Kinetics of phase separation in two-dimensional systems with competing interactions

Structural transitions in two-dimensional modulated systems under triangular confinement

We study numerically the structural transitions of two-dimensional systems of classic particles with competing interactions under a triangular confinement with two different types of soft-wall potentials. We observe a variety of novel confinement-induced equilibrium configurations as a function of particle density and confinement steepness for each considered confinement potential. The specific role played by the confining potentials on the ordering of the particle clusters is revealed. These findings allow us to control the self-organization of modulated systems through using external confinements.

Nonequilibrium pattern formation in circularly confined two-dimensional systems with competing interactions

We numerically investigate the nonequilibrium behaviors of classic particles with competing interactions confined in a two-dimensional logarithmic trap. We reveal a quench-induced surprising dynamics exhibiting rich dynamic patterns depending upon confinement strength and trap size, which is attributed to the time-dependent competition between interparticle repulsions and attractions under a circular confinement. Moreover, in the collectively diffusive motions of the particles, we find that the emergence of dynamic structure transformation coincides with a diffusive mode transition from superdiffusion to subdiffusion. These findings are likely useful in understanding the pattern selection and evolution in various chemical and biological systems in addition to modulated systems, and add a new route to tailoring the morphology of pattern-forming systems.

Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations

Phase field crystal (PFC) theory is extensively used for modeling the phase behavior, structure, thermodynamics, and other related properties of solids. PFC theory can be derived from dynamical density functional theory (DDFT) via a sequence of approximations. Here, we carefully identify all of these approximations and explain the consequences of each. One approximation that is made in standard derivations is to neglect a term of form ∇·[n∇Ln], where n is the scaled density profile and L is a linear operator. We show that this term makes a significant contribution to the stability of the crystal, and that dropping this term from the theory forces another approximation, that of replacing the logarithmic term from the ideal gas contribution to the free energy with its truncated Taylor expansion, to yield a polynomial in n. However, the consequences of doing this are (i) the presence of an additional spinodal in the phase diagram, so the liquid is predicted first to freeze and then to melt again as the density is increased; and (ii) other periodic structures, such as stripes, are erroneously predicted to be thermodynamic equilibrium structures. In general, L consists of a nonlocal convolution involving the pair direct correlation function. A second approximation sometimes made in deriving PFC theory is to replace L with a gradient expansion involving derivatives. We show that this leads to the possibility of the density going to zero, with its logarithm going to -∞ while being balanced by the fourth derivative of the density going to +∞. This subtle singularity leads to solutions failing to exist above a certain value of the average density. We illustrate all of these conclusions with results for a particularly simple model two-dimensional fluid, the generalized exponential model of index 4 (GEM-4), chosen because a DDFT is known to be accurate for this model. The consequences of the subsequent PFC approximations can then be examined. These include the phase diagram being both qualitatively incorrect, in that it has a stripe phase, and quantitatively incorrect (by orders of magnitude) regarding the properties of the crystal phase. Thus, although PFC models are very successful as phenomenological models of crystallization, we find it impossible to derive the PFC as a theory for the (scaled) density distribution when starting from an accurate DDFT, without introducing spurious artifacts. However, we find that making a simple one-mode approximation for the logarithm of the density distribution lnρ(x) rather than for ρ(x) is surprisingly accurate. This approach gives a tantalizing hint that accurate PFC-type theories may instead be derived as theories for the field lnρ(x), rather than for the density profile itself.

Kinetics of fluid demixing in complex plasmas: Domain growth analysis using Minkowski tensors

A molecular dynamics simulation of the demixing process of a binary complex plasma is analyzed and the role of distinct interaction potentials is discussed by using morphological Minkowski tensor analysis of the minority phase domain growth in a demixing simulated binary complex plasma. These Minkowski tensor methods are compared with previous results that utilized a power spectrum method based on the time-dependent average structure factor. It is shown that the Minkowski tensor methods are superior to the previously used power-spectrum method in the sense of higher sensitivity to changes in domain size. By analysis of the slope of the temporal evolution of Minkowski tensor measures, qualitative differences between the case of particle interaction with a single length scale compared to particle interactions with two different length scales (dominating long-range interaction) are revealed. After proper scaling the graphs for the two length scale scenarios coincide, pointing toward universal behavior. The qualitative difference in demixing scenarios is evidenced by distinct demixing behavior: in the long-range dominated cases demixing occurs in two stages. At first, neighboring particles agglomerate, then domains start to merge in cascades. However, in the case of only one interaction length scale only agglomeration but no merging of domains can be observed. Thus, Minkowski tensor analysis is likely to become a useful tool for further investigation of this (and other) demixing processes. It is capable to reveal (nonlinear) local topological properties, probing deeper than (linear) global power-spectrum analysis, however, still providing easily interpretable results founded on a solid mathematical framework.

Starvation-induced collective behavior in C. elegans

We describe a new type of collective behavior in C. elegans nematodes, aggregation of starved L1 larvae. Shortly after hatching in the absence of food, L1 larvae arrest their development and disperse in search for food. In contrast, after two or more days without food, the worms change their behavior—they start to aggregate. The aggregation requires a small amount of ethanol or acetate in the environment. In the case of ethanol, it has to be metabolized, which requires functional alcohol dehydrogenase sodh-1. The resulting acetate is used in de novo fatty acid synthesis, and some of the newly made fatty acids are then derivatized to glycerophosphoethanolamides and released into the surrounding medium. We examined several other Caenorhabditis species and found an apparent correlation between propensity of starved L1s to aggregate and density dependence of their survival in starvation. Aggregation locally concentrates worms and may help the larvae to survive long starvation. This work demonstrates how presence of ethanol or acetate, relatively abundant small molecules in the environment, induces collective behavior in C. elegans associated with different survival strategies.

Real-time observation of domain fluctuations in a two-dimensional magnetic model system

Domain patterns of perpendicularly magnetized ultra-thin ferromagnetic films are often determined by the competition of the short range but strong exchange interaction favouring ferromagnetic alignment of magnetic moments and the long range but weak antiferromagnetic dipolar interaction. Detailed phase diagrams of the resulting stripe domain patterns have been evaluated in recent years; however, the domain fluctuations in these pattern forming systems have not been studied in great detail so far. Here we show that domain fluctuations can be observed in ultra-thin two-dimensional ferromagnetic Fe/Ni/Cu(001) films with perpendicular magnetization in the stripe domain phase. Non-stroboscopic time-resolved threshold photoemission electron microscopy with high temporal resolution allows analysing the dynamic fingerprint of the topological excitations in the nematic domain phase. Furthermore, proliferation of domain ending defects in the vicinity of the spin reorientation transition is witnessed.

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.

Nonlinear dynamics of island coarsening and stabilization during strained film heteroepitaxy

Nonlinear evolution of three-dimensional strained islands or quantum dots in heteroepitaxial thin films is studied via a continuum elasticity model and both perturbation analysis of the system and numerical simulations of the corresponding nonlinear dynamic equation governing the film morphological profile. Three regimes of island array evolution are identified and examined, including a film instability regime at early stage, a nonlinear coarsening regime at intermediate times, and the crossover to a saturated asymptotic state, with detailed behavior depending on film-substrate misfit strains but not qualitatively on finite system sizes. The phenomenon of island array stabilization, which corresponds to the formation of steady but nonordered arrays of strained quantum dots, occurs at later time for smaller misfit strain. It is found to be controlled by the strength of film-substrate wetting interaction which would constrain the valley-to-peak mass transport and hence the growth of island height, and also determined by the effect of elastic interaction between surface islands and the high-order strain energy of individual islands at late evolution stage. The results are compared to previous experimental and theoretical studies on quantum dot coarsening and stabilization.

Kinetics of fluid demixing in complex plasmas: role of two-scale interactions

Using experiments and combining theory and computer simulations, we show that binary complex plasmas are particularly good model systems to study the kinetics of fluid-fluid demixing at the "atomistic" (individual particle) level. The essential parameters of interparticle interactions in complex plasmas, such as the interaction range(s) and degree of nonadditivity, can be varied significantly, which allows systematic investigations of different demixing regimes. The critical role of competition between long-range and short-range interactions at the initial stage of the spinodal decomposition is discussed.

Morphology and interaction between lipid domains

Cellular membranes are a heterogeneous mix of lipids, proteins and small molecules. Special groupings enriched in saturated lipids and cholesterol form liquid-ordered domains, known as "lipid rafts," thought to serve as platforms for signaling, trafficking and material transport throughout the secretory pathway. Questions remain as to how the cell maintains small fluid lipid domains, through time, on a length scale consistent with the fact that no large-scale phase separation is observed. Motivated by these examples, we have utilized a combination of mechanical modeling and in vitro experiments to show that membrane morphology plays a key role in maintaining small domain sizes and organizing domains in a model membrane. We demonstrate that lipid domains can adopt a flat or dimpled morphology, where the latter facilitates a repulsive interaction that slows coalescence and helps regulate domain size and tends to laterally organize domains in the membrane.

Self-tuning phase separation in a model with competing interactions inspired by biological cell polarization

We present a theoretical study of a system with competing short-range ferromagnetic attraction and a long-range antiferromagnetic repulsion, in the presence of a uniform external magnetic field. The interplay between these interactions, at sufficiently low temperature, leads to the self-tuning of the magnetization to a value which triggers phase coexistence, even in the presence of the external field. The investigation of this phenomenon is performed using a Ginzburg-Landau functional in the limit of an infinite number of order parameter components (large N model). The scalar version of the model is expected to describe the phase separation taking place on a cell surface when this is immersed in a uniform concentration of chemical stimulant. A phase diagram is obtained as a function of the external field and the intensity of the long-range repulsion. The time evolution of the order parameter and of the structure factor in a relaxation process is studied in different regions of the phase diagram.

How Pb-overlayer islands move fast enough to self-assemble on Pb-Cu surface alloys

Low-energy electron microscopy reveals that two-dimensional, approximately 50 000 atom, Pb-overlayer and vacancy islands both have diffusion coefficients of 25.6+/-0.8 nm2/sec at 400 degrees C on Pb-Cu surface alloys. This high mobility, key to self-assembly in this system, results from the fast transport of Pb atoms on the surface alloy and of Cu through the Pb overlayer. A high Pb vacancy concentration, predicted by ab initio calculations, facilitates the latter.

Morphologies of expansion ridges of elastic thin films onto a substrate

A model of a thin film elastically attached to a rigid substrate is considered. In the case in which the film expands relative to the substrate and assuming certain nonlinear elastic behavior of the film, expansion ridges may appear, in which the material collapses, and the density is higher on average. By studying numerically this process, the possible morphologies of these collapsed regions are presented. They range from circular spots and straight stripes, to wiggle polygonal patterns and ring-shaped domains. The similarity of some of these results with patterns observed in delamination of thin films and biphase epitaxial growth is emphasized.

Thermodynamic equilibrium of domains in a two-component Langmuir monolayer

This [corrected] article outlines the results from a combined experimental and theoretical study on the properties of circular domains in a mixed Langmuir monolayer at thermodynamic equilibrium. The mixed monolayer consisted of a binary mixture of dimyristoyl-phosphatidyl-choline and dihydrocholesterol. A long-term fluorescence microscopy study of these domains was carried out over the course of approximately 60 h. Image analysis of the domains over time revealed that the domains ripened slowly with an [corrected] increase in mean domain radius and a [corrected] decrease in domain number density. At the end of the measurement, the domains remained polydisperse, and true thermodynamic equilibrium was not reached. Theoretically, collective thermodynamic equilibrium properties such as mean domain size and size distribution were calculated by combining micelle self-assembly theory and the "equivalent dipole" model for the self-energy of two-dimensional domains. The calculations predicted existence of finite-sized circular domains at equilibrium. This suggests that equilibrium circular monolayer domains of single- or multicomponent lipids with a finite size distribution should form only at very limited experimental conditions. Both the predicted mean domain size and size distribution are strongly affected by line tension and dipole moment density difference. A comparison between the theoretical and experimental results is made.

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 simulations of two-dimensional magnetic domain patterns

I show that a model for the interaction of magnetic domains that includes a short range ferromagnetic and a long range dipolar antiferromagnetic interaction reproduces very well many characteristic features of two-dimensional magnetic domain patterns. In particular bubble and stripe phases are obtained, along with polygonal and labyrinthine morphologies. In addition, two puzzling phenomena, namely, the so called "memory effect" and the "topological melting" observed experimentally, are also qualitatively described. Very similar phenomenology is found in the case in which the model is changed to be represented by the Swift-Hohenberg equation driven by an external orienting field.

Dynamics of growth in a three-component mixture with competing interactions

We study numerically the dynamics, in two dimensions, of phase separation in ternary mixtures with dipolar interactions which lead to the formation of modulated phases. We distinguish three different modulated phases: a hexagonal phase of droplets, a lamellar phase, and a hexagonal phase of bubbles. Inside the crystal structures an additional phase separation occurs "coloring" the texture. The dynamics in the droplet phase mixes the two kinds of droplets of different composition. The lamellar phase does not evolve toward parallel lamellae, and the phase separation inside the channels proceeds until they reach a grain boundary. The hexagonal bubble phase is never formed due to the phase separation that forms an interface of bubbles which blocks the contact between the two phases. In its place we find an unsuspected lamellar phase.

Patterns in melting snow and vapor deposited layers

We have observed a natural periodic pattern occurring in partially melted snow lying on the ground under certain atmospheric conditions. We explain this phenomenon quantitatively by considering heat flow through this layer coexisting in two phases (snow and water). Our model equations exhibit a range of patterns depending on the average density of snow. Strikingly similar patterns have been observed by Plass and co-workers in monolayer depositions of Pb on heated PbCu substrates. We argue that the physics of the two phenomena, differing in length scales by 7 orders of magnitude, is similar.

Coarsening dynamics of ternary amphiphilic fluids and the self-assembly of the gyroid and sponge mesophases: Lattice-Boltzmann simulations

By means of a three-dimensional amphiphilic lattice-Boltzmann model with short-range interactions for the description of ternary amphiphilic fluids, we study how the phase separation kinetics of a symmetric binary immiscible fluid is altered by the presence of the amphiphilic species. We find that a gradual increase in amphiphile concentration slows down domain growth, initially from algebraic to logarithmic temporal dependence, and, at higher concentrations, from logarithmic to stretched-exponential form. In growth-arrested stretched-exponential regimes, at late times we observe the self-assembly of sponge mesophases and gyroid liquid-crystalline cubic mesophases, hence confirming that (a) amphiphile-amphiphile interactions need not be long-ranged in order for periodically modulated structures to arise in a dynamics of competing interactions, and (b) a chemically specific model of the amphiphile is not required for the self-assembly of cubic mesophases, contradicting claims in the literature. We also observe a structural order-disorder transition between sponge and gyroid phases driven by amphiphile concentration alone or, independently, by the amphiphile-amphiphile and the amphiphile-binary fluid coupling parameters. For the growth-arrested mesophases, we also observe temporal oscillations in the structure function at all length scales; most of the wave numbers show slow decay, and long-term stationarity or growth for the others. We ascribe this behavior to a combination of complex amphiphile dynamics leading to Marangoni flows.

Thermal motion and energetics of self-assembled domain structures: Pb on Cu(111)

Low energy electron microscope measurements of the thermal motion of 50-200 nm diameter Pb islands on Cu(111) are used to establish the nature and determine the strength of interactions that give rise to self-assembly in this two-dimensional, two-phase system. The results show that self-assembled patterns arise from a temperature-independent surface stress difference of approximately 1.2 N/m between the two phases. With increasing Pb coverage, the domain patterns evolve in a manner consistent with models based on dipolar repulsions caused by elastic interactions due to a surface stress difference.

Weakly nonlinear theory of grain boundary motion in patterns with crystalline symmetry

We study the motion of a grain boundary separating two otherwise stationary domains of hexagonal symmetry. Starting from an order parameter equation, a multiple scale analysis leads to an analytical equation of motion for the boundary that shares many properties with that of a crystalline solid. We find that defect motion is generically opposed by a pinning force that arises from nonadiabatic corrections to the standard amplitude equations. The magnitude of this force depends sharply on the misorientation angle between adjacent domains: the most easily pinned grain boundaries are those with a low angle (typically 4 degrees < or =theta;< or =8 degrees ). Although pinning effects may be small, they can be orders of magnitude larger than those present in smectic phases.

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.

Nanostructures. Self-assembled domain patterns

The ordered domain patterns that form spontaneously in a wide variety of chemical and physical systems as a result of competing interatomic interactions can be used as templates for fabricating nanostructures. Here we describe a new self-assembling domain pattern on a solid surface that involves two surface structures of lead on copper. The evolution of the system agrees with theoretical predictions, enabling us to probe the interatomic force parameters that are crucial to the process.

Instabilities of concentration stripe patterns in ferrocolloids

Equations describing the kinetics of the phase separation in ferrocolloids in a Hele-Shaw cell under the action of a rotating magnetic field are proposed. Numerical simulation on the basis of a pseudospectral technique demonstrates that upon the action of a rotating field on a magnetic colloid which undergoes the phase separation a periodical system of stripes parallel to the plane of a rotating magnetic field stripes is created. The period of a structure found numerically satisfactorily corresponds to the one calculated on the basis of the energy minimum. Thus, the undulation instability leading to the formation of chevron structures takes place if the tangential component of a rotating magnetic field is eliminated, whereas the normal component is increased at the same time. If during the development of the undulation deformations of a concentration pattern the magnetic Bond number is large enough the secondary instabilities may occur leading to the fingering of stripes to bring about merging and break-up of stripes. It is shown that an increase in the magnetic Bond number leads to the onset of the instability at the boundaries between the regions with homogeneous orientation of stripes as well as to formation of the characteristic hairpin patterns.

Phase separation with multiple length scales in polymer mixtures induced by autocatalytic reactions

A ternary polymer blend with two components photo-cross-linked independently in its miscible region undergoes phase separation, exhibiting morphology with multiple length scales. Contrary to the case of thermally induced phase separation, the morphology exhibits a unimodal-->multimodal transition. It is shown that these multiple length scales are caused by the inhomogeneous freezing kinetics of the cross-linking process. This inhomogeneity arises from the autocatalytic feedback driven by the couplings between concentration fluctuations and the photo-cross-linking reactions.

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.