Scaling functions, self-similarity, and the morphology of phase-separating systems

Characterization of Zr-Containing Dispersoids in Al-Zn-Mg-Cu Alloys by Small-Angle Scattering

The characterization of Zr-containing dispersoids in aluminum alloys is challenging due to their broad size distribution, low volume fraction, and heterogeneous distribution within the grains. In this work, small-angle X-ray scattering (SAXS) and small-angle neutron scattering (SANS) were compared to scanning electron microscopy (SEM) and transmission electron microscopy (TEM) regarding their capability to characterize Zr-containing dispersoids in aluminum alloys. It was demonstrated that both scattering techniques are suitable tools to characterize dispersoids in a multi-phase industrial 7xxx series aluminum alloy. While SAXS is more sensitive than SANS due to the high electron density of Zr-containing dispersoids, SANS has the advantage of being able to probe a much larger sample volume. The combination of both scattering techniques allows for the verification that the contribution from dispersoids can be separated from that of other precipitate phases such as the S-phase or GP-zones. The size distributions obtained from SAXS, SANS and TEM showed good agreement. The SEM-derived size distributions were, however, found to significantly deviate from those of the other techniques, which can be explained by considering the resolution-limited restrictions of the different techniques.

Evolution of single gyroid photonic crystals in bird feathers

Vivid, saturated structural colors are conspicuous and important features of many animals. A rich diversity of three-dimensional periodic photonic nanostructures is found in the chitinaceous exoskeletons of invertebrates. Three-dimensional photonic nanostructures have been described in bird feathers, but they are typically quasi-ordered. Here, we report bicontinuous single gyroid β-keratin and air photonic crystal networks in the feather barbs of blue-winged leafbirds (Chloropsis cochinchinensis sensu lato), which have evolved from ancestral quasi-ordered channel-type nanostructures. Self-assembled avian photonic crystals may serve as inspiration for multifunctional applications, as they suggest efficient, alternative routes to single gyroid synthesis at optical length scales, which has been experimentally elusive.

Exploring the free energy gain of phase separation via Markov state modeling

The gain of free energy upon unmixing is determined via application of Markov state modeling (MSM), using an Ising model with a fixed number of up- and down-spins. MSM yields reasonable estimates of the free energies. However, a closer look reveals significant differences that point to residual non-Markovian effects. These non-Markovian effects are rather unexpected since the typical criteria to study the quality of Markovianity indicate complete Markovian behavior. We identify the sparse connectivity between different Markov states as a likely reason for the observed bias. By studying a simple five state model system, we can analytically elucidate different sources of the bias and thus explain the different deviations that were observed for the Ising model. Based on this insight, we can modify the determination of the count matrix in the MSM approach. In this way, the estimation of the free energy is significantly improved.

Combined Experimental and Theoretical Approach to the Kinetics of Magnetite Crystal Growth from Primary Particles

It is now recognized that nucleation and growth of crystals can occur not only by the addition of solvated ions but also by accretion of nanoparticles, in a process called nonclassical crystallization. The theoretical framework of such processes has only started to be described, partly due to the lack of kinetic or thermodynamic data. Here, we study the growth of magnetite nanoparticles from primary particles-nanometer-sized amorphous iron-rich precursors-in aqueous solution at different temperatures. We propose a theoretical framework to describe the growth of the nanoparticles and model both a diffusion-limited and a reaction-limited pathway to determine which of these best describes the rate-limiting step of the process. We show that, based on the measured iron concentration and the related calculated concentration of primary particles at the steady state, magnetite growth is likely a reaction-limited process, and within the framework of our model, we propose a phase diagram to summarize the observations.

Instability-induced pattern formation of photoactivated functional polymers

Since the pioneering work of Turing on the formation principles of animal coat patterns [Turing AM (1952) Phil Trans R Soc Lond B 237(641):37-72], such as the stripes of a tiger, great effort has been made to understand and explain various phenomena of self-assembly and pattern formation. Prominent examples are the spontaneous demixing in emulsions, such as mixtures of water and oil [Herzig EM, et al. (2007) Nat Mater 6:966-971]; the distribution of matter in the universe [Kibble TWB (1976) J Phys A: Math Gen 9(8):1387]; surface reconstruction in ionic crystals [Clark KW, et al. (2012) Nanotechnol 23(18):185306]; and the pattern formation caused by phase transitions in metal alloys, polymer mixtures and binary Bose-Einstein condensates [Sabbatini J, et al. (2011) Phys Rev Lett 107:230402]. Photoactivated pattern formation in functional polymers has attracted major interest due to its potential applications in molecular electronics and photoresponsive systems. Here we demonstrate that photoactivated pattern formation on azobenzene-containing polymer films can be entirely explained by the physical concept of phase separation. Using experiments and simulations, we show that phase separation is caused by an instability created by the photoactivated transitions between two immiscible states of the polymer. In addition, we have shown in accordance with theory, that polarized light has a striking effect on pattern formation indicated by enhanced phase separation.

Nucleation and growth of magnetite from solution

The formation of crystalline materials from solution is usually described by the nucleation and growth theory, where atoms or molecules are assumed to assemble directly from solution. For numerous systems, the formation of the thermodynamically stable crystalline phase is additionally preceded by metastable intermediates . More complex pathways have recently been proposed, such as aggregational processes of nanoparticle precursors or pre-nucleation clusters, which seem to contradict the classical theory. Here we show by cryogenic transmission electron microscopy that the nucleation and growth of magnetite-a magnetic iron oxide with numerous bio- and nanotechnological applications-proceed through rapid agglomeration of nanometric primary particles and that in contrast to the nucleation of other minerals, no intermediate amorphous bulk precursor phase is involved. We also demonstrate that these observations can be described within the framework of classical nucleation theory.

Spontaneous formation of stable capillary bridges for firming compact colloidal microstructures in phase separating liquids: a computational study

Computer modeling and simulations are performed to investigate capillary bridges spontaneously formed between closely packed colloidal particles in phase separating liquids. The simulations reveal a self-stabilization mechanism that operates through diffusive equilibrium of two-phase liquid morphologies. Such mechanism renders desired microstructural stability and uniformity to the capillary bridges that are spontaneously formed during liquid solution phase separation. This self-stabilization behavior is in contrast to conventional coarsening processes during phase separation. The volume fraction limit of the separated liquid phases as well as the adhesion strength and thermodynamic stability of the capillary bridges are discussed. Capillary bridge formations in various compact colloid assemblies are considered. The study sheds light on a promising route to in situ (in-liquid) firming of fragile colloidal crystals and other compact colloidal microstructures via capillary bridges.

Structural characterization of surfactant aggregates adsorbed in cylindrical silica nanopores

The self-assembly of nonionic surfactants in the cylindrical pores of SBA-15 silica with a pore diameter of 8 nm was studied by small-angle neutron scattering (SANS) at different solvent contrasts. The alkyl ethoxylate surfactants C(10)E(5) and C(12)E(5) exhibit strong aggregative adsorption in the pores as indicated by the sigmoidal shape of the adsorption isotherms. The SANS intensity profiles can be represented by a sum of two terms, one accounting for diffuse scattering from surfactant aggregates in the pores and the other for Bragg scattering from the pore lattice of the silica matrix. The Bragg reflections are analyzed with a form factor model in which the radial density profile of the surfactant in the pore is approximated by a two-step function. Diffuse scattering is represented by a Teubner-Strey-type scattering function which indicates a preferred distance between adsorbed surface aggregates in the pores. Our results suggest that adsorption starts with formation of discrete surface aggregates which increase in number and eventually merge to interconnected patches as the plateau value of the adsorption isotherm is approached. A grossly different behavior, viz. formation of micelles as in solution, is found for the maltoside surfactant C(10)G(2), in agreement with the observed weak adsorption of this surfactant in SBA-15.

Phase Separation Kinetics of Binary Systems: Effects of Hydrodynamic Interaction And Surfactants

Cell dynamics computer simulation method is used to investigate effects of hydro-dynamic interaction as well as effects of added surfactants on phase separation kinetics. In the former the obtained scattering structure function for 3-dimensional system reproduces the experimental results for polymer blends remarkably well. In the latter we display self-assembling process in two-dimensional system on computer.

Incipient criticality in ecological communities

In ecology, there have been attempts to establish links between the relative species abundance (RSA), the fraction of species in a community with a given abundance, and a power-law form of the species area relationship (SAR), the dependence of species richness on sampling area. However the SAR and other patterns in ecology often do not exhibit power-law behavior over an appreciable range of scales. This raises the question whether a scaling framework can be applied when the system under analysis does not exhibit power-law behavior. Here, we derive a general finite-size scaling framework applicable to such systems that can be used to identify incipient critical behavior and links the scale dependence of the RSA and the SAR. We confirm the generality of our theory by using data from a serpentine grassland plot, which exhibits a power-law SAR, and the Barro Colorado Island plot in Panama, whose SAR shows deviations from power-law behavior at every scale. Our results demonstrate that scaling provides a model-independent framework for analyzing and unifying ecological data and that, despite the absence of power laws, ecosystems are poised in the vicinity of a critical point.

Spinodal decomposition of polymer solutions: a parallelized molecular dynamics simulation

In simulations of phase separation kinetics, large length and time scales are involved due to the mesoscopic size of the polymer coils, and the structure formation on still larger scales of length and time. We apply a coarse-grained model of hexadecane dissolved in supercritical carbon dioxide, for which in previous work the equilibrium phase behavior has been established by Monte Carlo methods. Using parallelized simulations on a multiprocessor supercomputer, large scale molecular dynamics simulations of phase separation following pressure jumps are presented for systems containing N=435136 coarse-grained particles, which correspond to several millions of atoms in a box with linear dimension 447 A . Even for large systems the phase separation can be observed up to the final, macroscopically segregated, equilibrium state. It is shown that in the segregation process the two order parameters of the system (density and concentration) are strongly coupled. The system does not follow the predicted growth law for the characteristic domain size l(t) proportional, variant t in binary fluid mixtures for the range of times accessible in the simulation. Instead, it exhibits a distinctly slower growth, presumably due to the dynamic asymmetry of the constituents.

Tilings of space and superhomogeneous point processes

We consider the construction of point processes from tilings, with equal-volume tiles, of d -dimensional Euclidean space R;{d} . We show that one can generate, with simple algorithms ascribing one or more points to each tile, point processes which are "superhomogeneous" (or "hyperuniform")-i.e., for which the structure factor S(k) vanishes when the wave vector k tends to zero. The exponent gamma characterizing the leading small- k behavior, S(k-->0) proportional, variant k(gamma), depends in a simple manner on the nature of the correlation properties of the specific tiling and on the conservation of the mass moments of the tiles. Assigning one point to the center of mass of each tile gives the exponent gamma=4 for any tiling in which the shapes and orientations of the tiles are short-range correlated. Smaller exponents in the range 4-d<gamma<4 (and thus always superhomogeneous for d< or =4 ) may be obtained in the case that the latter quantities have long-range correlations. Assigning more than one point to each tile in an appropriate way, we show that one can obtain arbitrarily higher exponents in both cases. We illustrate our results with explicit constructions using known deterministic tilings, as well as some simple stochastic tilings for which we can calculate S(k) exactly. Our results provide an explicit analytical construction of point processes with gamma>4 . Applications to condensed matter physics, and also to cosmology, are briefly discussed.

Two-point correlation properties of stochastic splitting processes

We study how the two-point density correlation properties of a point particle distribution are modified when each particle is divided, by a stochastic process, into an equal number of identical "daughter" particles. We consider generically that there may be nontrivial correlations in the displacement fields describing the positions of the different daughters of the same "mother" particle and then treat separately the cases in which there are, or are not, correlations also between the displacements of daughters belonging to different mothers. For both cases exact formulas are derived relating the structure factor (power spectrum) of the daughter distribution to that of the mothers. An application of these results is that they give explicit algorithms for generating, starting from regular lattice arrays, stochastic particle distributions with an arbitrarily high degree of large-scale uniformity. Such distributions are of interest, in particular, in the context of studies of self-gravitating systems in cosmology.

Phase separation in a compressible 2D Ising model

We perform a high precision Monte Carlo study of asymptotic domain growth in a compressible two-dimensional spin-exchange Ising model with continuous particle positions and zero total magnetization, and we investigate the effects of compressibility and lattice mismatch on the late-time domain growth law, R(t) = A + Bt(n). For mismatched systems, we measure significant deviations from the theoretically expected n = 1/3 late-time growth (n = 0.224 +/- 0.004), and for a compressible model with no mismatch, we measure only a slight deviation from n = 1/3. These results strongly suggest that the current understanding of the classes of domain growth is incomplete.

Isotropic-nematic spinodal decomposition kinetics

The initial stage of isotropic-nematic spinodal demixing kinetics of suspensions of very long and thin, stiff, repulsive rods is analyzed on the basis of the N-particle Smoluchowski equation. Equations of motion for the reduced probability density function of the position and orientation of a rod are expanded up to second order in spatial gradients and leading order in orientational order parameter. The resulting equation of motion is solved analytically, from which the temporal evolution of light-scattering patterns are calculated. It is shown that inhomogeneities in number density are enslaved by the temporal development of inhomogeneities in orientational order. Furthermore, demixing due to rotational diffusion is shown to be much faster as compared to translational diffusion. This results in an instable mode that is rotational, for which the corresponding eigenvector remains finite at zero wave vector. The scattered intensity nevertheless exhibits a maximum at a finite wave vector due to the wave-vector dependence of time-exponential prefactors. The wave vector where the intensity exhibits a maximum is therefore predicted to be a function of time even during the initial stage of demixing.

Stochastic lattice model for bone remodeling and aging

We investigate the remodeling process of trabecular bone inside a human vertebral body using a stochastic lattice model, in which the ability of living bone to adapt to mechanical stimuli is incorporated. Our simulations show the emergence of a networklike structure similar to real trabecular bone. With time, the bone volume fraction reaches a steady state. The microstructure, however, coarsens with a typical length in the system following a power law. The simulation results suggest that a coarsening of the trabecular structure should occur as a natural aging phenomenon, not related to disease.

Topological complexity and the dynamics of coarsening

Coarsening or Ostwald ripening occurs in a vast array of two-phase systems. Coarsening results in a decrease in the interfacial area per unit volume and a concomitant increase in the size scale of the interfacial morphology. Much is known about the coarsening process in two-phase mixtures consisting of a polydisperse array of spherical particles. In contrast, in many two-phase mixtures, such as those found in two-phase polymers, ceramics, dendritic solid-liquid mixtures and order-disorder transformations, the interfaces are both interconnected and have a spatially varying mean curvature. Here we show that the morphological evolution of these topologically complex systems during coarsening can be quantified by measuring the probability of finding a patch of interface with a given curvature tensor. We find that the morphological evolution is described by the flow of probability density in this curvature space that is induced by the coarsening process. The hallmark of our approach is a close coupling between experiment and theory; we use the experimentally measured three-dimensional microstructure as an input to a phase-field calculation that then determines the flow in curvature space. The methodology is general, and applicable to many systems undergoing coarsening, regardless of their topology.

Surface-directed phase separation with off-critical composition: analytical and numerical results

We study the interplay of wetting and phase separation in an unstable binary mixture (AB) with off-critical composition, placed in contact with a surface which prefers the component A. We consider surface potentials V(z) approximately z(-n), where z is the distance from the surface, and present analytical arguments and detailed numerical results to elucidate wetting-layer kinetics for arbitrary mixture compositions. If the preferred component is the minority phase, the wetting-layer thickness exhibits a potential-specific behavior at early times tau, R1 approximately tau(1/(n+2)), before crossing over to the universal growth law, R1 approximately tau(1/3). On the other hand, if the preferred component is the majority phase, there is a crossover from potential-specific growth (as before) to a slower growth regime.

Phase ordering with a global conservation law: Ostwald ripening and coalescence

Globally conserved phase ordering dynamics is investigated in systems with short range correlations at t=0. A Ginzburg-Landau equation with a global conservation law is employed as the phase field model. The conditions are found under which the sharp-interface limit of this equation is reducible to the area-preserving motion by curvature. Numerical simulations show that, for both critical and off-critical quench, the equal-time pair correlation function exhibits dynamic scaling, and the characteristic coarsening length obeys l(t) approximately t(1/2). For the critical quench, our results are in excellent agreement with earlier results. For off-critical quench (Ostwald ripening) we investigate the dynamics of the size distribution function of the minority phase domains. The simulations show that, at large times, this distribution function has a self-similar form with growth exponent 1/2. The scaled distribution, however, strongly differs from the classical Wagner distribution. We attribute this difference to coalescence of domains. A theory of Ostwald ripening is developed that takes into account binary coalescence events. The theoretical scaled distribution function agrees well with that obtained in the simulations.

Surface-directed spinodal decomposition in binary fluid mixtures

We consider the phase separation of binary fluids in contact with a surface, which is preferentially wetted by one of the components of the mixture. We review the results available for this problem and present numerical results obtained using a mesoscopic level simulation technique for the three-dimensional problem.

Scaled structures in late stages of microphase separation of binary paraffin mixtures

Since the dynamic properties in paraffin mixtures are different from those in alloys and polymer blends, paraffin mixtures are believed to be another class of substances establishing the "universality" of the dynamical scaling behavior of phase separation. In this paper, the scaled structures of microphase separation in paraffin mixtures are described using a scaling function based on a two-phase model with kinetics based on the Cahn-Hilliard equation. It has been demonstrated that at the late stages of the microphase separation the scaling behavior in the paraffin mixtures agrees well with the universal features predicted by the scaling function. The scaled structures of microphase separation in paraffin mixtures can be directly calculated from the volume fraction of the minority phase without any adjustable parameter.

Abnormal bone mineralization after fluoride treatment in osteoporosis: a small-angle x-ray-scattering study

Sodium fluoride treatment of osteoporosis is known to stimulate bone formation and to increase bone mass, but recent clinical trials failed to prove its antifracture effectiveness. The formation of bone with abnormal structure and, therefore, increased fragility is discussed as a possible explanation. Until now, however, exact information on the mineral structure of osteoporotic bone after fluoride treatment has been lacking. Bone biopsies were taken from three patients with postmenopausal osteoporosis before and after fluoride treatment (60 mg NaF/day for 1-2 years), from one patient with iatrogenic fluorosis, as well as from three normal controls. The mineral in these samples was investigated by a combination of backscattered electron imaging and small-angle x-ray scattering. Depending on the total dose of fluoride, an increasing amount of new bone is laid down on the surface of preexisting trabeculae. Its mineral structure is identical to that of heavy fluorosis and is characterized by the presence of additional large crystals, presumably located outside the collagen fibrils. These large crystals, which are not present in the controls or in osteoporotic bone before fluoride treatment, contribute to increase the mineral density without significantly improving the biomechanical properties of the bone. The possible success of fluoride treatment depends not only on the amount of newly formed bone but also on the rate of bone turnover. Indeed, as soon as significant amounts of fluoride are present, bone turnover leads to the replacement of old (normal) bone by new (pathologically mineralized) bone.(ABSTRACT TRUNCATED AT 250 WORDS)