Uniaxial Order Parameter Research Articles

Deformation localisation in stretched liquid crystal elastomers

We model within the framework of finite elasticity two inherent instabilities observed in liquid crystal elastomers under uniaxial tension. First is necking, which occurs when a material sample suddenly elongates more in a small region where it appears narrower than the rest of the sample. Second is shear striping, which forms when the in-plane director rotates gradually to realign and become parallel with the applied force. These phenomena are due to the liquid crystal molecules rotating freely under mechanical loads. To capture necking, we assume that the uniaxial order parameter increases with tensile stretch, as reported experimentally during polydomain-monodomain transition. To account for shear striping, we maintain the uniaxial order parameter fixed, as suggested by experiments. Our finite element simulations capture well these phenomena. As necking in liquid crystal elastomers has not been satisfactorily modelled before, our theoretical and numerical findings related to this effect can be of wide interest. Shear striping has been well studied, yet our computed examples also show how optimal stripe width increases with the nematic penetration depth measuring the competition between the Frank elasticity of liquid crystals and polymer elasticity. Although known theoretically, this result has not been confirmed numerically by previous nonlinear elastic models.

A theoretical liquid crystal elastomer model that mimics the elasticity of cat skin

A mathematical model for nematic liquid crystal elastomers is proposed that mimics the elastic response of cat skin where reorientation of dermal fibres produces an increase in the thickness direction under tensile stretch. To capture this unusual effect, the uniaxial order parameter in the nematic elastomer model is allowed to decrease then increase again, and the critical stretch at which this change of monotonicity occurs and where the director also rotates suddenly is predicted. In addition, the model parameters are described by probability density functions and their uncertainty is propagated numerically to the predicted mechanical results.

Direct Observation of Biaxial Nematic Order in Auxetic Liquid Crystal Elastomers

Auxetic materials exhibit a negative Poisson's ratio, i.e., they become thicker rather than thinner in at least one dimension when strained. Recently, a nematic liquid crystal elastomer (LCE) was shown to be the first synthetic auxetic material at a molecular level. Understanding the mechanism of the auxetic response in LCEs is clearly important, and it has been suggested through detailed Raman scattering studies that it is related to the reduction of uniaxial order and emergence of biaxial order on strain. In this paper, we demonstrate direct observation of the biaxial order in an auxetic LCE under strain. We fabricated ~100 μm thick LCE strips with complementary geometries, exhibiting either planar or homeotropic alignment, in which the auxetic response is seen in the thickness or width of the sample, respectively. Polarized Raman scattering measurements on the planar sample show directly the reduction in the uniaxial order parameters on strain and suggest the emergence of biaxial order to mediate the auxetic response in the sample thickness. The homeotropic sample is studied via conoscopy, allowing direct observation of both the auxetic response in the width of the sample and increasing biaxiality in the LCE as it is strained. We verified that the mechanism of the auxetic response in auxetic LCEs is due to the emergence of the biaxial order and conclude such materials can be added to the small number of biaxial nematic systems that have been observed. Importantly, we also show that the mechanical Frèedericksz transition seen in some LCEs is consistent with a strain-induced transition from an optically positive to an optically negative biaxial system under strain, rather than a director rotation in a uniaxial system.

Electrical modification of order parameters and director fluctuations in a dielectrically negative nematic doped with a positive additive

Electric effects of nematic liquid crystals (NLCs) have enabled the revolution of the display industry. Recently, the electro-optic switching of NLCs has been accelerated by applying electric fields in such a way that the field does not change the director orientation, but modifies the optical tensors of NLCs due to three effects: induced biaxial orientational order, enhanced uniaxial orientational order, and quenching of director fluctuations. Till now, these three effects have been explored only in NLCs with negative dielectric anisotropy. Here, we experimentally study the electrical modification of order parameters and director fluctuations in a dielectrically negative/positive composite NLC. The dielectrically positive dopant causes the following changes: (a) a faster nanosecond switching for optical responses attributed to the change of order parameters, (b) an increase of the biaxial order parameter, (c) a reduction of the uniaxial order parameter, and (d) a slight reduction in the field-induced birefringence change. This work provides a basis for further studies in nanosecond electro-optics.

Impact of Weak Nanoparticle Induced Disorder on Nematic Ordering

Dilute mixtures of nanoparticles (NPs) and nematic liquid crystals (LCs) are considered. We focus on cases where NPs enforce a relatively weak disorder to the LC host. We use a Lebwohl-Lasher semi-microscopic-type modeling where we assume that NPs effectively act as a spatially-dependent external field on nematic spins. The orientational distribution of locally favoured “easy” orientations is described by a probabilistic distribution function P. By means of a mean field-type approach, we derive a self-consistent equation for the average degree of nematic uniaxial order parameter S as a function of the concentration p of NPs, NP-LC coupling strength and P. Using a simple step-like probability distribution shape, we obtain the S(p) dependence displaying a crossover behaviour between two different regimes which is in line with recent experimental observations. We also discuss a possible origin of commonly observed non-monotonous variations of the nematic-isotropic phase temperature coexistence width on varying p.

Molecular properties of a bent-core nematic liquid crystal A131 by multi-level theory simulations

ABSTRACTMulti-level theory simulations have been performed to model a number of important molecular properties of a bent-core nematic liquid crystal (LC) A131. These important properties include molecular conformations, molecular Raman spectra, differential polarisability ratios, molecular crystals packing, atomic LC structures, order parameters, and Raman depolarisation spectra. The simulations contain four theory levels, involving molecular quantum chemistry, molecular crystal packing, super cell density functional based tight binding optimisation, and super cell molecular dynamics calculations. To heat initial optimised super cell structures, molecular dynamics simulations reveal phase transitions to uniaxial and biaxial nematic phases from molecular crystals. LC atomic structures result in direct calculations on order parameters, which can be further applied to computations on Raman depolarisation spectra with differential polarisability ratios, obtained in the molecular quantum chemistry theory level. The good agreement of simulated Raman depolarisation spectra with the experiment provides a detailed analysis on the unusually low values of experimental uniaxial order parameters.

Order parameters and time evolution of mesophases in the lyotropic chromonic liquid crystal Sunset Yellow FCF by DNMR.

Uniaxial order parameters of the nematic and columnar mesophases in the lyotropic chromonic liquid crystal Sunset Yellow FCF have been determined from deuteron nuclear magnetic resonance, where random confinement of the system by the dispersion of aerosil nanoparticles has been performed to help obtain the angular dependent spectra. The long-time evolution study of the order parameters shows that the system requires tens of hours to stabilize after a deep change in temperature, in contrast with the very fast assembly process of the aggregates. Finally, the degree of order of the water molecules, forced by the uniaxial environment, has been determined.

Raman Scattering Study of Phase Biaxiality in a Thermotropic Bent-Core Nematic Liquid Crystal

Polarized Raman spectroscopy was used to investigate the development of orientational order and the degree of phase biaxiality in a bent-core mesogenic system. The values of the uniaxial order parameters <P200> and <P400>, and biaxial order parameters <P220>, <P420>, and <P440>, and their evolution with temperature were determined. The temperature dependence of almost all order parameters reveals a second order transition from the uniaxial to biaxial nematic phase with <P220> increasing to ∼0.22 before a first order transition to the smectic-C phase, upon cooling.

Universal Features in the Nematic Uniaxial-to-Biaxial Transition

Recent experimental findings about new nematogenic molecules have reported thermally driven transitions to biaxial phases, which has revived the interest for macroscopic biaxiality. In particular, the simplified interaction model proposed by Straley for molecules endowed with D2h symmetry has met renewed interest. In the scenario obtained for phases a unifying character can be captured. There exists a universal feature in the behaviour of one uniaxial order parameter, rebounding at the biaxial-to-uniaxial transition. This finding has been confirmed by means of a Monte-Carlo simulation. Moreover, the phase sequences and the order of the transitions are poorly influenced by one of the interaction parameters.

Uniaxial rebound at the nematic biaxial transition

Over the last few years, renewed interest has been raised by the simplified general interaction models proposed by Straley for mesogenic molecules possessing the D{2h} symmetry and capable of producing biaxial nematic order. It has already been shown that, in the presence of certain special symmetries, just two out of the four order parameters that are in general necessary, suffice for the description of a biaxial phase. For some other range of parameters, these reducing symmetries do not hold, and, moreover, a mean-field treatment has to be suitably changed into a minimax strategy, still producing a transition to a low-temperature biaxial phase. Upon studying the general parameter range, we identify as a common feature the behavior of a uniaxial order parameter, attaining a local minimum at the biaxial-to-uniaxial transition temperature, and recognizably increasing away from it. This finding is confirmed by a Monte Carlo simulation.

Symmetry of Nematic Phase and Oblique Axial Order

Nematic order is usually described by a uniaxial order parameter, or occasionally by a biaxial order parameter together with the uniaxial one, in a typical configuration. In a general case of a system exposed to two different types of external fields, e.g., an electric field and a magnetic field, an additional order parameter, here called an oblique axial order parameter, is shown to be necessary, in addition to the above order parameters, to describe transition phenomena of the system correctly. The symmetry of the phase with the triplet of the order parameters is discussed. Effect of the oblique axial order are demonstrated practically in two phenomena: (i) qualitative change of the phase transition when the oblique axial order is neglected; (ii) disappearance of phase transition between ordered phases with different ordering axes when an oblique axial field conjugate to the oblique axial order parameter exists.

A continuum model for flows of cholesteric liquid crystal polymers and permeation flows

We present a hydrodynamic theory for flows of cholesteric liquid crystal polymers (CLCPs) following the continuum mechanics formulation of McMillan’s second order tensor theory for liquid crystals [E.H. MacMillan, Ph.D. thesis, A Theory of Anisotropic Fluid Department of Mechanics, University of Minnesota, 1987], in which anisotropic distortional elastic coefficients are prescribed as functions of the local nematic order. We explore two classes of spatially inhomogeneous equilibrium solutions associated with the variation of the uniaxial order parameter and the helical director pattern. In the latter solution, we show that the chirality effectively couples the distortional elasticity and the cholesteric pitch to impact on the local nematic order and vice versa. When the model is used to study permeation flows of weakly sheared cholesteric liquid crystal polymers, we find that secondary flows are strongly coupled to the primary flow and the director dynamics. At the first order of the coarse-grained asymptotic scheme, the flow and orientational director dynamics dominate while the dynamics of the orientation order parameters show up at the next order. The leading order solutions exhibit boundary layers of thickness in the order of O ( q − 1 ) , where q is the dimensionless wave number in the base cholesteric equilibrium. In plane Couette flows, the velocity components are nearly constants outside the boundary layers and the director angle varies linearly across the shear cell while the apparent viscosity scales like O ( q ) ; whereas in weak Poiseuille flows, the angle is quadratic outside the boundary layers and velocity components barely change while the apparent viscosity scales like O ( q 2 ) .

Core hysteresis in nematic defects.

We study field-induced transformations in the biaxial core of a nematic disclination with strength m=1, employing the Landau-de Gennes order tensor parameter Q. We first consider the transition from the defectless escaped radial structure into the structure hosting a line defect with a negative uniaxial order parameter along the axis of a cylinder of radius R. The critical field of the transition monotonically increases with R and asymptotically approaches a value corresponding to xi(b)/xi(f) approximately 0.3, where the correlation lengths xi(b) and xi(f) are related to the biaxial order and the external field, respectively. Then, in the same geometry, we focus on the line defect structure with a positive uniaxial ordering along the axis, surrounded by the uniaxial sheath, the uniaxial cylinder of radius xi(u) with negative order parameter and director in the transverse direction. We study the hysteresis in the position of the uniaxial sheath upon increasing and decreasing the field strength. In general, two qualitatively different solutions exist, corresponding to the uniaxial sheath located close to the defect symmetry axis or close to the cylinder wall. This latter solution exists only for strong enough anchorings. The uniaxial sheath is for a line defect what the uniaxial ring is for a point defect: by resorting to an approximate analytic estimate, we show that essentially the same hysteresis exhibited by the uniaxial sheath is expected to occur at the uniaxial ring in the core structure of a point defect.

Molecular dynamics study of the isotropic-nematic quench.

Effects of cylindrical and spherical confinement on the kinetics of the isotropic-nematic quench is studied numerically. The nematic liquid crystal structure was modeled by a modified induced-dipole--induced-dipole interaction. Molecules were allowed to wander around points of a hexagonal lattice. Brownian molecular dynamics was used in order to access macroscopic time scales. In the bulk we distinguish between the early, domain, and late stage regime. The early regime is characterized by the exponential growth of the nematic uniaxial order parameter. In the domain regime domains are clearly visible and the average nematic domain size xi(d) obeys the dynamical scaling law xi(d)-t(gamma). The late stage evolution is dominated by dynamics of individual defects. In a confined system the qualitative change of the scaling behavior appears when xi(d) becomes comparable to a typical linear dimension R of the confinement. In the confining regime (xi(d)>or=R) the scaling coefficient gamma depends on the details of the confinement and also the final equilibrium nematic structure. The domain growth is well described with the Kibble-Zurek mechanism.

Role of surface elasticity in capillary instability of cylindrical jets of nematic liquid crystalline polymers

We study the capillary instability of infinite cylindrical jets of nematic liquid crystal polymers (LCPs) using an approximate Doi theory for rodlike LCPs with an emphasis on the role of surface elasticity, arising from the potential discrepancy between the averaged molecular orientation in the bulk and the preferred surface orientation. Our analyses focus on the linear stability of two equilibrium (constant), cylindrical jets of distinctive preferred surface orientation patterns: the equilibrium jet of the axially aligned uniaxial director corresponding to the preferred tangential (planar) aligning at the jet free surface and the one of the radially aligned uniaxial director corresponding to the preferred normal (homeotropic) aligning at the surface. For the axially aligning jet, our results establish that the capillary instability persists for a finite band of wave numbers and the surface elasticity lowers the upper bound of the unstable wave number band (cutoff wave number), reduces the growthrate across the entire band of unstable wave numbers, and lengthens the wave length of the most unstable mode compared to our previous results in the absence of the surface elasticity. We then extend the study to the radially aligning jet and show that it shares the same stability properties of the axially aligning jet qualitatively. Quantitatively, however, the radially aligning jet suffers from a much greater growthrate and shorter wave length at the most unstable mode with the same set of physical parameters. In both cases, we demonstrate explicitly that the cutoff wave number varies inversely with respect to the uniaxial order parameter of the underlying equilibrium jet. This study extends the 1-D asymptotic (slender body) analysis of Rey [18] to a full 3-D, axisymmetric analysis with variable degree of orientation and meanwhile our early work to include the surface elasticity [10].

Special cylindrical free surface jets of liquid crystalline polymers and their stability

We examine two special solution families of the governing equations for free surface, axisymmetric, cylindrical liquid crystalline polymer (LCP) jet flows, in which only the uniaxial orientation pattern is accommodated. From these ideal transient solutions, we analyze the linearized stability of the coupled flow and microstructure in the LCP slender jet approximation. Our analysis reveals a window of uniaxial order parameters for both jets in which the LCP microstructure yields a negative first normal stress difference that may completely stabilize the jets provided the negative first normal stress difference dominates the isotropic surface tension and viscous stress during some period of time. It is shown that the negative first normal stress difference in the stable jets is the result of the low degree of orientation in the microstructural material.

Lattice modelling of nematodynamics

The microstructures of textured nematics under shear are investigated by means of a director lattice model incorporating linear shear response as well as elastic interactions between neighbouring directors. The model can be understood as a lattice implementation of the so-called nematodynamics equation for a constant uniaxial order parameter. The dimensionless number governing the model is found to be a mesoscale Ericksen number, which scales with the square of the lattice cell size. It is shown that the predicted microstructure depends strongly on the scale of that number. In particular, disclination loops are found to grow for a range of mesoscale Ericksen numbers, while below or above that they disappear. We apply the model to investigate the director profiles of tumbling nematics. If the orientations are restricted to lying in the vorticity plane, we reproduce the director wind-up layers and distortion saturation predicted theoretically. In the full three-dimensional case an initially polydomain director field evolves to a vorticity-aligned state up to a critical Ericksen number, above which in-plane orientations with distortion saturation are found. The simulations hence reproduce the transition from log-rolling to flow aligning with increasing shear rate observed experimentally.

Phase Diagram and Orientational Order in a Biaxial Lattice Model: A Monte Carlo Study.

We have determined the phase diagram for a lattice system of biaxial particles interacting with a second rank anisotropic potential using Monte Carlo simulations for a number of values of the molecular biaxiality. We find increasing differences from mean field theory as the biaxiality increases. We have also calculated for the first time the full set of second rank biaxial and uniaxial order parameters and their temperature dependence, and on this basis we comment on the difficulties of measuring phase biaxiality by NMR.

Textural and calorimetric investigations on a homologous series of mesogenic azocompounds complexed with palladium(II)

Abstract A comparative study of some mesogenic azobenzene compounds and their organometallic palladium(II) derivatives is reported. On the basis of the textural properties, calorimetric data and uniaxial order parameters (as deduced approximately from infrared spectroscopy), the main features induced by palladium complexation are (i) the transition temperatures increase, (ii) the number of ordered mesophases increases, (iii) the nematic uniaxial order parameter decreases, (iv) the textures are typical of highly ordered phases, and (v) an optical biaxiality appears in the nematic phase. The calorimetric data as well as the order parameters, if discussed in terms of molecular biaxiality result are interpreted at least qualitatively. Optical investigations show that macroscopical biaxial ordering can be achieved.

Topological properties of cholesteric blue phases.

The topological properties of cubic structures which have been proposed as models for the cholesteric blue phases (BP) are considered. When the space of the order parameter is restricted to the standard one for biaxial nematics [SO(3)/${D}_{2}$] then, as a consequence of the cubic symmetry, these structures must have disclinations. The usual classification scheme, wherein defect lines are characterized by the conjugacy classes of the fundamental group appropriate to a biaxial system, is used; some exceptions are also noted. For each of the three groups studied [body-centered-cubic ${O}^{5}$ (I432) and ${O}^{8}$ (I${4}_{1}$32), and simple-cubic ${O}^{2}$ (P${4}_{2}$32)] two aspects are considered: general symmetry constraints on disclinations and the topological properties of equilibrium order-parameter distributions as found from Landau-theory calculations. For ${O}^{5}$ and ${O}^{2}$, there are two nonequivalent symmetry axes upon which there must be disclinations. When they are characterized by a uniaxial order parameter, one must be uniaxial positive and the other uniaxial negative in order for the topology to be consistent with the space-group symmetries. This is not the case for ${O}^{8}$ where only one type of defect line, located on the threefold axes, is topologically required. The order-parameter fields obtained from Landau--free-energy minimization have, except in the case of ${O}^{5}$, additional disclinations. In particular, the three structures described by the ${O}^{8}$ space group are found to have distinctly different networks of defect lines. These, as well as the additional disclination found in the ${O}^{2}$ structure, are examined and their group structures were determined. Finally, some open questions and directions for future work are examined.