Analysis of Experimental Grain Boundary Distributions Based on Boundary-Space Metrics
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ariety of properties of polycrystalline materials are affected by grain boundaries. To explore relationships between boundary structures and material properties, the boundaries need to be investigated at both atomic and ‘‘macroscopic’’ levels. Studies at the atomic scale are limited by experimental capabilities, but the macroscopic boundary parameters (i.e., misorientations between neighboring grains and directions of boundary plane normals[1]) can be relatively easily determined. Experimental methods of three-dimensional microstructure characterization have been improved greatly over the last decade, and large sets of boundary parameters are being collected, e.g., References 2, 3. The sizes of resulting data sets allow for statistical analyses of boundaries. One of the most basic statistical characteristics of a boundary network is the distribution of grain boundaries with respect to the macroscopic boundary parameters. In relevant reports published so far (e.g., References 4 through 8), the distributions have been computed using a method[4] based on partition of a certain domain in the boundary parameter space into equivolume bins. Although this method has been successfully applied to various materials, it has deficiencies leading to artifacts in computed distributions, and
KRZYSZTOF GLOWINSKI, Ph.D. Student, and ADAM MORAWIEC, Associate Professor, are with the Institute of Metallurgy and Materials Science, Polish Academy of Sciences, Reymonta 25, 30-059 Krakow, Poland. Contact e-mail: [email protected] Manuscript submitted November 7, 2013. Article published online May 14, 2014 METALLURGICAL AND MATERIALS TRANSACTIONS A
complicating estimation of the reliability of the distributions. This note presents an alternative approach to computation of the boundary distributions. Suggestions given in Reference 9 are followed to adapt the kernel density estimation technique and to replace the partition of the boundary space by probing the distributions at selected points and counting boundaries that are not farther from those points than an assumed limiting distance defined in the boundary space. It is shown that this change of the computation method leads to significant improvements in the quality of resulting distributions. The new method also allows for a direct estimation of the reliability of the distributions. In the following, deficiencies of the hitherto used approach are discussed. Then, the new approach is described and confronted with the old one. Both methods are applied to grain boundary data of a nickel-based superalloy. For simplicity, only cubic ðm3mÞ crystal symmetry is considered; similar analysis can be performed for other holohedral symmetries. The grain misorientations and boundary plane normals are usually parameterized by Euler angles u1 ; U; u2 and spherical (polar and azimuth) angles # and w, respectively. With cubic crystal symmetry, the parameter domain used in the partition-based approach is restricted by 0 deg u1 ; U; u2 ; # 90 deg and 0 deg w 360 deg. The ‘‘rectangular’’ box u1 cos
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