Diffusional Phase Transformations Research Articles

High-fidelity level-set modeling of diffusive solid-state phase transformations for polycrystalline materials

The formation of microstructures in metallic alloys during hot metal forming involves simultaneous metallurgical complex phenomena. Traditional high-fidelity numerical frameworks used on the polycrystalline scale tend to focus on single-phase microstructures or isolate phase transformations from grain boundary migration mechanisms. The level-set method is highlighted as effective in proposing a global framework for modeling multiphase polycrystalline materials and diffusive solid-state phase transformations. This framework includes novel techniques for efficient large-scale microstructural representation, strong coupling with ThermoCalc software for real-time thermodynamic data, application for ternary alloys and beyond by taking solute drag aspects, and the use of advanced nucleation models. Numerous applications are then illustrated.

Structural pathway for nucleation and growth of topologically close-packed phase from parent hexagonal crystal

The solid diffusive phase transformation involving the nucleation and growth of new phase particles is universal and frequently employed but has not yet been fully understood at the atomic level. Here, our first-principles calculations reveal a structural formation pathway of a series of topologically close-packed (TCP) phases within the hexagonally close-packed (hcp) matrix. The results show that the nucleation follows a nonclassical nucleation process, and apart from atomic diffusion, the entire hcp→TCP transformation only involves shuffle-based displacements, with a specific 3-layer unstable hcp-ordering serving as the basic structural transformation unit (BSTU). The thickening of plate-like TCP phases relies on the formation of these unstable hcp-orderings at their coherent TCP/matrix interface to nucleate a ledge, but the ledge lacks the dislocation characteristics which is commonly considered in the conventional view. Additionally, the atomic structure of the critical nucleus for the Mg2Ca and MgZn2 Laves phases was predicted in terms of Classical Nucleation Theory (CNT). The formation of polytypes and off-stoichiometry in TCP precipitates is found to be related to the nonclassical nucleation behavior. These insights contribute to a deeper understanding of solid diffusive phase transformations at the atomic level and provide a foundation for investigating other technologically significant transformations.

On simulating diffusional phase transformation in diffusion triple

Diffusion triple technique (stand-alone or composed diffusion multiples) is crucial for high-throughput construction of phase diagrams or extracting diffusion coefficients, taking advantage of the broad two-dimensional (2D) composition variations. However, currently there is a lack of methods available for simulating 2D diffusion accompanied by phase transformations inside diffusion triples. In the present work, we take a noticeable step forward by simulating diffusional phase transformation between the contacting interfaces, especially in the tri-junction of diffusion triple. The present method is an extension of the homogenization model (H. Larsson and L. Höglund, Calphad, 33 (2009) 495–501) to 2D simulations, allowing for disclosure of microstructure evolutions in diffusion triples. The practicability of the present method is demonstrated by comparing the simulation results with the experiment of a Fe-Cr-Ni diffusion triple (as a part of a multiple) from the literature.

3D Model of Carbon Diffusion during Diffusional Phase Transformations.

The microstructure plays a crucial role in determining the properties of metallic materials, in terms of both their strength and functionality in various conditions. In the context of the formation of microstructure, phase transformations that occur in materials are highly significant. These are processes during which the structure of a material undergoes changes, most commonly as a result of variations in temperature, pressure, or chemical composition. The study of phase transformations is a broad and rapidly evolving research area that encompasses both experimental investigations and modeling studies. A foundational understanding of carbon diffusion and phase transformations in materials science is essential for comprehending the behavior of materials under different conditions. This understanding forms the basis for the development and optimization of materials with desired properties. The aim of this paper is to create a three-dimensional model for carbon diffusion in the context of modeling diffusional phase transformations occurring in carbon steels. The proposed model relies on the utilization of the LBM (Lattice Boltzmann Method) and CUDA architecture. The resultant carbon diffusion model is intricately linked with a microstructure evolution model grounded in FCA (Frontal Cellular Automata). This manuscript provides a concise overview of the LBM and the FCA method. It outlines the structure of the developed three-dimensional model for carbon diffusion, details its correlation with the microstructure evolution model, and presents the developed algorithm for simulating carbon diffusion. Demonstrative examples of simulation results, illustrating the growth of the emerging phase and affected by various model parameters within particular planes of the 3D calculation domain, are also presented.

A kinetic model of diffusional phase transformations in ternary line compounds and determination of diffusion coefficients: Application to the Nb–Mo–Si system

The diffusional phase transformations in inhomogeneous multicomponent systems play an important role in affecting the material properties. Ternary line compounds, which are ternary phases with components that have both large and small solid solubility, are often observed in applications. They are different from solid solutions with variable composition ratios and stoichiometric compounds with fixed composition ratios. However, the existing models cannot quantitatively characterize the diffusional phase transformations in ternary line compounds. Therefore, by introducing the state equation related to the chemical potential and mole fraction, a kinetic model of diffusional phase transformations was established for the first time to describe the mole fraction evolution and phase growth in ternary line compounds. Furthermore, a method to determine the tracer diffusion coefficients of components related to the mole fraction and crystal phase structure in ternary line compounds was proposed, and applied to the diffusion of Nb-Mo-Si ternary system composed of the MoSi2 and Nb. During the high-temperature diffusion process of Nb-MoSi2, the silicide-poor phase (Mo, Nb)5Si3 is formed, which is a typical ternary line compound. Based on the diffusion experiments and inverse calculations, the tracer diffusion coefficients of components in ternary line compound (Mo, Nb)5Si3 were experimentally obtained. Moreover, ab-initio calculations of defect energetics clarified the microscopic diffusion mechanisms in the silicide-poor phase, and explained the reason why that the diffusion coefficient of Si decreases significantly with the increase of Nb mole fraction in ternary line compound (Mo, Nb)5Si3. The research provides valuable guidance for modeling and analyzing the diffusional phase transformation of multi-component systems.

“Anomalous” diffusion in multi-layer silicide systems

The transition metal silicide coatings have excellent oxidation resistance. In the high-temperature oxidization environment, the “anomalous” diffusion phenomenon of the reverse concentration gradient occurs in the multilayer silicide coating structure, which has a significant impact on the coating degradation. This study is to explore the physical mechanism of “anomalous” diffusion phase transformation in the MoSi2–NbSi2 bilayer silicide coating on the Nb alloy substrate. Through vacuum annealing experiments, combined with micro-Raman spectroscopy and electron probe microanalysis measurements, the diffuisonal phase transformation of multilayer silicide coating in the oxygen-free environment at high temperatures was studied. By decoupling the oxidation and diffusion, the experiments indicate that high-temperature oxidation is not the dominant driving factor for the “anomalous” diffusion of atoms. The thermodynamic analysis reveals that the reduction in the nucleation barrier of the silicide-poor layer due to multicomponent solid solution and the non-uniform distribution of component chemical potential provide the driving force for the “anomalous” diffusion growth. Based on the diffusion kinetic modeling, the simulation of diffusion-controlled phase transformation in multilayer silicide coating was carried out, and the effect of tracer diffusion coefficients on the growth of the silicide-poor phase was analyzed. The research will have guiding significance for the recognition of failure mechanisms of silicide coating systems and performance improvement.

Numerical treatment of reactive diffusion using the discontinuous Galerkin method

AbstractThis work presents a new finite element variational formulation for the numerical treatment of diffusional phase transformations using the discontinuous Galerkin method (DGM). Steep concentration and property gradients near phase boundaries require particular focus on a sound numerical treatment. There are different ways to tackle this problem ranging from (i) the well-known phase field method (PFM) (Biner et al. in Programming phase-field modeling, Springer, Berlin, 2017, Emmerich in The diffuse interface approach in materials science: thermodynamic concepts and applications of phase-field models, Springer, Berlin, 2003), where the interface is described continuously to (ii) methods that allow sharp transitions at phase boundaries, such as reactive diffusion models (Svoboda and Fischer in Comput Mater Sci 127:136–140, 2017, 78:39–46, 2013, Svoboda et al. in Comput Mater Sci 95:309–315, 2014). Phase transformation problems with continuous property changes can be implemented using the continuous Galerkin method (GM). Sharp interface models, however, lead to stability problems with the GM. A method that is able to treat the features of sharp interface models is the discontinuous Galerkin method. This method is well understood for regular diffusion problems (Cockburn in ZAMM J Appl Math Mech 83(11):731–754, 2003). As will be shown, it is also particularly well suited to model phase transformations. We discuss the thermodynamic background by review of a multi-phase, binary system. A new DGM formulation for the phase transformation problem with sharp interfaces is then introduced. Finally, the derived method is used in a 2D microstructural evolution simulation that features a binary, three-phase system that also takes the vacancy mechanism of solid body diffusion into account.

An elastoplastic phase-field study of the precipitation behaviors of Mg17Al12 phase in Mg-Al-based alloys: Part II. Precipitation under various loadings

An elastoplastic phase-field model, developed in Part I, was employed to study stress-assisted diffusional phase transformation, taking plastic deformation into consideration. Different loading conditions were applied to a single crystal of Mg-Al-based alloys during precipitation, aiming at investigating the effects of elastoplastic deformation associated with both phase transformation and external loadings. The simulation results show that even when external loadings are below the materials’ yield limits, plastic deformation may occur due to stress accumulation triggered by the volume expansion of β-Mg17Al12 precipitates and the lattice mismatch between the precipitate and the matrix. Under both axial and shear loading conditions, plasticity plays a significant role in the morphological evolution of β-Mg17Al12 precipitate with Burgers orientation. In contrast to the lath morphology observed in the absence of external loadings, a rhombic-like precipitate is formed when applying an external tensile loading of 30 MPa along [112̅0]α direction, while a mirror-symmetrical-parallelogram-like morphology emerges under a shear loading of + 30 MPa. Additionally, plasticity plays a similar dual role during precipitation under different loading conditions, which would promote the growth of the precipitate at the initial stage but tend to retard it afterward. These findings illustrate that the plastic strain generated under external loadings has a great impact on the precipitate microstructure, which provides guidelines for designing the precipitate microstructure to obtain better mechanical properties.

Short-range order and compositional phase stability in refractory high-entropy alloys via first-principles theory and atomistic modeling: NbMoTa, NbMoTaW, and VNbMoTaW

Using an all-electron, first-principles, Landau-type theory, we study the nature of short-range order and compositional phase stability in equiatomic refractory high-entropy alloys, NbMoTa, NbMoTaW, and VNbMoTaW. We also investigate selected binary subsystems to provide insight into the physical mechanisms driving order. Our approach examines the short-range order of the solid solutions directly, infers disorder/order transitions, and also extracts parameters suitable for atomistic modeling of diffusional phase transformations. We find a hierarchy of relationships between the chemical species in these materials which promote ordering tendencies. The most dominant is a relative atomic size difference between the $3d$ element, V, and the other $4d$ and $5d$ elements which drives a $B32$-like order. For systems where V is not present, ordering is dominated by the difference in filling of valence states; pairs of elements that are isoelectronic remain weakly correlated to low temperatures, while pairs with a valence difference present $B2$-like order. Our estimated order-disorder transition temperature in VNbMoTaW is sufficiently high for us to suggest that SRO in this material may be experimentally observable.

Atomistic simulations of diffusive phase transformations with non-conservative point defects

Atomistic simulations of diffusive phase transformations with non-conservative point defects

Precipitation evolution induced by electron beam irradiation in Mg-RE alloys

At present, it is of important significance for the microstructure design of rare earth magnesium (Mg-RE) alloys to comprehend the evolutionary mechanisms of metastable precipitation phases. In this study, a phase transition from β S ′ to β 1 caused by electron beam irradiation is studied particularly in a Mg-Y-Ce alloy by means of atomic scale scanning transmission electron microscopy. Results show that the structural evolution from β S ′ to β 1 precipitates can be accomplished thoroughly within two minutes for a nanosized particle. It is mainly ascribed to the fact that the increasing heat and vacancy concentration generated by electron beam irradiation can greatly accelerate the atomic diffusion, especially along c -axis direction. It is confirmed that the phase transition from β S ′ to β 1 is segmented into the two stages, including the preceding diffusive and the subsequent displacive phase transitions. The findings will provide some useful clues to control effectively the morphologies of precipitates and the precipitation behaviors of Mg-RE alloys. By means of electron beam irradiation, a phase transformation from β S ′ to β 1 in a Mg-Y-Ce alloy is studied by atomic scale scanning transmission electron microscopy. It is found that the evolution is accomplished thoroughly within two minutes. The whole process involves not only diffusive but also displacive phase transformations. • The β 1 precipitate is derived from the merging of β S ′ in Mg-Y-Ce alloy. • Electron beam irradiation can accelerate the merging process of β S ′. • The transition involves both diffusive and displacive phase transformations.

Grain refinement tailoring the kinetics of phase transformation in nanograined Fe alloy: Smaller is slower

The conventional belief for diffusive phase transformations (PTs), based on the role of grain boundaries (GBs) in promoting heterogeneous nucleation, has been that a reduction in grain size at the micrometer scale tends to accelerate the overall transformation kinetics, i.e., a classical phenomenon known as smaller is faster. Here, by investigating the ferrite (α) to austenite (γ) transformation upon heating in nanograined Fe-Ni-based alloys, interestingly, we demonstrate that refining the grain size at the nanoscale leads to slower kinetics, that is, smaller is slower. Based on experimental analysis and phase-field crystal simulations, such slower kinetics in nanograined materials is attributed to the inhibiting effect of GBs on new phase growth being stronger than their facilitating effect on nucleation. This new finding reshapes our view on the mechanism of the size dependence of PTs and provides useful guidance for designing PTs by grain refinement.

Evaluation of Phase Transformation Strain and Its Influence on Residual Stress Generation in Laser Welded Ti–6Al–4V Alloy

Abstract The assessment of welding-induced residual stress is always of significant interest due to its adverse effect on the structural stability and mechanical performance of the welded structure. Incorporation of intricate phenomena, in particular, thermal gradient, phase development, volumetric dilation, and phase transformation strain during FE modelling, not only acts as a reliable method for residual stress calculation but also serves as a directive to reduce tensile residual stress in a weldment which is sensitive to the selection of materials. In order to investigate the same for continuous and pulse laser-welded Ti–6Al–4V alloy, the sequentially coupled thermal–metallurgical–mechanical models are established. The internal state-dependent variables (SDVs) are implemented to capture the growth of phase evolved during diffusionless β → α′ transformation using Koistinen–Marburger (K–M) theory in the cooling cycle. The role played by a phase transformation induced strain on the generation of residual stress is systematically investigated. The volumetric dilation and associated phase fraction form the basis for the estimation of phase transformation strain in the present study. The accomplishment of highest martensitic fraction (∼95%) produced a phase transformation strain of 7.95 × 10−3 in pulse mode of operation. As a result, reversal of residual stress from tensile to compressive is perceived for pulse laser-welded specimen. Similarly, a sign of enriched martensitic transformation is noticed that puts the weld surface into a compression state and mitigates the overall tensile residual stress. The assumption of diffusional phase transformation during heating cycle and non-diffusional transformation during cooling phase in laser welding is more appropriate to predict the residual stress using thermal–metallurgical–mechanical model.

A Sharp-Interface Model of the Diffusive Phase Transformation in a Nickel-Based Superalloy

A sharp-interface model employing the extended finite element method is presented. It is designed to capture the prominent γ-γ′ phase transformation in nickel-based superalloys. The novel combination of crystal plasticity and sharp-interface theory outlines a good modeling alternative to approaches based on the Cahn–Hilliard equation. The transformation is driven by diffusion of solute γ′-forming elements in the γ-phase. Boundary conditions for the diffusion problem are computed by the stress-modified Gibbs–Thomson equation. The normal mass balance of solute atoms at the interface yields the normal interface velocity, which is integrated in time by a level set procedure. In order to capture the influence of dislocation glide and climb on interface motion, a crystal plasticity model is assumed to describe the constitutive behaviour of the γ-phase. Cuboidal equilibrium shapes and Ostwald ripening can be reproduced. According to the model, in low γ′ volume-fraction alloys with separated γ′-precipitates, interface movement does not have a significant effect on tensile creep behaviour at various lattice orientations.

Modeling diffusive phase transformation and fracture in viscoplastic materials

In recent years, the phase-field method has attracted a lot of attention and has emerged as one of the most efficient tools for modeling fracture and phase transformation. The phase-field method is based on the regularization of discontinuities which allows the resolution of complex problems using classical numerical methods. A multiphase-field model which is essential for describing crack propagation in non-static heterogeneous materials is still missing. In this work, we fill this gap by combining the fracture phase-field approach based on Griffith’s theory with the phase-field approach for diffusive phase transformation in viscoplastic materials. Several benchmark examples of increasing complexity demonstrate, the effect of viscoplasticity and diffusive phase transformation on fracture behavior based on the critical loading, energy density until failure and crack topology. We show that viscoplastic activity at the crack tip mitigates the effect of pre-existing defects on mechanical strength. In a heterogeneous case, it is shown that viscoplastic activity introduces an intermediate fracture response between the reference behavior, which does not take into consideration the phase transformation-induced residual stresses, and the elastic behavior, which integrates these residual stresses by taking into account only the elasticity. The interaction between the heterogeneities and the crack topology is also highlighted.

Recrystallization behavior and phase transformation in a hot-rolled pure cobalt during annealing at the elevated temperature

The evolution of microstructures and properties of the hot rolled cobalt (Co) during the isothermal annealing at different temperature was investigated. Recrystallization only occurred in hexagonal closed-packed (HCP) Co below martensitic transformation temperature (400 °C), the proportion of face-centered cubic (FCC) Co decreased dramatically and the Co sample became an HCP-Co single-phase with increasing annealing time due to the diffusive phase transformation promoted by high angle boundaries (HABs). For samples annealed at 400 °C, recrystallization occurred in both of HCP-Co and FCC-Co. The proportion of FCC-Co reached the highest value of 50% in sample annealed at 400 °C for 4 h, and the average grain size was less than 2.5 μm even after 12 h annealing due to the formation of special boundaries, annealing twins and the alternate permutation of nano FCC-Co lamellae and HCP-Co lamellae. The recrystallization mechanism was analyzed by the Johnson-Mehl-Avrami-Kolmogorov model and modified Arrhenius relationship. The nucleation and grain growth through the migration of HABs promoted the FCC→HCP diffusive phase transformation below martensitic transformation temperature. At temperature above martensitic transformation temperature, recrystallization kinetics of HCP-Co and migration of HABs was hindered by reverse martensitic transformation.

Modeling of the precipitation kinetics and morphology evolution of lamellar α in Ti-alloys during non-isothermal treatments

The non-isothermal diffusional phase transformation plays an important role in adjusting materials microstructure. In the modeling of non-isothermal transformation, actual temperature history has a remarkable effect on the precipitation kinetics of new phase. When morphology anisotropy effect is considered, taking actual temperature history effect into account is very difficult for guaranteeing the accuracy of kinetics prediction. In order to solve this problem, a new non-isothermal transformation model in combination with cellular automaton (CA) method with mixed-controlled mode was proposed. In this new model, actual temperature history effect was characterized by the effects of cooling path and additive isothermal path on the nucleation and growth of new phase. Firstly, the cooling path with the consideration of supercooling effect was introduced into the created isothermal transformation theory model. Secondly, the temperature-time path (i.e. additive isothermal path) in CA model was calibrated by using the solute concentration model from experiments. With the use of this new model, the precipitation kinetics and morphology evolution of the lamellar α for IMI834 titanium alloy during continuous cooling from single-phase region was predicted. The predicted results were in good agreement with experiments. It was also revealed that the dominant role of mixed-controlled mode for lamellar α precipitation was gradually changed from the diffusion control to the interface control with the increase of cooling rate.

Progress in research on diffusional phase transformations of Fe–C alloys under high magnetic fields

Progress in research on diffusional phase transformations of Fe–C alloys under high magnetic fields

On the variability of critical size for homogeneous nucleation in a solid-state diffusional transformation

In solid-state diffusional phase transformations involving nucleation and growth, the size of the critical nucleus for a homogeneous process (rhomo*=r∗) has been assumed to be time invariant (constant). The strain associated with the process leads to an increase in the value of r*, with respect to that for nucleation from a liquid. The strain energy stored in the matrix increases with transformation and the nuclei forming at a later stage encounter a strained matrix. Using devitrification of a bulk metallic glass as a model system, we demonstrate that r* is not a cardinal time invariant constant for homogeneous nucleation and can increase or decrease depending on the strain energy penalty. We show that the assumption regarding the constancy of r* is true only in the early stages and establish that with progress of the transformation leads to an altered magnitude of r*, which is a function of the microstructural details, geometrical variables and physical parameters. Using, HRLFI and computations, we argue that, ‘liquid-like' homogeneous nucleation can occur and that the conclusions are applicable to a broad set of solid-state diffusional transformations. The above effect ‘opens up' a lower barrier transformation pathway arising purely from the internal variables of the system.

Atomic structure and evolution of a precursor phase of Ω precipitate in an Al-Cu-Mg-Ag alloy

Precipitates in an aged Al-Cu-Mg-Ag alloy have been characterised using atomic-resolution energy-dispersive X-ray spectroscopy (EDXS) scanning transmission electron microscopy (STEM) and high-angle annular dark-field (HAADF) STEM. A metastable precursor phase, designated Ω′, which forms as thin plates on {111}α planes of the α-Al matrix phase, is found for Ω precipitate which is the key strengthening constituent in the Al-Cu-Mg-Ag alloy. The Ω′ phase has a hexagonal structure that is distinguishably different from that of the Ω precipitate, and this hexagonal structure is isomorphous to that of the single unit-cell thick T1 precipitate plates in Al-Cu-Li alloys. Our proposed hexagonal structure is also supported by density functional theory (DFT) calculations. One interesting feature of Ω′ phase is that its thickness is invariably one unit-cell height, ∼0.915 nm thick. In contrast to the commonly accepted view that shear is not involved in diffusional phase transformations, the formation of Ω′ phase involves an appreciably large shear component, which is equivalent to two 30° Shockley partial dislocations, and the magnitude of this shear component is significantly larger than the dilatational strain associated with the α-Al → Ω′ structural transformation. With continued ageing, the metastable Ω′ phase transforms in-situ into the Ω phase. Compared to the direct α-Al → Ω structural transformation, the existence of Ω′ considerably lowers the energy barrier to the nucleation of Ω via the reduction of the shear strain.