Molecular field theory for biaxial nematics formed from liquid crystal dimers and inhibited by the twist-bend nematic.

Liquid crystal dimers with odd spacers are good candidates as materials for biaxial nematic phases (NB). The dimers are flexible molecules sustaining biaxial conformations, and couplings between the conformational and orientational distributions could be expected to stabilise NB. We apply a molecular field theory for flexible molecules developed elsewhere to study a simple system made up of dimers composed of two cylindrically symmetric mesogenic groups. Our model allows for two idealised conformations: one linear and one bent at a tetrahedral angle. For a restricted set of chain lengths, the model predicts a first-order reentrant phase transition from the NB phase into a low temperature uniaxial nematic phase (NU). However the formation of the biaxial nematic could be blocked by the appearance of a twist-bent nematic.

Cloaking by shells with radially inhomogeneous anisotropic permittivity.

We model electromagnetic cloaking of a spherical or cylindrical nanoparticle enclosed by an optically anisotropic and optically inhomogeneous symmetric shell, by examining its electric response in a quasi-static uniform electric field. When the components of the shell permittivity are radially anisotropic and power-law dependent (ε~r(m)) whereris distance to the shell center, and m a positive or negative exponent which can be varied), the problem is analytically tractable. Formulas are calculated for the degree of cloaking in the general case, allowing the determination of a dielectric condition for the shells to be used as an invisibility cloak. Ideal cloaking is known to require that homogeneous shells exhibit an infinite ratio of tangential and radial components of the shell permittivity, but for radially inhomogeneous shells ideal cloaking can occur even for finite values of this ratio.

Cooperation and punishment in community-structured populations with migration.

The stable presence of punishing strategies in various cooperative species is a persistent puzzle in the study of the evolution of cooperation. To investigate the effect of group competition, we study the evolutionary dynamics of the Public Goods Game with punishment in a metapopulation that consists of separate communities. In addition to (a) well-mixed non-interacting communities, we model three distinct types of interaction between communities, (b) Migration independent of fitness; (c) Competition between whole communities, where entire communities replace each other depending on average fitness; (d) Migration where the probability of an offspring replacing an individual in another community depends on fitness. We use stochastic simulations to study the long-run frequencies of strategies with these interactions, subject to high mutation and migration rates. In cases (a) and (b), the transition between cooperation/punishment and defection regimes occurs for similar parameter values; with migration (b), the transitions are steeper due to higher total mixing. Fitness-based migration (d) by contrast can help support cooperation, changing the locations of transitions, but while group selection (c) does stabilise cooperation over much of the parameter space, fitness-based migration (d) acts as a proxy for group selection only in a smaller region.

Long-range dispersal, stochasticity and the broken accelerating wave of advance.

Rare long distance dispersal events are thought to have a disproportionate impact on the spread of invasive species. Modelling using integrodifference equations suggests that, when long distance contacts are represented by a fat-tailed dispersal kernel, an accelerating wave of advance can ensue. Invasions spreading in this manner could have particularly dramatic effects. Recently, various authors have suggested that demographic stochasticity disrupts wave acceleration. Integrodifference models have been widely used in movement ecology, and as such a clearer understanding of stochastic effects is needed. Here, we present a stochastic non-linear one-dimensional lattice model in which demographic stochasticity and the dispersal regime can be systematically varied. Extensive simulations show that stochasticity has a profound effect on model behaviour, and usually breaks acceleration for fat-tailed kernels. Exceptions are seen for some power law kernels, K(l)∝|l|-ß with ß<3, for which acceleration persists despite stochasticity. Such kernels lack a second moment and are important in 'accelerating' phenomena such as Lévy flights. Furthermore, for long-range kernels the approach to the continuum limit behaviour as stochasticity is reduced is generally slow. Given that real-world populations are finite, stochastic models may give better predictive power when long-range dispersal is important. Insights from mean-field models such as integrodifference equations should be applied with caution in such circumstances.

Nonequivalence of updating rules in evolutionary games under high mutation rates.

Moran processes are often used to model selection in evolutionary simulations. The updating rule in Moran processes is a birth-death process, i. e., selection according to fitness of an individual to give birth, followed by the death of a random individual. For well-mixed populations with only two strategies this updating rule is known to be equivalent to selecting unfit individuals for death and then selecting randomly for procreation (biased death-birth process). It is, however, known that this equivalence does not hold when considering structured populations. Here we study whether changing the updating rule can also have an effect in well-mixed populations in the presence of more than two strategies and high mutation rates. We find, using three models from different areas of evolutionary simulation, that the choice of updating rule can change model results. We show, e. g., that going from the birth-death process to the death-birth process can change a public goods game with punishment from containing mostly defectors to having a majority of cooperative strategies. From the examples given we derive guidelines indicating when the choice of the updating rule can be expected to have an impact on the results of the model.

Molecular field theory for biaxial smectic A liquid crystals.

Thermotropic biaxial nematic phases seem to be rare, but biaxial smectic A phases less so. Here we use molecular field theory to study a simple two-parameter model, with one parameter promoting a biaxial phase and the second promoting smecticity. The theory combines the biaxial Maier-Saupe and McMillan models. We use alternatively the Sonnet-Virga-Durand (SVD) and geometric mean approximations (GMA) to characterize molecular biaxiality by a single parameter. For non-zero smecticity and biaxiality, the model always predicts a ground state biaxial smectic A phase. For a low degree of smectic order, the phase diagram is very rich, predicting uniaxial and biaxial nematic and smectic phases, with the addition of a variety of tricritical and tetracritical points. For higher degrees of smecticity, the region of stability of the biaxial nematic phase is restricted and eventually disappears, yielding to the biaxial smectic phase. Phase diagrams from the two alternative approximations for molecular biaxiality are similar, except inasmuch that SVD allows for a first-order isotropic-biaxial nematic transition, whereas GMA predicts a Landau point separating isotropic and biaxial nematic phases. We speculate that the rarity of thermotropic biaxial nematic phases is partly a consequence of the presence of stabler analogous smectic phases.

Biaxiality-induced magnetic field effects in bent-core nematics: molecular-field and Landau theory.

Nematic liquid crystals composed of bent-core molecules exhibit unusual properties, including an enhanced Cotton-Mouton effect and an increasing isotropic (paranematic)-nematic phase transition temperature as a function of magnetic field. These systems are thought to be good candidate biaxial liquid crystals. Prompted by these experiments, we investigate theoretically the effect of molecular biaxiality on magnetic-field-induced phenomena for nematic liquid crystals, using both molecular field and Landau theory. The geometric mean approximation is used in order to specify the degree of molecular biaxiality using a single parameter. We reproduce experimental field-induced phenomena and predict also an experimentally accessible magnetic critical point. The Cotton-Mouton effect and temperature dependence of the paranematic-nematic phase transition are more pronounced with increased molecular biaxiality. We compare our theoretical approaches and make contact with recent relevant experimental results on bent-core molecular systems.

Two-beam energy exchange in a hybrid photorefractive-flexoelectric liquid-crystal cell.

We develop a semiquantitative theory to describe the experimentally observed energy gain when two light beams intersect in hybrid organic-inorganic photorefractives. These systems consist of a nematic liquid-crystal (LC) layer placed between two photorefractive windows. A periodic space-charge field is induced by the interfering light beams in the photorefractive windows. The field penetrates into the LC, interacting with the nematic director and giving rise to a diffraction grating. LC flexoelectricity is the principal physical mechanism driving the grating structure. Each light beam diffracts from the induced grating, leading to an apparent energy gain and loss within each beam. The LC optics is described in the Bragg regime. In the theory the exponential gain coefficient is a product of a beam interference term, a flexoelectricity term and a space-charge term. The theory has been compared with results of an experimental study on hybrid cells filled with the LC mixture TL 205. Experimentally the energy gain is maximal at much lower grating wave numbers than is predicted by naïve theory. However, if the director reorientation is cubic rather than linear in the space-charge field term, then good agreement between theory and experiment can be achieved using only a single fitting parameter. We provide a semiquantitative argument to justify this nonlinearity in terms of electric-field-induced local phase separation between different components of the liquid crystal.

Theory of surface-potential-mediated photorefractivelike effects in liquid crystals.

We make a phenomenological model of optical two-beam interaction in a model planar liquid crystal cell. The liquid crystal is subject to homeotropic anchoring at the cell walls, is surrounded by thin photosensitive layers, and is subject to a variable potential across the cell. These systems are often known as liquid crystal photorefractive systems. The interference between the two obliquely incident beams causes a time-independent periodic modulation in electric field intensity in the direction transverse to the cell normal. Our model includes this field phenomenologically by supposing an effect on the electric potential at the cell walls. The transverse periodic surface potential causes spatially periodic departures from a pure homeotropic texture. The texture modulation acts as a grating for the incident light. The incident light is both directly transmitted and also subject to diffraction. The lowest order diffracted beams correspond to energy exchange between the beams. We find that the degree of energy exchange can be strongly sensitive to the mean angle of incidence, the angle between the beams, and the imposed potential across the cell. We use the model to speculate about what factors optimize nonlinear optical interaction in liquid crystalline photorefractive systems.

Zigzagging: theoretical insights on climbing strategies.

Human and animal trails on steep hillsides often exhibit dramatic switchbacks and shortcuts. Helbing et al. have recently examined the emergence of human trail systems on flat terrains while Minetti and Margaria established the effect of gradients on human metabolic efficiency. In this paper we use these ideas to develop a semi-quantitative theoretical model of the behaviour of humans moving on a terrain with relief. The model determines the direction of movement by minimising metabolic cost per unit of distance in a desired direction. The structure of the theory resembles the Landau Theory of Phase Transitions, much used in theoretical physics. We find that both hairpin bends (switchbacks) and shortcuts appear as efficient strategies for downhill walkers, while uphill walkers retain switchbacks. For weakly inclined slopes, the best strategy involves walking directly uphill or downhill. For sufficiently steep slopes, however, we find that the best strategy should undergo a transition to a broken symmetry solution corresponding to the switchback trail patterns typical of rugged environments. The critical slope at which this transition takes place should be less steep for uphill and downhill walkers. The theory should be amenable to empirical investigation. Amongst other applications, this model will enable us to generalize the work of previous authors to real landscapes, eventually permitting the reconstruction of ancient patterns of movement in archaeological landscapes.

Molecular model for de Vries type smectic- A -smectic- C phase transition in liquid crystals.

We develop both phenomenological and molecular-statistical theory of smectic- A -smectic- C phase transition with anomalously weak smectic layer contraction. Using a general mean-field molecular model, we demonstrate that a relatively simple interaction potential suffices to describe the transition both in conventional and de Vries type smectics. The theoretical results are in excellent agreement with experimental data. The approach can be used to describe tilting transitions in other soft matter systems.

Waves at the nematic-isotropic interface: thermotropic nematogen-non-nematogen mixtures.

We develop a theory for surface modes at the nematic-isotropic interface in thermotropic nematogen-non-nematogen mixtures. We employ the dynamical generalization of the Landau-de Gennes model for the orientational (nonconserved) order parameter, coupled with the Cahn-Hilliard equation for concentration (conserved parameter), and include hydrodynamic degrees of freedom. The theory uses a generalized form of the Landau-de Gennes free-energy density to include the coupling between the concentration of the non-nematogen fluid and the orientational order parameter. Two representative phase diagrams are shown. The method of matched asymptotic expansions is used to obtain a generalized dispersion relation. Further analysis is made in particular cases. Orientational order parameter relaxation dominates in the short-wavelength limit, while in the long-wavelength limit viscous damping processes become important. There is an intermediate region (depending on the temperature) in which the interaction between conserved parameter dynamics and hydrodynamics is important.

Twist of cholesteric liquid crystal cells: stability of helical structures and anchoring energy effects.

We consider helical configurations of a cholesteric liquid crystal (CLC) sandwiched between two substrates with homogeneous director orientation favored at both confining plates. We study the CLC twist wave number q characterizing the helical structures in relation to the free twisting number q(0) which determines the equilibrium value of CLC pitch P(0) = 2 pi/ q(0) . We investigate the instability mechanism underlying transitions between helical structures with different spiral half-turn numbers. Stability analysis shows that for equal finite anchoring strengths this mechanism can be dominated by in-plane director fluctuations. In this case the metastable helical configurations are separated by the energy barriers and the transitions can be described as the director slippage through these barriers. We extend our analysis to the case of an asymmetric CLC cell in which the anchoring strengths at the two substrates are different. The asymmetry introduces two qualitatively different effects: (a) the intervals of twist wave numbers representing locally stable configurations with adjacent helix half-turn numbers are now separated by instability gaps; and (b) sufficiently large asymmetry, when the difference between azimuthal anchoring extrapolation lengths exceeds the thickness of the cell, will suppress the jumplike behavior of the twist wave number.

Annihilation of edge dislocations in smectic-A liquid crystals.

This paper presents a theoretical study of the annihilation of edge dislocations in the same smectic plane in a bulk smectic-A phase. We use a time-dependent Landau-Ginzburg approach where the smectic ordering is described by the complex order parameter psi( r--> ,t) =eta e(iphi) . This quantity allows both the degree of layering and the position of the layers to be monitored. We are able to follow both precollision and postcollision regimes, and distinguish different early and late behaviors within these regimes. The early precollision regime is driven by changes in the phi ( r--> ) configuration. The relative velocity of the defects is approximately inversely proportional to the interdefect separation distance. In the late precollision regime the symmetry changes within the cores of defects also become influential. Following the defect collision, in the early postcollision stage, bulk layer order is approached exponentially in time. At very late times, however, there seems to be a long-time power-law tail in the order parameter fluctuation relaxation.

Surface modes at the nematic-isotropic interface.

We examine surface modes at the nematic-isotropic interface using the generalized dynamical Landau-de Gennes theory. We assume an isothermal, infinite, unbounded nematic-isotropic system characterized by a scalar order parameter, both phases having the same density and viscosity, respectively. The generalized dispersion relation is obtained and analyzed in particular cases. Order parameter relaxation dominates in the short wavelength limit, while in the long wavelength limit viscous damping becomes important. We study the crossover between the two regimes and estimate the extent of this region for the liquid crystal 8CB.

Light scattering by optically anisotropic scatterers: T-matrix theory for radial and uniform anisotropies.

We extend the T-matrix approach to light scattering by spherical particles to some simple cases in which the scatterers are optically anisotropic. Specifically, we consider cases in which the spherical particles include radially and uniformly anisotropic layers. We find that in both cases the T-matrix theory can be formulated using a modified T-matrix ansatz with suitably defined modes. In a uniformly anisotropic medium we derive these modes by relating the wave packet representation and expansions of electromagnetic field over spherical harmonics. The resulting wave functions are deformed spherical harmonics that represent solutions of the Maxwell equations. We present preliminary results of numerical calculations of the scattering by spherical droplets. We concentrate on cases in which the scattering is due only to the local optical anisotropy within the scatterer. For radial anisotropy we find that nonmonotonic dependence of the scattering cross section on the degree of anisotropy can occur in a regime to which both the Rayleigh and semiclassical theories are inapplicable. For uniform anisotropy the cross section is strongly dependent on the angle between the incident light and the optical axis, and for larger droplets this dependence is nonmonotonic.

Shear-induced structural changes of a smectic-A phase: a computer simulation study.

We have carried out a Monte Carlo simulation of a thin sample of the smectic-A phase of the Gay-Berne mesogen GB(4.4,20.0,1,1) sandwiched between two plates and subject to shear. The smectic layers are perpendicular to the confining plates and are pinned at the boundaries. The thickness of the samples studied ranges from about three to twenty molecules. The layers tilt progressively with increasing shear, but rearrange themselves at a critical shear. At this critical shear the layers melt near the center of the sample and reform with a reduced tilt consistent with the layer pinning at the walls. The pseudodynamics of this process as the smectic layers melt and are reformed have been followed during the simulation. The critical layer tilt at which slippage takes place tends to a constant value for thick samples, but for very thin samples the critical shear tends toward half a smectic layer, with a significantly reduced translational order near the sample center just before the critical shear. The simulation results are consistent with the predictions of the mean field theory of this phenomenon developed by Mottram et al.

Surface depinning of smectic-A edge dislocations.

Using a Landau-de Gennes approach, we model the formation of an edge dislocation in a smectic-A cell initially in the bookshelf structure. The driving force is the mismatch between the layer thickness in a bulk smectic-A liquid crystal and that imposed by confining plates. The core structure of the dislocation is calculated taking into account spatial variations of the smectic translational order parameter. We numerically determine the critical condition for the surface-driven formation and depinning of the dislocation. By exploiting this phenomenon, we show how the value of the positional anchoring strength at the surface can be determined.

Nematic director slippage: role of the angular momentum of light.

We propose a theoretical model of the light-induced director slippage effect. In this effect the bulk director reorientation contributes to the surface director reorientation. It is found that the director and ellipticity profiles, obtained in the geometric optics approximation, are dependent on the ellipticity of the incident light wave. The director distribution is spatially modulated in linearly polarized light but grows monotonically in circularly polarized light. The surface director deviation has been examined, and comparison made with existing experimental data, which then permits the magnitude of the orientational nonlinearity coefficient to be calculated.