“FORTUNATELY crystals are seldom, if ever, perfect.”

—Anonymous

2.1

Crystal Structure

Engineering materials (metals and alloys and ceramics) used in nuclear applica­tions are almost always crystalline. That is why the understanding of crystal struc­ture basics is very important in the context of nuclear materials. As a matter of fact, an overwhelming majority of materials crystallize when they solidify, that is, atoms get arranged in a periodic three-dimensional pattern leading to a long-range order and symmetry, while minimizing the overall free energy of the solid. However, before we start discussing the details of a crystal structure, let us assess broadly what are the different length scales in a material system. It has long been estab­lished that only a single length scale cannot adequately describe all the behaviors of a material, and that is why multiscale methodologies (Figure 2.1) are being increas­ingly implemented in modeling various materials systems, including nuclear mate­rials. For instance, subatomic scale involves the interaction between the subatomic particles (neutrons, protons, and electrons). On the other hand, a single crystal entails an ensemble of several atoms (the smallest unit being called a unit cell), while several such single crystals can create a polycrystalline material (microscale), and, finally, the macrostructure (basically the components, machines, etc.) can be seen by our bare eyes at the top of the length scale. Hence, to describe the behavior of a material, one needs to rely on several length scales, not just one.

An underlying theme of the materials science and engineering (MSE) field is to understand the interrelationships of processing-structure-property, which gives one greater opportunity for predicting to a reasonable degree the materials perform­ance under real service conditions. This is exemplified by the materials science tetrahedron, as depicted in Figure 2.2.

This theme is equally applicable to the nuclear materials. For example, materials scientists and engineers study microstructural features (grain size, type of second phases and their relative proportions, grain boundary character distribution to name a few) to elucidate the behavior of a material. These are structural features

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.

that are influenced by the nature of the processing techniques (casting, rolling, forging, powder metallurgy, and so forth) employed, leading to changes in propert­ies. This understanding will be very helpful as we wade through the subsequent chapters.

A lattice is an array of points in three dimensions such that each point has identi­cal surroundings. When such lattice point is assigned one or more atoms/ions (i. e., basis), a crystal is formed. In this chapter, we present a simple treatment of the crystal structure and relate it to its importance with respect to nuclear materials. There are 7 basic crystal systems and a total of 14 unique crystal structures (Bravais lattices) that can be found in most elemental solids. These are based on the crystal symmetry and the arrangement of atoms as described in the following sections.

Although Fick’s first law is helpful, it cannot express concentration profile at a point as a function of time. It is because Fick’s first law cannot link the concentra­tion gradient at a point to the rate at which the concentration changes within a fixed volume element of a material. That is why Fick’s second law was derived from embodying Fick’s first law and the law of mass conservation. The resulting law can deal with nonsteady-state diffusion phenomena satisfactorily. Nonsteady-state

diffusion refers to a situation when the diffusion fluxes at various cross-sectional planes across the diffusion distance are not constant; they vary with time. Thus, the concentration profile changes over time. In fact, this situation is found in most practical cases.

Let us derive Fick’s second law using a simple method. At the onset, one needs to assume that the concentration change in a volume element can be calculated if one knows how much flux is getting in (/in) and how much is getting out (/out) in a given volume element. Also, consider two parallel planes in a volume element of thickness dx. Then, the flux through the first plane is

tin = -0■ <218>

On the other hand, the flux from the second plane is

By subtracting Eq. (2.18) from Eq. (2.19), we get the following relation:

dJ^_ D

dx dx dx

Comparing Eqs (2.20) and (2.21), we get

do d / do d2o

dt dx D dx D dx2 .

One can only take diffusivity (D) term out of the del operator as constant if it is assumed that the diffusivity does not change with concentration, and thus with dis­tance. One of the great significances of Fick’s second law is that the equation can be solved to describe the concentration profile c(x, t) as a function of time and position given the appropriate initial and boundary conditions. A couple of examples are given in the following sections. More cases can be found in standard diffusion textbooks.

Thin-Film Solution

If a finite thin-film source contains an initial amount of solute M (in mol m~2), the concentration profile of an infinite volume at both sides is given by the following relation (see Figure 2.44 for details):

This has significance in determining D. If the solute present in concentration M in the thin film is a radioactive isotope of the matrix element, the diffusion equa­tion can help in determining self-diffusion characteristics by taking logarithm on both sides of Eq. (2.23):

(2.24)

If one plots ln (c) against x2 using experimental data, a straight line with a slope of (4Dt)-1 is easily obtained and the diffusivity D is calculated since t would be known. Remember that D does depend on temperature, and suitable experimental data collection at different temperatures will give more details like activation energy for diffusion.

Carburizing/Decarburizing

If the composition of the surface is changed from the initial composition c0 to cs and is maintained at that level, the concentration profile at any point in the semi­infinite volume is given by

c(x, t) = cs — (cs — C0) erf (^pDt) ’ (2-25)

where “erf” or the error function is an integral defined by the following expression:

z

erf (z)=p—j exp (- j2)dj. (2.26)

0

Equation (2.25) is applicable for describing the concentration-distance profile during carburization (surface hardening through addition of carbon atoms) and decarburization (taking away carbon atoms from the surface layer of a material) ofsteels.

For z > 1, erf (z) =~ 1; for 1 > z > — 1, erf (z) = z; and for z < —1, erf (z) =~ — 1.

Furthermore, relations such as erf(0) = 0, erf(i) = 1, erf(—z) = —erf(z), and, for very small values of x, erf (z) = 2z/v— are helpful in determining the concentra­tion profiles. Table 2.5 lists the error function values for different x values. Detailed error function tables are generally available in the books that deal exclusively with diffusion and can be found in the references cited at the end of this chapter.

One extended case would be the determination of the concentration profile of two semi-infinite plates of different solute concentrations of c1 and c2 joined together (also known as diffusion couple). In this case, the appropriate equation is

x

2 VDtJ’

2.3.2

Frank (1949) described another type of partial dislocation of (a0/3)(111) type (Fig­ure 4.27). This type of dislocation is called Frank partial dislocation and its Burgers vector is perpendicular to the plane ofthe fault, and thus Frank partial cannot move by glide in the plane of the loop (i. e., a prismatic or a sessile dislocation). It can move only by climb. But as it requires higher homologous temperatures, sessile disloca­tions generally provide obstacles to the movement of other dislocations, and may aid in the strain hardening effect. A method by which a missing row of atoms can be created in the {111} plane is by the collapse of vacancy clusters on that plane. These dislocation loops are frequently observed in irradiated materials. Also refer to the discussion on prismatic loops in Section 4.2.1.

4.3.1.2

Lomer-Cottrell Barriers

Sessile dislocations (such as Lomer-Cottrell barrier) that obstruct the normal dislocation movement are created in an FCC crystal by the glide of dislocations on intersecting {111} planes during duplex slip. Two perfect dislocations, (a0/2)[101] and (a0/2)[110], lying on the intersecting slip planes {111} attract each other and move toward their line of intersection, as illustrated in Figure 4.28a. Lomer (1951) suggested that the following reaction can happen to create a new dislocation of reduced energy:

f [101]+a° [тю] ^ a° [011]. (4.26)

Try applying Frank’s rule to show that the above dislocation reaction is energeti­cally favorable. The new dislocation (called Lomer lock) with a Burgers vector (a0/2)[011] lies parallel to the line of the intersection but on a plane (100). Its Bur­gers vector lying in the (100) plane is perpendicular to the dislocation line (so, it is a pure edge dislocation), but as (100) slip plane is not a close-packed plane in FCC, this dislocation does not glide freely and is thus known as Lomer Lock.

This dislocation is an edge dislocation since b3 is perpendicular to t3 (since [011] ■ [0Tl] =0). To see if this dislocation is a sessile or glissile dislocation, we need to find the glide plane of this dislocation that contains both the Burgers vector and the line vector (i. e., normal to the plane given by the cross-product of b3 and t3):

111

111

і [101]

r 211

Stacking fault

.211

Figure4.28 (a) Formation of Lomer lock. (b) and (c) Formation of Lomer-Cottrell lock [3].

Cottrell (1952) reexamined Lomer’s reaction and suggested that it can be made strictly immobile if the perfect dislocations are considered to have dissociated into respective partial dislocations. Only the leading partials would interact with each other following the reaction shown in Eq. (4.24) to produce a truly imperfect, ses­sile dislocation with Burgers vector, (a0/6)[011] (Eq. (4.27)). The formation of this dislocation, called Lomer-Cottrell lock, is shown in Figure 4.28b and c. This new dislocation lies parallel to the line of intersection of the slip plane and has a pure edge character in the (100) plane. The dislocation is sessile because the product dislocation does not lie on either of the planes of its stacking faults. In analogy with the carpet on stair steps, this sessile dislocation is called a stair-rod dislocation. This type of sessile dislocation offers resistance to the dislocation movement and con­tributes to strain hardening, but it is, by no means, the major contributor. It is to be noted that the Lomer-Cottrell barriers can be surmounted at high-enough stresses and/or temperatures.

[Ї2Т] + a° [112] ^ a° м (4.27)

6 6 6

4.3.2

About 90% of engineering failures are attributed to fatigue of materials under cyclic/fluctuating loading. Fatigue failure occurs after a lengthy period of stress or strain reversals and is brittle-like in nature even in normally ductile metals. Gener­ally, the fracture surface turns out to be perpendicular to the applied stress. An example of fatigue failure in a shaft keyway is shown in Figure 5.41.

Various types of fatigue loading are shown in Figure 5.42. Figure 5.42a shows a fully reversed tension-compression stress cycle, while Figure 5.42b depicts a tension-tension cycle. No compression-compression cycle is shown since fatigue cracks do not open up under compression; so no fatigue failure is possible under such loading. However, in reality, such a regular loading cycle is often not the norm; the fluctuating stress cycle may look like a random or irregular stress cycle, as shown in Figure 5.42c.

Let us now start defining some common terms used in fatigue. Mean stress (am) is the algebraic mean or average of the maximum and minimum stresses in the

cycle:

am = (amax V amin)/2. (5-6°)

The range of stress (ar) is just the difference between amax and amin, that is,

ar = amax smin* (5-61)

Stress amplitude or alternating stress (aa) is just half the stress range, that is,

aa or S = sr/2 = (amax amin)/2. (5-62)

The stress ratio (r) is the ratio of minimum and maximum stresses, that is,

r = smin/smax* (5-63)

Amplitude ratio : Ar — aa/am = (1 — r)/(1 + r). (5.64)

We note that Figure 5.42a is a case with r — —1.

There are many different ways to carry out fatigue testing. One of them is a canti­lever beam fatigue test, as illustrated in Figure 5.43. In this setup, the specimen is

in the form of a cantilever loaded at one end. It is rotated at the same time by means of a high-speed motor to which it is connected. At any instant, the upper surface of the specimen is in tension and the lower surface is in compression with the neutral axis at the center. During each revolution, the surface layers pass through a full cycle of tension and compression. Other types of fatigue testing include rotating-bending test and uniaxial tension-compression test in universal test machines.

5.1.7.1 Fatigue Curve

If the stress amplitude (S) is plotted against the number of cycles to failure (N), S-N curves are created (Figure 5.44). The S-N curve indicates that the higher the mag­nitude of the stress, the smaller the number of cycles that material is capable of sustaining before failure and vice versa. For some ferrous metals and titanium alloys, the S-N curve becomes horizontal at higher N values or there is a limiting stress level called the “fatigue limit” or “endurance limit,” which is the largest value of fluctuating stress that will not cause failure for essentially infinite number of cycles. For many steels, fatigue limits range between 35% and 60% of the tensile strength. However, most other metals/alloys show a gradually sloping S-N curve. For these materials, fatigue limit is not clearly delineated and fatigue strength in these cases is obtained at an arbitrary number of cycles (e. g., 108 cycles).

The process of fatigue failure consists of three distinct steps: (a) crack initiation, (b) crack propagation, and (c) final failure. The fatigue life (Nf) is thus the sum of the number of cycles for crack initiation (N;) and the number for the crack propaga­tion (Np):

Nf = Ni + Np

Even when the cyclic stress is less than the yield strength, microscopic plastic deformation on a localized scale can take place. The cyclic nature of the stress causes slip to appear as extrusions and intrusions on the surface, as shown in Figure 5.45. Fatigue cracks can also initiate at other surface discontinuities or stress raisers. During the tensile cycle, slip occurs on a plane with maximum shear stress on it. During the compression part of the cycle, slip may occur on a nearby parallel slip plane with the slip displacement on the opposite direction. These act as the nucleation sites for the fatigue cracks.

The S-N curves as depicted in Figure 5.44 are classified into low and high cycle fatigue with low cycle fatigue (LCF) for N < 5 x 105 cycles and high cycle fatigue (HCF) for N > 5 x 105 cycles. These tests are usually performed on smooth speci­mens in strain-controlled mode and the total strain range is divided into elastic and plastic regions:

Basquin equation is usually considered to describe the HCF with stress range (An = EAe):

NCT? = C;

where N is the number of cycles to failure at stress amplitude sa (As/2) and p and C are material constants. Similarly, the LCF is characterized by Coffin-Manson equation:

Aep. . c

= A(2N)

with c ranging from —0.5 to —0.7 and A being a material constant that is propor­tional to the tensile ductility (total strain to fracture in a tensile test). HCF region

can also be represented in a similar fashion:

Dr = B(2N)b, (5-69)

where b has a value between —0.05 and —0.12 and B is proportional to the tensile or fracture strength of the material. Thus, we note that LCF is controlled by ductil­ity and HCF by strength so that cold working (or exposure to intense neutron irradiation) resulting in higher strength with corresponding decreased ductility will exhibit detrimental effect (i. e., lower N values) during LCF, while being benefi­cial to HCF.

Figure 5.46 shows typical fatigue life plot as strain range (Ae) against number of failure cycles (Nf) along with the corresponding stress-strain loops (broad in LCF and narrow in HCF). The complete fatigue curve can be described by combining the LCF and the HCF formulations by either the universal slopes (Eq. (5.70a)) or the characteristic slopes (Eq. (5.70b)):

Ae = 3.5 ^ N—0-12 + e°-6N—°’6, (5.70a)

Ae = sf (2N)b + ef (2N)c, (5.70b)

2 E

where Su is ultimate tensile strength, ef is true fracture strain, of is the true fracture stress, and b and c are material constants. In terms of the characteristic slopes, the value of fatigue life at which the transition from low cycle (plastic) to high cycle (elastic) occurs is given b

The concept of neutron flux is very similar to heat flux or electromagnetic flux. Neutron flux (W) is simply defined as the density of neutrons n (i. e., number of neutrons per unit volume) multiplied by the velocity of neutrons v. Hence, nv rep­resents the neutron flux, which is the number of neutrons passing through a unit cross-sectional area per second perpendicular to the neutron beam direction. How­ever, sometimes this is called current if we consider that neutrons moving in one direction only. Neutron flux, in general terms, should take into account all the neu­trons moving in all directions and be defined as the number of neutrons crossing a sphere of unit projected area per second. Total neutron flux (W) is expressed by the following integral:

W = W(Ei)dEi, (1.4)

J0

where W(Ei)dEi is the flux of neutrons with energies between Ei and Ei + dEi.

The term “nvt" represents the neutron fluence, that is, neutrons per unit cross­sectional area over a specified period of time (here t). Thus, the units of neutron flux and fluence are n cm-2 s-1 and n cm-2, respectively.

1.7 Neutron Cross Section j 7

1.7

Carbon exhibits two major polymorphic forms — diamond and graphite. Dia­mond structure consists of an FCC unit cell of carbon atoms with half of its tetrahedral positions filled by additional carbon atoms (Figure 2.18). Thus, the effective number of atoms in a diamond unit cell is (4 + 4) = 8. Due to the pres­ence of strong directional covalent bonding (sp3 hybridization) between carbon atoms, diamond is the hardest natural material known, although with atomic packing factor of only 0.34.

Ш Metal atoms О Octahedral interstices

0 Metal atoms О Tetrahedral interstices

Figure 2.18 Diamond crystal structure (from Olander [15]).

In graphite, carbon atoms are arranged in hexagonal arrays (basal planes) and bonded by strong directional covalent bonding (sp2 type). However, the bonding between the hexagonal layers is much weaker “van der Waals” inter­action. Figure 2.19a illustrates the graphite crystal structure. The presence of highly delocalized electrons makes graphite electrically conductive. Conse­quently, the graphite structure has very strong directional properties. Hexago­nal boron nitride (h-BN) also has a similar structure. Graphite is a known moderator used in many past thermal reactors and high-temperature graphite reactors (HTGRs), and also used as a coating for TRISO (TRI-structural oxide) fuels for the VHTR. A single hexagonal layer of graphite is known as “gra­phene.” When the graphene layer is rolled up, it can form a structure known as “carbon nanotube” (Figure 2.19b). Boron nitride nanotubes have a similar structure (Figure 2.19c).

2.1.8

In Section 2.2, we have already discussed the various types of precipitates (coher­ent, semicoherent, and incoherent). The understanding of these precipitates will prove to be helpful in the next discussion. When particles are small (typically less than 5 nm, however dependent on the particular alloy system) and/or soft and coherent, dislocations can cut and deform the particles. In this type of situation, six particle features become important to find out how easily they can be sheared: (a) coherency strains, (b) stacking fault energy, (c) ordered structure, (d) modulus effect, (e) interfacial energy and morphology, and (f) lattice friction stress.

As you can imagine, there are many aspects to the particle shearing mechanism, which is outside of the scope of this book. When the dislocation enters the particle, it produces a step at the interface, and when it leaves the particle, it produces another step at the opposite interface, as shown in Figure 4.35. The step size is generally on the order of the Burgers vector of the dislocation.

In the models of coherency strain and ordered structure, the strength increment due to the particle shearing mechanism varies directly with the square root of the product of the particle radius and volume fraction (i. e., / Vfr since ‘ ~ (1/f). Note that deformation occurring through the shearing of particles does not entail a lot of strain hardening — rather it produces little strain hardening. The slip bands produced are planar and coarse. The steps produced may be detrimental to the cor­rosion properties ofthe materials involved.

Concentration cells are formed due to the variation in the electrolyte composition. As noted before, according to the Nernst equation (Eq. (5.90)), an electrode in

contact with a dilute electrolyte is more anodic compared to that in a concentrated one. Concentration cell aggravates corrosion where the electrolyte concentration is dilute. This concept can be explained by taking a simple example that involves a cell setup where one copper electrode is submerged in a dilute copper sulfate solution while another copper electrode in a concentrated solution. In this case, at the con­centrated solution end, a higher concentration of Cu2+ will drive the reaction, Cu2+ + 2e! Cu0, with copper plating the electrode. Thus, that electrode becomes cath­ode. The electrode in the dilute solution corrodes and acts as anode following the reaction: Cu2+ + 2e~ ! Cu. This type of concentration cells is of interest in chemi­cal plants and some flow-corrosion problems.

Oxidation-type concentration cells are of more importance. The oxidation cell aggravates corrosion where the oxygen concentration is low, for example, crevice corrosion. An example would be as simple as leaving a stone on an iron plate under a moist environment. After a few days, if you come back and check, you will find a rust formation in a much more aggressive manner under the region that the stone occupied. This happens because the cathodic reaction O2 + 2H2O + 4e~ ! 4(OH)~ occurs due to the preponderance of oxygen and moisture outside the region blocked by the stone. But such cathodic reactions would not happen until electrons are supplied from the anode. The region underneath the stone does not have enough oxygen for the above reaction to happen and thus acts as anode and sup­plies electrons to the cathode regions. Thus, the area below the stone corrodes more. Similar oxidation cells can be formed in situations with steel bolts holding steel plates. The regions inside bolt head and shank do not get exposed to oxygen as much as the plate surface as shown in Figure 5.62. Thus, the region deficient in oxygen acts as anode and gets corroded.

“We physicists can dream up and work out all the details of power reactors based on dozens of combinations of the essentials, but it’s only a paper reactor until the metallurgist tells us whether it can be built and from what.

Outer shim cylinder drives

In-pile tubes

Safety rod drives

Reactor core

Discharge chute

In-pile tubes entrance/exit piping

Neck shim and regulating rod drives

A view section of the Advanced Test Reactor. Courtesy: Idaho National Laboratory.

Then only can we figure whether there is any hope that they can produce power at a price.”

—Dr. Norman Hillberry (former director ofArgonne National Laboratory, 1957-1961) (excerpt from the book by C. O. Smith)

The above statement by Dr. Hillberry (a physicist by training) says it all! Erstwhile metallurgical engineering field has now largely morphed into the field of materials science and engineering (MSE). MSE as a field of study is

based on a common theme of finding out the interlinkage between process­ing, structure, and properties in various types of materials. If the interlinkage is clearly understood and established, the performance of these materials under service conditions could be ensured. Hence, the materials selection process in any structure design is a very important step. It is very common to encounter various tradeoffs during the materials selection process for a given application, and most times compromise is called for. Moreover, a nuclear reactor design entails complex procedures in itself given the multiple chal­lenges. Different components of a reactor may require different types of materials. This is mostly done with the help of engineering expertise (experi­ence and judgment).

There could be two broader types of materials selection considerations — gen­eral and special. General considerations involve factors such as mechanical strength, ductility, toughness, dimensional stability, fabricability, cost and avail­ability, heat transfer properties, and so on. General properties come from the general engineering considerations as they would be applicable in most engi­neering designs. On the other hand, special properties are considered solely because the materials are to be used in a nuclear reactor. These include prop­erties like the neutron absorbing characteristics, susceptibility to induced radioactivity, radiation damage resistance, and ease of reprocessing of materi­als. Each of the material characteristics is evaluated following standard (some­times nonconventional) testing procedures. The knowledge of service conditions and broader goals of the future reactor is a must for a successful materials selection process. This information may come from utilizing predic­tive capabilities (modeling and simulation tools) and/or known data/experien — ces from previous reactor design and operations, if available. Brief discussions on these properties have been made in the following sections. Some of these properties will be again elaborated in the subsequent chapters.

1.9.1