The relative free energies of grain boundaries in magnesia as a function of five macroscopic parameters
Introduction
Grain boundaries in materials with centrosymmetric crystal structures can be distinguished on the basis of five macroscopically observable parameters [1]. Three parameters are needed to specify the transformation that brings two adjacent misoriented crystals into coincidence; the remaining parameters specify the orientation of the boundary plane separating the crystals. Based on measurements of dihedral angles at triple junctions where three grain boundaries meet [2], [3], [4], [5], [6] or at thermal grooves where a grain boundary meets a free surface [7], [8], [9], [10], [11], [12], [13], [14], it has been shown that the grain boundary free energy per unit area varies with the macroscopically observable parameters. However, the relative grain boundary energy has not yet been measured as a function of all five degrees of freedom. The basic problem is that there are so many physically distinct boundaries in the five-dimensional domain that if one applies traditional microscopies under human control, it was not feasible to make enough dihedral angle measurements to characterize all of the distinguishable interfaces.
In a companion paper referred to as Part I, an experimental method was described for mapping the geometry of the grain boundary network and measuring the frequency with which different types of grain boundaries occur [15]. This semi-automated technique has been applied to MgO and was used to measure the grain boundary dihedral angles of 1.9×104 triple junctions assumed to be in local thermodynamic equilibrium; these measurements are used here to determine the grain boundary energies. Thus, the purpose of the present paper is to report on the relative grain boundary energy of MgO as a function of the five macroscopic grain boundary parameters. After describing the methods for determining the relative energies in the next section, the characteristics of the five-dimensional energy distribution are presented in Section 3. In Section 4, the observed energies are compared to existing models for the anisotropy of the grain boundary energy and our findings are summarized in Section 5.
Section snippets
Sample
The polycrystalline magnesia sample has been described in Part I [15]. As part of the earlier analysis, we measured the function λ(Δg, n), which is the relative frequency of occurrence of a grain boundary with misorientation (Δg) and boundary plane normal (n), in units of multiples of a random distribution (MRD). These data serve as a basis for the present analysis, whose goal is to determine γ(Δg, n), the relative grain boundary energy per unit area as a function of misorientation and boundary
Results
Results for the low angle boundaries formed by rotations about <1 1 0> are illustrated in Fig. 3. When the relative energies (a) are compared to the population (b), it is clear that there is an approximate inverse correlation between the energies and the populations. The RS [20] model predicts that the energies of grain boundaries with low angle misorientations will vary with the density and energy of the dislocations that compensate for the misfit. While the famous RS dependence of the energy
Discussion
For the case of low misorientation angle grain boundaries, the conventional RS [20] model provides a reasonable explanation for the relatively low energy of the symmetric tilt boundaries on and planes. This is because in crystals with the rock salt crystal structure in general, and in MgO in particular, dislocations with Burgers vectors of are most common. While any boundary can be formed by three dislocations with non-coplanar Burgers vectors, boundaries with the minimum
Summary
In the five-dimensional domain of grain boundary types, the boundary energy and population exhibit an inverse correlation. This implies that during grain growth, higher energy boundaries are preferentially eliminated in favor of low energy boundaries. At low misorientation angles, the distribution and energy of the observed grain boundaries can be explained by the conventional Read–Shockley dislocation model. One overarching characteristic of the distribution in MgO is that at all
Acknowledgements
This work was supported primarily by the MRSEC program of the National Science Foundation under Award Number DMR-0079996.
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Present address: National Institute of Standards and Technology, Gaithersburg, MD, USA.