Domain coarsening in electroconvection

Active smectics on a sphere

The dynamics of active smectic liquid crystals confined on a spherical surface is explored through an active phase field crystal model. Starting from an initially randomly perturbed isotropic phase, several types of topological defects are spontaneously formed, and then annihilate during a coarsening process until a steady state is achieved. The coarsening process is highly complex involving several scaling laws of defect densities as a function of time where different dynamical exponents can be identified. In general the exponent for the final stage towards the steady state is significantly larger than that in the passive and in the planar case, i.e. the coarsening is getting accelerated both by activity and by the topological and geometrical properties of the sphere. A defect type characteristic for this active system is a rotating spiral of evolving smectic layering lines. On a sphere this defect type also determines the steady state. Our results can in principle be confirmed by dense systems of synthetic or biological active particles.

Dislocations and metastable chevrons in the electroconvective inplane normal roll state of a bent core nematic liquid crystal

We report experimental results on the formation, dynamics, and annihilation of edge dislocations of opposite topological charge in the electroconvective inplane vortex state of a bent core nematic liquid crystal. The approach of paired, oppositely charged defects toward each other is a two-step process. Near constant velocity at large separation and accelerated motion close to annihilation are found, as in the case of nematic rolls belonging to standard electroconvection. Periodic arrays of dislocations of alternating polarity form upon a sudden, strong elevation of the control parameter. Chevron structures that appear between undulatory defect chains are metastable, and their decay with time is accompanied by an exponential reduction in the dislocation density. The initial periodicity of defect chains also drops exponentially with increasing field.

Effect of noise on ordering of hexagonal grains in a phase-field-crystal model

We present a quantitative analysis of grain morphology of self-organizing hexagonal patterns based on the phase-field crystal model to examine the effect of stochastic noise on grain coarsening. We show that the grain size increases with increasing noise strength, resulting in enhanced hexagonal orientation due to noise up to some critical noise level above which the system becomes disordered.

Coarsening in potential and nonpotential models of oblique stripe patterns

We study the coarsening of two-dimensional oblique stripe patterns by numerically solving potential and nonpotential anisotropic Swift-Hohenberg equations. Close to onset, all models exhibit isotropic coarsening with a single characteristic length scale growing in time as t1/2. Further from onset, the characteristic lengths along the preferred directions x and ŷ grow with different exponents, close to 1/3 and 1/2, respectively. In this regime, one-dimensional dynamical scaling relations hold. We draw an analogy between this problem and model A in a stationary, modulated external field. For deep quenches, nonpotential effects produce a complicated dislocation dynamics that can lead to either arrested or faster-than-power-law growth, depending on the model considered. In the arrested case, small isolated domains shrink down to a finite size and fail to disappear. A comparison with available experimental results for electroconvection in nematics is presented.

Grain boundary dynamics in stripe phases of nonpotential systems

We describe numerical solutions of two nonpotential models of pattern formation in nonequilibrium systems to address the motion and decay of grain boundaries separating domains of stripe configurations of different orientations. We first address wave-number selection because of the boundary, and possible decay modes when the periodicity of the stripe phases is different from the selected wave number for a stationary boundary. We discuss several decay modes including long wavelength undulations of the moving boundary, as well as the formation of localized defects and their subsequent motion. We find three different regimes as a function of the distance to the stripe phase threshold and initial wave number, and then correlate these findings with domain morphology during domain coarsening in a large aspect ratio configuration.

Impact of noise on domain growth in electroconvection

The growth and ordering of striped domains has recently received renewed attention due in part to experimental studies in diblock copolymers and electroconvection. One surprising result has been the relatively slow dynamics associated with the growth of striped domains. One potential source of the slow dynamics is the pinning of defects in the periodic potential of the stripes. Of interest is whether or not external noise will have a significant impact on the domain ordering, perhaps by reducing the pinning and increasing the rate of ordering. In contrast, we present experiments using electroconvection in which we show that a particular type of external noise decreases the rate of domain ordering.

Early time evolution of Freédericksz patterns generated from states of electroconvection

We report on the early time ordering in a nematic liquid crystal subjected to a sudden change in an external ac electric field. We compare time evolution for two different initial states of electroconvection. Electroconvection is a highly driven state of a nematic liquid crystal involving convective motion of the fluid and periodic variations of the molecular alignment. By suddenly changing either the voltage or the frequency of the applied ac field, the system is brought to the same thermodynamic conditions. The time ordering of the system is characterized by the evolution of features of the power spectrum, including the average wave number, total power, and shape of the power spectrum. We observe that ordering of the system occurs faster after a sudden change in frequency than it does after a sudden change in voltage.

Coarsening of topological defects in oscillating systems with quenched disorder

We use large scale simulations to study interacting particles in two dimensions in the presence of both an ac drive and quenched disorder. As a function of ac amplitude, there is a crossover from a low drive regime where the colloid positions are highly disordered to a higher ac drive regime where the system dynamically reorders. We examine the coarsening of topological defects formed when the system is quenched from a disordered low ac amplitude state to a high ac amplitude state. When the quench is performed close to the disorder-order crossover, the defect density decays with time as a power law with alpha = 1/4 to 1/3. For deep quenches, in which the ac drive is increased to high values such that the dynamical shaking temperature is strongly reduced, we observe a logarithmic decay of the defect density into a grain boundary dominated state. We find a similar logarithmic decay of defect density in systems containing no pinning. We specifically demonstrate these effects for vortices in thin film superconductors, and discuss implications for dynamical reordering transition studies in these systems.

Hierarchical structure description of spatiotemporal chaos

We develop a hierarchical structure (HS) analysis for quantitative description of statistical states of spatially extended systems. Examples discussed here include an experimental reaction-diffusion system with Belousov-Zhabotinsky kinetics, the two-dimensional complex Ginzburg-Landau equation, and the modified FitzHugh-Nagumon equation, which all show complex dynamics of spirals and defects. We demonstrate that the spatial-temporal fluctuation fields in the above-mentioned systems all display the HS similarity property originally proposed for the study of fully developed turbulence [Phys. Rev. Lett. 72, 336 (1994)]]. The derived values of a HS parameter beta from experimental and numerical data in various physical regimes exhibit consistent trends and characterize the degree of turbulence in the systems near the transition, and the degree of heterogeneity of multiple disorders far from the transition. It is suggested that the HS analysis offers a useful quantitative description for the complex dynamics of two-dimensional spatiotemporal patterns.

Dislocation dynamics in Rayleigh-Bénard convection

Theoretical results on the dynamics of dislocations in Rayleigh-Bénard convection are reported both for a Swift-Hohenberg model and the Oberbeck-Boussinesq equations. For intermediate Prandtl numbers the motion of dislocations is found to be driven by the superposition of two independent contributions: (i) the Peach-Koehler force and (ii) an advection force on the dislocation core by its self-generated mean flow. Their competition allows to explain the experimentally observed bound dislocation pairs.

Rayleigh-Bénard convection in large-aspect-ratio domains

The coarsening and wave number selection of striped states growing from random initial conditions are studied in a nonrelaxational, spatially extended, and far-from-equilibrium system by performing large-scale numerical simulations of Rayleigh-Bénard convection in a large-aspect-ratio cylindrical domain with experimentally realistic boundaries. We find evidence that various measures of the coarsening dynamics scale in time with different power-law exponents, indicating that multiple length scales are required in describing the time dependent pattern evolution. The translational correlation length scales with time as t0.12, the orientational correlation length scales as t0.54, and the density of defects scale as t(-0.45). The final pattern evolves toward the wave number where isolated dislocations become motionless, suggesting a possible wave number selection mechanism for large-aspect-ratio convection.

Dynamics of laser-induced electroconvection pulses

We first report that, for planar nematic 4-methoxy-benzilidene-4-butylaniline (MBBA), the electroconvection threshold voltage has a nonmonotonic temperature dependence, with a well-defined minimum, and a slope of about -0.12 V/degrees C near room temperature at 70 Hz. Motivated by this observation, we have designed an experiment in which a weak continuous-wave absorbed laser beam with a diameter comparable to the pattern wavelength generates a locally supercritical region, or pulse, in dye-doped MBBA. Working 10-20 % below the laser-free threshold voltage, we observe a steady-state pulse shaped as an ellipse with the semimajor axis oriented parallel to the nematic director, with a typical size of several wavelengths. The pulse is robust, persisting even when spatially extended rolls develop in the surrounding region, and displays rolls that counterpropagate along the director at frequencies of tenths of Hz, with the rolls on the left (right) side of the ellipse moving to the right (left). Systematic measurements of the sample-voltage dependence of the pulse amplitude, spatial extent, and frequency show a saturation or decrease when the control parameter (evaluated at the center of the pulse) approaches approximately 0.3. We propose that the model for these pulses should be based on the theory of control-parameter ramps, supplemented with new terms to account for the advection of heat away from the pulse when the surrounding state becomes linearly unstable. The advection creates a negative feedback between the pulse size and the efficiency of heat transport, which we argue is responsible for the attenuation of the pulse at larger control-parameter values.

Numerical study of domain coarsening in anisotropic stripe patterns

We study the coarsening of two-dimensional smectic polycrystals characterized by grains of oblique stripes with only two possible orientations. For this purpose, an anisotropic Swift-Hohenberg equation is solved. For quenches close enough to the onset of stripe formation, the average domain size increases with time as t(1/2). Further from onset, anisotropic pinning forces similar to Peierls stresses in solid crystals slow down defects, and growth becomes anisotropic. In a wide range of quench depths, dislocation arrays remain mobile and dislocation density roughly decays as t(-1/3), while chevron boundaries are totally pinned. We discuss some agreements and disagreements found with recent experimental results on the coarsening of anisotropic electroconvection patterns.

Dislocation dynamics in an anisotropic stripe pattern

The dynamics of dislocations confined to grain boundaries in a striped system are studied using electroconvection in the nematic liquid crystal N4. In electroconvection, a striped pattern of convection rolls forms for sufficiently high driving voltages. We consider the case of a rapid change in the voltage that takes the system from a uniform state to a state consisting of striped domains with two different wave vectors. The domains are separated by domain walls along one axis and a grain boundary of dislocations in the perpendicular direction. The pattern evolves through dislocation motion parallel to the domain walls. We report on features of the dislocation dynamics. The kinetics of the domain motion is quantified using three measures: dislocation density, average domain wall length, and total domain wall length per area. All three quantities exhibit behavior consistent with power-law evolution in time, with the defect density decaying as t(-1/3), the average domain wall length growing as t(1/3), and the total domain wall length decaying as t(-1/5). The two different exponents are indicative of the anisotropic growth of domains in the system.

Dependence of domain wall dynamics on background wave number

We report on the growth of domains of standing waves in electroconvection in a nematic liquid crystal, focusing on the evolution of domain walls. An ac voltage is applied to the system, forming an initial state that consists of traveling striped patterns with two different orientations, zig and zag rolls. The standing waves are generated by suddenly applying a periodic modulation of the amplitude of the applied voltage that is approximately resonant with the traveling frequency of the pattern. By varying the modulation frequency, we are able to vary the steady-state, average wave number. We characterize the evolution of the domain walls as a function of the average background wave number by measuring the total area and length of domain walls present in the system as a function of time. We find that as the background wave number is varied away from the "natural" wave number for the pattern, the evolution of the domain walls occurs at a faster rate.

Defect formation in the Swift-Hohenberg equation

We study numerically and analytically the dynamics of defect formation during a finite-time quench of the two-dimensional Swift-Hohenberg (SH) model of Rayleigh-Bénard convection. We find that the Kibble-Zurek picture of defect formation can be applied to describe the density of defects produced during the quench. Our study reveals the relevance of two factors: the effect of local variations of the striped patterns within defect-free domains and the presence of both pointlike and extended defects. Taking into account these two aspects we are able to identify the characteristic length scale selected during the quench and to relate it to the density of defects. We discuss possible consequences of our study for the analysis of the coarsening process of the SH model.

Dynamics of pattern coarsening in a two-dimensional smectic system

We have followed the coarsening dynamics of a single layer of cylindrical block copolymer microdomains in a thin film. This system has the symmetry of a two-dimensional smectic. The orientational correlation length of the microdomains was measured by scanning electron microscopy and found to grow with the average spacing between +/-1/2 disclinations, following a power law xi2(t) approximately t(1/4). By tracking disclinations during annealing with time-lapse atomic force microscopy, we observe dominant mechanisms of disclination annihilation involving tripoles and quadrupoles (three and four disclinations, respectively). We describe how annihilation events involving multiple disclinations result in similarly reduced kinetic exponents as observed here. These results map onto a wide variety of physical systems that exhibit similarly striped patterns.

Coarsening dynamics of striped patterns in thin granular layers under vertical vibration

The process of pattern formation in granular layers was experimentally studied. Ten layers of granular materials inside a vacuum container were placed under a vertical vibration of A sin2pi f t. Control parameters were the dimensionless acceleration Gamma = A(2pi f)(2)/g and vibration frequency f. When the system was quenched from a flat pattern state to a striped pattern state by instantly increasing Gamma, there were more than 10(4) periods before a full steady striped pattern appeared. This nonequilibrium and nonsteady process showed dynamic scaling behavior. The growth exponent of the characteristic length scale of the ordered domain was 0.25, which agrees with that of the Swift-Hohenberg system.

Hydrodynamic interactions in ordering process of two-dimensional quenched block copolymers

The hydrodynamic coarsening of microphase separation in two-dimensional diblock copolymers is studied using numerical simulations. Results for symmetric and asymmetric block copolymers are compared. In contrast to the formation of the hexagonal phase where hydrodynamic flow appears not to be effective in enhancing domain coarsening, the late-time evolution of the lamellar phase proceeds faster, thus leading to a different power-law scaling with the addition of coupling of the velocity field to the order parameter.

Grain boundary pinning and glassy dynamics in stripe phases

We study numerically and analytically the coarsening of stripe phases in two spatial dimensions, and show that transient configurations do not achieve long ranged orientational order but rather evolve into glassy configurations with very slow dynamics. In the absence of thermal fluctuations, defects such as grain boundaries become pinned in an effective periodic potential that is induced by the underlying periodicity of the stripe pattern itself. Pinning arises without quenched disorder from the nonadiabatic coupling between the slowly varying envelope of the order parameter around a defect, and its fast variation over the stripe wavelength. The characteristic size of ordered domains asymptotes to a finite value R(g) approximately lambda(0)epsilon(-1/2)exp(absolute value of a/square root of epsilon), where epsilon<<1 is the dimensionless distance away from threshold, lambda(0) the stripe wavelength, and a a constant of order unity. Random fluctuations allow defect motion to resume until a new characteristic scale is reached, function of the intensity of the fluctuations. We finally discuss the relationship between defect pinning and the coarsening laws obtained in the intermediate time regime.