Mobility of small clusters of self-interstitial atoms in dilute Fe–Cr alloy studied by means of atomistic calculations

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Abstract

Atomistic simulations have been used to characterize the interaction and mobility of small clusters of self-interstitial atoms (SIAs) in dilute Fe–Cr alloys. The variety of migration mechanisms for Di- and Tri-SIA clusters in the bcc Fe matrix were studied using the nudged elastic band method. The corresponding binding and migration energies for the SIA clusters interacting with isolated Cr atoms and Cr–Cr close pairs were calculated using the two-band model interatomic potential. The obtained results are discussed in the light of available experimental data for dilute Fe–Cr alloys and are compared with results obtained using ab initio calculations.

Introduction

The mobility and stability of point defect clusters that are formed under irradiation essentially determine the evolution of the microstructure. This work is devoted to the study of the mobility of self-interstitial atom (SIA) defects in Fe in the presence of Cr solutes. Here, we apply atomistic calculations to consider properties of small SIA clusters in dilute Fe–Cr alloys.

Resistivity recovery experiments suggest that in electron-irradiated Fe–Cr alloys containing less than 1%Cr, self-interstitial atoms in the form of stable mixed Fe–Cr dumbbells are created and migrate slightly faster than Fe–Fe dumbbells in pure Fe [1], [2]. This was confirmed by recent ab initio calculations, which have shown that an Fe–Cr 〈1 1 0〉 dumbbell is stable and its migration energy (via movement of Cr) is lower than that of the Fe–Fe dumbbell by ∼0.1 eV [3]. In dilute alloys no significant shift of the position of the peak corresponding to stage II (attributed to the long-range migration of small interstitial clusters) was observed in Fe–Cr alloys containing up to ∼0.1%Cr [1], [2], even though the total resistivity recovered during stage II was slightly higher in Fe [2]. By increasing the Cr concentration up to 3%, the damage retained above stage IE (attributed to the onset of long-range migration of self-interstitial atoms) up to the beginning of stage III (associated with free migration of vacancies) is observed to be higher than in pure Fe and dilute alloys [1]. This effect was ascribed to the trapping of self-interstitials in specific atomic configurations involving more than one Cr atom, which prevents recombination with vacancies [1]. In the concentrated alloys, the features of the stage II (i.e. number of peaks, their amplitudes and positions) strongly depend on the Cr content [4]. Whereas in pure Fe, the onset of Di-SIA migration is believed to determine the stage II [5]. The strength and concentration of traps for single SIAs is believed to be different in concentrated Fe–Cr alloys containing at least up to 16%Cr [4], which is in line with the atomistic calculations presently available [6]. It is therefore clear that the mobility and clustering of small self-interstitial defects differ significantly in dilute and concentrated alloys. In our previous work [7] we have carried out a set of atomistic calculations to validate the ability of the existing two-band model Fe–Cr empirical potential (EP) [8] to predict some important features of self-interstitial – Cr interaction. EP results were compared to data obtained by density functional theory (DFT) and were discussed in accordance with experimental indications. The applied EP was found to provide reasonable and sometimes unexpectedly good agreement with DFT data, we therefore use it in the present work to study the stability and mobility of small interstitial clusters interacting with Cr atoms. The main goal of this work is to see up to what extent the presence of Cr in dilute solution may affect the mobility of Di- and Tri-interstitial clusters that are believed to be responsible for the appearance of stage II, at least in Fe.

According to the atomic-level studies of self–interstitial defects in α-Fe performed up today [9], [10], [11], [12], [13], it is recognized that: (i) the migration of a single SIA in its ground state (i.e. 〈1 1 0〉 dumbbell) occurs via translation–rotation mechanism and the corresponding migration energy (Em) is 0.34–0.37 eV; (ii) Di-SIA (I2), which is a set of two parallel 〈1 1 0〉 dumbbells situated as first nearest neighbours, migrates via the same mechanism with Em = 0.42 eV [10]. In a recent work, focused on the simulation of resistivity recovery during isochronal annealing of electron-irradiated Fe, a combination of event kinetic Monte Carlo and DFT calculations was applied [13]. According to that work, the migration energy and migration mechanisms of the Tri-SIA cluster (I3) are essentially the same as for the I2. Larger clusters were assumed to be immobile in this work, which was essential for the reproduction of the stages III and V. Later on it has been shown that this assumption is not unreasonable because I3, I4, I5 and probably larger clusters may occupy practically immobile configurations [14]. With the help of molecular dynamics simulations, later on confirmed by more accurate DFT calculations, it was found that except for the high symmetry configurations of SIA clusters, some low symmetry (so-called ‘non-parallel’) configurations for I2, I3, I4 and probably larger clusters may also exist. The formation energy of the low symmetry configurations can be even lower than that of the canonical configurations [14]. In addition, the ‘non-parallel’ configurations are furthermore stabilized at finite temperature due to the excess of the vibration entropy [14].

Given that the occurrence of stage II in Fe can be explained by the onset of the long-range migration of the I2 clusters, we do not consider the above-mentioned unusual SIA clusters for the moment. Here, we study the effect Cr atoms on the migration mechanism and on the corresponding migration energy of single SIA and small ‘canonical’ SIA clusters (shown in the upper row of Fig. 1 in [14]) containing up to three self-interstitial defects. The main focus of this work is put on dilute Fe–Cr alloys, therefore we do not consider the interaction of SIA clusters with Cr clusters and include only pairs of Cr that might be formed while the Fe–Cr dumbbell, capable of dragging Cr [3], is migrating in the lattice. To do so we carry out a set of molecular static (MS) calculations to identify ground states and to estimate their migration energy and path applying the nudged elastic band method. In this work, calculations were performed using the two-band model EP from [8], as the number of configurations to be explored is significant and therefore the performance of this parametric study with DFT would be a heavily time consuming task. Some important configurations were however cross-checked against DFT calculations. The EP applied here was already extensively tested in terms of the description of different SIA-Cr and SIA-Cr–Cr configurations [7] and fitted to a number of key properties of the Fe–Cr system obtained from DFT calculations [8].

Section snippets

Simulation technique

The present MS calculations were carried out using the EP of embedded atom method type from [15] and [8] for Fe–Fe and Fe–Cr, Cr–Cr interactions, respectively. The calculation of migration paths was performed at constant volume (the lattice unit of Fe crystal was set to 2.8553 A) in supercells originally containing 1024 Fe atoms. The size of the supercells was varied to establish the convergence of the obtained migration energies, that in turn were calculated using the nudged elastic band (NEB)

Properties in pure Fe

Following the investigation carried out in [9], where a similar potential [19] to the one used here was applied, the migration mode found for a single SIA in Fe is a translation–rotation jump proposed by Johnson [20]. The corresponding size-independent migration energy calculated here using constant volume relaxation conditions is 0.36 eV, in a close agreement with the DFT value (see introduction). The on-site rotation energy from the 〈1 1 0〉 state into 〈1 0 1〉/〈0 1 1〉 or 〈1 1 1〉 orientation is Er = 0.41 

Summary and conclusive remarks

The stability of the Fe–Cr mixed dumbbell and saddle point energy for different migration modes predicted by the used EP are clearly in line with DFT calculations [3] and with the interpretation given for resistivity recovery experiments in dilute Fe–Cr alloys [1], [2], where the shift of stage IE to a lower temperature has been observed. It is important to note that according to the obtained results, the effective energy characterizing the stage IE is the migration energy of an Fe–Fe dumbbell,

Acknowledgements

DT thanks P. Klaver and P. Olsson for providing unpublished DFT data and discussions. DT also thanks C.C. Fu and S. Dudarev for the discussion concerning movement of Di-interstitial cluster. This work was performed in the framework of the FP7 collaborative project GETMAT. This work, supported by the European Community under the contract of Association between EURATOM/Confédération Suisse, was also carried out within the framework of the European Fusion Development Agreement.

References (24)

  • H. Abe et al.

    J. Nucl. Mater.

    (1999)
  • P. Olsson

    J. Nucl. Mater.

    (2009)
  • D. Terentyev et al.

    Comput. Mater. Sci.

    (2008)
  • P. Olsson et al.

    Phys. Rev. B

    (2005)
  • C.C. Fu et al.

    Phys. Rev. Lett.

    (2004)
  • P. Olsson et al.

    Phys. Rev. B

    (2007)
  • F. Maury et al.

    J. Phys. F: Met. Phys.

    (1987)
  • A.L. Nikolaev et al.

    J. Phys.: Condens. Matter

    (1999)
    A.L. Nikolaev

    Phil. Mag.

    (2007)
  • S. Takaki et al.

    Radiat. Eff.

    (1983)
  • D. Terentyev et al.

    J. Nucl. Mater.

    (2009)
  • F. Willaime et al.

    Nucl. Instr. Meth. Phys. Res. B

    (2005)
  • E. Vincent et al.

    J. Nucl. Mater.

    (2006)
  • Cited by (0)

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