Tensor Order Parameter Research Articles

A good and computationally efficient polynomial approximation to the Maier–Saupe nematic free energy

A new computational strategy is proposed to approximate, with a simple but accurate expression, the Maier–Saupe free energy for nematic order. Instead of the traditional approach of expanding the free energy with a truncated Taylor series, we employ a least-squares fitting to obtain the coefficients of a polynomial expression. Both methods are compared, and the fitting with at most five polynomial terms is shown to provide a satisfactory fitting, and to give much more accurate results than the traditional Taylor expansion. We perform the analysis in terms of the tensor order parameter, so the results are valid in uniaxial and biaxial states.

Characterising texture formation in fibre lattices embedded in a nematic liquid crystal matrix

A two-dimensional computational study is performed on the texturing of fibre-filled nematic liquid crystals using the Landau-de Gennes model describing the spatio-temporal evolution of the second moment of the orientation distribution function or quadrupolar tensor order parameter. The investigation is performed on a consistent computational domain comprising a square array of four circular fibres embedded within a unit square containing a uniaxial low molar mass calamitic liquid crystal. Interest is focused on the role of temperature, boundary conditions and their effect on the nucleation and evolution of defect structures. Thermal effects are characterised below and above the temperature at which the nematic state is stable. Simulations in the stable nematic state serves as a scenario for investigating the effect of imposing different external boundary conditions, namely periodic and Dirichlet; the former describes a square lattice array of fibres embedded in a nematic liquid crystal, and the latterdescribes a four-fibre arrangement in an aligned nematic material. In each case, the influence of temperature is characterised, with defect structures forming and either remaining or splitting into lower strength defects. For fibre lattices, splitting transitions of defects at the centre of the domain occur at a critical temperature, but for the four-fibre arrangement, defect transitions occur continuously over a temperature range. The discontinuous defect splitting transition in fibre arrays occurs at lower temperatures than the continuous defect transformation in the four-fibre arrangement. At sufficiently low temperatures, the four-fibre arrangement and the fibre lattice give the same texture consisting of two disclination lines close to each fibre. The evolution of the texture with respect to temperature can be characterised as a change from single-fibre mode at low temperature to a collective mode with a centre-located heterogeneity at higher temperature. At higher temperatures, in the stable isotropic state, it is shown that surface-induced ordering arising from the fibre/liquid crystal interaction propagates into the bulk forming thin disclination lattices around the four-fibre configuration.

One order parameter tensor mean field theory for biaxial liquid crystals

In this paper, we present a simple one tensor mean eld model of biaxial nematic liquid crystals. The salient feature of our approach is that ma- terial parameters appear explicitly in the order parameter tensor. We construct the free energy from a mean eld potential based on anisotropic dispersion in- teractions, identify the order parameter tensor and its elements, and obtain self-consistent equations, which are then solved numerically. The results are illustrated in a 3D ternary phase diagram. The phase behavior can be sim- ply related to molecular parameters. The results may be useful for designing molecules that show a thermotropic biaxial phase.

Fluctuating dynamics of nematic liquid crystals using the stochastic method of lines

We construct Langevin equations describing the fluctuations of the tensor order parameter Q(alphabeta) in nematic liquid crystals by adding noise terms to time-dependent variational equations that follow from the Ginzburg-Landau-de Gennes free energy. The noise is required to preserve the symmetry and tracelessness of the tensor order parameter and must satisfy a fluctuation-dissipation relation at thermal equilibrium. We construct a noise with these properties in a basis of symmetric traceless matrices and show that the Langevin equations can be solved numerically in this basis using a stochastic version of the method of lines. The numerical method is validated by comparing equilibrium probability distributions, structure factors, and dynamic correlations obtained from these numerical solutions with analytic predictions. We demonstrate excellent agreement between numerics and theory. This methodology can be applied to the study of phenomena where fluctuations in both the magnitude and direction of nematic order are important, as for instance, in the nematic swarms which produce enhanced opalescence near the isotropic-nematic transition or the problem of nucleation of the nematic from the isotropic phase.

Structural modeling of carbonaceous mesophase amphotropic mixtures under uniaxial extensional flow

The extended Maier-Saupe model for binary mixtures of model carbonaceous mesophases (uniaxial discotic nematogens) under externally imposed flow, formulated in previous studies [M. Golmohammadi and A. D. Rey, Liquid Crystals 36, 75 (2009); M. Golmohammadi and A. D. Rey, Entropy 10, 183 (2008)], is used to characterize the effect of uniaxial extensional flow and concentration on phase behavior and structure of these mesogenic blends. The generic thermorheological phase diagram of the single-phase binary mixture, given in terms of temperature (T) and Deborah (De) number, shows the existence of four T-De transition lines that define regions that correspond to the following quadrupolar tensor order parameter structures: (i) oblate (perpendicular, parallel), (ii) prolate (perpendicular, parallel), (iii) scalene O(perpendicular, parallel), and (iv) scalene P(perpendicular, parallel), where the symbols (perpendicular, parallel) indicate alignment of the tensor order ellipsoid with respect to the extension axis. It is found that with increasing T the dominant component of the mixture exhibits weak deviations from the well-known pure species response to uniaxial extensional flow (uniaxial perpendicular nematic-->biaxial nematic-->uniaxial parallel paranematic). In contrast, the slaved component shows a strong deviation from the pure species response. This deviation is dictated by the asymmetric viscoelastic coupling effects emanating from the dominant component. Changes in conformation (oblate <==> prolate) and orientation (perpendicular <==> parallel) are effected through changes in pairs of eigenvalues of the quadrupolar tensor order parameter. The complexity of the structural sensitivity to temperature and extensional flow is a reflection of the dual lyotropic/thermotropic nature (amphotropic nature) of the mixture and their cooperation/competition. The analysis demonstrates that the simple structures (biaxial nematic and uniaxial paranematic) observed in pure discotic mesogens under uniaxial extensional flow are significantly enriched by the interaction of the lyotropic/thermotropic competition with the binary molecular architectures and with the quadrupolar nature of the flow.

Segmental Order Parameters and Swelling in Polymer Networks

AbstractWe study segmental order parameters in polymer networks in good solvent. In particular we consider the tensor order parameter which characterizes the degree of orientational order of segments and which is directly related to the residual coupling constant obtained in NMR‐experiments. Using a simple relation between the order parameter and the force acting on the chain end at a given extension we derive an universal relation for the order parameter for concentrations above the overlap threshold. Considering polymer gels at the equilibrium state of swelling we predict a unique relation between the tensor order parameter and the correlation length (blob size) of the gel. Experiments applying multi‐quantum NMR‐methods to both end‐linked and randomly cross‐linked polymer networks are in excellent agreement with this prediction. We critically discuss the widely used Flory‐Rehner model of network swelling and show that initial decay of the tensor order parameter as observed in experiments at low degrees of swelling can be explained as a solvent effect without making additional assumptions about constraint release processes during swelling.

Effects of strong anchoring on the dynamic moduli of heterogeneous nematic polymers II: oblique anchoring angles

We extend previous work on the linear viscoelastic moduli of heterogeneous nematic polymers in a small-amplitude oscillatory shear flow, focusing on the role of the orientational anchoring conditions at the plates. When tangential or normal anchoring conditions are applied, the Doi–Marrucci–Greco orientation tensor-flow model effectively reduces to the Leslie–Ericksen director-flow model, predicting that director distortions control the dynamic moduli with negligible contributions from tensor-order parameters. In this paper, we examine oblique anchoring angles. We use a combination of analysis and numerical simulation on the generalized tensor-flow system for arbitrary anchoring conditions to show that any oblique anchoring condition induces a nontrivial order parameter contribution to the dynamic moduli, which vanishes only in the limit of tangential or normal anchoring. Our approach reveals that the storage and loss moduli admit an approximate decomposition in terms of two reduced problems that are exactly solvable: the heterogeneous director–flow response plus the monodomain tensor response to an imposed shear. The importance of this result is that we gain scaling properties of the moduli with respect to material parameters and experimental conditions without having to compute and assimilate across the full parameter space. These results provide insight into the relative importance of the distortional vs bulk nematic elastic stress in determining the viscoelastic moduli, predicting that anchoring conditions tune the relative contributions.

Collective Modes and Biaxial Ordering Observed by Deuterium NMR in the Nematic Phases of an Organosiloxane Tetrapode

Calculations of deuterium NMR spectra were performed using a model of slow motions based on the collective modes present in liquid crystalline systems and evaluated within the Landau de Gennes free energy expansion on the order parameter tensor. Simulations obtained with this model are applied to the case of deuterium NMR spectra collected in static and rotating samples of organosiloxane tetrapodes exhibiting uniaxial and biaxial nematic phases. The analysis of the slow motions influence on deuterium NMR spectra shows that molecular motions within a time-scale of the order of magnitude of NMR observation times are particularly effective on modulating the NMR line-shape in the case of the liquid crystalline system investigated.

Tensorial Form of Leslie-Ericksen Equations and Applications

We propose a system of dynamical equations coupling a tensor Qij related to the molecular order with the flow velocity, whose structure is very similar to the Leslie-Ericksen equations. In particular, the Leslie viscosities and the Frank elastic constants appear explicitly, which contrasts with constitutive equations derived from the microscopic Doi theory and using the full order parameter tensor Sij. The description in terms of the tensor Qij may be seen as intermediate between the director of Leslie-Ericksen theory and the order parameter tensor Sij of microscopic theories. With our approach we take advantage of the fact that the Leslie viscosities and the Frank elastic constants are well-established for several real systems (as for example 5CB, 8CB, or MBBA). Our equations, that become manifestly equivalent to Leslie-Ericksen equations when a director may be defined, are used in order to regularise the director field at the position of the field singularities in a simple way. Our objective is to simulate the macroscopic behaviour of nematics containing a large assembly of defects under several experimental conditions (under shear and/or magnetic field). Since we are essentially concerned by the macroscopic response, the exact structure of the defects is secondary, it is why we believe this approach to be pertinent in studying the dynamics of the texture induced by defects. The model is applied to a spinning nematic (5CB) subjected to a magnetic field, and a relaxation mechanism involving topological defects is evidenced.

Shape-dynamic growth, structure, and elasticity of homogeneously oriented spherulites in an isotropic/smectic-A mesophase transition

A Landau-de Gennes model that integrates the nematic quadrupolar tensor order parameter and complex smectic-A order parameters is used to simulate the two-dimensional growth of an initially homogeneous smectic-A spherulite in an isotropic matrix. These simulations are performed in the shape-dynamic (nano-scale) regime of growth under two material conditions: isotropic nematic elasticity and equal splay-bend nematic elasticity. A comparison of the growth kinetics, spherulite morphology, and interfacial/bulk energy landscapes between both cases is made showing that equal nematic splay–bend elasticity is required to reproduce past experimental and theoretical observations. In addition, a previously unknown undulation instability during spherulite growth is found which, in conjunction with preferred planar anchoring and defect shedding mechanisms at micrometer length scales, could explain the formation mechanism of focal conic curvature defects and ultimately smectic-A ‘batonnet’ structures observed experimentally.

Faceted nanoparticles in a nematic liquid crystal: defect structures and potentials of mean force

We investigate the defect structures and the potentials of mean force (PMF) that arise when faceted nanoparticles, namely cubes and triangular prisms, are immersed in a nematic liquid crystal (NLC). Using a mesoscale theory for the tensor order parameter Q of the nematic, we have determined the thermodynamic stability of different orientations of one nanoparticle with respect to the far-field director n(r). A nanocube with perpendicular anchoring of the nematic at its surfaces tends to align in such a way that none of its faces is parallel or perpendicular to n(r); the most stable defect structure consists of a distorted Saturn ring with sharp bends, which covers six of the edges of the cubic particle. In contrast, a triangular nanoprism with homeotropic anchoring of the nematic at its surfaces tends to align with its long axis perpendicular to the far-field director n(r), and with one of its rectangular faces perpendicular to n(r). For such a configuration, the defect structure consists of two large disclination regions covering the two triangular faces of the prism, and two narrow disclination regions surrounding two of the rounded edges of the prism. We also studied the thermodynamic stability of different arrays of two particles, finding that for two nanocubes that approach each other keeping their orientations fixed, the nematic forms a distorted ‘entangled hyperbolic’ defect structure around the particles, in analogy to what was observed for pairs of spherical and spherocylindrical nanoparticles in close proximity. The NLC-mediated interactions between the nanocubes in this case are of the order of − 85k B T, which are weaker than those observed for spherical nanoparticles of comparable diameter (∼ − 110k B T). For systems of two nanoprisms having their long axes perpendicular to the far-field director, we considered three particle arrays: linear (the long axes of the particles are collinear and the particles have the same orientation), parallel (the long axes of the particles are parallel and the particles have the same orientation) and inverted parallel (the long axes of the particles are parallel, and one of the prisms is inverted with respect to the other one). Our results suggest that inverted parallel arrays are thermodynamically more stable than linear arrays, which in turn are more stable than parallel arrays. The minima observed in the PMF curves for the inverted parallel (∼ − 1050k B T) and linear arrays (∼ − 525k B T) are significantly deeper than that observed for the parallel array (∼ − 150k B T). In comparison, a pair of nanospheres with a diameter comparable to the size of the triangular faces of our nanoprisms has a PMF minimum of ∼73k B T when the spheres are in close proximity. These NLC-mediated, anisotropic interparticle interactions can make the particles bind together at specific locations, and thus could be used to assemble the particles into ordered structures with different morphologies.

Kinetics of photoinduced ordering in azo-dye films: Two-state and diffusion models

We theoretically study the kinetics of photoinduced ordering in azo-dye photoaligning layers and present the results of modeling performed using two different phenomenological approaches. A phenomenological two-state model is deduced from the master equation for the one-particle distribution functions of an ensemble of two-level molecular systems by specifying the angular redistribution probabilities and by expressing the order parameter correlation functions in terms of the order parameter tensor. Using an alternative approach that describes light-induced reorientation of azo-dye molecules in terms of a rotational Brownian motion, we formulate the two-dimensional diffusion model as the free energy Fokker-Planck equation simplified for the limiting regime of purely in-plane reorientation. The models are employed to interpret the irradiation time dependence of the absorption order parameters defined in terms of the principal extinction (absorption) coefficients. Using the exact solution to the light transmission problem for a biaxially anisotropic absorbing layer, these coefficients are extracted from the absorbance-vs-incidence angle curves measured at different irradiation doses for the probe light linearly polarized parallel and perpendicular to the plane of incidence. It is found that, in the azo-dye films, the transient photoinduced structures are biaxially anisotropic whereas the photosteady and the initial states are uniaxial.

Quasiaverages and classification of equilibrium states of condensed media with spontaneously broken symmetry

Classification of equilibrium states of condensed media with spontaneously broken symmetry is carried out. Conditions of residual symmetry and spatial symmetry are formulated. The connection between these symmetry conditions and equilibrium states of various media with tensor order parameter is found out. Superfluid 3He, liquid crystals, quadrupolar magnetics are considered in detail. Possible homogeneous and heterogeneous states are found out. Discrete and continuous thermodynamic parameters, which define an equilibrium state, allowable form of order parameter, residual symmetry, and spatial symmetry generators are established.

Quadrupolar particles in a nematic liquid crystal: Effects of particle size and shape

We investigate the effects of particle size and shape on the quadrupolar (Saturn-ring-like) defect structures formed by a nematic liquid crystal around nm-sized and mum -sized particles with spherical and spherocylindrical shapes. We also report results for the potentials of mean force in our systems, calculated using a mesoscale theory for the tensor order parameter Q of the nematic. Our results indicate that for pairs of nm-sized particles in close proximity, the nematic forms "entangled hyperbolic" defect structures regardless of the shape of the nanoparticles. In our calculations with nanoparticles we did not observe any other entangled or unentangled defect structures, in contrast to what was reported for pairs of mum -sized spherical particles. Such a finding suggests that the "entangled hyperbolic" defect structures are the most stable for pairs of nanoparticles in close proximity. For pairs of mum -sized particles, our results indicate that the nematic forms entangled "figure-of-eight" defect structures around pairs of spheres and spherocylinders. Our results suggest that the transition between "entangled hyperbolic" and figure-of-eight defect structures takes place when the diameter of the particle is between D=100 nm and 1 microm . We have also calculated the torques that develop when pairs of spherocylindrical nanoparticles in a nematic approach each other. Our calculations suggest that the nematic-mediated interactions between the nm-sized particles are fairly strong, up to 5700 k{B}T for the case of pairs of spherocylindrical nanoparticles arranged with their long axis parallel to each other. Furthermore, these interactions can make the particles to bind together at specific locations, and thus could be used to assemble the particles into ordered structures with different morphologies.

Theory and computation of directional nematic phase ordering

A computational study of morphological instabilities of a two-dimensional nematic front under directional growth was performed using a Landau-de Gennes-type quadrupolar tensor order parameter model for the first-order isotropic-nematic transition of 5CB (pentyl-cyanobiphenyl). A previously derived energy balance, taking anisotropy into account, was utilized to account for latent heat and an imposed morphological gradient in the time-dependent model. Simulations were performed using an initially homeotropic isotropic-nematic interface. Thermal instabilities in both the linear and nonlinear regimes were observed and compared to past experimental and theoretical observations. A sharp-interface model for the study of linear morphological instabilities, taking into account additional complexity resulting from liquid-crystalline order, was derived. Results from the sharp-interface model were compared to those from full two-dimensional simulation identifying the specific limitations of simplified sharp-interface models for this liquid-crystal system. In the nonlinear regime, secondary instabilities were observed to result in the formation of defects, interfacial heterogeneities, and bulk texture dynamics.

Effect of excluded volume on segmental orientation correlations in polymer chains

We study the impact of excluded volume interactions on the orientation statistics of chain segments in polymer gels, and show that nuclear magnetic resonance (NMR) experiments provide a direct and unique measure of excluded-volume effects on the chain statistics. In particular we consider the tensor order parameter, which can be expressed as the second Legendre polynomial of the segment orientation with respect to a fixed end-to-end distance vector and which is directly related to the residual coupling constant obtained in NMR experiments. We provide analytical results for the case of single chains in a good solvent and for semidilute solutions. Computer simulations using the bond fluctuation model are applied to compare with the analytical predictions. Considering polymer gels at the equilibrium state of swelling we predict a unique relation between the tensor order parameter and the correlation length (blob size) of the gel. Experiments applying multiple-quantum NMR methods to both end-linked and randomly cross-linked polymer networks are in excellent agreement with this prediction. The initial decay of the tensor order parameter as observed in experiments at low and intermediate degrees of swelling can be explained as a solvent effect without making additional assumptions about constraint release processes during swelling.

Landau theory for biaxial nematic liquid crystals with two order parameter tensors

We present a phenomenological theory for the homogeneous phases of nematic liquid crystals constituted by biaxial molecules. We propose a general polynomial potential in two macroscopic order parameter tensors that reproduces the mean-field phase diagram confirmed by Monte Carlo simulations [De Matteis et al. in Phys Rev E 72:041706 (2005)] and recently recognized to be universal [Bisi et al. in Phys Rev E 73:051709 (2006)] for dispersion force molecular pair-potentials enjoying the D 2h symmetry. The requirement that the phenomenological theory comply uniquely with this phase diagram reduces considerably the admissible phenomenological coefficients, both in their number and in the ranges where they can vary.

Non-Isothermal Model for Nematic Spherulite Growth

A computational study of the growth of two-dimensional nematic spherulites in an isotropic phase was performed using a Landau-de Gennes-type quadrupolar tensor order parameter model for the first-order isotropic/nematic transition of 5CB (pentylcyanobiphenyl). An energy balance, taking anisotropy into account, was derived and incorporated into the time-dependent model. Growth laws were determined for two different spherulite morphologies of the form t(n), with and without the inclusion of thermal effects. Results show that incorporation of the thermal energy balance correctly predicts the transition of the growth law exponent from the volume driven regime (n=1) to the thermally limited regime (approaching n = 1/2), agreeing well with experimental observations. An interfacial nematodynamic model is used to gain insight into the interactions that result in the progression of different spherulite growth regimes.

Biomechanical ordering of dense cell populations

The structure of bacterial populations is governed by the interplay of many physical and biological factors, ranging from properties of surrounding aqueous media and substrates to cell-cell communication and gene expression in individual cells. The biomechanical interactions arising from the growth and division of individual cells in confined environments are ubiquitous, yet little work has focused on this fundamental aspect of colony formation. We analyze the spatial organization of Escherichia coli growing in a microfluidic chemostat. We find that growth and expansion of a dense colony of cells leads to a dynamical transition from an isotropic disordered phase to a nematic phase characterized by orientational alignment of rod-like cells. We develop a continuum model of collective cell dynamics based on equations for local cell density, velocity, and the tensor order parameter. We use this model and discrete element simulations to elucidate the mechanism of cell ordering and quantify the relationship between the dynamics of cell proliferation and the spatial structure of the population.

Octupolar ordering of classical kagome antiferromagnets in two and three dimensions

Classical Heisenberg antiferromagnets on two-dimensional kagome and three-dimensional hyperkagome lattices are investigated by Monte Carlo simulations. For both models the symmetry-breaking states at low temperatures are described by nonzero octupole moments or third-rank spin tensor order parameters. In the case of the two-dimensional kagome antiferromagnet, a sharp crossover into a coplanar state takes place at ${T}_{k}\ensuremath{\approx}0.004J$, which we attribute to proliferation of fractional vortices. The three-dimensional model exhibits a first-order transition at ${T}_{c}\ensuremath{\approx}0.002J$ into a phase with critical spin correlations, which possesses a long-range order of octupole moments.