Frank Free Energy Research Articles

Free energy formulae for confined nematic liquid crystals based on analogies with Kirchhoff–Routh theory in vortex dynamics

Active nematics are influenced by alignment angle singularities called topological defects. The localization of these defects is of major interest for biological applications. The total distortion of alignment angles due to defects is evaluated using Frank free energy, which is one of the criteria used to determine the location and stability of these defects. Previous work used the line integrals of a complex potential associated with the alignments for the energy calculation (Miyazako and Nara 2022 R. Soc. Open Sci . 9 , 211663 (doi: 10.1098/rsos.211663 )), which has a high computational cost. We propose analytical formulae for the free energy in the presence of multiple topological defects in confined geometries. The formulae derived here are an analogue of Kirchhoff–Routh functions in vortex dynamics. The proposed formulae are explicit with respect to the defect locations and conformal maps, which enable the explicit calculation of the energy extrema. The formulae are applied to calculate the locations of defects in so-called doublets and triplets by solving simple polynomial formulae. A stability analysis is also conducted to detect whether defect pairs with charges ± 1 / 2 are stable or unstable in triplet regions. Our numerical results are shown to match the experimental results (Ienaga et al. 2023 Soft Matter 19 , 5016–5028. (doi: 10.1039/d3sm00071k )).

Emergent gauge symmetry in active Brownian matter.

We investigate a two-dimensional system of interacting active Brownian particles. Using the Martin-Siggia-Rose-Janssen-de Dominicis formalism, we built up the generating functional for correlation functions. We study in detail the hydrodynamic regime with a constant density stationary state. Our findings reveal that, within a small density fluctuations regime, an emergent U(1) gauge symmetry arises, originated from the conservation of fluid vorticity. Consequently, the interaction between the orientational order parameter and density fluctuations can be cast into a gauge theory, where the concept of "electric charge density" aligns with the local vorticity of the original fluid. We study in detail the case of a microscopic local two-body interaction. We show that, upon integrating out the gauge fields, the stationary states of the rotational degrees of freedom satisfy a nonlocal Frank free energy for a nematic fluid. We give explicit expressions for the splay and bend elastic constants as a function of the Péclet number (Pe) and the diffusion interaction constant (k_{d}).

Computational Analysis on the Performance of Elongated Liquid Crystal Biosensors.

Elongated ellipsoidal liquid crystal microdroplet reorientation dynamics are discussed in this paper for biosensor applications. To investigate the effect of elongated droplets on nematic liquid crystal droplet biosensors, we simulated a model of a liquid crystal droplet using ellipse geometry. Director reorientation is examined in relation to the elongated droplet shape. In addition, we examined aspect ratio as a factor affecting biosensor response time in relation to surface viscosity and anchoring energy. Finally, the findings suggest that the aspect ratio should be taken into account when designing biosensors. These results can be used to develop more effective biosensors for a variety of applications. This model then predicts the director reorientation angle, which is dependent on the anchoring energy and surface viscosity. This model further suggests that both surface viscosity and homeotropic anchoring energy play an important role when it comes to the director reorientation angle. We developed and applied a nonlinear unsteady-state mathematical model utilizing torque balance and Frank free energy according to the Leslie-Ericksen continuum theory for simulating elongated nematic liquid crystal biosensor droplets with aqueous interfaces. Using the Euler-Lagrange equation, a transient liquid crystal-aqueous interface realignment is modeled by changing the easy axis when surfactant molecules are added to the interface. The realignment at the surface of the droplet is assumed to be driven by the effect of the surfactant, which causes an anchoring transition. According to the results, the response time of the biosensor depends on the aspect ratio. Therefore, the elongation has the potential to control biosensing response time. The result of our study provides a better understanding of director reorientation in elongated liquid crystal droplets in biosensing applications through the numerical results which are presented in this paper.

An Abelian Higgs model for disclinations in nematics

Topological defects in elastic media may be described by a geometric field akin to three-dimensional gravity. From this point of view, disclinations are line defects of zero width corresponding to a singularity of the curvature in an otherwise flat background. On the other hand, in two dimensions, the Frank free energy of a nematic liquid crystal may be interpreted as an Abelian Higgs Lagrangian. In this work, we construct an Abelian Higgs model coupled to ``gravity'' for the nematic phase, with the perspective of finding more realistic disclinations. That is, a cylindrically symmetric line defect of finite radius, invariant under translations along its axis. Numerical analysis of the equations of motion indeed yield a +1 winding number ``thick'' disclination. The defect is described jointly by the gauge and the Higgs fields, that compose the director field, and the background geometry. Away from the defect, the geometry is conical, associated to a dihedral deficit angle. The gauge field, confined to the defect, gives a structure to the disclination while the Higgs field, outside, represents the nematic order.

Computational Analysis to Optimize the Performance of Thin Film Liquid Crystal Biosensors

A nonlinear unsteady-state mathematical model employing torque balance and Frank free energy according to the Leslie-Ericksen continuum theory is developed and implemented to simulate the performance of nematic liquid crystal biosensor films with aqueous interfaces. A transient liquid crystal-aqueous interface realignment is modeled using the Euler–Lagrange equation by changing the easy axis when the surfactant molecules at the interface are introduced. In our study, we evaluated the dynamics between bulk and interface by controlling surface properties of the interface, such as homeotropic anchoring energy and surface viscosity. In addition, transient optical interference and response time have been examined in this study. Our parametric study results indicated that both homeotropic anchoring energy and surface viscosity at the interface contribute to bulk reorientation. Furthermore, the obtained numerical results indicate that as homeotropic anchoring strength increases, the effective birefringence decreases more gradual due to the increasing surfactant concentration at the aqueous interface, consistent with available experimental observations. Our results have been validated and compared to experimental results from thin-film liquid crystal biosensors in this study.

Chromonic liquid crystals and packing configurations of bacteriophage viruses.

We study equilibrium configurations of hexagonal columnar liquid crystals in the context of characterizing packing structures of bacteriophage viruses in a protein capsid. These are viruses that infect bacteria and are currently the focus of intense research efforts, with the goal of finding new therapies for bacteria-resistant antibiotics. The energy that we propose consists of the Oseen-Frank free energy of nematic liquid crystals that penalizes bending of the columnar directions, in addition to the cross-sectional elastic energy accounting for distortions of the transverse hexagonal structure; we also consider the isotropic contribution of the core and the energy of the unknown interface between the outer ordered region of the capsid and the inner disordered core. The problem becomes of free boundary type, with constraints. We show that the concentric, azimuthal, spool-like configuration is the absolute minimizer. Moreover, we present examples of toroidal structures formed by DNA in free solution and compare them with the analogous ones occurring in experiments with other types of lyotropic liquid crystals, such as food dyes and additives. This article is part of the theme issue 'Topics in mathematical design of complex materials'.

Twisted loxodromes in spindle-shaped polymer nematics.

We develop an energetic model that captures the twisting behavior of spindle-shaped polymer microparticles with nematic ordering, which display remarkably different twisting behavior to ordinary nematics confined to spindles. We have previously developed a geometric model of the twisting, based on experimental observations, in which we showed that the twist pattern follows loxodrome spirals [Ansell, et al., Phys. Rev. Lett., 2019, 123, 157801]. In this study, we first consider a spindle-shaped surface and show that the loxodrome twisting behavior of our system can be captured by the Frank free energy of the nematic with an additional term constraining the length of the integral curves of the system. We then extend the ideas of this model to the bulk and explore the parameter space for which the twisted loxodrome solution is energetically favorable.

Weak-strong uniqueness for measure-valued solutions to the Ericksen–Leslie model equipped with the Oseen–Frank free energy

We analyze the Ericksen–Leslie system equipped with the Oseen–Frank energy in three space dimensions. Recently, the author introduced the concept of measure-valued solutions to this system and showed the global existence of these generalized solutions. In this paper, we show that suitable measure-valued solutions, which fulfill an associated energy inequality, enjoy the weak-strong uniqueness property, i.e. the measure-valued solution agrees with a strong solution if the latter exists. The weak-strong uniqueness is shown by a relative energy inequality for the associated nonconvex energy functional.

Stability of disclination loop in pure twist nematic liquid crystals

In this work, the annihilations dynamics and stability of disclination loop in a bulk pure twist nematic liquid crystal are investigated. This work is based on the Frank free energy and the nematodynamics equations. The energy dissipation is calculated by using two methods. In the first method, the energy dissipation is obtained from the Frank free energy. In the second method, it is calculated by using the nematodynamics equations. Finally, we derive a critical radius of disclination loop that above this radius, loop creation is energetically forbidden.

Role of anchoring energy on the texture of cholesteric droplets: Finite-element simulations and experiments.

We present a numerical method to compute defect-free textures inside cholesteric domains of arbitrary shape. This method has two interesting properties, namely a robust and fast quadratic convergence to a local minimum of the Frank free energy, thanks to a trust region strategy. We apply this algorithm to study the texture of cholesteric droplets in coexistence with their isotropic liquid in two cases: when the anchoring is planar and when it is tilted. In the first case, we show how to determine the anchoring energy at the cholesteric-isotropic interface from a study of the optical properties of droplets of different sizes oriented with an electric field. This method is applied to the case of the liquid crystal CCN-37. In the second case, we come back to the issue of the textural transition as a function of the droplet radius between the double-twist droplets and the banded droplets, observed for instance in cyanobiphenyl liquid crystals. We show that, even if this transition is dominated by the saddle-splay Gauss constant K_{4}, as was recently recognized by Yoshioka etal. [Soft Matter 12, 2400 (2016)1744-683X10.1039/C5SM02838H], the anchoring energy does also play an important role that cannot be neglected.

Spontaneous emergence of twist and the formation of a monodomain in lyotropic chromonic liquid crystals confined to capillaries

Having a well-aligned monodomain is critical to performing fundamental studies on liquid crystals as well as exploiting them for technological applications. Unlike conventional nematics, lyotropic chromonic liquid crystals are notoriously hard to align. Here, we report on the homogeneous planar alignment of Sunset Yellow in a flat rectangular capillary. Remarkably, the in-plane director aligns perpendicular to the long axis of the rectangular capillary resulting in a counterintuitive configuration. We rationalize the evolution of this configuration from a metastable twisted configuration by considering the coupling of the curvature of the edges of the rectangle to the Frank free energy via the saddle-splay contribution. In contrast, for a square capillary, the doubly-twisted configuration is the final ground state. Using a simple scaling argument, we show that the aspect ratio of the confining geometry determines the ground state.

Effects of surface anchoring on the electric Frederiks transition in ferronematic systems

Abstract The effects of anchoring phenomenon on the electric Frederiks transition threshold field in a nematic liquid crystal doped with ferroelectric nanoparticles are discussed. The polarizability of these nanoparticles in combination with confinement effects cause the drastic effects on the ferronematic systems. This study is based on Frank free energy and Rapini–Papoular surface energy for ferronematic liquid crystal having finite anchoring condition. In the case of different anchoring boundary conditions, the Euler–Lagrange equation of the total free energy is numerically solved by using the finite difference method together with the relaxation method and Maxwell construction to select the physical solutions and therefore investigate the effects of different anchoring strengths on the Frederiks transition threshold field. Maxwell construction method is employed to select three periodic solutions for nematic liquid crystal director at the interfaces of a slab. In the interval from zero to half- π , there is only one solution for the director orientation. In this way, NLC director rotates toward the normal to the surface as the applied electric field increases at the walls. Our numerical results illustrate that above Frederiks transition and in the intermediate anchoring strength, nematic molecules illustrate the different orientation at slab boundaries. We also study the effects of different anchoring strengths, nanoparticle volume fractions and polarizations on the Frederiks transition threshold field. We report that decreasing in the nanoparticle polarization results in the saturation Frederiks threshold. However, this situation does not happen for the nanoparticles volume fraction.

Texture transformation in circular domain of polar smectic films: Chiral elasticity induced by coupling of flexoelectric and spontaneous polarizations

Both pure tangential and radial textures in circular domains of polar SmC* LCs are shown to be solutions satisfying the equations for bulk- and boundary-orientation, and for domain shape derived via the Frank free energy, with pure tangential and normal line tensions. The stability of both textures is studied in terms of the elastic anisotropy. Results show that, at a critical domain diameter, both textures switch to reversing spiral textures. By considering the chiral elasticity contributed by the coupling of flexoelectric and spontaneous polarizations the existence of an inverse relation of the critical radius and the spontaneous polarization is revealed.

Disclination lines at homogeneous and heterogeneous colloids immersed in a chiral liquid crystal.

In the present work we perform Monte Carlo simulations in the isothermal-isobaric ensemble to study defect topologies formed in a cholesteric liquid crystal due to the presence of a spherical colloidal particle. Topological defects arise because of the competition between anchoring at the colloidal surface and the local director. We consider homogeneous colloids with either local homeotropic or planar anchoring to validate our model by comparison with earlier lattice Boltzmann studies. Furthermore, we perform simulations of a colloid in a twisted nematic cell and discuss the difference between induced and intrinsic chirality on the formation of topological defects. We present a simple geometrical argument capable of describing the complex three-dimensional topology of disclination lines evolving near the surface of the colloid. The presence of a Janus colloid in a cholesteric host fluid reveals a rich variety of defect structures. Using the Frank free energy we analyze these defects quantitatively indicating a preferred orientation of the Janus colloid relative to the cholesteric helix.

Nonlinear smectic elasticity of helical state in cholesteric liquid crystals and helimagnets

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 Anderson-Higgs mechanism that locks the spiral orthonormal (director or magnetic moment) frame to the cholesteric (helical) layers. This shows explicitly the reduction of three orientational modes of a cholesteric down to a single-phonon Goldstone mode that emerges on scales longer than the pitch. At a harmonic level our result reduces to that derived many years ago by Lubensky and collaborators.

Periodic splay-twist Fréedericksz transition for nematics confined between two concentric cylinders

This paper derives theoretical results for the periodic splay-twist Fréedericksz transition in nematic liquid crystals confined between two infinite concentric cylinders. The calculation of Lonberg and Meyer [Phys. Rev. Lett. 55, 718 (1985)], for nematics sandwiched between two infinite planes, is extended to annular domains. The phase transition is triggered by an applied voltage between the outer and the inner delimiting walls. The critical threshold behavior is analyzed via the linearized Euler-Lagrange equations related to the Frank's free energy. It is found that, the threshold depends on both the ratio between the twist and the splay elastic constants, and the sample radii ratio. Results for planar samples are recovered in the thin cell limit. With respect to the planar geometry, our analysis predicts that for annular geometries the periodic Fréedericksz transition is also allowed for elastic anisotropies K2/K1>0.303.

Tunable NIR filter based on a free-standing porous silicon film containing nematic liquid crystal

We present the design of a thermally tunable interference filter for the NIR range. The filter is based on a free-standing porous silicon multilayer filled with the nematic liquid crystal mixture E7. The porous silicon stack is constituted by a quarter-wave microcavity sandwiched between two distributed Bragg reflectors. Reversible thermal tuning of the microcavity resonance is obtained in the range of 13 nm. Thermal dependence of the effective refractive index of the liquid crystal in the sample is characterized using variable angle spectroscopic ellipsometry. The non-axial liquid crystal director configuration is found inside silicon mesopores. The distribution of the liquid crystal director within pores is simulated using the Frank's free energy approach. Assuming the molecular configuration of the liquid crystal to be escaped radial, we have determined the value of the surface anchoring strength in the mesoporous silicon to be less than 10−6 J m−2. By fitting the transmittance spectra, using an effective medium approximation and a transfer matrix method, the liquid crystal is found to fill 84% of the void inside the porous sample. The good agreement between the experimental data and the theoretical curves justifies the Frank's approach and the hypothesis of escape radial configuration of nematics confined in silicon mesopores.

Electrical reorientation of liquid crystal molecules inside cylindrical pores for photonic device applications

We present the simulated distribution of the local director of a nematic liquid crystal inside cylindrical macropores under the influence of an electric field. The Frank free energy approach is used to describe the nematic behavior. The influence of both molecular anchoring strength and pore radius is investigated. The results of this analysis are applied for simulation of an electrically tunable microcavity based on porous silicon infiltrated with a liquid crystal. The Bruggeman approximation is used while calculating the effective refractive index of each layer in the porous silicon multilayer structure. The reflectivity spectrum of the latter is simulated using the transfer matrix approach. The electrical tuning range of a microcavity designed for near-infrared waves is found to vary from 10.5 up to 23 nm for weak and strong surface anchoring conditions, respectively.

Annihilation Dynamics of the Disclination Loop in Bulk Nematic Liquid Crystals

We investigate the annihilation dynamics of the disclination loop in bulk nematic liquid crystals. This work is based on the Frank free energy and the nematodynamics equations. The dissipation of energy is calculated by two methods. In the first method, the dissipation of energy is obtained from the Frank free energy. In the second method, it is calculated by the nematodynamics equations. Finally, we obtain the equation of motion and annihilation velocity of the disclination loop.

Disclinations dynamics in confined nematic liquid crystals: strong anchoring

We have investigated the dynamics of pair annihilation of disclination lines in strong anchoring. This work is based on the Frank free energy. The director angle, φ(x, y, z), is obtained by the continuous theory. We show that the form of the viscous force in a confined nematic liquid crystal and in strong anchoring is a function of the initial distance between the two disclination lines. The asymptotic velocity, v asy, is also a function of the initial distance. Our theoretical result on the asymptotic velocity is in good agreement with previous experimental results.