An FFT based approach to account for elastic interactions in OkMC: Application to dislocation loops in iron

Abstract

Object kinetic Montecarlo (OkMC) is a fundamental tool for modeling defect evolution in volumes and times far beyond atomistic models. The elastic interaction between defects is classically considered using a dipolar approximation but this approach is limited to simple cases and can be inaccurate for large and close interacting defects. In this work a novel framework is proposed to include "exact" elastic interactions between defects in OkMC valid for any type of defect and anisotropic media. In this method, the elastic interaction energy of a defect is computed by volume integration of its elastic strain multiplied by the stress created by all the other defects, being both fields obtained numerically using a FFT solver. The resulting interaction energies reproduce analytical elastic solutions and show the limited accuracy of dipole approaches for close and large defects. The OkMC framework proposed is used to simulate the evolution in space and time of self-interstitial atoms and dislocation loops in iron. It is found that including the anisotropy has a quantitative effect in the evolution of all the type of defects studied. Regarding dislocation loops, it is observed that using the "exact" interaction energy result in higher interactions than using the dipole approximation for close loops.

Figures (9)

• Figure 1: Elastic interaction energy of a straight dislocation and a SIA in Fe as a function of their relative distance along the $x$ axis using the analytical expressions of Volterra for the straight dislocation field (red line) or using the field obtained numerically by solving the dislocation mechanics problem in FFT for both isotropic (blue squares) and anisotropic (green triangles) matrix.
• Figure 2: Left: Configuration of the straight dislocation and the prismatic loop in the simulation box. Right: Elastic interaction energy of the defects in iron as a function of their relative distance along the $x$ axis using the dipole approximation and analytical expressions of Volterra for the straight dislocation field (red line), or using a full numerical evaluation of the interaction energy based on the FFT approach for both isotropic (blue squares) and anisotropic (green triangles) matrix.
• Figure 3: Left: Configuration of the two dislocation loops in the simulation box. Right: Elastic interaction energy of the two DL in iron as a function of their relative distance along the $x$ axis using an analytical expression from Ref. Dudarev2017 (red line), or using a full numerical evaluation of the interaction energy based on the FFT approach for both isotropic (blue squares) and anisotropic (green triangles) matrix.
• Figure 4: Energy barriers of a moving defect (a) without interaction energy (b) with interaction elastic energy increasing linearly to the right direction. In red the elastic energy, in blue the total energy and the green arrow indicates the effective barriers. In this case, the migration of the particle to position $X_{k-1}$ is more probable.
• Figure 5: Left: Evolution of the coordinates of the loops along the $x$ axis as function of time in the case where they are in the potential well ($x_{relat} = 100$ Å) seen in Fig. \ref{['fig:Eint_DL-DL']}. Right: Evolution of the coordinates of the loops along the $x$ axis as function of time in the case where they are outside the potential well ($x_{relat} = 200$ Å) seen in Fig. \ref{['fig:Eint_DL-DL']}. Symbols represent the mean trajectories obtained over the 10 runs. The envelope of the trajectories of all the loops is illustrated by the corresponding colored areas.