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Flower-Petal Arrangements of Focal Conic Domains: Friedel’s Laws in Reverse
Daniel A. Beller,1,* Mohamed A. Gharbi,1,2,3 Apiradee Honglawan,2 Kathleen J. Stebe,2 Shu Yang,2,3 and
Randall D. Kamien1 1 Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, USA
2 Department of Chemical and Biomolecular Engineering, University of Pennsylvania, Philadelphia, USA 3 Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, USA
Focal conic domains (FCDs) in smectic-A liquid crystals have drawn much attention, both for their exquisitely structured internal form and for their ability to direct the assembly of micromaterials and nanomaterials in a variety of patterns. Two key properties for determining FCD assembly are the domains’ eccentricity and orientation, which determine their packing arrangement through geometrical rules elucidated by Friedel’s Law of Corresponding Cones. Here, we demonstrate control over the spatial variation of FCD eccentricity and orientation by designing mismatched boundary orientations for a hybrid-aligned smectic. In particular, we manipulate interface behavior with colloidal particles in order to experimentally produce two examples of what has recently been dubbed the flower texture [1], where the focal hyperbolæ diverge radially outward from the center of the texture, rather than inward as in the canonical éventail (or fan) texture studied by Friedel. We explain how this unconventional assembly can arise from appropriately curved interfaces. Finally, we present a geometric model for this system that applies the Law of Corresponding Cones iteratively, showing how FCDs may be embedded smoothly within a ‘‘background texture’’ of large FCDs and concentric spherical layers, in a manner consistent with the qualitative features of the smectic flower.
Flower texture of focal conic domains in a thin-film smectic with a colloidal inclusion in the center [2]. Scale bar is 10 µm.
Geometric model of the flower texture consistent with Friedel’s Law of Corresponding Cones [2].
References: [1] C. Meyer et al., Focal Conic Stacking in Smectic A Liquid Crystals: Smectic Flower and Apollonius Tiling, Materials 2 (2): 499-513 (2009). [2] D. Beller et al., Focal Conic Flower Textures at Curved Interfaces, Physical Review X 3 (4): 041026 (8 pages) (2013). _____________________________________________ * presenting author; E-mail: [email protected]
