Stereological Corrections for Grain Boundary Number Fractions in Three Dimensions
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Stereological Corrections for Grain Boundary Number Fractions in Three Dimensions Bryan W. Reed and Mukul Kumar Materials Science and Technology Division, Lawrence Livermore National Laboratory Livermore, CA 94550 ABSTRACT Two-dimensional (2D) cross sections through three-dimensional (3D) polycrystalline materials present a biased picture of the statistical properties of grain boundary networks. These properties are essential to many practical applications such as grain boundary engineering. We show a simple correction that will partly correct for the sampling biases by removing the effect of the correlation between grain boundary type and grain boundary area. This correction alters number fraction estimates by as much as ~60% for Σ3 boundaries in the highly-twinned copper samples we consider. We also estimate the bias introduced by the correlation between boundary type and boundary shape, which for many materials represents perhaps a 10% shift in the measured statistics, so that the simple method we propose should correct for the majority of the bias in favorable cases. This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor the University of California nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or the University of California, and shall not be used for advertising or product endorsement purposes. INTRODUCTION The statistical laws governing networks of grain boundaries in polycrystalline materials are essential to the large-scale mechanical properties of these materials; advances in grain boundary engineering and related topics in the statistics of polycrystals [1-6] are testament to this fact. With the rise of high-throughput electron backscatter diffraction (EBSD) imaging [7], it is now possible to generate vast amounts of data concerning these statistical laws. This enables empirical testing of ever more sophisticated models of the origin, controlling factors, and consequences of these statistical laws. For example, it has been found that the crystallographic compatibility constraint at a junction of three grain boundaries has very significant effects on long-range properties such as weak boundary percolation thresholds [5]. Still, the great majority of the effort has been restricted to the statistics governing 2D cross sections of 3D materials--in fact in some segments of
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