At the outset, surface diffusion is generally assumed to be simple. On the contrary to the belief, it is rather difficult to precisely estimate. This problem arises from the difficulty in describing the surface as “atomically flat,” as surfaces also have grooves, scratches, steps, and so on. Moreover, surfaces are easily susceptible to contamination. The general view is that the activation energy for surface diffusion is considerably less than that for the bulk diffusion as a surface atom has almost half the nearest-neighbor atoms compared to an atom in the bulk. So, it is relatively easier for a surface atom to hop from one lattice position to another equivalent atom position.

2.3.7

Summary

This section discussed a very important kinetic phenomenon in materials known as diffusion. Diffusion has a vital role to play in several nuclear-specific materials, which will be elucidated in later chapters. Both phenomenological and atomistic theories of diffusion are discussed. Microstructural paths of diffusion have an important bearing on the activation energy and thus on diffusion rates; activation energy values in a material would be sequenced from smaller to higher magnitudes as shown in the following: surface (Qf) < grain boundary (Qgb) < dislocation core (Qc) < lattice self-diffusion (Ql).

Problems

2.1 Given the following information on tantalum (Ta): atomic number = 73, atomic mass = 180.95 amu, atomic radius = 0.1429 nm, density = 16.6 g/cc m~3. (a) Find out the number of atoms per mm3. (b) What is the atomic pack­ing factor? (c) If it is cubic, what is its Bravais lattice?

2.2 What is polymorphism? Give four examples.

2.3 The unit cell of uranium has orthorhombic symmetry with a, b, and c lattice parameters of 0.286, 0.587, and 0.495 nm, respectively. If its density, atomic weight, and atomic radius are 19.05 gcm~3, 238.03 gmol-1, and 0.1385 nm, respectively. Calculate the atomic packing factor.

2.4 Calculate the density of uranium carbide (UC) given the atomic radii of U and C species.

2.5 Copper has an FCC structure with an atomic radius of 0.128 nm and atomic weight of 63.5 g mol-1.

a) Calculate its density in g cm-3?

b) Draw a neat sketch of a unit cell with axes appropriately shown and depict the planes and directions: (100), (110) and (111); [110] and [111].

c) Calculate the planar density (per cm2) of atoms on the above planes?

d) Calculate the linear density (per cm) of atoms along these directions?

2.6 What is the angle between the planes (011) and (001) in a cubic crystal?

2.7 a) What are the crystal structures of UC and UO2 (draw a neat sketch of a

unit cell and show the ionic positions) and what are the advantages of UO2 over UC as a nuclear fuel?

b) In UO2, what are the coordination numbers of U and O ions?

c) Given that the ionic radii of U and O are 0.97 and 1.32 A, respectively, cal­culate the lattice constant of UO2.

2.8 Draw a neat sketch of a unit cell of Fe (BCC) and depict a close-packed plane and close-packed direction (i. e., the densest plane and direction) showing clearly the choice of coordinates.

a) What are the Miller indices of the specific plane and direction you chose?

b) Calculate the planar and linear atomic densities for the plane and direction above?

2.9 a) Compute and compare the linear densities of the [110] and [111] directions

for BCC.

b) Calculate and compare the planar densities of the (100) and (111) planes for FCC.

2.10 Show that the ideal c/a ratio for HCP crystals is 1.633?

2.11 On a neat sketch of a HCP crystal, show the following (clearly depict the axes):

a) Prism plane (1010) and a close-packed direction in that plane — What are the Miller indices of the direction chosen and what is the interatomic distance?

b) Pyramidal plane (1122) and a close-packed direction in that plane — What are the Miller indices of the direction chosen?

c) A basal plane and a close-packed direction in that plane — What are the Miller indices of the plane and direction chosen?

2.12 a) If the activation energy for vacancy formation in Fe is 35 kcal mol-1, what

is the vacancy concentration at (i) 900 °C and (ii) 400 °C? b) Calculate the number of vacancies per unit cell at 900 °C in the above problem.

2.13 In alkali halides, such as NaCl, one finds intrinsic and extrinsic vacancies (cat­ion vacancies) due to divalent impurity such as Ca. On an Arrhenius plot show (schematically) the temperature variation of vacancy (cation) concentra­tion indicating the extrinsic and intrinsic regions. Note down the equations relevant to these regions.

2.14 Compare and contrast the edge and screw dislocations.

2.15 Show how the dislocation model of low-angle grain boundaries can explain the creation of high-angle grain boundaries.

2.16 Show that Hume-Rothery’s rules apply to copper-nickel alloy system. Why is the solubility of carbon in FCC iron more than in BCC iron?

2.17 Steel surfaces can be hardened by carburization, the diffusion of carbon into the steel from a carbon-rich atmosphere. During one such treatment at 1000 °C, there is a drop in carbon concentration from 5.0% to 4.0% carbon between 1 and 2 mm from the surface of the steel. Estimate the flux of carbon into the steel in this near-surface region to be 2.45 x 1019 atoms per m2 s (The density of y-Fe at 1000 °C is 7.63 gcm-3.).

2.18 A steel with 0.2% C is to be carburized in a carburizing atmosphere to reach a carbon concentration of 1.1% at the surface. After 10 h at 890 °C, at what depth below the surface one would find 0.4% C concentration? (For diffusion of C in austenite, D0 = 2.0 x 10-5 s-1, and Q = 140 kJ mol-1)

2.19 a) Given the diffusion data (D in cm2 s-1) for yttrium in chromium oxide at

different temperatures, find the activation energy and diffusion coefficient

(D0): 1.2 x 10-13 @800 °C; 5.4 x 10-13 @850 °C; 6.7 x 10-13 @900 °C;

1.8

x 10-12 @950 °C; and 4.6 x 10-12 @1000 °C. b) Find D at 925 °C.

This interaction is produced only in superlattices (ordered alloys), not in conven­tional disordered alloys. In a superlattice, there is a long-range periodic arrange­ment of dissimilar atoms, such as in Cu3Au. The movement of dislocations through a superlattice causes regions of disorder called antiphase boundary (APB).

The dislocation dissociates into two ordinary dislocations separated by an APB. As the slip proceeds, more APBs are created. Ordered alloys with a fine domain size are stronger than the disordered state.

2.4.1.2 Stacking Fault Interactions

Stacking fault interactions are important as the solute atoms preferentially segre­gate to the stacking faults (contained in extended dislocation). This effect is also known as Suzuki effect or chemical interaction. The stacking fault energy gets reduced due to the increasing concentration of solutes in the SF, and thus the sepa­ration between the partial dislocations increases making it increasingly harder for the partial dislocations to move.

Heat is transported through solid materials from the high-temperature region to the low-temperature region. Thermal conduction is a principal mode of heat trans­fer in solid materials. In nuclear reactors, the heat is conducted away by the clad­ding materials from the fuel interior. Thermal conductivity (k) is defined by the following equation known as Fourier’s law:

dT

q = ~k dx ’

where q is the heat flux (the heat flow per unit perpendicular area per unit time) and dT/dx is the thermal gradient. This equation is applicable for the steady-state heat flow. The minus sign comes due to the heat being conducted from the hot region to cold region, that is, down the temperature gradient. The SI unit of ther­mal conductivity is Wm-1K-1. The above equation is much similar to Fick’s steady-state flow (Eq. (2.21)). Thermal diffusivity is another term that is often used. It is given by the following expression:

k

D = —, (5.87)

Cp Q

where CP is the constant pressure specific heat and q is the physical density.

Heat conduction takes place by both phonons and free electrons. Thus, the ther­mal conductivity (k) is given by

k = kl + ke, (5.88)

where kl is the phonon contribution and ke is the electronic contribution to thermal conductivity.

In high-purity metals, thermal energy transport through free electrons is much more effective compared to the phonon contribution as free electrons are readily available as they are not easily scattered by atoms and imperfections in the crystal and they have higher velocities. The thermal conductivities of metals can vary from 20 to 400 W m-1 K-1. Silver, copper, gold, aluminum, and tungsten are some of the common metals with high thermal conductivities. Metals are generally much better thermal conductors than nonmetals because the same free electrons that partici­pate in electrical charge transport also take part in the heat conduction. For metals, the thermal conductivity is quite high, and those metals that are the best electrical conductors are also the best thermal conductors. At a given temperature, the ther­mal and electrical conductivities of metals are proportional, but interestingly the temperature increases the thermal conductivity while reducing the electrical con­ductivity. This behavior can be explained with the help of Wiedemann-Franz law:

LWF = CT ’ (5.89)

where T is the temperature in K, Ce is the electrical conductivity, LWF is a constant (Lorenz number) that is ~2.44 x 10-8 V W K-2. The above relation is based on the

fact that both heat and electrical (charge) transport are associated with free elec­trons in metals. The thermal conductivity in metals increases with the average electron velocity as that increases the forward transport of energy. However, the electrical conductivity decreases with increasing electron velocity because the scattering or collisions divert the electrons from forward transport of charge. For more details on thermal properties, readers may consult an excellent text by Ziman [14]. The presence of grain boundaries and other crystal defects reduces the ther­mal conductivity. Researchers have observed a marked reduction in nanocrystalline materials (i. e., with grain size less than 100 nm).

Alloying generally acts upon the thermal conductivity of metals. The alloying atoms act as scattering centers for free electrons and thus reduce the effective ther­mal conductivity. Brass has a lower thermal conductivity compared to pure copper for that reason. Specifically, for a 70Cu-30Zn, thermal conductivity at room tempera­ture is 120 Wm-1 K-1, whereas the thermal conductivity of pure copper at the same temperature is 398Wm-1K-1. Figure 5.57 illustrates the point. For the same rea­son, stainless steels are generally poor conductors of heat compared to pure iron.

The thermal conductivity ofglass and ceramics is generally smaller compared to that of metals. They range between 2 and 50 W m-1 K-1. As free electrons are not available in these materials, phonon is the main mode of heat transport (at least at lower temperatures). Glass and other noncrystalline ceramics have much lower thermal conductivity compared to crystalline ceramics as phonons are more sus­ceptible to scattering in the materials lacking definite atomic order. With increasing temperature, thermal conductivity of materials decreases; but at higher tempera­tures, another mode of heat transport known as infrared radiant heat transport becomes active and its contribution increases as the temperature increases, espe­cially in transparent ceramics. Porosity in ceramics also contributes to the reduc­tion in thermal conductivity. Figure 5.58 shows the variation of thermal

Temperature (°С)

Figure 5.58 Dependence of thermal conductivity of different ceramic materials on temperature. From Ref. [1].

conductivity of certain ceramics as a function of temperature, and compared against that of graphite. The still air generally present in the pores has extremely low thermal conductivity, on the order of 0.02 W m-1 K-1, thus giving the porous material low thermal conductivity. That is why thermal insulating materials are made porous. Figure 5.59 shows thermal conductivity of some oxide ceramics (some of nuclear importance) as a function of temperature.

5.2.4

Summary

Thermophysical properties play an important role in the selection of materials as well as in their service performance in nuclear reactors. Here, three thermophysi­cal properties (specific heat, thermal expansion coefficient, and thermal conductiv­ity) are discussed. The effects of structure and composition on these properties are also highlighted. It also elucidates the effect of temperature on thermal conductiv­ity, specific heat, and coefficient of thermal expansion in metals and nonmetals. However, it should be noted that there are various specific exceptions where the foregoing discussion may not apply.

5.3

Most of the commercial nuclear power plants operating today are of Genera — tion-II type. Also, the reactors employed in naval vessels (such as aircraft carri­ers and submarines) and many research/test reactors are of this type. The Generation-II reactors incorporated improved design and safety features and productivity over Generation-I reactors. In the Western Hemisphere, a majority of commercial nuclear power plants have light water reactor (LWR), both pres­surized water reactor (PWR) and boiling water reactor (BWR). It is important to remember that LWRs were also built as Generation-I reactors (such as Ship — pingport facility with 60 MWe power capacity), however most of them are no longer in operation. Another variety is the CANDU (Canadian Deuterium Ura­nium) reactor, which is basically a pressurized heavy water reactor (PHWR). There are a few different versions of pressurized water reactors (e. g., RBMK type) in Russia and former Soviet-block countries, but discussion on those reactors is outside the scope of this book.

As described in Section 2.1, crystals are hardly perfect even when there is no radiation damage. The deviations from the ideal crystal structure are instrumen­tal in influencing various structure-sensitive properties of crystals. There could be several types of these defects, and collectively they are called crystal or lattice defects. Interestingly, a perfect crystal could be composed of atoms at rest with only zero-point oscillation at the absolute zero temperature. However, as the temperature increases, the amplitude of the lattice vibration also increases. This lattice vibration basically manifests itself as elastic waves and can influence some very important physical properties (such as thermal conductivity). This type of lattice vibration is called phonon because of its similarity in behavior with the light photons (mainly because of the relationship between their fre­quency expressions). Electrons can jump to higher orbits creating electron holes. This can also affect electronic properties (recall the semiconductor theo­ries). However, electronic properties are not of pressing importance in the con­text of nuclear reactor materials. Henceforth, in this section, our focus would be to give the readers an introduction to various types of crystal defects like point defects (vacancies, self-interstitials, substitutional, or interstitial impurity atoms), line defects (dislocations), surface defects (grain boundaries), and vol­ume defects (voids, cavities, and precipitates).

2.2.1

Dislocation velocity depends on the purity of the crystal, applied shear stress, tem­perature, and dislocation type. Johnston and Gillman (1959) developed an expres­sion for the dislocation velocity in freshly grown lithium fluoride crystals. It was found that the edge dislocations travel about 50 times faster than screw disloca­tions. Studies on the close-packed FCC and HCP metals have revealed that the dis­location velocity approaches ~1 ms-1 at the critical resolved shear stress of the specific crystal. Dislocation velocities have been found to be a very strong function of applied shear stress as shown in Eq. (4.9):

Vd = Atm’, (4.9)

where m’ is a material constant with values ranging from 1.5 to 40 for various types of materials and A is a constant. However, for pure crystals, m’ generally remains ~1 at 300 K and 4-12 at 77 K. At lower temperatures, dislocation velocity is higher compared to that at higher temperatures because phonons (lattice vibra­tions) are more at higher temperatures obstructing the dislocation velocity. The the­oretical maximum velocity of dislocation in a crystalline solid is the velocity of the transverse shear wave propagation. However, damping forces (related to phonons) are enhanced as the dislocation velocity reaches 1/1000 of the theoretical limit. Figure 4.6 illustrates the variation of dislocation velocity as a function of applied stress in an iron alloy containing 3.5% Si.

4.2

Temperature strongly affects stress-strain curves. Generally, strength decreases and ductility increases. However, this trend does change according to the micro­structural evolution such as precipitation, strain aging, or recrystallization that may take place during testing. Thermally activated processes help in deformation pro­cess and result in reducing the strength. Figure 5.9 shows the stress-strain curves of a mild steel at three different temperatures.

Understanding of thermally activated deformation is important for structural materials serving at elevated homologous temperatures. The flow stress (shown as shear component) of a pure metal is composed of two parts:

t = t* + Tg,

where t* and tG are the thermally activated stress and athermal (temperature — independent) stress components, respectively. There are two types of obstacles in a material: long-range obstacles and short-range obstacles. The influence of long-range obstacles occurs over several atom distances and is difficult to surmount through pure thermal fluctuations. Hence, the athermal stress component comes from the long — range obstacles. The long-range stress field is not generally affected by temperature or strain rate except for the change in modulus due to temperature change (which is purely due to reduction in interatomic bonding forces). On the other hand, the short- range obstacles (less than 10 atom diameters) for which dislocations can surmount

these barriers with thermal fluctuations result in temperature-sensitive strengthening. These short-range obstacles are also known as thermal barriers and their influence on flow stress strongly depends on temperature and strain rate. Figure 5.10 shows the variation of normalized flow stress (tested at a strain rate of 3 x 10~4 s-1) as a function of temperature in a high-purity titanium.

Figure 6.10 illustrates schematically all the point defect and other higher order defects that may occur. Besides stable faulted Frank loops, SIAs may form meta­stable arrangement of SIAs that do not reorganize into a stable, glissile form by the end of thermal spike. They are important because they do not migrate away from the cascade and act as precursors to extended defects. Mobile clusters can interact with other clusters or with impurity atoms. Vacancy clusters that form perfect dislo­cation loops are also intrinsically glissile. Formation of dislocation loops and voids occurs from defect clusters. Interstitial-type dislocation loops in deuterium — irradiated molybdenum and tungsten are shown in Figure 6.11.

In the subsequent chapters, we will see more detailed accounts of how energetic radiation plays a significant role in modifying the microstructure of the materials involved. Radiation damage under the fast neutron flux involves atomic displace­ments (i. e., displacement damage) leading to the creation of a host of defects in the material. The effects of radiation can be diverse, including radiation hardening, radiation embrittlement, void swelling, irradiation creep, and so forth, with all hav­ing significant effects on the performance of the reactor components. Another interesting effect of radiation is the radiolytic decomposition of coolant (e. g., water molecule is radiolyzed into more active radicals) that may definitely affect the corro­sion behavior of the reactor components. Fission fragments also cause damage, but they are mostly limited to the fuel. So, for selecting materials for a nuclear reactor, we must know the concomitant radiation effects on these materials. That is why millions of dollars are spent to wage materials irradiation campaigns in test reactors followed by careful postirradiation examination to ascertain fitness-for-ser- vice quality of the materials to be used in nuclear reactors.

1.9.3

No kinetic aspect in materials is as fundamental and important as diffusion! Diffu­sion can be defined as the effective movement of atoms/molecules relative to their neighbors under the influence of a gradient. The process is assisted by the intrinsic thermal or kinetic energy of atoms. The driving force or the gradient can be of vari­ous types. It can be chemical potentials arising from the concentration gradient or gradients in electrical field, mechanical stresses, or even gravitational field. The movement of atoms could be over a large number of interatomic distances (i. e., long-range diffusion) or over one or two interatomic distances (i. e., short-range dif­fusion). Although diffusion in liquid and gaseous states is easier to visualize, diffu­sion of atoms in solids is not so. Diffusion is regarded as one of the most important mechanisms of mass transport in materials. It is sometimes difficult to assume that atoms remain “diffusible” until the temperature of the solid is brought down to the absolute zero (still a hypothetical situation though)! It is again amazing to know how many well-known materials phenomena are influenced by diffusion. Here are some examples: phase transformations, precipitation, high-temperature

creep, high-temperature oxidation of metals, metal joining by diffusion bonding, impurity transistors, grain growth, and radiation damage defects and their migration.

There are two general ways by which diffusion can be categorized. If one consid­ers diffusion of atoms in a pure metal, the diffusion happens basically between its own lattice atoms. This diffusion is called self-diffusion. On the other hand, diffusion of alloying elements or impurities may well be occurring in the parent lattice and then the diffusion is termed as heterodiffusion. In the following section, the macro­scopic diffusion theories are first dealt with and then the topics of atomic diffusion.

2.3.1