Frustrated liquid crystalline phases: Compatibility conditions for three-dimensional director fields

The geometry of the space in which a director-field lies imposes restrictions on the admissible orientational order fields; for example, a field of constant bend and vanishing splay and twist cannot reside in R³. These restrictions give rise to compatibility conditions relating the quantities describing the director to the geometry of the embedding space. It was previously shown that in two dimensions, the splay and bend fields suffice to uniquely determine a director and are subject to a single compatibility condition relating their first derivatives to the Gaussian curvature of the embedding space. In this talk, we will show that in three dimensions five local fields are required to uniquely describe a director field and that these fields are related to each other and to the curvature tensor of the embedding space through six compatibility conditions. Characterizing and understanding the compatibility conditions allows us to construct previously unknown solutions, and predict the super-extensive energy growth rate as a frustrated phase grows in size without the need to solve for its ground state.