Full length articleModeling radiation induced segregation in iron–chromium alloys
Graphical abstract
Introduction
Ferritic Fe–Cr steels are good candidates as structural materials for the next generation of nuclear power plants (generation IV and fusion reactors) [1], [2]. The addition of Cr prevents corrosion and the Fe–Cr ferritic form is weakly sensitive to the swelling phenomenon. However, the irradiation flux induces an increase of point defect concentrations – vacancies and self-interstitial atoms (I) – that migrate toward sinks where they are eliminated. These defect fluxes may induce a variation in the alloy composition in the vicinity of sinks called Radiation Induced Segregation (RIS) [3]. If this causes a Cr depletion, the alloy can become sensitive to corrosion. On the other hand, if this leads to a Cr enrichment, the local concentration can reach the solubility limit, causing Radiation Induced Precipitation (RIP), which could lead to embrittlement. RIS has been extensively studied in austenitic steels where Cr depletion and Ni enrichment are frequently observed at sinks. These tendencies are relatively well-understood, even if the interstitial contribution is still under discussion [3]. In ferritic steels, the situation is far from being clear: approximately 15 experiments have been reviewed by Lu et al. [4] in very different materials and conditions, showing both depletions and enrichments in the vicinity of sinks. However, it is difficult to draw clear conclusions on how irradiation conditions and materials properties control the tendency towards enrichment or depletion. A systematic study, focusing on industrial steels with Cr content between 8 and 12% has been performed by Was et al. [5]. They observed that for given irradiation conditions, sinks tend to be enriched in Cr at low temperature and to be depleted in Cr at high temperature. Moreover, the Cr enrichment tends to decrease with the alloy Cr content. These tendencies are reproduced with the inverse Kirkendall model of Wharry et al. [6] and are attributed to a positive coupling of Cr with self-interstitials (dominant at low temperatures) and a negative coupling with vacancies (dominant at high temperatures). Other factors, such as grain boundary misorientation [7], interactions with C or other impurities [7], [8] may influence the segregation of Cr at sinks.
In the present study, we limit ourselves to the Fe–Cr binary system in order to understand basic mechanisms controlling RIS at the atomic scale. Thermodynamics of Irreversible Processes (TIP) shows that a complete description of RIS requires a full determination of the Onsager coefficients Lij, or alternatively, of partial diffusion coefficients dij [3]. Such a consistent description remains to be established for concentrated Fe–Cr alloys. Existing models are based on Density Functional Theory (DFT) calculations and multi-frequency models that are only valid in the dilute limit [9], [10], or have been extended to concentrated alloys using additional approximations [9], [5], [6]. Others use Molecular Dynamics (MD) simulations that rely on empirical potentials and only determine the self-interstitial contribution [11], [12].
We present here a multiscale approach of RIS, starting from DFT calculations of the point defect migration energies. A diffusion model describing the variation of point defect properties with local composition is developed. Atomistic Kinetic Monte Carlo (AKMC) simulations are performed in order to (i) determine the Lij coefficients and predict the RIS tendencies; and (ii) model the evolution of point defect and Cr concentration profiles under irradiation in the vicinity of a grain boundary (GB). AKMC simulations include the effect of non-homogeneous concentration fields and account for the stochastic nature of point defect migration. It also enables the simulation of RIP, when the RIS leads to a strong enrichment of Cr near GBs and therefore to nucleation of α’ precipitates.
In Section 2 we present a reminder of RIS principles in the framework of TIP. In Section 3, the diffusion model and the AKMC method are introduced. The fitting of parameters to DFT calculations is detailed and relevant point defects properties are discussed. In Section 4, the point defect tracer diffusion coefficients and Onsager coefficients are analyzed. The effect of composition and temperature on these Onsager coefficients is studied. Simulations of RIS and RIP near grain boundaries are also shown. We discuss these results by comparing them to previous models and experiments. We conclude by commenting on the limitations of our model, possible improvements and perspectives.
Section snippets
Thermodynamics of irreversible processes and radiation induced segregation
RIS is due to fluxes of vacancies (JV) and self-interstitials (JI) towards point defect sinks that, in an A-B alloy, induce fluxes of chemical species (JA and JB). TIP describes the fluxes J as linear combinations of thermodynamic driving forces, i.e. chemical potential gradients [3]:with Xβ=∇μβ/(kB T). Lαβ are the Onsager coefficients, μβ the chemical potentials, kB the Boltzmann constant and T the temperature.
In steady state, under the conditions where there is no precipitation and
Diffusion model and effective pair interactions
The calculation of point defects jump frequencies is based on a rigid lattice model using effective pair interactions hereafter referred to as the “AKMC model”. We start from a previous version [17] that only includes vacancy diffusion. It assumes that the free enthalpy of a given atomic configuration can be written as a sum of effective pair interactions on a rigid bcc lattice.
Such “broken bond” models with constant pair interactions are widely used in AKMC simulations to model diffusive phase
Point defect tracer diffusion coefficients
We present first diffusion coefficients of point defects which determine the steady state point defect concentration under irradiation. The tracer diffusion coefficients of one vacancy and one interstitial in alloys at nominal concentrations up to 15% Cr and at temperatures between 300 K and 1500 K are presented in Fig. 3 . The vacancy diffusion coefficients slightly increase with the Cr concentration, due to the low Cr–V exchange barrier in Fe.
Self-interstitial diffusion coefficients are
Conclusions
In this study we have analyzed radiation-induced segregation (RIS) in the binary bcc Fe–Cr alloy with a new Atomistic Kinetic Monte Carlo model. The previous RIS model of Wharry et al. [41] uses partial diffusion coefficients defined by single effective diffusion barriers for each element-defect exchange fitted to a mean of migration barriers in a dilute configuration. This model does not exactly take into account the correlations between successive point defect jumps. It is nevertheless
Acknowledgments
We would like to thank E. Clouet for many fruitful discussions. This research has received partial funding from the Eurofusion IReMEV program and from the European Atomic Energy Community 7th Framework Program (FP7/2007-2011), under Grant agreement No. 212175 (GetMat project). This work also contributes to the Joint Program on Nuclear Materials (JPNM) of the European Energy Research Alliance (EERA). DFT calculations were performed using resources from DARI within project x2015096020. E. M.
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