1.9.2.1 Neutronic Properties

Neutronic properties are of significant consideration in the design and develop­ment of nuclear reactors. As discussed earlier, the fission chain reaction requires continued supply of neutrons for it to proceed and that is why neutron economy plays an important role. Fuel cladding materials need to have a lower neutron absorption cross section and that is why zirconium alloys are used in LWRs (see Appendix 1.A). On the other hand, to control the chain reaction, the control materi­als should have high neutron absorption cross section. The same consideration would also apply to shielding materials.

The close-packed lattices have a specific atom stacking sequence. For example, FCC crystal has ABCABC… stacking sequence, as shown in Figure 2.39a. Any disrup­tion in this stacking sequence causes local region to get out of perfect sequence, as shown in Figure 2.39b. Locally, this region is improper or faulty, and so this region is called a stacking fault. Stacking faults possess surface energy. The surface tension due to this tends to minimize the area of the faulted region. Stacking fault energies of some typical metals/alloys are as follows: Al: 0.2 J m-2, Cu: 0.04 J m-2, Ni: 0.03 J m-2, Cu-25Zn: 0.007 Jm-2, Fe-18Cr-8Ni: 0.007 Jm-2. This topic will be revisited in detail in Chapter 4.

Generally, two dislocations with the same sign on the same slip plane would repel each other, whereas dislocations with the opposite signs would attract each other.

Let us assume that two parallel edge dislocations of the same sign (both positive edge in the case) are on the same slip plane, as illustrated in Figure 4.16a. If the dislocations come very close together, the configuration can be assumed to be a dislocation with double the Burgers vector (i. e., 2b) of the individual dislocations, and the elastic strain energy would be given by aG(2b)2 or 4aGb2. However, when they are separated by a large distance, they have a total energy of 2aGb2. The dislo­cation configuration with a smaller Burgers vector is more stable. Thus, the disloca­tions would repel each other. However, when two edge dislocations of the opposite signs lie on the same slip plane (one positive and another negative) as illustrated in Figure 4.16b, the effective Burgers vector is zero, and thus the elastic strain energy becomes zero. As in this way the dislocation configuration reduces their energy, the two dislocations will be attracted to each other and will get annihilated. Similarly, while the like screw dislocations on the slip plane will repel each other, the unlike dislocations will attract each other.

Now let us consider the case of the interaction between two parallel screw dislo­cations (not necessarily on the same slip plane). As the screw dislocation has a radi­ally symmetrical stress field, the force between the two dislocations is given by the following equation and depends only on the distance of separation.

Gb2

Fi = tgzb = -—. (4.18)

2pr

This interaction force is repulsive between two like screw dislocations and attractive between two unlike dislocations. This can easily be shown by evaluating the force on a screw dislocation using Peach-Koehler formula (Eq. (4.17b)) due to the stress fields from the second dislocation using Eq. (4.10a) (or using forces in Cartesian coordinates).

The interaction between two edge dislocations is much more complex because of their more complicated stress fields. If we consider one of the edge dislocations to be lying parallel to the z-direction with Burgers vector parallel to the x-direction, the interaction would change from attractive to repulsive and vice versa depending on how the two dislocations are positioned with respect to each other. The various cases are shown in Figure 4.17 in terms of coordinates.

The arrows in Figure 4.17 correspond to forces in the x-direction, while the forces along the y-direction are repulsive between two like edge dislocations although they cannot glide in the y-direction. The edge dislocations can move along the y-direction only by nonconservative climb that involves vacancies and/or atoms move away/to the dislocation core. From Figure 4.17, certain simple cases can be derived. Figure 4.18a shows an edge dislocation parallel to another like edge dislocation just vertically above its slip plane. This is a stable configuration. This type of configuration is found in the low angle tilt boundaries (see Figure 2.37). Two stable forms of dipole pairs are shown in Figure 4.18b where two edge dislocations of opposite signs glide past each other in parallel slip planes at low applied stresses.

& Example 4.6

Derive the force between two like dislocations lying along the z-axis. Solution

?1 : b1 — b1fe, t1 — k and?2 : b2 — b2k, t2 — k.

Force on dislocation 2 due to dislocation 1 : F2 — (a(1) ■ b2)xt2.

0 0 aXz

Recall stresses around screw dislocations : a(1) =

4.2.6

It is important to have fracture toughness parameter as an inherent mechanical property of materials just like yield stress. The mode-I fracture toughness (KIc) meets that definition with certain limitations. As the analysis involved still depends on the linear elastic fracture mechanics theories, the testing procedures followed are applicable to materials with limited ductility, such as high-strength steels, some titanium and aluminum alloys, and, of course, other brittle materials like ceramics.

The elastic stress field around the crack tip can be described by a single parame­ter known as “stress intensity factor (K).’’ This factor depends on many factors such as the geometry of the crack-containing solids, the size and location of the crack, and the magnitude and distribution of the loads applied. It can be reasonably assumed that an unstable rapid failure would occur if a critical value of K is reached. There are three modes of testing, as shown in Figure 5.22. In the opening mode (mode-I), the displacement is perpendicular to the crack faces. In mode-II (sliding mode), the displacement is made parallel to the crack faces, but perpendic­ular to the leading edge. In mode-III (tearing mode), the displacement is parallel to the crack faces and the leading edge.

In reality, opening mode (i. e., mode-I) is most important. That is why the conven­tional tests for fracture toughness are done under mode-I (i. e., opening mode of loading), and the critical value of K is called KIc, the plain strain fracture toughness. For a given type of loading and geometry, the relation is

KIc — Y opPOc, (5.32)

where Y is a parameter that depends on the specimen and crack geometry, ac is the critical crack length, and o is the applied stress. If KIc and applied stress are known, one can compute the maximum crack length tolerable. In other words, maximum

allowable stress can be computed for a given crack size provided one knows the KIc value of the material. KIc generally decreases with decreasing temperature and increasing strain rate, and vice versa. It also strongly depends on metallurgical vari­ables such as heat treatment, crystallographic texture, impurities, inclusions, grain size, and so on.

A notch in a thick plate is far more damaging than that in a thin plate because it leads to a plane strain state of stress with a high degree of triaxiality. The fracture toughness measured under plane strain conditions is obtained under maximum constraint or material brittleness. The plane strain fracture toughness is thus desig­nated as KIc and is a true material property. A mixed mode (ductile-brittle) fracture occurs with thin specimens. Once the specimen has the critical thickness, the frac­ture surface becomes flat and the fracture toughness reaches a constant minimum value with increasing specimen thickness (Figure 5.23). The minimum thickness (B for breadth) to achieve plane strain condition is given by

B > 2.5(KIC/s,)2, (5.33)

where s0 is 0.2% yield stress.

“Someday man will harness the rise and fall of the tides, imprison the power of the sun, and release atomic power.’’

—Thomas Alva Edison

1.1

Introduction

There is no doubt that energy has been driving and will drive the technological prog­ress of the human civilization. It is a very vital component for the economic devel­opment and growth, and thus our modern way of life. Energy has also been tied to the national security concerns. It has been projected that the world energy demand will almost double by the year 2040 (based on 2010 energy usage), which must be met by utilizing the energy sources other than the fossil fuels such as coal and oil. Fossil fuel power generation contributes to significant greenhouse gas emissions into the atmosphere and influences the climate change trend. Although several research and development programs (e. g., carbon sequestration and ultrasupercrit­ical steam turbine programs) have been initiated to make the fossil power genera­tion much cleaner, they alone will not be enough to fend off the bigger problem. Therefore, many countries worldwide have recognized the importance of clean (i. e., emission-free) nuclear energy, and there are proven technologies that are more than ready for deployment. The use of nuclear energy for the power genera­tion varies widely in different parts of the world. The United States produces about 19% (2005 estimate) of its total energy from nuclear sources, whereas France pro­duces ~79% and Brazil and India rely on the nuclear energy for only about 2.5% and 2.8% of their energy needs, respectively. Japan, South Korea, Switzerland, and Ukraine produce 30%, 35%, 48%, and 40%, respectively, of their energy require­ments from the nuclear sources. It is important to note that the fast growing econo­mies like China, India, and Brazil produce relatively less electricity from the nuclear sources. Hence, there are tremendous opportunities for nuclear energy growth in these emerging economies as well as many other countries. Nuclear reactors have been built for the primary purpose of electricity production, although they are used for desalination and radioisotope production.

An Introduction to Nuclear Materials: Fundamentals and Applications, First Edition.

K. Linga Murty and Indrajit Charit.

© 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA.

There are now about 440 nuclear power reactors worldwide generating almost 16% of the world electricity needs; among them, 104 nuclear reactors are in the United States. Since the first radioactive chain reaction that was successfully initiated at the University of Chicago research reactor in the 1940s, the field has seen an impressive growth until Three Mile Island and Chernobyl accidents happened. Following these incidents, the public confi­dence in the nuclear power dwindled, and the nuclear power industry saw a long stagnation. However, the US government’s decision to increase energy security and diversity by encouraging nuclear energy generation (as laid out in the US government’s Advanced Energy Initiative in 2005) has rekindled much hope for the revival of the nuclear power industry in the United States, and as a matter of fact, the Nuclear Renaissance has already begun — the US Nuclear Regulatory Commission (NRC) approved an early site permit application for the Clinton Power Station in Illinois (Exelon Power Corporation) in March 2007. As the scope of the nuclear energy is expanded, the role of materials is at the front and center. Recent (2011) accidents in Japan due to earthquakes and tsunami are now pointing toward further safeguards and development of more resistant materials. Thus, this book is devoted to addressing various important fundamental and application aspects of materials that are used in nuclear reactors.

1.2

So far, we have noted FCC and HCP metals to be close-packed structures with the highest atomic packing factor (0.74), but we have not discussed how a close-packed structure results. Both FCC and HCP crystals can be constructed by stacking close — packed layers of atoms on top of one another. There are two ways the stacking can be achieved leading to the creation of two different stacking sequences and thus two different crystal structures. Figure 2.8a shows a close-packed plane A where six equal-sized spheres surround a central atom. To create a close-packed three-dimen­sional structure, more spheres need to be placed on top of the A layer to fill up the triangular cavities created by the first layer. There are two sets of cavities on the A layer, B and C positions, as shown in the Figure 2.8a. A second plane of atoms can be placed on top of either B or C position. If one assumes that the second atom layer is over B, there are two ways the third close-packed layer can be stacked either vertically above C positions or directly above A positions. This scheme leads to the possibility of two types of close-packed packing. An FCC crystal shows a stacking sequence of ABCABCABC … (Figure 2.8d). On the other hand, a HCP crystal shows a stacking sequence of ABABAB… as shown in Figure 2.9. The lower and upper basal plane layers constitute the A layer and middle layer forms the B layer.

ABAB… stacking sequence

3D projection

Top layer

Middle layer

Bottom layer

Figure 2.9 Stacking of close-packed planes resulting in a HCP structure.

Due to this kind of stacking, a set of planes known as close-packed planes in FCC and HCP crystals are the densest (i. e., the planar atom density is the greatest). We will discuss this further in a later section.

2.1.4

A microstructure is not as homogeneous as we think from a larger length scale. If it is possible to delve into the microscale, we can encounter various features like grain boundaries and dislocations. Diffusion through these features would be dif­ferent from the diffusion that takes place through the lattice interior or the bulk of the crystal.

2.3.6.1 Grain Boundary Diffusion

We have discussed the characteristics of grain boundaries in Section 2.2. Even though we have not fully covered grain boundary models, we can easily develop a picture of grain boundary where atoms are more loosely packed compared to the crystal interior. Grain boundary in itself is typically only a few atomic diameters in thickness. Naturally, it leads us to believe that atomic migration rates are greater in grain boundaries than in the grain interior or single crystals. In order to under­stand the effect of grain boundaries, one needs to compare the diffusion results between a single crystal and a polycrystalline material. Let us take an example involving the diffusion in single crystal and polycrystalline silver. The data were plotted as log(D) versus 1/T (a schematic plot is shown in Figure 2.51 without actual experimental data). The plot is a straight line for the single crystal across the temperature range studied, but the curve for the polycrystal coincides with the sin­gle crystal at the higher temperature range. However, with decreasing temperature, the diffusivity in the polycrystal silver becomes higher. It can only happen if the

grain boundary activation energy is lower than that for bulk diffusion. Equa­tion (2.50) expresses the diffusion data in single-crystal silver crystal, whereas Eq. (2.51) shows the diffusivity data in polycrystalline silver.

D = 0.9 exp(—1.99 eV/fcT) cm2 s—1 (2.50)

and

D = 0.9 exp(—1.01 eV/fcT) cm2 s—1. (2.51)

The equations above reflect the fact that at lower temperatures, the lower acti­vation energy for grain boundary diffusion lowers the overall activation energy and thus grain boundary diffusion becomes more dominant at lower tempera­tures. However, as the temperature is increased, the contribution of bulk diffu­sion becomes more dominant. That is why at higher temperatures, the grain boundary diffusion contribution becomes negligible compared to the bulk diffu­sion. Thus, one cannot see much change at all in the position of the curves (Figure 2.49) in the higher temperature range. For example, the grain boundary activation energy for bulk diffusion in aluminum is taken as 142kJmol—1, whereas the grain boundary activation energy for aluminum is only 84 kJ mol—1. However, grain boundary activation energy has been variously taken as 0.35­0.60 times the activation energy for lattice diffusion. It is worth noting that poly­crystals are generally composed of small grains oriented randomly with one another. Thus, when the diffusion rates are measured, it gives an average value of measurement over several grains, thus simulating a macroscopic isotropy of diffusion, even though diffusion in each grain (single crystal) in essence is highly anisotropic.

2.3.6.2 Dislocation Core Diffusion

Dislocation core structure is quite different from the lattice crystal structure. Diffusion along the dislocation core may provide a faster diffusion path and contribute to the overall diffusion, especially at lower temperatures. This type of diffusion is also known as pipe diffusion. Activation energy for the dislocation core diffusion is generally close to the activation energy for grain boundary diffusion.

This interaction takes place due to the mutual interaction of elastic stress fields surrounding misfitting solute atoms and the dislocation cores. The hardening effect related to the elastic interaction is directly proportional to the solute-lattice misfit. Substitutional atoms only obstruct the movement of edge dislocations. On the other hand, interstitial solutes impede both screw and edge dislocations.

2.4.1.1 Modulus Interaction

This occurs if the presence of a solute atom locally alters the modulus of the crystal.

If the solute has a smaller shear modulus than the matrix, the energy of the strain field of the dislocation will be reduced and there will be an attraction between the solute and the matrix. The modulus interaction is quite similar to the elastic inter­action, however, since it is accompanied by a change in the bulk modulus, both edge and screw dislocations will be subject to this interaction.

The specific heat (molar heat capacity) is the amount of heat needed to raise the temperature of a g mol of material by 1 degree at constant pressure (Cp) or constant volume (CV). Thermodynamically, they can be written as

where U and H are the internal energy and enthalpy of the system, respectively. It is experimentally more suitable to evaluate the specific heat at constant pressure Cp (atmospheric). However, it is theoretically easier to predict Cv. Cp is always greater than Cv; at room temperature or below; the difference is very small for solids

(whereas for ideal gas, CP — CV = R). The SI unit of specific heat is generally given in J mol—1 K—1.

Petit and Dulong [12] conducted experiments and suggested an empirical rule that states that all solid elements have CV of 3R (where R is the universal gas con­stant, that is, ~24.9 J mol—1 K—1). Kopp [13] introduced an empirical rule that the molar heat capacity of a solid chemical compound is approximately equal to the sum of molar heat capacities of its constituent elements. Even though the molar heat capacity values for elements tend to be close to 3R near room temperature, they increase with temperature and become smaller than 3R at lower temperatures. Figure 5.52 shows CV for four pure elements (lead, copper, silicon, and diamond) as a function of temperature. Only metals, lead, and copper show specific heat close to 3R. However, for other two elements, they are quite different.

The classical physics could not explain why the specific heat changes with increasing temperature. During the early part of the twentieth century, the concept of specific heat was explained with the help of quantum theory. The problem was first treated by Einstein in 1907 by assuming that atoms in a crystal (called Einstein crystal) behave like independent harmonic oscillators. But Einstein’s derivation could not explain the experimental data near the absolute zero. Later in 1912, Debye assumed that the range of frequencies available to the oscillators is the same as that available in a homogeneous elastic continuum. Basically, Debye assumed that the vibration occurs in a coordinated way (because of the presence of atomic bonding) as opposed to Einstein’s assumption of independent oscillation. They were considered as elastic lattice waves (much like sound waves) that have high frequency with small amplitude. The lattice vibrational energy can be quantized (phonon) in a similar way as done for light waves (photon). An expression for con­stant volume specific heat developed by Debye provided good agreement with

where x = hv/kT is dimensionless, 0D = hvDfkT is Debye temperature, T is the tem­perature in K, R is the universal gas constant, and vD is the maximum frequency of atomic vibration (called Debye frequency). We should be cognizant of two facts regarding the function given in Eq. (5.80): (i) At higher temperatures up to room temperature (i. e., a low value of 0D/T), Cv tends to reach the value as predicted by Dulong and Petit. (ii) The expression also gives good accord to the experimental data at very low temperatures when Debye’s original expression is reduced to

(5.81)

This equation correctly captures the rapid approach toward zero as the temperature reaches the absolute zero. Figure 5.53 shows the Cv values as a function of temper­ature for four different elements. If the data are normalized with respect to 0D, all data come together and fall on a single master curve.

It should be noted that Cp rather than Cv is measured more often experimentally. In order to relate the theories to experiments, (Cp — Cv) needs to be known from an expression as shown in Eq. (5.82):

a2VT

Cp — Cv = ь ; (5.82)

where a is the coefficient of thermal expansion, ((dln V)/@T)p, and b is the coefficient of compressibility, — ((d ln V)=@P) T.

However, at a temperature much above the room temperature, Cv gets larger than that predicted by Dulong and Petit. The nonapplicability of Debye’s theory in

this temperature range stems from the fact that electrons of the atoms not only absorb energy but also increase their energy and this electronic contribution was not considered in Debye’s analysis. However, this is only possible for free electrons; that is, the electrons that have been excited from the filled states above the Fermi energy level. This type of situation contribution is certainly valid for metals/ alloys that have free electrons. However, they are still a small fraction of total elec­trons. In insulating and semiconducting materials, the electronic contribution is quite insignificant. The electronic contribution to specific heat bears a linear pro­portionality with temperature (i. e., CV/ T). There are also few other contributions to specific heat depending on the system, such as randomization of electron spins in a ferromagnetic material (like alpha-iron) as it is heated through the Curie tem­perature. A large spike in the specific heat can be observed at that temperature in a differential scanning calorimetry (DSC) curve.

Often the experimentally measured variation in the constant pressure specific heat (CP) is described by empirical fitting of the form shown below:

CP = a + bT + cT-2, (5.83)

where a, b, and c are curve fitting constants. Figure 5.54 shows a number of CP versus temperature curves for various elements and compounds that exhibit poly­morphism. Note the sharp changes at the temperature where the polymorphic transformations take place.

Generation-I reactors were built in the initial period of nuclear power expan­sion and generally had primitive design features. Most of these reactors have either been shut down or will be soon done so. Examples of such reactors are Magnox reactor (Calder Hall reactor in the United Kingdom) and first com­mercial power reactor at Shippingport in 1957 (in the state of Pennsylvania in the United States).

Magnox Reactor

This is a notable Generation-I gas-cooled reactor. Early breed of this reactor was used for the purpose of plutonium production (for atomic weapons) as well as elec­tricity generation. Figure 1.8a shows a cross section of a typical Magnox reactor. The

Calder Hall station in the United Kingdom was a Magnox type of reactor starting successful operation in 1956. Following that, several of these reactors were built and operated in the United Kingdom and a few elsewhere (e. g., Italy, France, and Japan). Generally, Magnox reactors were graphite moderated, and used the natural uranium as fuel clad in thin cylindrical tubes of a magnesium alloy (Magnox comes from the name of the magnesium-based alloy with a small amount of aluminum and other minor elements, magnesium nonoxidizing, for example, Mg-0.8Al — 0.005Be) and carbon dioxide (CO2) as coolant (heat transfer medium). Magnesium — based alloy was chosen since Mg has a very low thermal neutron capture cross sec­tion (0.059 b; lower than Zr or Al). The fuel elements were impact extruded with the integral cooling fins or machined from finned extrusions (Figure 1.8b). Also,

the alloy was resistant to creep and corrosion from CO2 atmosphere in the operat­ing temperature range, and contrary to Al, the alloy did not react with the uranium fuel. The addition of Al in the alloy provided solid solution strengthening, while the presence of minor amounts of Be helped improve the oxidation resistance. CO2 was circulated under pressure through the reactor core and sent to the steam gener­ator to produce steam that is then passed through a turbo generator system gener­ating electricity. These reactors could sustain lower temperatures (maximum coolant temperature of 345 °C) and, thus, has a limited plant efficiency and power capacity. This was mainly out of the concern of the possible reaction of CO2 with graphite at higher temperatures and the lower melting point of uranium fuel (1132 °C). Another problem was that the spent fuel from these reactors could not be safely stored under water because of its chemical reactivity in the presence of water. Thus, the spent fuels needed to be reprocessed immediately after taking out of the reactor and expensive handling of equipment was required. Only two Mag — nox reactors still operating in the United Kingdom are scheduled to be decommis­sioned soon. Magnox reactors were followed by an improved version of gas-cooled reactor known as Advanced Gas-Cooled Reactor (AGR) operating at higher temper­atures and thus improving the plant efficiency. The magnesium fuel element was replaced by stainless steels.