Ab initioscaling laws for the formation energy of nanosized interstitial defect clusters in iron, tungsten, and vanadium

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 1/2111 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 1/2111 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.