Atomistic to continuum modeling of solidification microstructures
Introduction
The quantitative prediction of solidification microstructures has been a long standing computational challenge [15], [5]. It generally requires accurate modeling of the complex dynamics of the solid–liquid interface during the entire solidification process. The past two decades have witnessed major progress in modeling this dynamics through atomistic- and continuum-scale simulations, and a handshake between those methods.
On the one hand, atomistic simulations have yielded accurate predictions of fundamental anisotropic properties of the solid–liquid interface for various metallic systems, such as the interface free-energy [65], [66], [5], [48] and kinetic coefficient [69], [66], [130], [53], [101] that affect microstructural pattern formation over a wide range of solidification conditions. Those simulations typically use a million atoms and interatomic potentials modeled by the embedded-atom method (EAM). They have also yielded useful quantitative insights into solute-trapping in rapidly solidified alloys [158], [70], as well as interface coalescence phenomena that control crystal cohesion during the late stages of solidification [67], [46], [111] and are relevant to hot tearing, reviewed in the article by Rappaz in this issue [118].
On the other hand, phase-field simulations have been extensively used to model the formation of monophase and polyphase solidification microstructures in a wide range of dendritic, eutectic and peritectic alloys [16], [5]. Progress has been made to formulate quantitative phase-field models of alloy solidification for a wider range of phase diagrams [113] and diffusive transport properties [19], [20] within the theoretical framework of the thin interface limit [79], [38], [77]. Results of atomistic modeling have been incorporated into phase-field simulations to model complex observations. For example, the quantitative characterization of interface free-energy anisotropy derived from atomistic simulations has been useful to model changes of dendrite growth orientation [63], [35]. Moreover, boosted by steady increases in computing speed and massively parallel code implementations on multicore architectures such as Graphic Processing Units (GPUs), phase-field simulations have become increasingly 3D and able to access experimentally relevant length and time scales [124], [12].
In the following section, we review recent advances in atomistic modeling of solid–liquid interface properties and coalescence, focusing on the prediction of key parameters influencing microstructure formation or hot tearing on a larger scale. In Section 3, the core section of this article, we review progress in phase-field modeling focusing on cellular and dendritic microstructures. We leave out two-phase microstructures, reviewed in the separate article by Plapp in this issue [114]. In Section 4, we discuss recent progress to bridge the gap between phase-field modeling on the microstructure scale and grain structure modeling [118]. This gap stems from the fact that, while 3D volumes up to about a mm3 are within reach of today’s large scale phase-field computations, those volumes are still minute on the scale of a casting. In contrast, Cellular Automata coupled with Finite Elements (CAFE) models [119], [51], [118] can access those much larger scales and predict grain structures of castings [26]. However, those models do not resolve dynamical interactions between branches of hierarchical dendritic networks, which can strongly influence both intra-grain microstructure selection and the growth competition of different grains [139]. Section 4 describes a Dendritic Needle Network (DNN) approach designed to quantitatively model those interactions on the grain scale, beyond the current reach of phase-field modeling. While this approach is not a substitute for grain structure models, it opens new avenues to test and improve those models by investigating complex branch interactions that shape the grain structure [138].
Section snippets
Solid–liquid interface properties
The advent of microscopic solvability theory during the 1980s [86], [81] led to the understanding that anisotropic properties of the solid–liquid interface have a profound influence on dendrite growth. Those properties generally determine the selection of both the dendrite tip operating state and growth direction. Predictions of solvability theory were validated by phase-field simulations during the 1990s [78], [79] and the first decade of this millennium witnessed major progress in
Quantitative phase-field modeling
Due to their simplicity and potential in handling complex free-boundary problems, phase-field (PF) models have become ubiquitous in materials science [16], [29]. They are well adapted to study solid–liquid interface dynamics during solidification. Quantitative predictions have long been limited by the requirement of a diffuse interface width of comparable size to the physical interface, i.e. a few atomic planes. However, the development of asymptotic analysis of PF equations, now referred to as
Towards the grain scale
In concentrated alloys, solidification usually produces grains in the form of hierarchical networks of sharp needle-like branches, like in Fig. 5a. Thus, several orders of magnitude separate the scale of the tip radius ρ and the diffusion length where D is the solute diffusivity and V the growth velocity. Such morphologies are challenging to model with PF simulations that need to resolve the solid–liquid interface of each branch. To overcome this limitation, we recently developed a
Outlook
The last two decades have witnessed major progress in atomistic simulations methods to compute key interfacial properties impacting solidification microstructures. However, quantitative predictions of those properties remain scarce for alloys. Atomistic modeling is needed to predict interfacial energy anisotropies in the Al–Zn system in order to link more directly continuum scale modeling to experimentally observed changes of dendrite growth directions as a function of Zn concentration. Current
Acknowledgements
A.K. acknowledges support of grant DEFG02-07ER46400 from the US Department of Energy, Office of Basic Energy Sciences, for support of atomistic-scale phase-field-crystal modeling of interfacial properties, and NASA for continuum-scale phase-field and dendritic-needle-network modeling of cellular/dendritic microstructures. D.T. is supported by Amy Clarke’s Early Career award from the U.S. DOE, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering and Los Alamos National
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