We present a generalized van der Waals theory for a lyotropic cholesteric system of chiral spherocylinders based on the classical Onsager theory for hard anisometric bodies. The rods consist of a hard spherocylindrical backbone surrounded with a square-well potential to account for attractive (or soft repulsive) interactions. Long-ranged chiral interactions are described by means of a simple pseudo-scalar potential which is appropriate for weak chiral forces of a predominant electrostatic origin. Based on the formalism proposed by Straley [Phys. Rev. A {\bf 14}, 1835 (1976)] we derive explicit algebraic expressions for the twist elastic modulus and the cholesteric pitch for rods as a function of density and temperature. The pitch varies non-monotonically with density, with a sharp decrease at low packing fractions and a marked increase at higher packing fractions. A similar trend is found for the temperature dependence. The unwinding of the helical pitch at high densities (or low temperatures) originates from a marked increase in the local nematic order and a steep increase of the twist elastic resistance associated with near-parallel local rod configurations. This contrasts with the commonly held view that the increase in pitch with decreasing temperature as often observed in cholesterics is due to layer formation resulting from pre-smectic fluctuations. The increase in pitch with increasing temperature is consistent with an entropic unwinding as the chiral interaction becomes less and less significant than the thermal energy. The variation of the pitch with density, temperature and contour length is in qualitative agreement with recent experimental results on colloidal {\em fd} rods.
We present a microscopic analysis of the instability of the nematic phase to chirality when molecular chirality is introduced perturbatively. We show that previously neglected short-range biaxial correlations play a crucial role in determining the cholesteric pitch. We propose an order parameter which quantifies the chirality of a molecule.
We report the first experimental realization of a chiral artificial microswimmer exhibiting the helical motion. We found that a cholesteric liquid crystal droplet with a helical director field swims in a helical path driven by the Marangoni flow in an aqueous surfactant solution. We confirmed that the handedness of the droplet determines that of the helical path. This result strongly suggests that the helical motion is originated from the chirality of the cholesteric liquid crystal. To study the mechanism of the emergence of the helical motion, we propose a coupled time-evolution equations in …
Many nanoparticle-based chiral liquid crystals are composed of polydisperse rod-shaped particles with considerable spread in size or shape, affecting the mesoscale chiral properties in, as yet, unknown ways. Using an algebraic interpretation of Onsager-Straley theory for twisted nematics, we investigate the role of length polydispersity on the pitch of nanorod-based cholesterics with a continuous length polydispersity, and find that polydispersity enhances the twist elastic modulus, $K_{2}$, of the cholesteric material without affecting the effective helical amplitude, $K_{t}$. In addition, fo…
Heliconical cholesteric liquid crystals are expected to be more sensitive to torque induced by light field since their structure allows both bend and twist in molecular orientations, differently from the conventional cholesterics in which only twist deformation is involved, requiring much higher fields. We report here a demonstration of tuning the helical pitch in heliconical cholesterics induced by an optical torque. Experimental observations are in agreement with expectations of the classical theory extended to include the effect of the optical field. A dual control of the helical pitch is a…
We have written expressions for the free energy of a cholesteric liquid crystal in an approximation using the elasticity constants K_1, K_2, K_3 and the energy variation and the corresponding energy and energy gradient along the direction of the director vector n at each lattice point for the lattice model. A feature of this lattice model is the shift of the n components in space relative to each other. An assessment has been made of the error associated with the respective substitution. Such an approach can increase the accuracy of calculations of the energy and dynamics of localized states i…
General symmetry arguments, dating back to de Gennes dictate that at scales longer than the pitch, the low-energy elasticity of a chiral nematic liquid crystal (cholesteric) and of a Dzyaloshinskii-Morya (DM) spiral state in a helimagnet with negligible crystal symmetry fields (e.g., MnSi, FeGe) is identical to that of a smectic liquid crystal, thereby inheriting its rich phenomenology. Starting with a chiral Frank free-energy (exchange and DM interactions of a helimagnet) we present a transparent derivation of the fully nonlinear Goldstone mode elasticity, which involves an analog of the Ande…
We consider the interaction potential between two chiral rod-like colloids which consist of a thin cylindrical backbone decorated with a helical charge distribution on the cylinder surface. For sufficiently slender coiled rods a simple scaling expression is derived which links the chiral `twisting' potential to the intrinsic properties of the particles such as the coil pitch, charge density and electrostatic screening parameter. To predict the behavior of the macroscopic cholesteric pitch we invoke a simple second-virial theory generalized for weakly twisted director fields. While the handedne…
Solids with spatially varying photonic structure offer gaps to light of a wider range of frequencies than do simple photonic systems. We solve numerically the field distribution in a solid cholesteric with a linearly varying inverse pitch (helical wavevector) using equations we derive for the general case. The simple idea that the position where the Bragg condition is locally satisfied is where reflection takes place is only true in part. Here, reflection is due to a region where the waves are forced to become evanescent, and the rate of variation of structure determines over which distance th…
We experimentally study how the cholesteric pitch, $P$, depends on the equilibrium one, $P_0$, in planar liquid crystal (LC) cells with both strong and semistrong anchoring conditions. The cholesteric phase was induced by dissolution in the nematic LC the right-handed chiral dopant 7-DHC (7-dehydrocholesterol, provitamin $D_3$) which transforms to left-handed tachysterol under the action of UV irradiation at the wavelength of 254 nm. By using the model of photoreaction kinetics we obtain dependencies of isomers concentrations and thus the equilibrium pitch on UV irradiation dose. The cholester…
We report measurements of the cholesteric pitch and twist elastic constant $(K_{22})$ in monodisperse suspensions of the rod-like virus filamentous bacteriophage {\it fd}. Measurements were taken for concentrations spanning the entire cholesteric region at several ionic strengths and temperatures. In the limit of high ionic strength the cholesteric pitch $(P_0)$ scales with concentration (c) as $P_0 \propto c^{-1.66}$. As the ionic strength decreases the scaling exponent systematically changes to lower values.
Recent numerical simulations of hard helical particle systems unveiled the existence of a novel chiral nematic phase, termed screw-like, characterised by the helical organization of the particle C$_2$ symmetry axes round the nematic director with periodicity equal to the particle pitch. This phase forms at high density and can follow a less dense uniform nematic phase, with relative occurrence of the two phases depending on the helix morphology. Since these numerical simulations were conducted under three-dimensional periodic boundary conditions, two questions could remain open. Firstly, the r…
Building a general theoretical framework to describe the microscopic origin of macroscopic chirality in (colloidal) liquid crystals is a long-standing challenge. Here, we combine classical density functional theory with Monte Carlo calculations of virial-type coefficients, to obtain the equilibrium cholesteric pitch as a function of thermodynamic state and microscopic details. Applying the theory to hard helices, we observe both right- and left-handed cholesteric phases that depend on a subtle combination of particle geometry and system density. In particular, we find that entropy alone can ev…
We have investigated the relationship between the morphology of helical particles and the features of the cholesteric (N$^\ast $) phase that they form. Using an Onsager-like theory, applied to systems of hard helices, we show that the cholesteric handedness and pitch depend on both the pitch and the curliness of the particles. The theory leads to the definition of pseudoscalars that correlate the helical features of the phase to the chirality of the excluded volume of the constituent particles.
We consider helical configurations of a cholesteric liquid crystal (CLC) sandwiched between two substrates with homogeneous director orientation favored at both confining plates. We study the CLC twist wavenumber $q$ characterizing the helical structures in relation to the free twisting number $q_0$ which determines the equilibrium value of CLC pitch, $P_0=2π/q_0$. We investigate the instability mechanism underlying transitions between helical structures with different spiral half-turn numbers. Stability analysis shows that for equal finite anchoring strengths this mechanism can be dominated b…
We investigate the pitch sensitivity of cholesteric phases of helicoidal patchy cylinders as a generic model for chiral (bio-)polymers and helix-shaped colloidal rods. The behaviour of the macroscopic cholesteric pitch is studied from microscopic principles by invoking a simple density functional theory generalised to accommodate weakly twisted director fields. Upon changing the degree of alignment along the local helicoidal director we find that cholesteric phases exhibit a sudden sense inversion whereby the cholesteric phase changes from left- to right-handed and vice versa. Since the local …
We investigate the statistical mechanics of chirality and biaxiality in liquid crystals through a variety of theoretical approaches, including Monte Carlo simulations, lattice mean-field theory, and Landau theory. All of these calculations show that there is an important interaction between cholesteric twist and biaxial order: The twist acts as a field on the biaxial order, and conversely, the biaxial order increases the twist, i.e. reduces the pitch. We model the behavior of chiral biaxial liquid crystals as a function of temperature, and discuss how the predictions can be tested in experimen…
Active elastomeric liquid crystal particles with initial cylindrical shapes are obtained by means of soft lithography and polymerization in a strong magnetic field. Gold nanocrystals infiltrated into these particles mediate energy transfer from laser light to heat, so that the inherent coupling between the temperature-dependent order and shape allows for dynamic morphing of these particles and well-controlled stable shapes. Continuous changes of particle shapes are followed by their spontaneous realignment and transformations of director structures in the surrounding cholesteric host, as well …
The upper critical temperature Tc2 for the phase transition between the Cholesteric phase (N*) and the Twist Grain Boundary C phase with the layer inclination tilted to the pitch axis (TGBct) in thermotropic liquid crystals is determined by the mean field Chen-Lubensky approach. We show that the N*-TGBct phase transition is split in two with the appearance of either the TGBA or the TGB2q phase in a narrow temperature interval below Tc2. The latter phase is novel in being superposed from two degenerate TGBct phases with different (left and right) layers inclinations to the pitch axis.
Cholesteric liquid crystals (CLCs) have a characteristic length scale given by the pitch of the twisted stacking of their constituent rod-like molecules. Under homeotropic anchoring conditions where the molecules prefer to orient perpendicular to an interface, cholesteric interfaces exhibit striped phases with stripe widths commensurate with the pitch. Conversely, planar anchoring conditions have the molecules remain in the plane of the interface so that the CLC twists perpendicular to it. Recent work [L. Tran et al. Phys. Rev. X 7, 041029 (2017)] shows that varying the anchoring conditions dr…
The application of a sufficiently strong strain perpendicular to the pitch axis of a monodomain cholesteric elastomer unwinds the cholesteric helix. Previous theoretical analyses of this transition ignored the effects of Frank elasticity which we include here. We find that the strain needed to unwind the helix is reduced because of the Frank penalty and the cholesteric state becomes metastable above the transition. We consider in detail a previously proposed mechanism by which the topologically stable helical texture is removed in the metastable state, namely by the nucleation of twist disclin…
In a nematic gel, the appearance of nematic order is accompanied by a spontaneous elongation of the gel parallel to the nematic director. If such a gel is made chiral, it has a tendency to form a cholesteric helical texture, in which local elongation of the gel parallel to the nematic director is suppressed due to the requirement of elastic compatibility. We show that a conical helix in which the director makes an oblique angle with respect to the helix axis serves as an energy minimizing compromise between the competing tendencies for elongation and twisting. We find the dependence of the hel…
A mechanical strain field applied to a monodomain cholesteric elastomer will unwind the helical director distribution. There is an analogy with the classical problem of an electric field applied to a cholesteric liquid crystal, but with important differences. Frank elasticity is of minor importance unless the gel is very weak. The interplay is between director anchoring to the rubber elastic matrix and the external mechanical field. Stretching perpendicular to the helix axis induces the uniform unwound state via the elimination of sharp, pinned twist walls above a critical strain. Unwinding th…
When a chiral isotropic elastomer is brought to low temperature cholesteric phase, the nematic degree of freedom tends to order and form a helix. Due to the nemato-elastic coupling, this also leads to elastic deformation of the polymer network that is locally coaxial with the nematic order. However, the helical structure of nematic order is incompatible with the energetically preferred elastic deformation. The system is therefore frustrated and appropriate compromise has to be achieved between the nematic ordering and the elastic deformation. For a strongly chiral elastomer whose pitch is much…
We propose a construction of a cholesteric pitch axis for an arbitrary nematic director field as an eigenvalue problem. Our definition leads to a Frenet-Serret description of an orthonormal triad determined by this axis, the director, and the mutually perpendicular direction. With this tool we are able to compare defect structures in cholesterics, biaxial nematics, and smectics. Though they all have similar ground state manifolds, the defect structures are different and cannot be, in general, translated from one phase to the other.
We study the effects of chirality at the segment scale on the thermodynamics of block copolymer melts using self consistent field theory. In linear diblock melts where segments of one block prefer a twisted, or cholesteric, texture, we show that melt assembly is critically sensitive to the ratio of random coil size to the preferred pitch of cholesteric twist. For weakly-chiral melts (large pitch), mesophases remain achiral, while below a critical value of pitch, two mesocopically chiral phases are stable: an undulated lamellar phase; and a phase of hexagonally-ordered helices. We show that the…
Lyotropic cholesteric liquid crystal phases are ubiquitously observed in biological and synthetic polymer solutions, characterized by a complex interplay between thermal fluctuations, entropic and enthalpic forces. The elucidation of the link between microscopic features and macroscopic chiral structure, and of the relative roles of these competing contributions on phase organization, remains a topical issue. Here we provide theoretical evidence of a novel mechanism of chirality amplification in lyotropic liquid crystals, whereby phase chirality is governed by fluctuation-stabilized helical de…
Unique electro-optical properties of the oblique helicoidal cholesteric (ChOH) stem from its heliconical director structure. An applied electric field preserves the single harmonic modulation of the director while tuning the ChOH period and the corresponding Bragg peak wavelength within a broad spectral range. We use the response of ChOH to the electric field to measure the elastic constants of twist K22 and bend K33 directly in the cholesteric phase. The temperature dependencies of K22 and K33 allow us to determine the range of the electric tunability of the ChOH pitch and the heliconical ang…
Molecular chirality controls droplet division and helical fiber formation in liquid crystal emulsions
Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学与物理学院）；Condensed Matter Physics Department, Jožef Stefan Institute（约热·斯蒂芬研究所凝聚态物理系）；Dynamics of Complex Fluids, Max Planck Institute for Dynamics and Self-Organization（马克斯·普朗克动力学与自组织研究所复杂流体动力学）；Department of Chemistry, Faculty of Science, Tokyo University of Science（东京理科大学理学部化学系）
Molecular chirality, and the chiral symmetry breaking of resulting macroscopic phases, can be topologically imprinted and manipulated by crosslinking and swelling of polymer networks. We present a new experimental approach to stereo-specific separation of chiral isomers by using a cholesteric elastomer in which a helical director distribution has been topological imprinted by crosslinking. This makes the material unusual in that is has a strong phase chirality, but no molecular chirality at all; we study the nature and parameters controlling the twist-untwist transition. Adding a racemic mixtu…
Stiff, elongated biomolecules such as filamentous viruses, DNA or cellulose nanocrystals are known to form liquid crystals often exhibiting a helical supramolecular organization. Little is known about the microscopic origin, size and handedness of the helical pitch in these, so-called cholesteric phases. Experimental observations in chiral lyotropics suggest that long-ranged chiral forces of electrostatic origin acting between the mesogens are responsible for such organization. Using large-scale computer simulation we study the sensitivity of the pitch imparted by soft microscopic helices and …
We consider a one-dimensional variational problem arising in connection with a model for cholesteric liquid crystals. The principal feature of our study is the assumption that the twist deformation of the nematic director incurs much higher energy penalty than other modes of deformation. The appropriate ratio of the elastic constants then gives a small parameter $\varepsilon$ entering an Allen-Cahn-type energy functional augmented by a twist term. We consider the behavior of the energy as $\varepsilon$ tends to zero. We demonstrate existence of the local energy minimizers classified by their o…
We theoretically investigate the pitch of lyotropic cholesteric phases composed of slender rods with steric chirality transmitted via a weak helical deformation of the backbone. In this limit, the model is amenable to analytical treatment within Onsager theory and a closed expression for the pitch versus concentration and helical shape can be derived. Within the same framework we also briefly review the possibility of alternative types of chiral order, such as twist-bend or screw-like nematic phases, finding that cholesteric order dominates for weakly helical distortions. While long-ranged or …
We derive formulae for the electrostatic interaction between two long, rigid macromolecules that may have arbitrary surface charge patterns and cross at an arbitrary interaxial angle. We calculate the dependence of the interaction energy on the interaxial angle, on the separation, and on the precise alignment of the charge pattern on one molecule with respect to the other. We focus in particular on molecules with helical charge patterns. We report an exact, explicit expression for the energy of interaction between net-neutral helices in a nonpolar medium as well as an approximate result for ch…
Even though chiral nematic phases were the first liquid crystals experimentally observed more than a century ago, the origin of the thermodynamic stability of cholesteric states is still unclear. In this Letter we address the problem by means of a novel density functional theory for the equilibrium pitch of chiral particles. When applied to right-handed hard helices, our theory predicts an entropy-driven cholesteric phase, which can be either right- or left-handed, depending not only on the particle shape but also on the thermodynamic state. We explain the origin of the chiral ordering as an i…
Periodic structures of phases with orientational order of molecules, but homogenous electron density distribution: a short pitch cholesteric, blue phase and twist-bend nematic phase, were probed by a resonant soft x-ray scattering (RSoXS) at the carbon K-edge. The theoretical model shows that in case of a simple heliconical nematic structure two resonant signals corresponding to the full and half pitch band should be present, while only the full pitch band is observed in experiment. This suggests that the twist-bend nematic phase has complex structure with a double-helix, built of two interloc…
We study elastic interactions between colloidal particles and deformation landscapes of undulations in a cholesteric liquid crystal under an electric field applied normal to cholesteric layers. The onset of undulation instability is influenced by the presence of colloidal inclusions and, in turn, layers' undulations mediate the spatial patterning of particle locations. We find that the bending of cholesteric layers around a colloidal particle surface prompts the local nucleation of an undulations lattice at electric fields below the well-defined threshold known for liquid crystals without incl…
We present an investigation of the phase diagram of cholesteric liquid crystals within the framework of Landau - de Gennes theory. The free energy is modified to incorporate all three Frank elastic constants and to allow for a temperature dependent pitch in the cholesteric phase. It is found that the region of stability of the cubic blue phases depends significantly on the value of the elastic constants, being reduced when the bend elastic constant is larger than splay and when twist is smaller than the other two. Most dramatically we find a large increase in the region of stability of blue ph…
We introduce a comprehensive numerical framework to generically infer the emergent macroscopic properties of uniaxial nematic and cholesteric phases from that of their microscopic constituent mesogens. This approach, based on the full numerical resolution of the Poniewierski-Stecki equations in the weak chirality limit, may expediently handle a wide range of particle models through the use of Monte-Carlo sampling for all virial-type integrals. Its predictions in terms of equilibrium cholesteric structures are found to be in excellent agreement with previous full-functional descriptions, thereb…
Telephone-cord patterns have been recently observed in smectic liquid crystal capillaries. In this paper we analyse the effects that may induce them. As long as the capillary keeps its linear shape, we show that a nonzero chiral cholesteric pitch favors the SmA*-SmC* transition. However, neither the cholesteric pitch nor the presence of an intrinsic bending stress are able to give rise to a curved capillary shape. The key ingredient for the telephone-cord instability is spontaneous polarization. The free energy minimizer of a spontaneously polarized SmA* is attained on a planar capillary, char…
Imprinting of cholesteric textures in a polymer network is a method of preserving a macroscopically chiral phase in a system with no molecular chirality. By modifying the elastics properties of the network, the resulting stored helical twist can be manipulated within a wide range since the imprinting efficiency depends on the balance between the elastics constants and twisting power at network formation. One spectacular property of phase chirality imprinting is the created ability of the network to adsorb preferentially one stereo-component from a racemic mixture. In this paper we explore this…
We study the effective chiral interaction between molecules arising from quantum dispersion interactions within a model in which a) the dominant excited states of a molecule form a band whose width is small compared to the average excitation energy and b) biaxial orientational correlation between adjacent molecules can be neglected. Previous treatments of quantum chiral interactions were based on a multipole expansion of the intermolecular interaction. However, because real liquid crystals are composed of elongated molecules, we utilize an expansion in terms of only coordinates transverse to t…
We investigate the emergence of chiral meso-structures in one-dimensional fluids consisting of stacked discotic particles and demonstrate that helical undulations are generated spontaneously from internal elastic stresses. The stability of these helical conformations arises from an interplay between long-ranged soft repulsions and nanopore confinement which is naturally present in columnar liquid crystals. Using a simple mean-field theory based on microscopic considerations we identify generic scaling expressions for the typical buckling radius and helical pitch as a function of the density an…
Using dynamic simulations and analytic methods, we study the elastic response of a helical filament subject to uniaxial tension over a wide range of bend and twist persistence length. A low-pitch helix at low temperatures exhibits a stretching instability and the force-extension curve consists of a sequence of spikes. At elevated temperature (i.e. small persistence lengths) the helix melts and a pronounced force plateau is obtained in the fixed-extension ensemble. The torque boundary condition significantly affects the resulting elastic properties.
The recently discovered twist-bend nematic phase, $\rm N_{tb}$, is a non-uniform equilibrium nematic phase that presents a spontaneous bend with a precession of the nematic director, $\textbf{n}$, on a conical helix with a tilt angle $θ$ and helical pitch $P$. The stability of the $\rm N_{tb}$ phase has been recently demonstrated from the elastic point of view by extending the Frank elastic energy density of the nematic phase to include the symmetry element of helical axis, $\bf t$. In the present communication, we investigate the influence of an external bulk field (magnetic or electric) on t…
A systematic computational and scaling analysis of defect textures in polygonal arrangement of cylindrical particles embedded in a cholesteric (Ch) liquid crystal matrix is performed using the Landau de Gennes model for chiral self assembly, with strong anchoring at the particles' surface. The defect textures and LC phases observed are investigated as a function of chirality, elastic anisotropy, particle polygonal arrangement and particle size. The presence of a polygonal network made of N circular inclusions results in defect textures of a net charge of -(N-2)/2 per unit polygonal cell. As th…
We study the non-monotonic force-extension behaviour of helical ribbons using a new model for inextensible elastic strips. Unlike previous rod models our model predicts hysteresis behaviour for low-pitch ribbons of arbitrary material properties. Associated with it is a first-order transition between two different helical states as observed in experiments with cholesterol ribbons. Numerical solutions show non-uniform uncoiling with hysteresis also occurring under controlled tension. They furthermore reveal a new uncoiling scenario in which a ribbon of very low pitch shears under tension and suc…
Using the framework of potential energy landscape theory, we describe two in silico designs for self-assembling helical colloidal superstructures based upon dipolar dumbbells and Janus-type building blocks, respectively. Helical superstructures with controllable pitch length are obtained using external magnetic field driven assembly of asymmetric dumbbells involving screened electrostatic as well as magnetic dipolar interactions. The pitch of the helix is tuned by modulating the Debye screening length over an experimentally accessible range. The second design is based on building blocks compos…
We report fabrication of highly flexible micron-sized helices from nanometer-thick ribbons. Building upon the helical coiling of such ultra-thin ribbons mediated by surface tension, we demonstrate that the enhanced creep properties of highly confined materials can be leveraged to shape helices into the desired geometry with full control of the final shape. The helical radius, total length and pitch angle are all freely and independently tunable within a wide range: radius within $\sim$ 1-100 micrometer, length within $\sim$ 100-3000 micrometer, and pitch angle within $\sim$ 0-70°. This fabrica…
A series of asymmetric dimers with an odd number of atoms in the spacer were found to form different types of twisted structures despite being achiral. The formation of a variety of helical structures is accompanied by a gradual freezing of molecular rotation. The tight pitch heliconical nematic (NTB) phase and heliconical tilted smectic C (SmCTB) phase are formed. In the lowest temperature smectic phase, HexI, the twist is expressed through the formation of hierarchical structure: nano-scale helices and mesoscopic helical filaments. The short pitch helical structure in smectic phases is confi…
Minimal surfaces arise as energy minimizers for fluid membranes and are thus found in a variety of biological systems. The tight lamellar structures of the endoplasmic reticulum and plant thylakoids are composed of such minimal surfaces in which right and left handed helical motifs are embedded in stoichiometry suggesting global pitch balance. So far, the analytical treatment of helical motifs in minimal surfaces was limited to the small-slope approximation where motifs are represented by the graph of harmonic functions. However, in most biologically and physically relevant regimes the inter-m…
