Correlation between structure, energy, and ideal cleavage fracture for symmetrical grain boundaries in fcc metals
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The misorientation phase space for symmetrical grain boundaries is explored by means of atomistic computer simulations, and the relationship between the tilt and twist boundaries in this three-parameter phase space is elucidated. The so-called random-boundary model (in which the interactions of atoms across the interface are assumed to be entirely random) is further developed to include relaxation of the interplanar spacings away from the grain boundary. This model is shown to include fully relaxed free surfaces naturally, thus permitting a direct comparison of the physical properties of grain boundaries and free surfaces, and hence the determination of ideal cleavage-fracture energies of grain boundaries. An extensive comparison with computer-simulation results for symmetrical tilt and twist boundaries shows that the random-boundary model also provides a good description of the overall structure-energy correlation for both low- and high-angle tilt and twist boundaries. Finally, the role of the interplanar spacing parallel to the grain boundary in both the grain-boundary and cleavage-fracture energies is elucidated. I. INTRODUCTION
One of the motivations for the continued investigation of grain boundaries (GBs) during the past four decades has been to explore the correlation between their physical properties and both the underlying geometrical degrees of freedom (DOFs) and the detailed atomic structure. As is well known, five macroscopic and three microscopic DOFs must be specified to characterize a single GB.1 The three microscopic DOFs are usually represented by a vector, T, which defines translations of the two halves of a bicrystal relative to one another, parallel and perpendicular to the GB plane. These translations are usually ones which minimize the interface energy. By contrast with the macroscopic DOFs, little control can be exerted over these microscopic DOFs or the atomic structure, and the fundamental question really concerns the correlation of physical properties with the five macroscopic DOFs. To understand ideal cleavage fracture of GBs in the Griffith sense,2 the related structure-energy correlation for free surfaces must be understood also. With only two macroscopic and no translational DOFs, free surfaces represent the simplest of all planar defects; yet, with the exception of the densest surface orientations, relatively little work has been reported on the related two-parameter phase space. The problem of exploring phase space for the ideal cleavage-fracture energy of grain boundaries obviously requires the treatment of free surfaces and grain boundaries from a common point of view, a task complicated by the fact that the coincident-site-lattice (CSL) based terminology developed for grain boundaries, including the distinction be1708
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tween tilt and twist boundaries, is not usually applied to free surfaces. One of the goals of this paper is to develop a model for high-angle grain boundaries which includes free surfaces naturally. Combined with a novel way of sampli
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