arXiv subjects

Alain Karma

Publications and source records attributed to Alain Karma.

At least 37 records · Page 2Linked to original sources

arXiv2019

Phase Field Modeling of Chemomechanical Fracture of Intercalation Electrodes: Role of Charging Rate and Dimensionality

We investigate the fracture of Li-ion battery cathodic particles using a thermodynamically consistent phase-field approach that can describe arbitrarily complex crack paths and captures the full coupling between Li-ion diffusion, stress, and fracture. Building on earlier studies that introduced the concept of electrochemical shock, we use this approach to quantify the relationships between stable or unstable crack propagation, flaw size, and C-rate for 2D disks and 3D spherical particles. We find that over an intermediate range of flaw sizes, the critical flaw size for the onset of crack propagation depends on charging rate as an approximate power-law that we derive analytically. This scaling law is quantified in 2D by exhaustive simulations and is also supported by 3D simulations. In addition, our results reveal a significant difference between 2D and 3D geometries. In 2D, cracks propagate deep inside the particle in a rectilinear manner while in 3D they propagate peripherally on the surface and bifurcate into daughter cracks, thereby limiting inward penetration and giving rise to complex crack geometries.

cond-mat.mtrl-sci↗

arXiv2019

Influence of morphological instability on grain boundary trajectory during directional solidification

The interplay between the diffusion-controlled dynamics of a solidification front and the trajectory of a grain boundary groove at the solid-liquid interface is studied by means of thin-sample directional solidification experiments of a transparent alloy, and by numerical simulations with the phase-field method in two dimensions. We find that low-angle grain boundaries (subboundaries) with an anisotropic interfacial free energy grow tilted at an angle $θ_t$ with respect to the temperature gradient axis. $θ_t$ remains essentially equal to its value imposed at equilibrium as long as the solidification velocity $V$ remains low. When $V$ increases and approaches the cellular instability threshold, $θ_t$ decreases, and eventually vanishes when a steady-state cellular morphology forms. The absence of mobility of the subboundary in the solid is key to this transition. These findings are in good agreement with a recent linear-stability analysis of the problem.

cond-mat.mtrl-sci↗

arXiv2018

Phase-Field Models for Fatigue Crack Growth

We introduce a class of models based on near crack tip degradation of materials that can account for fracture growth under cyclic loads below the Griffith threshold. We incorporate the gradual degradation due to a cyclic load through a flow equation that decreases spatially varying parameters controlling the fracture toughness in the vicinity of the crack tip, with the phase and displacement fields relaxed to an energy minimum at each time step. Though our approach is phenomenological, it naturally reproduces the Paris law with high exponents that are characteristic of brittle fatigue crack growth. We show that the exponent decreases when the phase field dynamics is of the Ginzburg-Landau type with a relaxation time comparable to the cyclic loading period, or when degradation occurs on a scale larger than the process zone. In addition to reproducing the Paris law, our approach can be used to model the growth of multiple cracks in arbitrarily complex geometries under varied loading conditions as illustrated by a few numerical examples in two and three dimensions.

cond-mat.mtrl-sci↗

arXiv2018

Configurational stability of a crack propagating in a material with mode-dependent fracture energy - Part I: Mixed-mode I+III

In a previous paper (Leblond et al., 2011), we proposed a theoretical interpretation of the experimentally well known instability of coplanar crack propagation in mode I+III. The interpretation relied on a stability analysis based on analytical expressions of the stress intensity factors for a crack slightly perturbed both within and out of its original plane, due to Gao and Rice (1986) and Movchan et al. (1998), coupled with a double propagation criterion combining Griffith's energetic condition and principle of local symmetry. Under such assumptions instability modes were indeed evidenced for values of the mode mixity ratio of the mode III to mode I stress intensity factors applied remotely larger than some threshold depending only on Poisson's ratio. Unfortunately, the predicted thresholds were much larger than those generally observed for typical values of this material parameter. While the subcritical character of the nonlinear bifurcation from coplanar to fragmented fronts has been proposed as a possible explanation for this discrepancy (Chen et al., 2015), we propose here an alternative explanation based on the introduction of a constitutive relationship between the fracture energy and the mode mixity ratio, which is motivated by experimental observations. By reexamining the linear stability analysis of a planar propagating front, we show that such a relationship suffices, provided that it is strong enough, to lower significantly the threshold value of the mode mixity ratio for instability so as to bring it in a range more consistent with experiments. Interesting formulae are also derived for the distributions of the perturbed stress intensity factors and energy release rate, in the special case of perturbations of the crack surface and front obeying the principle of local symmetry and having reached a stationary state.

cond-mat.soft↗

arXiv2018

Universality and stability phase-diagram of two-dimensional brittle fracture

The two-dimensional oscillatory crack instability, experimentally observed in a class of brittle materials under strongly dynamic conditions, has been recently reproduced by a nonlinear phase-field fracture theory. Here we highlight the universal character of this instability by showing that it is present in materials exhibiting widely different near crack tip elastic nonlinearity, and by demonstrating that the oscillations wavelength follows a universal master curve in terms of dissipation-related and nonlinear elastic intrinsic length scales. Moreover, we show that upon increasing the driving force for fracture, a high-velocity tip-splitting instability emerges, as experimentally demonstrated. The analysis culminates in a comprehensive stability phase-diagram of two-dimensional brittle fracture, whose salient properties and topology are independent of the form of near tip nonlinearity.

cond-mat.soft↗

arXiv2017

Phase-field model of vapor-liquid-solid nanowire growth

We present a multi-phase-field model to describe quantitatively nanowire growth by the vapor-liquid-solid (VLS) process. The free-energy functional of this model depends on three non-conserved order parameters that distinguish the vapor, liquid, and solid phases. The evolution equations for those order parameters describe basic kinetic processes including the rapid (quasi-instantaneous) equilibration of the liquid catalyst to a droplet shape with constant mean curvature, the slow incorporation of growth atoms at the droplet surface, and crystallization within the droplet. The standard constraint that the sum of the phase fields equals unity and the conservation of the number of catalyst atoms, which relates the catalyst volume to the concentration of growth atoms inside the droplet, are handled via separate Lagrange multipliers. An analysis of the model is presented that rigorously maps the phase-field equations to a desired set of sharp-interface equations for the evolution of the phase boundaries under the constraint of force-balance at three-phase junctions given by the Young-Herring relation that includes torque term related to the anisotropy of the solid-liquid and solid-vapor interface excess free energies. Numerical examples of growth in two dimensions are presented. The simulations reproduce many of the salient features of nanowire growth observed experimentally, including growth normal to the substrate with tapering of the side walls, transitions between different growth orientations, and crawling growth along the substrate.

cond-mat.mtrl-sci↗

arXiv2017

Instability in dynamic fracture and the failure of the classical theory of cracks

Cracks, the major vehicle for material failure, tend to accelerate to high velocities in brittle materials. In three-dimensions, cracks generically undergo a micro-branching instability at about 40% of their sonic limiting velocity. Recent experiments showed that in sufficiently thin systems cracks unprecedentedly accelerate to nearly their limiting velocity without micro-branching, until they undergo an oscillatory instability. Despite their fundamental importance and apparent similarities to other instabilities in condensed-matter physics and materials science, these dynamic fracture instabilities remain poorly understood. They are not described by the classical theory of cracks, which assumes that linear elasticity is valid inside a stressed material and uses an extraneous local symmetry criterion to predict crack paths. Here we develop a model of two-dimensional dynamic brittle fracture capable of predicting arbitrary paths of ultra-high-speed cracks in the presence of elastic nonlinearity without extraneous criteria. We show, by extensive computations, that cracks undergo a dynamic oscillatory instability controlled by small-scale elastic nonlinearity near the crack tip. This instability occurs above a ultra-high critical velocity and features an intrinsic wavelength that increases proportionally to the ratio of the fracture energy to an elastic modulus, in quantitative agreement with experiments. This ratio emerges as a fundamental scaling length assumed to play no role in the classical theory of cracks, but shown here to strongly influence crack dynamics. Those results pave the way for resolving other long-standing puzzles in the failure of materials.

cond-mat.soft↗

arXiv2016

Elastically-mediated interactions between grain boundaries and precipitates in two-phase coherent solids

We investigate analytically and numerically the interaction between grain boundaries and second phase precipitates in two-phase coherent solids in the presence of misfit strain. Our numerical study uses amplitude equations that describe the interaction of composition and stress [R. Spatschek and A. Karma, Phys. Rev. B {\bf 81}, 214201 (2010)] and free-energies corresponding to two-dimensional hexagonal and three-dimensional BCC crystal structures that exhibit isotropic and anisotropic elastic properties, respectively. We consider two experimentally motivated geometries where (i) a lamellar precipitate nucleates along a planar grain boundary that is centered inside the precipitate, and (ii) a circular precipitate nucleates inside a grain at a finite distance to an initially planar grain boundary. For the first geometry, we find that the grain boundary becomes morphologically unstable due to the combination of long-range elastic interaction between the grain boundary and compositional domain boundaries, and shear-coupled grain boundary motion. We characterize this instability analytically by extending the linear stability analysis carried out recently [P.-A. Geslin, Y.-C. Xu, and A. Karma, Phys. Rev. Lett. {\bf 114}, 105501 (2015)] to the more general case of elastic anisotropy. The analysis predicts that elastic anisotropy hinders but does not suppress the instability. Simulations also reveal that, in a well-developed non-linear regime, this instability can lead to the break-up of low-angle grain boundaries when the misfit strain exceeds a threshold that depends on the grain boundary misorientation. For the second geometry, simulations show that the elastic interaction between an initially planar grain boundary and an adjacent circular precipitate causes the precipitate to migrate to and anchor at the grain boundary.

cond-mat.mtrl-sci↗

arXiv2015

Crack Front Segmentation and Facet Coarsening in Mixed-Mode Fracture

A planar crack generically segments into an array of "daughter cracks" shaped as tilted facets when loaded with both a tensile stress normal to the crack plane (mode I) and a shear stress parallel to the crack front (mode III). We investigate facet propagation and coarsening using in-situ microscopy observations of fracture surfaces at different stages of quasi-static mixed-mode crack propagation and phase-field simulations. The results demonstrate that the bifurcation from propagating planar to segmented crack front is strongly subcritical, reconciling previous theoretical predictions of linear stability analysis with experimental observations. They further show that facet coarsening is a self- similar process driven by a spatial period-doubling instability of facet arrays with a growth rate dependent on mode mixity. Those results have important implications for understanding the failure of a wide range of materials.

cond-mat.mtrl-sci↗

arXiv2014

Ginzburg-Landau theory of the bcc-liquid interface kinetic coefficient

We extend the Ginzburg-Landau (GL) theory of atomically rough bcc-liquid interfaces [Wu {\it et al.}, Phys. Rev. B \textbf{73}, 094101 (2006)] outside of equilibrium. We use this extension to derive an analytical expression for the kinetic coefficient, which is the proportionality constant $μ(\hat n)$ between the interface velocity along a direction $\hat n$ normal to the interface and the interface undercooling. The kinetic coefficient is expressed as a spatial integral along the normal direction of a sum of gradient square terms corresponding to different nonlinear density wave profiles. Anisotropy arises naturally from the dependence of those profiles on the angles between the principal reciprocal lattice vectors $\vec K_i$ and $\hat n$. Values of the kinetic coefficient for the$(100)$, $(110)$ and $(111)$ interfaces are compared quantitatively to the prediction of linear Mikheev-Chernov (MC) theory [J. Cryst. Growth \textbf{112}, 591 (1991)] and previous molecular dynamics (MD) simulation studies of crystallization kinetics for a classical model of Fe. Additional MD simulations are carried out here to compute the relaxation time of density waves in the liquid in order to make this comparison free of fit parameter. The GL theory predicts a similar expression for $μ$ as the MC theory but yields a better agreement with MD simulations for both its magnitude and anisotropy due to a fully nonlinear description of density wave profiles across the solid-liquid interface. GL theory is also used to derive an inverse relation between $μ$ and the solid-liquid interfacial free-energy. The general methodology used here to derive an expression for $μ(\hat n)$ also applies to amplitude equations derived from the phase-field-crystal model, which only differ from GL theory by the choice of cubic and higher order nonlinearities in the free-energy density.

cond-mat.mtrl-sci↗

arXiv2014

Spatiotemporal Dynamics of Calcium-Driven Cardiac Alternans

We investigate the dynamics of spatially discordant alternans (SDA) driven by an instability of intracellular calcium cycling using both amplitude equations [P. S. Skardal, A. Karma, and J. G. Restrepo, Phys. Rev. Lett. 108, 108103 (2012)] and ionic model simulations. We find that, close to the alternans bifurcation, SDA is manifested as a smooth wavy modulation of the amplitudes of both repolarization and calcium transient (CaT) alternans, similarly to the well-studied case of voltage-driven alternans. In contrast, further away from the bifurcation, the amplitude of CaT alternans jumps discontinuously at the nodes separating out-of-phase regions, while the amplitude of repolarization alternans remains smooth. We show that node motion of discontinuous SDA patterns is strongly hysteretic even in homogeneous tissue due to the novel phenomenon of "unidirectional pinning": node movement can only be induced towards, but not away from, the pacing site in response to a change of pacing rate or physiological parameter. In addition, we show that the wavelength of discontinuous SDA patterns scales linearly with the conduction velocity restitution length scale, in contrast to the wavelength of smooth patterns that scales sub-linearly with this length scale. Those results are also shown to be robust against cell-to-cell fluctuations owing to the property that unidirectional node motion collapses multiple jumps accumulating in nodal regions into a single jump. Amplitude equation predictions are in good overall agreement with ionic model simulations. Finally, we briefly discuss physiological implications of our findings. In particular, we suggest that due to the tendency of conduction blocks to form near nodes, the presence of unidirectional pinning makes calcium-driven alternans potentially more arrhythmogenic than voltage-driven alternans.

q-bio.TO↗

arXiv2013

Unified Theoretical Framework for Polycrystalline Pattern Evolution

The rate of curvature-driven grain growth in polycrystalline materials is well-known to be limited by interface dissipation. We show analytically and by simulations that, for systems forming modulated phases or non-equilibrium patterns with crystal ordering, growth is limited by bulk dissipation associated with lattice translation, which dramatically slows down grain coarsening. We also show that bulk dissipation is reduced by thermal noise so that those systems exhibit faster coarsening behavior dominated by interface dissipation for high Peierls barrier and high noise. Those results provide a unified theoretical framework for understanding and modeling polycrystalline pattern evolution in diverse systems over a broad range of length and time scales.

cond-mat.mtrl-sci↗

arXiv2012

Structural short-range forces between solid-melt interfaces

We predict the structural interaction of crystalline solid-melt interfaces using amplitude equations which are derived from classical density functional theory or phase-field-crystal modeling. The solid ordering decays exponentially on the scale of the interface thickness at solid-melt interfaces; the overlap of two such profiles leads to a short range interaction, which is mainly carried by the longest-range density waves, which can facilitate grain boundary premelting. We calculate the tail of these interactions, depending on the relative translation of the two crystals fully analytically and predict the interaction potential, and compare it to numerical simulations. For grain boundaries the interaction is predicted to decay twice faster as for two crystals without misorientation.

cond-mat.mtrl-sci↗

arXiv2012

Phase-field-crystal study of grain boundary premelting and shearing in bcc iron

We use the phase-field-crystal (PFC) method to investigate the equilibrium premelting and nonequilibrium shearing behaviors of $[001]$ symmetric tilt grain boundaries (GBs) at high homologous temperature over the complete range of misorientation $0<θ<90^\circ$ in classical models of bcc Fe. We characterize the dependence of the premelted layer width $W$ as a function of temperature and misorientation and compute the thermodynamic disjoining potential whose derivative with respect to $W$ represents the structural force between crystal-melt interfaces due to the spatial overlap of density waves. The disjoining potential is also computed by molecular dynamics (MD) simulations, for quantitative comparison with PFC simulations, and coarse-grained amplitude equations (AE) derived from PFC that provide additional analytical insights. We find that, for GBs over an intermediate range of misorientation ($θ_{\rm min}<θ<θ_{\rm max}$), $W$ diverges as the melting temperature is approached from below, corresponding to a purely repulsive disjoining potential, while for GBs outside this range ($θ<θ_{\rm min}$ or $θ_{\rm max}<θ<90^\circ$), $W$ remains finite at the melting point, with its value corresponding to a shallow attractive minimum of the disjoining potential. In response to a shear stress parallel to the GB plane, GBs in PFC simulations exhibit coupled motion normal to this plane, with a discontinuous change of the coupling factor as a function of $θ$ that reflects a transition between two coupling modes, and/or sliding (shearing of the two grains). The coupling factor for the two coupling modes is in excellent quantitative agreement with previous theoretical predictions [J. W. Cahn, Y. Mishin, and A. Suzuki, Acta Mater. 54, 4953 (2006)].

cond-mat.mtrl-sci↗

arXiv2012

Unidirectional Pinning and Hysteresis of Spatially Discordant Alternans in Cardiac Tissue

Spatially discordant alternans is a widely observed pattern of voltage and calcium signals in cardiac tissue that can precipitate lethal cardiac arrhythmia. Using spatially coupled iterative maps of the beat-to-beat dynamics, we explore this pattern's dynamics in the regime of a calcium-dominated period-doubling instability at the single cell level. We find a novel nonlinear bifurcation associated with the formation of a discontinuous jump in the amplitude of calcium alternans at nodal lines separating discordant regions. We show that this jump unidirectionally pins nodal lines by preventing their motion away from the pacing site following a pacing rate decrease, but permitting motion towards this site following a rate increase. This unidirectional pinning leads to strongly history-dependent nodal line motion that is strongly arrhythmogenic.

nlin.PS↗

arXiv2010

Amplitude equations for polycrystalline materials with interaction between composition and stress

We investigate the ability of frame-invariant amplitude equations [G. H. Gunaratne, Q. Ouyang, and H. Swinney, Phys. Rev. E {\bf 50}, 2802 (1994)] to describe quantitatively the evolution of polycrystalline microstructures and we extend this approach to include the interaction between composition and stress. Validations for elemental materials include studies of the Asaro-Tiller-Grinfeld morphological instability of a stressed crystal surface, polycrystalline growth from the melt, grain boundary energies over a wide range of misorientation, and grain boundary motion coupled to shear deformation. Amplitude equations with accelerated strain relaxation in the solid are shown to model accurately the Asaro-Tiller-Grinfeld instability. Polycrystalline growth is also well described. However, the survey of grain boundary energies shows that the approach is only valid for a restricted range of misorientations as a direct consequence of an amplitude expansion. This range covers approximately half the complete range allowed by crystal symmetry for some fixed reference set of density waves used in the expansion. Over this range, coupled motion to shear is well described by known geometrical rules and a transition from coupling to sliding motion is also reproduced. Amplitude equations for alloys are derived phenomenologically in a Ginzburg-Landau spirit. Vegard's law is shown to be naturally described by seeking a gauge invariant form of those equations under a transformation that corresponds to a lattice expansion and deviations from Vegard's law can be easily incorporated. Those equations realistically describe the dilute alloy limit and have the same flexibility as conventional phase-field models for incorporating arbitrary free-energy/composition curves...

cond-mat.mtrl-sci↗

arXiv2010

Phase-field-crystal model for fcc ordering

We develop and analyze a two-mode phase-field-crystal model to describe fcc ordering. The model is formulated by coupling two different sets of crystal density waves corresponding to <111> and <200> reciprocal lattice vectors, which are chosen to form triads so as to produce a simple free- energy landscape with coexistence of crystal and liquid phases. The feasibility of the approach is demonstrated with numerical examples of polycrystalline and (111) twin growth. We use a two-mode amplitude expansion to characterize analytically the free-energy landscape of the model, identifying parameter ranges where fcc is stable or metastable with respect to bcc. In addition, we derive analytical expressions for the elastic constants for both fcc and bcc. Those expressions show that a non-vanishing amplitude of [200] density waves is essential to obtain mechanically stable fcc crystals with a non-vanishing tetragonal shear modulus (C11 - C12)/2. We determine the model parameters for specific materials by fitting the peak liquid structure factor properties and solid density wave amplitudes following the approach developed for bcc [K.-A. Wu and A. Karma, Phys. Rev. B 76, 184107 (2007)]. This procedure yields reasonable predictions of elastic constants for both bcc Fe and fcc Ni using input parameters from molecular dynamics simulations. The application of the model to two-dimensional square lattices is also briefly examined.

cond-mat.mtrl-sci↗

arXiv2010

Structural disjoining potential for grain boundary premelting and grain coalescence from molecular-dynamics simulations

We describe a molecular dynamics framework for the direct calculation of the short-ranged structural forces underlying grain-boundary premelting and grain-coalescence in solidification. The method is applied in a comparative study of (i) a Sigma 9 <115> 120 degress twist and (ii) a Sigma 9 <110> {411} symmetric tilt boundary in a classical embedded-atom model of elemental Ni. Although both boundaries feature highly disordered structures near the melting point, the nature of the temperature dependence of the width of the disordered regions in these boundaries is qualitatively different. The former boundary displays behavior consistent with a logarithmically diverging premelted layer thickness as the melting temperature is approached from below, while the latter displays behavior featuring a finite grain-boundary width at the melting point. It is demonstrated that both types of behavior can be quantitatively described within a sharp-interface thermodynamic formalism involving a width-dependent interfacial free energy, referred to as the disjoining potential. The disjoining potential for boundary (i) is calculated to display a monotonic exponential dependence on width, while that of boundary (ii) features a weak attractive minimum. The results of this work are discussed in relation to recent simulation and theoretical studies of the thermodynamic forces underlying grain-boundary premelting.

cond-mat.mtrl-sci↗