Dynamic scaling and quasiordered states in the two-dimensional Swift-Hohenberg equation

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.

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.

Pathways to equilibrium orientation fluctuations in finite stripe-forming systems

Small-angle orientation fluctuations in ordered stripe-forming systems free of topological defects can exhibit aging and anisotropic growth of two length scales. In infinitely extended systems, the stripe orientation field develops a dominant modulation length λ_{∥}^{*}(t) in the direction parallel to the stripes, which increases with time t as λ_{∥}^{*}(t)∼t^{1/4}. Simultaneously, the orientation correlation length ξ_{⊥}(t) in the direction perpendicular to the stripes increases as ξ_{⊥}(t)∼t^{1/2} [Riesch et al., Interface Focus 7, 20160146 (2017)2042-889810.1098/rsfs.2016.0146]. Here we show that finite systems of size L_{⊥}×L_{∥} with periodic boundary conditions reach equilibrium when the dominant modulation length λ_{∥}^{*}(t) reaches the system size L_{∥} in the stripe direction. The equilibration time τ_{eq}^{∥} is solely determined by L_{∥}, with τ_{eq}^{∥}∼L_{∥}^{4}. In systems with L_{⊥}<L_{∥}^{2}/2πλ_{p}, where λ_{p} is the undulation penetration length, the initial aging and coarsening dynamics changes at the crossover time τ_{C}^{⊥}∼L_{⊥}^{2} to an aging and coarsening dynamics described by the one-dimensional Mullins-Herring equation, before reaching equilibrium at τ_{∥}^{eq}. Our work reveals the two pathways to equilibrium in stripe phases with periodic boundary conditions, the finite-size scaling behavior of equilibrium orientation fluctuations, and the characteristic exponents associated with the influence of a finite system size.

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.

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.

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.

Nanostructures with long-range order in monolayer self-assembly

A key ingredient for continued expansion of nanotechnologies is the ability to create perfectly ordered arrays on a small scale with both site and size control. Self-assembly-i.e., the spontaneous formation of nanostructures-is a highly promising alternative to traditional fabrication methods. However, efforts to obtain perfect long-range order via self-assembly have been frustrated in practice as ensuing patterns contain defects. We use an idea based on the fundamental physics of pattern formation to introduce a strategy to consistently obtain perfect patterns.

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.

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.

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.

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.

Nature of slow dynamics in a minimal model of frustration-limited domains

We present simulation results for the dynamics of a schematic model based on the frustration-limited domain picture of glass-forming liquids. These results are compared with approximate theoretical predictions analogous to those commonly used for supercooled liquid dynamics. Although model relaxation times increase by several orders of magnitude in a non-Arrhenius manner as a microphase separation transition is approached, the slow relaxation is in many ways dissimilar to that of a liquid. In particular, structural relaxation is nearly exponential in time at each wave vector, indicating that the mode-coupling effects dominating liquid relaxation are comparatively weak within this model. Relaxation properties of the model are instead well reproduced by the simplest dynamical extension of a static Hartree approximation. This approach is qualitatively accurate even for temperatures at which the mode-coupling approximation predicts loss of ergodicity. These results suggest that the thermodynamically disordered phase of such a minimal model poorly caricatures the slow dynamics of a liquid near its glass transition.

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.

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.

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.

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.

Kinetics of ordering in fluctuation-driven first-order transitions: simulation and theory

Many systems involving competing interactions or interactions that compete with constraints are well described by a model first introduced by Brazovskii [Zh. Eksp. Teor. Fiz. 68, 175 (1975) [Sov. Phys. JETP 41, 85 (1975)]]. The hallmark of this model is that the fluctuation spectrum is isotropic and has a maximum at a nonzero wave vector represented by the surface of a d-dimensional hypersphere. It was shown by Brazovskii that the fluctuations change the free energy structure from a straight phi(4) to a straight phi(6) form with the disordered state metastable for all quench depths. The transition from the disordered phase to the periodic lamellar structure changes from second order to first order and suggests that the dynamics is governed by nucleation. Using numerical simulations we have confirmed that the equilibrium free energy function is indeed of a straight phi(6) form. A study of the dynamics, however, shows that, following a deep quench, the dynamics is described by unstable growth rather than nucleation. A dynamical calculation, based on a generalization of the Brazovskii calculations, shows that the disordered state can remain unstable for a long time following the quench.

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.

A method to Fourier filter textured images

An algorithm is introduced to extract an underlying image from a class of textures. It is assumed that the image is bandwidth limited and the noise is broad-band. The initial step of the algorithm extends the signal to a larger periodic image using "Distributed Approximating Functionals." The second step introduces a low-pass filter which allows the identification and elimination of the high-frequency components of the noise. The periodicity of the resulting image allows it to be Fourier filtered without aliasing. The feasibility of the algorithm is demonstrated on several noisy patterns generated in experiments and model systems. (c) 2000 American Institute of Physics.

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.

Domain Shapes and Patterns: The Phenomenology of Modulated Phases

A wide variety of two- and three-dimensional physical-chemical systems display domain patterns in equilibrium. The phenomenology of these patterns, and of the shapes of their constituent domains, is reviewed here from a point of view that interprets these patterns as a manifestation of modulated phases. These phases are stabilized by competing interactions and are characterized by periodic spatial variations of the pertinent order parameter, the corresponding modulation period generally displaying a dependence on temperature and other external fields. This simple picture provides a unifying framework to account for striking and substantial similarities revealed in the prevalent "stripe" and "bubble" morphologies as well as in commonly observed, characteristic domain-shape instabilities. Several areas of particular current interest are discussed.