Phase-field model of solute trapping during solidification

Atomistic Insights into Ultrafast SiGe Nanoprocessing

Controlling ultrafast material transformations with atomic precision is essential for future nanotechnology. Pulsed laser annealing (LA), inducing extremely rapid and localized phase transitions, is a powerful way to achieve this but requires careful optimization together with the appropriate system design. We present a multiscale LA computational framework that can simulate atom-by-atom the highly out-of-equilibrium kinetics of a material as it interacts with the laser, including effects of structural disorder. By seamlessly coupling a macroscale continuum solver to a nanoscale superlattice kinetic Monte Carlo code, this method overcomes the limits of state-of-the-art continuum-based tools. We exploit it to investigate nontrivial changes in composition, morphology, and quality of laser-annealed SiGe alloys. Validations against experiments and phase-field simulations as well as advanced applications to strained, defected, nanostructured, and confined SiGe are presented, highlighting the importance of a multiscale atomistic-continuum approach. Current applicability and potential generalization routes are finally discussed.

A Phase-Field Model for In-Space Manufacturing of Binary Alloys

The integrity of the final printed components is mostly dictated by the adhesion between the particles and phases that form upon solidification, which is a major problem in printing metallic parts using available In-Space Manufacturing (ISM) technologies based on the Fused Deposition Modeling (FDM) methodology. Understanding the melting/solidification process helps increase particle adherence and allows to produce components with greater mechanical integrity. We developed a phase-field model of solidification for binary alloys. The phase-field approach is unique in capturing the microstructure with computationally tractable costs. The developed phase-field model of solidification of binary alloys satisfies the stability conditions at all temperatures. The suggested model is tuned for Ni-Cu alloy feedstocks. We derived the Ginzburg-Landau equations governing the phase transformation kinetics and solved them analytically for the dilute solution. We calculated the concentration profile as a function of interface velocity for a one-dimensional steady-state diffuse interface neglecting elasticity and obtained the partition coefficient, k, as a function of interface velocity. Numerical simulations for the diluted solution are used to study the interface velocity as a function of undercooling for the classic sharp interface model, partitionless solidification, and thin interface.

Simple Method of Including Density Variation in Quantitative Continuum Phase-Change Models

We present a simple, quantitative, and thermodynamically self-consistent method of capturing density and pressure variation in continuum phase-change models. The formalism shows how the local state of homogenous dilation may be entirely given by species concentration in an Eulerian formulation. A hyperelastic contribution to the thermodynamic potential generalizes the lattice constraint while permitting composition, temperature, and phase-dependent specific volumes. We compare the results of models implementing this paradigm to those with the lattice constraint by examining the composition and size-dependent equilibrium of a Ni-Cu nanoparticle in its melt and free dendritic growth.

Phase field modeling for the morphological and microstructural evolution of metallic materials under environmental attack

The complex degradation of metallic materials in aggressive environments can result in morphological and microstructural changes. The phase-field (PF) method is an effective computational approach to understanding and predicting the morphology, phase change and/or transformation of materials. PF models are based on conserved and non-conserved field variables that represent each phase as a function of space and time coupled with time-dependent equations that describe the mechanisms. This report summarizes progress in the PF modeling of degradation of metallic materials in aqueous corrosion, hydrogen-assisted cracking, high-temperature metal oxidation in the gas phase and porous structure evolution with insights to future applications.

Diffusion equations expressed in molar fractions: Theory and application to ionic diffusion and demixing

Molar fractions are used in applied diffusion kinetics for incorporating thermodynamic and kinetic databases. Molar quantities (molar concentration and molar flux) and reduced molar quantities (molar fraction and reduced molar flux) usually are regarded to be equivalent; i.e., molar quantities are replaced with their reduced forms. However, as the fluxes are related to material properties, the diffusion equations expressed in molar fractions are not consistent with the normalization condition of molar fractions. We develop diffusion kinetics consistent with this condition. Our method is applicable to diffusion with total reduced flux, such as diffusion in an external field or flow field. As two case studies, the developed method is used to investigate ionic diffusion in an electrolyte solution and ionic demixing in a semiconductor oxide.

Kinetic Phase Diagrams of Ternary Al-Cu-Li System during Rapid Solidification: A Phase-Field Study

Kinetic phase diagrams in technical alloys at different solidification velocities during rapid solidification are of great importance for guiding the novel alloy preparation, but are usually absent due to extreme difficulty in performing experimental measurements. In this paper, a phase-field model with finite interface dissipation was employed to construct kinetic phase diagrams in the ternary Al-Cu-Li system for the first time. The time-elimination relaxation scheme was utilized. The solute trapping phenomenon during rapid solidification could be nicely described by the phase-field simulation, and the results obtained from the experiment measurement and/or the theoretical model were also well reproduced. Based on the predicted kinetic phase diagrams, it was found that with the increase of interface moving velocity and/or temperature, the gap between the liquidus and solidus gradually reduces, which illustrates the effect of solute trapping and tendency of diffusionless solidification.

Linearization-based method for solving a multicomponent diffusion phase-field model with arbitrary solution thermodynamics

This article describes a phase-field model for an isothermal multicomponent, multiphase system which avoids implicit interfacial energy contributions by starting from a grand potential formulation. A method is developed for incorporating arbitrary forms of the equilibrium thermodynamic potentials in all phases to determine an explicit relationship between chemical potentials and species concentrations. The model incorporates variable densities between adjacent phases, defect migration, and dependence of internal pressure on object dimensions ranging from the macro- to nanoscale. A demonstrative simulation of an overpressurized nanoscopic intragranular bubble in nuclear fuel migrating to a grain boundary under kinetically limited vacancy diffusion is shown.

Thermodynamic coarsening arrested by viscous fingering in partially miscible binary mixtures

We study the evolution of binary mixtures far from equilibrium, and show that the interplay between phase separation and hydrodynamic instability can arrest the Ostwald ripening process characteristic of nonflowing mixtures. We describe a model binary system in a Hele-Shaw cell using a phase-field approach with explicit dependence of both phase fraction and mass concentration. When the viscosity contrast between phases is large (as is the case for gas and liquid phases), an imposed background flow leads to viscous fingering, phase branching, and pinch off. This dynamic flow disorder limits phase growth and arrests thermodynamic coarsening. As a result, the system reaches a regime of statistical steady state in which the binary mixture is permanently driven away from equilibrium.

A diffuse-interface modeling for liquid solution-solid gel phase transition of physical hydrogel with nonlinear deformation

A diffuse-interface model is presented in this paper for simulation of the evolution of phase transition between the liquid solution and solid gel states for physical hydrogel with nonlinear deformation. The present domain covers the gel and solution states as well as a diffuse interface between them. They are indicated by the crosslink density in such a way that the solution phase is identified as the state when the crosslink density is small, while the gel as the state if the crosslink density becomes large. In this work, a novel order parameter is thus defined as the crosslink density, which is homogeneous in each distinct phase and smoothly varies over the interface from one phase to another. In this model, the constitutive equations, imposed on the two distinct phases and the interface, are formulated by the second law of thermodynamics, which are in the same form as those derived by a different approach. The present constitutive equations include a novel Ginzburg-Landau type of free energy with a double-well profile, which accounts for the effect of crosslink density. The present governing equations include the equilibrium of forces, the conservations of mass and energy, and an additional kinetic equation imposed for phase transition, in which nonlinear deformation is considered. The equilibrium state is investigated numerically, where two stable phases are observed in the free energy profile. As case studies, a spherically symmetrical solution-gel phase transition is simulated numerically for analysis of the phase transition of physical hydrogel.

Scaling of alloy interfacial properties under compositional strain

Complex morphologies and microstructures that emerge during materials growth and solidification are often determined by both equilibrium and kinetic properties of the interface and their crystalline anisotropies. However, limited knowledge is available on alloying and, particularly, compositionally generated elastic effects on these interface characteristics. Here we systematically investigate such compositional effects on the interfacial properties of an alloy model system based on a phase-field-crystal analysis, including the solid-liquid interfacial free energy, kinetic coefficient, and lattice pinning strength. Scaling relations for these interfacial quantities over various ranges of material parameters are identified and predicted. Our results indicate the important effects of couplings among mesoscopic and microscopic length scales of alloy structure and concentration, and the influence of compressive and tensile interface stresses induced by composition variations. The approach developed here provides an efficient way to systematically identify these key material properties beyond the traditional atomistic and continuum methods.

The formation mechanism of eutectic microstructures in NiAl–Cr composites

Based on a thermodynamic foundation, a phase field model is developed for describing eutectic composites. The inherent complexity of NiAl–Cr microstructure is rationalized by modeling eutectic solidification that precedes conditional spinodal decomposition.

Multicomponent phase-field model for extremely large partition coefficients

We develop a multicomponent phase-field model specially formulated to robustly simulate concentration variations from molar to atomic magnitudes across an interface, i.e., partition coefficients in excess of 10±23 such as may be the case with species which are predominant in one phase and insoluble in the other. Substitutional interdiffusion on a normal lattice and concurrent interstitial diffusion are included. The composition in the interface follows the approach of Kim, Kim, and Suzuki [Phys. Rev. E 60, 7186 (1999)] and is compared to that of Wheeler, Boettinger, and McFadden [Phys. Rev. A 45, 7424 (1992)] in the context of large partitioning. The model successfully reproduces analytical solutions for binary diffusion couples and solute trapping for the demonstrated cases of extremely large partitioning.

Phase-field-crystal study of solute trapping

In this study we have incorporated two time scales into the phase-field-crystal model of a binary alloy to explore different solute trapping properties as a function of crystal-melt interface velocity. With only diffusive dynamics, we demonstrate that the segregation coefficient, K as a function of velocity for a binary alloy is consistent with the model of Kaplan and Aziz where K approaches unity in the limit of infinite velocity. However, with the introduction of wavelike dynamics in both the density and concentration fields, the trapping follows the kinetics proposed by Sobolev [Phys. Lett. A 199, 383 (1995)], where complete trapping occurs at a finite velocity.

Recovering thermodynamic consistency of the antitrapping model: a variational phase-field formulation for alloy solidification

The phenomenological antitrapping phase-field model has attained much success in describing alloy solidification. The heuristically introduced antitrapping current enables removing artificial effects due to the use of large interfacial width. Nevertheless, such a model is not thermodynamically consistent and has not been fitted into a variational framework. Here we present two approaches to develop a variational phase-field model to describe patten formation in alloys. Following the principles of linear irreversible thermodynamics we build in the cross-coupling between the phase transition rate and solute diffusion current. Our formulation not only naturally incorporates the antitrapping current but also predicts the conjugated mesoscopic solute drag effect. A more general form of the antitrapping current is obtained by thin-interface analysis. Benchmark simulations on isothermal dendrite growth are carried out to show the capability of our model to quantitatively characterize the interface evolution and solute profile even with a large interface width used. Importantly, our theory also provides general insights on how to obtain the genuine dynamic coupling between nonconserved and conserved order parameters. This leads to a thermodynamically consistent generalization of the celebrated model C proposed by Hohenberg and Halperin [Rev. Mod. Phys. 49, 435 (1977)].

Quantitative investigation of cellular growth in directional solidification by phase-field simulation

Using a quantitative phase-field model, a systematic investigation of cellular growth in directional solidification is carried out with emphasis on the selection of cellular tip undercooling, tip radius, and cellular spacing. Previous analytical models of cellular growth are evaluated according to the phase-field simulation results. The results show that cellular tip undercooling and tip radius not only depend on the pulling velocity and thermal gradient, but also depend on the cellular interaction related to the cellular spacing. The cellular interaction results in a finite stable range of cellular spacing. The lower limit is determined by the submerging mechanism while the upper limit comes from the tip splitting instability corresponding to the absence of the cellular growth solution, both of which can be obtained from phase-field simulation. Further discussions on the phase-field results also present an analytical method to predict the lower limit. Phase-field simulations on cell elimination between cells with equal spacing validate the finite range of cellular spacing and give deep insight into the cellular doublon and oscillatory instability between cell elimination and tip splitting.

Solute trapping in rapid solidification of a binary dilute system: a phase-field study

The phase-field model of Echebarria, Folch, Karma, and Plapp [Phys. Rev. E 70, 061604 (2004)] is extended to the case of rapid solidification in which local nonequilibrium phenomena occur in the bulk phases and within the diffuse solid-liquid interface. Such an extension leads to the fully hyperbolic system of equations given by the atomic diffusion equation and the phase-field equation of motion. This model is applied to the problem of solute trapping, which is accompanied by the entrapment of solute atoms beyond chemical equilibrium by a rapidly moving interface. The model predicts the beginning of complete solute trapping and diffusionless solidification at a finite solidification velocity equal to the diffusion speed in bulk liquid.

Thermodynamic phase-field model for microstructure with multiple components and phases: the possibility of metastable phases

A diffuse-interface model for microstructure with an arbitrary number of components and phases was developed from basic thermodynamic and kinetic principles and formalized within a variational framework. The model includes a composition gradient energy to capture solute trapping and is therefore suited for studying phenomena where the width of the interface plays an important role. Derivation of the inhomogeneous free energy functional from a Taylor expansion of homogeneous free energy reveals how the interfacial properties of each component and phase may be specified under a mass constraint. A diffusion potential for components was defined away from the dilute solution limit, and a multi-obstacle barrier function was used to constrain phase fractions. The model was used to simulate solidification via nucleation, premelting at phase boundaries and triple junctions, the intrinsic instability of small particles, and solutal melting resulting from differing diffusivities in solid and liquid. The shape of metastable free energy surfaces is found to play an important role in microstructure evolution and may explain why some systems premelt at phase boundaries and phase triple junctions, whereas others do not.

Phase-field simulations of nuclei and early stage solidification microstructures

To investigate the local properties of heterogeneous nuclei on substrates, a phase-field model is extended to incorporate volume constraints and a third order line tension in the gradient free energy density formulation. The new model is applied to sessile drop simulations of Cu nuclei on Ni substrates to precisely analyse 3D equilibrium shapes and diffusion processes across the phase boundaries. In particular, the formalism with higher order potentials is used to investigate the length-scale dependent effect of the line tension on Young's force balance at triple lines in 3D. The employment of parallel and adaptive simulation techniques is essential for three-dimensional numerical computations. Early stage solidification microstructures of cubic Ni crystals are simulated by scale-bridging molecular dynamics (MD) and phase-field (PF) simulations. The domain of the PF computations is initialized by transferring MD data of the atomic positions and of the shape of the nuclei. The combined approach can be used to study the responses of microstructures upon nucleation.

Quantitative phase-field modeling for dilute alloy solidification involving diffusion in the solid

An antitrapping current scheme for quantitative phase-field model [A. Karma, Phys. Rev. Lett. 87, 115701 (2001)] is extended to solidification process in a dilute binary alloy system involving diffusion in the solid. It is demonstrated in an asymptotic analysis that in the case of an arbitrary value of the solid diffusivity, five types of constraints exist between interpolating functions used in the phase-field model, which need to be satisfied simultaneously to eliminate all anomalous interface effects. Then, the authors present an appropriate form of the antitrapping current term for the two-sided case to remove all the spurious effects. The convergence test of the output with respect to the interface thickness was carried out for the isothermal dendrite growth process, which demonstrates an excellent performance of the present model.

Solute trapping and diffusionless solidification in a binary system

Numerous experimental data on the rapid solidification of binary systems exhibit the formation of metastable solid phases with initial (nominal) chemical composition. This fact is explained by complete solute trapping leading to diffusionless (chemically partitionless) solidification at a finite growth velocity of crystals. Special attention is paid to developing a model of rapid solidification which describes a transition from chemically partitioned to diffusionless growth of crystals. Analytical treatments lead to the condition for complete solute trapping which directly follows from the analysis of the solute diffusion around the solid-liquid interface and atomic attachment and detachment at the interface. The resulting equations for the flux balance at the interface take into account two kinetic parameters: diffusion speed VDI on the interface and diffusion speed VD in bulk phases. The model describes experimental data on nonequilibrium solute partitioning in solidification of Si-As alloys for the whole range of solidification velocity investigated.

Perturbative approach to the structure of a planar interface in the Landau-de Gennes model

The structure of nearly static planar interfaces is studied within the framework of the Landau-de Gennes model with the dynamics governed by the time-dependent Ginzburg-Landau equation. To account for the full elastic anisotropy the free energy expansion is extended to include a third order gradient term. The solutions corresponding to the in-plane or homeotropic director alignment at the interface are sought. For this purpose a consistent perturbative scheme is constructed which enables one to calculate successive corrections to the velocity and the order parameter of the interface. The implications of the solutions are discussed. The elastic anisotropy introduces asymmetry into the order parameter and free energy profiles, even for the high symmetry homeotropic configuration. The velocity of the interface with the homeotropic or in-plane alignment is enhanced or reduced, respectively. There is no reorientation of the optical axis in the boundary layer. For the class of nematogens with approximate splay-bend degeneracy the temperature dependence of the interface velocity is weakly affected by the remaining twist anisotropy.

Multiscale model for microstructure evolution in multiphase materials: Application to the growth of isolated inclusions in presence of elasticity

We present a multiscale model based on the classical lattice time-dependent density-functional theory to study microstructure evolution in multiphase systems. As a first test of the method, we study the static and dynamic properties of isolated inclusions. Three cases are explored: elastically homogeneous systems, elastically inhomogeneous systems with soft inclusions, and elastically inhomogeneous systems with hard inclusions. The equilibrium properties of inclusions are shown to be consistent with previous results: both homogeneous and hard inclusions adopt a circular shape independent of their size, whereas soft inclusions are circular below a critical radius and elliptic above. In all cases, the Gibbs-Thomson relation is obeyed, except for a change in the prefactor at the critical radius in soft inclusions. Under growth conditions, homogeneous inclusions exhibit a Mullins-Sekerka shape instability [W. Mullins and R. Sekerka, J. Appl. Phys. 34, 323 (1963)], whereas in inhomogeneous systems, the growth of perturbations follows the Leo-Sekerka model [P. Leo and R. Sekerka, Acta Metall. 37, 3139 (1989)]. For soft inclusions, the mode instability regime is gradually replaced by a tip-growing mechanism, which leads to stable, strongly out-of-equilibrium shapes even at very low supersaturation. This mechanism is shown to significantly affect the growth dynamics of soft inclusions, whereas dynamical corrections to the growth rates are negligible in homogeneous and hard inclusions. Finally, due to its microscopic formulation, the model is shown to automatically take into account phenomena caused by the presence of the underlying discrete lattice: anisotropy of the interfacial energy, anisotropy of the kinetics, and preferential excitation of shape perturbations commensurate with the rotational symmetry of the lattice.

Sharp-interface projection of a fluctuating phase-field model

We present a derivation of the sharp-interface limit of a generic fluctuating phase-field model for solidification. As a main result, we obtain a sharp-interface projection which presents noise terms in both the diffusion equation and in the moving boundary conditions. The presented procedure does not rely on the fluctuation-dissipation theorem, and can therefore be applied to account for both internal and external fluctuations in either variational or nonvariational phase-field formulations. In particular, it can be used to introduce thermodynamical fluctuations in nonvariational formulations of the phase-field model, which permit to reach better computational efficiency and provide more flexibility for describing some features of specific physical situations. This opens the possibility of performing quantitative phase-field simulations in crystal growth while accounting for the proper fluctuations of the system.

Diffuse-interface model for rapid phase transformations in nonequilibrium systems

A thermodynamic approach to rapid phase transformations within a diffuse interface in a binary system is developed. Assuming an extended set of independent thermodynamic variables formed by the union of the classic set of slow variables and the space of fast variables, we introduce finiteness of the heat and solute diffusive propagation at the finite speed of the interface advancing. To describe transformations within the diffuse interface, we use the phase-field model which allows us to follow steep but smooth changes of phase within the width of the diffuse interface. Governing equations of the phase-field model are derived for the hyperbolic model, a model with memory, and a model of nonlinear evolution of transformation within the diffuse interface. The consistency of the model is proved by the verification of the validity of the condition of positive entropy production and by outcomes of the fluctuation-dissipation theorem. A comparison with existing sharp-interface and diffuse-interface versions of the model is given.

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 [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 [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 [Phys. Rev. E 53, R3017 (1996)]]. The performance of the model is demonstrated by calculating accurately 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.

Efficient phase field simulation of a binary dendritic growth in a forced flow

Efficient quantitative phase field simulation using an adaptive finite volume method with an antisolutal trapping scheme is presented for a binary dendritic growth in a forced flow. For the case of no convection, the calculated results with different interface thickness are examined. It is found that with a proper antisolutal trapping flux, a thick interface, but smaller than the diffusion boundary layer, could be used and the solution could approach to the sharp-interface Gibbs-Thompson equation limit in almost all aspects quantitatively. Based on the concentration driving force obtained from the sharp-interface limit of the Wheeler-Boettinger-McFadden (WBM) model, the calculated results are in good agreement with the classic Oseen-Ivantsov solution for the concentration-driven growth in a forced flow. And the selection scaling factor also increases with the external flow as the theoretical prediction.

Interface dynamics and solute trapping in alloy solidification with density change

We present a phase-field model for the solidification of a binary alloy, which incorporates hydrodynamic effects due to the different densities of the solid and liquid phases. We start from a generalized thermodynamic potential with squared gradient terms for the associated fields; the condition of local positive entropy production is then utilized to derive a set of equations that drive the system towards equilibrium. The model has been numerically solved in one dimension, to investigate the effects of the flow field on the interface dynamics. We observed that solute trapping is almost unaffected by the fluid advection, while the interface mobility is strongly reduced as the fluid velocity increases. This reflects on the dependence of the interface temperature T(I) on the growth rate v(I): the region of the unstable (ascending) branch is reduced, and the maximum of the T(I)(v(I)) curve is shifted towards lower velocities.

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. 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 lamellae 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.

Nucleation and bulk crystallization in binary phase field theory

We present a phase field theory for binary crystal nucleation. In the one-component limit, quantitative agreement is achieved with computer simulations (Lennard-Jones system) and experiments (ice-water system) using model parameters evaluated from the free energy and thickness of the interface. The critical undercoolings predicted for Cu-Ni alloys accord with the measurements, and indicate homogeneous nucleation. The Kolmogorov exponents deduced for dendritic solidification and for "soft impingement" of particles via diffusion fields are consistent with experiment.

Density change effects on crystal growth from the melt

When a crystal grows from its undercooled melt the local density changes, driving a convective flow in the liquid phase. Then, the purely diffusional description of the process ceases to be satisfactory. Moreover, the dynamic pressure associated with the flow field may affect the melting temperature (and the effective undercooling) of the system. Both these effects have been addressed in recent experimental work. In the present study we derive a thermodynamically consistent phase-field model that accounts for the density change effects in the solidification of a pure substance. Starting from a thermodynamic potential that includes squared gradient terms for both the order parameter and the density, the field equations are derived assuming positive local entropy production. The model is numerically solved in one dimension to show deviations from the classic phase-field description of the same phenomenon.

Phase-field formulation for quantitative modeling of alloy solidification

A phase-field formulation is introduced to simulate quantitatively microstructural pattern formation in alloys. The thin-interface limit of this formulation yields a much less stringent restriction on the choice of interface thickness than previous formulations and permits one to eliminate nonequilibrium effects at the interface. Dendrite growth simulations with vanishing solid diffusivity show that both the interface evolution and the solute profile in the solid are accurately modeled by this approach.

Curvature effects in rapid alloy solidification

The growth of a cylindrical or spherical crystal into its undercooled melt is a process whose description is complicated by the lack of a stationary regime. A simple approach to the problem, justified for low growth rates and widely used in the past for both pure substances and alloy solidification, is based on a quasistatic approximation which assumes an instantaneous adaptation of the diffusional field to the interface configuration. For alloy solidification, assuming isothermal conditions and local interface equilibrium, this simplified model predicts a diffusion controlled growth, with the radius of the crystal increasing asymptotically as ~t(1/2). However, as pointed out by recent investigations, thermal diffusion and nonequilibrium effects enter as essential ingredients in rapid alloy solidification. In the present paper we use the phase-field model to simulate the cylindrical and spherical growth of a solid germ into a supersaturated alloy melt. The problem is treated in its full time-dependent characteristics, accounting for nonequilibrium effects as well as for the rejection of both heat and solute away from the advancing front. We observe a complex behavior and a rich variety of dynamic regimes: in different regions of parameter space the growth rate is limited by diffusion (either thermal or chemical) or is kinetic controlled. Traversing the boundaries which limit these regions, the process undergoes sharp transitions which leave a trace in the solidified alloy. For realistic values of the Lewis number, thermal effects drive the process into a a diffusive regime, in which the rate limiting mechanism is the rejection of solute.

Phase-field modeling of stress-induced instabilities

A phase-field approach describing the dynamics of a strained solid in contact with its melt is developed. Using a formulation that is independent of the state of reference chosen for the displacement field, we write down the elastic energy in an unambiguous fashion, thus obtaining an entire class of models. According to the choice of reference state, the particular model emerging from this class will become equivalent to one of the two independently constructed models on which brief accounts have been given recently [J. Müller and M. Grant, Phys. Rev. Lett. 82, 1736 (1999); K. Kassner and C. Misbah, Europhys. Lett. 46, 217 (1999)]. We show that our phase-field approach recovers the sharp-interface limit corresponding to the continuum model equations describing the Asaro-Tiller-Grinfeld instability. Moreover, we use our model to derive hitherto unknown sharp-interface equations for a situation including a field of body forces. The numerical utility of the phase-field approach is demonstrated by reproducing some known results and by comparison with a sharp-interface simulation. We then proceed to investigate the dynamics of extended systems within the phase-field model which contains an inherent lower length cutoff, thus avoiding cusp singularities. It is found that a periodic array of grooves generically evolves into a superstructure which arises from a series of imperfect period doublings. For wave numbers close to the fastest-growing mode of the linear instability, the first period doubling can be obtained analytically. Both the dynamics of an initially periodic array and a random initial structure can be described as a coarsening process with winning grooves temporarily accelerating whereas losing ones decelerate and even reverse their direction of motion. In the absence of gravity, the end state of a laterally finite system is a single groove growing at constant velocity, as long as no secondary instabilities arise (that we have not been able to see with our code). With gravity, several grooves are possible, all of which are bound to stop eventually. A laterally infinite system approaches a scaling state in the absence of gravity and probably with gravity, too.

Thermal and chemical diffusion in the rapid solidification of binary alloys

Solidification of binary alloys is characterized by the necessity to reject away from the advancing front two conserved quantities: the latent heat released at the solid-liquid interface and the solute atoms that cannot be accommodated in the solid phase. As thermal diffusion is much faster than chemical diffusion, the latter is generally assumed to be the rate limiting mechanism for the process, and the problem is addressed through the isothermal approximation. In the present paper we use the phase-field model to study the planar growth of a solid germ, nucleated in its undercooled melt. We focus on the effects of a noninstantaneous thermal relaxation. The steady growth predicted at large supersaturation in the isothermal limit is prevented. Depending on the value of the Lewis number the growth rate is limited by either mass or heat diffusion; in the latter case we observe a sharp transition between two different regimes, in which originates a nonmonotonic time dependence of the interface temperature. The effects of this transition reflect in the composition of the solidified alloy.

Phase-field model for binary alloys

We present a phase-field model (PFM) for solidification in binary alloys, which is found from the phase-field model for a pure material by direct comparison of the variables for a pure material solidification and alloy solidification. The model appears to be equivalent with the Wheeler-Boettinger-McFadden (WBM) model [A.A. Wheeler, W. J. Boettinger, and G. B. McFadden, Phys. Rev. A 45, 7424 (1992)], but has a different definition of the free energy density for interfacial region. An extra potential originated from the free energy density definition in the WBM model disappears in this model. At a dilute solution limit, the model is reduced to the Tiaden et al. model [Physica D 115, 73 (1998)] for a binary alloy. A relationship between the phase-field mobility and the interface kinetics coefficient is derived at a thin-interface limit condition under an assumption of negligible diffusivity in the solid phase. For a dilute alloy, a steady-state solution of the concentration profile across the diffuse interface is obtained as a function of the interface velocity and the resultant partition coefficient is compared with the previous solute trapping model. For one dimensional steady-state solidification, where the classical sharp-interface model is exactly soluble, we perform numerical simulations of the phase-field model: At low interface velocity, the simulated results from the thin-interface PFM are in excellent agreement with the exact solutions. As the partition coefficient becomes close to unit at high interface velocities, whereas, the sharp-interface PFM yields the correct answer.

Phase and solute fields across the solid-liquid interface of a binary alloy

Solidification of binary alloys is characterized by a sharp structural transition across the solid-liquid interface. It is a typical situation which suggests a nonlocal dependence of the appropriate thermodynamic potential on its associate fields. Phase-field models with a gradient contribution for the order parameter have proved to account for nonequilibrium effects, as solute trapping and kinetic undercooling of the solid-liquid interface. The inclusion of a gradient term for the concentration field has also been theoretically investigated, but in this case the correspondence between predicted phenomena and experimental results is still rather unexplored. In the present study we analyze numerical solutions of a phase-field model in both steady and transient conditions. We focus on effects which are critically dependent on the composition field across the solid-liquid interface; the extent of the concentration gradient correction is related to measurable quantities to suggest methods for a comparison between the model predictions and experimental data.