0

Chapter 16. Oxide fuels 16.1 LWR Fuel: UO2..................................................................................1

16.1.1 Fabrication of Fuel Pellets.....................................................................2 16.1.2 Microstructure of fuel.............................................................................5 16.1.3 Varieties of fuel pellets...........................................................................6 16.2 Chemistry of Nuclear Oxides............................................................8 16.2.1 Experimental methods............................................................................8 16.2.2 Nanochemistry.........................................................................................9 16.2.3 Macrochemistry.....................................................................................10 16.3 Microchemistry of nuclear oxides.................................................13 16.3.1 Characteristics of point defects in nuclear oxides............................13 16.3.2 Measures of concentrations in oxides................................................14 16.3.3 Point defects in UO2±x...........................................................................16 16.3.4 Structural defect equilibria...................................................................18 16.3.5 Electronic disorder - U4+ disproportionation......................................18 16.3.6 Gas-phase/defect equilibrium..............................................................19 16.3.7 Nonstoichiometry in UO2± x fixed by

2Op ............................................20 16.4 Mixed Oxides...................................................................................22 16.4.1 Mean valence rule..................................................................................23 16.4.2 Oxygen pressure of z2xx1 O)QU(

QQ ±− ..................................................23 16.4.3 Defects in Irradiated UO2.......................................................................26 16.5 Point-defect clustering in UO2+x.....................................................28 16.6 Properties of UO2 dependent upon point defects........................32

1

16.1 LWR Fuel: UO2 There is no leeway in selecting the element that provides energy by fission; 235U is the only naturally occurring fissile nuclide, and undergoes fission by reaction with neutrons according to:

235U + nth → 2FP + 2nf (16.1) where nth denotes a neutron in thermal equilibrium with the water coolant (E ~ 0.1 eV) and nf represents a “fast”, or high-energy neutron produced by the fission process (E ~ 1 MeV). The fluxes of the thermal and fast neutrons in an LWR are about 4x1013 n/cm2-s each. The two fission products (FP in Eq (16.1)) are born with about 100 MeV of energy each. The energy they lose by interaction with the electrons in the fuel material is converted to heat which is conducted through the fuel and ultimately is deposited in the flowing coolant. Natural uranium contains only 0.71% of the fissile isotope 235U, an amount that is insufficient for sustaining a chain reaction in the presence of ordinary water. As a result, the uranium in LWRs is enriched to 235U concentrations up to 5%. The other uranium isotope, 238U, does not fission readily with thermal neutrons. However, it can be converted to 239Pu by absorbing thermal neutrons. The sequence is:

238U + nth → 239U → 239Np → 239Pu (16.2a) The initial absorption product, 239U, decays to the neptunium isotope with a 23-minute half life, and 239Np decays to 239Pu with a half life of 56 hours. The plutonium isotope is relatively stable (lifetime of 24,000 years), but fissions efficiently in a thermal neutron flux:

235Pu + nth → 2FP + 2nf (16.2b)

The thermal energy produced by the fission products from the combination of reactions (16.2a) and (16.2b) increases with time. At the end of life of the fuel (up to 6 years), this route contributes as much as reaction (16.1) to the reactor power. By growing in slowly, 239Pu fission partially offsets the loss of nuclear reactivity caused by the exponential decrease in the 235U concentration with time. Spent fuel removed from the core of a large LWR generates approximately 1000 kg/yr of fission products and 250 kg/yr of plutonium. The chemical form of uranium in the fuel is the oxide, UO2. The oxygen in this compound serves no nuclear purpose, but neither is it detrimental to neutron economy. Its main purpose is to provide a chemically inert fuel form that is also relatively resistant to radiation damage, has a high melting point, and maintains the same cubic crystal phase throughout its entire solid range. Compared to all of these criteria, UO2 is superior to uranium metal except for the uranium atom density and thermal conductivity. On the negative side, the oxygen in UO2 decreases the density of uranium in the fuel by a factor of two compared to the metallic form, with a corresponding increase in the size of the reactor core. Plutonium (as PuO2) can be added to UO2 during fabrication to produce a mixed oxide (MOX) fuel. The plutonium for this purpose originates either from reprocessed spent UO2 fuel or from decommissioned nuclear weapons.

2

The quantity that most directly controls the reactor power, the fuel temperature, and the rate of production of fission products and neutrons is the fission density, F . This is the rate at which reactions (16.1) and (16.2) proceed. It is proportional to the concentrations of the fissile nuclides and the neutron concentration expressed as the thermal neutron flux φth:

F = (σfU235NU235 + σfPu239NPu239)φth (16.3)

The rate constants are the fission cross sections for the reaction of the two fissile species with thermal neutrons. Because the thermal neutron flux varies with position in the reactor core, so does the fission density. A typical value of F is 1013 fissions/cm3-s. The fission density is a measure of the rate of fission. The cumulative fissions for an irradiation time t is called the burnup, β. This quantity has three equivalent definitions. 1. The fractional burnup is the ratio of the number of fissions to the number of initial uranium atoms (of both isotopes):

β = F t/NU (16.4a) where NU = 2.5x1022 atoms/cm3 is the uranium density in UO2. In LWRs, 1% burnup is accumulated per year of full-power operation. 2. The fissions per initial metal atom, or fima, is defined by:

)NN/(tFfima QU += (16.4b) where NQ is the concentration of another oxide Q, mixed with uranium oxide. 3. The energy produced per unit mass of initial uranium This measure is the product of the fractional burnup of Eq (16.4a) and the energy per fission:

Ukgfissionedkg

NtF

×fissionedkgMWd

950=βU

(16.4c)

A typical burnup of fuel discharged from LWRs is 4%, or 40 MWd/kgU.

16.1.1 Fabrication of Fuel Pellets The standard fuel pellet is a solid cylinder of polycrystalline uranium dioxide 1 cm or less in diameter. The two variants of the LWR, the boiling water reactor (BWR) and the pressurized water reactor (PWR) use slightly different pellet sizes, with that for the BWR being larger in diameter. The pellet height in both fuels is between 1 and 2 cm.

Fabrication of fuel for LWRs consists of two distinct processes. In the first, uranium hexafluoride is converted chemically into uranium dioxide powder. In the second, high-density pellets are produced from the starting UO2 powder.

3

A flowsheet of the conversion process is shown in Fig. 16.1. The feed material is UF6, which is the chemical form that is used in the isotope enrichment plants. It arrives as a liquid at elevated pressure and is vaporized by reducing the pressure and warming slightly. The resulting UF6 gas is fed into a chemical reactor along with steam and hydrogen in nitrogen as an inert carrier gas. The following chemical reaction produces a fine power of UO2 as reaction product:

UF6(g) + 2H2O(g) + H2(g) → UO2(s) + 6HF(g) (16.5)

The steam serves as the source of oxygen for the oxide product. Hydrogen reduces hexavalent U in UF6 to the tetravalent state in UO2. The hydrofluoric acid gaseous product is neutralized by NaOH, leaving relatively benign NaF as the sole waste stream. The remainder of the flow sheet in Fig. 16.1 consists of steps that modify the UO2 into a powder form that is easily made into a pellet in subsequent processing. In these steps, dry nitrogen cover gas prevents oxidation of UO2. A lubricant such as stearic acid is added to aid in pressing pellets from the powder. A substance called a poreformer is also added to help control the quantity and shape of the voids that remain in the finished pellet.

Fig. 16.1 Conversion of UF6 to UO2 powder

The product of the conversion process constitutes the feed to the pellet fabrication process depicted in Fig. 16.2. the first step is pressing the powder into “green” pellets that are 50 – 60

4

percent of the theoretical density of the crystalline material (this is abbreviated as %TD). The following high-temperature sintering step serves four functions. First, it drives off the lubricant added to the powder to assist pressing (care must be taken not to decompose the lubricant to carbon, which, along with fluorine, is an undesirable impurity in the final product). Second, the void spaces between the particles in the green pellets are nearly completely eliminated by annealing at high temperature for several hours. This process is called sintering. The porosity of the sintered pellets consists almost exclusively of closed spherical cavities called closed porosity with negligible open porosity. The latter takes the form of interconnected channels that communicate with free surfaces of the pellet. Most fuels are 95 - 96% TD. Retention of few percent porosity is desirable. The pores serve as sinks for fission gases and lessen their release. Swelling of the fuel due to solid fission products is also reduced by filling in the internal voidage.

Fig. 16.2 Pellet Fabrication flow sheet

Third, the time and temperature of the sintering step also affect the size of the grains in the final polycrystalline pellets. Simply eliminating porosity during sintering leaves grains of the same size as the starting powder particles. More extensive annealing increases the average size of the grains by a process termed grain growth. The usual pellet fabrication process produces polycrystals with grain diameters about 8 μm. Both sintering (porosity reduction) and grain growth occur naturally at temperatures high enough to give reasonably rapid kinetics (see Chap.8). However, both processes can be enhanced by additives introduced in the preparation of the feed powder.

The fourth function accomplished during sintering is control of the oxygen-to-uranium ratio. This characteristic of the fuel is controlled by the temperature and the ratio of H2 to H2O in the gas fed to the sintering furnace (see Chap 7). Oxygen in excess of the 2:1 ratio of stoichiometric UO2 is to be avoided. The excess oxygen in UO2+x (even when x ≅ 0.01) is corrosive towards the cladding. Hyperstoichiometry also produces undesirable physical property changes in the fuel. Excess oxygen reduces the thermal conductivity, thereby increasing the fuel temperature during operation. Hyperstoichiometric fuel also enhances the mobility of the fission products, making their release easier. The composition of the oxide in typical reactor fuel is UO2.005.

5

The step following sintering in Fig. 16.1b is a grinding operation designed to smooth the outer surface of the pellet and closely control its diameter. The latter is important because, along with the cladding inside diameter, it determines the thickness of the gap between the fuel and the cladding in the finished fuel rod. This gap is gas-filled and constitutes a significant thermal resistance to heat removal from the pellet during operation.

The final step in Fig. 16.1b is a low-temperature vacuum drying step intended to reduce the concentration of adsorbed water on fuel surfaces to less than 10 ppm by weight. Improperly dried fuel can cause cladding failure by reaction of desorbed water with Zr to produce both ZrO2 and ZrH2. The latter embrittles the cladding and has been responsible for numerous fuel failures during the past several decades.

16.1.2 Microstructure of fuel Figure 16.3 shows a photomicrograph of polycrystalline UO2 after sintering. To obtain this

picture, the pellet is first cut with a diamond saw, then polished with fine abrasives, and finally briefly exposed to an etching solution. Chemical etching reveals most of the grain boundaries as the thick irregular lines enclosing individual single-crystal grains. The grain boundaries are the intersections of the planar cut through the specimen with the faces of the polycrystalline grains that make up the solid. (the straight lines that cross grain boundaries are polishing scratches). The grain diameters are roughly 8 μm. The junctions almost always consist of three grain boundaries intersections called triple points. The third grain-boundary trace is sometimes missing from a few triple points in Fig. 16.2, possibly because of inadequate etching during specimen preparation. The porosity visible in the photomicrograph is all of the closed variety and is visible as roughly circular black spots. The pores are less than one micron in diameter and occupy 7% of the total specimen volume. In UO2 destined for reactor fuel, the size of the pores is as important as the total porosity. In operation, very fine pores (< 1 μm) are eliminated by a process called radiation densification. In essence, the pores are converted to their component vacancies, which are then removed by sinks in the microstructure such as dislocations and grain boundaries. This process causes an

Fig. 16.3 Photomicrograph of sintered UO2 showing pores and grains. The dimension of the grains are about 8 μm

6

initial rapid reduction of the pellet diameter and concomitant collapse of the cladding on the shrunken fuel. This potential problem is solved by removing fine pores during the sintering step of fuel fabrication. 16.1.3 Varieties of fuel pellets Over time, the standard cylindrical fuel pellet design has been modified in numerous ways, as shown by the gallery in Fig. 16.3. The standard pellet is shown in (a). During operation, the temperature distribution in the pellet and the resultant nonuniform thermal expansion causes its shape to change to the "hourglass” shape shown in (b). The edges of the top and bottom pellet surfaces deform the cladding with high local strains and stresses, risking perforation. Cladding subjected to this type of deformation resembles a stalk of bamboo. To avoid this type of cladding deformation and to minimize chipping during manufacturing and handling, the fuel pellets are chamfered as shown in (c). Additional void space to accommodate fission product swelling and axial thermal expansion of hot pellet centers is provided by “dishing” the top and bottom pellet surfaces.

In a solid pellet, fission heat generation and removal creates differences in temperature between the centerline and the pellet surface as large as 1000oC. This is undesirable for a number of reasons, the principal one being release of fission gases from the hot center. To reduce the centerline temperatures at the same linear power (i.e., power produced per unit fuel height), the designs shown in (d) and (e) have been developed. Both move the heat source closer to the heat sink (the coolant) than in the standard fuel pellet. In (d), the outer radial zone of the pellet is more highly enriched in U-235 than the inner zone, thus reducing heat production near the center and lowering the centerline temperature. In (e), the center is removed entirely, resulting in an annular pellet. To produce the same power, this pellet must contain a higher enrichment than the standard solid cylindrical pellet. However, the temperature at the inner surface of the annular pellet is significantly lower than that of the standard cylindrical pellet for the same linear power. Annular pellets are standard in the Russion VVER reactors. In two designs, urania is mixed with other oxides for reasons related to the nuclear processes. In (f), the additive is Gd2O3. Gadolinium acts as a burnable poison that allows a longer irradiation time without excessive reactivity held in control rods at beginning-of-life.

In (g), UO2 is mixed with PuO2 to produce mixed oxide (MOX) fuel. The plutonia is in the form of small particles in a matrix of UO2. Since Pu is the principal fissionable nuclide, these particles are hotter than the rest of the fuel. Despite this nonuniformity of temperature, MOX fuel behaves very similarly to standard UO2 fuel in reactor. In (h), the microstructure of the fuel is purposely altered to increase the grain size from 8 μm to as much as 40 μm by using oxides such as Nb2O5 as a fuel additive. Niobium promotes grain growth during the sintering process. The objective of the large grains is to reduce fission product release by increasing the length of the diffusion path from the grain interior to the grain boundary. So-called “nonfertile” or “inert matrix” fuel depicted in panel (i) is a response to the desire to burn excess plutonium without simultaneously producing this element from a U-238 component, as is the case with MOX fuel. Nonfertile fuels contain PuO2 dissolved in a matrix such as (Zr,Eb)O2. Erbium also serves as a burnable poison like gadolinium.

7

(g) MOX

PuO2

particles

(f) Urania-gadolinia(e) annular(b) hourglassing at power

(c) dished and chamfered

(a) classic (d) zoned enrichment

high U-235 low U-235

(high U-235)

(h) large grain size

Fig. 1.3 Ceramic Oxide Pellet Designs

(i) non-fertile

Fig. 16.4

8

16.2 Chemistry of Nuclear Oxides Even without accounting for the effects of radiation, the chemistry of uranium oxide and mixed oxides has been theoretically and experimentally investigated for over half of a century. The aim of these efforts is to understand how the partial pressure of oxygen in equilibrium with the solid,

2Op , depends on temperature and composition, as defined by the oxygen-to-metal ratio O/M and the cation fraction of the other metal ion, y. The symbol for such a mixed oxide is (U1-yQ y)O2 ± x, where Q is the second cation. The latter is typically an actinide such as Pu or Th, one or more fission products that are soluble in the oxide (La, Y, Eu, Sr) or a neutron absorber such as Gd. The deviation from stoichiometry is quantified by the quantity x; the subscript 2-x signifies a hypostochiometric oxide and 2+x means hyperstoichiometry.. The zirconium oxide corrosion scale that grows on Zircaloy cladding as a result of exposure to high-temperature water is an example of a non-fissile oxide whose behavior is critical to the lifetime of LWR fuel elements. This section is primarily devoted to the theoretical aspects of oxide chemistry, but a brief description of the experimental techniques employed is given below. 16.2.1 Experimental methods The wide array of techniques employed to determine the chemical behavior of nuclear oxides include the following 1. Thermogravimetry [Ref 1, Sect. 11.4.1, Refs 2 - 10] A sample is suspended inside a quartz or alumina tube in a furnace. The wire holding the specimen is attached to a microbalance. A gas of known oxygen partial pressure flows through the tube past the specimen. Examples are: - O2 in an inert gas - a gas mixture such as CO2/CO, which fixes

2Op by the equilibrium: CO(g) + 21 O2 = CO2(g)

The microbalance measures weight changes as temperature or oxygen pressure is changed. The specimen mass is then converted to the O/M ratio, or x = O/M - 2. 2. Galvanic Cell [Ref 1, Sect. 11.4.2, Refs. 11 and 12] An all-solid assembly consisting of a metal/metal oxide mixture (e.g., Ni/NiO) that establishes the oxygen pressure by the equilibrium Ni(s) + 2

1 O2 = NiO(s). Oxygen ions are transferred to or from the test oxide via a piece of cubic zirconia until the oxygen potential (RTln

2Op ) is the same at both electrodes 3. Electrical Conductivity [Refs 6, 11, 12] Standard conductivity techniques are used, usually in conjunction with thermogravimetry. Conductivity measurements provide additional insight into the defect structure of the oxides. 4, Neutron scattering [Refs 13,14] These studies were the first (and only ones) that revealed the existence of the now well-established cluster of point defects in high-temperature, high-hyperstoichiometry uranium dioxide.

9

16.2.2 Nanochemistry By the term nanochemistry, we mean theoretical investigation of equilibrium states of the oxides at the atomic level. In the ab-initio method [15], the Schrodinger wave equation is solved for the many-electron/atomic nucleus structure of the interacting ions. The potential energy of the system is calculated for a number of configurations, such as along the path of a diffusing ion. To reduce the computational demands of the ab-initio method, the interactions between entire ions are instead represented by empirical potentials. Only two-body interactions are considered; many-body effects are neglected. A method considerably less detailed than the ab-initio technique goes by the generic name molecular dynamics. The basic information needed for the simulation by this method is the potential between pairs of ions, φ(r). The interionic force is the derivative of φ with respect to r, the distance between them. Common to all potential functions is the long-range Coulomb potential,

ijo

jiij r

14

zz)r(

πε=φ (16.6)

rij is the distance separating ions i and j, zi and zj are the charges of the ions (e.g., 4+ for a U4+

cation) and εo is the The other components of the interionic potential are determined either from ab-initio computations or by fitting to crystal properties. The short-range potential contains a repulsive component expressed by the Born-Meyer potential and an attractive term representing the Lennard-Jones potential:

6r

/rij ij

ij /BAe)r( −=φ ρ− (16.7)

where A, ρ and B are empirical constants, obtained by using the model to compute known crystal properties such as the lattice parameter, the elastic constants and the specific heat. Sophisticated molecular dynamic methods also include the energy contained in covalent bonds between atoms or ions. There are various techniques for minimizing computer demands by dividing the system into regions in which appropriate simplifications are made. For example, in the rigid ion model, the ions interact with one another via long-range electrostatic (i.e., Coulomb) forces. The shell model adds the polarization of the ions due to these forces. Another simplification restricts the individual ion-ion potential calculation to a certain radius around the defect. Beyond this sphere, the medium is treated as a continuum, that is, a charge distribution without discrete charges. The number of particles in the system ranges from a few thousand to millions, depending on the level of detail with which the chosen method calculates energies. Whatever mathematical simplification is employed, the equilibrium state of the crystal with a particular point defect is the one for which the sum of the pair potentials is a minimum. In addition to treating thermal systems characterized by the temperature, molecular dynamics is the standard method for treating collision cascades generated by collisions of neutrons or energetic ions such as fission fragments with the stationary ions of the crystal (Chap. 19). In this application, both potential and kinetic energies are involved.

10

The outcomes of these calculations include: - The intrinsic point defects that minimize the system energy (e.g., anion Frenkel defect in UO2) - The charge states of the defects in nonstoichiometric oxides (O/M ≠ 2) - The extent of electronic disorder (e.g., transfer of an electron from one U4+ ion to another to produce a U3+ ion and a U5+ ion) - relaxation of lattice ions surrounding a defect. - The energies and entropies of formation of defects - migration energies - the activation energies of diffusion of O2- and U4+ ions Table 16.1 shows the results of applying the molecular dynamic method to UO2. It is clear that from both experiment and modeling that the anion Frenkel pair is the lowest energy defect. The Schottky defect, although much less stable than the anion Frenkel, must be considered because it provides the cation vacancies by which uranium diffusion occurs. However, the low migration energy for anion vacancies means that O2- is much more mobile in UO2 than U4+.

Table 16.1 Energies of defect processes in UO2 from Table 17 of Ref. 16

Energy, eV* Process Experiment Calculation

Anion Frenkel 3.7±0.5 4.8 Cation Frenkel 9.0 19.4

Schottky 7.0 7.3 Anion vacancy migration 0.5 0.5

Anion interstitial migration 1.0 0.6 Cation vacancy migration 2.4 6.0

Cation interstitial migration 2.0 8.8 * 1 eV = 96.5 kJ/mole 16.2.3 Macrochemistry By macrochemistry is meant treatment of chemical equilibria without delving into the behavior of the atomic species involved, as was presented in Sect. 2.8. As an example, the equilibrium between metallic uranium, oxygen and the hypostoichiometric oxide UO2-x' is described by the reaction:

U(s) + (1-x'/2)O2(g) = UO2-x'(s) (16.8) for which the law of mass action is:

)2/'x1(O2

p]1[]1[K −= (16.9)

The activities of the two solids are unity because both are pure phases that coexist at opposite boundaries of the two-phase region of the U-O phase diagram (Fig. 16. ). The macrochemical approach to reaction (16.8) merely says that if K were known, the O2 pressure in equilibrium with the oxide at the lower boundary of this phase (O/M = 2-x') could be determined. However, this approach provides no means for knowing K, nor does it provide a method for determining

2Op in equilibrium with UO2-x where x < x' (i.e., in the region of Fig. 16.

11

labeled UO2 ± x). For this purpose, a more sophisticated approach, such as species representation proposed by Lindemer and Besmann [17], is needed. This method represents nonstoichiometric urania (or other oxide) by a solution of two species of different O/U ratios. The two components are in equilibrium with gaseous oxygen according to the "reaction":

αUO2 + O2(g) = βUaOb (16.10)

The two hypothetical species forming the solid are the "solvent" UO2 and the "solute" UaOb. In fitting the data to the model, the subscripts a and b are varied and the oxygen pressure computed as a function of the O/U ratio for several temperatures. The a,b combination hat produces oxygen pressures that best agrees with experimental results is chosen. The first step is to determine the coefficients α and β in reaction (16.10). Balancing U on the two sides of the reaction gives α = aβ. The analogous O balance is 2α + 2 = bβ. These two equations are solved for α and β in terms of a and b:

a2b2 and

a2ba2

−=β

−=α (16.11)

In analyzing UO2+x, Lindemer and Besmann tested the following "solutes": UO3; U2O5; U3O7; U10/3O23/3 and U2O9. Note that none of these "compounds" (including the UO2 "solvent") are real chemical entities. They are constructs intended to mimic the actual chemistry of the oxide. Different solute species are required for different ranges of hyperstoichiometry. 2.01 < O/U < 2.2 In this approximate nonstoichiometry range, the best fit was achieved with the "solute" U3O7. With α and β determined from Eq (16.11) for a = 3, b = 7, the equilibrium "reaction" is

6UO2 + O2(g) = 2U3O7 (16.10a) According to Eq (2.57), the equilibrium condition is given by the following relation between the chemical potentials:

2273 OUOOU 62 μ+μ=μ (16.12)

Assuming the solution to be ideal, the chemical potentials of the two uranium-oxygen components are given by Eq (2.36) with the activity of each replaced by the mole fraction. The chemical potential of O2 is given by Eq (2.44). Substituting these into Eq (16.12) yields:

7322 OUOUO

o xlnRT2plnRTxlnRT6g −+=Δ (16.13) Δgo is the standard-state free energy change of reaction (16.10a), R = 8.314 J/mole-K and T is the temperature in Kelvins. The sum of the mole fractions of the "solution" components is:

1xx732 OUUO =+ (16.14a)

In 1 mole of the "solution", there are: 732 OUUOU x3xn += moles U and

732 OUUOO x7x2n += moles O so the O/U ratio of the oxide is:

12

732

732

OUUO

OUUO

U

O

x3xx7x2

x2nn

+

+=+= (16.14b)

Solving for the two mole fractions from Eqs (16.14a) and (16.14b) produces:

x21x x and

x21x31x

732 OUUO −=

−−

= (16.14c)

Substituting these results into Eq (16.13) yields:

⎥⎦

⎤⎢⎣

⎡−−

+Δ

−Δ

= 3

2oo

O )x31()x21(xln2

Rs

RThpln

2 (16.15)

where the standard free energy change of the reaction has been expressed in terms of the corresponding enthalpy and entropy changes according to Eq (2.62). Equation (16.15) is the

⎥⎦

⎤⎢⎣

⎡−−

3

2

)x31()x21(xln2

Figure 16.5 Fit of Eq (16.15) to a compendium of data at 1500 K (Ref. 17)

13

desired result. In addition to choosing the chemical "components" that result in the stoichiometry dependence in the square brackets of Eq (16.15), fitting the model to the data involves choosing the "equilibrium" parameters Δho and Δso. Figure 16.5 shows the fit of Eq (16.15) to a compendium of UO2+x data for x > 0.01. The thermodynamic parameters are: Δho = - 316 kJ/mole and Δso = 126 J/mole-K. There is a distinct change in slope at x just a bit below 0.01. This required choosing a solute species different from U3O7, for which U2O4.5 provided the best agreement with ln

2Op vs x data. The analog of Eq (16.15) is:

⎥⎦

⎤⎢⎣

⎡−−

+−= 2O )x41()x21(x2ln4

R214

RT360pln

2

(16.16) Although not noted by the authors of Ref. 17, the data in the range 0.01 < x < 0.1 lie along a line with a slope that is less than that deduced for the U3O7 case discussed above. It will be shown in Sect. 16.4 that such slope changes has been recognized by later analyses of oxygen pressure vs stoichiometry data. PuO2-x and mixed oxides The species representation model has been applied to PuO2-x, (U,Pu)O2-x[18,19] and (U,RE)2±x[20] (RE includes the rare earths La, Eu, Dy,...Y). The significant advantage of the model is that for the two-cation systems, the proper solutes and the reaction parameters (Δho and Δso) of the single-cation oxides (e.g., the results for UO2-x and PuO2-x are directly applicable to (U,Pu)O2-x). Even for multi-cation oxides such as (U,Pu,Am)O2-x [21], the "solutes" AmO2, UO2, PuO2 and Am5/4O2 sufficed. The reactions involved in the modeling of this system were:

4Am5/4O2 + O2 = 5AmO2 3Pu4/3O2 + O2 = 4PuO2

41 Am5/4O2 + 8

5 UO2 + O2 = 1615 Am1/3U2/3O4

32 Am5/4O2 + 6

5 PuO2 + O2 = 35 Am1/2Pu1/2O3

In addition, in the ternary (U,Q)O2±x or quarternary (U,Q,W)O2±x oxides, the assumption of ideal behavior of the solid "solution" is usually not sufficient to achieve a good fit to the

2Op vs x data. Simple nonideality formulations such as regular-solution theory (Eq (2.35)) are often used for this purpose. Of course, this introduces another adjustable parameter into the analysis. 16.3 Microchemistry of nuclear oxides Between the nanochemical approach of Sect. 16.2.2 and the macrochemical treatment such as that described in Sect. 16.2.3 is a methodology that analyzes the chemical aspects of nuclear oxides at an intermediate scale. This "microchemical" method combines the atomic picture of point defects with equilibrium thermodynamics for characterizing their formation and the effect of the environment on them. The output of this method are quantitative relations between oxygen pressure, temperature and oxide stoichiometry. 16.3.1 Characteristics of point defects in nuclear oxides(See also Sects. 4.3 and 4.4) Several criteria distinguish point defects:

14

1. They are either randomly distributed in the crystal as isolated entities or the individual defects agglomerate into clusters. 2. Their concentrations are determined by thermodynamics (intrinsic) or by the presence of impurity cations (extrinsic). 3. They are present on the anion (oxygen) sublattice and/or the cation (metal) sublattice 4. Structural defects (a) a structural point defect is an ion missing from a sublattice (vacancy) or an ion in a non-regular location (interstitial). (b) the Frenkel defect is a vacancy and interstitial of the same ionic species; (c) the Schottky defect is a vacancy on both the anion and cation sublattices 5. Electrical Defects Electrical defects are cations of valences different from the normal valence in the stoichiometric compound. For example, in oxides, oxygen is always O2-and uranium is usually in the 4+ oxidation state. However, under appropriate external environments, other oxidation states of uranium appear. If the oxide is in air, U5+ and U6+ form; the product of mining of uranium ore is U3O8, in which the uranium cations consist of two U5+ for every U6+ (to balance the negative charges on O2-, viz, 2×5 + 1×6 = 8×2). The U3+ state is not as easily stabilized; exposing UO2 to high-purity hydrogen gas at high temperature produces UO2-x, but U2O3 does not exist. The ease with which nonstoichiometric oxides are stabilized is reflected in the metal-oxygen phase diagram. The U-O phase diagram shown in Fig. 16.6 shows a wide range of single-phase UO2+x ( to the right of the vertical dashed line at O/U = 2) that starts to open at ~ 500 K. Hypostoichiometric urania, UO2-x does not appear until ~ 1500 K, an indication of the less stable U3+ ion. The corresponding phase diagram for the zirconium-oxygen system is shown in Fig. 28.16. There is no hyperstoichiometric region because stable valence states greater than Zr4+ do not exist. The very thin hypostoichiometric region shown hatched at the extreme right of the diagram indicates that Zr3+ is difficult to produce. Some metal ions have only one stable state. The only stable ion of aluminum is Al3+ ; barium and strontium possess only divalent ions, and the alkali metals exist in compounds only as Na+, K+ and Cs With the exception of cerium, the rare earths (La, etc) exist only in the 3+ oxidation state. The growth of rare-earth fission products in UO2 nuclear fuel is an important example of 3+ valence cations replacing a tetravalent metal ion in an oxide (Sect. 16.4.3). +. 16.3.2 Measures of concentrations in oxides So far, five measures of defect concentrations in nonstoichiometric binary ionic solids have been introduced. With O2- as the anion and U4+ as the cation, they are: - oxygen-to-metal ratio, O/U, which appears as the abcissa in Fig 16.6 - deviation from stoichiometry For oxides such as UO2±x these are related to O/U by:

O/U = 2-x (hypo), 2+x (hyper) (16.17a)

- atom fraction (or percent) of oxygen, as on the Zr-O phase diagram of Fig. 28.16:

15

atom fraction O = )hyper(x3x2),hypo(

x3x2

++

−− (16.17b)

- Kroger-Vink Notation (Ref. 18, pp 6 - 8) This system applies to volumetric concentrations (moles or molecules per unit volume) and is fully explained in Sect. 4.3.2. For UO2, this system is written as:

••OV = doubly-positively charged anion vacancy ""

UV = vacancy on the cation (uranium) sublattice. ••••

IU = cation on a cation interstitial site. "IO = oxygen ion on an anion interstitial site

U'U = trivalent uranium ion (i.e., U3+) on a normal cation sublattice site •UU = U5+ on a cation sublattice site

UU = normal uranium ion (U4+) on a cation sublattice site OO = oxygen ion on an anion sublattice site Fixed-valence impurity ions in UO2 are designated in a similar manner:

•MQ = pentavalent cation (Q5+) on a regular cation site.

M'Q = trivalent cation (Q3+) on a regular cation site.

Fig. 16.6 The uranium-oxygen phase diagram

16

In the Kroger-Vink system, electrical charges are deviations from the normal valences on the two sublattices. Superscript dots are relative positive charges and superscript apostrophes indicate relative negative charges. The subscript letter indicates the sublattice on which the defect is located: sub O for the anion sublattice and sub U for the cation sublattice. The concentrations are denoted by the symbol enclosed in brackets. The concentrations of regular anion and cation sites are [OO]* and [UU]*, respectively, whereas the concentrations of O and U ions on these sites are [OO] and [UU], respectively. In addition to the regular sublattices, there are sublattices for anion and cation interstitials. These would usually be designated by sub IO and sub IU, respectively. However, the fluorite unit cell shown in Fig. 3.12a consists of 8 oxygen simple cubes only half of which are occupied by uranium (Fig. 3.12b shows a single empty anion-cornered cube). The empty cubes are sites for uranium interstitials, which are stabilized by the nearest-neighbor oxygen ions. The sites for O2- interstitials are probably the same as those for the U4+ ions, namely the empty cubes with corners occupied by oxygen anions. Thus the interstitial site density is denoted by [I]* for both anions and cations. The relations between the site densities are:

[I]* = [UU]* = 21 [OO]* (16.18)

- Site fraction is the fraction of the lattice sites available for a particular defect or ion that are filled with that defect. Site fractions are related to Kroger-Vink concentrations as explained in Sect. 16.3.2. The site fractions for the point defects in UO2±x are: - anion vacancy: xVO = [ ••

OV ]/[OO]* (16.19a)

- anion interstitial: xIO = [ "IO ]/[I]* (16.19b)

- cation vacancy: xVU = [""

UV ]/[UU]* (16.19c)

- cation interstitial: xIU = [ ••••IU ]/[I]* (16.19d)

- aliovalent cation xU' = [U'U]/[UU]* or xU. = [ •

UU ]/[UU]* (1619e)

- aliovalent impurity xQ' = [ U'Q ]/[UU]* or xQ. = [ •UQ ]/[UU]* (16.19f)

The Kroger-Vink concentration unit is required for writing the condition of electrical neutrality in a defected crystal. The equilibria relating the partners in a defect are usually expressed in terms of site fractions. 16.3.3 Point defects in UO2±x site-filling:

[OO]* = [OO] + [ ••OV ] (16.20a)

[UU]* = [UU] + [U'U] + [ •UU ] (16.20b)

electrical neutrality: 2[ "IO ] + [U'U] = 2[ ••

OV ] + [ •UU ] (16.21)

Dividing by the cation site density [UU]* and with the aid of Eq (16.18) and the appropriate site-fraction definitions of Eqs (16.19), Eq (16.21) equation becomes:

17

2 xIO + xU' = 4xVO + xU

. (16.21a)

composition/stoichiometry: x2*]U[

]O[]V[*]O[UO

U

"I

..OO +=+−

=

16.22a)

using Eqs (16.19a) and (16.19b), this becomes:: xx2x VOIO =− (16.22b)

Figure 16.7 shows the three principal point defects in uranium oxide. The top cube contains an anion Frenkel defect, the middle cube a Schottky defect and the lower cube represents the main electrical defect

18

16.3.4 Structural defect equilibria1 Anion-Frenkel defects in UO2 are produced by moving an anion from a regular site on the anion sublattice to a site on the interstitial sublattice:

OO = ••OV + "

IO (16.23) for which the law of mass action is:

RT/R/sIOVOFO

FOFO eexxK ε−== (16.24) where the subscript FO denotes Frenkel defects on the anion (oxygen) sublattice. R = 8.314 J/mole-K is the gas constant and T is in Kelvins. Substituting Eq (16.22b) into Eq (16.24) gives:

( )1x/K81xx 2FO4

1VO −+= and ( )1x/K81xx 2

FO21

IO ++= (16.25)

In the limit as x →0, these concentrations reduce to:

2/Kx FOVO = FOIO K2x = (16.25a) Example Estimates of the thermodynamic parameters for KFO in UO2 are sFO ~ 4 J/mole-K and εFO ~ 360 kJ/mole K. What is KFO at 1500oC and what are the site fractions of the anion point defects? Eq (16.24) becomes:

KFO =1.7exp(-43000/T) (16.24a)

at 1500oC (1773 K), Eq (16.24a) gives: KFO =5×10-11. If the oxide is exactly stoichiometric, the site fractions of anion defects from Eq (16.25a) are: xVO = 5×10-6, xIO = 1.1×10-5. Schottky defects:

UU + 2OO = ••OV2 + ''''

UV + (UU + 2OO)surface (16.26) are minor species in UO2, where anion Frenkel defects predominate. Because of the dominance of the latter, their analyses above remain valid. All that is needed to determine the cation vacancy fraction is to substitute the appropriate expression for xVO into the Schottky equilibrium constant:

VU2VOS xxK = (16.26a)

and solve for xVU. UO2 - xVO is given by Eq (16.25a) FOSVU K/K2x = (16.26b) UO2-x - xVO ≅ x/2 from Eq (16.22b) xVU = 4KS/x2 (16.26c)

UO2+x - xIO ≅ x from Eq (16.22b) xVU = x2 KS/ 2

FOK (16.26d) 1 Cation Frenkel defects are neglected; the formation energy is too large (Table 16.1)

Fig. 16.7 Electrical and structural point defects in uranium dioxide

19

16.3.5 Electronic disorder - U4+ disproportionation U4+ ions spontaneously donate an electron to a nearby cation, thereby changing the valence states of the donor and the recipient. The equilibrium constant for the reaction:

2UU = U'U + •UU (16.27)

is: .U'Udis xxK = (16.27a)

.sublatticecation on the Uand U of fractions theare xand x where 53U'U .

++

Kdis is poorly known but the best estimate is:

Kdis = 0.04exp(-17000/T) (16.27b) Example: What are the fractions of the aliovalent cations in stoichiometric UO2 at 1500 oC? For x = 0, Eq (16.22b) gives xIO = 2xVO, which, when substituted into Eq (16.21a), yields xU' = xU. From Eq (16.27b), Kdis = 3×10-6, so Eq (16.27a) gives:

3U'U 107.1 x x .

−×== 16.3.6 Gas-phase/defect equilibrium In order to arrive at a connection between stoichiometry and oxygen potential, a final relation between the point-defect concentrations and the environment is needed. This is supplied by the equilibrium between the solid and the oxygen partial pressure in the gas phase, for which the reaction is:

OO + 2UU = 21 O2(g) + 2U'U + ••

OV (16.28)

In the reverse of this reaction, an oxygen atom ( 21 O2) enters a vacant anion sublattice site ( ••

OV ) at the same time extracting an electron from each of two U3+ cations (2U'U). The result is an O2- ion on a previously-vacant anion sublattice site (OO) and two U4+ cations on regular cation sites (2UU). The equilibrium condition for the reaction is expressed by:

2UO

O2

U..o

RE ]U[]O[

p]'U[]V[K 2= (16.29)

the subscript RE means that the equilibrium constant refers to a "REDOX", or oxidation/reduction reaction. KRE is given by:

KRE = 2×106exp(-1×105/T) (16.29a)

The numerical values correspond to an entropy of reaction sRE = 120 J/mole-K and a reaction enthalpy εRE = 830 kJ/mole. Both of these large positive values correspond to the high-enthalpy, disordered (especially O2(g)) right-hand side of reaction (16.28) compared to the stable, tightly-bound normal lattice ions on the left-hand side. At 1500oC. KRE = 7×10-19.

2Op in Eq (16.29) is fixed by one of the methods described in Sect. 16.2.1. Because the point defect site fractions are < 0.002 at 1500oC (see examples above), Eqs (16.20a) and (16.20b) can be approximated by [OO]* = [OO] and [UU]* = [UU] and Eq (16.29) reduces to:

20

2OVO2

'URE pxxK = (16.29b) 16.3.7 Nonstoichiometry in UO2± x fixed by

2Op

The principal point defects in hyperstoichiometric urania are anion interstitials "IO and

pentavalent uranium, UU.. In Eq (16.29b), xVO is eliminated using Eq (16.24) and xU' is

expressed in terms of .U x via Eq (16.27a), yielding:

2OUIO pB.xx = (16.30) where:

RE2disFO K/KKB = (16.31)

In order to determine the relation between the nonstoichiometry parameter x in UO2+x and the oxygen partial pressure, a relation between xU

. and xIO is needed to accompany Eq (16.30). This

relation is supplied by Eq (16.21a) in which the same substitutions yield:

2xIO + Kdis/xU. = 4KFO/xIO

+ xU. (16.21b)

Eliminating xIO between Eqs (16.21b) and (16.30) yields.

..

2..

2

U2U

O

FO

U

dis2U

O xxpB

K4xK

x

pB2+=+ (16.32)

or:

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛±

−+−= 1

.x/K.xK32

1.x/K.x.xpB 2UdisU

FOUdisU

2U4

1O2

(16.32a)

The + sign applies when xU. > Kdis/xU., or in the hyperstoichiometric region where U5+ electrically balances the anion interstitials that represent the excess oxygen. Combining Eqs (16.21a) and (16.22b) and replacing xU' using Eq (16.27) produces:

x = 21 (xU

. - Kdis/xU.) (16.33)

and solving for xU. yields:

( )2disU x/K11x.x ++= (16.33a)

Specifying the temperature fixes the three equilibrium constants. Two applications of these equations are often encountered: Case I x is specified. What is ?p

2O xU. is

calculated from Eq (16.33a) then substituted into Eq (16.32a) for the oxygen pressure. Case II

2Op is fixed. What is x? Eq (16.32) is solved for xU

. (numerically) which is then substituted into Eq (16.33) to give x.

21

The results of the computations for these two cases are plotted as ln2Op vs log(x) in Figure

16.8. Of equal interest as the curve proper are the three regions of the variable x in which the curve can more-or-less be represented by straight lines. The literature on this topic is invariably expressed in these terms. Slope 0 - Below x ~ 10-5, the oxide is effectively stoichiometric. From Eq (16.22b) at x = 0, xIO = 2xVO, from which Eq (16.21a) gives xU' = xU

. . Equation (16.25a) is FOIO K2x = and

from Eq (16.27a), xU. = √Kdis. Substituting these into Eq (16.30) yields:

( ) atm102.2)103)(105(

)107(2KK

K2p 152611

219

2disFO

2RE

0xO2

−−−

−

=×=

×××

==

(16.34)

which corresponds to the horizontal dashed line in the lower left-hand corner of Fig. 16.8. Slope 1/2 - Following the slope 0 portion of the curve is a region 10-5 ≤ x ≤ ~6×10-4 in which the curve can be roughly represented by a line of slope 1/2. This line originates from the values of the terms in the electrical neutrality equation in the general solution for this zone. Here, xVO and xIO are much smaller than xU

. and xU', so Eq (16.21a) reduces to xU. ~ xU' = √Kdis. Equation

(16.30) becomes:

22 OORE

disFOIO p260p

KKK

xx ≅=≅

(16.34b)

Slope 1/6 - in the range ~6×10-4 ≤ x ≤ ~ 4×10-3, the oxide is sufficiently hyperstoichiometric that the only intrinsic defects of significance are the anion interstitial "

IO and the pentavalent cation .U . Substituting what remains of the charge balance (Eq (16.21a)), xU

. = 2xIO and xIO = x (Eq(16.22b)) into Eq (16.30) yields the approximation:

6/1O

6/1O

3/1

RE

2disFO

22p054.0p

K4KK

x ≅⎟⎟⎠

⎞⎜⎜⎝

⎛≅

(16.34c)

which corresponds to the dashed line in this range. The

2Opln vs stoichiometry x plots of Figs 16.5 and 16.8 are distinctly different in shape, although the data they represent are for temperatures that differ by ~ 300o. The entire curve in Fig. 16.8 corresponds to the data to the left of the dividing line in Fig. 16.5, which roughly fits:

4/1O2

px∝ . This dependence of the oxygen pressure on stoichiometry is intermediate between

those shown in Fig. 16.8. However, neither plot shows the very steep decrease in2Op as exact

stoichiometry is approached (Fig. 16.9a). The reason for this failure is the lack of modeling the chemistry (macro- or micro-) of the hypostoichiometric oxide.

22

log(x)

-6 -5 -4 -3 -2

lnp O

2

-40

-35

-30

-25

-20

-15

-10

slope 0

slope 1/2

slope 1/6

Fig. 16.8 O2 pressure in equilibrium with hyperstoichiometric urania at 1500oC Kdis = 3×10-6; KFO = 5×10-11; KRE = 7×10-19

16.4 Mixed Oxides It has long been known that the oxygen potential of mixed oxides such as (U,Th)O2+x consisting of cations of the same valence is very little different from that of UO2+x. On the other hand, the oxygen pressure over (U,La)O2+x is very different from that of UO2+x of the same x. Thorium is a very stable 4+ cation, and so does not affect the uranium valence in the mixed oxide. However, trivalent lanthanum on the cation sublattice is effectively negative (in Kroger-Vink notation) and so must be balanced by an effective positive charge. This is supplied during fabrication of the mixed oxide by conversion of some U4+ to U5+ (UU

.) or by creation of anion vacancies, ••

OV , which produces a large increase in the equilibrium oxygen pressure. The oxygen potentials of mixed oxides wherein one cation is fixed-valence and the other possesses a variable valence are functions of the latter's average valence. Moreover, the oxygen potential is independent of the fixed-valence element. Examples are: - (U,Ce)O2-x - uranium has a fixed valence of 4+ (because of the difficulty of stabilizing U3+ except at high temperature) and the cerium valence lies between 3+ and 4+[20]. - (U,Gd)O2+x - Gd has a fixed valence of 3+ and the uranium valence varies between 4+ and 5+[21]. - (U,Th)O2+x - Th valence is 4+ and U valence lies between 4+ and 5+[22]. -(U,Y)O2+x - Y3+ is mixed with either U4+ and U5+ or U4+ and anion vacancies [23]. 16.4.1 Mean valence rule

23

In the cases where the positive charge consists of a mixture of U3+, U4+ and U5+, the oxygen potentials are uniquely a function of the average valence of the uranium ions. This approximation is known as the mean valence rule [24], or valence-control rule [25,26]. Moreover, this rule applies to the pure oxide, UO2+x as well, as to mixed oxides, and herein lies its utility. Because so much effort has been expended in understanding the thermochemistry of the pure oxide, the ability to apply this knowledge base to mixed oxides is a great advantage. 16.4.2 Oxygen pressure of z2xx1 O)QU(

QQ ±−

In this general case, Q is a fixed-valence ion (either Q3+ or Q5+) mixed in the oxide at a cation site fraction of xQ. In the following development, xVO is eliminated in terms of xIO using Eq (16.24) and xU

. replaces xU' via Eq (16.27a) The cation site-filling condition that replaces Eq (16.20b) is:

1 = xQ + xU + xU

. + Kdis/xU.

(16.20c) where xU is the cation site fraction of U4+. The equation of electrical neutrality that replaces Eq (16.21b) is:

Kdis/xU. + 2xIO = 4KFO/xIO + xU

. + nxQ (16.21c) where n = +1 if the impurity ion is pentavalent and n = -1 if it is trivalent. Equation (16.22b) is

zx/K2x IOFOIO ±=− (16.22c)

Given the temperature (which fixes the equilibrium constants), the cation site fraction xQ, and the oxygen-to-metal ratio of the mixed oxide (z), the procedure for determining

2Op is as follows: 1. Solve Eq (16.22c) for xIO

( )1z/K81zx 2FO2

1IO ±+= (16.22d)

The + sign is used for z2xx1 O)QU(

QQ +− and the - sign for z2xx1 O)QU(QQ −−

2. Solve Eq (16.21c) for xU

. ( )1b/K41bx 2

dis21

U. ∓+= (16.21d) where

b = nxQ +4KFO/xIO - 2xIO (16.35) The + sign in Eq (16.21d) applies for b < 0 and the - sign for b > 0. 3. The mean uranium valence is:

24

.xxx.x5x4x3

VUU'U

UU'UU

.

.

++

++≡ (16.36)

Eliminating xU between this equation and Eq (16.20c) and expressing xU' in terms of xU. using

Eq (16.27) yields:

Q

UdisUU x1

.x/K.x4V

−−

+= (16.36a)

4. The mean-valence rule is invoked and applied to UO2±x , where xU. is determined from VU by

first solving Eq (16.36a) (with xQ removed):

⎥⎥⎦

⎤

⎢⎢⎣

⎡±

−+−= 1

)4V(K4

14V.x 2U

disU2

1U (16.37)

The + sign applies when VU > 4 and the - sign when VU < 4. 5. then determining

2Op from Eq (16.32a).

Example At 1500oC, what is the equilibrium oxygen pressure for (U0.99Q0.01)O2± z? The impurity ion is Q3+ . The equilibrium constants are given in the caption of Fig. 16.8.

Following the above steps produces the plots in Figs.16.9a- 16.9c.

The extremely rapid decrease in 2Op as O/M approaches 2.00 is characteristic of all mixed oxides

involving uranium as the major constituent. The drop-off in the hypostoichiometric region is slower than in UO2± x because of the presence of the trivalent impurity ion.

Fig. 16.9a Oxygen pressure in equilibrium with (U0.99Q0.01)O2±z at 1500oC; O/M = 2+z

25

The structural defects (anion vacancy and interstitial) are not affected by the trivalent impurity and the curves for xIO and xVO cross at O/M = 2, as they would in pure UO2 ± x. The charge defects (U3+ and U5+), however, are sensitive to the presence of Q3+, which causes the U5+ site fraction to begin increasing well before exact stoichiometry. These two point-defect curves cross at ~ 1.995.

The U valence decreases as the oxide becomes hypostoichiometric. Because of the trivalent impurity cation, VU remains > 4.00 well below O/M < 2.00.

Fig. 16.9c Mean uranium valence in (U0.99Q0.01)O2±z at 1500oC as a function of O/M = 2+z

Fig. 16.9b Point defect site fractions in (U0.99Q0.01)O2±z at 1500oC

26

16.4.3 Defects in Irradiated UO2 Irradiation of nuclear fuel (UO2) produces a vast array of fission products that interact with the remaining fuel in a very complicated way. Because of its importance to the performance of fuel elements as the burnup increases, the chemistry of irradiated fuel has been extensively studied [27 - 30]. For the purposes of this book, this complex system is simplified in order to illustrate how burnup changes the oxygen potential of the fuel. The following are neglected: 1. Molybdenum partitioning between an oxide phase (e.g. BaMoO4 - Mo is not soluble in UO2) and the noble-metal phase (with Rh, Ru, Pd) [Ref. 1, Sect. 12.4.2] 2. Oxygen tied up in ternary oxide phases, zirconates, molybdates, uranates. 3. Clusters formed by binding of oxygen vacancies to soluble fission products. 4. Blockage of lattice sites by insoluble fission products (e.g., Xe) 5. Schottky defects - consequently, the cation sublattice is structurally (but not electrically) perfect 6. Uranium Frenkel defect formation (only oxygen Frenkel defects are treated) 7. Details of the retention of fission products in solution - all soluble fps (especially the rare-earths and Zr) are lumped into a single pseudo-species with fission yield Yfp (<1) and valence Vfp (<4). The remaining fission products are insoluble in UO2 and are rejected from the solid. 8. The effect of irradiation on point-defect concentrations. This is a very significant simplification, but it permits thermodynamics to be applied. 9. Decrease of the free volume in the fuel rod due to gap closure caused by fuel swelling. 10. Reaction of oxygen released from the fuel with the cladding inner surface - The fresh fuel consists of [UU]* ions of U4+ per unit volume filling all cation lattice sites and [OO]* oxygen ions in all anion lattice sites. - F t fissions per unit volume2 replace U4+ on cation lattice sites with Yfp Ft fission-product ions, leaving (1 - Yfp) F t empty cation sites. - the remaining cations collapse to

[UU]o = [UU]* - (1 - Yfp) F t = [UU]*[1 - (1-Yfp)×β] (16.38)

structurally-perfect sublattice sites per unit volume. β is the burnup expressed as: β = *]U/[tF U (16.39)

which is the fissions per initial metal atom, or fima. The cation site fraction of soluble fission products is:

β−−

β=

−−=

)Y1(1Y

tF)Y1(*]U[tFY

xfp

fp

fpU

fpfp (16.40)

cation site-filling The cation sublattice is structurally perfect, but contains soluble fission product ions and uranium ions of valences 3+, 4+ and 5+: 1 = xfp + xU' + xU + xU

. (16.41) 2 F is fissions/s per unit volume and t is time in seconds

27

anion site-filling The crystal structure of the solid is still fluorite, so the concentration of anion sublattice sites in the collapsed structure is:

[OO]o = 2[UU]o (16.42) where [UU]o is given by Eq (16.38). Filling of the anion sublattice sites gives:

[OO]o = [OO] + [ ••OV ] (16.43)

Oxygen conservation The collapse of the cation sublattice to eliminate the empty sites results in rejection of oxygen from the solid. This oxygen reappears in the gas phase of the free volume in the fuel element:

[OO]* = [OO] + [ "IO ] + [Og] (16.44)

where Og represents gas-phase oxygen (actually as O2). Its concentration is negligibly small and Og can henceforth be omitted from the oxygen balance. Eliminating [OO] by Eq (16.43), [OO]o by Eq (16.42) and [UU]o by Eq (16.38) yields:

2 = 2[1 - (1-Yfp)×β] - [ ••OV ]/[UU]* + [ "

IO ]/[UU]* (16.45) The site fraction of vacancies on the anion sublattice is

β×−−===

)Y1(1*]U/[]V[

]U[2]V[

]O[]V[

xfp

U..O

21

oU

..O

oO

..O

VO

and the site fraction of anion interstitials on the interstitial sublattice is:

β×−−==

)Y1(1*]U/[]O[

]U[]O[x

fp

U''I

oU

''I

IO

Using the above two equations and Eq (16.24) in Eq (16.45) yields:

1 = (1 - KFO/xIO + 21 xIO)[1 - (1-Yfp)×β] (16.46)

Electrical neutrality In Kroger-Vink concentration notation, electrical neutrality in the irradiated fuel is: 2[ ••

OV ] + [UU.] = (4-Vfp) [fp] + [UU'] + 2[ "

IO ] (16.47) or, with Eqs (16.24) and (16.27a):

4KFO/xIO + xU. = (4-Vfp)xfp + Kdis/xU

. + 2xIO (16.48)

xVO and xU' are expressed by Eqs (16.24) and (16.27a), respectively. The solution is straightforward. Equation (16.46) is solved for xIO as a function of burnup β and the result used in Eq (16.48) to calculate xU

.. The oxygen pressure is given by Eq (16.30). Figure 16.10 shows these results plotted as functions of burnup (bu), expressed in the MWd/kgU, units more common than fima for LWRs. β and bu are related by: β= 1.22×10-3bu.

28

Fig. 16.10 Point defect site fractions and oxygen pressure as functions of burnup.

Equilibrium constants from the caption of Fig. 16.8

It is clear from this graph that fission is an oxidizing process, although because of neglecting O2 reaction with the cladding inside surface, the buildup of

2Op on the graph is exaggerated. The relative negative charges introduced by the fission products (xfp) and the anion interstitials (xIO) together electrically neutralize the positively-charged pentavalent uranium (xU

.).

16.5 Point-defect clustering in UO2+x

In 1964, Willis' neutron-diffraction measurements [13] on single-crystal UO2.12 at 800oC definitively showed that anion vacancies and interstitials existed in distinct complexes, or clusters, rather than as individual point defects scattered randomly in the fluorite lattice, as has been assumed in the preceding sections. Subsequently, the structure of the cluster was shown [14] to be that depicted in Fig. 16.11. This cluster contains two oxygen interstitial ions, which are labeled "2" in Fig. 16.11. The two ions are located along a <110> direction that passes through a side of the inner cube in the diagram. Because of the close packing of anions in this region, the two closest lattice anions ("1") move away from the interstitials along <111> directions. Simultaneously, four nearby U4+ ions are oxidized to U5+ to balance the negative charges introduced by the two oxygen interstitials. The cluster consists of 4U. +2 OV +2 1IO +2 2IO point defects. This cumbersome combination is more conveniently denoted by (4:2:2:2). An intrinsic equilibrium that relates the cluster concentration to those of the other point defects is:

4UU. + 2OO +2 "

IO = (4:2:2:2) (16.49) The four U5+ ions supply this component of the cluster; the two normal anions (OO) become the No. 2 interstitials and the two anion interstitials enter the cluster without change.

xU'

xU.

29

The next equilibrium involving the cluster involves extraction of an electron from a neighboring U4+ ion:

UU + (4:2:2:2) = (4:2:2:2)' + UU. (16.50)

Unlike the structure of the cluster deduced by Willis, the electrical characteristic contained in the above equilibrium has not been determined by experiment; rather, as shown below, it is needed to produce the observed dependence of the equilibrium oxygen pressure on stoichiometry. Adding the preceding two reactions gives:

UU + 3UU. + 2OO +2 "

IO = (4:2:2:2)' (16.51) for which the law of mass action (in terms of site fractions) is:

2IO

3U

'WW xx

xK.

= (16.52)

Fig. 16.11 Willis (4:2:2:2) cluster

30

where xW is defined as the concentration of defects divided by the cation site density:

*]U[])'2:2:2:4[(x

U'W =

(16.53)

Expressing the equilibrium that determines the cluster concentration by Eq (16.51) does not imply that this reaction is the actual mechanism of cluster formation. For example, an alternative formation reaction is [31- 34]:

2OO + O2 + 5UU = (4:2:2:2)' + UU. (16.54)

In this formulation of the equilibrium, the No. 2 oxygen interstitials in the cluster are obtained directly from gaseous oxygen. The law of mass action for this reaction is given by:

2O

U'WD p

.xxK = (16.55)

The connection between the equilibrium constants KW and KD is obtained by adding to reaction (16.51): twice reaction (16.23):

2OO = 2 "IO + 2 ••

OV 2×(16.23) and 4 times reaction (16.27):

8UU = 4UU' + 4UU. 4×(16.23)

and subtracting twice reaction (16.28):

2 ••OV + 4UU + O2 = 2OO + 4UU -2×(16.28)

This combination produces reaction (16.54). The corresponding relation between the equilibrium constants is:

4dis

2FO

D2RE

W2RE

4dis

2FOW

D KKKKK or

KKKK

K == (16.56)

Using the values of KRE, KFO and Kdis that were employed in the preceding analysis of point defects in the absence of clustering and an approximate value of KD ~ 105 at x = 0.1 and 1500oC [Ref. 31, Fig.3] gives:

KW ~ 1146211

5219

102)103()105(

)10()107(×=

×××

−−

−

(16.56a)

In the following, conversion of concentrations [i] to site fractions xi employs the identifications [OO] ≈ [OO]* and [UU] ≈ [UU]*. These approximations are justified because the concentrations of the point defects are small compared to those of the normal ions. The anion site-filling condition that modifies Eq (16.20a) is:

[OO]* = [OO] + [ ••OV ] +2 [(4:2:2:2)'] (16.57)

The last term on the right accounts for the two vacant anion sites in the cluster. The total concentration of oxygen ions in the solid is:

31

[O]tot = [OO] + [ "IO ] +4[(4:2:2:2)'] (16.58)

The last term on the right accounts for the four interstitials in the cluster. The cation sublattice is perfect (structurally). In the presence of clusters, hyperstoichiometry is expressed by: 2+x = [O]tot/[UU]* or, eliminating [OO] from the above two equations, Eq (16.22b) becomes:

x = xIO - 2KFO/xIO + 2xW' (16.59)

The condition of electrical neutrality that replaces Eq (16.21a) (neglecting the U3+ contribution) is:

2KFO/xIO + xU. = 2xIO + xW'' + Kdis/xU

. (16.60) in which xVO has been eliminated by Eq (16.24) and xU' by Eq (16.27a). Simultaneous solution of Eqs (16.52), (16.59) and (16.60) yield xW', xIO and xU

. as functions of the nonstoichiometry variable x. These functions are displayed in Fig. 16.12.

Fig. 16.12 Point defects in hypostoichiometric UO2 at 1500oC

This plot reveals how far towards stoichiometric UO2 the cluster survives before decomposing into individual anion vacancies. The concentration of the [4:2:2:2] cluster drops rapidly with decreasing x while the anion vacancy site fraction xIO slowly grows. The two curves cross at x ~ 10-3, thereby attesting to the tenacity of the cluster. At about the same concentration, xIO reaches a maximum and begins to decrease as well. This is due to the reduced need to provide negative charges to balance the U5+ concentration, which as shown by the dashed curve in the figure, is also decreasing.

32

16.6 Properties of UO2 dependent upon point defects Anion Frenkel defects greatly influence the diffusivity of oxygen ions in UO2. The vacancies on the cation sublattice produced by the Schottky process are responsible for the diffusion coefficient of uranium ions in UO2. Another property affected by the creation of point defects in ionic crystals is the heat capacity. At relatively low temperatures (but not approaching absolute zero), the heat capacity is nearly temperature-independent with a value of 3R per gram atom (R is the gas constant). Per mole of UO2, this classical value of the heat capacity is (CV)lattice = 9R. This contribution to the heat capacity arises from the vibrations of the atoms in the lattice. The creation of structural point defects by the anion Frenkel process (Eq (16.23)) provides an additional component to the heat capacity, as does the electrical defect arising from disproportionation of U4+ (Eq (16.27)):

CV = (CV)lattice + (CV)struct+ (CV)elect. (16.63)

The extra energy in the solid due to creation of the anion Frenkel defects is the concentration times the energy εFO (J/mole of defects ). The excess energy due to the point defects is:

eFO = xIO×εFO = εFO× FOK2 J /mole UO2

The excess heat capacity arising from the point defects is the derivative of eFO with respect to temperature, or, using Eq (16.24) for KFO:

( )FO

2FOFOstructV K

RT21

dTde

R1

RC

⎟⎠⎞

⎜⎝⎛ ε== (16.64)

with εFO = 360 kJ/mole and KFO given by Eq (16.24a). The analogous contribution from U4+ disproportionation is:

( )dis

2disdis.electV K

RTdTde

R1

RC

⎟⎠⎞

⎜⎝⎛ ε==

(16.65) where εdis = 140 kJ/mole and Kdis is given by Eq (16.27b). The two defect contributions and the total heat capacity are shown in Fig. 16.13. The electrical defect contribution is larger than the structural component, but with the equilibrium constants used here, together they add very little to the classical heat capacity. At 3000 K, the combined effect is only 7% of the total.

33

Fig. 16.13 Point defect components of the specific heat of UO2