How liquid crystals handle conflicting boundary conditions

Two weeks ago, we discussed nematic films: thin layers of liquid crystals in solution, dropped on a substrate. These thin films are pretty to look at in polarized light, but they also teach us a lot about the behavior of the molecules in solution. Because the thin films can be easily manipulated with electric and magnetic fields, or changes in temperature and boundary conditions, they are excellent experimental territory.

This is interesting physics not only because we use liquid crystals and other types of soft matter in many every-day applications, but also because of their closeness to biological systems: Most of your body is molecules in solution, and most of your body’s processes depend on the organization and interaction of these molecules. Granted, most molecules in biological systems are larger and more complex than the molecules in these thin layers, but one has to start somewhere.

So let us look again at the image of the nematic film that we have seen earlier. Note the not quite regular stripes. Interesting.

Image source: arXiv:1010.0832 [cond-mat.soft]

This, it turns out, is not the only type of regular structure that one can find in nematic films if they are thin enough. Sometimes one also finds little squares.

Source. Image Credits: Oleg Lavrentovich

This behavior has puzzled physicists since it was first observed, almost 20 years ago. Especially it has not been understood theoretically at which thickness of the film such modulations start to appear and what determines their size. The width of the film is typically at least 20 times or so larger than the length of the molecules, so this cannot be the relevant scale.

This puzzle is what Oksana Manyuhina, now a postdoc at Nordita, and collaborators studied in their paper

Instability patterns in ultrathin nematic films: comparison between theory and experiment
O. V. Manyuhina, A.-M. Cazabat and M. Ben Amar
Eur. Phys. Lett. 92, 16005 (2010)
arXiv:1010.0832 [cond-mat.soft] 

The neat thing is how straight-forward their analysis is.

The nematic films are described by a vector field for the molecules orientation. The direction of the vector field does not matter, only its orientation. The system tries to minimize energy, which depends on the orientation of neighboring molecules relative to each other. Up to 2nd order in derivatives of the vector field there is a handfull of terms that can be written down with constants to parameterize their relative strength.

The relevant new ingredient to understand the structures in the thin films are boundary terms. The substrate below the film and the air above it have different chemical properties that lead to conflicting preferences for the molecules: At the liquid interface the molecules want to be parallel to the surface while at the air interface they want to be orthogonal to the surface.

Surface terms had been investigated before to account for the appearance of quasi-periodic structures, but without success. It was found instead that there should be structures at arbitrarily long wavelengths, in conflict with what the experiments show. In the above mentioned paper now a new term was added that introduces an energy penalty for relative angles between neighboring molecules at the plane of the interface. This has the effect that solutions for the vector field are no longer isotropic in the plane.

Once the expression for the energy is written down, one considers a perturbation of the system by rotating each molecule by a small angle, and does a linear stability analysis. This way one finds the energetically preferred configuration, at least as long as the linear approximation is good. And indeed, these configurations show a quasi-periodic behavior that sets in at some specific width of the film! Exactly when it sets in depends on the coupling constant in front of the new term, which can be nicely fitted with the data.

Below is a schematic image of how the molecules try to arrange themselves with the conflicting boundary conditions. You have to imagine the liquid substrate on bottom and air on top. The little rods represent the molecules of the liquid crystal. Note how, at the bottom, they are parallel to the surface while at the top they alternate between trying to remain parallel to the lower layers and trying to be orthogonal to the air interface – that is what causes the quasi-periodic structures.

Image credits: Oksana Manyuhina

This is such a nice example for how theoretical physics is supposed to work: An experimental result that can’t be explained. A mathematical model for the system, and an analysis that shows it can correctly describe the observations. We learned in this process about the relevance of boundary conditions, and that one should keep in mind configurations of a system need not respect the symmetries of the Hamiltonian (here: isometry in the plane parallel to the substrate).