Geometric frustration and compatibility conditions for two-dimensional director fields.

Area of Science:

• Materials Science
• Liquid Crystals
• Geometry

Background:

• Bent core liquid crystals possess intrinsic molecular geometries favoring zero splay and constant bend.
• These molecular tendencies create geometric frustration when assembled in a plane, as they are mutually contradictory.
• Geometric frustration in materials leads to diverse morphologies and unique response properties.

Purpose of the Study:

• To establish necessary and sufficient conditions for scalar functions describing splay and bend in a director field.
• To generalize these conditions for surfaces with non-zero constant Gaussian curvature.
• To develop a reconstruction formula for the director field based on splay and bend.

Main Methods:

• Mathematical analysis of director fields in liquid crystals.
• Derivation of compatibility conditions for splay and bend functions.
• Development of a reconstruction formula for the director field.

Main Results:

• Defined conditions for scalar functions s (splay) and b (bend) in planar director fields.
• Generalized compatibility conditions for surfaces with constant Gaussian curvature.
• Provided a reconstruction formula for the director field from splay and bend data.

Conclusions:

• The study provides a mathematical framework for understanding geometric frustration in bent core liquid crystals.
• The findings enable the reconstruction of director fields, aiding in the analysis of complex liquid crystalline phases.
• Optimal compromises for achieving desired splay and bend values in frustrated systems are discussed.