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The development of self-organized surface patterns on Si due to noble gas ion irradiation has been studied extensively in the past. In particular, Ar ions are commonly used and the pattern formation was analyzed as function of ion incidence angle, ion fluence, and ion energies between 250 eV and 140 keV. Very few results exist for the energy regime between 1.5 keV and 10keV and it appears that pattern formation is completely absent for these ion energies. In this work, we present experimental data on pattern formation for Ar ion irradiation between 1 keV and 10 keV and ion incidence angles between 50° and 75°. We confirm the absence of patterns at least for ion fluences up to 1018 ions/cm². Using the crater function formalism and Monte Carlo simulations, we calculate curvature coefficients of linear continuum models of pattern formation, taking into account contribution due to ion erosion and recoil redistribution. The calculations consider the recently introduced curvature dependence of the erosion crater function as well as the dynamic behavior of the thickness of the ion irradiated layer. Only when taking into account these additional contributions to the linear theory, our simulations clearly show that that pattern formation is strongly suppressed between about 1.5 keV and 10 keV, most pronounced at 3 keV. Furthermore, our simulations are now able to predict whether or not parallel oriented ripple patterns are formed, and in case of ripple formation the corresponding critical angles for the whole experimentally studied energies range between 250eV and 140keV. [ABSTRACT FROM AUTHOR]

Coherent wave groups are not only characterized by the intrinsic shape of the wave packet, but also by the underlying phase evolution during the propagation. Exact deterministic formulations of hydrodynamic or electromagnetic coherent wave groups can be obtained by solving the nonlinear Schrödinger equation (NLSE). When considering the NLSE, there are two asymptotically equivalent formulations, which can be used to describe the wave dynamics: the time- or space-like NLSE. These differences have been theoretically elaborated upon in the 2016 work of Chabchoub and Grimshaw. In this paper, we address fundamental characteristic differences beyond the shape of wave envelope, which arise in the phase evolution. We use the Peregrine breather as a referenced wave envelope model, whose dynamics is created and tracked in a wave flume using two boundary conditions, namely as defined by the time- and space-like NLSE. It is shown that whichever of the two boundary conditions is used, the corresponding local shape of wave localization is very close and almost identical during the evolution; however, the respective local phase evolution is different. The phase dynamics follows the prediction from the respective NLSE framework adopted in each case. [ABSTRACT FROM AUTHOR]

Molecular dynamics simulations and integral equation calculations of a simple equimolar mixture of diatomic molecules and monomers interacting via attractive and repulsive short-range potentials show the existence of pattern formation (microheterogeneity), mostly due to depletion forces away from the demixing region. Effective site-site potentials extracted from the pair correlation functions using an inverse Monte Carlo approach and an integral equation inversion procedure exhibit the features characteristic of a short-range attractive and a long-range repulsive potential. When charges are incorporated into the model, this becomes a coarse grained representation of a room temperature ionic liquid, and as expected, intermediate range order becomes more pronounced and stable. [ABSTRACT FROM AUTHOR]

Extending previous work [Pusztai et al., Phys. Rev. E 87, 032401 (2013)], we have studied the formation of eutectic dendrites in a model ternary system within the framework of the phase-field theory. We have mapped out the domain in which two-phase dendritic structures grow. With increasing pulling velocity, the following sequence of growth morphologies is observed: flat front lamellae ? eutectic colonies ? eutectic dendrites ? dendrites with target pattern ? partitionless dendrites ? partitionless flat front. We confirm that the two-phase and one-phase dendrites have similar forms and display a similar scaling of the dendrite tip radius with the interface free energy. It is also found that the possible eutectic patterns include the target pattern, and single- and multiarm spirals, of which the thermal fluctuations choose. The most probable number of spiral arms increases with increasing tip radius and with decreasing kinetic anisotropy. Our numerical simulations confirm that in agreement with the assumptions of a recent analysis of two-phase dendrites [Akamatsu et al., Phys. Rev. Lett. 112, 105502 (2014)], the Jackson-Hunt scaling of the eutecticwavelength with pulling velocity is obeyed in the parameter domain explored, and that the natural eutecticwavelength is proportional to the tip radius of the two-phase dendrites. Finally, we find that it is very difficult/virtually impossible to form spiraling two-phase dendrites without anisotropy, an observation that seems to contradict the expectations of Akamatsu et al. Yet, it cannot be excluded that in isotropic systems, two-phase dendrites are rare events difficult to observe in simulations. [ABSTRACT FROM AUTHOR]

Self‐organized pattern formation is widespread and functionally significant. Scale‐dependent feedback in space (short‐distance positive feedback coupled with long‐distance negative feedback) has been embraced as an arguably universal mechanism of ecological self‐organization. Recently, intraspecific territorial competition has been proposed as a complementary mechanism contributing to spatial self‐organization in ecology. In geomorphology, regular patterning is also widespread and has often been attributed to competition among geomorphic features. This mechanism has never been integrated into the framework of ecological pattern formation. Using the regularly patterned landscape of Big Cypress National Preserve in South Florida as a case study, we formalize a third mechanism of spatial self‐organization: competition among pattern elements of finite amplitude stabilized by scale‐dependent feedback in time. Depressions first accelerate their expansion rate via the positive feedback between depression volume and weathering rate. Later negative feedbacks become stronger, and eventually stabilize the size of depressions. While scale‐dependent feedback in time provides a mechanism to stabilize individual depressions, it is the competition among depressions that induces spatial regularity. A relatively smaller depression could have a greater expansion rate than larger ones in its development. Higher weathering rate on the side of a divide toward the smaller depression causes migration of the divide to the larger depression. Consequently, the smaller depression expands its catchment area while the catchment area of the neighboring larger depression contracts, resulting in depressions achieving similar size and distance from each other. The diversity of regular patterns dictates the need to integrate perspectives from multiple disciplines. Plain Language Summary: Regular patterns in ecosystems formed by self‐organization are widespread and have significant ecological consequences. Coupled locally positive, distantly negative feedbacks (i.e., scale‐dependent feedbacks) have been embraced in ecology as a unifying mechanism to explain formation of regular patterns. In geomorphology, regular patterns are also widespread, and many have been attributed to competition among geomorphic features. This geomorphic mechanism has not heretofore been integrated into the framework of ecological pattern formation. We demonstrated that it is competition, instead of scale‐dependent feedbacks, that leads to the formation of the evenly spaced limestone depressions in the Big Cypress National Preserve in Florida. Growth of the size of limestone depressions requires water. As a depression grows, it holds more water, which extends the wet period, resulting in a higher weathering rate. Meanwhile, negative feedbacks triggered by ecohydrologic processes become stronger over time and eventually stabilize the size of depressions. During this process, depressions compete with each other for catchment area (proxy for water), by repeatedly moving the divide between the catchments caused by difference in their weathering rate. Such spatial interactions eventually result in similar depression size and distance of depressions to each other, giving rise to spatial regularity. Key Points: Competition among limestone depressions for surface water leads to the emergence of landscape‐scale regular patterning of depressionsScale‐dependent feedbacks in time, instead of in space, induces the spatial interactions, competition, among depressionsDegree of regularity of the spatial patterns enhances over time by competition at the boundary between limestone depressions [ABSTRACT FROM AUTHOR]

In the present paper, the spectral meshless radial point interpolation (SMRPI) tech- nique is applied to the solution of pattern formation in nonlinear reaction-diffiusion systems. Firstly, we obtain a time discrete scheme by approximating the time derivative via a finite difference formula, then we use the SMRPI approach to approximate the spatial derivatives. This method is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. In the current work, the thin plate splines (TPS) are used as the basis functions and in order to eliminate the nonlinearity, a simple predictor-corrector (P-C) scheme is performed. The effect of parameters and conditions are studied by considering the well known Brusselator model. Two test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

Pattern formation in uniaxial polymeric liquid crystals is studied for different dynamic closure approximations. Using the principles of mesoscopic non-equilibrium thermodynamics in a mean-field approach, we derive a Fokker-Planck equation for the single-particle non-homogeneous distribution function of particle orientations and the evolution equations for the second and fourth order orientational tensor parameters. Afterwards, two dynamic closure approximations are discussed, one of them considering the relaxation of the fourth order orientational parameter and leading to a novel expression for the free-energy like function in terms of the scalar order parameter. Considering the evolution equation of the density of the system and values of the interaction parameter for which isotropic and nematic phases coexist, our analysis predicts that patterns and traveling waves can be produced in lyotropic uniaxial nematics even in the absence of external driving. [ABSTRACT FROM AUTHOR]

The kinetics of mesoscopic pattern formation is studied for a reversible A+B<=>0 reaction between mobile oppositely charged molecules at the interface. Using formalism of the joint correlation functions, non-equilibrium charge screening and reverse Monte Carlo methods, it is shown that labyrinth-like percolation structure induced by (even moderate-rate) reaction is principally non-steady-state one and is associated with permanently growing segregation of dissimilar reactants and aggregation of similar reactants into mesoscopic size domains. A role of short-range and long-range reactant interactions in pattern formation is discussed. [ABSTRACT FROM AUTHOR]

A 30 cm long plasma column is excited by a surface wave, which acts as a plasma antenna. Using plasma properties (pattern formation/striations in plasmas) single plasma antenna can be transformed into array, helical, and spiral plasma antenna. Experiments are carried out to study the power patterns, directivity, and half power beam width of such different plasma antennas. Moreover, field properties of plasma and copper antenna are studied. Further, wireless communication and jamming capability of plasma antenna are tested. Findings of this study suggest that directivity and communication range can be increased by converting single plasma antenna in to array/helical/spiral plasma antenna. Field frequencies of plasma antenna determine the communication and jamming of radio frequency waves. Therefore, this study invokes applications of pattern formation or striations of plasmas in plasma antenna technology. [ABSTRACT FROM AUTHOR]

An effective medium theory is employed to derive a simple qualitative model of a pattern forming chemical reaction in a microemulsion. This spatially heterogeneous system is composed of water nanodroplets randomly distributed in oil. While some steps of the reaction are performed only inside the droplets, the transport through the extended medium occurs by diffusion of intermediate chemical reactants as well as by collisions of the droplets. We start to model the system with heterogeneous reaction-diffusion equations and then derive an equivalent effective spatially homogeneous reaction-diffusion model by using earlier results on homogenization in heterogeneous reaction-diffusion systems [S.Alonso, M.Bär, and R.Kapral, J. Chem. Phys. 134, 214102 (2009)]. We study the linear stability of the spatially homogeneous state in the resulting effective model and obtain a phase diagram of pattern formation, that is qualitatively similar to earlier experimental results for the Belousov-Zhabotinsky reaction in an aerosol OT (AOT)-water-in-oil microemulsion [V.K.Vanag and I.R.Epstein, Phys. Rev. Lett. 87, 228301 (2001)]. Moreover, we reproduce many patterns that have been observed in experiments with the Belousov-Zhabotinsky reaction in an AOT oil-in-water microemulsion by direct numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper we explore the spontaneous formation of micropatterns in thin prestressed polygonal films using finite element simulations. We study films with different size, thickness, and shape, including square, rectangular, pentagonal, and hexagonal films. Patterns form when the films release the internal eigenstrain by buckling-up, after which the films bond-back to the substrate. After an initial symmetric evolution of the buckling profile, the symmetry of the deflection pattern breaks when the wavelength of wriggles near the film edges decreases. During bond back the deflection morphology converges to a fourfold, fivefold, and sixfold ridging pattern for the square, pentagonal and hexagonal films, respectively, showing a close resemblance with experimental film systems of similar size and shape. Rectangular films of large length to width ratio go through a transition in buckling shapes from the initial Euler mode, through the varicose mode into the antisymmetric telephone-cord mode. For all the film shapes, the ratio of the film height to the effective film width scales with the square root of eigenstrain and is independent of thickness. The bond-back mechanism determines the final wrinkle morphology and is governed by the eigenstrain value at the end of the buckling-up stage and the dimensionless parameter (Γ/EWeq)(Weq/t)3, relating the interface energy to the strain energy in the film. [ABSTRACT FROM AUTHOR]

Effects of time-delayed feedback on pattern formation are studied both numerically and theoretically in a bistable reaction-diffusion model. The time-delayed feedback applied to the activator and/or the inhibitor alters the behavior of the nonequilibrium Ising-Bloch (NIB) bifurcation. If the intensities of the feedbacks applied to the two species are identical, only the velocities of Bloch fronts are changed. If the intensities are different, both the critical point of the NIB bifurcation and the threshold of stability of front to transverse perturbations are changed. The effect of time-delayed feedback on the activator opposes the effect of time-delayed feedback on the inhibitor. When the time-delayed feedback is applied individually to one of the species, positive and negative feedbacks make the bifurcation point shift to different directions. The time-delayed feedback provides a flexible way to control the NIB bifurcation and the pattern formation. [ABSTRACT FROM AUTHOR]

Pattern formation in the anode layer at a water electrode in atmospheric pressure glow discharges in air is studied. With increasing current a sequence of different anode spot structures occurs from a constricted homogeneous spot in the case of small currents to a pattern consisting of small distinct spots for larger currents. The dependence of the number of spots on the current is (18.5+0.4N) mA with N the number of spots. The spots group together and form stripe patterns. Disappearance of the spots with increasing conductivity of the electrode seems to indicate that mechanisms of electrical nature govern the phenomenon. Similarities and differences with other experiments and models revealing pattern formation in glow discharges are discussed. [ABSTRACT FROM AUTHOR]

We commenced basic research on the three-dimensional (3D) pattern formation of microgel beads for applications in biological tissue engineering. In this new technique, microgel beads are premagnetized by doping them with magnetic nanoparticles. Living cells will be included in the beads for actual use. If a nonuniform magnetic field is applied to a solution containing these magnetized beads, the beads will align, contact, and form a 3D structure. The structure is controlled by the seed pattern of the magnetic particles plugged in a substrate and the profile of the magnetic field distribution. We constructed tubes, which imitate blood vessels, for demonstration using gel beads whose diameters are of the order of several tens of micrometers. The diameter of the demonstrated tube was less than 0.5 mm and its length was 6.6 mm, although living cells were not included in the beads. Numerical calculations by using the discrete element method were conducted to confirm the formation of the tube and to predict the effect of centrifugal force, which will be applied to fill cells in the space between magnetically patterned beads. Although this unique technology is in the nascent stage, this 3D pattern formation technique by the control of the magnetic field has potential to be one of the effective engineering technologies for manufacturing 3D patterned biological tissues in the future. [ABSTRACT FROM AUTHOR]

Heteroepitaxial nucleation and growth of twin-free diamond on Ir(111) is reported. The bias enhanced nucleation (BEN) technique was applied in a microwave plasma chemical vapor deposition setup to induce diamond nucleation on the new multilayer stack Ir/YSZ/Si(111). We demonstrate that the gathering of the diamond nuclei in so-called “domains,” a pattern formation process unique for diamond nucleation on iridium, is also present on Ir(111). The 1–2 nm thick carbon layer deposited by BEN does not show any crystalline diamond structures in electron diffraction and high resolution lattice imaging microscopy. In contrast, x-ray photoelectron diffraction (XPD) measurements yield C 1s diffractograms with clear threefold symmetry. The main features are comparable to measurements on diamond (111) single crystal surfaces. The weaker fine structure in the XPD patterns of the BEN layers is attributed to some disorder due to the harsh ion bombardment. However, this ion bombardment did not induce any measurable amount of twinning as deduced from the threefold symmetry. After 3 h diamond growth, the signal due to twins in the x-ray diffraction pole figures is still below the noise level of ∼1%. Negligible twinning and the low mosaic spread of 0.96° (tilt) and 1.85° (twist) indicate that these films mark a breakthrough toward heteroepitaxial diamond films with (111) orientation. They provide interesting growth substrates, e.g., for phosphorous doped diamond or for the formation of heterojunction devices by deposition of wurtzite-type wide band gap semiconductor materials. [ABSTRACT FROM AUTHOR]

Ion bombardment of crystalline silicon targets induces pattern formation by the creation of mobile surface species that participate in forming nanometer-scale structures. The formation of these mobile species on a Si(001) surface, caused by sub-keV argon ion bombardment, is investigated through molecular dynamics simulation of Stillinger-Weber [Phys. Rev. B 31, 5262 (1985)] silicon. Specific criteria for identifying and classifying these mobile atoms based on their energy and coordination number are developed. The mobile species are categorized based on these criteria and their average concentrations are calculated. [ABSTRACT FROM AUTHOR]

We present a general formalism for scattering of electron bunches used in ultrafast electron diffraction experiments. To perform the scattering calculation, we associate the classical distribution function that describes the electron bunch just before scattering with the asymptotic-in Wigner distribution. Using single scattering and far-field approximations, we derive an expression for the diffracted signal and discuss the effects of the different bunch parameters on the measured diffracted flux. We identify the transverse and longitudinal coherence lengths and discuss the importance of these length scales in diffraction pattern formation. We present sample numerical calculations for scattering by nanosize particles based on our model and discuss the results in terms of bunch and scattering target parameters. [ABSTRACT FROM AUTHOR]

Low-symmetry distortions are present in cubanes such as Fe4S4, but their effects on electron delocalization properties are not well-understood. Mixed-valence cubanes often exhibit experimentally measurable “pair delocalization” of a delocalizable electron. An important question is, what is the role of physical interactions (vibronic, electronic, exchange) and symmetry distortions in determining the electron delocalization pattern? Semiclassical models are used to explore the electron delocalization patterns of S=12 tetragonally (D2d) distorted mixed-valence cubanes comprising four metal centers with bridging ligands, a single delocalizable “excess” electron, and either closed-shell or open-shell ion cores. Phase diagrams show that distorted S=12 ground state cubanes with antiferromagnetic exchange (as found in nature) have delocalization patterns qualitatively similar to those of an S=12 model with no Heisenberg exchange, suggesting that exchange is not necessarily a dominant factor in determining electron delocalization properties. The open-shell model reveals two types of pair delocalization for the S=12 ground state, with differing dimer subunit spins for compressed and elongated geometries. Previous studies emphasize the importance of exchange interactions for pair delocalization. Here, it is shown that electron exchange is not always necessary for pair delocalization and that it can be achieved with relatively small tetragonal distortions from tetrahedral (Td) symmetry. The results contradict those of an earlier theoretical study of distorted Fe4S4 clusters, which concluded that distortions of lower symmetry than D2d are necessary to induce a transition to pair delocalization. [ABSTRACT FROM AUTHOR]

We have applied the atomic force microscopy (AFM) lithography technique to electrically phase-separated (La,Pr,Ca)MnO3 (LPCMO) thin films. It was found that the maximum pattern height of 10.5 nm for the as-deposited film was much higher than that of 5.3 nm for the annealed film, suggesting that AFM lithography of LPCMO thin film was highly influenced by oxygen nonstoichiometry that enhances ion migration. The nanochannel with 150 nm width was fabricated using this technique. The nanostructure shows insulating behavior under H=0 T and large hysteretic behavior accompanied with metal-insulator transition temperature Tp=140 and 170 K upon cooling and warming under 5 T. In contrast, the unpatterned film has Tp=170 K (H=0 T) and 240 K (H=5 T) without large hysteretic behavior. The difference between the nanochannel and the unpatterned film can be related to the electronic phase separation. [ABSTRACT FROM AUTHOR]

The control of Turing pattern formation by weak spatial perturbation is investigated. The weak spatial perturbation added before Turing pattern stabilization is found to show prominent spatial orientation effect. The control process of perturbation to Turing patterns is tracked. The effect of perturbation factors, such as amplitude and imposing time are also discussed. [ABSTRACT FROM AUTHOR]

We study a model of pattern formation in an excitable medium with concentration-dependent diffusivities. The reaction terms correspond to a two-variable Gray–Scott model in which the system has only one stable steady state. The diffusion coefficients of the two species are assumed to have a functional relationship with the concentration of the autocatalyst. A transition from self-replicating behavior to stationary spots is observed as the influence of the local autocatalyst concentration on the diffusion process increases. Notably, the transition occurs even though there is no change in the relative diffusivities of the activator and inhibitor. The observed time-independent patterns exhibit an unusual dependence on the size and geometry of an initial perturbation. Initial perturbations with a large spatial size, for example, sometimes revert to the homogeneous equilibrium state, whereas perturbations of smaller spatial extent develop into stable spots at the same parameter values. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The spontaneous pattern formation and morphological transitions in thin liquid films on chemically heterogeneous, periodic patterned surfaces are studied based on nonlinear simulations. Conditions are identified for the creation of desired mesostructures in soft materials by spontaneous dewetting on patterned substrates. On a surface consisting of alternating less and more wettable stripes, dewetting is suppressed on some less wettable stripes when their spacing is below a characteristic length scale of instability (λ[sub h]), which is smaller than the spinodal length scale of instability. Ideal templating, i.e., replication of the substrate surface energy pattern in the thin film morphology occurs only when (a) the periodicity of substrate pattern is greater than λ[sub h], (b) width of the less wettable stripe is within a range bounded by a lower critical length, below which no heterogeneous rupture occurs, and an upper transition length above which complex morphological features bearing little resemblance to the substrate pattern are formed, (c) the contact line eventually rests close to the stripe boundary, and (d) the liquid cylinders that form on the more wettable stripes remain stable. Conditions (a) and (b) ensure the onset of dewetting at the center of every less wettable stripe, and conditions (c) and (d) ensure full coverage of every more wettable site by liquid cylinders. The above conditions for ideal templating on a periodic heterogeneous substrate can be engineered by modulating the stripe-periodicity and width, film thickness and the wettability gradient across the stripe boundary. For example, ideal templating on a given substrate always occurs at an intermediate film thickness, since dewetting is suppressed at some less wettable stripes at higher thickness and multiple dewetting sites form on less wettable stripes at lower thickness. © 2002 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Studies pattern formation in unstable thin liquid films under the influence of antagonistic short- and long-range forces. Reorganization of initial random nonhomogeneities into a small amplitude undulating structure; Characteristics of a pathway of evolution and the morphology at the onset dewetting.

Pattern formation is one of the most surprising natural phenomena in real life. Analysis of spatiotemporal reaction-diffusion system can lead to understanding the pattern dynamics. However, the periodic traveling wave solutions resulting from the reaction-diffusion system can play an important role to explain the pattern dynamics. In this study, we analyze a system of nonlinear reaction-diffusion equations called the Brusselator model. We establish a parameter plane to investigate the existence of periodic traveling waves as well as stability results of the model using the method of continuation. We also find an Eckhaus type stability boundary where we confirm the stability change by calculating the essential spectra of the solutions of the model. As a result, we obtain a pattern transition from stripe pattern to spot pattern of the model in the two spatial dimensions numerically. [ABSTRACT FROM AUTHOR]

Abstract Effect of side wall boundary conditions and aspect ratio in very small boxes on patterns formed during Raleigh-Benard (RB) convection is studied using an indigenously developed flow solver based on the finite difference method. While keeping the bottom (hot) and top (cold) walls isothermal, the side wall thermal conditions are kept either constant heat flux or isothermal type. For constant heat flux side wall condition, the considered value of heat flux lies between 0 and maximum corresponding to conductive heat flux between top and bottom walls. For isothermal side wall condition, the chosen value of temperature is between the top and bottom wall temperatures. The computational domain is a cube or a square prism with aspect ratio in longitudinal direction (AR x) varied between 1.2 and 2 in steps of 0.2. For fixed value of AR x = 1.6, aspect ratio in transverse direction (AR z) has been varied for adiabatic side wall conditions. Effect of side wall condition on orientation of roll axis and the number of rolls has been investigated with observations well supported by stream traces and temperature contours. Moreover, for selected parameters, based on nature of pattern formation for steady flow, onset of oscillatory convection is explained. Highlights • Patterns formed are found to be sensitive to different sidewall thermal conditions. • Non-mean side wall temperature shifts roll axis up or down and makes it circular. • Heat leak at side walls turns the roll axis parallel to diagonal. • Effect of aspect ratio on pattern/roll formation has been studied. • Onset of oscillatory convection is explained based on the patterns/rolls formed. [ABSTRACT FROM AUTHOR]

Morphogenetic patterns are highly sophisticated dissipative structures. Are they governed by the same general mechanisms as chemical and hydrodynamic patterns? Turing's symmetry breaking and Wolpert's signalling provide alternative mechanisms. The current evidence points out that the latter is more relevant, but reality is still far more complicated. [ABSTRACT FROM AUTHOR]

In this essay we explore analogies between macroscopic patterns, which result from a sequence of phase transitions/instabilities starting from a homogeneous state, and similar phenomena in cosmology, where a sequence of phase transitions in the early universe is believed to have separated the fundamental forces from each other, and also shaped the structure and distribution of matter in the universe. We discuss three distinct aspects of this analogy: (i) Defects and topological charges in macroscopic patterns are analogous to spins and charges of quarks and leptons; (ii) Defects in generic 3+1 stripe patterns carry an energy density that accounts for phenomena that are currently attributed to dark matter; (iii) Space-time patterns of interacting nonlinear waves display behaviors reminiscent of quantum phenomena including inflation, entanglement and dark energy. [ABSTRACT FROM AUTHOR]

Geophysical turbulence is observed to self-organize into large-scale flows such as zonal jets and coherent vortices. Previous studies of barotropic $\unicode[STIX]{x1D6FD}$ -plane turbulence have shown that coherent flows emerge from a background of homogeneous turbulence as a bifurcation when the turbulence intensity increases. The emergence of large-scale flows has been attributed to a new type of collective, symmetry-breaking instability of the statistical state dynamics of the turbulent flow. In this work, we extend the analysis to stratified flows and investigate turbulent self-organization in a two-layer fluid without any imposed mean north–south thermal gradient and with turbulence supported by an external random stirring. We use a second-order closure of the statistical state dynamics, that is termed S3T, with an appropriate averaging ansatz that allows the identification of statistical turbulent equilibria and their structural stability. The bifurcation of the statistically homogeneous equilibrium state to inhomogeneous equilibrium states comprising zonal jets and/or large-scale waves when the energy input rate of the excitation passes a critical threshold is analytically studied. Our theory predicts that there is a large bias towards the emergence of barotropic flows. If the scale of excitation is of the order of (or larger than) the deformation radius, the large-scale structures are barotropic. Mixed barotropic–baroclinic states with jets and/or waves arise when the excitation is at scales shorter than the deformation radius with the baroclinic component of the flow being subdominant for low energy input rates and insignificant for higher energy input rates. The predictions of the S3T theory are compared with nonlinear simulations. The theory is found to accurately predict both the critical transition parameters and the scales of the emergent structures but underestimates their amplitude. [ABSTRACT FROM AUTHOR]

Highlights • Ion beam induced surface nanopatterning on mica surface. • Scaling behavior of nanopatterns is investigated. • Growth dynamics of anisotropic ripples on mica surface is studied. Abstract The scaling behavior of ion beam induced ripple patterns on mica surface is investigated by means of atomic force microscopy (AFM). The ripples were produced by 12 keV Ar+ ion beam erosion at an angle 60° with respect to the surface normal. Ripple wave-vectors are found to align parallel to the ion beam projection direction. The root-mean-square (rms) roughness and lateral correlation length of the rippled surface are observed to increase with ion fluence with a growth exponent β = 0.51 ± 0.04 and dynamic exponent 1/z x = 0.42 ± 0.07 along the ripple direction as well as 1/z y = 0.34 ± 0.09 normal to the ripple direction respectively. Ripple wavelength is also found to increase slowly with ion fluence upto ϕ ∼ 6 × 1017 ions.cm−2 and then shows a rapid coarsening behavior with an exponent n = 0.43 ± 0.04. The two different roughness exponents, α x and α y , along and normal to the ripple direction, are also calculated from their corresponding power spectral density function and found the values 1.25 ± 0.09 and 1.20 ± 0.08 respectively. The scaling properties of ripple growth for long time ion bombardment and the overall morphological changes are explored according to calculated erosive and redistribution coefficients in the framework of recently advanced continuum models of pattern formation by ion beam sputtering. The production of ripples is found to be driven by mass-redistribution effects rather than curvature dependent sputter erosion. This is an worth addition to the understanding of ion beam nanopatterning process as it ascertains mass redistribution as indispensable phenomena for pattern formation not only for ion energies less than 1 keV but also for ion irradiation of energy range 10 keV and above. Graphical abstract Image, graphical abstract [ABSTRACT FROM AUTHOR]

We propose a second-order accurate energy-stable time-integration method that controls the evolution of numerical instabilities introducing numerical dissipation in the highest-resolved frequencies. Our algorithm further extends the generalized- α method and provides control over dissipation via the spectral radius. We derive the first and second laws of thermodynamics for the Swift–Hohenberg equation and provide a detailed proof of the unconditional energy stability of our algorithm. Finally, we present numerical results to verify the energy stability and its second-order accuracy in time. [ABSTRACT FROM AUTHOR]

A chemostat model with maintenance energy and crossdiffusion is considered, and the formation of patterns is caused by the cross-diffusion. Fiist, through linear stabiliiy analysis, the necessary conditions for the formation of the spatial patterns are given. Then numerical simulations by changing the values of crossdifusions in the unstable domain are performed. The results show that the cross-diffusion coefficient plays an important role in the formation of the pattern, and the diferent values of the crossdifusion coefficients may lead to different types of pattern formation. [ABSTRACT FROM AUTHOR]

A rich variety of spatiotemporal patterns exists on reactive rings having a nonuniform activity. Simulations of a case with a nonuniformity indicate that when the local phase-plane features change along the ring, novel motions evolve which cannot form on a uniformly active ring. An example is a rotating pulse which can move only in one direction but not in the other. The nonuniformity increases the number of possible patterns which can exist on a specific ring relative to that existing on a uniformly active one. Thus the nonuniformity increases the sensitivity of the observed patterns to changes in the initial conditions or perturbations. © 1996 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Global interaction introduces a new mode of communication among all surface elements and may lead to the formation of a very rich variety of spatiotemporal patterns on a disk some of which cannot exist in its absence. While some of the motions on a disk are analogous to those observed in one-dimensional systems, some, such as targets and spirals, exist only in two-dimensional systems. The global interaction stabilizes motions such as target patterns, which are not stable in its absence. It may, however, destabilize the spiral motion and cause the spiral tip to meander in a rather complex fashion and in some cases to exit the disk. The global interaction also increases the sensitivity of the system to the initial conditions so that qualitatively different patterns may be obtained when different initial conditions are used. © 1995 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The theory of competitive coarsening is formulated and analyzed for applications to the formation of macroscopic precipitation patterns. Previous work is extended to account for ‘‘local coarsening’’ which results if processes on smaller length scales are properly eliminated. Formally the theory includes the width of the local particle size distribution as a dynamical variable and avoids the assumption of monodispersivity. The linear analysis shows that long-wavelength instabilities of the spatially homogeneous system are suppressed in comparison with the results of previous work. Numerical simulations of the nonlinear model are applied to locally perturbed homogeneous systems and produce one precipitation band surrounded by a growing region essentially void of precipitate. Recurrent banding and satellite induction which result under the assumption of monodispersivity are not explained by this theory. The random patterns found experimentally in supercooled PbI2 solutions containing a gel are interpreted as dissolution patterns driven by exceptionally large particles which act as local perturbations. Using an analogy to percolation problems this point of view can rationalize part of the varying appearance of the patterns as well as the sensitivity to the initial salt concentration. The theory also qualitatively describes the ‘‘focusing effect’’ found in recent experiments on Liesegang band formation. [ABSTRACT FROM AUTHOR]

A theory is presented which explains recurrent precipitation bands (‘‘Liesegang rings’’) as dissolution patterns in a previously produced essentially homogeneous and continuous colloid. This picture is consistent with recent experimental evidence that the formation of Liesegang bands is a postnucleation phenomenon. Mathematically we formulate a model which contains a suitable diffusion equation plus a threshold condition for band formation and repetitive boundary conditions at the location of the previous band. The theory predicts the familiar exponential (spacing) law for the band positions but is sufficiently flexible to account for deviations from this law if additional physico-chemical details are taken into account. [ABSTRACT FROM AUTHOR]