Fluid jets and polar domains, on the relationship between electromechanical instability and topology in ferroelectric nematic liquid crystal droplets

Abstract

Ferroelectric nematic liquid crystals are a class of recently discovered fluid materials formed by highly polar molecules that spontaneously align along a common direction, giving rise to a macroscopic polarization P. Since the polarization vector is locally collinear to the optical axis n, the study of the spatial patterns of n enables deducing the structure of P. We have carried on such topological study on ferroelectric nematic droplets confined between two solid ferroelectric substrates both when the droplet is in equilibrium and during a jet-emission phase that takes place when the solid surfaces become sufficiently charged. We find that in equilibrium the droplet splits in striped domains in which P has alternating directions. When these domains extend close to the droplets’ perimeter, P adopts a π-twisted structure to minimize accumulation of polarization charges. As the substrate surface charge is increased above threshold, fluid jets are emitted with a quasi-periodic pattern, a behaviour suggesting that their location is governed by an electrofluidic instability on the droplets’ rim, in turn indicating the absence of specific trigger points. Soon after their emission, the jet periodicity is lost; some jets retract while other markedly grow. In this second regime, jets that grow are those that more easily connect to polar domains with P along the jet axis. Occasionally, ejection of isolated spikes also occurs, revealing locations where polarization charges have accumulated because of topological patterns extending on length scales smaller than the typical domain size.

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Introduction

The discovery of ferroelectric nematic (N_F) liquid crystals is certainly one of the top novelties in condensed matter science of the last years.^1–4 This phase is formed by rod-like molecules with large permanent longitudinal electric dipoles, which not only align parallel to each other as in conventional nematics, but also share the dipole direction, thus producing a spontaneous macroscopic polarization P along the molecular nematic director n. In this new phase, the molecular arrangement breaks the n → −n mirror symmetry characterizing the conventional, apolar nematic phase, a feature at the core of many new properties. The N_F liquid crystalline materials represent the first fully liquid ferroelectric systems ever found and their peculiar combination of fluidity and polarity has opened the gate to a whole world of new phenomena, which are rapidly becoming the focus of the liquid crystals and soft material scientific communities.^5–13

In N_F, P is locally parallel to the optical axis n and polarized optical microscopy observations acquire the new meaning of an exploration of the electrostatic ground state of this phase, which is achieved in terms of the spatial arrangement of P at the micron scale. Indeed, in N_F, electrostatic interactions dominate over their fluid elasticity, so that the observed optical textures are mainly determined by long-range electrostatic forces and topological constrains,^5,14–17 as also reported here.

In this work, we carried on a topological study on ferroelectric nematic droplets confined between two solid ferroelectric substrates. This investigation follows the recent observation of the electromechanical instability of sessile N_F droplets positioned on a bare lithium niobate (LN) ferroelectric crystal surface, exposed to either the LN pyroelectric^7,18 or photovoltaic^11,19 field. We observed that upon entering the N_F phase, the liquid crystal droplets disintegrate by the explosive emission of fluid jets that exhibit a cascade of branching out into smaller streams to eventually disrupt, forming secondary droplets. We interpreted this behaviour as a manifestation of a Rayleigh-type instability of electrically charged fluid droplets, expected when the electrostatic repulsion exceeds the surface tension of the fluid. In the case of ferroelectric droplets, the charges are due to the bulk polarization of the ferroelectric fluid which couples to the LN polarization through the fringing field and via solid-fluid interface coupling. In our observations, jets start from the droplet rim, which suggests that the edges of the droplets are likely location where N_F accumulates charges.

Previous observation of N_F electro fluid instability could not detail the onset and development of jet emission because of the rapidity of the phenomenon and the sessile droplet geometry. With the aim of shading new light on the relationship between instability and the presence of charges and topological defects on the droplet rims, we adopted the double LN thin cell geometry. The additional friction from confinement and the smaller droplet volumes inherent to this geometry slow down the fluid motions with respect to sessile droplets, a feature that conveniently combines with the access to optical textures both in equilibrium and during the emission of fluid jets.

Noteworthy, N_F droplets confined between two solid substrates have recently been studied, although in a totally different experimental configuration.²⁰ The study reported there deals with thicker cells with screened ITO conductive glasses as bounding substrates and an externally applied uniform electric field either perpendicular or parallel to the bounding surfaces. Moreover, the higher cell thickness gives rise to the formation of a thick fluid bridge, quite different from the thin LC layer of the present work where the easy access to the topology and to the instability details is the main requirement.

Materials and methods

The ferroelectric liquid crystal used in this work is 4-[(4-nitrophenoxy)carbonyl]phenyl2,4-dimethoxybenzoate (RM734). It was synthesized as described in ref. 4 and its structure and phase diagram have already been reported.^4,5 In this compound the ferroelectric nematic phase (N_F) appears through a weak first order phase transition upon cooling from the conventional higher temperature nematic (N) phase and exists in the range 133–80 °C.^5,7 The spontaneous polarization P of RM734 is either parallel or antiparallel to the molecular director n, defining the average orientation of the molecular axis, and exceeds 6 μC cm⁻² at the lowest temperature in the N_F phase.⁴

As ferroelectric solid substrates we used 500 μm thick z-cut lithium niobate (LN) undoped crystals. The bulk spontaneous polarization of LN crystals along the [0001] z-axis is of the order of 70 μC cm⁻² and does not depend significantly on T in the explored range since its Curie temperature is much higher (≈1140 °C). The huge bulk polarization of LN does not however translate in a huge surface charge density because of very efficient but slow compensation mechanisms at the z-cut surfaces, lowering the equilibrium surface charge to only about 10⁻² μC cm⁻².²¹ When temperature variations are induced on timescales shorter than free charge migration, the surface charge of LN can significantly increase because of the so-called pyroelectric effect, a transient phenomenon observable during and shortly after the variation.²² The pyroelectric coefficient of undoped LN is of the order of 10⁻⁴ C m⁻² K at room temperature,^23,24 and increases by one order of magnitude around 100 °C.²⁵ Given the temperature used in our experiments, dictated by the RM734 phase diagram, we can thus expect an induced surface charge density of the order of 1 μC cm⁻², for T variations of a few degrees ramped in a short time compared to the LN charge relaxation. To match the conditions of the previous work on sessile droplets,⁷ the substrates were left uncoated and were unidirectionally mechanically rubbed with optical paper. We found this process to produce partially aligned cells, with quality that fluctuate from cell to cell (Fig. S1, ESI†).

To prepare the samples used in this work, a RM734 droplet, obtained as described in the ESI,† was deposited on a rubbed LN substrate at a temperature T = 170 °C (i.e. in the N phase) and then covered by a second LN slab previously heated at the same T. This procedure gives rise to not perfectly circular droplets. The gap between the substrates was fixed through silica microspheres cell spacers having diameter around 2.5 μm, leading to cell thickness in the range 3–6 μm. A sketch of the cell's side view is reported in the top panel of Fig. 1c. Because of the finite size of the LN crystals, their pyroelectric charging produces a fringing electric field E_f in the region between the two, which is a fraction f of the internal field σ_LN/ε₀, where f depends on the crystal finite size and has been estimated to be f ≈ 10⁻³ in our experimental conditions.⁷ With respect to the RM734 sessile droplets in which LN and droplet were in direct contact with the much colder environment, the thin cell geometry enables smaller thermal gradients. To partially compensate this condition, we arranged the two LN crystals to realize cells in which they had equal or opposite polarization. In principle, the condition leading to largest LN field is when the two substrates expose equally charged surfaces at the interfaces with the N_F droplet, a configuration where the in-plane component of the total fringing field is expected to be maximum (see Fig. S2, ESI†). However, we noticed that the two LN arrangement have no effect on the optical textures and have only a very weak effect on the observed instability. This is likely because in our experimental conditions the top substrate is in direct contact with the environment and this makes the temperature gradient much higher on this substrate, which thus dominates in terms of fringing field generation.

Fig. 1 Evolution of the droplet texture on cooling from the N into the N_F phase until equilibrium is reached. (a) Evolution of the texture upon slowly decreasing T. As the polarization gradually grows from P = 0 in the N phase (panel 1) to P > 0 in the N_F phase (panels 2–4), the portions of the droplets next to the edges that run perpendicularly to R (white arrow) become distorted to avoid accumulation of charges at the droplet edge. Lines that originate in such defected regions propagate in the droplet interior approximately following R. The formation of defect lines stops at about T = 125 °C and the droplet texture reaches a stationary condition, in which the droplet is split in elongated domains with P parallel to R, in a way that each domain is surrounded by domains with opposite P (red arrows in panels 3 and 4). (b) Structure of the splay-bend walls separating polar domains with opposite P (top view). Because of splay, polarization charges of opposite sign accumulated on the two sides of the wall. As shown in the sketch, the walls can take two symmetric topologies with either “down” or “up” concavity (top and bottom sketches, respectively). The shadowing marks the POM transmitted intensity given the direction of the polarizers as in the top right corner of the panel. (c) Side (top) and in-plane view (bottom) of the cell and of the equilibrium texture of the N_F droplets. Red arrows indicate P (the specific up vs. down attribution is arbitrary). (d) When droplet edges are in a direction that cuts across P perpendicularly or with a large angle, the N_F fluid develops triangular π-twisted domains, are compatible with rubbing induced surface anchoring while minimizing the accumulation of bound charges (orange dashed arrows). (e) Droplet boundaries with P parallel to the edge exhibit no twisted areas (blue arrow). When P and droplet edge form a small angle, the local P slightly tilts giving a modulation of transmitted intensity (black dashed arrow). Cusp-like structures formed by two merging wall lines (green dashed arrows) are always accompanied by a distortion on one or both sides of the cusp, which in some cases propagates in the cell as a linear distortion. (f) “Webbed foot” domains formed at the intersection among four striped polar domains (purple dashed arrows) exhibit a twisted structure similar to the one observed in panel d. Fig. 1 has been extracted from Videos S1 and S2 (ESI†).

Cells were placed in a small oven suitable for the microscope stage and the temperature was lowered to T = 90 °C. To study the droplets texture at equilibrium, temperature has been decreased slowly. Instability and jets ejection were instead induced by using a large cooling rate – of the order of 0.1 °C s⁻¹ – that produces strong LN pyroelectric charging.

The optical textures of the cells were studied at equilibrium and during the electromechanical instability by polarized optical microscopy (POM) and with the aid of a compensator. Videos were recorded with a rate of 25 frames per second.

Results and discussion

Equilibrium droplet texture

When a droplet exhibiting planar N alignment along the rubbing direction R (see panel 1 of Fig. 1a and top panels of Fig. S1, ESI†) is cooled in the N_F phase, the uniform planar nematic alignment is generally maintained, within which elongated domains with opposite P progressively emerge. The way this happens is portrayed in panels 2–4 of Fig. 1a. As the polarization grows from P = 0, the portions of the droplets next to the edges that run perpendicularly to R immediately lose the uniform nematic alignment, a behaviour that we understand as aimed at avoiding the accumulations of charges at the droplet edge. We observe lines originating in such defected regions and propagating in the droplet interior approximately following the rubbing direction. Upon cooling the samples slowly enough (cooling rate ≤ 1 °C min⁻¹), the formation of defect lines stops at about T = 125 °C and the polar domains reach a stationary (possibly equilibrium) condition which holds unless the sample is heated to the N phase. We understand these lines as marking the boundaries between regions of antiparallel P. Indeed, after the sample has reached equilibrium, the droplet appears split in tiles, whose shape is elongated and directed along R, that can be attributed to +P and −P domains in a way that each is surrounded only by the opposite type (the specific attribution is arbitrary). This is sketched in the bottom panel of Fig. 1c, where red arrows indicate P. Boundaries between the N_F polar domains are splay-bend walls, whose structure is sketched in Fig. 1b. Although these walls involve a significant amount of elastic distortion, they are stabilized and kept thin by the polar nature of the N_F phase, which leads to an accumulation of polarization charges ρ = div P of opposite sign on the two sides of the wall. As recently suggested, the in-plane splay-bend of P could be accompanied by out-of-plane tilts, which introduce the twist of P and reduce the overall energy of the domain wall.¹⁵

Splay-bend walls can take two symmetric topologies. Namely, to connect a +P domain to a −P domain, the bend of the optical axis is U-shaped with either “down” or “up” concavity, as in top and bottom panels of Fig. 1b, respectively. In these sketches the shadowing marks the POM transmitted intensity, which takes the shape of a double line. When two defect lines propagating from opposite droplet edges merge with opposite concavity, a narrowing or kink is observed in the line. This corresponds to a trapped +1 defect point, the presence of which becomes apparent when the liquid crystal is brought back into the N phase and lines broaden, as described below (see the regions marked by red dashed circles in Fig. 2d).

Fig. 2 Evolution of the droplet topology on heating to the N phase. (a) Equilibrium texture in the N_F phase. Black arrow indicates one of the triangular twisted domains that develop at the droplet edges in conditions of P perpendicular to the rim. (b) Upon increasing T the twisted triangular domain structures on the droplet edge disappear, followed by the breaking of the domain walls delimiting the polar domains. Disconnected lines shrink, leaving behind a weakly distorted region that smoothly anneals into a planarly aligned nematic. The domain wall lines that remain become broader than in the N_F phase. (c) and (d) Zoom view of the region marked on panel b, differing in the orientation of the polarizers. The red circle in d marks a defect kink in the wall line. (e) and (f) Texture and domain walls close to the isotropic phase. The defect kinks further expand and develop into Schlieren textures type black bands, which we interpret as in panel f. Fig. 2 has been extracted from Videos S3–S5 (ESI†).

Splay-bend wall lines terminate either when they approach the droplet rim, or when they converge in multi-line vertices, often containing impurities. The polar domain tips never extend their uniform alignment up to where the droplet's edge runs perpendicular to P, a condition which would lead a surface charge density = P·u (u being the edge normal unit vector). In these regions the fluid systematically develops triangular π-twisted domains (see Fig. 1d), which are compatible with rubbing induced anchoring on the two surfaces while minimizing the accumulation of bound charges. No twisted areas are instead found between polar domains and droplet boundaries when P runs parallel to the edge (Fig. 1e, blue arrow), a condition where no charge accumulation is expected. In intermediate inclination conditions, the polarity slightly tilts following the droplet rim and without producing a complete π-twisted domain (Fig. 1e, black arrow). All these observations agree with the notion that the arrangement of N_F domains is dominated by the minimization of the electrostatic costs associated to the spontaneous polar order.

Twisted structure very similar to that of the triangular regions at the droplet edges, are found inside the webbed foot domains at the intersection among four striped domains (Fig. 1f, purple arrows). Here also, developing of twist distortions is a way to connect domains of opposite polarity with minimal charge accumulation. These patterns are remarkably similar to those recently observed in the SmA ferroelectric phase of DIO mixtures.¹⁷ In that case the π-twist domains are N_F regions surrounded by four SmA polar domains, their boundaries consisting in polarization-stabilized kinks which mediate small changes in the orientation of P along the thickness of the cell. Those reported in Fig. 1f mediate instead the complete reversal of P in adjacent nematic domains with opposite polarity, their boundaries being less rigid, as indicated by their less regular shape.

Typical features observed in these cells are also the convergence of two wall lines in a cusp. When such structures are present, we observe a distortion on one or both sides of the cusp (as marked by green arrows in Fig. 1e), in close analogy with the dragonfly deformations reported in ref. 14. In some cases, this deformation propagates in the cell as a linear distortion. These distorted regions probably arise from the charge accumulated close to the cusp tip, in which the wall has a component perpendicular to P. Such charge accumulation calls for a compensating charge of opposite sign on the other side of the wall, which can be accumulated only by a tilt or a splay of the polarization close to the cusp, provoking the observed effect.

The droplet topology on heating is reported in Fig. 2. Fig. 2a shows the droplet texture in N_F phase. The first effect observed upon entering the N phase, is the disappearance of the twisted triangular domain structures, probably a result of the freedom to rotate of the nematic director at the droplet edge. This is followed by the breaking of the walls delimiting the polar domains. Break events occur mostly at the defect points trapped in kinks (see Fig. S3, ESI†), a sign that these are locations where the energy associated to line connectivity is lower. Disconnected lines shrink, leaving behind a weakly distorted region that smoothly anneals into a planarly aligned nematic (Fig. 2b) so to enable domains with former opposite polarity, and now with equal nematic director, to merge with continuity. Not all lines manage to break. Those that instead remain broaden, their width becoming larger than in the N_F phase – as shown in Fig. 2c and d for different alignment of the crossed polarizers – a behavior that confirms the role of polar charges (vanished in the N phase) in maintaining the walls thin against the tendency to relax the elastic energy. As T is further raised toward the isotropic phase, the coupling with the rubbed LN substrate weakens and remaining lines further expand. We could observe that kinks in the splay-bend wall lines – marked by red circles in Fig. 2d – that we attributed to reversal of the U shape director distortions, indeed develop into a topology analogous to Schlieren textures with a +1 defect in the center (red circle in Fig. 2e and sketch in Fig. 2f). The red circle in Fig. 2e indicates a region corresponding to one of the two kinks highlighted in Fig. 2d.

While the former discussion has been developed with reference to a droplet in which rubbing was effective, the same features are observed also when rubbing is less effective, but on a local scale only, whereas on a global scale the lines separating polar domains are provided with greater freedom of movement, typically favouring polar domains to be tangent to the droplet rim thus avoiding charge accumulation (Fig. S4, ESI†).

Instability and jet ejection

Upon cooling to the N_F phase with a fast cooling rate (5 °C min⁻¹), the liquid crystal droplets become electromechanically unstable and eject fluid jets from the rim that spread along the LN substrates.⁷ This phenomenon is induced by the coupling of P to the pyroelectric polarization of the two lithium niobate substrates through their combined fringing fields and solid-fluid interface coupling.

We find that most of the observed instabilities start with the formation of many small jets that bulge out from everywhere on the droplet rim with an approximately uniform spatial density. The quasi-periodic structure of such jet emission is an indication that the initial stage of ferroelectric electromechanic jets is due to some form of line instability, possibly an electrostatic analog of the Rosenweich-type instability reported for ferromagnetic liquid crystals.²⁶ Moreover, the simultaneous appearance of jets with a similar density everywhere around the droplet, regardless of the neighboring domain structure, suggests that the droplet rim is itself electrically uniform and does not host local charge accumulation sites. While being the one that more frequently occurs, the uniform periodic jet emission is not the only instability pattern we observed. We also found emission of groups of quasi periodic jets from a fraction of the droplet perimeter, suggesting these as regions of larger accumulation of bound charges, and emission of isolated jets from specific locations on the droplets perimeter, which we understand as resulting from “trigger points” where polarization charges are not efficiently compensated. Typical examples of the three situations are reported in Fig. S5 (ESI†) (quasi-periodic all around the droplet), Fig. 3 (quasi-periodic in a fraction of the droplet) and Fig. 4 (isolated jet).

Fig. 3 Evolution of N_F droplet texture on cooling, leading to the ejection of a set of quasi-periodic fluid jets. (a) Texture before the onset of the electromechanical instability at T = 85 °C. Red arrows indicate P, blue line n and white arrow the rubbing direction R. In most of the area shown, P lies either along or opposite to R (we remind that the attribution of the specific direction of P is arbitrary). (b) Right before instability, the polar domains whose P points away from the droplet centre rotate their polarization, revealing that the growing electric field E_f (green arrow) due to the increased LN pyroelectric charges has become large enough to affect the droplet alignment. (c) The growing E_f triggers the emission of a quasi-periodic set of fluid spikes (λ_E ≈ 24 μm), while the polarization of the rotating domains approaches the fringing field direction. (d) and (e) As the instability proceeds, the periodicity is lost due to the growth of only a fraction of the initial spikes, while the others retract. (f) The jets that grow become connected to polar domains having P oriented as E_f. Domains that rotated their P show the same colouring of the twisted areas on the droplet periphery, suggesting an inversion of the polar surface anchoring in one of the surfaces. Fig. 3 has been extracted from Video S6 (ESI†).

Fig. 4 Examples of emission of an isolated fluid jet from the two sides of a droplet whose textures are evolved by cooling from a N phase where the planar alignment was not uniform, yielding polar domains with irregular shape, except when they contact the droplet boundaries, where P is generally tangential to the rim. Red arrows mark the direction of P determined with the aid of a compensator. In green areas P is uniform across the cell thickness, while orange domains are regions where P twists along the cell thickness. Blue dots mark two highly distortion points (cusp junction of two splay-bend walls in panel 1 of a and +1 vortex in panel 1 of b) that appear to trigger the instability and the emission of single fluid jets. In panels 2 of both a and b a faint orange line appears, connecting the distortion points to the droplet rim, in the position where the jet is formed (panels 2–3 of a and panel 3 of b). As the instability proceeds lines and walls are displaced and the jets become connected to droplet domains either through a spread-out bend and splay deformation enabling joining the jet to a perpendicularly oriented polarity (panel 4 of a) or through a uniform connection with the closest domain of equal polarity (panel 4 of b). Fig. 4 has been extracted from Video S7 (ESI†).

Fig. 3 reports six frames, extracted from Video S6 (ESI†), showing the evolution of the instability in the ejection of a set of quasi-periodic jets. In Fig. 3a, a portion of the droplet before the onset of the instability is reported. Red arrows indicate the polarization vector, blue line the director and white arrow the rubbing direction. In most of the area shown in the figure, P lies either along or opposite to R (we remind that the attribution of the specific direction of P is here arbitrary). Fig. 3b captures the condition that develops right before the emission of fluid jets: all polar domains with polarity away from the droplet center rotate their polarization, revealing that the field E_f (green arrow) due to the increased LN pyroelectric charges is becoming large enough to affect the droplet alignment. Immediately after, the growing E_f triggers the emission of a quasi-periodic array of small liquid crystal spikes of separation λ_E ≈ 24 μm (Fig. 3c) while P in the rotating domains have approached the direction of the field. If we read this phenomenon as an instability, the jet periodicity can be understood as λ_E = 2πl_E where l_E is an electrostatic capillary length that can be defined as , being γ the surface tension, ρ the volume charge density and E_f the modulus of the fringing field generated by the charged LN substrates. By assuming γ ≈ 2 × 10⁻² J m⁻².⁷ and a pyroelectric surface charging of the order of 1 μC cm⁻², which results in E_f ≈ 10⁶ V m⁻¹, we obtain ρ ≈ 3 × 10³ C m⁻³. This rough estimate agrees with the volume charge density expected for a cylinder of height d ≈ 5 μm (average distance between the LN substrates) and radius R ≈ 500 μm having a surface charge of the order of 0.1 μC cm⁻².

As the instability proceeds, a fraction of the initial jets disappears, while other significantly grow, as visible by comparing panels c–e of the figure, resulting in a loss of the periodicity. Inspection of panels e and f reveals a general behavior characterizing all the growing jets: they connect and become one with domains that are stabilized by E_f, i.e. domains with P oriented as E_f. Thus, the rotation of P in a fraction of the polar domains and the emerging connectivity of jets to polar domains that do not rotate, coherently combine indicating that the whole observed phenomenology derives from the electric field on the LN surface. The growing jets have the same thickness of the mother droplet, as suggested by their color, which indicates that they occupy all the space between the confining surfaces. It can also be noted that, after jets emission, the polar domains that rotated their P in Fig. 3b and c, show a color equal to the one of the twisted areas on the droplet periphery.

As for the ejection of isolated jets, we inspected videos portraying emission of single isolated jets searching for common features providing a general understanding. Two examples of such emissions from the same droplet are provided in Fig. 4. In this droplet the regular polar striped domains observed in Fig. 3 are replaced by domains having irregular shape, separated by domain walls. Such a different texture developed on cooling from a N phase where the planar alignment was not uniform and Schlieren textures were present (see Fig. S4, ESI†), in turn due to a less effective mechanical rubbing. With the aid of a compensator, we could establish that in green areas P is uniform across the cell thickness. Generally, when such green domains contact the droplet boundaries, P is found to be tangential to the rim. We marked with red arrows the direction of P where we could determine it. The orange domains cannot instead be extinguished between crossed polarizers and represent regions where the polarization vector twists along the third dimension. In the two cases reported in the figure, jet emission appears to be triggered by high distortion points in the proximity of – but at some distance from – the droplet perimeter. We identified such points by blue dots in the frames preceding the emission. These points mark a cusp junction of two splay-bend walls and a +1 vortex-type texture, respectively (first panels of Fig. 4a and b). Inspection of the video frames reveals that right before the jet emission (by one or two frames time) a linear distortion, recognizable in the pictures as a faint orange line, develops to connect the distortion points with the droplet rim (panels 2 of Fig. 4a and b). While we do not have a solid interpretation of this phenomenon, we speculate that some charge accumulation in the distorted point becomes unstable under the growing fringing field and is pushed in the direction of the field itself becoming distributed along the line toward the droplet edge. In both cases, we observe that jet emission is localized where such distortion line meets the border of the droplet (panels 3 of Fig. 4a and b). As the jet grows, the texture connecting it to the droplet evolves with the displacement of lines and walls that involve a whole region in proximity of the jet itself. This behavior agrees with the notion that jet emission is caused by an instability occurring at some distance from the edge. In later stages, jets align their polarity along the direction of flow, which must be equal to the average direction of the fringing field. In doing this, they somehow force the area close to the ejection site to adjust either through the formation of a topological defect at the interface with the ejection site connecting two regions with orthogonal polarity, as in panel 4 of Fig. 4a, or through a uniform alignment of P, as in panel 4 of Fig. 4b.

Noteworthy, the specific texture of the N_F droplet does not seem to affect the observed kind of instability. Both the formation of quasi periodic fluid spikes and the emission of isolated jets have been observed independently on the quality of the liquid crystal alignment, that is due to the effectiveness of the rubbing.

In conclusion, we analyzed the relationship between the topological texture of ferroelectric nematic droplets and the electromechanical instability occurring upon pyroelectric charging of the LN substrates. Droplets are here confined between two LN crystals, which allows a good control on the sample thickness, necessary for an easy access to the domains shape and polarity, and guarantees weak thermal gradients, resulting in slow instability events. We found that mechanical rubbing of the confining surfaces produces partially aligned cells, with fluctuating quality. In the case of uniformly aligned N droplets, the transition to the N_f phase results in the formation of striped domains with opposite polarity, separated by splay-bend walls running approximately along the rubbing direction R. When these domains extend close to the droplets’ perimeter, P adopts a π-twisted structure to minimize the accumulation of polarization charges. In case of less effective rubbing, the lines separating polar domains possess greater freedom of movement, which typically favours polar domains tangent to the droplet rim to avoid charge accumulation. The observed tendency of the N_F phase to minimize charge accumulation, which strongly affects the droplets texture, fully agrees with the notion that the polarization field of this fluid polar phase can easily rotate to adopt configurations aimed at compensating any accumulation of electrostatic charge.¹²

The efficiency of this mechanism is particularly evident when the surface charging increases above threshold and gives rise to the droplet electromechanical instability. Indeed, near threshold, the ejected fluid jets mainly consist in quasi periodic thin spikes that are approximately uniformly distributed along the droplet perimeter, indicating the absence of specific sites of charge accumulation. Although the emission of isolate jets triggered by topological distortions in specific locations of the droplet – that thus become areas of local charge accumulation – are also observed, they appear more rarely.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

L. L. and R. B. acknowledge the European Union—Next Generation EU, project code: ECS00000041; project title: Innovation, digitalization, and sustainability for the diffused economy in Central Italy—VITALITY. F. C. acknowledges Cariplo Foundation, grant number 2023-1095.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4sm00317a

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