Guillaume Durey

Yoko Ishii

Teresa Lopez-Leon
email: teresa.lopez-leon@espci.fr

Temperature-driven anchoring transitions at liquid crystal / water interfaces

Controlling the anchoring of liquid crystal molecules at an interface with a water solution influences the entire organization of the underlying liquid crystal phase, which is crucial for many applications. The simplest way to stabilize such interfaces is by fabricating droplets of liquid crystal in water; however, a greater sensitivity to interfacial effects can be achieved using liquid crystal shells, i.e. spherical films of liquid crystal suspended in water. Anchoring transitions on those systems are traditionally triggered by the adsorption of surfactant molecules onto the interface, which is neither an instantaneous nor a reversible process. In this study, we report the ability to change the anchoring of 4-cyano-4'-pentylbiphenyl (5CB), one of the most widely used liquid crystals, at the interface with dilute water solutions of polyvinyl alcohol (PVA), a polymer commonly used for stabilizing liquid crystal shells, simply by controlling the temperature in the close vicinity of the liquid crystal clearing point. A quasi-static increase in temperature triggers an instantaneous reorientation of the molecules from parallel to perpendicular to the interfaces, owing to the local disordering effect of PVA on 5CB, prior to the phase transition of the bulk 5CB. We study this anchoring transition on both flat suspended films and spherical shells of liquid crystals. Switching anchoring entails a series of structural transformations involving the formation of transient structures in which topological defects are stabilized. The type of defect structure depends on the topology of the film. This method has the ability to influence both interfaces of the film nearly at the same time, and can be applied to transform an initially polydisperse group of nematic shells into a monodisperse population of bivalent shells.

Introduction

Nematic liquid crystals are a fascinating class of soft matter systems. They combine the solid-like property of long-range orientational order with the fluid-like ability to reorganize under very weak external stimuli. Liquid crystal molecules exhibit preferential orientation at interfaces, a phenomenon called anchoring. These boundary conditions are transmitted to the bulk molecules via the elasticity of the liquid crystal. Therefore, by tuning the anchoring at the interfaces, one can influence the organization of the bulk of a liquid crystal sample. Hence control over anchoring is one of the most valuable tools for investigating the fundamental principles that govern the organization of liquid crystals systems. It also lies at the core of many applications, in particular in display technologies, where liquid crystals are in con-tact with a solid substrate. Consequently, anchoring effects on a solid boundary have been extensively studied. [START_REF] Jérôme | Surface effects and anchoring in liquid crystals[END_REF][START_REF] Cognard | Alignment of nematic liquid crystals and their mixtures[END_REF] Two parameters govern the anchoring of a liquid crystal: the preferred molecular orientation at the interface (known as the easy axis) and the energetic penalty associated to violating this orientation (known as the anchoring strength). One speaks of planar anchoring when the easy axis lies in the plane of the substrate; homeotropic anchoring when it is perpendicular to it; or conical anchoring when the angle is between 0 • and 90 • .

Interfaces between liquid crystals and other immiscible liquids, such as water, have recently attracted special attention. From a fundamental point of view, fluid interfaces have remarkable properties: being free from chemical or roughness defects, they are much more spatially homogeneous than solid boundaries. In terms of anchoring, they only prescribe the out-of-plane angular component of the preferred orientation: the in-plane component is degenerate, enabling the bulk liquid crystal a greater freedom of organization. In addition, they have recently shown great potential for building chemical or biological sensors. [START_REF] Miller | Introduction to optical methods for characterizing liquid crystals at interfaces[END_REF][START_REF] Lockwood | Self-assembly of amphiphiles, polymers and proteins at interfaces between thermotropic liquid crystals and aqueous phases[END_REF][START_REF] Wang | Liquid crystals: emerging materials for use in real-time detection applications[END_REF] Indeed, microscopic events such as the adsorption of a protein at the liquid crystal / water interface can trigger a change in anchoring, which propagates a different orientation to the bulk molecules, in turn inducing a macroscopic change in the optical appearance of the sample. Most of these sensors take the form of flat nematic/water interfaces; [START_REF] Brake | Biomolecular Interactions at Phospholipid-Decorated Surfaces of Liquid Crystal[END_REF][START_REF] Sadati | Liquid Crystal Enabled Early Stage Detection of Beta Amyloid Formation on Lipid Monolayers[END_REF] however, some recent studies have shown the advantages of curved interfaces. They can be obtained by confining liquid crystals in capillaries, [START_REF] Zhong | Nematic liquid crystals confined in microcapillaries for imaging phenomena at liquid-liquid interfaces[END_REF][START_REF] Khan | Liquid crystalbased biosensor with backscattering interferometry: A quantitative approach[END_REF] droplets, [START_REF] Gupta | Characterization of adsorbate-induced ordering transitions of liquid crystals within monodisperse droplets[END_REF][START_REF] Lin | Endotoxin-induced structural transformations in liquid crystalline droplets[END_REF][START_REF] Verma | Poly(Llysine)-Coated Liquid Crystal Droplets for Sensitive Detection of DNA and Their Applications in Controlled Release of Drug Molecules[END_REF] or spherical shells. 13 Nematic droplets are the simplest systems whose topology stabilizes defects in the liquid crystalline order -points where the director, i.e. a nonpolar vector indicating the average molecular orientation at a mesoscopic scale, is undefined. [START_REF] Lavrentovich | Topological defects in dispersed words and worlds around liquid crystals, or liquid crystal drops[END_REF][START_REF] Lopez-Leon | Drops and shells of liquid crystal[END_REF] An anchoring transition on a drop therefore translates to a modification of its topological constraints, leading to a striking change between well-known optical patterns. Yet, the low surface-to-volume ratio of a droplet makes its behavior mostly governed by bulk effects. Finally, liquid crystal shells -spherical films of liquid crystal suspended in an aqueous solution -provide a system that combines the sensitivity to interfacial events of thin films with the stabilization of topological defects of droplets. [START_REF] Lopez-Leon | Drops and shells of liquid crystal[END_REF][START_REF] Fernández-Nieves | Novel defect structures in nematic liquid crystal shells[END_REF][START_REF] Lopez-Leon | Frustrated nematic order in spherical geometries[END_REF][START_REF] Urbanski | Liquid crystals in micron-scale droplets, shells and fibers[END_REF] The study of anchoring transitions in shells has a fundamental interest besides the development of applications. The two interfaces confining the shell allow for four different types of boundary conditions: planar-planar, planar-homeotropic, homeotropic-planar and homeotropic-homeotropic. Switching from one type of boundary condition to another usually triggers dynamical transitions involving the formation and recombination of topological defects. [START_REF] Lopez-Leon | Topological transformations in bipolar shells of nematic liquid crystals[END_REF][START_REF] Lopez-Leon | Defect coalescence in spherical nematic shells[END_REF][START_REF] Noh | Influence of interface stabilisers and surrounding aqueous phases on nematic liquid crystal shells[END_REF][START_REF] Tran | Change in stripes for cholesteric shells via anchoring in moderation[END_REF][START_REF] Noh | Sub-second dynamic phototuning of alignment in azodendrimer-doped nematic liquid crystal shells[END_REF][START_REF] Sharma | Realignment of Liquid Crystal Shells Driven by Temperature-Dependent Surfactant Solubility[END_REF] Transitions from purely planar or homeotropic states to hybrid states have been used to obtain information about the fundamental nature of topological defects, [START_REF] Lopez-Leon | Topological transformations in bipolar shells of nematic liquid crystals[END_REF][START_REF] Tran | Change in stripes for cholesteric shells via anchoring in moderation[END_REF] and to make them interact in unusual ways. [START_REF] Lopez-Leon | Defect coalescence in spherical nematic shells[END_REF] In contrast, direct transitions between purely planar and purely homeotropic states remain elusive when using traditional methods to control the orientation of the liquid crystal molecules at fluid interfaces.

The traditional method for inducing anchoring transitions at liquid crystal / water interfaces, whether in flat films, [START_REF] Brake | An experimental system for imaging the reversible adsorption of amphiphiles at aqueous-liquid crystal interfaces[END_REF] droplets, [START_REF] Poulin | Novel colloidal interactions in anisotropic fluids[END_REF][START_REF] Poulin | Inverted and multiple nematic emulsions[END_REF] or shells 13, [START_REF] Lopez-Leon | Topological transformations in bipolar shells of nematic liquid crystals[END_REF][START_REF] Lopez-Leon | Defect coalescence in spherical nematic shells[END_REF][START_REF] Noh | Influence of interface stabilisers and surrounding aqueous phases on nematic liquid crystal shells[END_REF][START_REF] Tran | Change in stripes for cholesteric shells via anchoring in moderation[END_REF][START_REF] Sharma | Realignment of Liquid Crystal Shells Driven by Temperature-Dependent Surfactant Solubility[END_REF][START_REF] Sharma | Influence of head group and chain length of surfactants used for stabilising liquid crystal shells[END_REF] consists in dissolving surfactant molecules in the water phase. Commonly used liquid crystals such as cyanobiphenyls display planar anchoring when in contact with water. The adsorption of surfactants onto the interface typically forces the liquid crystal molecules to reorient in the direction of the surfactant alkyl chains, therefore inducing homeotropic anchoring. This method comes with several shortcomings, especially when considering transitions on shells. Firstly, one cannot switch instantaneously between anchoring states in a controlled way. Indeed, the dynamics of the transition is governed by surfactant adsorption, whose spatiotemporal evolution cannot be controlled accurately. Secondly, the switching process is not reversible. While adding surfactants to the continuous phase without altering the shell position is tricky, flushing the shell with water to remove 2 the surfactants without losing it is impracticable. Thirdly, one cannot induce an anchoring transition on both interfaces of a shell at the same time, as the outer interface is the only one where the concentration of surfactants can be tuned directly. Indeed, although these molecules can diffuse through the liquid crystal film, from the continuous aqueous phase to the inner water droplet, it typically takes them about a day to do so -leaving the shell with ill-defined boundary conditions for an extended period of time. Despite recent efforts to induce controlled anchoring transitions in liquid crystal shells, [START_REF] Sharma | Realignment of Liquid Crystal Shells Driven by Temperature-Dependent Surfactant Solubility[END_REF][START_REF] Noh | Dynamic tuning of the director field in liquid crystal shells using block copolymers[END_REF] a powerful alternative to traditional surfactant-induced techniques is still missing.

In this article, we report a new method to induce a change in boundary conditions at liquid crystal / water interfaces that addresses those issues. It relies on coating the interface with polyvinyl alcohol (PVA), a polymer widely used for stabilizing double emulsions and for enforcing strong planar anchoring. We show that PVA enables a richer control over the molecular anchoring at a fluid interface than what was previously thought: it is actually possible to tune its influence on the underlying liquid crystal phase through a precise control of the temperature of the system. We focus on the interface between water solutions of PVA and 4cyano-4'-pentylbiphenyl (5CB), one of the most widely used liquid crystals. We demonstrate that the usual planar alignment at room temperature is replaced by perpendicular alignment to the interfaces when the temperature is brought just a few tenths of degrees Celsius below the clearing point of the liquid crystal. With this easily controllable experimental parameter, one can therefore have the system switch between two antagonist anchoring states. This anchoring transition is nearly instantaneous, as well as spatially homogeneous, and affects nearly simultaneously all the interfaces of the liquid crystal. We first study the transition in flat suspended nematic films, which provide a new experimental platform extremely sensitive to interfacial phenomena. We then demonstrate its usefulness by applying it on liquid crystal shells, in which the change of boundary conditions triggers a series of structural transformations, imposing the formation of topological defects with integer topological charge.

Materials and methods

For this study, we consider two different geometries: spherical nematic shells and flat suspended nematic films. Shells provide a simple way to stabilize a liquid crystal film in water without the assistance of any solid boundary. While surface tension tends to preserve such a spherical geometry, it makes flat films unstable. Flat suspended films are more challenging to stabilize, but provide a simpler geometry to interpret anchoring transitions in a straightforward way, since curvature and topological constraints are not present. It thus constitutes a good starting point for our study. Here we propose a strategy to stabilize such films.

A commonly used method for stabilizing water / liquid crystal interfaces, developed by Abbott and coworkers, [START_REF] Brake | An experimental system for imaging the reversible adsorption of amphiphiles at aqueous-liquid crystal interfaces[END_REF] relies on confining the liquid crystal to a transmission electron microscopy (TEM) grid -a small metal disk 3.05 mm in diameter and 18 µm in thickness, having an array of holes with a given geometric shape. In Abbott's system, the grid is placed on top of a glass substrate, so that the pores of the grid become micro-sized observation cells. Confining the liquid crystal to these cells prevents it from dewetting the water interface by pinning it to a solid substrate. When the cells are immersed in an aqueous solution, the liquid crystal is sandwiched between a fluid and a solid interface. The glass solid substrate is usually coated with octadecyltrichlorosilane (OTS) in order to enforce homeotropic anchoring. These strong anchoring cues on the bottom interface constrain the system, reducing its sensitivity to subtle anchoring changes occurring at the free interface.

Here, we propose a novel adaptation of Abbott's system, which addresses those issues. Indeed, we produced suspended films in water, in which the anchoring at the two interfaces is solely controlled by the aqueous solution.

3

A) D) C) E) B)
Figure 1: Films and shells of nematic liquid crystals in water. In this study, two complementary experimental systems were used: thin nematic films (top row) and nematic shells (bottom row). Nematic films were produced by fixing a copper double TEM grid in mid-height of an observation chamber filled with a water solution. One half of the double grid was embedded into the walls of the chamber, made of double-sided tape, while the other single grid was filled with a sub-microliter volume of liquid crystal. A) Photograph (left) and schematic representations (top right: top view, bottom right: side view) of the experimental chamber mounted on a microscope glass slide. B) Close-up photograph of a double TEM grid. Each grid is 3.05 mm in diameter. C) Cross-section of a cell of the grid filled with liquid crystal. The cell is 430 µm in width, 18 µm in thickness. The film is about 5 µm in the center. The vertical scale has been exaggerated ten times with respect to the horizontal scale. Nematic shells were produced by using a microfluidic device made of two tapered glass capillaries facing opposite directions, nested in a larger square capillary. D) Microphotograph of shell production. The tip of the capillary is about 150 µm wide. E) Schematic representation of a vertical cross-section of a nematic shell. The thickness is heterogeneous due to a slight difference in density between the water phases and the 5CB. The typical outer diameter is 100 µm while the typical mean thickness is 10 µm.

To do this, we used double "oyster" grids, [START_REF] Popov | Accurate optical detection of amphiphiles at liquidcrystal-water interfaces[END_REF] consisting of two laterally-fused standard TEM grids. Fig. 1A shows the observation chamber in which they were placed, and Fig. 1B one of these grids. A 500 µm high cylindrical observation chamber was built by stacking ten layers of double-sided tape (Gudy 804, Filmolux) on a glass slide. A double oyster grid (GD50/100-Cu, Gilder) was deposited on the tape, in such a way that one of the single grids was stuck to the tape while the second one was suspended in mid-air. Ten more layers of tape were added on top of the first ones, bringing the suspended grid in the middle plane of the one-millimeter-high chamber. 0.1 µL of 5CB (Synthon Chemicals) or 6CB (Sigma-Aldrich) were deposited on the grid. The observation chamber was then filled with the aqueous phase, which could either be pure deionized water (Millipore Synergy), or a PVA solution (13-23 kg • mol -1 , 87-89% hydrolyzed, Sigma-Aldrich). Filling the chamber also detached an excess of liquid crystal from overfilled cells of the grid, generating a few microdroplets which adsorbed to the hydrophobic walls of the chamber. The chamber was finally sealed with a glass cover slip. For observation, we focused on a single square hole in the grid, a vertical cross-section of which is

4 Page 4 of 17 ACS Paragon Plus Environment Langmuir i) B) i) A) ii) ii) iii) iii) iv) iv)
Figure 2: 5CB film in a 0.1 % wt PVA solution undergoing a slow temperature ramp Top: series of top-view micrographs through crossed polarizers. Bottom: series of vertical cross-sections schematically depicting the molecular arrangement at the location of the black and white rectangle. Throughout, the film is 430 µm wide, about 10 µm thick close to the edges, about 5 µm thick in the center, and is attached to a suspended copper TEM grid. The film undergoes a temperature ramp with a rate of 0.01 • C • min -1 . i) In most of the nematic range, the film displays strong birefringence colors, an indication of the planar anchoring that aligns the molecules parallel to the interfaces. This particular film includes a topological defect; director field lines are sketched in its vicinity. The vertical cross-section is taken far from the defect, along a direction where the director is in the plane of the cross-section and uniform in this initial stage. ii) A few tenths of degrees Celsius below the phase transition temperature of pure 5CB, a molecular reorientation perpendicularly to the interfaces occurs, evidenced by a dark film with bright edges, likely due to the interfacial PVA-5CB mixture becoming isotropic. iii) When the temperature reaches the clearing point of pure 5CB, islands of bulk isotropic phase nucleate from the top and bottom interfaces of the film, inducing conical anchoring of the remaining nematic pure 5CB in the bulk. iv) The isotropic islands grow in size until the whole film turns isotropic. These four micrographs, taken respectively at -0.05 • C, -0.01 • C, 0.02 • C and 0.11 • C of the apparition of the first island of bulk isotropic phase, are frames of the timelapse provided in Supplementary Movie 1. sketched in Fig. 1C.

Nematic shells of 5CB were produced using a glass capillary device. [START_REF] Utada | Monodisperse double emulsions generated 15 Page 15 of 17 ACS Paragon Plus Environment Langmuir from a microcapillary device[END_REF] In this type of device, a water / liquid crystal compound jet is sheared by a water-based outer solution using a combination of co-flow and flow-focusing geometries, and breaks up into double emulsions, as shown in Fig. 1C. To reach production regimes at reasonable flow rates, the viscosity of the outer aqueous phase was increased by adding 60 % wt glycerol (VWR Chemicals). The inner aqueous phase was either pure deionized water or a 1 % wt PVA solution. The shells were collected in a 200 µm high cylindrical observation chamber built by stacking two layers of double-sided tape. The continuous collection phase was iden-tical to the inner aqueous phase, with traces of glycerol originating from the outer aqueous phase in the capillary device. Due to a slight density difference between 5CB and the aqueous phases, the inner aqueous droplet typically rises to the top of the shell, making the shell heterogeneous in thickness.

We achieved an extremely fine level of temperature control in order to witness temperature-driven anchoring transitions on these experimental systems. The observation chambers were inserted into a TS62 thermal stage driven by a mK1000 unit, both from Instec. They were subjected to a 0.01 • C • min -1 temperature ramp close to the clearing point of the liquid crystal (around 34.5 • C for 5CB, 5 ACS Paragon Plus Environment Langmuir around 29.5 • C for 6CB). Temperature was controlled with an accuracy better than a tenth of degree Celsius. The sample's evolution was recorded using a Nikon DSLR camera (D5300 or D300s) mounted on an upright polarized optical microscope (Nikon Ni-U).

Results and discussion

Temperature-driven anchoring transitions in nematic films

Parallel state

As widely reported, 5CB displays planar anchoring when in contact with an aqueous solution containing PVA. This is consistent with the strong birefringence colors displayed by our suspended nematic films at room temperature, as shown in Fig. 2-i, which is a cross-polarised image of a 5CB film suspended in a 0.1 % wt PVA solution. The colors become more and more vivid towards the center of the cell, indicating that the film is concave. Using a Michel-Lévy chart, [START_REF] Sørensen | A revised Michel-Lévy interference colour chart based on firstprinciples calculations[END_REF] we estimate that its thickness ranges from 10 µm on the edges to 5 µm in the center. Besides, since the contact with the metallic substrate provides homeotropic anchoring, [START_REF] Lockwood | Self-assembly of amphiphiles, polymers and proteins at interfaces between thermotropic liquid crystals and aqueous phases[END_REF] the liquid crystal molecules are arranged perpendicularly to the edges of the film and parallel to the interfaces with water, yielding a radial defect in the center of the cell in the specific case of the film featured in Fig. 2. Defects can be characterized by their topological charge, which quantifies the number of times the director rotates around the singularity. The defect in our film has a [+1] topological charge, indicating a 2π-rotation of the director around the defect core. This can be distinguished by the number of dark brushes stemming from the defect, which is equal to four times the absolute value of its topological charge.

Interestingly, we observe that the molecular anchoring at the interfaces with the aqueous solution can be changed by accurately controlling the film temperature in the vicinity of its clearing point. By quasi-statically increasing the temperature of the system with a linear rate of 0.01 • C • min -1 , we were able to sweep sequen-tially through a series of distinct states with different boundary conditions until the entire sample turns isotropic. This evolution, recorded in Supplementary Movie 1, is as follows: the usual parallel state (Fig. 2-i), a new perpendicular state (Fig. 2-ii) and a coexistence state (Fig. 2-iii), before the isotropic state (Fig. 2-iv).

Perpendicular state

A few tenths of degrees Celsius below the phase transition temperature of the bulk 5CB, an unexpected abrupt reorientation to a perpendicular state occurs. Indeed, the appearance of the cell changes abruptly to a black film with a bright rim (Fig. 2-ii), a pattern which is recognized as the signature of homeotropic anchoring at a liquid crystal / water interface. [START_REF] Brake | An experimental system for imaging the reversible adsorption of amphiphiles at aqueous-liquid crystal interfaces[END_REF] The loss of birefringence is explained by the reorientation of the long axis of the molecules in the bulk perpendicularly to the interfaces: the optic axis of the liquid crystal becomes aligned with the optical axis of the microscope. The bright rim corresponds to the region where the director transitions to accommodate the homeotropic anchoring cues on the edges of the grid with the homeotropic anchoring cues on the top and bottom interfaces. We were able to maintain the film in this state for more than eight hours by keeping the temperature constant.

The perpendicular orientation of the liquid crystal molecules was confirmed by studying 5CB films doped with an anisotropic fluorescent dye.

We used the anisotropic dye N,N'-Bis(2,5-di-tert-butylphenyl)-3,4,9,10perylenedicarboximide (BTBP), which aligns with the 5CB molecules. [START_REF] Smalyukh | Focused laser beams and liquid crystals: fast three-dimensional imaging of structures and topological defects[END_REF] The intensity of the fluorescence signal is minimum when the dye molecules are oriented along the direction of light propagation. When imaging the transition from the birefringence colors to the dark state, we observed that the fluorescence in the homeotropic region was indeed less intense than in the planar region, as shown in Fig. S1. Also as expected, the isotropic state at the end of the evolution of the film went back up to intermediate intensities, about two-thirds of the way from homeotropic to planar levels. We considered the fluorescence emission as be- ing temperature-independent over such a small temperature interval. Besides, if the temperature were to influence the fluorescence intensity, we would expect a continuous decrease. Yet we report non-monotonic behavior.

Coexistence state

For a temperature even closer to the clearing point, the film enters a state of coexistence between the bulk nematic and isotropic phases. When the temperature is increased from the perpendicular state, the isotropic phase nucleates from the grid walls, creating a homeotropically-aligned nematic domain no longer in contact with the walls, as shown in Fig. 2-iii. The most striking feature of this state is the reappearance of birefringence in the form of a number of bright disk-like islands embedded in the homeotropic region, introduced in Fig. 2-iii and studied in detail in Fig. 3. As the isotropic phase grows, the homeotropic domain shrinks -in Fig. 3, the homeotropic domain has shrunk more than in Fig. 2-iii. It eventually breaks up into multiple smaller domains, before the whole film turns isotropic. When the temperature is lowered from the isotropic state, at the same rate of -0.01 • C • min -1 , homeotropically-aligned domains of nematic phase nucleate from the isotropic phase, then merge and grow until the whole film is homeotropic. Contrary to what happens upon heating the film, there are no birefringent islands formed at the coexistence state when the film is cooled.

The structure of the islands can be revealed by investigating their properties, both static and dynamic (Fig. 3, Supplementary Movie 1 and Fig. S2). They stem from the nucleation of flattened regions of isotropic phase at the liquid crystal / water interface, extending into the bulk of the nematic phase. This situation, where pure isotropic 5CB coexists with pure nematic 5CB, has been described in the literature. [START_REF] Faetti | Nematic-isotropic interface of some members of the homologous series of 4-cyano-4-(n-alkyl)biphenyl liquid crystals[END_REF][START_REF] Faetti | Anchoring at the Interface Between a Nematic Liquid Crystal and an Isotropic Substrate[END_REF] The isotropic phase of 5CB was reported to enforce out-of-plane anchoring of its nematic phase, at an angle of 26.5 • ± 6 • from the surface. [START_REF] Faetti | Nematic-isotropic interface of some members of the homologous series of 4-cyano-4-(n-alkyl)biphenyl liquid crystals[END_REF] This explains the re-emergence of birefringence around the islands of isotropic 5CB. In bright field, all islands exhibit a point defect at their center and a sharp boundary with the surrounding homeotropic domain (Fig. 3A). Under crossed polarizers, they display first-order blue-gray interference colors, meaning that the liquid crystal region contributing to the birefringence is very thin (Fig. 3B). The defects are located at the intersection of four dark brushes which follow both polarizers when rotated, indicating that their topological charge is [+1]. When a first-order retardation plate is added, the liquid 7 crystal molecules are found to be arranged radially from the defect (Fig. 3C). The absence of [+1/2] defects -which induce a π-rotation of the director, and thus, are only allowed when the boundary conditions are planar -is consistent with the existence of conical anchoring cues at the boundaries. [START_REF] Kim | Suspended, one-side anchored, or double-side anchored nematic droplets in an isotropic medium[END_REF][START_REF] Kim | Oligomers as Triggers for Responsive Liquid Crystals[END_REF] Monitoring the evolution of the islands in time reveals additional information about their nature. The islands nucleate in the homeotropic domain. As temperature rises, they slowly grow in diameter, their color remaining unchanged. Then they meet one of two possible outcomes, or a combination of both. On the one hand, they can disappear by coalescence: two islands fuse from the side, creating a [-1] defect at the junction, which quickly annihilates with one of the [+1] defects from the original islands (Fig. S2A). Yet islands can sometimes be seen to overlap without coalescing. In these instances, the interference colors add up but the geometry of the patterns remains unperturbed (Fig. S2B). This demonstrates that while the islands are located within the bulk of the sample, they do not extend far from the liquid crystal / water interfaces in terms of depth. In most experimental realizations, they only form at either the bottom or the top interface, but sometimes they appear on both, in which case they can only interact with islands on the same interface. On the other hand, they can collide with the boundary between the homeotropic domain and the isotropic phase. In this case, they fuse with the isotropic phase (Fig. S2A). We were able to trigger the formation of identical islands in a thin nematic film (approximately 30 µm thick) confined between two glass plates treated to induce strong homeotropic anchoring undergoing a phase transition (see Supplementary Information and Supplementary Movie 2). This confirmed that their presence was indicative of a simple phase coexistence in the bulk between pure nematic and isotropic 5CB. Finally, we note that the same type of islands were reported by Abbott et al. very recently, [START_REF] Kim | Oligomers as Triggers for Responsive Liquid Crystals[END_REF] and that their analysis is consistent with ours.

The role of PVA: experimental clues and mechanistic insights

Upon repeating these experiments for PVA solutions with decreasing concentrations, we were able to bring out the fact that a PVA concentration above a certain threshold was a necessary requirement for the temperature-driven anchoring transition to take place. Indeed, no anchoring transitions can be observed for samples with PVA below 0.1 % wt in the aqueous solution. Films go directly from planar nematic to isotropic, as can be seen in Supplementary Movie 3 and Fig. 4A, where the aqueous solution is simply pure water.

We were also able to prove that the value of the anchoring transition temperature was linked to the liquid crystal -not to the PVA -by conducting additional experiments with a different liquid crystal of the same series. While the clearing point of 5CB is around 34.5 • C, 4cyano-4'-hexylbiphenyl (6CB) has a phase transition temperature at around 29.5 • C. Performing the same experiments with 6CB yielded identical results, albeit uniformly shifted in temperature: an anchoring transition was observed a few tenths of degrees Celsius below the clearing point when the liquid crystal was in contact with a 0.1 % wt PVA solution; no anchoring transition was observed prior to the phase transition when in contact with pure water.

In the rest of this subsection, we offer suggestions into the nature of the mechanism at play in our experiments, although a conclusive experimental validation is out of the scope of the present paper.

From the experiments mentioned above, it seems that the temperature-induced anchoring transition observed in the presence of PVA stems from a change of interfacial order in the film. Since the PVA chains lie at the liquid crystal / water interface, they disrupt the nematic arrangement of the liquid crystal molecules in the vicinity. Therefore, the order parameter of the nematic phase is expected to be lower near the interface than in the bulk. With greater disorder comes a lower phase transition temperature: hence, such an interfacial mixture of PVA chains and 5CB molecules would have a slightly lower clearing point than the pure 5CB bulk. This suggests that our system undergoes an interfacial melting of the PVA-5CB mixture before the bulk pure 5CB turns isotropic. Thus, we hypothesize that once this interfacial layer melts, a new interface appears in our films: it separates the isotropic PVA-5CB surface layer from the bulk nematic phase of pure 5CB. Such a localized phase transition in the presence of adsorbates has been reported in recent studies. [START_REF] Kim | Oligomers as Triggers for Responsive Liquid Crystals[END_REF][START_REF] Ramezani-Dakhel | Amphiphile-Induced Phase Transition of Liquid Crystals at Aqueous Interfaces[END_REF] How does the presumed existence of this new interface cause the reorientation of the underlying liquid crystal molecules? The most probable hypothesis is that it induces a change in the easy axis accompanied by a decrease in anchoring strength. Indeed, it seems reasonable to think that the easy axis prescribed by the isotropic PVA-5CB surface layer is close to the one reported in the literature in the case of pure isotropic 5CB, [START_REF] Faetti | Nematic-isotropic interface of some members of the homologous series of 4-cyano-4-(n-alkyl)biphenyl liquid crystals[END_REF][START_REF] Faetti | Anchoring at the Interface Between a Nematic Liquid Crystal and an Isotropic Substrate[END_REF] thus imposing out-of-plane anchoring to the underlying bulk nematic film. Such boundary conditions would generate splay and bend distortions of the director field in the nematic phase. Yet, as the anchoring strength between the nematic and isotropic phases of a single compound is relatively weak, it may be energetically favorable for the system to relax these distortions immediately, through the uniform reorientation of the molecules perpendicularly to the interfaces, violating the new easy axis in the process. A similar mechanism has been used to explain recent experimental findings. [START_REF] Noh | Dynamic tuning of the director field in liquid crystal shells using block copolymers[END_REF][START_REF] Kim | Oligomers as Triggers for Responsive Liquid Crystals[END_REF][START_REF] Ramezani-Dakhel | Amphiphile-Induced Phase Transition of Liquid Crystals at Aqueous Interfaces[END_REF] However, as far as we know, the easy axis at the interface of a bulk nematic phase with an isotropic PVA/5CB mixture has not yet been measured. An alternative possibility to the indirect mechanism above would be that the interfacial layer directly enforces homeotropic anchoring. Quantitative in-depth experiments are required to tell the two mechanisms apart.

Application to nematic shells

This new mechanism, characterized in thin nematic films, can be profitably applied in liquid crystal shells, altering the topological con-

9 i) ii)
iii) iv) straints they are subjected to. Shells were prepared with planar anchoring at both of their interfaces. As exemplified by the lines of latitude or longitude on a globe, nematic ordering under planar anchoring is frustrated on a sphere, and has to incorporate topologicallyprotected defects in the director. As a consequcence of the Poincaré-Hopf theorem, [START_REF] Poincaré | Sur les courbes définies par les équations différentielles[END_REF][START_REF] Hopf | Vektorfelder in n-dimensionalen Mannigfaltigkeiten[END_REF] the topological charges of the defects on such a sphere have to add up to [+2]. In nematics, topological charges higher than [+1] are unstable, and pairs of defects with opposite topological charges annihilate. Therefore, planar nematic shells can either be bivalent (2×

[+1]), trivalent (1×[+1] + 2×[+1/2]) or tetravalent (4×[+1/2]
). Yet a three-dimensional shell is not a two-dimensional sphere: the defects are not surface points but disclination lines (singular or escaped) that span the shell thickness. Their equilibrium positions result from the balance between the repulsion of positivelycharged defects and the tendency of the lines to minimize their length by migrating to the region where the shell is thinner. [START_REF] Fernández-Nieves | Novel defect structures in nematic liquid crystal shells[END_REF] Liquid crystal shells are usually stabilized by adding PVA to the aqueous phase; however, those shells can be also produced without adding any stabilizer. While a regular waterin-oil-in-water double emulsion destabilizes in a matter of seconds without a surfactant, this is not the case when the middle phase is a liquid crystal, since rupturing the shell would trigger the nucleation of additional topological defects. This process has a large energy barrier associated to it, which stabilizes the double emulsion. [START_REF] Lavrentovich | Topological defects in dispersed words and worlds around liquid crystals, or liquid crystal drops[END_REF] For this reason, we were able to produce 5CB shells with pure water as the inner and outer solutions. These shells were subjected to a 0.01 • C • min -1 temperature ramp, as shown in Supplementary Movie 4 10 and Fig. 4B. We witnessed the nucleation and growth of the isotropic phase within the planar nematic phase, until it took over the whole shell, the planar texture remaining unchanged throughout the entire process.

However, when the shells were produced with 1 % wt PVA in water as the inner and outer solutions, under the same experimental conditions, two anchoring transitions occurred ahead of the phase transition, as depicted in Supplementary Movie 5 and Fig. 5. As the temperature slowly rose, three states could be unambiguously identified -a planar state (Fig. 5i), a hybrid state (Fig. 5-ii), and a radial state (Fig. 5-iii) -before the shell turned isotropic (Fig. 5-iv). The planar state is the one that was discussed above; the boundary conditions are planar at both interfaces. The hybrid state corresponds to a situation where the anchoring cues are planar at one interface and non-planar at the other one. Topological defects are not required on the surface if it has homeotropic anchoring. As the director has to adapt smoothly to the conflicting planar boundary condition on the other surface, it is necessarily tilted in the bulk. In this situation, half-integer defects are no longer allowed: while the orientation of rodlike molecules lying in a plane is unchanged by a π rotation, this is no longer the case when the molecules acquire polarity by tilting out of the plane. For this reason, hybrid shells can only display surface defects with integer topological charge, and are necessarily bivalent. The two defects are diametrically opposed to one another, as they only feel the inter-defect repulsion. These structural features, along with a characteristic vividly-colored and twisted texture, enable one to recognize hybrid shells under the microscope. [START_REF] Lopez-Leon | Topological transformations in bipolar shells of nematic liquid crystals[END_REF][START_REF] Lopez-Leon | Defect coalescence in spherical nematic shells[END_REF] Lastly, in the radial state, shells are defect-free, with a uniform, radiallyoriented director. Their optical signature is a series of colored rings barred with a large Maltese cross. It is in fact the same pattern obtained by illuminating a flat nematic cell with a uniform, vertically-oriented director in convergent light -known as a uniaxial conoscopic interference figure. [START_REF] Lopez-Leon | Drops and shells of liquid crystal[END_REF] These results show that temperature-driven anchoring transitions enable the rapid trans-formation between shell states with matching anchoring at both interfaces. This is a clear improvement from traditional surfactantdriven anchoring transitions, which are limited to transformations between planar to hybrid or hybrid to homeotropic shell states. On the other hand, while in suspended films the anchoring transition occurred simultaneously at the top and bottom interfaces, this is no longer the case in shells: the transition at the inner interface is slightly delayed compared to the one at the outer interface, leading to stable hybrid shells. This last result was unexpected: we hypothesize that the convex and concave natures of the outer and inner interfaces, respectively, play a role in shifting the interfacial melting temperature of the 5CB/PVA mixture. Additionally, another difference between those two systems is the absence of a stable coexistence state between the isotropic and the nematic phase: the islands from the flat films were not observed in shells.

This new result enabled our new technique to be used with a new goal in mind: controlling the valency of a population of shells. Typically, shells are produced in an out-ofequilibrium fashion, often yielding a polydisperse population -i.e. composed of shells with various metastable defect configurations, or valencies -such as depicted in Fig. 6-i. This particular collection of shells was subjected to a 0.01 • C • min -1 temperature ramp: the shells went through all of the previously described states, until all of them turned isotropic. The sample was then cooled down with a -0.01 • C • min -1 temperature ramp. The shells went through all of the states again, this time in reversed order, until the planar texture was retrieved. However, at this stage, all of the shells had the same defect configuration, as seen in Fig. 6-ii. This evolution can additionally be witnessed in Supplementary Movie 6.

Instead of exhibiting the lowest-energy configuration, with four [+1/2] defects, [START_REF] Seyednejad | Confined nematic liquid crystal between two spherical boundaries with planar anchoring[END_REF][START_REF] Koning | Spherical nematic shells with a threefold valence[END_REF][START_REF] Zhou | Mesoscale Structure of Chiral Nematic Shells[END_REF] they displayed a higher-energy configuration, with two [+1] defects. Indeed, our study shows that the phase transition is accompanied by an anchoring transition, which alters the topological constraints on the shell. As the hybrid state 11 forbids the presence of half-integer defects, the shell is topologically prevented from reaching its equilibrium state. Therefore, our method turns an initially polydisperse shell population into a monodisperse population of bivalent shells. This new ability to uniformize the defect configurations of a population of planar shells could prove very useful in the perspective of shell selfassembly through defect functionalization -a long-term goal which has motivated the experimental investigation of nematic shells since the very beginning. 44 Temperature-induced anchoring transitions in shells have been the object of a growing interest in the past few years. [START_REF] Noh | Influence of interface stabilisers and surrounding aqueous phases on nematic liquid crystal shells[END_REF][START_REF] Tran | Change in stripes for cholesteric shells via anchoring in moderation[END_REF][START_REF] Sharma | Realignment of Liquid Crystal Shells Driven by Temperature-Dependent Surfactant Solubility[END_REF][START_REF] Noh | Dynamic tuning of the director field in liquid crystal shells using block copolymers[END_REF] In particular, we note that in, [START_REF] Noh | Influence of interface stabilisers and surrounding aqueous phases on nematic liquid crystal shells[END_REF] Lagerwall et al. performed a similar experiment on planar shells of 5CB, dispersed in 1 % wt PVA in a 50:50 volume ratio water/glycerol mixture. However, no anchoring transition was reported when the shells were subjected to a 0.01 • C/ min temperature ramp. We were able to attribute experimentally the difference in behavior compared to our results to the large concentration of glycerol in the water phases, which is known to affect the surface activity of surfactant molecules.

Conclusion

We report a temperature-driven method for inducing molecular reorientation in a liquid crystal film, which is immediate, spatially homogeneous, and can affect both interfaces of the film. It is achieved with a precise control over the temperature of an aqueous interface of 5CB in the vicinity of its clearing point. The anchoring transition takes place if PVA is dissolved in water in concentrations greater than 0.1 % in weight. We explain this behavior by the melting of the disordered PVA/5CB mixture at the interfaces of the shell, which occurs at a lower critical temperature than that of pure 5CB. The presence of this new isotropic interfacial PVA-5CB mixture could be responsible for the reorientation of the underlying pure nematic 5CB. This new method allows us to control the boundary conditions of the most widely researched shell system, shells of 5CB in aqueous solutions of PVA, without changing the compo-sition of the system, just by using an external stimulus. It remarkably enables rapid transformations between planar and radial shell states. We find that the two interfaces in a nematic shell are not affected at exactly the same temperature: this yields a hybrid transitory state that can be used to transform an initially polydisperse population of shells, i.e. displaying all possible defect configurations, into a monodisperse population of bivalent shells.

This work raises a number of new questions beyond its scope, which would constitute an interesting follow-up. In particular, quantitative measurements of the anchoring strength [START_REF] Yesil | Anchoring Energy Measurements at the Aqueous Phase/Liquid Crystal Interface with Cationic Surfactants Using Magnetic Fréedericksz Transition[END_REF] and precise imaging of the director throughout the bulk of the films 33 at various stages of the transition would be of great interest. Our study expands the number of available parameters one can tune to trigger transitions in shells of complex liquid crystal phases. [START_REF] Tran | Change in stripes for cholesteric shells via anchoring in moderation[END_REF] Acknowledgement The authors acknowledge Christophe Blanc and Reviewer 1 for valuable feedback that helped us narrow down the nature of the mechanism at play in this phenomenon; Lisa Tran and the rest of Randall Kamien's group for experimental support and fruitful discussions; Kunyun He, Ashley Guo and Martina Clairand for experimental support; Bohdan Senyuk and Ivan Smalyukh for providing BTBP; Ye Zhou for useful discussions. This work was supported by the Agence Nationale de la Recherche (ANR) Grant No. 13-JS08-0006-01.

Supporting Information Available

The following data is available free of charge.

• Temperature ramp on a homeotropic 5CB cell with solid boundaries.

• Temperature-driven anchoring transition of a 5CB film in a 0.1 % PVA solution in water (Supplementary Movie 1).

• Temperature ramp on a homeotropic 5CB cell with solid boundaries (Supplementary Movie 2).

13

• Phase transition of a 5CB film in a solution with less than 0.1 % PVA in water (Supplementary Movie 3).

• Phase transition of a 5CB shell without PVA in the inner and outer water phases (Supplementary Movie 4).

• Temperature-driven anchoring transition of typical 5CB shells, with 1 % PVA in the outer and inner phases (Supplementary Movie 5).

• Temperature-driven anchoring transitions of typical 5CB shells, with 1 % PVA in the water phases as the sample is heated to isotropic then cooled back to nematic (Supplementary Movie 6).

• 5CB films doped with BTBP in a 0.1 % wt PVA solution undergoing a slow temperature ramp (Supplementary Figure 1).

• Additional clues pertaining to the isotropic islands (Supplementary Figure 2).

6 PageFigure 3 :

 63 Figure 3: Coexistence of regions of pure isotropic phase embedded into a homeotropic nematic domain in the bulk of a 5CB film, close to the clearing point. A) Bright field (single polarizer). From the edges to the center of the cell, one can make out the isotropic domain, the homeotropic nematic domain, and the circular islands with their central point defect. B) Crossed polarizers. The [+1] charge of all defects is revealed. First-order birefringence colors indicate that only a small thickness of liquid crystal has tilted back towards the planes of the interfaces. C) Crossed polarizers with first-order retardation plate. The radial orientation of the long axis of the molecules is revealed. Throughout, the film is 430 µm wide. D) Schematic of a typical isotropic droplet, top and side view.

8 ACS

 8

Figure 4 :

 4 Figure 4: 5CB film and shell in pure water solutions undergoing a slow temperature ramp As temperature of a film (A) or a shell (B) is increased, the nematic phase is progressively replaced by the isotropic phase, without any significant change in the shrinking planar texture. Throughout both experiments, the temperature ramp has a rate of 0.01 • C • min -1 . The film is 430 µm wide and the shell is about 200 µm in outer diameter. The first series of micrographs corresponds to frames of the timelapse provided in Supplementary Movie 3, and the second series to frames of the real-time film provided in Supplementary Movie 4.

Figure 5 :

 5 Figure 5: 5CB shell in a 1 % wt PVA solution undergoing a slow temperature ramp Top: micrographs of a shell under crossed polarizers. Bottom: cross-section of the shell schematically depicting the molecular arrangement. Throughout, the shell is 110 µm in outer diameter and has a 15 µm average thickness. The temperature ramp has a rate of 0.01 • C • min -1 . i) For most of the nematic state, the shell displays three defects, indicating that molecules within the shell are oriented parallel to the interfaces. ii) A few tenths of degrees Celsius below the phase transition temperature of the bulk 5CB, the shell transitions to a structure having only two [+1] defects, only one of them being visible here. This is a signature of hybrid anchoring, where the molecules have planar anchoring on one interface and homeotropic anchoring on the other. iii) Even closer to the phase transition temperature, the shell displays a classic Maltese cross, which shows that the anchoring has switched to being perpendicular on both interfaces. iv) In the end, the entire shell turns isotropic. These four micrographs, taken respectively at -0.43 • C, -0.18 • C, -0.03 • C and 0.05 • C of the first apparition of bulk isotropic phase, are close-ups of frames of the timelapse shown in Supplementary Movie 5.

Figure 6 :

 6 Figure 6: Controlling shell valency by going through the hybrid state i) Group of polydisperse nematic shells, i.e. displaying all three possible planar defect configurations. ii) Same group of nematic shells, showing only bivalent configurations after being subjected to a 0.01 • C/ min temperature ramp up to the isotropic phase, then to a -0.01 • C/ min temperature ramp back to the planar phase. Residual flow within the observation cell shifted the shells' position during the temperature ramps. Those two micrographs are frames of the timelapse provided in Supplementary Movie 6.

ACS Paragon Plus Environment


Yoko Ishii

Ye Zhou

Kunyun He

Yoichi Takanishi

Jun Yamamoto

Juan De Pablo
email: depablo@uchicago.edu

Teresa Lopez-Leon
email: teresa.lopez-leon@espci.fr

Structural transformations in tetravalent nematic shells induced by a magnetic field †

The role of applied fields on the structure of liquid crystals confined to shell geometries has been studied in past theoretical work, providing strategies to produce liquid crystal shells with controlled defect structure or valence. However, the predictions of such studies have not been experimentally explored yet. In this work, we study the structural transformations undergone by tetravalent nematic liquid crystal shells under a strong uniform magnetic field, using both experiments and simulations. We consider two different cases in terms of shell geometry and initial defect symmetry: i) homogeneous shells with four s = +1/2 defects in a tetrahedral arrangement, and ii) inhomogeneous shells with four s = +1/2 defects localized in their thinner parts. Consistently with previous theoretical results, we observe that the initial defect structure evolves into a bipolar one, in a process where the defects migrate towards the poles. Interestingly, we find that the defect trajectories and dynamics are controlled by curvature walls that connect the defects by pairs. Based on the angle between B s , the local projection of the magnetic field on the shell surface, and n + 1 2 , a vector describing the defect orientations, we are able to predict the nature and shape of those inversion walls, and therefore, the trajectory and dynamics of the defects. This rule, based on symmetry arguments, is consistent with both experiments and simulations and applies for shells that are either homogeneous or inhomogeneous in thickness. By modifying the angle between B s and n + 1 2 , we are able to induce, in controlled way, complex routes towards the final bipolar state. In the case of inhomogeneous shells, the specific symmetry of the shell allowed us to observe a hybrid splay-bend Helfrich wall for the first time.

Introduction

Topological defects are central to many areas of science, from particle physics to cosmology or materials engineering 1,2 . They can be described from symmetry breaking considerations and, to a large extent, they determine the structure and physical properties of a material. Liquid crystals offer a unique playground for the study of topological defects because of the larger length-scales typically involved. In nematic liquid crystals, rod-like molecules exhibit long range orientational order, with the long axis of the molecules aligned along the director, defined by a unit vector n with head-tail symmetry (n = -n). This symmetry of the director enables nematic liquid crystals (NLCs) to accommodate a variety of topological defects, which can be easily produced and observed in the laboratory 3,[START_REF] Lavrentovich | Defects in Liquid Crystals: Computer Simulations, Theory and Experiments[END_REF] .

Besides their fundamental interest, topological defects in NLCs have also attracted attention for a wide range of applications [START_REF] Lavrentovich | Defects in Liquid Crystals: Computer Simulations, Theory and Experiments[END_REF][START_REF] Lagerwall | [END_REF] . The defect-mediated self-assembly of colloidal particles has emerged as a promising strategy to create nano/microstructured materials with emergent new properties [6][7][START_REF] Lavrentovich | Proc. Natl. Acad. Sci[END_REF][START_REF] Smalyukh | [END_REF] . Embedding a micro-sized particle in a uniform director field causes the disruption of the field, leading to the formation of topological defects in the vicinity of the particle 10 . The anisotropic elastic interactions between defects associated with different particles induce the formation of colloidal structures, with a complexity that depends on the liquid crystal symmetry and the molecular anchoring at the particle surface. Additionally, nanoparticles can be trapped at the cores of topological defects, and patterned surfaces can therefore be engineered to introduce NLC defects as targeting sites for the assembly of colloidal particles into precisely controlled configurations [11][12][13] .

Topological defects can also induce anisotropic colloidal interactions in a completely different way. Coating the spherical surface of a colloidal particle with a thin nematic shell induces the formation of an irreducible set of defects, as a result of geometrical frustrations in the orientational order of the liquid crystal 14 . The ground state of very thin nematic shells has four defects sitting at the vertices of a tetrahedron. The coated sphere can then be viewed as a patchy colloidal particle with tetravalent coordination [14][15][16] . The bonds between patches could be provided by chemical linkers attached at the four defects present in each colloid. The idea of using liquid crystals to produce colloids with a valence has fuelled the growth of research on liquid crystal shells . These shells have been produced by confining a nematic phase between two spherical aqueous interfaces, as schematically shown in Fig. 1(a). The three-dimensional nature of these experimental shells enables a larger number of defect structures / particle valences. The predicted tetravalent defect structure has been observed in experimental shells, coexisting with a bivalent and a trivalent defect configurations 57 . Important efforts have been done in the last years to engineer shells with a controlled valence. Shell thickness and shape, molecular anchoring at the boundaries, or elasticity of the liquid crystal are just a few examples of parameters that have been studied theoretically, numerically and experimentally, with the goal of achieving such control 16,[18][19][20][21][22][22][23][24][25][26][27][28][29][30][33][34][35][38][39][40][41][42][43][44][45][46][47]55,56 . Despite the progress, we are still far from having a robust approach to produce shells with a given defect structure.

The use of external fields has been proposed as a promising strategy to produce, in a controlled way, shells with a large spectrum of defect structures 58,59 . Typical nematic liquid crystals have a larger dielectric constant (magnetic susceptibility) along the long molecular axis, and thus, the application of an electric (magnetic) field leads to the alignment of the director n along the direction of the field. Numerical studies have shown that the application of a strong uniform electric field causes structural changes in the tetrahedral configuration, which develops a bipolar structure with two surface defects or "boojums" at each spherical boundary. Remarkably, simulations reveal the formation of high-valence structures, such as an eight-defect structure, when quadrupolar fields are applied. Despite these promising predictions, the effect of external fields on liquid crystals shells has not been examined before in experiments.

In this work, we study the structural modifications undergone by tetravalent nematic shells under a strong and uniform magnetic field B using both experiments and simulations. The alignment of n with B triggers a series of structural transformations in the shells, which eventually adopt a bipolar configuration, as suggested by previous simulations. We uncover different scenarios in which the four +1/2 defects (inducing a π-rotation of n) recombine by pairs to form two +1 boojums (inducing a 2π-rotation of n) on each shell boundary. The recombination process is mediated by the presence of inversion walls, which dynamically bring the defects together towards the poles. We show that the shape and nature of the walls depend on the relative orientation of the +1/2 defects with respect to the surface projection of the field B s .

Experiment details

The experimental shells correspond to double emulsions produced in a conventional glass capillary microfluidic device, as described in 60 . The middle phase is 4-cyano-4'-pentylbiphenyl (5CB), a liquid crystal that forms a nematic phase at room temperature. The inner and outer liquids are aqueous solutions containing 1wt% of polyvinyl alcohol (PVA), which stabilizes the dou- ble emulsion and enforces planar anchoring of the liquid crystal at the inner and outer interfaces. The inset of Fig. 1(a) shows a schematic representation of a shell, where a and R are the radii of the inner and outer droplets respectively. Typical values of the shell radius are R = 50 -100 µm. The shell thickness, h = Ra, is of the order of several micrometers. Because of buoyancy effects and nematic elasticity, the experimental shells are inhomogeneous in thickness 57 . To obtain homogeneous shells, we produce extremely thin shells (h < 1 µm) by making the inner droplet bigger through osmotic swelling 16 .

After fabrication, the shells are collected in a 1 mm inner diameter square capillary, which is placed in a sample holder made with a 3D printer. The capillary is then sandwiched between two permanent neodymium magnets, as schematically shown in Fig. 1(a). This set-up facilitates the manual positioning of the magnets and allows us to control their separation distance. In this way, we are able to apply strong (≈ 0.5 T) and uniform magnetic fields in a direction perpendicular to gravity in a time scale of a second. To apply magnetic fields parallel to the gravity direction, we place the sample inside a hollow cylindrical neodymium magnet, which also induces a strong (≈ 0.5 T), uniform field, see Fig. 1(b). The very strong magnetic fields (≈ 4 T) used for some specific experiments were applied using a superconducting magnet. All experiments were performed at room temperature.

Simulation details

We use a Landau-de Gennes continuum model for the order tensor Q, which is defined by Q i j = S(n i n j - 1 3 δ i j ) [START_REF] De Gennes | The Physics of Liquid Crystals[END_REF] . The scalar order parameter is denoted by S. The total free energy is given by

F(Q) = bulk ( A 2 (1 - U 3 )Q i j Q ji - AU 3 Q i j Q jk Q ki + AU 4 (Q i j Q ji ) 2 )dV + bulk L 2 ∂ Q i j ∂ x k ∂ Q i j ∂ x k dV - bulk 1 3 µ 0 ∆χQ i j B i B j dV + surf W ( Qi j -Q⊥ i j ) 2 dΣ ( 1
)
where A is a material constant that sets the energetic scale of the system, and U is a dimensionless parameter that establishes the equilibrium order parameter S eq and represents the inverse temperature. A one-constant representation is adopted here, where L denotes the elastic constant of the liquid crystal. The magnetic permeability of free space is denoted by µ 0 , while ∆χ is the magnetic susceptibility anisotropy. The anchoring strength is denoted by W . The preferred orientation of the liquid crystal at the boundary is given by Q = Q + S eq I/3. The projection of Q onto the surface is denoted by Q⊥ = P QP, where the projection operator P is defined by P i j = δ i jv i v j , and ν ν ν is the unit vector normal to the surface [START_REF] Fournier | [END_REF] . The first term in Equation ( 1), which corresponds to enthalpic contributions to the free energy, serves to control the equilibrium value of the order parameter. The second term represents the elastic contributions to the free energy. It governs long-range director distortions and penalizes elastic deformations in the bulk [START_REF] Doi | Soft Matter Physics[END_REF] . The third term represents the energy due to the magnetic field [START_REF] Kralj | [END_REF] . The last term corresponds to the surface energy, which enforces degenerate planar anchoring on the shell surface. An iterative Ginzburg-Landau relaxation with finite differences on a cubic mesh is adopted to minimize the free energy 65 . To reveal the fine structure of defects, we use the splay-bend parameter S SB constructed from second derivatives of the order parameter tensor Q

S SB = ∂ 2 Q i j ∂ x i ∂ x j . (2)
Large positive (negative) values of S SB imply strong splay (bend) deformation 66 . Polarization micrographs are calculated using the Jones 2 + 2 matrix formalism, in which light travels along a chosen direction and the total phase shift is accumulated. In this work, simulations are carried out with typical values for 5CB such as the elastic constant L = 6 pN and nematic coherence length ξ = 7.15 nm 67 at the inverse temperature U = 5, setting the energetic scale to A = 1.067 × 10 5 J/m 3 . Strong surface alignment is imposed at the boundaries with anchoring strength W = 1 × 10 -3 J/m 2 . Mesh resolution is commensurate with the nematic coherence length to capture relevant phenomena. The shells have an outer radius R = 1 µm. For homogenous shells, the inner radius is a = 0.786 µm. For inhomogeneous shells, the inner radius is a = 0.643 µm, and the shift between the centers of the inner and outer droplets is d = 0.257 µm.

Results and discussion

Tetravalent shells exhibit four +1/2 defects, whose spatial distribution depends on the shell thickness gradient. In the next subsections, we study the structural transformations undergone by tetravalent shells in the presence of magnetic fields, stressing the role of the field direction with respect to the shell symmetries.

Homogeneous shell

Because of the elastic repulsion between like-charged topological defects, in homogeneous shells, the four +1/2 defects are located at the vertices of a tetrahedron, see Fig. 2(a) [14][15][16] . The experimental realization of such a structure is shown in Fig. 2(c) and (d), which are cross-polarized images of a thin, homogeneous nematic shell. Upon the application of a sufficiently strong magnetic field, B, the tetrahedral structure evolves into a bipolar one, see Fig. 2(b), as predicted by previous simulations 58 , the bipolar axis being aligned with the magnetic field direction. Interestingly, we observe that the route towards the final bipolar state can be very different depending on the orientation of B with respect to the axes of symmetry of the tetrahedron.

A regular tetrahedron has the symmetries of the C 2v point group along three equivalent orthogonal directions. These three directions can be easily visualized by inscribing the tetrahedron in a cube, since they connect pairs of opposite cube faces. One of these directions is represented in Fig. 2(e), showing the two-fold rotational axis C 2 (colored in red) and the two orthogonal mirror planes σ v and σ v (colored in purple) of the C 2v point group. In a nematic shell, some of these symmetries are broken due to the director field around the defects. In Figs. 2(f) and (g), we have represented the director streamlines connecting pairs of +1/2 defects to better illustrate this. There is only one direction, shown in Fig. 2(f), along which the shell has all the symmetries of the C 2v point group (C 2 , σ v and σ v ). For the sake of simplicity, we will refer to this direction as the C 2v direction. Along the other two orthogonal directions, such as the one shown in Fig. 2 are not parallel, leading to the formation of Helfrich walls with a non-trivial shape (not represented). (h) These curvature walls connect defects from different pairs and shrink to eventually yield the bipolar configuration depicted in (f).

but not mirror planes. In the following, we will refer to these two less symmetric degenerate directions as C 2 directions. Here, we explore the effect of applying a strong uniform magnetic field to homogeneous nematic shells along the C 2v and C 2 directions. When B is applied approximately along the C 2v direction, see Figs. 3(a)-(e), two inversion walls suddenly appear, connecting the +1/2 defects by pairs. These inversion walls, also called πwalls or Helfrich walls, separate domains in which n is inversely aligned with the external field: n and B are parallel in one domain, while they are antiparallel in the other domain [START_REF] De Gennes | The Physics of Liquid Crystals[END_REF]68,[START_REF] Chandrasekhar | Liquid Crystals[END_REF] . Across the wall, which extends over a narrow area, n rotates by π, leading to the optical texture shown in the inset of Fig. 3(b), where dark and bright stripes alternate. With time, the π-walls become shorter bringing the two associated +1/2 defects to the poles, see Figs. 3(c)-(e). The +1/2 defects approach each other by following the shorter path (geodesic): the π-walls run along great circles, where curvature is minimal. At the end of the process, which lasts several minutes, the shell adopts a bipolar configuration, with the bipolar axis aligned with the external field, see Fig. 3(e). Depending on the strength of B, the two +1/2 defects of each pair can either merge to form a +1 defect (strong fields) or stay close together (moderate fields). For the sake of simplicity, in the following, we will use the term "merging" for both cases.

When B is applied approximately along a C 2 direction, see Simulations allow us to extract additional information about the structural modifications in the shells under the effect of the magnetic field. Before diving in, we take a glance at the possible structural transformations that can be induced by B on single +1/2 defects, see Figs. 5(a)-(c). Here, we denote the defect orientation as the direction pointed by the arrow n + 1 2 in Fig. 5(a). When B is applied parallel to n + 1 2 , a bend-splay Helfrich wall appears in the n + 1 2 direction, as shown in Fig. 5(b). In this type of wall, the transition from +n to -n occurs mainly through a bend deformation (yellow stripe), although some splay is also present. When B is perpendicular to n + 1 2 , see Fig. 5(c), it produces a splaybend Helfrich wall instead, where the transition from +n to -n occurs mainly through a splay deformation (blue stripe). The wall appears again along the -n + 1 2 direction, and thus, it is perpendicular to B. For the sake of simplicity, here we will name these two types of inversion walls as "bend wall" and "splay wall", respectively.

When the external field B is applied along the C 2v direction, its projection on the surface (B s ) at the defect position is parallel to n + 1 2 , see Fig. 5(d). In this situation, we expect to see the formation of bend walls between +1/2 defects, represented as a yellow stripe in Figs. 5(d) and(e). This is confirmed by our simulations: Figs. 3(g) and (j) show one of these inversion walls with major bend distortions in yellow (S SB < -0.005). Since the wall bears high elastic energy, it shrinks and makes the defects approach until they merge into a +1 defect, see Fig. dicate that the formed wall is a bend wall. As the wall shrinks, the defects are pulled along, see Figs.

Inhomogeneous shell

Due to buoyancy and nematic elasticity, nematic LC shells are usually heterogeneous in thickness. In this geometry, the four +1/2 defects are located in the thinner part of the shell to reduce the bulk elastic energy 57 , as shown in Fig. 6(a), which is a crosspolarized image of an inhomogeneous shell. In this geometry, the director field connects the +1/2 defects in two asymmetric pairs, as schematically represented in Figs. 6(b) and (c): one pair is connected along the longest geodesic path (blue pair), while the other one is connected along the shortest geodesic path (red pair). This defect structure is less symmetric than the tetrahedral one: there is still a direction where the shell has the symmetries of the C 2v point group, indicated in Fig. 6(c), but the other two orthogonal C 2 axes no longer exist. In the following, we will denote the direction along which the shell has the symmetries of the C 2v point group as the z axis, and the directions connecting the blue and red defect pairs, see Fig. 6(b), as the x and y axes, respectively.

We first examine the effect of B when it is applied along the z axis. In this case, B s is again roughly parallel to the orientations of the four defects, see Fig. 6(b). The transformation, shown in Figs. 7(a)-(e) for experiments and Figs.7(f)-(o) for simulations, is very similar to the one observed in homogeneous shells, except that here the two bend inversion walls are not equal in length. As a result, the two pairs of defects move asymmetrically: the defects of the blue pair get closer before merging together, while the defects of the red pair move away from each other to reach the opposite hemisphere of the shell, where they eventually merge. This asymmetric behavior becomes evident when plotting the evolution of the angular distance between defects in each pair, θ 1 (red pair) and θ 2 (blue pair), as a function of time, see Fig. 7(p).

When the magnetic field B is applied along the y axis, the defect orientation of the red pair is parallel to B s while that of the blue pair is perpendicular to B s , see Fig. (m) and (r), with the bend part mostly parallel to B s and the splay part mostly perpendicular to B s . To our knowledge, such type of hybrid inversion wall has not been reported before, since its formation requires curved substrates and non-trivial defect configurations, as those present in heterogeneous tetravalent shells. We would like to note this configuration, where B is parallel to the y axis, is experimentally unstable. Indeed, applying B along the y axis triggers a solid rotation of the shell around its C 2 axis, so that B gets aligned with the x axis. The resulting shell orientation, which will be discussed later, is energetically more favorable than the initial one, since the director in the thick hemisphere of the shell is aligned with the magnetic field. The results shown in Figs. 8(a)-(e) were obtained by bringing the shell into contact with the glass walls of the observation capillary, where friction forces prevent the shell from rotating. This was done using buoyancy. Despite the images being blurry because of the light interaction with the the capillary walls, this trick allowed us to follow the position of the defects throughout the process. As in previous experiments, after the formation of the hybrid inversion walls, the defects move along the wall to eventually merge together (Fig. 8(e)). Here again the behavior of the two pairs of defects is asymmetric: the blue pair recombines before the red one, see the temporal evolution of θ 1 and θ 2 in Fig. 8(t). Besides, the pair connected by the longest geodesic path (blue pair) undergoes a π/2 rotation with respect to the y axis, while the pair connected by the shorter geodesic path (red pair) keeps its initial orientation, see the evolution of ϕ 1 and ϕ 2 in Fig. 8(u). This operation aligns the defect orientations of the two defect pairs. When B is applied along the x axis, the general rule for single defects fails: it predicts a splay wall connecting the red defect pair, which is not observed either in experiments or simulations. This is probably due to the crowding of defects in the thinner part of shell. Instead, the bend wall arising from the blue pair winds back and attaches itself to the red pair, as shown in Fig. 9(c). Here again, the walls connect defects from different pairs. The walls have an open-loop shape that evolves into a u-shape as the walls shrink, see Figs.9(b)-(d). Figs.9(p) and (q) show the evolution of each defect pair in terms of θ and ϕ during the transition. The blue pair moves along the B s direction to the pole directly. While the red defects first approach towards each other, and then move apart when the curved bend wall forms. The red pair keeps rotating until it becomes parallel to the blue pair, while the distance between defects progressively shortens. The experimental observations are in good agreement with our simulation calculations, shown in Figs. 9(f)-(o).

Finally, we would like to note that while the presence of a magnetic field induces structural re-arrangements that eventually lead to the formation of a bipolar structure, the +1 defects at the poles can be either +1 boojums or pairs of close +1/2 disclinations. Although both situations yield the same far field, the structures are topologically non-equivalent. The intensity of the magnetic field dictates whether the +1/2 disclinations completely fuse together to give a +1 boojum or not . Fig. 10 shows how imposing a 0.5T magnetic field on a shell that initially has four +1/2 disclinations makes it adopt a bipolar structure, which relaxes towards the initial state when the field is off, indicating that the +1/2 disclinations did not fuse together. Conversely, applying a 4T magnetic field leads to the irreversible formation of +1 boojums that relocate in the thin part of the shell when the field is off, demonstrating the ability of magnetic fields to change the valence of the shell.

Conclusion

In this work, we study the structural transformations that arise in tetravalent nematic shells under a magnetic field, using both experiments and simulations. The presence of a sufficiently strong magnetic field makes the +1/2 defects move in pairs towards the poles of the shell. The defect trajectories and dynamics are controlled by inversion walls, which appear when the field is applied, connecting the +1/2 defects by pairs. These inversion walls are unstable and shrink over time, pulling the two +1/2 defects attached to their ends together. Depending on the strength of the field, the two defects either fuse together (strong fields) or just stay close by (moderate fields) when they reach the poles. At the end of the transformation, the shells adopt a bipolar configuration, with the bipolar axis aligned along the field. The nature and shape of the inversion walls, and thus the defect trajectories and dynamics, depend on the relative orientation between B s , the local projection of the magnetic field on the shell surface, and n + 1 2

, a vector describing the orientation of the defects. By analyzing the motion of the defects in homogeneous shells, we observed that, when B s is parallel to n + 1 2 , straight (running along geodesics) bend Helfrich walls emerge between defect pairs. This situation arises when the field is applied along the shell C 2v direction. In contrast, when there is some angle between B s and n + 1 2 , curved (deviating from geodesics) Helfrich bend walls form, inducing a rotation in the defect motion. This situation occurs when the field is applied along any of the two orthogonal C 2 directions of the shell. This general behavior is reproduced in inhomogeneous tetravalent shells, whose specific symmetry allowed us to observe hybrid splay-bend Helfrich walls for the first time. Our experimental observations are confirmed by numerical results, validating the role of the inversion walls in the mechanism of defect reorganisation under the effect of external fields. A better understanding of this mechanism may provide new methods to not only control the defect position, number and valence, but also to manipulate defect dynamics.

Fig. 1 Fig. 2

 12 Fig. 1 Experimental set-ups used to apply the magnetic field in a direction (a) perpendicular to gravity and (b) parallel to gravity.b a

Fig. 3

 3 Fig. 3 Structural transformations undergone by a nematic shell with tetrahedral defect structure upon applying a uniform magnetic field along the C 2v direction. (a)-(e) Experimental cross-polarized images. (f)-(j) Simulated director field on the outer surface of the shell. (a), (f) Initial tetrahedral structure before applying the magnetic field. In the experimental images, defects of the same pair (connected by director streamlines) are represented with the same color. In the simulation plots, all the defects are represented in red (isosurface for S = 0.5), while regions of high splay and bend elastic distortions are represented in blue (S SB > 0.005) and yellow (S SB < -0.005), respectively. (b), (g) Formation of two inversion walls after applying the magnetic field. They run along geodesic lines, connecting defects of the same pair. The inset in (b) shows the change of birefringence near one of the walls, indicating a π-rotation of the director. The way in which the director rotates across the walls, shown in (j), indicates that they are bend Helfrich walls. (c), (d) and (h) The walls are unstable and shrink over time while bringing the defects to the poles, eventually leading to the bipolar configuration shown in (e), (i).

Fig. 4

 4 Fig. 4 Structural transformations undergone by a nematic shell with tetrahedral defect structure upon applying a uniform magnetic field along a C 2 direction. (a)-(e) Experimental polarized images. (f)-(m) Simulated director field on the outer surface of the shell. (a), (f) and (j) Initial tetrahedral structure before applying the magnetic field. In the experimental images, defects of the same pair (connected by director streamlines) are represented with the same color. In the simulation plots, all the defects are represented in red (isosurface for S = 0.5), while the splay and bend elastic distortions are represented in blue (S SB > 0.005) and yellow (S SB < -0.005), respectively. (b), (g) and (k) Formation of two inversion walls after applying the magnetic field. They run along curly paths (non-geodesic) and connect defects of different pair. The way in which the director rotates across the walls, shown in (k), indicates that they are bend Helfrich walls. (c), (d), (h) and (l) The walls are unstable and shrink over time. This makes the defects rotate and move to the poles, eventually leading to the bipolar configuration shown in (e), (i) and (m).

Fig. 5 2 ,

 52 Fig. 5 (a) Definition of the n + 1 2

Fig. 6

 6 Fig. 6 (a) Cross-polarized image of an inhomogeneous tetravalent shell. The four +1/2 disclinations are placed at the top of the shell, where it is thinner. (b) Schematic director field in an inhomogeneous tetravalent shell. (c) The blue defect pair is connected along the longest geodesic path, while the red defect pair is connected along the shortest geodesic path. This structure has the symmetries of the C 2v point group along the direction indicated by the arrow.

Figs. 4

 4 Figs.4(a)-(e), we observe an unexpected defect rotation. The two π-walls connecting the +1/2 defects by pairs do not run over geodesic lines but bend into wavy paths. The lines unwind while they shrink to eventually disappear when the defects reach the poles, see Fig.4(e).Simulations allow us to extract additional information about the structural modifications in the shells under the effect of the magnetic field. Before diving in, we take a glance at the possible structural transformations that can be induced by B on single +1/2 defects, see Figs.5(a)-(c). Here, we denote the defect orientation as the direction pointed by the arrow n + 1

3

 3

1 2,Fig. 7

 17 Fig. 7 Structural transformations undergone by an inhomogeneous tetravalent nematic shell upon applying a uniform magnetic field along the z axis. (a)-(e) Experimental cross-polarized images. (f)-(j) Simulated cross-polarized images. (k)-(o) Simulated director field on the outer surface of the shell. In the simulations, the defects are shown in red (isosurface for S = 0.5), while the splay and bend elastic distortions are shown in blue (S SB > 0.005) and in yellow (S SB < -0.005), respectively. (a), (f) and (k) Initial state with the four defects placed at the top of the shell. (b), (g) and (l) Formation of two bend Helfrich walls after applying the filed, which run along geodesic paths, connecting defects of the same pair. (c), (h) and (m) The defects connected by the longest streamlines (shortest wall) get closer and coalesce at the top of the shell. (d), (i), (n) The defects connected by the shortest streamlines (longest wall) move away from each other to eventually coalesce at the bottom of the shell. (p) Time evolution of the angular distance between defects (θ ) in experiments, showing that the two defect pairs behave asymmetrically. The lower density of experimental points around θ = 180 • is due to the fact that the defects cannot be easily visualized when they are close to the shell equator.

1 2

 1 4(h) and (l), and eventually align their orientation n + with B s . The two +1/2 defects reach the pole and fuse together when the wall disappears, see Figs.4(i) and (m).

  6(b). According to our previous discussion about single +1/2 defects on the plane, Figs.5(a)-(c), we expect to observe the emergence of a bend wall along the n + 1 2 direction of the red pair, and a splay wall along the -n + 1 2 direction of the blue pair. These two walls are presumed to grow orthogonally and joint at a given point, leading to the formation of a hybrid inversion wall. The results obtained experimentally and numerically are shown in Figs.8(a)-(e) and Figs.8(f)-(s), respectively. In both experiments and simulations, we observe the formation of two inversion walls with hyperbolic shape, see Figs.8(b) and (c), for experiments, and Figs.8(h) and (i), for simulations. The nature of these walls becomes clear when simulating the evolution of the director field after applying B, Figs.8(k)-(s). As predicted, two bend and a splay walls nucleate and propagate from different defect pairs, Figs.8(l) and (q), until they join each other and form two hybrid inversion walls, Figs.8

Fig. 8

 8 Fig. 8 Structural transformations undergone by an inhomogeneous tetravalent nematic shell upon applying a uniform magnetic field along the y axis. (a)-(e) Experimental cross-polarized images. (f)-(j) Simulated cross-polarized images. (k)-(s) Simulated director field on the outer shell surface: (k)-(o) top view, (p)-(s) bottom view. In the simulations, the defects are shown in red (isosurface for S = 0.5), while the splay and bend elastic distortions are shown in blue (S SB > 0.005) and in yellow (S SB < -0.005), respectively. (a), (f), (k) and (p) Initial state with the four defects placed at the top of the shell. (b), (g), (l) and (q) Formation of two parabolic Helfrich walls after applying the field, connecting defects of different pairs. (c), (h),(m) and (r) The hyperbolic walls have a hybrid nature: as highlighted in (r), they stem from the junction of a bend Helfrich wall with a splay Helfrich wall. (d), (i), (n) and (s) The walls are unstable and shrink while pulling the associated defects towards opposite poles in the B direction. (e), (j) and (o) Final bipolar configuration. (t) Time evolution of the angular distance between defects (θ ) in experiments, showing that the two defect pairs behave asymmetrically. (u) Defect rotation in the xy plane (ϕ) as a function of time in experiments. The red pair of defects rotates by π/2 to get aligned with the blue pair.

Fig. 9

 9 Fig. 9 Structural transformations undergone by an inhomogeneous tetravalent nematic shell upon applying a uniform magnetic field along the x axis. (a)-(e) Experimental cross-polarized images. (f)-(j) Simulated cross-polarized images. (k)-(o) Simulated director field on the outer shell surface. In the simulations, the defects are shown in red (isosurface for S = 0.5), while the splay and bend elastic distortions are shown in blue (S SB > 0.005) and in yellow (S SB < -0.005), respectively. (a), (f) and (k) Initial state with the four defects placed at the top of the shell. (b), (g) and (l) Formation of two bend Helfrich walls after applying the field, connecting defects of different pairs. (c), (h) and (m) The initial open-loop shape of the walls evolves into a u-shape with time. (d), (i), (n) and (s) The walls are unstable and shrink while bringing the associated defects towards opposite poles in the B direction. (e), (j) and (o) Final bipolar configuration. (p) Time evolution of the angular distance between defects (θ ) in experiments, showing that the two defect pairs behave asymmetrically. (q) Defect rotation in the xy plane (ϕ) as a function of time in experiments. The red pair of defects rotates by π/2 to get aligned with the blue pair.

Fig. 10

 10 Fig. 10 Effect of the magnetic field strength. (a) Initial state with four 1 + /2 disclinations placed at the top o the shell. (b) Applying a 0.5T magnetic field makes the shell adopt a bipolar structure, with a pair of close +1/2 disclinations at each pole. (c), (d) When the field is off, the structure relaxes back to the initial state. This process is reversible. (e) Applying a 4T magnetic field also makes the shell adopt a bipolar structure, but in this case the two +1/2 disclinations at the poles fuse together and give rise to two +1 boojums, one located above the other on the inner and outer surfaces of the shell. (f), (g) This transformation is irreversible: when the field is off, the boojum defect structure remains.

J o u r n a l N a me , [ y e a r ] , [ v o l . ] ,

Acknowledgements

We thank V. Palacio-Betancur and I. Smalyukh for fruitful discussions and support. This work was supported by the French National Research Agency under grant 13-JS08-0006-01, the National Science Foundation of the United States, Division of Materials Research, under grant DMR-1710318, and the Japan Science and Technology Agency, CREST Program, under grant JP-MJCR1424. Y. I. was partially supported by a Joliot Chair from the ESPCI Paris.