Comprehensive nuclear materials 3 20 modeling of fission gas induced swelling of nuclear fuels

Comprehensive nuclear materials 3 20 modeling of fission gas induced swelling of nuclear fuels Comprehensive nuclear materials 3 20 modeling of fission gas induced swelling of nuclear fuels Comprehensive nuclear materials 3 20 modeling of fission gas induced swelling of nuclear fuels Comprehensive nuclear materials 3 20 modeling of fission gas induced swelling of nuclear fuels Comprehensive nuclear materials 3 20 modeling of fission gas induced swelling of nuclear fuels Comprehensive nuclear materials 3 20 modeling of fission gas induced swelling of nuclear fuels

3.20 Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels* J Rest Argonne National Laboratory, Argonne, IL, USA ß 2012 Elsevier Ltd All rights reserved 580 3.20.1 Introduction 3.20.2 Intragranular Bubble Nucleation: Uranium-Alloy Fuel in the High-Temperature Equilibrium g-Phase Introduction A Multiatom Nucleation Mechanism Calculation of the Fission-Gas Bubble-Size Distribution Bubble Coalescence Analysis of U–10Mo High-Temperature Irradiation Data Conclusions Intergranular Bubble Nucleation: Uranium-Alloy Fuel in the Irradiation-Stabilized g-Phase Introduction Calculation of Evolution of Average Intragranular Bubble-Size and Density Calculation of Evolution of Average Intergranular Bubble-Size and Density Calculation of Intergranular Bubble-Size Distribution Comparison Between Model Calculations and Intragranular Data Comparison Between Model Calculations and Intergranular Data Calculation of Gas-Driven Fuel Swelling Safety Margins Conclusions Irradiation-Induced Re-solution Introduction Flux Algorithm Grain-Boundary-Bubble Growth Analysis of Bubble Growth on Grain Boundaries Discussion and Conclusions Irradiation-Induced Recrystallization Introduction Model for Initiation of Irradiation-Induced Recrystallization Model for Progression of Irradiation-Induced Recrystallization Theory for the Size of the Recrystallized Grains Calculation of the Cellular Network Dislocation Density and Change in Lattice Parameter Calculation of Recrystallized Grain Size Evolution of Fission-Gas Bubble-Size Distribution in Recrystallized U–10Mo Fuel Effect of Irradiation-Induced Recrystallization on Fuel Swelling Discussion and Conclusions Final Thoughts 3.20.2.1 3.20.2.2 3.20.2.3 3.20.2.4 3.20.2.5 3.20.2.6 3.20.3 3.20.3.1 3.20.3.2 3.20.3.3 3.20.3.4 3.20.3.5 3.20.3.6 3.20.3.7 3.20.3.8 3.20.4 3.20.4.1 3.20.4.2 3.20.4.3 3.20.4.4 3.20.4.5 3.20.5 3.20.5.1 3.20.5.2 3.20.5.3 3.20.5.4 3.20.5.5 3.20.5.6 3.20.5.7 3.20.5.8 3.20.5.9 3.20.6 References *The submitted manuscript has been authored by a contractor of the US Government under contract NO W-31-109-ENG-38 Accordingly, the US government retains a nonexclusive royaltyfree license to publish or reproduce the published form of this contribution, or allow others to so, for US Government purposes 581 581 582 584 586 586 591 591 591 591 593 593 595 596 597 599 601 601 601 603 604 608 610 610 610 611 614 614 616 620 621 624 625 625 Abbreviations ATR EOS PIE Advanced test reactor Equation of state Postirradiation examination 579 580 Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels RERTR Reduced Enrichment for Research and Test Reactors SEM Scanning electron microscope TEM Transmission electron microscope 3.20.1 Introduction This chapter addresses various aspects of modeling fission-gas-induced swelling in both oxide and metal fuels The underlying theme underscores the similarities and differences in gas behavior between these two classes of nuclear materials The discussion focuses more on a description of key mechanisms than on a comparison of existing models Three interrelated critical phenomena that dominate fission-gas behavior are discussed: the role of intra- and intergranular gas-bubble nucleation, irradiation-induced re-solution, and irradiation-induced recrystallization on gas-driven swelling in these materials The results of calculations are compared to experimental observations A clarifying comparison of existing models is clouded by the fact that many of the models employ different values for critical parameters and materials properties This condition is fueled by the difficulty in measuring these quantities in a multivariate irradiation environment Examples of such properties are gas-atom and bubble diffusion coefficients, bubble nucleation rates, re-solution rate, surface energy, defect formation and migration enthalpies, creep rates, and so on The behavior of fission gases in a nuclear fuel is intimately tied to the chemical and microstructural evolution of the material The complexity of the phenomena escalates when one considers the possibility that microstructure is dependent on the fuel chemistry Some of the key behavioral mechanisms, such as gas-bubble nucleation, are affected by fuel microstructure Likewise, mechanisms such as the diffusion of gas atoms and irradiation-produced defects are affected by fuel chemistry Thus, a realistic description of the phenomena entails an accurate representation of the evolving fuel chemistry and microstructure A simple example of this is the dependence of fission-gas release on the grain size: the larger the grains, the lower the fractional release at a given dose On the other hand, grain growth occurs as a result of time at temperature as well as by irradiation effects and fuel chemistry (e.g., stoichiometry) As the grain boundaries move, they encounter fission products and gas bubbles that impede their motion All aspects of this synergistic process need to be accounted for and modeled correctly in order to obtain a model that can accurately predict fission-gas release On a different level, below temperatures at which defect annealing occurs, at relatively high doses, fuel materials such as UO2 and uranium alloys such as U–10Mo undergo irradiation-induced recrystallization wherein the as-fabricated micron-size polycrystalline grains are transformed to submicron-sized grains As a result of this transformation, fission gases are moved from within the grain to the grain boundaries, transferring the materials response to gasdriven swelling from intragranular to intergranular In addition, gas-bubble/precipitate complexes can act as pinning sites that immobilize potential recrystallization nuclei, and thus affect the dose at which recrystallization is initiated The synergy between these different forces needs to be realistically captured in order to accurately model the phenomena Given the current uncertainties in materials properties, critical parameters, and proposed behavioral mechanisms, a key issue in modeling of fission-gas behavior in nuclear fuels is realistic validation In general, most of the model validation is accomplished by adjusting/predicting these properties and parameters to achieve agreement with measured gas release and swelling, and with mean values of the bubble-size distribution However, the uncertainties in these properties and parameters generate an inherent uncertainty in the validity of the underlying physics and the physical reality of proposed behavioral mechanisms This inherent uncertainty clouds the predictive aspects of any mechanistic approach to describe the phenomena Thus, more detailed data are required to help clarify these issues The shape of bubble-size distribution data contains information on the nature of the behavioral mechanisms underlying the observed phenomena that are not present in the mean or average values of the distribution This is due to information contained in the first and second derivates of the bubble density with respect to bubble size Literature descriptions of measured intragranular bubble-size distributions are few and far between, and measured intergranular bubble distributions are all but nonexistent In Sections 3.20.2 and 3.20.3, recently measured intra- and intergranular bubble-size distributions obtained from U–Mo alloy fuel are used for model validation, and the robustness of this technique in reducing uncertainties in proposed mechanisms and materials properties as compared Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels to employing average values is underscored In this regard, it will be shown in Section 3.20.3 that a substantial increase in validation leverage is secured with the use of bubble-size distributions compared with the use of mean values The results of a series of calculations made with paired values of critical parameters, chosen such that the calculation of average quantities remains unchanged, demonstrate that the calculated distribution undergoes significant changes in shape as well as position and height of the peak As such, a capacity to calculate bubble-size distributions along with the availability of measured distributions goes a long way in validating not only values of key materials properties and model parameters, but also proposed fuel behavioral mechanisms Sections 3.20.2 and 3.20.3 contain discussions of gas-bubble nucleation in the high-temperature equilibrium g-phase, and in the low-temperature irradiation-stabilized g-phase of uranium alloy fuel, respectively The connection between these regimes is that while intergranular multiatom nucleation appears to dominate at low temperature, intragranular multiatom nucleation is the dominant nucleation mechanism at high temperature Although the discussion on gas-bubble nucleation focuses on uranium alloys (because of the availability of measured bubble-size distributions), there is no reason to believe that they would not be applicable to oxide fuel as well Section 3.20.4 presents an analysis of irradiationinduced re-solution Specifically, the analysis presents a rationale for why gas-atom re-solution from grain-boundary bubbles is a relatively weak effect as compared to that for intragranular bubbles One of the arguments is that intergranular bubble nucleation results in bubble densities that are far smaller than observed in the bulk material For example, an intergranular bubble density of  1013 mÀ2 is equivalent to a bubble density of  1018 mÀ3 for a grain size of  10À6 m This is to be compared to observed intragranular bubble densities that are on the order of 1023 mÀ3 In addition, typical intergranular bubble sizes of tenths of a micron are to be compared to nanometer-sized intragranular bubbles This consideration is supported not only by the experimental results presented in Section 3.20.3, but also by the results of the multiatom nucleation theory that form the basis of the analysis Finally, in Section 3.20.5, models for the initiation and progression of irradiation-induced recrystallization are reviewed, and a theory for the size of the recrystallized grains is discussed The role of bubble nucleation and gas-atom re-solution in the 581 recrystallization story is clarified Calculations are compared to data for the dislocation density and change in lattice displacement in UO2 as a function of burnup In addition, calculations are compared to available data for the recrystallized grain-size distribution in UO2 and in U–10Mo Models such as those described in this chapter are stories that remain just stories until validated by experiment Fission-gas behavior in nuclear fuels has been studied since the early 1950s, and although almost 60 years have elapsed, a definitive picture of these phenomena is still unavailable today The reason, as stated above, is the difficulty in obtaining reliable single-effects data in a multivariate irradiation environment, coupled with the highly synergistic nature of the beast 3.20.2 Intragranular Bubble Nucleation: Uranium-Alloy Fuel in the High-Temperature Equilibrium g-Phase 3.20.2.1 Introduction Figure shows scanning electron microscope (SEM) micrograph of g-U–Zr–Pu alloy fuel.1 The microstructure shown in Figure is typical of most 10.0 mm Figure Scanning electron microscope micrograph of g-U–Zr–Pu alloy fuel Reproduced by permission of the experimentor, G L Hofman 582 Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels uranium metal alloys irradiated in the equilibrium g-phase Swelling of this material is predominantly due to the growth of fission-gas bubbles Its fissiongas behavior is characterized by high mobility at relatively high temperatures at which it exists at the equilibrium g-U–Zr–Pu phase As seen in Figure 1, the bubbles in this material comprise a relatively broad size range Some of the larger bubbles have a sinuous plastic-like appearance, indicative of high mobility A number of coalescence events are apparent, and some of the larger bubbles appear to be growing into the smaller neighboring bubbles Most attempts at describing intragranular gas-bubble nucleation in nuclear fuels at higher temperatures have relied on a homogeneous2 or heterogeneous3 two-atom mechanism In general, it is assumed that two atoms that come together in the presence of vacancies or vacancy clusters become a stable nucleus At lower temperatures, because of the relatively strong effect of irradiation-induced resolution, the number of nucleated bubbles increases due to the increase in the effective gas generation rate.4 In theory, the number of nuclei will increase until newly created gas atoms are more likely to be captured by an existing nucleus than to meet other gas atoms and form new nuclei.2 In practice, because of the coarsening of the bubble-size distribution, the two-atom nucleation process continues throughout the irradiation If both bubble motion and coalescence are neglected, the rate equation describing the time evolution of the density of gas in intragranular bubbles is given by dẵmb t ịcb t ị ẳ 16pfn Dg rg cg t ịcg t ị dt ỵ 4prb t ịDg cg ðt Þcb ðt Þ À bmb ðt Þcb ðt Þ ½1Š where cg , cb are the densities of gas atoms and bubbles, respectively, mb is the average number of gas atoms per bubble, Dg is the gas-atom diffusion coefficient, b is the gas-atom re-solution rate from bubbles, and fn , the so-called nucleation factor, is the probability that two gas atoms that come together stick long enough to form a stable bubble nucleus Often, fn is interpreted as the probability that there are sufficient vacancies or vacancy clusters in the vicinity of the two-atom to form a stable nucleus For example, for heterogeneous bubble nucleation along fission tracks in UO2, fn is approximately the average volume fraction of fission tracks %10À4 The three terms on the right-hand-side of eqn [1] represent, respectively, the change in the density of gas in intragranular bubbles because of bubble nucleation, gasatom diffusion to bubbles of radius, rb , and the loss of gas atoms from bubbles because of irradiationinduced re-solution An implicit assumption in eqn [1] is that once a two-atom nucleus forms, it grows instantaneously to an m-atom bubble Values of fn ranging from 10À7 to 10À2 have been proposed, which makes the nucleation factor little more than an adjustable parameter.5 A substantial contribution to the spread of reported values for fn is that most models describe the time evolution of mean values of cb and rb which are compared to the respective mean values of the measured quantities (comparing model predictions with average quantities is by far the dominant validation technique reported in the literature) In this regard, as will be demonstrated in the following section, the use of bubble-size distributions goes a long way toward the reduction of such uncertainties.6 As an approach to circumventing the deficiencies thus described, in what follows a multiatom bubble nucleation mechanism is proposed and implemented into a mechanistic calculation of the intragranular fission-gas bubble-size distribution The results of the calculations are compared to a measured bubblesize distribution in U–10Mo irradiated at relatively high temperature to 4% U-atom burnup The multiatom nucleation model is compared to the two-atom model within the context of the data, and the implications of each mechanism for the observable quantities are discussed In the next section, a multiatom nucleation mechanism is formulated Section 3.20.3 presents an outline for a calculation of the time evolution of the bubble-size distribution In Section 3.20.4, a discussion is presented of processes that lead to coarsening of the as-nucleated bubble distribution In Section 3.20.5, model calculations are used to interpret a measured distribution in U–8Mo uranium alloy fuel irradiated to 4% U burnup at 850 K In addition, in this section, a comparison between the multiatom and two-atom nucleation mechanisms is attempted Finally, conclusions are presented in Section 3.20.6 3.20.2.2 A Multiatom Nucleation Mechanism Fission gases Xe and Kr are generated in a nuclear fuel at a rate of about 0.25–0.30 atoms per fission as a result of decay of the primary fission products About seven times more Xe is produced than Kr These gas Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels atoms are very insoluble in the fuel in that they not react chemically with any other species Thus, left in the interstices, because of their relatively large size, they produce a strain in the material In order to lower the energy of the system and to minimize the strain, the gas atoms tend to relocate in areas of decreased density, such as in vacancies and/or vacancy clusters For example, in UO2, gas atoms have been calculated to sit in neutral trivacancy sites consisting of two oxygen ions and one uranium ion.7 Given enough energy via thermal fluctuations, and/or via irradiation, the gas atoms can hop randomly from one site to another and thus diffuse through the material The gas atom/vacancy complexes can combine forming clusters of gas atoms and vacancies If enough gas atoms come together, they become transformed into a gas bubble which, under equilibrium conditions, sits in a strain-free environment This process of forming gas bubbles is termed gas-bubble nucleation According to phase transition theory, at relatively large supersaturations, a system transforms not by atom-to-atom growth, but simultaneously as a whole In other words, the system is unstable against transformations into a low free energy state, and the new phase will have a certain radius defined by the supersaturation Solubility of rare-gas atoms in uranium alloys or ceramics is so low that it has not been measured In perfect crystals, the order of magnitude of the solubility has been estimated to be 10À10.8 This figure may be increased up to %10À5 in the vicinity of dislocations In addition, there may be a substantial effect from gas in dynamic solution, that is, as a result of irradiation-induced re-solution Thus, in regions of nuclear fuels that are near irradiation-produced defects and/or various microstructural irregularities, the solubility of the gas can be substantially higher than in the bulk material The gas concentration in these regions will increase until the solubility limit is reached, whereupon the gas will precipitate into bubbles Subsequently, nucleation is limited because of the gas concentration in solution falling below the solubility limit The trapping of the gas by the nucleated bubble distribution damps the increase in gas concentration Eventually, the gas in solution may reach the solubility limit at which time the nucleation event repeats Thus, assuming that all the gas precipitates into bubbles of equal size r 0, the concentration of gas in the bubble at nucleation is given by mr ị ẳ bv cgcrit 4=3pr 03 cb r ị ẵ2 583 where cgcrit is the concentration of gas at the solubility limit, bv is the volume per atom (van der Waals constant), and cb ðr Þ is the concentration of bubble nuclei at the unrelaxed radius r 0, that is, the initial stage of bubble nucleation is a volume-conserving process Subsequently, in order to lower the free energy of the system, the overpressurized nuclei relax by absorbing vacancies until the bubbles reach equilibrium At equilibrium, the bubble radius is r and, in the absence of significant external stress, the pressure in the bubble is given by 2g ½3Š r where g is the surface energy per unit area If it is assumed that the average gas bubble size r is a function of the equilibrium bubble size, then differentiating eqn [2] with respect to the equilibrium radius r and rearranging terms yields Pe ẳ dcb r ị dmr Þ dr ¼À À 0 cb ðr Þ dr mðr Þ dr r dr ½4Š Let us assume that during the relaxation phase there is no interaction between the nucleated bubbles, that is, r ! r ; mr ị ! mr ị ẳ mr Þ; cb ðr Þ ! cb ðr Þ ẳ cb r ị ẵ5 The nucleation problem thus consists of determining the two terms on the RHS of eqn [4] The first term on the RHS of eqn [4] can be determined from the equation of state (EOS), the capillarity relation, and the conditions expressed in eqn [5] Using the van der Waals EOS, PðV À mbv Þ ¼ mkT ½6Š where V ¼ 4/3pr is the bubble volume Recognizing that at nucleation the bubble size is small such that 2g/r ) s , where s is the external stress, and differentiating eqn [6] with respect to the equilibrium radius r one obtains ! dmr ị rkT ẵ7 ẳ mr ị dr r 3rkT ỵ 2gbv ị The remaining term on the RHS of eqn [4] can be determined by invoking energy minimization as the driving force for bubble equilibration The change in the Gibbs free energy due to bubble expansion is given by ÁG ¼ pr 03 ÁGv þ 4pr 02 g ½8Š 584 Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels where ÁGv is the free energy driving bubble equilibration, which, in analogy with the treatment of the nucleation of liquid droplets in a vapor,9 can be expressed as kT ẵ9 lnPr ịe =Pr ịị O where O is the atomic volume The critical bubble radius at equilibrium is given by the condition ÁGv ¼ @ÁG À2g ¼ ! r ¼ rcrit ¼ @r ÁGv ½10Š Inserting the expressions for Pe and P from eqns [3] and [6], respectively, into eqn [10], differentiating with respect to the bubble radius r, and applying a little algebra results in   dr O kT ¼À þ ½11Š 4pr 02 dr X r 2g 3.20.2.3 Calculation of the Fission-Gas Bubble-Size Distribution where X¼ kTr   03 2g pr À mbv ½12Š Making use of eqn [2] in eqn [11] results in   ! dr m2 c b 2gO=rkT rkT dm kT O bv dm ẳ ỵ ỵ e bv c g m dr r dr 2gm dr 2g r ½13Š Finally, substituting eqn [5], eqn [7], and eqn [13] into eqn [4] yields dcb dm mcb ẳ ỵ cb dr m dr bv cg   ! rkT dm mkT mO dm ỵ ỵ bv e2gO=rkT 2gm dr 2g r dr Definition of variables in eqn [15], The model consists of a set of coupled nonlinear differential equations for the intragranular concentration of fission product atoms and gas bubbles of the form10 dCi ¼ Ci Ci bi Ci ỵ ci i ẳ 1; ; N ị ẵ15 dt where Ci is the number of bubbles in the ith size class per unit volume; and the coefficients , bi , and ci obey functional relationships of the form ¼ ðCi Þ bi ¼ bi ðC1 ; ; Ci1 ; Ciỵ1 ; ; CN ị ẵ14 The as-nucleated bubble-size distribution is then obtained by the simultaneous solution of eqns [7] and [14] Table Subsequent to the nucleation event, the asnucleated bubble-size distribution evolves under the driving forces of gas diffusion to bubbles, gas-atom re-solution from bubbles, and bubble coalescence due to bubble–bubble interaction via bubble motion and geometrical contact As stated earlier, additional nucleation events are delayed because of the gas in solution remaining below the solubility limit, as the gas generated by continuing fission events is trapped within the existing bubble-size distribution This last point is facilitated by the relatively high gas-atom diffusivities at the temperatures of interest (i.e., those under which the equilibrium g-phase of the alloy exists) Eventually, the gas in solution may again reach the solubility limit at which time the nucleation event repeats The variables in eqn [15] are defined in Table represents the rate at which bubbles are lost from (grow out of) the ith size class because of coalescence with bubbles in that class; bi represents the rate at which bubbles are lost from the ith size class because of coalescence with bubbles in other size classes and dCi ¼ Àai Ci Ci bi Ci ỵ ci i ẳ 1; ; NÞ dt i Ci aiCiCi biCi ci Concentration of intragranular gas atoms 2, .,N Concentration of intragranular gas bubbles Rate of gas atom loss due to gas-bubble nucleation Rate of gas bubble loss due to bubble coalescence with bubbles within the same size class Rate of gas atom loss due to diffusion into gas bubbles Rate of gas bubble loss due to coalescence with bubbles in other size classes Rate of gas atom gain due to atom re-solution and fission of uranium nuclei Rate of gas bubble gain due to bubble nucleation and coalescence, and diffusion of gas atoms into bubbles Source: Rest, J J Nucl Mater 2010, 402(2–3), 179–185 Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels re-solution; and ci represents the rate at which bubbles are being added to the ith size class because of fissiongas generation, bubble nucleation, bubble growth resulting from bubble coalescence, and bubble shrinkage due to gas-atom re-solution The bubbles are classified by an average size, where size is defined in terms of the number of gas atoms per bubble This method of bubble grouping significantly reduces the number of equations needed to describe the bubble-size distributions The bubble classes are ordered so that the first class refers to bubbles that contain only one gas atom If Si denotes the average number of atoms per bubble for bubbles in the ith class (henceforth called i-bubbles), then the bubble-size classes are defined by 585 given by Tijk ẳ Skỵ1 ỵ Si Sj Sj ẳ1 Skỵ1 Sk Skỵ1 Sk ½20Š and the probability that the coalescence will result in a k ỵ bubble is given by Tijkỵ1 ẳ S i ỵ S j Sk Sj ẳ Skỵ1 Sk Skỵ1 Sk ẵ21 The array Tijk may be considered the probability that an i-bubble will become a k bubble as a result of its coalescence with a j-bubble The rate Nijk at which i-bubbles become k bubbles is given by X ij Tijk ẵ22 Nijk ẳ j i Si ẳ nSi1 ẵ16 p where the integer n ! 0:5 ỵ 1:25, i ! 2, and Si ¼ The i ¼ class is assumed to consist of a single gas atom associated with one or more vacancies or vacancy clusters In general, the rate of coalescence Àij of i-bubbles with j-bubbles is given by Àij ¼ Pij Ci Cj ½17Š where Pij is the probability in m3 sÀ1 of an i-bubble coalescing with a j-bubble For i ẳ j, ij becomes ẵ18 ii ẳ Pii Ci Ci so that each pair-wise coalescence is counted only once Coalescence between bubbles results in bubbles growing from one size class to another The probability that a coalescence between an i-bubble and a j-bubble will result in a k bubble is given by the array Tijk The number of gas atoms involved in one such coalescence is Si ỵ Sj The array Tijk is defined by three conditions: P k Tijk ¼ (the total probability of producing a bubble is unity) P k Tijk Sk ẳ Si ỵ Sj (the number of gas atoms, on average, is conserved) For a given pair ij, only two of the Tijk array elements are nonzero These elements correspond to k and k ỵ 1, where Sk Si ỵ Sj Skỵ1 The j-bubble is assumed to disappear because gas atoms are absorbed into the i-bubble The rate of disappearance wj is given by X Àij ½23Š wj ¼ j !i The rate Nik at which i-bubbles become k bubbles, with k ẳ i ỵ 1, is reduced by various processes such as the re-solution of gas atoms Re-solution is the result of collisions (direct and/or indirect) between fission fragments and gas bubbles From eqns [21] and [22], X Àij Tijk Nik ¼ j i ¼ X j i ¼ Sj S k À Si X Pij Cj Sj Pij Ci Cj Ci S k À Si ½24Š j i P The expression j i Pij Cj Sj is the rate at which gas atoms are added to an i-bubble Re-solution causes an i-bubble to lose gas atoms at a rate given by bi Si, where bi is the probability that a gas atom in an i-bubble is redissolved The reduced Nik becomes Ci X ðPij Cj Sj bi Si ị ẵ25 Nik ẳ S k À Si j i ½19Š If the expression within the parentheses is negative, then Nik is zero, and Nik , the rate at which i-bubbles become i À bubbles, with k ¼ i À 1, is defined as ! X Ci Nik ¼ bi S i À Pij Cj Sj ½26Š S i À Sk j i Thus, the probability that a coalescence between an i-bubble and a j-bubble will result in a k bubble is Equations [25] and [26] are proportional to the probabilities that any particular i-bubble will become an From these three conditions, it follows that k ¼ i, and Tijk Sk ỵ Tijk ịSkỵ1 ẳ Si ỵ Sj 586 Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels i ỵ or an i À bubble, respectively; the ratio of the probabilities is equal to the ratio of the rates The aforementioned definition of Nik and Nik is consistent with the conservation of the total number of gas atoms 3.20.2.4 Bubble Coalescence gc The bubbles are assumed to diffuse randomly through the solid alloy by a volume diffusion mechanism The bubble diffusion coefficient Di of a bubble having radius Ri is given by Di ¼ 3a03 Dvol 4pR3i j 6¼i The interaction cross-section represented in eqn [28] is based on an analysis of colloidal suspensions within the framework of the continuum theory.11 Fission-gas bubbles can also interact due to mobility from biased motion within a temperature gradient This aspect of the problem is handled in an analogous manner and will not be considered here As the bubbles grow and interact, the average spacing between bubbles shrinks In addition, as seen from eqn [27] for the volume diffusion mechanism, bubble mobility falls off as the inverse of the radius cubed such that, for all practical purposes, relatively large bubbles are immobile As the larger bubbles grow because of accumulation of the continual production of gas due to fission, the bubbles intercept other bubbles and coalesce This process is here termed geometrical coalescence For spherical bubbles that are all the same size and that are uniformly distributed, contact is reached when 2Rb 2Cb =3ị1=3 ẳ ẵ29 In analogy with percolation theory, the probability of an i-bubble contacting a j-bubble is given by  h pffiffiffiffiffiffi i gc 1=3 Pij ¼ 0:5 À erf À Rij Cij 0:5=s ½30Š where Rij ẳ Ri ỵ Rj ; ẳ Ci ỵ Cj ị ẳ 4pDvol Ri Pii X gc bi ẳ 4pDvol Ri Cj Pij ẵ32 j 6ẳi where Pij is given by eqn [30] In what follows, it is assumed that DXe ẳ Dvol ẵ27 where a0 is the lattice constant and Dvol is the volume self-diffusion coefficient of the most mobile species in the alloy The coefficients and bi (e.g., the first and second terms on the RHS of eqn [15]) are represented, respectively, by X Ri ỵ Rj ịDi ỵ Dj ịCj ẵ28 ¼ 16pRi Di ; bi ¼ 1=3 Cij and s is the width of the distribution that characterizes divergences from spherical bubbles and the uniform distribution assumption In principle, s is a measurable parameter The and bi coefficients in eqn [28] now have an additional term given by !1=3 ½31Š 3.20.2.5 Analysis of U–10Mo High-Temperature Irradiation Data Figure shows the as-nucleated bubble-size distribution made with the simultaneous solution of eqns [7] and [14] for a gas solubility of 10À7 at a fuel temperature of 850 K At a fission rate of  1020 fissions mÀ3 sÀ1, the solubility limit is reached in $140 s Subsequently, nucleation is limited as a result of the gas concentration in solution falling below the solubility limit The trapping of gas in solution by the nucleated gas bubbles damps the rate at which the generated gas increases the gas concentration in dynamic solution It is important to point out that here the solubility limit is an unknown parameter If the solubility limit was 10À6 or 10À5, the initial bubble nucleation event would occur after 1400 or 14 000 s of irradiation, respectively Figure shows m versus r obtained from the solution of eqn [7] for T ¼ 850 K and g ¼ 0:5 JmÀ2 As expected from the form of eqn [7], the number of gas atoms grows exponentially with bubble size Figure shows the amount of gas in bubbles as a function of bubble size corresponding to Figures and As is evident from Figure 3, although the bubble-size distribution shown in Figure is relatively broad, the majority of the gas generated prior to the nucleation event (i.e., within the first 140 s of irradiation) exists in bubbles having radii 140 s, the bubble distribution follows from the evolution of the as-nucleated distribution shown in Figure because of bubble–bubble coalescence and diffusion of generated gas to the existing bubble Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels 587 1E + 21 1E + 20 1E + 19 Cb (m−3) 1E + 18 1E + 17 1E + 16 1E + 15 1E + 14 1E + 13 1E + 12 1E + 11 10 12 r (nm) Figure As-nucleated bubble-size distribution made with the simultaneous solution of eqns [7] and [14] for a gas solubility of 10À7 Reproduced from Rest, J J Nucl Mater 2010, 402(2–3), 179–185 50 000 40 000 m 30 000 20 000 10 000 0 10 12 r (nm) Figure Number of gas atoms in a freshly nucleated bubble versus bubble radius corresponding to Figure Reproduced from Rest, J J Nucl Mater 2010, 402(2–3), 179–185 population When the solubility limit is again exceeded, additional nucleation events occur within the evolving bubble population, and this complex of bubbles again evolves under the driving forces of bubble coalescence, gas-atom diffusion to, and gasatom re-solution from bubbles Figure shows the calculated bubble-size distribution for an irradiation in U–8Mo at 850 K to 4% U-atom burnup using eqn [15] and the multiatom nucleation model described in Section 3.20.2 for three values of the rare-gas solubility The calculations shown in Figure were made using a gas-atom diffusivity, and re-solution rate given by Dvol ¼  10À4 eÀ33000=kT cm2 sÀ1 b ¼  10À18 f_ ½33Š 588 Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels 0.12 Fraction of generated gas in bubbles 0.12 0.10 0.10 0.08 0.08 0.06 0.04 0.06 0.02 0.04 0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.02 0.00 10 12 r (nm) Figure Fraction of generated gas in bubbles versus bubble radius corresponding to Figures and Reproduced from Rest, J J Nucl Mater 2010, 402(2–3), 179–185 1E + 20 U–8Mo data Multiatom nucleation solubility limit = 2.5 ϫ 10−9 Multiatom nucleation solubility limit = 2.5 ϫ 10−8 Multiatom nucleation solubility limit = 2.5 ϫ 10−7 1E + 19 1E + 18 Cb (r) (m−3) 1E + 17 1E + 16 1E + 15 1E + 14 1E + 13 1E + 12 1E + 11 1E + 10 Bubble radius (µm) 10 12 14 Figure Calculated bubble-size distributions for an irradiation in U–8Mo at 850 K to 4% U-atom burnup using eqn [15] and the multiatom nucleation model described in Section 72.2 for three values of the rare-gas solubility compared with irradiation data Reproduced from Rest, J J Nucl Mater 2010, 402(2–3), 179–185 where f_ is the fission rate The value for Dvol given in eqn [33] is about a factor of 10 less than the out-of-pile measured U self-diffusion coefficient in U–10Mo.12 On the other hand, it is not clear what diffusion mechanism dominates gas behavior in these alloys For example, the Mo self-diffusion coefficient in U–10Mo is about an order of magnitude less than the U self-diffusion coefficient.13 In addition, it is not at all clear how these diffusion couple measurements extrapolate to lower temperatures (lowest diffusion couple temperature was 1073 K) and to an irradiation environment The value for b is Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels 613 1.2 Theory Data Volume fraction of recrystallized fuel 1.0 0.8 0.6 0.4 0.2 0.0 40 50 60 Burnup (%) 70 80 Figure 25 Volume fraction of recrystallized fuel in U–10Mo calculated with eqn [106] using B2 =  10À29 compared with estimated data Reproduced from Rest, J J Nucl Mater 2005, 346(2–3), 226–232 0.6 Volume fraction of recrystallized fuel Theory Data 0.5 0.4 0.3 0.2 0.1 0.0 10 20 30 40 Grain diameter (μm) 50 60 70 Figure 26 Calculated grain-size dependence of the volume fraction of recrystallized fuel in UO2 fuel calculated with eqn [106] compared with data Reproduced from Nogita, K.; Une, K.; Hirai, M.; Ito, K.; Shirai, Y J Nucl Mater 1997, 248, 196; Rest, J J Nucl Mater 2005, 346(2–3), 226–232 B2 ¼  10À29 Equation [91b] was used to calculate the initiation of recrystallization at 45% burnup as shown in Figure 25 Although the data shown in Figure 25 allow an estimate of the value of the parameter B2, the data are not sufficient to validate the burnup dependence exhibited by eqn [110] Figure 26 shows the calculated grain-size dependence of the volume fraction of recrystallized fuel in UO2 fuel calculated with eqn [110] compared with data.61 The grain-size dependence of eqn [110], that is, % 1=dg 614 Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels as shown by the approximation given in eqn [111], is consistent with the trend of the data 3.20.5.4 Theory for the Size of the Recrystallized Grains dgx ẳ 12ị3 gs f nị p p 1=2 2pbv ECA Cr ị r tx ị ỵ 212ị2 bv1 f nịT Sr1=2 tx ị ẵ116 The increase in the lattice parameter is the driving force for irradiation-induced recrystallization The recrystallized grain size can be calculated by equating the total energy change during the transition with the energy required to create the new surfaces In addition to the volumetric strain energy, there is an entropy increase in going from the untransformed to the transformed state that should, in principle, be taken into account The total free energy change upon recrystallization consists of the decrease in volumetric stain energy ÁU and the increase in the volumetric configurational entropy ÁS Thus, the diameter of the newly recrystallized grains dgx can be calculated by equating the total free energy change ÁU þ T ÁS with the energy required to create the new surfaces, that is, if the recrystallized grain shape is taken to be spherical, then  x 3   3ggb 4p dgx 4p dg U ỵ T Sị ¼ x ½113Š dg 3 where  pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ggb ¼ 2gs tanÀ1 bv rN =2 % 2gs bv rN =2 ½117Š and where gs is the surface energy, and bv is the magnitude of the burgers vector In eqn [116], tx is the time at which irradiationinduced recrystallization occurs, that is, from eqns [91] and [112] tx ẳ 1024 = f_ ị13=15 ẵ118a tx ẳ 1024 = f_ ị13=15 ẵ118b where eqn [118a] corresponds to UO2 and eqn [118b] to U–10Mo The recrystallized grain size as given by eqn [116] depends on the ratio of the materials surface energy to the elastic modulus, the network dislocation density, and the change in configurational entropy The calculation of the cellular network dislocation density is presented in the next section and solving for dgx dgx ẳ 3ggb U ỵ T S ẵ114 where ggb is the energy per unit area of the subgrain boundary and the volumetric strain energy is given by   a E ẵ115 U ẳ a0 where E is the elastic modulus of the material Recrystallization occurs after the cellular dislocation network has formed, that is, the interstitial-loop line length is conserved, and the lowest energy configuration is a cellular dislocation network with cell size given by eqn [105] Equation [105] is to be understood in the context of a relationship between the cellular network dislocation density rN and the interstitial-loop density nl, that is, it is the loops that are causing the lattice displacement and not the dislocations that make up the cellular dislocation structure This relationship between rN and nl will be treated explicitly in the next section Substituting eqns [103]–[105] and eqn [115] into eqn [114] results in the following expression for dgx : 3.20.5.5 Calculation of the Cellular Network Dislocation Density and Change in Lattice Parameter In Section 3.20.5.2, the progression of recrystallization was described in terms of annuli located initially adjacent to the original grain boundary that transform to defect-free regions via the creation of the new recrystallized surface when the volumetric strain energy exceeds that necessary to create the new surface.55 The defects in the region interior to the defect-free annulus consist of a cellular dislocation network (see Chapter 1.03, Radiation-Induced Effects on Microstructure) The model for the time-dependent cellular network dislocation density rN is given by the following equations: drN 4nl nl f uị=pị1=2 3=2 ẳ pnl nl rN rN dt dg CA Cr pffiffiffi 2 5=3 dnl Di ci =O À nl nl =dl ¼ dt nl ẳ 2Ziv Di ci bv ẵ119 ẵ120 ẵ121 Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels In eqns [119–121], nl and dl are the interstitial-loop density and diameter, respectively, nl is the interstitialloop climb controlled glide velocity, Di and ci are the interstitial diffusivity and concentration, respectively, Ziv is the relative bias between interstitials and vacancies, and O is the atomic volume In eqn [120], the first term on the RHS represents the nucleation rate of interstitial loops due to di-interstitials; and the second term is the loss of loops to the dislocation network In eqn [119], the first term on the RHS represents dislocation line length creation due to the accumulation of interstitial loops, the second term represents the loss of line length due to the capture of dislocations by the grain boundary, and the third term represents loss of line length due to self annihilation via correlated movement of adjacent edge dislocations.62 The form of the third term in eqn [119] can also be deduced from the combined requirements imposed by eqns [104] and [105], that is, that the dislocation line length is generated from interstitial loops and that the underlying structure is that of a cellular dislocation network Equations [104] and [105] can be combined ð f uị=pị1=2 =pCA Cr An identical to yield nl ẳ r3=2 N expression to this one is also obtained from the steady-state solution of eqn [119] in the limit of large grain size On the basis of a more detailed numerical solution of the coupled, time-dependent defect and gas-bubble rate equations,45 the component of the interstitial-loop density that is generated from defect nucleation and diffusion reaches steady state relatively early in the irradiation, that is, from eqn [105] and eqn [120] with dnl =dt ¼ pffiffiffi 2 5=3 pnl n2l Di ci =O ẳ0 ẵ122 rN Using eqn [121] in eqn [122] and solving for the interstitial-loop density yields pffiffiffi !1=2 bv ci rN ½123Š nl ¼ 4pZiv O5=3 Equation [123] establishes a connection between the network dislocation density rN and the interstitialloop density nl, that is, at any given time within the steady-state regime there is a balance between nl and rN specified by eqn [123] Using eqn [123], the equation for rN, eqn [119] is now solved analytically to obtain pffiffiffiffiffi !2 c 1 À e c c2 t pffiffiffiffiffi ½124Š rN t ị ẳ c ỵ e c c2 t 615 where pffiffiffi !1=2 2 5=3 c1 ẳ pnl Di ci =O ẵ125 and c2 ẳ nl f uị=pị1=2 CA Cr ẵ126 where the steady-state concentration of interstitials is given by   ODv K 1=2 ẵ127 ci ẳ Di 4priv where K is the damage rate in atomic displacements per atom (K % f_ =1023 ), Dv the vacancy diffusivity, and riv the defect recombination distance The temperature dependence of ci in eqn [127] is contained in the interstitial and vacancy diffusivities In general, these diffusivities are expressed as Di ¼ Di0 expðÀei =kT ị and Dv ẳ Dv0 expev =kT ị Figure 27 shows the cellular network dislocation density at T ¼ 953 K calculated with eqn [121] and eqns [124] –[127] as a function of burnup The calculations incorporated the properties listed in Table Also shown is a fit to the measured dislocation density from Nogita and Une.65 The calculations shown in Figure 27 tend to underpredict the data in the lower burnup range This result is consistent with the steady-state assumption invoked for the interstitial-loop density given by eqn [123] As shown in Figure 27, the calculated network dislocation density approaches the data as the fuel burnup approaches 80 GWd tMÀ1 (giga-watt days per metric ton) Figure 28 shows the change in UO2 lattice parameter at T ¼ 923 K calculated with eqns [103], [108], and [124]–[127] versus burnup relative to a fit to the experimental data.66 In Figure 28, the initiation of recrystallization, given by eqn [91a], occurs at the peak of the calculated curve (Áaðt Þ=a0 % 10À3 ) Subsequently, Áaðt Þ=a0 decreases because of the progression of recrystallization, as given by eqn [110] As pointed out by Spino and Papaioannou,66 there are many other factors apart from interstitial loops that give rise to a change in lattice parameter (e.g., chemical effects) As such, the quantitative agreement between the calculated and measured Áaðt Þ=a0 should not be taken too seriously However, as shown in Figure 28, the calculated change in Áaðt Þ=a0 follows the trend of the data As such, it appears that lattice distortion induced by the presence of interstitial loops provides a relatively large contribution 616 Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels 1E + 16 1E + 15 1E + 14 r (m−2) 1E + 13 1E + 12 1E + 11 1E + 10 Theory Fit to data 1E + 1E + 20 40 60 Burnup (GWd tM–1) 80 100 Figure 27 Calculated dislocation density for T ¼ 953 K as a function of burnup Also shown is a fit to measured dislocation density from Nogita and Une Reproduced from Rest, J J Nucl Mater 2006, 349(1–2), 150–159.65 Table Values of various parameters used in the calculation for UO2 Parameter D0i D0v ei ev riv Ziv a bv E gs B2 ÁS Value References À1 5m s 75 m2 sÀ1 0.6 eV 2.4 eV 82.5 A˚  10À4 ˚ 5.47 p ffiffiffi A (UO2) 2a=4  1011(1–1.09154  10À4T ) N mÀ2, (UO2, 0.95 TD) J mÀ2  10À34 m5 NÀ1  10À6 J mÀ3 KÀ1 60 60 60 63 60 60 2 64 55 57 Source: Rest, J J Nucl Mater 2006, 349(1–2), 150–159 3.20.5.6 Calculation of Recrystallized Grain Size Using eqns [116], [118], and [124]–[127], and the properties listed in Table 4, the average recrystallized grain size can now be calculated Figure 29 shows the calculated recrystallized grain size in UO2 at T ¼ 923 K as a function of the average fission rate f_ As is evident from Figure 29, the calculated recrystallized grain size increases as the average fission rate in the fuel increases Figure 30 shows the calculated recrystallized grain size in UO2 for f_ ¼  1019 mÀ3 sÀ1 as a function of the fuel temperature for various values of the volumetric configurational entropy change ÁS upon recrystallization As demonstrated in Figure 30, the calculated recrystallized grain size decreases as the fuel temperature increases For fuel temperatures