AKMC simulations can be used to follow the atomic configuration of a finite-sized system, starting from a given initial condition, by performing successive

Figure 9 Comparison of Cr segregation profiles as a function of irradiation dose in FeNi12Cr19 at T = 635 K. Left cell represents typical experimental results of Busby et al. ,36 right cell is mean-field predicted result starting from the experimental profile observed just before irradiation. Reproduced from Nastar, M. Philos. Mag. 2005, 85, 641-647.

point defect jumps.16’17’126’127 Then, the migration barriers are exactly computed (in the framework of the used diffusion model) for each atomic configura­tion using equations such as [22] or [23], without any mean-field averaging. The jump to be performed can be chosen with a residence time algorithm,128,129 which can also easily integrate creation and annihila­tion events.16

Correlation effects between successive point defect jumps, as well as thermal fluctuations, are naturally taken into account in AKMC simulations; this pro­vides a good description of diffusion properties and of nucleation events (the latter being especially impor­tant for the modeling of RIP). The downside is that they are more time consuming than mean-field models, especially when correlation or trapping effects are significant. These advantages and drawbacks explain why AKMC is especially useful to model the first stages of segregation and precipitation kinetics.

AKMC simulations were first used to study RIS and RIP in model systems,16,17 to highlight the con­trol of segregation by point defect migration mecha­nism, and to test the results of classical diffusion equations. These studies show that it is possible — in favorable situations, where correlation and trapping effects are not too strong — to simulate microstruc­ture evolution with realistic dose rates and point defect concentrations, up to doses of typically 1 dpa.

In simple cases, AKMC simulations can validate the predictions of continuous models, on the basis of simple diffusion equations17,16: an example is given in Figure 10 for an ideal solid solution, that is, when RIS can be studied without any clustering or ordering tendency.16 In this simulation, the diffusion of

A atoms by vacancy jumps is more rapid than that of B atoms, and one observes an enrichment of B atoms at the sinks due to the IK effect (A and B atoms diffuse at the same speed by interstitial jumps; those jumps therefore do not contribute to the segre­gation). One can notice the nonmonotonous shape of the concentration profile, which corresponds to the prediction of Okamato and Rehn29 when the partial diffusion coefficients are dBV/dAV < dBI/du. In the more complicated case of a regular solution, Rottler et a/.17 have shown that the RIS profiles of AKMC simulations can be reproduced by a continu­ous model using a self-consistent formulation, which gives the dependence of the partial diffusion coeffi­cients with the local composition.

In alloys with a clustering tendency, AKMC simu­lations have been used to study microstructure evo­lution when RIS and precipitation interact, either in under — or supersaturated alloys. The evolution of the precipitate distribution can be quite complicated as the kinetics of nucleation, growth, and coarsening depend on both the local solute concentrations and the point defect concentrations (which control solute diffusion), concentrations that evolve abruptly in the vicinity of the sinks.16 A case of homogeneous RIP, due to a mechanism similar to the one proposed by Cauvin and Martin52 (see Section 1.18.2.4), has been simulated with a simplified interstitial diffusion model.130131

The application of AKMC to real alloys has just been introduced. Copper segregation and precipitation in a-iron has been especially studied, because of its role in the hardening of nuclear reactor pressure vessel steels.118’126’127’132’133 These studies are based on rigid

10-4

10-5

10-6

10-7

10-8

10-9

10-10

10-11

10-12

10-13

Figure 10 (a) Evolution of the B concentration profile in an A10B90 ideal solid solution under irradiation at 500 K and 10—3dpas—1 when dAV > dBV and dAi = dBi; (b) Point defect concentration profiles in the steady state. Reproduced from Soisson, F. J. Nucl. Mater. 2006, 349, 235-250.

Figure 11 Concentration profile and formation of small copper clusters near a grain boundary, in a Fe-0.05%Cu alloy under irradiation at T = 500 K and K0 = 10—3dpas—1

lattice approximations, using parameters fitted to DFT calculations. The point defect formation energies are found to be much smaller in copper-rich coherent clusters than in the iron matrix,79,118 and there is a strong attraction between vacancies and copper atoms in iron, up to the second-nearest neighbor posi — tions.70,125 AKMC simulations show that in dilute Fe-Cu alloys, the Lcuv = — [Lcucu + LcuFe] is positive at low temperatures, because of the diffusion of Cu-V pairs. At higher temperatures, Cu-v pairs dissociate and LcuV becomes negative.118,119 The resulting segre­gation behaviors have been simulated, with homoge­neous formation of Frenkel pairs (i. e., conditions corresponding to electron irradiations). Only vacancy fluxes are found to contribute to RIS; copper concen­tration increases near the sinks at low temperatures and decreases at high temperatures.118 In highly supersatu­rated alloys, RIS does not significantly modify the evo­lution of the precipitate distribution, except for the acceleration proportional to the point defect supersat­uration. Figure 11 illustrates a simulation of RIS in a very dilute Fe-Cu alloy, performed with the para­meters of Soisson and Fu118,125; the segregation of copper at low temperatures produces the preferen­tial formation of small copper-rich clusters near the sinks, which could correspond to the beginning of a
heterogeneous precipitation. However, simulations are limited to very short irradiation doses because of the trapping of defects as soon as the first clusters are formed. This makes it difficult to draw conclusions from these studies.

Coprecipitation of copper clusters with other solutes (Mn, Ni, and Si) has been modeled by Vincent et a/.126,127 and Wirth and Odette133 under irradiation at very high radiation fluxes and with formation of point defects in displacement cascades. AKMC simulations display the formation of vacancy clusters surrounded by copper atoms, which could result both from the Cu-V attraction (a purely thermodynamic factor) and from the dragging of Cu by vacancies (effect of kinetic coupling). The forma­tion of Mn-rich clusters is favored by the positive coupling between fluxes of self-interstitials and Mn (DFT calculations show that the formation of mixed Fe-Mn dumbbells is energetically favored).

1.18.4 Conclusion

We started this review with a summary of the exper­imental activity on RIS. Intensive experimental work has been devoted to austenitic steels and its model fcc alloys (Ni-Si, Ni-Cr, and Ni-Fe-Cr) and, more recently, to ferritic steels. A strong variation of RIS with irradiation flux and dose, temperature, composition, and the grain boundary type was observed. One study tried to take advantage of the sensitivity of RIS to composition to inhibit Cr depletion at grain bound­aries. A small addition of large-sized impurities, such as Zr and Hf, was shown to inhibit RIS up to a few dpa in both austenitic and ferritic steels. On the other hand, the Cr depletion at grain boundaries was observed to be delayed when the grain boundary was enriched in Cr before irradiation. A ‘W-shaped’ transitory profile could be maintained until a few dpa before the grain boundary was depleted in Cr. The mechanisms involved in these recent experiments are still not well understood, although RIS model development was closely related to the experimental study.

The main RIS mechanisms had been understood even before RIS was observed. From the first models, diffusion enhancement and point defect driving forces were accounted for. The kinetic equations are based on general Fick’s laws. While in dilute alloys one knows how to deduce such equations from atomic jump frequencies, in concentrated alloys more empir­ical methods are used. In particular, the definition of the partial diffusion coefficients of the Fick’s laws in terms of the phenomenological L-coefficients of TIP has been lost over the years. This can be explained by the lack of diffusion data and diffusion theory to deter­mine the L-coefficients from atomic jump frequencies. For years, the available diffusion data consisted mainly of tracer diffusion coefficients, and the RIS models employed empirical Manning relations, which approximated partial diffusion coefficients based on tracer diffusion coefficients. However, recent improve­ments of the mean-field diffusion theories, including short-range order effects for both vacancy and intersti­tial diffusion mechanism, are such that we can expect the development of more rigorous RIS models for concentrated alloys. It now seems possible to overcome the artificial dichotomy between dilute and concen­trated RIS models and develop a unified RIS model with, for example, the prediction of the whole kinetic coupling induced by an impurity addition in a con­centrated alloy. Meanwhile, first-principles methods relying on the DFT have improved so fast in the last decades that they are able to provide us with activation energies of both vacancy and interstitial jump frequencies as a function of local environment. Therefore, it now seems easier to calculate the L — coefficients and their associated partial diffusion coef­ficients from first-principles calculations rather than estimating them from diffusion experiments.

An alternative approach to continuous diffusion equations is the development of atomistic-scale simu­lations, such as mean-field equations or Monte Carlo simulations, which are quite appropriate to study the nanoscale RIS phenomenon. Although the mean-field approach did not reproduce the whole flux coupling due to the neglect ofcorrelation effects, it predicted the main trends of RIS in austenitic steels, with respect to temperature and composition, and was useful to under­stand the interplay between thermodynamics and kinetics during the formation of an oscillating transi­tory profile. Monte Carlo simulations are now able to embrace the full complexity of RIS phenomena, including vacancy and split interstitial diffusion mechanisms, the whole flux coupling, the resulting segregation, and eventual nucleation at grain bound­aries. But these simulations become heavy time­consuming when correlation effects are important.

roS

1 + oS + ko(1 — S) 3

_______ rp3________

1 + oS + ko(1 — S) r(1 + oS)

1 + oS + ko(1 — S) 3r(1 + oS)

1 + oS + ko(1 — S) 3r(1 + oS)

1 + oS + ko(1 — S) r(1 + o)S

[2] + oS + ko(1 — S)

ro(1 — S)S

1 + oS + ko(1 — S)

6r2u m (1 — S)

[6] Chiang, Y. M.; Birnie, D.; Kingery, W. D. Physical Ceramics: Principles for Ceramic Science and Engineering; MIT Press: Cambridge, 1997.

[7] Kittel, C. Introduction to Solid State Physics; Wiley:

New York, 1996.

[8] keV, the exponent is 0.75. This is marginally lower than the value in eqn [2], possibly because the 20 keV data were used in the current fitting. An exponent of 1.03 was found in the range above

20 keV, which is dominated by subcascade formation. Only in the highest energy range do the MD results

approach the linear energy dependence predicted by the NRT model. The range of plus or minus

one standard error is barely detectable around the

data points, indicating that the change in slope is statistically significant.

The data from Figure 9(a) are replotted in

Figure 9(b) where the number of surviving displace­ments is divided by the NRT displacements at each energy. The rapid decrease in this MD defect survival

Dvexp(—Evcl/kB T) (k( + Zdpd) — £gGv

[16]

[17]

where Evcl is an effective binding energy of vacancies with the vacancy clusters and x[[d м are the mean size of the vacancy and SIA glissile clusters (see eqn [8]).

It should be noted that the SDF of sessile intersti­tial clusters, the sink strength of which is described by eqn [132], is limited by the maximum size of the clusters produced in displacement cascades (see Figure 2 in Singh eta/.22). This is because the clusters

[20] mm

Figure 12 Three-dimensional view for the formation of

dislocation channels on glide planes which have low

stacking fault tetrahedron (SFT) density, for irradiated

copper at a dose of 0.01 dpa. For clarity of visualization, the

apparent SFT density has been reduced by a factor of 100.

Note that other dislocation segments are inactive as a result

of the high density of surrounding clusters.