The coupled motion of grain boundaries and exterior surfaces, which join along "groove roots," is a common phenomena which occurs to some degree in all finite polycrystalline specimens and can be critical in determining the physical properties of the resultant materials. A longstanding, commonly used geometry in studying this coupled effect is the " half-loop" geometry.
The "half loop" geometry consists of a U-shaped, half-loop grain extending entirely though the thickness of an otherwise single crystal.
The two grains are of the same material and differ only in their relative crystalline orientation. The interface between the two grains contacts the exterior surface along a "groove root" where various balance laws hold.This geoemtry, in which one grain grows at the expense of the other, contains two types of motion; one is motion by mena curvature of the grain boundary, and the other is moiton by surface diffusion of the exterior surfaces. The coupling of these moitions can be modelsd by a three-dimenional syst of time-dependent, nonlinear partial diferential equations.
There is a parameter m, defined as the ration of the free energy of the grain boundary to the free energy of the exterior surface, and typically 0