Saturn Ring Defect Research Articles

Defect-influenced particle advection in highly confined liquid crystal flows.

We study the morphology of the Saturn ring defect and director structure around a colloidal particle with normal anchoring conditions and within the flow of the nematic host phase through a rectangular duct of comparable size to the particle. The changes in the defect structures and director profile influence the advection behaviour of the particle, which we compare to that in a simple Newtonian host phase. These effects lead to a non-monotonous dependence of the differential velocity of particle and fluid, also known as retardation ratio, on the Ericksen number.

Statics and dynamics of point boojums, line and modified Saturn ring topological defects in nematic confined geometry.

In this paper we study the static structure and the dynamics of topological defects associated with isotropic droplets in nematic environment. Investigations were made in confined geometry of optical cells when the droplet size was of the order of or larger than the gap of the cell. We observed the coexistence of point boojums and Saturn ring or modified Saturn ring defects. We found transformation of the Saturn ring defect to two localized broad defects at increasing the droplet size. At droplet coalescence antipodes of point and localized broad defects were born and the dynamics of their annihilation with existing defects was investigated. We found strong difference in the process of annihilation of point and localized broad defects. Microscope images of isotropic droplets in nematic environment in a planar cell. The director orientation far from the droplets is in horizontal direction. The photographs were taken with crossed vertical and horizontal polarizers (a) and with a single horizontal polarizer (b). The cell thickness is 100μm. Droplet diameter is less than the cell thickness. 1 and 2 are point boojums, L is the Saturn ring defect.

The yielding of defect-entangled dispersions in a nematic solvent

Oscillatory rheology, at both small and large (LAOS) amplitudes, was performed to measure the dynamic response of a soft solid, formed on dispersing colloids into a thermotropic nematic liquid crystal at volume fractions of ϕ>18%. Due to weak homeotropic anchoring of nematogens at colloid surfaces, a Saturn-ring defect line, known as a “disclination,” encircles each particle and entangles with neighboring Saturn-ring disclinations [Wood et al., Science 334, 79–83 (2011)]. We present the first experimental investigation of the yielding behavior of the resulting gel to reveal the underpinning physics. Results reveal that the frequency response of the composite is independent of volume fraction ϕ, an indication that the dispersed phase simply increases the density of disclinations spanning the composite without further effect. Beyond the linear viscoelastic regime (LVR), LAOS experiments indicate the composite is an elastoplastic fluid exhibiting both strain-hardening and shear-thinning behaviors with Chebyshev coefficients e3>0 and ν3<0, respectively. We deduce that the disclination number density nd is constant until the strain amplitude is sufficient to break disclinations leading to shear-thinning behavior beyond the LVR. A simple theory is introduced revealing that the Ericksen number Er determines the onset of flow, when Er>1, generating a strain-hardening response since Frank elasticity resists reorientation of molecular alignment within confined nematic domains. Above a critical frequency ωc, the loss modulus G″ increases slowly due to enhanced viscosity within confined nematic domains, G″∝ω1/2 [Larson, R. G., The Structure and Rheology of Complex Fluids (Oxford University, 1999), p. 463]. Observation of this behavior in a small-molecule nematic solvent provides insights into the physics of flow behavior in other, more complex, defect-mediated liquid crystalline structures exhibiting similar properties [Colby, Europhys. Lett. 54, 269–274 (2001); Sahoo and Dhara, Liq. Cryst. 44, 1582–1591 (2017); and Romo-Uribe, Polym. Adv. Technol. 32, 651–662 (2021)].

Saturn ring defect around a spherical particle immersed in a nematic liquid crystal

We consider a nematic liquid crystal occupying the three-dimensional domain in the exterior of a spherical colloid particle. The nematic is subject to Dirichlet boundary conditions that enforce orthogonal attachment of nematic molecules to the surface of the particle. Our main interest is to understand the behavior of energy-critical configurations of the Landau–de Gennes Q-tensor model in the limit of vanishing correlation length. We demonstrate existence of configurations with a single Saturn-ring defect approaching the equator of the particle and no other line or point defects. We show this by analyzing asymptotics of energy minimizers under two symmetry constraints: rotational equivariance around the vertical axis and reflection across the horizontal plane. Energy blow-up at the ring defect is a significant obstacle to constructing well-behaved comparison maps needed to eliminate the possibility of point defects. The boundary estimates we develop to address this issue are new and should be applicable to a wider class of problems.

Nonequilibrium Dynamics of a Magnetic Nanocapsule in a Nematic Liquid Crystal.

Colloidal particles in nematic liquid crystals show a beautiful variety of complex phenomena with promising applications. Their dynamical behaviour is determined by topology and interactions with the liquid crystal and external fields. Here, a nematic magnetic nanocapsule reoriented periodically by time-varying magnetic fields is studied using numerical simulations. The approach combines Molecular Dynamics to resolve solute–solvent interactions and Nematic Multiparticle Collision Dynamics to incorporate nematohydrodynamic fields and fluctuations. A Saturn ring defect resulting from homeotropic anchoring conditions surrounds the capsule and rotates together with it. Magnetically induced rotations of the capsule can produce transformations of this topological defect, which changes from a disclination curve to a defect structure extending over the surface of the capsule. Transformations occur for large magnetic fields. At moderate fields, elastic torques prevent changes of the topological defect by tilting the capsule out from the rotation plane of the magnetic field.

Defects around nanocolloids in nematic solvents simulated by Multi-particle Collision Dynamics

Colloids of nanometric size dispersed in a nematic environment are simulated using a hybrid algorithm that combines rules from Multi-particle Collision Dynamics and Molecular Dynamics. Coupling between flow and orientation fields is incorporated through the application of the Leslie–Ericksen theory for reorientation of slender rods under flow. It is found that Multi-particle Collision Dynamics sustains Saturn ring defects in the solvent when homeotropic anchoring on the colloids is considered. Changes on these topological structures induced by different anchoring strengths and spontaneous fluctuating flows are quantified. Stable defect structures around two close colloidal particles are observed that coincide with those obtained in experiments and alternative numerical techniques. Simulations of uniform nematic flow around the suspended colloid are conducted. It is found that Saturn rings are displaced along the stream direction. Large flows can separate the Saturn ring from the colloid, though the motion of the defect and the suspended particle reaches a steady state where disclinations are aligned with the streamlines.

Modifying Nobili-Durand surface energy for conically degenerate anchorings at the interface of liquid crystal colloids.

We propose a surface energy for conically degenerate anchorings of uniaxial liquid crystal mesogens by modifying tensorial Nobili-Durand surface energy that is usually employed for fixed anchoring orientations with preferred polar angles. By minimizing Landau-de Gennes free energy and the proposed surface energy, we obtain the equilibrium director configuration around a spherical colloid in the uniform nematic liquid crystal. Our calculations show that the proposed surface energy can cause boojum or/and Saturn-ring defect textures depending on the equilibrium conic angle. We also study the interactions between two spherical colloids with the equilibrium conic angle 45^{∘}, where the surface energy provides both boojum and Saturn-ring defects on the surface of particles. We compare the calculated anisotropic colloidal interactions with experimental observations [B. Senyuk et al., Nat. Commun. 7, 10659 (2016)2041-172310.1038/ncomms10659]. In agreement with experiment, our results show two stable angular assemblies in the close particle-particle separations. Also, the long-range elastic interactions are almost consistent with the hexadecapolar elastic distortion.

Spherical Particle in Nematic Liquid Crystal Under an External Field: The Saturn Ring Regime

We consider a nematic liquid crystal occupying the exterior region in R^3 outside of a spherical particle, with radial strong anchoring. Within the context of the Landau-de Gennes theory, we study minimizers subject to a strong external field, modelled by an additional term which favors nematic alignment parallel to the field. When the external field is high enough we obtain a scaling law for the energy. The energy scale corresponds to minimizers concentrating their energy in a boundary layer around the particle, with quadrupolar symmetry. This suggests the presence of a Saturn ring defect around the particle, rather than a dipolar director field typical of a point defect.

The Ericksen model of liquid crystals with colloidal and electric effects

We present a robust discretization of the Ericksen model of liquid crystals with variable degree of orientation coupled with colloidal effects and electric fields. The total energy consists of the Ericksen energy, a weak anchoring (or penalized Dirichlet) energy to model colloids, and an electrical energy for a given electric field. We describe our special discretization of the total energy along with a method to compute minimizers via a discrete quasi-gradient flow algorithm which has a strictly monotone energy decreasing property. Numerical experiments are given in two and three dimensions to illustrate that the method is able to capture non-trivial defect patterns, such as the Saturn ring defect. We conclude with a rigorous proof of the Γ-convergence of our discrete energy to the continuous energy.

The Impact of Colloidal Surface-Anchoring on the Smectic A Phase.

Liquid-crystalline phases are known for their unique properties, i.e., the combination of fluidity and long-range orientational and/or positional order. The presence of a colloidal particle gives rise to perturbations of this order locally. These perturbations are the origin of intercolloidal forces driving the colloidal self-assembly in a directed manner. Hence, the understanding of these perturbations is the first step in understanding and controlling the self-assembly process. Here, we perform Monte Carlo simulations to investigate the perturbations of orientational and positional order in a smectic A phase caused by a spherical colloid. We model the host phase via an interaction potential that reproduces characteristic features of phase behavior, structure, dynamics, and elasticity [S. Püschel-Schlotthauer et al. J. Chem. Phys. 2016, 145, 164903]. For strong homeotropic anchoring conditions, we find a Saturn ring defect and an onion structure in the smectic A phase; the latter has never been reported for colloids so far. For strong planar anchoring conditions, we find Boojum defects that become elongated at low temperature, similar to what is observed in experiments. However, for weak planar anchoring conditions, a double surface ring defect is exhibited in the smectic A phase.

Motion of a colloidal particle in a nonuniform director field of a nematic liquid crystal.

We investigate the dynamics of a single spherical particle immersed in a nematic liquid crystal. A nonuniform director field is imposed on the substrate by a stripe alignment pattern with splay deformation. The particle of homeotropic anchoring at the surface is accompanied by hyperbolic hedgehog or Saturn-ring defects. The particle motion is dependent on the defect structure. We study the two types of motions theoretically and confirm the obtained results experimentally. The particle accompanied by a hyperbolic hedgehog defect is pulled to a deformed region to relax the elastic deformation energy. The motion occurs in the direction heading the hyperbolic hedgehog defect of a particle in a twist region. The position exhibits a weak S-shaped change as a function of time. The particle accompanied by a Saturn-ring defect shows insignificant motion due to its relatively small deformation energy.

Ordering of ferromagnetic nanoparticles in nematic liquid crystals.

Dispersions of magnetic nanoparticles in a nematic liquid crystal were investigated as magnetic fields were applied in three different boundary condition geometries: (i) planar substrates and B⊥n, (ii) planar substrates and B‖n, and (iii) homeotropic substrates and B⊥n. Particle chaining is observed when a magnetic field is applied, with a periodicity perpendicular to the chains. Furthermore, linear chains are observed for the magnetic field applied perpendicular to the director, while zigzag chains are formed when the magnetic field direction is parallel to the director field. This is attributed to a change from a dipolar defect configuration around dispersed nanoparticles, to a quadrupolar one, i.e. the change from satellite to Saturn-ring defects. This effect is largely independent of the sample thickness. The dynamic development of the chain length, as well as their two-dimensional order parameter was studied in all cases. Chain lengths increased rapidly until saturation at approximately l = 30 μm after a time of about t = 10 s. Similarly, the chain order parameters increased until saturation between S = 0.8-0.9, independent of sample geometry.

Controlled deformation of vesicles by flexible structured media.

Liquid crystalline (LC) materials, such as actin or tubulin networks, are known to be capable of deforming the shape of cells. Here, elements of that behavior are reproduced in a synthetic system, namely, a giant vesicle suspended in a LC, which we view as a first step toward the preparation of active, anisotropic hybrid systems that mimic some of the functionality encountered in biological systems. To that end, we rely on a coupled particle-continuum representation of deformable networks in a nematic LC represented at the level of a Landau-de Gennes free energy functional. Our results indicate that, depending on its elastic properties, the LC is indeed able to deform the vesicle until it reaches an equilibrium, anisotropic shape. The magnitude of the deformation is determined by a balance of elastic and surface forces. For perpendicular anchoring at the vesicle, a Saturn ring defect forms along the equatorial plane, and the vesicle adopts a pancake-like, oblate shape. For degenerate planar anchoring at the vesicle, two boojum defects are formed at the poles of the vesicle, which adopts an elongated, spheroidal shape. During the deformation, the volume of the topological defects in the LC shrinks considerably as the curvature of the vesicle increases. These predictions are confirmed by our experimental observations of spindle-like shapes in experiments with giant unilamellar vesicles with planar anchoring. We find that the tension of the vesicle suppresses vesicle deformation, whereas anchoring strength and large elastic constants promote shape anisotropy.

Analytical description of the Saturn-ring defect in nematic colloids.

We derive an analytical formula for the Saturn-ring configuration around a small colloidal particle suspended in nematic liquid crystal. In particular we obtain an explicit expression for the ring radius and its dependence on the anchoring energy. We work within Landau-de Gennes theory: Nematic alignment is described by a tensorial order parameter. For nematic colloids this model had previously been used exclusively to perform numerical computations. Our method demonstrates that the tensorial theory can also be used to obtain analytical results, suggesting a different approach to the understanding of nematic colloidal interactions.

Spherical microparticles with Saturn ring defects and their self-assembly across the nematic to smectic-A phase transition.

We report experimental studies on the Saturn ring defect associated with a spherical microparticle across the nematic (N) to smectic-A (SmA) phase transition. We observe that the director distortion around the microparticle changes rapidly with temperature. The equilibrium interparticle separation and the angle between two quadrupolar particles in the N phase are larger than those of the SmA phase. They are almost independent of the temperature in both phases, except for a discontinuous jump at the transition. We assembled a few particles using a laser tweezer to form a two-dimensional colloidal crystal in the N phase. The lattice structure of the crystal dissolves irreversibly across the N-SmA phase transition. The results on the pretransitional behavior of the defect are supported by the Landau-de Gennes Q-tensor modeling.

Effects of flow on topological defects in a nematic liquid crystal near a colloid

We perform molecular dynamics simulations of a nematic liquid crystal flowing around a colloidal particle. We study the flow-induced modifications of the topological defects in the liquid crystal due to the presence of the colloid. We show that flow distorts Boojum defects into an asymmetrically larger downstream lobe, and that Saturn ring defects are convected downstream along the flow direction, which is in agreement with experimental observations. Additionally, for a Janus colloid with both parallel and perpendicular patches, exhibiting a Boojum defect and a Saturn ring defect, we find that the Boojum defect facing the upstream direction is destroyed and the Saturn ring is convected downstream.

Light-induced rewiring and winding of Saturn ring defects in photosensitive chiral nematic colloids

We study the winding and unwinding of Saturn ring defects around silica microspheres with homeotropic surface anchoring in a cholesteric liquid crystal with a variable pitch. We use mixtures of a nematic liquid crystal 5CB and various photoresponsive chiral dopants to vary the helical pitch and sense of the helical winding by illuminating the mixtures with UV or visible light. Upon illumination, we observe motion of the Grandjean-Cano disclination lines in wedge-like cells. When the line touches the colloidal particle, we observe topological reconstruction of the Grandjean-Cano line and the Saturn ring. The result of this topological reconstruction is either an increase or decrease of the degree of winding of the Saturn ring around the colloidal particle. This phenomenon is similar to topological rewiring of -1/2 disclination lines, observed recently in chiral nematic colloids.

Particle size effects on nanocolloidal interactions in nematic liquid crystals

We study the interactions of submicrometer diameter silica particles, surface functionalized with DMOAP (N,N-dimethyl-n-octadecyl-3-aminopropyl-trimethoxysilyl chloride), in the nematic liquid crystal 5CB (pentylcyanobiphenil). Using the methods of video-tracking dark-field microscopy, we have measured the pair-binding energy of 35- to 450-nm-diameter silica particles, which is in the range between 100 and 1000 ${k}_{B}T$. It is therefore high enough for the formation of thermally stable nanocolloidal pairs of 35 nm diameter. We find that smaller colloids with the diameter around 22 nm do not form thermally stable pairs, which seems to be currently the lower limit for nanocolloidal assembly in the nematic liquid crystals. We also study the particle interactions with point and Saturn-ring defects and discuss the possibility of hierarchical structures generated by particles of different sizes assembled by topological defects.

Defect topologies in a nematic liquid crystal near a patchy colloid

Using isothermal-isobaric Monte Carlo simulations we investigate defect topologies due to a spherical colloidal particle immersed in a nematic liquid crystal. Defects arise because of the competition between the preferential orientation at the colloid's surface and the far-field director ̂n(0). Considering a chemically homogeneous colloid as a special case we observe the well-known surface and saturn ring defect topologies for weak and strong perpendicular anchoring, respectively; for homogeneous, strong parallel anchoring we find a boojum defect topology that has been seen experimentally [see P. Poulin and D. A. Weitz, Phys. Rev. E 57, 626 (1998)] but not in computer simulations. We also consider a heterogeneous, patchy colloid where the liquid-crystal molecules anchor either preferentially planar or perpendicular at the surface of the colloid. For a patchy colloid we observe a boojum ring defect topology in agreement with recent experimental studies [see M. Conradi, M. Ravnik, M. Bele, M. Zorko, S. Žumer, and I. Muševič, Soft Matter 5, 3905 (2009)]. We also observe two other novel defect topologies that have not been reported thus far neither experimentally nor theoretically.

Defect structures in nematic liquid crystals around charged particles

We numerically study the orientation deformations in nematic liquid crystals around charged particles. We set up a Ginzburg-Landau theory with inhomogeneous electric field. If the dielectric anisotropy epsilon 1 is positive, Saturn-ring defects are formed around the particles. For epsilon 1< 0 , novel "ansa" defects appear, which are disclination lines with their ends on the particle surface. We find unique defect structures around two charged particles. To lower the free energy, oppositely charged particle pairs tend to be aligned in the parallel direction for epsilon 1> 0 and in the perpendicular plane for epsilon 1< 0 with respect to the background director. For identically charged pairs the preferred directions for epsilon 1> 0 and epsilon 1< 0 are exchanged. We also examine competition between the charge-induced anchoring and the short-range anchoring. If the short-range anchoring is sufficiently strong, it can be effective in the vicinity of the surface, while the director orientation is governed by the long-range electrostatic interaction far from the surface.