**Nematic Order of Handle Droplets**

**Ekapop Pairam, Jayalakshmi Vallamkondu and Alberto Fernández-Nieves**

It is simple to pack things on a flat space. For example: crystalline order made out of
hexagons can be arranged into a honeycomb pattern to completely
filled up a space on a plane without geometrical frustration, as shown in
A crystalline packing structure on spherical surface can be fabricated experimentally by
the self-assembly of colloidal particles onto the interface of water in oil emulsion
droplets [3]. We can think of a location of a panel of polygon with \(m\) sides
as the location of particle surrounded that surrounded by \(m\) nearest neighbors.
For example a sphere particle that has 6 nearest neighbors is a hexagon, while a sphere
particle that has only 5 nearest pentagon is considered a pentagon. If every particle has
6 nearest neighbors then we have a honeycomb structure, as shown in In analogous to packing spherical particles to represent the crystalline structure; we can pack rods to represent the director/vector field, by characterizing the oriented direction of a rod as a director, \(\textbf{n}\). The total number of defects for a director field on a surface with arbitrary number of handles is dictated by the Poincaré-Hopf theorem [2]: $$\sum\limits_{i}s_i=\chi=2(1-g)\,$$ where \(s\) measures how much \(\textbf{n}\) rotates about the defect core by the multiples of \(\pi\) radians, for instance, \(s=+1\) when \(\textbf{n}\) rotates by \(2\pi\) radians around a defect core.
Just like crystalline packing, it is possible to arrange the packing of a rod such their \(\textbf{n}\)
are well define everywhere on a flat space, as shown in The experimental results has shown that the ground state in the physical system is not always the simplest solution possible in compliance with the topological constraints. This brings the physics into scene. For the case of crystalline packing on spherical surfaces, the experimental results in [7] has shown that the additional pairs of pentagon and heptagon defects may emerge in addition to the 12 pentagon defects depending on the ratio of the spherical droplet radius over the packing particles radius. These extra defects pair emerges to relived the excess strain causes by the isolated pentagon defects as the number of particles on a sphere grows. Ground states of toroidal crystals are also predicted to contain pentagon defects on the exterior with equal number of heptagon defects on the interior of the torus [8,9]. Here, the total number and structure of defects pairs are characterized by the aspect ratio of the torus and the relative size of crystal lattices to the size of torus.
Note that the addition of a pair of pentagon and heptagon defects does not change the net
Euler's characteristic. This can be intuitively understood by creating disclination out of
paper by folding hexagon shape (made out of 6 equilateral triangles) into pentagon or hexagon
by removing or adding a triangle, respectively, as shown in figure 4;
note the positive curvature of pentagon shape in
For the case of rods packing on spherical surfaces, depending on the intrinsic elastic properties of the liquid crystal,
the bipolar drop can develop some twist deformation and give rise to twisted bipolar drop [10].
The concentric structure in There have been a lot of experimental, theoretical and simulation on research of ordered media in confined spherical volumes [13-21] and their intriguing technological potential for divalent nano-particle assembly has been already demonstrated [22]. In contrast, there are virtually no controlled experiments with ordered media in confined volumes with handles. A notable exception is the optically induced formation of cholesteric toroidal droplets inside a nematic host [23]. This largely reflects the difficulties in generating stable handled objects with imposed order. While the sphere is relatively easy to achieve in liquids due to surface tension, the generation of stable droplets with handles remains a formidable challenge. |

Our method of generating toroidal droplets relies on the viscous forces exerted by a rotating continuous phase
of a high viscosity fluid over a liquid which is extruded from an injection needle (see
While increasing the viscosity of a simple liquid surrounding the toroidal droplet can slow down the breakup time,
it cannot stop the breakup causes by surface tension stress. To address the instabilities issue of the
viscous toroidal droplets, we replaced the outer simple liquid medium with a yield stress material,
which can withstand stresses causes by surface tension and external perturbations
and stabilize the toroidal shape, provided that these
stresses are less than the yield stress of the material. This also allows us to join two toroidal droplets
generated side by side together and manipulating its curvature using micro-surgery technique, as shown in
While nematic toroid droplet doesn't have any defect, double toroidal droplet usually has two -1 defect, as
required by the Poincaré-Hopf theorem. By employing optical polarizing microscopy, we can, map
the location of these defects. From our experiment, these defects are always found to located on the side where
the two tori meets as indicated by the points in
Through micro-surgery, we have a full control over the curvature double toroid.
For instance, we can make the flatten out the regions where the two single toroids meet
on both side, as shown in
It is possible to have different arrangement for droplet with three or more handles, for example,
three handles aligned along a common axis, as shown by the top view
image in Using our method, it is possible to generate droplets with one or multiple handles as well as controlling their curvature. By demonstrating that this method can be use to probe the nematic order of stabilized nematic liquid crystal handle droplets, we hope to show that our method can open up a versatile approach to generate topological soft materials that exploits nematic self-assembly within macroscopic droplets with handles, stabilized using a yield stress material as the outer fluid. |

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