Mesoscale Defect Motion in Binary Systems: Effects of Compositional Strain and Cottrell Atmospheres

The velocity of dislocations is derived analytically to incorporate and predict the intriguing effects induced by the preferential solute segregation and Cottrell atmospheres in both two-dimensional and three-dimensional binary systems of various crystalline symmetries. The corresponding mesoscopic description of defect dynamics is constructed through the amplitude formulation of the phase-field crystal model, which has been shown to accurately capture elasticity and plasticity in a wide variety of systems. Modifications of the Peach-Koehler force as a result of solute concentration variations and compositional stresses are presented, leading to interesting new predictions of defect motion due to effects of Cottrell atmospheres. These include the deflection of dislocation glide paths, the variation of climb speed and direction, and the change or prevention of defect annihilation, all of which play an important role in determining the fundamental behaviors of complex defect network and dynamics. The analytic results are verified by numerical simulations.

Figure 1

Profiles of phase segregation and strain around a dislocation in a 2D triangular crystal with $\overrightarrow{b}=(b_x,0)$: (a)–(c) $\psi (y-y_{\mathrm{core}})$, $\psi (\overrightarrow{r})$, and $U_V=U_{xx}+U_{yy}$ distributions for $\alpha =0.02$ and $\overline{\psi }=0$; (d)–(f) $\psi (y-y_{\mathrm{core}})$, $\psi (\overrightarrow{r})$, and $U_V$ distributions for $\alpha =0.02$ and $\overline{\psi }=0.05$.

Figure 2

Segregation-induced dislocation velocity $v_y(\alpha ,\overline{\psi })$ evaluated from APFC simulations and the analytic result Eq. (11), for (a) triangular and (b) bcc and fcc symmetries.

Figure 3

(a) Trajectories of two 2D edge dislocations in the $G$ configuration, for $\alpha =0.02$ and various values of $\overline{\psi }$, with $\overline{\psi }=0$ corresponding to pure glide. (b) Time evolution of the $y$ position of the upper dislocation in configuration $C$ with $\alpha =0.02$. $\overline{\psi }=0$ corresponds to pure climb. (c) Dislocation velocity as a function of $\overline{\psi }$ as identified from (b).

Figure 4

(a) A network of dislocations at the boundary of an inclusion rotated by 10° about the $[110]$ direction in a bcc crystal [regions shown: $\mathrm{\Phi }<0.85\max (\mathrm{\Phi })$]. (b) Normalized area of the grain boundary as a function of time, for $\alpha =0.02$. (c),(d) Concentration segregation at defects for the dislocation network of (a). The middle inset shows spatial profiles of quantities of (a), (c), and (d) in a defect cross section.