Equation of Motion for a Grain Boundary

Grain boundary (GB) migration controls many forms of microstructural evolution in polycrystalline materials. Recent theory, simulations, and experiments demonstrate that GB migration is controlled by the motion of discrete line defects or disconnections. We present a continuum equation of motion for grain boundary derived from the underlying discrete disconnection mechanism. We also present an equation of motion for the junctions where multiple grain boundaries meet—as is always the case in a polycrystal. The resulting equation of motion naturally exhibits junction drag—a widely observed phenomena in junction dynamics in solids and liquids.

Figure 1

A GB with disconnections (blue curve) and its continuum representation $y=h(x,t)$ (red curve). The GB velocity $v$ (in the $y$ direction) results from disconnection glide characterized by $(\mathbf{b},H)$ and $(-\mathbf{b},-H)$ in the $x$ direction.

Figure 2

(a) Numerical solution for the evolution of a GB from an initially sinusoidal profile for no externally applied force $\tau =0$ and $\mathrm{\Psi }=0$ and $B=0$. The GB profile is shown for $t=0$, $2t_0$, $6t_0$, $15t_0$, and $\infty$, where $t_0=L/(M_d\gamma )$. (b) The evolution of a GB pinned at two junctions for $\tau =5\times {10}^{-2}\mu$ at $t=0$, $5t_0$, $10t_0$, $15t_0$, and $\infty$ for $B=0.01$ (blue) and $t=0$, $t_0$, $2t_0$, $3t_0$, and $\infty$ for $B=0.1$ (red).

Figure 3

Illustration of TJ motion (red arrow) through disconnection fluxes from three GBs.

Figure 4

(a) Equilibrium GB profiles for $A/(M_{d}BH/d)=0$ (blue), 67.6 (green), 135.2 (red), and $\infty$ (black), at an applied shear stress $\tau =5\times {10}^{-2}\mu$. (b) The steady-state GB velocity (blue line) and angles, ${\theta }_{\wedge }$ and ${\theta }_{\vee }$, as functions of $A$. The red solid (dashed) lines are the angles at equilibrium states for $\tau =5\times {10}^{-2}\mu$ (${10}^{-2}\mu$).