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The Tutte polynomial of oriented matroids
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
In a graph G, a cyclic-bipolar orientation with respect to p is a strongly connected orientation where each directed cycle contains p. This corresponds to having a p-bounded dual. By Remark 31.15, if G has at least two edges, 2β(G)=#{cyclic-bipolar orientations ofGwith respect top}.
Azodendrimers as a photoactive interface for liquid crystals
Published in Liquid Crystals, 2018
Alexey Eremin, Hajnalka Nádasi, Pemika Hirankittiwong, Jarinee Kiang-Ia, Nattaporn Chattham, Osamu Haba, Koichiro Yonetake, Hideo Takezoe
LC droplets were prepared using LC compounds, which exhibit the nematic (N), cholesteric (Ch), smectic A (SmA) and twist-bend nematic (NTB) phases, and a small amount of azodendrimers (D2-6Azo5) in glycerol solution [10]. The photoisomerisation of the azodendrimers, shown in Figure 3 at LC/glycerol interfaces, induces different changes in each LC phase. In the N phase, a radial director orientation with a hedgehog defect at the centre changes to bipolar orientation. In the Ch phase, the helical structure with the helical axis along radial directions is fairly resistant to the reorienting force from the surface particularly inside of the droplets. However, the deformation of the helical structure can be recognised by the emergence of many dark lines suggesting the development of defects. In the SmA phase, the photo-induced change is more drastic as shown in Figure 4 [10]. Under crossed polarisers, we observed that concentric rings with an extinction cross the same as in the N phase suggesting an onion-like layer structure change to structures with many radial defect lines. Under UV light, layers are forced to be curved because of the tangential orientation of molecules at the interface.
Functional liquid crystalline particles and beyond
Published in Liquid Crystals, 2019
A straightforward method for the fabrication of such large particles are microfluidic approaches which were developed in our group. In our general microfluidic setup (Figure 6(a)) droplets of the monomer mixture are formed, at first, at the tip of a thin glass capillary and then transported by a highly viscous silicon oil further downstream where they are photopolymerized. This microfluidic process not only has the advantage that a multitude of equally sized particles can be prepared in a short time, but it also induces an orientation of the liquid crystalline director inside these droplets. This works due to a shear between the monomer mixture and the high viscous silicon oil. Two different mechanisms were found by Ohm et al. for the orientation of the mesogens depending on the shear rates, which can either be increased by raising the flow rate ratio between the monomer (dispersed) and oil (continuous) phase or by decreasing the diameter of the polymerization tube. If no shear is applied, also no orientation of the mesogens can be observed. This was experimentally proven by investigating particles which were polymerized while they were standing still inside the tube [38]. In a series of experiments, Ohm et al. varied the diameter of the polymerization tube in order to influence the shear rate (Figure 6(b)). While for large diameters of 1000 µm almost spherical particles were obtained, highly shape anisotropic fibres could be observed for small inner diameters of 250 µm. In between disc-like and rod-like morphologies emerged [35]. Wide angle X-ray scattering (WAXS) measurements were performed in order to investigate the director field of these particles. A determination by POM was not possible because the thick particles (several 100 μm) always appeared opaque in their LC phase. A concentric orientation was found for the disc-like particles, whereas a bipolar orientation was found for the fibres and rod-like particles. Latter appeared for high capillary numbers (high shear rates) and thus arises from stretching of the droplet by the high viscous silicone oil during the polymerization. This mechanism is equal to the orientation of the mesogens in main-chain LCP films [91,92]. Many other examples are known in which the fabrication of LCE particles, fibres or even tubes leads to a bipolar director orientation [36,37,45,93,94]. In the case of the disc-like particles, which are formed at smaller capillary numbers (smaller shear rate), the mechanism isn’t as obvious. It was assumed that the orientation originates from a log-rolling process which is illustrated in Figure 6(c). Due to the shear flow caused by the surrounding oil, a flow inside the droplet is induced leading to an orientation of the director perpendicular to the flow field. The same director orientation was found for the preparation of LCE core-shell particles and photoresponsive particles [39,46].