The five-dimensional parameter space of grain boundaries

Abstract

To specify a grain boundary at a macroscopic length scale requires the specification of five degrees of freedom. We use a specification in which three degrees of freedom associated with the boundary misorientation are in an orthogonal subspace from two associated with the mean boundary plane. By using Rodrigues vectors to describe rotations, we show how paths through these subspaces may be characterized. Some of these paths correspond to physical processes involving grain boundaries during microstructural evolution. Exploiting the orthogonality of the subspaces, a metric to measure ‘distance’ between two boundaries is defined in terms of the minimum set of rotations required to map one boundary on to the other. We compare our metric with others that have appeared. The existence of rotational symmetry in face-centred cubic crystals leads to as many as 2304 equivalent specifications of a boundary. We illustrate this multiplicity of descriptions for the (111) twin and a more general boundary. We present an algorithm to evaluate the geodesic distance between two boundaries, and apply it to identify the path along which the distance between these two boundaries is minimized. In general, the shortest path does not involve descriptions of boundary misorientations with the smallest misorientation angles.

Response to commentary by Morawiec

“…There is no need, and it would be quite wrong, to consider a further contribution to the metric from changes in the boundary normals in this case because those changes have been accounted for already in eqns (2.1) and (2.2). It should be noted that the information content of eqns (2.1) and (2.2) of ref [2] is exactly the same as that embodied in n = ρ ⋆ n ′ ⋆ (−ρ) = Rn ′ , where R is the rotation matrix corresponding to ρ. But the advantage of eqns (2.1) and (2.2) is the explicit vector relationships they provide between the boundary normals and the five degrees of freedom associated with the normal to the mean boundary plane and the misorientation between the adjoining crystals.…”

“…In reality there are boundaries at θ = π with their normals at any angle to the rotation axis. They do not appear to be described by eqns (2.1) and (2.2) of ref [2] because as |ρ| → ∞ either the boundary normals become parallel and anti-parallel to N × ρ, or they are parallel to ρ if N is parallel to ρ. This apparent behaviour is illustrated by the example of Morawiec's Appendix A in ref [1].…”

“…In section 5(a) of [2] we noted that Morawiec's metric does not separate these two operations. To achieve this separation we have to replace…”

“…In ref [2] changes in the inclination are effected through changes in the mean boundary plane N. Changes in the misorientation (both axis and angle) are effected through changes in the Rodrigues vector ρ. All possible boundaries with a given misorientation ρ are generated by allowing N to range over the radius vectors of a sphere 1 .…”

“…In eqns (2.1) and (2.2) of ref [2] the boundary normals n and n ′ are parametrically related to an arbitrary mean boundary plane normal N and the Rodrigues vector ρ describing the rotation axis and angle. As we vary the two degrees of freedom associated with the direction of N and/or the three degrees of freedom associated with ρ the changes in the boundary normals are prescribed by eqns (2.1) and (2.2) of ref [2]. In this way eqns (2.1) and (2.2) define paths through the 5D space.…”