The triple junction hull: Tools for grain boundary network design

Grain boundary engineering (GBE) studies have demonstrated significant materials properties enhancements by modifying the populations and connectivity of different types of grain boundaries within the grain boundary network. In order to facilitate rigorous design and optimization of grain boundary networks, we develop theoretical tools that are based upon a spectral representation of grain boundary network statistics. We identify the connection between a local length scale, embodied by triple junctions, and a global length scale, associated with the grain boundary network configuration as a whole. We define the local state space for triple junctions, A(3)A(3), and enumerate its symmetries. We further define the design space for grain boundary networks, MH(3), characterize its important geometric properties, and discuss how its convexity permits grain boundary network design. We also investigate the extent to which the control of texture alone allows one to probe the full design space.