Domain coarsening in systems far from equilibrium

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.

Effect of heterogeneous environmental conditions on labyrinthine vegetation patterns

Self-organization is a ubiquitous phenomenon in Nature due to the permanent balance between injection and dissipation of energy. The wavelength selection process is the main issue of pattern formation. Stripe, hexagon, square, and labyrinthine patterns are observed in homogeneous conditions. In systems with heterogeneous conditions, a single wavelength is not the rule. Large-scale self-organization of vegetation in arid environments can be affected by heterogeneities, such as interannual precipitation fluctuations, fire occurrences, topographic variations, grazing, soil depth distribution, and soil-moisture islands. Here, we investigate theoretically the emergence and persistence of vegetation labyrinthine patterns in ecosystems under deterministic heterogeneous conditions. Based on a simple local vegetation model with a space-varying parameter, we show evidence of perfect and imperfect labyrinthine patterns, as well as disordered vegetation self-organization. The intensity level and the correlation of the heterogeneities control the regularity of the labyrinthine self-organization. The phase diagram and the transitions of the labyrinthine morphologies are described with the aid of their global spatial features. We also investigate the local spatial structure of labyrinths. Our theoretical findings qualitatively agree with satellite images data of arid ecosystems that show labyrinthinelike textures without a single wavelength.

Orientational Correlations in Active and Passive Nematic Defects

We investigate the emergence of orientational order among +1/2 disclinations in active nematic liquid crystals. Using a combination of theoretical and experimental methods, we show that +1/2 disclinations have short-range antiferromagnetic alignment, as a consequence of the elastic torques originating from their polar structure. The presence of intermediate -1/2 disclinations, however, turns this interaction from antialigning to aligning at scales that are smaller than the typical distance between like-sign defects. No long-range orientational order is observed. Strikingly, these effects are insensitive to material properties and qualitatively similar to what is found for defects in passive nematic liquid crystals.

Dynamical Crystallites of Active Chiral Particles

One of the intrinsic characteristics of far-from-equilibrium systems is the nonrelaxational nature of the system dynamics, which leads to novel properties that cannot be understood and described by conventional pathways based on thermodynamic potentials. Of particular interest are the formation and evolution of ordered patterns composed of active particles that exhibit collective behavior. Here we examine such a type of nonpotential active system, focusing on effects of coupling and competition between chiral particle self-propulsion and self-spinning. It leads to the transition between three bulk dynamical regimes dominated by collective translative motion, spinning-induced structural arrest, and dynamical frustration. In addition, a persistently dynamical state of self-rotating crystallites is identified as a result of a localized-delocalized transition induced by the crystal-melt interface. The mechanism for the breaking of localized bulk states can also be utilized to achieve self-shearing or self-flow of active crystalline layers.

Coarsening dynamics in the Swift-Hohenberg equation with an external field

We study the Swift-Hohenberg equation (SHE) in the presence of an external field. The application of the field leads to a phase diagram with three phases, i.e., stripe, hexagon, and uniform. We focus on coarsening after a quench from the uniform to stripe or hexagon regions. For stripe patterns, we find that the length scale associated with the order-parameter structure factor has the same growth exponent (≃1/4) as for the SHE with zero field. The growth process is slower in the case of hexagonal patterns, with the effective growth exponent varying between 1/6 and 1/9, depending on the quench parameters. For deep quenches in the hexagonal phase, the growth process stops at late stages when defect boundaries become pinned.

Separation of Elastic and Plastic Timescales in a Phase Field Crystal Model

A consistent small-scale description of plasticity and dislocation motion in a crystalline solid is presented based on the phase field crystal description. By allowing for independent mass motion and lattice distortion, the crystal can maintain elastic equilibrium on the timescale of plastic motion. We show that the singular (incompatible) strains are determined by the phase field crystal density, while the smooth distortions are constrained to satisfy elastic equilibrium. A numerical implementation of the model is presented and used to study a benchmark problem: the motion of an edge dislocation dipole in a triangular lattice. The time dependence of the dipole separation agrees with continuum elasticity with no adjustable parameters.

Adhesion dynamics of confined membranes

We report on the modeling of the dynamics of confined lipid membranes. We derive a thin film model in the lubrication limit which describes an inextensible liquid membrane with bending rigidity confined between two adhesive walls. The resulting equations share similarities with the Swift-Hohenberg model. However, inextensibility is enforced by a time-dependent nonlocal tension. Depending on the excess membrane area available in the system, three different dynamical regimes, denoted as A, B and C, are found from the numerical solution of the model. In regime A, membranes with small excess area form flat adhesion domains and freeze. Such freezing is interpreted by means of an effective model for curvature-driven domain wall motion. The nonlocal membrane tension tends to a negative value corresponding to the linear stability threshold of flat domain walls in the Swift-Hohenberg equation. In regime B, membranes with intermediate excess areas exhibit endless coarsening with coexistence of flat adhesion domains and wrinkle domains. The tension tends to the nonlinear stability threshold of flat domain walls in the Swift-Hohenberg equation. The fraction of the system covered by the wrinkle phase increases linearly with the excess area in regime B. In regime C, membranes with large excess area are completely covered by a frozen labyrinthine pattern of wrinkles. As the excess area is increased, the tension increases and the wavelength of the wrinkles decreases. For large membrane area, there is a crossover to a regime where the extrema of the wrinkles are in contact with the walls. In all regimes after an initial transient, robust localised structures form, leading to an exact conservation of the number of adhesion domains.

Scaling properties of ageing orientation fluctuations in stripe phases

We investigate the non-equilibrium dynamics of an ordered stripe-forming system free of topological defects. In particular, we study the ageing and the coarsening of orientation fluctuations parallel and perpendicular to the stripes via computer simulations based on a minimal phase-field model (model B with Coulomb interactions). Under the influence of noise, the stripe orientation field develops fluctuations parallel to the stripes, with the dominant modulation length λ*∥ increasing with time t as λ*∥ ∼ t1/4 and the correlation length perpendicular to the stripes ξ⊥θ increasing as ξ⊥θ ∼ t1/2. We explain these anisotropic coarsening dynamics with an analytic theory based on the linear elastic model for stripe displacements first introduced by Landau and Peierls. We thus obtain the scaling forms and the scaling exponents characterizing the correlation functions and the structure factor of the stripe orientation field. Our results reveal how the coarsening of orientation fluctuations prevents a periodically modulated phase free of topological defects from reaching equilibrium.

Strip-Pattern-Spheres Self-Assembled from Polypeptide-Based Polymer Mixtures: Structure and Defect Features

We found that poly(γ-benzyl-L-glutamate)-block-poly(ethylene glycol) (PBLG-b-PEG) rod-coil block copolymers and polystyrene (PS) homopolymers can cooperatively self-assemble into nano-spheres with striped patterns on their surfaces (strip-pattern-spheres) in aqueous solution. With assistance of dissipative particle dynamics simulation, it is discovered that the PS homopolymers form a spherical template core and the PBLG-b-PEG block copolymers assemble into striped patterns on the spherical surface. The hydrophobic PBLG rods are packed orderly in the strips, while the hydrophilic PEG blocks stabilize the strip-pattern-spheres in solution. Defects such as dislocations and disclinations can be observed in the striped patterns. Self-assembling temperature and sphere radius are found to affect defect densities in the striped patterns. A possible mechanism is proposed to illustrate how PBLG-b-PEG and PS cooperatively self-assemble into hierarchical spheres with striped patterns on surfaces.

Hexagonal phase ordering in strongly segregated copolymer films

Strongly segregated copolymer mixtures with uneven composition ratio can form hexagonally ordered thin films. A simplified model describing the size and position of micellelike clusters is derived, allowing for investigation of much larger domain sizes than in previous studies. Simulations of this model are performed to study the generation of large scale order and evolution of pattern defects. We find three temporal regimes exhibiting different scaling laws for orientational correlation length and defect number. In the early stage, topological defects are rapidly eliminated by pairwise annihilation. A slower intermediate stage is characterized by the migration of grain boundaries and the elimination of small grains. In the final stage, grain boundaries become pinned and the evolution halts. A scaling law for defect interaction is proposed which is consistent with the crossover between the first and second stages.

Rotation-limited growth of three-dimensional body-centered-cubic crystals

According to classical grain growth laws, grain growth is driven by the minimization of surface energy and will continue until a single grain prevails. These laws do not take into account the lattice anisotropy and the details of the microscopic rearrangement of mass between grains. Here we consider coarsening of body-centered-cubic polycrystalline materials in three dimensions using the phase field crystal model. We observe, as a function of the quenching depth, a crossover between a state where grain rotation halts and the growth stagnates and a state where grains coarsen rapidly by coalescence through rotation and alignment of the lattices of neighboring grains. We show that the grain rotation per volume change of a grain follows a power law with an exponent of -1.25. The scaling exponent is consistent with theoretical considerations based on the conservation of dislocations.

Coarsening of stripe patterns: variations with quench depth and scaling

The coarsening of stripe patterns when the system is evolved from random initial states is studied by varying the quench depth ε, which is a measure of distance from the transition point of the stripe phase. The dynamics of the growth of stripe order, which is characterized by two length scales, depends on the quench depth. The growth exponents of the two length scales vary continuously with ε. The decay exponents for free energy, stripe curvature, and densities of defects like grain boundaries and dislocations also show similar variation. This implies a breakdown of the standard picture of nonequilibrium dynamical scaling. In order to understand the variations with ε we propose an additional scaling with a length scale dependent on ε. The main contribution to this length scale comes from the "pinning potential," which is unique to systems where the order parameter is spatially periodic. The periodic order parameter gives rise to an ε-dependent potential, which can pin defects like grain boundaries, dislocations, etc. This additional scaling provides a compact description of variations of growth exponents with quench depth in terms of just one exponent for each of the length scales. The relaxation of free energy, stripe curvature, and the defect densities have also been related to these length scales. The study is done at zero temperature using Swift-Hohenberg equation in two dimensions.

Stripe patterns: role of initial state and boundary conditions

This paper presents results on stripe patterns by numerical solution of the Swift-Hohenberg equation. The focus is on the role of initial state and boundary conditions. We choose initial states which generate simple defect configurations and study their evolution. Various classes of defects are identified and their motion and relaxation is studied numerically. We first study the dynamics of a straight front and present a comparison of numerical results with some analytical results. We then study the domain-wall dynamics in configurations containing two and three domains and identify some mechanisms of their relaxation. Rates of domain-wall relaxation depend on several features like incommensuration, dislocations and orientations in neighboring domains, in addition to the curvature of the walls. For a generic class of domain walls the relaxation process has an intrinsic frustration which leads to generation of dislocations. This process also generates stripe curvature thereby making relaxation nonmonotonic. We have also generated some other topological defects and studied their evolution and the effect of boundary conditions on their stability.

Unified theoretical framework for polycrystalline pattern evolution

The rate of curvature-driven grain growth in polycrystalline materials is well known to be limited by interface dissipation. We show analytically and by simulations that, for systems forming modulated phases or nonequilibrium patterns with crystal ordering, growth is limited by bulk dissipation associated with lattice translation, which dramatically slows down grain coarsening. We also show that bulk dissipation is reduced by thermal noise and that this reduction leads to faster coarsening behavior dominated by interface dissipation for a high Peierls-Nabarro barrier to dislocation motion and high noise. Those results provide a unified theoretical framework for understanding and modeling polycrystalline pattern evolution in diverse systems over a broad range of length and time scales.

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.

Phase instability and coarsening in two dimensions

Instabilities and pattern formation is the rule in nonequilibrium systems. Selection of a persistent length scale or coarsening (increase in the length scale with time) are the two major alternatives. When and under which conditions one dynamics prevails over the other is a long-standing problem, particularly beyond one dimension. It is shown that the challenge can be defied in two dimensions, using the concept of phase diffusion equation. We find that coarsening is related to the lambda dependence of a suitable phase diffusion coefficient, D11(lambda) , depending on lattice symmetry and conservation laws. These results are exemplified analytically on prototypical nonlinear equations.

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.

Domain chaos puzzle and the calculation of the structure factor and its half-width

The disagreement of the scaling of the correlation length xi between experiment and the Ginzburg-Landau (GL) model for domain chaos was resolved. The Swift-Hohenberg (SH) domain chaos model was integrated numerically to acquire test images to study the effect of a finite image size on the extraction of from the structure factor (SF). The finite image size had a significant effect on the SF determined with the Fourier-transform (FT) method. The maximum entropy method (MEM) was able to overcome this finite image-size problem and produced fairly accurate SFs for the relatively small image sizes provided by experiments. Correlation lengths often have been determined from the second moment of the SF of chaotic patterns because the functional form of the SF is not known. Integration of several test functions provided analytic results indicating that this may not be a reliable method of extracting xi. For both a Gaussian and a squared SH form, the correlation length xi[triple bond]1/sigma, determined from the variance sigma2 of the SF, has the same dependence on the control parameter epsilon as the length xi contained explicitly in the functional forms. However, for the SH and the Lorentzian forms we find xi approximately xi (1/2). Results for xi determined from new experimental data by fitting the functional forms directly to the experimental SF yielded xi approximately epsilon(-v) with v approximately equal to 1/4 for all four functions in the case of the FT method, but v approximately equal to 1/2, in agreement with the GL prediction, in the case of the MEM. Over a wide range of epsilon and wave number k, the experimental SFs collapsed onto a unique curve when appropriately scaled by xi.

Statistical characterizations of spatiotemporal patterns generated in the Swift-Hohenberg model

Two families of statistical measures are used for quantitative characterization of nonequilibrium patterns and their evolution. The first quantifies the disorder in labyrinthine patterns, and captures features like the domain size, defect density, variations in wave number, etc. The second class of characteristics can be used to quantify the disorder in more general nonequilibrium structures, including those observed during domain growth. The presence of distinct stages of relaxation in spatiotemporal dynamics under the Swift-Hohenberg equation is analyzed using both classes of measures.

Self-assembled patterns and strain-induced instabilities for modulated systems

The self-assembled domain patterns of modulated systems are characteristic of a wide variety of chemical and physical systems, and are the result of competing interactions. From a technological point of view, there is considerable interest in these domain patterns, as they form suitable templates for the fabrication of nanostructures. We have analyzed the domains and instabilities that form in modulated systems, and show that a large variety of patterns--based on long-lived metastable or glassy states--may be formed as a compromise between the required equilibrium modulation period and the strain present in the system. The strain results from topologically constrained trajectories in phase space, that effectively preclude the equilibrium configuration.

Hydrodynamic coarsening in striped pattern formation with a conservation law

We observed in numerical simulations that the interaction of striped-pattern-forming instability and a neutrally stable zero mode induces patterns of domains of upflow hexagons coexisting with domains of downflow hexagons. They appear only when hydrodynamic flow is present.

New and exotic self-organized patterns for modulated nanoscale systems

The self-assembled domain patterns of modulated systems are the result of competing short-range attractive and long-range repulsive interactions found in diverse physical and chemical systems. From an application point of view, there is considerable interest in these domain patterns, as they form templates suitable for the fabrication of nanostructures. In this work we have generated a variety of new and exotic patterns, which represent either metastable or glassy states. These patterns arise as a compromise between the required equilibrium modulation period and the strain resulting from topologically constrained trajectories in phase space that effectively preclude the equilibrium configuration.

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.

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.

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.

Regular patterns generated by self-organization of ammonium-modified polymer nanospheres

Macroscopic regular stripe-crack patterns have been observed in the course of drying the aqueous suspensions of ammonium-modified polymer nanospheres. These forms emerged because the evaporation of dispersed water and self-assembly of nanospheres originates shrinkage during drying the aqueous suspensions. The drying condition plays an important role as well as the nature of the ammonium-modified polymer nanospheres for the stripe-crack pattern formation. By means of the vertical deposition method, directional stripe-crack patterns have been achieved in the macroscopic scale. Surprisingly, we have still noted an interesting secondary stripe pattern occurred spontaneously on the stripes.

Model for striped growth

We introduce a model for describing the defected growth of striped patterns. This model, while roughly related to the Swift-Hohenberg model, generates a quite different mixture of defects during phase ordering. We find two characteristic lengths in the system: the scaling length L(t), and the average width of the domain walls. The growth law exponent is larger than the value of 1/2 found in typical point defect systems.

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.

Defect structures in the growth kinetics of the Swift-Hohenberg model

The growth of striped order resulting from a quench of the two-dimensional Swift-Hohenberg model is studied in the regime of a small control parameter and quenches to zero temperature. We introduce an algorithm for finding and identifying the disordering defects (dislocations, disclinations, and grain boundaries) at a given time. We can track their trajectories separately. We find that the coarsening of the defects and lowering of the effective free energy in the system are governed by a growth law L(t) approximately t(x) with an exponent x near 1/3. We obtain scaling for the correlations of the nematic order parameter with the same growth law. The scaling for the order parameter structure factor is governed, as found by others, by a growth law with an exponent smaller than x and near to 1/4. By comparing two systems with different sizes, we clarify the finite-size effect. We find that the system has a very low density of disclinations compared to that for dislocations and fraction of points in grain boundaries. We also measure the speed distributions of the defects at different times and find that they all have power-law tails and the average speed decreases as a power law.

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.

Spiral-defect chaos: Swift-Hohenberg model versus Boussinesq equations

Spiral-defect chaos (SDC) in Rayleigh-Bénard convection is a well-established spatio-temporal complex pattern, which competes with stationary rolls near the onset of convection. The characteristic properties of SDC are accurately described on the basis of the standard three-dimensional Boussinesq equations. As a much simpler and attractive two-dimensional model for SDC generalized Swift-Hohenberg (SH) equations have been extensively used in the literature from the early beginning. Here, we show that the description of SDC by SH models has to be considered with care, especially regarding its long-time dynamics. For parameters used in previous SH simulations, SDC occurs only as a transient in contrast to the experiments and the rigorous solutions of the Boussinesq equations. The small-scale structure of the vorticity field at the spiral cores, which might be crucial for persistent SDC, is presumably not perfectly captured in the SH model.

Stable droplets and growth laws close to the modulational instability of a domain wall

We consider the curvature driven dynamics of a domain wall separating two equivalent states in systems displaying a modulational instability of a flat front. An amplitude equation for the dynamics of the curvature close to the bifurcation point from growing to shrinking circular droplets is derived. We predict the existence of stable droplets with a radius R that diverges at the bifurcation point, where a curvature driven growth law R(t) approximately t(1/4) is obtained. Our general analytical predictions, which are valid for a wide variety of systems including models of nonlinear optical cavities and reaction-diffusion systems, are illustrated in the parametrically driven complex Ginzburg-Landau equation.

Domain coarsening of stripe patterns close to onset

We study domain coarsening of two-dimensional stripe patterns by numerically solving the Swift-Hohenberg model of Rayleigh-Bénard convection. Near the bifurcation threshold, the evolution of disordered configurations is dominated by grain-boundary motion through a background of largely immobile curved stripes. A numerical study of the distribution of local stripe curvatures, of the structure factor of the order parameter, and a finite size scaling analysis of the grain-boundary perimeter, suggest that the linear scale of the structure grows as a power law of time t(1/z), with z=3. We interpret theoretically the exponent z=3 from the law of grain-boundary motion.

Domain coarsening in electroconvection

We report on experimental measurements of the growth of regular domains evolving from an irregular pattern in electroconvection. The late-time growth of the domains is consistent with the size of the domains scaling as t(n). We use two isotropic measurements of the domain size: the structure factor and the domain wall length. Measurements using the structure factor are consistent with t(1/5) growth. Measurements using the domain wall length are consistent with t(1/4) growth. One source of this discrepancy is the fact that the distribution of local wave numbers is approximately independent of the domain size. In addition, we measure the anisotropy of the growing domains.

Grain-boundary motion in layered phases

We study the motion of a grain boundary that separates two sets of mutually perpendicular rolls in Rayleigh-Bénard convection above onset. The problem is treated either analytically from the corresponding amplitude equations, or numerically by solving the Swift-Hohenberg equation. We find that if the rolls are curved by a slow transversal modulation, a net translation of the boundary follows. We show analytically that although this motion is a nonlinear effect, it occurs in a time scale much shorter than that of the linear relaxation of the curved rolls. The total distance traveled by the boundary scales as epsilon(-1/2), where epsilon is the reduced Rayleigh number. We obtain analytical expressions for the relaxation rate of the modulation and for the time-dependent traveling velocity of the boundary, and especially their dependence on wave number. The results agree well with direct numerical solutions of the Swift-Hohenberg equation. We finally discuss the implications of our results on the coarsening rate of an ensemble of differently oriented domains in which grain-boundary motion through curved rolls is the dominant coarsening mechanism.

Spiral patterns in oscillated granular layers

Cell-filling spiral patterns are observed in a vertically oscillated layer of granular material when the oscillation amplitude is suddenly increased from below the onset of pattern formation into the region where stripe patterns appear for quasistatic increases in amplitude. These spirals are transients and decay to stripe patterns with defects. A transient spiral defect chaos state is also observed. We describe the behavior of the spirals, and the way in which they form and decay. Our results are compared with those for similar spiral patterns in Rayleigh-Bénard convection in fluids.

Mechanisms of ordering in striped patterns

We have studied the ordering dynamics of the striped patterns of a single layer of cylindrical block copolymer microdomains in a thin film. By tracking disclinations during annealing with time-lapse atomic force microscopy, we observe a dominant mechanism of disclination annihilation involving three or four disclinations (quadrupoles). Pairwise disclination annihilation events are suppressed as a result of the topological constraints in this system. The kinetic scaling laws with exponents observed here are consistent with topologically allowed annihilation events involving multiple disclinations. The results provide insight into two-dimensional pattern formation and may lead to the successful application of block copolymer lithography.

Ordering process in quenched block copolymers at low temperatures

We have studied domain growth of symmetric diblock copolymers undergoing microphase separation at low temperatures. We introduce a phenomenological nonlinear diffusion model with order-parameter-dependent mobility. Performing two-dimensional simulations, we find that the time-dependent scattering function exhibits dynamical scaling with a logarithmic growth law in the strong segregation limit where surface diffusion is the relevant mechanism for coarsening.

Emergence of order in textured patterns

A characterization of textured patterns, referred to as the disorder function delta(beta), is used to study properties of patterns generated in the Swift-Hohenberg equation (SHE). It is shown to be an intensive, configuration-independent measure. The evolution of random initial states under the SHE exhibits two stages of relaxation. The initial phase, where local striped domains emerge from a noisy background, is quantified by a power-law decay delta(beta) approximately t-(1/2)beta. Beyond a sharp transition, a slower power-law decay of delta(beta), which corresponds to the coarsening of striped domains, is observed. The transition between the phases advances as the system is driven further from the onset of patterns, and suitable scaling of time and delta(beta) leads to the collapse of distinct curves. The decay of delta(beta) during the initial phase remains unchanged when nonvariational terms are added to the underlying equations, suggesting the possibility of observing it in experimental systems. In contrast, the rate of relaxation during domain coarsening increases with the coefficient of the nonvariational term.