Pinning of crack fronts by hard and soft inclusions: A phase field study.

Through tridimensonal numerical simulations of cracks propagating in material with an elastic moduli heterogeneity, it is shown that the presence of a simple inclusion can dramatically affect the propagation of the crack. Both the presence of soft and hard inclusions can lead to the arrest of a crack front. Here the mechanism leading to the arrest of the crack are described and shown to depend on the nature of the inclusion. This is also the case in regimes where the presence of the inclusion leads to a slowdown of the crack.

Regularized anisotropic motion-by-curvature in phase-field theory: Interface phase separation of crystal surfaces.

The kinetic equation for anisotropic motion-by-curvature is ill posed when the surface energy is strongly anisotropic. In this case, corners or edges are present on the Wulff shape, which span a range of missing orientations. In the sharp-interface problem the surface energy is augmented with a curvature-dependent term that rounds the corners and regularizes the dynamic equations. This introduces a new length scale in the problem, the corner size. In phase-field theory, a diffuse description of the interface is adopted. In this context, an approximation of the Willmore energy can be added to the phase-field energy so as to regularize the model. In this paper, we discuss the convergence of the Allen-Cahn version of the regularized phase-field model toward the sharp-interface theory for strongly anisotropic motion-by-curvature in three dimensions. Corners at equilibrium are also compared to theory for different corner sizes. Then we investigate the dynamics of the faceting instability, when initially unstable surfaces decompose into stable facets. For crystal surfaces with trigonal symmetry, we find the following scaling law L∼t^{1/3}, for the growth in time t of a characteristic morphological length scale L, and coarsening is found to proceed by either edge contraction or cube removal, as in the sharp-interface problem. Finally, we study nucleation of crystal surfaces in a two-phase system, as for a terrace-and-step surface. We find that, as compared with saddle-point nucleation, ridge crossing is dynamically favored. However, the induced nucleation mechanism, when a facet induces at its wake formation of additional facets, is not evidenced with a type-A dynamics.

Kinetics of coarsening have dramatic effects on the microstructure: Self-similarity breakdown induced by viscosity contrast.

The viscous coarsening of a phase separated mixture is studied and the effects of the viscosity contrast between the phases are investigated. From an analysis of the microstructure, it appears that for moderate departure from the perfectly symmetric regime the self-similar bicontinuous regime is robust. However, the connectivity of one phase decreases when its volume fraction decreases or when it is becoming less viscous than the complementary phase. Eventually self-similarity breakdown is observed and characterized.

Grain coarsening in two-dimensional phase-field models with an orientation field.

In the literature, contradictory results have been published regarding the form of the limiting (long-time) grain size distribution (LGSD) that characterizes the late stage grain coarsening in two-dimensional and quasi-two-dimensional polycrystalline systems. While experiments and the phase-field crystal (PFC) model (a simple dynamical density functional theory) indicate a log-normal distribution, other works including theoretical studies based on conventional phase-field simulations that rely on coarse grained fields, like the multi-phase-field (MPF) and orientation field (OF) models, yield significantly different distributions. In a recent work, we have shown that the coarse grained phase-field models (whether MPF or OF) yield very similar limiting size distributions that seem to differ from the theoretical predictions. Herein, we revisit this problem, and demonstrate in the case of OF models [R. Kobayashi, J. A. Warren, and W. C. Carter, Physica D 140, 141 (2000)PDNPDT0167-278910.1016/S0167-2789(00)00023-3; H. Henry, J. Mellenthin, and M. Plapp, Phys. Rev. B 86, 054117 (2012)PRBMDO1098-012110.1103/PhysRevB.86.054117] that an insufficient resolution of the small angle grain boundaries leads to a log-normal distribution close to those seen in the experiments and the molecular scale PFC simulations. Our paper indicates, furthermore, that the LGSD is critically sensitive to the details of the evaluation process, and raises the possibility that the differences among the LGSD results from different sources may originate from differences in the detection of small angle grain boundaries.

Topological defects in two-dimensional orientation-field models for grain growth.

Standard two-dimensional orientation-field-based phase-field models rely on a continuous scalar field to represent crystallographic orientation. The corresponding order parameter space is the unit circle, which is not simply connected. This topological property has important consequences for the resulting multigrain structures: (i) trijunctions may be singular; (ii) for each pair of grains there exist two different grain boundary solutions that cannot continuously transform to one another; (iii) if both solutions appear along a grain boundary, a topologically stable, singular point defect must exist between them. While (i) can be interpreted in the classical picture of grain boundaries, (ii) and therefore (iii) cannot. In addition, singularities cause difficulties, such as lattice pinning in numerical simulations. To overcome these problems, we propose two formulations of the model. The first is based on a three-component unit vector field, while in the second we utilize a two-component vector field with an additional potential. In both cases, the additional degree of freedom introduced makes the order parameter space simply connected, which removes the topological stability of these defects.

An individual-based model for biofilm formation at liquid surfaces.

The bacterium Bacillus subtilis frequently forms biofilms at the interface between the culture medium and the air. We present a mathematical model that couples a description of bacteria as individual discrete objects to the standard advection-diffusion equations for the environment. The model takes into account two different bacterial phenotypes. In the motile state, bacteria swim and perform a run-and-tumble motion that is biased toward regions of high oxygen concentration (aerotaxis). In the matrix-producer state they excrete extracellular polymers, which allows them to connect to other bacteria and to form a biofilm. Bacteria are also advected by the fluid, and can trigger bioconvection. Numerical simulations of the model reproduce all the stages of biofilm formation observed in laboratory experiments. Finally, we study the influence of various model parameters on the dynamics and morphology of biofilms.

Macropore formation in p-type silicon: toward the modeling of morphology.

The formation of macropores in silicon during electrochemical etching processes has attracted much interest. Experimental evidences indicate that charge transport in silicon and in the electrolyte should realistically be taken into account in order to be able to describe the macropore morphology. However, up to now, none of the existing models has the requested degree of sophistication to reach such a goal. Therefore, we have undertaken the development of a mathematical model (phase-field model) to describe the motion and shape of the silicon/electrolyte interface during anodic dissolution. It is formulated in terms of the fundamental expression for the electrochemical potential and contains terms which describe the process of silicon dissolution during electrochemical attack in a hydrofluoric acid (HF) solution. It should allow us to explore the influence of the physical parameters on the etching process and to obtain the spatial profiles across the interface of various quantities of interest, such as the hole concentration, the current density, or the electrostatic potential. As a first step, we find that this model correctly describes the space charge region formed at the silicon side of the interface.

Tensorial mobilities for accurate solution of transport problems in models with diffuse interfaces.

The general problem of two-phase transport in phase-field models is analyzed: the flux of a conserved quantity is driven by the gradient of a potential through a medium that consists of domains of two distinct phases which are separated by diffuse interfaces. It is shown that the finite thickness of the interfaces induces two effects that are not present in the analogous sharp-interface problem: a surface excess current and a potential jump at the interfaces. It is shown that both effects can be eliminated simultaneously only if the coefficient of proportionality between flux and potential gradient (mobility) is allowed to become a tensor in the interfaces. This opens the possibility for precise and efficient simulations of transport problems with finite interface thickness.

Nonequilibrium ribbon model of twisted scroll waves.

We formulate a reduced model to analyze the motion of the core of a twisted scroll wave. The model is first shown to provide a simple description of the onset and nonlinear evolution of the helical state appearing in the sproing bifurcation of scroll waves. It then serves to examine the experimentally studied case of a medium with spatially varying excitability. The model shows the role of sproing in this more complex setting and highlights the differences between the convective and absolute sproing instabilities.

Dynamics of conduction blocks in a model of paced cardiac tissue.

We study numerically the dynamics of conduction blocks using a detailed electrophysiological model. We find that this dynamics depends critically on the size of the paced region. Small pacing regions lead to stationary conduction blocks while larger pacing regions can lead to conduction blocks that travel periodically towards the pacing region. We show that this size-dependence dynamics can lead to a novel arrhythmogenic mechanism. Furthermore, we show that the essential phenomena can be captured in a much simpler coupled-map model.

Dynamic instabilities of fracture under biaxial strain using a phase field model.

We present a phase-field model of the propagation of fracture under plane strain. This model, based on simple physical considerations, is able to accurately reproduce the different behavior of cracks (the principle of local symmetry, the Griffith and Irwin criteria, and mode-I branching). In addition, we test our model against recent experimental findings showing the presence of oscillating cracks under biaxial load. Our model again reproduces well observed supercritical Hopf bifurcation and is therefore the first simulation which does so.

Spiral wave drift in an electric field and scroll wave instabilities.

Here, I present the numerical computation of speed and direction of the drift of a spiral wave in an excitable medium in the presence of an electric field. The drift speed presents a strong variation close to the parameter value where the drift-speed component along the field direction from parallel becomes antiparallel. Using a simple phenomenological model and results from a numerical linear stability analysis of scroll waves, I show that this behavior can be attributed to a resonance of the meander modes with the translation modes of the spiral wave. Extending this phenomenological model to scroll waves also clarifies the link between the drift and long wavelength instabilities of scroll waves.

The role of M cells and the long QT syndrome in cardiac arrhythmias: simulation studies of reentrant excitations using a detailed electrophysiological model.

In this numerical study, we investigate the role of intrinsic heterogeneities of cardiac tissue due to M cells in the generation and maintenance of reentrant excitations using the detailed Luo-Rudy dynamic model. This model has been extended to include a description of the long QT 3 syndrome, and is studied in both one dimension, corresponding to a cable traversing the ventricular wall, and two dimensions, representing a transmural slice. We focus on two possible mechanisms for the generation of reentrant events. We first investigate if early-after-depolarizations occurring in M cells can initiate reentry. We find that, even for large values of the long QT strength, the electrotonic coupling between neighboring cells prevents early-after-depolarizations from creating a reentry. We then study whether M cell domains, with their slow repolarization, can function as wave blocks for premature stimuli. We find that the inclusion of an M cell domain can result in some cases in reentrant excitations and we determine the lifetime of the reentry as a function of the size and geometry of the domain and of the strength of the long QT syndrome.

Wave nucleation rate in excitable systems in the low noise limit.

Motivated by recent experiments on intracellular calcium dynamics, we study the general issue of fluctuation-induced nucleation of waves in excitable media. We utilize a stochastic Fitzhugh-Nagumo model for this study, a spatially extended nonpotential pair of equations driven by thermal (i.e., white) noise. The nucleation rate is determined by finding the most probable escape path via minimization of an action related to the deviation of the fields from their deterministic trajectories. Our results pave the way both for studies of more realistic models of calcium dynamics as well as of nucleation phenomena in other nonequilibrium pattern-forming processes.

Scroll waves in isotropic excitable media: linear instabilities, bifurcations, and restabilized states.

Scroll waves are three-dimensional analogs of spiral waves. The linear stability spectrum of untwisted and twisted scroll waves is computed for a two-variable reaction-diffusion model of an excitable medium. Different bands of modes are seen to be unstable in different regions of parameter space. The corresponding bifurcations and bifurcated states are characterized by performing direct numerical simulations. In addition, computations of the adjoint linear stability operator eigenmodes are also performed and serve to obtain a number of matrix elements characterizing the long-wavelength deformations of scroll waves.