We’ve studied pattern formation theoretically in a variety of liquid crystal systems; this post describes one example.
Pentacyanobiphenyl (5CB) is a typical liquid crystal compound: it comprises a rigid core with a flexible tail. At sufficiently high temperatures, 5CB is a normal isotropic fluid. Upon cooling, however, it enters a nematic phase where the molecules tend to align on average.
Other compounds in the same homologous series as 5CB, such as Octa-cyanobiphenyl (8CB) also exhibit smectic phases, where the molecules align and also tend to form layers. The layer spacing is comparable to the molecular length (a few nanometres).
The molecular alignment is illustrated schematically in the following diagrams, where the individual molecules are represented by lines. Notice the alignment present in both schematics and additionally the layers present in the smectic phase.
The local alignment orientation of a nematic liquid crystal can be controlled at a surface and, moreover, can vary in space in an arbitrary manner.
The molecular layers in a smectic are nearly incompressible which means that, unlike for nematics, there is a very strong constraint on the structures the liquid crystal can adopt: the only permitted structures are those with constant layer spacing. Only a few families of mathematical surfaces are compatible, which are illustrated below.
One suitable structure, the Focal Conic Domain, is illustrated in the following video. The structure is assembled from a family of torii. Note that only a few layers are depicted here; real smectic structures contain many thousands of layers.
Some patterns formed by these materials are shown here. Spatial variation of the local alignment orientation of a nematic liquid crystal be seen in this thin (5micron) nematic film viewed with a polarizing microscope. The changes in color and intensity occur because the liquid crystal is birefringent and therefore rotates the plane of polarization of incident polarized light depending on its orientation. The film is viewed through crossed-polarizers to observe these variations.
The same region of the nematic film as above is now cooled to just above the nematic-smectic transition. Small regions of pre-transitional smectic order form in the liquid crystal film. These tend to resist certain types of spatial variation of the alignment orientation that are forbidden in the smectic phase due to the layer constraint discussed above. The system is frustrated, and the frustration is resolved by the adopting the beautiful striped pattern shown.
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