Searcharxiv⌕ Search

arXiv subjects

Alain Karma

Publications and source records attributed to Alain Karma.

At least 55 records · Page 3Linked to original sources
arXiv2008

Phase-field crystal study of grain-boundary premelting

We study the phenomenon of grain-boundary premelting for temperatures below the melting point in the phase-field crystal model of a pure material with hexagonal ordering in two dimensions. We investigate the structures of symmetric tilt boundaries as a function of misorientation for two different inclinations and compute in the grand canonical ensemble the disjoining potential V(w) that governs the fundamental interaction between crystal-melt interfaces as a function of the premelted layer width w. The results reveal qualitatively different behaviors for high-angle grain boundaries that are uniformly wetted, with w diverging logarithmically as the melting point is approached from below, and low-angle boundaries that are punctuated by liquid pools surrounding dislocations, separated by solid bridges. This qualitative difference between high and low angle boundaries is reflected in the w-dependence of the disjoining potential that is purely repulsive (V'(w)<0 for all w) above a critical misorientation, but switches from repulsive at small w to attractive at large w for low angles. In the latter case, V(w) has a minimum that corresponds to a premelted boundary of finite width at the melting point. Furthermore, we find that the standard wetting condition (the grain boundary energy is equal to twice the solid-liquid free energy) gives a much too low estimate of the critical misorientation when a low-temperature value of the grain boundary energy is used. In contrast, a reasonable estimate is obtained if the grain boundary energy is extrapolated to the melting point, taking into account both the elastic softening of the material at high temperature and local melting around dislocations.

cond-mat.mtrl-sci↗
arXiv2008

Method for Computing Short-Range Forces between Solid-Liquid Interfaces Driving Grain Boundary Premelting

We present a molecular dynamics based method for computing accurately short-range structural forces resulting from the overlap of spatially diffuse solid-liquid interfaces at wetted grain boundaries close to the melting point. The method is based on monitoring the fluctuations of the liquid layer width at different temperatures to extract the excess interfacial free-energy as a function of this width. The method is illustrated for a high energy Sigma 9 twist boundary in pure Ni. The short-range repulsion driving premelting is found to be dominant in comparison to long-range dispersion and entropic forces and consistent with previous experimental findings that nanometer-scale layer widths may only be observed very close to the melting point.

cond-mat.mtrl-sci↗
arXiv2008

Line-Defect Patterns of Unstable Spiral Waves in Cardiac Tissue

Spiral wave propagation in period-2 excitable media is often accompanied by line-defects, the locus of points with period-1 oscillations. Here we investigate spiral line-defects in cardiac tissue where period-2 behavior has a known arrhythmogenic role. We find that the number of line defects, which is constrained to be an odd integer, is three for a freely rotating spiral, with and without meander, but one for a spiral anchored around a fixed heterogeneity. We interpret analytically this finding using a simple theory where spiral wave unstable modes with different numbers of line-defects correspond to quantized solutions of a Helmholtz equation. Furthermore, the slow inward rotation of spiral line-defects is described in different regimes.

q-bio.TO↗
arXiv2008

Laws of crack motion and phase-field models of fracture

Recently proposed phase-field models offer self-consistent descriptions of brittle fracture. Here, we analyze these theories in the quasistatic regime of crack propagation. We show how to derive the laws of crack motion either by using solvability conditions in a perturbative treatment for slight departure from the Griffith threshold, or by generalizing the Eshelby tensor to phase-field models. The analysis provides a simple physical interpretation of the second component of the classic Eshelby integral in the limit of vanishing crack propagation velocity: it gives the elastic torque on the crack tip that is needed to balance the Herring torque arising from the anisotropic interface energy. This force balance condition reduces in this limit to the principle of local symmetry in isotropic media and to the principle of maximum energy release rate for smooth curvilinear cracks in anisotropic media. It can also be interpreted physically in this limit based on energetic considerations in the traditional framework of continuum fracture mechanics, in support of its general validity for real systems beyond the scope of phase-field models. Analytical predictions of crack paths in anisotropic media are validated by numerical simulations. Simulations also show that these predictions hold even if the phase-field dynamics is modified to make the failure process irreversible. In addition, the role of dissipative forces on the process zone scale as well as the extension of the results to motion of planar cracks under pure antiplane shear are discussed.

cond-mat.mtrl-sci↗
arXiv2007

Amplitude equation approach to spatiotemporal dynamics of cardiac alternans

Amplitude equations are derived that describe the spatiotemporal dynamics of cardiac alternans during periodic pacing of one- and two-dimensional homogeneous tissue and one-dimensional anatomical reentry in a ring of homogeneous tissue. These equations provide a simple physical understanding of arrhythmogenic patterns of period-doubling oscillations of action potential duration with a spatially varying phase and amplitude as well as explicit quantitative predictions that can be compared to ionic model simulations or experiments. The form of the equations is expected to be valid for a large class of ionic models but the coefficients are only derived analytically for a two-variable ionic model and calculated numerically for the original Noble model of Purkinje fiber action potential.In paced tissue, the main result is the existence of a linear instability that produces a periodic pattern of discordant alternans. Moreover, the patterns of alternans can be either stationary, with fixed nodes, or travelling, with moving nodes and hence quasiperiodic oscillations of action potential duration, depending on the relative strength of the destabilizing effect of CV-restitution and the stabilizing effect of diffusive coupling. In both the paced geometries and the ring, the onset of alternans is different in tissue than for a paced isolated cell. The implications of these results for alternans dynamics during two-dimensional reentry are briefly discussed.

q-bio.TO↗
arXiv2007

Phase-field crystal modeling of equilibrium bcc-liquid interfaces

We investigate the equilibrium properties of bcc-liquid interfaces modeled with a continuum phase-field crystal (PFC) approach [K. R. Elder and M. Grant, Phys. Rev. E 70, 051605 (2004)]. A multiscale analysis of the PFC model is carried out which exploits the fact that the amplitudes of crystal density waves decay slowly into the liquid in the physically relevant limit where the freezing transition is weakly first order. This analysis yields a set of coupled equations for these amplitudes that is similar to the set of equations derived from Ginzburg-Landau (GL) theory [K.-A. Wu et al., Phys. Rev. E 73, 094101 (2006)]. The two sets only differ in the details of higher order nonlinear couplings between different density waves, which is determined by the form of the nonlinearity assumed in the PFC model and by the ansatz that all polygons with the same number of sides have equal weight in GL theory. Despite these differences, for parameters (liquid structure factor and solid density wave amplitude) of Fe determined from molecular dynamic (MD) simulations, the PFC and GL amplitude equations yield very similar predictions for the overall magnitude and anisotropy of the interfacial free-energy and density wave profiles. These predictions are compared with MD simulations as well as numerical solutions of the PFC model.

cond-mat.mtrl-sci↗
arXiv2006

Ginzburg-Landau theory of crystalline anisotropy for bcc-liquid interfaces

The weak anisotropy of the interfacial free-energy $γ$ is a crucial parameter influencing dendritic crystal growth morphologies in systems with atomically rough solid-liquid interfaces. The physical origin and quantitative prediction of this anisotropy are investigated for body-centered-cubic (bcc) forming systems using a Ginzburg-Landau theory where the order parameters are the amplitudes of density waves corresponding to principal reciprocal lattice vectors. We find that this theory predicts the correct sign, $γ_{100}>γ_{110}$, and magnitude, $(γ_{100}-γ_{110}) / (γ_{100}+γ_{110})\approx 1%$, of this anisotropy in good agreement with the results of MD simulations for Fe. The results show that the directional dependence of the rate of spatial decay of solid density waves into the liquid, imposed by the crystal structure, is a main determinant of anisotropy. This directional dependence is validated by MD computations of density wave profiles for different reciprocal lattice vectors for $\{110\}$ crystal faces. Our results are contrasted with the prediction of the reverse ordering $γ_{100}<γ_{110}$ from an earlier formulation of Ginzburg-Landau theory [Shih \emph{et al.}, Phys. Rev. A {\bf 35}, 2611 (1987)].

cond-mat.mtrl-sci↗
arXiv2005

Low temperature dynamics of kinks on Ising interfaces

The anisotropic motion of an interface driven by its intrinsic curvature or by an external field is investigated in the context of the kinetic Ising model in both two and three dimensions. We derive in two dimensions (2d) a continuum evolution equation for the density of kinks by a time-dependent and nonlocal mapping to the asymmetric exclusion process. Whereas kinks execute random walks biased by the external field and pile up vertically on the physical 2d lattice, then execute hard-core biased random walks on a transformed 1d lattice. Their density obeys a nonlinear diffusion equation which can be transformed into the standard expression for the interface velocity v = M[(gamma + gamma'')kappa + H]$, where M, gamma + gamma'', and kappa are the interface mobility, stiffness, and curvature, respectively. In 3d, we obtain the velocity of a curved interface near the <100> orientation from an analysis of the self-similar evolution of 2d shrinking terraces. We show that this velocity is consistent with the one predicted from the 3d tensorial generalization of the law for anisotropic curvature-driven motion. In this generalization, both the interface stiffness tensor and the curvature tensor are singular at the <100> orientation. However, their product, which determines the interface velocity, is smooth. In addition, we illustrate how this kink-based kinetic description provides a useful framework for studying more complex situations by modeling the effect of immobile dilute impurities.

cond-mat.stat-mech↗
arXiv2004

Crack Path Prediction in Anisotropic Brittle Materials

A force balance condition to predict quasistatic crack paths in anisotropic brittle materials is derived from an analysis of diffuse interface continuum models that describe both short-scale failure inside a microscopic process zone and macroscopic linear elasticity. The derivation exploits the gradient dynamics and translation symmetry properties of this class of models to define a generalized energy-momentum tensor whose integral around an arbitrary closed path enclosing the crack tip yields all forces acting on this tip, including Eshelby's configurational forces, cohesive forces, and dissipative forces. This condition is validated quantitatively by numerical simulations.

cond-mat.mtrl-sci↗
arXiv2004

Quantitative Phase Field Model of Alloy Solidification

We present a detailed derivation and thin interface analysis of a phase-field model that can accurately simulate microstructural pattern formation for low-speed directional solidification of a dilute binary alloy. This advance with respect to previous phase-field models is achieved by the addition of a phenomenological "antitrapping" solute current in the mass conservation relation [A. Karma, Phys. Rev. Lett 87, 115701 (2001)]. This antitrapping current counterbalances the physical, albeit artificially large, solute trapping effect generated when a mesoscopic interface thickness is used to simulate the interface evolution on experimental length and time scales. Furthermore, it provides additional freedom in the model to suppress other spurious effects that scale with this thickness when the diffusivity is unequal in solid and liquid [R. F. Almgren, SIAM J. Appl. Math 59, 2086 (1999)], which include surface diffusion and a curvature correction to the Stefan condition. This freedom can also be exploited to make the kinetic undercooling of the interface arbitrarily small even for mesoscopic values of both the interface thickness and the phase-field relaxation time, as for the solidification of pure melts [A. Karma and W.-J. Rappel, Phys. Rev. E 53, R3017 (1996)]. The performance of the model is demonstrated by calculating accurately for the first time within a phase-field approach the Mullins-Sekerka stability spectrum of a planar interface and nonlinear cellular shapes for realistic alloy parameters and growth conditions.

cond-mat.mtrl-sci↗
arXiv2004

Coupled dynamics of voltage and calcium in paced cardiac cells

We investigate numerically and analytically the coupled dynamics of transmembrane voltage and intracellular calcium cycling in paced cardiac cells using a detailed physiological model and its reduction to a three-dimensional discrete map. The results provide a theoretical framework to interpret various experimentally observed modes of instability ranging from electromechanically concordant and discordant alternans to quasiperiodic oscillations of voltage and calcium.

physics.bio-ph↗
arXiv2003

Phase-Field Approach for Faceted Solidification

We extend the phase-field approach to model the solidification of faceted materials. Our approach consists of using an approximate gamma-plot with rounded cusps that can approach arbitrarily closely the true gamma-plot with sharp cusps that correspond to faceted orientations. The phase-field equations are solved in the thin-interface limit with local equilibrium at the solid-liquid interface [A. Karma and W.-J. Rappel, Phys. Rev. E53, R3017 (1996)]. The convergence of our approach is first demonstrated for equilibrium shapes. The growth of faceted needle crystals in an undercooled melt is then studied as a function of undercooling and the cusp amplitude delta for a gamma-plot of the form 1+delta(|sin(theta)|+|cos(theta)|). The phase-field results are consistent with the scaling law "Lambda inversely proportional to the square root of V" observed experimentally, where Lambda is the facet length and V is the growth rate. In addition, the variation of V and Lambda with delta is found to be reasonably well predicted by an approximate sharp-interface analytical theory that includes capillary effects and assumes circular and parabolic forms for the front and trailing rough parts of the needle crystal, respectively.

cond-mat.mtrl-sci↗
arXiv2001

Eutectic colony formation: A phase field study

Eutectic two-phase cells, also known as eutectic colonies, are commonly observed during the solidification of ternary alloys when the composition is close to a binary eutectic valley. In analogy with the solidification cells formed in dilute binary alloys, colony formation is triggered by a morphological instability of a macroscopically planar eutectic solidification front due to the rejection by both solid phases of a ternary impurity that diffuses in the liquid. Here we develop a phase-field model of a binary eutectic with a dilute ternary impurity and we investigate by dynamical simulations both the initial linear regime of this instability, and the subsequent highly nonlinear evolution of the interface that leads to fully developed two-phase cells with a spacing much larger than the lamellar spacing. We find a good overall agreement with our recent linear stability analysis [M. Plapp and A. Karma, Phys. Rev. E 60, 6865 (1999)], which predicts a destabilization of the front by long-wavelength modes that may be stationary or oscillatory. A fine comparison, however, reveals that the assumption commonly attributed to Cahn that lamella grow perpendicular to the envelope of the solidification front is weakly violated in the phase-field simulations. We show that, even though weak, this violation has an important quantitative effect on the stability properties of the eutectic front. We also investigate the dynamics of fully developed colonies and find that the large-scale envelope of the composite eutectic front does not converge to a steady state, but exhibits cell elimination and tip-splitting events up to the largest times simulated.

cond-mat.mtrl-sci↗
arXiv2001

Linking Phase-Field and Atomistic Simulations to Model Dendritic Solidification in Highly Undercooled Melts

Even though our theoretical understanding of dendritic solidification is relatively well developed, our current ability to model this process quantitatively remains extremely limited. This is due to the fact that the morphological development of dendrites depends sensitively on the degree of anisotropy of capillary and/or kinetic properties of the solid-liquid interface, which is not precisely known for materials of metallurgical interest. Here we simulate the crystallization of highly undercooled nickel melts using a computationally efficient phase-field model together with anisotropic properties recently predicted by molecular dynamics simulations. The results are compared to experimental data and to the predictions of a linearized solvability theory that includes both capillary and kinetic effects at the interface.

cond-mat.mtrl-sci↗
arXiv2001

Instability and Spatiotemporal Dynamics of Alternans in Paced Cardiac Tissue

We derive an equation that governs the spatiotemporal dynamics of small amplitude alternans in paced cardiac tissue. We show that a pattern-forming linear instability leads to the spontaneous formation of stationary or traveling waves whose nodes divide the tissue into regions with opposite phase of oscillation of action potential duration. This instability is important because it creates dynamically an heterogeneous electrical substrate for inducing fibrillation if the tissue size exceeds a fraction of the pattern wavelength. We compute this wavelength analytically as a function of three basic length scales characterizing dispersion and inter-cellular electrical coupling.

cond-mat.stat-mech↗
arXiv2001

Phase-field modeling of microstructural pattern formation during directional solidification of peritectic alloys without morphological instability

During the directional solidification of peritectic alloys, two stable solid phases (parent and peritectic) grow competitively into a metastable liquid phase of larger impurity content than either solid phase. When the parent or both solid phases are morphologically unstable, i.e., for a small temperature gradient/growth rate ratio ($G/v_p$), one solid phase usually outgrows and covers the other phase, leading to a cellular-dendritic array structure closely analogous to the one formed during monophase solidification of a dilute binary alloy. In contrast, when $G/v_p$ is large enough for both phases to be morphologically stable, the formation of the microstructurebecomes controlled by a subtle interplay between the nucleation and growth of the two solid phases. The structures that have been observed in this regime (in small samples where convection effect are suppressed) include alternate layers (bands) of the parent and peritectic phases perpendicular to the growth direction, which are formed by alternate nucleation and lateral spreading of one phase onto the other as proposed in a recent model [R. Trivedi, Metall. Mater. Trans. A 26, 1 (1995)], as well as partially filled bands (islands), where the peritectic phase does not fully cover the parent phase which grows continuously. We develop a phase-field model of peritectic solidification that incorporates nucleation processes in order to explore the formation of these structures. Simulations of this model shed light on the morphology transition from islands to bands, the dynamics of spreading of the peritectic phase on the parent phase following nucleation, which turns out to be characterized by a remarkably constant acceleration, and the types of growth morphology that one might expect to observe in large samples under purely diffusive growth conditions.

cond-mat.mtrl-sci↗
arXiv2001

Phase-Field Model of Mode III Dynamic Fracture

We introduce a phenomenological continuum model for mode III dynamic fracture that is based on the phase-field methodology used extensively to model interfacial pattern formation. We couple a scalar field, which distinguishes between ``broken'' and ``unbroken'' states of the system, to the displacement field in a way that consistently includes both macroscopic elasticity and a simple rotationally invariant short scale description of breaking. We report two-dimensional simulations that yield steady-state crack motion in a strip geometry above the Griffith threshold.

cond-mat.soft↗