5.2.1. Chirality and optics

As is clear from the discussion thus far, optics played a key role in Pasteur’s discovery of molecular chirality. His discovery benefited immensely from the illustrious 19th century French traditions in the fields of crystallography, optics and theories of the arrangement of atoms in organic molecules. Hence it would seem beneficial to understand the key events that were instrumental for Pasteur’s discovery. As already described, Pasteur measured the rotation of the plane of polarization of light on passing through a chiral medium. George Wald (Wald 1957), in a paper titled, “The origin of optical activity” begins by stating: “no other chemical characteristic is as distinctive of living organisms as is optical activity”. Wald was referring to the phenomenon that has been studied since the early 1800s, when the French physicist Jean Baptiste Biot discovered that some substances of biological origin displayed optical rotation (Applequist 1987). Instruments designed to measure such optical rotations, referred to as polarimeters (key components are illustrated in Figure 5.2), have been used to study the interaction of polarized light with chiral or optically active materials. Some of the key events that led to the science of polarimetry and to the study of chirality (or optical activity) in various disciplines are described below:

different wavelengths are rotated to a different extent when the beam exits the crystal.

Scientists of the time were fascinated by the blue color of the sky and were having trouble identifying the reasons for this color until John Strutt (also known as Lord Rayleigh) in 1871 provided an explanation based on molecular scattering, refuting Tyndall’s argument that the blue color of the sky is due to foreign matter in the atmosphere. The color of the sky, coupled with the polarization of the sky remained unexplained and John Tyndall, in 1869 (six decades after Arago’s discovery of the polarization of the sunlit sky), stated, “these questions constitute in the opinion of our most eminent authorities, the two great standing enigmas of meteorology” (Young 1982).

The discovery of optical activity in organic molecules by Biot posed two major questions, the answers to which were not obvious at the time: what is the nature of light that makes the phenomena of polarization and optical rotation possible, and what property of the organic molecules enables them to rotate the plane of polarization of the light?

1821

In an effort to demonstrate this conjecture, Frensel constructed a three-piece prism, consisting of one piece of levorotatory quartz and two pieces of dextrorotatory quartz, cut in a way that light travels through the prism along the optic axes of all three parts. Linearly polarized light entering the prism was found to split into two circularly polarized rays of opposite sense when they exited the three-piece prism, thus demonstrating that the two components experience different velocities. It is remarkable that Fresnel’s conjecture of optical rotation as being equivalent to “circular double refraction” still forms the basis for modern theories of optical rotation.

Fresnel also offered an alluring explanation for why the refractive index might be different for the two circularly polarized rays: “this may result from a particular constitution of the refracting medium or of its integral molecules which establishes a difference between the sense of right to left and that of left to right; such would be, for example, a helical arrangement of the molecules of the medium which would present opposite properties according as these helices were right-handed or left-handed” (italics added by the author for emphasis).

Helices? One could ponder how Fresnel came to view certain matter to be present in a helical arrangement, as very little was known about the geometrical arrangement of molecules in matter at that time. Nature, of course, provides a number of examples of helices, such as in snail shells, the climbing habits of plants and whirlpools. In the case of quartz, Fresnel’s ideas, while wading into new areas of knowledge, also turn out to be correct, as schematically illustrated in Figure 5.5.

Viewing along the optic axis of quartz-crystal model, one finds columns of silicon and oxygen atoms built up in spirals. The known chemical structure of crystalline quartz is SiO2. Modern X-ray diffraction has provided the structure of crystalline quartz, where the SiO₄ tetrahedra are arranged along a helical axis, with three tetrahedral per turn. To propose “a helical arrangement of molecules” must have been natural to Fresnel, as it is a shape that can occur in two forms, right-handed and left-handed, being mirror images of each other, just as the two directions of the rotation of polarization are mirror images of each other.

Almost two centuries after Fresnel’s remarkable experiment, Ghosh and Fischer (2006) demonstrated that a chiral liquid also splits a linearly polarized light beam into two circularly polarized beams of opposite handedness. Figure 5.6(a) depicts the experimental arrangement and the results. In order for the splitting of the beams to be large enough to be seen by the detector, Ghosh and Fischer used multiple refractions through prismatic cuvettes filled with optically active solutions (limonene, in the experiments), a scheme similar to that used by Fresnel. In Figure 5.7(b), the splitting of the beams of a 405 nm diode laser beam is clearly demonstrated and the two beams are indeed circularly polarized with opposite handedness.

5.2.2. Chiral symmetry breaking and its misuse

The general phenomenon of the spontaneous emergence of chiral structures or objects from achiral or “racemic” initial states is often referred to as spontaneous chiral symmetry breaking. As has been noted (Walba 2003; Ávalos et al. 2004), this term can be befuddling at best and incorrect at worst. This befuddlement arises because an object or a system with “chiral symmetry” is achiral (i.e. it is congruent with its mirror image) in the physics community, while chemists, utilizing point groups to describe congruence with its mirror image, however, use the term “reflection symmetry” to describe an achiral object. It seems useful at this point to quote from Mislow’s text (Mislow 2002) from the section entitled “Reflection Symmetry-Point Groups”:

An object is said to have reflection symmetry if it is superimposable on its reflection or mirror image. Reflection symmetry is revealed by the presence of a rotation–reflection axis (also called mirror axis, improper axis or alternating axis) S_n: in a molecule having such an axis, reflection in a plane perpendicular to that axis followed by rotation by 360°/n will convert the object into itself.

It is obvious, then, for the case where n = 1, the rotation–reflection axis is simply a mirror plane.

Should an object lack reflection symmetry in its point group, then it is a chiral object. Hence, in this context, the phrase chiral symmetry most naturally suggests the antonym of reflection symmetry, i.e. a chiral object – precisely the opposite of what is meant by chiral symmetry in “chiral symmetry breaking”, as used in physics. Walba notes that from the perspective of chemists it will seem reasonable to use the term “achiral symmetry breaking” to describe the breaking of reflection symmetry, though this term is also rife with confusion. It has therefore been suggested by David Walba (2003) and others (Ávalos et al. 2004) that a compromise to bridge differences between different communities is to use the term “reflection symmetry breaking” as the least confusing terminology describing the phenomenon formerly known as chiral symmetry breaking. Barron (2009) provides an illuminating discussion on the use and misuse of the term “chiral symmetry breaking” and eventually settles on the term “mirror-symmetry breaking”. Herein, I use the term “spontaneous emergence of chirality” to denote the appearance of chiral structures from achiral systems.

5.2.3. Spontaneous emergence of chirality or chiral structures in liquid crystals

A well-known system where reflection symmetry is broken in achiral LCs is the twisted nematic (TN) state, where the spontaneous reflection symmetry breaking is made possible by specific boundary conditions. Nematic LCs possess long-range orientational order, and are characterized by an average molecular orientation along a given direction referred to as the director, specified by a non-polar unit vector or director () (de Gennes 1974). Consider two uniaxial solid substrates, for example, mica, with the optic axis parallel to the plates, that provide strong “planar anchoring” for the nematic director, n̂. When a nematic LC is confined between two such plates that are parallel, it is found that the director aligns along the optic axis of the uniaxial substrate, resulting in a uniform achiral director structure, as shown in Figure 5.7. Now consider that one plate is rotated by 90° relative to the other, a chiral, uniformly twisted director configuration results. If the twist is low, coupled with high birefringence of the LC, then the plane of polarization of a linearly polarized light incident on the cell parallel to the optic axis of the first plate will rotate to follow the twist of the director, and will exit the second plate with its polarization along the optic axis of the second plate. In other words, the plane of polarization will have rotated by 90°, as if the light had transited through a chiral medium.

In 1911, Mauguin reported on precisely this experiment using thin mica sheets as the solid substrates. While not much was known about the structure and dynamics of LCs in 1911, Mauguin nonetheless reasoned, quite correctly, that the director must be uniformly rotating through the cell, and established that the exiting polarization should rotate with the second plate. This is often referred to as the “wave-guiding mode” or the “adiabatic limit” of plane-polarized light in a TN cell with a 90^o twist, and it occurs when Δnd >λ/2π, where Δn is the birefringence of the nematic fluid and d the thickness of the cell. This is also referred to as the Mauguin limit in the literature.

Now consider an experiment that Mauguin did not report on in his 1911 article, where one takes the TN cell, with a 90^o twist, and heats the system to its isotropic phase, and lets it cool back down to the nematic phase. In this case, in the isotropic phase, the system is achiral (still with the two plates at 90^o) but when it cools to the nematic phase it must have a twisted structure, but since, in the nematic phase, n is equivalent to –n, domains of opposite handedness must form with equal probability. This is precisely what is observed in such an experiment, as can be seen from Figure 5.8.

When we observe these macroscopic domains by polarized light microscopy, we can easily observe the disclination lines that separate the heterochiral domains, though the output polarizaiton is identical for the pair of heterochiral domains. This is the simplest case of the spontaneous emergence of chirality from an achiral system.

Another example of the emergence of chiral structures is during a temperature-driven anchoring transition (Amundson and Srinivasarao 1998; Zhou et al. 2002; Zhou et al. 2006); for example, a nematic LC confined between two plates where, at lower temperatures, the surfaces dictate a condition where the director is normal to the substrates (homeotropic alignment) and, at a higher temperature, adopts a director configuration parallel to the bounding substrates (planar alignment). In such a system, cycling through the anchoring transition temperature, say, from the high temperature planar state to a lower temperature homeotropic state, allows for the rotation of the director in two opposite directions [n = − n], thus leading to a 180° inversion wall, in this case, a pure twist wall. Such inversion walls have been observed in a number of experiments (Leger 1973; Ryschenkow and Kleman 1976). Figure 5.10 schematically illustrates the director configuration of an inversion wall consisting of 180° twist deformation along the x direction, the so-called Bloch wall (Kleman 1983), where d is the thickness of the Bloch wall and h is the sample thickness. Helfrich first theoretically described the director configuration of inversion walls formed due to the application of a magnetic field (Helfrich 1968). Such wall defects in nematic phases are usually unstable and collapse on themselves in a short time, but can be stabilized when confined by the bounding surfaces. Such a Bloch wall, observed between crossed polarizers, is shown in Figure 5.9, where the director configuration on either side of the wall is homeotropic and we can see the symmetric and parallel color bands with respect to the center yz plane of the wall when a white light source is used in the polarized light microscope.

The color sequence follows the Michel–Levy birefringence chart from either edge of the wall to the central plane. When a monochromatic light source is used, the wall between crossed polarizers shows interference fringes parallel to the wall. Should the Bloch walls be long enough, it is often observed that the wall is composed of two different handednesses, separated by what is referred to as a Neél wall, that mediates the difference in the two distinct handednesses of the Bloch walls, as illustrated in Figure 5.10. These structures are often not considered as spontaneous emergence of chiral structures, but, in fact, these are clear examples of the emergence of a chiral structure from an otherwise achiral system.

5.2.4. Spontaneous emergence of chirality due to confinement

It is often convenient to study LCs in flat geometries, as it allows for easy manipulation of the director and the measurement of many characteristic properties using simple experiments. However, the confinement of LCs to curved geometry results in a much richer phenomenology. Curvature induces deformation of the director that costs energy. The terms describing the free energy of LCs are all of the order (∇n)² to leading order. For this reason, Frank (1958), in his original description, uses the term “curvature elasticity” to describe the nematic elasticity. The presence of curvature can also result in competing effects of surface and bulk elastic constants which result in several interesting director configurations. The richness is further enhanced if the nematic fluid has significant anisotropy of its elastic constants, as is often the case with lyotropic nematic LCs. In this section, we discuss the spontaneous emergence of chiral structures from a class of LCs referred to lyotropic chromonic liquid crystals (LCLCs), as well as some lyotropic polymeric LCs.

LCLCs have gained increasing attention in the last two decades as an interesting but still poorly understood class of lyotropic LCs (Attwood et al. 1986; Vasilevskaya et al. 1989; Lydon 1998a, 1998b, 2004). LCLCs consist of many dyes, drugs, nucleic acids, antibiotics and anti-cancer agents (Cox et al. 1971; Hartshorne and Woodard, 1973; Attwood et al. 1986; Vasilevskaya et al. 1989; Lydon 1998a, 1998b, 2004; Horowitz et al. 2005; Park et al. 2008). The main structural feature of the molecules that comprise a chromonic liquid crystalline phase is either a disk-like or plank-like rigid aromatic core surrounded by either ionic or hydrogen-bonding groups (Lydon 1998b). When these molecules are dissolved in water, they stack against each other into rod-like aggregates with polydisperse lengths, even at very low concentrations (Attwood et al. 1986; Vasilevskaya et al. 1989; Lydon 1998b, 2004). The π-π interactions among the molecules are thought to be the driving force for the aggregation of these molecules into rod-like shapes.

As the concentration reaches about 5–30 wt% of the dye in solution, a liquid crystalline phase forms (Attwood et al. 1986; Vasilevskaya et al. 1989; Lydon 1998b, 2004). There are two principal, well-characterized chromonic mesophases: a nematic N phase and a hexagonal M phase, as illustrated in Figure 5.11. The M phase is a fairly concentrated phase with a hexagonal array of molecular stacks, also referred to as the columnar phase.

In the recent past, the study of LCLCs has led to a number of observations that have been quite surprising and unexpected, and has inspired re-examination of a number of studies dealing with nematic LCs. We will start our description of them with a rather intriguing list of new observations that lead to the emergence of chiral structures from achiral systems which are quite exciting, and illustrate the rich phenomenology arising, primarily, from the fact that these materials posses anisotropic elastic properties, coupled with confinement to curved geometries.

As early as 1949, Onsager (1949) pointed out that a rod-like object in solution can form an ordered state, the so-called nematic phase (N), where the rods are parallel to each other, when their concentration exceeded some critical value.

Tobacco Mosaic Virus (TMV) is a rod-like particle with a length of 3,000 Å and 180 Å in diameter, and is monodisperse. Monodisperse aqueous solutions of TMV form a nematic phase between the concentrations of 30 and 70 mg/ml in pure water. Onsager provided an explanation for the obersvations of Bawden, Pirie, Bernal and Fankuchen (Bawden et al. 1936), dealing with aqueous suspensions of TMV that phase separated into an isotropic phase (I) and a nematic phase (N). It should be noted that the N-phase emerges as spindle-like objects called “tactoids”, and such N-tactoids have been shown to exist for a number of other materials. In these earlier studies, there was no mention of chirality in the tactoids, although it might seem prudent to re-examine these systems in light of things that are discussed in the following pages.

Recently, Tortora and Lavrentovich (2011), when studying solutions of disodium chromoglycate (DSCG, see Figure 5.11(a) for the chemical structure) in water with a finite concentration of polyethylene glycol (PEG), observed tactoids that displayed “chiral symmetry breaking”, meaning that the tactoids displayed both left-handed and right-handed twist structures, as illustrated in Figure 5.12. As already noted above, one should not use the term “chiral symmetry breaking” for these situations.

Figure 5.13 illustrates a chiral tactoid that was observed by Srinivasarao (Srinivasarao and Berry 1992) during the preparation of monodomains of a polymeric nematic in solution, but, at that time, was not recognized for the spontaneous emergence of chirality. This appears to be the first observation of the emergence of chiral structures from an achiral nematic solution. Incidentally, such polymeric nematics have a large anisotropy in their elastic constants, with twist elastic constant being at least an order of magniture lower than either splay or bend (Desvignes et al. 1993).

The rationale for the emergence of chiral structures in these tactoids is that the twist deformation replaces the energetically costly splay packing of the aggregates within the curved bipolar tactoids. This provides a simple prescription for creating chiral structures: that the material has a certain anisotropy of the elastic properties and that it has non-flat confinement!

It is well known that nematic LCs, when confined to spherical droplets with planar anchroing, form what are known as bipolar droplets with a point defect at the poles of the sphere. In their studies dealing with droplets of LCLCs, Yodh and co-workers (Jeong et al. 2014) used solutions of Sunset Yellow (SSY) in water and found that the droplets have a twisted bipolar structure, a consequence of a very low twist elastic constant. This leads to a very large twist in the droplets, despite the lack of chirality in the system, and are similar to those observed when the droplets are made with LCs possessing intrinsic chirality.

5.2.5. Spontaneous emergence of chirality due to cylindrical confinement

5.2.5.1. Director normal to the surfaces

In 1973, R.B. Meyer (Meyer 1973; de Gennes 1974) and Williams, Pieranski and Cladis (Williams et al. 1972) demonstrated that when a nematic fluid is confined to a cylindrical capillary, where the director is normal everywhere on the capillary surface, the director (n) escapes along the length of the capillary, therefore leading to a continous structure (plus a few point defects) and thus avoiding having a line defect throughout the capillary. This has been referred to as “escape in the third dimension” (or sometimes as an escaped radial) and has been experimentally verified over the course of five decades for many thermotropic LCs. Figure 5.14 illustrates such a situation where a well-known and well-studied thermotropic nematic LC, 5CB, is confined to a cylinder with homeotropic boundary conditions. We can clearly see the continous structure under crossed polarizers and a schematic of the director configuration is also shown in the figure. When such experiments are done with LCLCs the results are quite strikingly different with a new ground state.

Jeong et al. (2015) reported on such an experiment where the SSY solutions were confined to a cylindrical capillary with homeotropic boundary conditions. The initial configuration of the director was the well-known escape in the third dimension. However, this configuration quickly turned into a twisted configuration where the splay and bend deformations in the capillary were replaced by a twist (due to the low twist elastic constant). This results in a twisted and escaped radial configuration (referred to as TER, see Figure 5.15), where the nematic director is parallel to the cylindrical axis near the center; but to possess the homeotropic boundary condition near the capillary wall, the director escapes along the radius through bend and twist distrotions. This also leads to a panopoly of defects that will not be described in detail here, but has been discussed by Jeong et al. (2015). This TER structure is eventually replaced by a set of two singular lines forming a double helix (as shown in Figure 5.16) which remians stable for an extended period of time. It would appear that this structure, with two singular lines forming the double helix, is the new ground state for these lyotropic materials confined to a cylinder with homeotropic boundary conditions.

It appears that this report of a TER structure eventually leading to two line defects forming a double helical structure down the long axis of the cylinder, reported by Jeong et al., is not the first of its kind. Such a structure was reported by Jin-Hua Wang (1996), a student of R.B. Meyer. Wang observed such a structure in solutions of TMV in capillaries with homeotropic alignment, as early as 1996. That TMV forms a liquid crystalline phase was known as early as 1936, and led to the formulation of a theory for the phase separation by Lars Onsager in 1949, as noted above. In the experiments of Wang, TMV solutions in the nematic phase were confined to cylindrical capillaries with homeotropic boundary conditions. Wang also noted that the two bright “spiral lines” running through the entire length of the capillary were “clearly visible even without a microscope”. It was also noted that the lines did not lie on the surface of the quartz capillary, and that they were at a distance of 35 μm from the capillary walls. These observations are strikingly similar to those reported by Jeong et al. (2015), but for a system that is polymeric in nature.

It should be mentioned that we have used another polymeric nematic material, poly benzyl glutamate with homeotropic boundary conditions, and the observations mimic what has been seen for LCLCs, TMV and also some micellar nematics (Dietrich et al. 2017). It appears that the observations described thus far, that the escape in the third dimension giving rise to a TER configuration, followed by the transformation of the TER to a system with two spiraling line defects down the length of the capillary, seems to be common for all nematic fluids that are lyotropic in nature, as most if not all have a very low twist elastic constant (Dietrich et al. 2020).

The presence of TER structures can be rationalized on the basis of a low twist elastic constant for this class of LCs. The free energy cost of splay, twist, bend and saddle-splay distortions of a nematic fluid is described by the well-known Frank-Oseen elastic free energy:

[5.1]

Jeong et al. (2015) performed numerical calculations to understand the rationale for the observation of TER structures using the known values for lyotropic LCs (Zhou et al. 2012). From their calculations they came to the conclusion that the TER director configuration lowers the elastic free energies of the “escape in the third dimension” or the escaped radial configuration, with a degenerate twist deformation, the twist being either left- or right-handed. They noted that total elastic free energy of the twisted (TER) configuration is smaller than that of the escaped radial configuration when K₂₂< K₂₂~0.27K, where K = K₁₁=K₃₃. So, the condition for observing a twisted structure in lieu of the escaped radial structure is that when the twist elastic constant is about an order of magnitude smaller than the corresponding splay and bend elastic constants, a condition that many lyotropic LCs satisfy. It should be noted, for cylinders with homeotropic boundary conditions, where is normal to the boundaries of the cylinder, the saddle-splay (K₂₄) contribution to the total free energy takes on a constant value, and does not depend on what is in the bulk, and hence does not play a major role in determining the director configuration.

5.2.5.2. Director parallel to the surfaces

Now we consider the case of confining lyotropic nematics in a cylinder with planar degenerated (gliding) boundary conditions. As already indicated, when nematic LCs are confined within flat boundaries with planar boundary conditions, the ground state is characterized by an average molecular orientation along a common direction referred to as the easy axis of the director, denoted by n. Curvature for nematics costs energy and the coupling of the curvature with the free energy can be appreciated when one notes that the elastic free energy expression of the n is of the order of (∇n)². For this reason, the Frank free energy is often referred to as the curvature elasticity of nematic LCs (Frank 1958). Pairam et al. (2013) confined prototypical nematics, the thermotropic NLC 4-Cyano-4’-pentylbiphenyl (5CB), in a toroid with the director parallel to the surfaces everywhere, and demonstrated that the, achiral, axial state is unstable to the doubly twisted, chiral state that depended both on K₂₂/K₂₄ and the aspect ratio of the torus ξ =R₀/ a, where R₀ is the central ring radius and a the tube radius. They also demonstrated that the magnitude of the twist distortion in a toroid grew with decreasing aspect ratio due, in large part, to the K₂₄ contributions to the overall free energy of the system, thus effectively screening the energy penalty of an increasing twist distrotion.

We now confine the LCLCs in a cylinder with planar boundary conditions and interrogate the resulting director configuration. In most experiments with conventional nematics, the director lies along the long axis of the capillary, the axial configuration, as illustrated in Figure 5.17(a). When such a configuration is observed under crossed polarizers, the intensity profile of the transmitted light beam is of the form: where ∅_p is the angle between the easy axis of the polarizer and the director. The observed intensity is a minimum when the director is either parallel or perpendicular to the incident polarization, leading to complete extinction of the light, with the maximum intensity being at an angle of 45° with respect to the incident polarization. This is exactly what is observed when one confines a thermotropic LC, say 5CB for example, with planar boundary conditions.

However, when the experiments are repeated with solutions of DSCG, the observations point to a different result. Images under a polarized light microscope for two angular positions of a capillary filled with DSCG solutions are shown in Figure 5.18(a) and 5.18(b), corresponding to the long axis of the capillary being parallel and at 45° with respect to the incident polarization, respectively. It is quite clear in both cases that the expected extinction of the incident light is not achieved. Should one have an axial configuration, the image corresponding to Figure 5.18(a) would be one of complete extinction of light. Instead, incident light is transmitted by propagating through the sample and they appear to possess comparable intensities in both Figure 5.18(a) and Figure 5.18(b). This simple observation makes it abundantly clear that the plane of polarization of the linearly polarized incident light is rotated by the lyotropic LC, DSCG, and exits at an angle to the analyzer that is neither 0° nor 90°. As already noted above when dealing with Pasteur’s discovery of optical activity, this represents a classic signature of twists. A doubly twisted configuration would explain the observed images and is illustrated in Figure 5.17(b). The director of this doubly twisted configuration is axial at the center of the cylinder, twisting progressively as it approaches the surface.

Validation of the hypothesis of a double twist can be done by carrying out Jones matrix simulation with a simple doubly twisted director ansatz, as such simulations quantify the change in the polarization state of the light as it traverses the sample. The doubly twisted director is specified by e_z are the orthonormal vectors in cylindrical coordinates, with The twist parameter ω determines the amount of twist in the system; ω = 0 corresponds to an axial configuration, r is the radial distance from the center of the circular cross-section and R is the cylinder radius. The simulated optical microscopy textures conform closely to the experimental observations that light is indeed not extinguished with the capillary long-axis parallel to the polarization, of the incident polarization as well as reproducing all the aspects of the measured intensity profiles. These lend credibility to the idea that we must, indeed, have a doubly twisted director configuration in a cylindrical capillary.

Based on the experimental observations and the Jones matrix simulations, we can then ask a number of questions that are of consequence: What drives the spontaneous twist deformation of the director field? How twisted is the structure of the double twist? Is there a size dependence of this twist? All of these questions are addressed in detail elsewhere (Davidson et al. 2015; Nayani et al. 2015), but we provide the essential arguments below.

In order to understand the driving force for the doubly twisted configuration, we start by considering the contribution of the saddle-splay (K₂₄) term in the free energy expression given by equation [5.1], where K₁₁, K₂₂ and K₃₃ are the Frank elastic constants associated with splay, twist and bend bulk deformations. We ignore the splay-bend (K₁₃) contribution as we are dealing with a case of planar anchoring, though the influence of K₁₃ has been quite contentious (Lavrentovich and Pergramenshchik 1995). It should also be noted that several studies have neglected the contribution of saddle-splay under strong anchoring conditions, the rationale for which stems from the fact that the bulk integral of the saddle-splay can be reduced to a surface integral using Stoke’s theorem, and that the contribution of saddle-splay on the surface would be negligible in comparison to the cost of the anchoring violation. However, this argument is troublesome as the saddle-splay contribution can only be neglected if the director depends only on one Cartesian coordinate, and for strong anchoring, which is not the case in the experiments described (Kralj and Zumer 1995). For planar anchoring, when the confining boundaries are curved, the contribution of K₂₄ cannot be neglected, which would indicate that planar anchoring may lay at an angle to the confining cylinder long-axis. Further, in geometries like cylinders where the two principal curvatures are different, the contribution of saddle splay plays a prominent role in determining the director configuration.

For the case of planar anchoring, the saddle-splay term can align the director along the direction of the largest principal curvature. To better understand this, the K₂₄ contribution to the free energy (per unit length) can be written as: where k₁ and k₂ are the principal curvatures at a point on the surface and n₁ and n₂ are the director components along the corresponding directions. For a cylinder, n₁ = n_θ and n₂ = n_z hence, k₁ = 1/R and k₂ = 0. As a result, for the case of cylindrical geometry, the integral of F₂₄ is minimized when the director at the surface is along the e_θ direction, thus driving the system to twist and provided that there is sufficient anisotropy of the elastic constants, the twisted structure is always stable.

Using the same director ansatz outlined for the Jones matrix simulation, we obtain an expression in leading order of ω for the free energy per unit length:

[5.2]

It is, in fact, the sign of the quadratic term that determines whether the system adopts an axial configuration or breaks reflection symmetry, resulting in a chiral, doubly twisted director configuration. This theory is generic and would describe the reflection symmetry breaking of any nematic fluid. Provided K₂₄ > K₂₂, the director is always going to be twisted. Specifically, for LCLCs the low value of K₂₂ almost always leads to the possibility of satisfying the twisting criterion. From the observation of twisted director configurations, it is self-evident that K₂₄ for LCLCs studied must always be greater than K₂₂. While the model is relatively simplistic (in terms of the ansatz used), the essential physics is captured with the criterion that K₂₄ > K₂₂ for a twisted structure.

For ascertaining the criterion for the twist, we can calculate the value of K₂₄ by comparing the value of the twist parameter ω measured experimentally, and that obtained after minimizing the free energy expression with the experimental measurement of with ω The twist parameter is obtained by measuring the twist angle as the beam traverses the diameter of the cylinder. In order to do this, we used the Mauguin limit, where , with ∅ the total twist angle and Γ the retardation due to the anisotropy of the refractive indices, and the director field serves as a waveguide to the incident polarized light. Due to the low birefringence, n_e – n_o, (~0.02) for DSCG, capillaries of diameter 400 µm are necessary to satisfy the waveguiding requirement. Using waveguiding measurements, we estimate the total twist angle for the solutions used in the study to be (150 ± 10)^o.

In the ansatz chosen, ∅ = 2 sin^-1(ω), we find that ω = 0.95 ± 0.03. Substituting the value of ω thus determined into the theoretical expression below:

[5.3]

we find that the range of K₂₄/K₃₃ for DSCG varies from 0.75 to 1.75 under the assumption that K₂₂/K₃₃ ~ 0.1. We performed the same experiment with SSY to obtain a twist angle close to 170^o. This would correspond to a value of K₂₄/K₃₃ of 6.75. It should, however, be noted that the exact value of K₂₄/K₃₃ is extremely sensitive to the value of the twist angle for large twists (ω ~ 1). By way of comparison to small molecule LCs, say 5CB, we note that the value of K₂₄ is significantly higher in comparison to the other elastic constants. For all the capillary dimensions (50, 100, 400 and 500 µm) used in the experiments, the twist angle measured is independent of the system size, pointing to another interesting aspect of the results thus far. This is in contrast to measurements made by Pasteur where the specific rotation [α],

[5.4]

was +12° for tartaric acid, where α (in degrees) is the observed optical rotation, l is the length of the sample tube (in decimeters) and c is the concentration (in gm/ml) of the solution.

Since the twist angle is independent of the system size, we can confine LCLCs in as small a capillary as experimentally possible and still obtain the same values of the twist angle. It should be noted that the twist angle depends only on the ratio of the elastic constants. It should also be evident that the obtained values of K₂₄/K₃₃ seemingly violate Ericksen’s inequalities (Ericksen 1966); these emerge on the basis that the Frank free energy is positive definite, with the minimum being assumed to be a deformation free director. In the systems chosen for the experiments, the anisotropy of the elastic constants results in the formation of a ground state that is lower in energy than the deformation free ground state. Therefore, we conclude that Ericksen’s inequalities no longer apply. In a recent paper, Selinger considers the role of K₂₄, providing a nice discussion of its implications on various experimental observations (Selinger 2018), while also noting the circular nature of the argument leading to Ericksen’s inequalities.

In prior studies, we have formed monodomains of LCLCs using a rectangular capillary but had noticed the presence of two line defects confined to the edges of the capillary. We have also previously studied the dynamics of such line defects that were formed in the capillary (Shams et al. 2015); however, we had not paid enough attention to the pathway of the LC to the monodomain state. In an attempt to understand the pathway to a monodomain, we re-evaluate the experiments in rectangular capillaries (Fu et al. 2017). When LCLC is confined to a rectangular capillary with planar boundary conditions, one expects an axial director profile pointing along the long axis of the capillary, corresponding to a deformation free ground state. As already discussed, for such a director configuration one expects complete extinction of light when the capillary is either parallel or perpendicular to the polarizers. However, when a sample is cooled from the isotropic to the nematic phase, we find that light is transmitted even when the capillary is parallel to the polarizer, as shown in Figure 5.19(a). Such an observation implies that the director is twisted. It is also evident that there are disclination lines that separate different domains; these are in fact domains of opposite twists as can be seen from uncrossing the polarizers, where in Figure 5.19(b) and Figure 5.19(c) correspond to images with polarizers making an angle of 70° and 110° with each other. Figure 5.19(d) is a schematic illustration of the doubly twisted director profile that would be consistent with the experimental results.

On cooling the sample from the isotropic phase, it is seen that the twist, in fact, originates at the edges of the capillary (Figure 5.20(a)) with the twisted fronts approaching each other as they engulf the isotropic regions (Figure 5.20(b–e)). The twisted domains are spatially random with equal probability for either handedness. When the two twisted domains meet, those with the same handedness merge, while those with opposite handedness are separated by disclination lines.

We can use rather simplistic scaling arguments for the preference of a doubly twisted director configuration as opposed to the axial configuration. Let us start with the assumption that the rectangular capillary can be treated as a pair of flat plates capped with semi-cylindrical edges. In this case, it is evident that the director would favor a doubly twisted profile driven by K₂₄, as was the case for a cylinder. For a twist angle close to 180°, the K₂₄ contribution to the free energy per until length would be –πK₂₄. While twist is favorable at the edges, the twist in the flat region of the capillary results in an energy penalty estimated as K₂₂ (w/h) (~10K₂₂, for the geometry used), where w is the width of the rectangle and h is the height between the plates. The energy cost for having a doubly twisted director (F_DT) for a rectangular capillary is F_DT ~ (10K₂₂−πK₂₄), and K₂₄ for SSY is about 50K₂₂, implying F_DT < 0, and hence a doubly twisted configuration is favored in the rectangular capillary.

Although it is favorable to twist at the edges, twisting in the flat region costs energy. A simple scaling argument would estimate the energy cost per unit length in the flat region to be of the order K₂₂ (w/h) (~10K₂₂, for our geometry), where w is the width of the rectangle and h is the height, as shown in Figure 5.20d. Hence, the deformation cost of having a doubly twisted director (F_DT) in a rectangular capillary is: F_DT ~ (10K₂₂−πK₂₄). Previous experimental estimates of K₂₄ for SSY are about the order 50K₂₂ (Davidson et al. 2015; Nayani et al. 2015). Clearly this implies, F_DT<0, and the doubly twisted director is preferable when compared to axial configuration (F_axial=0). This explains the initial doubly twisted director configuration in the rectangular capillary. However, in a few hours the doubly twisted configuration leads to a monodomain in the flat regions of the capillary but with the twist confined to the edges, mediated by two line defects that separate the twisted and the axial configuration. It should be noted that the defect lines are in the midplane running along the entire length of the capillary, and the twist being confined to the hemispherical caps of the capillary.

The monodomain configuration in the flat region does not suffer any energy penalty; however, there is a penalty to be paid for the two half-strength disclination lines. A simple estimate of the energy cost of the disclination core per unit length is of the order of πKm, where m = 1/2 is the strength of the line. Since twists are the dominant deformation around the core, the energy cost for the two lines is π(K₂₂/2). Clearly, the cost of having two disclination lines is an order of magnitude lower than the doubly twisted flat regions, and the monodomain-like configuration is stable for months and is likely the ground state for SSY in rectangular capillaries.

In an effort to ascertain the above analysis, experiments were done with square capillaries with widths of 50 and 100 μm. In this case, one does form the doubly twisted configuration and this remains stable for months without ever reverting to a monodomain, as in the case of a rectangular capillary. Again, using the previous scaling arguments, we note F_DT ~ (K₂₂ (w/h) − πK₂₄) for a doubly twisted configuration; and F_Mono ~ (π(K₂₂/2) − πK₂₄) for a monodomain-like configuration in a rectangular capillary. For a square capillary, w/h is 1, such that F_DT is smaller than F_Mono, thus leading to the conclusion that the doubly twisted configuration is the desired one for SSY in a square capillary.

5.2.6. Some misconceptions about optical rotation

Having discussed the spontaneous emergence of chirality or chiral structures in the previous sections, dealing primarily with ordered nematic fluids, to conclude this chapter, we now turn our attention to a common misconception about optical rotation and its link to molecular chirality; thus reconnecting back to the historical beginnings outlined in this chapter.

Chemistry students know that the presence of optical activity implies a chiral, or dissymmetric, arrangement, the appreciation of this fact was possible only after many years of painstaking work by some of the most noted scientists of the 19th century, as discussed in section 5.2.2. Perhaps the first exposure of a chemistry student to the rotation of the plane of polarization of a linearly polarized light beam is often in introductory courses dealing with organic chemistry. One is taught in such courses, that the observation of optical rotation is a confirmation of chirality of molecules in isotropic liquids. Often a corollary of this lesson is that enantiomorphism is a necessary condition for optical rotation. As Kahr notes in his minireview on “Optical rotation of achiral compounds” (Claborn et al. 2008), a premier textbook in organic chemistry states that “the same non-superimposability of mirror images that gives rise to enantiomerism is also responsible for optical activity” (Boyd and Morrison 1976). With a sense of frustration, Kahr notes, “Unfortunately, the link between optical activity and enantiomorphism is not only introduced early and reinforced relentlessly in a chemist’s education, it is wrong”. It has been pointed out quite elegantly by Barron (2009), O’Loane (1980) and Nye (1985), among others, that oriented systems belonging to non-enatiomorphous point groups will show optical rotation or be optically active for certain directions of the incident light; however, this fact has not made significant inroads in the science of molecular chirality.

The term nonenantiomorphous refers to crystals that can be superposed on to their mirror image. From his experiments Pasteur concluded that enantiomorphism is a necessary condition for optical activity or optical rotation in crystals, an erroneous conclusion to be sure. In 1882, J. Willard Gibbs, in a paper entitled, “Double refraction in perfectly transparent media which exhibit the phenomena of circular polarization”, in the American Journal of Science – utilizing Maxwell’s electromagnetic theory – proposed that crystals belonging to two non-enatiomorphous point groups [S₄(4) and D_2d- (42m)] could be optically active and rotate the plane of polarized light. In a footnote, Gibbs imagines crystals composed of spiral forms and writes:

There is no difficulty in conceiving of the constitution of a body which would have the properties described above. Thus, we may imagine a body with molecules of a spiral form, of which one half are right-handed and one half left-handed, and we may suppose that the motion of electricity is opposed by a less resistance within them than without. If the axes of the right-handed molecules are parallel to the axis of X, and those of the left-handed molecules to the axis of Y, their effects would counterbalance one another when the wave-normal is parallel to the axis of Z. But when the wave-normal (of a beam of polarized light) is parallel to the axis of X, the left-handed molecules would produce a left-handed (negative) rotation of the plane of polarization, the right-handed molecules having no effect; and when the wave-normal is parallel to the axis of Y, the reverse would be the case.

This implies that an incident light beam might encounter paths with differing optical activity along different directions, within a given material. Validation of the Gibbs conjecture had to wait for about 85 years until Hobden (1967) was able to measure optical activity in two nonenatiomorphous crystals – silver gallium sulfide belonging to point group D_2d (42 m), and cadmium gallium sulfide – of point group S4(4), and these measurements made possible by the material becoming fortuitously isotropic for light at a wavelength of 487.2 nm. With these measurements, Hobden wrote in 1967, “It is hoped that this observation will finally eradicate the notion that optical activity is exclusively related to enatiomorphism”. It is our hope that the above discussion rights some of the misconceptions about optical activity in various materials.