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Category Archives: Comprehensive nuclear materials
Kinetic Monte Carlo Simulations of Irradiation Effects
C. S. Becquart
Ecole Nationale Superieure de Chimie de Lille, Villeneuve d’Ascq, France; Laboratoire commun EDF-CNRS Etude et Modelisation des Microstructures pour le Vieillissement des Materiaux (EM2VM), France B. D. Wirth
University of Tennessee, Knoxville, TN, USA © 2012 Elsevier Ltd. All rights reserved.
1.14.1 Introduction 393
1.14.2 Modeling Challenges to Predict Irradiation Effects on Materials 394
1.14.3 KMC Modeling 395
1.14.4 KMC Modeling of Microstructure Evolution Under Radiation Conditions 396
1.14.4.1 Irradiation Rate 397
1.14.4.2 Transmutation Rate 397
1.14.4.3 Diffusion Rate 397
1.14.4.4 Emission/Dissociation Rate 398
1.14.5 Atomistic KMC Simulations of Microstructure Evolution in
Irradiated Fe-Cu Alloys 399
1.14.6 OKMC Example: Ag Fission Product Diffusion and Release in
TRISO Nuclear Fuel 404
1.14.7 Some Limits of KMC Approaches 406
1.14.8 Advanced KMC Methods 407
1.14.9 Summary and a Look at the Future of Nuclear Materials Modeling 408
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Abbreviations |
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AKMC |
Atomistic kinetic Monte Carlo |
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BKL |
Boris, Kalos, and Lebowitz |
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EKMC |
Event kinetic Monte Carlo |
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FP |
Frenkel pairs |
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HTR |
High-Temperature Reactor |
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KMC |
Kinetic Monte Carlo |
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MC |
Monte Carlo |
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MD |
Molecular dynamics |
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NEB |
Nudged elastic band |
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NRT |
Norgett, Robinson, and Torrens |
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OKMC |
Object kinetic Monte Carlo |
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PKA |
Primary knockon atom |
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RPV |
Reactor pressure vessel |
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RTA |
Residence time algorithm |
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SIA |
Self-interstitial atoms |
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TRISO |
Tristructural isotropic fuel particle |
Many technologically important materials share a common characteristic, namely that their dynamic behavior is controlled by multiscale processes. For example, crystal growth, plasma processing of materials, ion-beam assisted growth and doping of electronic materials, precipitation in structural materials, grain boundary and dislocation evolution during mechanical deformation, and alloys driven by high-energy particle irradiation all experience cluster nucleation, growth, and coarsening that impact the evolution of the overall microstructure and, correspondingly, property changes. These phenomena involve a wide range of length and time scales. While the specific details vary with each material and application, kinetic processes at the atomic to nanometer scale (especially related to nucleation phenomena) are largely responsible for materials evolution, and typically involve a wide range of characteristic times. The large temporal diversity of controlling processes at the atomic to nanoscale level makes experimental identification of the governing mechanisms all but impossible and clearly defines the need for computational modeling. In such systems, the potential benefits of modeling are at a maximum and are related to reduction in time and expense ofresearch and development and introduction of novel materials into the marketplace.
Systems in which the materials microstructure can be represented by multiple particles experiencing
Brownian motion and occasional collisions against one another and systems with other defects (dislocations, grain boundaries, surfaces, etc.) are in particular amenable to multiscale modeling. Within a multiscale approach, atomistic simulations (utilizing either electronic structure calculations or semiempirical potentials) investigate controlling mechanisms and occurrence rates of diffusional and reactive interactions between the various particles and defects of interest, and inform larger length scale kinetic (Monte Carlo, phase field, or chemical reaction rate theory) models, which subsequently lead to the development of constitutive models for predictive continuum scale models. Simulating long-time materials dynamics with reliable physical fidelity, thereby providing a predictive capability applicable outside limited experimental parameter regimes is the promise of such a computational multiscale approach.
A critical need is the development of advanced and highly efficient algorithms to accurately model nucleation, growth, and coarsening in irradiated alloys that are kinetically controlled by elementary (diffusive) processes involving characteristic time scales between
10~12 and ^10~3 s. The goal of this chapter is to describe the state of the art in kinetic Monte Carlo (KMC) simulation, as well as to identify a number of priority research areas, moving toward the goal of accelerating the development ofadvanced computational approaches to simulate nucleation, growth, and coarsening of radiation-induced precipitates and defect clusters (cavities and/or dislocation loops). It is anticipated that the approaches will span from atomistic molecular dynamics (MD) simulations to provide key kinetic input on governing mechanisms to fully three-dimensional (3D) phase field and KMC models to larger scale, but spatially homogeneous cluster dynamics models.
The CALPHAD Approach and Free Energy Minimization
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The overall development of a consistent thermochemical representation for the phase equilibria and thermodynamics of a system utilizing all available information has been termed the CALPHAD (computer coupling of phase diagrams and thermochemistry) approach.1 Whether free energy and heat |
Austenitic steelsMost of the RIS models for concentrated alloys were devoted to the ternary Fe-Cr-Ni system, which is a model alloy of austenitic steels. The diffusion ratios used in the fitting process are the ones extracted from the tracer diffusion coefficients measured by Rothman et al. (referenced in Perks and Murphy104 and Allen and Was11 ). In most of the studies, the input parameters are taken from Perks model.104 The more recent MIK model, which was initially based on the Perks model, used the CALPHAD database to fit its concentration-dependent migration energy model.11 A significant improvement in the predictive capabilities of RIS modeling was concluded after a systematic comparison with RIS, observed by Auger spectroscopy in Fe-Cr-Ni as a function of temperature, nominal composition, and irradiation dose.1 However, all the models were proved to be powerful enough to reproduce the correct tendencies of RIS in austenitic steels: a depletion of Cr and an enrichment of Ni near a grain boundary. When the binding energies of point defects with atoms are not so strong and the ratios between the tracer diffusion coefficients of the major elements are large enough (larger than 2-3), a rough estimation of the partial diffusion coefficients from tracer diffusion coefficients seems to be sufficient to reproduce the main tendencies. The interstitial contribution to RIS is usually neglected due to the lack of diffusion data. Stepanov eta/.120 observed an electron-irradiated foil at a temperature low enough so that only interstitials contributed to the RIS. Segregation profiles were similar to the typical ones at higher temperature. Parameters of the interstitial diffusion model were estimated in such a way that the experimental RIS was reproduced. The migration energy of interstitials was assumed to be equal to 0.2 eV, which is quite low in comparison with the effective migration energy deduced from recovery resistivity measurements.121 The attempts of the MIK model to reproduce the characteristic ‘W-shaped’ Cr profiles observed at intermediate doses were not conclusive36; transitory profiles disappeared after a dose of 0.001 dpa, while the experimental threshold value was around 1 dpa. A possible explanation may be the approximation used to calculate the chemical driving force. Indeed, a thermodynamic factor equal to 1 pushes the system to flatten the concentration profile in reaction to the formation of the RIS profile. A study based on a lattice rate theory pointed out that an oscillating profile was the signature of a local equilibrium established between the surface plane and the next plane.13 This kind of mean-field model predicts that the local enrichment of Cr at a surface persists at larger irradiation dose (0.1 dpa), though not as large as the experimental value. The role of impurities as point defect traps has been explored since the 1970s.122 In those models, impurities do not contribute to fluxes but to the sink population as immobile sinks with an attachment parameter depending on an impurity-point defect binding energy.123 Other models account for immobile vacancy traps by renormalizing the recombination coefficient with a vacancy-impurity binding energy. Whether by vacancy or by interstitial trapping, the result is a recombination enhancement and a decrease of point defect concentrations, leading to a reduction of RIS and swelling. Hackett et a/.12 estimated some binding energies between a vacancy and impurities, such as Pt, Ti, Zr, and Hf in fcc Fe, using ab initio calculations. The energy calculations seem to have been performed without relaxing the structure, probably because fcc Fe is not stable at 0 K. Although the absolute values of the binding energies should be used with caution, one can expect the strong difference between the binding energies of a vacancy with Zr (1.08 eV) and Hf (0.71 eV) to persist after a relaxation of atomic positions. In a more rigorous way, the trapping of dumbbells could be modeled using the high migration energy associated with dumbbells bound to an impu — rity.12 Such a model would allow the impurity to migrate and change the sink density with dose. The same model could explain the recent experimental results observing that, after a few dpa, RIS of the major elements starts again.123 RIS in austenitic steels was observed to be strongly affected by the nature of the grain boundary, that is, by the misorientation angle and the S value.40 Differences between the observed RIS are explained by a so-called grain boundary efficiency, introduced in a modified rate equation model.40,109,114,124 Calculations of vacancy formation energies at different grain boundaries, for example, in nickel in which atomic interactions are described by an embedded atom method, have been used to model sink strength as a function of misorientation angle. The resulting RIS predictions around several tilt grain boundaries were found to be in good agreement with RIS data.124 Grain boundary displacement and its effect on RIS were considered too. Grain boundary displacement was explained by an atomic rearrangement process due to recombination of excess point defects at the interface. New kinetic equations including an atomic rearrangement process after recombination of point defects at the interface predict asymmetrical concentration profiles, in agreement with experiments.114 Radiation Damage Using Ion Beams
Radiation effects research has been conducted using a variety of energetic particles: neutrons, electrons, protons, He ions, and heavy ions. Energetic ions can be used to understand the effects of neutron irradiation on reactor components, and interest in this application of ion irradiation has grown in recent years for several reasons including the avoidance of high residual radioactivity and the decline of neutron sources for materials irradiation. The damage state and microstructure resulting from ion irradiation, and thus the degree to which ion irradiation emulates neutron irradiation, depend upon the particle type and the damage rate. This chapter will begin with a summary of the motivation for using ion irradiation for radiation damage studies, followed by a brief review of radiation damage relevant to charged particles. The contribution of ion irradiation to our understanding of radiation damage will be presented next, followed by an account of the advantages and disadvantages of the various ion types for conducting radiation damage studies, and wrapping up with a consideration of practical issues in ion irradiation experiments. Point defect assembliesIn this class, one can include the calculation of interstitial assemblies as well as the complexes built with impurities and vacancies. One then has access to the binding of monoatomic defects to the complexes,2 possibly with the associated kinetic energy barriers. As for perfect crystals, the information obtained by ab initio calculations can be gathered and integrated in larger scale modeling, especially, kinetic models. Many kinetic Monte-Carlo models were thus parameterized with ab initio calculations (see e. g., the works on pure iron25 or FeCu26and Chapter 1.14, Kinetic Monte Carlo Simulations of Irradiation Effects). Even ifthe cell sizes accessible by ab initio calculations are small, it is possible to deal with some extended defects. Calculations then often need some tricks to accommodate the extended defect in the small cells. Some examples are given in the next section on studies on dislocations. Hard Spheres and Binary Collision ApproximationThe simplest atomistic model is the binary collision approximation (BCA), which simulates cascades as a series of atomic collisions. The ‘potential’ is then basically a hard sphere, with collisions either elastic or inelastic and the possibility of adding friction to describe an ion’s progress through an electron field. There is no binding energy, so the condensed phase is stabilized due to confinement by the boundary conditions. This is a poor approximation, since the structures and energies of the equilibrium crystal and defects do not correspond to real material. However, it allows for quick calculation and the scaling of defect production with energy of the primary knock-on atom. The best-known code for this type of calculation is probably Oak Ridge’s MARLOWE.1 Pairwise potentials are the next level in complexity beyond BCA. They allow soft interactions between particles and simultaneous interaction between many atoms. Pair potentials always have some parameters that relate to a particular material, requiring fitting to experimental data. This immediately introduces the question ofwhether the reference state should be free atoms (e. g., in argon), free ions (e. g., NaCl), or ions embedded in an electron gas (e. g., metals). The two classic pair potentials used for modeling are the LennardJones (LJ) and Morse potential. Each consists of a short-range repulsion and a long-range attraction and has two adjustable parameters.
The (y) 6 cohesion is on the basis of van der Waals interactions, while the e-ar is motivated by a screened — Coulomb potential. The repulsive terms were invented ad hoc. Because they have only two parameters, all simulations using just LJ or Morse are equivalent, the values of e, s, or a , which simply rescale energy and length. Hence, these potentials cannot be fitted to other properties of particular materials. Both LJ and Morse potentials stabilize close- packed crystal structures, and both have unphysically low basal stacking fault energies. Equivalently, the energy difference between fcc and hcp is smaller than for any real material. For materials modeling, this introduces a problem that the (110) dislocation structure is split into two essentially independent partials. For radiation damage, this means that configurations such as stacking fault tetrahedra are overstabilized, and unreasonably large numbers of stacking faults can be generated in cascades, fracture, or deformation. If the energy scale is set by the cohesive energy of a transition metal, then the vacancy and interstitial formation energy tend to be far too large; if the vacancy energy is fitted, then the cohesive energy is too small. For binary systems, these potentials can stabilize a huge range of crystal structures, even without explicit temperature effects; some progress has been made to delineate these, but it is far from complete.2 Once one moves to N-species systems, there are e and а parameters for each combination of particles, that is, N(N + 1) parameters. Now it is possible to fit to properties of different materials, and the rapid increase in parameters illustrates the combinatorial problem in defining potentials for multicomponent systems. Secondary Factors Influencing Cascade Damage Formation
The results of simulations such as those presented above should not be viewed as being quantitatively accurate. As already mentioned, subtle changes in the fitting of the interatomic potential can alter the cascade simulations both qualitatively and quantitatively. Even if a sufficiently accurate potential can be identified, the results represent a certain limiting case of what may be observed experimentally. This is because all the simulations mentioned so far were carried out in perfect material — computer-pure material. Nowhere in nature can such perfect metal be found, particularly for iron, which is easily contaminated with minor interstitial impurities such as carbon. In this section, a few examples will be discussed to illustrate how reality may influence cascade damage production relative to the perfect material case. The examples include the influence of preexisting defects, free surfaces, and grain boundaries. 1.11.4.4.1 Influence of preexisting defects Even if a well-annealed, nearly defect-free, single crystal material is selected for irradiation, radiation — induced defects will rapidly change the state of the material. A simple calculation employing typical elastic scattering cross-sections for fast neutrons and the cascade volumes observed in MD simulations will demonstrate that by the time a dose of ~0.01 dpa is reached, essentially the complete volume will have experienced at least one cascade. There have been relatively few studies on how cascade damage production may be different in material with defects.95-97 The results of cascade simulations reported in Stoller and Guiriec97 were carried out at 10keV and 100 K to expand the range of previous work carried out using 1 keV simulations in copper95 and 0.40, 2.0, and 5.0 keV simulations in iron.96 A 10 keV cascade energy is high enough to initiate in-cascade clustering, is near the plateau region of the defect survival curve, and involves a limited degree of subcascade formation. For these conditions, the database discussed above (see Figure 11) includes two independent sets of cascades, seven in a 128 k atom cell and eight in a 250 k atom cell that can be used to provide a basis of comparison. A cell size of 250 k atoms was used for the cascade simulations with preexisting damage. The study in Stoller and Guiriec97 involved three simple configurations of preexisting damage that were all derived from cascade debris. This is perhaps the simplest possible damage structure, a collection of point defects and point defect clusters. The first configuration was simply the as-quenched debris from a 10 keV cascade in perfect crystal. A total of 30 vacancies and interstitials were present, including one di — and one 7-interstitial cluster. The second case was similar, but the point defects were reconfigured so that the 30 vacancies included a 6-vacancy void and a 9-vacancy loop, and the interstitial clusters included four di-, one tri-, and one 8-interstitial cluster. The third configuration contained only a single 30-vacancy void. These configurations are shown in Figure 23. Eight simulations were carried out with different initial PKAs and the same <135> PKA direction. The selected PKAwere 15—20 lattice parameters from the center of the cascade debris and located such that the < 135 > direction pointed them toward the center of the debris field. The same set of PKAs was used for all three defect configurations. As expected, substantial variation was observed between the different simulations for any given preexisting defect configuration; in some cases the cascade produced more defects than in perfect crystal,
while in others fewer were produced. The most dramatic visible effects were observed for the 30-vacancy void. In one case, the void was completely intact after the second cascade, while in the others it was destroyed to varying degrees. The impact of preexisting damage on stable defect formation in the 10keV cascades is shown in Figure 24, where results from the three different defect configurations are compared with those obtained in perfect crystal. The variation between two sets of perfect crystal simulations is provided for comparison purposes. The statistical information from analysis of defect survival
and interstitial clustering is summarized in Table 3. On average, a significant reduction in defect formation was observed for the two configurations most typical of random cascade debris. A slight increase (that may not be statistically significant) in defect production was observed when the cell contained only a small void. Only the second defect configuration led to a significant change in interstitial clustering. Although the approach in Stoller’s investigation of preexisting damage was slightly different, the results are consistent with previous studies by Foreman and coworkers95 and Gao and coworkers.96 They observed substantial reductions in defect production when a cascade was initiated in material containing defects. The reductions in defect production observed by Stoller (Figure 24 and Table 3) are somewhat smaller. This difference may partially be due to the higher cascade energy employed here (10 keV vs. 0.4— 5 keV), but the statistical nature of cascade evolution is also a factor. Gao and coworkers analyzed the results of several simulations as a function of distance between the center of mass (COM) of the new cascade and that of the preexisting damage. A good correlation was found between this spacing and the number of defects produced. In the work of Stoller and Guiriec,97 the distance between PKA location and the preexisting damage was nearly constant. As the morphology of each cascade is quite different, the COM spacings varied. This is certainly part of the reason for the variety of behaviors mentioned above for the case of the small void. The average behavior for a fixed initial separation cannot be Early Radiation Damage Theory Model
the corresponding PD-sink elastic interactions. Thus, the preferential absorption of SIAs by dislocations (i. e., the dislocation bias) is the only driving force for microstructural evolution in this model, which is a variant of the FP3DM. It should be emphasized that, in the framework of the FP3DM, no distinction is made between different types of irradiation: ^1MeV electrons, fission neutrons, and heavy-ions. It was believed that the initial damage is produced in the form of FPs in all these cases. Now we understand the mechanisms operating under different conditions much better and make clear distinction between electron and neutron/heavy-ion irradiations (see Singh et a/.,1,22 Garner et al.,33 Barashev and Golubov,35 and references therein for some recent advances in the development of the so-called PBM). However, the FP3DM is the simplest model for damage production and it correctly describes 1 MeV electron irradiation. It is therefore useful to consider it first. The more comprehensive PBM includes the FP3DM as its limiting case. following section, we present examples of such a treatment based on the so-called lossy-medium approximation. Phase Field MethodsP. Bellon Electronic and atomistic processes often dictate the pathways of phase transformations and microstructural evolution in solid materials. For quantitative modeling of these transformations and evolution, it is thus effective, and sometime necessary, to rely on methods using some representation of atoms and of their dynamics, as for instance in molecular dynamics simulations (see Chapter 1.09, Molecular Dynamics) and atomistic Monte Carlo simulations (see Chapter 1.14, Kinetic Monte Carlo Simulations of Irradiation Effects). While these atomistic methods can now simulate quite accurately the evolution of specific alloy systems, these simulations are nevertheless limited to small length scales, from a few to 100 nm. Molecular dynamics is furthermore limited to small time scales, typically in the nanosecond range, although in some cases, new developments have made it possible to obtain atomistic simulations at much longer times (see Chapter 1.14, Kinetic Monte Carlo Simulations of Irradiation Effects). An alternative modeling approach is to replace the many microscopic degrees of freedom of the system of interest by the few mesoscopic variables that are sufficient to provide a realistic description. This approach has been widely used in many disciplines, and well-known examples are the Fourier and Fick equations, which describe the diffusive transport of heat and chemical species, respectively. This approach is also commonly used in modeling the evolution of point defects, in particular, during irradiation (see Chapter 1.13, Radiation Damage Theory and Sizmann1). The work of Cahn and Hilliard2- 5 and Landau and Lifshitz (see for instance Tole — dano and Toledano6) provided a way to include the contributions of interfaces to chemical evolution, thus making it possible to model heterogeneous and multiphase materials. Kinetic models based on these descriptions are broadly referred to as phase field (PF) methods, since the microstructure of a material is fully characterized by a few mesoscopic field variables such as concentration, magnetization, chemical order, or temperature. One key assumption of this approach is that the variables chosen to describe the state of the system vary smoothly across any interface or, in other words, that interfaces are diffuse. This assumption finds a natural justification in the theory of critical phenomena, since the interface thickness diverges at the critical temperature.7 Diffuse interface models offer some advantages over sharp interface models,8 in particular, for the modeling of complex microstructures. Furthermore, the PF approach can be extended to include macroscopic variables other than the local composition, making it possible to describe chemical order-disorder transitions, solid-liquid reactions, displacive transformations, and more recently dislocation glide. PF methods and applications have been recently reviewed by Chen,9 Emmerich,1 and Singer-Loginova and Singer.11 This chapter focuses on solid-solid phase transformations, with a particular emphasis on transformations and microstructural evolution relevant to irradiated materials. While conventional PF modeling lacks atomic resolution, the main interest in this technique comes from the fact that it can provide the evolution of large systems, exceeding the micrometer scale, over very long time scales, from seconds to centuries. Recent developments have led to the introduction of PF models (PFMs) that possess atomic resolution,12-26 the so-called PF crystal models. This model, which can be seen as a density functional theory for atoms, appears very promising, although at this time it is not clear whether it can reproduce correctly the discrete nature of point — defect jumps from one lattice site to a neighboring lattice site. The PF crystal model is not covered in this chapter, so the interested reader should consult the above references. This chapter is organized as follows. Section 1.15.2 introduces the key concepts and steps employed in conventional, that is, phenomenological PF modeling, and provides some illustrative examples. Section 1.15.3focuses on important recent developments toward quantitative PF modeling, whereby evolution equations are rigorously derived by coarse-graining a microscopic model. This approach provides a full treatment of fluctuations and thus makes it possible to study fluctuation-controlled reactions, such as nucle — ation of a second phase. The capability of PFMs to reach large time and length scales makes them an attractive tool for simulating the evolution of materials relevant to nuclear applications, in particular, for alloys subjected to irradiation. Applying PF modeling to these nonequilibrium materials, however, raises new challenges, as is discussed in Section 1.15.4.1. Some selected results of PF modeling applied to irradiated materials are presented in Section 1.15.4.2. Finally, conclusions and perspectives are given in Section 1.15.5. Experimental Observations1.18.2.1 Anthony’s Experiments RIS was predicted by Anthony,18 in 1969, a few years before the first experimental observations: a rare case in the field of radiation effects. The prediction stemmed from an analogy with nonequilibrium segregation observed in aluminum alloys quenched from high temperature. Between 1968 and 1970, in a pioneering work in binary aluminum alloys, Anthony and coworkers18-22 systematically studied the nonequilibrium segregation ofvarious solute elements on the pyramidal cavities formed in aluminum after quenching from high temperature. They explained this segregation by a coupling between the flux of excess vacancies toward the cavities and the flux Anthony suggested that similar coupling should produce nonequilibrium segregation in alloys under irradiation.18,19 He predicted that the segregation should be much stronger than after quenching because under irradiation, the excess vacancy concentration and the resulting flux can be sustained for very long times.19,25 As for the cavities formed by vacancy condensation in alloys under irradiation, which result in the swelling phenomenon (Chapter 1.03, Radiation-Induced Effects on Microstructure and Chapter 1.04, Effect of Radiation on Strength and Ductility of Metals and Alloys), he pointed out that with solute and solvent atoms of different sizes, segregation should generate strains around the voids.25 Finally, he predicted intergranular corrosion in austenitic steels and zirconium alloys, resulting from possible solute depletion near grain boundaries.25 Anthony also presented a detailed discussion on nonequilibrium segregation mechanisms, in the framework of the TIP,18-21 showing that the nonequilibrium tendencies are controlled by the phenomenological coefficients Lj of the Onsager matrix, which can be — in principle — computed from vacancy jump frequencies (see below Section 1.18.3). Clarifying previous discussions on nonequilibrium segregation mechanisms,23,24 he considered two limiting cases for the coupling between solute and vacancy fluxes in an A-B alloy (at the time, he did not apparently consider the coupling between solute and interstitial fluxes and its possible contribution to RIS). In both cases, the total flux of atoms must be equal and in the direction opposite to the vacancy flux:
Figure 2 Radiation-induced segregation mechanisms due to coupling between point defect and solute fluxes in a binary A-B alloy. (a) An enrichment of B occurs if dBV < dAV and a depletion if dBV > dAV. (b) When the vacancies drag the solute, an enrichment of B occurs. (c) An enrichment of B occurs when dBI > dAI. 1. If both A and B fluxes are in the direction opposite to the vacancy flux (Figure 2(a)), one can expect a depletion of B near the vacancy sinks if the vacancy diffusion coefficient of B is larger than that of A (dBV > dV); in the opposite case (dBV < dAV), one can expect an enrichment of B (it is worth noting that this was essentially the explanation proposed by Kuczynski eta/23 in 1960). 2. But A and B fluxes are not necessarily in the same direction. If the B solute atoms are strongly bound to the vacancies and if a vacancy can drag a B atom without dissociation, the vacancy and solute fluxes can be in the same direction (Figure 2(b)): this was the explanation proposed by Aust et at24 In such a case, an enrichment of B is expected, even if dBV > dAV. |