Progress and Prospect of Liquid Crystal Droplets

Abstract

Liquid crystal (LC) droplets are highly attractive for applications in privacy windows, optical switches, optical vortices, optical microresonators, microlenses, and biosensors due to their ease of fabrication and easy alignment at surfaces. This review presents the latest advancements in LC droplets, which have nematic, chiral nematic, and twist–bend nematic and ferroelectric nematic phases, or blue phases. Finally, it discusses the challenges and opportunities for applications based on LC droplets. The main challenges encompass the precise control of internal structures and defects to meet diverse application requirements, enhancing stability and durability across various environments, reducing large-scale production costs to improve commercial feasibility, increasing response speeds to external stimuli to adapt to rapidly changing scenarios, and developing tunable LC droplets to achieve broader functionalities.

1. Introduction

Liquid crystals (LCs), fascinating materials in soft matter, primarily have nematic phases, chiral nematic (cholesteric) phases, smectic phases or blue phases (BPs) [1]. LC molecules in the nematic phase exhibit a long-range orientational order along a common direction called director $\overrightarrow{\mathit{n}}$, but they lack a long-range positional order. When confined to droplets, some bulk-translational symmetries break down, potentially stabilizing intermediate phases unseen in bulk systems [2]. For instance, nematic droplets with parallel anchoring can display various configurations derived from size and boundary conditions [3]. Researchers employ multiple methods to control LC orientations, such as chemical treatments or polymer coatings for specific anchoring, surface rubbing to alter molecular orientation, external electric or magnetic fields to influence molecular arrangement, substrate material selection, and temperature control to modify phase states [4]. These techniques, often combined, allow for the manipulation of the LC droplet boundaries [5]. This confinement offers a unique platform for studying fundamental physics and developing applications, providing insights into topological defects, phase transitions, and molecular organizations inaccessible in bulk systems [4,5]. This research has implications for display technologies [6], sensors [7], and biomimetic materials [8] while also contributing to our understanding of self-assembly processes in nature and the interplay between surface effects and bulk properties in confined systems.

When LCs are confined to droplets with a radius ranging from less than 100 nm to over 100 μm, their arrangement is impacted by elastic forces, external fields (such as electric, magnetic, temperature, and light), and surface interaction [9,10,11,12]. Nematic defects, present at interfaces, have universal structures. Confined nematic LCs may appear bipolar under tangential anchoring (with the director near the surface perpendicular to the surface normal) and radial or axial structures under homeotropic anchoring (where the surface director is parallel to the surface as normal) [13].

As LC complexity increases, droplet configurations become more intricate [14,15]. Chirality manifests in various LC phases (Figure 1), such as the chiral nematic phase (Figure 1a) [16], the twist–bend nematic phase (N_TB) (Figure 1b) [17], and BPs (Figure 1c,d) [18]. The competition between elastic forces and surface tension results in droplets that may have anisotropic shapes [19]. For chiral nematic LC droplets with parallel surface anchoring, the director configuration is related to the R–p ratio, where p is the pitch, and R is the droplet radius [20]. When p > R, the structure appears twisted and bipolar; whereas, when p < R, it assumes the Frank–Pryce structure, and when p << R, it is radial nematic [21]. Typically, a radial spherical structure (RSS) is observed in chiral nematic LC droplets with planar anchoring (Figure 1g) [22]. Twisted bipolar structures (TBSs) are more likely to form as R or p decreases because of increasing frustration and a twist in the LC alignment [22]. As the anchoring strength decreases, planar bipolar structures (PBSs) may appear (Figure 1h) [23]. If there are discontinuities in the LC alignment, a diametrical spherical structure (DBS) is seen in LC droplets [24].

BPs exist between the cholesteric phase and the normal isotropic liquid phase, including blue phase I (BPI) with a body-centered cubic (bcc) structure (Figure 1c), blue phase II (BPII) with a simple cubic structure (Figure 1d), and the amorphous blue phase III (BPIII) [25]. Due to high chirality, unique structures exist under the combined effects of frustrated BP arrangements and surface anchoring [26]. BPs confined to spherical droplets can alter Bragg diffraction and induce the formation of single-domain BPs [27]. Additionally, confined BPs undergo continuous and reversible deformations from spherical to oblate and prolate shapes [28]. Moreover, the phase transition temperatures ($T_{iso}\to T_{BPI}\to T_{BPII}\to T_{chol}$) of confined BP droplets differ from those of bulk BPs, when the radius, R, is small [29]. When BPs are confined to thin layers, many exotic structures can be observed, such as double-helix forms, disclination loops, undulating disclinations, and Skyrmion structures [30]. The rich variety of structures and behaviors observed in confined LCs continues to inspire new technological innovations and deepen our understanding of complex fluid systems.

LC droplets are a well-studied system as a result of their ease of preparation and the rich phenomena they exhibit [31]. The simplest method to create them is mixing a small amount of LC with a medium that does not dissolve it, such as combining organic molecule-based LCs with polar solvents like water [32]. LC droplets can also form through phase separation (like polymer-dispersed liquid crystals, abbreviated as PDLCs) or be precisely controlled using microfluidic techniques [33]. LC droplets have broad application potential in various fields, such as display technologies, optical devices, sensors, biomedicine, privacy windows, optical vortices [34], tunable optical microresonators with high Q-factors [35], and microlenses [36]. By adjusting light transmittance, LC droplets can be used to create privacy windows for enhanced privacy protection [37]. They demonstrate their versatility and wide-ranging potential uses in generating optical vortices and manufacturing microlens arrays [38]. Moreover, they offer significant applications in drug delivery and biosensing [39]. As technology advances, the potential applications of LC droplets in more fields will continue to be explored and utilized.

2. Nematic LC Droplets

When LCs are confined to a spherical surface, the orientation of the director field, $\overrightarrow{\mathit{n}}$, is influenced by geometric frustration induced via spherical constraints [40,41]. The droplets’ size significantly affects the stability of LC configurations. For instance, small droplets tend to adopt radial configurations (Figure 2a–h) as surface anchoring becomes more pronounced, and anchoring energy dominates at smaller scales, thereby restricting the formation of complex configurations [42,43]. In contrast, larger droplets allow for a greater variety of configurations, such as bipolar or twisted bipolar structures (Figure 2b), on account of the increasing internal space that facilitates the emergence of complex defect structures, which, in turn, affects overall stability [44]. Bipolar LC drops moving inside microchannels present periodic field transformation due to the induced circulating flows inside them [45]. Changes in size may lead to configuration transitions, for example, from radial to bipolar (Figure 2i), depending on the anchoring type and external conditions [46]. In summary, droplet size profoundly influences the stability of LC configurations by affecting molecular alignment and defect formation [47].

For nematic LC droplets with tangential anchoring, the LC molecules align parallel to the droplet surface, leading to two-point defects (“boojums”) at the poles, known as bipolar structures [48]. Conversely, with perpendicular anchoring, the nematic LC molecules radiate outward from the center of the droplet, creating a radial structure that features a single-point defect at the center [49]. Additionally, several other configurations exist, depending on the strengths and types of anchoring [50]. Transitions between these different director configurations and changes in topological defects are driven by variations in droplet sizes, boundary conditions, temperature, electric fields, magnetic fields, and other chemical compounds [51,52]. In nematic LC droplets with planar anchoring, various configurations can emerge, relying on the droplet sizes and boundary conditions [53]. For the bipolar configuration within an LC droplet, the LC director divides the droplet into two regions with opposite alignment directions, resembling two distinct domains [54]. For the radial configuration, molecules radiate outward from the center, with their alignment becoming more parallel to the droplet surface as the distance from the center increases [55]. This configuration typically occurs in small droplets with strong surface anchoring [56]. The structure of the twisted bipolar configuration contains regions with opposite twisting directions; at the droplet center, molecules are perpendicular to the surface, while at the poles, they twist in opposite directions [56]. This configuration is characteristic of larger droplets with weaker perpendicular anchoring [56]. In this arrangement of axial configuration, LC molecules align parallel to the central axis of the droplet at the surface, while at the poles, they are perpendicular to the surface [57]. This configuration usually appears in elongated droplets or those with cylindrical symmetry [57]. Escaped radial configuration is like radial configuration, but the LC director deviates from the purely radial pattern, leading to the formation of defects either on the droplet surface or internally, which disrupts the radial alignment [58,59,60].

When an electric field is applied, and the LC director adopts planar anchoring at the surface (resulting in bipolar structures; Figure 2j), the droplet, including its two-point defects, rotates to align the director predominately along the direction of the applied electric field [61,62,63]. This reorientation process has a minimal effect on the bipolar configuration of the droplet, indicating that the geometric shape of the confined LC phase remains largely unchanged [63]. Conversely, under perpendicular anchoring (resulting in radial structures), a first-order phase transition is expected. The droplet adopts a configuration aligned with the electric field direction, and it develops an equatorial line distortion [63]. In other words, the radial “hedgehog” structure, which contains a central point defect, transforms into an axially symmetric structure with a circular defect [63]. The director field distortion caused by this circular defect is more significant than that in the bipolar structure, thereby increasing the effective elastic deformation constant, $\mathit{K}$, of the radially confined LC phase.

Figure 2k shows a polarized optical microscopy (POM) snapshot of a 25 µm droplet in relation to the horizontal magnetic field ($\mathit{H}$) [64]. Figure 2k shows the radial configuration of the typical cross pattern of radial NLC droplets at $\mathit{H}=0$ [60]. Between crossed polarizers, this configuration appears as four black brushes (horizontal and vertical) joined together at one point [64]. When the magnetic field strength is relatively low but gradually increasing, the deformation of the radial structure is observed [64]. This configuration evolves gradually and is exemplified by the asymmetry in the POM pattern. When the nematic LC is aligned with the magnetic field, the spherical symmetry of the director configuration is broken [64]. Due to the phase delay of light passing through the nematic LC, the concentric circular fringes around the point defect contract horizontally, taking on an oval shape with short axis parallel to H [64]. Importantly, in the deformed radial structure, the directional field in the droplet is distorted, but the point defect is not influenced [64]. As a result, the point defect in the center of the four bright lobes is preserved. Above the critical field, the defect in the center is transformed into a disclination ring, perpendicular to the axis of the $\mathit{H}$ [64].

Figure 2. (a–h) Configurations of nematic droplets: (a) bipolar, (b) twisted bipolar, (c) super-twisted bipolar, and (d) toroidal structures for tangential boundary conditions; (e) radial, (f) twisted radial, (g) escaped radial, and (h) axial configurations for homeotropic anchoring [41]; (i) radial configuration and bipolar configuration and their related POM images [51]; (j) schematic depiction of the nematic director field confined to spherical droplets under the influence of an external electric field (${\mathrm{E}}_{\mathrm{a}}$) for bipolar (top) and radial (bottom) droplet configurations, copyright 2014, RSC Publication [63]. (k) With an increasing strength of the magnetic field, experimental POM images (top row), simulated POM images (second row), and the director configuration (bottom row) of a nematic LC droplet are presented, copyright 2023, APS Publication [64].

Figure 2. (a–h) Configurations of nematic droplets: (a) bipolar, (b) twisted bipolar, (c) super-twisted bipolar, and (d) toroidal structures for tangential boundary conditions; (e) radial, (f) twisted radial, (g) escaped radial, and (h) axial configurations for homeotropic anchoring [41]; (i) radial configuration and bipolar configuration and their related POM images [51]; (j) schematic depiction of the nematic director field confined to spherical droplets under the influence of an external electric field (${\mathrm{E}}_{\mathrm{a}}$) for bipolar (top) and radial (bottom) droplet configurations, copyright 2014, RSC Publication [63]. (k) With an increasing strength of the magnetic field, experimental POM images (top row), simulated POM images (second row), and the director configuration (bottom row) of a nematic LC droplet are presented, copyright 2023, APS Publication [64].

3. Chiral Nematic LC Droplets

Chiral nematic (cholesteric) LC droplets have garnered significant interest over the years, with studies focusing on their formation in PDLCs, structural stability, transitions between different structures, the influence of external fields, and the assembly of nanoparticles at surface or bulk defects [65]. The LC configuration within the droplet is governed by the anchoring conditions at the droplet surface [66]. These droplets exhibit a rich variety of structures dictated by the interplay of chirality (which enforces an intrinsic periodicity), elasticity, and anisotropic surface energy. In comparison with their nematic counterparts, chiral nematic droplets possess an additional degree of freedom: the chiral pitch, denoted as $p₀$. The helical twist of the director field can be accommodated within the spherical confinement of the droplets in a multitude of ways. Consequently, for a given set of parameters, several director configurations with distinct elastic energies can arise. Notably, structures with non-minimal energy can also be stabilized due to energy barriers associated with unwinding the twisted configurations. These structures, characterized by their stability and non-minimal energy, are termed metastable. The droplet diameter and the $p₀$ are as the key parameters determining the energy and stability of different structures. The helical pitch of the LCs/chiral dopant mixture would adopt in a bulk, unconfined phase. It is often convenient to express the pitch in units of the droplet diameter (2R) as a measure of the relative droplet size in comparison to the chiral pitch. This is represented as $p₀=4\mathrm{R}/\mathrm{N}$, where $\mathrm{N}$ effectively represents the number of π-turns the director field would make along a distance equal to the droplet diameter in an unconfined cholesteric phase.

In cholesteric LC droplets with planar degenerate anchoring, six (meta)stable orientational structures have been identified, and each is characterized by its distinct director field configuration and topological defects: the diameter spherical structure (DSS) (Figure 3a), Frank–Pryce or an RSS (Figure 3b), a bipolar structure (BS) (Figure 3c), a PBS (Figure 3d), a lyre structure (Figure 3e), and a yeti structure (Figure 3f) [24]. The DSS represents the most symmetrical configuration in cholesteric LC droplets with degenerate planar surface anchoring. It exhibits cylindrical symmetry around the z direction, for instance, through the center of the defects (red arrow in Figure 3a) [24]. The director field forms curved cholesteric layers whose normal layer is in the radial direction (except along the +z and −z directions). The field can be visualized as a series of concentrically placed, twisted loops of double-twist cylinders (the cross section of the twisted cylinder is banana-shaped, rather than circular). If the director field is tangentially anchored and the pitch $p₀$ is smaller than the radius, R, the LC droplet exhibits a Frank–Pryce spherulite texture with concentric circles (such as an RSS) with a distance between two consecutive rings of half the pitch. In this structure, the helical axis is perpendicular, and the isopitch surfaces (tilted with the same director) are parallel to the surface. Due to the radial distribution of the helical axis and the approximately spherical nature of the isopitch surfaces, this structure is sometimes also called radial or spherulitic. The RSS (Figure 3b) [24], the most commonly observed structure experimentally, effectively consists of twisted double-twist loops with a variable (angle-dependent) small radius. Compared to the diameter spherical structure, the symmetry of the double-twist loops with a uniform small radius is broken, effectively expanding one side of the loop at the expense of the other. In the RSS structure, there are no singular disclination lines in the nematic director field. The BS (Figure 3c) [24] possesses cylindrical symmetry and is characterized by two surface defects located diametrically opposite to each other. This structure evolves smoothly from the bipolar structure in a non-chiral nematic droplet when chirality is effectively switched on, and the chiral pitch becomes finite. The PBS is closely associated with the bipolar structure, and it also exhibits two surface defects, situated diametrically opposite to each other. The director field in the central region of the droplet forms cholesteric layers whose normal layer is in the x direction (see Figure 3d and the top right inset) [24]. This PBS is a unique configuration in cholesteric droplets, closely related to the bipolar structure but distinguished by its distinct symmetry and the arrangement of cholesteric layers. The formation of this structure arises from the specific arrangement of chiral molecules within the droplet and the influence of strong planar surface anchoring. The other two structures found in Figure 3e,f are highly metastable and belong to the bipolar type with only two surface defects positioned diametrically along the z-axis [24]. The yeti structure possesses higher symmetry due to its additional two-fold symmetry along an arbitrary symmetry axis in the x–y plane. However, contrary to the DSS structure, in the lyre and yeti structures, the cholesteric disclinations in the form of rings are non-singular. In the case of perpendicular anchoring, the structures and texture of chiral droplets have been less studied. In confined geometries, the periodicity of the layered structure may deviate from this equilibrium value, as the structure can distort elastically to accommodate the boundary conditions. When cholesteric LC is confined in a droplet, defect lines may form if frustrated states arise. Orientational structures in cholesteric LC droplets with perpendicular surface anchoring exhibit toron-like and low-symmetry intermediate layer structures with varying helical parameters. In frustrated chiral nematic droplets, there are numerous complex, free-standing, metastable, micron-scaled topological structures in frustrated chiral nematic droplets [67]. Free-standing knotted and linked disclination structures in confined chiral nematic LCs is predicted [67]. Various types of external fields (thermal, electrical, and optical) are used to achieve topological remote control, which may foster the development of new devices based on topologically structured soft media [67]. When confined to spherical droplets with homeotropic surface anchoring, the creation of free-standing knotted and linked disclination loops in the cholesteric ordering fields is demonstrated, as the cholesteric pitch, p, is short on the confinement scale, and the knotted structures switches to practically unperturbed cholesteric structures with disclinations expelled close to the surface [68].

Spherical cholesteric LC droplets possess omnidirectional reflectivity, and they are capable of selectively reflecting light with a wavelength in the band gap of the cholesteric LC [69]. In addition, the apparent band gap counts on the angle with the helical axis, with a blue shift for larger angles of incidence. When a parallel beam is incident on a droplet, various wavelengths are reflected in different directions. When many spherical cholesteric LC droplets are dispersed within a polymer film, the film often appears opaque and white. This is because the selectively reflected portion of light from each spherical cholesteric LC droplet is minimal, and light scattering from the droplets dominates, as depicted in Figure 3g [69]. When the spherical cholesteric LC droplets are deformed into a flattened shape due to external stress, such as stretching, the point defect at the center of each droplet evolves into a ring defect. Within this ring defect, the helical axis of the cholesteric LC aligns perpendicularly to the plane of the horizontal film. This realignment significantly expands the area of selective reflection. As a result, structural color emerges within the film, exhibiting a mechanochromic response, as illustrated in Figure 3h [69]. CLC particles suspended in a liquid can be trapped by a laser beam, and depending on their size, orientation, refractive index contrast with the solvent, as well as the polarization of the laser beam, they may start to rotate in a static laser beam [70]. For nanoparticle (NP) assemblies with cholesteric LC as a host, when the content of NP is low, cellulose nanocrystal (CNC) droplets display a flat-ellipsoidal packing of pseudolayers and, then a toroidal ring- or two cone-shaped assemblies at the droplet poles [71]. For nematic droplets that are used in biological sensing, the introduction of a target analyte on their surface can change the initial director configuration. Chiral nematic droplets have the advantage that the reflective wavelength can be matched with the visible range by using the appropriate concentration of a chiral dopant, enabling the visual detection of the target analyte [72]. Additionally, switching between the highly reflective Frank–Pryce structures with planar anchoring and weak reflective structures with perpendicular anchoring makes chiral LC droplets applicable to sensor systems [73].

Figure 3. Numerically computed structures in chiral nematic LC droplets with planar anchoring. (a) DSS in a droplet with $N=10$; copyright 2012, RSC Publication [24]. (b) RSS in a droplet with $N=10$; copyright 2012, RSC Publication. (c) BS in a droplet with $N=6$; copyright 2012, RSC Publication [24]. (d) PBS in a droplet with $N=8$. The insets present cross-sectional views of the structures and simulated transmission images under crossed polarizers, with the viewing direction along or perpendicular to the symmetry axis; copyright 2012, RSC Publication [24]. (e,f) Two exotic numerically computed structures in chiral nematic LC droplets with planar anchoring: lyre (e) in a droplet with $N=4$ and yeti (f) in a droplet with $N=5$; copyright 2012, RSC Publication [24]. (g) spherical cholesteric LC droplets dispersed in a polymer matrix (opaque and white); copyright 2022, ACS Publication [69]. (h) When spherical cholesteric LC droplets are compressed into an oblate shape due to external strain, such as stretching, the selective Bragg reflection of light via oblate cholesteric LC droplets gives the film its color; copyright 2022, ACS Publication [69].

Figure 3. Numerically computed structures in chiral nematic LC droplets with planar anchoring. (a) DSS in a droplet with $N=10$; copyright 2012, RSC Publication [24]. (b) RSS in a droplet with $N=10$; copyright 2012, RSC Publication. (c) BS in a droplet with $N=6$; copyright 2012, RSC Publication [24]. (d) PBS in a droplet with $N=8$. The insets present cross-sectional views of the structures and simulated transmission images under crossed polarizers, with the viewing direction along or perpendicular to the symmetry axis; copyright 2012, RSC Publication [24]. (e,f) Two exotic numerically computed structures in chiral nematic LC droplets with planar anchoring: lyre (e) in a droplet with $N=4$ and yeti (f) in a droplet with $N=5$; copyright 2012, RSC Publication [24]. (g) spherical cholesteric LC droplets dispersed in a polymer matrix (opaque and white); copyright 2022, ACS Publication [69]. (h) When spherical cholesteric LC droplets are compressed into an oblate shape due to external strain, such as stretching, the selective Bragg reflection of light via oblate cholesteric LC droplets gives the film its color; copyright 2022, ACS Publication [69].

Nematic shells present defects, including disclinations, which are line-like singularities in the LC molecular orientation [74]. By utilizing double-emulsion drops, the defect structures of spherical shells of nematic LCs have been investigated [74]. For thicker shells, the defect structure may be similar to the boojum structure of a nematic drop. When the shell is thinner, the structure switches to a tetrahedral structure [74]. By varying the thickness of a nematic LC shell, the number and orientation of defects can be systematically controlled [75]. When the boundary conditions for $\overrightarrow{\mathit{n}}$ are tangential, bipolar shells are characterized by two pairs of boojums, one pair for each spherical surface [59]. When $\overrightarrow{\mathit{n}}$ is tangential at the inner surface and perpendicular at the outer surface, only the innermost pair of defects remains [59]. For confined cholesteric shells, a tetravalent structure, a bipolar structure, a bent structure, and an RSS can be found [76].

4. Blue-Phase LC Droplets

BPs are highly ordered phases of LCs that exist within a narrow temperature range [77,78]. They are arranged in a cubic structure, reflecting blue light due to Bragg reflection within the visible spectrum. BP LCs exhibit fast response characteristics, making them suitable for high-performance displays and optical devices [79]. Their unique self-organizing structures have garnered significant attention in technological innovation. BPs with sub-micron cubic lattices demonstrate tunable Bragg reflection and a sub-millisecond response time to external stimuli, such as electric fields, enhancing their appeal in advanced photonic materials [25]. These LCs possess high viscosity and a limited shear modulus, producing Bragg reflection in the visible spectrum. These properties originate from their highly ordered defect structures. In bulk form, BPs exist only within a narrow temperature range between the cholesteric phase and isotropic phase, which limits their practical applications. Through confinement, the corresponding number of unit cells and their characteristics can be altered. It has been shown that the BPs in microdroplets expand the stable temperature range slightly. In spherical droplets, various new morphologies emerge under confinement, in addition to the so-called BPI and BPII. The complexity of these structures increases with the chirality of the medium, and their properties are modified through surface anchoring. Furthermore, the surface anchoring and physical confinement of BPs can enhance their thermal and mechanical stability.

In the low chirality region, the twisted cylindrical (TC) structure is stable [26]. As shown in Figure 4a, the orientation vector field of this structure forms a double-twisted cylinder [26]. For strong planar anchoring, a bipolar structure can be observed in this region of the phase diagram. As chirality increases, additional double-twisted cylinders emerge, leading to the formation of distinct dislocation lines; however, these structures still maintain well-defined symmetry. The extent of divergence and curvature deformation can be quantified using the divergence–curvature order parameter ($S_{SB}$) [26]. Figure 4b illustrates the divergence–curvature isosurfaces of the frustrated radial spherical morphology (FRSS) structure [26]. In the high-chirality region, the TC structure reappears, but now it is somewhat deformed, primarily in the central area of the droplet, due to the curvature and divergence distortions induced via the high chirality of the system (Figure 4c) [26]. In this case, the divergence–curvature isosurfaces show that the orientation vector field is deformed into a twisted paraboloid. The BP appears in the region where $N>3.5$, close to the isotropic phase. The simulated BPII structure corresponds to a confined BP II unit cell (Figure 4d) [26]. Like in the bulk phase, BPI (Figure 4e) exhibits a broader range of existence in terms of temperature and chirality compared to these of BPII [26]. As expected, the number of unit cells increases with the increase in N (Figure 4f) [26]. Figure 4g displays the optical appearance of a 25 μm high-chirality LC droplet in polyvinyl alcohol (PVA) solution, cooled from 50 °C to 25 °C at a rate of 0.2 °C/min and observed in cross-polarized reflection mode [22]. In the presence of a high concentration of chiral dopant (S-811), BP structures emerge between the isotropic phase and the cholesteric phase. During cooling, the BPII region, reflecting blue light, nucleates and grows first, characterized by a simple cubic lattice structure. As the temperature decreases further, BPII transitions to BPI, which features a body-centered cubic lattice and is identifiable through the appearance of green light [22]. This color change is ascribed to an increase in the helical pitch length during cooling, revealing how temperature finely adjusts LC structures and demonstrating the phase transition sequence: isotropic phase→BPII→BPI→cholesteric phase. Figure 4h shows the micrographs of droplets captured under crossed polarizers, illustrating their states in BPII and BPI [80]. Figure 4i,j present how the electric field induces symmetry changes in BPI LC droplets, as well as variations in the emission intensities and linewidths of the corresponding laser lines [79]. These observations reveal the significant impact of the electric field on the structure and optical properties of BPI LC droplets [80]. The electric field causes a rearrangement of the internal structure of BPI LC droplets, altering their symmetry, which may result from the reorientation of LC molecules under the influence of an electric field [68]. By applying an electric field, changes in the intensity of the laser lines emitted from the microdroplets occur, likely due to alterations in the optical resonance conditions from structural rearrangement [79]. The linewidth of the laser lines also varies with the electric field, with the changes in linewidth typically correlating to the coherence and the quality of the laser emission. These observations are crucial for the development of tunable micro-laser sources. By adjusting the electric field, it is possible to control the output characteristics of the microdroplet laser, such as intensity and linewidth, paving the way for the design of highly tunable optical devices.

Figure 4. Typical morphologies of chiral LC droplets. (a–c) Divergence–bend isosurfaces are shown in red ($S_{SB}$ = −0.0001) and green ($S_{SB}$ = 0.0001). It displays the z-view (top) and side view (bottom) of the director field for TC (a), FRSS (b), and DTC (c) morphologies. The DTC configuration exhibits a defect region represented by the isosurface $S_{SB}$ = 0.57. For BPII (d) and BPI (e), divergence–bend isosurfaces correspond to red ($S_{SB}$ = −0.001) and green ($S_{SB}$ = 0.001), and they rotate along the defect lines ($S_{SB}$ = 0.35). (f) BPI dislocation lines in a droplet with a diameter of D ≈ 1.5 μm, presenting an increase in the number of unit cells with increasing chirality [26]. (g) Cross-polarized reflection mode micrographs demonstrating the phase transition process of a 25 μm chiral LC droplet in a 10 wt.% high-molecular-weight (HM_w) PVA solution when cooled from the isotropic phase at a rate of 0.2 °C/min [22]. (h) Dispersion of BPII and BPI microdroplets in a glycerol solution (scale bar: 100 μm). The effect of an external electric field on the symmetry of BPI microdroplets and their emitted laser lines [80]. (i) 0 V/μm; (j) 0.20 V/μm; (k) 0.30 V/μm (scale bar: 70 μm); copyright 2020, ACS Publication [80].

Figure 4. Typical morphologies of chiral LC droplets. (a–c) Divergence–bend isosurfaces are shown in red ($S_{SB}$ = −0.0001) and green ($S_{SB}$ = 0.0001). It displays the z-view (top) and side view (bottom) of the director field for TC (a), FRSS (b), and DTC (c) morphologies. The DTC configuration exhibits a defect region represented by the isosurface $S_{SB}$ = 0.57. For BPII (d) and BPI (e), divergence–bend isosurfaces correspond to red ($S_{SB}$ = −0.001) and green ($S_{SB}$ = 0.001), and they rotate along the defect lines ($S_{SB}$ = 0.35). (f) BPI dislocation lines in a droplet with a diameter of D ≈ 1.5 μm, presenting an increase in the number of unit cells with increasing chirality [26]. (g) Cross-polarized reflection mode micrographs demonstrating the phase transition process of a 25 μm chiral LC droplet in a 10 wt.% high-molecular-weight (HM_w) PVA solution when cooled from the isotropic phase at a rate of 0.2 °C/min [22]. (h) Dispersion of BPII and BPI microdroplets in a glycerol solution (scale bar: 100 μm). The effect of an external electric field on the symmetry of BPI microdroplets and their emitted laser lines [80]. (i) 0 V/μm; (j) 0.20 V/μm; (k) 0.30 V/μm (scale bar: 70 μm); copyright 2020, ACS Publication [80].

5. Twist–Bend Nematic LC Droplets

The twist–bend nematic phase is a novel liquid crystalline state characterized by a helical molecular arrangement [81]. This structure results from the spontaneous twisting and bending of molecules, forming a nanoscale pitch. N_TB LCs exhibit unique electro-optical properties, showing potential applications in flexible displays and advanced optical materials. The N_TB phase represents a significant advancement in LC science, bridging the gap between conventional nematics and chiral phases. Its helical nanostructure, typically with a pitch of 8–10 nm, arises from the bent-core or dimeric nature of its constituent molecules. This unique molecular organization leads to fascinating optical and electrical responses that differ markedly from those of traditional nematic phases. Research into N_TB LCs is rapidly evolving, with ongoing efforts to understand their fundamental properties and explore their potential in various technological applications. In addition, within these N_TB LC droplets, the structural complexity increases notably. The striations observed in N_TB LC droplets are a manifestation of the underlying helical structure interacting with surface anchoring conditions. These patterns provide valuable insights into the internal organization of the phase and its response to external influences such as substrate treatment and applied fields. The complex behavior of N_TB LC droplets, particularly their interface with conventional nematic phases and their response to confinement, continues to be an area of active investigation in the field of soft-matter physics.

By utilizing a photoresponsive bent dimer that exhibits the N-N_TB phase transition, Yoshioka, et al. have prepared spherical cap-shaped droplets, via temperature control or UV irradiation, and structural switching between the N phase and N_TB phase has been successfully performed [79]. On bare and PVA-coated substrates, striped patterns with periods ranging from a few to tens of micrometers have been observed (Figure 5a) [82]. These patterns likely result from geometric frustration induced via planar anchoring surfaces, causing periodic bending of the N_TB pseudo-layers, like that seen in spherical smectic A (SmA) shells or sandwiched N_TB LCs. On vertical PI surfaces, small droplets exhibited SmA-like fan textures. For larger droplets, a circular region of a few micrometers with an N-type smooth texture appeared at the center. When two collinear, antiparallel dipoles form via two equal-sized N_TB LC droplets propelling towards each other, pushing one satellite nematic drop against the other, the satellite droplets often move away from each other, drifting in opposite directions around their companion N_TB LC droplets. Eventually, at equilibrium, these antiparallel dipoles align along opposite sides of a rhombus, with droplets centered at its four corners, as shown in Figure 5b,c [83]. In the N-N_TB coexistence region, the N_TB phase nucleates sporadically as the droplets in the homogeneous nematic matrix. These droplets grow into large, flattened spherulites, often extending to hundreds of micrometers in diameter within the layer plane. These droplets act as radial hedgehogs, with the N_TB helix axis radially disposed around a point singularity of charge +1. Overall, topological charge neutrality is achieved, as in typical nematic colloids, through the generation of a hyperbolic nematic hedgehog of strength −1 near each N_TB droplet (Figure 5d,e) [84]. The resulting dipole-associated director field is cylindrically symmetric about the dipole axis x. While the components of P_f in the plane orthogonal to the dipole axis cancel out due to cylindrical symmetry, the axial components do not fully compensate due to left–right asymmetry. Consequently, a net flexoelectric dipole moment, $p_e$, exists along the x-axis, opposite to the topological dipole ${\mathrm{p}}_{\mathrm{t}}$. Under crossed polarizers, the PFCD (planar focal conic defect) network in large droplets displays regularly disposed extinction crosses of alternating positive and negative types along a line through the defects (Figure 5f,g) [85]. In planform, negative sites occur where confocal parabolas intersect, while positive sites are where four parabolas meet. In thin samples, positive sites are located at the substrates and negative ones in the midplane region $\mathrm{z}=0$. This complex behavior of N_TB LC droplets showcases the rich interplay between molecular ordering, surface interactions, and topological constraints in these advanced LC systems. The observed patterns and structures provide valuable insights into the fundamental physics of twist–bend nematics and their potential for novel applications in soft-matter technology.

Eremin, et al. reported on the behavior of the nematic director for LC droplets with a photoswitchable dendrimer dispersed in an isotropic fluid. In this case, the transition into the N_TB phase is visible only in freely suspended droplets with a radius of over 20 μm [86]. With UV light, the regular striped texture is destroyed [86]. When the temperature is reduced (2° below the N-N_TB transition), the response to UV light is significantly slower [86].

Figure 5. Complex structures and dynamics of N_TB LC droplets. (a) Model of striped domains formed by cap-shaped N_TB LC droplets on bare glass and rubbed PVA substrates; copyright 2019, RSC Publication [82]. (b,c) Satellite droplets carrying −1 defects move in opposite directions around their companion N_TB LC droplets. As collinear antiparallel dipoles approach, the satellite droplets are squeezed together; copyright 2020, RSC Publication [83]. (d) A typical topological dipole observed in the N-N_TB coexistence region of C-TA5 in a planar-aligned LC cell, viewed through crossed polarizers p and A. The N_TB LC droplet’s twist axis radiates outward from a +1 strength point singularity, generating a −1-defect satellite in the surrounding nematic region (dark area) with hyperbolic field lines. The far-field director $\overrightarrow{\mathbf{n}}$ is oriented along the x-direction. Copyright 2020, RSC Publication [84]. (e) Schematic representation of the director field around an N_TB droplet. The elastic dipole $p_{\mathrm{t}}$ points along the −x direction, while the associated flexoelectric dipole $p_{\mathrm{e}}$ acts in the opposite direction, along + x. Copyright 2021, APS Publication [84]. (f,g) A large N_TB droplet in its PFCD (planar focal conic defect) geometry, formed in the isotropic liquid phase within a 5-μm-thick layer of C-OP7 at 88.4 °C. The plus and minus signs in (f) indicate the sense of rotation of the extinction crosses at these PFC (planar focal conic) sites relative to changing azimuth of crossed polarizers; copyright 2023, APS Publication [85]; (h,i) N_TB droplets of ER geometry in a 5-μm-thick layer of C-OP7 at 88 °C for + 6 V (h) and −6 V (i) of the applied square-wave field of frequency 50 mHz. Note that the extinction crosses (encircled in white lines) in all ER drops (except the composite one at bottom right) switch between opposite poles as the field polarity reverses; copyright 2023, APS Publication [85].

Figure 5. Complex structures and dynamics of N_TB LC droplets. (a) Model of striped domains formed by cap-shaped N_TB LC droplets on bare glass and rubbed PVA substrates; copyright 2019, RSC Publication [82]. (b,c) Satellite droplets carrying −1 defects move in opposite directions around their companion N_TB LC droplets. As collinear antiparallel dipoles approach, the satellite droplets are squeezed together; copyright 2020, RSC Publication [83]. (d) A typical topological dipole observed in the N-N_TB coexistence region of C-TA5 in a planar-aligned LC cell, viewed through crossed polarizers p and A. The N_TB LC droplet’s twist axis radiates outward from a +1 strength point singularity, generating a −1-defect satellite in the surrounding nematic region (dark area) with hyperbolic field lines. The far-field director $\overrightarrow{\mathbf{n}}$ is oriented along the x-direction. Copyright 2020, RSC Publication [84]. (e) Schematic representation of the director field around an N_TB droplet. The elastic dipole $p_{\mathrm{t}}$ points along the −x direction, while the associated flexoelectric dipole $p_{\mathrm{e}}$ acts in the opposite direction, along + x. Copyright 2021, APS Publication [84]. (f,g) A large N_TB droplet in its PFCD (planar focal conic defect) geometry, formed in the isotropic liquid phase within a 5-μm-thick layer of C-OP7 at 88.4 °C. The plus and minus signs in (f) indicate the sense of rotation of the extinction crosses at these PFC (planar focal conic) sites relative to changing azimuth of crossed polarizers; copyright 2023, APS Publication [85]; (h,i) N_TB droplets of ER geometry in a 5-μm-thick layer of C-OP7 at 88 °C for + 6 V (h) and −6 V (i) of the applied square-wave field of frequency 50 mHz. Note that the extinction crosses (encircled in white lines) in all ER drops (except the composite one at bottom right) switch between opposite poles as the field polarity reverses; copyright 2023, APS Publication [85].

6. Ferroelectric Nematic LC Droplets

As a special kind of recently discovered nematic phase, the ferroelectric nematic (N_F) phase has the characteristics of spontaneous polarization [87]. Because the molecules have an aligned, permanent electric dipole moment, they exhibit a fast response under an electric field [87]. This characteristic makes it important for applications in displays and optical switches with a high response speed. The ferroelectric nematic phase is a recently discovered variant of the nematic phase with polarity [88]. In the traditional uniaxial (N) phase, unlike isotropic liquids, LCs are non-polar due to head-to-tail invariance, which has been widely used in liquid crystal displays (LCDs) and other electro-optical devices. Unlike the conventional N phase, the molecular orientation in the N_F phase is extremely sensitive to electric fields, making them promising materials for thin, flexible, and reconfigurable ferroelectric devices. Nishikawa, et al. and Mandle, et al. respectively synthesized highly polar rod-like compounds 2, 3’, 4’, 5’ -tetrafluorobiphenyl-4yl 2,6-difluoro-4-(5-propyl-1,3-dioxan-2-yl)benzoate (DIO, Iso-174 °C-N-84 °C-SmZ_A-69 °C-N_F-34 °C-Cr) [89] and 4-[(4-nitrophenoxy) carbonyl] phenyl 2, 4-dimethoxybenzoate (RM734, Iso-188 °C-N-133 °C-N_F-70 °C-Cr) [90]. Both RM734 and DIO have large longitudinal molecular dipole moments and side groups, such as methoxyl groups or highly fluorinated groups.

Figure 6 shows polarizing optical microscope images of RM734 submillimeter diameter-fixed droplets heated from the bottom at four different temperatures near the N–N_F phase transition (Figure 6a–d,a”–d”) [91]. In phase N, the director is perpendicular to the solid substrate (vertical orientation), which is achieved through a thin polyimide coating. The four-brush texture seen in Figure 6a corresponds to a radial vector structure with a central top defect that is parallel (or tilted) at the upper curved interface, as shown in Figure 6a’,a” [91]. As the cooler top approaches the temperature of N_F (abbreviated as T_NF), the elastic constant ${\mathrm{K}}_{11}$ decreases, possibly due to the presence of flexoelectric (${\mathrm{F}}_{\mathrm{l}}$) polarization caused by the bending of the radial vector structure. When the radial flexural polarization can no longer be shielded by the free charge, an internal electric field is formed, forcing a helical distribution of vectors at the top to prevent polarization from terminating on the insulating surface, resulting in the texture and vector structure seen in Figure 6b,b’,b” [91]. Further cooling at the cooler top through the N→N_F phase transition creates ferroelectric polarization, resulting in an internal electric field, and the polarization field becomes tangential. A sketch of this texture and the corresponding vector configuration is shown in Figure 6c,c’,c” [91]. When the ferroelectric phase reaches the bottom, the polar vertical surface orientation causes an internal electric field that rotates the vector and the polarization tangentially throughout the droplet, except near the central defect. This texture and the corresponding vector structure are shown in Figure 6d,d’,d”. Perera, et al. have demonstrated electromechanical effects in ferroelectric nematic LC droplets coexisting with isotropic melt [92]. Figure 6e presents the structures and positions of two nuclei without an electric field, and 3 s after the application of a 0.6 mV/μm electric field in the x direction, two synchronous motions are observed [92]. Figure 6f shows that a N_F droplet moves toward the illuminated area or away from it, relying on the direction of the lithium niobite (LN) bulk polarization with respect to the direction of the incoming light [93]. Second harmonic generation (SHG) microscopy and SHG interference (SHG-I) microscopy data (Figure 6g,h) help us distinguish the relative polarization orientation in N_F droplets. The SHG-I signal shows a π-phase shift between the upper and lower halves of the CV and L-shaped structures, indicating that the polarization on both sides of the distortion is opposite. The current difference between CV and L-shaped structures is that there is a visible distortion line in the L-shaped structure. The distortion line separates two regions with opposite polarizations [94].

Nonlinear light generation with an imprinted geometric phase has been demonstrated in ferroelectric nematic droplets, with second-order nonlinear susceptibility leading to a spin-dependent nonlinearity [95]. By utilizing linear and nonlinear light–matter interactions, second harmonic optical vortices with spin-locked topological charges are presented [95]. The dynamic tunability of second harmonic structured light through temperature, electric field, and twist elastic forces is presented, with potential applications in optical communications, high-dimensional quantum information processing with a large capacity and high security, high-resolution imaging, and spatiotemporal optical vortices [95].

Figure 6. (a) POM images and structures of a fixed LC droplet near the phase transition of nematic→N_F. Upper temperature: T-δT; lower temperature: T. (a) Radial vector structure in N phase (T-ΔT > T_NF); (b) the upper temperature is close to T_NF; (c) the temperature above is lower than TN_F; (d) the entire droplet is in the N_F phase. The diagrams with single and double apostrophes show the side and top cross sections of the vector structure, respectively; copyright 2022, APS Publication [91]. (e) POM images of N_F nuclei formed at the Iso–N_F transition at zero electric field and via the application of an electric field in the x direction; copyright 2023, RSC Publication [92]; (f) sketches of the experimental arrangements in the case of droplet attraction (I) and repulsion (I′, II, III, and IV) video frames taken at various instants presenting an RM734 N_F droplet moving toward the center of the illuminated area; II′, III′, and IV′ video frames taken at various instants showing an RM734 N_F droplet moving away from the center of the illuminated area; copyright 2023, Wiley Publication [93]. (g,h) Microscopic images of SHG (second harmonic generation) and SHG-I (second harmonic generation interference) of N_F droplets, corresponding to CV (i) and L (j) structures, respectively. The SHG-I image is obtained under two interference conditions, in which the second harmonic signal phase of the reference quartz plate shifts π. The wavelength of the incident polarized fundamental field (represented by green double arrows) is 1064 nm. Scale: 10 μm. (i,j) Polarization fields of CV (i) and L-type (j) structures reconstructed based on the 4 × 4 matrix method and SHG-I experiment. Scale: 10 μm; copyright 2024, RSC Publication [94].

Figure 6. (a) POM images and structures of a fixed LC droplet near the phase transition of nematic→N_F. Upper temperature: T-δT; lower temperature: T. (a) Radial vector structure in N phase (T-ΔT > T_NF); (b) the upper temperature is close to T_NF; (c) the temperature above is lower than TN_F; (d) the entire droplet is in the N_F phase. The diagrams with single and double apostrophes show the side and top cross sections of the vector structure, respectively; copyright 2022, APS Publication [91]. (e) POM images of N_F nuclei formed at the Iso–N_F transition at zero electric field and via the application of an electric field in the x direction; copyright 2023, RSC Publication [92]; (f) sketches of the experimental arrangements in the case of droplet attraction (I) and repulsion (I′, II, III, and IV) video frames taken at various instants presenting an RM734 N_F droplet moving toward the center of the illuminated area; II′, III′, and IV′ video frames taken at various instants showing an RM734 N_F droplet moving away from the center of the illuminated area; copyright 2023, Wiley Publication [93]. (g,h) Microscopic images of SHG (second harmonic generation) and SHG-I (second harmonic generation interference) of N_F droplets, corresponding to CV (i) and L (j) structures, respectively. The SHG-I image is obtained under two interference conditions, in which the second harmonic signal phase of the reference quartz plate shifts π. The wavelength of the incident polarized fundamental field (represented by green double arrows) is 1064 nm. Scale: 10 μm. (i,j) Polarization fields of CV (i) and L-type (j) structures reconstructed based on the 4 × 4 matrix method and SHG-I experiment. Scale: 10 μm; copyright 2024, RSC Publication [94].

7. Conclusions

LC droplets can function as microcavity resonators, in which light undergoes total internal reflection, forming whispering gallery modes suitable for micro-lasers or highly sensitive sensors. External electric fields and temperature changes can significantly alter the optical properties of LC droplets, enabling fast optical switches or tunable optical elements. Moreover, these droplets can exhibit nonlinear optical effects and unique light scattering properties, even displaying optical bistability under certain conditions. Chiral LC droplets can selectively reflect or transmit circularly polarized light, offering potential for circular polarization filters. Due to the nonlinear optical behavior of N_F LC droplets, they show potential applications in structured light for optical communication, high-dimensional quantum information processing with a large capacity and high security, and high-resolution imaging. These characteristics make LC droplets a focus of research and applications, with the potential to drive advancements in novel display technologies, tunable lasers, optical sensors, etc. With the growing interest and continuous innovation in production and characterization techniques, exciting developments in LC droplets can be anticipated in the coming years.