Ab initio scaling 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 $\langle 100\rangle$ and $1/2\langle 111\rangle$ 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 $1/2\langle 111\rangle$ 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.

Figure 1

(a) Formation energies of the $1/2\langle 111\rangle$ and $\langle 100\rangle$ dislocation loops in Fe against the number of SIAs, computed (i) from atomic-scale simulations using an EAM interatomic potential [18]; (ii) using predictions based on the law $P_0+P_1{n}^{2/3}$ (see Ref. [25]); (iii) using the anisotropic elasticity theory [see Eq. (5)], which is parametrized with atomic-scale calculations up to $n=21$ SIAs. $P_0$ and $P_1$ are fitted while term $T$ is computed directly from the elastic tensor associated with the EAM potential. (b) The ratio of the convex hull perimeter of the dislocation loops to the perimeter deduced from the discrete number $n$ of SIAs contained in the cluster. The full curve was fitted using the function $\{1-[1/(x^{a_1}+a_2)]\}$, with values of 0.70 and 0.88 for the exponent $a_1$ of $1/2\langle 111\rangle$ and $\langle 100\rangle$ loops, respectively.

Figure 2

Structure of a $\langle 100\rangle$ loop in the $\left\{001\right\}$ habit plane containing 15 SIAs showing the number of first and second nearest neighbors of all dumbbells. Note: The loops were constructed such that each dumbbell has at least one dumbbell in the first-nearest-neighbor position. As a result, the possible number of first nearest neighbors varies from 1 to 4, while the number of second nearest neighbors varies from 0 to 4. There exists just one exception, namely the case of mono-SIA.

Figure 3

Formation energies of $n$ SIAs of (a) $1/2\langle 111\rangle$ and (b) $\langle 100\rangle$ dislocation loops in Fe, computed with an EAM potential [18] with different configurations in cubic simulation cells containing $250+n$ (red circle), $432+n$ (blue square), $686+n$ (orange triangle up), and $1024+n$ (green rhombus) atoms. All energies are normalized to the asymptotic limit, taken as the formation energy in a simulation cell containing $207646+n$ atoms. Lines are guides for the eyes, obtained using a fit to a fourth-order polynomial. Formation energies have been corrected using the elastic dipole correction method to account for the finite size of the simulation cells [47].

Figure 4

(a)–(d) Top: Structure of small C15 interstitial clusters in a bcc lattice of the di-, tetra-, hexa-, and octo-interstitial clusters, in a skeleton representation, i.e., only SIAs are represented as orange spheres without any representation of vacancies and cubic lattice sites. (a)–(d) Bottom: centers of the Z16 Frank-Kasper polyhedron corresponding to the top C15 skeletons are represented by green spheres. (e) The 11 SIA C15 cluster, the lowest size that forms a closed ring with the centers of Z16 Frank-Kaspers polyhedra. This ring is emphasized by blue bonds connecting the centers of Z16 polyhedra.

Figure 5

Cluster formation energies as a function of cluster size for (a) $1/2\langle 111\rangle$ and (b) $\langle 100\rangle$ loops and for (c) C15 SIA clusters in Fe calculated using the Ackland-Mendelev potential for Fe [18]. Open diamonds represent the direct EAM results derived from simulations using large cells, while blue full diamonds represent values predicted by the discrete-continuum model. The relative errors are plotted as insets. Note that for the nanometric clusters, the relative error is less than $3\%$.

Figure 6

(a) DFT formation energies of $1/2\langle 111\rangle ,\langle 100\rangle$, and C15 clusters in Fe (empty circles, squares, and diamonds, respectively), and the DFT-based predictions made using the discrete-continuum model (full circles, squares, and diamonds, respectively). (b) Extrapolation of the formation energies at large sizes for the $1/2\langle 111\rangle$ loops, $\langle 100\rangle$ loops, and C15 clusters in Fe, empty symbols. Full lines represent the elastic model [Eq. (5)] parametrized using the points predicted by the present discrete-continuum model. This extrapolation can be done without size limitation. Note the crossover between $1/2\langle 111\rangle$ loops and the C15 clusters at 51 SIAs, and between $\langle 100\rangle$ loops and C15 clusters at 91 SIAs.

Figure 7

(a) DFT formation energies of $1/2\langle 111\rangle ,\langle 100\rangle$, and C15 clusters in W (empty circles, squares, and diamonds, respectively) and the DFT-based extrapolation from the discrete-continuum model (colored lines). (b) Extrapolation of formation energies at large sizes for the $1/2\langle 111\rangle$ loops, $\langle 100\rangle$ loops, and C15 clusters in W, empty symbols. Full lines represent the elastic model [Eq. (5)] parametrized on the points predicted by the present discrete-continuum model.

Figure 8

DFT and discrete-continuum model predictions for the formation energies of $1/2\langle 111\rangle$ interstitial loops and C15 SIA clusters in V. The same conventions as in Figs. 6 and 7 are used.

Figure 9

Histograms showing the number of occurrence of pairs of dumbbells with respect to the type $(n_1,n_2)$ for all (a) training (small clusters) and (b) validation (small up to large clusters) $\langle 100\rangle$ configurations, where $n_1$ is the number of first nearest neighbors and $n_2$ refers to the number of second nearest neighbors. Possible $(n_1,n_2)$ dumbbell pairs of the type $(0,n)$, where $n=0$–4 were not included due to absence of such pairs in both training and validation configuration sets.

Figure 10

Database configurations of $\langle 100\rangle$ loop type. The cluster dumbbells are projected (and represented) in the $\left\{001\right\}$ habit plane. The color of each dumbbell is assigned according to the number of first and second neighbors, $(n_1,n_2)$. The color assignment map is shown in Fig. 2.

Figure 11

Database configurations of $1/2\langle 111\rangle$ loop type in $\left\{110\right\}$ habit plane, projected in the $\left\{110\right\}$ plane. The same color convention is applied as in Fig. 10.