The most noticeable change from the radial profile of a nematic droplet is the displacement of the defect from the centre towards the surface. This can be understood by looking at the twist throughout the droplet, which we show in Fig. There is then an energetic drive to expand the region with the correct handedness blue at the expense of that with the wrong sense of twist red and the balance of this with the increased elastic distortion determines the position of the point defect.
The broken spherical symmetry from the displacement of the defect defines an axis through the droplet and the chiral distortion of the director field turns this into the axis of a double twist cylinder. Increasing N for small values continues this trend; the defect moves further towards the surface to reduce the size of the region with the wrong handedness and the axis it defines becomes increasingly recognisable as a double twist cylinder. We describe this structure now. Generic linear vector fields fail to be chiral because their curl is non-zero. In the generic case, the critical point is described by a quadratic function, referred to as Morse-type.
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The twist of the director field on a slice through the droplet is shown in Fig. Throughout the interior, including at the defect, the twist is right-handed blue but close to the surface there are regions where it is left-handed red. This behaviour is topological, as we describe presently. However, this structure is not preserved in a chiral droplet and exhibits fundamental frustration with a state of uniform twist; there is no choice of higher order term m c that makes the radial hedgehog chiral.
Straight round the twist: frustration and chirality in smectics-A | Interface Focus
The twist over each disc can be related to the nature of the director field on its boundary. This is based on the properties of integral curves of the plane field orthogonal to n , i. This applies in particular to the boundary of any disc. When the twist is non-zero the director is no longer perpendicular to the boundary. In this case, the boundary can be lifted — pushed up or down along the surface normal direction — to create an integral curve. Importantly, if the twist is right-handed then the displacement is along the positive normal direction, while if it is left-handed the displacement will be in the negative direction.
Applying this to the separation of the spherical level set into two hemispheres, the displacement around the boundary of each hemisphere is equal and opposite, since the two boundaries are the same equator but traversed in opposite directions. As a result, if the twist is right-handed in one hemisphere it will be left-handed in the other, and in equal measure.
We see from this that radial hedgehogs are incompatible with the preferred handedness of cholesterics. Precisely the same argument applies to a boundary layer at the droplet surface where the director becomes radial to match the boundary conditions. In the case of the Morse index 2 defect Fig. In those experiments the surface is the flat plane of a glass slide, as opposed to being spherical.
It is only spherical surfaces that have a topological requirement for regions of reversed twist. The gradient vector field can be made chiral in a neighbourhood of the origin by adding the germ. This points to the importance of defect core structure, or elastic anisotropy, for stability in physical systems.
This is a codimension 9 singularity with multiplicity 11, characteristics which convey its high complexity. The codimension of a singularity is the number of independent parameters, or perturbations, that resolve it into simpler pieces, while the multiplicity is the number of Morse critical points that it splits into as a complex polynomial under a generic perturbation. The gradient field of 5 can be perturbed into a chiral point defect via a generic method that we describe in the following section.
In particular, the unfoldings of singularities provide models for the combination and splitting of chiral point defects. The annihilation of two chiral defects with opposite degree is given generically by the unfolding of the A 2 singularity and the germ. The resulting pattern of point defects within the droplet is shown in Fig. At the same time, not all aspects of singularity theory have an immediate realisation in cholesterics.
For instance, it is not clear whether the full list of singularities all occur, or in an order predicted by their codimension. In Fig. In the latter, the surface defects are maximally separated, which lowers the elastic free energy. Each term, m j c , is determined up to the addition of a harmonic gradient.
Thus the entire structure of such a chiral point defect is determined by the gradient field of a harmonic function. They can all be perturbed so as to be chiral following the prescription for the germs of curl eigenfields. Nonetheless, it is well-known that not all types of critical point can occur in a harmonic function: By the maximum principle, there are no local maxima or minima.
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In contrast, it is usually argued that in the liquid state the energetic difference between the conformers is too small to give rise to any significant bias of molecular conformations and that the entropy of mixing disfavours enantiomeric segregation [ 36 Takezoe H. Ferrochirality: a simple theoretical model of interacting, dynamically invertible, helical polymers, 2 a molecular field approach: supports and the details.
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solar-yug.ru.u6667.th6.vps-private.net/modules/qabom-hydroxychloroquine-sulphate-store.php Figure 9. This is indicated by the circular shape of the chiral domains see Figure 9 b,d and the easy flow under gravity see videos attached to ref. Nematic phases of bent-core mesogens. The cybotactic nematic phase of bent-core mesogens: state of the art and future developments. It was postulated that the chirality in these liquids results from a synchronisation of the helix sense of molecular conformers, which allows a slightly denser packing of molecules with uniform helix sense, thus leading to an enthalpy gain for homochiral packing [ 8 Tschierske C , Ungar G.
Strategies to create hierarchical self-assembled structures via cooperative non-covalent interactions. Cooperativity in noncovalent interactions. This is provided by a locally ordered structure in the cybotactic liquids and sets in at a certain temperature when the cybotactic clusters exceed a certain critical size. Below this temperature the enthalpy gain of chirality synchronisation exceeds the entopic penalty of the separation of the enantiomorphic conformers.
Chirality synchronisation is additionally favoured by a coupling between the molecular conformational chirality and the chirality provided by the helical organisation of the molecules in the cybotactic clusters diastereomerism. Macroscopic chirality results from network formation by random connections between the clusters Figure 7 [ 92 Kauffman S.
Network formation in the liquid state is dynamic, i. In contrast, chirality synchronisation in the crystalline state is associated with a much stronger restriction of molecular and conformational motions, leaving mainly thermal vibrations and thus leading to a larger entropic disadvantage of this process. As the contribution of entropy increases with temperature, the classical non-dynamic mode of chirality synchronisation requires low temperatures and is typically observed in low temperature crystalline or semicrystalline mesophases, like HNF phases [ 36 Takezoe H.
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