Geometrical model for the energy of semicoherent interphase interfaces

Abstract

The basis for the considerations given in this paper is the 0-lattice description of crystalline interfaces of Bollmann. In the development of his approach presented here, all possible interfacial planes between two crystalphases having a defined orientation relationship are considered. The energies of these interfaces are then computed in terms of the energies of the primary intrinsic dislocations. A number of modeling interactions are incorporated into this approach, and a better agreement with experimental data is thus obtained.The O-lattice theory of crystal interfaces of Bollmann (1) is the most general geometrical treatment to have been developed. A natural extension of the Coincidence Site Lattice approach to the structure of homophase boundaries, it may also be used to consider the structure of interphase interfaces. However, because coincidence site lattices are not necessarily generated in the two-phase case, the O-lattice theory is at present basically restricted to the consideration of semicoherent (2) interphase interfaces, described in terms of the primary O-lattice. These interfaces are of considerable importance (3). Bollmann has suggested a geometrical parameter (1) that varies monotonically with the energy of such semicoherent interfaces. His parameter, P, has been used to compare the relative favorabilities of the three special interfacial planes that may occur between two crystal phases in a particular orientation relationship (O.R.). This particular O.R. defines the 0-lattice cell, and it is the faces of this cell that are taken to be the three special interfacial planes. The present study extends these arguments such that the assumption that one of these special interfacial planes has lowest energy is not made. Rather, the modeling has been performed in such a way as to include the possibility of computing the energy of any interfacial plane between two phases having a defined O.R. We show that this extension to the Bollmann approach removes some of the difficulties in his predictions and produces results that more closely match experimental observations. The 0-lattice method The 0-lattice is formed from the interpenetration, with the relevant O.R., of the two crystal lattices. The coincidences of equivalent interstitial points (rather than only lattice points as in the coincidence site lattice approach) are designated O-lattice points. The O-points constitute regions of good geometrical fit between the two crystal lattices. The misfit between O-points is considered to be localized into cell walls, which form dislocations when sectioned by the interfacial plane.Mathematically, the 0-lattice is produced as follows. Let the transformation between the crystal lattices 1 and 2 be the matrix A so that X2 = Ax, (1 = xl + t = xO and (I -A-') xo = Tx. = t (I is the identity matrix). Now the vectors t represent the discontinuities in the match between the interpenetrating crystal lattices, and it is therefore at this juncture that the Burgers vectors have to be chosen. The determinant of the...

Computational design of patterned interfaces using reduced order models

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Crystallography and Energetics of Second Phases and Interfaces

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Models and observations of fcc/bcc interfaces

“…However, a consideration of the experimental and theoretical work of Matthews (1974Matthews ( , 1976 on the energy of [OlO] twist boundaries between orthorhombic Moo3 smoke crystals by Ecob & Ralph (1980) shows that P does not in fact vary in the same way as the interfacial energy. In particular, the P parameter does not reflect the way in which the energy is minimized at misorientations of 90" by two arrays of edge dislocations and at (90kfl (where-the angle f is such as to produce an invariant line parallel to [loll in one crystal and [loll in the other) by a single array of screws albeit of finer spacing than the edge configurations at 90".…”

“…In particular, the P parameter does not reflect the way in which the energy is minimized at misorientations of 90" by two arrays of edge dislocations and at (90kfl (where-the angle f is such as to produce an invariant line parallel to [loll in one crystal and [loll in the other) by a single array of screws albeit of finer spacing than the edge configurations at 90". A different parameter, R , is proposed by Ecob & Ralph (1980) which, in the MOO, case, does vary like the interfacial energy, and moreover renders the assumption of favoured interfaces being parallel to 0-lattice unit cell surfaces self-consistent. It is to be noted that the predicted morphology is thus expected to be the same as the shape of the 0-lattice unit cell.…”