arXiv subjects

Mathis Plapp

Publications and source records attributed to Mathis Plapp.

At least 37 records · Page 2Linked to original sources

arXiv2011

Theoretical and numerical study of lamellar eutectic three-phase growth in ternary alloys

We investigate lamellar three-phase patterns that form during the directional solidification of ternary eutectic alloys in thin samples. A distinctive feature of this system is that many different geometric arrangements of the three phases are possible, contrary to the widely studied two-phase patterns in binary eutectics. Here, we first analyze the case of stable lamellar coupled growth of a symmetric model ternary eutectic alloy, using a Jackson-Hunt type calculation in thin film morphology, for arbitrary configurations, and derive expressions for the front undercooling as a function of velocity and spacing. Next, we carry out phase-field simulations to test our analytic predictions and to study the instabilities of the simplest periodic lamellar arrays. For large spacings, we observe different oscillatory modes that are similar to those found previously for binary eutectics and that can be classified using the symmetry elements of the steady-state pattern. For small spacings, we observe a new instability that leads to a change in the sequence of the phases. Its onset can be well predicted by our analytic calculations. Finally, some preliminary phase-field simulations of three-dimensional growth structures are also presented.

cond-mat.mtrl-sci↗

arXiv2010

Controlling crystal symmetries in phase-field crystal models

We investigate the possibility to control the symmetry of ordered states in phase-field crystal models by tuning nonlinear resonances. In two dimensions, we find that a state of square symmetry as well as coexistence between squares and hexagons can be easily obtained. In contrast, it is delicate to obtain coexistence of squares and liquid. We develop a general method for constructing free energy functionals that exhibit solid-liquid coexistence with desired crystal symmetries. As an example, we develop a free energy functional for square-liquid coexistence in two dimensions. A systematic analysis for determining the parameters of the necessary nonlinear terms is provided. The implications of our findings for simulations of materials with simple cubic symmetry are discussed.

cond-mat.mtrl-sci↗

arXiv2010

Defects and multistability in eutectic solidification patterns

We use three-dimensional phase-field simulations to investigate the dynamics of the two-phase composite patterns formed upon during solidification of eutectic alloys. Besides the spatially periodic lamellar and rod patterns that have been widely studied, we find that there is a large number of additional steady-state patterns which exhibit stable defects. The defect density can be so high that the pattern is completely disordered, and that the distinction between lamellar and rod patterns is blurred. As a consequence, the transition from lamellae to rods is not sharp, but extends over a finite range of compositions and exhibits strong hysteresis. Our findings are in good agreement with experiments.

cond-mat.mtrl-sci↗

arXiv2010

Remarks on some open problems in phase-field modelling of solidification

Three different topics in phase-field modelling of solidification are discussed, with particular emphasis on the limitations of the currently available modelling approaches. First, thin-interface limits of two-sided phase-field models are examined, and it is shown that the antitrapping current is in general not sufficient to remove all thin-interface effects. Second, orientation-field models for polycrystalline solidification are analysed, and it is shown that the standard relaxational equation of motion for the orientation field is incorrect in coherent polycrystalline matter. Third, it is pointed out that the standard procedure to incorporate fluctuations into the phase-field approach cannot be used in a straightforward way for a quantitative description of nucleation.

cond-mat.mtrl-sci↗

arXiv2009

Phase-field simulations of viscous fingering in shear-thinning fluids

A phase-field model for the Hele-Shaw flow of non-Newtonian fluids is developed. It extends a previous model for Newtonian fluids to a wide range of shear-dependent fluids. The model is applied to perform simulations of viscous fingering in shear- thinning fluids, and it is found to be capable of describing the complete crossover from the Newtonian regime at low shear rate to the strongly shear-thinning regime at high shear rate. The width selection of a single steady-state finger is studied in detail for a 2-plateaux shear-thinning law (Carreau law) in both its weakly and strongly shear-thinning limits, and the results are related to previous analyses. In the strongly shear-thinning regime a rescaling is found for power-law (Ostwald-de-Waehle) fluids that allows for a direct comparison between simulations and experiments without any adjustable parameters, and good agreement is obtained.

cond-mat.soft↗

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

Phase-field modeling of the discontinuous precipitation reaction

A multi-phase-field model for the description of the discontinuous precipitation reaction is formulated which takes into account surface diffusion along grain boundaries and interfaces as well as volume diffusion. Simulations reveal that the structure and steady-state growth velocity of spatially periodic precipitation fronts strongly depend on the relative magnitudes of the diffusion coefficients. Steady-state solutions always exist for a range of interlamellar spacings that is limited by a fold singularity for low spacings, and by the onset of tip-splitting or oscillatory instabilities for large spacings. A detailed analysis of the simulation data reveals that the hypothesis of local equilibrium at interfaces, used in previous theories, is not valid for the typical conditions of discontinuous precipitation.

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

Stability of hexagonal solidification patterns

We investigate the dynamics of cellular solidification patterns using three-dimensional phase-field simulations. The cells can organize into stable hexagonal patterns or exhibit unsteady evolutions. We identify the relevant secondary instabilities of regular hexagonal arrays and find that the stability boundaries depend significantly on the strength of crystalline anisotropy. We also find multiplet states that can be reached by applying well-defined perturbations to a pre-existing hexagonal array.

cond-mat.mtrl-sci↗

arXiv2003

A mean-field kinetic lattice gas model of electrochemical cells

We develop Electrochemical Mean-Field Kinetic Equations (EMFKE) to simulate electrochemical cells. We start from a microscopic lattice-gas model with charged particles, and build mean-field kinetic equations following the lines of earlier work for neutral particles. We include the Poisson equation to account for the influence of the electric field on ion migration, and oxido-reduction processes on the electrode surfaces to allow for growth and dissolution. We confirm the viability of our approach by simulating (i) the electrochemical equilibrium at flat electrodes, which displays the correct charged double-layer, (ii) the growth kinetics of one-dimensional electrochemical cells during growth and dissolution, and (iii) electrochemical dendrites in two dimensions.

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

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↗

arXiv2000

Eutectic Colony Formation: A Stability Analysis

Experiments have widely shown that a steady-state lamellar eutectic solidification front is destabilized on a scale much larger than the lamellar spacing by the rejection of a dilute ternary impurity and forms two-phase cells commonly referred to as `eutectic colonies'. We extend the stability analysis of Datye and Langer for a binary eutectic to include the effect of a ternary impurity. We find that the expressions for the critical onset velocity and morphological instability wavelength are analogous to those for the classic Mullins-Sekerka instability of a monophase planar interface, albeit with an effective surface tension that depends on the geometry of the lamellar interface and, non-trivially, on interlamellar diffusion. A qualitatively new aspect of this instability is the occurence of oscillatory modes due to the interplay between the destabilizing effect of the ternary impurity and the dynamical feedback of the local change in lamellar spacing on the front motion. In a transient regime, these modes lead to the formation of large scale oscillatory microstructures for which there is recent experimental evidence in a transparent organic system. Moreover, it is shown that the eutectic front dynamics on a scale larger than the lamellar spacing can be formulated as an effective monophase interface free boundary problem with a modified Gibbs-Thomson condition that is coupled to a slow evolution equation for the lamellar spacing. This formulation provides additional physical insights into the nature of the instability and a simple means to calculate an approximate stability spectrum. Finally, we investigate the influence of the ternary impurity on a short wavelength oscillatory instability that is already present at off-eutectic compositions in binary eutectics.

cond-mat.mtrl-sci↗

arXiv2000

Multiscale Finite-Difference-Diffusion-Monte-Carlo Method for Simulating Dendritic Solidification

We present a novel hybrid computational method to simulate accurately dendritic solidification in the low undercooling limit where the dendrite tip radius is one or more orders of magnitude smaller than the characteristic spatial scale of variation of the surrounding thermal or solutal diffusion field. The first key feature of this method is an efficient multiscale diffusion Monte-Carlo (DMC) algorithm which allows off-lattice random walkers to take longer and concomitantly rarer steps with increasing distance away from the solid-liquid interface. As a result, the computational cost of evolving the large scale diffusion field becomes insignificant when compared to that of calculating the interface evolution. The second key feature is that random walks are only permitted outside of a thin liquid layer surrounding the interface. Inside this layer and in the solid, the diffusion equation is solved using a standard finite-difference algorithm that is interfaced with the DMC algorithm using the local conservation law for the diffusing quantity. Here we combine this algorithm with a previously developed phase-field formulation of the interface dynamics and demonstrate that it can accurately simulate three-dimensional dendritic growth in a previously unreachable range of low undercoolings that is of direct experimental relevance.

cond-mat.mtrl-sci↗

arXiv2000

Multiscale Random-Walk Algorithm for Simulating Interfacial Pattern Formation

We present a novel computational method to simulate accurately a wide range of interfacial patterns whose growth is limited by a large scale diffusion field. To illustrate the computational power of this method, we demonstrate that it can be used to simulate three-dimensional dendritic growth in a previously unreachable range of low undercoolings that is of direct experimental relevance.

cond-mat.mtrl-sci↗

arXiv1999

Three-dimensional dendrite tip morphology at low undercooling

We investigate the three-dimensional morphology of the dendrite tip using the phase-field method. We find that, for low undercoolings, this morphology is ostensibly independent of anisotropy strength except for a localized shape distortion near the tip that only affects the value of the tip radius $ρ$ (which is crudely approximated by $ρ\approx (1-α)ρ_{Iv}$ where $ρ_{Iv}$ is the Ivantsov tip radius of an isothermal paraboloid with the same tip velocity and $α$ is the stiffness anisotropy). The universal tip shape, which excludes this distortion, is well fitted by the form $z=-r^2/2+A_4 r^4\cos 4ϕ$ where $|z|$ is the distance from the tip and all lengths are scaled by $ρ_{Iv}$. This fit yields $A_4$ in the range $0.004-0.005$ in good quantitative agreement with the existing tip morphology measurements in succinonitrile [LaCombe et al., Phys. Rev. E {\bf 52}, 2778 (1995)], which are reanalyzed here and found to be consistent with a single $\cos 4ϕ$ mode non-axisymmetric deviation from a paraboloid. Moreover, the fin shape away from the tip is well fitted by the power law $z=-a |x|^{5/3}$ with $a\approx 0.68$. Finally, the characterization of the operating state of the dendrite tip is revisited in the light of these results.

cond-mat.mtrl-sci↗

arXiv1999

Spinodal decomposition of an ABv model alloy: Patterns at unstable surfaces

We develop mean-field kinetic equations for a lattice gas model of a binary alloy with vacancies (ABv model) in which diffusion takes place by a vacancy mechanism. These equations are applied to the study of phase separation of finite portions of an unstable mixture immersed in a stable vapor. Due to a larger mobility of surface atoms, the most unstable modes of spinodal decomposition are localized at the vapor-mixture interface. Simulations show checkerboard-like structures at the surface or surface-directed spinodal waves. We determine the growth rates of bulk and surface modes by a linear stability analysis and deduce the relation between the parameters of the model and the structure and length scale of the surface patterns. The thickness of the surface patterns is related to the concentration fluctuations in the initial state.

cond-mat.mtrl-sci↗