We present three dimensional continuum models for the energy and equilibrium dislocation structure of low angle grain boundaries and for the dynamics of low angle grain boundaries incorporating the dislocation structure. The orientation-dependent continuous distributions of dislocation lines on curved grain boundaries are described conveniently using the dislocation density potential functions. In the first part, we present a continuum model to determine the dislocation structure and energy of fixed low angle grain boundaries in three dimensions. The equilibrium dislocation structure is obtained by minimizing the grain boundary energy that is associated with the constituent dislocations subject to the constraint of Frank's formula. In the second part, we develop a continuum model...[ Read more ]

We present three dimensional continuum models for the energy and equilibrium dislocation structure of low angle grain boundaries and for the dynamics of low angle grain boundaries incorporating the dislocation structure. The orientation-dependent continuous distributions of dislocation lines on curved grain boundaries are described conveniently using the dislocation density potential functions. In the first part, we present a continuum model to determine the dislocation structure and energy of fixed low angle grain boundaries in three dimensions. The equilibrium dislocation structure is obtained by minimizing the grain boundary energy that is associated with the constituent dislocations subject to the constraint of Frank's formula. In the second part, we develop a continuum model for the dynamics of grain boundaries in three dimensions that incorporates the motion and reaction of the constituent dislocations. The continuum model includes evolution equations for both the motion of the grain boundary and the evolution of dislocation structure on the grain boundary. The critical but computationally expensive long-range elastic interaction of dislocations is replaced by a projection formulation that maintains the constraint of the Frank's formula describing the equilibrium of the strong long-range interaction. This continuum model is able to describe the grain boundary motion and grain rotation due to both coupling and sliding effects, to which the classical motion by mean curvature model does not apply. Comparisons with atomistic simulation results show that our continuum models are able to give excellent predictions of the energy, dislocation structure, and dynamics of low angle grain boundaries.