Nonlinear waves in a quintic FitzHugh-Nagumo model with cross diffusion: Fronts, pulses, and wave trains.

We study a tristable piecewise-linear reaction-diffusion system, which approximates a quintic FitzHugh-Nagumo model, with linear cross-diffusion terms of opposite signs. Basic nonlinear waves with oscillatory tails, namely, fronts, pulses, and wave trains, are described. The analytical construction of these waves is based on the results for the bistable case [Zemskov et al., Phys. Rev. E 77, 036219 (2008) and Phys. Rev. E 95, 012203 (2017) for fronts and for pulses and wave trains, respectively]. In addition, these constructions allow us to describe novel waves that are specific to the tristable system. Most interesting is the pulse solution with a zigzag-shaped profile, the bright-dark pulse, in analogy with optical solitons of similar shapes. Numerical simulations indicate that this wave can be stable in the system with asymmetric thresholds; there are no stable bright-dark pulses when the thresholds are symmetric. In the latter case, the pulse splits up into a tristable front and a bistable one that propagate with different speeds. This phenomenon is related to a specific feature of the wave behavior in the tristable system, the multiwave regime of propagation, i.e., the coexistence of several waves with different profile shapes and propagation speeds at the same values of the model parameters.

Phase-field study of crystal growth in three-dimensional capillaries: Effects of crystalline anisotropy.

Phase-field simulations are performed to explore the thermal solidification of a pure melt in three-dimensional capillaries. Motivated by our previous work for isotropic or slightly anisotropic materials, we focus here on the more general case of anisotropic materials. Different channel cross sections are compared (square, hexagonal, circular) to reveal the influence of geometry and the effects of a competition between the crystal and the channel symmetries. In particular, a compass effect toward growth directions favored by the surface energy is identified. At given undercooling and anisotropy, the simulations generally show the coexistence of several growth modes. The relative stability of these growth modes is tested by submitting them to a strong spatiotemporal noise for a short time, which reveals a subtle hierarchy between them. Similarities and differences with experimental growth modes in confined geometry are discussed qualitatively.

Protocol-independent granular temperature supported by numerical simulations.

A possible approach to the statistical description of granular assemblies starts from Edwards's assumption that all blocked states occupying the same volume are equally probable [Edwards and Oakeshott, Physica A 157, 1080 (1989)]. We performed computer simulations using two-dimensional polygonal particles excited periodically according to two different protocols: excitation by pulses of "negative gravity" and excitation by "rotating gravity." The first protocol exhibits a nonmonotonous dependency of the mean volume fraction on the pulse strength. The overlapping histogram method is used in order to test whether the volume distribution is described by a Boltzmann-like distribution and to calculate the inverse compactivity as well as the logarithm of the partition sum. We find that the mean volume is a unique function of the measured granular temperature, independently of the protocol and of the branch in ϕ(g), and that all determined quantities are in agreement with Edwards's theory.

Crystal growth in a channel: pulsating fingers, merry-go-round patterns, and seesaw dynamics.

We perform phase-field simulations of unsteady crystal growth in a three-dimensional capillary. Motivated by the appearance of chirality-symmetry breaking periodic states in our preceding study, we here focus on more general dynamic states. Most of these are obtained in the limit of isotropic surface tension, but we test genericity by looking at a few cases with weak anisotropy. Whereas steady states are similar for all channel shapes studied so far, including channels with circular, hexagonal, quadratic, and triangular cross sections, there is a stronger dependence on the cross section for time-dependent states. Various oscillatory modes are identified and discussed, including rotating and swinging patterns as well as pulsating modes containing one, two, and four fingers, respectively.

Selection theory of free dendritic growth in a potential flow.

The Kruskal-Segur approach to selection theory in diffusion-limited or Laplacian growth is extended via combination with the Zauderer decomposition scheme. This way nonlinear bulk equations become tractable. To demonstrate the method, we apply it to two-dimensional crystal growth in a potential flow. We omit the simplifying approximations used in a preliminary calculation for the same system [Fischaleck, Kassner, Europhys. Lett. 81, 54004 (2008)], thus exhibiting the capability of the method to extend mathematical rigor to more complex problems than hitherto accessible.

Phase-field study of solidification in three-dimensional channels.

Three-dimensional solidification of a pure material with isotropic properties of the solid phase is studied in cylindrical capillaries of various cross sections (circular, hexagonal, and square). As the undercooling is increased, we find, depending on the width of the capillary, a number of different growth modes and dynamical behaviors, including stationary symmetric single fingers, stationary asymmetric fingers, and oscillating double and quadruple fingers. Chaotic states are also observed, some of them in unexpected parameter regions. Our simulations suggest that the bifurcation from symmetric to asymmetric fingers is supercritical. We discuss the nature of the oscillatory states, one of which is chirality breaking, and the origin of the unexpected chaotic finger. Bifurcation diagrams are given comparing three different ratios of capillary length to channel width in the hexagonal channel as well as the three different geometries.

Comparison of phase-field models for surface diffusion.

The description of surface-diffusion controlled dynamics via the phase-field method is less trivial than it appears at first sight. A seemingly straightforward approach from the literature is shown to fail to produce the correct asymptotics, albeit in a subtle manner. Two models are constructed that approximate known sharp-interface equations without adding undesired constraints. Numerical simulations of the standard and a more sophisticated model from the literature as well as of our two models are performed to assess the relative merits of each approach. The results suggest superior performance of the models in at least some situations.

Nonlinear study of symmetry breaking in actin gels: implications for cellular motility.

Force generation by actin polymerization is an important step in cellular motility and can induce the motion of organelles or bacteria, which move inside their host cells by trailing an actin tail behind. Biomimetic experiments on beads and droplets have identified the biochemical ingredients to induce this motion, which requires a spontaneous symmetry breaking in the absence of external fields. We find that the symmetry breaking can be captured on the basis of elasticity theory and linear flux-force relationships. Furthermore, we develop a phase-field approach to study the fully nonlinear regime and show that actin-comet formation is a robust feature, triggered by growth and mechanical stresses. We discuss the implications of symmetry breaking for self-propulsion.

Influence of external flows on crystal growth: numerical investigation.

We use a combined phase-field-lattice-Boltzmann scheme [Medvedev and Kassner, Phys. Rev. E 72, 056703 (2005)] to simulate nonfaceted crystal growth from an undercooled melt in external flows. Selected growth parameters are determined numerically. For growth patterns at moderate to high undercooling and relatively large anisotropy, the values of the tip radius and selection parameter plotted as a function of the Péclet number fall approximately on single curves. Hence, it may be argued that a parallel flow changes the selected tip radius and growth velocity solely by modifying (increasing) the Péclet number. This has interesting implications for the availability of current selection theories as predictors of growth characteristics under flow. At smaller anisotropy, a modification of the morphology diagram in the plane of undercooling versus anisotropy is observed. The transition line from dendrites to doublons is shifted in favor of dendritic patterns, which become faster than doublons as the flow speed is increased, thus rendering the basin of attraction of dendritic structures larger. For small anisotropy and Prandtl number, we find oscillations of the tip velocity in the presence of flow. On increasing the fluid viscosity or decreasing the flow velocity, we observe a reduction in the amplitude of these oscillations.

Phase field modeling of fast crack propagation.

We present a continuum theory which predicts the steady state propagation of cracks. The theory overcomes the usual problem of a finite time cusp singularity of the Grinfeld instability by the inclusion of elastodynamic effects which restore selection of the steady state tip radius and velocity. We developed a phase-field model for elastically induced phase transitions; in the limit of small or vanishing elastic coefficients in the new phase, fracture can be studied. The simulations confirm analytical predictions for fast crack propagation.

Phase-field approach to three-dimensional vesicle dynamics.

We extend our recent phase-field [T. Biben and C. Misbah, Phys. Rev. E 67, 031908 (2003)] approach to 3D vesicle dynamics. Unlike the boundary-integral formulations, based on the use of the Oseen tensor in the small Reynolds number limit, this method offers several important flexibilities. First, there is no need to track the membrane position; rather this is automatically encoded in dynamics of the phase field to which we assign a finite width representing the membrane extent. Secondly, this method allows naturally for any topology change, like vesicle budding, for example. Thirdly, any non-Newtonian constitutive law, that is generically nonlinear, can be naturally accounted for, a fact which is precluded by the boundary integral formulation. The phase-field approach raises, however, a complication due to the local membrane incompressibility, which, unlike usual interfacial problems, imposes a nontrivial constraint on the membrane. This problem is solved by introducing dynamics of a tension field. The first purpose of this paper is to show how to write adequately the advected-field model for 3D vesicles. We shall then perform a singular expansion of the phase field equation to show that they reduce, in the limit of a vanishing membrane extent, to the sharp boundary equations. Then, we present some results obtained by the phase-field model. We consider two examples; (i) kinetics towards equilibrium shapes and (ii) tanktreading and tumbling. We find a very good agreement between the two methods. We also discuss briefly how effects, such as the membrane shear elasticity and stretching elasticity, and the relative sliding of monolayers, can be accounted for in the phase-field approach.

Lattice Boltzmann scheme for crystal growth in external flows.

A composite phase-field lattice-Boltzmann scheme is used to simulate dendritic growth from a supercooled melt, allowing for heat transport by both diffusion and convection. The phase-transition part of the problem is modelled by the phase-field approach of Karma and Rappel, whereas the flow of the liquid is computed by the lattice-Boltzmann-BGK (LBGK, referring to Bhatnagar, Gross, and Krook) method into which interactions with solid and thermal convection are incorporated. For simplicity, we have so far restricted ourselves to the symmetric model. Heat transport is simulated via the multicomponent LBGK method. Depending on the level of anisotropy and undercooling, dendrites or doublons are obtained in our simulations. Crystal growth in a shear flow is considered for different flow velocities and undercoolings. Doublons turn out to be robust against the perturbation imposed by a shear flow and display interesting dynamic behavior, quite different from that of dendrites. In addition, the influence of a parallel flow on the operating state of the tip of dendrites is studied. To complement information from selection theories such as the one presented by Bouissou and Pelcé, we measure selected growth characteristics of dendrites as a function of a flow imposed parallel to the growth direction, for intermediate undercoolings. The observed dependencies are compatible with power law behavior, if the undercooling is not too high. It is shown that for sufficiently large flow velocities, an oncoming flow can lead to tip oscillations of the dendrite and, consequently, to the generation of coherent side branches.

Growth patterns in a channel for singular surface energy: phase-field model.

We study solidification in a two-dimensional channel for faceted materials whose facets correspond to cusps in the gamma plot. The main result is the existence of three growth modes, according to the anisotropy strength: a single faceted finger at high anisotropies, two faceted fingers in the intermediate range, and an oscillating mode at low anisotropies. Simple geometrical and dynamical models are proposed to explain the nature of the observed modes. In particular, the one-finger patterns are shown to be similar to free dendrites while the two-finger patterns correspond to confined solidification fingers.

Pattern selection in biaxially stressed solids.

We analyze the morphological instability of the surface of a solid which is subject to a biaxial stress. The stability calculation reveals a new favored pattern: a diamond morphology. This occurs if the stress is tensile in one direction and compressive in the orthogonal one and the ratio exceeds a certain value. A nonlinear analysis shows that the bifurcation is subcritical and hints to a nontrivial competition between tilted stripes and diamonds. This study opens a new line of inquiries in the field of stress-induced pattern selection.

Amplitude equations for systems with long-range interactions.

We derive amplitude equations for interface dynamics in pattern forming systems with long-range interactions. The basic condition for the applicability of the method developed here is that the bulk equations are linear and solvable by integral transforms. We arrive at the interface equation via long-wave asymptotics. As an example, we treat the Grinfeld instability, and we also give a result for the Saffman-Taylor instability. It turns out that the long-range interaction survives the long-wave limit and shows up in the final equation as a nonlocal and nonlinear term, a feature that to our knowledge is not shared by any other known long-wave equation. The form of this particular equation will then allow us to draw conclusions regarding the universal dynamics of systems in which nonlocal effects persist at the level of the amplitude description.

Influence of ohmic heating on the flow field in thin-layer electrodeposition.

In thin-layer electrodeposition the dissipated electrical energy leads to a substantial heating of the ion solution. We measured the resulting temperature field by means of an infrared camera. The properties of the temperature field correspond closely with the development of the concentration field. In particular, we find that the thermal gradients at the electrodes act similar to a weak additional driving force to the convection rolls driven by concentration gradients.

Experimental investigation of the initial regime in fingering electrodeposition: dispersion relation and velocity measurements.

Recently a fingering morphology, resembling the hydrodynamic Saffman-Taylor instability, was identified in the quasi-two-dimensional electrodeposition of copper. We present here measurements of the dispersion relation of the growing front. The instability is accompanied by gravity-driven convection rolls at the electrodes, which are examined using particle image velocimetry. While at the anode the theory presented by Chazalviel et al. [J. Electroanal. Chem. 407, 61 (1996)] describes the convection roll, the flow field at the cathode is more complicated because of the growing deposit. In particular, the analysis of the orientation of the velocity vectors reveals some lag of the development of the convection roll compared to the finger envelope.