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Nematic Order of Handle Droplets

Ekapop Pairam, Jayalakshmi Vallamkondu and Alberto Fernández-Nieves

Motivation & Background

Figure 1: (a) Crystalline lattices arranged in honeycomb pattern. (b) Red panels indicate pentagon defects on typical soccer ball.

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 figure 1(a). However, on the surface of a sphere, for a familiar example of this fact; a soccer ball has the total number of 12 pentagon panels dispersed among hexagons, as shown in figure 1(b). Another example of structure that has twelve pentagons defects is Buckminsterfullerene (or buckyball) [1]. The 12 pentagons defects on a sphere is a direct consequence of the invariant known as Euler characteristic [2]: $$\chi = \frac{1}{2\pi}\oint_{area} KdS=V-E+F=2(1-g),$$ where \(K\) is the Gaussian curvature, \(V\) is the number of vertices, \(E\) is the number of edges, \(F\) is the number of faces and \(g\) is the number of handle. For a sphere \(g=0\), and thus \(\chi=2\). This also applies identically to spherical polyhedra, such as Tetrahedron, Hexahedron or cube, Octahedron, Dodecahedron and Icosahedron etc. In fact, any network that can be deformed into a sphere without cutting the edges or changing the vertices has the same Euler characteristic of a sphere. This means that any network with the topology of a sphere will inevitably posses the Euler characteristic of 2.

Figure 2: Spheres packed into crystalline order in (a) flat 2D and (b) spherical surfaces.

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 figure 2(a). An equivalence of soccer ball structure would include both spheres with 5 nearest neighbors and spheres with 6 nearest neighbor, as shown in figure 2(b); note the analogy to a soccer ball in figure 1(b).

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.

Figure 3: Example of rods arranged in parallel on (a) flat and on (b)&(c) spherical surfaces. The arrangements on spherical surface is called bipolar in (b) and concentric in (c).

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 figure 3(a). However, it is impossible for \(\textbf{n}\) to be well define everywhere when rods are packed onto a spherical surface. Familiar examples to this are director field resemble to the North and South Pole configuration (bipolar) and concentric structure, as shown in figure 3(b)&(c), respectively. Note that there are two \(s=+1\) defects for the director patterns in figure 3(b)&(c) each as required by Poincaré-Hopf theorem. Packing orders of rods can be model directly using liquid crystals drops [4,5] and shells [6] with strong parallel anchoring between the liquid crystal molecules at the surface of the drop, here \(\textbf{n}\) is in the direction of the preferred molecular orientation in liquid crystals.

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.

Figure 4: Sequences of hexagon folded into (a)-(c) pentagon and (d-h) heptagon.

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 figure 4(c) and the negative curvature of heptagon shape in figure 4(h). Here the pentagon contribute to the positive Gaussian curvature while the heptagon contribute to the negative Gaussian curvature.

Figure 5: Schematic of rods packing on toroidal surface in (a) axial and (b) twisted configuration.

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 figure 3(c) is actually unstable and the disclination line that connects between the two defects typically escapes into the third dimention and leads to the escaped concentric drop [11]. Beside the axial configuration (see figure 5(a)) of toroidal liquid crystal drops, theoretical work by [12] also predicted a transition to twisted structure (see figure 5(b)), depending on the aspect ratio of a torus and ratio of the twist and bend elastic constant of the liquid crystal. For thin spherical nematic shells, the structures of four \(s=+1/2\) and one \(s=+1\) with two \(s=+1/2\) both maximizing their separation throughout the surface resulting in a tetrahedral and trigonal geometries of \(sp^2\) and \(sp^3\) carbon bonds, respectively, were also observed beside the bipolar configuration [6].

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.

Overcoming Surface Tension and Probing Nematic Order on Handle Droplets

Figure 6: (a) A curved jet is formed due to a viscous drag force resulting from the rotation of the continuous viscous bath. (b) A toroidal droplet is formed once a jet completed one whole revolution.

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 figure 6(a)). The torus is form once the resultant jet closed in on it self (see figure 6(b)). Using this method allows us to free control the aspect ratio of the torus. We can either vary the amount of the injected liquid or the location of needle that delivers the inner liquid with respect to the center of rotation to vary the thickness of the torus or change to overall size of the torus, respectively. If both the inner and the outer liquid are Newtonian liquid then the toroidal droplet is unstable because of surface tension driven stress/instabilities. Depending on the aspect ratio of the torus, it will either breaks in a similar way as a long fluid cylinder or will shrink towards its center to coalesce onto itself [24].

Figure 7: (a) Two toroidal droplets side by side. (b) The continuous yield stress material is being sucked out from the center between the two droplets. (c) Two toroidal droplets joined to form a double toroid droplet once sufficient material between the two original drops is removed. (d)&(e) More continuous phase is being removed from both side where the two original drops joined to widen the joint area. (f) The area around the joint is stirred so that the thickness of the joint is the same as the thickness of handle on both sides. (g) Double toroid droplet with wide joint.

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 figure 7.

Figure 8: (a-c) Top view of a double toroid in bright field. Solid dark circles indicate the location of the two -1 surface defects. (d-f) The same image under cross-polarizers. (g-i) and (j-l) Side view of the double toroid under cross-polarizers when focused at its back from the upper and lower side of (a-c), respectively. The insets are the magnified view of the regions where the topological defect with charge \(s=-1\) are located. Scale bar: 100 \(\mu\)m.

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 figure 8(a)-(c); the corresponding cross-polarized image is shown in figure 8(d)-(f). Note the four black brushes in the region where the two tori meets (see figure 8(g)&(j)) indicate the presence of a topological charge with \(|s|=1\). The sign of this charge is determined by rotating the double torus. Since the brushes rotate in the same sense as the rotation, we conclude the defect has charge \(s=-1\) (see supplementary video in [25]). Stabilizing toroidal droplets inside a yield stress material allows us to freely manipulate the curvature of a droplet landscape, which allows us to explore how the defects arrangement interacts with landscape curvature.

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 figure 8(b), or make only one side has positive curvature while leaving the other side with a curvature of a saddle ,as shown in figure 8(c). Even with these changes, the defects remain roughly at the same locations (where the two toroids joined) as indicated in figure 8(d)-(f). However, note different in visibility (see figure 8(g)-(l)) and vertical locations of defects when view from the side (see figure 8(g)-(l)).

Figure 9: (a) Top view image of a triple toroid with a side-by-side arrangement of the handles. (b) The same image under cross-polarizers. (c) Side view of a triple toroid with a side-by-side arrangement of the handles viewed under cross-polarizers. The defects are located in the outer regions where the individual toroids meet. (d) Top view image of a triple toroid with a triangular arrangement of the handles. Solid circles show the defect locations found by looking at the droplets between cross-polarizers along different viewing directions (e,f,g). The red circle in (e,f,g) indicate the locations of defects when viewed from the side. Scale bar: 100 \(\mu\)m.

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 figure 9(a); the corresponding cross-polarized image is shown in figure 9(b), or arranged in a triangle, as shown in figure 9(d). In the first case, there are four defects, each of topological surface charge -1, located in the regions where the individual tori meet, as shown when the droplet is viewed along its side between cross-polarizers in figure figure 9(c). In contrast, when the handles are arranged in a triangle, there are two -1 defects that cluster together in one of the three regions where the single tori meet, as indicated in figure 9(d). In this situation, in addition to the natural frustration imposed by the bounding surface, there is an addition frustration arising from the lack of a sufficient number of negative-curvature regions between the single torus to position the defects. There are only three natural regions for the defects to be located and four defects. We find that a possible solution to this problem is to cluster two of the four defects together in one of the three natural regions for them to be located. figure 9(e)&(f) show the magnified view of the individual defects as view from the side as indicates by the dashed lines in figure 9(d). Figure 9(g) show the magnified side view of the remaining area where the two defects are clustered together.

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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