PHYSICAL REVIEW B 94, 024103 (2016) Ab initio scaling laws for the formation energy of nanosized interstitial defect clusters in iron, tungsten, and vanadium R. Alexander, 1 M.-C. Marinica, 1 L. Proville, 1 F. Willaime, 2 K. Arakawa, 3 M. R. Gilbert, 4 and S. L. Dudarev 4 1 DEN-Service de Recherches de M´ etallurgie Physique, CEA, Universit´ e Paris–Saclay, F-91191, Gif-sur-Yvette, France 2 DEN-D´ epartement des Mat´ eriaux pour le Nucl´ eaire, CEA, Universit´ e Paris–Saclay, F91191, Gif-sur-Yvette, France 3 Department of Materials Science, Faculty of Science and Engineering, Shimane University, 1060 Nishikawatsu, Matsue 690-8504, Japan 4 Culham Centre for Fusion Energy, Culham Science Centre, Abingdon, Oxfordshire OX14 3DB, United Kingdom (Received 14 April 2016; published 6 July 2016) The size limitation of ab initio calculations impedes first-principles simulations of crystal defects at nanometer sizes. Considering clusters of self-interstitial atoms as a paradigm for such crystal defects, we have developed an ab initio–accuracy model to predict formation energies of defect clusters with various geometries and sizes. Our discrete-continuum model combines the discrete nature of energetics of interstitial clusters and continuum elasticity for a crystalline solid matrix. The model is then applied to interstitial dislocation loops with 〈100〉 and 1/2〈111〉 Burgers vectors, and to C15 clusters in body-centered-cubic crystals Fe, W, and V, to determine their relative stabilities as a function of size. We find that in Fe the C15 clusters were more stable than dislocation loops if the number of self-interstitial atoms involved was fewer than 51, which corresponds to a C15 cluster with a diameter of 1.5 nm. In V and W, the 1/2〈111〉 loops represent the most stable configurations for all defect sizes, which is at odds with predictions derived from simulations performed using some empirical interatomic potentials. Further, the formation energies predicted by the discrete-continuum model are reparametrized by a simple analytical expression giving the formation energy of self-interstitial clusters as a function of their size. The analytical scaling laws are valid over a very broad range of defect sizes, and they can be used in multiscale techniques including kinetic Monte Carlo simulations and cluster dynamics or dislocation dynamics studies. DOI: 10.1103/PhysRevB.94.024103 I. INTRODUCTION The ability of materials to sustain extreme conditions, encountered in fusion-plasma confinement reactors or in space exploration, depends on the formation and mobility of clusters of vacancies and interstitial atoms. As such, a study of defects in body-centered-cubic refractory metals and iron provides a foundation for future research in structural materials, and it paves the way for a better understanding of materials ageing. Over the lifetime of reactor components, the mobility of individual defects gives rise to clustering and growth of defect clusters. Vacancies and self-interstitial atoms (SIAs) form either two- or three-dimensional clusters, depending on their size, as a result of competition between the interface and bulk energies, as described by the Gibbs theory of wetting [1]. Vacancy cluster morphology of various bcc metals is fairly well known and exhibits similar behavior. There is a competition between planar loops and voids, as confirmed by experimental observations [2,3]. However, SIA clusters show acutely different properties depending on the bcc material under consideration. Density functional theory (DFT) calculations and other ab initio methods provide quantitative insight into the nature of clusters containing a small number of defects. DFT calculations show that the most stable single SIA in Fe adopts a configuration that corresponds to a 〈110〉 dumbbell, whereas in other bcc transition metals, a single SIA forms a defect aligned along the 〈111〉 direction, known as a crowdion [4–7]. These DFT predictions broadly agree with experiment [8], which makes it desirable to extend predictions to clusters larger than a single SIA. Dumbbells can be packed together in bundles to form small dislocation loops. DFT predicts that in Fe the orientation of these dumbbells changes from 〈110〉 to 〈111〉 depending on the number of SIAs involved. The transition occurs at around five SIAs [9,10]. In Fe, observation of nanometric-sized clusters of SIAs by transmission electron microscopy (TEM) techniques reveals the presence of planar loops, which can adopt either the 1/2〈111〉 (highly mobile) or 〈100〉 (immobile) configurations, depending on temperature [8,11,12]. At high temperature, the magnetic excitations induce elastic instabilities, near the temperature of the α-γ transition, which play a crucial role in the relative stability of the two types of loops. It has been shown that at low temperature 1/2〈111〉 loops are more stable, while at high temperatures (over 700 K) 〈100〉 loops are more stable [13,14]. TEM observations show that in all other bcc metals, dislocation loops with a 1/2〈111〉 Burgers vector are dominant, which suggests that they are the most stable configurations for bundles of dumbbells. Recently, much progress has been made in the experimental field, enabling observation of small 〈100〉 loops in W under heavy-ion irradiation at low temperatures, which vanish at high temperatures [15,16]. The reason why the 〈100〉 loops form in W is still under debate. In the intermediate defect cluster size range, spanning the interval between individual self-interstitial atoms and nanometric-sized dislocation loops, it is difficult to generate experimental data because of the high resolution of observations required to characterize such small objects. According to recent DFT calculations [10], SIA clusters can also form three-dimensional structures with symmetry corresponding to the C15 Laves phase. In Fe, these C15 aggregates are stable, immobile, and exhibit large antiferromagnetic moments. These C15 clusters have been found to form directly inside atomic displacement cascades, and they are able to grow by capturing self-interstitial atoms from the surrounding material. 2469-9950/2016/94(2)/024103(15) 024103-1 ©2016 American Physical Society