Percolation theory is now standard in the analysis of polycrystalline materials where the grain boundaries can be divided into two distinct classes, namely “good” boundaries that have favorable properties and “bad” boundaries that seriously degrade the material performance. Grain-boundary engineering (GBE) strives to improve material behavior by engineering the volume fraction $c$ and arrangement of good grain boundaries. Two key percolative processes in GBE materials are the onset of percolation of a strongly connected aggregate of grains, and the onset of a connected path of weak grain boundaries. Using realistic polycrystalline microstructures, we find that in two dimensions the threshold for strong aggregate percolation $c_{\mathrm{SAP}}$ and the threshold for weak boundary percolation $c_{\mathrm{WBP}}$ are equivalent and have the value $c_{\mathrm{SAP}}=c_{\mathrm{WBP}}=0.38\left(1\right)$, which is slightly higher than the threshhold found for regular hexagonal grain structures, $c_{RH}=2\sin (\pi ∕18)=0.347\dots$ . In three dimensions strong aggregate percolation and weak boundary percolation occur at different locations and we find $c_{\mathrm{SAP}}=0.12\left(3\right)$ and $c_{\mathrm{WBP}}=0.77\left(3\right)$. The critical current in high $T_c$ materials and the cohesive energy in structural systems are related to the critical manifold problem in statistical physics. We develop a theory of critical manifolds in GBE materials, which has three distinct regimes: (i) low concentrations, where random manifold theory applies, (ii) critical concentrations where percolative scaling theory applies, and (iii) high concentrations, $c>c_{\mathrm{SAP}}$, where the theory of periodic elastic media applies. Regime (iii) is perhaps most important practically and is characterized by a critical length $L_c$, which is the size of cleavage regions on the critical manifold. In the limit of high contrast $ϵ\to 0$, we find that in two dimensions $L_c\propto gc∕(1-c)$, while in three dimensions $L_c\propto g\exp [b_{0}c∕(1-c)]∕{\left[c(1-c)\right]}^{1∕2}$, where $g$ is the average grain size, $ϵ$ is the ratio of the bonding energy of the weak boundaries to that of the strong boundaries, and $b_0$ is a constant which is of order 1. Many of the properties of GBE materials can be related to $L_c$, which diverges algebraically on approach to $c=1$ in two dimensions, but diverges exponentially in that limit in three dimensions. We emphasize that GBE percolation processes and critical manifold behavior are very different in two dimensions as compared to three dimensions. For this reason, the use of two dimensional models to understand the behavior of bulk GBE materials can be misleading.

• Received 9 August 2004

DOI:https://doi.org/10.1103/PhysRevE.71.026102