The triple junction hull: Tools for grain boundary network design

Abstract

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) , and enumerate its symmetries. We further define the design space for grain boundary networks, M (3)H , 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.

Texture mediated grain boundary network design in two dimensions

“…An ODF can be expanded as a harmonic series 6,7,20,21,24 and, in the present case, may be expressed as:…”

“…For this reduced set of microstructures each texture (ODF) associates with exactly one grain boundary network (TJDF). This mapping has been published previously for the full 3D case 20,21 ; here, we provide the formula for the 2D case relevant to the present design problem:…”

“…Using a previously developed relationship between crystallographic texture and grain boundary network structure, 20,21 we develop the mathematical tools necessary for a texture mediated approach to grain boundary network design. This mathematical apparatus is exercised in the context of a relatively simple, but illustrative, materials design problem.…”

“…The crystallographic structure of the grain boundary network can be quantified via the triple junction distribution function (TJDF), 20,21,25 denoted T(x 12 , x 23 ), where the quantity T(x 12 , x 23 )dx 12 dx 23 provides the probability of observing a triple junction in the microstructure that is coordinated by an ordered pair of grain boundary misorientations infinitesimally close to (x 12 , x 23 ) (see also Refs. 26 and 27, which describe a related function).…”

“…As a convention we take the first two grain boundary misorientations, (x 12 , x 23 ), as the independent ones that characterize a given triple junction. Like the ODF, the TJDF can be expressed in spectral form, 20,21 and in the present two-dimensional case, may be represented in the basis of bipolar complex exponentials:…”