Quantitative phase-field modeling of two-phase growth

Microgravity studies of solidification patterns in model transparent alloys onboard the International Space Station

We review recent in situ solidification experiments using nonfaceted model transparent alloys in science-in-microgravity facilities onboard the International Space Station (ISS), namely the Transparent Alloys (TA) apparatus and the Directional Solidification Insert of the DEvice for the study of Critical Liquids and Crystallization (DECLIC-DSI). These directional-solidification devices use innovative optical videomicroscopy imaging techniques to observe the spatiotemporal dynamics of solidification patterns in real time in large samples. In contrast to laboratory conditions on ground, microgravity guarantees the absence or a reduction of convective motion in the liquid, thus ensuring a purely diffusion-controlled growth of the crystalline solid(s). This makes it possible to perform a direct theoretical analysis of the formation process of solidification microstructures with comparisons to quantitative numerical simulations. Important questions that concern multiphase growth patterns in eutectic and peritectic alloys on the one hand and single-phased, cellular and dendritic structures on the other hand have been addressed, and unprecedented results have been obtained. Complex self-organizing phenomena during steady-state and transient coupled growth in eutectics and peritectics, interfacial-anisotropy effects in cellular arrays, and promising insights into the columnar-to-equiaxed transition are highlighted.

Asymptotic analysis of multi-phase-field models: A thorough consideration of junctions

The solutions of multi-phase-field models exhibit boundary layer behavior not only along the binary interfaces but also at the common contacts of three or more phases, i.e., junctions. Hence, to completely determine the asymptotic behavior of a multi-phase-field model, the inner analysis of both types of layers has to be carried out, whereas, traditionally, the junctions part is ignored. This is remedied in the current work for a phase-field model of simple grain growth in two spatial dimensions. Since the junction neighbourhoods are fundamentally different from those of the binary interfaces, pertinent matching conditions had to be derived from scratch, which is also accomplished in a detailed manner. The leading-order matching analysis of the junctions exposed the restrictions present on the interfacial arrangement at the common meeting point, while the next-to-the-leading one uncovered the law governing the instantaneous motion of the latter. In particular, it is predicted for the considered model that the Young's law is always satisfied at a triple point, whether or not it is at rest. Surprisingly, the mobilities and the curvatures of the involving interfaces as well as the driving forces on the them do not affect this result. However, they do play a significant role in determining the instantaneous velocity of the junction point. The study has opened up many new directions for future research.

Microstructural Pattern Formation during Far-from-Equilibrium Alloy Solidification

We introduce a new phase-field formulation of rapid alloy solidification that quantitatively incorporates nonequilibrium effects at the solid-liquid interface over a very wide range of interface velocities. Simulations identify a new dynamical instability of dendrite tip growth driven by solute trapping at velocities approaching the absolute stability limit. They also reproduce the formation of the widely observed banded microstructures, revealing how this instability triggers transitions between dendritic and microsegregation-free solidification. Predicted band spacings agree quantitatively with observations in rapidly solidified Al-Cu thin films.

Asymptotic analysis of multi-phase-field models: A thorough consideration of binary interfaces

Although multi-phase-field models are applied extensively to simulate various pattern formations, their asymptotic analysis is not typically performed at a level of rigor common to their scalar counterparts. Most of the time, arguments given, such as for the justification of the selection of the bulk phases or the phasal composition of the interfaces between them, are only heuristic in nature. In particular, the reduction of the multi-phase-field models to two-phase ones, so as to ascertain the dynamical laws captured by them, can only be termed as hand waving, at best. It is also common to land the starting point of the analysis directly at a point where the binary interfaces have already formed and continue therefrom with the prediction of their instantaneous evolution. However, exactly how a given initial filling transitions to a state characterized by the presence of bulk phases separated by internal layers, and with what distribution, is rarely addressed. Moreover, a detailed and systematic study, focused on the numerical realization of the asymptotics predicted laws, has never been reported before for multi-phase-field models. In the current article, endorsing against these undesirabilities of the common presentations, a full-fledged asymptotic analysis of a multi-grain-growth phase-field model is put forth and numerically verified. However, the consideration is only limited to the analysis of binary interfaces; that of junctions (triple points, quadruple points, etc.) is deferred to a later work.

Geometry of Triple Junctions during Grain Boundary Premelting

Grain boundaries (GBs) whose energy is larger than twice the energy of the solid-liquid interface exhibit the premelting phenomenon, for which an atomically thin liquid layer develops at temperatures slightly below the bulk melting temperature. Premelting can have a severe impact on the structural integrity of a polycrystalline material and on the mechanical high-temperature properties, also in the context of crack formation during the very last stages of solidification. The triple junction between a dry GB and the two solid-liquid interfaces of a liquid layer propagating along the GB cannot be defined from macroscopic continuum properties and surface tension equilibria in terms of Young's law. We show how incorporating atomistic scale physics using a disjoining potential regularizes the state of the triple junction and yields an equilibrium with a well-defined microscopic contact angle. We support this finding by dynamical simulations using a multiphase field model with obstacle potential for both purely kinetic and diffusive conditions. Generally, our results should provide insights on the dynamics of GB phase transitions, of which the complex phenomena associated with liquid metal embrittlement are an example.

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.

Phase-Field Modeling of Biomineralization in Mollusks and Corals: Microstructure vs Formation Mechanism

While biological crystallization processes have been studied on the microscale extensively, there is a general lack of models addressing the mesoscale aspects of such phenomena. In this work, we investigate whether the phase-field theory developed in materials' science for describing complex polycrystalline structures on the mesoscale can be meaningfully adapted to model crystallization in biological systems. We demonstrate the abilities of the phase-field technique by modeling a range of microstructures observed in mollusk shells and coral skeletons, including granular, prismatic, sheet/columnar nacre, and sprinkled spherulitic structures. We also compare two possible micromechanisms of calcification: the classical route, via ion-by-ion addition from a fluid state, and a nonclassical route, crystallization of an amorphous precursor deposited at the solidification front. We show that with an appropriate choice of the model parameters, microstructures similar to those found in biomineralized systems can be obtained along both routes, though the time-scale of the nonclassical route appears to be more realistic. The resemblance of the simulated and natural biominerals suggests that, underneath the immense biological complexity observed in living organisms, the underlying design principles for biological structures may be understood with simple math and simulated by phase-field theory.

Dynamics of grain boundary premelting

The mechanical strength of a polycrystalline material can be drastically weakened by a phenomenon known as grain boundary (GB) premelting that takes place, owing to the so-called disjoining potential, when the dry GB free energy [Formula: see text] exceeds twice the free energy of the solid-liquid interface [Formula: see text]. While previous studies of GB premelting are all limited to equilibrium conditions, we use a multi-phase field model to analyze premelting dynamics by simulating the steady-state growth of a liquid layer along a dry GB in an insulated channel and the evolution of a pre-melted polycrystalline microstructure. In both cases, our results reveal the crucial influence of the disjoining potential. A dry GB transforms into a pre-melted state for a grain-size-dependent temperature interval around [Formula: see text], such that a critical overheating of the dry GBs over [Formula: see text] should be exceeded for the classical melting process to take place, the liquid layer to achieve a macroscopic width, and the disjoining potential to vanish. Our simulations suggest a steady-state velocity for this transformation proportional to [Formula: see text]. Concerning the poly-crystalline evolution, we find unusual grain morphologies and dynamics, deriving from the existence of a pre-melted polycrystalline equilibrium that we evidence. We are then able to identify the regime in which, due to the separation of the involved length scales, the dynamics corresponds to the same curvature-driven dynamics as for dry GBs, but with enhanced mobility.

Effect of In Situ Grown SiC Nanowires on the Pressureless Sintering of Heterophase Ceramics TaSi2-TaC-SiC

To ascertain the influence of SiC nanowires on sintering kinetics of heterophase ceramics, two composite powders (TaSi₂-TaC-SiC and TaSi₂-TaC-SiC-SiC_nanowire) are fabricated by mechanically activated combustion synthesis of Ta-Si-C and Ta-Si-C-(C₂F₄) reactive mixtures. Remarkable compressibility is achieved for the TaSi₂-TaC-SiC-SiC_nanowire composition (green density up to 84% as compared with 45.2% achieved for TaSi₂-SiC-TaC) which is attributed to the lubricating effect of residual adsorbed fluorinated carbon (most likely C₄F₈). The outcomes of pressureless sintering of TaSi₂-TaC-SiC and TaSi₂-TaC-SiC-SiC_nanowire compositions are vastly different; the former experiences no significant densification or grain growth and does not attain structural integrity, whereas the latter achieves relative density up to 93% and hardness up to 11 GPa. The SiC nanowires are not retained in consolidated ceramics, but instead, act as a sintering aid and promote densification and grain growth. Sintering mechanisms of TaSi₂-TaC-SiC and TaSi₂-TaC-SiC-SiC_nanowire powders are analyzed using thermodynamic and ab initio grand potential calculations, as well as the analysis of grain size versus relative density relations. In the case of solid-state sintering, the densification and grain growth in heterophase non-oxide ceramics are governed by the same mechanisms as previously investigated single-phase oxides. The presence of SiC nanowires enhances grain-boundary related diffusion processes due to the high specific surface and aspect ratio of the nanowires. At 1500 °C, where the formation of the transient Si-based liquid phase is thermodynamically viable, only the SiC nanowire-containing composition demonstrated the intense grain coarsening and densification associated with liquid-assisted sintering. This effect can be attributed both to the presence of SiC nanowires and purification of residual oxide impurities due to C₂F₄-activated combustion synthesis employed for the in situ formation of SiC nanowires.

Regulating lamellar eutectic trajectory through external perturbations

The present understanding of asymmetric lamellar eutectics focuses on pure diffusive transport, and how the external perturbations cause asymmetric pattern transitions remains unclear. In this work, the effect of external perturbations is discussed in terms of both thermal and convective effects via phase-field modeling. The presence of thermal perturbation distorts eutectic lamellae, while the convective perturbation causes a tilt band. Both can adjust the eutectic trajectory to accommodate newly established thermodynamics by reconstructing the transport equilibrium. Furthermore, how to regulate the eutectic growth (eutectic colony, zigzag, and snakelike patterns) by altering external perturbations is investigated, which provides information on how to control eutectic evolution.

Archimedean lattices emerge in template-directed eutectic solidification

Template-directed assembly has been shown to yield a broad diversity of highly ordered mesostructures^1,2, which in a few cases exhibit symmetries not present in the native material^3-5. However, this technique has not yet been applied to eutectic materials, which underpin many modern technologies ranging from high-performance turbine blades to solder alloys. Here we use directional solidification of a simple AgCl-KCl lamellar eutectic material within a pillar template to show that interactions of the material with the template lead to the emergence of a set of microstructures that are distinct from the eutectic's native lamellar structure and the template's hexagonal lattice structure. By modifying the solidification rate of this material-template system, trefoil, quatrefoil, cinquefoil and hexafoil mesostructures with submicrometre-size features are realized. Phase-field simulations suggest that these mesostructures appear owing to constraints imposed on diffusion by the hexagonally arrayed pillar template. We note that the trefoil and hexafoil patterns resemble Archimedean honeycomb and square-hexagonal-dodecagonal lattices⁶, respectively. We also find that by using monolayer colloidal crystals as templates, a variety of eutectic mesostructures including trefoil and hexafoil are observed, the former resembling the Archimedean kagome lattice. Potential emerging applications for the structures provided by templated eutectics include non-reciprocal metasurfaces⁷, magnetic spin-ice systems^8,9, and micro- and nano-lattices with enhanced mechanical properties^10,11.

A High-Order Convex Splitting Method for a Non-Additive Cahn–Hilliard Energy Functional

Various Cahn–Hilliard (CH) energy functionals have been introduced to model phase separation in multi-component system. Mathematically consistent models have highly nonlinear terms linked together, thus it is not well-known how to split this type of energy. In this paper, we propose a new convex splitting and a constrained Convex Splitting (cCS) scheme based on the splitting. We show analytically that the cCS scheme is mass conserving and satisfies the partition of unity constraint at the next time level. It is uniquely solvable and energy stable. Furthermore, we combine the convex splitting with the specially designed implicit–explicit Runge–Kutta method to develop a high-order (up to third-order) cCS scheme for the multi-component CH system. We also show analytically that the high-order cCS scheme is unconditionally energy stable. Numerical experiments with ternary and quaternary systems are presented, demonstrating the accuracy, energy stability, and capability of the proposed high-order cCS scheme.

Grand-potential-based phase-field model for multiple phases, grains, and chemical components

Grand-potential-based phase-field model for multiple phases, grains, and chemical components is derived from a grand-potential functional. Due to the grand-potential formulation, the chemical energy does not contribute to the interfacial energy between phases, simplifying parametrization and decoupling interface thickness from interfacial energy, which can potentially allow increased interface thicknesses and therefore improved computational efficiency. Two-phase interfaces are stable with respect to the formation of additional phases, simplifying implementation and allowing the variational form of the evolution equations to be used. Additionally, we show that grand-potential-based phase-field models are capable of simulating phase separation, and we derive conditions under which this is possible.

Quantitative phase-field lattice-Boltzmann study of lamellar eutectic growth under natural convection

The influence of natural convection on lamellar eutectic growth was determined by a comprehensive phase-field lattice-Boltzmann study for Al-Cu and CBr_{4}-C_{2}Cl_{6} eutectic alloys. The mass differences resulting from concentration differences led to the fluid flow and a robust parallel and adaptive mesh refinement algorithm was employed to improve the computational efficiency. By means of carefully designed "numerical experiments", the eutectic growth under natural convection was explored and a simple analytical model was proposed to predict the adjustment of the lamellar spacing. Furthermore, by alternating the solute expansion coefficient, initial lamellar spacing, and undercooling, the microstructure evolution was presented and compared with the classical eutectic growth theory. Results showed that both interfacial solute distribution and average curvature were affected by the natural convection, the effect of which could be further quantified by adding a constant into the growth rule proposed by Jackson and Hunt [Jackson and Hunt, Trans. Metall. Soc. AIME 236, 1129 (1966)].

Nonequilibrium Thermodynamics of Hydrate Growth on a Gas-Liquid Interface

We develop a continuum-scale phase-field model to study gas-liquid-hydrate systems far from thermodynamic equilibrium. We design a Gibbs free energy functional for methane-water mixtures that recovers the isobaric temperature-composition phase diagram under thermodynamic equilibrium conditions. The proposed free energy is incorporated into a phase-field model to study the dynamics of hydrate formation on a gas-liquid interface. We elucidate the role of initial aqueous concentration in determining the direction of hydrate growth at the interface, in agreement with experimental observations. Our model also reveals two stages of hydrate growth at an interface-controlled by a crossover in how methane is supplied from the gas and liquid phases-which could explain the persistence of gas conduits in hydrate-bearing sediments and other nonequilibrium phenomena commonly observed in natural methane hydrate systems.

Classical nucleation theory in the phase-field crystal model

A full understanding of polycrystalline materials requires studying the process of nucleation, a thermally activated phase transition that typically occurs at atomistic scales. The numerical modeling of this process is problematic for traditional numerical techniques: commonly used phase-field methods' resolution does not extend to the atomic scales at which nucleation takes places, while atomistic methods such as molecular dynamics are incapable of scaling to the mesoscale regime where late-stage growth and structure formation takes place following earlier nucleation. Consequently, it is of interest to examine nucleation in the more recently proposed phase-field crystal (PFC) model, which attempts to bridge the atomic and mesoscale regimes in microstructure simulations. In this work, we numerically calculate homogeneous liquid-to-solid nucleation rates and incubation times in the simplest version of the PFC model, for various parameter choices. We show that the model naturally exhibits qualitative agreement with the predictions of classical nucleation theory (CNT) despite a lack of some explicit atomistic features presumed in CNT. We also examine the early appearance of lattice structure in nucleating grains, finding disagreement with some basic assumptions of CNT. We then argue that a quantitatively correct nucleation theory for the PFC model would require extending CNT to a multivariable theory.

Three-dimensional mesostructures as high-temperature growth templates, electronic cellular scaffolds, and self-propelled microrobots

Recent work demonstrates that processes of stress release in prestrained elastomeric substrates can guide the assembly of sophisticated 3D micro/nanostructures in advanced materials. Reported application examples include soft electronic components, tunable electromagnetic and optical devices, vibrational metrology platforms, and other unusual technologies, each enabled by uniquely engineered 3D architectures. A significant disadvantage of these systems is that the elastomeric substrates, while essential to the assembly process, can impose significant engineering constraints in terms of operating temperatures and levels of dimensional stability; they also prevent the realization of 3D structures in freestanding forms. Here, we introduce concepts in interfacial photopolymerization, nonlinear mechanics, and physical transfer that bypass these limitations. The results enable 3D mesostructures in fully or partially freestanding forms, with additional capabilities in integration onto nearly any class of substrate, from planar, hard inorganic materials to textured, soft biological tissues, all via mechanisms quantitatively described by theoretical modeling. Illustrations of these ideas include their use in 3D structures as frameworks for templated growth of organized lamellae from AgCl-KCl eutectics and of atomic layers of WSe₂ from vapor-phase precursors, as open-architecture electronic scaffolds for formation of dorsal root ganglion (DRG) neural networks, and as catalyst supports for propulsive systems in 3D microswimmers with geometrically controlled dynamics. Taken together, these methodologies establish a set of enabling options in 3D micro/nanomanufacturing that lie outside of the scope of existing alternatives.

Variational formulation of a quantitative phase-field model for nonisothermal solidification in a multicomponent alloy

A variational formulation of a quantitative phase-field model is presented for nonisothermal solidification in a multicomponent alloy with two-sided asymmetric diffusion. The essential ingredient of this formulation is that the diffusion fluxes for conserved variables in both the liquid and solid are separately derived from functional derivatives of the total entropy and then these fluxes are related to each other on the basis of the local equilibrium conditions. In the present formulation, the cross-coupling terms between the phase-field and conserved variables naturally arise in the phase-field equation and diffusion equations, one of which corresponds to the antitrapping current, the phenomenological correction term in early nonvariational models. In addition, this formulation results in diffusivities of tensor form inside the interface. Asymptotic analysis demonstrates that this model can exactly reproduce the free-boundary problem in the thin-interface limit. The present model is widely applicable because approximations and simplifications are not formally introduced into the bulk's free energy densities and because off-diagonal elements of the diffusivity matrix are explicitly taken into account. Furthermore, we propose a nonvariational form of the present model to achieve high numerical performance. A numerical test of the nonvariational model is carried out for nonisothermal solidification in a binary alloy. It shows fast convergence of the results with decreasing interface thickness.

High-Operating-Temperature Direct Ink Writing of Mesoscale Eutectic Architectures

High-operating-temperature direct ink writing (HOT-DIW) of mesoscale architectures that are composed of eutectic silver chloride-potassium chloride. The molten ink undergoes directional solidification upon printing on a cold substrate. The lamellar spacing of the printed features can be varied between approximately 100 nm and 2 µm, enabling the manipulation of light in the visible and infrared range.

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.

Variational formulation and numerical accuracy of a quantitative phase-field model for binary alloy solidification with two-sided diffusion

We present the variational formulation of a quantitative phase-field model for isothermal low-speed solidification in a binary dilute alloy with diffusion in the solid. In the present formulation, cross-coupling terms between the phase field and composition field, including the so-called antitrapping current, naturally arise in the time evolution equations. One of the essential ingredients in the present formulation is the utilization of tensor diffusivity instead of scalar diffusivity. In an asymptotic analysis, it is shown that the correct mapping between the present variational model and a free-boundary problem for alloy solidification with an arbitrary value of solid diffusivity is successfully achieved in the thin-interface limit due to the cross-coupling terms and tensor diffusivity. Furthermore, we investigate the numerical performance of the variational model and also its nonvariational versions by carrying out two-dimensional simulations of free dendritic growth. The nonvariational model with tensor diffusivity shows excellent convergence of results with respect to the interface thickness.

A phase-field approach to nonequilibrium phase transformations in elastic solids via an intermediate phase (melt) allowing for interface stresses

A phase-field approach for phase transformations between three different phases at nonequilibrium temperatures with mechanics and interfacial stresses is developed.

Template-Directed Directionally Solidified 3D Mesostructured AgCl-KCl Eutectic Photonic Crystals

3D mesostructured AgCl-KCl photonic crystals emerge from colloidal templating of eutectic solidification. Solvent removal of the KCl phase results in a mesostructured AgCl inverse opal. The 3D-template-induced confinement leads to the emergence of a complex microstructure. The 3D mesostructured eutectic photonic crystals have a large stop band ranging from the near-infrared to the visible tuned by the processing.

Quantitative phase-field modeling of dendritic electrodeposition

A thin-interface phase-field model of electrochemical interfaces is developed based on Marcus kinetics for concentrated solutions, and used to simulate dendrite growth during electrodeposition of metals. The model is derived in the grand electrochemical potential to permit the interface to be widened to reach experimental length and time scales, and electroneutrality is formulated to eliminate the Debye length. Quantitative agreement is achieved with zinc Faradaic reaction kinetics, fractal growth dimension, tip velocity, and radius of curvature. Reducing the exchange current density is found to suppress the growth of dendrites, and screening electrolytes by their exchange currents is suggested as a strategy for controlling dendrite growth in batteries.

Ternary eutectic dendrites: Pattern formation and scaling properties

Extending previous work [Pusztai et al., Phys. Rev. E 87, 032401 (2013)], we have studied the formation of eutectic dendrites in a model ternary system within the framework of the phase-field theory. We have mapped out the domain in which two-phase dendritic structures grow. With increasing pulling velocity, the following sequence of growth morphologies is observed: flat front lamellae → eutectic colonies → eutectic dendrites → dendrites with target pattern → partitionless dendrites → partitionless flat front. We confirm that the two-phase and one-phase dendrites have similar forms and display a similar scaling of the dendrite tip radius with the interface free energy. It is also found that the possible eutectic patterns include the target pattern, and single- and multiarm spirals, of which the thermal fluctuations choose. The most probable number of spiral arms increases with increasing tip radius and with decreasing kinetic anisotropy. Our numerical simulations confirm that in agreement with the assumptions of a recent analysis of two-phase dendrites [Akamatsu et al., Phys. Rev. Lett. 112, 105502 (2014)], the Jackson-Hunt scaling of the eutectic wavelength with pulling velocity is obeyed in the parameter domain explored, and that the natural eutectic wavelength is proportional to the tip radius of the two-phase dendrites. Finally, we find that it is very difficult/virtually impossible to form spiraling two-phase dendrites without anisotropy, an observation that seems to contradict the expectations of Akamatsu et al. Yet, it cannot be excluded that in isotropic systems, two-phase dendrites are rare events difficult to observe in simulations.

The strong influence of internal stresses on the nucleation of a nanosized, deeply undercooled melt at a solid-solid phase interface

The effect of elastic energy on nucleation and disappearance of a nanometer size intermediate melt (IM) region at a solid-solid (S1S2) phase interface at temperatures 120 K below the melting temperature is studied using a phase-field approach. Results are obtained for broad range of the ratios of S1S2 to solid-melt interface energies, k(E), and widths, k(δ). It is found that internal stresses only slightly promote barrierless IM nucleation but qualitatively alter the system behavior, allowing for the appearance of the IM when k(E) < 2 (thermodynamically impossible without mechanics) and elimination of what we termed the IM-free gap. Remarkably, when mechanics is included within this framework, there is a drastic (16 times for HMX energetic crystals) reduction in the activation energy of IM critical nucleus. After this inclusion, a kinetic nucleation criterion is met, and thermally activated melting occurs under conditions consistent with experiments for HMX, elucidating what had been to date mysterious behavior. Similar effects are expected to occur for other material systems where S1S2 phase transformations via IM take place, including electronic, geological, pharmaceutical, ferroelectric, colloidal, and superhard materials.

Interphase anisotropy effects on lamellar eutectics: a numerical study

In directional solidification of binary eutectics, it is often observed that two-phase lamellar growth patterns grow tilted with respect to the direction z of the imposed temperature gradient. This crystallographic effect depends on the orientation of the two crystal phases α and β with respect to z. Recently, an approximate theory was formulated that predicts the lamellar tilt angle as a function of the anisotropy of the free energy of the solid(α)-solid(β) interphase boundary. We use two different numerical methods-phase field (PF) and dynamic boundary integral (BI)-to simulate the growth of steady periodic patterns in two dimensions as a function of the angle θ(R) between z and a reference crystallographic axis for a fixed relative orientation of α and β crystals, that is, for a given anisotropy function (Wulff plot) of the interphase boundary. For Wulff plots without unstable interphase-boundary orientations, the two simulation methods are in excellent agreement with each other and confirm the general validity of the previously proposed theory. In addition, a crystallographic "locking" of the lamellae onto a facet plane is well reproduced in the simulations. When unstable orientations are present in the Wulff plot, it is expected that two distinct values of the tilt angle can appear for the same crystal orientation over a finite θ(R) range. This bistable behavior, which has been observed experimentally, is well reproduced by BI simulations but not by the PF model. Possible reasons for this discrepancy are discussed.

Extending the nonequilibrium square-gradient model with temperature-dependent influence parameters

Nonequilibrium interface phenomena play a key role in crystallization, hydrate formation, pipeline depressurization, and a multitude of other examples. Square gradient theory extended to the nonequilibrium domain is a powerful tool for understanding these processes. The theory gives an accurate prediction of surface tension at equilibrium, only with temperature-dependent influence parameters. We extend in this work the nonequilibrium square gradient model to have temperature-dependent influence parameters. The extension leads to thermodynamic quantities which depend on temperature gradients. Remarkably the Gibbs relation proposed in earlier work is still valid. Also for the extended framework, the "Gibbs surface" described by excess variables is found to be in local equilibrium. The temperature-dependent influence parameters give significantly different interface resistivities (∼9%-50%), due to changed density gradients and additional terms in the enthalpy. The presented framework facilitates a more accurate description of transport across interfaces with square gradient theory.

Phase-field modeling of grain-boundary premelting using obstacle potentials

We investigate the multiorder parameter phase field model of Steinbach and Pezzolla [Physica D 134, 385 (1999)] concerning its ability to describe grain boundary premelting. For a single order parameter situation solid-melt interfaces are always attractive, which allows us to have (unstable) equilibrium solid-melt-solid coexistence above the bulk melting point. The temperature-dependent melt layer thickness and the disjoining potential, which describe the interface interaction, are affected by the choice of the thermal coupling function and the measure to define the amount of the liquid phase. Due to the strictly finite interface thickness the interaction range also is finite. For a multiorder parameter model we find either purely attractive or purely repulsive finite-ranged interactions. The premelting transition is then directly linked to the ratio of the grain boundary and solid-melt interfacial energy.

Interface kinetics in phase-field models: isothermal transformations in binary alloys and step dynamics in molecular-beam epitaxy

We present a unified description of interface kinetic effects in phase-field models for isothermal transformations in binary alloys and steps dynamics in molecular-beam-epitaxy. The phase-field equations of motion incorporate a kinetic cross-coupling between the phase field and the concentration field. This cross-coupling generalizes the phenomenology of kinetic effects and was omitted until recently in classical phase-field models. We derive general expressions (independent of the details of the phase-field model) for the kinetic coefficients within the corresponding macroscopic approach using a physically motivated reduction procedure. The latter is equivalent to the so-called thin-interface limit but is technically simpler. It involves the calculation of the effective dissipation that can be ascribed to the interface in the phase-field model. We discuss in detail the possibility of a nonpositive definite matrix of kinetic coefficients, i.e., a negative effective interface dissipation, although being in the range of stability of the underlying phase-field model. Numerically we study the step-bunching instability in molecular-beam-epitaxy due to the Ehrlich-Schwoebel effect, present in our model due to the cross-coupling. Using the reduction procedure we compare the results of the phase-field simulations with the analytical predictions of the macroscopic approach.

Kinetic cross coupling between nonconserved and conserved fields in phase field models

We present a phase field model for isothermal transformations of two-component alloys that includes Onsager kinetic cross coupling between the nonconserved phase field and the conserved concentration field ø C. We also provide the reduction of the phase field model to the corresponding macroscopic description of the free boundary problem. The reduction is given in a general form. Additionally we use an explicit example of a phase field model and check that the reduced macroscopic description, in the range of its applicability, is in excellent agreement with direct phase field simulations. The relevance of the newly introduced terms to solute trapping is also discussed.

Quantitative phase-field modeling of nonisothermal solidification in dilute multicomponent alloys with arbitrary diffusivities

A quantitative phase-field model is developed for simulating microstructural pattern formation in nonisothermal solidification in dilute multicomponent alloys with arbitrary thermal and solutal diffusivities. By performing the matched asymptotic analysis, it is shown that the present model with antitrapping current terms reproduces the free-boundary problem of interest in the thin-interface limit. Convergence of the simulation outcome with decreasing the interface thickness is demonstrated for nonisothermal free dendritic growth in binary alloys and isothermal and nonisothermal free dendritic growth in a ternary alloy.

Solidification in syntectic and monotectic systems

We present theoretical studies of syntectic and monotectic solidification scenarios. Steady-state solidification along the liquid-liquid interface in a syntectic system is considered by means of a boundary-integral technique developed previously. We study the case of small asymmetry of the pattern and extract from the results the scaling relations in terms of the undercooling and the asymmetry parameter. We also investigate monotectic solidification using the phase-field method. We present two kinds of two-phase fingers, with the solid phase being either the exterior phase or the interior phase, and the pattern corresponding to the growth along the solid-liquid interface. We finally analyze the asymptotic shape of these new morphologies far behind their tip.

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

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

Unified derivation of phase-field models for alloy solidification from a grand-potential functional

In the literature, two quite different phase-field formulations for the problem of alloy solidification can be found. In the first, the material in the diffuse interfaces is assumed to be in an intermediate state between solid and liquid, with a unique local composition. In the second, the interface is seen as a mixture of two phases that each retain their macroscopic properties, and a separate concentration field for each phase is introduced. It is shown here that both types of models can be obtained by the standard variational procedure if a grand-potential functional is used as a starting point instead of a free energy functional. The dynamical variable is then the chemical potential instead of the composition. In this framework, a complete analogy with phase-field models for the solidification of a pure substance can be established. This analogy is then exploited to formulate quantitative phase-field models for alloys with arbitrary phase diagrams. The precision of the method is illustrated by numerical simulations with varying interface thickness.

Phase-field investigation of rod eutectic morphologies under geometrical confinement

Three-dimensional phase-field simulations are employed to investigate rod-type eutectic growth morphologies in confined geometry. Distinct steady-state solutions are found to depend on this confinement effect with the rod array basis vectors and their included angle (α) changing to accommodate the geometrical constraint. Specific morphologies are observed, including rods of circular cross sections, rods of distorted (elliptical) cross sections, rods of peanut-shaped cross-sections, and lamellar structures. The results show that, for a fixed value of α > 10°, the usual (triangular) arrays of circular rods are stable in a broad range of spacings, with a transition to the peanut-shaped cross sectioned rods occurring at large spacings (above 1.5 times the minimum undercooling spacing λ(m)), and the advent of rod eliminations at low spacings. Furthermore, a transition from rod to lamellar structures is observed for α < 10° for the phase fraction of 10.5% used in the present paper.

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.

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 geometry, 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.

Solidification along the interface between demixed liquids in monotectic systems

The steady-state solidification along the liquid-liquid interface in the monotectic system is discussed. A boundary-integral formulation describing the diffusion in the two liquid phases is given and the corresponding equations for the three interfaces (two solid-liquid interfaces and one liquid-liquid interface) are solved. Scaling relations are extracted from the results and supported by analytic arguments in the limit of small deviation from the monotectic temperature. We present also a complementary phase-field simulation, which proves the stability of the process.

Growth of a two-phase finger in eutectics systems

We present a theoretical study of the growth of a two-phase finger in eutectic systems. This pattern was observed experimentally by Akamatsu and Faivre [Phys. Rev. E 61, 3757 (2000)]. We study this two-phase finger using a boundary-integral formulation and we complement our investigation by a phase-field validation of the stability of the pattern. The deviations from the eutectic temperature and from the eutectic concentration provide two independent control parameters, leading to very different patterns depending on their relative importance. We propose scaling laws for the velocity and the different length scales of the pattern.

Ginzburg-Landau-type multiphase field model for competing fcc and bcc nucleation

We address crystal nucleation and fcc-bcc phase selection in alloys using a multiphase field model that relies on Ginzburg-Landau free energies of the liquid-fcc, liquid-bcc, and fcc-bcc subsystems, and determine the properties of the nuclei as a function of composition, temperature, and structure. With a realistic choice for the free energy of the fcc-bcc interface, the model predicts well the fcc-bcc phase-selection boundary in the Fe-Ni system.

Three-dimensional chemotaxis model for a crawling neutrophil

Chemotactic cell migration is a fundamental phenomenon in complex biological processes. A rigorous understanding of the chemotactic mechanism of crawling cells has important implications for various medical and biological applications. In this paper, we propose a three-dimensional model of a single crawling cell to study its chemotaxis. A single-cell study of chemotaxis has an advantage over studies of a population of cells in that it provides a clearer observation of cell migration, which leads to more accurate assessments of chemotaxis. The model incorporates the surface energy of the cell and the interfacial interaction between the cell and substrate. The semi-implicit Fourier spectral method is applied to achieve high efficiency and numerical stability. The simulation results provide the kinetic and morphological traits of a crawling cell during chemotaxis.

Phase-field-crystal calculation of crystal-melt surface tension in binary alloys

A phase field crystal (PFC) density functional for binary mixtures is coarse grained and a formalism for calculating the simultaneous concentration, temperature, and density dependence of the surface energy anisotropy of a solid-liquid interface is developed. The methodology systematically relates bulk free energy coefficients arising from coarse graining to thermodynamic data, while gradient energy coefficients are related to molecular properties. Our coarse-grained formalism is applied to the determination of surface energy anisotropy in two-dimensional Zn-Al films, a situation relevant for quantitative phase field simulations of dendritic solidification in zinc coatings.

Effective model hierarchies for dynamic and static classical density functional theories

The origin and methodology of deriving effective model hierarchies are presented with applications to solidification of crystalline solids. In particular, it is discussed how the form of the equations of motion and the effective parameters on larger scales can be obtained from the more microscopic models. It will be shown that tying together the dynamic structure of the projection operator formalism with static classical density functional theories can lead to incomplete (mass) transport properties even though the linearized hydrodynamics on large scales is correctly reproduced. To facilitate a more natural way of binding together the dynamics of the macrovariables and classical density functional theory, a dynamic generalization of density functional theory based on the nonequilibrium generating functional is suggested.

Multi-phase-field analysis of short-range forces between diffuse interfaces

We characterize both analytically and numerically short-range forces between spatially diffuse interfaces in multi-phase-field models of polycrystalline materials. During late-stage solidification, crystal-melt interfaces may attract or repel each other depending on the degree of misorientation between impinging grains, temperature, composition, and stress. To characterize this interaction, we map the multiphase-field equations for stationary interfaces to a multidimensional classical mechanical scattering problem. From the solution of this problem, we derive asymptotic forms for short-range forces between interfaces for distances larger than the interface thickness. The results show that forces are always attractive for traditional models where each phase-field represents the phase fraction of a given grain. Those predictions are validated by numerical computations of forces for all distances. Based on insights from the scattering problem, we propose a multi-phase-field formulation that can describe both attractive and repulsive forces in real systems. This model is then used to investigate the influence of solute addition and a uniaxial stress perpendicular to the interface. Solute addition leads to bistability of different interfacial equilibrium states, with the temperature range of bistability increasing with strength of partitioning. Stress in turn, is shown to be equivalent to a temperature change through a standard Clausius-Clapeyron relation. The implications of those results for understanding grain boundary premelting are discussed.

Phase-field study of three-dimensional steady-state growth shapes in directional solidification

We use a quantitative phase-field approach to study directional solidification in various three-dimensional geometries for realistic parameters of a transparent binary alloy. The geometries are designed to study the steady-state growth of spatially extended hexagonal arrays, linear arrays in thin samples, and axisymmetric shapes constrained in a tube. As a basis to address issues of dynamical pattern selection, the phase-field simulations are specifically geared to identify ranges of primary spacings for the formation of the classically observed "fingers" (deep cells) with blunt tips and "needles" with parabolic tips. Three distinct growth regimes are identified that include a low-velocity regime with only fingers forming, a second intermediate-velocity regime characterized by coexistence of fingers and needles that exist on separate branches of steady-state growth solutions for small and large spacings, respectively, and a third high-velocity regime where those two branches merge into a single one. Along the latter, the growth shape changes continuously from fingerlike to needlelike with increasing spacing. These regimes are strongly influenced by crystalline anisotropy with the third regime extending to lower velocity for larger anisotropy. Remarkably, however, steady-state shapes and tip undercoolings are only weakly dependent on the growth geometry. Those results are used to test existing theories of directional finger growth as well as to interpret the hysteretic nature of the cell-to-dendrite transition.

Pattern formation during diffusional transformations in the presence of triple junctions and elastic effects

We compare different scenarios for dendritic melting of alloys with respect to the front propagation velocity. In contrast to conventional dendritic growth, selection can here be also due to the presence of a grain boundary or coherence strains, and the propagation speed is higher. The most favorable situation is partial melting, where two parabolic fronts, one melting and one solidifying interface, are moving together, since the process is then determined by diffusion in the thin liquid layer. There, and also in phase field simulations of melting in peritectic and eutectic systems, we observe a rotation of the triple junction relative to the growth direction. Finally, we discuss the role of elastic effects due to density and structural differences on solid-state phase transformations, and we find that they significantly alter the selection principles. In particular, we obtain free dendritic growth even with isotropic surface tension. This is investigated by Green's function methods and a phase field approach for growth in a channel and illustrated for the formation of a twin phase.

Phase-field simulation of peritectic solidification closely coupled with directional solidification experiments in an Al-36 wt% Ni alloy

In this work we present experimental and theoretical investigations of the directional solidification of Al-36 wt% Ni alloy. A phase-field approach (Folch and Plapp 2005 Phys. Rev. E 72 011602) is coupled with the CALPHAD (calculation of phase diagrams) method to be able to simulate directional solidification of Al-Ni alloy including the peritectic phase Al(3)Ni. The model approach is calibrated by systematic comparison to microstructures grown under controlled conditions in directional solidification experiments. To illustrate the efficiency of the model it is employed to investigate the effect of temperature gradient on the microstructure evolution of Al-36 wt% Ni during solidification.