Materials Science › Materials Chemistry

Solidification and crystal growth phenomena

Works Count: 35970

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Description

This cluster of papers focuses on the application of phase-field models to study microstructure evolution, solidification, crystal growth, and elasticity in materials science. It covers topics such as diffuse interface methods for simulating complex fluids, modeling of multicomponent alloys, dendritic growth, and thermodynamically consistent numerical simulations.

Keywords

Phase-Field Models; Solidification; Microstructure Evolution; Crystal Growth; Diffuse Interface Method; Elasticity; Multicomponent Alloys; Dendritic Growth; Thermodynamic Consistency; Numerical Simulation

Overview no. 65

1988-03-01

H.V. Atkinson

Fundamentals of Solidification

2012-03-01

W. Kurz , David J. Fisher

Abstract The solidification process has a major influence on the microstructure and mechanical properties of metal casting as well as wrought products. This appendix covers the fundamentals of solidification. It … Abstract The solidification process has a major influence on the microstructure and mechanical properties of metal casting as well as wrought products. This appendix covers the fundamentals of solidification. It discusses the formation of solidification structures, the characteristics of planar, cellular, and dendritic growth, the basic freezing sequence for an alloy casting, and the variations in cooling rate, heat flow, and grain morphology in different areas of the mold. It also describes the types of segregation that occur during freezing, the effect of solidification rate on secondary dendrite arm spacing, and the factors that contribute to porosity and shrinkage.

Diffuse interface model for incompressible two-phase flows with large density ratios

2007-07-06

Hang Ding , Peter Spelt , C. Shu

Dendrite growth at the limit of stability: tip radius and spacing

1981-01-01

W. Kurz , David J. Fisher

Growth mechanism and growth habit of oxide crystals

1999-05-01

Wenjun Li , Er‐Wei Shi , Weizhuo Zhong +1 more

A coupled finite element-cellular automaton model for the prediction of dendritic grain structures in solidification processes

1994-07-01

Charles‐André Gandin , M. Rappaz

Modeling and numerical simulations of dendritic crystal growth

1993-03-01

Ryo Kobayashi

Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals

2004-11-19

K. R. Elder , Martin Grant

A continuum field theory approach is presented for modeling elastic and plastic deformation, free surfaces, and multiple crystal orientations in nonequilibrium processing phenomena. Many basic properties of the model are … A continuum field theory approach is presented for modeling elastic and plastic deformation, free surfaces, and multiple crystal orientations in nonequilibrium processing phenomena. Many basic properties of the model are calculated analytically, and numerical simulations are presented for a number of important applications including, epitaxial growth, material hardness, grain growth, reconstructive phase transitions, and crack propagation.

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Phase-Field Formulation for Quantitative Modeling of Alloy Solidification

2001-08-27

Alain Karma

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 … 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.

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Quantitative phase-field model of alloy solidification

2004-12-17

Blas Echebarria , R. Folch , Alain Karma +1 more

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 … 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.

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Fundamental aspects of VLS growth

1975-12-01

E. I. Givargizov

Solidification microstructures and solid-state parallels: Recent developments, future directions

2008-12-27

Mark Asta , C. Beckermann , Alain Karma +6 more

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Interface morphology development during stress corrosion cracking: Part I. Via surface diffusion

1972-07-01

R.J. Asaro , William A. Tiller

An introduction to phase-field modeling of microstructure evolution

2007-12-05

Nele Moelans , Bart Blanpain , Patrick Wollants

Phase-field model for binary alloys

1999-12-01

Seong Gyoon Kim , Won Tae Kim , Toshio Suzuki

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 … 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.

Calculation of Two-Phase Navier–Stokes Flows Using Phase-Field Modeling

1999-10-01

David Jacqmin

Quantitative phase-field modeling of dendritic growth in two and three dimensions

1998-04-01

Alain Karma , Wouter‐Jan Rappel

We report the results of quantitative phase-field simulations of the dendritic crystallization of a pure melt in two and three dimensions. These simulations exploit a recently developed thin-interface limit of … We report the results of quantitative phase-field simulations of the dendritic crystallization of a pure melt in two and three dimensions. These simulations exploit a recently developed thin-interface limit of the phase-field model [A. Karma and W.-J. Rappel, Phys. Rev. E 53, R3017 (1996)], which is given here a detailed exposition. This limit makes it possible to perform efficient computations with a smaller ratio of capillary length to interface thickness and with an arbitrary interface kinetic coefficient. Simulations in one and two dimensions are first carried out to test the accuracy of phase-field computations performed within this limit. Dendrite tip velocities and tip shapes are found to be in excellent quantitative agreement with exact numerical benchmarks of solvability theory obtained by a boundary integral method, both with and without interface kinetics. Simulations in three dimensions exploit, in addition to the asymptotics, a methodology to calculate grid corrections due to the surface tension and kinetic anisotropies. They are used to test basic aspects of dendritic growth theory that pertain to the selection of the operating state of the tip and to the three-dimensional morphology of needle crystals without sidebranches. For small crystalline anisotropy, simulated values of ${\ensuremath{\sigma}}^{*}$ are slightly larger than solvability theory predictions computed by the boundary integral method assuming an axisymmetric shape, and agree relatively well with experiments for succinonitrile given the uncertainty in the measured anisotropy. In contrast, for large anisotropy, simulated ${\ensuremath{\sigma}}^{*}$ values are significantly larger than the predicted values. This disagreement, however, does not signal a breakdown of solvability theory. It is consistent with the finding that the amplitude of the $\mathrm{cos}4\ensuremath{\varphi}$ mode, which measures the departure of the tip morphology from a shape of revolution, increases with anisotropy. This departure can therefore influence the tip selection in a way that is not accurately captured by the axisymmetric approximation for large anisotropy. Finally, the tail shape at a distance behind the tip that is large compared to the diffusion length is described by a linear law $r\ensuremath{\sim}z$ with a slope $dr/dz$ that is nearly equal to the ratio of the two-dimensional and three-dimensional steady-state tip velocities. Furthermore, the evolution of the cross section of a three-dimensional needle crystal with increasing distance behind the tip is nearly identical to the evolution of a two-dimensional growth shape in time, in accord with the current theory of the three-dimensional needle crystal shape.

Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics

1996-04-01

Alain Karma , Wouter‐Jan Rappel

We present mathematical results which dramatically enhance the computational efficiency of the phase-field method for modeling the solidification of a pure material. These results make it possible to resolve a … We present mathematical results which dramatically enhance the computational efficiency of the phase-field method for modeling the solidification of a pure material. These results make it possible to resolve a smaller capillary length to interface thickness ratio and thus render smaller undercooling and three-dimensional computations accessible. Furthermore, they allow one to choose computational parameters to produce a Gibbs-Thomson condition with an arbitrary kinetic coefficient. The method is tested for dendritic growth in two dimensions with zero kinetic coefficient. Simulations yield dendrites with tip velocities and tip shapes which agree within a few percent with numerical Green's function solutions of the steady-state growth problem.

A phase field concept for multiphase systems

1996-07-01

Ingo Steinbach , F. Pezzolla , Britta Nestler +4 more

Phase-field model for isothermal phase transitions in binary alloys

1992-05-01

A. A. Wheeler , W. J. Boettinger , G. B. McFadden

In this paper we present a phase-field model to describe isothermal phase transitions between ideal binary-alloy liquid and solid phases. Governing equations are developed for the temporal and spatial variation … In this paper we present a phase-field model to describe isothermal phase transitions between ideal binary-alloy liquid and solid phases. Governing equations are developed for the temporal and spatial variation of the phase field, which identifies the local state or phase, and for the composition. An asymptotic analysis as the gradient energy coefficient of the phase field becomes small shows that our model recovers classical sharp-interface models of alloy solidification when the interfacial layers are thin, and we relate the parameters appearing in the phase-field model to material and growth parameters in real systems. We identify three stages of temporal evolution for the governing equations: the first corresponds to interfacial genesis, which occurs very rapidly; the second to interfacial motion controlled by diffusion and the local energy difference across the interface; the last takes place on a long time scale in which curvature effects are important, and corresponds to Ostwald ripening. We also present results of numerical calculations.

Overview 12: Fundamentals of dendritic solidification—I. Steady-state tip growth

1981-05-01

Shenyan Huang , M. E. Glicksman

Applications of semi-implicit Fourier-spectral method to phase field equations

1998-02-01

Long‐Qing Chen , Jie Shen

Stability of a Planar Interface During Solidification of a Dilute Binary Alloy

1964-02-01

W. W. Mullins , Robert F. Sekerka

The stability of the shape of a moving planar liquid-solid interface during the unidirectional freezing of a dilute binary alloy is theoretically investigated by calculating the time dependence of the … The stability of the shape of a moving planar liquid-solid interface during the unidirectional freezing of a dilute binary alloy is theoretically investigated by calculating the time dependence of the amplitude of a sinusoidal perturbation of infinitesimal amplitude introduced into the planar shape. The calculation is accomplished by using gradients of the steady-state thermal and diffusion fields satisfying the perturbed boundary conditions (capillarity included) to determine the velocity of each element of interface, a procedure justified in some detail. Instability occurs if any Fourier component of an arbitrary perturbation grows; stability occurs if all components decay. A stability criterion expressed in terms of growth parameters and system characteristics is thereby deduced and is compared with the currently used stability criterion of constitutional supercooling; some very marked differences are discussed.

Interfacial Density Profile for Fluids in the Critical Region

1965-10-11

Frank P. Buff , Ronald Lovett , Frank H. Stillinger

Received 16 September 1965DOI:https://doi.org/10.1103/PhysRevLett.15.621©1965 American Physical Society Received 16 September 1965DOI:https://doi.org/10.1103/PhysRevLett.15.621©1965 American Physical Society

Theory of Microstructural Development During Rapid Solidification

1986-01-01

R. Trivedi , W. Kurz

An analysis of a phase field model of a free boundary

1986-09-01

Gunduz Caginalp

A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening

1979-06-01

Samuel M. Allen , John W. Cahn

Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method

1995-02-01

James A. Warren , W. J. Boettinger

Probabilistic modelling of microstructure formation in solidification processes

1993-02-01

M. Rappaz , Charles‐André Gandin

Numerical approximations of Allen-Cahn and Cahn-Hilliard equations

2010-01-01

Jie Shen , Xiaofeng Yang

Stability analyses and error estimates are carried out for a number of commonly usednumerical schemes for the Allen-Cahn and Cahn-Hilliard equations. It is shown thatall the schemes we considered are … Stability analyses and error estimates are carried out for a number of commonly usednumerical schemes for the Allen-Cahn and Cahn-Hilliard equations. It is shown thatall the schemes we considered are either unconditionally energy stable, orconditionally energy stable with reasonable stability conditions in thesemi-discretized versions. Error estimates for selected schemes with aspectral-Galerkin approximation are also derived. The stability analyses and errorestimates are based on a weak formulation thus the results can be easily extended toother spatial discretizations, such as Galerkin finite element methods, which arebased on a weak formulation.

Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance

1996-05-01

Morton E. Gurtin

Steady state columnar and equiaxed growth of dendrites and eutectic

1984-07-01

J.D. Hunt

Modeling Melt Convection in Phase-Field Simulations of Solidification

1999-09-01

C. Beckermann , H.-J. Diepers , Ingo Steinbach +2 more

The redistribution of solute atoms during the solidification of metals

1953-07-01

William A. Tiller , Kenneth A. Jackson , J. W. Rutter +1 more

Note on the conduction of heat in crystals

1938-06-01

H. B. G. Casimir

Quasi–incompressible Cahn–Hilliard fluids and topological transitions

1998-10-08

John Lowengrub , Lev Truskinovsky

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It … One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

Instabilities and pattern formation in crystal growth

1980-01-01

J. S. Langer

Several common modes of crystal growth provide particularly simple and elegant examples of spontaneous pattern formation in nature. Phenomena of interest here are those in which an advancing nonfaceted solidification … Several common modes of crystal growth provide particularly simple and elegant examples of spontaneous pattern formation in nature. Phenomena of interest here are those in which an advancing nonfaceted solidification front suffers an instability and subsequently reorganizes itself into a more complex mode of behavior. The purpose of this essay is to examine several such situations and, in doing this, to identify a few new theoretical ideas and a larger number of outstanding problems. The systems studied are those in which solidification is controlled entirely by a single diffusion process, either the flow of latent heat away from a moving interface or the analogous redistribution of chemical constituents. Convective effects are ignored, as are most effects of crystalline anisotropy. The linear theory of the Mullins-Sekerka instability is reviewed for simple planar and spherical cases and also for a special model of directional solidification. These techniques are then extended to the case of a freely growing dendrite, and it is shown how this analysis leads to an understanding of sidebranching and tip-splitting instabilities. A marginal-stability hypothesis is introduced; and it is argued that this intrinsically nonlinear theory, if valid, permits aone to use results of linear-stability analysis to predict dendritic growth rates. The review concludes with a discussion of nonlinear effects in directional solidication. The nonplanar, cellular interfaces which emerge in this situation have much in common with convection patterns in hydrodynamics. The cellular stability problem is discussed briefly, and some preliminary attempts to do calculations in the strongly nonlinear regime are summarized.

Phase-Field Models for Microstructure Evolution

2002-07-28

Long‐Qing Chen

▪ Abstract The phase-field method has recently emerged as a powerful computational approach to modeling and predicting mesoscale morphological and microstructure evolution in materials. It describes a microstructure using a … ▪ Abstract The phase-field method has recently emerged as a powerful computational approach to modeling and predicting mesoscale morphological and microstructure evolution in materials. It describes a microstructure using a set of conserved and nonconserved field variables that are continuous across the interfacial regions. The temporal and spatial evolution of the field variables is governed by the Cahn-Hilliard nonlinear diffusion equation and the Allen-Cahn relaxation equation. With the fundamental thermodynamic and kinetic information as the input, the phase-field method is able to predict the evolution of arbitrary morphologies and complex microstructures without explicitly tracking the positions of interfaces. This paper briefly reviews the recent advances in developing phase-field models for various materials processes including solidification, solid-state structural phase transformations, grain growth and coarsening, domain evolution in thin films, pattern formation on surfaces, dislocation microstructures, crack propagation, and electromigration.

Phase-field models in materials science

2009-07-30

Ingo Steinbach

The phase-field method is reviewed against its historical and theoretical background. Starting from Van der Waals considerations on the structure of interfaces in materials the concept of the phase-field method … The phase-field method is reviewed against its historical and theoretical background. Starting from Van der Waals considerations on the structure of interfaces in materials the concept of the phase-field method is developed along historical lines. Basic relations are summarized in a comprehensive way. Special emphasis is given to the multi-phase-field method with extension to elastic interactions and fluid flow which allows one to treat multi-grain multi-phase structures in multicomponent materials. Examples are collected demonstrating the applicability of the different variants of the phase-field method in different fields of materials science.

Computation of multiphase systems with phase field models

2003-07-31

Vittorio Badalassi , Héctor D. Ceniceros , S. Banerjee

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Phase-Field Simulation of Solidification

2002-07-28

W. J. Boettinger , James A. Warren , C. Beckermann +1 more

▪ Abstract An overview of the phase-field method for modeling solidification is presented, together with several example results. Using a phase-field variable and a corresponding governing equation to describe the … ▪ Abstract An overview of the phase-field method for modeling solidification is presented, together with several example results. Using a phase-field variable and a corresponding governing equation to describe the state (solid or liquid) in a material as a function of position and time, the diffusion equations for heat and solute can be solved without tracking the liquid-solid interface. The interfacial regions between liquid and solid involve smooth but highly localized variations of the phase-field variable. The method has been applied to a wide variety of problems including dendritic growth in pure materials; dendritic, eutectic, and peritectic growth in alloys; and solute trapping during rapid solidification.

Model for solute redistribution during rapid solidification

1982-02-01

Michael J. Aziz

A microscopic model for impurity uptake at a sharp crystal-liquid interface during alloy solidification is presented in terms of the bulk properties of the liquid and solid phases. The results … A microscopic model for impurity uptake at a sharp crystal-liquid interface during alloy solidification is presented in terms of the bulk properties of the liquid and solid phases. The results for stepwise growth and continuous growth at the same interface velocity differ quantitatively but exhibit the same qualitative features. A transition from equilibrium segregation to complete solute trapping occurs as the velocity surpasses the diffusive speed of solute in the liquid. The location of the transition varies little with equilibrium segregation coefficient, and a kinetic limit to solute trapping is found to be quite unlikely. Comparison is made with other models; critical differences are pointed out. Coupled with a growth velocity equation and with macroscopic heat- and solute-diffusion equations, the model forms a complete description of one-dimensional crystal growth. The steady-state solution to this system is indicated for the case of a planar interface. The results are applied to describe regrowth from laser-induced melting. Preliminary comparison with experiment is made. The steady-state solution for thermal and impurity transport is suggested for use whenever detailed computer calculations are unavailable or are unnecessarily involved.

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A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method

2003-05-01

Chun Liu , Jie Shen

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PHYSICAL REVIEW LETTERS

1988-01-01

Richard C. Brewer , David A. Kessler , Joel Koplik +1 more

Crystal Defects and Crystalline Interfaces

1970-01-01

W. Bollmann

Crystal Lattice Defects

1962-07-01

Theory of microstructural development during rapid solidification

1986-05-01

W. Kurz , Bruno Giovanola , R. Trivedi

Thermohaline Convection with Two Stable Regimes of Flow

1961-05-01

Henry Stommel

Accelerating phase-field simulation of coupled microstructural evolution using autoencoder-based recurrent neural networks

2025-06-25

Aidan Gesch , Chongze Hu | Journal of Materials Informatics

Accelerated phase-field frameworks leveraging time-dependent neural networks have recently been developed to accelerate microstructure-based phase-field simulations in both temporal and spatial domains. However, most of these frameworks have been designed … Accelerated phase-field frameworks leveraging time-dependent neural networks have recently been developed to accelerate microstructure-based phase-field simulations in both temporal and spatial domains. However, most of these frameworks have been designed for phase-field problems involving a single variable field, such as spinodal decomposition. In this study, we developed an accelerated framework for predicting the microstructural evolution of Ostwald ripening, a classical phase-field problem involving multiple interdependent parameter fields. This framework integrates various components: high-throughput phase-field simulations for generating high-quality microstructure database, autoencoder-based dimensionality reduction to transform 2D microstructure images into latent representations, and long short-term memory (LSTM) networks serving as the microstructure learning engine. Our results demonstrate that autoencoder techniques can effectively reduce the large dimension of microstructure images into 16 key values, while maintaining high accuracy in reconstructing these reduced representations back to their original space. Using these latent representations, LSTM models are employed to capture the key microstructural features of Ostwald ripening and predict their evolution over future time sequences, with a speedup of approximately 3.35 × 10<sup>5</sup> times compared to the high-fidelity phase-field simulations. The accelerated framework presented in this work is the first data-driven emulation specifically designed for coupled phase-field problems, and it can be easily extended to predict other evolutionary phenomena with more complex microstructural features.

Weak solutions to the generalized nonhomogeneous incompressible Kelvin–Voigt–Cahn–Hilliard system

2025-06-25

A. Ndongmo Ngana , T. Tachim Medjo , P. M. Tchepmo Djomegni | Journal of Evolution Equations

Simulation of dendritic growth on a spherical surface using a multi-component phase-field model

2025-06-24

Sangkwon Kim , Soobin Kwak , Seokjun Ham +2 more | International Communications in Heat and Mass Transfer

A novel linear relaxation explicit-invariant energy quadratization numerical scheme for the Allen-Cahn and Cahn-Hilliard type models

2025-06-24

Jincheng Pu , Jun Zhang | Journal of Applied Mathematics and Computing

Optimal error estimates of the local discontinuous Galerkin method and high-order energy stable RK-SAV scheme for the phase field crystal equation

2025-06-20

Ruihan Guo , Yuqing Yan , Xiaobin Liu | Journal of Computational and Applied Mathematics

Investigation of thermal and solidification behaviors to predict dendrite microstructure formation during directed energy deposition of AlSi10Mg

2025-06-20

Tingyu Chang , Enjie Dong , Linjie Zhao +2 more | CIRP journal of manufacturing science and technology

A unified framework for N-phase Navier–Stokes Cahn–Hilliard Allen–Cahn mixture models with non-matching densities

2025-06-19

M.F.P. ten Eikelder | Journal of Fluid Mechanics

Over the past few decades, numerous N-phase incompressible diffuse-interface flow models with non-matching densities have been proposed. Despite aiming to describe the same physics, these models are generally distinct, and … Over the past few decades, numerous N-phase incompressible diffuse-interface flow models with non-matching densities have been proposed. Despite aiming to describe the same physics, these models are generally distinct, and an overarching modelling framework is absent. This paper provides a unified framework for N-phase incompressible Navier–Stokes Cahn–Hilliard Allen–Cahn mixture models with a single momentum equation. The framework emerges naturally from continuum mixture theory, exhibits an energy-dissipative structure, and is invariant to the choice of fundamental variables. This opens the door to exploring connections between existing N-phase models and facilitates the computation of N-phase flow models rooted in continuum mixture theory.

Grain growth in thin films – When the third dimension matters

2025-06-19

Dana Zöllner , Wolfgang Pantleon | Computational Materials Science

Symmetry breaking bifurcation and the stability of stationary solutions of a phase-field model

2025-06-18

Yasuhito Miyamoto , Tatsuki Mori , Sohei Tasaki +2 more | Journal of Differential Equations

Energy-decreasing implicit-explicit Runge-Kutta methods for the modified phase field crystal equation

2025-06-18

Xin Qi , Xian‐Ming Gu | Journal of Applied Mathematics and Computing

Advances in Solidification Processing in Steady Magnetic Field

2025-06-18

Shengya He , Chenglin Huang , Chuanjun Li | Materials

As a contactless physical field, a steady magnetic field (SMF) is capable of acting on substances, which leads to changes in physical and/or chemical properties and to further influencing thermodynamic … As a contactless physical field, a steady magnetic field (SMF) is capable of acting on substances, which leads to changes in physical and/or chemical properties and to further influencing thermodynamic and kinetic behaviors at macroscopic, mesoscopic, and microscopic scales. The application of the SMF to material science has evolved into an important interdisciplinary field-the Electromagnetic Processing of Materials (EPM). Therein, the implementation of the SMF for the solidification of metals and alloys has been increasingly given attention. The SMF was found to regulate nucleation, crystal growth, the distribution of solutes and structure morphology during alloy solidification via various magnetic effects, such as magnetic damping, the thermoelectric magnetic effect, magnetic orientation and magnetically controlled diffusion. In this review, we briefly summarize the main SMF effects and review recent progress in magnetic field-assisted solidification processing, including nucleation, dendritic growth, solute segregation and interfacial phenomena. Finally, future perspectives regarding controlling alloys' solidification using an SMF are discussed.

A new decoupled unconditionally stable scheme and its optimal error analysis for the Cahn-Hilliard-Navier-Stokes equations

2025-06-18

Haijun Gao , Xi Li , Minfu Feng | Computers & Mathematics with Applications

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An Efficient Computational Algorithm for the Nonlocal Cahn–Hilliard Equation with a Space-Dependent Parameter

2025-06-15

Zhengang Li , Xinpei Wu , Junseok Kim | Algorithms

In this article, we present a nonlocal Cahn–Hilliard (nCH) equation incorporating a space-dependent parameter to model microphase separation phenomena in diblock copolymers. The proposed model introduces a modified formulation that … In this article, we present a nonlocal Cahn–Hilliard (nCH) equation incorporating a space-dependent parameter to model microphase separation phenomena in diblock copolymers. The proposed model introduces a modified formulation that accounts for spatially varying average volume fractions and thus captures nonlocal interactions between distinct subdomains. Such spatial heterogeneity plays a critical role in determining the morphology of the resulting phase-separated structures. To efficiently solve the resulting partial differential equation, a Fourier spectral method is used in conjunction with a linearly stabilized splitting scheme. This numerical approach not only guarantees stability and efficiency but also enables accurate resolution of spatially complex patterns without excessive computational overhead. The spectral representation effectively handles the nonlocal terms, while the stabilization scheme allows for large time steps. Therefore, this method is suitable for long-time simulations of pattern formation processes. Numerical experiments conducted under various initial conditions demonstrate the ability of the proposed method to resolve intricate phase separation behaviors, including coarsening dynamics and interface evolution. The results show that the space-dependent parameters significantly influence the orientation, size, and regularity of the emergent patterns. This suggests that spatial control of average composition could be used to engineer desirable microstructures in polymeric materials. This study provides a robust computational framework for investigating nonlocal pattern formation in heterogeneous systems, enables simulations in complex spatial domains, and contributes to the theoretical understanding of morphology control in polymer science.

An implicit finite element phase-field technique with temperature-dependent properties for liquid-solid interface evolution

2025-06-12

P. Areias , Afonso Gusmão , Frederico Castelo Ferreira +1 more | Proceedings of the Institution of Mechanical Engineers Part L Journal of Materials Design and Applications

We present a fully implicit, finite-element phase-field framework for simulating ice–water solidification in both two and three dimensions. Our model couples a modified heat conduction equation with a regularized Ginzburg–Landau … We present a fully implicit, finite-element phase-field framework for simulating ice–water solidification in both two and three dimensions. Our model couples a modified heat conduction equation with a regularized Ginzburg–Landau phase-field evolution law, explicitly accounting for anisotropic diffusion and temperature-dependent constitutive properties (density, specific heat and thermal conductivity). After verifying accuracy on a classical 2D undercooling benchmark, we carry out systematic parametric studies to assess the effects of time-step size and mesh resolution on interface dynamics. Three demonstrative cases—two in 2D and one in 3D—show that dendritic morphologies emerge naturally on uniform meshes, without any special adaptive remeshing. We further quantify how temperature-dependent parameters accelerate mid-stage growth kinetics. The proposed approach combines robustness, versatility and computational efficiency, and is released alongside documented Mathematica/AceGen source code to support reproducibility and further extensions.

Why grain growth is not curvature flow

2025-06-12

Caihao Qiu , David J. Srolovitz , Gregory S. Rohrer +2 more | Proceedings of the National Academy of Sciences

Grain growth in polycrystals is traditionally considered a capillarity-driven process, where grain boundaries (GBs) migrate toward their centers of curvature (i.e., mean curvature flow) with a velocity proportional to the … Grain growth in polycrystals is traditionally considered a capillarity-driven process, where grain boundaries (GBs) migrate toward their centers of curvature (i.e., mean curvature flow) with a velocity proportional to the local curvature (including extensions to account for anisotropic GB energy and mobility). Experimental and simulation evidence shows that this simplistic view is untrue. We demonstrate that the failure of the classical mean curvature flow description of grain growth mainly originates from the shear deformation naturally coupled with GB motion (i.e., shear coupling). Our findings are built on large-scale microstructure evolution simulations incorporating the fundamental (crystallography-respecting) microscopic mechanism of GB migration. The nature of the deviations from curvature flow revealed in our simulations is consistent with observations in recent experimental studies on different materials. This work also demonstrates how to incorporate the mechanical effects that are essential to the accurate prediction of microstructure evolution.

Smoothed boundary method modified phase-field modeling of growth dynamics of thin films on the surface of the planar substrate and the etch pit during physical vapor deposition

2025-06-12

Xinlei Du , Hui Xing , Hanxu Jing +2 more | Computational Materials Science

Modelling Microsegregation of Binary Alloy During Solidification

2025-06-04

Tongzhao Gong , Shuting Cao , Weiye Hao +4 more | Acta Metallurgica Sinica (English Letters)

Modulated (spinodal) alloys

2025-06-03

| Periodicals of Engineering and Natural Sciences (PEN)

In this work, after defining spinodal reactions experimental investigation on spinodal decomposition are overviewed for the last five decades. Also, future developments in spinodal decomposition for modulated alloys are forecasted … In this work, after defining spinodal reactions experimental investigation on spinodal decomposition are overviewed for the last five decades. Also, future developments in spinodal decomposition for modulated alloys are forecasted and criticized in an outlook.

A decoupled, unconditionally stable numerical algorithm for phase transition in aluminum under electromagnetic levitation

2025-06-01

Qixing Chen , Li Cai , Feifei Jing +2 more | Physics of Fluids

In this paper, we provide an efficient decoupled numerical algorithm for solving the phase transition in aluminum (Al) under electromagnetic levitation (EML) conditions. This work includes threefolds: (i) Models. The … In this paper, we provide an efficient decoupled numerical algorithm for solving the phase transition in aluminum (Al) under electromagnetic levitation (EML) conditions. This work includes threefolds: (i) Models. The Maxwell equations, heat transfer equations, and the incompressible Navier–Stokes (NS) equations are involved in a whole electromagnetic–heat-flow system. (ii) Algorithm (main contribution). We solve the electromagnetic equation based on the magnetic potential vector A; the time-filter (TF) technique is employed to the heat equations to improve the temporal step. For the fluid part, a consistent splitting scheme to decouple the velocity and pressure with the TF technique is utilized. The algorithm exhibits unconditional stability. The finite element method is dragged into spatial discretization. (iii) Numerical experiments. We first present a numerical test to check the second-order temporal accuracy of the incompressible NS equations. We then compare the results with those by the proposed algorithm and the commercial software COMSOL, which confirm the accuracy of our algorithm. Finally, the phase transition processes of aluminum under EML are simulated, and also we compare them with the data from the literature [Cai et al., Int. J. Heat Mass Transfer 151, 119386 (2020)] to further demonstrate the robustness and efficiency of the proposed algorithm.

A uniquely solvable and positivity-preserving finite difference scheme for the Flory–Huggins-Cahn-Hilliard equation with dynamical boundary condition

2025-06-01

Yunzhuo Guo , Cheng Wang , Steven M. Wise +1 more | Journal of Computational and Applied Mathematics

Thermodynamically consistent phase-field modeling for polycrystalline multi-phase continua

2025-06-01

Hendrik Westermann , Rolf Mahnken | International Journal of Solids and Structures

Densification behavior of EBCO superconducting target by phase-field simulation

2025-06-01

W. Zhang , Benshuang Sun , Xiaokai Liu +7 more | Ceramics International

An unconditionally stable hybrid discontinuous Galerkin scheme for a Cahn–Hilliard phase-field model of two-phase incompressible flow with variable densities

2025-06-01

Changlun Ye , Hai Bi Hai Bi , Xianbing Luo | Mathematics and Computers in Simulation

The driving force for twin boundary migration in phase field model coupled to crystal plasticity finite element

2025-06-01

Linfeng Jiang , Guisen Liu , Peipeng Jin +2 more | International Journal of Plasticity

Impacts of linear coupling on exact solutions and spectral classification in the 1D Allen–Cahn system

2025-06-01

Mostafa Fazly | Communications in Nonlinear Science and Numerical Simulation

Proactive Lithium Dendrite Regulation Enabled by Manipulating Separator Microstructure Using High‐Fidelity Phase‐Field Simulation (Adv. Energy Mater. 24/2025)

2025-06-01

Yajie Li , Yiping Wang , Bin Chen +4 more | Advanced Energy Materials

Interaction of gas bubbles and dendrite interfaces during directional solidification: Modeling and experiment

2025-06-01

Mengdan Hu , Dongke Sun , Qingyu Zhang +1 more | Materials & Design

Three-dimensional neutron diffraction study of dendrite growth behavior in Ni-based single-crystal turbine blades

2025-05-31

Peng Sheng , Jiachen Long , Nan Li +4 more | Scripta Materialia

Enhanced Purification Efficiency of High-Purity Tellurium via Sustainable Rotary Crystallization: A Focus on Impurity Migration Behavior

2025-05-30

Qijin Luo , Jingbo Zhai , Guolong Chen +6 more | JOM

Cahn–Hilliard equations with singular potential, reaction term and pure phase initial datum

2025-05-27

Maurizio Grasselli , Luca Scarpa , Andrea Signori

We consider local and nonlocal Cahn–Hilliard equations with constant mobility and singular potentials including, e.g., the Flory–Huggins potential, subject to no-flux (or periodic) boundary conditions. The main goal is to … We consider local and nonlocal Cahn–Hilliard equations with constant mobility and singular potentials including, e.g., the Flory–Huggins potential, subject to no-flux (or periodic) boundary conditions. The main goal is to show that the presence of a suitable class of reaction terms allows to establish the existence of a weak solution to the corresponding initial and boundary value problem even though the initial condition is a pure state. This fact was already observed by the authors in a previous contribution devoted to a specific biological model. In this context, we examine the essential assumptions required for the reaction term to ensure the existence of a weak solution. Also, we explore the scenario involving the nonlocal Cahn–Hilliard equation and provide some illustrative examples that contextualize within our abstract framework.

A Finite Element Method for Anisotropic Crystal Growth on Surfaces

2025-05-26

Harald Garcke null , Robert Nürnberg

Degenerate Area Preserving Surface Allen-Cahn Equation and Its Sharp Interface Limit

2025-05-26

Michal Beneš , Miroslav Kolář , Jan Magnus Sischka null +1 more