Equation [35] can be used to obtain the rate of recombination reactions between vacancies and SIAs. In a coordinate system where the vacancy is immobile, the SIAs migrate with the diffusion coeffi­cient Di V Dv and, hence, the total recombination rate is

R = 4жгеЄ(Di V Dy) Ci«v « mRDi Ci Cv [67]

where nv = Cv/Q and the fact that Di ^ Dv at any temperature is used. In this equation, reff is the effective capture radius of a vacancy, defining an effective volume where recombination occurs spon­taneously (athermally). The recombination constant, mR, in eqn [67] is, hence, equal to

MD calculations show that a region around a vacancy, where such a spontaneous recombination takes place, consists of ^100 lattice sites.100,101 From 4preff/3 = 100Q, one finds that reff is approximately two lattice parameters, hence mR ~ 1021m-2.

1.15.4.1 Challenges Specific to Alloys Under Irradiation

The PFMs discussed so far are broadly applied to materials as they relax toward some equilibrium state. In particular, the kinetics of evolution is given by the product of a mobility by a linearized driving force, see for instance eqns [2] and [3]. In the context of the thermodynamics of irreversible processes,28 the mobility matrix is the matrix of Onsager coefficients. Irradiation can, however, drive and stabilize a material system into a nonequilibrium state,108 owing to ballis­tic mixing and permanent defect fluxes, and so it may appear questionable at first whether linearized relax­ation kinetics is applicable. A sufficient condition, however, is that these different fields undergo linear relaxation locally, and this condition is often met even under irradiation. A complicating factor arises from the presence of ballistic mixing, which adds a second dynamics to the system on top of the thermally acti­vated diffusion of atoms and point defects. A superpo­sition oflinearized relaxations for these two dynamics is valid as long as they are sufficiently decoupled in time and space, so that in any single location, the system will evolve according to one dynamic at a time. KMC simulations indicate that, for dilute alloys, this decoupling is valid except for a small range of kinetic parameters where events from different dynamics interfere with one another.109

A second issue is that PFMs, traditionally, do not include explicitly point defects. Vacancies and inter­stitials are, however, essential to the evolution of irradiated materials, and it is thus necessary to include them as additional field variables. The situa­tion is more problematic with point-defect clusters, which often play a key role in the annihilation of free point defects. Since the size of these clusters cover a wide range of values, it would be quite difficult to add a new field variable for each size, for example, for vacancy clusters of size 2 (divacancies), size 3 (triva­cancies), size 4, etc. Moreover, under irradiation con­ditions leading to the direct production of defect clusters by displacement cascades, additional length scales are required to describe the distribution of defect cluster sizes and of atomic relocation dis­tances. These new length scales are not physically related to the width of a chemical interface at equi­librium, we, and therefore, they cannot be safely rescaled by we. This analysis clearly suggests that one needs to rely on a PFM where the atomic scale has been retained. This is, for instance, the case in the quantitative PFM reviewed in Section 1.15.3. Another possible approach is to use a mixed continu­ous-discrete description, as illustrated below in Sec­tion 1.15.4.2.4. We note that information on defect cluster sizes and relocation distances should be seen as part of the noise imposed by the external forcing, here the irradiation, on the evolution of the field variables. The difficulty is thus to develop a model that can correctly integrate this external noise. It is
well documented that, for nonlinear dissipative sys­tems, the external noise can play a determinant role and, for large enough noise amplitude, may trigger nonequilibrium phase transformations.110-113

One last and important challenge in the develop­ment of PFMs for alloys under irradiation is the fact that in nearly all traditional models the mobil­ity matrix is oversimplified, for instance Mirr = C(1 — C)Dirr/kBT, which is a simple extension to eqn [17] where D has been replaced by Dirr to take into account radiation-enhanced diffusion. In the common case of multidimensional fields, for instance for multicomponent alloys, or for alloys with conserved and nonconserved field variables, the mobility matrix is generally taken as a diagonal matrix, thus eliminating any possible kinetic cou­pling between these different field variables. As discussed at the end of Section 1.15.2, this approxi­mation raises concerns because it misses the fact that these kinetic coefficients are related since they origi­nate from the same microscopic mechanisms. This is, in particular, the case for the coupled evolution of point defects and chemical species in multicompo­nent alloys. This coupling is of particular relevance to the case of irradiated alloys since irradiation can dramatically alter segregation and precipitation reac­tions owing to the influence of local chemical envir­onments on point-defect jump frequencies. While new analytical models have been developed recently using mean field approximations to obtain expressions for correlation factors in concentrated alloys,114-117 work remains to be done to integrate these results into PFMs.

Segregation concentration profiles induced by irradi­ation display some specific features. They can spread over large distances — a few tens of nanometers (see examples in Russell1 and Okamoto and Rehn29) — while equilibrium segregation is usually limited to a few angstroms. This is due to the fact that they result from a dynamic equilibrium between RIS fluxes and the back diffusion created by the concentration gra­dient at the sinks, while the scale of equilibrium segregation profiles is determined by the range of atomic interactions. Equilibrium profiles are usually monotonic, except for the oscillations, which can appear — with atomic wavelengths — in alloys with ordering tendencies.35 Segregation profiles observed in transient regimes are often nonmonotonic because of the complex interaction between concentration gradients of point defects and solutes. A typical example is shown in Section 1.18.5.3, where an enrichment of solute is observed near a point defect sink, followed by a smaller solute depletion between the vicinity of sink and the bulk. In this particular case, the depletion is due to a local increase in vacancy concentration, which results from the lower interstitial concentration and recombination rate.

Other kinds of nonmonotonic profiles are some­times observed, with typical ‘W-shapes.’ In some austenitic or ferritic steels, a local enrichment of Cr at grain boundaries survives during the Cr depletion induced by irradiation (see below). This could result from a competition between opposite equilibrium and RIS tendencies. However, the extent of the Cr enrichment often seems too wide to be simply due to an equilibrium property (around 5 nm, see, e. g., Sections 1.18.2.5 and 1.18.5.3).

RIS profiles at grain boundaries are sometimes asymmetrical, which has been related to the migra­tion of boundaries resulting from the fluxes of point defects under irradiation.37,38 The segregation is affected by the atomic structure and the nature of the sinks. It has been clearly shown that RIS in

austenitic steels is much smaller at low angles and special grain boundaries than at large misorientation angles,39’40 the latter being much more efficient point defect sinks than the former.

1.18.2.3.2 Temperature effects

RIS can occur only when significant fluxes of defects towards sinks are sustained, which typically happens only at temperatures between 0.3 and 0.6 times the melting point. At lower temperatures, vacancies are immobile and point defects annihilate, mainly by mutual recombination. At higher temperatures, the equilib­rium vacancy concentration is too high; back diffusion and a lower vacancy supersaturation completely sup­press the segregation. Temperature can also modify the direction ofthe RIS by changing the relative weight of the competing mechanisms, which do not have the same activation energy. In Ni—Ti alloys, for exam­ple, the enrichment of Ti at the surface below 400 °C has been attributed to the migration of Ti—V com­plexes, and the depletion observed at higher tempera­tures should result from a vacancy IK effect.41

Measurements of mechanical properties on irra­diated materials usually require bulk samples and therefore neutron irradiation. Ion beams, however, can be employed for some measurements, such as plastic deformation. Typically, these experiments employ high energy protons, E > « 2 MeV, or He ions, E > 7 MeV, as these particles can penetrate through thin foils, such as Fe or steel, that are greater than 15 pm in thickness. Moreover, displacement rates «10~5dpas-1 are obtainable without excessive beam heating.33 Deformation experiments have also been performed using GeV heavy ions, as these pen­etrate targets several microns in thickness. The dis­placement rates, however, are low as most ofthe beam energy is lost through electronic excitations. Heavy ions with lower energies, E « 1-4 MeV, have also been used in deformation studies; for these, however, specimen must be very thin, «200 nm, and effects of the surface must be taken into account.34,35

1.07.4.2.3 Multiple ion beams

One of the difficulties in using ion beams to simulate neutron irradiation damage is the potential for miss­ing certain synergistic behaviors in the damage evo­lution. For example, neutron irradiation leads to transmutation products and the generation of He and fission gases in addition to displacement damage. Generation of gas is particularly relevant to 14 MeV neutron irradiation for which large amounts of He and H are produced. Ion beams, however, offer the opportunity of using two or even three beams simultaneously and thus to tailor test irradiations to meet expected reactor conditions; see, for example, Serruys et al.36 This is often not possible in existing test reactor facilities, and the building of new test facilities for fusion machines has been formidably expensive. The application of multiple ion beams is illustrated in Figure 17 in a study of void swelling in vanadium. Here, the synergistic effects of simulta­neously implanting 350 keV H and 1 MeV He, while irradiating with 12 MeV Ni ions are shown. Without the He beam, swelling is negligible, even with the implantation of H, but with it, the H greatly enhances the swelling. H implantation, on the other hand, is seen to reduce the density of cavities.

The first DFT calculations of point defects in silicon carbide,98 dating back to 1988, were burdened by strong limitations in computing time. For this reason, they were performed with relatively small supercells (16 and 32 atoms), largely insufficient basis sets (plane waves with energy up to 28 Ry), and further approximations, namely for the relaxation of atomic positions. Moreover, they were limited to high sym­metry configurations. The results were only qualita­tive; however, it was already clear that vacancies and antisites could be relatively abundant, at equilibrium, with respect to interstitial defects. The authors dared to approach some defect complexes and could predict that antisite pairs and divacancies were bound.

Vacancies were thoroughly studied at the turn of the century.97’99-101 The most prominent result may be the metastability of the silicon vacancy. Indeed, following a suggestion coming from a self-consistent DFT-based tight-binding calculation by Rauls and coworkersy 2 the electron paramagnetic resonance (EPR) spectra of annealed samples of irradiated SiC were measured103 and compared with calculated hyperfine parameters. This showed that silicon vacancies are metastable with regard to a carbon vacancy-carbon antisite complex (Vc—Csi); a fact that has since been consistently confirmed by the other calculations.

Interstitials were less studied than vacancies. One should however mention a study104 devoted to carbon and silicon in interstitials in silicon carbide. Beyond these studies dedicated to one type of defect, very complete and comprehensive work on both vacancies and interstitials was also published. One should cite Bernardini et a/.105 devoted to the formation energies of defects, while Bockstedte eta/.106 goes further as it also covers migration energetics of basic intrinsic defects (vacancies, interstitials, antisites). It is worth noting that in such covalent compounds there are many possible atomic structures for defects as simple as a monointerstitial and that all these structures must be considered in the calculation (see Figure 11).

As examples, the results of these various studies on what concerns formation energies and CTL of vacan­cies are summarized in the following tables.

One can see a general agreement in the formation energies of the neutral defects, especially in the re­cent references. The small differences are related to k-point sampling or cell size in the calculations. Larger discrepancies appear between the various pre­dicted CTL. They relate to the inaccuracy of standard DFT calculations in treating empty or defect states.

A simple example relates directly to the underestima­tion of the band gap: the silicon interstitial (in the Itc configuration) in the neutral state shows up as metallic in standard calculations, the defect states lying inside the conduction band. This fact, on one hand, calls for a better description of the exchange — correlation potential for these configurations; on the other, it makes the convergence with k points and cell size very slow, as has recently been pointed out.107 This drawback of standard DFT-LDA/GGA supercell calculations is common to other defects in SiC. Even when calculated defect states fall within the band gap, their position inside it can be grossly mis­calculated with standard DFT calculations.

The errors produced by standard DFT calcula­tions for the CTL are well known nowadays. The determination of an accurate method to calculate these CTL is an active field of research with works on advanced methods such as GW (e. g., the results on SiO2108) or hybrid functionals.109 For what concerns nuclear materials, and especially SiC, GW correc­tions and excitonic effects will allow further compar­isons with experiments (Table 1).

It is worth noting that the ‘glue’ type potentials cannot be expanded in a sum of two-body, three — body, four-body, etc. terms. Three-body terms enter into the free electron picture through nonlinear response of the electron gas, and into the tight — binding picture in the fourth moment description and beyond.

At constant second moment, increasing the fourth and higher even moments of the DOS tends to lead to a bimodal distribution. Bimodal distributions will be favored by materials with half-filled bands. Thus, three-body terms are likely to be important in struc­tures with few small-membered rings of atoms, and hence, small low moments. The bcc structure is a borderline example of this, but the classic is the dia­mond structure. Diamond has no rings of less than six atoms, resulting in a strongly bimodal DOS. This bimodal structure in the tight-binding representation is also interpreted as bonding and antibonding states in a covalent picture.

Interatomic potentials for carbon and silicon fall into this category. However, once a band gap is opened, the Fermi energy and perfect screening are lost, and the rigid band approximation is less appropriate.

The IAP is a crucial property of a model for it determines all the physical properties of the sim­ulated system. Discussion on modern IAPs is pre­sented in Chapter 1.10, Interatomic Potential Development and so we do not elaborate on this subject here.

Another important property is the spatial scale of the simulated system. The periodic spacing, Lg, in the direction of the dislocation glide has to be large enough to avoid unwanted effects due to interaction between the dislocation and its periodic neighbors in the PAD; 100-200b is usually sufficient.10 Further­more, the model should be large enough to include all direct interactions between the dislocation and obstacle and the major part of elastic energy that may affect the mechanism under study. MD simula­tions have demonstrated that a system with a few million atoms is usually sufficient to satisfy condi­tions for simulating interaction between a dislocation and an obstacle of a few nanometers in size. The biggest obstacles considered to date are 8 nm voids,23 10 nm DLs,25 and 12 nm SFTs26 in crystals containing ^6-8 million mobile atoms. It should be noted that static simulation (T = 0 K) usually requires the largest system because most obstacles are stronger at low T and the dislocation may have to bend strongly and elongate before breaking free.2

Simulation of a dynamic system, that is, T> 0 K, introduces another important and limiting factor for atomic-scale study ofdislocation behavior, namely the simulation time, t, which can be achieved with the computing resource available. Under the action of increasing strain applied to the model, the time to reach a given total strain determines the minimum applied strain rate, є, that can be considered. This parameter defines in turn the dislocation velocity. Consider a typical simulation of dislocation-obstacle interaction in an Fe crystal, for which b = 0.248 nm. For L = 41 nm, a model containing 2 x 106 mobile atoms would have a cross-section area of 5.73 x 10-16m2; that is, a dislocation density pD = 1.75 x 1015m-2. For Lg = 120b = 29.8 nm, the model height per­pendicular to the glide plane would be 19 nm. At є = 5 x 106s-1, the steady state velocity, vd, of a single dislocation estimated from the Orowan relation vd = _/Pob is 11.6 m s-1. The time for the dislocation to travel a distance Lg at this velocity would be 2.6 ns. Thus, even if the dislocation breaks away from the void without traversing the whole of the glide plane, the total simulated time would be ~1 ns.

The lowest strain rate for dislocation-obstacle interaction reported so far27 is 105 s-1 and it resulted in vo = 48 cm s- . This strain rate is about six to ten orders of magnitude higher than that usually applied in laboratory tensile experiments and more than ten orders higher than that for the creep regime. This presents an unresolvable problem for atomic-scale modeling and even massive parallelization gains only three or four orders in є or vo We conclude that the possibilities of modern atomic-scale modeling are lim­ited to dislocation velocity of at least ~0.1 cms — .

Nevertheless, atomic-scale modeling, particularly using MD (T> 0 K), is a powerful, and sometimes the only, tool for investigating processes associated with lattice defect interactions and dynamics. The main advantage of MD is that, if applied properly to a large enough system, it includes all classical phe­nomena such as evolution of the phonon system and therefore free energy, rates of thermally activated defect motion, and elastic interactions. It is, therefore, one of the most accurate techniques for investigating the behavior of large atomic ensembles under differ­ent conditions. We reemphasize that the realism of atomic-scale modeling is limited mainly by the valid­ity of the IAP and restricted simulation time.

In the case where 1D migrating SIA clusters are generated during irradiation in addition to PDs, the ME has to account for their interaction with the immobile defects. In the simplest case where the mean-size approximation is used for the clusters, Ggl(x) = Gg8(x — xg), the ME for the defects such as voids or vacancy and SIA loops takes a form22

dfs (x, t) /dt = Gs (x)+ J(x — 1, t)

— J (x, t) — P1D (x)fs (x)

± P1d(x T xg)f s(x T xg), x > 2 [126]

where P1d(x) is the rate of glissile loop absorption by the defects. The ± and T in eqn [126] are used to distinguish between vacancy-type defects (voids and vacancy loops/SFT) and SIA type because capture of SIA glissile clusters leads to a decrease in the size in the former case and an increase in the latter one.

The rate P1d (x) depends on the type of immobile defects. In the case of voids, their interaction with the SIA clusters is weak and therefore the cross-sections may be approximated by the corresponding geomet­rical factor equal to nR2vNY. The rate P1d (x) in this case is given by (see eqn [11c] in Singh eta/.22) 3PPY/3ADgCg 2/3

4 O1/3 x where A = J kg/2. Note that the factor 2 in eqn

[127] was missing in Singh eta/.22

In the case of dislocation loops, the situation is more complicated as the cross-section is defined by long-range elastic interaction. A fully quantitative evaluation is rather difficult because of the compli­cated spatial dependence of elastic interactions, in particular, for elastically anisotropic media. For loops of small size, the effective trapping radii turn out to be large compared with the geometrical radii of the loops and hence the ‘infinitesimal loop approx­imation’ may be applied. It is shown (see Trinkaus et a/.20) that in this case the cross-section is propor­tional to (xxg)1/3 thus the rate P1d (x) is equal to 2=3

ADg Cgx2/3

where T and Tm are temperature and melting tem­perature, the multiplier q is a correction factor which is introduced because eqn [4] in Trinkaus et a/.20 was obtained using some approximations ofthe elastically isotropic effective medium and, consequently, it can be considered as a qualitative estimate of the cross­section rather than a quantitative description. The factor q is of order unity and was introduced as a fitting parameter. Since sessile SIA and vacancy clus­ters have different structures (loops in the case of the SIA clusters and frequently SFTs in the case of vacancy clusters), the multiplier q and, consequently, the appropriate cross-sections may be slightly differ­ent. Also note that qO = Tm has been used in

Trinkaus eta/.20 as an estimate on a homologous basis.

In the case of large size dislocation loops, the cross-section of their interaction with the SIA glissile clusters can be calculated in a way similar to that of edge dislocations. Namely, it is proportional to the product of the length of dislocation line, that is, 2kRi, and the capture radius, by The rate P1d(x) in that case is given by

P1D(x)= Vopbb1ADgCgx1/2 [129]

Note that in the more general case where differ­ent sizes of the SIA glissile clusters are taken into account, the last term on the right side of eqn [126] has to be replaced with the sum

p=*“x

prn(x T y)f (x T y).

j=xm"

Penetration of activated F-R sources into a 3D field of destructible SFTs can be viewed as a per­colation problem, first considered by Foreman19 on a single glide plane, and extended here to complex 3D climb/glide motion. The critical stress above which an equilibrium dislocation configuration is unsustain­able corresponds to the percolation threshold, and is considered here to represent the flow stress of the radiation-hardened material. Since activated F-R sources may encounter nearby dislocations, such interactions should be considered in estimates of the flow stress. The effects of interplanar F-R source interactions on the flow stress in copper irradiated and tested at 100°C are shown in Figure 7. A plastic stress-strain curve is constructed from the computer

simulation data, where the local strain is measured in terms of the fractional area swept by expanding F—R sources on the glide plane. It is shown that, while the majority of the increase in applied stress of irradiated copper can be rationalized in terms of dislocation interaction with SFTs on a single glide plane, dislocation-dislocation dipole hardening can have an additional small component on the order of 15% for very close dislocation encounters on neighboring slip planes (e. g., separated by ~20й). For larger sepa­ration (e. g., ^500й) the additional effects of dipole hardening is negligible. Dislocation forest hardening does not seem to play a significant role in determina­tion of the flow stress, as implicitly assumed in earlier treatments of radiation hardening (e. g., Seeger eta/40 and Foreman19).

The influence of the irradiation dose on the local stress-strain behavior of copper irradiated and tested at 100oC is shown in Figure 8. In the present calculations, we do not consider strain hardening by dislocation — dislocation interactions, and make no attempt to reproduce the global stress-strain curve of irradiated copper. Computed values of the flow stress are in general agreement with the experimental measure­ments of Singh et a/.38 as can be seen from Figure 9. A more precise correspondence with experimental data depends on the value of the critical interaction angle Fc, which is the only relevant adjustable param­eter in the present calculations. In Figure 8, Fc = 165°. Determination of the two adjustable parameters (Fc and B) requires atomistic computer simulations
beyond the scope of the present investigation. Addi­tional calculations for the flow stress for OFHC copper, irradiated at 47 °С and tested at 22 °С, are given in Table 1 and compared with the experimental data of Singh et a/.38 The flow stress value at a dose of 0.001 dpa has been extrapolated from the experimental results of Dai39 for single-crystal copper irradiated with 600-MeV protons. It is noted that, while the general agreement between the computer simulations and the experimental data of irradiated Cu is reason­able at both 22 and 100 °С, the dose dependence of radiation hardening indicates that some other mechan­isms may be absent from the current simulations.

Investigations of dislocation interactions with full or truncated SFTs considered by Sun et a/.37 indi­cated that local heating may be responsible for the dissolution of SFTs by interacting dislocations, and that their vacancy contents are likely to be absorbed by rapid pipe diffusion into the dislocation core. The consequence of this event is that the dislocation climbs out of its glide plane by the formation of atomic jogs, followed by subsequent glide motion of jogged dislocation segments on a neighboring plane. In Figure 10, we present results of computer simulations of this glide/climb mechanism of jogged F-R source dislocations. Figure 11 shows a side view of the glide/climb motion of a dislocation loop pileup, consisting of three successive loops, by projecting dislocation lines on the plane formed by the vectors [111] and [ТІ2]. SFTs have been removed for visualization clarity. It is noted that

for this simulated three-dislocation pileup, the first loop reaches the boundary and is held there, while the second and third loops expand on different slip planes. We assume that the simulation boundary is rigid, and no attempt is made to simulate slip trans­mission to neighboring grains. However, the force field of the first loop stops the motion of the second and third loop, even though the stress is sufficient to penetrate through the field of SFTs. This glide/ climb mechanism of jogged dislocations in a pileup can be used to explain two aspects of dislocation channel formation.

As the group of emitted dislocations expands by glide, their climb motion is clearly determined by the size of an individual SFT. For the densities consid­ered here, a climb step of nearly one atomic plane results from the destruction of a single SFT. The climb distance is computed from the number of vacancies in an SFT and the length of contacting dislocation segments. The jog height is thus variable, but is generally of atomic dimensions for the condi­tions considered here. The width of the channel is a result of two length scales: (1) the average size of an SFT (^2.5 nm); and (2) the F-R source-to-boundary distance (^1-10 pm). Secondary channels, which are activated from a primary channel (i. e., source point), and which end up in a nearby primary channel (i. e., boundary) are thinner than primary channels. Fur­ther detailed experimental observations of the chan­nel width dependencies are necessary before final conclusions can be drawn. The second aspect of experimental observations, and which can also be explained by the present mechanism, is that a small degree of hardening occurs once dislocation channels
have been formed. Dislocation-dislocation interac­tion within the noncoplanar jogged pile up requires a higher level of applied stress to propagate the pileup into neighboring grains.

While the initiation of a dislocation channel is simulated here, full evolution of the channel requires successive activation of F-R sources within the vol­ume weakened by the first F-R source, as well as forest hardening within the channels themselves. The possibility of dislocation channel initiation on the basis of the climb/glide mechanism is further investigated by computer simulation of OFHC cop­per, irradiated to 0.01 dpa and tested at 100 °C, is shown in Figure 12. Formation of clear channels is experimentally observed at this dose level (Singh eta/.38). The figure is a 3D representation for the initial stages of multiple dislocation channel for­mation. For clarity of visualization, the apparent SFT density has been reduced by a factor of 100, since the total number of SFTs in the simulation volume is 3.125 x 10[20] [21] [22] [23] [24] [25] [26] [27]. The initial dislocation den­sity is taken as p = 1013m~ . To show the impor­tance of spatial SFT density variations, a statistical spatial distribution within the simulation volume has
been introduced such that lower SFT densities are assigned near ten glide planes. All dislocation seg­ments are inactive as a result of high density of surrounding SFTs, except for those on the specified glide planes. Search for nearby SFTs is performed only close to active channel volumes, which in this case totals 174 846. It is observed that within a 5-pm volume, the number of loops within a pileup does not exceed 5. It is expected that if the pileup continues across an entire grain (size ~10 pm), a higher number of loops would be contained in a jogged dislocation pileup, and that the corresponding channel would be wider than in the present calculations. We have not attempted to initiate multiple F-R sources within the volume swept by the dislocation pileup to simu­late the full evolution of channel formation. As a result, the channel shape created by a single active F-R source is of a wedge nature. In future simula­tions, we plan to investigate the full evolution of dislocation channels.

The present investigations have shown that two possible mechanisms of dislocation unlocking have been identified: (1) an asymmetric unzipping-type instability caused by partial decoration of dislocations; (2) a fluctuation-induced morphological instability, when the dislocation line is extensively decorated by defect clusters. Estimated unlocking stress values are in general agreement with experimental observa­tions, which show a yield drop behavior. It appears that unlocking of heavily decorated dislocations will be most prevalent in areas of stress concentration (e. g., precipitate, grain boundary, triple point junc­tion, or surface irregularity). Computer simulations of the interaction between unlocked F-R sources and a 3D random field of SFTs have been used to esti­mate the magnitude of radiation hardening and to demonstrate a possible mechanism for the initiation of localized plastic flow deformation and cleared channels. Reasonable agreement with experimental hardening data has been obtained with the critical angle Fc in the limited range of 158-165°. Both the magnitude and dose dependence of the increase in flow stress by neutron irradiation at 50 and 100 °C are reasonably well predicted. In spatial regions of inter­nal high stress, or on glide planes of statistically low SFT densities, unlocked dislocation sources can expand and interact with SFTs. Dislocations drag atomic-size jogs and/or small glissile SIAs when an external stress is applied. High externally applied stress can trigger point-defect recombination within SFT volumes resulting in local high temperatures. A fraction of the vacancies contained in SFTs can
therefore be absorbed into the core of a gliding dis­location segment, producing atomic-size jogs and segment climb. The climb height is a natural length scale dictated by the near-constant size of the SFT in irradiated copper. It is shown by the present com­puter simulations that the width of a dislocation channel is on the order of 200-500 atomic planes, as observed experimentally and is a result of a stress — triggered climb/glide mechanism. The atomic details of the proposed dislocation-SFT interaction and ensuing absorption of vacancies into dislocations need further investigation by atomistic simulations. Finally, it should be pointed out that at relatively high neutron doses, dense decorations of dislocations with SIA loops and a high density of defect clusters/ loops in the matrix are most likely to occur. As shown here, these conditions can lead to the phenomena of yield drop and flow localization.

In the last decade, especially since the development of the density functional theory (DFT), first-principle methods have dramatically improved our knowl­edge of point defect and diffusion properties in metals.69 They provide a reliable way to compute the formation and binding energies of defects, their equilibrium configuration and migration barriers, the influence of the local atomic configuration in alloys, etc. Migration energies are usually com­puted by the drag method or by the nudged elastic band methods. The DFT studies on self-interstitial properties — for which few experimental data are available — are of particular interest and have recently contributed to the resolution of the debate on self-interstitial migration mechanism in a-iron.70,71 However, the knowledge is still incomplete; calcula­tions of point defect properties in alloys remain scarce (again, especially for self-interstitials), and, in general, very little is known about entropic contributions. Above all, DFT methods are still too time consuming to allow either the ‘on-the-fly’ calculations of the migration barriers, or their prior calculations, and tab­ulation for all the possible local configurations (whose
number increases very rapidly with the range of inter­actions and the number of chemical elements). More approximate methods are still required, based on para­meters which can be fitted to experimental data and/or ab initio calculations.

1.18.3.4.1.1 Interatomic potentials

Empirical or semiempirical interatomic potentials, currently developed for molecular dynamics simula­tions, can be used for the modeling of RIS, but two problems must be overcome:

• To get a reliable description of an alloy, the inter­atomic potential should be fitted to the properties that control the flux coupling of point defects and chemical elements. A complete fitting procedure would be very tedious and, to our knowledge, has never been achieved for a given system.

• The direct calculation of migration barriers with an interatomic potential, even though much sim­pler than DFT calculations, is still quite time con­suming. Full calculations of vacancy migration barriers have indeed been implemented in Monte Carlo simulations,72 using massively parallel cal­culation methods, but they are still limited to rela­tively small systems and short times, for example, for the study of diffusion properties rather than microstructure evolution. It is possible to simplify the calculation of the jump frequency, for example, by not doing the full calculation of the attempt frequencies (their impact on the jump frequency must be less critical than that of the migration barriers involved in the exponential term) or by the relaxation of the saddle-point position.73 Malerba et al. have recently proposed another method where the point defect migration barriers of an interatomic potential are exactly computed for a small subset of local configurations, the others being extrapolated using artificial intelli­gence techniques. This has been successfully used for the diffusion of vacancies in iron-copper alloys.74,75 Such techniques have not yet been used to model RIS phenomena, but this could change in the future.