V V 

1/2

.

(13.92)

4(k I2 D I ) 2 K IV Multiplying the first term in the brackets by 2 2 and rearranging 4(k I D I ) K IV yields k I2 D I CV = ( 1 + ξ − 1) (13.93) 2 K IV and where

kV2 DV CI = ( 1 + ξ − 1) 2 K IV 4 kK IV I V I DV

ξ= 2 2 kk D

(13.94) (13.95)

Microstructure Evolution Under Irradiation 623

is the dimensionless defect annihilation number, which measures the balance between defect production/recombination and annihilation in small clusters versus annihilation at extended sinks.5 Taking the limit where ξ >> 1 (equivalent to assuming that production/recombination [numerator] is much stronger than annihilation at extended sinks [denominator]) makes the term in brackets in Equations (13.93) and (13.94) approximately equal to ξ1/2. In this defect cluster or recombination-dominated regime, the steady-state concentrations are

and

k I2 D I CV ∼ 2 K IV

1/2 2D k k   I I ξ1/2 =  kV2 DV K IV 

(13.96)

kV2 DV CI ∼ 2 K IV

 ξ1/2 =

kV2 DV k  1/2 .  k I2 D I K IV 

(13.97)

In the other limit, when sinks are the main annihilation mechanism, ξ

(so (1 + ξ)1/2 ∼ 1 + . Using Equation (13.95), it can be shown that 2 the steady-state concentrations are ξ > DV, Equation (13.115) implies τ I > [Na+] and the only ions at significant concentration are H + and Cl −. Equation (14.87) reduces to (14.96) [ H + ] ≈ [Cl − ] = [ NaCl ]0 exp(Φ) ,

Fundamentals of Aqueous Corrosion 703

or, with Equation (14.91), [ NaCl ]0 exp(Φ) . (14.97) [ H + ]0 Substituting Equation (14.97) into Equation (14.88) gives dΦ (14.98) exp(Φ) = A . dη With the boundary condition Φ = 0 at η = 0, the solution is (14.99) Φ = ln(1 +Aη) . Despite the electrical neutrality approximation in Equation (14.96), Equation (14.99) is correct at η = 0 and a good approximation as η → 1 as long as A is large. The dimensionless constant A is 10 3 L  i Fe  A= . (14.100) [ NaCl ]0 D H  z Fe F ′ Substituting Equation (14.86) into the above, along with ΦL = ln(1 + A), yields: 10 3 L  i oFe  exp(α Fe z Fe Φ M ) (14.101) A= [ NaCl ]0 D H  z Fe F ′ (1 + A) α Fe z Fe α=

The distribution of H + in the crevice is determined by the full electrical neutrality condition without the approximation in Equation (14.96). Assuming that the coolant is neutral water, Equation (14.88) becomes B + exp(Φ) = Bα + exp(−Φ) (14.102) α

where

B = [ H + ]0 /[NaCl]0 .

(14.103)

Equation (14.102) approaches the correct limit as η = 0 (where Φ = 0 and α = 1). As η → 1, α and Φ become large, so that the first and the fourth terms become negligible and Equation (14.102) reduces to Equation (14.97).

704 Light Water Reactor Materials

The distributions in the crevice are determined as follows: 1. A is determined from Equation (14.101) for the system conditions and properties. 2. The electrode potential Φ is determined as a function of location in the crevice from Equation (14.99). 3. The Cl − and Na+ concentrations are computed from Equation (14.95). 4. The H + concentration (α) is calculated as a function of η from the solution of Equation (14.102): α=

  1 f (Φ)  1 + f 2 (Φ) + 1  

(14.104)

1 (14.105) (exp(Φ) − exp(−Φ)) . 2B At the crevice mouth, Φ → 0, and f(Φ) → 0, so α → 1, in agreement with Equation (14.94). At the crevice tip, Φ >> 1, f(Φ) → exp (Φ)/2B >> 1, so α → 2f(Φ) = exp (Φ)/B, which is Equation (14.96). where

f (Φ ) =

Example #8: Ion distributions in a crevice in iron at 25oC Crevice depth L = 3 mm Metal potential φM = −0.2 V DH = 1 × 10−4 cm2/s Temperature T = 25oC Neutral bulk water [H+]0 = 10−7 M Salt concentration [NaCl]0 = 10−4 M exchange-current density ioFe = 107 A/cm2 anodic symmetry factor αFe = 0.5 valence zFe = 2

Fundamentals of Aqueous Corrosion 705

From Equation (14.92), the dimensionless electrode potential of the iron structure in which the crevice is located is φM 0.2 −0.2 = = −7.7 . ΦM = = 3 R g T/F 8.314 × 300 / 96.5 × 10 0.026 The parameter A is determined by Equation (14.101): 10 3 × 0.3  10 7  A(1 + A) = −4 exp(−7.7) = 6 × 10 8 ,  −4  × 2 96,500 10 × 10   A ≅ 6 × 10 8 = 2 × 10 4 .

which yields Equation (14.99) becomes

Φ = ln(1 + 2 × 10 4 η)

. Converting to the dimensional potential by Equation (14.92), the distribution of φ is plotted in Figure 14.25. 0.30

Potential in crevice (V)

0.25 0.20 0.15 0.10 0.05 0.00 0.0 Mouth

0.2

0.4

0.6

0.8

Depth in crevice (x/L)

FigurE 14.25: Electrode potential in crevice solution.

1.0 Tip

706 Light Water Reactor Materials 1 [CI–]

[H+]

Molarity

10–4

10–8

[Na+]

10–12 [OH–] 10–16

0.0 Mouth

0.2

0.4

0.6

Depth in crevice (x/L)

0.8

1.0 Tip

FigurE 14.26: Concentration profiles of ions in crevice for η > 0.01

The concentrations of the four ionic species in Equations (14.88) to (14.90) are determined next. [Cl −] and [Na+] follow from Equation (14.95) and the last two of Equations (14.91): β = exp( Φ) = 1 + 2 × 10 4 η ~ 2 × 10 4 η

; [Cl − ] = 10− 4 β = 2 η ;

5 × 10 1 + −4 χ = exp( −Φ) = ≅ ; and [Na ] = 10 χ = η 1 + 2 × 10 4 η 2 × 10 4 η 1

α

−9

.

is obtained from Equations (14.104) and (14.105) and from Equation (14.103), B = 10−7/10−4 = 10−3. Equation (14.105) becomes: f(Φ) = 500(exp[Φ] − exp[−Φ] ). Except for η very close to zero, where Φ → 0, f(Φ) ∼ 500exp[Φ] = 500(1 + 2 × 104 η) ∼ 107 η. With f(Φ) >> 0, Equation (14.104) reduces to α = 2f(Φ) = 2 × 107 η. Converting this result to concentration units (molarity) using the first of Equations (14.91), we obtain [H + ] = 2η . [OH −] follows from Equation (14.85).

Fundamentals of Aqueous Corrosion 707

The concentration profiles are shown in Figure 14.26. The approximation of Equation (14.96) applies for η > 0.01. At η = 0, [Cl −] = [Na+] = 10−4 M and [H +] = [OH −] = 10−7 M.

14.7.3 Computational results Figures 14.27 and 14.28 result from a more complete analysis of the chemistry of the crevice [10]. The calculations underlying the curves include active corrosion along crevice walls, which was neglected in the simple model presented in the previous section. Also, two iron ions, Fe2+ and FeOH+, are included among the ionic products of corrosion of the metal. In the previous section, these ions were assumed to react rapidly with water to produce the solid oxide (Eq. [14.75] and [14.76]). The electrode potentials are similar in shape to that in Figure 14.25 (the directions of the abscissas are reversed), rapid changes near the mouth decreasing greatly on approaching the tip. The increase in electrode potential in the crevice and the low pH leads to active corrosion (see Fig. 14.1). –0.35

Active walls

Potential (Volts)

–0.30 Passive walls

–0.25 –0.20 –0.15 –0.10 –0.05 0 1.0 Tip

0.85

0.7

0.55 0.4 x/L

0.25

0.1 0 Mouth

FigurE 14.27: Potential profiles in a 2-mm-long crevice with the following conditions: φo = 0 V; [NaCl]o = 0.02 M; pH = 7. From [10].

708 Light Water Reactor Materials CI–

Log10 concentration (M)

[CI] = 0.48 M 2.5

Fe2+ FeOH+

0

[NaCl]o = 0.02 M

–2.5

pH = 4.3

H+

–5.0

–7.5

–10.0 1.0 Tip

Na+

pH = 7

0.85

0.7

0.55

0.4 X/L

0.25

0.1

0 Mouth

FigurE 14.28: Concentration profiles in a 2-mm-long crevice. Active corrosion on walls after [10].

The following are important features of the curves in Figure 14.28: a) Sharp changes occur within 5% of the crevice mouth; by x/L ~ 0.1, the curves become essentially flat, indicating only minor concentration changes. b) The pH is ~4.5 over most of the length of the crevice, falling to the bulk water pH = 7 only very near the crevice mouth. Due to neglect of the iron ions in Figure 14.26, the solution in the crevice approaches pH = 0 at the tip. The consequence of the simultaneous decreases in pH and increases in electrode potential from the crevice mouth to the tip can be seen on the Pourbaix diagram (Fig. 14.1). The thermodynamically stable chemistry changes from an oxide to Fe2+, which promotes active corrosion. c) The combination of the concentrations of Fe2+ and Fe(OH)+ along the crevice causes a high Cl− concentration (~0.5 M) to provide solution neutrality. d) Na+ has been pushed out of the crevice.

Fundamentals of Aqueous Corrosion 709

14.7.4 improvements of the model • • •

• •

• •

Cathodic half-cell reactions (e.g., H + + e − → 1/ 2 H 2) along the crevice walls Aggressive anions other than Cl − in the bulk water (e.g., SO 24− ) Restricted diffusion of ions in the water in the crevice by precipitation of oxide (e.g., by Eq. [14.76]); the solubilities in water of ionic species of iron, nickel, and chromium are very low Concentration and electric potential gradients in the bulk water, as in [9] Trapezoidal rather than the rectangular crevice shape shown in Figure 14.24, this modification is necessary when the method is applied to a crack in the metal rather to a crevice formed by a gap between two metal pieces [11] Transient crevice corrosion (i.e., growth of a crack) Inclusion of solubility limits of all metal ions in solution in the crevice water [12]

References 1. J. Scully, The Fundamentals of Corrosion, 2nd ed. (Oxford: Pergamon Press, 1981). 2. J. Bischoff and A. T. Motta, “Oxidation behavior of ferriticmartensitic and ODS steels in supercritical water,” J Nuc Mater 424 (2012): 261–276. 3. G. Bohnsack, The Solubility of Magnetite in Water and in Aqueous Solutions of Acid and Alkali (Washington, DC: Hemisphere Publishing Corp., 1988). 4. F. Sweeton and C. Baes Jr., “The solubility of magnetite and hydrolysis of ferrous ions in aqueous solutions at elevated temperatures,” J. Chem. Thermodynamics 2 (1970): 479. 5. A. Couet, A. T. Motta, and A. Ambard. “The coupled current charge compensation model for zirconium alloy fuel cladding

710 Light Water Reactor Materials

6. 7. 8. 9. 10. 11. 12.

oxidation: I. parabolic oxidation of zirconium alloys,” Corrosion Science 100 (2015): 73–84. A. T. Fromhold, “Space-charge modification of the ionic currents for oxide growth,” Solid-State Ionics 75 (1995): 229. P. Marcus, ed., Corrosion Mechanisms in Theory & Practice, 2nd ed., (New York: Marcel Dekker, 2002). J. Newman, Electrochemical Systems (Englewood Cliffs, New Jersey: Prentice-Hall, 1973): 12–21. G. Engelhardt, et al. “Fast algorithms for estimating stress-corrosion crack-growth rate,” Corr Sci 41 (1999): 2267. S. Sharland, “A mathematical model of crevice and pitting corrosion II,” Corrosion Sci 28 (1989): 621. S. Sharland, “A review of theoretical modeling of crevice and pitting corrosion,” Corrosion Sci 27 (1987): 289. A. Tucker and J. Thomas, “A model of crack electrochemistry for steels in the active state based on mass transport by diffusion and ion migration,” J Electrochem Soc 129 (1982): 1412.

Problems 14.1 Look up the Gibbs energy of formation of H2O(liq) from the website of the National Institute of Standards (kinetics.nist.gov/janaf/). At 25oC, show that this value is consistent with the standard electrode potential number 1 in Table 2.2. 14.2 Water at pH 7 fills a glass container. The pertinent half-cell reactions and their standard electrode potentials and exchange current densities are: j 1. 2. 3.

Half-cell Reaction ½O2 + 2H+ + 2e = H2O H2O2 + 2H+ + 2e = 2H2O 2H+ + 2e = H2

φo (V)

1.23 1.77 0

io(A/cm2) 2 × 10−6 4 × 10−8 4 × 10−8

Fundamentals of Aqueous Corrosion 711

Water is saturated with O2 at 0.5 atm and H2 at 0.1 atm and has an H2O2 concentration of 10−3 M. The symmetry parameters (Sec. 14.4.4) are 0.5 for all half-cell reactions. What is the electrochemical potential (ECP) of the water? There are no reactions with the container surface. 14.3 A piece of iron is immersed in water containing: [Fe2+] = 0.1 M [H2] = 10 ppm (wt) pH = 2 The Tafel lines are i oH = 10 −6 A /cm 2; i oFe = 10 −7 A /cm 2 ; and βH = βΦε = 0.5 (see Eq. [14.16] and [14.17]) (a) What is the electrochemical potential of the system? (b) According to the Pourbaix diagram (Fig. 14.1), should iron corrode? (c) What is the corrosion rate in mm/yr? 14.4 O2 dissolves in water according to Henry’s law: KHen = 0.03 (25°C), and K Hen = p/[i ] . p is the pressure of the gaseous species in atm. [i] is the concentration in water in ppm by weight. (a) What is the electrochemical potential of neutral water containing 5 ppm by weight of O2? (b) According to Figure 14.1, what is the stable phase of iron in neutral water at 25°C? 14.5

2 8 2 Fe 3 O 4 + 4H + + H 2 = 2Fe 2+ + H 2 O 3 3 3 (a) What is the half-cell reaction corresponding to the Fe2O3/ Fe3O4 line on the Pourbaix diagram of Figure 14.1?

712 Light Water Reactor Materials

(b) What is the standard electrode potential of the half-cell reaction of (a)? (c) What is the value of the equilibrium constant for the above reaction? 14.6 Construct the Pourbaix diagram for the cadmium-water system based on the following: 1. Cd = Cd 2+ + 2 e φ1o = −.403 2. Cd + 2 H 2O = HCdO 2− + 3 H + + 2e

φ o2 = .583

3. Cd + H 2O =2CdO + 2 H++ + 2e + 2e

φ o3 = .005

+ + −13.8 4. Cd 2+ + H 2+O = CdO 2 + 2 HO + 2KH = 1 K 4 = 10 5. CdO + H 2O = HCdO 2− + H + + 2e K 5 = 10 −19.5

Include the water decomposition lines (a) and (b) in Figure 14.1. Indicate regions of corrosion, immunity, and passivity. 14.7 The corrosion situation described in Problem 14.3 takes place in a liquid volume V and an iron specimen of surface area A. The Fe2+ and H+ concentrations are initial values. As corrosion proceeds, the former increases and the latter decreases, so icorr decreases with time. What are the equations from which [Fe2+], [H+], and icorr can be calculated? Do not attempt to solve. 14.8 Cathodic reduction of O2 dissolved in water is limited by concentration polarization. Calculate the mass-transfer limited corrosion rate (mm/yr) of iron in an aqueous solution containing 40 ppm (by wt) dissolved oxygen. The diffusion coefficient of O2 in water is 10−8 cm2/s, and the stagnant film in the specimen is 100 mm thick. 14.9 The pH of the coolant water in a PWR affects corrosion of zirconiumbased cladding in part by controlling the aqueous concentration of zirconium ions according to the reaction: ZrO 2 (s) + 4 H + = Zr 4+ + 2 H 2O

Fundamentals of Aqueous Corrosion 713

(a) What is the equilibrium constant for this reaction at 400°C obtained from the following information: Zr = Zr 4+ + 4e

ε o = −1.53 eV

Zr (s) + O 2 = ZrO 2 (s)

∆ µ o = −227

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

∆µ o = −9

kcal /mole

kcal /mole

(b) If the coolant is made basic by addition of 10−4 mole/l of LiOH, what is the equilibrium concentration of zirconium ions in the coolant? The dissociation of water at 400°C is 10−11. (c) At what pH would “significant corrosion” (in the sense used in the Pourbaix diagrams) of oxide-covered Zircaloy occur?

15

Chapter Computational Modeling of Nuclear Fuels Brian Wirth 15.1 Introduction The development and qualification of nuclear fuels, cladding, and structural components has traditionally involved decades-long testing and examination. The long lead times derive from the reliance on nuclear test reactor irradiations that drive microstructural and material property changes and the desire to have materials with long lifetimes. This goal is driven by the economic, national security, and fuel utilization gains that it enables. The emphasis on reactor irradiation for materials design, development, and qualification is due to the difficulty in reproducing the degradation of materials observed in reactors outside of such environments, and the inability to project the results from accelerated out-of-pile materials experiments (e.g., ion irradiation) into reliable predictions of in-pile performance.

716 Light Water Reactor Materials

Over the past two to three decades, advances in computational modeling, coupled with increasingly powerful high-performance computing and improved experimental tools, have led to a multiscale modeling paradigm for understanding and predicting material behavior that is multidisciplinary and offers the promise of developing models over all relevant length and time scales. The promise of high-fidelity performance models is not only to predict the lifetime and failure of fuels and components in a wide variety of advanced nuclear energy systems, but also to enable the design of new materials tailored for such aggressive environments. The expectation is that high-fidelity predictive performance models will explicitly incorporate all relevant physical mechanisms controlling material behavior revealed by subscale physics modeling, parameterized by targeted laboratory experiments, and validated by full-scale test-reactor irradiations. The realization of highfidelity, physically-based nuclear materials performance models will safely increase the burnup and performance of nuclear fuels in a wide range of light water reactor (LWR) designs, assist the licensing and waste stewardship, and decrease the time necessary for new materials development and qualification. A key consideration in the development of a comprehensive understanding of nuclear fuels is the evolving microstructure of multicomponent oxides, nitrides, carbides, and alloys containing U and Pu, as well as minor actinides including Np, Am, and Cm. Along with these actinide elements, the in-growth of fission products such as Xe, Ba, Cs, Sr, I, and Tc must also be considered. In reactor cores, the fuels, cladding, and structural materials (pressure vessels, pipes, ducts, etc.) are subject to severe irradiation, along with chemical and thermomechanical environments that continuously alter their physical properties. The physics and chemistry of such materials become more complex with increasing reactor exposure. It is well documented [1] that ceramic fuels develop radial and azimuthal cracks, and the severity of the structural damage

Computational Modeling of Nuclear Fuels 717

µs - s

Long-range defect transport & annihilation at sinks Gas diffusion & trapping

ps - ns

ns - µs

Time scale

Nano/microstructure & local chemistry changes; nucleation & growth of extended defects, bubbles & precipitates

Irradiation temperature, fission density, n/γ energy spectrum, flux, fluence, temperature history & initial microstructure inputs

s - year

decades

increases with burnup. Root-cause analysis of cladding failures indicates that the overarching cause of rupture is the degradation of fracture properties resulting from complex microstructural changes from exposure to high-energy neutrons. Such irradiation displaces atoms from their lattice sites many times, in addition to altering the chemistry of the material. The effect of irradiation on nuclear fuel or structural materials is a classic example of an inherently multiscale phenomenon in both time and dimension, as schematically illustrated in Figure 15.1. Pertinent processes span in excess of 10 orders of magnitude in length scale from the subatomic nuclear to structural component level, and span 22 orders of

Gas production (fission and/or transmutation)

Radiation enhanced diffusion and induced segregation of solutes

Cascade aging & diffusion/redistribution of solutes

Defect recombination, clustering & migration

Underlying microstructure (pre-existing & evolving) impacts defect - fission product – solute fate

Primary defect production & short-term annealing atomic - nm

nm - µm

µm - mm

mm - m

Length scale

FIGURE 15.1: Illustration of the length and time scales (and inherent feedback) involved in the multiscale processes responsible for microstructural changes in irradiated nuclear fuel or structural material. Inset images come from References [3] and [4].

718 Light Water Reactor Materials

magnitude in time from subpicosecond nuclear collisions to decade-long component service lifetimes [2, 3]. The micrographs in Figure 15.1 show a scanning electron microscope image of fission gas bubbles in UO2 after a base irradiation of ~9 GWd/ tHM (Gigawatt day per ton of heavy metal), followed by a ramp test to ~1900°C (left) [4] and a transmission electron microscope image of the intragranular gas bubbles (lighter spheres) and second-phase precipitates (darker spheres) formed in UO2 following a burnup to ~83 GWd/tHM at ~800°C (right) [5]. Many different variables control the mix of nano/ microstructural features formed and the corresponding degradation of physical and mechanical properties of nuclear fuels, cladding, and structural materials. The most important variables include the initial material composition and microstructure, the thermomechanical loads, and the irradiation history. While the initial material state and thermomechanical loading are of concern in all materials performance-limited applications, the added complexity introduced by the effects of radiation is the differentiating and overarching concern for materials in nuclear energy systems. The inherently wide range of time scales of the mechanisms that control the kinetic evolution of irradiation-damaged nuclear fuel or materials makes modeling irradiation effects in materials extremely challenging, and experimental characterization of individual processes governing degradation is often unattainable. Indeed, accurate models of microstructure (i.e., point defects, dislocations, and grain boundaries), evolution of nuclear fuel, and cladding, as well as structural materials during service, are still lacking. Understanding the irradiation effects and microstructure evolution to the extent required for a high-fidelity nuclear materials performance model requires a combination of experimental, theoretical, and computational tools. A hierarchy of models is employed in the theory and simulation of complex systems in materials science and condensed matter physics: macroscale continuum mechanics, mesoscale models of defect evolution,

Computational Modeling of Nuclear Fuels 719

molecular-scale models based on classical mechanics, and various techniques for representing quantum-mechanical effects. These models are classified according to the spatial and temporal scales that they resolve, as indicated in Figure 15.2. Individual modeling techniques are indicated within a series of linked process circles showing the overlap of relevant length and time scales. The modeling methodology includes ab initio electronic structure calculations, molecular dynamics (MD), accelerated molecular dynamics, kinetic Monte Carlo (KMC), phase field equations, Meso-scale

days - yr

Atomistic

ms - s

Phase field

µs - ms

Time scale

ns - µs

Finite element simulations

Reaction-diffusion kinetic rate theory – cluster dynamics

Kinetic Monte Carlo

um

u tin on c k n,” or ow mew d p fra “To

Accelerated molecular dynamics

Molecular dynamics ps - ns

Continuum

Electronic structure atomic - nm

n

tio

,” iga -up vest m o n ott d i “B ase b icist

m

ato

nm - µm

µm - mm

mm - m

Length scale

FIGURE 15.2: Illustration of the multiscale materials modeling paradigm, where the simulation techniques are shown corresponding to the length and time scales where they are most appropriate, and which are described in more detail in Section 15.2. Within a hierarchical modeling approach, the atomistic-based models would provide constitutive properties to continuum methods, whereas boundary conditions would be provided from the continuum-level simulations, including fuel performance modeling tools.

720 Light Water Reactor Materials

as well as kinetic-rate-theory-based approaches including the use of cluster dynamics, and are ideally linked by passing information about the controlling physical mechanisms between the modeling techniques over the relevant length and time scales. The key objective is to track the fate of defects, impurities, and other chemical species during irradiation, and thereby, to predict microstructural evolution. Detailed microstructural information serves as a basis for modeling the mechanical behavior through meso- (represented by kinetic Monte Carlo, phase field, or kineticrate-theory-based methods) and continuum-scale models, which must be incorporated into constitutive models at the continuum finite element modeling scale to predict performance limits. To span length and temporal scales, these methods are linked into a multiscale simulation. In Figure 15.2, the passing of information between the scales is represented by arrows. The objective of the lower-length scale modeling is to provide constitutive properties, such as how thermal conductivity, porosity, or creep rate vary as a function of burnup or radiation damage, and to provide them to higher length-scale, continuum-level simulations. These higher length-scale simulations can provide boundary conditions such as appropriate temperature and stress distributions to the lower length-scale models. However, there is not currently a robust way to link these single-scale methods into a multiscale simulation that includes error control across the scales. Errors can be introduced in passing information from fine-grained to coarse-grained models because of the accompanying loss of physical detail. Even without error control, multiscale simulations are computationally intensive. Massively parallel computers frequently offer a way out of most length-scale constraints, as simulations focused on different spatial regions can often be done in parallel. Typically, molecular dynamics simulations with realistic forces calculated from electronic structure theory can reliably simulate up to approximately 1 nanosecond using so-called ab initio molecular dynamics (MD), while more simplified empirical force fields can reliably simulate

Computational Modeling of Nuclear Fuels 721

hundreds of nanoseconds but generally cannot describe bond breaking or charge transfer. Thus, MD simulations alone are unable to treat long-time dynamic behavior of nuclear materials. For example, an MD simulation of a region within a fuel pellet whose volume is approximately ¼ µm3, would involve Newton’s equations of motion for slightly less than 1.2 × 109 (billion) atoms. In a prototypic fission reactor, this volume of material would experience about 0.5 fissions/sec and generate about 0.125 Xe noble gas atoms per second assuming a typical LWR fission rate density of 3 × 1019 fissions/m3.sec. Assuming a typical MD time step of 10−15 seconds, ~1016 time steps are required just to introduce a single Xe into the simulation volume at prototypic reactor rate, and well in excess of 1024 time steps to simulate the fuel pellet’s entire time in the reactor. Therefore, modeling approaches that more easily reach meso- to continuum scales are required to be able to compare to experimentally relevant conditions for validation. Recently developed accelerated molecular dynamics [6–8] or adaptive kinetic Monte Carlo [9] methods can extend time scales to approach milliseconds and beyond through the use of high-performance computing, but with restrictions on the system size. However, because of the current limitations of long time simulations, there is a restricted ability to describe slow processes (e.g., phase transitions) or rare events, such as defect cluster dissolution, both of which play an important role in nuclear environments. Specific modeling and advanced experimental characterization components of the multiscale materials modeling approach, illustrated in Figure 15.2, are briefly described in Section 15.2, with representative examples of these techniques discussed in the following sections of this chapter.

15.2 Brief Description of Modeling Techniques 15.2.1 Ab initio electronic structure calculations Basic alloy thermodynamic and defect properties, which include the formation, binding, and migration energies for the interaction of solutes

722 Light Water Reactor Materials

and impurities with point defects and small defect clusters, can be obtained from ab initio methods involving plane-wave pseudopotentials to describe the core electrons and the nucleus codes using the generalized gradient approximations (GGA) and local density approximations (LDA) within the density functional theory (DFT), which was developed based on pioneering work performed by Pierre Hohenberg and Walter Kohn in the early 1960s [10]. Hohenberg and Kohn showed that the ground state properties of a many-electron system can be specified by the spatially dependent electron density through the use of an energy functional (the calculated total energy of the system). It can be shown that the correct ground state electron density is the one that minimizes the energy functional [10]. Compared to empirical or semi-empirical interatomic potentials, DFT calculations provide higher accuracy and the ability to treat chemical changes, e.g., oxidation and reduction, known to occur during reactor operation. Over the past two decades, a substantial fraction of electronic structure calculations, covering a variety of topics such as magnetic materials, vacancies and impurities, surfaces, clusters, and biomolecules, relied on the pseudopotential framework. The pseudopotential is an approximation of Schroedinger’s equation to describe the interatomic interactions of complex systems. This approach is based on the observation that most physical and chemical properties of atoms are determined by the atomic valence states. This is true not only for the formation of chemical bonds, but also for the magnetic behavior and for low-energy excitations. On the other hand, these properties are affected only indirectly by the electrons filling the deeper-lying levels, the so-called core electrons. The main reason for the limited role of the core electrons is the spatial separation of the core and valence shells, which originates from the comparatively strong binding of the core electrons to the nucleus [11]. However, more recently, the majority of DFT calculations have begun to use the projector augmented wave (PAW) method [12].

Computational Modeling of Nuclear Fuels 723

Studying materials at the atomic scale is often a matter of choosing efficiency versus accuracy. For example, when modeling ionic solids like uranium dioxide, empirical potentials of the Buckingham form allow for the study of larger system sizes, which can easily exceed millions of atoms with spatial dimensions approaching 50 nm or more that are necessary for simulating complex structures such as grain boundaries, dislocations, and nanoparticles embedded in a matrix. Such large size simulations cannot be done with more accurate approaches such as DFT. At the same time, such Buckingham-type potentials are quite simple in form but are not nearly as accurate as DFT calculations. Correspondingly, an important question arises: When are empirical potentials suitable, and when should studies rely on DFT? According to Uberuaga and co-workers [13], the use of empirical potentials versus DFT for modeling oxides essentially comes down to a question as to whether charge transfer occurs in the material. In cases where charge transfer is minimal, migration energies for even complex clusters in oxides calculated with empirical potentials agree very well with DFT values [14]. Even in cases where charge transfer occurs, as long as comparisons are made between structures where the charge transfer is similar, the trends revealed by potentials are typically very reasonable. For example, substitutional fission products in UO2 can cause oxidation of nearby U4+ ions, with the number of oxidized U ions depending on the valence of the fission product. The empirical potential method does not capture oxidation, yet the energetics of a fission product in the bulk and at a grain boundary calculated with an empirical potential agree very favorably with DFT calculations [15]. This is a consequence of the fact that the charge transfer in both environments is very similar and effectively cancels when comparing the energies. However, when the charge transfer reactions are critical to the defect property being investigated, one cannot trust the results obtained from interatomic potentials. For example, neglecting the oxidation of

724 Light Water Reactor Materials

U4+ ions when interstitial oxygen is introduced into UO2 + x leads to inaccurate predictions of the properties of interstitial oxygen. Advanced semi-empirical potential formalisms that allow for variable charge, or valence, states offer the promise of bridging between regimes where oxidation of uranium cations is important. Examples of these approaches include the potentials known as ReaxFF [16], and the charge-optimized many-body (COMB) potentials [17]. Returning to the topic of ab initio electronic structure calculations, the use of pseudo-potentials has tremendously simplified DFT calculations. A pseudo-potential is an effective potential used in DFT calculations, which incorporates the atomic nucleus and the tightly bound core electrons, so that the solution to the Schrodinger electron wave equation only has to consider the valence electrons [11]. However, more recently, DFT calculations are evolving toward the use of a PAW approach to describe the wave function of valence electrons [12]. In DFT calculations, it is typical to use supercells containing a few hundred (typically 108 to 512) atoms, which can be simulated with current laboratory-scale computer clusters as well as high-performance computing. However, this system size only allows studying point defects or very simple defect and/or solute/fission product configurations. Most DFT studies of UO2 use the Vienna Ab initio Simulation Package (VASP; https://www.vasp.at/) code [18–21]. It is again important to point out that most actinide oxides, like UO2, exhibit mixed valence character and, thus, interactions between ionic and electronic defects belong to the class of strongly correlated materials. For such materials, the standard approximations to the exchange-correlation potential within DFT fail to capture the properties of the partially filled d and f electron orbitals, which typical results in an underestimate of the band gap between the top of the electron valence band and the bottom of the conduction band. The strong electron-electron correlations of the uranium f

Computational Modeling of Nuclear Fuels 725

electrons have been successfully modeled using the Hubbard+U methodology [22, 23], which provides accurate predictions while still exhibiting good computational efficiency compared to more sophisticated ab initio techniques, which involve the use of hybrid functionals. The interested reader is directed to References 24 and 25 to learn more about the use of hybrid functionals in ab initio calculations, and to References 22 and 23 to learn more about the Hubbard+U, or so-called DFT+U methodology. The DFT+U method is by far the most popular for studying point defects and fission gas in UO2 due to its better computational efficiency than hybrid functional methods. Section 15.3.1 will provide an example of the use of DFT calculations for UO2.

15.2.2 Molecular dynamics (MD) MD computer simulations of radiation damage were first performed by George Vineyard and co-workers in 1960 [26]. Since that time, numerous groups have utilized MD simulations to study the physics of primary defect production in high-energy displacement cascades [26–28] and defect diffusion mechanisms [29, 30] important to understanding radiation damage in solids. Molecular dynamics provides an excellent tool for simulating the atomistic mechanisms of radiation damage, processes associated with defect diffusion, the interactions between defects and microstructural features, and dislocation-obstacle interactions. A brief, introductory description of molecular dynamics simulation will be provided here. For additional information, the reader is referred to excellent tutorials by Allen and Tildesley [31] and Hoover [32]. MD computer simulation generates atomic trajectories for an N-particle system by direct numerical integration of Newton’s equations of  motion (F = ma) based on prescribed boundary and initial conditions. The  atomic forces are derived from U, the potential energy function (Fi = − ∂U/ ∂ ri ) , which is, in general, a complicated function of the

726 Light Water Reactor Materials

positions of all of the atoms in the system. The classical equations of atomic motion can be written as   ∂U (r1 ,… , rN ) d 2ri i = 1,… N (15.1) Fi = mi 2 = −  ∂ ri dt  where mi is the mass and ri is the position vector of atom i. Starting from an initial configuration of the N-particle system (N is a function of available computing power) and a specified potential energy function, Equations 15.1 are numerically integrated to provide the atomic trajectories in an MD simulation. The basic MD solution algorithm can be summarized as follows: 1. Specify the initial simulation model, e.g., the number of particles, the potential energy function, the initial system configuration, boundary conditions, and the system state (pressure, temperature, and volume). 2. Initialize the system by assigning each atom an initial kinetic energy, generally from a Boltzmann distribution for the specified temperature. 3. Calculate the forces on each atom based on summing over the neighboring atoms, either through numerical differentiation of the potential energy function or an analytical expression, and integrate the equations of motion (Eq. [15.1]) to determine each atom’s acceleration, velocity, and change in position. 4. Repeat step 3 until the desired time has elapsed. Whereas MD simulates the dynamic evolution of an ensemble of atoms at a prescribed temperature and external applied stress from a given initial condition, molecular statics (MS) is used to determine relaxed structures and energetics of static (zero Kelvin) configurations by minimizing the potential energy U({rN}) of the system. Potential energy minimization is performed through standard approaches, including steepest-descent and

Computational Modeling of Nuclear Fuels 727

conjugate gradient quenches. LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator, http://lammps.sandia.gov) [33] is a common MD simulation package for which numerous interaction potentials have been implemented. A key consideration of atomistic MD simulations involves the description of the interatomic potentials. Additional background information about interatomic potentials is provided in Chapter 6 in the discussion of Figure 6.6, but here we will provide a brief description relevant to simulations of UO2. Interatomic potentials describe the interaction between lattice U4+ and O2− ions as well as lattice ions and gas atoms. Since the first empirical potential was developed for UO2 in 1962 [34], a multitude of parameter sets and potential forms have been developed with the goal of improving the description of some given aspect of UO2; see Govers et al. for a summary of some of these potentials and properties [35]. Unfortunately, due to the complexity of UO2, which is a function of both the f electron nature of the uranium ions and their flexibility in adopting different charge, or valence, states, none of these potentials provides a perfect description of the material. However, they should provide accurate qualitative behavior that gives fundamental insight into physical mechanisms that can be refined using density function theory. Here, we briefly identify three interatomic potentials to describe UO2. These involve a traditional pair potential originally developed by Basak et al. [36], along with a newer version developed by Morelon and coworkers [37], and an even more recent potential developed by Cooper, Rushton, and Grimes (CRG) [38]. While the Basak and Morelon potentials are both of the Buckingham form, the CRG potential overcomes limitations of this form by adding an embedded-atom method (EAM) [39] interaction term to effectively incorporate many-body interactions, which greatly improves the description of elastic behavior over a wide temperature range.

728 Light Water Reactor Materials

The potential energy, Ei, of an atom i with respect to all other atoms can be calculated for UO2, as 1 E i (ri ) = ∑ Φ αβ (rij ) − Gα ∑ Σ β (rij )  j  2 j

1/2

(15.2)

where rij is the distance between two species i and j, Φαβ is the pairwise potential energy interaction at a distance rij between two species designated as α and β, the Σβ are the effective pairwise interactions, and Gα alpha is a proportionality constant. This potential energy interaction combines a long-range electrostatic Coulomb interaction (ΦC) with short-range interaction terms that include interatomic interactions via both Morse (ΦM) and Buckingham (ΦB) potentials, such that Φ αβ (rij ) = Φ C (rij ) + Φ B (rij ) + Φ M (rij ) . (15.3) The Coulomb, Buckingham, and Morse interatomic potentials have the functional forms given by the following equations, respectively: q α qβ Φ C (rij ) = , (15.4) 4 πε orij ΦB (rij ) =

 rij  C Aαβ exp  −  − αβ6 , and  ρ αβ  rij

(

)

(

(15.5)

)

  − 2exp −γ r − r o  . (15.6)   αβ ij αβ     In Equations (15.4) to (15.6), εo represents the vacuum permittivity constant, qα and qβ are the atomic charges of the two species designated alpha and beta, respectively, and Aαβ, ραβ, Cαβ, Dαβ, γαβ, and rαβo are the empirically derived parameterized constants that describe the Buckingham and Morse interaction potentials between atoms i and j. The last term in Equation (15.2) represents the many-body, embedding term consistent with the EAM formalism [39]. The effective many-body o Φ M (rij ) = D αβ exp −2 γ αβ rij − rαβ 

Computational Modeling of Nuclear Fuels 729

contribution to the dominant pair potential interaction resulting from the superposition of the Coulomb, Buckingham, and Morse interactions is achieved by combining a set of pairwise interactions, Σβ(rij), between atom i and its neighboring atoms j. The resulting Σβ(rij), summed over all neighbors, is then passed through a nonlinear embedding function, which scales as the square root of Σβ(rij), and is multiplied by a constant of proportionality Gα. In the CRG potential [38], the effective pairwise interactions in the beta phase, Σβ(rij), are calculated as

∑ β (rij ) =

nβ rij8

(15.7)

where nβ is a constant of proportionality, and Equation (15.7) is cut off for distances (rij) shorter than 0.15 nm to prevent nonphysical forces from the embedded atom method (EAM) term, which could overwhelm the short-range repulsive interactions. The upper cut-off distance for the interactions described in Equation (15.7) is set to 1.1 nm, whereas a slightly shorter cut-off distance of 1.0 nm is used for the pairwise interactions described in Equations (15.5) and (15.6). Table 15.1 provides the values of the constants for the three interatomic potentials that describe UO2 [36–38].

15.2.3 Accelerated molecular dynamics Accelerated molecular dynamics encompasses a range of approaches to extend the typical simulated time by approximately 100 nanoseconds to reach milliseconds or beyond. These methods have been developed in particular by Voter and colleagues beginning in the 1990s [6–8]. Of these accelerated molecular dynamics approaches, the most commonly used are referred to as hyperdynamics, parallel replica dynamics (ParRep), and temperature-accelerated dynamics (TAD). Accelerated molecular dynamics methods can extend the time covered by atomistic simulations, but they are generally limited to systems containing a few thousand atoms and are best able to provide an increase in simulated time when the system does

730 Light Water Reactor Materials Parameter qU (eV) qO (eV) AU-O (eV) ρU-O (nm) CU-O (eV nm6) DU-O (eV) γU-O (nm−1) ρoU-O (nm) AO-O (eV) ρO-O (nm) CO-O (eV nm6) AU-U (eV) ρU-U (nm) CU-U (eV nm6) GU (eV nm1.5) nU (nm5) GO (eV nm1.5) nO (nm5)

Basak et al. [36]

Morelon et al. [37]

2.40

3.22724

−1.20

−1.61362

693.9297 0.0327022 0.0 0.57745 16.5 0.2369 1633.6666 0.0327022 3.95063 × 106 294.7593 0.0327022 0.0 0.0 0.0 0.0 0.0

566.498 0.042056 0.0 0.0 0.0 0.0 11,272.6 0.01363 134.0 × 106 0.0 0.1 0.0 0.0 0.0 0.0 0.0

Cooper, Ruston, and Grimes [38] 2.2208 −1.1104 448.779 0.03878 0.0 0.6608 20.58 0.2381 830.283 0.03529 3.8843 × 106 18,600 0.02747 0.0 57.1107 3450.995 × 105 21.82 106.856 × 105

TABLE 15.1: Parameter values used to represent UO2, as described by Equations (15.2) through (15.7). Note that the large number of significant figures provided in the Table for select parameters are required to ensure that the interatomic potential is continuous in both energy and derivative.

not have multiple activation barriers that control the rate of time evolution. The three main techniques will be briefly summarized here. Interested readers are referred to the references provided here, although we will not further discuss examples of the use of these techniques in this chapter. Hyperdynamics [6] attempts to increase the simulated time by providing a bias potential, δU, to the normal interatomic potential to increase the relative probability of overcoming an activation barrier between two adjacent

Computational Modeling of Nuclear Fuels 731

minima. Hyperdynamics assumes that no correlated activation barrier re-crossing events occur—i.e., that transition state theory (TST) is exact. A modified dynamical system is defined by the addition of a bias potential to the interatomic potential energy function(s) describing the system. This raises the energy in the potential basin but does not change the energy at the dividing surface at the location of the saddle point. Consequently, the accelerated trajectory executes a sequence of state-to-state transitions indistinguishable from that of a single long trajectory on the original potential surface. In the ParRep method [7], infrequent-event processes are first-order (Markovian), which offers a way to parallelize time. ParRep considers M replicas of the entire system running independently on M equivalent processors. The key point is that these M replicas behave like a new physical system with M times as many escape paths and a total escape rate that is Mktot, where the time is defined as that accumulated on all the processors up to the instant that a transition is detected. Because this replication does not disrupt the relative probabilities of the available escape paths, repeating this procedure gives a sequence of states and transition times that are indistinguishable from those of a long trajectory on a single processor. Therefore, this approach can accelerate infrequent-event dynamics nearly M times faster than a single-processor trajectory. Implemented properly, ParRep gives the classically exact dynamical evolution for an infrequent-event system. In TAD [8], the rapid escape from each potential basin is stimulated by raising the temperature of the system from the desired temperature (Tlow) to some higher temperature (Thigh). Simply raising the temperature does not give the correct state-to-state evolution for Tlow, because the relative escape probabilities change with temperature. However, TAD has developed techniques to ensure that the correct Tlow transitions are selected, so the system evolves appropriately from state to state. While TAD makes more approximations than hyperdynamics, it can provide significant accelerations to the calculations without having to construct a

732 Light Water Reactor Materials

bias potential. TAD results are accurate to within the desired confidence level if harmonic transition state theory [111] remains accurate up to Thigh and an appropriate lower bound on the pre-exponential factors is chosen. Defect and solute diffusion, interactions, clustering, and second-phase formation occur over seconds and microns and are inaccessible to MD, and often to accelerated MD. These must thus be modeled using either rate theory or kinetic Monte Carlo. Both methods solve kinetic reactiondiffusion equations. Rate theory generally solves the equations in a spatially independent continuum (see Ch. 13) without any spatial correlations between defects. Kinetic Monte Carlo solves the defect balance equations in three-dimensional space and thus captures all of the inherent spatial correlations and fluctuations.

15.2.4 Kinetic Monte Carlo The Monte Carlo method was originally developed by von Neumann, Ulam, and Metropolis to study the diffusion of neutrons in fissionable material on the Manhattan Project [40] and first applied to simulating radiation damage of metals by Doran [41] and later Heinisch and co-workers [42]. Monte Carlo utilizes random numbers to select from probability distributions and generates atomic configurations in a stochastic, or random, process [32], rather than the deterministic manner of MD simulations. While different Monte Carlo applications are used in computational materials science, this brief description is focused on object kinetic Monte Carlo (KMC) or kinetic lattice-based Monte Carlo (KLMC) methods. Additional information on Monte Carlo simulation techniques can be found in Allen and Tildesley [31]. Object KMC modeling of radiation damage involves tracking the location and fate of all defects, impurities, and solutes as functions of time to predict microstructural evolution. The starting point in these simulations is the primary damage state, i.e., the locations of vacancies and interstitials produced in displacement cascades resulting from irradiation and obtained from MD simulations, along with the displacement or damage rate, which

Computational Modeling of Nuclear Fuels 733

sets the time scale for defect introduction. The appropriate activation energies for diffusion and dissociation as well as the reactions that occur between species are also key inputs. In many cases, these activation energies can be obtained from lower-length-scale simulations using either DFT or atomistic MD simulation methods. The defects execute random diffusion jumps (in one, two, or three dimensions depending on the nature of the defect) with a probability (rate) proportional to their diffusivity. Similarly, cluster dissociation rates are governed by a dissociation probability that is proportional to the binding energy of a particle to the cluster. The basic steps in a KMC simulation are summarized in the following: 1. Calculate the probabilities (rates) of all possible events in the system. In object-oriented KMC simulations, the probabilities are generally from reaction rate tables. In KLMC simulations, the probabilities are based on semi-empirical interatomic interactions coupled to transition state theory. 2. Generate a random number to select an event from all possible events. 3. Increment the simulation time based on the residence time, or the Bortz-Kalos-Lebowitz algorithm [43], in which the inverse sum of the rates of all possible events is ( ∆t = χ/Σi N i Ri ), where χ is a random deviate that assures a Poisson distribution in time steps, and Ni and Ri are the number and rate of each event i. 4. Perform the selected event and all spontaneous events as a result of the event performed. 5. Repeat steps 1 through 4 until the desired simulation condition is reached.

15.2.5 Reaction rate theory and spatially dependent cluster dynamics model Modeling the spatially dependent nucleation, growth, re-solution, and coarsening kinetics of gas bubbles in nuclear fuel as well as the defect cluster evolution in nuclear fuel and cladding or core structural materials can be performed using reaction-rate-theory models, either in simplified form or

734 Light Water Reactor Materials

as part of cluster dynamics. The classical reaction-diffusion kinetic theory defines a defect/cluster by its character (e.g., atomic configuration, and size or, more specifically, the number of gas atoms and vacancies contained within the cluster) but not by its spatial position since the theory assumes that the concentration of each cluster is homogeneous through the volume of the sample. However, as is often the case in nuclear fuel and materials, the spatial dependence of the microstructural evolution is an important aspect to incorporate. Thus, we will include spatial dependence in the description here. The partial differential equation (PDE) governing reaction-diffusion for each species i within a kinetic-rate-theory-based cluster dynamics model has the general form  ∂c i ( r , t )   ɺ i ( r ) + ∇ i [Di ∇c i ( r , t ) − vc i ( r , t )] = FY ∂t + GRT + GRE − ART − ARE , (15.8) where ci refers to the volumetric concentration of the i-th cluster at the spatial position r , F is the fission rate density, Yi ( r ) is the yield, or production probability, of the i-th cluster by fission (e.g., 0.25 formation  probability for Xe), Di is the diffusivity of the i-th cluster, ∇ denotes the gradient operator, and v is the bulk velocity driving advective flow of species i or biased diffusion resulting from gradients in elastic interactions. The rates of reaction among species can lead to addition or subtraction of clusters of type i through a combination of first and second order reaction kinetics, including: 1. GRT, the rate of generation of the i-th cluster by trapping reactions (A + B → i) among other clusters; 2. GRE, the rate of generation of the i-th cluster by emission processes (e.g., decomposition by C → i + B) of other clusters; 3. ART, the rate of annihilation of the i-th cluster by its trapping reaction with other clusters (decomposition) (i + B → C); and 4. ARE, the rate of annihilation of the i-th cluster by its own emission (decomposition) process (i → A + B).

Computational Modeling of Nuclear Fuels 735

The system of PDEs represented by the coupled equations defined by Equation (15.8) are converted to ordinary differential equations (ODEs) by discretizing the spatial gradient terms and solving for each cluster evolution at each spatial grid point. The resulting coupled ODEs include variation within the reaction terms to ensure species conservation, i.e., that all the reactants and products of any reaction (trapping or emission) are accounted for and are constrained within the prescribed phase space. Equations similar to Equation (15.8) can also be defined for the reaction and diffusion evolution of point defects in cladding or core structural materials, as documented by Wirth and co-workers in Reference [3]. Further discussion on the use of rate-theory-based equations to model void nucleation and growth are presented in Chapter 19. Also, simplified versions of diffusion-reaction models are commonly used to model fission gas bubble evolution as initially derived by Speight [44]. As discussed recently by Pastore and co-workers [45], in such reduced models of noble fission gas behavior, which are typical of the types of models implemented in fuel performance codes, it is assumed that only single gas atoms diffuse through the crystalline lattice, that gas bubbles are immobile, and that gas atoms are absorbed in growing bubbles at a fixed rate, Jbub (atoms/sec), which is derived from a standard analysis of diffusioncontrolled growth as J bub = 4 πDfg Rig N ig (15.9) where Dfg is the Xe gas diffusion coefficient in the UO2 lattice, and Rig and Nig are the mean radius and number density of the intragranular gas bubbles, respectively. Since energetic fission fragments can collide with fission gas bubbles and knock Xe out of the bubbles, this model also includes re-solution that occurs at a rate, b (atoms/sec), as b = 3.03 Fπl f ( Rig + Z 0 ) 2

(15.10) . where F is the fission rate density, lf is the length of the fission fragment track, and Z0 is the radius of influence of a fission fragment track [45].

736 Light Water Reactor Materials

The processes of Xe fission gas absorption at the intragranular bubbles and Xe re-solution are assumed to be in equilibrium, which then leads to the following, which are used to model fission gas swelling in the UO2, and fission gas release (see also Sect 20.7.2): Cs b = , (15.11) Cb J bub 1 ∂  2 ∂c t  r = D eff 2 +β ∂t r ∂r  ∂r 

∂c t

and

Deff

=

b D . b + J bub s

,

(15.12) (15.13)

In Equations (15.11) through (15.13), cs is the volumetric concentration of fission gas in the form of single, mobile gas atoms, cb is the concentration of fission gas contained in intragranular bubbles, and ct is the total concentration of fission gas (= cs + cb); b and Jbub are the re-solution parameter and gas bubble absorption rate of single gas atoms mentioned in the preceding paragraph, respectively; β is the volumetric fission gas generation rate; and Deff is the effective diffusion coefficient of the mobile fission gas resulting from a balance of gas that is mobile and distributed within the UO2 versus that trapped in growing bubbles. For a fixed volumetric concentration of intragranular bubbles, Nig, the number of gas atoms in each bubble, m, can be obtained by partitioning the concentration of gas in bubbles, cb, among the intragranular bubbles, and thus c (15.14) m= b . N ig The bubble size increases according to 1/3

 3Ω ig  1/3 Rig =  m  4 π 

(15.15)

Computational Modeling of Nuclear Fuels 737

where Rig is the radius of the intragranular bubbles, and Ωig is the volume per gas atom contained in the bubble. Correspondingly, the fractional volume swelling due to intragranular bubbles is 3   1/3    ∆V  = N  4 πR 3  = N 4  3Ω ig c b    π ig  V  ig  3 ig  ig 3  4 πN ig         J bub = Ω ig c b = Ω ig c . (15.16) J bub + b t Thus, the volumetric swelling due to fission gas in this model is determined by the partitioning of fission gas to the growing bubbles relative to the amount of fission gas re-solutioned, where both g and b are functions of many parameters including the gas diffusivity, the bubble number density and radius, as well as the fission rate density and spatial size of the fission fragment tracks. The impact of this model for fission gas swelling and release incorporated in Equations (15.9) to (15.16), as well as extended to incorporate grain boundary bubbles and fission gas release, will be described later in Sections 15.4.1 and 15.5.1.

15.2.6 Phase field modeling The phase field method emerged as a powerful computational approach in the early 1990s to model and predict morphological and microstructure evolution in materials at the mesoscale, encompassing dimensions from 100s of nanometers to 100s of microns [46]. Phase field models a microstructure using field variables and a Gibbs energy functional to describe both the compositional/structural domains and the interfaces between distinct domains. The field variables are continuous across the interfacial regions and, hence, the interfaces in a phase field model are diffuse (not atomically sharp). The time and spatial evolution of these field variables are governed by solving the nonlinear Cahn-Hilliard diffusion

738 Light Water Reactor Materials

equation coupled to an Allen-Cahn relaxation equation. In the diffuseinterface description [46], the total Gibbs energy of an inhomogeneous microstructure system described by a set of conserved (c1, c2, …) and nonconserved (η1, η2, …) field variables is then given by n   2 ( , , , , , , , ) ( ) η η η + α ∇   g c c c c ∑ n 1 2 p i i   1 2 i =1  d 3r + G=∫ 3 3 p   + β ∇ η ∇ η ∑ ∑ ∑ ij i k j k   i =1 j =1 k =1  

∫∫ H (r − r ′)d 3rd 3r ′

(15.17)

where g is the local Gibbs-energy density that is a function of field variables ci and ηi, while αi and βij are the gradient energy coefficients [46]. In Equation (15.17), the first volume integral represents the local contribution to the Gibbs energy from relatively short-range chemical interactions. The interfacial energy contributions to the Gibbs energy result from gradient energy terms that are nonzero only at, and around, the interfaces. The second integral in Equation (15.17) represents a nonlocal term that contains the contributions to the total Gibbs energy from any of a number of long-range interactions, including electric dipole interactions, electrostatic interactions, and elastic interactions, all of which depend on the field variables. The temporal and spatial evolution of the field variables is governed by the nonlinear Cahn-Hilliard diffusion equation and the Allen-Cahn relaxation equation. Based on using fundamental thermodynamic and kinetic information as input, phase field models are able to predict the evolution of arbitrary morphologies and complex microstructures without explicitly tracking the positions of interfaces. Phase field models have successfully modeled numerous microstructural processes including solidification, solid-state structural

Computational Modeling of Nuclear Fuels 739

phase transformations, grain growth and coarsening, domain evolution in thin films, pattern formation on surfaces, dislocation microstructures, crack propagation, and electromigration, as reviewed by Chen [46]. More recently, phase field models have been applied to nuclear fuel modeling, and the interested reader is referred to References [47] and [48].

15.2.7 Finite element–based fuel performance models Nuclear fuel is most succinctly described as a multicomponent system required to (i) produce thermal energy through fission, (ii) efficiently transfer that energy to the coolant, and (iii) retain fission products by maintaining structural integrity under all operational conditions and anticipated accidents [49]. As shown schematically in Figure 15.3, these requirements lead to various indicators of fuel performance, the controlling mechanisms that dictate performance, and the atomistic processes responsible for the thermal, mechanical, and chemical response to the extreme nuclear fission environment. These steps are highly interrelated, and vary with time and position, and ultimately integrate hierarchically in time and space to determine nuclear fuel performance. Ideally, at each level, processes modeled analytically are verified based on dedicated separate effects or integral tests. This poses certain problems for treating atomistic-scale processes, as the attainment of experimental verification and validation at such a resolution level are difficult to obtain and apply on irradiated material. As one example, the modeling of a single Xe bubble in an infinite and perfectly ordered UO2 matrix, while highly complex in itself, is still too simple in comparison to the modeling of the generation and spatial diffusion of Xe (and Kr) under actual high burnup fuel conditions, which in reality is much more complex than described by Equations (15.9) to (15.16). Therefore, extreme caution is necessary in scaling up atomistic processes to represent behavior at the global level, and the development of fuel performance models that are grounded in atomistic-level modeling remains a work in progress.

740 Light Water Reactor Materials Requirements Produce thermal energy, retain fission products, maintain structural integrity

Performance indicators Temperature, stress, and strain

Behavioral mechanisms Power, burnup, corrosion, fission gas release, swelling, thermal expansion, creep, irradiation growth, cracking, etc.

Atomistic processes Atomic fission, lattice impurities, atom diffusion, defect formation and annihilation, chemical reactions, etc.

FIGURE 15.3: Schematic illustration of fuel behavior modeling and simulation, as reproduced from Rashid et al. [49].

Finite element–based fuel performance models generally incorporate constitutive models to describe fuel behavior and thermal-mechanical response of the fuel and cladding during the complex, time-dependent operational history. Within a finite element method-based model, the governing partial differential equations are formulated as boundary value problems in which the larger problem is divided into smaller domains. Each of these domains is referred to as a finite element. For example, for modeling nuclear fuel performance, the governing partial differential equations provide for energy, species, and momentum conservation, in which the energy balance expressed in terms of the production and dissipation of heat, the species conservation solves a diffusion equation, and momentum conservation involves assuming static equilibrium with

Computational Modeling of Nuclear Fuels 741

Cauchy’s equation relating the material stress to the body forces. The individual equations within each finite element domain are then assembled into a larger system of equations that provides the simultaneous solution to the governing equations for the entire nuclear fuel domain. The finite element method typically uses linear algebra approaches to approximate a solution by minimizing the error. Until the development of three-dimensional fuel performance codes in the late 2000s, the state of the art in numerical simulation of fuel performance predominantly involved an axisymmetric, axially stacked onedimensional (1-D) radial representation of the fuel rod; this is sometimes referred to in the literature as a 1-D simulation. The best-known examples of such codes are the FRAPCON and FRAPTRAN steady-state and transient codes, respectively, used by the Nuclear Regulatory Commission (NRC) as audit codes and under development by Pacific Northwest National Laboratory (PNNL) [50, 51]. Industry codes in this class include COPERNIC by AREVA France, RODEX by AREVA USA, PRIME by GNF, PAD by Westinghouse, TRANSURANUS by ITU [52], and ENIGMA by British Energy. The FALCON code, developed by EPRI [49, 53] is capable of performing a fully coupled thermal-mechanical fuel performance simulation in two dimensions (2-D), using either an R-Z axisymmetric simulation or an R-θ slice at a particular axial location. FALCON is capable of performing both steady-state and transient simulations using finite elements to solve the thermal and mechanical response of the fuel. As an example of threedimensional (3-D) fuel performance codes, the ALCYONE code, which was developed by the CEA [54], and the BISON code developed at the Idaho National Laboratory [55] are used for 3-D fuel performance calculations. As noted previously, many of the fuel behavior mechanisms represented in Figure 15.3 exist in fuel performance codes as simple correlational approximations of the governing atomic-scale processes. These include the description of the burnup-dependent fuel thermal conductivity; fission gas generation, diffusion, and release; solid fission product swelling; thermal and irradiationinduced densification; and the evolution of the high-burnup microstructure,

742 Light Water Reactor Materials

as well as the neutron-induced irradiation damage of the cladding. These correlations have evolved over decades of analytical and experimental development. The associated physical-chemical processes such as fission gas generation and release, fuel restructuring, fuel thermal conductivity dependency on burnup and microstructure evolution, pellet cracking and relocation, irradiation growth, cladding corrosion and hydriding, and so on, are added as behavioral models to the set of governing equations in the fuel performance code, and subsequently benchmarked within experimental limitations. While efforts continue to develop mechanistic-based models that extend beyond these empirical correlations, the challenges to doing so are numerous. Figure 15.4 demonstrates the complex interactions that govern fuel performance modeling. Despite the progress made in recent years in the understanding of the various physical phenomena governing LWR fuel behavior and their interdependence. Figure 15.4 shows some of the phenomena, with processes in the fuel in dark gray, processes in the cladding in light gray, and processes at the interface in mixed color. The arrows represent some of the most obvious interdependences. For example, as fission gas release occurs, the internal rod pressure increases and void swelling decreases. We note that not all dependencies are shown; one could almost draw lines between every box. Shown in dashed boxes are results from the phenomena, while power conditions are in white boxes and coolant/assembly conditions in textured background. Many fundamental issues remain to improve the fidelity with which we can model fuel rod behavior and performance. These include the modeling of fuel gaseous swelling and of the evolution of the high-burnup structure (HBS); pellet-cladding interfacial friction and bonding; fuel pellet and cladding deformation as they affect fuel rod stress and failure probability; the evolution of cladding hydriding during corrosion; and the mechanistic characterization of fission product interactions with the cladding stress state at failure.

15.2.8 Summary The overarching scientific challenge is to develop a realistic, predictive capability for modeling the complex microstructure evolution under

Computational Modeling of Nuclear Fuels 743 Coolant C Co olan ol an ntt Pressure, n Pressure, Temperature, Tem mp perat rature Ch Chemistry, hemistry r , Void d Fraction Fracttion F Deformation: Creep and growth

Gap width & composition

Irradiation hardening and embrittlement Cladding failure Cladding corrosion and hydriding

Pellet-Cladding mechanical interaction

Fuel/Cladding bonding

Fission gas release

Creep and swelling

Fuel pellet cracking

Fuel restructuring

Thermal conductivity decrease

High burnup rim formation

Densification and porosity Cladding temperature and gradient

Fission density and Particle Flux

Internal rod pressure

Total Power and power History

Fuel temperature and gradient

Exposure time inreactor and Fuel Burnup

FIGURE 15.4: Schematic illustration of the complex phenomena and behavioral evolution and their interactions, which must be incorporated in modeling fuel rod behavior, after Rashid [49] and Olander [1]. The dark gray boxes represent processes in the fuel, the light gray in the cladding and the mixed boxes at their interface. Dashed outline boxes are responses of the fuel rod to external processes. The coolant and assembly conditions are represented in the textured boxes up top while the reactor power conditions are in the white boxes below.

irradiation of nuclear fuels as well as the alloys that are used as cladding or structural materials. This requires the further development of predictive theories of kinetics, nucleation, and coarsening. This degree of predictive capability requires the use of multiscale models that inform macroscale properties with resolved microstructure evolution. A further challenge is to combine the multiple concurrent processes into a comprehensive computational model to provide an accurate description of the co-evolution of various interacting elements of microstructure— dislocations, grain boundaries, radiation defects, and alloy phases—to yield the required net thermomechanical response. For a hierarchical

744 Light Water Reactor Materials

multiscale simulation approach to become a useful and reliable tool for material design, and certification, the models at every single-scale level have to be computationally efficient to allow for error propagation and quantification-margin-uncertainty analysis and for a thorough exploration of the relevant parameter space in order to identify the most informative validation experiments.

15.3 Irradiation Behavior of UO2 The remainder of this chapter will provide a select set of examples of modeling techniques associated with xenon (Xe) behavior in nuclear fuel. The solubility of noble gases in the UO2 matrix is extremely low, with most estimates of maximum solubility well below a part per billion [56, 57]. Xe diffusion and precipitation in fission gas bubbles influences both the amount of fuel swelling and the quantity of fission gas released to the fuel rod plenum. Despite decades of investigation, significant uncertainties exist regarding the underlying mechanisms of Xe diffusion, precipitation, and release. These impact predictions of fission gas release during both normal operation and transient conditions in accidents, and therefore impact integrated fuel performance models [49]. Most fission gas behavior models trace back to the 1957 formulation by Booth [58], the mid-to-late 1960s formulation by Speight and co-workers [44, 59]; or the slightly more recent work by Turnbull et al. [57], White and Tucker [60], and Forsberg and Massih [61, 62]. These models typically rely on a small number of spatially-independent partial differential equations, or even a single partial differential equation, along with the concept of an effective diffusivity to treat the partitioning of fission gas between bubbles within the grain interior versus at the grain boundaries, combined with a dependence on fission gas re-solution. Further, it has long been postulated that Xe fission gas release occurs as a result of Xe diffusion to grain boundaries at sufficient concentration [45, 62, 63], but it is not

Computational Modeling of Nuclear Fuels 745

entirely certain that grain boundaries provide higher gas diffusivities, or interpercolating pathways, since an extensive network of grain boundary bubbles is commonly observed in irradiated nuclear fuel [64]. Finally, fission gas retention and release impact nuclear fuel performance by, for example, reducing the fuel thermal conductivity, causing fuel swelling that leads to mechanical interaction with the cladding, thereby increasing the plenum pressure and reducing the fuel-cladding gap thermal conductivity. In fission gas-release models, the bulk fission gas diffusion rates, e.g., Dfg from Equation (15.13), are taken from the analysis of Turnbull [57], which is meant to include both intrinsic and radiation-enhanced diffusion. This model divides the diffusivity into three regimes, as shown in Figure 15.5. At a high temperature (T >≈ 1700 K), intrinsic diffusion dominates, and in the intermediate temperature range (~1400 < T < ~1700  K), radiation-enhanced diffusion is the main contribution to fission gas transport. The rate depends on the concentration of radiationinduced vacancies, which is presumed to be proportional to the square root of the fission rate based on a rate theory analysis that include the simplest vacancy and interstitial defects and effective diffusion rates derived from experiments. Below about 1400 K, Xe diffusion is measured to be essentially athermal and proportional to the fission rate. It is important to point out that Turnbull defined these regimes of Xe diffusion based on analysis of fission gas release during or following in-reactor exposure, and thus, the transition temperatures between regimes are likely subject to significant uncertainty. This section concludes with examples of using electronic structure calculations to calculate the diffusional behavior of Xe in UO2, including a brief comparison to the diffusional behavior of other fission products (Sec. 15.3.1), as well as atomistic MD simulations to calculate defect production in displacement cascades (Sec. 15.3.2) and the thermal conductivity of UO2 (Sec. 15.3.3). Section 15.3.3 also includes an example of using atomistic data on thermal conductivity within the

746 Light Water Reactor Materials

10

–17

Intrinsic diffusion (from [28]) 10–18 D (m2 s–1)

Key:Contribution due to enhanced cation vacancy concentration caused by irradiation damage. Irradiation induced athermal contribution.

10–19

10–20

10–21

4

6

8

10

12 14 104 / T(°K)

16

18

FIGURE 15.5: Illustration of the length and time scales (and inherent feedback) involved in the multiscale processes responsible for microstructural changes in irradiated materials, as reproduced from Reference [57].

BISON fuel performance code to evaluate the contributions of Xe to thermal conductivity degradation at the center versus the periphery of a fuel pellet.

15.3.1 Electronic structure calculations of fission product and gas behavior in UO2 Many authors have used DFT calculations to calculate the behavior of intrinsic Frenkel and Schottky defects in UO2, as well as the thermodynamics of notable fission products, based on using the plane wavebased VASP [18–21] or ABINIT (http://www.abinit.org) [65, 66]. This section includes an example of fission gas incorporation and diffusion in UO2.

Computational Modeling of Nuclear Fuels 747

The strongly correlated nature of the uranium 5f electrons requires special attention in DFT calculations, as discussed briefly in Section 15.2.1. Standard local or gradient-corrected exchange-correlation potentials for describing the electron-electron interaction fail to capture the electron band gap responsible for the semi-conducting properties of UO2. There are several extensions to standard DFT addressing this issue, such as hybrids [24, 67] and the Hubbard+U [23, 68] methodology that more explicitly capture the Coulomb interaction among 5f electrons. The latter, the so-called DFT+U method, is by far the most popular for studying point defects and fission gas due to its relative computational efficiency, allowing the complex defects of critical importance for nuclear fuel performance to be investigated. Early work using the DFT+U methodology produced an unexpectedly wide range of defect formation energies (several eVs), which was ascribed to metastable electronic solutions for uranium f orbital occupation [69]. Several schemes have since been developed to resolve this issue and reduce the uncertainty for the predicted defect energies, including the occupation matrix control [69, 70] and dynamic changes to the Hubbard U term [71]. Despite this progress, challenges remain to capture the strongly correlated behavior of uranium f electrons and, correspondingly, the properties of defects at a truly quantitative level. The behavior of fission products in UO2 is determined by the location of the fission products in the lattice and the interaction of the fission products with thermal and irradiation-induced defects. The preferred location of fission products is governed by the availability of trap sites, typically expressed by an effective defect formation energy through an Arrhenius relation, and the energy associated with adding the fission product to the trap site (incorporation energy) [72–74]. In thermal equilibrium the concentration of trap sites, such as single uranium vacancies or vacancy complexes containing several uranium and/or oxygen vacancies, is controlled by a combination of the Frenkel pair formation reaction, the Schottky defect formation reaction, and the binding energy

748 Light Water Reactor Materials

of vacancy clusters [72–74]. The resulting relationship is even more complicated under irradiation. The defect concentration is also a function of stoichiometry and varies strongly between UO2−x, in which the favored defects are oxygen vacancies and uranium interstitials, and that of UO2 and UO2+x, in which the favored defects are oxygen interstitials and uranium vacancies [72–76], as discussed in more detail in Chapter 4. The large, noble fission gas atoms Xe and Kr prefer to occupy the trap site with the largest atomic volume, which leads to increasing incorporation energy from a tri-vacancy Schottky defect (containing one uranium and two oxygen vacancies, to the di-vacancy of one uranium and one oxygen vacancy) to the single uranium vacancy. Xe interstitials are quite high in energy compared to the situation where Xe is able to occupy one or more vacancies, which are referred to as vacancy trap sites. The preferred trap site is determined by the Xe solution energy, defined as the sum of the defect formation energy and the incorporation energy, and most calculations agree that the preferred site for Xe is dependent on the stoichiometry of the fuel, in which the preferred trap site is a bound Schottky defect in UO2 x (in a bound Schottky the uranium and oxygen vacancies are close by and interact), but is either a bound Schottky or a di-vacancy in UO2, and is a single uranium vacancy in UO2+x [74–76]. The impact of irradiation, or other impurities, as well as the charge state of the defect and the uranium valence state, on the preferred trap site has not been extensively explored. In stoichiometric UO2, the formation energy of a uranium vacancy trap site is given by ES−EF (where ES is the Schottky energy and EF is the anion Frenkel energy), while that of a di-vacancy trap site is given by ES−1/2EF−Bdv (where Bdv is the di-vacancy binding energy) and that of a neutral tri-vacancy is given by ES−Bnt (where Bnt is the neutral tri-vacancy binding energy) [74–76]. The corresponding vacancy configuration energies, and binding energies, can be estimated by either DFT or empirical potential calculations. For example, using periodic −

Computational Modeling of Nuclear Fuels 749 Defect type Schottky: ES Frenkel (anion): EF Di-vacancy binding: Bdv Tri-vacancy defect: Bnt

Energy (eV) 7.65 4.26 1.54 2.15

TABLE 15.2: Defect reaction and binding energies defining the trap site formation energies in stoichiometric UO2 (see Table 15.3), with results obtained from Reference [75].

boundary conditions within a supercell approach (a supercell is an expansion of the unit cell used to isolate individual defects in simulations assuming periodic repetition of the simulation cell), the Frenkel energy can be defined as E F = E tot(UO 2 + O vacancy ) + E tot(UO 2 + O interstitial ) − 2 E tot(UO 2 ) , (15.18) where Etot(UO2 + O vacancy) equals the energy of a supercell of UO2 containing an oxygen vacancy; Etot(UO2 + O interstitial) equals the energy of a supercell of UO2 containing an oxygen interstitial atom; and Etot(UO2) is the energy of the perfect UO2 lattice without any defects. The defect and binding energies for UO2 obtained from DFT calculations performed in Reference [75] are summarized in Table 15.2. The Xe incorporation energies are obtained by similar supercell calculations, such that the incorporation energy (Einc) of Xe in a uranium vacancy trap site (EFVU) is defined as E inc = E FVU = E tot(UO 2 + Xe in VU ) − E tot(UO 2 + U vacancy ) − E ref( Xe ) , (15.19) where Etot(UO2 + Xe in VU) equals the energy of a supercell of UO2 with one Xe atom occupying a uranium vacancy site; Etot(UO2 + U vacancy) equals the energy of a supercell of UO2 with a uranium vacancy; and

750 Light Water Reactor Materials

Eref(Xe) represents the reference energy of a Xe atom, which is taken to be an isolated atom. The incorporation energies for common trap sites are listed in Table 15.3 [75]. Based on this analysis, the preferred Xe trap site for stoichiometric UO2 is the di-vacancy, as given by the sum of the defect formation and the Xe incorporation energy (solution energy). The diffusion of fission gas atoms that are trapped in one or more vacancies has been shown to occur by first binding another uranium vacancy to the Xe trap site. The resulting cluster, which in most cases has a fairly high binding energy, then migrates according to a concerted mechanism in which the intracluster migration of a uranium vacancy is the rate-limiting step [76]. Even though Xe interstitials have very high formation energies, the interstitial Xe is expected to migrate rapidly as a result of a fast mechanism involving displacements of ions on the oxygen sublattice and, thus, are rapidly expected to trap in vacant sites containing one or more vacancies [77]. The activation energy for Xe diffusion by a vacancy mechanism is a sum of the vacancy formation energy (EVf), the binding energy of the vacancy to the trap site (Eb) and the rate-limiting intracluster migration step (Em). This picture of Xe diffusion qualitatively reproduces the experimentally observed change in activation energy and diffusivity as function of the nonstoichiometry in UO2 ± x. Further, the value for Xe migration in stoichiometric UO2 corresponds to the hightemperature intrinsic diffusion regime in Turnbull’s model [57]. However, the predicted diffusion coefficients have been reported to be lower than observed in experiments [78], which is probably a combined effect of the complex clusters involved, including the charge state of clusters (the number of bound electrons and holes), and the sensitivity of experimental results to the chemistry controlling nonstoichiometry. The activation energy for Xe diffusion in stoichiometric UO2, under thermal conditions that do not include any radiation-induced defects, can be estimated by first calculating the binding energy of a vacancy to the Xe trap

Computational Modeling of Nuclear Fuels 751

site and then the barrier for cluster migration. Thus, the activation energy for diffusion (EXem) can be defined as (15.20) E Xem = EVf + E b + E m , where EVf is the vacancy formation energy; Eb is the binding energy of a vacancy to the Xe trap site, and Em is the migration barrier of the Xe trap site + vacancy complex. Again, by using a supercell approach, the binding energy can be calculated as E b = E tot(UO 2 + bound Xe vac cluster ) + E tot(UO 2 ) (15.21) − E tot(UO 2 + Xetrap) − E tot(UO 2 + vacancy ) , where Etot(UO2 + bound Xe vac cluster) is the energy of a supercell containing a bound cluster, Etot(UO2) is the energy of a perfect supercell, Etot(UO2 + Xe trap) is the energy of a supercell containing the Xe trap site, and Etot(UO2 + vacancy) is the energy of a supercell containing a vacancy. The vacancy formation energy is obtained by a similar formula as Equation (15.19) for the vacancy trap site formation energy. The nudged elastic band method maps the energy landscape (migration barrier) between the original cluster configuration and that after the completed migration process, which for some clusters may involve several steps. This vacancy diffusion mechanism is schematically illustrated in Figure 15.6. The binding energies, migration barriers, and resulting activation energies have been calculated from a combination of DFT modeling predictions from References [78] and [79], and are listed in Table 15.3. Note that, even though the activation energy for the uranium vacancy trap site is the lowest, the di-vacancy is still the most important because it is a more stable trap site compared to the uranium vacancy (the value within parentheses accounts for the different solution energy between trap sites). For in-reactor conditions, it is not sufficient to model the behavior of fission gases under equilibrium conditions, but the effect of the irradiation environment must be taken into account. These effects are most

752 Light Water Reactor Materials (a)

E = E vF U

(b)

E = E vF U – EB

(c)

E = E vF – EB

(d)

E = E vF – EB

Xe

U

U

+Em

FIGURE 15.6: Schematic illustration of Xe diffusion via a vacancy-mediated mechanism. For simplicity, the cubic oxygen sublattice is omitted and only the fcc uranium sublattice is shown. Xe atoms are shown as large, light gray balls, uranium as mid-sized dark gray balls, and vacancies are represented by squares. (a) [101] projection of the uranium sublattice with Xe occupying a uranium vacancy site and a second vacancy located several lattice distances away from the Xe trap site (unbound). (b) The vacancy is moved to the Xe trap site and the Xe atom occupies the central position of the cluster created by the original trap site, and the second bound vacancy. (c) Three-dimensional view of the fcc uranium sublattice with a Xe atom occupying a vacancy and a second vacancy bound to this trap site. The highlighted uranium atom can migrate into one of the cluster vacancies, thus giving rise to net Xe diffusion. (d) Equivalent to the defect cluster in (c) but with the highlighted uranium atom translated from its original position in (c) into the nearest neighbor vacancy site along the [1/2 0 1/2] lattice vector. This figure is reproduced from Reference [75].

significant at low to intermediate temperatures, while the damage quickly anneals out at higher temperatures and the intrinsic equilibrium behavior is therefore recovered. The main consequence of irradiation for fission gas diffusion is believed to relate to increasing the availability of uranium

Computational Modeling of Nuclear Fuels 753

Defect formation energy: Ef Xe incorporation energy: Einc Solution energy (Ef + Einc): Es Vacancy binding energy to the Xe trap site: Eb Migration barrier for the Xe trap site + vacancy complex: Em Activation energy for Xe diffusion (Ef + Eb + Em)

Uranium Uranium + Uranium + two vacancy oxygen vacancy oxygen vacancies 3.39 4.00 5.50 3.54 1.24 0.28 6.93 5.24 5.78 –0.99 –1.82 –3.77 3.50 4.94 5.25 6.84 (8.53)

7.12 (7.12)

6.98 (8.52)

TABLE 15.3: Defect energies controlling trap site occupancy and diffusion of Xe in stoichiometric UO2, as obtained from References [78] and [79].

vacancies above the thermodynamic limit, which translates into increased diffusion due to fission gas atoms finding more vacancies to bind with and thus increasing the fraction of mobile clusters. A simple rate theory treatment of the irradiation effects of Xe diffusion in UO2, however, does not reproduce the known experimental diffusion behavior of a low activation energy and low pre-exponential factor [78]. This is likely the result of the single uranium vacancy, as well as Xe clusters involving a single mobile uranium vacancy, which are calculated by DFT to not be sufficiently mobile. The best explanation for this deficiency seems to be that for in-reactor conditions, both vacancies and Xe trap sites may form larger clusters than those considered previously, and this is schematically illustrated in Figure 15.6. The simplest example to demonstrate this would involve two uranium vacancies binding together to form a di-vacancy cluster with a migration barrier that is almost 2 eV lower than that of a single vacancy, and which would result in much faster diffusion rates in irradiated UO2. The increased mobility competes with the lower concentration of the extended defects, but at least under irradiation, the concentration of di-vacancies may be sufficiently high to dominate Xe migration behavior. A similar reduction of the cluster migration barrier

754 Light Water Reactor Materials

occurs for Xe clusters involving several uranium vacancies, although the details are not yet fully understood. Finally, MD simulations by Cooper and co-workers [80] indicate that Xe diffusion during the damage process of displacement cascade evolution could be consistent with the low temperature, athermal Xe diffusion, noted by Turnbull [57] and shown in Figure 15.5. Solid fission products have much higher solubility in the UO2 matrix than noble fission gas atoms, and generally also have smaller atomic radii, which means that these fission products can occupy substitutional sites on the uranium sublattice, meaning that these solid fission products are located on uranium vacancies. Substitutional solutes will diffuse by a vacancy mechanism similar to uranium cations or Xe trapped at a vacancy. The resulting activation energy for diffusion of solid fission products may be higher than for Xe, however, mainly because of the lower binding energy of vacancies enabling diffusion to the fission product trap site [79]. The highest activation energies are for fission products that are similar in size to uranium, while lower activation energies are observed for smaller ions with ruthenium in certain charge states having the lowest energy. The activation energies of diffusion of select fission products have been calculated by DFT [78, 79, 81], and are compared to Xe in Figure 15.7.

15.3.2 Displacement cascade simulations in UO2 Martin and collaborators [28] have used a rigid ion model for UO2 developed by Morelon et al. [37], to which the universal Ziegler-Biersack-Littmark potential [82] was fit to describe high-energy elastic collisions that occur at distances less than 0.16 nm. MD was used on a volume containing about 3.8 million atoms to simulate displacement cascade evolution caused by uranium primary knock-on atoms (PKAs) with energy from approximately 1 keV to 100 keV.

Computational Modeling of Nuclear Fuels 755 1.42 5

Ionic radius (Å)

0.62

Activation energy (eV)

4

3

2

1

0 Xe0

Ba2+ Sr2+ La3+ Y3+ U4+ Ce4+ Zr 4+ Ru3+ Ru4+

FIGURE 15.7: The activation energy for diffusion of a range of solid fission products, as compared to the uranium cation or noble fission gas Xe, as reproduced from Reference [79].

Due to the fluorite crystal structure and ionic bonding of UO2, it is important to calculate the number of defects produced on both the uranium and the oxygen sublattices. The total number of defects were calculated on the uranium, oxygen, and interstitial octahedral sites. The defects are then calculated as the total number of vacancies and atoms at interstitial positions irrespective of whether they form clusters or not. For instance, Martin et al. noted that a Schottky defect composed of a uranium and two oxygen vacancies are counted as three defects [28]. As with MD simulations of cascade evolution in metals, the displacement cascade structure induced by high-energy PKAs is observed to develop into several distinguishable subcascades. Figure 15.8 illustrates this for the case of a 100-keV uranium PKA in UO2, in which at least three distinct subcascade volumes can be distinguished. Figure 15.9 shows the total number of Frenkel pair defects produced on the uranium and oxygen sublattices as a function of PKA energy [28].

756 Light Water Reactor Materials

FIGURE 15.8: Displaced atoms (by dx ≥ 0.15 nm) at a time of 15 picoseconds following initiation of a 100-keV displacement cascade in UO2 at 700 K, as reproduced from Reference [28].

As shown, there is a fairly clear linear dependence of the total number of defects produced, which is in agreement with the Norgett, Robinson, and Torrens (NRT) [85] model of primary defect production although the absolute number of defects is less than predicted. Also shown for comparison are the results from a database of cascades in body-centered cubic iron obtained by Stoller [83]. It is important to note that the damage energy threshold on the oxygen sublattice is significantly lower than on the uranium sublattice, with quoted values of approximately 20 eV for oxygen and 40 to 50 eV for the uranium sublattice, as evaluated by Meis and Chartier for this interatomic potential [84]. However, Cooper and co-workers [38] have indicated that the absolute number of Frenkel pairs produced by the Morelon et al. potential [37] is below that of both the Basak [36] and the CRG [38] potentials listed in Table 15.1 for a 1-keV U

Computational Modeling of Nuclear Fuels 757 1000

Total number of defects

Defects in UO2 D = D U + DO

27

DNRT [85] for UO2

100

5

6.

E0 12

E0

E0

Defects in iron [83]

10 1

10 Initial PKA energy E 0 (keV)

100

FIGURE 15.9: Total number of defects in UO2 (hexagons) as a function of PKA (E0) energy, compared to the NRT model and to prior MD simulations of cascades in iron, as reproduced from Reference [28].

primary knock-on atom energy. Assuming an approximate average value of 30 eV for the displacement threshold energy of urania, the NRT model would predict an average linear slope of about 27 times the PKA energy dependence (in keV). However, Figure 15.9 indicates a reduced value for the slope as 12E0 (keV). Presumably, this results from the recombination of vacancy and interstitial defects on the respective uranium and oxygen sublattices, due to the high local point defect concentrations during the cascade evolution.

15.3.3 Thermal conductivity of UO2 (see also Ch. 9 and 16) At the crystal lattice level, thermal transport in UO2 and other semiconductors in which the heat transport occurs by phonon vibrations can be investigated using MD simulations. Phonons are the collective vibrations of atoms and, thus, MD simulations can assess thermal conductivity. This is not the case for metals, in which heat transport occurs by electrons. There are two widely used methods to compute thermal conductivity using MD: (i) the equilibrium Green-Kubo method; and (ii) the

758 Light Water Reactor Materials

nonequilibrium MD method in which a temperature gradient is imposed on the system. Phillpot and co-workers have used the nonequilibrium MD simulation method to evaluate the thermal conductivity of UO2. In this method, a heat source and sink are included in the simulation cell, which effectively introduces an imposed heat flux and leads to the establishment of a temperature gradient which can then be directly related to the thermal conductivity through Fourier’s law [86]. For comparison, the Green-Kubo method, in which the time correlation of the microscopic heat current is calculated during an equilibrium simulation, and the interested reader is referred to simulations performed on SiC [87]. Figure 15.10 compares the temperature dependence of the thermal conductivity determined from MD simulations performed by Phillpot and co-workers [86] (solid line and circles) to a commonly used empirical correlation (dashed line) [88]. The MD simulations used an empirical interatomic potential for UO2 similar to those described earlier; however, this potential was fit by Busker [89]. Over the temperature range of interest, MD simulations agree with the empirical mode to within ~25%. The results shown in Figure 15.10 are for isotopically pure and defectfree UO2 with a very large grain size. To assess the effects of defects and microstructure, Lee and co-workers [90] have more recently evaluated the effect of 0.8-nm-diameter pores by using the nonequilibrium MD approach with a Buckingham-style interatomic potential fit by Grimes and co-workers [91]. Figure 15.11 shows the MD results of the dependence of the thermal conductivity as a function of the volume fraction of porosity. While the MD simulations indicate a relatively minor decrease in thermal conductivity for 0.01% porosity, quite significant decreases in porosity are predicted at higher values of porosity, with an ~28% decrease in thermal conductivity for a porosity of 2.2% [90]. The MD results for the thermal conductivity are compared to many different empirical formulations in Figure 15.11a. The model due to Alvarez

Computational Modeling of Nuclear Fuels 759

Thermal conductivity (W/mk)

11 10 9 8 7 6 5 4 3 2 200

400

600

800 1000 1200 Temperature (K)

1400

1600

FIGURE 15.10: Temperature dependence of the thermal conductivity obtained from nonequilibrium MD simulations by Phillpot and co-workers (solid lines and circles) versus an empirical model commonly used in fuel performance codes (dashed line), as reproduced from Reference [86].

et al. [92] is apparently the best of the empirical models, and predicts only a slightly greater thermal conductivity than the MD data, while all the other models considered significantly underestimate the reduction of the thermal conductivity. It is important to note that a likely source of disagreement between MD simulations and other approaches relates to the imposition of periodic boundary conditions. As such, the pores are not randomly distributed, but they effectively arrange themselves in a periodic array of pores in the direction perpendicular to the heat flow. Fortunately, the models due to Loeb and Alvarez can be adjusted to include not only a random distribution of pores, but also a specific geometry. Lee and co-workers made this modification to the empirical model of Alvarez, as well as that of Loeb. It is now clear that both models predict a greater reduction of thermal conductivity with a periodic distribution of pores than models using a random pore distribution. In particular, the results of the Alvarez model fall within the error bars from the MD simulations for all the considered porosities. Clearly, accounting for the

760 Light Water Reactor Materials 1.00 (a)

0.95 0.90 0.85

k/ks

0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.000

MD Loeb Maxwell-Eucken Ondracek Alvarez Alvazov Van craeynest 0.005

0.010 0.015 0.020 Porosity volume fraction

0.025

FIGURE 15.11: Effect of porosity on the effective thermal conductivity from analytical models described in the text and MD simulations. The graph shows models that describe random pore positions. As reproduced from Reference [90].

dimension of the pore relative to the phonon mean free path is an important feature for the Alvarez model that makes it stand out from all of the other approaches. These simulations highlight the application of MD to calculate engineering-relevant material properties, namely, the temperature and porosity-dependent thermal conductivity of UO2. In particular, the results from Lee and co-workers indicate that the smallest of intragranular pores may have a more significant impact on thermal transport in UO2 than previously realized. Also, theoretical models, provided they incorporate appropriate geometrical details and phonon mean free paths, can provide a rigorous extrapolation from atomistic calculations to the experimental domain. The use of atomistic results such as those in larger length-scale modeling of temperature distributions in a nuclear fuel rod has recently been demonstrated by Tonks and co-authors [93]. Atomistic MD simulation of the degradation of UO2 thermal conductivity resulting from isolated Xe atoms trapped in a Schottky defect as well as intragranular or grain boundary fission gas bubbles have been shown by Tonks et al.

Computational Modeling of Nuclear Fuels 761

Tonks and co-workers used the BISON fuel performance code to simulate a ten-pellet fuel rodlet for typical pressurized water reactor conditions for a duration of 2.5 years. The fuel rodlet was represented as a smeared stack of fuel pellets and was simulated with a 2-D, axisymmetric r-z geometry. Xe fission gas was introduced at an appropriate rate, and allowed to diffuse and partition to either intragranular or grain boundary fission gas bubbles following the model developed by Pastore and co-workers [45], and partially described in Equations (15.9) to (15.16), as well as discussed later in Section 15.4.1. This resulted in Xe either being dispersed within the UO2 matrix trapped at Schottky defects, or in intragranular or grain boundary bubbles. A model for the thermal conductivity degradation based on the population of Xe was implemented into BISON, and the results are shown in Figure 15.12. Figure 15.12a shows the reduction in thermal conductivity, relative to the unirradiated value k0, at the periphery of the fuel pellet, where the majority of the fission gas remains dispersed in the fuel due to the lower fuel temperature that limits Xe diffusion. After 2.5 years of operation, the dispersed fission gas accounts for approximately 70% of the overall decrease in the thermal conductivity. Figure 15.12b shows the thermal conductivity decrease with time in the fuel center, by only about 25%, which is less than the much larger decrease at the fuel periphery. In both locations, the largest contribution to the decrease in the fuel thermal conductivity is the presence of dispersed gas atoms. However, at the fuel center, the higher temperature leads to more Xe partitioning to the intragranular and grain boundary fission gas bubbles, and thus less dispersed gas concentration and a correspondingly lower decrease in thermal conductivity. While much work remains to effectively include atomistic-scale simulation results in fuel performance models, in particular with incorporating the effect of fission products, second-phase precipitates, and porosity evolution during reactor operation on the thermal conductivity degradation, the results presented in Figure 15.12 represent one example of incorporating atomistic input into a fuel performance-level simulation.

762 Light Water Reactor Materials (a)

1

keff /k0

0.9 Total 0.8

Dispersed Intragranular GB

0.7 0.6 0.5 0

0.5

1

2

1.5

2.5

Time (yrs)

(b)

1

keff /k0

0.9 0.8 0.7

Total

0.6

Dispersed Intragranular GB

0.5 0

0.5

1 1.5 Time (yrs)

2

2.5

FIGURE 15.12: Effect of fission gas on thermal conductivity, as predicted by Tonks and co-authors [93], in which a ten-pellet fuel rodlet was simulated; (a) shows the effective thermal conductivity as a function of time at the fuel outer edge, whereas (b) shows the behavior at the fuel center.

This example indicates that Xe fission gas has a substantial effect on the decrease in thermal conductivity of UO2.

15.4 Mesoscale Simulations As noted previously, the use of mesoscale modeling techniques is required to capture time scale and microstructural processes that occur beyond the

Computational Modeling of Nuclear Fuels 763

1 µs that can easily be reached by MD simulations. Such mesoscale simulations are also required to compare calculations to experiments for the validation of mechanistic models used for engineering fuel performance theory. This section includes an example of applying a rate-theory-based model of fission gas partitioning developed by Pastore et al. [45], to model the fission gas swelling and release in UO2.

15.4.1 Rate-theory modeling of fission gas Fission gas bubbles have a very strong driving force for forming due to the extremely low solubility of Xe and Kr in UO2, which are generated in approximately one of every four fission. Accurately modeling of spatial dependence of bubble formation and the processes controlling fission gas release is complicated by the irradiation environment, which includes neutrons and radiation damage from the energetic fission fragments. These radiation effects drive an evolving UO2 microstructure and oxygen-to-metal stoichiometry. Furthermore, the basic processes of gas diffusion, bubble nucleation, and bubble growth are mediated by interactions of the gas bubbles with high-energy fission fragments, which can re-solve otherwise insoluble noble gas atoms from the bubbles back into the ceramic matrix. Re-solution occurs in competition with processes of radiation-enhanced diffusion, bubble nucleation, growth, and coarsening, in addition to grain boundary migration and grain growth. These processes ultimately result in spatially dependent gas bubble populations. Figure 15.13 shows scanning electron micrographs of a UO2 fuel pellet that experienced a base irradiation to approximately 9 GWd/tHM followed by a ramped power increase to a peak temperature of approximately 1900°C [4]; this power history produces spatially varying gas bubble populations, in which very different bubble size distributions are clearly observed on grain boundaries relative to grain interiors. More specifically, Figure 15.13a shows a relatively low magnification scanning electron micrograph of intragranular fracture surfaces showing small

764 Light Water Reactor Materials

(a) 10 µm

(b) 1 µm

FIGURE 15.13: Scanning electron microscopy (SEM) images of fission gas bubbles in UO2 following a base irradiation of ~9 GWd/tHM and a ramp test to ~1900°C. (a) Low magnification image of fractured grain boundaries indicating large (intergranular) bubbles on the grain boundaries, and (b) higher magnification image of the (intragranular) bubble population within a single grain that exhibits a strong bi-modal size distribution, as reproduced from Reference [4].

intragranular bubbles with diameters of hundreds of nanometers, along with larger lenticular, multilobed intergranular bubbles that are located on the grain boundary surfaces. It is known that a bi-modal size distribution of intragranular bubbles can develop during power transients, with the coexistence of a coarse bubble population with diameters of hundreds of nanometers along with numerous smaller bubbles [5, 81, 94]. Figure 15.13b shows a magnification of the grain interior region, which reveals a bi-modal size distribution with many small bubbles in addition to some larger, coarse bubbles. Coarse intragranular bubbles can be responsible for the majority of the fuel swelling [4], as well as strongly affect the rate of fission gas release [94]. The micrograph shown in Figure 15.13 clearly demonstrates a very complex fission gas bubble distribution, which is beyond the ability of a simplified model, like that described previously in Equations (15.9) through (15.16), to capture. Pastore and co-workers have developed an extension

Computational Modeling of Nuclear Fuels 765

of the model to incorporate the partitioning of fission gas to grain boundary bubbles [40]. The model itself is beyond the scope of this textbook, and the interested reader is referred to Reference [40], although we will conclude this section with a brief review of the results of Pastore’s model for fission gas partitioning to grain boundaries and fission gas release. Figure 15.14 shows the results of a comparison of the volumetric expansion, or swelling, at grain boundaries in UO2 as calculated by Pastore [45] with experimental data from White [4]. Overall, the agreement is quite acceptable, although the modeling tends to underpredict the amount of grain boundary swelling obtained from the experiments. It should again be noted that in this set of experiments, UO2 fuel pellets underwent a base irradiation to a burnup of ~9 GWd/tHM followed by a power ramp test. In his calculation, Pastore assumed that all fission gas was initially

Volumetric swelling, Calculated (%)

10

1

4000 4004 4005 4064 4065 4159 4160 4162 4163

0.1

0.01 0.01

0.1

1

10

Volumetric swelling, Measured (%)

FIGURE 15.14: Comparison between values of grain-face swelling calculated using a model for partitioning of fission gas atoms to grain boundary bubbles and available experimental data, noted by the different experiment reference numbers in the figure key, as reproduced from Reference [45].

766 Light Water Reactor Materials

within the grain interiors either in the form of trapped Xe gas atoms or in intragranular bubbles following the base irradiation, based on the model presented in Equations (15.9) to (15.16). Subsequently, during the ramp test to a temperature of ~1900°C, the gas is able to diffuse to the grain boundaries along with vacancies, resulting in the formation and growth of large (intergranular) bubbles. This leads to significant grain-face swelling and fission gas release. Pastore subsequently implanted his model of fission gas diffusion, bubble formation in grain interiors and at grain boundaries, and fission gas release into the fuel performance code TRANSURANUS, and performed a systematic comparison of integral fission gas release predicted with his model to experimental values and a standard fission gas-release model due to Lassman [52]. Figure 15.15 shows a comparison of the integral fission gas release (FGR) versus experimental results obtained for 28 rods that were tested in the Super-ramp or Inter-ramp projects [84, 86]. Figure 15.15 includes results from the standard model of fission gas release in TRANSURANUS due to Lassman [52], shown as open symbols, and the new model of Pastore [45], shown as filled symbols. In evaluating the standard Lassman model, a systematic underestimation of the released fission gas is observed in comparison to the experimental results, with the predicted values deviating by a factor of about 2 from the experimental values. The new model of Pastore also underestimates the amount of fission gas release, but the overall under-prediction is reduced to a value of about 1.6, thereby leading to some improvement of the TRANSURANUS predictions. However, Figure 15.15 also shows that there is a substantial overestimation for the model of Pastore for FGR values below 10%. Thus, we can conclude that the Pastore model, as fully described in Reference [45], represents a step forward over more simplified Boothstyle models of fission gas release, but there remains many open avenues for future research, including improved treatment of modeling the intragranular bubbles beyond the simple model presented in Equations (15.9)

Computational Modeling of Nuclear Fuels 767 100 Standard model, Super-Ramp cases Standard model, Inter-Ramp cases Integral FGR, Calculated (%)

New model, Super-Ramp p cases New model, Inter-Ramp cases 10

1

0.1 0.1

1

10

100

Integral FGR, Measured (%)

FIGURE 15.15: Comparison between calculated values of integral fission gas release at end of life, and the experimental data from Super-Ramp and Inter-Ramp experiments [95, 96], as reproduced from Reference [45].

to (15.16), improving the model of Xe re-solution from bubbles, explicitly modeling the bubble nucleation phenomena, and incorporating high burnup mechanisms of fission gas swelling and release.

15.5 Fuel Performance Simulations As noted earlier in Section 15.2.7, nuclear fuel is a complex, multicomponent system with numerous thermal and mechanical interactions that control fuel performance and, in particular, the ability of the fuel cladding to satisfy the performance requirement because of fission gas release. The remainder of this section will briefly review two key aspects of fission gas retention related to the amount of fission gas released from the UO2 fuel into the gas plenum and, in particular, the key uncertainties in modeling fission gas release determined through integrated, finite element–based

768 Light Water Reactor Materials

fuel performance modeling, and maintaining cladding integrity against cracking induced by pellet cladding interactions. As noted throughout this chapter, modeling fission gas behavior is critically important to successfully quantifying fuel performance, since gas retention within the fuel in the form of bubbles either within the grain interior or at grain boundaries is a significant contributor to fuel swelling that is responsible for pellet cladding interaction, while the concomitant fission gas release to the gas plenum and fuel rod free volume (e.g. the fuel cladding gap) increase the rod internal pressure, which also influences the stress state of the fuel cladding. Moreover, as demonstrated in Section 15.3.3, fission gas precipitation in bubbles versus being isolated at vacancy trapping sites within the UO2 matrix influences the fuel thermal conductivity, while fission gas release impacts the thermal conductivity of the fuel pellet– cladding gap, both of which determine the temperature distribution within the fuel pellet.

15.5.1 Modeling uncertainty of fission gas release As noted in Section 15.4.1, the computational analysis of fission gas release and swelling involves the numerical treatment of multiple intricate and mutually dependent phenomena, and inevitably depends on uncertain parameters. Indeed, many internal parameters of fission gas behavior models are difficult to measure and prone to large uncertainties. In addition, the dependence of the relevant processes on the general solution variables (e.g., temperature) implies that uncertainties pertaining to the global fuel analysis turn into uncertainties in fission gas behavior calculations [97]. Comparative benchmarks of numerous fuel performance codes [98–100] have shown that difficulties remain in predicting fission gas release and fuel swelling, since a wide spectrum of predictions are observed for identical conditions [97], which can be ascribed to the complexity and nonlinearity involved with the physical processes controlling fission gas behavior in nuclear fuel.

Computational Modeling of Nuclear Fuels 769

Indeed, Pastore et al. [97] recently performed an engineering-scale fuel performance simulation study and extensive sensitivity analysis to assess the impact of the uncertainty associated with fission gas behavior. The model for fission gas release implemented in the BISON fuel performance code is the same as that discussed in Section 15.4.1, which provides for the formation of grain interior and grain boundary fission gas bubbles and for fission gas release upon reaching a saturation coverage of grain boundary bubbles. Pastore and co-workers [97] explicitly included the effect of variations in the temperature, grain size, gas diffusion coefficient within UO2 grains and at the grain boundary, and the re-solution parameter. Figure 15.16 shows the relative importance of these five variables on the fission gas release calculated for a representative 3-year irradiation history for a fuel rod with a linear heat rate of 30 or 40 kW/m, respectively. Pastore et al. [97] notes that all uncertain parameters appear to have a significant effect on the predicted fission gas release. The highest influence is associated with the intragranular diffusion coefficient and the re-solution parameter. Also, the impact of the uncertainties in the calculated fuel temperature and in the grain boundary diffusion coefficient is remarkable. Note that the diffusion coefficients and the grain radius are temperature-dependent; hence, the effects of these parameters are partly due to the intrinsic uncertainties, and partly associated with temperature uncertainty. As shown in Figure 15.16, the effect of grain size (radius) can either increase or decrease the predicted fission gas release, with larger grain sizes generally reducing the fission gas release. Figure 15.17 shows BISON predictions for an irradiation experiment performed in the Riso-3 ramp test irradiation campaign [101, 102], in which the upper bound prediction for fission gas release matches the lone experimental data point following the linear heat rate ramp shown in Figure 15.17. However, a large scatter exists within the parameter variation performed, as evidenced by the comparison between the lower and upper bound calculated values of fission gas release shown in the figure.

770 Light Water Reactor Materials Temperature

Grain radius

Intra-granular diff. coeff.

0.07

Mean fission gas release (/)

0.06 0.05 0.04 0.95

1.00

1.05

Resolution parameter

0.4

1.0

1.6

0.1

1.0

10.0

Grain-boundary diff. coeff.

0.07

LHR = 30 kW/m 3 years

0.06 0.05 0.04 0.1

1.0

10.0

0.1

1.0

10.0

X-axis values are scaling factors multiplying the reference parameter values Temperature

Grain radius

Intra-granular diff. coeff.

0.240 0.225

Mean fission gas release (/)

0.210 0.195 0.180 0.95

1.00

1.05

Resolution parameter

0.4

1.0

1.6

0.1

1.0

10.0

Grain-boundary diff. coeff.

0.240 LHR = 40 kW/m 3 years

0.225 0.210 0.195 0.180 0.1

1.0

10.0

0.1

1.0

10.0

X-axis values are scaling factors multiplying the reference parameter values

FIGURE 15.16: Parameter variation influence on calculated fission gas release following a 3-year irradiation time with a linear heat rate of 30 kW/m (top) and 40 kW/m (bottom), as reproduced from Reference [97].

Computational Modeling of Nuclear Fuels 771 40 Calculated FGR, lower bound Calculated FGR, upper bound Experimental FGR Linear heat rate

0.15

30

0.1

20

0.05

10

Linear heat rate (kW/m)

Fission gas release (/)

0.2

0

0 0

5

10

15

Time (hours)

FIGURE 15.17: Fission gas release and rod average linear heat rate as functions of time for the Riso-3 GE7 experiment [91, 92]. Both lower and upper bounds of the calculated fission gas release are shown along with the postirradiation experimental value, and a time of zero corresponds to the beginning of the transient test, as reproduced from Reference [97].

This difference between the lowest and highest prediction of fission gas release is about 3.5 times at the end of the irradiation test. Thus, the Pastore study concludes that there is at least a factor of 2 uncertainty in the current engineering predictions of fission gas release. This level of uncertainty, combined with the uncertainty in fission gas-induced fuel swelling, had a significant impact on predicted fuel cladding deformation. Pastore et al. concluded that “a better characterization of the parameters through experimental and theoretical research may reduce the uncertainty in fission gas behavior calculations and in the multiple related aspects of fuel performance analysis” [97]. There is a critical need to both identify and characterize the diffusion behavior of Xe on the atomic scale, and then use this diffusivity in higher-level models that can integrate numerous competing mechanisms, which in turn depend on irradiation conditions (temperature, grain size, stoichiometry, and fission density), to predict the

772 Light Water Reactor Materials

gas bubble formation and evolution both within the UO2 grains and at the grain boundaries.

15.5.2 Fuel performance simulations of pellet-cladding interaction Pellet-cladding interaction (PCI) in LWR fuel is a coupled thermalchemical-mechanical process that can lead to cladding breach and release of radioactive fission products into the coolant [103–106]. Reactor operating restrictions, which limit power maneuvering, have been established to mitigate PCI, but they restrain operational flexibility and lead to reduced power generation. PCI failures generally occur following an increase in the local power over a short period of time, and in fuel that has been previously exposed to irradiation. Classical PCI is driven by the localized strains in the vicinity of a pellet crack, as well as the presence of a chemical species, such as iodine, that drive stress corrosion-induced cracking of the cladding [107]. Cracks that form in brittle ceramic pellets by large temperature gradients are believed to be important in the PCI failure mechanism. During a local power increase, pellet expansion produces a high contact force between the fuel pellet and cladding material, when a reduced or eliminated residual pellet-clading gap is present because of previous irradiation. Furthermore, during the rapid thermal expansion of the pellet, the fuel cracks can further open, which transfers tangential shear forces onto the cladding. These tangential shear forces are a function of the equilibrium pellet-cladding gap or residual contact pressure at the start of the power increase, the power level at gap closure, the interfacial friction, and the maximum local power. Nonclassical PCI failure is associated with the presence of a missing pellet surface (MPS) defect [107]. These MPS defects form through mishandling during the manufacturing process, where the pellet is chipped leaving a flaw on the outer surface. The presence of an MPS defect during a localized power

Computational Modeling of Nuclear Fuels 773

ramp can cause severe bending moments in the cladding in the vicinity of the MPS when the fuel undergoes rapid thermal expansion due to this increase in local power. Furthermore, the localized region near the MPS also experiences a different temperature distribution compared to when the MPS is not present. The result is a localized hot spot in the fuel and cold spot in the cladding. Both classical and nonclassical PCI are significantly influenced by the geometry of fuel pellet flaws (i.e., pellet cracks and MPS) and the state of the fuel pellet, typically referred to as conditioning of the fuel. A “conditioned” fuel rod is one that can withstand rapid power changes without cladding failure due to PCI. Conditioning of the fuel is a consequence of the fundamental changes in fuel pellets and cladding as a function of irradiation, which is both time and exposure dependent. These changes include pellet cracking and relocation, fuel densification and/or swelling, which is driven by both solid and gaseous fission products, as well as grain growth in the fuel and creep deformation of both the fuel and cladding. During the first rise to power, ceramic fuel pellets are subject to thermal gradients due to the poor thermal conductivity of UO2, which results in differential thermal expansion that induces tensile stresses in the outer regions of the pellet. When the local stress state exceeds the fracture stress, or more specifically, the fracture toughness is exceeded, the fuel pellet cracks. Due to this thermally induced stress state, the majority of cracks are radial with crack propagation from the pellet periphery inward. Some cracking is circumferentially or transversely oriented, although most cracks are radial, as shown in Figure 15.18 [108]. This cracking allows fuel fragments to move outward, consuming part of the original fuel-cladding gap, and increasing the compliance of the pellet when it makes contact with the cladding. Further, at the early stage of irradiation, some initial porosity disappears as the pellets initially densify and shrink; however, eventually, the accumulation of fission products causes the ceramic fuel to swell.

774 Light Water Reactor Materials

FIGURE 15.18: Micrographs of UO2 fuel pellets irradiated under normal operating conditions, with centerline temperature of ~1000°C and periphery temperature of ~500°C, denoting a network of cracks, as reproduced from Reference [108].

Simultaneously with changes in the fuel, the cladding creeps by irradiationinduced mechanisms inward under the influence of the external coolant pressure differential with the fuel pin gas loading. Eventually, with the fuel swelling, the cladding creeps down, the inside surface of the cladding makes contact with the outer surface of the fuel pellets, and both achieve mechanical equilibrium. When fuel and cladding are in contact, a power increase can cause the cladding to experience a significant tensile stress. At the same time, increased fuel temperatures may allow for increased mobility of volatile fission products, such as iodine, to travel along cracks from the pellet interior to the cladding surface. The local stress in the cladding is intensified in the presence of fuel pellet cracks, when the cladding is in hard contact with the fuel, and a fuel radial crack opens, or when there is a sufficiently large fuel or cladding defect. In the presence of elevated tensile stress (above some critical threshold) and sufficient iodine activity, zirconium alloys experience stress corrosion cracking, PCI, which is controlled by pellet-cladding mechanical interaction (PCMI). With sufficiently high stresses, which can result from MPS or radial cracks, cracking in zirconium alloys can propagate mechanically purely because of PCMI.

Computational Modeling of Nuclear Fuels 775

In order to reliably model PCI fuel failures, it is important to identify accurate metrics that signify the probability of a through-wall crack developing in the cladding. Historically, the more common approach has been to develop a correlation to a threshold hoop stress that should not be exceeded to ensure cladding integrity. Such a threshold stress can be based on estimates of the peak stresses experienced by failed rods. These methods, although effective in practice, impose operational constraints that can be overly conservative. This is because they cannot fully segregate failures from nonfailures above the best-estimate stress threshold. In order to reduce this conservatism, alternative analytical methods based on more mechanistic approaches have been explored (see, e.g., [49, 54, 55, 107–110]). These methods are typically based on detailed local-effects analyses, which involve modeling a cross-sectional “r-theta” slice of the fuel in 2-D, or full 3-D of the PCMI-induced stresses, combined with cladding failure model correlated to failed fuel rod tests or operational failures. The correlation to cladding failure can be based on stress corrosion cracking–based cumulative damage index (CDI) or strain energy density (SED) failure models [109]. Cumulative damage index (CDI) model An example of the implementation of the CDI approach to fuel cladding failure is in EPRI’s FALCON fuel performance code [53, 109]. The CDI is a cumulative damage model based on time-to-failure experimental data. Cladding failure by the stress corrosion cracking (SCC) mechanism is a two-stage process: (i) an incubation stage, which is the time taken to nucleate a small surface crack at a local stress concentration point or a surface defect, and (ii) a crack propagation stage. The duration of the first stage can vary from a few seconds to a few hours, depending upon the temperature and the stress level above the SCC threshold. During the second stage, crack extension proceeds in a stable manner until it reaches a critical size, (KI ≥ KISCC), suddenly extending to a through-wall fracture. In a fuel rod, fission products, assumed to be either iodine or cesium,

776 Light Water Reactor Materials

assist the failure process through chemical attack at the crack tip. The CDI model in FALCON treats both stages of the SCC mechanism in which damage accumulation is linear with time, such that a damage fraction D can be defined as follows: dt , D= ∫ (15.22) t where t is the time-to-failure in a material experiment conducted under SCC conditions, defined as follows:

t = f (σ , σ y , σ ref , bu , T ) .

(15.23)

The independent variables in Equation (15.23) are defined as follows: σ = applied hoop stress (MPa) σy = yield stress (MPa) σref = burnup-dependent function for Zry-2 and Zry-4 materials (MPa) bu = burnup (MWd/MTU) T = temperature (K). In Equation (15.23), t is the time-to-failure in an SCC out-of-pile test, for a material with burnup level bu, in which the applied stress σ and the temperature T are held constant. Equation (15.22) applies to a single continuous power event during a reactor cycle, such as a power ramp followed by a hold at constant power, but is not carried from cycle to cycle of reactor operation. D is calculated incrementally as n

∆t D=∑ t i , i

(15.24)

i =1

where, ti is the value of t for the stress and material conditions in the time step. Experience with the SCC damage parameter defined above shows logarithmic behavior with stress and lends itself to a probabilistic interpretation as follows: D = 1 implies 50% probability of failure, with < 5% and > 95% failure probabilities assigned to D = 0.1 and D = 10, respectively.

Computational Modeling of Nuclear Fuels 777

Application of the above formulation to fuel rods requires special interpretation of the laboratory tests with respect to power maneuvers, where the cladding stress can drop below the SCC threshold due to power reduction or stress relaxation during power holds. There is also the question of differences between the fission product environment of the fuel rod and the out-of-pile SCC-test environment. To account for these, a deterministic form of Equation (15.24) is applied as follows: n

D=∑ i =1

∆ti

(βti )

,

(15.25)

where β is a single-valued parameter determined by benchmarking the model against SCC power ramp test data. Essentially, this term accounts for the translation of the out-of-pile time-to-failure measurements of cladding tubes to the in-pile behavior of a fuel rod at the time of failure. This approach then provides a definable, probabilistic relationship between the value of D, the computed damage fraction (CDI), and fuel rod failure probability calibrated to actual power ramp test data. Additional information and a demonstration of the successful application of the CDI to PCI-related fuel failures is available in Reference [109]. 2-D and 3-D examples of PCI failure modeling Capps and co-workers [110] have used the BISON fuel performance code to assess the impact of fuel cracking and missing pellet surfaces on the stress development in the fuel cladding. An r-theta geometric model was developed, based on an AP-1000 fuel rod design, as shown in Figure 15.19, and for which a 70% reduction in the as-fabricated gap size was used to simulate the pellet swelling and cladding creep-down at a local burnup of  approximately 20 GWd/tU. Furthermore, an initial fast fluence of 5  ×  1025 n/m2 was applied to the cladding to account for the material property changes at this exposure. To evaluate the impact of discrete radial pellet cracks on the local cladding hoop stress, we have used

778 Light Water Reactor Materials

a quarter-symmetry, or 90-degree, model containing two, three, or four radial cracks with a crack length of 50% of the pellet radius, as shown in Figure 15.19a. A similar method was used to assess the role of the crack length on the maximum cladding hoop stress. The sides of the cracks in Figure 15.19 are considered to be free surfaces. Their simulations involved performing a power increase representative of a ramp test of pressurized water reactor fuel. This involved ramping up to a linear power of 25 kW/m over 2.8 hours, followed by a 24-hour hold at power and then a final ramp to 40 kW/m over about 2 hours. Figure 15.20 shows the stresses calculated within the fuel cladding for the geometry of Figure 15.19a during the ramp test, as a function of the length of the radial cracks. The hoop stress contours shown in Figure 15.20 are at the time of the maximum cladding stress (~30 hours into the power history) for a crack length of 30%, 50%, and 70% of the fuel radius, respectively. These contours clearly indicate that the maximum hoop stress occurs at the position where the pellet crack impinges on the cladding. The magnitude of the stress intensity associated with the crack (e.g., maximum/minimum cladding hoop stress) (a)

(b)

Cladding tube

Cladding tube

Fuel pellet

Fuel pellet

Discrete pellet crack

Discrete pellet crack

Missing pellet surface

FIGURE 15.19: Finite element meshes for fuel performance simulations of a 2-D, r-θ model showing (a) a representative fuel rod where the fuel pellet contains radially oriented cracks (shown as bold lines), and (b) a fuel pellet containing a missing pellet surface defect, in addition to the radially oriented cracks (b), as reproduced from Reference [110].

Computational Modeling of Nuclear Fuels 779 (a) Hoop stress (MPa) 167.31 150 125 100 75 57.42

(b) Hoop stress (MPa) 240.15 240 200 160 120 90.78

(c) Hoop stress (MPa) 258.85 240 200 160 120 92.69

FIGURE 15.20: Hoop stress contour plots in the cladding at the time of maximum hoop stress for the 2-D, r-θ model shown in Figure 15.19a, subject to a representative pressurized water reactor (PWR) power ramp, in which the fuel crack length is (a) 30%, (b) 50%, and (c) 70% of the fuel pellet radius, as reproduced from Reference [110].

on the cladding inner surface displays a dependence on the crack length, varying from 2.2 in the case of the 30% crack to 2.7 in the case of the 70% pellet radius crack. Further, high tensile hoop stresses are observed at the pellet crack tip for both the 30% and 50% cases.

780 Light Water Reactor Materials

Capps subsequently modeled the effect of the missing pellet surface on the cladding hoop stress during the same power ramp [99]. Two different MPS defect geometries were included in this assessment: a flatfaced defect and a concave defect. Post-irradiation examinations of MPS defects have found both types present in LWR fuel. The width of the MPS defect was varied from a small size (1.1 mm) to a large size (3.8 mm). These values span the range from below an acceptable MPS defect to the size observed to cause cladding failure. It is important to note that because of the 2-D representation, the length of the MPS defect is infinitely long (in the out-of-plane direction), which produces an exaggerated effect on the temperature and stress condition. The MPS defect width or angle subtended by the missing circumferential surface affects the length of the unsupported cladding tube once contact occurs elsewhere along the interface. Both the width and the curvature also influence the distance between the pellet surface within the MPS defect and the cladding inner surface, as shown in Figure 15.21. This distance or gap has an impact on the heat transfer from the pellet to the cladding, and results in a hotter region in the pellet adjacent to the MPS defect and a cooler cladding region. Figure 15.21 summarizes the results of BISON calculations that compare the impact of flat and concave MPS defects, as functions of MPS width. From Figure 15.21a, it can be seen that as the MPS defect width increases, the maximum cladding hoop stress increases, and corresponds to the increase in the maximum fuel temperature, as shown in Figure 15.21b. Both the bending moment due to the unsupported cladding tube and the large thermal expansion of the pellet are responsible for the increase in maximum hoop stress with increasing MPS size. The fuel temperature is also increased by the concave MPS defect geometry. These results clearly indicate that a curved MPS with concave geometry will produce both higher fuel centerline temperatures and larger maximum hoop stresses, for a fixed MPS size than will a flat MPS.

Computational Modeling of Nuclear Fuels 781 500 Flat MPS Curved MPS

Maximum cladding hoop stress (MPa)

480 460 440 420 400 380 360 340 320 300 10

15

20

25

30

35

40

45

50

55

60

MPS width (Degrees)

(a)

1600 Flat MPS Curved MPS

Maximum fuel temperature (K)

1580

1560

1540

1520

1500

1480 1460 10 (b)

15

20

25

30

35

40

45

50

55

60

MPS width (Degrees)

FIGURE 15.21: (a) Cladding stress calculated for various-size missing pellet surface (MPS) defects, including whether the MPS geometry is flat or curved, and (b) the corresponding maximum temperature in the fuel pellet, as reproduced from Reference [110].

782 Light Water Reactor Materials

Capps also demonstrated 3-D fuel performance modeling of PCI using a five-pellet fuel rodlet, in which the central pellet contained an MPS defect extending partially down the length of the fuel pellet, in addition to radial fuel cracks that extended 50% of the fuel radius [99]. Figure 15.22 shows the temperature distribution in the fuel pellets and cladding, as well as the hoop stress contour on the inner cladding surface, respectively, for the five-pellet 3-D model containing a 105 mm (squared) MPS defect at a time corresponding to the maximum hoop stress. These results illustrate the complex effects caused by the MPS defect, such as cold spots in the cladding, localized regions of increased hoop stress in the cladding near the region of the MPS defect as well as the radial fuel crack and increased domain of high temperatures in the fuel. The pellet– pellet interface and the MPS defect location are easily discernable from the contour plots. These results highlight the use of thermal-mechanical finite element methods to simulate the development of high tensile stresses in the cladding as a result of pellet–cladding interaction, and to demonstrate an ability to model temperature and stress distributions in three dimensions. Although not discussed explicitly here, the simulations also indicate the importance of improving the fidelity of the models for fuel creep, densification, and swelling that occur during fuel operation, as well as the importance of fission gas behavior noted in Section 15.5.1.

15.6 Summary This chapter has attempted to demonstrate a framework for multiscale materials modeling of nuclear fuel, as well as provide a short introduction and summary of the specific modeling techniques. Over the past two to three decades, advances in computational modeling, coupled with increasingly powerful high-performance computing and improved experimental tools, have led to this framework, which provides the promise of developing predictive models of material behavior that capture all relevant length and time scales.

Computational Modeling of Nuclear Fuels 783 (a)

MPS defect location Temperature (K) 1529.465 1400 1200 1000 800 702.233

(b) MPS defect location Temperature (K) 677.7771 660 640 620 600 580.0013

(c)

MPS defect location

Radial crack location

Hoop stress (MPa) 418.3633 400 200 0 –200 –400 –407.185

FIGURE 15.22: (a) Pellet temperature distributions in the vicinity of the MPS defect, (b) inner cladding surface temperature, and (c) inner cladding hoop stress distribution in the vicinity of the MPS defect, as reproduced from Reference [110].

This chapter subsequently provided several examples of the application of computational modeling to nuclear fuel, including the behavior of Xe and point defects in UO2, displacement cascade damage, and the thermal conductivity of UO2 from electronic structure and atomistic modeling. A mesoscale rate-theory-based model of fission gas diffusion

784 Light Water Reactor Materials

and precipitation into intragranular and grain boundary bubbles was described, along with the results of that model for predicting fission gas release and fuel swelling. The chapter then described results from integrated fuel performance modeling by the BISON code to assess the uncertainty associated with fission gas behavior, and the stress levels that develop in fuel cladding due to pellet–cladding interaction. This chapter has also indicated several challenging issues requiring further modeling, which will certainly be the subject of advances in understanding and modeling nuclear fuel performance.

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Problems 15.1 Plot the pair potential portions of the 3 interatomic potentials for U, O, and U-O that are described in Table 15.1. Discuss the differences between these three. 15.2 Based on the ZBL potential described in Reference [70], plot the interatomic potential energy between U and O, and compare to the values of the potentials listed in Table 15.1. 15.3 Calculate the migration energy of a Xe-Schottky defect in UO2, based on the values provided in Tables 15.2 and 15.3. 15.4 Discuss three limitations to the fission gas swelling model described by Equations (15.10–15.17). 15.5 Calculate the amount of swelling produced from intragranular fission gas bubbles at a burnup of 6 at% (generating 1.5 atom% Xe) for intragranular bubble densities of 1.5 × 1023 or 1.5 × 1022 bubbles per m3. Which produces the most swelling?