Introduction

The phase-field method is one of the mainstream computational methods in materials research at the mesoscale. It is the most powerful method for modeling and predicting spatial and temporal evolution of materials microstructures and their local and overall microstructure-dependent properties.1,2,3,4,5,6,7,8,9,10,11 It has been applied to a wide range of materials processes from solidification,12,13 grain growth,14,15,16,17 nucleation,18 coherent precipitate coarsening,19,20,21 martensitic transformations22,23,24 to the formation and evolution of domains in ferroelectrics25,26 and ferromagnetics.27 More recently, it has been applied to processes in composites,28,29 nuclear materials,30 corrosion,31 metal extraction,32 batteries,33 two-dimensional (2D) materials,34,35,36 etc. The phase-field method has also found applications in several other closely related fields such as fluid flow37,38 and fracture39,40,41,42 in mechanics, cell structure and dynamics in biology,43,44,45,46 etc.

Thermodynamically, the phase-field method is a density-functional-based approach at the mesoscale domain, grain, and phase microstructure level, similar to the electron density functional theory at the electron level47 and the atomic density functional theory at the atom level, on which the phase-field crystal is based.48

A microstructure in the phase-field method is described by two types of fields:49 (1) fields of basic thermodynamic variables such as the density fields of chemical species, including atoms and molecules, electronic/ionic charges, electric displacements, magnetic flux density, etc., or potential fields such as temperature, pressure, electric potential, chemical potential, etc.; and (2) fields of internal process parameters or phase fields such as degrees of chemical phase separation and phase transformations, order parameters of atomic order–disorder, ferroic/antiferroic, or electronic phase transitions, extents of chemical reactions, or orientations of grains in a polycrystal.

All of the fields in the phase-field method are continuous across the interfaces between domains or grains with the same phase or between different phases throughout a computational domain. Such a continuum description is also referred to as a diffuse-interface description.50,51,52,53

The thermodynamics of a microstructure in the phase-field method is described by its total free energy expressed as a functional of a set of independent density/potential and internal process order parameter fields and their gradients for a specific material system of interest. It could include contributions from elastic, electric, and magnetic energy contributions depending on the system of interest, in addition to the common thermal and chemical energy contributions in simple systems. An advantage of the phase-field method is that the effect of specific types of energy contributions can be considered separately to understand their individual importance to stability and evolution of a particular mesoscale structure.

The partial differential equations describing the temporal and spatial evolution of these density/potential and internal process parameter fields can be derived from irreversible thermodynamics using a combination of fundamental equations of thermodynamics for nonequilibrium systems and the kinetic relations relating the thermodynamic driving forces and the rates of changes for the density, potential, and internal order parameter fields.49 The numerical solutions to the evolution equations under a set of thermal, chemical, mechanical, electric, and magnetic boundary conditions yield the temporal and spatial evolution of the density, potential, and internal process order parameter fields, and thus of the microstructures including their local and overall physical properties. Such a continuous or diffuse-interface description incorporating contributions from the gradients of density fields to the thermodynamics automatically takes into account the interfacial chemical energy contributions and thus avoids the explicit tracking of interfacial positions during phase, grain, or domain shape evolution and coalescence or splitting. Therefore, another advantage of the phase-field method is its ability to handle arbitrarily complex morphologies and microstructures at finite temperatures.

Figure 1 illustrates the general procedure of a typical phase-field simulation. It starts with an initial microstructure, which can be generated assuming a high-temperature disordered state where all the internal process parameters are zero with small fluctuations, or from the output of a previous phase-field simulation, or a digitized microstructure from experimental measurement. The environmental conditions are described by a combination of boundary conditions and thermodynamic potentials such as temperature and chemical potential or fields such as stress, electric, and magnetic fields. It requires the input for parameters describing the fundamental equation of thermodynamics and rate or time constants for different processes. To evolve the microstructures in the phase-field method, a general strategy is to evolve only those fields describing materials processes with time constants of our interest using relaxation kinetic equations while treating those with much shorter time constants at quasistatic equilibrium (i.e., by solving the quasistatic equilibrium equations), and those fields with much longer time constants as frozen or not evolving. Microstructure evolution processes subject to frequency-dependent external stimuli can be obtained by solving the dynamic version of phase-field equations,54 which is essentially the Newton’s equation of motion with dissipation at the mesoscale. The temporal outputs from a phase-field simulation are the spatial distributions of various chemical, mechanical, electric, and magnetic variables or fields as well as local and average properties. Finally, the obtained temporal evolution of microstructures and averaged properties can be interpreted and used to assist and inspire new material design through microstructure optimization.

Figure 1
figure 1

Schematic illustration of a typical phase-field simulation procedure (X.-X. Cheng).

Full size image

The main purpose of the articles in this issue of MRS Bulletin on the phase-field method is not to provide a set of extensive reviews on the important advances in the development of the phase-field method to illustrate its capability of modeling materials microstructure evolution phenomena for different materials processes. Instead, the main objective of this issue is to provide a selected set of recent success stories in illustrating the application of the phase-field method to understanding, discovering, and designing mesoscale structures that could potentially possess novel or dramatically enhanced properties.

Selected highlights of applications of the phase-field method

We have a total of nine articles in this issue.

Ji et al.55 describe how phase-field models are able to model experimentally relevant length and time scales, including nonequilibrium effects at the rapidly evolving solid–liquid interfaces, yielding unprecedented insights into high-velocity interface dynamics.

Based on the phase-field method, Wang et al.56 illustrate a new alloy design strategy to obtain compositionally modulated materials for controlled strain release during deformation, which guided the new experimental designs of microcomposition-modulated alloys with outstanding properties, such as quasi-linear superelasticity with an exceptionally low apparent Young’s modulus, as well as Invar and Elinvar anomalies.

Steinbach et al.57 present examples for the use of phase-field simulations to gain new understandings in complex materials processes, such as solidification in additive manufacturing, carbon redistribution during bainitic transformation, and prediction of the onsets of damages during high-temperature creep of superalloys, solving some of the long-standing controversies in these areas.

Li et al.58 illustrate the use of phase-field simulations in combination with experiments to gain new understandings in the effects of microstructure heterogeneities on boundary migration during recrystallization, which could not have been obtained based on experiments alone. The phase-field simulations quantitatively correlate the presence of spatial variations in stored deformation energy, such as dislocation boundaries, with the velocity and shape of the recrystallization boundary. They revealed a clear deviation from the generally accepted relation between grain-boundary velocity and driving force for recrystallization due to the heterogeneities.

Martínez-Pañeda59 discusses the recent successes of phase-field models to simulate corrosion front propagation and related phenomena. The integration of the phase-field simulations of microstructure evolution within a multiphysics framework, considering electrochemistry, plastic deformation, and cracking, provides powerful tools to answer key questions related to biodegradation of Mg alloys, the role of secondary phases in the preferential corrosion path, intergranular corrosion, electrochemistry, localized corrosion, and corrosion cracking or mechanical failure as a consequence of corrosion.

Zhao et al.60 combine macrocasting process data, precipitation and recrystallization conditions, and phase-field modeling to optimize the casting parameters, leading to the improvement of product qualification rate from 40% to more than 80 percent. Phase-field modeling of microstructure evolution of Mg–Li-based alloys guided the design of the strengthening mechanism of the Mg–Li–Al alloy for achieving ultrahigh specific strength (470–500 kN m kg−1).

Li et al.61 describe the design of relaxor ferroelectric ceramics and single-crystals with record-high piezoelectricity guided by the fundamental understanding achieved through phase-field simulations. The new insights from phase-field simulations of alternative-current (AC) field poling of relaxor ferroelectric crystals helped discover the simultaneous light transparency and high piezoelectricity of AC-poled crystals. Based on these new relaxor ferroelectric materials, high-performance electromechanical and electro-optical devices are being developed and fabricated, including higher performance ultrasonic transducers and energy harvesters with ultrahigh output power density.

Hu62 highlights several examples of applying phase-field simulations to predict new physical phenomena and design new-concept magnetomechanical devices by computationally identifying the desirable combination of the composition, size, and geometry of monolithic materials as well as the device structures.

Andrews and Thornton63 discuss the applications of phase-field modeling to electrochemical systems, with a focus on battery electrodes using a composite battery cathode with an intercalation compound (LixFePO4) as the electrochemically active material and a displacement reaction compound (Li–Cu–TiS2) as examples. With the input parameters mostly from atomistic calculations and experimental measurements, they discussed how phase-field simulations are employed to untangle the mutual interactions among transport, reaction, electricity, chemistry, and thermodynamics leading to highly complex evolution of the materials within battery electrodes.

Comments and outlook

Because the phase-field method is based on a combination of irreversible thermodynamics and diffuse-interface description of inhomogeneous systems, it can be employed to model essentially all types of reaction, phase transformation, and thermal, chemical, and electric transport processes as either individual or coupled processes as well as the accompanying microstructure evolution. For example, it has also been extended to systems involving electronic processes in functional materials and electronic phase transitions in quantum materials such as magnetic phase transitions, metal–insulator transition,64 and superconducting transitions. It can incorporate defects such as electronic and ionic defects (e.g., oxygen vacancies)65 and their associated electronic and lattice energy states as well as extended defects such as dislocations.66,67,68

The phase-field method can be employed for processes at multiple time scales ranging from long-range diffusion-controlled processes to processes taking place at terahertz and even optical frequencies by extending the relaxational kinetics to dynamic evolution equations54 and from linear to nonlinear kinetics relating the rate of responses to thermodynamic driving forces.69

Phase-field simulations can be formulated at different spatial scales for the same microstructure evolution process, which is similar to an experimental microscopic technique that can be used to obtain microstructure images at different spatial resolutions. For example, the phase-field method formulated at the atomic scale, the phase-field crystal, which is not highlighted in this issue, can be applied to resolve atomic-level structures based on the atomic density functional theory. Even within the more traditional phase-field method without specifically resolving the atomic-level structures, one can choose to spatially resolve the physical order parameter profiles across a solid interface of nanometer thickness in a phase-field model or formulate a phase-field model with a spatial discretization grid much wider than the physical interface. For the latter, the model will have to be formulated in such a way that the smearing of the interface over a distance much larger than its physical width does not quantitatively change the interfacial kinetics.70,71,72,73,74,75 Methods with a very steep, but still continuous, transition at interfaces are being developed.76,77 However, it is extremely challenging to resolve the vastly different spatial and time scales within a single phase-field simulation.

The phase-field method can be applied to modeling and predicting the evolution of not only microstructures but also properties and thus the relation between materials processing, (micro)structure, and properties.78 Therefore, it has been considered as an important component of Integrated Computational Materials Engineering (ICME). With thermodynamic and kinetic parameters from a combination of experimental measurements, DFT calculations, and CALPHAD-type of approaches, it can be naturally applied to multicomponent and multiphase systems79,80 to make quantitative predictions of complex diffusion and phase-transformation paths and the accompanying microstructure evolution.81,82

Similar to any other computational approaches, depending on the system and processes to be studied, phase-field simulations can still be computationally expensive, even when using advanced models and numerical techniques. For these cases, neural network and other data-driven models trained using outputs from phase-field simulations could help accelerate microstructure predictions.83,84,85,86 High throughput phase-field simulations, on the other hand, can be used to generate volumes of reliable data on microstructures and properties under different processing conditions for machine learning models.

There are ongoing concentrated efforts in phase-field developments including two US DOE Centers: the PRISMS Center focused on the phase-field models and software package (PRISMS) development for magnesium alloys [http://prisms-center.org/#/home] and the Center for Computational Mesoscale Materials Science (COMMS) focused on phase-field model and software package (Q-POP) development for functional and quantum materials [https://sites.psu.edu/doecomms/]. There is also the Center for Hierarchical Materials Design for formulating standard problems testing phase-field codes [https://pages.nist.gov/pfhub/; https://chimad.northwestern.edu/news-events/articles/2016/PhaseField_BenchMark.html].

This issue is the first of its kind focused on the successes of applying the phase-field method in basic understanding of mesoscale materials processes and for guiding the design of experiments to optimize properties or discover new phenomena or functionalities. The examples collected in this issue are by no means the only existing success examples of phase-field applications to understanding, discovering, and designing materials microstructures and properties. There are many that are unable to be included in this issue due to limited space. We have no doubts there will be many more success stories of phase-field applications in understanding and designing materials yet to come. We also expect to have increasingly wider adoption of the phase-field method, as both open-source such as PRISMS [https://prisms-center.github.io/phaseField/], OpenPhase [https://openphase.rub.de/], Q-POP [https://github.com/DOE-COMMS], and MOOSE phase-field module [https://mooseframework.inl.gov/modules/phase_field/] and commercial software such as MICRESS [https://micress.rwth-aachen.de/], OPStudio [www.OpenPhase-Solutions.com], Mu-PRO [https://mupro-software.com], etc., are available or become available.