Acta metall, mater. Vol. 41, No. 12, pp. 3525-3534, 1993 0956-7151/93 $6.00 + 0.00 Printed in Great Britain. All rights reserved Copyright © 1993 Pergamon Press Ltd EVALUATION OF COINCIDENCE LATTICE GRAIN BOUNDARY AND INTERFACE GEOMETRY IN OXIDES BY STRUCTURAL THERMODYNAMICS I. SINITSKY, A. MEN and D. G. BRANDON Department of Materials Engineering, Technion Israel Institute of Technology, Haifa 32000, Israel (Received 26 March 1993) Abstract--Theoretical calculations of the relative grain boundary energy for ceramics are presented. A geometric representation consisting of several boundary layers is considered based on specific ionic distributions in the bulk crystals which are in contact at the boundary. The model assumes charge compensation by charged (point) vacancies on each site of an effective lattice. Within the framework of a simple structural description, the influence of both the misorientation angle and the ionic distribution (including the order parameters) on the relative boundary energy has been evaluated for the case of unrelaxed, symmetrical tilt boundaries in MgO-doped v-AI203 . The configurations with minimal energy in the model correspond to selected low Y, (inverse coincidence site density) orientations for typical aluminium ion distributions. A simple twist boundary with (100) rotation (E = 5) has also been simulated. Using the structural thermodynamics approach, the atomic redistribution in the twist boundary layers has been evaluated and a specific spatial ionic configuration is proposed to fill the intermediate boundary layers. 1. INTRODUCTION The theoretical treatment of a grain boundary may be considered through a combination of structural and thermodynamic approaches. A preliminary step of this procedure is the choice of a periodic lattice for the description of the most probable configurations in the boundary layer. This must then be supported by an evaluation of the energy of each configuration by statistical thermodynamics. Possible relaxation of the derived structure could change the whole energy hierarchy of boundary structures significantly, but the lack of reliable inter- atomic potentials for an oxide interface justifies the initial application of a simplified model of a rigid, unrelaxed lattice for these boundary problems. For a joint geometric [1-3] and struc- tural/thermodynamic approach [4, 5], the y-A120 3 system has been chosen. This cubic system has a complex spinel structure [6] with vacancies present in several types of atomic positions, even in the stable state. Dopants may change significantly the details of this structure [7, 8]. It is possible to evaluate the main features of the thermodynamics of doped v-alumina and its boundaries by describing the crystallography in terms of a high-symmetry cubic structure. The same statistical geometry approach can be used to simulate ionic redistribution in the near-boundary layers of a pure twist boundary. 2. EFFECTIVE REPRESENTATION y-alumina is a defect oxide with the spinel structure Fd3m and has six non-equivalent crystallographic positions per unit cell (Fig. 1 and Table 1). For the ideal y-spinel structure, the 8b, 16c, 48f positions are occupied only by vacancies V, the 32e sites only by 02 ions, and the 8a and 16d positions are filled by cations and vacancies in different ways. The effective representation of this system includes (1) mapping of the initial complex lattice onto a more primitive one, and (2) substitution of ions by effective "particles", situated on the lattice sites. By mapping a-, b-, and f-positions onto the tetrahedral sites of a /~-brass (body centered cubic) structure and c-, d-, e-positions onto the corresponding octahedral sites, one can obtain a "uniform matrix" description of ?-alumina, AI~/30l/2 V7/6 A1 O V [ : Oc -6 +~ 2 Te 6+-~ 0 --6+~] where 6 is the long range order parameter which determines the nonequivalence in the AI3+ distri- bution over tetrahedral and octahedral sites: 6 = 0 corresponds to an equal probability of filling tetrahe- dral and octahedral positions by aluminium ions. To simulate the MgO-AI 2 03 crystal lattice in terms of a b.c.c, structure, one needs to represent the NaC1 structure (Fm3m) of MgO as a set of occupied cube vertices and empty cube centers Mg O V E 101 Oc ~ ~ Te 0 0 1 In both cases, each matrix element is the pro- bability, p ~(re) of Occupancy for a given type of position (octahedral or tetrahedral) by either a given ion or a vacancy (~ =A1, Mg, O, V). The close 3525