
==== Front
Proc Natl Acad Sci U S A
Proc Natl Acad Sci U S A
PNAS
Proceedings of the National Academy of Sciences of the United States of America
0027-8424
1091-6490
National Academy of Sciences

38513103
202313629
10.1073/pnas.2313629121
videoVideoresearch-articleResearch ArticlephysPhysics426
Physical Sciences
Physics
Fluid fibers in true 3D ferroelectric liquids
Jarosik Alexander a https://orcid.org/0009-0002-9796-0018

Nádasi Hajnalka a
Schwidder Michael b https://orcid.org/0000-0001-5098-574X

Manabe Atsutaka c https://orcid.org/0000-0002-8481-0216

Bremer Matthias d https://orcid.org/0000-0003-3615-8953

Klasen-Memmer Melanie d https://orcid.org/0009-0002-3209-477X

Eremin Alexey alexey.eremin@ovgu.de
a 1 https://orcid.org/0000-0001-9743-6895

aDepartment of Nonlinear Phenomena, Institute of Physics, Otto von Guericke University, Magdeburg 39106, Germany
bDepartment Industrial Chemistry, Institute of Chemistry, Otto von Guericke University, Magdeburg 39106, Germany
cIndependent Researcher, Bensheim, Germany
dMerck Electronics KGaA, Darmstadt 64293, Germany
1To whom correspondence may be addressed. Email: alexey.eremin@ovgu.de.
Edited by Noel Clark, University of Colorado, Boulder, CO; received August 15, 2023; accepted February 8, 2024

21 3 2024
26 3 2024
21 9 2024
121 13 e231362912115 8 2023
08 2 2024
Copyright © 2024 the Author(s). Published by PNAS.
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

Significance

A recent discovery that 3D fluids can form phases with vector symmetries, such as ferroelectric and ferromagnetic nematics, has sparked intense studies of their behavior in confinement. We demonstrate one of the most remarkable consequences of the high spontaneous polarization in fluid ferroelectrics—the formation of static liquid fibers. Polarization-driven confinement of the electric field stabilizes polar liquid cylinders suppressing surface fluctuations in systems without translational symmetry breaking. Such ferroelectric liquid fibers exhibit exceptional nonlinear optical properties and a strong response to electric fields. The potential impact of these new materials on technology cannot be overstated. Light manipulation, actuation, and energy harvesting are just a few examples of potential applications in developing smart multifunctional materials.

We demonstrate an exceptional ability of a high-polarization 3D ferroelectric liquid to form freely suspended fluid fibers at room temperature. Unlike fluid threads in modulated smectics and columnar phases, where translational order is a prerequisite for forming liquid fibers, recently discovered ferroelectric nematic forms fibers with solely orientational molecular order. Additional stabilization mechanisms based on the polar nature of the mesophase are required for this. We propose a model for such a mechanism and show that these fibers demonstrate an exceptional nonlinear optical response and exhibit electric field-driven instabilities.

soft condensed matter
ferroelectrics
liquid crystals
nonlinear optics
complex fluids
Deutsche Forschungsgemeinschaft (DFG) 501100001659 ER 467/8-3 Alexander JarosikHajnalka NádasiAlexey Eremin Deutsche Forschungsgemeinschaft (DFG) 501100001659 NA 1668/1-2 Alexander JarosikHajnalka NádasiAlexey Eremin
==== Body
pmcFreely suspended fluid fibers in non-Newtonian fluids are a fascinating topic with many practical applications. From living organisms to technological applications, such as spider silk, textiles, and photonic devices, they are observed in a wide range of systems (1–3). Fiber-based materials are particularly desirable for wearable electronics due to their flexibility and stretchability, providing advantages over solid-state materials for designing the next generation of electronic devices (4, 5). Advancements in smart materials and wearable technology have paved the way for the development of structured and multifunctional materials, leading to further research in this field.

Fibers can be formed from solidified liquids or glasses, which initially start in a liquid state (6, 7). In Newtonian liquids, long liquid filaments cannot form due to the Rayleigh–Plateau instability (8). However, non-Newtonian fluids such as polymeric solutions and melts can form cylindrical filaments and jets during the thinning process of fluid bridges suspended between two supports or during droplet detachment (9).

The capillary-induced thinning of liquid filaments has been demonstrated in multiple studies as an effective method for characterizing rheological properties of materials (10, 11). Flow and deformation characteristics of materials under various conditions can be extracted in a rheometric device such as a thinning fiber (11).

Fiber formation also occurs in some structured fluids such as particular types of liquid crystals (12–14).

Experiments with pure thermotropic liquid crystals have shown that the orientational anisotropy of nematics alone does not lead to the filament stability (12). However, the columnar (15–20) or modulated smectic order (21–25) is the prerequisite for stabilizing the filament structure.

Nematics are liquids exhibiting orientational molecular order. They find various applications, from displays to electro-optic and photonic devices, as well as in sensorics (26). Most nematics show quadrupolar uniaxial and rarely biaxial types of order. The possibility to stabilize the nematic phase assuming dipolar correlations was proposed by Max Born as early as ref. 27. The symmetry of such a liquid corresponds to that of the ferroelectric nematic. In contrast to the common uniaxial nematics with quadrupolar symmetry, the symmetry of the NF phase is dipolar since the head–tail invariance of the nematic director is broken in NF. Although such symmetry breaking is quite common at interfaces, the experimental realization of the bulk ferroelectricity in nematics remained elusive until recently.

A few polymeric and lyotropic systems showed indications of polar order in bulk (28–30). Only in 2017, Nishikawa et al. synthesized a thermotropic liquid crystal with a 1,3-dioxane unit in the mesogenic core (DIO) exhibiting two nematic phases with a spontaneously polar low-temperature phase where the polarization is aligned parallel to the nematic director (31). Around the same time, Mandle et al. designed a series of mesogens exhibiting polymorphic nematic phases with a low-temperature ferroelectric nematic phase (32, 33).

Ferroelectric nematics are distinguished by their lower than nematics symmetry and unusually high spontaneous polarization, typically within the range of a few µC cm−2 (34–38). Sebastián et al. have demonstrated that in materials with a low-temperature NF phase, the splay deformations soften in the nematic phase (N) upon approaching the transition into the NF phase, which is expressed in reduction of the splay elastic constant K11 in the N phase (39). This is also accompanied by the growth of polar correlations. In contrast, in the NF phase, splay elasticity stiffening is observed where the electrostatic interactions are essential in defining the materials’ macroscopic properties. The director stiffening occurs due to the electrostatically unfavorable polarization splay (40). On the other hand, the vector symmetry of the nematic phase allows for the development of the spontaneous director splay, as theoretically demonstrated by Pleiner and Brand (41). Thus, the equilibrium state is given by the balance between the two contributions.

In this paper, we experimentally demonstrate that ferroelectric nematics spontaneously form liquid filaments. Although the field stabilized NF thread-like structures were found by Nishimura et al. (42) in experiments on electrostatic actuators, spontaneous formation of fibers remained unexplored. The filament formation occurs regardless of whether the compound undergoes a direct isotropic-NF or nematic-NF transition. These filaments can be created by mechanically pulling from a droplet or through the electrocapillary instability of a droplet in a vertical electric field. We use nonlinear optical microscopy to show that those filaments exhibit a remarkably efficient optical second harmonic generation. A model based on the polar structure of the phase is proposed to explain filament formation.

Results

Filament Stability and Optical Properties.

A unique feature of ferroelectric nematic is observed while attempting to extract a small amount of liquid crystal with a spatula from a vial. Long and occasionally, multiple fibers form attached to the spatula (Fig. 2A). On further pulling, the fiber remains at a nearly constant radius, where the necessary material is supplied from the menisci.

We explored this behavior in two compounds: Compound 1 is a single-component mesogen exhibiting a direct transition from the isotropic to the ferroelectric nematic (NF) phase (Fig. 1). The details of the mesomorphic behavior are reported in ref. 43. The phase transitions on cooling are:

Fig. 1. (A) A sketch of a nematic phase with the quadrupolar order; (B) Molecular arrangement in the ferroelectric nematic phase (NF) with the polar order; (C) Chemical formula of compound 1 with an optimized 3D structure. We have used the density functional B3LYP and the 6-31G(d) basis set for the calculation.

Cr19.6°CNF44.0°Cisotropic recrystallization−7.0°C

where the brackets (...) indicate a monotropic transition. At room temperature, the material is crystalline, and upon heating, it melts directly to an isotropic liquid at 44°C. Upon cooling, a direct transition occurs from isotropic to ferroelectric nematic at 19.6°C, followed by crystallization at −7.0°C. However, if compound 1 is heated from the NF phase, before crystallization has occurred, the phase transition from NF to the isotropic occurs at 21.8°C, this is 2.2° higher than the isotropic—NF phase transition on cooling.

Material 2 is a mixture exhibiting the transitions:

crystal <-20°C NF 45.8°CM 57.9°C N 87.6°C isotropic

The coexistence range of the N and isotropic phases does not exceed 2K. The transition values correspond to the maxima of the heat flux in the differential scanning calorimetry (DSC) curve with a cooling rate of 5K min−1. Both 1 and 2 exhibit the room temperature NF with a high spontaneous polarization in the range of 5 to 6 μC cm−2. The mesogen in compound 1 has a strong dipole moment μ=11.3D, and its properties are described in refs. 40 and 43.

A drop of isotropic liquid spanned between two supports forms a catenoid-shaped profile of a liquid bridge formed by two menisci. Surface tension and the wetting conditions at the support determine its shape. As the distance between the supports increases, the bridge collapses due to the Plateau–Rayleigh instability. The bridge becomes birefringent when cooling to the nematic phase N (Fig. 2B). A grainy character of the microscopic texture observed between crossed polarizers suggests a disordered polydomain arrangement in the absence of the alignment within the meniscus. A slightly better alignment was observed in the central part of the bridge. Strong fluctuations of the nematic director result in the scattering of light appearing as a typical flickering of the microscopic texture. Those fluctuations become quenched upon the transition into the intermediate M phase (Fig. 2C). The microscopic texture remains grainy in the meniscus and partially aligned in the central part of the bridge. Labyrinthine structures with a typical periodicity of 2.8 μm ± 0.1 μm appear at the surface of the fluid bridge (Fig. 2C). Such structures are universal for a wide range of systems and indicate breaking of the translational symmetry often occurring as a result of the frustration between two competing interactions (44). Chen et al. observed occurrence of intermediate phases between N and NF in several similar compounds and mixtures. Using exhaustive X-ray studies, they concluded that the intermediate phase is the uniaxial antiferroelectric smectic-ZA phase (45). The presence of the in-plane modulated structure of polarization reversal areas can be responsible for the labyrinthine patterns. Labyrinthine patterns were observed in polarization-modulated ferroelectric smectic-C phases of bent-shaped mesogens (24, 46).

Fig. 2. Filament formation in Material 2: (A) Spontaneously formed freely suspended filament pulled at room temperature using a spatula. Short birefringent bridges can be formed below the Rayleigh–Plateau limit in (B) the nematic, and (C) the M phases; (D) Long filaments appear in the NF phase. (E) Transient current accompanying the thinning results from the thickness dependence of the polarization-induced charges in a filament structure shown in (F).

The stable bridge length in the N and M phases is not greater than that in the isotropic phase.

When entering the ferroelectric nematic phase NF, the situation changes drastically. The director fluctuations become visible again. Stretching the bridge above the critical length does not lead to immediate collapse. Rather, a cylindrical filament forms between the pair of separated menisci (Fig. 2D). Dust particles occasionally trapped at the filament surface help to visualize the material flow at the surface. The particles move slowly during the filament’s extension, now and then exhibiting slow circulating motion around the filament axis. These observations suggest that there is no significant flow present at the filament surface in the steady state.

Optical anisotropy of the filaments indicates the anisotropic molecular order typical for most liquid crystals. In nematics, the birefringence is related to the orientational order parameter. In the NF filaments, the slow optical axis (with the largest refractive index) pointed along the filament axis, as determined using a variable retarder. The experiments on aligned samples in planar cells showed that the same axis was parallel to the director in the N and the NF phases. This suggests that the molecular orientation in the filament is parallel to the filament axis. This phenomenon is akin to the case of spider silk formed by the hardening of silk fibers through the extensional flow (1).

The thinning dynamics comprises three stages. At the first stage, the liquid bridge transforms into a cylindrical filament. In the second stage, the filament is slowly thinning on the time scale between 10 s and 100 s. This is marked by a plateau in (Fig. 3A). The final stage is the terminal collapse, occurring on a scale of 100 ms. Connecting the ends of the filament between two electrodes, we observe a transient current accompanying the filament thinning (Fig. 2 E and F). This current is the direct consequence of the macroscopic polarity of the filament with the polarization charge at the ends being thickness dependent.

Fig. 3. Behavior of fluid bridges and filaments in an external electric field (compound 2): (A) Thinning dynamics in filaments prepared with/without applied voltage. (B) Electrically stabilized filament becomes destabilized on the inversion of the field polarity when the new bunches of filaments appear (C). (D) A thick LC bridge observed between crossed polarizers exhibit electro-optical switching manifested by the field-dependence of the optical transmittance shown in (E).

Behavior in Electric Field.

Spontaneous polarization in the nematic phase gives rise to the whole zoo of remarkable effects in electric fields, such as explosive electrostatic instabilities in droplets (47), formation of dendritic structure in droplets (48), and behavior of soliton walls in NF (49), light-driven propulsion of ferroelectric droplets (50). Fluid filaments in NF also show remarkable behavior in an external electric field applied axially along the filament axis.

In thick bridges, application and sign reversal of moderate electric fields resulted in the optical switching (as shown in Fig. 3 D and E). To demonstrate this, we measured the optical transmittance of a bridge between crossed polarizers as the field was varied. When the voltage was reversed, there was a temporary decrease in transmittance, followed by an increase. However, if the same voltage was applied again after the field was removed, there was only a negligible change in transmittance. In high fields, in the case of nonisolated electrodes, convective patterns developed, and the nematic alignment became disturbed.

Optical switching was not observed in thin filaments, but the electric field had stabilizing or destabilizing effects depending on polarity. In the stabilizing case, stronger fields slowed down or reversed thinning dynamics through the supply of material from the meniscus (Fig. 3C). However, field reversal could cause the collapse of the filament. For asymmetric metal/glass interfaces, voltage reversal led to a destabilizing field after the collapse of the filament, resulting in a bursting instability and a cluster of sideways-injected filaments (Fig. 3 B and C).

However, if the filament was pulled from a droplet in an electric field, the field effect had a stabilizing character independent of the polarity. As shown in Fig. 3A, applying the field as low as 2V mm−1 increased the thinning time by order of magnitude compared to the field-free case. If a filament was pulled between two conducting electrodes, rupturing occurred when the circuit was closed. Therefore, maintaining the electric potential difference between the ends is necessary for filament stability.

Nonlinear Optical Behavior.

Ferroelectric nematics exhibit an exceptionally high second harmonic generation efficiency (SHG) (51–53). This is a direct consequence of the polar symmetry of the mesophase and high values of the second-order molecular hyperpolarizability responsible for the conversion of the infrared light (λ=880nm) of the primary beam to the second harmonic, SH (λ=440nm). As a result, the filaments prepared in the NF exhibit strong SH generation.

Fig. 4A exemplifies an image of a filament captured by SHG-microscopy. The angular dependence of the SH signal shows that the highest signal is obtained when the polarization of the primary laser beam is parallel to the filament axis (Fig. 4B), suggesting that the polar axis is aligned parallel to the nematic director in the NF phase. Due to C∞h symmetry of the NF phase, the SHG was determined by two nonlinear second-order optical coefficients d31 and d33 with axis “3” (Z) directed along the polar axis of the nematic.

Fig. 4. Nonlinear optical behavior of the ferroelectric fibers: (A) SHG microscopy image superimposed with the bright-field image of the filament at T = 21°C; (B) Angular dependence of the SHG signal I2ω; (C) Temperature dependence of the SHG signal of a rapidly thinning filament. (D) Maker fringes in the thickness dependence of a filament. The red curve is a fit with Eq. 1.

Typical Maker fringes occur as a result of the energy exchange between the primary and the SH rays. In the case of a plane parallel slab, the Maker fringes can be described by SI Appendix, Eq. S2. The periodic dependence of I2ω(D) is determined by the effective NLO coefficient deff and the dispersion of the refractive index Δnd=n2ω−nω (SI Appendix).

Analysis of Maker fringes allows us to estimate the second-order nonlinear optical (NLO) coefficients of the LC material (54–56). To determine the NLO coefficients, we can use the periodicity of the fringe pattern to calculate the difference in refractive indices, even if we don’t know the exact values of n2ω and nω. By comparing the maximal intensities to those of a reference like α-Quartz, we can extract the NLO coefficients for the material under study. In the case of filaments, the thickness dependence of I2ω can be recorded during filament thinning. However, the fringe pattern is different from that of the plane parallel slab (SI Appendix, Fig. S3B and Fig. 4 C and D). The periodic fringe pattern is superimposed with a baseline growing with increasing filament diameter. In the range of rather small diameters D, the fringe pattern can be approximated by the function[1] I2ω(D)=CbaseD+CAsin22πλΔndD,

where Cbase and CA are fit parameters for the baseline and the amplitude, respectively.

The different form of the size dependence occurs due to the circular shape of the filament cross-section allowing the rays with different path lengths to contribute to the net SH power. A simulation of the SH generation in a filament using the finite element method is shown in SI Appendix, Fig. S3 qualitatively confirms this behavior. The net power exhibits fringes within the filament cross-section as shown in SI Appendix, Fig. S3A.

Even in this case, the period of the fringe pattern equals λ/Δnd. This allows us to determine Δnd=0.093 from the thickness dependence of the I2ω(D). The slope of the baseline allows us to roughly estimate the nonlinear optical coefficient d33,F of the filament material. The simulation in SI Appendix, Fig. S3B compares the thickness dependences of the SHG signal of a filament and a reference slab with identical coefficients d33. Taking the mean intensity of the slab as a reference, the baseline of the normalized filament intensity is proportional to the diameter D, Ifil2ω=κD, where κ≈1×10−9μm−1. The same approach can be applied to the experimental data giving the ratio[2] ⟨IF2ω⟩⟨Iquartz2ω⟩=d33,F2d33,Q2κD,

where d33,Q=0.4pmV−1 is the nonlinear optical coefficient of α-Quartz. The slope of the baseline determined in the experiment is ≈9μm−1 giving an estimation d33(F)≈2.2pmV−1.

This value is more than five times higher than that of α-Quartz, making this material stand out. At the same time, it is smaller than the values reported by Folcia et al. (51) for another ferroelectric nematogen mesogen, RM734, aligned in an electric field. This difference can be attributed to the different chemical structures of our compounds containing several fluorine substituents.

Discussion

Polar, SHG-active filaments were initially discovered in the polarization-modulated smectic phase of bent-core liquid crystals (22, 24). These filaments display exceptional stability for a few days. In contrast, the stability range of the nematic filaments is restricted to 10 to 30 s in compound 1 to several minutes in material 2. The filaments thin retaining their cylindrical shape until they eventually break. What is the reason for the stability of the nematic filaments?

In polymeric systems, the nonlinear rheological behavior is responsible for filament formation. Intensive theoretical and experimental studies of the filament formation and jet break-up showed that extensional rheological response dictates whether or not a stable filament is formed (9, 57, 58). During the necking process, as a fluid bridge is thinning, elastic tensile stresses resist the pinching driven by the capillary forces (11). In our case, however, the mesogens are low-weight molecules. In the whole temperature range of the liquid crystal phases, the rheological behavior is Newtonian with a linear flow curve (SI Appendix, Fig. S1A), excluding this mechanism of filament stabilization. The viscosity exhibits Arrhenius dependence in the conventional and ferroelectric nematic phases (N and NF) (SI Appendix, Fig. S1B). Non-Arrhenius behavior occurs in the intermediate M phase. At low temperatures, there is a strong increase of the viscosity which also slows down the thinning dynamics of the filaments (SI Appendix, Fig. S1B). It appears that nematic filaments form only in the ferroelectric nematic phase regardless of the presence of the mesophase above NF. The same behavior is observed in compounds with the direct iso-NF and iso–N–M–NF transitions. These observations suggest that ferroelectric polarization is essential for filament stability. Indeed, the instability of a fluid cylinder occurs due to the growth of surface perturbation modes when the aspect ratio of the cylinder exceeds a critical value. Those perturbations are accompanied by a decrease in the surface energy, driving the system to a state where the liquid is confined to a set of spherical droplets. A sinusoidal perturbation of the fluid cylinder of an initial radius R0 is given by the radius dependence r(z)=R0∗+δRsinqz, where δR is the perturbation amplitude, q=2π/λ is the wave number, and λ is the perturbation wavelength. The constant volume constraint results in a reduction of the mean radius to R0∗=R0−δR2/4R0. The surface energy gain with respect to the unperturbed state is given by[3] ΔEsurf=∫0λ2πσr(z)dz−2πσλR0.

resulting in[4] ΔEsurf=πλσ2R0(q2R02−1)δR2.

The coefficient πλσR0(q2R02−1) is equal to the second derivative of the surface energy d2Esurf/dδR2, and, in the linear stability analysis, determines the stability of the unperturbed state. The positive values of d2Esurf/dδR2 correspond to the stable cylinder. However, when the aspect ratio R0/L exceeds π, the lowest mode q=2πn/L, n=1 becomes unstable, leading to the Rayleigh–Plateau instability. This is marked by the sign inversion of d2Esurf/dδR2.

Perturbation in the form of surface undulations results in the deformation of the nematic director within the filament (Fig. 5B). Such deformations are driven by the strong anchoring at the liquid/air interface. The distortion of the director field in bulk is controlled by the nematic elasticity and determined by the splay, twist, and bend elastic constants (K11, K22, K33, respectively) (26). Assuming planar anchoring at the filament’s interface and parametrizing the nematic director using the angle u(r,z) as n=(cos(u),sin(u)), we obtain u(r,z) in one-constant approximation (K11=K22=K33=K) from a solution of the Laplace equation (Eq. 6).

Fig. 5. Numerical simulations of electrostatic energy of a deformed filament: (A) Electric potential φ in an exemplary filament with L=40μm, initial radius r0=100μm undulated with sine waveform (amplitude A=10μm, mode number n=3. (B) Electric potential φ(r,z) superimposed with the vector diagram of Dr0(r,z) in a cross-section of the filament in (A). (C) Dependence of the electrostatic energy on the undulation amplitude δR for the first four modes (L=2,000μm, r0=100μm); The inset shows the d2Eel/dδR2. (D) The dependence of the d2Eel/dδR2 and d2Esurf/dδR2 on the aspect ratio a for a filament with the initial radius r0=100μm. The Inset shows the absolute values in the double logarithmic presentation. The negative and positive ranges of the interfacial contribution are marked in color.

In the ferroelectric nematic with Ps∝n, surface undulations are accompanied by the splay of the director and, as a result, by undulations of the spontaneous polarization. Since the polarization splay results in the bound electric charge density ρb=−∇·Ps, the surface undulations will cost electrostatic energy. As Bellini et al. (59) demonstrated the ferroelectric nematic phase confined in microchannels exhibits a unique property known as electric “super-screening.” This phenomenon enables the polarization in the NF to be restricted to a designated channel or filament, guiding the electric field. Specifically, any transverse electric field that exists with respect to the filament axis is rapidly compensated for by the induced bound electric charges. This compensation process leads to a confinement of the electric field of the polarized filament within the filament itself. Electric displacement in a ferroelectric is given by the equation D=ε0εE+Dr0, where E is the electric field and Dr0 is the remanent displacement, equivalent to the spontaneous polarization Ps. Assuming no free charges, introducing the electric potential φ with E=−∇φ we can derive φ from the Poisson equation ε0ε∇2φ=∇·Dr0, where the term ∇·Dr0=−ρb represents the density of the bound charges ρb and is determined by the deformation of the polar director field as Dr0(u)=Dr0[cos(u),sin(u)]. The solutions of Eqs. 5 and provides required potential:[5] ∇2φ=1ε0ε∇·Dr0(u).

[6] ∇2u=0.

The electrostatic energy Eel is determined by the integration over the volume of the filament:[7] Eel=∫ρbφdv.

Numerical solutions of the Eqs. 5 and 6 for an exemplary filament of the length L=40μm and the initial radius r0=100μm are shown in Fig. 5C. The filament’s surface is undulated with r(z)=r0∗+δRsin2πnz/L, n=3. The potential φ(r,z) has an axial symmetry and exhibits a modulation as shown in the cross-section in Fig. 5D together with the residual displacement Dr0(r,z).

By keeping the modulation amplitude below 10% of r0, the electrostatic energy increases quadratically with A and is mostly unaffected by the mode number n (Fig. 5E). The slope sel=(1/2)d2Eel/dδR2 only slightly increases with increasing wave numbers. As a result, it is adequate to only consider the lowest mode with n=1. With the elastic constant K in the range of piconewtons, the elastic energy given by the energy density term felastic=K∇2u is several orders of magnitude smaller than the surface (60) and electrostatic contributions.

When the length of a filament increases while the radius r0 is fixed, the slope sel linearly increases with the aspect ratio a=L/r0 (black curve in Fig. 5D). Conversely, the surface energy decreases as the aspect ratio increases, and the slope ssurf=dEsurf/dA2 changes sign from positive to negative, ultimately making long filaments unstable (red curve in Fig. 5D).

However, the electrostatic contribution exceeds the surface one by almost an order of magnitude, which explains why the breaking instability is suppressed. A drastic slowing down of the thinning dynamics in filaments prepared in an electric field is another confirmation that the stabilization is of electrostatic origin. Indeed, the field-guiding property of the NF filament (59) results in an electric field induced by the polarization charges at the junctions to the menisci. The filament acts as a capacitor, and the electric field couples to the spontaneous polarization within the fiber. Nonetheless, this stabilization mechanism does not prevent the filament’s thinning from occurring due to the transversal tension and material transport into the menisci. Another structural feature contributing to filament stabilization is required to sustain the transversal stresses. One possible mechanism is the conjectured spontaneous polarization splay creating a smectic or columnar-type superstructure that can balance the transversal stresses. Such splay is of flexoelectric nature and is the consequence of the symmetry of NF. The linear splay term is proposed in the theoretical models of the NF phase (33, 41, 61, 62). The experimental observations of the spontaneous splay were suggested by Sebastián et al. in ref. 62 using photo-patterned substrates. A detailed theoretical description accounting for the conductivity, free charges, and polarization fluctuations is required to accurately describe the stability of the NF filaments.

Conclusion

In summary, we demonstrated that the recently discovered ferroelectric nematic phase can form freely suspended filaments in addition to freely suspended films. The filaments exhibit a remarkable efficiency of the optical second harmonic conversion and their stability is strongly dependent on the external electric field.

Although those features are rather typical for the (modulated) smectic and columnar phases, they are yet observed in the nematic phase suggesting the suppression of the surface fluctuations and the presence of the polarization-driven secondary structure such as a periodic director modulation.

Materials and Methods

The experimental observations were performed using polarizing optical microscopy. The fibers were prepared in the heating stage (INSTEC, USA) with a custom-made fixture. Nonlinear optical properties were characterized using multiphoton (SHG) microscopy (SI Appendix). Rheological studies were made using the Anton Paar MCR301 rheometer equipped with a temperature controller in plate-cone geometry using 50mm cone (Anton Paar CP50-0.5) with a cone angle of 0.5°. Data analysis and numerical simulations were made using Matlab (Mathworks) and Comsol software (SI Appendix).

Supplementary Material

Appendix 01 (PDF)

Movie S1. Thinning filament of compound 1 observed bright field microscopy. (Compound 2, RT).

Movie S2. Thinning filament of compound 1 observed in optical/SHG microscopy. The SHG signal is shown in green channel. (Compound 1, T = 16°C, the colours/brightness were adjusted for optimal presentation).

Movie S3. Bursting instability of filaments appearing upon poling from 1.5 kV to -1.5 kV V (Bright field/Polarising Microscopy, compound 2).

We would like to thank Antal Jákli, Joseph Maclennan, Michail Osipov, and Tommaso Bellini for fruitful discussions. NF and their electromechanical properties were also demonstrated by A. Jákli at the European Liquid Crystal Conference in Rende (Italy) Jákli et al. (2023). We acknowledge the financial support of Deutsche Forschungsgemeinschaft (Project ER 467/8-3).

Author contributions

A.M., M.K.-M., and A.E. designed research; A.J., H.N., M.S., and A.E. performed research; A.M., M.B., and M.K.-M. contributed new reagents/analytic tools; A.J., H.N., and M.S. analyzed data; and H.N., M.K.-M., and A.E. wrote the paper.

Competing interests

The authors declare no competing interest.

Data, Materials, and Software Availability

All study data are included in the article and/or supporting information and have been deposited at the Repository for Research Data and Publications of OVGU (DOI: https://doi.org/10.24352/ub.ovgu-2024-049) (63).

Supporting Information

This article is a PNAS Direct Submission.
==== Refs
1 F. Vollrath, D. Knight, Liquid crystalline spinning of spider silk. Nature 410 , 541–548 (2001).11279484
2 K. Kerkam, C. Viney, D. Kaplan, S. Lombardi, Liquid crystallinity of natural silk secretions. Nature 349 , 596–598 (1991).
3 M.-Y. Wang , Dry-spinning of artificial spider silk ribbons from regenerated natural spidroin in an organic medium. Macromol. Rapid Commun. 44 , 2300024 (2023).
4 M. Stoppa, A. Chiolerio, Wearable electronics and smart textiles: A critical review. Sensors 14 , 11957–11992 (2014).25004153
5 J. S. Heo, J. Eom, Y. Kim, S. K. Park, Recent progress of textile’ based wearable electronics: A comprehensive review of materials, devices, and applications. Small 14 , 1703034 (2018).
6 C. Mercader , Kinetics of fiber solidification. Proc. Natl. Acad. Sci. U.S.A. 107 , 18331–18335 (2010).20937910
7 J. Sparkes, C. Holland, The requirements for flow-energy induced solidification of silk. Macromol. Biosci. 19 , 1800229 (2019).
8 L. Rayleigh, On the capillary phenomena of jets. Proc. R. Soc. London A: Math. Phys. Eng. Sci. 28 , 190–195 (1879).
9 M. Goldin, J. Yerushalmi, R. Pfeffer, R. Shinnar, Breakup of a laminar capillary jet of a viscoelastic fluid. J. Fluid Mech. 38 , 689–711 (1969).
10 C. Clasen, J. Eggers, M. A. Fontelos, J. Li, G. H. Mckinley, The beads-on-string structure of viscoelastic threads. J. Fluid Mech. 556 , 283–308 (2006).
11 S. L. Anna, G. H. McKinley, Elasto-capillary thinning and breakup of model elastic liquids. J. Rheol. 45 , 115–138 (2001).
12 M. Mahajan, M. Tsige, P. Taylor, C. Rosenblatt, Liquid crystal bridges. Liquid Cryst. 26 , 443–448 (1999).
13 A. G. Cheong, A. D. Rey, P. T. Mather, Capillary instabilities in thin nematic liquid crystalline fibers. Phys. Rev. E 64 , 041701 (2001).
14 A. G. Cheong, A. D. Rey, Temperature effects on capillary instabilities in a thin nematic liquid crystalline fiber embedded in a viscous matrix. Euro. Phys. J. E - Soft Matter 9 , 171–193 (2002).
15 M. Gharbia, A. Gharbi, M. Cagnon, G. Durand, Capillary oscillations and instabilities of discotic liquid crystal threads. J. Phys. 51 , 1355–1365 (1990).
16 E. Fontes, P. Heiney, W. de Jeu, Liquid-crystalline and helical order in a discotic mesophase. Phys. Rev. Lett. 61 , 1202–1205 (1988).10038728
17 C. R. Safinya, N. A. Clark, K. S. Liang, W. A. Varady, L. Y. Chiang, Synchrotron X-ray scattering study of freely suspended discotic strands. Mol. Cryst. Liquid Cryst. 123 , 205–216 (1985).
18 V. G. Kamenskii, E. I. Kats, Stability of filaments of discotic liquid crystals. Soviet J. Exp. Theor. Phys. Lett. 37 , 261 (1983).
19 D. H. van Winkle, N. A. Clark, Freely suspended strands of tilted columnar liquid-crystal phases: One-dimensional nematics with orientational jumps. Phys. Rev. Lett. 48 , 1407–1410 (1982).
20 M. H. Kwok, C. A. Bohannon, R. Li, B. Zhao, L. Zhu, Ordered hexagonal columnar liquid crystalline self-assembly of mesogen-free sulfonylated side-chain chiral polyethers and their high dielectric property. Giant 8 , 100079 (2021).
21 C. Bailey, E. C. Gartland, A. Jákli, Structure and stability of bent core liquid crystal fibers. Phys. Rev. E 75 , 031701 (2007).
22 A. Jákli, D. Krüerke, G. G. Nair, Liquid crystal fibers of bent-core molecules. Phys. Rev. E 67 , 051702 (2003).
23 A. Eremin , Structure and mechanical properties of liquid crystalline filaments. Phys. Rev. E 71 , 031705 (2005).
24 A. Eremin , Pattern-stabilized decorated polar liquid-crystal fibers. Phys. Rev. Lett. 109 , 017801 (2012).23031131
25 M. G. Tamba , A fibre forming smectic twist-bent liquid crystalline phase. RSC Adv. 5 , 11207 (2015).
26 P. G. de Gennes, J. Prost, The Physics of Liquid Crystals (Clarendon Press, 1995).
27 M. Born, Über anisotrope Flüssigkeiten. Theorie der flüssigen Kristalle und des elektrischen Kerr-Effekts in Flüssigkeiten. Sitzungsber. Preuss. Akad Wiss. 30 , 614–650 (1916).
28 B. Park, Y. Kinoshita, H. Takezoe, J. Watanabe, Ferroelectricity in the lyotropic cholesteric phase of poly l-glutamate. Jpn. J. Appl. Phys. 37 , L136 (1998).
29 T. Watanabe , Nematic liquid crystals with polar ordering formed from simple aromatic polyester. Japan. J. Appl. Phys. 35 , L505 (1996).
30 M. Koike , Unusual nematic liquid crystal with polar Cs symmetry formed from aromatic polyesters with head-tail character. Macromolecules 40 , 2524–2531 (2007).
31 H. Nishikawa , A fluid liquid-crystal material with highly polar order. Adv. Mater. 29 , 1702354 (2017).
32 R. J. Mandle, S. J. Cowling, J. W. Goodby, A nematic to nematic transformation exhibited by a rod-like liquid crystal. Phys. Chem. Chem. Phys. 19 , 11429–11435 (2017).28422219
33 A. Mertelj , Splay nematic phase. Phys. Rev. X 8 , 041025 (2018).
34 O. Lavrentovich, Ferroelectric nematic liquid crystal, a century in waiting - commentary. Proc. Natl. Acad. Sci. U.S.A. 117 , 14629–14631 (2020).32541021
35 X. Chen , First-principles experimental demonstration of ferroelectricity in a thermotropic nematic liquid crystal: Polar domains and striking electro-optics. Proc. Natl. Acad. Sci. U.S.A. 117 , 14021–14031 (2020).32522878
36 R. J. Mandle, N. Sebastián, J. Martinez-Perdiguero, A. Mertelj, On the molecular origins of the ferroelectric splay nematic phase. Nat. Commun. 12 , 4962 (2021).34400645
37 S. Brown , Multiple polar and non-polar nematic phases. Chem. Phys. Chem 22 , 2506–2510 (2021).34623724
38 N. Sebastián, R. J. Mandle, A. Petelin, A. Eremin, A. Mertelj, Electrooptics of mm-scale polar domains in the ferroelectric nematic phase. Liq. Cryst. 48 , 2055–2071 (2021).
39 N. Sebastián , Ferroelectric-ferroelastic phase transition in a nematic liquid crystal. Phys. Rev. Lett. 124 , 037801 (2020).32031856
40 E. Zavvou, M. Klasen-Memmer, A. Manabe, M. Bremer, A. Eremin, Polarisation-driven magneto-optical and nonlinear-optical behaviour of a room-temperature ferroelectric nematic phase. Soft Matter 18 , 8804–8812 (2022).36354279
41 H. Pleiner, H. R. Brand, Spontaneous splay phases in polar nematic liquid crystals. Europhys. Lett. 9 , 243–249 (1989).
42 S. Nishimura , Lowering of electrostatic actuator driving voltage and increasing generated force using spontaneous polarization of ferroelectric nematic liquid crystals. Adv. Phys. Res. 1 , 2200017 (2022).
43 A. Manabe, M. Bremer, M. Kraska, Ferroelectric nematic phase at and below room temperature. Liq. Cryst. 48 , 1–8 (2021).
44 M. Seul, D. Andelman, Domain Shapes and Patterns: The Phenomenology of Modulated Phases. Science 267 , 476–483 (1995).17788780
45 X. Chen , Observation of a uniaxial ferroelectric smectic A phase. Proc. Natl. Acad. Sci. U.S.A. 119 , e2210062119 (2022).36375062
46 A. Eremin , Labyrinthine instability in freely suspended films of a polarization-modulated smectic phase. Phys. Rev. E 88 , 062512 (2013).
47 R. Barboza , Explosive electrostatic instability of ferroelectric liquid droplets on ferroelectric solid surfaces. Proc. Natl. Acad. Sci. U.S.A. 119 , e2207858119 (2022).35914148
48 M. T. Máthé , Electric field-induced interfacial instability in a ferroelectric nematic liquid crystal. Sci. Rep. 13 , 6981 (2023).37117269
49 B. Basnet , Soliton walls paired by polar surface interactions in a ferroelectric nematic liquid crystal. Nat. Commun. 13 , 3932 (2022).35798735
50 S. Marni, G. Nava, R. Barboza, T. G. Bellini, L. Lucchetti, Walking ferroelectric liquid droplets with light. Adv. Mater. 35 , 2212067 (2023).
51 C. L. Folcia, J. Ortega, R. Vidal, T. Sierra, J. Etxebarria, The ferroelectric nematic phase: An optimum liquid crystal candidate for nonlinear optics. Liq. Cryst. 1–8 (2022).
52 H. Nishikawa, F. Araoka, A new class of chiral nematic phase with helical polar order. Adv. Mater. 33 , 2101305 (2021).
53 J. Li , Development of ferroelectric nematic fluids with giant-epsilon dielectricity and nonlinear optical properties. Sci. Adv. 7 , eabf5047 (2021).33883139
54 P. Vivek , Determination of SHG deff by Maker fringes studies on unidirectional grown guanidinium chlorochromate single crystal for NLO device applications. J. Opt. 50 , 77–82 (2021).
55 N. Okamoto, Y. Hirano, O. Sugihara, Precise estimation of nonlinear-optical coefficients for anisotropic nonlinear films with C∞V symmetry. J. Opt. Soc. A. B 9 , 2083 (1992).
56 J. Jerphagnon, S. K. Kurtz, Maker fringes: A detailed comparison of theory and experiment for isotropic and uniaxial crystals. J. Appl. Phys. 41 , 1667–1681 (1970).
57 D. B. Bogy, Drop formation in a circular liquid jet. Annu. Rev. Fluid Mech. 11 , 207–228 (1979).
58 J. Eggers, Nonlinear dynamics and breakup of free-surface flows. Rev. Mod. Phys. 69 , 865–930 (1997).
59 F. Caimi , Fluid superscreening and polarization following in confined ferroelectric nematics. Nat. Phys. 19 , 1658–1666 (2023).
60 R. Stannarius, A. Nemes, A. Eremin, Plucking a liquid chord: Mechanical response of a liquid crystal filament. Phys. Rev. E 72 , 020702 (2005).
61 E. I. Kats, Stability of the uniform ferroelectric nematic phase. Phys. Rev. E 103 , 01270 (2021).
62 N. Sebastián , Polarization patterning in ferroelectric nematic liquids via flexoelectric coupling. Nat. Commun. 14 , 3029 (2023).37230977
63 A. Eremin, Fluid Fibers in Ferroelectric Liquids. Repository for Research Data and Publications of OVGU. 10.24352/ub.ovgu-2024-049. Deposited 7 March 2024.
