Multiphase-field Model Research Articles

Multi-Phase-Field Model of Intergranular Corrosion Kinetics in Sensitized Metallic Materials

Improperly heat-treated metals exhibit preferential corrosion along sensitized grain boundaries when exposed to a corrosive electrolyte. This localized corrosion process is commonly known as Intergranular Corrosion (IGC). A multi-phase-field (MPF) model is presented to quantitatively predict IGC kinetics in metallic materials. The total free energy of the system is defined in terms of chemical, gradient and electromigration energy. The system is defined by a set of phase field variables which evolve due to the minimization of Gibbs free energy of the system. The simulation results show that IGC predicted by two-dimensional MPF model agrees well with the experimental results. The model also predicts plane-direction-dependent IGC in rolled sheets, commonly observed in the experimental studies. It is also observed that the corrosion process becomes transport controlled even at lower values of applied potentials due to the saturation of the metal ions in the corroded grain boundaries region. A three-dimensional study is also presented to show the practical applications of using this MPF model for complex three-dimensional geometries.

Phase field modeling of deformation twinning in β-metastable titanium alloys

A multi-phase field model describing the anisotropic growth of the {332}<113> twinning system in β-metastable titanium alloys is hereby presented considering the geometrically linear and non-linear frameworks. Phase fields are associated with each twin variant of the {332}<113> system. The lattice transformation describing the twinning systems is incorporated in the model via relevant eigenstrain tensors. We illustrate the capability of our model by investigating the formation of twins networks, within both geometrically linear and non linear frameworks. In particular, it is shown that the predicted microstructures exhibit strong differences, resulting from a difference in the selection of the variant habit planes.

Phase-field modelling of dynamic recrystallization process during friction stir welding of aluminium alloys

ABSTRACTNumerical investigation on dynamic recrystallization in friction stir welds is of great significance to control the microstructure evolution and grains size in joints. The recrystallization and grain growth in the nugget centre of friction stir welded 6061 aluminium alloy are numerically simulated by combining the multiphase-field model with Kocks–Mecking dislocation model and employing the calculated temperature and strain rate variations with time in FSW process. The reliability of the model is verified by electron backscattered diffraction measurement results of grains size distribution at the same position. The specific reasons for different grain sizes under different levels of welding speed are quantitatively analysed. The ratios of recrystallization duration to deformation period and incubation period to deformation time are determined.

Integrated crystal plasticity and phase field model for prediction of recrystallization texture and anisotropic mechanical properties of cold-rolled ultra-low carbon steels

A multi-scale approach integrating a polycrystal plasticity finite element (CPFEM) and a multi-phase field model (PFM) is developed in this study and employed to the prediction of anisotropic mechanical properties of annealed ultra-low carbon steel. The CPFEM simulates inhomogeneous local deformation and orientation distribution of ultra-low carbon steel, while the PFM predicts microstructure evolution originating from nucleation and growth. The present study highlights a systematic modeling process including reduction of the stored energy by the recovery process, nucleation of grains with preferred orientation based on generalized strain energy release maximization theory (GSERM), and grain growth with consideration of the stored energy difference and misorientation between neighboring grains, which are provided from local effects in the CPFEM. The two computational approaches are numerically mapped by the Wigner-Seitz cell method to efficiently transfer state variables between the domains of finite element and finite difference schemes for CPFEM and PFM, respectively. The proposed multiscale model is quantitatively validated through experimentally measured microstructural characteristics of annealed ultra-low carbon steel after the rolling deformation, i.e., its recrystallization texture in the aspect of pole figures and ODF, grain size, and distributions. Finally, the predicted microstructural information after the thermo-mechanical process is used as input for virtual mechanical tests simulating orientation-dependent tensile properties. The comparison of predicted orientation-dependent yield stresses and Lankford coefficients as measures for anisotropic mechanical properties with those of experiments confirm the validity and accuracy of the proposed CPFEM-PFM approach as an efficient tool for understanding the relationship between the thermo-mechanical process, microstructure, and mechanical properties in the material design stage.

Phase-Field Simulation of Grain Boundary Evolution In Microstructures Containing Second-Phase Particles with Heterogeneous Thermal Properties

Understanding the interaction between complex thermal fields and metallic structures at the meso-scale is crucial for the prediction of microstructural evolution during thermomechanical processing. The competitive growth of crystal grains, driven by thermodynamic forces at the grain boundaries, is one of the most fundamental phenomena in metallurgy and solid state physics. The presence of second phase particles, which act as pinning sites for boundaries, drastically alters the coarsening behaviour of the system; particularly when considering that these particles have different thermal properties to the primary phase. In this work a multi-phase field model, incorporating thermal gradient and curvature driving forces, is used to predict grain growth in a Ti6Al4V alloy system with second phase particle inclusions representative of oxide and carbide precipitates. The multi-phase field framework is fully coupled to the heat equation. The incorporation of the thermal gradient driving force enables the detailed behaviour of the grain boundaries around the particles to be predicted. It is shown that the inclusion of particles with a lower thermal conductivity has a significant influence on the coarsening behaviour of various systems of grains, due to the combined effects of thermal shielding and the generation of thermal gradient driving forces between the boundaries and pinning particles.

In Situ Observation and Phase-Field Modeling of Peritectic Solidification of Low-Carbon Steel

In the present study, both in situ experiment and multiphase field modeling are adopted to investigate the peritectic solidification of a low-carbon steel. The results show that the peritectic reaction occurs at the temperature of 3.4 K lower than the equilibrium peritectic temperature, the γ-austenite first nucleates at the δ/L boundary, and then rapidly propagates along the δ/L boundary by the advance of the L/γ/δ triple point till the δ-ferrite is encircled by the γ austenite. The whole peritectic reaction process is very fast, and the measured and predicted average propagation velocity of L/γ/δ triple point along the δ/L boundary are, respectively, 1.36 and 1.09 mm/s. This small difference between the measurement and prediction means that the developed multiphase field model is capable of predicting the peritectic reaction and peritectic transformation during the peritectic solidification process of Fe-C alloy, and the peritectic reaction can be regarded as a solute diffusion-controlled process. With the increase of cooling rate and undercooling, the advancing velocities of L/γ/δ triple point, L/γ interface and γ/δ interface increase, and thus the γ-austenite between the liquid phase and δ phase becomes longer and thicker for the same elapsed time.

Numerical simulation of microstructure evolution in Ni-based superalloys during P-type rafting using multiphase-field model and crystal plasticity

During the service process, Ni-based superalloys undergo microstructure evolutions such as rafting, which greatly affect its service performance. In the present study, a multiphase-field model coupled with crystal plasticity was established to simulate the γ/γ' microstructure evolution of Ni-based superalloys during P-type rafting. The anisotropic plasticity considering 12 octahedral slip systems was defined in γ matrix channel. The PanNickel© database was used to calculate thermodynamic and kinetic data for phase transition and diffusion in multicomponent superalloys. First, the model was validated by two benchmarks with and without consideration of external load respectively. Then, the anisotropic of slip systems was analyzed. It was found that eight slip systems A2, A3, B2, B4, C1, C3, D1, D4 were activated in gable channel first when the structure was compressed in [0 0 1] direction. Besides, the elements distribution was studied. It can be seen that Al, Ti, Ta diffused into precipitates while Co, Cr and Mo diffused away the roof channel when the structure was compressed in [0 0 1] direction. Re and W mainly distributed in the γ/γ' interface because their slow diffusion rate. Finally, the P-type raft of CMSX-4 superalloy was simulated, and the plasticity distribution and plastic behavior of the 12 slip systems were discussed. The current model can be used to simulate phase transformation of multicomponent superalloys where plasticity is strongly coupled.

Multiphase-field approach with parabolic approximation scheme

A parabolic approximation scheme is implemented to calculate the phase composition and driving force for the numerical application of multiphase-field model. Using an algorithm of solving linear equations instead of iteration, this scheme is more efficient than the traditional Newton-Raphson method in binary or multi-binary system. In order to verify the rationality of this parabolic approximation scheme, we simulate the peritectic solidification of Fe-C binary alloy as an example of numerical application. The study not only compares the driving forces derived from different schemes, but also reveals the interaction of ferrite, austenite and liquid phase during the isothermal solidification, which proves the practicability of the parabolic approximation scheme.

Growth competition between columnar dendrite and degenerate seaweed during directional solidification of alloys: Insights from multi-phase field simulations

In this paper, the growth competition between columnar dendrite and degenerate seaweed during directional solidification of a dilute alloy was investigated by using the quantitative multi-phase field model of dilute binary alloys developed by Ofori-Opoku et al. [Acta Materialia, 2010, 58: 2155–2164]. Our recent experiments of degenerate seaweed in directional solidification of an Al-4.5 wt% Cu alloy by employing (100)[011] orientated planar-front seeds motivate this work. We found that the way of symmetrical columnar dendrites impinging is the lateral movement of the grain boundary by adjusting the dendritic primary spacing. The elimination rate of degenerate seaweed at the converging grain boundary increases rapidly as the misorientation angle is increased. This can be explained by the increase of the lateral component of the real tilted dendritic direction with the increase of misorientation angle. This proves the finding from our experiments. Predicted results also show that the degenerate seaweed can overgrow the tilted dendrite at the diverging grain boundary. This is not consistent with experimental results. Moreover, our results show that grain boundary angles at the two grain boundary angles increase with the misorientation angle increasing.

Phase-field simulation of multi-phase interactions in Fe-C peritectic solidification

A multi-phase field model is implemented to simulate the microstructure evolution and carbon diffusion of the Fe-C peritectic solidification. Based on the ferrite phase dendritic structure and the initial carbon distribution, austenite phase first nucleates at the tips of the ferrite phase dendrites, and then changes the initial phase distribution and carbon distribution mainly through the solidification of austenite and peritectic transformation. Both simulation and experiment clarify the interaction of ferrite phase, austenite phase and liquid phase controlled by carbon diffusion. The present study contributes to the understanding of formation mechanism of microscopic morphology and micro-segregation.

Phase-field–lattice Boltzmann simulation of dendrite growth under natural convection in multicomponent superalloy solidification

The thermosolutal convection can alter segregation pattern, change dendrite morphology and even cause freckles formation in alloy solidification. In this work, the multiphase-field model was coupled with lattice Boltzmann method to simulate the dendrite growth under melt convection in superalloy solidification. In the isothermal solidification simulations, zero and normal gravitational accelerations were applied to investigate the effects of gravity on the dendrite morphology and the magnitude of melt flow. The solute distribution of each alloy component along with the dendrite tip velocity during solidification was obtained, and the natural convection has been confirmed to affect the microsegregation pattern and the dendrite growth velocity. In the directional solidification simulations, two typical temperature gradients were applied, and the dendrite morphology and fluid velocity in the mushy zone during solidification were analyzed. It is found that the freckles will form when the average fluid velocity in the mushy zone exceeds the withdraw velocity.

Multiphase-field and experimental study of solidification behavior in a nickel-based single crystal superalloy

Understanding the solidification behavior in nickel-based single crystal superalloy solidification is essential for determining the following heat treatment process, while the experimental findings have led to contradictory results. On the one hand, the fine γ/γ′ structure in the interdendritic zone suggests a eutectic reaction. On the other hand, the coarse γ′ precipitates indicate a peritectic reaction. In this work, the CALPHAD-based multiphase-field model was used to investigate the superalloy solidification behavior and complex γ/γ′ morphology generation. The fine γ/γ′ and coarse γ′ patterns were reproduced in the γ/γ′ colony, which were compared with the interdendritic microstructure observed by experiments. And the eutectic and peritectic reactions were verified by analyzing the chemical driving force at the phase interfaces. The simulation results suggest that the coarsening of γ′ in the γ/γ′ colony was due to the fast decrease of chemical driving force at the γ/l and γ'/l interfaces. Besides, remelting of the γ dendrite was found near the bulk γ′, which can be mainly ascribe to the increasing of Mo concentration rejected by the growing γ'. The simulated microsegregation patterns of the alloy components were also obtained and the results were in good agreement with the experimental findings.

Multi-Phase Field Model of Localized Corrosion Kinetics with Corrosion Products Formation

Localized corrosion is one of the complex forms of corrosion which makes it difficult to detect and design-against. During metal corrosion in the corrosive environment, corrosion products are also formed as a result of electrochemical reactions inside the electrolyte. These products can precipitate on the corroding surface and stop/ slowdown the overall corrosion process. In order to understand this complex phenomenon, a multi-phase field model is proposed to simulate metal corrosion with corrosion products formation. Both anodic (metal oxidation) and cathodic (oxygen reduction) reactions along with other electrochemical reactions in the electrolyte are considered. The free energy of the system is described in terms of its metal ion concentration and the order parameters. Rather than considering linear kinetics (Allen-Cahn equation), inspired from classical rate theory, non-linear (Butler-Volmer) kinetics is considered to describe the temporal evolution of order parameters. The time dependent evolution of ionic species is governed by Nernst-Plank equations while electrostatic potential is governed by Poisson equation. The model results are compared with experimental findings and several examples are presented to show the practical applications of this model. One interesting example, which can be studied in detail with this methodology, is the formation of metal oxide nanowires (zinc/ aluminum oxide).

On the multiphase-field modeling of martensitic phase transformation in dual-phase steel using J2-viscoplasticity

On the mesoscopic length scale, dual-phase steel (DP) consists of hard and brittle martensitic inclusions, which are embedded in a soft and ductile ferritic matrix. On the macroscopic scale, this composite-like structure is responsible for the attractive combination of good formability and high tensile strength. In order to improve the properties of DP, computer models can be used to investigate the effects of various influencing factors during the production chain. As an extension to our previous work (Schoof et al 2018 Int. J. Solids Struct. 134 181–94), we include a J2-viscoplasticity formulation into a multiphase-field framework to study plastic effects in an EBSD-based DP microstructure during the martensitic phase transformation. The hardening behavior of each phase is modeled using a nonlinear function. The effect of the plastic relaxation rate on the microstructure formation and the residual strains is investigated. Two different models for introducing a yield criterion into a diffuse interface framework are discussed, and the results are compared. In this context, a plastic driving force is formulated, and its influence on the microstructure evolution is evaluated. The model not only predicts high values of accumulated plastic strain in ferritic regions, close to ferrite-martensite grain boundaries, but also within martensitic islands.

Influence of post-carburizing heat treatment on the core microstructural evolution and the resulting mechanical properties in case-hardened steel components

The case hardening process (carburizing followed by quenching to room temperature) is one of the heat treatments routinely conducted in various industrial sectors. Depending on the parameters applied during the post-carburizing stage, different microstructural constituents may develop in the interior (core) section of a case-hardened component which substantially influence the overall mechanical properties. In the present work, the effects of different heat treatments following the carburizing stage on the core microstructure evolution, the resulting toughness and hardness properties were investigated. The heat treatments were carried out employing Navy C-ring specimens, the core of which were used for microstructural investigations and fabrication of mini-Charpy specimens. Besides, nanoindentation tests were also carried out in the same region to examine the local hardness of different constituents. Furthermore, the microstructural evolutions were also studied utilizing multi-phase field modeling. Based on the obtained results, it can be shown that the increase in fractions of bainite and retained austenite is in direct correlation with the impact toughness improvement, however, the increase in fraction of ferrite and martensite acts inversely. Moreover, despite the differences in fractions of constituents, the level of overall hardness is similar in all the developed microstructures. Additionally, the nanoindentation results revealed that the formation of large fraction of bainitic ferrite leads to high level of scattering in nanohardness of martensite. Phase field simulations show that this is attributed to the degree of carbon partitioning into the adjacent austenitic area where depending on the local morphology and size can be quite different.

A Multi-phase-field Approach for Solidification with Non-negligible Volumetric Expansion—Application to Graphite Growth in Nodular Cast Iron

Multi-phase field models have become powerful tools for prediction of microstructure evolution in technical alloys. Coupling to CALPHAD databases provides complex multicomponent multiphase thermodynamic data and diffusion matrices. Although many thermodynamic databases contain volume data, this information is by now hardly used for solidification, since a comprehensive modelling of volumetric expansion based on solid- and fluid-mechanics exceeds present computational capacities. A novel multicomponent multi-phase-field approach is presented which handles volumetric expansion in a simplified way. It assumes that temporary pressure caused by local expansion is instantaneously released by matter transport on a time scale much faster than diffusion. The new approach neglects any mechanical and kinetic aspects of expansion, but allows for a change in total volume and for thermodynamically consistent volume fractions. Moreover, expansion-driven matter fluxes and associated advective solute transport are considered. Application is illustrated by the example of spheroidal graphite growth in a hypoeutectic cast iron alloy.

Microstructure evolution under isothermal and continuous cooling conditions via a combined multiphase field and nucleation approach

We demonstrate the microstructure evolution involving the nucleation and grain growth of a binary alloy during the continuous cooling process via a combined multiphase field model and stochastic nucleation computational model. The nuclei are initiated by adding into the total free energy a term of the nucleation energy related to the variables that represents grains of different orientation. Moelans’ interpolation functions are implemented to construct the chemical bulk free energy from coexisting phases/grains for increased stability of multiphase/grain junctions. It was found that the temperature dependent nucleation rate and interface mobility are the main materials properties controlling the features of the resultant microstructures. More even-sized and fine grains can be formed under a high cooling rate given the specified temperature dependent nucleation rate. Solute trapping is the most prominent and solute segregation is the slightest at the highest cooling rate. Equipped with a grain-tracking method, this computational framework provides a viable and computationally efficiently pathway to investigate the large-scale microstructure evolution under various temperature histories that could occur in manufacturing processes.

On the globularization of the shapes associated with alpha-precipitate of two phase titanium alloys: Insights from phase-field simulations

In two-phase titanium alloys, α-precipitate exhibits plate morphology which spheroidizes during static globularization. The stability of the 3-dimensional plates to withstand perturbation is well-established. However, based on Rayleigh instabilities, the current understanding rendered by ‘the modified perturbation theory’ holds that the spheroidization of the plates is governed by the stable growth of the ridges. To address this counterintuitive view, a multiphase-field model, which recovers the sharp interface solutions, is employed to investigate the morphological transformation accompanying globularization of shapes that resemble α-precipitate. The present investigation unravels an alternate mechanism, wherein the plates globularize by the successive formation and decay, by lateral expansion, of multiple terminal perturbations. This unique mode of spheroidization, distinctly, characterizes the temporal evolution of the driving force which contradicts the hitherto assumed conventional monotonic-decrease in the curvature-difference. Particularly, it is identified that the ellipsoidal plates exhibit a step-wise change in the curvature-difference while non-monotonic series of peaks are observed during the evolution of the pancake structures. To aid the optimization of the thermal cycle involved in the globularization treatment, a simplified relation between the aspect ratio and spheroidization time, which encompasses the influence of the transformation mechanism, is derived.

Chemo-elastic phase-field simulation of the cooperative growth of mutually-accommodating Widmanstätten plates

A comprehensive understanding of the phase transformations is of primary importance to optimize the properties of a material. In the present study, a novel and thermodynamically consistent chemo-elastic multiphase-field model is formulated to simulate phase transformations which yield unique needle-like patterns, referred to as Widmanstätten structure. Owing to the critical role of the curvature in the growth of the Widmanstätten structure, the model is devised to recover the sharp interface solutions, despite the introduction of a diffuse interface. This condition is fulfilled by formulating the chemical and the elastic driving force based on the grand potential density and the mechanical jump conditions, respectively. Additionally, to render a quantitative chemical driving force, parameters from CALPHAD-database (TCFE8) are incorporated. The current work, through the multicomponent multiphase-field simulations, shows that at high temperatures, when the chemical driving force is insufficient to actuate the growth of a plate, the Widmanstätten structure evolves by the co-operative growth of self-accommodating plates. Furthermore, the growth of a single Widmanstätten plate at low temperature is also simulated, and the growth kinetics of the transformations are verified through existing analytical predictions.

Modeling fracture cementation processes in calcite limestone: a phase-field study

The present work investigates the influence of crack opening rates on the development of four important calcite vein morphologies, namely fibrous, elongate-blocky, partially open, and euhedral, as a result of bitaxial growth in syntaxial veins using a multiphase-field model. The continued fracturing that occurs during synkinematic cementation in these veins is simulated using the geometric shift algorithm. The stark resemblance of the numerically sealed vein microstructures with the natural samples in terms of structural characteristics as well as remaining pore space signifies a dominant role of crack opening rates in the resulting morphological patterns. Further, simulation results of slow crack opening rates reveal that non-uniform fibers of variable lengths are formed when initial crack aperture is small, due to suppression of growth competition and vice versa.