*Review* **Recent Progresses on Experimental Investigations of Topological and Dissipative Solitons in Liquid Crystals**

**Yuan Shen and Ingo Dierking \***

Department of Physics and Astronomy, School of Natural Sciences, University of Manchester, Oxford Road, Manchester M13 9PL, UK; yuan.shen@postgrad.manchester.ac.uk

**\*** Correspondence: ingo.dierking@manchester.ac.uk

**Abstract:** Solitons in liquid crystals have received increasing attention due to their importance in fundamental physical science and potential applications in various fields. The study of solitons in liquid crystals has been carried out for over five decades with various kinds of solitons being reported. Recently, a number of new types of solitons have been observed, among which, many of them exhibit intriguing dynamic behaviors. In this paper, we briefly review the recent progresses on experimental investigations of solitons in liquid crystals.

**Keywords:** liquid crystal; soliton; toron; skyrmion; nematic; cholesteric; smectic; micro-cargo transport; dissipative dynamics

#### **1. Introduction**

Solitons are self-sustained localized packets of waves in nonlinear media that propagate without changing shape. They are found everywhere in our daily life from nerve pluses in our bodies to eyes of storms in the atmosphere and even density waves in galaxies. They were first observed as water waves in a shallow canal by a Scottish engineer John Scott Russell in 1834 [1], which initiated the theoretical work of Rayleigh and Boussinesq and eventually led to the well-known KdV (Korteweg, de Vries) equation which has been broadly used as an approximate description of solitary waves [2]. However, the significance of soliton was not widely appreciated until 1965 when the word "soliton" was coined by Zabusky and Kruskal [3]. Nowadays, solitons have appeared in every branch of physics, such as nonlinear photonics [4], Bose-Einstein condensates [5], superconductors [6], and magnetic materials [7], just to name a few. Generally, solitons appear as self-organized localized waves that preserve their identities after pairwise collisions [8]. This ideal nonlinear property of solitons may enable distortion-free long-distance transport of matter or information and thus makes them considerably attractive to both fundamental research and technological applications [9–11].

Liquid crystals (LCs) are self-organized anisotropic fluids that are thermodynamically intermediate between the isotropic liquid and the crystalline solid, exhibiting the fluidity of liquids as well as the order of crystals [12,13]. Generally, LCs consist of anisotropic building blocks with rod- or disc-like shapes, which spontaneously orient in a specific direction on average, called director, **n**. As a typical nonlinear material, LCs have been broadly used as an ideal testbed for studying solitons, in which different kinds of solitons have been generated in the past five decades.

In this review, we first give a brief overview of the early works of solitons in LCs (Section 2), which is followed by a short discussion of investigations on nematicons (Section 3) and a discussion of recent progress in studies of topological solitons in chiral nematics (Section 4). The article then continues by overviewing the investigations of dynamic particle-like dissipative solitons, "director bullets" or "directrons", which were first reported by Brand et al. in 1997 but did not receive much attention until recently (Section 5). This review is mainly focused on the recent experimental investigations on

**Citation:** Shen, Y.; Dierking, I. Recent Progresses on Experimental Investigations of Topological and Dissipative Solitons in Liquid Crystals. *Crystals* **2022**, *12*, 94. https://doi.org/10.3390/ cryst12010094

Academic Editor: Borislav Angelov

Received: 15 December 2021 Accepted: 8 January 2022 Published: 11 January 2022

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solitons in liquid crystals, readers who are interested in this topic can find more detailed early experimental and theoretical investigations in the excellent book edited by Lam and Prost [14].

#### **2. Early Works**

The study of solitons in LCs was started in 1968 by Wolfgang Helfrich [15]. He theoretically modelled alignment inversion walls as static solitons in an infinite sample of nematic order. By applying a magnetic field, **H**, depending on the assumed orientation of the director at infinity, there are three types of possible walls, i.e., twist wall, splay-bend wall parallel to the applied field, and the splay-bend wall perpendicular to the field, which are analogous to the Bloch and Neel walls in ferromagnetics (Figure 1). Such a model was later improved by de Gennes who studied the boundary effects of the substrate and the movement of the walls [16].

**Figure 1.** Schematic diagrams of different alignment inversion walls. (**a**) Twist wall. (**b**) Splay-bend wall parallel to the magnetic field. (**c**) Splay-bend wall vertical to the magnetic field. Reprinted with permission from Ref. [15]. Copyright 1968 American Physical Society.

The first experimental investigation of these inversion walls was probably reported by Leger in 1972 [17]. In this work, a 180◦ twist wall was generated at the top free surface of a nematic droplet by rotating the applied magnetic field by 180◦ (Figure 2a,b). The migration time of the twist walls was also measured. However, instead of working with a free surface geometry, the measurements were carried out in nematic droplets sandwiched between two rubbed glass plates due to the difficulty of measuring the sample thickness. In this case, the twist walls were generated near each glass plate and then moved toward the midplane of the sample. Because the pair had opposite twists, they annihilated each other once they met. The measured migration time was in great agreement with theoretical prediction [16]. However, because the observation was from the top of the sample, one could not really see the propagation process of the twist walls in the experiment.

In the same year, a different type of wall was investigated theoretically by Brochard [18] and experimentally by Leger [19,20]. These walls separate domains in which LC molecules rotate in two different directions, i.e., the so-called reverse tilt domains [21]. They can be generated by increasing the applied magnetic field above the Freedericksz transition (Figure 2c,d). The walls can either form in a straight line state where they are stable and static or in a closed loop state where they continuously shrink inward and eventually annihilate to minimize the free energy. The authors investigated the static structure and the dynamic behavior of the walls. Unlike the twist walls mentioned above, the walls discussed here move in the plane of the LC cell and can be observed directly.

**Figure 2.** Typical image of the free surface of a MBBA droplet within which a twist wall was generated at (**a**) H = 3000 G and (**b**) H = 6000 G, where H represents the magnitude of the magnetic field. Reprinted with permission from Ref. [17]. Copyright 1972 Elsevier. (**c**) Schematic distortion of the director structure of different walls. (**d**) The microscopy of the walls corresponding to (**c**). Reprinted with permission from Ref. [19]. Copyright 1972 Elsevier.

The interactions between flows and director field in a nematic LC can introduce a nonlinear term in the director equation of motion. Thus, it is possible to induce solitons by shearing nematics without applying any external field [14]. In 1976, Cladis and Torza reported propagating solitary wave instabilities of a nematic in a Couette flow field [22]. They found that for small shears, a "tumbling" instability was observed, which was similar to the appearance of a solitary wave in a long torsion bar to which is attached a dense array of pendulums. By slightly increasing the shear rate, a cellular flow instability was induced (Figure 3a). Further increase of the shear rate, led to a dense mass of disclinations being formed, which aligned with their long axis parallel to the flow field. At very large shear rates, Taylor vortices were generated. In 1982, Zhu reported a soliton-like director wave in a nematic by mechanical shearing [23]. In his work, the nematic film was confined in a home-made cell with an exciter at the entrance. The nematic was homeotropically aligned. By moving the exciter, propagating director waves were observed through polarized white light as black lines, which travelled through the nematic bulk (Figure 3b). It should be noted that these solitons were first theoretically predicted and explained by Lin et al. [24–26]. Later, in 1987, C. Q. Shu et al. reported the generation of two-dimensional (2D) axisymmetric propagating solitons in a radial Poiseuille flow of homeotropic nematic LCs [27]. These solitons appear as dark rings in polarized white light and are large enough to be observed directly by the naked eye (Figure 3c). They are generated by periodically pressing the rim of the cell and can move through the nematic bulk at a constant speed with their shape remaining unchanged.

Convection as one of the simplest examples of hydrodynamic instability induces a rich variety of nonlinear phenomena in a fluid. In conventional Rayleigh-Benard and Taylor experiments, convective patterns emerge once the temperature gradient across the fluid layer, and the relative velocity field of the rotating planes, exceed some specific threshold (the Rayleigh number *Ra* and *Ta*). Generally, the convective pattern is composed of spatially periodic rolls (Williams rolls) with translational invariance, whose periodicity is of order of the thickness of the fluid layer. In LCs, the convective pattern exhibits similar characteristic features and can be generated by applying an electric field to a properly aligned nematic. It was reported that the nonlinear coupling between the convective flow and the director field may lead to the generation of a number of solitonic structures. In 1979, Ribotta measured the penetration length of a vortex into a subcritical region by using the electro-convective instability in a nematic LC [28]. In his experiment, the electrode on one plate is divided into two parts separated by a small gap of about 5 μm. Convective patterns are generated in one region (region 1) by applying an electric field. An electric field with the same frequency but a relatively low amplitude is applied in the other region (region 2), in which no convective pattern is generated. One then observes that small portions of individual rolls are "emitted" from the gap, and propagate into region 2 with a uniform group velocity (Figure 4a). The rolls behave like solitary waves with their shape and amplitude remaining constant during the motion. A different kind of soliton was later reported by Lowe and Gollub in 1985 in a convecting nematic subjected to spatially periodic forcing [29]. The solitons (also called domain walls or discommensurations) are regions of local compression of convective rolls (Figure 4b) and can be well described by the solutions to the Sine-Gordon equation. Generally, the convective pattern is stationary and spatially homogeneous with its velocity field and orientational field being time independent. However, in 1988, Joets and Ribotta reported a time-dependent localized state of electro-convection in a nematic [30]. These spatially localized domains exhibit elliptical shape and distribute randomly throughout the sample. Inside the domains, a periodic structure of Williams Rolls translates uniformly (Figure 4c). The velocity of the rolls has the same amplitude in different domains, but its sign changes randomly from one domain to another. The domains themselves do not show uniform translational motion, instead they fluctuate around an average location. Both the velocity of the rolls and the shape of the domains can be varied by tuning the applied electric field.

**Figure 3.** (**a**) Cellular flow in a nematic LC. Reprinted with permission from Ref. [22]. Copyright 1976 Elsevier. (**b**) Propagation process of the director wave. Reprinted with permission from Ref. [23]. Copyright 1982 American Physical Society. (**c**) Propagation of the 2D axisymmetric soliton. Reprinted with permission from Ref. [27]. Copyright 1987 Taylor & Francis.

**Figure 4.** (**a**) Portions of convective rolls "emitted" from region 1 to region 2. Reprinted with permission from Ref. [28]. Copyright 1979 American Physical Society. (**b**) Micrograph of a quasiperiodic convective structure. The solitons (indicated by arrows) are regions of compression of the rolls. Reprinted with permission from Ref. [29]. Copyright 1985 American Physical Society. (**c**) Micrograph of localized domains of travelling convective rolls. Reprinted with permission from Ref. [31]. Copyright 1988 American Physical Society.

#### **3. Nematicons**

Nematicons are self-focused light beams (spatial optical solitons) that propagate in nematic LCs. The beginning may date back to the early works by Braun et al. in which optical beams of complex structures, such as the formation of focal light spots, the onset of transverse beam undulations, and the development of multiple beam filaments, are realized by interacting a low-power laser beam with a nematic LC [32,33]. Compared to most materials, the nonlinear coefficient of nematic LC is extremely large (106 to 10<sup>10</sup> times greater than that of typical optical materials such as CS2), making it an ideal system for investigating spatial optical solitons [32]. As shown in Figure 5, a linearly polarized beam propagates along the *z*-axis and enters a nematic cell. The polarization of the beam is parallel to the *y*-axis. The nematic LC within the cell is homogeneously aligned with its director, **n**, being parallel to the cell substrates but making an angle *θ* with respect to the wave vector (**k**) of the beam. The extraordinary waves of the beam whose electric field **E** is in the **nk** plane propagate along the Poynting vector **S** which is deviated from **k** at the angle *δ* [34]. The light induces electric dipoles in the LC molecules which interact with the electric field and produce a torque Γ = *ε*0Δ*ε*(**n** · **E**)(**n** × **E**), where *ε*<sup>0</sup> is the dielectric susceptibility of vacuum, Δ*ε* = *n*<sup>2</sup> *<sup>e</sup>* − *<sup>n</sup>*<sup>2</sup> *<sup>o</sup>* is the optical anisotropy, *ne* and *no* are the extraordinary and ordinary refractive indices, respectively. For nematic LCs with *ne* > *no*, the torque reorients the director and increases the angle *θ*, leading to the increase of the extraordinary refractive index *ne*,*<sup>θ</sup>* = √ *neno n*2 *<sup>o</sup>* ·sin2(*θ*)+*n*<sup>2</sup> *<sup>e</sup>* ·cos2(*θ*) . Such an increase of the refractive index focuses the beam, and leads to the formation of a nematicon which propagates along **S**. The study of nematicons has attracted a great deal of interest since the beginning of the 21st century due to its promising applications in nonlinear optics and photonics [35]. Recently, different kinds of nematicons, such as vortex nematicons [36–38], have been reported. Since this review is mainly concentrated on topological and dissipative LC solitons, we refer the readers who are interested in nematicons, to a book and several reviews published by Assanto, et al., recommended here [35,39–41].

**Figure 5.** (**a**) Schematic of the formation of a nematicon in a planar nematic LC cell. (**b**) Photographs of a propagating ordinary light beam (top) and a nematicon (bottom). Reprinted with permission from Ref. [34]. Copyright 2019 Optical Society of America.

#### **4. Topological Solitons in Chiral Nematics**

Topological solitons are continuous but topologically nontrivial field configurations embedded in uniform physical fields that behave like particles and cannot be transformed into a uniform state through smooth deformations [42]. They were probably first proposed by the great mathematician Carl Friedrich Gauss, who envisaged that localized knots of physical fields, such as electric or magnetic fields, could behave like particles [43]. Kelvin and Tait noticed the importance of this concept in physics and proposed one of the early models of atoms, in which they tried to explain the diversity of chemical elements as different knotted vortices [43]. Based on these theories, Hopf proposed the celebrated mathematical Hopf fibration [44], which was later applied to three-dimensional physical fields by Finkelstein [45] and led to the increasing interest of topological solitons to mathematicians and physicists. Nowadays, topological solitons have been investigated in many branches of physics such as instantons in quantum theory [46,47], vortices in superconductors [48], rotons in Bose-Einstein condensates [49], and Skyrme solitons in particle physics [50], etc. The field of topological solitons in LCs started about 50 years ago with the discussion of static linear and planar solitons, which are actually the inversion walls discussed above. In this section, the attention will be mainly focused on the 3D particle-like topological solitons, i.e., the so-called "baby skyrmions", in chiral nematic LCs (CNLCs).

In CNLCs, topological solitons such as 2D merons and skyrmions (low-dimensional analogs of Skyrme solitons) can be generated and have recently received great attention. The molecules of a CNLC form a "layered" structure. In each molecular layer, the director, **n**, aligns in a specific direction. The director of different layers twists at a constant rate along a helical axis which is perpendicular to the layers. The distance over which **n** rotates by an angle of 2π is called the pitch, *p*. Generally, by applying an electric field to a CNLC or sandwiching it between surfaces of homeotropic anchoring, the helical superstructure of the CNLC will be deformed, leading to the formation of string-like cholesteric fingers [51] and/or nonsingular solitonic field configurations [52]. In 1974, Haas and Adams [52] reported the formation of densely packed particle-like director field configurations, called "spherulites" by the authors, which are now known as "skyrmions", following Skyrme who developed a 3D soliton model of nucleons [50]. In the experiment, the CNLC is confined in a cell with homeotropic anchoring. By applying an electric field to the sample, electro-hydrodynamic effects are induced, and the spherulites can be generated after removing the electric field. Almost at the same time, similar results were also reported by Kawachi et al. [53]. However, in their publication, the spherulites were called "bubble domains". These spherulites or bubble domains soon attracted a great deal of interest and fueled an explosive growth of studies in the next few years [54–62]. In 2009, with the help of laser tweezers, different kinds of spherulites or bubble domains were optically generated at will at a selected place of a homeotropically aligned CNLC by Smalyukh, et al. [63]. By characterizing and simulating the 3D director structure of these solitons, the authors recognized that these are low-dimensional analogs of Skyrme solitons. The solitons are composed of a double-twist cylinder closed on itself in the form of a torus and coupled to the surrounding uniform field by point or line topological defects and are called "torons" (Figure 6a). The authors successfully demonstrated the structure and stability of the torons by the basic field theory of elastic director deformations and obtained the equilibrium field configuration and elastic energy of torons through numerical simulations. Later, the same authors reported the generation of 2D reconfigurable photonic structures composed of ensembles of torons [64–66]. In 2013, Chen et al. reported the generation of Hopf fibration (Figure 6c) in a CNLC by manipulating the two point defects of torons [67]. They demonstrate the relationship between Hopf fibration and torons through a topological visualization technique derived from the Pontryagin-Thom construction. In the following years, a variety of different kinds of skyrmionic solitons, such as half-skyrmions, twistion, skyrmion bags, skyrmion spin ice, skyrmion-dressed colloidal particles, and more complicated structures composed of torons, hopfions, and various

disclinations, were realized and reported by different groups [68–76]. The self-assembly of torons (Figure 6d) [77–79] and hopfions in ferromagnetic LCs were later realized by Ackerman et al. [80]. Furthermore, the continuous transformation of 3D Hopf solitons [81] and the generation of 3D knots dubbed "heliknotons" (Figure 6e,f) [82] in CNLCs were reported by Tai et al. Due to the continuous twist of the director field within topological solitons, they can be used as optical devices for controlling and modulating the propagation of light [83–86]. For instance, Varanytsia et al. reported that the surface-assisted assembly of a two-dimensional toron array could be utilized as a spatial light modulator to control the light transmission and scattering [83,87]. Recently, Hess et al. showed that the skyrmionic solitons can act as lenses to steer laser beams [84]. Papic et al. showed that the topological LC solitons inserted in a Fabry-Perot microcavity can be used as a tunable microlaser to generate structured laser beams. The structure of the emitted light could be easily controlled by tuning the topology and geometry of the solitons [88]. In addition, Mai et al. recently showed that topological solitons, such as heliknotons, can be used as micro-templates for spatial reorganization of nanoparticles [89].

**Figure 6.** (**a**) Configuration of a toron. (**b**) Polarizing microscopy texture of different defectproliferated torons. Reprinted with permission from Ref. [63]. Copyright 2010 Nature. (**c**) Flow lines and preimage surfaces of Hopf fibration. Reprinted with permission from Ref. [67]. Copyright 2013 American Physical Society. (**d**) Self-assembly of skyrmions. Reprinted with permission from Ref. [77]. Copyright 2015 Nature. (**e**) Knotted co-located half-integer vortex lines in a heliknoton. (**f**) Polarizing microscopy texture of heliknotons. Reprinted with permission from Ref. [82]. Copyright 2019 Science.

In most investigations, these topological solitons are viewed as static field configurations in LCs. However, it is found that they can be driven into motion by applying electric fields. In 2017, Ackerman et al. reported an electrically driven squirming motion of baby skyrmions in a chiral nematic [90]. By applying a modulated electric field, the skyrmions behave like defects in active matter and move in directions orthogonal to the electric field. Such a motion stems from the non-reciprocal rotational dynamics of LC director fields. During motion, the periodic relaxation and tightening of the twisted region make the skyrmions expand, contract, and morph, resembling squirming motion (Figure 7a). Both the direction and speed of the moving solitons can be controlled by tuning the applied electric field. Such a controllable motion of skyrmions may enable versatile applications such as micro-cargo transport [91]. In 2019, Sohn et al. reported an electrically driven collective motion of skyrmions, in which thousands to millions of skyrmions started from random orientations and motions, but then synchronized their motions and developed polar ordering within seconds (Figure 7b) [92]. They also showed that such a collective motion could even be enriched and guided by light (Figure 7c) [93] and could be used as a model for studying the dynamics of topological defects and grain boundaries in crystalline solid systems [94]. As we mentioned above, the topological defects and solitons are usually generated through local relief of geometric frustration of the helical structure of CNLCs.

As a result, in most works, the topological defects are generated in CNLCs confined by homeotropic anchoring conditions. Very recently, Shen and Dierking reported the creation of 3D topological solitons, i.e., the torons, in a CNLC which is confined in cells of homogeneous anchoring by applying electric fields [95]. In that work, the authors demonstrate the transformation between the cholesteric fingers and the solitons and the formation of "skyrmion bags" with a tunable topological degree. The solitons exhibit different static geometric textures and dynamic behaviors by changing the pitch of the CNLC system (Figure 7d). They undergo anomalous diffusion at equilibrium and performed directional motion driven by electric fields. The solitons could even form aggregates of tunable shape, anisotropy, and fractal dimension through inelastic collisions with each other.

**Figure 7.** (**a**) Polarizing microscopy texture of a squirming skyrmion. Reprinted with permission from Ref. [90]. Copyright 2017 Nature. (**b**) Temporal evolution of skyrmion velocity order. Reprinted with permission from Ref. [92]. Copyright 2019 Nature. (**c**) Motion of skyrmions guided by light. Reprinted with permission from Ref. [93]. Copyright 2020 Optical Society of America. (**d**) Different dynamic behaviors of torons. Reprinted with permission from Ref. [95]. Copyright 2021 American Physical Society.

In this section, the recent experimental studies on 3D particle-like skyrmionic topological solitons were briefly introduced. For readers who are interested in more details concerning this topic, an excellent review written by Smalyukh is strongly recommended [96].

#### **5. Dynamic Dissipative Solitons in Liquid Crystals**

Dissipative solitons are stable localized solitary deviations of a state variable from an otherwise homogeneous stable stationary background distribution. They are generally powered by an external driver and vanish below a finite strength of the driver [97]. Experimentally, dissipative solitons were generated in the form of electric current filaments in a 2D planar gas-discharge system [98]. In LCs, different kinds of dissipative solitons have been generated and reported recently [99–105].

In 2018, Li et al. reported the formation of 3D dissipative solitons in an electrically driven nematic, which were called "director bullets" [99] or "directrons" [100] by the authors (Figure 8). These solitons were first reported by Brand et al. in 1997, and were called "butterflies", but did not receive much attention at that time. The directrons (we will refer to them as "directrons" to distinguish them from other solitons) are self-confined

localized director deformations. While the nematic aligns homogeneously outside the directrons, the director field is distorted and oscillates with the frequency of the applied AC electric field within the directrons. Such an oscillation breaks the fore-aft symmetry of the structure of the directrons and leads to the rapid propagation perpendicular to the alignment direction. The directrons can move with speeds as large as 1000 μm s−<sup>1</sup> through the homogeneous nematic bulk over a macroscopic distance thousands of times larger than their size. They survive collisions and pass through each other without losing their identities. Unlike the topological solitons, the directrons are topologically equivalent to a uniform state and disappear right after switching off the applied electric field. The nematic media in which the directrons were generated by Li et al., 4 -butyl-4-heptyl-bicyclohexyl-4-carbonitrile (CCN-47), is of the (−,−) type, which means that both the dielectric and conductivity anisotropies are negative, i.e., <sup>Δ</sup>*<sup>ε</sup>* = *<sup>ε</sup>* − *<sup>ε</sup>*<sup>⊥</sup> < 0 and <sup>Δ</sup>*<sup>σ</sup>* = *<sup>σ</sup>* − *<sup>σ</sup>*<sup>⊥</sup> < 0, respectively. The basic mechanism of many electro-hydrodynamic instabilities in nematics of the (−,+) type is now quite well understood and can be explained by the well-known "Carr-Helfrich" model, in which a subtle balance between the dielectric torque that stabilizes the initial planar director field and an anomalous conductive torque induced by space charges that breaks the planar state is reached [106,107]. However, the model cannot be used to explain the generation of the directrons observed in (−,−) nematics, in which case both the dielectric and conductivity torques can only stabilize the planar state. Instead, due to the equality of the frequency of the directron oscillation and that of the applied electric field, the main reason of the excitation of the directrons is attributed to the flexoelectric polarization [105].

**Figure 8.** Director bullets in a planar nematic cell. (**a**) Cell scheme. (**b**) Transmitted light intensity map and director distortions in the *xy* plane within a single bullet. (**c**–**e**) Polarizing microscopy of the director bullets at varied voltages. (**f**) Polarizing microscopy of the electro-hydrodynamic pattern. Scale bar 200 μm. Reprinted with permission from Ref. [99]. Copyright 2018 Nature.

The study carried out by Aya and Araoka showed that similar directrons can also be generated in nematics of the (−,+) type [101]. In order to systemically examine the influences of the material parameters on the generation and dynamics of the directrons, the authors used a mixture of two different nematics which are of the (−,−) type and the (+,+) type, respectively. By altering the concentrations of the two nematics, a continuous transition of the dielectric and conductivity anisotropies of the mixture could be realized, leading to the formation of different kinds of electro-hydrodynamic patterns. The authors found that the conductivity is vital in determining the stability of the directrons. These directrons can only exist in the limited range of moderate conductivity, 0.8 × <sup>10</sup>−<sup>8</sup> <sup>&</sup>lt; *<sup>σ</sup>* <sup>&</sup>lt; <sup>4</sup> × <sup>10</sup>−<sup>8</sup> <sup>Ω</sup>−1m−1. The unifying feature of the generation of the directrons in nematics of the (−,−) type and the (−,+) type is that the conductivity of the nematic host is relatively low compared to those typically used in exploring conventional Carr-Helfrich electrohydrodynamic phenomena (~10−<sup>7</sup> Ω−1m−1) [108]. Li et al. reported that the conductivity of the nematic host (CCN-47) for producing dissipative solitons was about (0.5–0.6) × 10−<sup>8</sup> Ω−1m−<sup>1</sup> [99], and the conductivity of the nematic (ZLI-2806) used in the experiments by Shen and Dierking was about (0.6–1.9) × <sup>10</sup>−<sup>8</sup> <sup>Ω</sup>−1m−<sup>1</sup> [102].

On the other hand, a recent study carried out by Shen and Dierking showed that similar directrons could even be produced in nematics of the (+,+) type (Figure 9) [103], which is unexpected by the "standard model" [109]. In their experiment, a nematic (5CB) with both positive dielectric and conductivity anisotropies was confined in a cell with homogeneous alignment. The alignment of the cells was induced through the photoalignment technique instead of the conventional rubbing method, the former providing a relatively weak azimuthal anchoring. The directrons are generated after applying an AC electric field with a relatively low frequency to the sample. The dynamic behavior of the directrons is similar to that reported by Li et al. [87,88] and Aya et al. [89]. Directrons can propagate either parallel or perpendicular to the alignment direction which can be switched by tuning the frequency and amplitude of the applied electric field [99–101]. The directrons behave like waves when they pass through each other without losing their identities after collisions. Electro-hydrodynamic instabilities in LCs have been investigated for decades. Early studies mainly focus on the electro-convection effects in nematics with opposite signs of anisotropies, e.g., nematics of the (−,+) type, in which electroconvection rolls, such as the well-known "Williams domains" [110] and "chevrons" [28], are observed. In most of the cases, the destabilization can be explained by the charge separation mechanism introduced by Carr [106] and Helfrich [107], which was later extended to a 3D theory, i.e., the "standard model" [109]. On the other hand, most nonstandard electroconvection phenomena observed in nematics of the (−,−) type can be explained by adding flexoelectricity effects to the standard model. The standard model predicts no electrohydrodynamic instability in nematics of the (+,+) type. However, complicated electroconvection patterns, such as fingerprint textures [111], Maltese crosses [111], and cellular patterns [112], were reported in nematic cells with homeotropic alignment. Different explanations, including isotropic ionic flows [111,113,114], charge injection (known as Felici-Benard mechanism) [115,116], flexoelectricity, and surface-polarization effects [117,118], etc., were proposed to account for the origin of the instabilities. However, a rigorous explanation is still to be found. In the case of homogeneous alignment, only stationary Williams domains were observed in nematics with small values of the dielectric anisotropy (0 < Δ*ε* < 0.4). For nematics with large dielectric anisotropy (just as the situation in ref [103]), electro-hydrodynamic instabilities are usually suppressed by the Freedericksz transition and thus are not expected [119]. The formation of the directrons in 5CB reported by Shen and Dierking is attributed to the special conditions of their experimental setup, i.e., a relatively high ion concentration of the nematic host and a relatively weak azimuthal anchoring of the cells. Both of these factors lead to the strong nonlinear coupling between the isotropic ionic flows and the director field which induces the directrons [103].

**Figure 9.** The structure of the dissipative solitons in 5CB. (**a**) Time series of polarizing micrographs of a soliton modulated by an AC electric field. Scale bar 20 μm. (**b**) The schematic structure of a soliton. **m** represents the alignment direction. *ϕ*m and *θ*m represent the azimuthal angle and the polar angle of the local mid-layer director. (**c**) Transmitted light intensity maps and the corresponding mid-layer director fields (black dashed lines) in the *xy* plane within solitons. **v** represents the velocity of the soliton. The color bar shows a linear scale of transmitted light intensity. Insets are the corresponding POM micrographs, scale bar 10 μm. Red squares 1, 2, and 3 are corresponding to the ones in (**b**). Reprinted with permission from Ref. [103]. Copyright 2020 Royal Society of Chemistry.

The formation of the dynamic directrons is not the privilege of nematics only. Shen and Dierking showed that the directrons can also form in chiral nematics of the (−,+) type and the (+,+) type, respectively [102,103]. In the experiments, the chiral nematic hosts are prepared by doping a chiral dopant into achiral nematics, and the mixtures are filled into LC cells with homogeneous alignment conditions. An AC electric field is applied parallel to the helical axis of the chiral nematics, and the directrons emerge as the amplitude of the electric field increases above some frequency-dependent thresholds. The dynamic behavior of the directrons was investigated and compared to those in achiral nematics. It was found that the directrons in the achiral nematics show a "butterfly-like" structure (Figure 10a), but the ones in chiral nematics exhibit a "bullet-like" structure (Figure 10b). In both cases (chiral and achiral nematics), the directrons move either parallel or perpendicular to the alignment direction, a behavior which is dependent on the applied electric field. Interestingly, the directrons in achiral nematics behave like waves in that they collide and pass through each other as reported by Li et al. [99]. The directrons in chiral nematics behave similar to a wave-particle dualism in so far that they either pass through each other without losing identity (solitary wave) (Figure 10c), or collide with each other and undergo reflection (hard particle) (Figure 10d). The authors also showed that the motion of the directrons can be controlled by the alignment. As shown in Figure 10e, the LC cell is divided into three regions with different alignment directions by using the photo-alignment technique. Tuning the applied electric field, the directrons either move parallel or perpendicular to the alignment direction in each individual region. However, once the directrons move across the boundaries of different regions, they change their directions continuously to fit the alignment condition. The directrons can therefore be used as vehicles for micro-cargo transport [102,103]. As shown in Figure 10f, a directron is induced around a dust particle once the electric field is applied. It then carries and translates the particle by moving it through the nematic bulk. Similar phenomena were also reported by Li et al. which was termed "directron-induced liquid crystal-enabled electrophoresis" [120].

**Figure 10.** Dissipative solitons in chiral nematics. Solitons in achiral (**a**) and chiral (**b**) nematics. (**c**) Two solitons pass through each other. (**d**) Two solitons collide and reflect into opposite directions. (**e**) Motion of solitons in a cell divided into three regions with different alignment directions. (**f**) Microcargo transport by a soliton. Reprinted with permission from Ref. [102]. Copyright 2020 Nature.

So far, most dynamic solitons have been reported only in nematics. However, recently, it was shown that particle-like dynamic dissipative solitons can be formed in LCs of the fluid smectic A phase [104]. A smectic A phase is characterized by the formation of a layered structure of elongated molecules with orientational and 1D positional order. Within each layer, the LC molecules are orientated perpendicular to the layer, but their molecular centers of mass are distributed randomly without any further in-plane positional order [121]. The smectic phase is usually characterized by the remarkable patterns of singular ellipses, hyperbolas, and parabolas known as the focal conic domains (FCDs). In this work, a smectic LC (8CB) is confined in a LC cell with homogeneous alignment condition. The sample is kept at a temperature slightly below the nematic-smectic phase transition point. The solitons are formed by applying a low-frequency field with a mediate voltage to the so-called "scattering state" of the smectic A phase. The solitons exhibit a swallow-tail like texture under crossed polarizers with a static structure analogous to the parabolic focal conic domains (PFCDs) (Figure 11). They are characterized by an elliptical contour, which is generated as a result of the localization of stress. The contour is composed of the loci of the cusps of smectic layers. Outside the solitons, the equidistant smectic layers align homogeneously perpendicular to the alignment direction. Within the solitons, the transmitted light intensity increases, indicating azimuthal deviations of the director field from the alignment. As a result, the smectic layers continuously deform into curves within the solitons. The curvature of the curves exhibits a maximum at one of the foci of the elliptical contour, where a singular defect line is located and acts as the core of the solitons. The size of the solitons is dependent on the applied voltage, and decreases with increasing voltage. When driven by a low-frequency electric field, the director within the solitons tilts up and down due to the dielectric torque, which leads to a periodic shape transformation of the soliton. The solitons move bidirectionally along the alignment direction with constant speed, which is attributed to the permeation ion flow perpendicular to the smectic layers. The solitons behave like particles when they collide with each other. They can also interact with colloidal micro-particles.

**Figure 11.** Swallow-tail solitons in smectics. (**a**) Polarizing micrographs of the swallow-tail soliton and its corresponding static structure. (**b**) Collision of two swallow-tail solitons. (**c**) Nucleation of swallow-tail solitons on a colloidal micro-particle. Reprinted with permission from Ref. [104]. Copyright 2021 Royal Society of Chemistry.

#### **6. Conclusions**

In summary, we have briefly discussed some important early works of solitons in LCs and the recent progresses made in the investigations of topological solitons and dissipative solitons in LCs. Although recent studies of topological solitons and dissipative solitons have received great attention, many fundamental questions remain unanswered. For instance, the existence of topological solitons with higher dimensions in biaxial liquid crystal systems, a systematic classification of the topological solitons, the stability of the topological and dissipative solitons, the transformation between different topological solitons, the influence of the topological structure on the dynamics and interactions of topological solitons, the formation mechanism of the directrons, the role of ions played in the formation and motion of dissipative solitons, the influence of surface anchoring on the stability, formation and dynamics of the solitons, the effect of chirality on the structure and dynamics of the solitons, the interactions between solitons and colloidal particles, the self-assembly and collective behavior of the solitons, the existence of topological and dissipative soliton in lyotropic and active LC systems, the relation between the solitons in LCs and the solitons in other physical systems, etc. All these questions remain elusive and require further experimental and theoretical investigations to answer.

After over five decades of research, various solitons have been created and described in different liquid crystalline systems. This not only broadens the research and understanding of LCs, but also enhances our understanding of solitons in other physical systems. Furthermore, the solitons in LCs may even lead to novel phenomena, such as emergent collective motion of solitons [92,93], and applications, such as micro-cargo transport [102,103,120], optic processing [84,85], or fast LC displays [14]. We hope this brief review can arouse more researchers' interest in the field of solitons in LC systems.

**Author Contributions:** Y.S. conceived and wrote the manuscript. I.D. contributed through discussing and writing the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** Yuan Shen would like to thank the China Scholarship Council (CSC number: 201806310129).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Ferroelectric Smectic Liquid Crystals as Electrocaloric Materials**

**Peter John Tipping and Helen Frances Gleeson \***

School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK; py13pjt@leeds.ac.uk **\*** Correspondence: h.f.gleeson@leeds.ac.uk

**Abstract:** The 1980s saw the development of ferroelectric chiral smectic C (SmC\*) liquid crystals (FLCs) with a clear focus on their application in fast electro-optic devices. However, as the only known fluid ferroelectric materials, they also have potential in other applications, one of which is in heat-exchange devices based on the electrocaloric effect. In particular, ferroelectric liquid crystals can be both the electrocaloric material and the heat exchanging fluid in an electrocaloric device, significantly simplifying some of the design constraints associated with solid dielectrics. In this paper, we consider the electrocaloric potential of three SmC\* ferroelectric liquid crystal systems, two of which are pure materials that exhibit ferroelectric, antiferroelectric, and intermediate phases and one that was developed as a room-temperature SmC\* material for electro-optic applications. We report the field-induced temperature changes of these selected materials, measured indirectly using the Maxwell method. The maximum induced temperature change determined, 0.37 K, is currently record-breaking for an FLC and is sufficiently large to make these materials interesting candidates for the development for electrocaloric applications. Using the electrocaloric temperature change normalised as a function of electric field strength, as a function of merit, the performances of FLCs are compared with ferroelectric ceramics and polymers.

**Keywords:** ferroelectric materials; smectic liquid crystals; electrocaloric effect

#### **1. Introduction**

The electrocaloric (EC) effect, discovered in 1930, is the induction of a reversible temperature change in a material via the adiabatic application of an electric field [1]. The EC effect has long been regarded as having potential as a cooling technology for cryogenic [2,3] and, more recently, for room temperature applications [4]. Interest has also grown as it is considered to be an environmentally friendly alternative to the ubiquitous vapour compression devices. This is because the typically high efficiency of vapour compression materials is offset by their large global warming impact. The unavoidable leakage of refrigerant results in a large environmental impact over the lifetime of the device [5]. In comparison, electrocaloric materials have a negligible direct impact on global warming as they are not volatile gases. Additionally, with the continuing rise in computing power, there is a growing demand for efficient and compact refrigeration technology in microelectronics. Vapour compression devices that have been proposed [6,7] are cumbersome with significantly reduced efficiency. Thus, there is a need for alternative refrigeration technologies that are efficient on a range of length scales and do not rely on the use of greenhouse gases.

For many years, EC materials have offered only small induced temperature (*T*) changes (Δ*T* ~ 2 K) in response to very large applied voltages (several hundred volts), so the phenomenon was considered to be far from practical applications. However, in 2006 [8], Mischenko et al. reported Δ*T* ~ 12 *K* in the ferroelectric ceramic lead zirconate titanate (PZT) for fields of 48 V μm−<sup>1</sup> at a temperature of 220 ◦C, closely followed by a similarly large EC temperature change in ferroelectric polymers near room temperature [9]. Significant research into solid inorganic ceramics and fluorinated polymers as potential EC materials followed, but there has been only one report of a ferroelectric liquid crystal considered [10], despite some exciting potential advantages, discussed further below [4,11–13]. This paper

**Citation:** Tipping, P.J.; Gleeson, H.F. Ferroelectric Smectic Liquid Crystals as Electrocaloric Materials. *Crystals* **2022**, *12*, 809. https://doi.org/ 10.3390/cryst12060809

Academic Editor: Ingo Dierking

Received: 10 May 2022 Accepted: 4 June 2022 Published: 8 June 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

examines three carefully selected FLCs for their electrocaloric potential, demonstrating both the current state-of-the-art research and offering insight into how to develop this exciting application area.

An EC material works as follows: the entropy of a dielectric material can be considered as being the sum of two contributions, one due to thermal vibrations and phonons and the second from the ordering of dipoles in the material. Upon adiabatic application of an electric field, the dipoles in the dielectric material align with the field, causing a decrease in the dipole entropy. The total entropy of the system is constant in an adiabatic process; therefore, the entropy due to molecular vibrations, and subsequently the temperature, increases. The converse occurs when the field is removed adiabatically, resulting in a temperature decrease. Clearly, for this to work in a device, heat exchange must also occur, and a common method of heat transfer is to pump a heat-exchanging liquid over the EC material and into a heat-exchange unit [14–18]. A major challenge with all electrocaloric device designs is the transfer of heat away from the refrigerated area. There are significant efficiency losses due to imperfect heat transfer between the heat exchange liquid and EC material. Alternative methods for heat exchange that do not use a liquid have been proposed, but there are still engineering challenges [19–22]. Therefore, the potential of using a dielectric liquid, rather than a solid, as the EC material is exciting as it could be pumped away from the refrigerated area. Liquid crystals (LCs) are obvious candidates worthy of serious consideration as novel EC materials; this paper specifically considers the potential of ferroelectric smectic liquid crystals as electrocaloric materials. In particular, we aim to: (i) deduce the electrocaloric performance of known ferroelectric, smectic, liquid crystals; (ii) elucidate the design rules for optimising the electrocaloric performance of new liquid crystal materials; and (iii) consider how their performance compares to solid-state ferroelectric electrocaloric materials (ceramics and polymers).

Most of the (rather few) measurements of the electrocaloric effect in LCs have explored commercially available materials and considered the phenomenon near the isotropic– nematic transition, with some promising results. Direct measurements of the EC effect in 5CB [23] showed an induced temperature change of Δ*T* = 0.36 K for an applied field of 19 V μm−1, while indirect measurements [24] suggest a peak change of Δ*T* = 5.26 K for a field of 90 V μm−1. A temperature change of Δ*T* ~ 1.4 K has also been directly measured in 8CB [25] using a 6 V μm−<sup>1</sup> field. The temperature change induced at the isotropic to SmA transition has been studied using 12CB, with a significant temperature change, Δ*T* ~ 6.5 K, measured near the transition using an 8 V μm−<sup>1</sup> field [25,26]. The results for 12CB suggest that the greatest EC effect is around a phase transition where the applied field can induce a comparatively large change in the order parameter. However, a disadvantage of measurements at an isotropic–LC phase transition is the very narrow temperature range over which the phenomenon can be exploited—typically a few tenths of a degree at most.

It seems an obvious step to consider the potential of ferroelectric liquid crystals as EC materials, given that the largest EC effect measured to date has been in solid ferroelectrics [12,13]. However, we are aware of only one report of the EC effect in ferroelectric liquid crystals, in which two commercial ferroelectric liquid crystal mixtures designed for electro-optic displays were investigated [10], with a peak temperature change Δ*T* = 0.16 K. We selected three materials as follows: one is a well-known commercial material, SCE13, which is rather similar to the materials already studied, allowing us to perform a direct comparison with published data. As is explained in Section 2, a larger electrocaloric effect occurs for materials that exhibit a larger spontaneous polarisation, so we have also selected two pure, smectic, ferroelectric LCs (LC 1 and LC 2) with large values of *PS*, each with different phase sequences. In all cases, we measured the EC effect indirectly using the Maxwell approach [27]. We also normalised the maximum EC absolute temperature change with respect to the applied field as <sup>Δ</sup>*<sup>T</sup>* <sup>Δ</sup>*<sup>E</sup>* , offering a "figure of merit" for the electrocaloric effect, which allows a meaningful comparison of the effect across LCs, polymers, and ferroelectric ceramics. We demonstrate that simple material selection criteria that include the consideration of the *PS* and the phase behaviour allowed us to record the largest EC

temperature change to date for a ferroelectric LC, both in terms of absolute temperature change and normalised with respect to the applied field.

As already stated, in this paper, the electrocaloric temperature change is measured indirectly using the Maxwell approach [27]. An expression for the isothermal entropy change per unit volume, <sup>Δ</sup>*<sup>S</sup> <sup>V</sup>* , as a function of electric field can be derived using the Maxwell relation between the electric field and temperature,

$$\frac{\Delta S}{m^3} = \int\_{E\_1}^{E\_2} \left(\frac{\partial P\_S}{\partial T}\right)\_E dE \tag{1}$$

where *<sup>E</sup>*<sup>1</sup> and *<sup>E</sup>*<sup>2</sup> are the initial and final field strengths and *∂PS ∂T <sup>E</sup>* is the rate of change of spontaneous polarisation with respect to temperature at a constant field strength. Assuming that the initial temperature and the volumetric heat capacity do not vary with the applied field, an estimate of the induced temperature change, Δ*T*, is

$$
\Delta T \cong -\frac{T\_1}{\mathcal{C}\_E(0, T\_1)} \int\_{E\_1}^{E\_2} \left(\frac{\partial P\_S}{\partial T}\right)\_E dE\_\prime \tag{2}
$$

where *T*<sup>1</sup> (K) is the temperature at which the field is applied and *CE*(0, *T*1) (J K−<sup>1</sup> m−3) is the volumetric heat capacity at zero field, measured at *T*1. Equation (2) offers a basis for indirectly measuring the temperature change of an FLC in response to an applied field, provided that both *∂PS ∂T <sup>E</sup>* and *CE*(0, *<sup>T</sup>*1) are known.

#### **2. Materials and Methodology**

The FLC materials chosen for this study were selected to (i) allow us to evaluate the influence of the magnitude of the *PS* and (ii) examine the influence of the phase behaviour at the transition to the FLC phase. SCE13 is a ferroelectric mixture supplied by Merck, with the phase sequence shown in Figure 1. As this mixture was designed for use in electro-optic devices, the phase sequence includes a chiral nematic phase with very large pitch at the chiral nematic (N\*)–smectic A (SmA) phase transition, and the material has a modest spontaneous polarization (*PS* <sup>≈</sup> <sup>25</sup> nC cm<sup>−</sup>2) at room temperature. Liquid crystals 1 (LC 1) and 2 (LC 2) are shown in Figure 1, together with their phase sequences; they were originally designed as novel antiferroelectric materials and have a larger spontaneous polarization (*PS* <sup>≈</sup> <sup>70</sup> nC cm<sup>−</sup>2). Full details of their properties are reported elsewhere [28,29]. Both LC 1 and LC 2 exhibit a narrow (~1 ◦C) smectic C alpha (SmC\*α) phase directly above the SmC\* phase. LC 1 also exhibits a twist-grain boundary A (TGBA) phase extending for ~5 ◦C above the SmC\* phase, before the material becomes isotropic. LC 2 has a relatively wide (~17 ◦C) SmA phase directly above the SmC\*α phase. The nature of the SmC\*α to SmC\* phase transition has been discussed in detail elsewhere [30] and can be first or second order, a factor that will influence the field and temperature dependence of the spontaneous polarization. The subphases exhibited by LC 1 and LC 2 are all several degrees below ferroelectric to paraelectric (SmC\*<sup>α</sup> to TGBA or SmA phase) and do not contribute to the measurements reported in this paper.

The heat capacity, *CE*(0,*T*), is measured via differential scanning calorimetry using a TA Instruments Q2000 Different Scanning Calorimeter. All heat capacity measurements were made at zero field (the option of applying a field during the calorimetry was not available) and taken on cooling at 10 Kmin−1. The data are quoted with respect to a critical temperature, TO, defined as the temperature where the ferroelectric phase was first observed. To determine the volumetric heat capacity of a sample, the specific heat capacity is multiplied by the density of the sample. For this work, the density of the materials was estimated from literature values for calamitic LCs in the SmC [31–33] or SmA phase [34] at the phase transition. The range of values reported for density span 0.96 to 1.02 g cm<sup>−</sup>3, with an average density of (0.97 ± 0.02) g cm−3. We employed the average value to indirectly determine the electrocaloric temperature change and estimate that this contributes to ~2% uncertainty in our final measurement of Δ*T*.

**Figure 1.** The phase transitions (measured on cooling) for all three materials studied, together with the chemical structure of LC 1 and LC 2. The notation of the phases is as defined in the text, with the following: Cr is crystal; SmC\*Fi1 is the 3-layer intermediate phase; SmC\*Fi2 is the 4-layer intermediate phase; SmC\*A is the antiferroelectric phase.

The spontaneous polarisation is measured using the current reversal technique with an accuracy of ±1 nC cm−<sup>2</sup> [35]. All measurements were taken in cells approximately 1.8 μm thick treated for planar alignment, purchased from AWAT (Poland). An electric field with a triangular wave was applied to the cell, and the current associated with the change in sign of the *Ps* of the ferroelectric LC, IP, was passed in series to a current-to-voltage amplifier. The resulting signal was then recorded on a Tektronix 2024C oscilloscope. *PS* is determined by analysing the current peak using Equation (3),

$$P\_S = \frac{1}{2A \cdot R} \int I\_P dt\tag{3}$$

where *A* is the surface area of the electrodes, *R* is the resistance of the current to voltage amplifier, and *t* is time.

Temperature control is achieved using a Linkam TMS 94 with an LTS 350 hot plate. The spontaneous polarisation is determined as a function of applied field at temperature intervals of 0.2 K spanning temperatures from 2 K above to 10 K below the SmA–SmC\* transition for SCE13, TGBA–SmC\*<sup>α</sup> for LC 1 and SmA–SmC\*<sup>α</sup> transition for LC 2. Results are plotted on a reduced temperature scale with the critical temperature, TO, defined as the temperature where the ferroelectric phase was first observed using polarised microscopy.

The temperature change, Δ*T*, that occurs in the FLC as a consequence of applying a field is determined using the volumetric heat capacity and spontaneous polarisation data as outlined in the Introduction. For each material, *Ps* measurements were taken at 0.2 K temperature intervals across the phase transition while varying the electric field strength from 3–19 V μm−<sup>1</sup> for SCE13 and LC 1 and from 5–19 V μm−<sup>1</sup> for LC 2 in steps of ~3 V μm<sup>−</sup>1. The maximum electric field strengths applied to the samples are sufficient to fully saturate the *Ps* measurements whilst not inducing a chevron to bookshelf transition [36,37].

In order to determine *∂PS ∂T <sup>E</sup>* and substitute into Equations (1) and (2), a numerical fit was applied to the experimental data to allow extrapolation between the data points as follows. The *Ps* measurements determined for SCE13 and Material 2 for each (constant) field strength were fit to a Curie–Weiss law (*PS* = *P*0(*T* − *TC*) *<sup>γ</sup>*) at temperatures up to the sample's respective critical temperature, TC. The *Ps* results for LC 1 were fit to a third-order polynomial because LC 1 shows a more discontinuous transition into the SmC\*<sup>α</sup> phase. *PS* values determined above the critical temperature for all materials were fit to an exponential decay curve.

The gradient of each of the numerical fits with respect to temperature *∂PS ∂T* in Equations (1) and (2) was thus found for each field strength, and this was then plotted as a function of temperature. Taking a vertical slice through the graph of *<sup>∂</sup>PS <sup>∂</sup><sup>T</sup>* as a function of temperature represents *<sup>∂</sup>PS <sup>∂</sup><sup>T</sup>* as a function of electric field strength at a constant temperature, *∂PS ∂T E* . Finally, a numerical fit can be made of *<sup>∂</sup>PS <sup>∂</sup><sup>T</sup>* as a function of electric field strength; integrating under the fitting curve results in the indirect measurement for isothermal entropy change per unit volume, as described by Equation (2). The electrocaloric temperature change was indirectly determined by multiplying the isothermal entropy change per unit volume at a given temperature, *T*1, by the temperature in Kelvin, and dividing by the volumetric heat capacity at *T*1.

#### **3. Results**

#### *3.1. Heat Capacity*

Figure 2 presents the volumetric heat capacity of SCE13, LC 1, and LC 2 around the transition into the ferroelectric phase of each material. The magnitude of the heat capacity is different for each of these materials, a factor that is important in their use as electrocaloric materials (Equation (2)). A peak is seen in the volumetric heat capacity around the ferroelectric to non-ferroelectric phase transitions for each of the materials. The relative magnitude of the peak is quite different for the three samples: LC 1 is the largest, 19% higher than the value 1 K above the transition; SCE13 shows less than a 1% increase, while LC 2 shows a 6% increase over the baseline value. The relative magnitude of the peaks is representative of the discontinuity at the phase transition. As will be explained further in Section 4, this subsequently affects the magnitude and applicable temperature range of the electrocaloric effect. The SmC\*α–SmC transition is seen in the DSC traces for LC 1 and LC 2; however, the transition is convoluted with the TGBA–SmC\*<sup>α</sup> transition in LC 1 and the SmA–SmC\*α transition in LC 2. The convolution broadens the peak; therefore, the heat capacity, *CE*, around the transition stays larger over a wider temperature range. Subsequently, from Equation (2), the electrocaloric temperature change at all temperatures is deduced. For these measurements, the critical temperature for each sample is defined as the peak in the heat capacity.

#### *3.2. Spontaneous Polarisation*

The spontaneous polarization determined using Equation (3) as a function of reduced temperature is shown in Figure 3 over the temperature range around the transition. SCE13 has the smallest absolute spontaneous polarization, taking a value of ~14 nC cm−<sup>2</sup> at 10 K below the transition. The field-induced *PS* above *TC* is very small, below the sensitivity of our experiment, as would be expected from the very small electroclinic effect [38] known for SCE13. Both LC 1 and LC 2 show evidence of a significant field-induced *PS* above the phase transition, an extremely desirable phenomenon for electrocaloric applications, because this can extend the applicable temperature range. Both materials have a PS of approximately 71 nC cm−<sup>2</sup> 10 K below the transition, but as the rate of change is greater near the phase change for LC 1, a larger EC effect is expected.

**Figure 2.** Volumetric heat capacity (*CE*(0, *T*1)) (MJ K−<sup>1</sup> m<sup>−</sup>3) measured from 2 K above to 10 K below the transition into the ferroelectric phase for each material. The critical temperature was defined as the peak in the heat capacity for each material. For SCE13 (black circles, the peak occurs across the SmA–SmC\* transition. For LC 1 (red triangles), the peak occurs across the TGBA–SmCα\* transition, and for LC 2 (blue squares), the peak is across the SmA–SmCα\* transition. In LC 1 and LC 2, the SmCα\* –SmC\* transition is marked using dotted and dashed lines (LC 1 and LC 2, respectively).

**Figure 3.** Maximum spontaneous polarisation (*PS*) measured as a function of reduced temperature relative to the SmA–SmC\* transition for SCE13 (black circles), TGBA–SmCα\* for LC 1 (red triangles), and SmA–SmCα\* for LC 2 (blue squares). The uncertainty in *PS* <sup>=</sup> <sup>±</sup>1 nC cm−2. LC 1 shows a discontinuous transition, while LC 2 and SCE13 show a continuous transition.

#### *3.3. Field-Induced Isothermal Entropy Change per Unit Volume*

Figure 4 shows the peak isothermal entropy change per unit volume, *V*, <sup>Δ</sup>*<sup>S</sup> <sup>V</sup>* , of the materials, determined using Equation (2), demonstrating the significantly better performance of LC 1 over that of the other systems considered. The maximum isothermal entropy change per unit volume of SCE13 is 0.4 kJ K−<sup>1</sup> m<sup>−</sup>3, less than 20% that of LC 1, which has a maximum of 2.3 kJ K−<sup>1</sup> m<sup>−</sup>3.

**Figure 4.** Isothermal entropy change per unit volume as a function of reduced temperature determined using Equation (1). The results for SCE13 and LC 1 are for an effective field of 16 V μm<sup>−</sup>1, and an effective field of 14 V <sup>μ</sup>m−<sup>1</sup> for LC 2. The peak position of SCE13 (black circles) is ~−0.3 K; for LC 1 (red triangles), the peak position is ~−1 K; for LC 2 (blue squares), the peak position is ~−2 K.

Although LC 1 and LC 2 have approximately the same value of *PS*, LC 1 performs ~25% better than LC 2. This is attributed to the more discontinuous TGBA–SmC\*<sup>α</sup> transition in LC 1, which results in a larger *<sup>∂</sup>PS <sup>∂</sup><sup>T</sup>* at each field strength. The peak isothermal entropy changes of LC 1 and LC 2 occur at ~1 K and ~2 K, respectively, below the transition temperature. The temperature at which the maximum value occurs corresponds to the region where the gradient of the spontaneous polarisation, with respect to temperature, reaches a maximum for every electric field strength measured. These observations confirm the importance of the value of the parameter *∂PS ∂T <sup>E</sup>* when considering which ferroelectric LCs will show the largest EC effect; although a large magnitude of *PS* is important, having a large gradient is vital. It is also important to consider the range over which a useful EC effect is available; real systems would typically need to operate over ~10 K. LC 1 maintains at least 90% of the value of its maximum entropy change over a 0.9 K range, while the useful range of LC 2 extends over 1.4 K, a noticeably wider temperature range.

#### *3.4. Electrocaloric Temperature Change*

An indirect measurement of the EC temperature change, Δ*T*, was obtained by multiplying the isothermal entropy change per unit volume by the scaling factor *<sup>T</sup>*<sup>1</sup> *CE*(0,*T*1) (Equation (2), Figure 5). The importance of a low volumetric heat capacity can be seen in the relative differences in the maximum values for isothermal entropy, Δ*S* (Figure 4), and the EC temperature change Δ*T* (Figure 5). The lower heat capacity of SCE13 means that the electrocaloric temperature change is comparatively larger than the isothermal entropy change per unit volume alone would suggest. It is nonetheless still much smaller Δ*Tmax* ∼ 0.1 K than the temperature change in LC 1 or LC 2 (Δ*Tmax* ∼ 0.37 K and Δ*Tmax* ∼ 0.22 K, respectively). The lower heat capacity of LC 1 serves to enhance the induced temperature change compared to LC 2.

**Figure 5.** Indirect measurement of the electrocaloric temperature change as a function of reduced temperature for a field strength of 16 V μm−<sup>1</sup> SCE13 (black circles) and LC 1 (red triangles) and 14 V <sup>μ</sup>m−<sup>1</sup> for LC 2 (blue squares). The peak position occurs at reduced temperatures of ~−0.3 K (SCE13), ~−1 K (LC 1), and ~−2 K (LC 2).

#### **4. Discussion**

Table 1 summarises the maximum values of the physical parameters relevant to the EC effect in the ferroelectric liquid crystals that are the subject of this paper and the few others reported elsewhere [10]. For comparison, the table also includes EC values for 12CB at the SmA–I transition [25] and for a solid-state ferroelectric ceramic device, which is currently considered state-of-the-art [39]. A figure of merit can be defined [40], <sup>Δ</sup>T*max* <sup>Δ</sup>*<sup>E</sup>* , where Δ*E* is the field required to induce the maximum temperature change, Δ*Tmax*, which offers a measure of the efficiency of the EC effect in different materials.

**Table 1.** The maximum spontaneous polarisation (*PS*), volumetric heat capacity (*CE*), maximum EC temperature change (*Tmax*), figure of merit, and temperature range over which the electrocaloric temperature change remains greater than 90% of the peak temperature change, for the materials studied in this paper and other systems chosen for comparison. The ferroelectric materials FELIX-017/000 and OB4HOB [10] were studied by Bsaibess et al., while Klemenˇciˇc et al. induced a temperature change at the isotropic to SmA phase transition in 12CB [25]. The ferroelectric ceramic, lead scandium tantalate, PST, [39] arranged in a multilayer capacitor MLC is also included. The temperature changes for 12CB and PST are direct measurements, and all other measurements are indirect.


LC 1 shows the largest EC temperature change reported to date for a ferroelectric liquid crystal, Δ*Tmax* ∼ 0.37 K, compared to Δ*Tmax* ∼ 0.17 K, which was the maximum reported by Bsaibess et al. for the ferroelectric liquid crystal OB4HOB [10]. It can be seen that the main reason for the significant improvement seen for LC 1 over OB4HOB is the much higher heat capacity (7.1 MJ K<sup>−</sup>1m−3) and slightly smaller spontaneous polarization (~60 nC cm−2) of the latter material. Interestingly, the peak isothermal entropy change of OB4HOB can be estimated from data reported by Bsaibess et al. to be 3.1 kJ K−<sup>1</sup> m−3, which is ~ 50% larger than that of LC 1. An additional important factor in the EC response is the gradient, *∂PS ∂T E* , which is considerably larger for OB4HOB due to its first-order isotropic to SmC\* phase transition (the *Ps* saturates 4 K below the transition temperature). LC 1 has a much smaller gradient, and subsequently, the induced entropy change over the same temperature range is smaller.

As mentioned, the heat capacity is an important factor. Indeed, the heat capacity of OB4HOB is over three-times larger than that of LC 1 at the temperature where the EC peak occurs, which results in a lower induced temperature change, Δ*T*, for a given isothermal entropy change Equation (2). Clearly, the heat capacity, saturated *PS* value, and the gradient of *PS* with respect to temperature must be compared in ferroelectric LCs to determine the overall suitability of materials with respect to their electrocaloric effect. It is also important that the indirect methodology employed here uses the heat capacity measured at zero field. As discussed in a recent review [12], not considering the field or temperature dependence results in the heat capacity being smeared and overestimated. Therefore, the indirectly measured temperature change reported in Figure 5 is an underestimate of the expected temperature change.

Both of the ferroelectric liquid crystal mixtures designed for display devices, SCE13 and FELIX-017/000, behave unsurprisingly modestly in terms of their EC potential. As materials that were designed for a completely different application, namely electro-optic devices, it was desirable to have a relatively low *Ps*, and the phase sequences were designed to allow good alignment to be obtained. The optical properties of such commercial mixtures were also important, with the ideal tilt angle of 22.5◦ being carefully engineered in them. Their figures of merit are very poor, a consequence of the low values of *Ps*.

Although having a large gradient, *∂PS ∂T E* , is clearly important in maximising Δ*T*, there are drawbacks to materials that reach maximum *Ps* over a shorter temperature range. Specifically, this will mean that the temperature span over which the electrocaloric effect decays is also smaller, giving such materials a poor useful range. Table 1 summarises this issue by considering the temperature range over which the electrocaloric temperature change remains greater than 90% of the peak temperature change, <sup>Δ</sup>*<sup>T</sup>* <sup>Δ</sup>*Tmax* <sup>&</sup>gt; 0.9. As is shown in Figure 6, while the figure of merit, <sup>Δ</sup>*<sup>T</sup>* <sup>Δ</sup>*<sup>E</sup>* of OB4HOB is comparable to that of LC 1, <sup>Δ</sup>*<sup>T</sup>* <sup>Δ</sup>*Tmax* <sup>&</sup>gt; 0.9 for OB4HOB is a factor of 9 smaller than LC 1. The rate of decay of the electrocaloric effect as a function of temperature is an important quantity to consider for engineering purposes, as any electrocaloric refrigeration device must be able to operate over a broad temperature span. For comparison, lead scandium tantalate, PST, a ferroelectric ceramic, arranged in a multilayer capacitor, is also shown in Figure 6. This ceramic demonstrates an electrocaloric temperature change of 3 K over one of the broadest temperature ranges reported, 73 K [39]. Although the figures of merit for ferroelectric liquid crystals are only around an order of magnitude lower than the very best ceramic EC devices, their useful temperature range is not yet comparable.

It is appropriate to discuss briefly the liquid crystal system that performs best in terms of Δ*Tmax*, 12CB. This system was mentioned in the Introduction, and it can be seen that the field-induced isotropic to SmA phase change offers an enormous figure of merit, Δ*T* <sup>Δ</sup>*<sup>E</sup>* <sup>=</sup> 0.8 K m MV<sup>−</sup>1, and a giant maximum induced temperature change, <sup>Δ</sup>*Tmax* <sup>=</sup> 6.5 K. Unfortunately, this exceptional EC performance is unsuitable for applications because of the very narrow useful temperature range, less than 0.1 K. Both the relatively large Δ*Tmax* and narrow temperature range are due the physical phenomena behind the EC effect in this material, i.e., factors that electrically drive the isotropic to smectic transition. The absorption of latent heat as the LC transitions dominates the electrocaloric effect when

inducing a liquid crystal phase [25]. Consequently, the effect is only significant across the isotropic–SmA coexistence region, which is extremely narrow in a pure material, with the effect reducing significantly on cooling further into the phase. Extending the very limited usable temperature range in this system to ~ 2 K has been achieved by extending the coexistence region by mixing nanoparticles into the LC [26].

**Figure 6.** The EC figure of merit, Δ*T*/Δ*E*, plotted as a function of the temperature at which the maximum EC effect was recorded. Results for LCs (triangles and labelled) and a sample of solid EC materials, including polymers (circles) and ceramics [12,39] (squares), are shown. Data for 8CB and 12CB [25] and previously reported ferroelectric LCs [10] are also shown for comparison. The inset graph is an expansion of the dotted rectangle between 330 K and 375 K that expands the region containing ferroelectric LCs.

One final, but important point for consideration in applications is the actual temperature at which the maximum electrocaloric effect occurs. Figure 6 shows Δ*T*/Δ*E* for the materials considered in this paper together with selected ferroelectric ceramics and polymers [12,39], the ferroelectric LCs previously reported [10,39], and the cyanobiphenyl nematic liquid crystals 8CB and 12CB (where the entropy changes at the isotropic to liquid crystal transitions were considered) [25]. The temperature on the ordinate axis is that where Δ*Tmax* was recorded. The specific application will determine whether or not a particular Δ*Tmax* and temperature range of the EC effect is suitable, but it is noteworthy that engineering phase transition temperatures and physical properties across wide temperature ranges is well known in liquid crystals. For example, SCE13 was designed to have a ferroelectric phase from ~−20–60 ◦C to make it suitable for display applications, a relatively low *Ps* to optimise switching speed, and a tilt angle of 22.5◦ over a wide temperature range to optimise the optical contrast of electro-optic devices. Thus, provided FLCs are considered promising for EC applications and the design rules are known, one might expect them to be serious contenders in the future. In this case, despite the fact that LC 1 shows the largest normalized EC temperature change of any ferroelectric LC to date, it is evident that these materials are currently an order of magnitude less efficient, in terms of <sup>Δ</sup>*<sup>T</sup>* <sup>Δ</sup>*<sup>E</sup>* , than solid EC materials. Although a disappointing outcome, it is not a surprising one as the *PS* of FLCs is two or three orders of magnitude lower than that of solid ferroelectrics.

#### **5. Conclusions**

This work showed that for the development of ferroelectric LCs for the EC effect, both the isothermal entropy change and volumetric heat capacity must be considered. A large *Ps* and small heat capacity are clearly important, but the maximum EC temperature change coincides with the maximum gradient in spontaneous polarisation with respect to temperature, *∂PS ∂T E* , bringing a new design rule to ferroelectric liquid crystals for this application. Furthermore, the variation in the gradient is the most significant factor affecting how the EC temperature change varies with temperature. Therefore, the development of new materials should focus on both maximising spontaneous polarisation and optimising the *PS* gradient to occur over the broadest temperature range without significantly reducing the EC temperature change. These are very different design considerations than were relevant to the development of FLC electro-optic devices with microsecond response times. We suggest that materials with a large *PS* designed for electroclinic devices would be interesting candidates for electrocaloric applications, but other systems have also been proposed, e.g., antiferroelectric bent-core liquid crystals [41]. However, it is clearly also important that the heat capacity of the LC material be considered; this changes by a factor of 3 even in the few liquid crystals we considered here. Finally, it is worth noting that although these current systems perform relatively poorly with respect to solid-state systems, the fact that we are considering fluids offers several significant advantages in electrocaloric applications. This is an exciting application area, especially in these times where sustainability and the efficiency of energy use are critical, and we demonstrated some important design considerations for developing liquid crystals for electrocaloric applications.

**Author Contributions:** Conceptualization, H.F.G. and P.J.T.; methodology, H.F.G. and P.J.T.; software, P.J.T.; validation, H.F.G. and P.J.T.; formal analysis, P.J.T.; investigation, P.J.T.; data curation, P.J.T.; writing—original draft preparation, P.J.T.; writing—review and editing, H.F.G. and P.J.T.; supervision, H.F.G.; project administration, H.F.G.; funding acquisition, H.F.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** P.J.T. and H.F.G. acknowledge funding from the Engineering and Physical Sciences Research Council and from Merck Performance Materials Ltd. through a CASE award.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data associated with this paper are available from University of Leeds at https://doi.org/10.5518/1149.

**Acknowledgments:** P.J.T. and H.F.G. acknowledge the supply of materials from Merck Performance Materials Ltd.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

