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We consider the process of precipitation in binary alloys in the presence of mechanical deformation. It is commonly observed that mechanical deformation prior to or during precipitation leads to microstructure with excess defects, which allows for enhanced precipitate nucleation and growth rates. To investigate this phenomenon, we employ a two-dimensional phase-field crystal alloy model endowed with a temperature dependent mobility, making it capable of recovering isothermal transformation (TTT) diagrams with a characteristic inflection point (nose) about a critical temperature. We examine the variation in the time-scale of precipitation and its connection to the time-scale of the applied deformation, focusing on the roles of atomic defects in the processes involved. Our results indicate that precipitation is initially delayed through application of a deformation until a critical strain is achieved, beyond which precipitation proceeds more rapidly, assisted by plastic deformation such as grain boundary buckling or dislocation nucleation. We show that the evolution of the precipitated fraction, $f(t)$, departs from classical Avrami behaviour. Specifically, $df/dt$ develops two peaks indicative of a ``plateau"-like inflection in $f(t)$, signalling the transition to defect assisted precipitate nucleation. We analyze these plateaus as a function of the deformation rate and demonstrate that the they exhibit a discontinuous bifurcation as the time-scale of applied deformation is increased. These findings are compared to and found to be consistent with experiments., Comment: Main text: 6 pages, 5 figures, Supplemental Material: 2 pages, 1 figure submitted to Physical Review Letters

In a recent class of phase field crystal (PFC) models, the density order parameter is coupled to powers of its mean field. This effectively introduces a phenomenology of higher-order direct correlation functions acting on long wavelengths, which is required for modelling solid-liquid-vapor systems. The present work generalizes these models by incorporating, into a single-field theory, higher-order direct correlations, systematically constructed in reciprocal space to operate across long {\it and} short wavelengths. The correlation kernels introduced are also readily adaptable to describe distinct crystal structures. We examine the three-phase equilibrium properties and phase diagrams of the proposed model, and reproduce parts of the aluminum phase diagram as an example of its versatile parametrization. We assess the dynamics of the model, showing that it allows robust control of the interface energy between the vapor and condensed phases (liquid and solid). We also examine the dynamics of solid-vapor interfaces over a wide range of parameters and find that dynamical artifacts reported in previous PFC models do not occur in the present formalism. Additionally, we demonstrate the capacity of the proposed formalism for computing complex microstructures and defects such as dislocations, grain boundaries, and voids in solid-liquid-vapor systems, all of which are expected to be crucial for investigating rapid solidification processes.

The development of novel materials in recent years has been accelerated greatly by the use of computational modelling techniques aimed at elucidating the complex physics controlling microstructure formation in materials, the properties of which control material function. One such technique is the phase field method, a field theoretic approach that couples various thermophysical fields to microscopic order parameter fields that track the phases of microstructure. Phase field models are framed as multiple, non-linear, partial differential equations, which are extremely challenging to compute efficiently. Recent years have seen an explosion of computational algorithms aimed at enhancing the efficiency of phase field simulations. One such technique, adaptive mesh refinement (AMR), dynamically adapts numerical meshes to be highly refined around steep spatial gradients of the PDE fields and coarser where the fields are smooth. This reduces the number of computations per time step significantly, thus reducing the total time of computation. What AMR doesn't do is allow for adaptive time stepping. This work combines AMR with a neural network algorithm that uses a U-Net with a Convolutional Long-Short Term Memory (CLSTM) base to accelerate phase field simulations. Our neural network algorithm is described in detail and tested in on simulations of directional solidification of a dilute binary alloy, a paradigm that is highly practical for its relevance to the solidification of alloys., Comment: 20 pages (2 pages of those being references), 10 figures, pre-print

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 modeling analysis consisting of bond-order potential based molecular dynamics (MD), phase field crystal based amplitude expansion (PFC-AE) simulations, and sequentially coupled phase field -- crystal plasticity (PF--CP) 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.

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.

A vector order parameter phase field model derived from a grand potential functional is presented as a new 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 new vector order parameter model can be used to describe nucleation in polycrystalline systems via thermal fluctuations of the vector order parameter, a feature which has been thus far lacking from multi-phase 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.

Mechanochemistry is becoming an established method for the sustainable, solid-phase synthesis of scores of nano-materials and molecules, ranging from active pharmaceutical ingredients to materials for cleantech. Yet we are still lacking a good model to rationalize experimental observations and develop a mechanistic understanding of the factors at play during mechanically assisted, solid-phase nanoparticle synthesis. We propose herein a structural-phase-field-crystal (XPFC) model with a ballistic driving force to describe such a process, with the specific example of the growth of gold nanoparticles in a two component mixture. The reaction path is described in the context of free energy landscape of the model, and dynamical simulations are performed based on phenomenological model parameters closely corresponding to the experimental conditions, so as to draw conclusions on nanoparticle growth dynamics. It is shown that the ballistic term lowers the activation energy barrier of reaction, enabling the reaction in a temperature regime compatible with experimental observations. The model also explains the mechanism of precipitated grain size reduction that is consistent with experimental observations. Our simulation results afford novel mechanistic insights into mechanosynthesis with implications for nanaparticle production and beyond., Comment: 47 pages, 11 figures

This work presents a consistent formulation of the structural phase-field-crystal model of substitutional binary alloys that allows for the description phases of unequal densities, a key feature in solidification. We further develop the dynamics of the model to be consistent with conserved Langevine dynamics in the true governing species densities. Additionally, this work expands on the ability to control pressure, so far only implemented in pure materials, to binary alloys by improving the control system that controls pressure from previous work. We study the equilibrium properties of the new model, and demonstrate that control of pressure can drive various kinematic microscopic processes in materials such as grain boundary pre-melting, phase instability, and grain or inter-phase boundary motion., Comment: 13 pages, 11 figures

In this work, the effect of building direction on the microstructure evolution of laser-powder bed fusion (LPBF) processed AlSi10Mg alloy was investigated. The building direction, as shown in experimentally fabricated parts, can influence the solidification behavior and promote morphological transitions in cellular dendritic microstructures. We develop a thermal model to systemically address the impact of laser processing conditions, and building direction on the thermal characteristics of the molten pool during laser processing of AlSi10Mg alloy. We then employ a multi-order parameter phase field model to study the microstructure evolution of LPBF-AlSi10Mg in the dilute limit, using the underlying thermal conditions for horizontal and vertical building directions as input. The phase field model employed here is designed to simulate solidification using heterogeneous nucleation from inoculant particles allowing to take into account morphological phenomena including the columnar-to-equiaxed transition (CET). The phase field model is first validated against the predictions of the previously developed steady-state CET theory of Hunt \cite{hunt1984steady}. It is then used under transient conditions to study microstructure evolution, revealing that the nucleation rate is noticeably higher in the horizontally built samples due to larger constitutional undercooling, which is consistent with experimental observations. We further quantify the effect of building direction on the local cooling conditions, and consequently on the grain morphology., Comment: 14 pages, 10 figures, two tables

We self-consistently derive a formalism that couples a Phase Field Crystal (PFC) density field to thermal transport. It yields a theory for non-uniform transient temperature and density evolution, and includes local latent heat release during atomic rearrangements of the PFC density field. The basic formalism is applied to the basic PFC model, demonstrating the approach's capacity to capture solidification and recalescence. With an aim towards linking physical temperature to PFC temperature, a new classical density field theory for solid/liquid/vapor systems is then derived. It presents a different approach to those traditionally used in the PFC literature while still retaining the major advantages that have become the hallmark of PFC modelling. The new model is based entirely on physical density and temperature scales. We end the paper by applying the thermal-density coupling formalism to this new multi-phase density functional theory/PFC model., Comment: 14 pages, 10 figures

Although the anisotropy of the solid-liquid interfacial free energy for most alloy systems is very small, it plays a crucial role in the growth rate, morphology and crystallographic growth direction of dendrites. Previous work posited a dendrite orientation transition via compositional additions. In this work we examine experimentally the change in dendrite growth behaviour in the Al-Sm (Samarium) system as a function of solute concentration and study its interfacial properties using molecular dynamics simulations. We observe a dendrite growth direction which changes from <100> to <110> as Sm content increases. The observed change in dendrite orientation is consistent with the simulation results for the variation of the interfacial free energy anisotropy and thus provides definitive confirmation of conjecture in previous works. In addition, our results provide physical insight into the atomic structural origin of the concentration dependent anisotropy, and deepens our fundamental understanding of solid-liquid interfaces in binary alloys., Comment: 8 pages, 5 figures

Solute trapping is an important phenomenon in rapid solidification of alloys, for which the continuous growth model (CGM) is a popular sharp interface theory. Using matched asymptotic analysis, we show how to quantitatively map the sharp interface behavior of a binary alloy phase field model onto the CGM kinetics of Aziz et al. [1], with a controllable partition coefficient k(V ). We demonstrate the parameterizations that allow the phase field model to map onto the corresponding CGM or classical sharp interface models. We also demonstrate that the mapping is convergent for different interface widths. Finally we present the effect that solute trapping can have on cellular growth in a directional solidification simulation. The treatment presented 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 rapid solidification regimes, such as those relevant to additive manufacturing.

While the microscopic structure of defected solid crystalline materials has significant impact on their physical properties, efficient and accurate determination of a given polycrystalline microstructure remains a challenge. In this paper we present a highly generalizable and reliable variational method to achieve this goal for two-dimensional crystalline and quasicrystalline materials. The method is benchmarked and optimized successfully using a variety of large-scale systems of defected solids, including periodic structures and quasicrystalline symmetries to quantify their microstructural characteristics, e.g., grain size and lattice misorientation distributions. We find that many microstructural properties show universal features independent of the underlying symmetries.

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 multi-variable theory.

We explore numerically the morphological patterns of thermo-diffusive instabilities in combustion fronts with a continuum fuel source, within a range of Lewis numbers and ignition temperatures, focusing on the cellular regime. For this purpose, we generalize the model of Brailovsky et al. to include distinct process kinetics and reactant heterogeneity. The generalized model is derived analytically and validated with other established models in the limit of infinite Lewis number for zero-order and first-order kinetics. Cellular and dendritic instabilities are found at low Lewis numbers thanks to a dynamic adaptive mesh refinement technique that reduces finite size effects, which can affect or even preclude the emergence of these patterns. This technique also allows achieving very large computational domains, enabling the study of system-size effects. Our numerical linear stability analysis is consistent with the analytical results of Brailovsky et al. The distinct types of dynamics found in the vicinity of the critical Lewis number, ranging from steady-state cells to continued tip-splitting and cell-merging, are well described within the framework of thermo-diffusive instabilities and are consistent with previous numerical studies. These types of dynamics are classified as "quasi-linear" and characterized by low amplitude cells and may follow the mode selection mechanism and growth prescribed by the linear theory. Below this range of Lewis number, highly non-linear effects become prominent and large amplitude, complex cellular and {\it{seaweed}} dendritic morphologies emerge., Comment: 14 pages, 17 figures

We extend the phase field crystal (PFC) framework to quantitative modeling of polycrystalline graphene. PFC modeling is a powerful multiscale method for finding the ground state configurations of large realistic samples that can be further used to study their mechanical, thermal or electronic properties. By fitting to quantum-mechanical density functional theory (DFT) calculations, we show that the PFC approach is able to predict realistic formation energies and defect structures of grain boundaries. We provide an in-depth comparison of the formation energies between PFC, DFT and molecular dynamics (MD) calculations. The DFT and MD calculations are initialized using atomic configurations extracted from PFC ground states. Finally, we use the PFC approach to explicitly construct large realistic polycrystalline samples and characterize their properties using MD relaxation to demonstrate their quality., Comment: 18 pages, 15 figures

This paper introduces a new structural phase field crystal (PFC) type model that expands the PFC methodology to a wider class of structurally complex crystal structures than previously possible. Specifically, our new approach allows for stabilization of graphene, as well as its coexistence with a disordered phase. It also preserves the ability to model the usual triangular and square lattices previously reported in 2D PFC studies. Our approach is guided by the formalism of the classical field theory, wherein the the free energy functional is expanded to third order in PFC density correlations. It differs from previous PFC approaches in two main features. First, it utilizes a hard-sphere repulsion to describe two-point correlations. Second, and more important, is that it uses a rotationally invariant three-point correlation function that provides a unified way to control the formation of crystalline structures that can be described by a specific bond angle, such as graphene, triangular or square symmetries. Our new approach retains much of the computational simplicity of previous PFC models and allows for efficient simulation of nucleation and growth of polycrystalline 2D materials. In preparation for future applications, this paper details the mathematical derivation of the model and its equilibrium properties, and uses dynamical simulations to demonstrate defect structures produced by the model., Comment: Updated figures, included comparison to experimental paper by Huang et al

The nonequilibrium dynamics of diffusion-mediated plasticity and creep in materials subjected to constant load at high homologous temperatures is studied atomistically using Phase Field Crystal (PFC) methods. Creep stress and grain size exponents obtained for nanopolycrystalline systems, $m \simeq 1.02$ and $p \simeq 1.98$, respectively, closely match those expected for idealized diffusional Nabarro-Herring creep. These exponents are observed in the presence of significant stress-assisted diffusive grain boundary migration, indicating that Nabarro-Herring creep and stress-assisted boundary migration contribute in the same manner to the macroscopic constitutive relation. When plastic response is dislocation-mediated, power law stress exponents inferred from dislocation climb rates are found to increase monotonically from $m \simeq 3$, as expected for generic climb-mediated natural creep, to $m \simeq 5.8$ as the dislocation density $\rho_d$ is increased beyond typical experimental values. Stress exponents $m \gtrsim 3$ directly measured from simulations that include dislocation nucleation, climb, glide, and annihilation are attributed primarily to these large $\rho_d$ effects. Extrapolation to lower $\rho_d$ suggests that $m \simeq 4-4.5$ should be obtained from our PFC description at typical experimental $\rho_d$ values, which is consistent with expectations for power law creep via mixed climb and glide. The anomalously large stress exponents observed in our atomistic simulations at large $\rho_d$ may nonetheless be relevant to systems in which comparable densities are obtained locally within heterogeneous defect domains such as dislocation cell walls or tangles., Comment: 14 pages, 10 figures

A new phase field crystal (PFC) type theory is presented, which accounts for the full spectrum of solid-liquid-vapor phase transitions within the framework of a single density order parameter. Its equilibrium properties show the most quantitative features to date in PFC modelling of pure substances, and full consistency with thermodynamics in pressure-volume-temperature space is demonstrated. A method to control either the volume or the pressure of the system is also introduced. Non-equilibrium simulations show that 2 and 3-phase growth of solid, vapor and liquid can be achieved, while our formalism also allows for a full range of pressure-induced transformations. This model opens up a new window for the study of pressure driven interactions of condensed phases with vapor, an experimentally relevant paradigm previously missing from phase field crystal theories., Comment: 5 pages, 4 figures, peer reviewed version

The long wavelength limit of a recent microscopic phase field crystal (PFC) theory of a binary alloy mix- ture is used to derive an analytical approximation for the segregation coefficient as a function of the interface velocity, and relate it to the two-point correlation function of the liquid and the thermodynamic properties of solid and liquid phases. Our results offer the first analytic derivation of solute segregation and solute drag de- rived from a microscopic model, and analytically support recent molecular dynamics and fully numerical PFC simulations. Our analytical result also provides an independent framework, motivated from classical density functional theory, from which to elucidate the fundamental nature of solute drag, which is still highly contested in the literature.

Strengthening in age-hardenable alloys is mainly achieved through nano-scale precipitates whose formation paths from the atomic-scale, solute-enriched entities are rarely analyzed and understood in a directly-verifiable way. Here, we discover a pathway for the earliest-stage precipitation in Al-Mg-Si alloys: solute clustering leading to three successive variants of FCC clusters, followed by the formation of non-FCC $GP$-$zones$. The clusters, which originally assume a spherical morphology ($C1$), evolve into elongated clusters and orient themselves on $\{111\}_{Al}$ ($C2$) and subsequently on $\{100\}_{Al}$ planes and $<$$100$$>_{Al}$ directions ($C3$). We also analyze the association of quenched-in dislocations with clustering phenomena. The results of this work can open a new frontier in advancing alloy-process-property design for commercially-important age-hardenable Al alloys.

This paper re-visits the weakly fourth order anisotropic Ginzburg-Landau (GL) theory of freezing. First we determine the anisotropy of the interfacial free energy in the Phase-Field Crystal (PFC) model analytically, and prove that it remains finite at the critical point as a direct consequence of the one-mode dominance of the model. Next, we derive the leading order PFC amplitude model and show the formal analogy to traditional weakly 4th order anisotropic GL theories. We conclude that the material-independent anisotropy appearing in emergent GL theory coincides with the remnant anisotropy of the generating PFC model. As a result, we show that the reduced temperature {\epsilon} does not enter into the interfacial free energy anisotropy for metallic materials in both the Phase-Field Crystal model and the emerging Ginzburg-Landau theories. Finally, we investigate the possible pathways of calibrating anisotropic Ginzburg-Landau theories., Comment: 13 pages, 2 figures