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Rapid solidification leads to unique microstructural features, where a less studied topic is the formation of various crystalline defects, including high dislocation densities, as well as gradients and splitting of the crystalline orientation. As these defects critically affect the material’s mechanical properties and performance features, it is important to understand the defect formation mechanisms, and how they depend on the solidification conditions and alloying. To illuminate the formation mechanisms of the rapid solidification induced crystalline defects, we conduct a multiscale modelling analysis consisting of bond-order potential-based molecular dynamics (MD), phase field crystal-based amplitude expansion simulations, and sequentially coupled phase field–crystal plasticity simulations. The resulting dislocation densities are quantified and compared to past experiments. The atomistic approaches (MD, PFC) can be used to calibrate continuum level crystal plasticity models, and the framework adds mechanistic insights arising from the multiscale analysis. This article is part of the theme issue ‘Transport phenomena in complex systems (part 2)’.

Rapid solidification experiments on thin film aluminum samples reveal the presence of lattice orientation gradients within crystallizing grains. To study this phenomenon, a single-component phase-field crystal (PFC) model that captures the properties of solid, liquid, and vapor phases is proposed to model pure aluminium quantitatively. A coarse-grained amplitude representation of this model is used to simulate solidification in samples approaching micrometer scales. The simulations reproduce the experimentally observed orientation gradients within crystallizing grains when grown at experimentally relevant rapid quenches. We propose a causal connection between formation of defects and orientation gradients.

Solute trapping is an important phenomenon in rapid solidification of alloys, for which the continuous growth model (CGM) of Aziz et al. [1] is a popular sharp interface theory. By modulating the so-called anti-trapping current and using asymptotic analysis, we show how to quantitatively map the thin interface behavior of an ideal dilute binary alloy phase field model onto the CGM kinetics. We present the parametrizations that allow our phase field model to map onto the sharp interface kinetics of the CGM, both in terms of partition coefficient k ( V ) and kinetic undercooling. We also show that the mapping is convergent for different interface widths, both in transient and steady state simulations. Finally we present the effect that solute trapping can have on cellular growth in directional solidification. The presented treatment for solute trapping can be easily implemented in different phase field models, and is expected to be an important feature in future studies of quantitative phase field modeling in quasi-rapid solidification regimes, such as those relevant to metal additive manufacturing.

A vector order parameter phase field model derived from a grand potential functional is presented as an alternative approach for modeling polycrystalline solidification of alloys. In this approach, the grand potential density is designed to contain a discrete set of finite wells, a feature that naturally allows for the growth and controlled interaction of multiple grains using a single vector field. We verify that dendritic solidification in binary alloys follows the well-established quantitative behavior in the thin interface limit. In addition, it is shown that grain boundary energy and solute back-diffusion are quantitatively consistent with earlier theoretical work, with grain boundary energy being controlled through a simple solid-solid interaction parameter. Moreover, when considering polycrystalline aggregates and their coarsening, we show that the kinetics follow the expected parabolic growth law. Finally, we demonstrate how this vector order parameter model can be used to describe nucleation in polycrystalline systems via thermal fluctuations of the vector order parameter, a process that cannot be treated consistently with multiphase or multi-order-parameter based phase field models. The presented vector order parameter model serves as a practical and efficient computational tool for simulating polycrystalline materials. We also discuss the extension of the order parameter to higher dimensions as a simple method for modeling multiple solid phases.

Rapid solidification gives rise to solute trapping, which decreases solute partitioning and alters equilibrium solidification velocity-undercooling relationships. These effects influence microsegregation, solidification morphology, and the emergent microstructure length scales. Here, we review solute trapping and solute drag in rapid solidification in terms of theory, simulation methods, and experimental techniques. The basic theory to describe solute trapping is contained in the continuous growth model. This model breaks down at high solidification velocities, where solidification transitions abruptly to complete trapping, a limit that can be captured with the local nonequilibrium model. Solute trapping theories contain unknown parameters. Their determination from atomistic simulations or pulsed laser melting experiments is discussed. Microstructural evolution in rapid solidification can be readily investigated with the phase-field method, various alternatives of which are presented here. Uncertainties related to kinetic parameters and heat transfer during rapid solidification can be studied by comparing phase-field simulations to dynamic transmission electron microscopy observations.

Selective laser melting (SLM) is a novel manufacturing technique for producing complex parts, with unique mechanical properties that result from rapidly solidified structures. For SLM of austenitic 316L stainless steel, the reported mechanical properties have a large scatter, which reflects an incomplete understanding and control of the process-structure-properties linkage. This paper demonstrates how material micromechanical behavior --- including length scale dependent yield strength and hardening --- is linked to rapid solidification microstructure, and how this structure emerges from typical SLM process conditions. This linkage is produced by concurrently coupling phase field simulations with crystal plasticity modeling. In particular, the evolution of rapidly solidified SLM microstructures is described with a quantitative phase field model with imposed solute trapping kinetics. A range of process conditions are considered by varying the thermal gradient and pulling speed driving solidification. The phase field model can predict morphological transitions (dendritic-cellular-planar), microsegregation, and emergent microstructure length scales, as function of process conditions. Both 2D and 3D phase field simulations are conducted, yielding microstructure morphologies and cell spacing Vs. cooling rate data that are in good agreement with the available experiments in the literature. Micromechanical response of domains characterized by cellular subgrain solidification structures, generated from phase simulations, are analyzed with a Cosserat crystal plasticity model which includes calibrated length-scale and hardening effects, and a realistic description of solid solution strengthening. It is shown that length scale characteristics (cell spacing and grain size) and solute segregation greatly influence the overall hardening behavior, and also affect plastic localization and evolution of geometrically necessary dislocation (GND) type hardening. We conclude with a study of the micromechanical behavior of a polycrystalline structure, focusing on the effects of cell spacing and grain size, and the relative orientation between crystalline lattices and cells. Our results suggest that the material strength is more sensitive to the cell spacing (microstructural length scale) than to the magnitude of solute segregation. The concurrently coupled phase field-crystal plasticity modeling scheme is presented as a proof-of-concept for systematically investigating and discovering new compositions, processes and microstructures for additive manufacturing of metals.

Phase field (PF) models are one of the most popular methods for simulating solidification microstructures due to their fundamental connections to the physics of phase transformations. However, these methods are numerically very stiff due to the multiple length scales in a solidifying material, from the nanoscopic solid-liquid interface, to dendritic structures on the order of hundreds of microns. While this problem can be greatly alleviated by thin-interface analytical treatments of the PF equations, additional numerical methods are required to explore experimentally relevant sample sizes and times scales. It was shown about 18 years ago that the use of dynamic adaptive mesh refinement (AMR) can alleviate this problem by exploiting the simple fact that the majority of the solidification kinetics occur at the solid-liquid interface, which scales with a lower dimensionality than the embedding system itself. AMR methods, together with asymptotic analysis, nowadays provide one of the most efficient numerical strategies for self-consistent quantitative PF modelling of solidification microstructure processes. This paper highlights the latest developments in the AMR technique for 3D modelling of solidification using classical phase field equations. This includes a move away from finite element techniques to faster finite differencing through the use of dynamic mini-meshes which are each associated with each node of a 3D Octree data structure, and distributed MPI parallelism that uses a new communication algorithm to decompose a 3D domain into multiple adaptive meshes that are spawned on separate cores. The numerical technique is discussed, followed by demonstrations of the new AMR algorithm on select benchmark solidification problems, as well as some illustrations of multi-phase modelling using a recently developed multi-order parameter phase field model.

A coupled phase field and crystal plasticity model is established to analyze formation of dislocation structures and residual stresses during rapid solidification of additively manufactured 316L stainless steel. The work focuses on investigating the role of microsegregation related to the intra-grain cellular microstructure of 316L. Effect of solidification shrinkage is considered along with dislocation mediated plastic flow of the material during solidification. Different cellular microstructures are analyzed and the characteristics of the cell core, boundary and segregation pools are discussed with respect to heterogeneity of dislocation density distributions and residual stresses. Quantitative comparison with experimental data is given to evaluate the feasibility of the modeling approach.

We describe herein a nanocellulose-alginate hydrogel suitable for 3D printing. The composition of the hydrogel was optimized based on material characterization methods and 3D printing experiments, and its behavior during the printing process was studied using computational fluid dynamics simulations. The hydrogel was biofunctionalized by the covalent coupling of an enhanced avidin protein to the cellulose nanofibrils. Ionic cross-linking of the hydrogel using calcium ions improved the performance of the material. The resulting hydrogel is suitable for 3D printing, its mechanical properties indicate good tissue compatibility, and the hydrogel absorbs water in moist conditions, suggesting potential in applications such as wound dressings. The biofunctionalization potential was shown by attaching a biotinylated fluorescent protein and a biotinylated fluorescent small molecule via avidin and monitoring the material using confocal microscopy. The 3D-printable bioactivated nanocellulose-alginate hydrogel offers a platform for the development of biomedical devices, wearable sensors, and drug-releasing materials.

A binary alloy multi-order parameter phase field model is used to study rapid solidification in Al-Cu under conditions corresponding to recent dynamic transmission electron microscopy (DTEM) experiments. The phase field model’s sharp interface limit is set through a recent matched asymptotic analysis to follow the solute trapping and interface undercooling kinetics of the Continuous Growth Model (CGM). The phase field model convergence to the CGM sharp interface model is investigated, and based on this an optimal interface width is chosen to simulate the DTEM experimental conditions. The temperature distribution used in the phase field simulations is taken from an analytic expression extracted from experiments. Simulated solidification structures are compared to experiments, including time-resolved DTEM images and post-mortem TEM-based image quality and orientation maps. We find that the large scale morphological features of the simulated microstructures are in good agreement with the experiments, and the corresponding concentration profiles that emerge are in qualitative agreement with experiments. These results show that phase field simulations, informed with DTEM experiments, provides a promising framework to investigate rapidly solidified microstructure evolution and solute segregation, and to calibrate hard-to-determine solidification parameters.