Geometry of focal conics in sessile cholesteric droplets

We create cholesteric droplets in glycerol, and because they are less dense than the solvent, they slowly cream under gravity. The anchoring conditions between the mesogen and either the glass or glycerol are planar, requiring the pitch axis to be perpendicular to the interfaces. When the glass is carefully cleaned, the droplets do not wet the glass and remain spherical even on contact; however, when the glass has only been cleaned with isopropanol, the droplets stick to the surface and are deformed with a contact angle close to 124°. We can see in Fig. 1 that a parabolic interface separates two regions. Even without direct observation of the cholesteric stripes through cross-polarizers as in Fig. 3B, this demonstrates the layered nature of the cholesteric indirectly, in the same spirit in which George Friedel deduced the layered structure of smectics by observing the FCD texture of confocal ellipses and hyperbola under the optical microscope. Spherical and flat layers join continuously across an interface if and only if it is a paraboloid, whose focus is the center of the sphere: Indeed, a standard definition of a parabola is the locus of points that are equidistant from the focus and the directrix as sketched in Fig. 4A. The congruent lines demonstrate the definition of the parabola: the locus of points (in blue) that are equidistant from the focus and the directrix (both in red). It follows by construction that the outer, equally spaced spheres (yellow) come into registry with the inner, equally spaced planes (green). With degenerate planar anchoring, these pseudolayers satisfy the boundary conditions and equal spacing. Following the diagram, let $R$ be the radius of the droplet, $\rho$ be the radius of the droplet’s disc-like interface with the glass, ${\theta }_0$ be the contact angle, and $\mathrm{\Delta }$ be the distance from the parabola focus to the slide. Then, applying the Pythagorean theorem, we find the height of the parabola $z(r)$ as a function of the radius in cylindrical coordinates:

Since $\mathrm{\Delta }=R_{cos}(\pi -{\theta }_0)$ and $\rho =R_{sin}(\pi -{\theta }_0)$, we find that the bottom of the paraboloid is $z(0)=\frac{1}{2}R(1-cos{\theta }_0)\approx 0.82R$, which agrees extremely well with the observations.

Further inspection of the interface reveals the presence of curves on the paraboloid that, when viewed from below are nearly circular with lines emanating from foci, in the manner of FCDs (Figs. 1 and 3A). From the side, they appear to be elliptical. Note that for a general paraboloid in cylindrical coordinates, $z=A(r^2-{\rho }^2)$ and the intersection of the paraboloid with the plane $z=\delta -A{\rho }^2+xtan\alpha$ (shifted along $z$ by $\delta$ from the bottom of the paraboloid and at an angle $\alpha$ with the $\mathit{xy}$-plane) projects to the circle $A{(x-A^{-1}\frac{1}{2}tan\alpha )}^2+A{y}^2-\delta -\frac{1}{4A}{tan}^2\alpha =0$ in the $\mathit{xy}$-plane (whenever ${tan}^2\alpha +4\delta \ge 0$) (Fig. 4B). To be concrete, if the circle in the $\mathit{xy}$-plane has radius $b$ and is centered at a distance $c$ from the centerline of the droplet, then the ellipse has minor axis $b$ and major axis $b\sqrt{1+4A^2{c}^2}$. On the tilted plane, this is an ellipse with eccentricity $e=sin\alpha$. Note that when $e=c=0$, the ellipse reduces to a circle centered on the paraboloid axis, while the hyperbola reduces to a straight line passing through the droplet center. We are led to conclude that the closed curves on the parabola are, indeed, ellipses and are the telltale signatures of focal conic domains. Indeed, in both the bottom-up (Fig. 3A) and side view (Fig. 1), we not only observe the ellipses but we also observe lines perpendicular to them that we will show are the hyperbolæ that are confocal with the ellipses of the FCDs.

Recall that FCDs have remarkable geometric properties in relation to spheres. As prominently noted by Kleman and Sethna (15), it is possible to excise a cone from a collection of equally spaced, concentric spheres (with the cone apex coincident with the common center) and replace it with equally spaced sections of cyclides with no discontinuity in the layer normal field. As a result, there is no discontinuity in the local dielectric complexion, and the interfaces between the FCD and spherical regions are not observed. In Fig. 5, we demonstrate the geometry of the cyclides and the associated hyperbola in the plane containing the hyperbola. The black circles are the largest and smallest meridional cycles on the cyclide, while the blue circles are cross-sections of spheres. How can we demonstrate that the tangency conditions shown in this plane extend to the remaining part of space, absent cylindrical symmetry? Recall that equally spaced layers have a hidden Poincaré symmetry (19) that can be used, for instance, to Lorentz transform a general cyclide to a symmetric torus. Under the same Lorentz transformation, spheres map to spheres. In the symmetric torus case, cylindrical symmetry insures that the blue circle tangent to the two black circles becomes a blue sphere (with center on the axis of symmetry) tangent to a black torus. Lorentz transforms preserve intersections and tangencies, and it follows that the general cyclide is tangent along a circle to a sphere centered on the hyperbola, the transform of the axis of symmetry. Note that in the limit that the blue sphere center moves off to infinity along real branch, it becomes a plane tangent to the cyclide, and the cone becomes an infinite cylinder. This construction is the basis for the construction of large angle grain boundaries (16) and can also give rise to spatially varying eccentricity in thick enough freestanding films (20–23) or by confining a hybrid-aligned smectic with curved interfaces (17) by manipulating interface behavior with colloidal particles. Note that a sphere centered on the virtual branch at the diametrically opposed infinity also is a plane tangent to the cyclide. Bringing that center in from infinity, we see that now there are spheres tangent to the positive curvature regions of the cyclide.

With this in hand, we propose the following structure for the cholesteric layers in the deformed droplet: Inside the paraboloid, the ellipses cap off circular cylinders that lie along $\widehat{z}$ and are filled with concentric, nonpositive Gaussian curvature regions of cyclides. The angle between the plane of the ellipse and the normal to the layers is $\alpha$, as above. In the region outside the paraboloid, right circular cones centered on the focus of the paraboloid intersect the ellipses and in them we replace the spherical layers of Fig. 4 with concentric, equally spaced pieces of cyclides. The ray emanating from the center of the cone to the droplet surface is at an angle $\alpha$ to the plane of the ellipse and so the ellipse renders a rotation of the layer normals by $2\alpha$ (16). Note that in this construction, both negative and positive Gaussian curvature regions contribute. This construction explains the “inverse flower” textures observed in refs. 17 and 24 and here where the hyperbolæ go through the outer foci of the ellipses, contrary to the hallowed law of corresponding cones. Fortunately, nothing is lost. We extend the law of corresponding cones to allow for the virtual branches of the hyperbola to meet at one point—here at the focus of the paraboloid or, equivalently, at the center of the original spherical droplet. We observe domains with both positive and negative Gaussian curvature. This is surprising since the formation of FCDs is often attributed to a preference for one particular sign.

The resulting internal structure of the droplet in terms of cholesteric layers in a cross-section perpendicular to the substrate and containing the center is depicted in Fig. 6. The mother spherical and flat layers in some places are replaced by FCDs. Blue corresponds to layers of the central toric FCD (TFCD), whose ellipse and hyperbola reduced, respectively, to the circle and straight line passing through the circle center. Red depicts layers of two other FCDs on both sides of the TFCD. Mother spherical layers concentric around the droplet center are shown in yellow, and mother flat layers are in green. Note that in all three FCDs, the sign of Gaussian curvature of the layers switches from negative to positive within each FCD. Namely, in the central domain, blue layers have negative Gaussian curvature inside the cylinder but have positive Gaussian curvature outside the cylinder yet remain sections of Dupin cyclides (nested tori in this case). In the left and right domains, the layers in red have negative curvature inside the cylinders and regions of positive curvature outside the cylinders depicted by the dashed red lines. The sketch of the corresponding 3D structure is depicted in Fig. 7A. The remaining volume of the droplet can again be broken into analogous cylinder/cone FCDs creating the hierarchical bouquet in Fig. 7B. Each “flower” demonstrates the energetic balance between two configurations: 1) The flat/sphere configuration has lower curvature energy than the cyclide sections but has a director singularity on the entire two-dimensional interface between the inner and outer regions; and 2) the cyclide configuration, with higher curvature energy, reduces the singular set to a one-dimensional set of curves, the ellipse and the hyperbola. As the angle of the director discontinuity decreases toward the top of the adsorbed droplet, the associated interfacial energy likewise decreases and, at some point, it is no longer economical to replace the flat/sphere regions with focal domains.

Not only does this model reproduce the broad geometrical structure of the droplet, but it provides, as well, numerical predictions. In addition to predicting the height of the parabola in terms of the contact angle, the geometric construction relates, for instance, the radius ($b$) and center of each circle ($c$) in the $\mathit{xy}$-plane (Fig. 3A) to the height of the center of the ellipse ($h$), shown in Fig. 1. Noting the equation for the paraboloid, we find $h=(z_{+}+z_{-})/2$ with $z_{\pm }=z(c\pm b)$. In the bottom view of the sample, we have a central circle, a primary ring of circles, and a secondary ring of circles. In the primary ring, the average values of $b$ and $c$ are $7.4\mathrm{\mu }$m and $21\mathrm{\mu }$m, while in the outer ring, they are, on average, $6.4\mathrm{\mu }$m and $33\mathrm{\mu }$m, respectively. Using $R=65\mathrm{\mu }$m, $\mathrm{\Delta }=36\mathrm{\mu }$m, and $\rho =54\mathrm{\mu }$m, we predict $h_{\mathrm{in}}=42\mathrm{\mu }$m and $h_{\mathrm{out}}=31\mathrm{\mu }$m. Measuring (with significant noise) the side image, we find ${\overline{h}}_{\mathrm{in}}=47\mathrm{\mu }$m and ${\overline{h}}_{\mathrm{out}}=34\mathrm{\mu }$m. Likewise, we can calculate the expected angle between the asymptotes of the outer hyperbolæ, $2\alpha$, via ${csc}^2\alpha =\sqrt{1+4b^2/{(z_{+}-z_{-})}^2}$ and find $2\alpha \approx {97}^{°}$ compared to the measured $2\overline{\alpha }\gtrsim {85}^{°}$. Since it is difficult to identify the circles in the bottom view with the ellipses in the side view, we cannot, at this time, offer more precise numbers. Suffice to say, the agreement between the model and measurement is encouraging if not convincing.

Contrary to smectic A textures, the normal to the layers is not the only parameter to describe the textures: The director field which is perpendicular to the layer normal has to be defined. This clearly increases the complexity of the texture. Fortunately, the geometry here allows us to make some progress. When the integral curves of the local pitch axis are straight lines, there is no obstruction to adding a director field everywhere perpendicular to the pitch axis that rotates smoothly (25). Since all the geometries we have used consist of equally spaced, parallel layers, this condition is satisfied, and we can build a cholesteric-like winding on the equally spaced cyclides shown in Fig. 8. However, topological considerations create a global obstruction in some cases. Consider, for instance, the spherical droplet shown in Fig. 3B. Were this a smectic A droplet, it could be filled with concentric spherical layers parallel to the boundary. However, if we try to decorate each spherical pseudolayer with a director field, perpendicular to the normal, we cannot. The Poincaré–Hopf–Brouwer theorem requires defects with total winding of $4\pi$. Indeed, the Robinson texture shown in Fig. 3B demonstrates this (18). The singular line running from the center to the surface is required by topology and captures a winding of $4\pi$. In a droplet with no focal conic domains, as described in Fig. 3B, this is the only necessary singularity and will still be required even when the spherical pseudolayers of the cholesteric are deformed to the squashed droplet shape. The introduction of the focal conic domains adds to the complexity of the cholesteric. When the domains are built from ring cyclides, there is no problem—ring cyclides have the topology of a torus and, as a result, the director field need not have any defects. Eventually, however, the holes pinch off, and we must build with sections of spindle cyclides (Fig. 5, Inset). When this happens, the director texture will be required to have a $+1$ winding defect at the cusp, leading to a line of defects along the confocal hyperbola. These lines can be seen emanating from the far foci of the ellipses in the bottom view of the sample. To see this, consider the spindle cyclide shown in the Fig. 8, Inset. The director field makes a constant angle with the meridians of the cyclide (this angle rotates smoothly along the pitch axes). The Burgers circuit therefore measures a winding of $+2\pi$.

The side-view of the droplet (Fig. 1) does not show any evidence of a disclination with winding number 2 either spanning the diameter of the droplet or even ending at the center. The dot in the center in Figs. 3A and 9 might be either a projection of the disclination to which a hyperbola reduces when an FCD reduces to a TFCD. Fig. 9 shows the presence of a circle defect centered at the droplet center in the central part of the droplet, thus suggesting that the dot in the center is a projection of the straight line of the TFCD. However, which of the scenarios realized is subject to further experimental verification.

Of course, if there were exactly one cylinder/cone FCD, this would account for the total required charge of $4\pi$—one defect line emanating from the parabolic focus ending on the spherical part of the droplet and a second $+1$ line, also emanating from the focus, ending on the flat part of the boundary. This is not the case: The bottom view of the droplet shows many domains. Note that $+1$ disclination lines can “escape into the third dimension” in pure nematics (26); in a cholesteric, this would create a defect in the pitch axis. For instance, in a disc of radius $\pi /2$, the nematic texture (in cylindrical coordinates) $\mathbf{n}(r)=\widehat{z}cosr+\widehat{\theta }sinr$ has winding $+2\pi$ at $r=\pi /2$ and no winding at $r=0$. However, at the origin, there is not a unique pitch axis around which the director rotates. We have only exchanged a $\chi$-disclination line for a $\lambda$-disclination line. Because of this, the core of a $+1$ disclination line is not only larger than the molecular scale, but it can also show additional structure (27). Indeed, as shown in Fig. 9, we observe the hyperbolæ splitting and twisting, presumably into a pair of disclination lines tracing out charge $-1/2$ defects in the pseudolayers as in Fig. 9.

If each FCD generates $2\pi$ winding in the director that lies in each pseudolayers, what happens to the topological constraint of total winding $4\pi$ in the textures we observe? Note that in a packing of circles or ellipses, typically three of them are mutually tangent. Since the director is continuous on these measuring circuits, it follows that in the interstitials between three tangent circles, there must be a winding of $-\pi$, charge $-1/2$. Isolated charge $-1/2$ disclinations in the pseudolayers have no means of escape or reconfiguration, and we expect their cores to be on the molecular scale. Topological constraints are incorruptible, and so we attribute our inability to observe these defect lines to their suboptical size (no bright field scattering) and the optical turbidity of the droplet (not visible under cross polarizers). Arduous freeze-fracture studies might be possible in the future.

We also note that, though common, the texture we find and describe here is not the only observed structure. When the droplet is smaller than $80\mathrm{\mu }$m in diameter, we only observe a single row of flowers, as in Fig. 7A. Even smaller droplets with a diameter less than $40\mathrm{\mu }$m exhibit no FCD cylinder/cone textures. Similarly, when we increase the pitch of the cholesteric, only larger droplets form the bouquet, suggesting that it is only the dimensional ratio of pitch to diameter that determines the equilibrium structure. We noticed that the history of the sample is crucial for the formation of focal conic flowers. Usually, a bouquet of FCDs similar to those shown in Figs. 1, 3A, and 9 forms some time (several days or weeks) after a freely suspended droplet with the Robinson texture (Fig. 3) attaches to the glass substrate. Liquid crystal material stuck to the substrate during the filling of the capillary might form a sessile droplet, but its texture often is cluttered, far from that of an FCD bouquet. Once such a sessile droplet with a cluttered texture is formed, heating it to the isotropic phase does not result in the formation of a FCD bouquet because of the strong anchoring of liquid crystal molecules at the glass substrate. Analogous textures appear in Janus particle systems of smectic A but with much lower regularity (28). With our model in mind, it would be interesting to return to that system to search for purely smectic bouquets.

The reader might notice that no energetic considerations were made in our construction. That the system is, indeed, finding a ground state is not only implied by the persistence of the droplet texture but also by noting that the geometry alone does not set the size of the cylinder/cone flowers. The fact that they are relatively uniform in radius strongly suggests that they are minimizing an energy and not the result of kinetic trapping, which would, as in typical FCD textures, present as a large assortment of sizes. Apollonius conceived of conics over two millennia ago only to now reveal themselves so perfectly in a state of matter with uniform density. What’s next?

The cholesteric liquid crystal (ChLC) phase was prepared by mixing a nematic phase E3100-100 (from Merck liquid crystals) with the chiral agent S811 (from XARLM, China). The optical indices measured in the nematic phases E3100-100 are $n_o=1.57$ (o=ordinary) and $n_e=1.72$ (e=extraordinary), and thus the birefringence is $\mathrm{\Delta }n=0.15$. Taking into account that the typically for the given chiral dopant HTP=12.77 $\mathrm{\mu }{\mathrm{m}}^{-1}$, the mixtures of the concentrations from 4 to 24 weight percent were prepared in order to obtain cholesteric between 0.33 $\mathrm{\mu }$m and 2 $\mathrm{\mu }$m. Suspensions of cholesteric droplets were prepared stirring the LC material in glycerol. After waiting a few minutes, the emulsion was gently sucked into a glass capillary (VitroCom, USA)—either a 0.4-mm $\times$ 4-mm flat rectangular capillary or a 1-mm square capillary. The dispersity of the droplet diameter depends on the stirring conditions and varies from 50 to 150 $\mathrm{\mu }$m. The capillary was then kept horizontal. After some time, the droplets move to the capillary surface and stick to the glass surface. The droplet structure was followed by optical microscopy (Microscope MEIJI MT9930L, Camera DeltaPix 12MP EXMOR) over several weeks.

Glycerol is known to promote degenerate planar anchoring of the director, which implies homeotropic anchoring of the pitch axis (the twist direction). With such anchoring, the droplets suspended in glycerol exhibit a Robinson or onion structure where the pitch direction is radial Fig. 3B. Quasi-spherical parallel equidistant cholesteric pseudolayers self-assemble in droplets. When the cholesteric liquid crystal is doped with a dye, this microdroplet geometry has been used to promote a tunable and omnidirectional microlaser (29). The director field necessarily contains defects, called disclinations, because of topological constraints due to the spherical geometry. Spherical symmetry of the cholesteric layers in such a droplet is broken by one (or two) radial disclination(s) of total topological strength +2 (+1 each). This (these) line disclination(s) is (are) required by the Poincaré–Hopf–Brouwer theorem since the director lies tangent to the quasi-spherical pseudolayers.

We thank Ivan Dozov (LPS, Paris-Saclay) for providing the nematic liquid crystal material and giving us pertinent advice on microscopy observations. We acknowledge Mélanie Lebental (C2N, Paris-Saclay) who brought our attention on the role of the glass surface on the flower architecture of the droplets. Y.N. thanks the Ministry of Education and Science of Ukraine and the Ministry of Defence of Ukraine for support. Y.N. is grateful to Institut des Sciences Moléculaires d’Orsay (ISMO), Laboratoire de Physique des Solides (LPS), and École Normale Supériere (ENS) Paris-Saclay for hospitality, with a special thanks to Q. Kou (ISMO). Y.N. received financial support from Fondation de Coopération Scientifique “Campus Paris-Saclay” through the MILACHOL3D project of Réseau Thématique de Recherche Avancée (RTRA) ( No. P2013-0561T) and from the Erasmus Mundus program Monabiphot. R.D.K. thanks the Institute for Theoretical Physics at Utrecht University for their hospitality when this work was initiated. R.D.K. was supported by a Simons Investigator Grant from the Simons Foundation.