Three-Dimensional Icosahedral Phase Field Quasicrystal.

We investigate the formation and stability of icosahedral quasicrystalline structures using a dynamic phase field crystal model. Nonlinear interactions between density waves at two length scales stabilize three-dimensional quasicrystals. We determine the phase diagram and parameter values required for the quasicrystal to be the global minimum free energy state. We demonstrate that traits that promote the formation of two-dimensional quasicrystals are extant in three dimensions, and highlight the characteristics required for three-dimensional soft matter quasicrystal formation.

Quasicrystalline order and a crystal-liquid state in a soft-core fluid.

A two-dimensional system of soft particles interacting via a two-length-scale potential is studied. Density functional theory and Brownian dynamics simulations reveal a fluid phase and two crystalline phases with different lattice spacing. Of these the larger lattice spacing phase can form an exotic periodic state with a fraction of highly mobile particles: a crystal liquid. Near the transition between this phase and the smaller lattice spacing phase, quasicrystalline structures may be created by a competition between linear instability at one scale and nonlinear selection of the other.

Depinning, front motion, and phase slips.

Pinning and depinning of fronts bounding spatially localized structures in the forced complex Ginzburg-Landau equation describing the 1:1 resonance is studied in one spatial dimension, focusing on regimes in which the structure grows via roll insertion instead of roll nucleation at either edge. The motion of the fronts is nonlocal but can be analyzed quantitatively near the depinning transition.

Forced symmetry breaking as a mechanism for rogue bursts in a dissipative nonlinear dynamical lattice.

We propose an alternative to the standard mechanisms for the formation of rogue waves in a nonconservative, nonlinear lattice dynamical system. We consider an ordinary differential equation (ODE) system that features regular periodic bursting arising from forced symmetry breaking. We then connect such potentially exploding units via a diffusive lattice coupling and investigate the resulting spatiotemporal dynamics for different types of initial conditions (localized or extended). We find that in both cases, particular oscillators undergo extremely fast and large amplitude excursions, resembling a rogue wave burst. Furthermore, the probability distribution of different amplitudes exhibits bimodality, with peaks at both vanishing and very large amplitude. While this phenomenology arises over a range of coupling strengths, for large values thereof the system eventually synchronizes and the above phenomenology is suppressed. We use both distributed (such as a synchronization order parameter) and individual oscillator diagnostics to monitor the dynamics and identify potential precursors to large amplitude excursions. We also examine similar behavior with amplitude-dependent diffusive coupling.

Homoclinic snaking in bounded domains.

Homoclinic snaking is a term used to describe the back and forth oscillation of a branch of time-independent spatially localized states in a bistable spatially reversible system as the localized structure grows in length by repeatedly adding rolls on either side. On the real line this process continues forever. In finite domains snaking terminates once the domain is filled but the details of how this occurs depend critically on the choice of boundary conditions. With periodic boundary conditions the snaking branches terminate on a branch of spatially periodic states. However, with non-Neumann boundary conditions they turn continuously into a large amplitude filling state that replaces the periodic state. This behavior, shown here in detail for the Swift-Hohenberg equation, explains the phenomenon of "snaking without bistability," recently observed in simulations of binary fluid convection by Mercader et al. Phys. Rev. E 80, 025201 (2009).

Eckhaus instability and homoclinic snaking.

Homoclinic snaking is a term used to describe the back and forth oscillation of a branch of time-independent spatially localized states in a bistable, spatially reversible system as the localized structure grows in length by repeatedly adding rolls on either side. This behavior is simplest to understand within the subcritical Swift-Hohenberg equation, but is also present in the subcritical regime of doubly diffusive convection driven by horizontal gradients. In systems that are unbounded in one spatial direction homoclinic snaking continues indefinitely as the localized structure grows to resemble a spatially periodic state of infinite extent. In finite domains or in periodic domains with finite spatial period the process must terminate. In this paper we show that the snaking branches in general turn over once the length of the localized state becomes comparable to the domain, and examine the factors that determine the location of the termination point or points, and their relation to the Eckhaus instability of the spatially periodic state.

Bifurcation structure of localized states in the Lugiato-Lefever equation with anomalous dispersion.

The origin, stability, and bifurcation structure of different types of bright localized structures described by the Lugiato-Lefever equation are studied. This mean field model describes the nonlinear dynamics of light circulating in fiber cavities and microresonators. In the case of anomalous group velocity dispersion and low values of the intracavity phase detuning these bright states are organized in a homoclinic snaking bifurcation structure. We describe how this bifurcation structure is destroyed when the detuning is increased across a critical value, and determine how a bifurcation structure known as foliated snaking emerges.

Spatiotemporal canards in neural field equations.

Canards are special solutions to ordinary differential equations that follow invariant repelling slow manifolds for long time intervals. In realistic biophysical single-cell models, canards are responsible for several complex neural rhythms observed experimentally, but their existence and role in spatially extended systems is largely unexplored. We identify and describe a type of coherent structure in which a spatial pattern displays temporal canard behavior. Using interfacial dynamics and geometric singular perturbation theory, we classify spatiotemporal canards and give conditions for the existence of folded-saddle and folded-node canards. We find that spatiotemporal canards are robust to changes in the synaptic connectivity and firing rate. The theory correctly predicts the existence of spatiotemporal canards with octahedral symmetry in a neural field model posed on the unit sphere.

Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system.

A subcritical pattern-forming system with nonlinear advection in a bounded domain is recast as a slow-fast system in space and studied using a combination of geometric singular perturbation theory and numerical continuation. Two types of solutions describing the possible location of stationary fronts are identified, whose origin is traced to the onset of convective and absolute instability when the system is unbounded. The former are present only for non-zero upstream boundary conditions and provide a quantitative understanding of noise-sustained structures in systems of this type. The latter correspond to the onset of a global mode and are present even with zero upstream boundary conditions. The role of canard trajectories in the nonlinear transition between these states is clarified and the stability properties of the resulting spatial structures are determined. Front location in the convective regime is highly sensitive to the upstream boundary condition, and its dependence on this boundary condition is studied using a combination of numerical continuation and Monte Carlo simulations of the partial differential equation. Statistical properties of the system subjected to random or stochastic boundary conditions at the inlet are interpreted using the deterministic slow-fast spatial dynamical system.

Soft-core particles freezing to form a quasicrystal and a crystal-liquid phase.

Systems of soft-core particles interacting via a two-scale potential are studied. The potential is responsible for peaks in the structure factor of the liquid state at two different but comparable length scales and a similar bimodal structure is evident in the dispersion relation. Dynamical density functional theory in two dimensions is used to identify two unusual states of this system: a crystal-liquid state, in which the majority of the particles are located on lattice sites but a minority remains free and so behaves like a liquid, and a 12-fold quasicrystalline state. Both are present even for deeply quenched liquids and are found in a regime in which the liquid is unstable with respect to modulations on the smaller scale only. As a result, the system initially evolves towards a small-scale crystal state; this state is not a minimum of the free energy, however, and so the system subsequently attempts to reorganize to generate the lower-energy larger-scale crystals. This dynamical process generates a disordered state with quasicrystalline domains and takes place even when this large scale is linearly stable, i.e., it is a nonlinear process. With controlled initial conditions, a perfect quasicrystal can form. The results are corroborated using Brownian dynamics simulations.

Long-wavelength instabilities of three-dimensional patterns.

Long-wavelength instabilities of steady patterns, spatially periodic in three dimensions, are studied. All potentially stable patterns with the symmetries of the simple-, face-centered- and body-centered-cubic lattices are considered. The results generalize the well-known Eckhaus, zigzag, and skew-varicose instabilities to three-dimensional patterns and are applied to two-species reaction-diffusion equations modeling the Turing instability.

Zigzag and varicose instabilities of a localized stripe.

A localized stripe solution to a reaction-diffusion equation can lose stability simultaneously to zigzag and varicose perturbations at a codimension-two point. The resulting mode interaction is described by O(2)xZ(2) equivariant amplitude equations. Stationary mixed-mode solutions are found which can undergo instabilities to breathing modes or to patterns which travel along the stripe.

Sectrofluorimetric micromethod for determining riboflavin in the blood of new-born babies and their mothers.

A microdetermination of riboflavin on the blood of new-born babies and their mothers was developed on the basis of hydrolysis of the blood in a trichloroacetic acid medium, separation of riboflavin and FMN on a Florisil column and measurement of the content spectrofluorimetrically by the standard addition method after elution with collidine buffer. The sensitivity of the method is 0.01 microgram/ml, the blood sample size 0.5--1.0 ml. The normal level in new-born babies in the first day of life was found to have an average value of 17.1+/-2.4 microgram/100 ml, in women 14.2+/-2.5 microgram/100 ml; the contents in cord blood and maternal vein blood were also determined.

Interaction of bilirubin with human lymphocytes and granulocytes--the effect on bilirubin metabolism.

An in vitro degradation of bilirubin by isolated human granulocytes and lymphocytes is described. Changes in the bilirubin concentration during interaction with these cells were determined spectrophotometrically. The dependence of the reaction velocity on the original bilirubin concentration followed an adsorption isotherm, typical of enzyme processes with a Km of 79 microM at a cell concentration of 500/microL. The primary event is the adsorption of bilirubin to the cell surface and, in addition, its detoxication. Cyanide and sonic treatment of cells inhibit the reaction, salicylic acid enhances the cell activity. The bilirubin-detoxicating effect is discussed with respect to the therapy of hyperbilirubinaemia in neonates.

[Photoisomers of bilirubin in the blood of neonates during phototherapy]. / Fotoizoméry bilirubinu v krvi novorozencu pri fototerapii.

In a group of mature neonates with hyperbilirubinaemia the authors investigated by chromatography, using the HPLC method, levels of bilirubin photoisomers before phototherapy, in the course of phototherapy and after its termination. The configuration isomer 4 Z, 15 E is detected in blood of all icteric neonates already before the onset of treatment in a mean concentration of 5.2 (s = 3.8) mumol/l, during phototherapy its mean concentration is 23.0 (s = 8.0) mumol/l. Photoisomers participate in non-conjugated bilirubinaemia on average by 10%: 92% are formed by isomer 4 Z, 15 E, 5% by isomer 4 E, 15 Z and 3% by the structural isomer lumirubin. On the day following discontinuation of treatment the mean photoisomer concentrations are significantly lower than during phototherapy and significantly higher than before its initiation.

Solidification in soft-core fluids: Disordered solids from fast solidification fronts.

Using dynamical density functional theory we calculate the speed of solidification fronts advancing into a quenched two-dimensional model fluid of soft-core particles. We find that solidification fronts can advance via two different mechanisms, depending on the depth of the quench. For shallow quenches, the front propagation is via a nonlinear mechanism. For deep quenches, front propagation is governed by a linear mechanism and in this regime we are able to determine the front speed via a marginal stability analysis. We find that the density modulations generated behind the advancing front have a characteristic scale that differs from the wavelength of the density modulation in thermodynamic equilibrium, i.e., the spacing between the crystal planes in an equilibrium crystal. This leads to the subsequent development of disorder in the solids that are formed. In a one-component fluid, the particles are able to rearrange to form a well-ordered crystal, with few defects. However, solidification fronts in a binary mixture exhibiting crystalline phases with square and hexagonal ordering generate solids that are unable to rearrange after the passage of the solidification front and a significant amount of disorder remains in the system.

Modeling the structure of liquids and crystals using one- and two-component modified phase-field crystal models.

A modified phase-field crystal model in which the free energy may be minimized by an order parameter profile having isolated bumps is investigated. The phase diagram is calculated in one and two dimensions and we locate the regions where modulated and uniform phases are formed and also regions where localized states are formed. We investigate the effectiveness of the phase-field crystal model for describing fluids and crystals with defects. We further consider a two-component model and elucidate how the structure transforms from hexagonal crystalline ordering to square ordering as the concentration changes. Our conclusion contains a discussion of possible interpretations of the order parameter field.

Solidification fronts in supercooled liquids: how rapid fronts can lead to disordered glassy solids.

We determine the speed of a crystallization (or, more generally, a solidification) front as it advances into the uniform liquid phase after the system has been quenched into the crystalline region of the phase diagram. We calculate the front speed by assuming a dynamical density functional theory (DDFT) model for the system and applying a marginal stability criterion. Our results also apply to phase field crystal (PFC) models of solidification. As the solidification front advances into the unstable liquid phase, the density profile behind the advancing front develops density modulations and the wavelength of these modulations is a dynamically chosen quantity. For shallow quenches, the selected wavelength is precisely that of the crystalline phase and so well-ordered crystalline states are formed. However, when the system is deeply quenched, we find that this wavelength can be quite different from that of the crystal, so the solidification front naturally generates disorder in the system. Significant rearrangement and aging must subsequently occur for the system to form the regular well-ordered crystal that corresponds to the free energy minimum. Additional disorder is introduced whenever a front develops from random initial conditions. We illustrate these findings with simulation results obtained using the PFC model.

Swift-Hohenberg equation with broken cubic-quintic nonlinearity.

The cubic-quintic Swift-Hohenberg equation (SH35) provides a convenient order parameter description of several convective systems with reflection symmetry in the layer midplane, including binary fluid convection. We use SH35 with an additional quadratic term to determine the qualitative effects of breaking the midplane reflection symmetry on the properties of spatially localized structures in these systems. Our results describe how the snakes-and-ladders organization of localized structures in SH35 deforms with increasing symmetry breaking and show that the deformation ultimately generates the snakes-and-ladders structure familiar from the quadratic-cubic Swift-Hohenberg equation. Moreover, in nonvariational systems, such as convection, odd-parity convectons necessarily drift when the reflection symmetry is broken, permitting collisions among moving localized structures. Collisions between both identical and nonidentical traveling states are described.