Liquid crystals in micron-scale droplets, shells and fibers

The long-range orientational order of LCs is quantified by an orientational order parameter (also called nematic order parameter) that in 3D takes the form:

$$\begin{eqnarray}&&S=\frac{1}{2}\langle 3{{\cos}^{2}}\beta -1\rangle.\end{eqnarray} \tag{ 1 }$$

The angle β describes the deviation in orientation away from $\boldsymbol{n}$ of a particular molecule in the averaging ensemble, see figure 1. The pointed brackets here express averaging over 3D space, thus integrating over the angles β and φ, defined in figure 1.

In 3D, the order parameter (1) is zero for total disorder, whereas for complete orientational order, S  =  1. The isotropic-nematic transition in thermotropics is first-order, i.e. S jumps discretely at the phase transition from zero to a non-zero value, typically around S  =  0.4. On further cooling, S increases, initially fast but rapidly approaching saturation around S  =  0.6–0.7.

The mesogen anisometry together with the long-range orientational order generally render LCs optically anisotropic, i.e. their refractive index ${{n}_{\parallel}}$ parallel to $\boldsymbol{n}$ is different from that perpendicular to the director, ${{n}_{\bot}}$. The director defines the optic axis. The magnitude of optical anisotropy, the birefringence, is defined as $\Delta n={{n}_{\parallel}}-{{n}_{\bot}}$. Consequently, if ${{n}_{\parallel}}$ is the maximum refractive index, $\Delta n>0$ and we call the LC positive uniaxial (normal case for rod-shaped mesogens). Light passing through the sample will experience ${{n}_{\bot}}$ as the ordinary refractive index n_o, while the extraordinary index n_e varies depending on the direction of light propagation, ${{n}_{\parallel}}$ being the limiting value.

Like any birefringent material, LCs can affect the polarization $\boldsymbol{E}$ of light, changing it continuously between linear, elliptical and circular as well as changing the main oscillation direction or reversing rotation handedness. The effect disappears, however, if the light propagates along $\boldsymbol{n}$, because a birefringent material appears isotropic to light traveling along the optic axis: since light is a transverse wave, $\boldsymbol{E}$ is then by necessity perpendicular to the optic axis, hence the light experiences only n_o. An LC sample that is prepared such that the director is perpendicular to its boundary is called homeotropic, a name that reflects the 'isotropic-like' appearance for light entering the sample at normal incidence, along the director. If a flat homeotropic LC sample is placed between crossed polarizers during observation and if the alignment persists throughout the sample (also the other boundary must thus be homeotropic), it appears black. When working with curved samples, like those considered in this review, the meaning of the term 'homeotropic' as director perpendicular to the boundary is carried over. Note, however, that the curvature then means that light will generally not enter the sample along the director, and the curved sample will thus not (except in certain points) appear black.

The geometry orthogonal to homeotropic, where $\boldsymbol{n}$ is in the plane of the sample boundary, is called planar. If the sample is flat and if also the opposite boundary promotes the same planar alignment, such that $\boldsymbol{n}$ is uniform throughout the sample, the full birefringence $\Delta n$ is experienced by light entering at normal incidence. Inside the LC the light is split up into two eigenmodes with polarization perpendicular and parallel to $\boldsymbol{n}$, respectively, experiencing the refractive indices ${{n}_{\text{o}}}={{n}_{\bot}}$ and ${{n}_{\text{e}}}={{n}_{\parallel}}$. Homeotropic and planar anchoring are the two limiting cases of anchoring, but the anchoring conditions can often promote an arbitrary tilt of $\boldsymbol{n}$ out of the interfacial plane. A tilt of 0° corresponds to planar alignment, a tilt of 90° is homeotropic, whereas we here refer to anchoring with any other inclination simply as tilted.

For curved samples the situation is again more complex. As for the term 'homeotropic', the terms 'planar' and 'tilted' are carried over in the sense of how $\boldsymbol{n}$ is oriented with respect to the sample boundary. A planar interface thus has the director in the boundary plane, regardless of how this curves. This means that, across the surface of a planar-aligned curved sample, $\boldsymbol{n}$ and thus the optic axis change direction with respect to the light propagation. The resulting optical behavior is very different compared to a planar flat sample. We refer the reader to an optics textbook for a deeper analysis, here settling by pointing out that the effective birefringence in most points is lower than for a flat planar sample, the difference depending on the inclination of $\boldsymbol{n}$ in each point. The reduction in $\Delta n$ is partially compensated by an increased length of light passage through the curving LC. For a shell, moreover, we frequently have top and bottom surfaces and—as discussed in section 3.3—both contribute to the birefringence, even if only one of them is in focus.

If light propagates perpendicular to a uniformly planar-aligned LC, the orientation that $\boldsymbol{n}$ has within the sample plane—which is then equal to the light's polarization plane—is also very important for the optical effect of the LC. For tilted alignment the same holds for the orientation of the projection of $\boldsymbol{n}$ onto this plane, which we here call $\boldsymbol{n}_{{p}}$. Whenever the light polarization $\boldsymbol{E}$ is along or perpendicular to $\boldsymbol{n}$ (or $\boldsymbol{n}_{{p}}$), one of the two eigenmodes has zero amplitude, and the effect of the birefringence is lost. Therefore, when a sample is observed between crossed polarizers and $\boldsymbol{E}\parallel \boldsymbol{n}$ or $\boldsymbol{E}\bot \boldsymbol{n}$ (or $\boldsymbol{E}\parallel \boldsymbol{n}_{{p}}$ or $\boldsymbol{E}\bot \boldsymbol{n}_{{p}}$), it appears black. For all other polarizations both components exist, the birefringence changes the polarization state, and the sample typically appears bright between crossed polarizers, see figure 2.

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Planar alignment means that the director is in the sample boundary plane, and tilted alignment that it has a component $\boldsymbol{n}_{{p}}$ in that plane, but the terms say nothing about which orientation in this plane $\boldsymbol{n}$ or $\boldsymbol{n}_{{p}}$ should adopt. With a solid substrate a preferred orientation can often be induced, for instance by rubbing the substrate in a specific direction, but many planar- or tilted-aligning interfaces induce what is called 'degenerate planar/tilted alignment', giving no preference for any particular direction within the interface plane. In a nematic LC subject to such boundary conditions, the direction of $\boldsymbol{n}$/$\boldsymbol{n}_{{p}}$ in the interface plane can vary smoothly as a function of location. Since $\boldsymbol{n}$ defines the optic axis, this means that such a spatially varying director field $\boldsymbol{n}\left(\boldsymbol{r}\right)$ is recognized in the polarizing microscope through a characteristic schlieren texture, named after the German word for streak. An example is shown in figure 2. Softly curving black 'brushes' appear, often emanating from crosses or cusps that are defects in the director field (see section 3.3). Along a black brush, $\boldsymbol{n}$ is parallel to one of the polarizers.

While nematics normally can flow easily, it makes a great difference if the flow is along or perpendicular to $\boldsymbol{n}$, and, for the latter case, it also makes a difference if the flow velocity gradient is along or perpendicular to $\boldsymbol{n}$. These three situations, depicted by three orthogonal director orientations $\boldsymbol{n}_{{1}}$, $\boldsymbol{n}_{{2}}$ and $\boldsymbol{n}_{{3}}$ in figure 3, were investigated by the Polish physicist Marian Mięsowicz [4], after whom the three corresponding viscosities, ${{\eta}_{1}}$, ${{\eta}_{2}}$ and ${{\eta}_{3}}$ are now called the Mięsowicz viscosities. The lowest viscosity is ${{\eta}_{2}}$, when the flow is along the director, whereas the highest is ${{\eta}_{1}}$, measured when the director is perpendicular to the flow but parallel to the flow velocity gradient. The third director orientation, $\boldsymbol{n}_{{3}}$, perpendicular to both flow and gradient, results in an intermediate viscosity ${{\eta}_{3}}$, generally much closer to ${{\eta}_{2}}$ than to ${{\eta}_{1}}$. The Mięsowicz viscosities are typically on the order of a Poise, or 1kg m⁻¹ s⁻¹ in SI units.

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The Mięsowicz viscosities constitute the first approximation for understanding the viscous flow of LCs. If we make no effort to control the director orientation during flow, $\boldsymbol{n}$ will align nearly parallel to the flow to minimize viscous losses. The minimum viscosity is actually obtained for a slight offset, as a more rigorous treatment shows [5]. Flow alignment can be important in the study of drops, shells and fibers of LCs, as the sample preparation always involves flow. If the samples are prepared with the material in its nematic phase, the viscosity anisotropy will lead to a relatively uniform alignment during the process. Note, however, that for smectics (see section 2.2) the minimum dissipation is not along $\boldsymbol{n}$ but along the smectic layers. Moreover, in many cases the LC is brought into an isotropic state during sample preparation, by heating or by adding a solvent.

Liquid crystals, with nematics forming the clearest example, exhibit a type of elasticity that is unique to this materials class. This 'nematic elasticity' is fundamentally different from solid or rubber elasticity, since in this case there is no restoring force arising in response to deformation of the sample shape, but instead a restoring torque that builds up in response to deformation of the director field $\boldsymbol{n}\left(\boldsymbol{r}\right)$.

In the elastic ground state of an achiral nematic (ignoring thermally activated fluctuations [1]) the director field is uniform, i.e. $\boldsymbol{n}\left(\boldsymbol{r}\right)$ is a constant, independent of location. A deviation from the uniform field costs energy, and this is what gives rise to nematic elasticity. This energy is quantified in the Oseen–Frank theory, based on Oseen's groundwork in the 1920s and the refinement, and introduction of modern terminology, by Frank in 1958 [2]. In the bulk of a nematic, any generic deformation from uniform $\boldsymbol{n}\left(\boldsymbol{r}\right)$ can be described as a linear combination of three elementary elastic deformations: splay, twist and bend. These three fundamental distortions, defined graphically in figure 4, are independent of each other, such that no combination of twist and/or splay can compensate for a bend etc. The total free energy F_b related to bulk elastic deformations of the director field can thus be calculated by integrating a bulk energy density f_b, containing one term each for splay, twist and bend, over the volume of the sample (if S is not uniform within the volume considered, a fourth term, saddle-splay, to be discussed in a moment, must be included as well):

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{F}_{\text{b}}}={\int}^{}\text{d}V{{f}_{\text{b}}}, \end{array}\end{eqnarray} \tag{ 2a }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{f}_{\text{b}}}=&\frac{1}{2}{{K}_{1}}{{\left(\nabla \centerdot \boldsymbol{n}\right)}^{2}}+\frac{1}{2}{{K}_{2}}{{\left[\boldsymbol{n}\centerdot \left(\nabla \times \boldsymbol{n}\right)\right]}^{2}} \\ &+\frac{1}{2}{{K}_{3}}{{\left[\boldsymbol{n}\times \left(\nabla \times \boldsymbol{n}\right)\right]}^{2}}. \end{array}\end{eqnarray} \tag{ 2b }$$

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Reflecting its character as a source, the splay term in (2b) contains the divergence of the director field ($\nabla \centerdot \boldsymbol{n}$), whereas the terms representing the two rotational distortions twist and bend contain the curl ($\nabla \times \boldsymbol{n}$). Because each term is squared, the bulk elastic constants, K₁ for splay, K₂ for twist and K₃ for bend, must all be positive, otherwise there would be no finite energy minimum.

When considering strongly confined LC samples such as drops, shells and fibers, one must also consider the influence from the bounding interfaces. There are two contributions to the interfacial energy. First, as with any interface, there is an energy term given by the product of interfacial tension and the bounding area. However, since this is an LC the interfacial tension may depend on how $\boldsymbol{n}$ is aligned with respect to the boundary. We say that $\boldsymbol{n}$ is anchored in a certain preferred orientation at the interface. This terminology should not be interpreted too literally, however, as we may have anything from very strong to very weak anchoring. The balance between surface anchoring and bulk elasticity will be discussed for the case of droplets in section 3.3.1. The anchoring energy is quantified by a parameter W. A common expression for this contribution to the interfacial energy is the Rapini–Papoular energy density (an area density, thus J/m²) [6, 7]:

$$\begin{eqnarray}&&{{f}_{\text{a}}}=\frac{W}{2}\left[1-{{\left(\boldsymbol{n}\centerdot \widehat{\boldsymbol{v}}\right)}^{2}}\right]\end{eqnarray} \tag{ 3 }$$

where the unit vector $\widehat{\boldsymbol{v}}$ denotes the preferred orientation of the director at the interface, and we let the index 'a' stand for 'anchoring'.

The second interfacial energy contribution is the just mentioned saddle-splay term, which actually is an elastic energy related to director field distortion in the Oseen–Frank theory. As the name suggests, the term, quantified by the saddle-splay constant K₂₄, relates to a deformation of the director field such that $\boldsymbol{n}\left(\boldsymbol{r}\right)$ is normal to a saddle (or potato chip) surface [8]. Such surfaces have negative Gaussian curvature, i.e. their two principal curvatures have opposites signs; a saddle curves upwards along the horse but downwards across the horse. The corresponding energy density is a pure divergence ($\nabla \centerdot$) term:

$$\begin{eqnarray}&&{{f}_{24}}=-\left({{K}_{24}}/2\right)\nabla \centerdot \left(\boldsymbol{n}\nabla \centerdot \boldsymbol{n}+\boldsymbol{n}\times \nabla \times \boldsymbol{n}\right).\end{eqnarray} \tag{ 4 }$$

This means that its volume integral, required to calculate the corresponding energy, can be converted into a surface integral via the divergence theorem (Gauss' theorem), provided that K₂₄ is constant within the integration volume:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{F}_{24}}={\int}^{}\text{d}V{{f}_{24}} \\ \quad \;=-\frac{{{K}_{24}}}{2}{\int}^{}\text{d}\boldsymbol{S}\centerdot \left(\boldsymbol{n}\nabla \centerdot \boldsymbol{n}+\boldsymbol{n}\times \nabla \times \boldsymbol{n}\right). \end{array}\end{eqnarray} \tag{ 5 }$$

Note that the integrand in the last term in (5) is not f₂₄, as defined in (4), which is a volume energy density, J/m³. Compiling the above, we can obtain the total free energy (in the absence of fields and for constant K₂₄ within the sample volume) as [7]:

$$\begin{eqnarray}&&F={{F}_{\text{b}}}+{{F}_{24}}+{\int}^{}\text{d}S{{f}_{\text{a}}}.\end{eqnarray} \tag{ 6 }$$

A key difference between the F₂₄ term and the other contributions to the free energy is that F₂₄ enters with negative sign. In other words, while any deformation described by F_b raises the free energy, as does any increase in interfacial area, captured by ${\int}^{}dS{{f}_{\text{a}}}$, a saddle-splay deformation of the bounding interface actually decreases the energy if K₂₄  >  0. This seems to be the case for all materials that have been experimentally studied so far with respect to saddle-splay deformation.

Because the saddle-splay term in case of constant K₂₄ does not influence the bulk energy far from boundaries it has often been ignored, as the sample size was reasonably large in many studies. Moreover, when the sample boundaries are flat, as in standard display-like devices, the F₂₄ contribution vanishes entirely. In contrast, in our focus on small samples with strongly curved boundaries, where the interface influence can be considerable, the F₂₄ contribution can play an important role. If K₂₄ varies within the sample volume, which should be the case if the order parameter S is non-uniform, Gauss' theorem cannot be applied and the saddle-splay contribution must be treated as a bulk elastic energy. Such a situation is not as exotic as one might at first think; at temperatures near the clearing point a temperature gradient within the sample would lead to varying S and thus varying K₂₄, but even at uniform temperature, S decreases in the close vicinity of a topological defect or an impurity. As droplets, shells and fibers frequently comprise topological defects, the saddle-splay term may need to be treated with its full bulk character².

The four elastic constants are significant material parameters, with key relevance for understanding the behavior of strongly confined LCs. When the boundaries are strongly curved, as in the cases considered in this review, the boundary conditions often impose elastic distortions within the LC, thereby raising the free energy. The elastic constants have the dimension J/m  =  N, i.e. a force, with a magnitude typically in the pN range. Note that the saddle-splay energy density (4) contains no squared terms, hence the saddle-splay constant K₂₄ can, in principle, take negative as well as positive signs, in contrast to the bulk elastic constants. For thermotropics, K₁ and K₂ often have about the same value whereas K₃ is about twice that value. A commonly applied convenient approximation is $K={{K}_{1}}={{K}_{2}}={{K}_{3}}$ and K₂₄  =  0, a simplification referred to as the 'one-constant approximation'.

Since a spatial derivative yields an inverse length, each term in (2b) is proportional to 1/R², where R is the characteristic length of the respective deformation. This is the pitch in case of a twist, the radius of curvature of a bend, and the distance to the 'source' of a splay deformation. Note that R is independent of location for a twist of constant magnitude, while, in contrast, the characteristic length of bend and splay is only defined once we have chosen a certain location. A bend or splay gets less apparent the further away from the center of curvature or 'source' that we are.

The 1/R² dependence and the small magnitude of the elastic constants means that elastic deformations with large characteristic length cost very little energy, so little in fact that large-scale deformations are thermally generated in a bulk nematic LC in its ground state. This means that large-scale non-uniformities arise spontaneously in bulk LC samples, giving them a characteristic turbid character. A uniform $\boldsymbol{n}\left(\boldsymbol{r}\right)$ actually has higher free energy, which we may provide, e.g. by applying a magnetic field that aligns the LC [1].

The stabilization of liquid crystal shells/droplets can be ensured by adding interface stabilizers, also called emulsifiers, most often to the aqueous phases. Compared to water a thermotropic LC can be considered an oil phase, immiscible with water, thus the stabilizers are needed to reduce the interfacial tension between the phases. Without stabilizers, droplets would coalesce and shells would collapse into droplets. Importantly, the stabilizers may also play another key role, namely to determine the liquid crystal alignment.

The common choice for emulsifier/stabilizer is either an ionic surfactant, e.g. sodium dodecyl sulfate (SDS), or a water soluble polymer, e.g. polyvinylalcohol (PVA). At the water–LC interface, a surfactant adsorbs with its apolar aliphatic chain penetrating into the LC, promoting homeotropic alignment (see section 2.1.2) and thus a radial orientation of the director field in the shell/droplet. In contrast, a dissolved polymer like PVA in aqueous solution has a disordered random coil configuration which cannot be expected to promote a particular type of ordering in the LC. The polymer thus most likely acts solely as a stabilizer, leaving the LC-water interaction to determine $\boldsymbol{n}$ at the interface, which ends up being planar or possibly tilted. This is because the mesogens typically contain an aromatic core, and often also a cyano group, which can participate in hydrogen bonding with the surrounding water, whereas the aliphatic chains at the mesogen end are avoided by the water. In case of the shells, we can additionally achieve hybrid director alignment [83, 84] by imposing asymmetric boundary conditions at the interfaces to the LC. In this case we may use a surfactant in the inner phase and PVA in the outer, or vice versa.

As a first approximation, we can consider stabilizers to be localized to the LC–water interface and the immiscible phases to be truly immiscible, i.e. the LC is in its pure state within the actual shell. Because the phase sequence of LCs is so sensitive to contaminants, however, it is easy to find evidence that this simplified picture is not quite true in practice. We recently reported that the clearing point of nematic shells of the commonly used mesogen 5CB is distinctly lower than that of the bulk LC [85]. Interestingly, the study revealed that all constituents of the surrounding aqueous phases can enter the shell to some extent, water, surfactant and glycerol, which is often added to the aqueous phases for density and viscosity tuning. Moreover, an alignment change was detected in a long-term experiment with shells that were initially hybrid-aligned thanks to preparation with surfactant in one aqueous phase and PVA in the other. Over the course of a few days, the alignment was transformed into fully homeotropic, demonstrating that the surfactant can pass through the LC shell and adsorb on the opposite side.

By varying the orifice sizes of capillaries, interfacial tensions, viscosities, flow rates etc during the microfluidic production, we can tune the diameter and thickness of LC shells over a considerable range. However, it is challenging to produce the shells with a diameter smaller than about a 100 μm or a thickness below a few μm. If a thinner and larger or smaller and thicker shell is desired, an elegant trick can be applied for post processing, introduced by Lopez-Leon and co-workers [87], involving osmosis. Because the LC is not impenetrable to water, the LC shell can act like a semipermeable membrane separating the inner and outer phases, and if these have non-equal compositions, an osmotic pressure will develop that drives water into or out of the shell's inner isotropic droplet.

Although it was not properly elucidated until the mid 19th century, osmosis is a very common phenomenon on Earth, playing an important role in many vital and central processes in biological systems [88]. Originally osmosis is used to describe the movement of a solvent across a semipermeable membrane towards a higher concentration of solute, which is driven by the chemical potential difference between the two sides of the membrane [88, 89]. The membrane can be solid or liquid and it can be formed by layers of cells, particles and molecules [88]. The permeability of the membrane depends not only on the size of holes in the membrane, compared to the size of the solvent molecules, but also on the polarity of the solvent molecules and their miscibility with the material making up the membrane.

Lopez-Leon and co-workers introduced osmosis into LC shell studies as a method to reduce the shell thickness by increasing the size of the inner droplets [87]. Water based solutions were used as the inner and outer phase with different salt concentrations. Compared with the salt, water is more miscible with the liquid crystal, so the water can flow from low salt concentration side to the side with high concentration by passing through the LC shell. The shell thickness change can happen in both directions: by giving the outer phase a higher salt concentration than the inner phase, water would flow from inside the LC to the outside, making the shell thicker and the diameter smaller. By tuning the thickness of the nematic LC layer, the elastic free energy related to the defect arrangement is changed, allowing the authors to control which defects are formed and how they are distributed across the shell, see section 3.3.4.

Osmosis was also used in our recent studies to help remove the oily streak defects in cholesteric shells with planar anchoring on both sides of the LC layer [86]. Instead of using salt content difference to establish an osmotic pressure, we used higher concentration of PVA in the inner droplets (15wt.%) than in the outer continuous phase (10wt.%). As a result, the shells were expanded and became thinner over the course of several hours, with a consequent rapid disappearance of oily streak defects (figure 11). Already after 12 h the shells are free of visible defects, which should be compared to about a week without osmosis. In the process of osmotic expansion, the shell diameter increased from about 180 μm–240 μm, and the shell thickness decreased from 27 μm to 11 μm.

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Observable textures in nematic droplets differ for various droplet sizes and anchoring conditions and the respective equilibrium director field configuration strongly depends on bulk and surface energy contributions. The bulk elastic energy F_b scales linearly with the droplet radius R [91], while the surface energy F_s scales as ${{F}_{s}}\propto W{{R}^{2}}$ [105], where the coefficent W is the amplitude of the angular dependent anchoring energy from (3). Consequently, the energetic contribution of the interface dominates for large droplet radii. This relation might seem counterintuitive, since the ratio of surface area to bulk volume increases upon decreasing droplet radius and one might expect a dominating surface contribution for small droplets. The reason is the 1/R² dependence of the bulk elastic energy density, and experimental observations confirm that influence of the topological constraints imposed by the interface vanishes for small droplet sizes. The director configuration is then determined solely by the bulk elasticity of the liquid crystal [105]. A critical radius R_c  =  K/W (in the one-constant approximation) with a transition from small droplet to large droplet behavior is, depending on the anchoring energy W and bulk elasticity K, in the order of 1 μm [91].

In small droplets (R  <  K/W) with homeotropic anchoring the bulk elastic energy enforces a defect and deformation-free uniaxial director configuration, which disregards the homeotropic boundary condition and the topological constraints related to it [105, 106]. A schematic representation of such a director configuration is shown on the top left side of figure 14. Upon increasing the droplet size, the surface contribution also increases and for R  >  R_c the homeotropic anchoring conditions impose a bulk charge Q of unity within the droplet [107]. The topological constraints imposed by the Gauss-Bonnet theorem are fullfilled by forming a radial director field with a hedgehog defect in the center of the droplet (see figure 14 top-right) [84, 107]. This centrosymmetric director field then typically relaxes away from the symmetric radial configuration to a lower energy twisted state, which appears as a spiraling cross between crossed polarizers [5, 107].

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Nanometer-sized droplets with planar or tilted anchoring conditions exhibit a radial director configuration in equilibrium [108, 109], while a bipolar configuration is frequently observed in larger droplets (see figure 14, bottom). Tomar et al showed by numerical simulations that this transition from bipolar to radial upon decreasing droplet dimension is connected to the saddle-splay term (see (4) in section 2.1) in the elastic free energy [105]. For large nematic droplets (R  >  K/W) with planar or tilted anchoring the topological constraints expressed by the Poincaré-Hopf theorem require a total topological surface charge of s  =  +2 at the droplet interface. The liquid crystal typically meets these constraints by forming bipolar drops, which are characterized by two boojums with topological charge s  =  +1 separated by the maximum distance of a drop diameter [84]. The director field within the droplet is then aligned parallel between these two radial half hedgehogs, corresponding to pure splay deformation close to the defects and bend in the droplet bulk [103].

A second possible director configuration that fullfills the topological constraints, the concentric configuration with two bend disclinations of surface charge s  =  +1, is only observed for unusually low splay elastic constant, in comparison to the twist and bend constants [103]. The surface is here decorated by line rather than point defects, but—as stated in section 3.3—integer line defects are not stable. The line defect 'escapes in the third dimension' via a twisted configuration, yielding a virtual line defect connecting the two surface defects. There is no defect core, but the bulk is still much more distorted than in the usual bipolar configuration [110]. As shown by Fernández-Nieves and coworkers, this unstable concentric configuration is easily transformed into the stable bipolar configuration by applying external electric fields to align the director field [103]. The authors also show that a retransformation from bipolar to escaped concentric does not occur and contribute this observation to a significant energy barrier [111] between the two configurations and an overall lower energy for the bipolar configuration. According to their findings, only the large forces occuring during circular flow inside an LC droplet allow a transition from bipolar back to the escaped concentric configuration.

A continous transition between the bipolar and the radial director field configuration can be observed upon changing the anchoring conditions from planar to homeotropic or vice versa, as demonstrated by Volovik and Lavrentovich for large drops [91, 97]. The transformation may take place via a so-called axial configuration, in which the surface is already homeotropic but an equatorial defect ring allows a non-radial internal director field [91].

The first description of cholesteric droplets dates back to Lehmann and Friedel [112] at the beginning of the 20th century, but only from the 1980s onwards a systematic study of the complex configurations in higher ordered liquid crystal droplets began [113]. In contrast to the nematic phase, the equilibrium director configuration in cholesteric droplets depends not only on the boundary conditions at the interface and the bulk elasticity, but it is also influenced by a factor qR, the product of the cholesteric helix wave vector (see section 2.3) and the droplet radius [113, 114]. The continuous twist allows for new types of disclinations in cholesteric liquid crystals, which can be categorized into the three classes ${{\lambda}^{\pm m}}$, ${{\tau}^{\pm m}}$ and ${{\chi}^{\pm m}}$ [93, 115]. Half-integer and integer values for m characterize the winding number of the respective defect type, schematic representations of which are shown in figure 15.

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Since a smooth unwinding of the cholesteric helix yields the uniaxial order of the nematic phase, a continuous transition from the director field configuration of a cholesteric droplet towards that of a nematic droplet can be expected. For cholesterics with a long pitch that are confined to small droplets (qR  <  1) with planar or tilted boundary conditions, Bezić and Žumer predicted an unwinding of the cholesteric helix and the formation of a bipolar structure as frequently observed for nematics [114]. Their model yielded that the director lies solely in the meridional planes of the droplet, which could not be confirmed by later experimental studies [116]. Instead, Xu and Crooker established a twisted bipolar model for cholesteric droplets, which differs from the bipolar model for nematics by higher twist within the droplet (see figure 16(a)). While in nematic bipolar droplets the twist is not intrinsic and decays exponentially from the surface towards the bulk, the model introduced by Xu and Crooker assumes a linear decay of twist [116].

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For larger cholesteric droplets with planar/tilted anchoring and/or short helix pitch (qR  >  1) the Frank-Pryce structure is the most commonly observed configuration. This structure is shown in figure 16(e) and consists of an s  =  +2 disclination line of length R that connects a hedgehog in the center of the droplet with its surface [114, 117]. The configuration is also referred to as radial spherical structure or spherulitic texture [115]. This structure also attracted attention beyond liquid crystal research, as it represents an intriguing analogy to an elementary magnetic charge known as Dirac monopole [91, 118]. Several studies report that the regular Frank-Pryce structure structure can relax into the so-called escaped Frank-Pryce structure, in which the director escapes along the direction of the disclination line [116, 119]. Bouligand and Livolant found that the Frank-Pryce structure with concentric layers can sometimes also occur with the defect line covering the whole diameter of the droplet [120]. Droplets with such structure show cylindrical symmetry as the disclination line connects two diametrical surface defects of topological charge s  =  +1 [114]. Frank-Pryce structures with a curved defect line connecting two non-antipodal s  =  +1 surface defects have only rarely been reported [120]. When a cholesteric droplet with Frank-Pryce structure is observed along the disclination line, one sees either a double spiral texture (usually for a defect line of length R) or concentric circles (typically for defect lines of length 2R or curved singular defect lines) [120].

Besides these experimentally observed structures in cholesteric droplets with planar or tilted anchoring, a more recent numerical simulation study by Seć and coworkers suggested that the s  =  +2 disclination line of the Frank-Pryce structure is actually unstable and can decompose into two s  =  +1 defect lines [115]. Based on their simulation results, the authors introduced two new director field configurations in cholesteric droplets: the Lyre-structure, which resembles the shape of a lyre instrument, symmetric around the vertical axis, and the Yeti structure, with two diametrically positioned boojums and higher symmetry than the Lyre structure [115].

For cholesteric droplets with homeotropic boundary conditions and high values qR  >  10, the conditions on the surface do not affect the packing of the helix. The textures in the droplets indicate that the Frank-Pryce structure prevails [113, 117]. For qR  <  5 the appearance of an equatorial disclination ring is observed [117]. For smaller qR values, this ring relaxes to a surface point defect, which then for even smaller qR  <  1 relocates to the center of the droplet, yielding the radial structure frequently observed in nematic droplets [117].

Director field configurations of smectic phases (SmA, SmC) in spherical confinement have been less intensively studied than for nematic or cholesteric phases. One reason for this is that smectic textures are typically highly complex with numerous macroscopic defects called focal conic domains [121, 122]. It is generally assumed that the smectic texture represents a packing of curved and equally spaced surfaces [123], as long as the dimension of confinement (typically in the μm range) is much larger than the spacing of smectic layers (typically  <  10 nm). Although Kleman and Lavrentovich showed that the filling of space by smectic layers is far from complete [124], this simplifying assumption still allows a reasonable description of smectic textures in spherical confinement [122].

For spherical confinement and planar boundary conditions, a bipolar configuration represents the simplest SmA texture [122]. As in the bipolar configuration for nematic droplets, two boojums (s  =  +1) at the surface are separated by the diameter and $\boldsymbol{n}\left(\boldsymbol{r}\right)$ connects the two defects along the shortest possible path throughout the sphere. The smectic layers are perpendicular to the director field and consequently run along the parallels of the sphere [125]. A topologically equivalent configuration is obtained by dividing the two boojums into four surface defects with topological charge $s=+\frac{1}{2}$, positioned along a great circle on the sphere [122]. Both configurations exhibit the same free energy [126] and can by principle be transformed one into another by cutting the bipolar configuration in two halves, through a plane containing both boojums, and subsequently rotating one half with respect to the other by an angle of $\frac{\pi}{2}$ [125]. A schematic representation of both configurations is shown in figure 17.

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Liquid crystal shells exhibit an even larger variety of equilibrium director field configurations than full droplets, as the presence of the inner interface imposes additional topological constraints to the director field. Beyond the classification of shells into homeotropic or planar/tilted shells, this second interface allows the formation of hybrid shells, i.e. shells with one interface imposing homeotropic alignment while the other interface imposes planar or tilted anchoring.

Early studies on the formation of topological defects in liquid crystal shells date back to the work by Lubensky et al [107] and Poulin et al [90] on multiple emulsions (i.e. water droplets dispersed in larger nematic droplets that in turn are dispersed in a continous aqueous phase). Due to a lack of a proper method for precise control over the shell dimensions during these early works, these studies derive information on $\boldsymbol{n}\left(\boldsymbol{r}\right)$ from topological constraints only and do not study the formation of defects in detail. The introduction of a microfluidic-based method by Fernández-Nieves and coworkers [127], as described in detail in section 3.2, allowed the preparation of droplet-in-droplet double emulsions with well- controlled radii of outer (LC) and inner droplet (e.g. water) in an immiscible continuous phase. This setup also allowed studying the effects of the frequently observed thickness heterogeneity of shells in more detail: Depending on elastic forces in the nematic shell and density differences between the LC and the fluid of the inner droplet, the latter is not located in the center of the shell but shifted along one direction, in case of a density mismatch along the direction of gravity. More advanced topological studies on the formation of defects in nematic shells then focused on the effect of shell thickness [87, 104], heterogeneity in thickness [128] or variations of elastic constants [87, 128]. As discussed in more detail in section 3.2.2, dynamic tuning of shell thickness and diameter was achieved by utilizing osmosis [87, 104].

Homeotropic-aligned nematic shells.

Planar-aligned nematic shells.

Nematic shells with planar anchoring at outer and inner interfaces show a larger variety of defect structures. For thick shells with small inner droplet a bipolar defect configuration with two pairs of opposing surface defects at the outer and inner shell surfaces is expected [129]. Experimental observations show that for such shells the defect pairs are either located along a shell diameter [87, 127], resembling the boojum structure of bipolar nematic droplets (see figure 18(i)), or situated close to each other at one side of the shell [84, 128], see figures 18(g)  +  (h). The latter defect configuration is observed for shells with high thickness heterogeneity, i.e. a large offset of the inner droplet from the center of the shell, and the defects are located at the thinnest part of the LC [84]. The equilibrium distance between the defects is given by the balance between elastic director field distortions in the bulk of the nematic and the repulsion between defects of same topological charge [130]. Within the shell, the bulk director deformation around each boojum core represents half hyperbolic hedgehogs on the inner surface, while the distortion close to the outer surface resembles half radial hedgehogs [84]. Each defect carries a topological surface charge of s  =  +1 and can be identified by four black brushes when observed between crossed polarizers [84].

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For thin planar shells a defect configuration with four surface defects constitutes the ground state [128, 131]. In the limit of infinitely thin shells the LC is confined to a two-dimensional surface and the disclinations reduce to points with no extension to the bulk. Each of the four defects carries a topological surface charge of s  =  +1/2, which reflects the π rotation of $\boldsymbol{n}\left(\boldsymbol{r}\right)$ along a path encircling each defect. The sum of topological charges is then  +2, as required by the Poincaré-Hopf theorem for the coinciding outer and inner interfaces [128]. Depending on the elastic anisotropy and the heterogeneity in thickness, the ground state with four defects can occur as tetrahedral arrangement of defects [131] (see figure 18(a)), as clustering of defects at the thinnest part of the shell [132] (figures 18(b)  +  (c)), or as arrangement of defects along a great circle [132, 133].

Fernández-Nieves et al [127] and Lopez-Leon et al [87] showed that the local defect clustering occurs in asymmetric shells only, as long as the average shell thickness is high enough that the reduction of the length of the defect cores can minimize the total free energy [132]. Later experimental studies showed that the four defects are then typically not symmetrically arranged around the thinnest part of the shell, but they adopt a skewed configuration of two pairs, where the defects in one pair are separated by a larger distance compared to the defects in the other pair [132, 133] (see figure 18(b)). By performing a basic polarization optics analysis, Liang et al noted that $\boldsymbol{n}\left(\boldsymbol{r}\right)$ between the short pair of defects mainly exhibits bend deformation, while $\boldsymbol{n}\left(\boldsymbol{r}\right)$ around the longer pair is mainly splay. Because bend defomations typically have greater energy densities than splay deformations (${{K}_{3}}\approx 2{{K}_{1}}$) the total elastic free energy is minimized by contracting the length of the bend disclinations at expense of the splay disclinations [132]. The difference in the free energy cost of director bend and splay thereby causes the frequently observed skewed distribution of surface defects.

Seč and coworkers later confirmed these observations by computer simulations [128]: The two closest defects are connected by a director field that runs along the shortest arc, whereas the other two defects are connected by the longest arc spanning a large fraction of a great circle (see figure 19). Increasing the anisotropy of elasticity further enhances the asymmetry between the two defect pairs. The simulations also showed that for larger ratios ${{K}_{3}}/{{K}_{1}}$ the defects connected by splay deformations move towards opposite sides of the sphere until they stabilize in the equatorial plane. Only after this first step, upon further increasing the elastic anisotropy, the defects of the bend pair progressively migrate towards the equatorial plane. This large ring configuration is expected to occur when the elastic splay deformation is strongly energetically favoured over bend deformation, as also confirmed by Monte Carlo simulations [126, 134]. Seč et al also showed that upon reducing the thickness heterogeneity of the shell, the defects increase their separation in a roughly symmetrical way and the ratio of separation between the two pairs of defects remains roughly constant [128].

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Experimental observations near the N-SmA transition, see figure 20, confirm this relationship between the defect configuration and the relative elastic constant values [128, 132, 133], as the ratio between K₃ and K₁ increases rapidly on approaching a second-order transition to the SmA phase. Moreover, we recently demonstrated that this relationship can be used to tune the defect location by moving closer to or further away from the N-SmA transition and then making the configuration permanent by polymer-stabilization [135]. However, it turns out that polymerization of the reactive component very close to the transition actually triggers the phase transition, even though the temperature rises due to the heat released during polymerization, an effect of the smectic-promoting character of acrylate reactive mesogens. In addition to defect configurations with two or four surface defects per interface, a third possible configuration is found for intermediate shell thickness which contains three topological defects forming an isosceles triangle [87], see figures 18(d)–(f).

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Buoyancy enhances the tendency for thickness heterogeneity in shells and favours vertical shifts along the field of gravity, so that only two scenarios (inner droplet shifted up or down) are experimentally observed. Seč and coworkers showed by numerical simulations that even in the absence of gravity heterogeneous shells are lower in energy compared to concentric shells [128]. These simulations also indicate that the total free energy would reach its minimum for a maximum displacement of the inner droplet. However, inner and outer surfaces of experimental shells have not been observed to coalesce [128]. The authors assume the existence of a disjoining pressure between outer and inner interface. This pressure is probably caused by the presence of PVA as a stabilizer. The PVA concentration increases as the nematic film gets thinner and this effect tends to homogenize the thickness [128]. In conclusion, buoyancy effects initiate and liquid crystal elasticity enhances a thickness heterogeneity of nematic shells by displacing the inner droplet from the center of the outer droplet. This symmetry breaking is irrespective of the elastic anisotropy [128].

Hybrid-aligned nematic shells.

In a recent experimental study on cholesteric shells with planar anchoring, Darmon and coworkers discussed bipolar and radial director field configurations with a higher complexity than director fields in cholesteric droplets [118]. A bivalent configuration with two topological defects of surface charge s  =  +1 at the outer and inner shell interface is observed for cholesteric liquid crystals with long pitch p₀. The pairs of boojums on one interface appear close to each other and the defects on the outer and inner interface are linked to a stack of disclination rings through the shell.

Liquid crystals with shorter p₀ lead to monovalent shells with a radial defect. This structure is reminiscent of the Frank-Pryce structure described above for cholesteric droplets. Darmon et al showed by polarising optical microscopy that the disclination line in this structure actually consists of two ${{\lambda}^{+1}}$ interwinded disclination lines forming a double helix [118]. The authors thereby provide an experimental confirmation for numerical simulations on cholesteric droplets by Seč et al, who were the first to suggest that the disclination line of the Frank-Pryce structure is not a ${{\chi}^{+2}}$ disclination, but is built from two ${{\lambda}^{+1}}$ disclinations, whose ends form a pair of close s  =  +1 defects at the outer surface while they form a monopole at the center of the droplet [115, 118]. A transition between the two possible configurations was shown to be of first-order. In a shell of thickness h the angular distance between the boojums is governed by the ratio h/R, while the transition itself depends on h/p₀ [118]. The optical behavior of cholesteric shells with sub-micron p₀ values will be discussed in section 3.5.

Experimental textures of planar SmA shells can be quite complex [104, 125, 132, 133], see figure 20. The equilibrium texture exhibits four topological defects of charge s  =  +1/2, located within the equatorial great circle, together with a set of additional lines which connect the defects by pairs [125]. The additional lines represent curvature walls that divide the sphere in crescent domains (spherical lunes) within which the smectic layers are undulated in a wavy fashion. Liang et al proposed that the undulation arises from a layer strain that is caused by the radially increasing spherical cross section area, coupled with the fact that the SmA phase is incompatible with director bend (see section 2.2) [132]. Dislocations would be needed to obtain a larger number of smectic layers at the outer interface compared to the inner interface, but the high free energy cost of such a solution renders it unfavoured.

Instead, the layers buckle increasingly on approaching the outer interface, increasing the effective smectic layer distance a₁ along the original director orientation compared to the true layer spacing a₀ (see figure 21). This reconciles the radially increasing surface area with the requirement for constant a₀. Lopez-Leon et al rather emphasized the possibility of a tilted instead of planar anchoring at the shell interfaces in order to release the layer strain [133]. In a subsequent theoretical treatment of SmA shells of varying thickness, Manyuhina and Bowick confirmed that both scenarios are possible [136]. In our recent study of polymer stabilization of LC shells [135] we demonstrated that the regular buckled geometry can be imprinted on the polymer network, rendering the instability-generated structure stable over a large temperature range, even after heating to the isotropic range of the host LC.

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In SmA shells with homeotropic anchoring at the inner and outer interfaces the characteristic cross-like texture as for nematic shells is observed, the only difference between nematic and SmA shells being a reduction of thermal fluctuations in SmA [104]. Liang et al also proposed that, compared to nematics, smectic LCs reduce the shell thickness heterogeneity and moves the inner droplet towards the shell centre [137]. In a homeotropic shell with high thickness heterogeneity, many smectic layers in the thicker part of the shell cannot be closed since fewer layers fit in the thinner part of the shell.

Smectic-A shells with hybrid alignment show focal conic domains as equilibrium director field configuration, mediating between planar/tilted alignment at the outer interface and homeotropic alignment at the inner interface, or vice versa. After the phase transition from the nematic phase, an unstable pattern of stripes, connecting the two poles of the shell, appears [104]. These stripes represent the boundaries of a folding membrane of remaining nematic order, which connects the smectic regimes with opposite alignment that grow from the two interface regions of the shell [83, 104]. Torza and Cladis observed the same type of texture much earlier for hybrid alignment in flat smectic samples, explaining it as a result of director bend being expelled from SmA [138]. The stripes are unstable since they come with an interfacial energy between two fluid phases. Hence they undergo a Rayleigh instability and break up into toric focal conic domains within which the layers bend, the SmA phase now spanning the full sample. In this configuration the SmA phase can accommodate the hybrid boundary conditions by only showing splay deformations of the director field [83].

A pecularity occurs for homeotropic SmC shells, as despite homeotropic anchoring the tilt in the smectic layers provides a tangential director field component at the interfaces and therefore requires the formation of topological defects [104]. Half-integer defects are ruled out because the C-director field is not sign invariant (see section 2.2), so a bipolar defect configuration with two opposing boojums at the outer and inner interface constitutes the ground state. Upon further cooling the SmC biaxiality becomes apparent and the texture distorts such that it does not allow to identify the number and charge of topological defects [104]. Liang et al propose that at low temperatures the SmC phase avoids the formation of defects by locally reducing the tilt to zero and shifting the system to a SmA state [104].

The attention early on focused on the interesting frustration caused when a nematic is confined in a narrow cylindrical space with homeotropic anchoring to the surface. The intuitive idea that an s  =  +1 wedge disclination line would form along the core, with a uniformly splayed, or Radial (R), director field within the Perpendicular (P) cross section (PR, see figure 28a) was early on recognized to be incorrect; the LC can in most cases reduce the elastic distortion energy by the escape into the third dimension (see section 3.3). In the cylinder this means that $\boldsymbol{n}$($\boldsymbol{r}$) bends into the cylinder axis as one moves from the periphery to the center. Indeed, this Escaped Radial (ER) configuration was confirmed by optical microscopy in 1972 by Williams, Pieranski and Cladis [110] and by Meyer [98]. Both studies used the nematic compound MBBA in glass capillaries where the inner wall was coated with a surfactant for homeotropic alignment. The capillary radius in these studies varied from 20 μm to 200 μm. Since the direction of escape ('up' or 'down' the cylinder axis) is arbitrary, Williams et al also found a regular set of $Q=\pm 1$ point defects where an upward escape encountered a downward escape and vice versa. Considering a certain length of the cylinder we may call this configuration Escaped Radial with Point Defects (ERPD, figure 28(b)).

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Some 20 years later the focus shifted to cylindrical cavities of radically smaller diameter. LCs were now filled into the pores of Nucleopore (polycarbonate) or Anopore (alumina) membranes, where the radius is frequently below 1 μm. Classic optical microscopy does not provide a resolution high enough to interrogate $\boldsymbol{n}$($\left. \boldsymbol{r}\right)$ in such thin channels, thus most studies were done using nuclear magnetic resonance (NMR). Crawford, Allender and Doane [175] filled selectively deuterated 5CB into a series of Nucleopore membranes, varying the pore radius from 0.05 μm to 0.5 μm. In some cases they left the pristine polycarbonate in contact with the LC, in others they coated the pore walls with lecithin, a classic agent for promoting homeotropic alignment. The key question was whether the ERPD configuration would still be energetically favorable in these narrow cylindrical cavities, as the required bend deformation would now take place over such a small range that the elastic distortion cost would be substantial.

The result depends on the anchoring strength: for strong homeotropic anchoring—surprisingly given by the naked polycarbonate—the ERPD configuration developed down to 0.05 μm pore radius. Lecithin actually gave weaker homeotropic anchoring, such that ERPD was found only at 0.5 μm radius. For thinner cavities another configuration developed, which we here refer to as Perpendicular cross section Polar (PP, figure 28(c))³. The homeotropic anchoring breaks down at two antipodes on the cylinder periphery. This allows a defect-free $\boldsymbol{n}\left(\boldsymbol{r}\right)$, entirely within the cross section perpendicular to the cylinder axis, with only moderate splay-bend deformation. Interestingly, the authors also succeeded in establishing the saddle-splay constant K₂₄ of 5CB, finding that it is comparable to the bulk elastic constants. Kralj and Zumer later proposed [176], based on numerical simulation, that a modification of the PP configuration may arise for strong homeotropic alignment. In this case two $s=+\frac{1}{2}$ disclination lines, developing along the cylinder axis, allows the homeotropic anchoring to be retained everywhere on the surface, while the deformations in the perpendicular circular cross section are moderate. We refer to this configuration as perpendicular cross section polar with line defects, PPLD (figure 28(d)).

In a complementary experimental study, Crawford et al studied Anopore membranes with 0.2 μm diameter pores, in which the alumina surface was coated with fatty acids of varying chain length, acting as surfactant [177]. The naked alumina promotes planar alignment whereas the fatty acid increasingly promotes homeotropic alignment the longer the chain. This allowed them to detect an alignment transition from the PP configuration for long fatty acids, to fully planar alignment for chain lengths of 6–7 carbon atoms. The LC thus adopts a uniform axial (UA) configuration, where $\boldsymbol{n}$ points along the cylinder axis everywhere (figure 30(a)).

Crawford, Stannarius and Doane conducted an interesting study where the high surface-to-volume ratio of Anopore membranes was used to induce nematic order at temperatures high above the bulk clearing point $T_{\text{NI}}^{B}$ [178]. Depending on the surface treatment slightly different results were obtained, but in all cases nematic order was detected close to the solid surfaces at least 15 K above $T_{\text{NI}}^{B}$. The same phenomenon was later detected by Lagerwall et al using Raman spectroscopy on a nematic mixture contained within electrospun fibers with a PVP-titania composite sheath [179]. Here anisotropic order was detected even 20 K above $T_{\text{NI}}^{B}$.

While the majority of experiments have been conducted on uniform confining cylinders, i.e. with constant radius (and the corresponding theory was developed for the same situation), Matthias and Kitzerow conducted a combined experimental-simulation study of the behavior of a nematic inside channels of sinusoidally varying diameter [180]. The channels were electrochemically etched out of a silicon substrate, and the walls were then coated with surfactant to ensure strong homeotropic alignment, hence the ERPD configuration was expected. The pore diameter was about an order of magnitude larger than in Anopore/Nucleopore membranes, varying between 2.2 μm at the necks and 3.3 μm at the bellies.

A smart choice was to work with a glass forming LC polymer. This allowed the researchers to fill the pores at high temperature, anneal the sample in the N phase until an equilibrium configuration is stable, and then quench-cool the sample into the glassy state. The silicon substrate was removed chemically, allowing a confocal microscopy characterization of the nematic glass. While the basic ER configuration was indeed confirmed, the hedgehog point defects split into disclination rings, the cross section of which is illustrated on the right in figure 29. Because of the sinusoidal diameter modulation the ring defects were regularly spaced, the positive-signed disclination rings appearing at the bellies and the negative-signed ones at the necks. These observations of LC behavior in cylinders with modulated diameter are highly relevant for the understanding of LCs in fibers, formed e.g. by electrospinning, since a certain modulation of fiber diameter is not uncommon, see section 4.2.

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This appears to be the first experimental report of hedgehog point defects splitting into disclination rings in cylinders, but it turns out that the modulated pore diameter is not necessary for this to occur. In fact, a number of theoretical papers predicted that point defects are unstable and should always be replaced by nanometer-sized disclination rings [95, 100, 105, 181–183]. In an elegant recent experimental study [184], the Abbott group verified that this is in fact the case for Q  =  +1 radial hedgehogs, replaced by an $s=+\frac{1}{2}$ line disclination bent into a nano-sized loop, but not for Q  =  −1 hyperbolic hedgehogs.

They filled a 400 μm diameter capillary, prepared with homeotropic-inducing aligning agent, with 5CB to which was added fluorescently labeled and photopolymerizable phospholipids, respectively. These lipids spontaneously aggregate within defect lines above a critical concentration [185], allowing the researchers to localize defects and even form solid objects templated by them. When studying the capillary using fluorescence microscopy the authors found lipid aggregation in the radial Q  =  +1 hedgehog but not in the hyperbolic Q  =  −1. Moreover, imaging the solids formed by polymerizing the aggregated lipids in transmission electron microscopy (TEM), they found the objects to be not spheres, but tori (rings). A calculation of the free energies of $Q=\pm 1$ point defects and $s=\pm \frac{1}{2}$ disclination rings confirmed that the positive-signed hedgehog was unstable with respect to the disclination ring, but not the negative-signed hedgehog. A question that may be interesting to investigate in the future is why these defect rings were orders of magnitude smaller than those found by Matthias et al [180].

With planar surface anchoring, the intuitive expectation is that the director aligns without deformation along the cylinder axis, in a so-called uniform axial (UA) configuration (figure 30(a)), at least for thin cylinders. Interestingly, Ondris-Crawford et al succeeded in preparing a sample where planar anchoring was promoted, yet with the preferred direction in the interface being azimuthal [186]. They coated the polycarbonate surface of Nucleopore membranes with polyimide, benefiting from a production-induced inner corrugated topography of the pores to create a preference for circumferential rather than axial alignment. Having achieved this unusual boundary condition for nematics in cylindrical confinement, they contemplated three possible configurations (figures 30(b)–(d)): perpendicular cross section concentric (PC), perpendicular cross section bipolar (PB) and escaped twisted (ET). Unsurprisingly, the PC configuration, with its extremely strong bend deformation at the core, was never observed. For small cavity sizes, with radii of 0.2–0.3 μm, the PB configuration was detected, whereas for radius greater than 0.3 μm a transition to the ET configuration was found.

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The ET configuration is intriguing, as it means that an achiral LC spontaneously develops a twist. As in any such case of spontaneous chiral symmetry breaking, the twisting sense can be left- or right-handed, each occurring with equal probability, hence interesting phenomena happen where left- and right-handed regimes meet [187]. The twist can either be unwound at the meeting point, resulting in a wall that locally violates the boundary condition, or a hedgehog point defect can form at the center. This costs much energy locally but it respects the boundary condition, hence it may be preferable in case of strong anchoring at the cylinder interface. Considering the results discussed above, suggesting that a Q  =  +1 point defect is unstable with respect to a nanoscopic ring-shaped  +$\frac{1}{2}$ defect line, it may be that—in practice—the point defect (PD) connection sketched towards the bottom of figure 30(d) should rather be a defect ring.

The saddle-splay term F₂₄ of the elastic energy, (5), was important for explaining the transition to the escaped twisted state in the study by Ondris-Crawford et al. Recently, in three separate studies [187–189], F₂₄ was found to generate twist also in planar-aligning cylinders of much greater thickness, and without any tangential symmetry breaking promoting concentric to axial alignment of $\boldsymbol{n}$. In the first case [188] the cylinder was bent into a torus, following an ingenious preparation procedure where 5CB nematic is injected into a yield-stress material subjected to continuous rotation. Because a torus has a topological Euler characteristic $\chi =0$ (see section 3.3) the total topological defect charge on the surface of a planar-aligned nematic torus must be zero. Hence no surface defect formation is expected, in contrast to the case of a planar-aligned shell or droplet. Indeed, no defects were formed on a simple torus, but the surprising observation was that $\boldsymbol{n}\left(\boldsymbol{r}\right)$ was always twisted, although no chiral dopant had been added.

It is perhaps less surprising that F₂₄ generates twist in a torus than in a cylinder, since the inside of a torus is indeed saddle-shaped. Importantly, $\boldsymbol{n}\left(\boldsymbol{r}\right)$ only experiences this saddle shape if the surface director is neither parallel nor perpendicular to the local symmetry axis of the torus. In these two special cases it would experience pure bend, but for any other orientation, i.e. in a twisted configuration, it will experience the saddle-splay on the inside. If this is difficult to visualize, it becomes clear by simply drawing lines in different directions on a donut. Thus, if K₂₄  >  0, making F₂₄ negative and thereby promoting saddle-splay, a twisted ground state seems plausible for a torus. Pairam et al complemented their experimental study with numerical simulations, finding that, indeed, the tighter the torus, the more a twist will reduce the free energy. However, even as the torus gets larger and larger, eventually to an infinite major radius, corresponding to a straight cylinder, they found that a twist can be promoted if ${{K}_{24}}>{{K}_{2}}$ [188].

The twist in straight cylinders with degenerate planar boundary conditions was found last year in two independent studies [187, 189]. In these cases, the thermotropic 5CB was replaced by lyotropic chromonic nematics, formed by aqueous solutions of the dyes sunset yellow (SSY) [187] and disodium chromoglycate (DSCG) [189], respectively. Chromonics are convenient because they have exceptionally low twist constant K₂. Thus, ${{K}_{24}}\gg {{K}_{2}}$, ensuring a twisted ground state in the cylinder although both chromonic nematics are non-chiral. The theoretical analysis of the effect of F₂₄ on $\boldsymbol{n}\left(\boldsymbol{r}\right)$ is quite involved and the two papers follow slightly different approaches, coming to qualitatively identical conclusions but a quantitative offset: while Nayani et al [189] reproduce the ${{K}_{24}}>{{K}_{2}}$ condition found in the study of tori by Pairam et al [188], Davidson et al [187] require ${{K}_{24}}>2{{K}_{2}}$ to stabilize a twisted $\boldsymbol{n}\left(\boldsymbol{r}\right)$.

Both author teams conclude that most thermotropics such as 5CB will generally not have a sufficiently large K₂₄ compared to K₂ to generate a twisted configuration in degenerate planar-aligning cylinders. However, very recently such twisted states appear to have been detected in electrospun fibers containing 5CB, without any chiral dopant present [190]. These fibers are substantially smaller (diameter on the order of a μm) than the glass capillaries used in the chromonics studies (100 and 400 μm, respectively) but both theoretical analyses predict that the effect should be scale invariant. While the results on the electrospun fibers are preliminary, and further investigations both of $\boldsymbol{n}$($\left. \boldsymbol{r}\right)$ and of possible deviations from a degenerate planar-aligning cylindrical cavity are required, the observations may indicate that K₂₄ is still slightly larger than K₂ in 5CB, and that this difference is sufficient to stabilize a twisted ground state.

While twisted structures arising in non-chiral nematics are surprising at first sight, this is what we expect when a chiral LC is used, whether by using chiral mesogens or by adding a chiral dopant. The non-trivial question is in what way $\boldsymbol{n}\left(\boldsymbol{r}\right)$ will twist within the cylinder, and again it depends sensitively on the boundary conditions. One of the first studies, by Cladis et al, focused on an N^* phase formed on heating from SmA^*, contained in a cylindrical capillary imposing radial boundary conditions [191]. The authors also studied the effect of an applied magnetic field. Because neither twist nor bend is compatible with the SmA/SmA^* geometry, the authors considered PR (figure 28(a)) as starting configuration (later theoretical studies, discussed below, indicate that this may not have been quite appropriate). As mentioned above, the PR configuration is unstable in N/N^* and consequently the authors detected a splitting directly after the phase transition into two $s=\frac{1}{2}$ line defects, essentially developing the PPLD configuration of figure 28(d), with the modification that the disclinations spiral due to the chirality. Moreover, they found an additional helically modulated line, which they postulated to be an s  =  +2 non-singular line [191].

Close to the N^*-SmA^* phase transition, heating/cooling yields strong simultaneous variations in elastic constants and helix pitch p₀ (on cooling towards a second order N^*-SmA^* transition p₀ must diverge and become infinite), rendering this study rather complex. Moreover, kinetic aspects are important, hence the speed of heating influenced the behavior strongly. An elegant way to systematically study $\boldsymbol{n}$($\left. \boldsymbol{r}\right)$ as a function of p₀ in the long-pitch regime without the complication of simultaneous changes in the elastic constants was presented by Kitzerow and co-workers [192]. They used a chiral dopant known to exhibit a helical twisting power inversion at a specific temperature, such that it induces helices with opposite handedness above and below this temperature. At the inversion point p₀ is infinite, thus there is no difference in $\boldsymbol{n}$($\left. \boldsymbol{r}\right)$ from a non-chiral nematic at exactly this temperature.

The authors prepared capillaries with diameters ranging from 25 to 200 μm with agents for hometropic and planar boundary conditions, respectively, in the latter case also imposing (weak) director anchoring along the cylinder by means of photoalignment. In homeotropic-aligning cylinders at the helix inversion temperature they reproduced the ERPD configuration (figure 28(b)) well known from studies of achiral nematics under the same type of confinement. A peculiar observation was the behavior of the hedgehog point defects as the temperature was slightly raised or lowered from the helix inversion point: while p₀ was still too long for any sign of helix formation to be detected in the capillary, the radial hedgehog defects moved towards one capillary wall whereas the hyperbolic defects stayed in the middle of the capillary, see figure 31. At the time of publication, the authors explained the difference by the inability of the hyperbolic defect to satisfy the homeotropic boundary conditions. In light of the recent experimental results of Wang et al [184] mentioned above, demonstrating that the radial hedgehog defects are in fact no point defects but nanoscopic defect rings, it would be worthwhile to revisit this difference between radial and hyperbolic hedgehog in weakly chiral systems.

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As p₀ decreased further on heating or cooling from the inversion point, the defect pattern gradually increased in complexity, first with a sinusoidal periodicity remaining visible, finally succeeded by a chaotic pattern. Representative snapshots for different p₀ are shown in figure 31. The pitch in this study was never very short, its minimum value being several tens of micrometers. Within the cylinder diameter range investigated, it appears that only the ratio of pitch and capillary diameter was important, hence one may anticipate a similar behavior for short-pitch cholesterics in very thin cylinders, such as those formed by electrospinning (see below), if strong homeotropic anchoring can be ensured.

In the planar-aligning cylinders a perfect UA configuration was found at the inversion point, without any sign of twist, see figure 32. While this may be interpreted as a sign of F₂₄ playing a less important role here than in the cylinder-enclosed chromonics discussed above [187, 189], we note that in this case the boundary conditions were not degenerate planar, but with anchoring for axial director alignment. It would be interesting to revisit also this experiment with degenerate planar anchoring conditions.

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Far away from the inversion point, where p₀ is much smaller than the capillary diameter, a periodically striped 'fingerprint' texture was observed (figure 32). The lines ran along the capillary, indicating that $\boldsymbol{n}\left(\boldsymbol{r}\right)$ adopted a radially twisted arrangement, as known from the so-called double-twist (DT) cylinders (figure 33(c)) characteristic of blue phases. The director must be aligned along the capillary axis at the boundaries and at the center, to comply with the boundary conditions, hence an integer number of helix pitches must fit within the diameter. As the temperature was shifted towards the inversion point the fingerprint lines therefore did not move away from each other continuously, but the pitch quantization imposed by the boundary conditions forced them to do so in discrete steps, each time with a pair of lines disappearing. The confinement locks the pitch into non-equilibrium values $p\ne {{p}_{0}}$. The same effect was later described by Enz et al [193] for selectively reflecting short-pitch cholesterics, confined inside the much thinner cylindrical cavity provided by electrospun fibers, as discussed in section 4.3.1.

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Surprisingly, when p₀ tightened again after having passed the inversion point, the fingerprint texture reflecting a double-twist cylinder configuration did not reappear. Instead, the texture transitioned into a complex but regular spiral pattern, see the right side of figure 32. Supported by computer- simulated polarizing microscopy textures the authors conjectured that this pattern was due to an Eccentric Double-twisted (ED) cylinder configuration (figure 33(b)). As p₀ decreased further, instabilities in the helical arrangement were detected once p₀ was below the radius of the cylinder. These helical instabilities, as well as the double-twist configurations at slightly longer p₀, were also detected in an earlier study by Lequeux and Kleman [195].

The first study of cholesterics in smaller cylindrical confinement than what glass capillaries offer was reported by Ondris-Crawford et al in 1994 [194]. NMR was again used to characterize the configurations in the cylindrical pores of alumina anopore membranes, with diameter of 0.1 μm. As mentioned above, the uncoated Anopore membrane induces planar alignment, and the anchoring is not degenerate but concentric, perpendicular to the cylinder axis. In these very thin cylinders the authors found an alignment transition for p₀  <  1 μm from the DT configuration to a configuration in which the helix develops along the cylinder. The Perpendicular cross section is of the Bipolar type, with the bipolar axis direction being helically modulated (figure 33(a)); we may call this configuration the axially twisted perpendicular cross section bipolar (ATPB) configuration.

This alignment transition was not seen by Kitzerow et al in their study of much thicker planar-aligning cylinders, for corresponding pitch-to-diameter ratios [192]. It has also not been detected in the first optical studies of cholesterics in very thin cylinders, conducted on coaxially electrospun fibers [193, 196, 197]. In these cases cholesterics with pitch short enough to yield selective reflection were confined inside channels as narrow as 0.5 μm [196], and selective reflection was always detected, suggesting a (possibly distorted) DT configuration rather than an ATPB configuration, which should not be reflective when studied in a direction orthogonal to the fiber axis.

Matthias et al studied also cholesterics in the cylinders with modulated diameter [198]. As with the study of non-chiral nematics they chose a glass-forming LC, which they could then investigate by fluorescence polarizing microscopy in the glassy state after the template had been removed. The experimental study was complemented with a thorough theoretical treatment. For very long pitch they again found disclination rings rather than point defects in an escaped radial structure, positioned regularly along the channel thanks to the sinusoidally modulated diameter. For pitch comparable to the pore size, a twisted escaped radial structure was found.

A variety of director fields might be expected when different smectics are confined in a cylindrical space. Here we will discuss only the response of SmA/SmA^*, with planar as well as homeotropic boundary conditions. While the chiral SmA^* phase has reduced symmetry and thus a different response to electric fields than non-chiral SmA [12], the structures of the chiral and non-chiral versions are identical. Some studies of other smectics exist and it may be a stimulating topic for future work to investigate different smectic phases within cylinders with varying boundary conditions.

The fact that SmA/SmA^* cannot sustain twist or bend means that no other deformations than splay should appear in an equilibrium director field configuration for pure SmA/SmA^*. Specifically, this would mean that the director escape in the third dimension that favors the ERPD over the PR configuration (figure 28) is no longer expected. As a consequence, the PR configuration was early on thought to be seen in homeotropic-aligned SmA/SmA^* in cylinders [191], as mentioned above.

Ondris-Crawford et al did a corresponding study with achiral 8CB in the thin pores of a Nucleopore polycarbonate membrane to follow the effect of diverging K₃ on cooling the N phase towards the second-order transition to SmA, potentially extracting information also about surface anchoring W and the saddle-splay constant K₂₄ [199]. Their theoretical analysis of the elastic deformations in these very narrow cylinders suggested that a direct transition from ERPD to PR should be found in case of strong anchoring, and that the temperature of the transition can be used to establish K₂₄. However, the homeotropic anchoring in their nucleopore membranes was too weak, bringing the system into a regime where also the PP configuration is stable. They were not able to give an absolute numerical value of the saddle-splay constant but they could conclude that ${{K}_{24}}>{{K}_{1}}$ in 8CB.

A more elaborate theoretical study of possible smectic arrangements inside narrow cylinders with homeotropic surface anchoring was carried out a few years later by Kralj and Zumer [200]. They expanded the viewpoint first by considering that the confinement-induced frustration may lead to an extremely thin prevailing nematic core that releases the elastic stress by letting $\boldsymbol{n}\left(\boldsymbol{r}\right)$ escape in the third dimension through a centrally localized ERPD configuration, although the temperature is below ${{T}_{\text{SmA-N}}}$ of a bulk sample. This is similar to the model of Cladis and Torza discussed in section 3.3.4, for explaining the striped patterns forming during a N-SmA transition of a sample subject to hybrid boundary conditions, with homeotropic and planar anchoring on the two surfaces [138].

Second, Kralj and Zumer contemplated that, at least for weak homeotropic anchoring, the core may adopt layers perpendicular to the fiber or tilted in a chevron geometry. Their analysis revealed that the PR configuration is in fact extremely unlikely, and that rather a supercooled nematic ERPD core exists in case of strong homeotropic anchoring, whereas weak anchoring yields chevron or perpendicular smectic layers. This would suggest that the core of any smectic arrangement with concentric layers, for instance in the Dupin cyclides that often form spontaneously in bulk samples with little control of alignment [5], should be nematic in an ERPD configuration. Considering today's possibilities to confirm such structures on the nanoscale, as demonstrated by Wang et al [184], this motivates new studies of smectics in frustrated geometries.

Using calorimetry, Iannacchione and Finotello studied the N-SmA transition of 8CB in the 0.2 μm pores of Anopore membrane (alumina), prepared for planar and homeotropic surface anchoring, respectively [201]. They found a profound influence on the N-SmA as well as N-isotropic transitions, both broadening considerably and reducing in transition enthalpy as an effect of the confinement, for either alignment geometry. Moreover, the clearing point was always reduced in confinement, whereas the N-SmA transition was shifted to lower temperatures with planar alignment but to higher temperatures with radial alignment. Unfortunately no information was given on sample history during the experiments, in particular concerning whether the measurements were done on cooling or heating, and the time for which the sample had stayed in the preceding phase prior to the experiment.

These aspects can be very important for the behavior of LCs in strong cylindrical confinement, as shown in a later calorimetry study by Enz et al of 8CB inside electrospun fibers with PVP sheath [202]. While this study was restricted to planar alignment (as confirmed by x-ray scattering on the fibers), increase as well as decrease of the N-SmA transition could be found, depending on the sample history. On heating, the SmA-N transition was considerably increased compared to the bulk value of ${{T}_{\text{SmA-N}}}$, up to some 5K for the thinnest fibers studied (outer fiber diameter less than 2 μm; inner core diameter was not assessed). However, when cooling back from the isotropic phase, through the nematic, the N-SmA transition was absent throughout the experiment. The N-SmA transition thus exhibits extreme supercooling inside the planar-aligning fibers. The encapsulated LC stayed in the N state well below room temperature, as confirmed by a subsequent heating experiment where only the clearing transition was detected. However, if the fibers were annealed at room temperature for several weeks, the smectic phase reformed and the SmA-N transition peak was recovered in the calorimetry curves, shifted to higher temperatures compared to the bulk value.

The authors explained these surprising observations as a result of the mesogens at the interface to the polymer sheath being strongly anchored at that interface, in an arrangement reflecting the history. After long annealing at room temperature, allowing the SmA phase to develop, the sheath-anchored mesogens are in a smectic arrangement, promoting the layering and thereby raising ${{T}_{\text{SmA-N}}}$. However, if the positional order is lost by heating into the nematic or isotropic state, then the mesogens close to the polymer sheath are anchored without a layered arrangement, promoting the N state in a narrow cylinder where the influence of surface molecules can penetrate throughout the volume of the sample. The broadening and weakening of transition peaks seen by Iannacchione and Finotello was reproduced in this study, but the clearing point on heating was always raised (the shift was much smaller than for ${{T}_{\text{SmA-N}}}$) in the fibers.

Electrospinning is the process by which solidified filaments are drawn from a viscous, continuously flowing polymer solution by the action of an electric field. While the modern term 'electrospinning' was popularized by Doshi, Reneker, and many other researchers in the latter end of the 20th century [203–207], the process had in fact already appeared in many patents filed during the period 1900–1940 by John Francis Cooley, and Anton Formhals. It was then referred to as 'electrostatic' or 'electrical spinning' [208–210].

During the mid-1900s, much of the world moved towards a fast paced industrially and commercially developing economy, and many were attempting to exploit the electrospinning process for commercial gains. This explains the appearance mainly of patents rather than fundamental research articles on the topic during this time [211–214]. It wouldn't be until the 1960s, and then again, in the 1990s, when the rigorous theoretical and experimental frameworks for truly understanding the process would be developed. Sir Geoffrey Ingram Taylor contributed to the former by first accurately proposing a model for the characteristic shape of the prolate polymer solution droplet (now referred to as the Taylor cone) known to emerge from the charged spinneret proceeding the fiber forming process [215, 216].

Since then, a large number of theoretical and experimental works for comprehensively explaining the intricate level of instabilities observed during electrospinning, such as the onset of jet formation and the final bending trajectories of the fiber formed from the jet, have been published [217–222]. At the forefront, Reneker and Yarin were the leaders in revitalizing modern 21st century interest in the process by not only further characterizing the bending instabilities of the fluid jet, but also systematically studying the parameters affecting various types of fiber formation [217, 223, 224]. Ultimately, while these parameters have succumbed to countless debates and scrutiny by both experimentalists and theoreticians over the years, it has all culminated into the identification of a select handful of both processing parameters (set-up and apparatus related), and solution specific parameters (such as concentration, polymer type, polymer molar mass M, and solvent type) which influence fiber production the most. These parameters will be discussed in section 4.2.2.

The 2000s saw an exponential increase in the research devoted to electrospinning [208]. The technique has proved to be extremely useful for the development of advanced materials, as well as the advanced functioning of other processes in many fields, from air filtration [225, 226] and biotechnology [227] to sensors [228] and waste remediation [229]. The introduction of LCs to the field of electrospinning is rather recent. Interestingly, the two discoveries—the electrospinning method and the liquid crystalline state of matter—were formally first reported by C.V. Boys, and Friedrich Reinitzer, respectively, within only one year of each other in 1887, and 1888. And yet, the first study on combining LCs with electrospinning was published only just nine years ago. Since then, though, it has been steadily growing in popularity.

The first electrospun LCE fibers were presented by Wendorff and Finkelmann et al [230]. About one year later Godinho and co-workers demonstrated electrospun fibers from lyotropic LCP solutions [231], and within the same year Lagerwall, Xia and co-workers demonstrated the incorporation of low molar mass LCs inside the core of coaxially electrospun fibers [179]. Our group continuously developed the coaxial approach [190, 193, 202, 232, 233], while two years later West and co-workers [234] demonstrated that LC core fibers could be spun from mixing LC and polymer in a common solvent, without separating them into two channels. This single-phase electrospinning relies on in situ phase separation during solvent evaporation for acquiring the final coaxial LC-polymer fiber morphology.

Ultimately, the realization that LC functionalized fibers may prove to be useful in the future for the development of highly functional, environmentally responsive textile fabrics is steadily moving toward the forefront. In the sections following the discussion of how the electrospinning process works, and the parameter considerations that should be considered during the fiber development process, we discuss how electrospun LC-fibers, and LC-elastomeric fibers are already showing high potential for becoming the next non-electronic functional gas sensors.

Despite the many complex and interdependent variables that must be accounted for in order to properly analyze an electrospinning experiment, it is extensively used by researchers in many independent scientific, and even design-related [235], fields. The simple reason is that the process provides immense flexibility for experimenting with fiber production more quickly than other fiber spinning processes and that the set-up (see figure 34) is small-scale, not very complex and comparatively inexpensive. An experiment starts with dissolving a chosen polymer with appropriate molar mass in a compatible solvent until it is homogenously dispersed. This solution is mechanically pumped via an external device (often a syringe pump) through a metal spinneret which is connected to a high voltage power supply. In case of coaxial electrospinning a thin second capillary runs at the center of the main spinneret, allowing ejection of a core fluid inside the polymer solution, see figure 35.

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A strong electric field (∼kV m⁻¹) is then established between spinneret and a grounded counter electrode some 10–20 cm away. As the solution in the spinneret becomes charged, the droplet protruding from the tip undergoes distortion into a conical shape (the Taylor cone) as a result of the conflict between surface tension—promoting a spherical droplet shape—and the electrostatic attraction to the grounded counter electrode acting on the high density of charges in the droplet. Once the voltage passes a threshold, usually 10–30 kV (depending on solution viscosity, type of polymer and solvent, and the ionic content), the combined action of attraction to the ground electrode and self-repulsion between the concentrated like-charged ions overpowers the surface tension and a thin jet is expelled from the tip of the Taylor Cone.

The polymer solution carried by the jet undergoes a series of helical whipping and bending instabilities as solvent evaporates. The initial jet formation and the subsequent instabilities (figure 36) are triggered, in part, by the fluid jet attempting to lower the charge density by extending [236], in part by the electrostatic self-repulsion between like-charged ions along the jet [217]. The equations describing the initial stage of the process have been derived by Bugarski et al [237, 238], Loeb et al [239] and Yarin et al [240], respectively. The final result is a solidified, continuous fiber collected as a non-woven mat on the grounded electrode or on another collector inserted between spinneret and counter electrode [217, 236].

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All components required to devise a basic electrospinning setup are shown in figure 34. Typically, the dissolved polymer solution is placed into a syringe, with an appropriately sized needle, hereafter referred to as the spinneret (often blunt ended needles between 22 gauge and 18 gauge are employed). The solution is pumped (via a syringe pump or similar device) to the needle orifice at constant rate on the order of 0.1– 1 ml h⁻¹. The viscosity of the polymer solution is often near that of corn syrup or molasses, about 30–100 Ps (or, 3–10 Ns m⁻²).

If the dissolved polymer has insufficient molar mass and/or concentration, the stretching within the jet is so large that polymer coils are disconnected (there are too few entanglements) and then nothing prevents the Rayleigh–Plateau instability (already mentioned in section 3.1) to break the jet up into droplets. Such an experiment results in the electrospraying phenomenon. Electrospray involves further interesting phenomena as the solvent evaporates, leading to break-up of the initial droplets into much smaller ones, but a deeper discussion of electrospray is outside the scope of this review.

However, if the polymer solution employed contains enough molecular chain entanglements, then electrospraying will not occur. The Rayleigh–Plateau instability is now counteracted by a continuous chain of stretched-out but entangled polymer molecules along the jet. To ensure continuous spinning, polymers of $M\approx 10$–100 kg mol⁻¹ are used, with concentrations varying from 5% (for high M polymers) up to 30% (for low M) [242, 243].

As the jet is stretched considerably due to the whipping instability, it will get considerably thinner before enough solvent has evaporated to solidify the polymer, and the diameter of electrospun fibers can thus reach values as low as tens of nanometers. In the case of electrospinning PAN (poly(acrylonitrile)) fibers from a DMF (N,N-dimethylformamide) solution, according to Wu et al [244], a jet emanating from a Taylor cone that is 50 μm in diameter, can be reduced during solvent evaporation to about 100 nm diameter in only a few seconds. For jets that start at sub-micron diameters, their reduction to the nanometer scale can occur within milliseconds.

While fibers collected onto the grounded collector have a landing speed of about 10 m s⁻¹, the initial acceleration of the straight part of the jet from the Taylor cone can be up to 600 m s⁻² [242]. The fact that this value is about two orders of magnitude larger than the acceleration due to the gravitational force plainly indicates that the effects gravity acting on the electrospinning jet are nearly negligible—it is also the reason why we have the flexibility of creating setups which can function horizontally or vertically without the jet being perturbed in the electrospinning process. However, in case of coaxial electrospinning, where the Taylor cone is comprised of two different fluids, generally with different density, gravity may have an influence on the symmetry of the Taylor cone. One may speculate that this could influence the structure of the produced fibers, but this is hardly investigated.

Viscoelastic shearing in the jet is largely absent as well since the speed at which fluid flows inside of the launching jet can be up to 5 m s⁻¹ at a 2 cm distance from the Taylor cone, much lower than the speed with which the whole jet is moved due to the stretching. Once the whipping jet impacts the collector it should by this point be a solidified fiber. Depending on the type of polymer and the solvent used, the fiber can have varying morphologies, but typically, continuous and bead free fibers with circular cross-section are desired [236, 242, 245, 246].

There are many polymers which, when dissolved in the right solvent and electrospun under certain conditions, can form a very large variety of fiber morphologies, from the standard cylindrical to fibers with internal pores and cavities [247, 248] to barbed fibers [249]. Although the processing related parameters, such as the spinneret—ground electrode distance, the polymer solution flow rate and the applied voltage all can have significant effects on the final fiber quality, they come secondary to the choices of polymer and solvent. These two components dictate the solution properties, determining whether or not fibers can be spun.

When we wish to incorporate a low molar mass LC in the fiber core, whether by using a coaxial spinneret or by relying on in situ phase separation, choosing the polymer for the sheath can be non-trivial. If we use a crystallizing polymer, for instance, the elongational forces acting on the solidifying jet during spinning will cause the polymer molecules to be much better aligned than if we used an amorphous polymer. While beneficial for fiber strength, this gives the sheath strong birefringence, competing with the birefringence of the incorporated LC. In case of coaxial electrospinning, also the polymer solvent can be a tricky decision to make: if the solvent is miscible with the LC it may significantly diffuse into the core during spinning, a loss of control that may have considerable impact on the end result.

In applied research, the choice of polymer is largely limited by the intended use of the fibers. Biodegradable polymers such as poly(ethylene glycol) (PEG), and poly(l-lactide) (PLLA) may be better suited for producing tissue scaffolds due to their lack of toxicity effects [227, 250], than polystyrene (PS) or polyurethane (PU). If high tensile strength is critical, then crystallizing polymers like Nylon-6 may be useful, although we then get problems with sheath birefringence. Moreover, electrospun nylon fibers tend to be very thin, most likely a consequence of the rather low maximum values of M that are commercially available. A large number of literature reviews, books and book chapters exist, discussing how choice of polymer and solvent influence the final fiber performance [217, 242, 251].

As far as the solvents go, polar aprotic or nonpolar solvents are often recommended. While it is possible to electrospin polymers dissolved in polar protic solvents, which tend to have lower vapor pressure (e.g. water: 17.5 hPa; aqueous acetic acid: 15.3 hPa), special care needs to be taken when working with such solvents [252]. Their strong tendency for hydrogen bonding increases ionic dissociation, thus raising the conductivity of the polymer solution, with the possible consequence of rapidly fluctuating charge densities during electrospinning. This can destabilize the Taylor cone, which may dissociate into multiple, small side cones and result in the formation of branched fibers [253]. Moreover, the limited rate of solvent evaporation can lead to swollen or wet fibers landing on the collector.

Even if no water is in the original polymer solution, the hydrophilic character of non-aqueous polar protic solvents can lead to accumulation of water from the vapor in the atmosphere [254], hence high humidity should generally be avoided. Usually, electrospinning at 20–40% relative humidity works well for most polymer solutions. Higher humidity can influence electrospinning dramatically, both for water-soluble and water-insoluble polymers. For instance, at 50% relative humidity PS, being hydrophobic, will tend to form porous fibers [248, 255, 256], whereas PEG and PLLA, when spun together, will create hydrophilic fibers that can easily bead or mesh together [227, 254].

Solvents with higher vapor pressures such as 1,1-dichloroethane (240 hPA), acetone (240 hPA), tetrahydrofuran (THF) (200 hPA), and ethyl acetate (97 hPA), are commonly used, often yielding smooth cylindrical fibers [236]. Many researchers use guides such as the Hildebrand solubility parameter for solvents [257], and the more comprehensive Teas Plots, first developed by Jean Teas [258] and made popular by John Burke [259]. A Teas plot is particularly useful if one is attempting to find multiple solvents for a particular polymer, as the graph is able to more specifically show the fractional contributions of dispersive, hydrogen, and polar bonding that make up the total Hildebrand solubility parameter [236, 259].

As already mentioned, the molar mass of the polymer is critical for successful electrospinning. The molecules must be large enough, and present at sufficient concentration, to ensure that a continuous chain of entangled molecules persist throughout spinning, otherwise the Rayleigh–Plateau instability will disrupt the jet and no fibers are produced. We thus have two critical values that must be considered for the polymer: the molar mass M and the concentration, which must both be greater than a respective minimum value [236, 260]. Raising the polymer concentration beyond the minimum will not necessarily have an increased positive effect in the spinning process. In fact, very high concentration could lead to premature drying at the spinneret tip, and a solidified Taylor cone before fibers can even be ejected [241, 261].

Typical M values range from 10 to 10⁴ kg mol⁻¹. To give a few reference values, the isoprene compound (commonly called Latex) that naturally pours out from rubber trees as a thick white syrup generally has M  >  100 kg mol⁻¹ [262], while Nylon ranges between 3 to 20 kg mol⁻¹ [263]. While molar mass and concentration are clearly related, no increase in concentration can compensate for too low molar mass. This precludes spinning of certain polymers, which may be impossible to obtain at the high molar mass required for electrospinning.

The viscosity of a polymer solution can provide a first indication of whether or not sufficient entanglements are present to allow stable fiber spinning [243, 264–266]. However, it can be misleading, since also the solvent can strongly influence the viscosity. According to Agarwal et al, for a certain polymer type and molar mass, low vapor pressure solvents tend to raise the viscosity of the solution, whereas high vapor pressure solvents yield lower viscosity [261].

Close to the critical values of molar mass and concentration, fibers may be produced but the Rayleigh–Plateau instability is not entirely hindered. Some polymer molecules disentangle from the overall chain, coiling up more or less spherically as they are no longer subject to the stretching force. This results in periodic bulging-out of the fibers, a phenomenon often referred to as 'beading', since the final fiber morphology may resemble beads on a string. Koski and co-workers found that electrospinning a solution of PVA with $M\approx 10$ kg mol⁻¹ yielded beaded fibers, while raising M to 30–50 kg mol⁻¹ produced non-beaded smooth long fibers, with all other parameters kept constant, and an equal concentration of 25% PVA in solvent for both situations [267].

Even though many aspects of the coaxial electrospinning process are insufficiently understood, this process for making core-shell nano- and micro-fibers has gained popularity as it can allow low viscosity fluids to be incorporated within polymer fibers. Numerical simulations of coaxial electrospinning suggest that charges rapidly escape to the outer surface of the compound flow, meaning that it is fundamentally the sheath that is acted upon by the electric field, and the extension of the core flow into the jet thus relies on traction between sheath and core flows during spinning [268–270].

As the sheath solution is forming the Taylor cone and the jet is ejected from it, the core fluid, flowing at a similar rate as that of the sheath solution, is being 'carried' inside. Processing parameters, such as applied voltage and spinneret design, influence coaxial fiber formation strongly. Too high applied voltage will result in jet splitting whereby both the core fluid and sheath fluid separate and spin as two Taylor cones side by side [271]. Experiments have shown that core incorporation into the jet is favored by having the inner capillary of the coaxial spinneret protruding slightly from the outer sheath fluid nozzle [268].

Coaxial electrospinning is a very convenient and adaptable technique for introducing LCs into micro- and nanosized fibers. The spinning components can be easily assembled, most of the materials are readily available, and small amounts are sufficient to produce macroscale non-woven fiber mats which can be easily handled. Moreover, by replacing syringe pumps with a pneumatic microfluidic control system [137, 193, 202], very small vials can be used, the dead volume minimized, and temperature control is greatly simplified. These aspects makes the coaxial electrospinning approach for incorporating LCs into fibers very attractive, as LCs can be very expensive. The set-ups used in our group offers a simple, low-cost and versatile solution for coaxially introducing nematic [179, 190, 202, 272], smectic [202] and cholesteric [193, 197, 232, 233, 272] LCs into polymer fiber cores for obtaining fibers with varying functionalities.

Since a non-crystallizing polymer is preferred when spinning together with LCs, to avoid competition in birefringence when they are characterized via polarized light microscopy, many experiments were done with polyvinylpyrrolidone (PVP) as sheath. This amorphous polymer is available at very high molar mass. The polymer sheath serves the double function of encapsulation and alignment layer, because it appears that the stretching during spinning gives a slight preferential axial alignment of the polymer chains (not so strong as to induce significant birefringence) that orients $\boldsymbol{n}$ along the fiber [196]. As the polymer sheath often becomes quite thin during the spinning, and as the core is filled with the LC component, responsive core-sheath fibers can be readily obtained, the macroscopic properties being governed by the type of LC used.

In the very first report by Lagerwall et al [179] core-sheath fibers were obtained by coaxially spinning a room temperature nematic LC mixture inside a solution of PVP in ethanol containing dissolved Ti(O iPr)₄, a sol-gel precursor for TiO₂. The diameters of the non-woven and aligned fibers were on the order 0.1-1 μm, see figure 37. Because the large amount of TiO₂ in the sheath precluded polarizing microscopy characterization, Raman spectroscopy was used to establish that LC was present in the fiber core, with $\boldsymbol{n}$ aligned along the fiber axis. As discussed in section 4.1.1 a considerable increase in LC clearing point as a result of encapsulation was detected.

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In a subsequent study this coaxial LC-polymer electrospinning approach was extended for spinning coaxial fibers with the smectic-forming 8CB in the core [202]. The fibers were spun at room temperature, with 8CB in the SmA phase. Finding a correct LC flow rate was found to be key to ensure continuous LC filling, yet maintaining good overall fiber morphology, see figure 38. At very low LC flow rates, the fibers were discontinuously filled, whereas at too high LC flow rate, the fibers became deformed and appeared 'smeared out'. X-ray diffraction on oriented fibers showed that the smectic layers were perpendicularly oriented to the fiber axis, a rather unexpected finding considering that SmA flows the easiest along the layers (section 2.1.3). As discussed in section 4.1.4, an interesting hysteresis effect was observed upon heating and cooling between the N and SmA phases.

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Scalia and co-workers [197] studied the effects of varying spinning distance, polymer concentration and the presence/absence of sol-gel precursors on the morphology of coaxially electrospun PVP fibers with LC core. The morphology changed considerably, with high polymer concentration and/or large spinning distance yielding premature solvent evaporation with a consequent flat ribbon fiber morphology, whereas the opposite conditions resulted in soft fibers landing on the substrate, where they collapsed into a hemicylinder shape. In some fibers a distinctly porous sheath was detected, a phenomeon that the authors explained as a result of the spinning atmosphere being saturated with solvent. The solvent then re-condenses as liquid droplets on the fiber, leaving a porous surface after final evaporation.

The optical properties and responsiveness of LC core polymer sheath fibers can be further improved by introducing short-pitch cholesterics in the core as shown by Enz et al [193]. The PVP fibers, with diameters in the range of a few microns, showed selective reflection in the visible spectrum. Similar to the case when smectic LCs were used as the core, the LC filling greatly depends on flow rates which in turn affect the overall fiber thickness. When the LC flow rate was high enough, close to that of the sheath solution, the selective reflection from the macroscopic fiber mats could be observed by the naked eye, see figure 39. For thinner fibers, less than  ∼7 μm in diameter, light scattering from the sheath dominates over the selective reflection, such that the reflected colors are only detectable with a microscope.

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Interestingly, the selective reflection color of the fibers with diameter in the micron range varied across the full visible spectrum, although the reflection color of the bulk N^* phase was blue. This was explained as a result of quantization of the helix pitch p (not necessarily at its equilibrium value p₀) within the fiber core, due to the cylindrical boundary conditions imposed by the polymer sheath [193]. The effect is identical to that described by Kitzerow and co-workers [192], discussed in section 4.1.3. Because there is a certain random variation in the diameter of the electrospun fibers, we see the full spectrum of colors. Even variations on the scale of tens of nanometers yield visible color changes, as p must always adjust to the actual fiber diameter. The relation between the selective reflection color and the core size and shape was investigated in much greater detail in a follow-up study by Scalia and co-workers [196], in which Focused Ion Beam (FIB) was used to dissect the core-sheath fibers at selected locations, allowing nanoscale imaging of the interior of the fibers.

Kye et al [272] further demonstrated that spinning with dual core channels containing two different types of LCs within the same polymer sheath held further potential to increase the multifunctionality of LC-fibers fabricated through the coaxial electrospinning method, see figure 40. The authors produced fibers with adjacent N^* and N cores, the former showing visible selective reflection, as well as fibers with adjacent cores filled with the nematic LCs 5CB and E7, respectively, clearing at distinctly different temperatures. The greatest challenge with this multi-core spinning concept was to counteract mixing of the two core fluids and to maintain a wall between the cores that was intact throughout the fiber, even after heating. To this end the fibers were collected on hydrophobic substrates, because Kim and Lagerwall showed in an earlier study that the wetting of water condensed from the atmosphere during spinning promotes fiber spreading and loss of core content [273].

This allowed Kye et al to maintain core integrity, see figure 40. Moreover, the authors noted that, if a surfactant had been added to the PVP sheath solution (to reduce surface tension), then even in fibers with intact wall between the cores the two LCs appeared to be contaminated. The authors proposed that this was due to diffusion of surfactant and condensed water from the sheath to the core solutions during spinning. This assumption was corroborated when single-core fibers without surfactant in the sheath solution were spun with 5CB and E7, and on heating clearing points were close to the bulk values of the pure LCs.

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Reyes et al [190] showed that coaxially electrospun fibers with PVP and 5CB could be used as effective optically responsive platforms for gas sensing, as will be discussed in detail in section 4.4.1. The authors investigated aligned and randomly-aligned mats of fibers with morphology varying from uniform cylindrical with continuous LC-filled core to strongly beaded character, where the LC is mainly in the beads. Beading was promoted either by spinning at high relative humidity or by increasing the LC flow rate in relation to that of the polymer sheath. The LC inside partially beaded fibers with continuous LC core show interesting textures, see figure 41. Defect lines running perpendicular to the fiber axis are seen repeatedly along the fibers, in particular at the widest diameter points. The investigation of these defects is still on-going. Considering that we here have essentially cylindrical fibers with modulated diameter in the μm range, but inducing planar rather than homeotropic LC alignment, it is interesting to compare with the related situation for homeotropic boundary conditions studied by Matthias et al [180] (see section 4.1.1).

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When co-electrospinning LCs that are crystalline at room temperature, other factors such as miscibility with the sheath's solvent must be considered. Although finding a solvent for a non-room temperature LC may be trivial, its compatibility with the solvent of the sheath fluid to be coaxially electrospun alongside it, may prevent fiber formation. If the core and sheath solvents are miscible, but the sheath solvent is a non-solvent for the LC, a rapid coagulation of the LC results, analogous to the coagulation of polymer fibers during wet spinning. Examples are shown in figure 42.

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An alternative way of introducing LCs into electrospun fibers was devised by Buyuktanir et al [234]. A chloroform-acetone solution of 5CB as LC and polylactide (PLA) was electrospun to obtain birefringent fibers (diameter  ∼0.5–2 μm). The PLA concentration was kept constant while 5CB to PLA ratio (7–86%) was varied for optimum spinnability. During spinning, the LC phase separates from the polymer, giving a core–sheath morphology to the fiber. However, phase separation was not always complete and the sheath was in some cases a mixture of PLA and 5CB. The director was generally aligned along the fiber but there was strong beading. Recently, the same group made a more systematic investigation of the balance between beaded and cylindrical morphology in electrospun LC core fibers produced via the phase separation route [274]. By controlling humidity and optimizing flow rate and solution composition, the authors succeeded in reproducibly spinning LC-filled fibers with morphologies varying from uniform cylindrical to beaded.

The simplicity of the phase separation route is attractive, as a standard spinneret can be used. The recent success is therefore important for the further development of LC-functionalized fibers. At the same time, coaxial electrospinning offers more detailed and dynamic control, and it is more versatile in the LCs and polymer solutions that can be combined. The two methods can be considered largely complimentary, each with its benefits and limitations.

The West and Jakli team also demonstrated air brushing as a facile (non-spinning) way of producing LC core–polymer sheath fibers with average diameter of 5–10 μm [275]. Two types of polymers, PLA and PVP, were dissolved together with E7 nematic LC, in a chloroform/acetone solvent mixture for PLA or in isopropanol for PVP. The obtained fibers were investigated for optical, electro-optical, and morphological characteristics by POM and SEM, respectively. The core-sheath morphology was confirmed as a pipe structure in SEM, demonstrating that LC was phase separated and forms a core.

The fibers obtained were uniformly birefringent with planar anchoring conditions of E7 at the polymer sheath and average LC alignment along the fiber axis. The authors also applied an electric field perpendicular to the fiber axis with the aim to control the optic axis direction; since E7 has positive dielectric anisotropy, $\boldsymbol{n}$ should reorient away from the fiber axis, reducing the effective birefringence, ultimately yielding a dark state between crossed polarizers. However, the latter condition could not be reached with the available fields, a result the authors explained with the voltage drop over the polymer sheath. Fibers were also produced with short-pitch cholesteric LC core, and they showed visible selective reflection.

The first approach to incorporate liquid crystallinity into fibers that might come to mind is to electrospin liquid crystal polymers (LCPs). Although there are only a few examples of lyotropic LCPs that have been known to form fibers using electrospinning, several types of thermotropic nematic, smectic, and cholesteric LCPs have been electrospun. For instance, Canejo et al [231] showed that electrospinning a lyotropic LC solution of 60 wt.% acetoxypropylcellulose (APC) with dimethylacetamide (DMAc) as solvent successfully formed chiral birefringent fibers with an average diameter of 3 μm and a helically twisted morphology. Fibers with different diameters had different helical pitch and the authors observed spontaneous helical twisting with both left- and right-handed twist. About 75% of the fibers were right-handed. Interestingly, according to the theory developed to support the experiments, the presence of more right-handed fibers did not come from the natural twisting sense of APC.

Rather, it was attributed to the spontaneous helical winding of the polymer chain in the presence of the electrical field, with the twisting sense determined by a chiral coupling constant depending on the enantiomer of the polymer chain, when the solution concentration is sufficiently high. This finding further establishes the importance of tuning electrospinning parameters; the properties of the solution do not only influence the choice of processing parameters, the processing parameters can also influence the development of the fibers at sub-molecular levels (refer to section 4.2.2).

Tsuboi et al [276] showed another example of fibers electrospun from a lyotropic LCP; poly(γ-benzyl- L-glutamate) (PBLG), a synthetic polypeptide that forms a rod-like α-helical conformation in organic solvents. Nanofibers were electrospun from an isotropic and biphasic PBLG /dichloromethane-pyridine solution under various spinning conditions. For fibers averaging sizes from 0.65 to 1.67 μm, it was counter- intuitively observed that $\Delta n$ increased with decreasing fiber diameter, which was attributed to improved uniaxial molecular orientation of PBLG from undulated molecular orientation. A further interesting finding was that the fiber orientation and internal structure could be controlled by changing the spinning solution from isotropic to biphasic solution, which was confirmed by wide-angle x-ray diffraction (WAXD) analysis, and polarized micro-Raman spectroscopy. The nanofibers spun from an isotropic solution generally showed lower orientational order than nanofibers spun from the biphasic solutions, in which the the hexagonal lattice of α helices was uniaxially aligned along the fibers.

Commercially available LCPs, such as Vectra and Kevlar, have also been used to obtain birefringent nanofibers via electrospinning [277–279]. In preliminary work shown by Reneker's group, an isotropic solution of poly(p-phenylene terephthalamide) (PPTA), commercially known as Kevlar 49^®, in concentrated sulfuric acid, was electrospun into a water-bath as a collection target [277]. Using transmission and scanning electron microscopy, TEM and SEM, respectively, they found the average fiber sizes ranged from 40 nm to a few hundreds of nanometers, much smaller than the fibers obtained from conventionally spun $\text{Kevla}{{\text{r}}^{\circledR}}$. Although this method produced thermally stable ordered fibers with smaller diameter, the surface roughness of the fibers was of the same order as of commercially obtained fibers.

For obtaining Vectra fibers, a melt electrospinning setup equipped with a CO₂ laser was used by Ogata et al [278] to obtain micron size fibers of polyalirate commercially known as $\text{Vectr}{{\text{a}}^{\circledR}}$. The microfibers analyzed by x-ray diffraction were found to have $\boldsymbol{n}$ oriented along the fibers. Interestingly, the diameter of the fibers decreased upon increasing the laser intensity. Due to limitation of the set-up fibers with diameters smaller than 1 μm could not be formed.

Araujo and co-workers [279] continued the work on electrospinning $\text{Vectr}{{\text{a}}^{\circledR}}$ and employed a host-guest approach using blends of Vectra with polyethylene oxide (PEO) to electrospin Vectra-PEO nanofibers. Rather than employing melt spinning and the required strong heating, a two-solvent system of chloroform and pentafluorophenol (70:30 ratio) was used to dissolve the commercial Vectra sample. This solution was mixed with a chloroform solution containing PEO, and then the authors electrospun fibers with a Vectra:PEO ratio of 8.19:1, by weight. After spinning, a heat treatment was used to remove the PEO portion of the fibers, and to impose mechanical stability and ordering onto the amorphously spun Vectra fibers.

The heat treatment also improved the fibers' thermal stability and increased their tensile strength by more than 300% and elongation at break points by around 150%. The tensile modulus of the fibers, however, was found to be similar to that of conventional Vectra microfibers. It was hypothesized that the increase in tensile strength is due to PEO acting as plasticizer during Vectra fiber spinning, and contributed to the ductile behavior of the untreated mats. Annealing after the complete removal of the PEO enhanced the structural order and resulted in stiffer fibers, hence, improved mechanical properties.

Summarizing the results from electrospinning of Vectra and Kevlar fibers, the overall conclusion is that the functionality of the produced fibers is limited, disappointing in the sense that the fibers do not demonstrate superior properties compared to current commercial Kevlar or Vectra fibers, produced following melt spinning or extrusion. These limitations have ultimately dissuaded many researchers from continuing studies to produce nanoscale fibers made from Kevlar and Vectra.

Rather than using these famous high-performance LCPs, which are difficult to dissolve in compatible solvents, Wu and co-workers chose to electrospin blends of polysiloxane based side-chain cholesteric LCPs with triethylamine and PEO, respectively, dissolved in chloroform [280]. The PEO was added to increase the spinnability of LCPs. In order to obtain uniform fibers another LC was added, triethylamine (Et₃N), and different parameters were optimized, such as the PEO concentration, along with the processing parameters like applied voltage and collector distance. Although the addition of a small amount of LC successfully increased the spinnability of the LCP solution, initially only beaded fibers were obtained.

The authors also studied the effect of conductivity of the spinning solution, which affects the net charge density of the jet and hence the quality of fibers produced. As observed by POM the addition of Et₃N in the presence, and absence, of trace amounts of water had a great effect on morphology and optical properties of the obtained fibers. It was ultimately found that at a low added volume of Et₃N (0.05 ml), uniform fibers formed, while beaded fibers were seen after the addition of more Et₃N (0.1 ml). In the presence of trace water Et₃N (0.05 ml) is protonated and converts to Et₃NH⁺ , which acts an electrolyte. This increases the birefringence and the morphology of the fibers improved as high enough electrolyte concentration increases the conductivity and hence the spinnability. However, the fibers, with an average size of 1 μm, were brittle and hence not suitable for mechanical characterization. When anhydrous Et₃N was used, without trace water present, the LCP solution could not be electrospun.

WXRD analysis showed that while the polymer chains within the fibers were aligned through the electrospinning process, the degree of orientational order was lower compared to fibers prepared by conventional drawing processes. A significant amount of annealing was necessary to increase the level of crystallinity and mechanical stability in the thin electrospun LCP fibers; the Young's modulus of a single fiber increased from 2.00 to 3.93 GPa. In addition to finding optimal processing parameters needed to electrospin LCPs, varying degrees of after treatment processes (annealing, drying...etc) may thus be required to reach satisfactory performance.

Finally, attempts were also made to electrospin main-chain smectic [281] and nematic [230] LCPs. Nakashima et al [281] studied the effects that changing fiber diameter had on the internal structure and molecular orientation of smectic LCP, BB-5(3-me) spun from HFIP solvent, or 1,1,1,1,3,3,3- hexafluro-2-propanol. With concentrations varying from 3–25wt.%, fibers with diameters ranging from 0.13 to 4.71 μm were observed. Under optimal spinning conditions, fibers with diameters ranging from 0.11 to 0.69 μm were found. Interestingly, this polymer was known to form an anticlinic tilted smectic phase, SmC_a, in melt spun fibers, but the pristine electrospun fibers showed only SmA order. However, this SmA phase was metastable, and after annealing it was found to transform into the stable SmC_a phase also in the electrospun fibers.

Alternatively, while electrospinning main-chain nematic LCPs with photocrosslinkable moieties, Krause et al [230] were able to obtain aligned LCE microfibers after exposing them with UV light for in situ crosslinking. A 5 wt.% solution of LCP in chloroform yielded beaded fibers with average diameter sizes of 450 nm, but upon increasing the concentration to 7 wt.%, uniform fibers were observed with average diameter 1.5 μm. Although the fibers were found to be insoluble in chloroform, proving that the fiber mat was successfully crosslinked, and had uniform director alignment along the fiber axis as confirmed by polarizing microscopy, no tests of actuation were reported. The authors tested actuation only on films prepared in parallel from the same LCE material, without commenting on the lack of actuation data for the fibers. One may speculate that this may have been due to the fibers being fixed to the substrate on which they were collected, thus precluding actuation. It remains an important challenge to electrospin free-hanging LCE fibers and study their actuation behavior.

Based on the observation that light scattering by an electrospun mat of PVP sheath–5CB core fibers is visibly reduced (change easily seen by naked eye) upon exposure to toluene or other organic solvent vapor, Kim et al proposed that LC-core fibers could be used as wearable sensors/detectors of hazardous gases [287]. A recent detailed follow-up study by Reyes et al [190] showed that the morphology of the fibers (beaded or uniformly cylindrical) is very important for the behavior. Interestingly, it was found that beaded fiber mats had visible scattering responses without the need for polarizers, while the response of non-beaded fiber mats could only be visualized with polarizers. Although the latter response thus requires additional components (polarizers) it is very interesting, because it is extremely fast (response and relaxation time below seconds) and highly sensitive [190]. While the alignment effects of the fiber sheath encapsulating the 5CB are still under investigation, as well as the limits to the fiber mats' functionality, the concept of integrating LCs into fibers for furthering ideas in wearable technology is steadily advancing. In terms of developing LC-fiber based gas sensors as a viable concept, the two key challenges are to ensure specificity (response only to the analyte gas) and enhance sensitivity. To reach these critical goals, a good starting point is to survey the work done on gas sensing with liquid crystals in flat non-fibrous samples.

Liquid crystals, on their own, have been used for gas sensing in a research context since the 1990s [288–291]. If issued patents on the usage of LCs for gas sensing are taken into account, then the idea was present decades before, since the 1970s [292–294]. Many of the early works regarding detection of aromatic and halogenated organic vapors at room temperature exploited the use of short-pitch cholesterics because of their ability to change p₀, and therefore their color, depending on how much organic vapor came in contact with the samples. The early pioneering works from Dickert and his group [288, 289, 295] demonstrated quantitative correlation between analyte concentration, determined in situ using quartz crystal microbalance (QCM), and p₀, calculated from the optical reflection band as determined by UV-VIS spectroscopy. More recent work from the TU Eindhoven group demonstrated that the principle can be used even with fluctuations in atmospheric humidity and temperature [296, 297]. These and other reports [291, 298, 299] have ensured that the idea that LCs could be used for optically detecting harmful volatile organic compounds (VOCs) has grown in popularity throughout the years.

The critical problem of specificity was, however, largely left unaddressed in the early work. Likewise, it remained an open question how LC-based gas sensors can be tailored to respond to different gaseous analytes of interest. During the last decade, the interest in LC based gas sensing applications has been revitalized, largely thanks to a new approach that solves the specificity problem, presented by the Abbott group at the University of Wisconsin-Madison. In their work, they characterized organophosphate and organic gas vapor responses in thin films of 5CB with one side exposed to the air, the other supported on a substrate decorated by perchlorate salt. Due to complexation of the salt by 5CB as well as by the analyte gas, the latter forming a much stronger complex than the former, a sensitive and specific response can be ensured. The substrates vary from polystyrene [300] to photolithography masked PDMS and polyurethane micro-wells [301–303]. The team successfully quantified the chemo-optical transitions induced upon gas exposure from as low as ppb level concentrations under various humidity settings [300, 302–305]. Thus, throughout the years Abbott's group has influenced other groups to extend, and they themselves have extended, their initial micro-level LC-VOC sensing concept. Apart from gas sensing they study sensing of biomolecular phenomena such as protein binding energies [305, 306] and enzymatic activities using nematic LCs [307].

While the Abbott group VOC sensing approach is based on an analyte-triggered change in boundary conditions, resulting in a change of LC alignment through the bulk, other groups have instead focused on sensitizing the bulk LC to specific analytes. This may be disadvantageous for the sensitivity and speed of the sensor, but it obviously gives more freedom to how the boundaries are designed. Moreover, it is an interesting alternative that serves to expand the range of VOCs that can be detected by LCs. Dirk Broer's group demonstrated high selectivity using functionalized chiral dopants added to the standard nematic mixture E7. Depending on the choice of chiral dopant they could detect carbon dioxide or oxygen [308]. Later on the same concept was used to sense increases in temperature [296] and humidity [296, 297]. The Broer group has also experimented with different form factors of LC containment in creating thin LCP responsive films for sensing [296].

If the above strategies, or alternative ones, to ensure specificity can be transferred to LCs in fibers, the new form factor should be most attractive, in particular in implementing LC sensors into non-laboratory environments. While Abbott et al demonstrated VOC sensitivity of perchlorate-sensitized 5CB films down to the ppb level, microscope and video equipment are still necessary to detect the anchoring rearrangement upon gas exposure. Since LC-filled fibers can yield a response that is visible to the naked eye, they may offer a means to take the concept out of the lab. Moreover, with a textile polymer sheath, a mat of appropriately LC-functionalized fibers can easily be incorporated in clothing, for instance uniforms of persons who risk exposure to hazardous gases in their working environment. Because the sensing requires no electrical power, the garment does not need to carry any power supply, battery or electrical wiring to drive the sensor. And with the large surface-to-volume ratio that thin fibers provide, we may expect both a sensitive and rapid response if all parameters are optimized.

Table 1 gives an overview of available and currently developing gas sensing devices, with their respective advantages and disadvantages. This can give a more succinct perspective of the progress and highlight the possibilities LCs have for integration into wearable sensing technology. The primary aspects of gas selectivity, robust form factor containment, and gas sensitivity are the main three issues which need to be addressed for creating commercially viable LC-based gas sensors. Fundamentally, as applied LC research has primarily targeted displays and related technologies, requiring electrodes and assuming hard impermeable (usually glass) substrates to encase the LCs, this novel type of LC application poses a challenge quite different from the ordinary.

There are a number of LCE materials and fabrication methods that have been explored for making actuators that are capable of either thermal, photo-mechanical, or photo-thermo-mechanical actuation. Undoubtedly, these types of LCE actuators show promise in fields like soft robotics [21, 309], and performance comparable to that of skeletal muscles has been demonstrated [16]. As discussed in section 2.4, a commonly used method for obtaining monodomain LCE networks is the two-step crosslinking approach pioneered by Küpfer and Finkelmann [40]. When the target is to make LCE fibers, this approach is convenient, the fibers being stretched during the second crosslinking stage. However, connected to the fiber production another unique way to form LCE fibers arises, based on mechanically orienting a non-crosslinked but reactive LC polymer by flow while drawing fibers, and then quickly crosslinking the fibers to prevent the orientational loss. While some have tried simply drawing fibers using tweezers [26], and toothpicks [310], others have used more sophisticated methods like melt extrusion [25], and even wet-spinning driven by microfluidic setups [311–313].

Naciri et al [26] showed the first examples of LCE fibers obtained through the simple approach of drawing from a melted viscous mixture of a nematic LC terpolymer and crosslinker. The samples were left in the stretched state for 120 hours for crosslinking to complete. The obtained fibers, with average diameter of 300 μm, were found to be birefringent with $\boldsymbol{n}$ along the fiber axis. Thermoelastic measurements to determine the mechanical properties were conducted on the elastomeric fibers in the nematic phase at 50 °C, yielding a Young's modulus of 4.37 MPa. Furthermore, under constant applied stress the length was monitored, and the resulting strain calculated, as the fibers were heated from the nematic into the istoropic phase. The length did not change in the nematic phase, it started to reduce on approaching the clearing transition at T_NI, where it sharply decreased, revealing a transition-induced strain up to 35%. The actuation behavior of the LCE fiber was also demonstrated by mounting it inside a heating coil and attaching a 200 mg weight to one end. When the fiber was heated above the N-isotropic transition, the fiber contracted and the weight was lifted.

Ohm et al [311] introduced a technique similar to the classic wet-spinning fiber formation process by instead using a microfluidics based mechanism (shown in figure 22) to obtain oriented main-chain LCE fibers from a main-chain photocrosslinkable smectic LCP. A solution of LCP (30 wt.%) and photo-initiator (3wt.%) in dichloromethane (DCM), was injected into a stream of silicone oil within a glass capillary. The co-flowing 'bath' of silicone oil, similar in one of its two functions to the precipitation bath that wet-spun fibers are extruded into prior to being collected, coagulated the fibers. Due to the miscibility of DCM and silicone oil, the solvent is removed from the LCP, which thus enters its LC state (the co-flowing system was kept at 40 °C, within the SmA phase). The result is a continuous LCP filament that is pulled and aligned by the silicone oil stream (its second function), prior to UV exposure for initiating the crosslinking.

The resulting LCE filaments were collected on a winding spool and checked for their optical and actuation behaviors. The fibers obtained had thickness between 20 and 50 μm, which was varied by changing the flow rate of the oil phase. With POM experiments it was also confirmed that the director orientation was along the fiber axis. After heating above 80 °C, to the isotropic phase, the fiber lost its birefringence and a reduction in fiber length was observed. Surprisingly, although the fiber regained its birefringence upon cooling back to 50 °C, the deformation in length was irreversible. The authors proposed that the irreversibility was due to the high activation energy required to form new smectic layers.

Theoretically, this activation energy could be overcome by an external force. To check this, the authors did another actuation experiment in which a single LCE fiber was glued at both ends, with a 'u'-shaped 25 mg weight hanging from the center, the entire setup kept inside a thermo-regulating box, see figure 43. Upon heating the air inside the box above T_NI, the weight was lifted as the fiber length reduced from 19.4 mm to 16.3 mm. On cooling, the fiber now regained its original length at a temperature of 50 °C, indicating a reversible actuation. This work showed that the continuous microfluidics based spinning method has the capability of making LCE fibers with reproducible and regular thickness which can act as actuators and lift objects more than one thousand times its own weight. Ultimately, it demonstrates a method for producing versatile and useful soft actuators.

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A more detailed mechanical and optical study on LCE fibers with thickness of 30  ±  1 μm, based on the same material, was done by Stannarius et al [312]. One of the findings was strain induced birefringence; $\Delta n$ could either be increased or decreased upon stretching and compressing of the samples in an axial direction. However, the effect was reversible only once the fiber had been heated into the SmA state at 40 °C. The slow dynamics of the room temperature glassy state does not allow the dynamic coupling between strain and orientational order parameter that the varying birefringence signifies.

Mechanical studies indicated that the relaxation of the fibers after stretching at room temperature shows glassy behavior and plastic deformation, as the initial length is not recovered. However, if the fiber is then heated above its glass transition temperature, it reaches its original relaxed length within one minute. When the fiber is further heated to T_NI, it shortens and a suspended weight is lifted accordingly. A very small stress of 80 μN is required to stretch the sample during the transition to the smectic phase. This shape memory effect is attributed to the formation of a polydomain structure in the SmA phase and a structural reorganization of smectic layers during cooling, similar to chiral SmC^* LCEs [328, 329] and nano-sized shape-changing LCE colloids [147]. However, the authors did not provide any x-ray data to confirm this particular hypothesis.

To the best of our knowledge, there has been only one report, so far, on side-chain nematic LCE fibers [313]. This was fabricated in a microfluidic set-up using a side-chain UV-crosslinkable LCP exhibiting a nematic phase between 40 and 56 °C. The LCP was mixed with 0.4 wt.% chlorobenzene and 5 wt.% photoinitiator and injected through a fused silica tube into co-flowing silicone. The set-up was immersed in a water bath for temperature control. Crosslinking was done at different temperatures within the nematic phase. POM analysis showed that fibers crosslinked deep in the N phase were uniaxially oriented whereas fibers crosslinked at a temperature closer to T_NI were not. Depending on the exact combination of precursor LCPs and on the crosslinking time, some fibers did not actuate upon heating above the clearing point, whereas others actuated several 100%. The relevant parameter is the crosslinking density, which must not be too high, as this prevents actuation. Thus, fibers obtained from LCPs with lower numbers of crosslinkable units showed the best actuation upon heating, see figure 44.

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Conclusively, for actuation, it seems like fiber orientation and a softer network that allows deformation are the key factors. The orientation of the LC phase in the fibers can also be optimized by crosslinking at low temperatures in the nematic phase, for example. Actuation can be tuned by either varying the degree of crosslinking, or varying the UV exposure time when the crosslinker amount is lowered.