Disclinations

A disclination is a defect involving a rotation operation, as opposed to the more familiar dislocation, which is associated with a translation given by its Burgers vector (Friedel 1964). For this reason, this defect, introduced by Volterra at the beginning of the twentieth century in his description of a continuous solid medium, is sometimes called a rotation-dislocation. A disclination can be generated by a so-called ‘Volterra’ process, by cutting the structure along a line and adding (or removing) a sector of material between the two lips of the cut. In two dimensions, this defect is point-like, while it is linear in three dimensions. The two lips of the sector should be equivalent under a rotation belonging to the structure symmetry group in order to get a pure topological defect confined near the apex of the cut (Kléman 1983).

A simple example of disclinations: wedge disclinations in two dimensions

It is possible to describe this defect, and the induced deformation, as a concentration of curvature (figure 4.1). This will be argued, in §4.4, from a differential geometry analysis, but it is possible, in two dimensions, to describe this relation more simply. Let us first do the Volterra construction with a sheet of paper. We first cut it along a straight segment up to its centre. Then, upon rotating around this centre, we can either add or remove a sector, and then glue again along the lips of the cut.