Experimental tests of stereological estimates of grain boundary populations

We present experimental validation of a method for estimating three-dimensional (3-D) relative numerical populations of grain boundaries from measurements of individual two-dimensional (2-D) cross-sections. Such numerical populations are relevant to network topology and the modeling of intergranular failure modes in grain boundary engineered materials, and are distinct from geometrical population measures such as area per volume. We examine 3-D reconstructions of stainless steel and copper, with varying populations of twin-related boundaries, generated by serial-section electron backscatter diffraction and high-energy X-ray diffraction microscopy. We show that 2-D length fractions, 2-D number fractions and 3-D number fractions are all distinct quantities when grain boundary type is correlated with grain boundary size. We also demonstrate that the last quantity may be reliably inferred from the first two, provided the experimental spatial resolution is much finer than the grain size, eliminating the need to use 3-D experimental methods to access at least some information about 3-D network properties. Many of the Σ3 boundaries are extremely complex, with highly re-entrant shapes that can intersect a sample plane many times, giving a false impression of multiple separate boundaries.