Carme Calderer

** Mathematical studies of liquid crystals**

**ABSTRACT**. Discoveries of new materials and their novel
applications have brought fascinating new perspectives into the
modeling and mathematical studies of liquid crystals. This lectures
will begin with a survey of the calculus of variations methods applied
to studying the non-convex energy of nematic configurations and their
defects. Riding on symmetry changes in the material as phase
transitions to increasingly ordered phases occur, we will explore
mathematical problems arising in studies of chiral liquid crystals and
the layered smectic phases. The underlying property of molecular
polarization and its macroscopic effects will follow our
developments. Some materials with bent-shape molecules present smectic
phases with polarization strengths comparable to solid ferroelectric
compounds. A new feature of such materials is the ability to change
shape upon decreasing the temperature. We will also explore liquid
crystal elastomers. These are materials consisting of polymeric
networks able to deform upon alignment, and by application of electric
field. Finally, we will address models of liquid crystal gels and
their role in biological systems.

**1. Nematic liquid crystals: Modeling and Mathematical
Surveys**

Some ordinary fluids have the capability of becoming orientationally ordered upon decreasing the temperature. Such materials consist of elongated and rigid molecules with some degree of polarization. The Van der Waals interaction between dipole moments together with steric effects, yields the orientationally aligned nematic phase. In this lecture, we will survey the modeling approaches to nematic, from the work of Oseen (1933) through the free energy formulation of Frank (1963), and the later contributions of de Gennes and Ericksen. We will present the mathematical studies dealing with existence of energy minimizing configurations corresponding to nematic phases and explore the role of defects. Stability analyses of nematic configurations with respect to applied electric fields make the modeling background of the Fredericks transition relevant to display applications. We will explore such topics in the context of composite polymer-liquid crystal systems.

**2. Phase transitions in smectic liquid crystals**

Nematic liquid crystals experience a transition to the positionally
ordered smectic~A phase upon lowering the temperature. In this
lecture, we will explore the phase transition from nematic to
smectic~A in the case of chiral molecules. We will discuss the
problem of minimizing the free energy emphasizing the competition
between the tendency of molecules to form chiral configurations and
that of becoming positionally ordered in layers. For certain values of
the temperature, the outcome of such a competition gives rise to the
twist grain boundary defects. These are analogs to the vortex phases
of superconductors. We will present mathematical studies of twist
grain boundaries within the modeling contexts of Landau, de Gennes and
Lubensky.

**3. Ferroelectric liquid crystals**

Liquid crystal molecules in the smectic~A phase are positionally arranged in layers with the average molecular orientation being parallel to the layer normal. A symmetry breaking takes place in the transition to smectic~C upon lowering the temperature: the average molecular orientation is at a nonzero angle with the layer normal. As a consequence, the smectic~C phase presents spontaneous polarization. The interaction between the polarization field with the applied electric field becomes much more complex. This is the underlying physical mechanism that allows for high speed optic switches and devices. We will examine the existence of energy minimizers and explore the stability of the

**4. Liquid crystal elastomers**

This lecture addresses the effects of change of shape of molecules as orientation takes place. Many of the optic phenomena involving liquid crystals can be modeled under the assumption that liquid crystals are fluids with stiff, rod-like molecules that change very little upon aligning. In particular, this is the case of polymeric nematic liquid crystals, whose molecules posses anisotropic components that are able to form nematic and smectic phases as the concentration of solvent changes. We now consider the case that the polymer chains link together into gel networks, and the melt becomes an elastic solid, very much like a rubber. We will explore the properties that emerge from the ability of molecules to change their shape while in the solid states. We will explore the nature of the free energy and study the coupling between elastic deformation and nematic alignment.

**5. Polymers and gels**

In this lecture, we will present some of the latest developments in studies of liquid crystal gels. This are polymeric materials that resemble elastomers in their capability of deforming the polymeric network and change shape, but also are very sensitive to ionic changes in the environment. We will present a survey of the thermodynamic theories of gel as developed by Doi and Tanaka, and the latest liquid crystal formulations of such materials.