Percolation of grain boundaries and triple junctions in three-dimensions: A test of theory

Percolation is a fundamental property of grain boundary (GB) networks that has so far been analyzed through synthetic three-dimensional (3D) microstructures alone. In this work, we visualize an unprecedented 10,265 GBs in a 3D Al-Cu sample using laboratory-based X-ray diffraction tomography (LabDCT). By applying finite-size scaling with the critical exponents from standard percolation theory, we determine from our experimental data a percolation threshold of 0.236 ± 0.039 for the GBs in the thermodynamic limit. We compare our results to those from past simulations, which model the 3D microstructure using space-filling polyhedra; the thresholds show good agreement when they are normalized by the topological characteristics of the GB network. We further investigate the percolation threshold of triple junction (TJ) lines, which we show to be necessarily lower than that of the GBs. The percolation behavior of TJs is also different from that of the theoretical diamond lattice due to two factors: a surprisingly higher coordination of nodes (6.184 vs. 4) and also a spatial clustering of TJs in the microstructure. The insights obtained herein can help inform the design of failure-resistant materials via GB engineering.