Compatible Director Fields in $\mathbb{R}3$
Source abstract
Abstract
Abstract The geometry and interactions between the constituents of a liquid crystal, which are responsible for inducing the partial order in the fluid, may locally favor an attempted phase that could not be realized in $\mathbb{R}^3$. While states that are incompatible with the geometry of $\mathbb{R}^3$ were identified more than 50 years ago, the collection of compatible states remained poorly understood and not well characterized. Recently, the compatibility conditions for three-dimensional director fields were derived using the method of moving frames. These compatibility conditions take the form of six differential relations in five scalar fields locally characterizing the director field. In this work, we rederive these equations using a more transparent approach employing vector calculus and characterize a wide collection of compatible phases.
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Research context
OpenAlex topics: Liquid Crystal Research Advancements · Nonlinear Dynamics and Pattern Formation · Advanced Materials and Mechanics
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References (27)
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27. Note that in the original derivation of the $2d$ compatibility equation, [19], the bend was defined as the geodesic curvature of the director integral lines, i.e. with an opposite sign to the more common definition of the bend, which we employ here. Consequently. The equations also differ by a minus sign