On the schlieren texture in nematic and smectic liquid crystals

Observations on schlieren textures of nematic and smectic-C liquid crystals in thin layers are reported. The theory of schlieren textures is discussed and forces between singularities are considered. A singularity can be characterized by a positive or negative number which is proportional to the number of brushes it shows between crossed polarizers. In smectic-C, only singularities with four brushes are observed, nematics may in addition show singularities with two brushes. The latter singularities lead to conclusions on the molecular order and alignment: without the formation of inversion walls they can only occur in the absence of polarity and when the molecules align parallel to the surface. Singularities of opposite sign attract each other, those of equal sign repel each other. The sum of the characteristic numbers of the singularities in an extended layer tends to be zero. Under special assumptions about the elastic constants, it can be shown that the forces between singularities are additive and inversely proportional to the distances and that, for an isolated pair of singularities with the same absolute characteristic number but with opposite sign, the curves of equal molecular alignment are circles through the singularities.