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Geometric frustration and compatibility conditions for two dimensional director fields


Bent core (or banana shaped) liquid-crystal-forming-molecules locally favor an ordered state of zero splay and constant bend. Such a state, however, cannot be realized in the plane \cite{Mey76} and the resulting liquid-crystalline phase is frustrated and must exhibit some compromise of these two mutually contradicting local intrinsic tendencies. This constitutes one of the most well studied examples in which the intrinsic geometry of the constituents of a material give rise to a geometrically frustrated assembly \cite{Mey76,Mey69,TT06,LR02,CNS+14,LMCG03,Doz01,PSB+16,SDS13,HSN+09}. Such geometric frustration, is not only natural and ubiquitous but also leads to a striking variety of morphologies of ground states \cite{Gra16,KS01} and exotic response properties \cite{LS16}. In this work we establish the necessary and sufficient conditions for two scalar functions, $s$ and $b$ to describe the splay and bend of a director field in the plane. We generalize these compatibility conditions for geometries with non-vanishing constant Gaussian curvature, and provide a reconstruction formula for the director field depending only on the splay and bend fields and their derivatives. Last, we discuss optimal compromises for simple incompatible cases where the locally preferred values of the splay and bend cannot be simultaneously achieved.

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Publication details

The article was received on 20 Aug 2017, accepted on 17 Nov 2017 and first published on 17 Nov 2017

Article type: Paper
DOI: 10.1039/C7SM01672G
Citation: Soft Matter, 2017, Accepted Manuscript
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    Geometric frustration and compatibility conditions for two dimensional director fields

    I. Niv and E. Efrati, Soft Matter, 2017, Accepted Manuscript , DOI: 10.1039/C7SM01672G

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