Hidden topological constellations and polyvalent charges in chiral nematic droplets

Abstract

Topology has an increasingly important role in the physics of condensed matter, quantum systems, material science, photonics and biology, with spectacular realizations of topological concepts in liquid crystals. Here we report on long-lived hidden topological states in thermally quenched, chiral nematic droplets, formed from string-like, triangular and polyhedral constellations of monovalent and polyvalent singular point defects. These topological defects are regularly packed into a spherical liquid volume and stabilized by the elastic energy barrier due to the helical structure and confinement of the liquid crystal in the micro-sphere. We observe, for the first time, topological three-dimensional point defects of the quantized hedgehog charge q=-2, -3. These higher-charge defects act as ideal polyvalent artificial atoms, binding the defects into polyhedral constellations representing topological molecules.

ARTICLE

Received 7 Sep 2016 | Accepted 17 Jan 2017 | Published 21 Feb 2017

Topology has an increasingly important role in the physics of condensed matter, quantum systems, material science, photonics and biology, with spectacular realizations of topological concepts in liquid crystals. Here we report on long-lived hidden topological states in thermally quenched, chiral nematic droplets, formed from string-like, triangular and polyhedral constellations of monovalent and polyvalent singular point defects. These topological defects are regularly packed into a spherical liquid volume and stabilized by the elastic energy barrier due to the helical structure and connement of the liquid crystal in the micro-sphere. We observe, for the rst time, topological three-dimensional point defects of the quantized hedgehog charge q 2, 3. These higher-charge defects act as ideal polyvalent articial

atoms, binding the defects into polyhedral constellations representing topological molecules.

DOI: 10.1038/ncomms14594 OPEN

Hidden topological constellations and polyvalent charges in chiral nematic droplets

Gregor Posnjak1, Simonopar2 & Igor Muevi1,2

1 Condensed Matter Department, Joef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia. 2 Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia. Correspondence and requests for materials should be addressed to I.M.(email: mailto:[email protected]

Web End [email protected] ).

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Topology currently emerges as a major research theme in a number of subelds of physics, and many new topological phenomena have been observed in very different contexts.

For example, nontrivial topology of the electronic structure of crystalline materials leads to exotic material properties, such as surface conductivity of topological insulators1,2 and peculiar electronic structure of Weyl and Dirac semimetals36. Topologically important phenomena can nowadays be traced in condensed matter1, quantum systems7, material science8, photonics9, active matter1012 and even cell division processes in biology13. Understanding the topological properties of matter and applying topological concepts across various subelds of physics not only helps us understand the physics of these systems, but can inspire design of new materials with unusual material properties.

Liquid crystals (LCs) are well known for their richness of topological defects and phenomena, which are observed and analysed on the micrometre scale using optical methods. During the past decade, several spectacular realizations of topological concepts have been demonstrated in LCs, such as topological charge creation and manipulation in nematic liquid crystals (NLCs)1416, and fascinating defect motion in active NLCs11,12. By using laser tweezers and uorescent confocal polarized microscopy (FCPM) it is possible to analyse the topological charges, knot and link the tensorial nematic ordering eld1719 and experimentally prove fundamental theorems, such as the Poincar-Hopf theorem20,21.

Closely related to topology are hidden states of matter, which can be stabilized by topology and refer to states that are not accessible under equilibrium conditions, but can be created if the system is rapidly quenched22. Optically induced hidden states were reported in metallic glasses23 and layered, quasi-two-dimensional electronic crystals24 by applying strong laser pulses. In LCs, transient hidden states of topological defects are obtained by a rapid temperature or pressure quench across the isotropic-to-nematic phase transition. This quench generates a random constellation of topological defects via the Kibble-Zurek mechanism2527, consisting of mutually compensating topological charges, which are unstable and annihilate into vacuum. Topological defects of unit charge can be stabilized in LCs by colloidal inclusions, appearing as micrometre-scale, loop-like structural imperfections, accompanying spheres17, bres14, handlebodies20 and knotted particles19. In chiral nematic liquid crystals, topological monopoles of unit charge are stabilized by the spontaneous winding of the nematic orientational eld, which forms skyrmion-like twisted three-dimensional (3D) structures called torons2830. Topological defects of higher than unit charge have not been observed before in nematic LCs, as an equivalent number of smaller charges usually achieves a lower free energy due to a lower elastic distortion31,32.

NLCs are characterized by spontaneous orientational ordering of long axes of rod-like molecules, which is combined with complete positional disorder of the centres of gravity of molecules. This orientation order is described with a tensorial orientational eld, where the largest eigenvalue of this Q-tensor is pointing into a direction called the director, also referred to as a headless vector n. If the constituent molecules are chiral, the nematic phase spontaneously twists along a direction perpendicular to the director and forms twisted or chiral nematic liquid crystalline phase. When such a spontaneously twisted LC is conned into a sphere, which enforces perpendicular orientation of the LC molecules on the interface, the orientational eld is frustrated and can form a variety of different topological states. It has recently been demonstrated experimentally that these states are rich in topological point defects30, where the core of the defect is molten and the orientational order is strongly depressed. Theory also predicted the existence of linked and knotted loop

defects in chiral nematic droplets33, but such topological states have not yet been conrmed.

Here we show that long-lived topological states in thermally quenched chiral nematic droplets are formed from string-like, triangular and polyhedral constellations of monovalent and polyvalent singular point defects. Topological defects are regularly packed into a spherical liquid volume and stabilized by the elastic energy barrier due to the helical structure and connement of the LC in the micro-sphere. We observe for the rst time the quantization of topological charge of 3D point defects in terms of a unit topological charge. In addition to the q 1 hedgehogs,

which are usually stable in LC droplets and colloids, we observe higher charges with a multiple hedgehog charge q 2, and 3.

These monster charges can be regarded as polyvalent articial atoms, and are able to bind the surrounding unit charge defects into polyhedral constellations representing topological molecules.

ResultsQuenched chiral nematic droplets. To obtain droplets with various topological constellations of point defects we mix chiral LCs with chiral pitch in the range 6 to 12 mm into a viscous liquid, which promotes perpendicular anchoring to form droplets with diameters 10 to 20 mm (see Methods for details). After the droplets are formed, the sample is heated to the isotropic phase and cooled at a rate of several degrees per second back to the chiral nematic phase. This produces topologically complex and higher free energy director structures, which are stable for several days33,34. The complexity of the topological states in chiral nematic droplets depends on the chirality parameter N 2d/p0,

where d is the diameter of the droplet and p0 is the intrinsic pitch of the chiral nematic LC. By far the most common are layered cholesteric structures with a single point defect, which is due to the topology of spherical connement, but for N42.5 structures with more point defects can appear. The droplets are examined by wide-eld optical and FCPM, and director elds are reconstructed from FCPM data using a recently developed method30 (details are described in Methods).

In samples with a chirality NB2.54 we obtain two 1 radial

hedgehogs residing at the surface of the droplet and facing each other on a symmetry axis, as shown in Fig. 1ah. Their topological charge amounts to 2 and another hyperbolic

hedgehog with negative charge 1 appears in between, adding

up the total topological charge inside the sphere to 1. This is

because the Poincar-Hopf theorem requires the net hedgehog charge q of the director eld n(r) with perpendicular surface anchoring along the closed bounding surface S with genus g to be equal to (up to sign) q (1 g) (refs 31,32). For a sphere with

g 0, having no handles, there should be a total hedgehog charge

of 1, where we choose the sign according to the convention

that the director eld, represented as a vector eld, points outwards from the surface of the droplet.

The space between the 1 and 1 hedgehogs is lled with a

twisted LC, having the characteristic shape of an elongated cholesteric bubble, anchored with one end to the 1 defect, as

shown in Fig. 1f. The cholesteric bubbles can be easily recognized in a streamline representation of the director eld in Fig. 1g which shows only the projection of the director eld on a plane. The white areas where the director is mostly perpendicular to the cross-section indicate the locations of the cholesteric bubbles. The bubbles are non-singular, axially symmetric, smoothly winding structures, which induce elastic repulsion between the attracting pairs of topological charges and stabilize the defect constellations. The cross-section of a cholesteric bubble is a two-dimensional skyrmion29 of the Bloch type35, as shown in Fig. 1h.

At a higher chirality parameter N43, we observe in different droplets many diverse constellations for the same value of N,

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1

0

a b c d

e

y

x

f g

h

i

j l m

k

y

Figure 1 | Unit topological charges separated by cholesteric bubbles in spherical chiral nematic droplets. (a) A non-polarized micrograph of a droplet with N 3. (be) Bleaching corrected, deconvolved and normalized FCPM intensities in an equatorial xy plane of the droplet in a, for each of the four

excitation/detection polarizations. Polarization for each panel is indicated with an arrow in top right corner. (f) Reconstructed director cross-section from the FCPM images showing a string of charge-alternating hedgehogs. The cylinders representing the director eld are coloured by the director projection to the cross-section plane. The green-shaded area represents a cholesteric bubble. The thin rods that connect point defects indicate the spatial relation between the defects. (g) Director eld from f in streamlines. Areas where the director is perpendicular to the cross-section have no streamlines, indicating the location of the cholesteric bubble. The schematic representation of the structure in the inset shows the relative positions of the point defects and the cholesteric bubbles. (h) Cross-section of director eld perpendicular to the 3-point string in the middle of a cholesteric bubble. (i) A non-polarized micrograph of a N 3.1 droplet with ve unit charge hedgehogs, forming a V-shaped planar constellation in the focal plane. (jm) Bleaching corrected,

deconvolved and normalized FCPM intensities in an equatorial xy plane of the droplet in i, for each of the four excitation/detection polarizations. (n,o) Director cross-section showing a V-shaped string of ve unit charge hedgehogs shown in n cylinders and o streamlines. The inset to o shows a schematic representation of the structure in io. All scale bars are 5 mm.

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n o

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which can be classied into two species: (i) strings of alternating point defects with unit hedgehog charges, (ii) constellations with a mixture of unit and multiple (that is, higher) hedgehog charges. In all cases, the 1 hedgehogs are found next to the surface,

each anchored to its corresponding cholesteric bubble, while the negatively charged hedgehogs arrange in the bulk between the bubbles.

String-like constellations. Strings of alternating unit-charge hedgehogs are presented in Fig. 1io for 5 and in Fig. 2a,b and Fig. 2c,d for 7 and 9 unit-charge hedgehogs, respectively. Figure 1i shows a non-polarized micrograph of a droplet with a chirality parameter N 3.1 and Fig. 1jm show the corresponding

experimental FCPM data. The reconstructed director eld in Fig. 1n reveals a V-shaped constellation of ve unit, charge-alternating, point defects and the streamline representation in Fig. 1o indicates the positions of three cholesteric bubbles between the point defects as is schematically shown in the inset. The three hedgehogs close to the surface are of the 1 radial type, whereas

the two hedgehogs in the midst of both arms of the V-shaped constellation are the 1 hyperbolic type, so the total topological

charge is 1. Similarly as in the droplet with three point

defects (Fig. 1ah), the cholesteric bubbles, anchored to the 1

hedgehogs, face the hyperbolic 1 hedgehogs between them.

Even longer strings with 7 and 9 unit charge hedgehogs are found at higher chirality (Fig. 2, Supplementary Movies 1 and 2), always containing an odd number of hedgehogs, with each added pair of 1 and 1 hedgehogs forming another bubble and

extending the string of hedgehogs with a new arm as shown in the schematic representations in insets to Fig. 2b,d. Whereas the 5 unit-charge hedgehogs in Fig. 1io can be neatly packed into the equatorial plane of the droplet, the constellations with 7 and 9 unit hedgehogs in Fig. 2b,d pack in string-like structures in 3D.

Higher topological charges. Some hidden states are not formed only of unit topological charge hedgehogs, but also contain

double (q 2) and triple (q 3) hedgehog charges as the

binding elements. A double hedgehog charge appears in a planar conguration in Fig. 3ac and Supplementary Movie 3 with a threefold optical symmetry containing three 1 hedgehogs close

to the surface of the droplet and the fourth hedgehog (cyan dot) located exactly in the centre of the droplet. The three near-surface

1 hedgehogs together with the central hedgehog must add up to the 1 total topological charge of the droplet, which means

that the central hedgehog has a 2 topological charge. Its

threefold symmetry of binding to the surface unit hedgehogs (Fig. 3b) resembles a trivalent atom with three symmetrically positioned orbitals (for example, a carbon atom with sp2 hybridized orbitals). This structure does not match any of the previously experimentally observed point defects in 3D nematics.

An even larger topological charge, carrying a hedgehog charge q 3, is shown in Fig. 3d,e and Supplementary Movie 4. Here

four unit-charge q 1 hedgehogs form a tetrahedral constella

tion around the tetravalent topological charge in the centre of the droplet (green dot). Its topological charge q 3 follows from

the conservation of the overall topological charge. The q 3

hedgehog forms a tightly squeezed director structure at the centre of the droplet, surrounded by four neighbouring cholesteric bubbles, which are compressing it towards the centre. The structure of this q 3 hedgehog defect is fundamentally

three-dimensional, reminiscent of a sp3 hybridized atomic orbital.

Like in molecular chemistry, a single charged unit can be replaced by a compound structure with the same charge, while not changing the valence of the conguration. For example, in Fig. 4a,b and Supplementary Movie 5, one of the 1 hedgehogs

from Fig. 3c is replaced by a string-like constellation of three hedgehogs, similar to the three-hedgehog string in Fig. 1ag and likewise carrying a 1 topological charge. The tetravalent

conguration from Fig. 3d,e also has a less symmetric variant, shown in Fig. 4c,d and Supplementary Movie 6 with six 1

hedgehogs in the vertices of an octahedron. Three of them are a part of the tetravalent conguration with a q 3 point defect at

the centre. The other three, together with two q 1 hedgehogs,

a

c

b

d

Figure 2 | String-like constellations of unit topological charges in spherical chiral nematic droplets. (a) Transmission images of a droplet (N 5.2) with

seven unit charge hedgehogs taken at different focusing depths. (b) Three-dimensional constellation of seven unit charge hedgehogs and a cross-section of the director eld. Tetrahedral symmetry of the constellation is highlighted by the geometrical representation in the inset. Experimental FCPM intensities and complete reconstructed director are shown in Supplementary Movie 1. (c) Transmission micrographs of a droplet with a nine unit charge hedgehog string constellation. (d) Three-dimensional constellation of nine unit charge-alternating hedgehogs and a cross-section of the director eld throughthe equator. Experimental FCPM intensities and complete reconstructed director are shown in Supplementary Movie 2. All scale bars are 5 mm.

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0

a

b

d

1

Figure 3 | Higher topological charges in triangular and tetrahedral constellations. (a) A micrograph of a droplet with three q 1 point defects (yellow)

and a higher charge q 2 point defect (cyan). (b) FCPM total intensity Itot in the equatorial cross-section of the droplet in a. (c) Reconstructed director

eld, shown in streamline projection to the equatorial plane. The thin rods connecting the point defects show the spatial relation of the defects which is highlighted in the geometrical representation in the inset. The other inset schematically shows the point defects and the cholesteric bubbles. Experimental FCPM intensities and complete reconstructed director are shown in Supplementary Movie 3. (d) Micrographs of a droplet with a tetrahedral conguration of q 1 hedgehogs and a triple charge q 3 in the centre at different focuses, showing the positions of defects. (e) A 3D representation of the

structure in a droplet with the 3 defect illustrating the tetrahedral conguration of q 1 hedgehogs, which is highlighted in the inset where the 1

defects are shown in vertices of a tetrahedron. The streamlines show the director eld in a plane which includes two 1 and the 3 defect, intersecting

two cholesteric bubbles. Experimental FCPM intensities and complete reconstructed director are shown in Supplementary Movie 4. All scale bars are 5 mm.

c

e

a

c

b

d

Figure 4 | Tetrahedral and octahedral topological molecules with higher charge defects. (a) Micrographs at different focusing depths showing the locations of point defects and (b) a 3D representation of the structure in a droplet similar to Fig. 3c, where one of the unit charges is replaced by a hedgehog molecule as can be seen in the schematic representation of the structure in bottom left inset. The streamlines show the reconstructed director in a plane which includes two 1, a 1 and the 2 defect. The thin rods connecting the point defects show the spatial relation of the defects, which is

highlighted by the geometrical representation in the top right inset where the 1 defects are shown in the vertices of a tetrahedron. Experimental FCPM

intensities and complete reconstructed director are shown in Supplementary Movie 5. (c) Non-polarized micrographs and (d) a 3D representation of a structure in which one of the 1 charges in the tetrahedral constellation from Fig. 3e is replaced by a V-shaped string of ve unit charges as shown in the

schematic inset. By this substitution the symmetry of the structure changes to octahedral, as illustrated in the lower right inset. Experimental FCPM intensities and complete reconstructed director are shown in Supplementary Movie 6. All scale bars are 5 mm.

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form a V-shaped string with a total charge q 1, which acts as

the fourth 1 hedgehog of the tetrahedral conguration. The

stability of these structures demonstrates that the high-charge defects do not occur strictly in a connement eld with just the right symmetry, but can also withstand exchanges of the surrounding hedgehogs with larger constellations.

DiscussionThe experiments shown in Figs 3 and 4 and Supplementary Movies 36 constitute the rst-ever observation of higher point topological charges in 3D. In NLCs, the topological charge is measured by counting the number of patches of the director eld that are piercing the sphere which is enveloping the defect36. Figure 5a,b show these patches, formed by the reconstructed director eld around the 2 defect, which is decorated

continuously with arrows; a patch is a region where the director eld points outwards (red regions) on a continuous background of inward-pointing director (blue regions; note that this choice of arrows is opposite to the one we chose for the calculation of topological charge and its sign to make patches visually more discernible). Each patch can be understood as a bundle of hyperbolic director streamlines in Fig. 5c that converge towards a neighbouring 1 hedgehog. A defect with M patches

has a hedgehog charge of q 1 M (ref. 36); therefore, the defect

in Fig. 5a,b,d,e has a q 2 charge and Fig. 5f shows a q 3

charge. The patch structure reveals that q 2 and q 3

defects are generalizations of the hyperbolic hedgehog q 1, and form at the conuence of streamlines from the 1 charges below the droplet surface, similar to how saddle points occur in an electric eld formed by a set of equally charged point charges.

Highly symmetric constellations are obtained for the higher charges, which reside in the centre of the droplet: the q 2

charge stabilizes the triangular molecule in Fig. 5g, the q 3

charge binds the tetrahedral molecule in Fig. 5h, and the q 5 the octahedral one in a model in Fig. 5i. The valence

properties of the higher topological charges in chiral nematic spheres are similar to the tetrahedral structure of the monovalent topological defects in nematic shells37 which were proposed to be used for realization of tetravalent colloids38.

It is quite interesting that we nd the 1 hedgehogs close to

the surface in all our experiments on topological hidden states in cholesteric droplets. This can be understood by considering the growth of the cholesteric regions (the cholesteric bubbles) after the quench. It is energetically favourable to expel the 1

hedgehogs towards the surface, where the helical order cannot exist because of the n (r n) 0 surface constraint. The

homeotropic surface itself naturally prefers a general radial-like structure of the near-surface defects, which is why the defects at

a b c

Experiment

d e f

Model

g h i

Figure 5 | Topological valence chemistry with higher topological charges. (a,b) Patch structure from the FCPM experimental data for a 2 defect, from

two perspectives. Notice the outgoing arrows at three valence directions. The direction of arrows on ac is for the sake of visual clarity of this gure inverted with respect to the directions we used for calculation of topological charge. (c) A single patch structure represented as streamlines as an attachment point for a 1 hedgehog. (d,e) A sketch of director streamlines for 2 defect, showing three patches. (f) A sketch of director streamlines for a

tetrahedrally symmetric 3 defect; the fourth patch is pointing away from the reader. (gi) Topological defect constellations with polyvalent charges.

The 1 defects near the droplet surface are bound by the polyvalent negative charge in the centre of the droplet. See Fig. 3c,e for realizations of g,h,

respectively. The scale bar is 2 mm.

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the surface all have the same form and the same unit charge 1.

Defects with positive topological charge greater than 1 cannot

form close to the interfacethey are not equivalent to hyperbolic saddle points, which occur naturally between other defects.

In quasi-two-dimensional systems, the higher topological charges are, in most cases, unstable, as an equivalent number of smaller defects or disclinations achieves a lower free energy due to a lower elastic distortion, but were observed in specic experiments3941. It was shown theoretically that in 3D achiral nematics the hedgehogs with higher charges than 1 are unstable42. Here they are stabilized by chirality (Fig. 6), which acts as a stabilizing spring inside the spherical connement, preventing dissociation of the higher-charge defects. As we can see from the stability chart Fig. 6, several different structures can appear at a given relative chirality N. This implies that the state which will emerge in a droplet after the quench is not determined by the helical pitch and droplet size; instead, the metastable states appear with probabilities which depend on their relative energies17,33. The most complex hidden states appear only in the middle of the stability range of structures with the same number of positive charges (Fig. 6). This shows there is an optimal ratio between chirality and connement, which promotes the formation of higher charge topological defects.

The higher charge defects could in general be true point defects, or have the internal structure of a defect loop, analogous to recently demonstrated defect loops in the cores of 1

defects43, but with a more complex structure of its cross-section44. Alternatively, higher charge defects could be composed of several tightly packed topological point defects, for example, the q 2 could be composed of three 1 defects and one 1

defect. However, the FCPM image in Fig. 3b shows that the q 2 defect is conned to a volume comparable to a 1

defect. Any internal structure would have to be conned to a volume smaller than the resolution of our microscope, which is B300 nm. This is an order of magnitude smaller than the helical period of the LC mixture, which sets the minimum separation between the defects for which the helical twist still acts as a repulsive force30. We can, therefore, safely conclude that the central topological defect is a single point defect, carrying the topological charge q 2.

Our experiments in chiral nematic droplets clearly reveal, for the rst time, the full anatomy of the hedgehog defects, as we are able to reconstruct the topology of the 3D director and perform a patch analysis of the streamlines. Higher topological charges,

which were previously considered unstable are in fact strongly stabilized by the chirality and spherical connement of the NLC droplets on a time scale of several days. Surprisingly, the topological defects can be arranged in a perfect polyhedral constellation inside a liquid sphere, which has implications far beyond the eld of LCs. Our strategy, which combines spherical connement and a chiral eld, offers guidance for the formation of hidden skyrmionic constellations in chiral magnets, cold atoms, and polyvalent soft-matter colloidal science. Regular packing of the topological defects into a sphere can be regarded as the synthesis of topological molecules, formed of polyvalent defects, where the skyrmion-like structures have the role of topological bonds.

From a practical perspective, chiral nematic droplets with higher topological charges in the centre could serve as polyvalent colloidal particles, providing directional bonds to the surrounding colloidal particles. For example, a q 3 topological charge in

the centre of the chiral nematic droplet provides a tetravalent coordination of unit charge q 1 hedgehogs located close to

the surface of the droplet. If the surroundings of these q 1

hedgehogs are decorated with polymer linkers, similar to the idea of Nelson for the nematic shells38, a cholesteric droplet would be similar to a 4-valent atom like carbon, silicon and germanium. Here the polymer linkers could provide articial bonds to neighbouring droplets for self-assembly into colloidal crystals with a diamond lattice.

Methods

Preparation of droplets. The LC droplets were prepared with a low-birefringence LC mixture of 1:1 weight ratio of 40-butyl-4-heptyl-bicyclohexyl-4-carbonitrile (CCN-47) and 4,40-dipentyl-bicyclohexyl-4-carbonitrile (CCN-55, both purchased from Nematel; refractive indices of the mixture no 1.47 and ne 1.50) doped

with 12% of chiral dopant S-811 (Merck) to get a mixture with a pitch in the range B612 mm. A small amount of dye N,N0-bis(2,5-di-tert-butylphenyl)-3,4,9, 10-perylenedicarboximide (BTBP, Sigma Aldrich) was added to the LC mixture to enable FCPM by rst dissolving the dye in acetone, adding the dye/acetone solution to the LC mixture and evaporating the solvent at room temperature to achieve a homogeneous mixture. Droplets with diameters in the range B1020 mm were produced by mixing the LC mixture in a glycerol medium (nglycerol 1.47) doped

with 4% wt. L-a-phosphatidylcholine (lecithin, Avanti Polar Lipids) to achieve perpendicular orientation of LC molecules on the droplet surface. The medium with the droplets was sandwiched between a microscopic cover glass of 150 mm thickness and a thicker 1 mm glass, which were separated by 30 mm mylar spacers. The cell was sealed along the perimeter with a fast curing two-component epoxy glue to prevent droplet movements because of glycerol ow. Constellations of point defects formed after the sample was quenched from the isotropic phase at a rate of several K s 1 to room temperature.

FCPM microscopy and director reconstruction. The structures in the droplets were imaged by uorescence confocal polarizing microscopy (FCPM)45 on a Leica TCS SP5 X confocal microscope with a Leica WLL laser light source. The microscope was modied by inserting a quarter-wave plate for the selected excitation wavelength (488 nm) in the slot below the objective to transform the polarization of the microscope from linear to circular. Linear polarizers were inserted above the waveplate into the same slot as the waveplate to select different linear polarizations of excitation. A full xyz-scan was preformed at four linear polarizations in the xy plane, separated by 45 to obtain intensities I0, Ip/4, Ip/2 and

I3p/4. The scan at the rst polarization was repeated in the end, to be used for linear correction of intensities because of bleaching between the scans. The xyz-stacks were then deconvolved with SVI Huygens Professional software. Orientation of the director in the xy plane was calculated from the intensities as21:

f

1

2 arctan

Number of positive charges

6

5

4

3

2

1

0 1 2 3 4

Relative chirality N = 2*d/p

Ip=4 I3p=4 I0 Ip=2

StringsStructures with 2 defect Structures with 3 defect

: 1

Total intensity Iexp was calculated as: Iexp 12(I0 Ip/4 Ip/2 I3p/4) and

z-dependent background Ioffset and normalization Inorm corrections were applied to calculate the out-of-plane angle y from equation30: IexpI

offset

Inorm cos4y.

The director eld was calculated from y and f as: n (cos y cos f, cos y sin f,

sin y). A simulated annealing algorithm30 was used to determine the sign of the z component of the director eld by randomly selecting a point in the droplet and calculating the elastic energy of director deformation in that point from:

fe

1

2 L

5 6 7

Figure 6 | Stability chart of topological defect strings and polyhedral constellations with higher charges. Note that for each number of defects, the higher hedgehogs are concentrated in the middle of the chirality range.

; 2

@Qij

@xk

@Qij

@xk 4qEiklQij

@Qlj

@xk

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where L is the elastic constant in the one constant approximation and q 2p/p0 is

the inverse cholesteric pitch, which makes the elastic energy chirality-dependent. The tensor Q is calculated from the director as

Qij

S2 3ninj dij

;

3

where the scalar order parameter S, which is not available from the FCPM experiment, is taken to be constant. This elastic energy is compared withthe elastic energy in that point with ipped z-component of director eld(nx, ny, nz). If the energy of the ipped state is lower than the starting energy,

this state is kept, but if it is higher, the ip is accepted with a probability given by a Boltzmann weight exp( DE/t), where DE is the energy difference between the

two states and t is a free parameter. The algorithm starts at a large t, iterating the ipping procedure over the whole volume of the droplet until a stable energy of the whole structure is reached, and then t is lowered and the procedure is repeated until thermalization of the structure is achieved. This procedure scans all possible congurations of signs of nz and nds the ones with the lowest elastic energy within the restraint of experimental data for y and f, which means itis the best approximation of the observed structure that can be found from experimental data30.

To generate the streamline representations of the director, the director eld in a cross-section of the sample was projected on the cross-section plane and smoothed to indicate the average local direction. Areas where the projection of the normalized director is smaller than 1/3 (where director is mostly perpendicular to the cross-section) are excluded from the nal streamline presentation.

Data availability. All the data relevant to the ndings of this study are available from the authors upon request.

References

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Acknowledgements

G.P. acknowledges the nancial support of Slovenian Research Agency (ARRS) through contract PR-05014. S.. acknowledges the nancial support of ARRS through contracts Z1-6725 and P1-0099. I.M. acknowledges the nancial support of ARRS through contracts J1-6723 and P1-0099.

Author contributions

G.P. conducted the experiments, G.P. and S.. analysed the results, I.M. conceived and supervised the experiments and wrote the main manuscript. All authors contributed to the nal version of the manuscript.

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How to cite this article: Posnjak, G. et al. Hidden topological constellations and polyvalent charges in chiral nematic droplets. Nat. Commun. 8, 14594doi: 10.1038/ncomms14594 (2017).

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