The Volterra process

Both dislocation and disclination can be produced by the well-known Volterra process. Take a cylinder of a medium and do the following operations in sequence on it  

There is a simple process to produce a disclination rotate the directors on two slips respectively by uq and wo and make lo — luo = w. Thus the same disclination line is produced. The process is named the de Gennes-Friedel process. One can prove that the de Gennes-Friedel process is equivalent to the Volterra process for nematic liquid crystals. The operation Pv of the Volterra process can in fact be divided into the translation and rotation steps, i.e., first, translate the directors (T) and then rotate them around themselves (IV). The latter is actually the de Gennes-Friedel process. In other words 

Because there is no dislocation in nematics which have only translational symmetry, consequently 

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The helical structure of the c-director in the smectic C phase makes the defects different from those in the smectic C phase. As the Volterra process produces a screw dislocation, for example, along the z axis and the Burger vector b = d, it must be accompanied by a parallel wedge disclination in the c-director, in the form... 

In terms of the Volterra process one can visualize the topological features of these disclinations in the following way. Cut the material by a plane that is parallel to the director. The limit of this cut is a line L called... 

The Volterra process for creating a loop, i.e., a closed disclination line, in a nematic is as follows. Let S be the surface enclosed by the loop L. Call the two sides of the surface and S . Rotate the molecules in contact with... 

The Volterra process for creating these disclinations is the same as for nematic disclinations. For the screw disclination the plane of cut is parallel to the cholesteric twist axis while for the edge disclination it is perpendicular to it. 

Several examples of the Volterra process in smectic A phases are indicated in Fig. 16 [2, 3]. The symmetry involved can be a translation normal to the layers, in which case the defect is said to be a dislocation. 

Figure 15. The Volterra process applied to a nematic liquid. A planar section limited by a line normal to the director n allows the two lips Si and S2 to be separated by an angle 7t (b), and nematic material to be added to obtain the disclination structure (c). An initial matter subtraction creates two lips Si and Sj (d), both rotated by an angle kH (e) and restuck to obtain another disclination (0-...

Examples of defect lines created in cholesterics according to the Volterra process are shown in Fig. 17. The core structure will be considered below this being replaced by a narrow cylinder. Figures 17 a and b represent translation dislocations in both cases. 

Figure 17. The Volterra process applied to cholesteric phases. The core structure is masked by a cylinder along the line L. (a, b) Edge and screw dislocation, (c-e) A section S limited by L, normal to the cholesteric axis, allows one to build either the edge dislocation (a) or a disclination (d), as in smectics (Fig. 16d and d"). (f, g) Construction of the opposite disclination. (Drawing made in collaboration with F. Livo-lant).

If we now apply the Volterra process, there are two possible ways to obtain a screw dislocation, as indicated in Fig. 22 b and c, when the length of the Burgers vector is p/2, the half-helicoidal pitch. In a simple model it is assumed that the directors... 

Along a circle nm, directors show a definite distribution that is coherent, with a uniform alignment within the horizontal plane (a"). When the Volterra process is applied along the cylinder axis to create screw dislocations (b or c), the rectangles ABA B are transformed into parallelograms (b or c ) and the distributions of the directors are changed along a transverse circle nm. This leads to two different patterns in the core (b" or c") when the molecules are assumed to remain horizontal. (Drawn in collaboration with F. Livolant.)... 

The core structure of cholesteric discli-nations was interpreted by K16man and Frie-del [2, 3]. The rotation vector considered in the Volterra process is normal to the cholesteric axis and is either parallel to the molecules or normal to them, this resulting in a core structure that is either continuous, with a longitudinal nematic alignment of directors in the core (A disclinations), or discontinuous (t disclinations), with a singular line of the type encountered in non-twisted nematic liquids. 

As for achiral disclinations, the Volterra process may be used to create screw or edge disclinations by cutting parallel or perpendicular, respectively, to the chiral nematic twist axis. 

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