Two-dimensional Phase-field Simulations Research Articles

Effect of Nb and B on the Precipitation Behaviors in Al-Ti-Nb Balanced-Ratio Ni-Based Superalloy: A Phase-Field Study

In this paper, quantitative two-dimensional (2-D) phase-field simulations were performed to gain insight into the effects of B and Nb for Al-Ti-Nb balanced-ratio GH4742 alloys. The microstructure evolution during the precipitation process was simulated using the MICRESS (MICRostructure Evolution Simulation Software) package developed in the formalism of the multi-phase field model. The coupling to CALPHAD (CALculation of PHAse Diagram) thermodynamic databases was realized via the TQ interface. The morphological evolution, concentration distribution, and thermodynamic properties were extensively analyzed. It is indicated that a higher Nb content contributes to a faster precipitation rate and higher amounts and the smaller precipitate size of the γ′ phase, contributing to better mechanical properties. The segregation of the W element in γ′ precipitate due to its sluggish diffusion effect has also been observed. Higher temperatures and lower B contents accelerate the dissolution of boride and reduce the precipitation of borides. With the increased addition of B, the formation of borides may have a pinning effect on the grain boundary to hinder the kinetic process. In addition, borides are prone to precipitate around the interface rather than in the bulk phase. Once the M3B2 borides nucleate, they grow in the consumption of γ′ phases.

Laser powder bed fusion of immiscible steel and bronze: A compositional gradient approach for optimum constituent combination

The microstructures and the mechanical properties of a laser powder bed fusion (LPBF) manufactured alloy coupons that were compositionally graded with austenitic stainless steel 316 L and bronze (Cu10Sn), which are immiscible, were investigated. While the microstructure of the pure 316 L contains only columnar γ-Fe grains, the formation of the equiaxed α-Cu, with α-Fe particles embedded in them, at the grain boundaries of the γ-Fe grains was observed upon alloying with Cu10Sn, for up to 50 wt.%. In the graded alloys with > 50 wt.% Cu10Sn, the microstructure inverts and contains α-Fe particle embedded α-Cu matrix along with some scattered γ-Fe grains. The two-dimensional phase-field simulations were employed to reproduced the phase transition and microstructure evolution in different compositions, revealing that the nucleation of γ-Fe occurs synchronously with the spinodal decomposition of liquid and that with further cooling, Cu-rich liquid solidifies and nanoscale spherical γ-Fe precipitates appear. Cross-sections with 10–40 wt.% Cu10Sn were found structurally unreliable owing to the formation of liquation cracks and shrinkage pores, whereas the rest of the build contained few pores and was crack-free. The Scheil solidification simulations together with the crack susceptibility index calculations over the entire compositions range of the CGA, reveal that when with < 50 wt.% Cu10Sn, liquid feeding to interdendritic regions of γ-Fe is limited, which is otherwise necessary to mitigate shrinkage-induced stresses. The hardness, yield and tensile strengths of the compositionally graded alloy in the cross-section with 50 wt.% Cu10Sn are higher than those of the parent constituents and the one with 80 wt.% Cu10Sn has the best combination of strength and ductility. The variations in strength and ductility are attributed to the microstructural and compositional changes in α-Cu matrix that were influenced by the formation of α-Fe particles. Finally, the appropriate proportions and the influence of the LPBF method in successfully mixing Cu10Sn and SS316L was discussed.

On the accuracy, stability and computational efficiency of explicit last-stage diagonally implicit Runge–Kutta methods (ELDIRK) for the adaptive solution of phase-field problems

Runge–Kutta algorithms with adaptive step size control provide reliable tools for the solution of initial value problems. Most widespread within the vast family of Runge–Kutta methods are the diagonally implicit Runge–Kutta (DIRK) methods. In particular, the new low-order explicit last-stage diagonally implicit Runge–Kutta (ELDIRK) methods are investigated, combining implicit schemes with an additional explicit evaluation as an explicit last stage. This results in Butcher tableaus with two solutions of different convergence orders suitable for embedded methods, where the higher-order solution is achieved by additional explicit evaluations. Thus, the iterative solution of non-linear systems is omitted for the additional stage, presenting a major reduction in computational cost for the determination of a local error estimate. The key contribution is the application of the novel Butcher tableaus to phase-field problems, solved with the finite-element method, leading to substantial numerical investigations with an efficient approach for diagonally implicit Runge–Kutta schemes. The most important aspects are the extension towards fourth-order methods, the study of the convergence orders for the ELDIRK schemes, their respective stability regions and computational efficiency. A local error estimator is presented capturing the evolution of the phase-field problems, such that adaptive step size control for the new low-order embedded schemes based on an empirical approach for error estimation is achieved. A suitable parallel algorithm is presented with conclusive two-dimensional phase-field simulations based on a Kobayashi–Warren–Carter model including benchmarks for computational efficiency. The higher-order convergence suggested by the novel schemes is confirmed, and their effective results are demonstrated, resulting in a valuable semi-explicit addition to the family of Runge–Kutta time integration schemes.

Evolution dynamics of premelted polycrystalline microstructures

Below their bulk melting temperature, polycrystalline materials may exhibit the premelting phenomenon, with the grain boundaries (GB) turning into liquidlike nanofilms. This structural change has several important consequences, for example in terms of mechanical properties. We study using phase field simulations the grain evolution in the premelted polycrystal of a pure substance. The dynamics is governed by heat fluxes across the liquid films, across the liquid pockets at the triple junctions, and within the grains. Depending on the dimensionless undercooling Δ and the grain size λ, we identify different regimes. When Δ≪1, the width ρ∼δ/Δ of the liquid pockets is much larger than the atomic distance δ and the dynamics becomes independent of the free energy of the dry GBs. Then, when λ≫ρ, the premelted GBs evolve according to a curvature-driven dynamics, while the grain evolution corresponds to a usual Ostwald coarsening when the liquid pockets overlap, i.e. when λ∼ρ. Within the curvature-driven regime, the triple junctions maintain their equilibrium configuration with 2π/3 angles when λ≫ρ/Δ, while a non-equilibrium configuration develops when λ∼ρ/Δ. We analyze and confirm these effects through two-dimensional phase field simulations. We finally relate our findings to experiments on ceramics exhibiting an anisotropic abnormal grain growth. Supported by additional phase field simulations, we propose a scenario for this phenomenon based on the competition between the velocity of the quasi-liquid layer propagating along the dry GBs and the curvature-driven velocity of the already pre-melted GBs.

Study on the solutal convection during dendrite growth of superalloy under directional solidification condition

The solutal convection is able to affect the dendrite growth and solute distribution during directional solidification. The solutal convection is raised by the solute difference-caused buoyancy. When solutal convection becomes intensive, solute plumes will occur, which may cause freckle defects in directional solidification of superalloys. The onset of solutal plumes depends on the solidification condition. In this study, the dendrite growth with solutal convection during directional solidification of nickel-based superalloy was investigated by two-dimensional phase-field simulations. The lattice Boltzmann method was used to solve the flow field driven by solute difference caused-buoyancy force. The pseudo-binary alloy approximation was adopted for simplification of the multicomponent alloy. The effects of the temperature gradient, pulling velocity, and isotherm inclination angle on dendrite growth and solutal convection were investigated. The dendrite growth dynamics and solutal convection pattern under different solidification conditions were analyzed. Dendrite growth velocity oscillation was observed. The effect of pulling velocity on dendrite growth velocity oscillation was analyzed.

Parallel GPU-accelerated adaptive mesh refinement on two-dimensional phase-field lattice Boltzmann simulation of dendrite growth

Simulations of dendritic solidification involving melt convection and solid motion usually require a considerably higher computational domain than the dendrite size, whose computational efficiency with a uniform mesh is extremely low. In this study, to accelerate those two-dimensional simulations using the phase-field and lattice Boltzmann (PF-LB) methods, we developed a parallel computing method with multiple graphics processing units (GPUs) for the adaptive mesh refinement (AMR) method with dynamic load balancing (parallel-GPU AMR). It was confirmed that parallel-GPU AMR simulations were faster than those with the uniform mesh when the number of grid points in the adaptive mesh was around 40% or less than those in the uniform mesh. We also demonstrate that the developed parallel-GPU AMR can greatly accelerate the PF-LB simulations of dendrite growth with melt convection and solid motion.

Coarse Columnar Dendrites of GH3039 Nickel-Based Alloy at Diverging Grain Boundaries during Wire and Laser Additive Manufacturing: A Phase Field Simulation

Coarse columnar dendrite greatly reduced the mechanical performance of GH3039 nickel-based alloy in the additive manufactured parts, which limited its application in the engineering fields. This study provides a comparison of overgrowth behaviors at diverging grain boundaries through two-dimensional phase field simulation, and the effect of dendrite orientation on overgrowth behavior was analyzed. Moreover, our results show that the primary spacing becomes larger as the increasing of dendrite orientation. The columnar dendrites branch new dendrites near grain boundaries to refine the primary arm spacing in the process of wire and laser additive manufacturing (WLAM).

Equiaxed dendritic growth in nearly isothermal conditions: A study combining in situ and real-time experiment with large-scale phase-field simulation

The equiaxed dendritic growth of Al-Cu alloys in nearly isothermal temperature field under continuous cooling condition is studied using in situ and real-time observation of experiments by synchrotron X-ray radiography and large-scale quantitative two-dimensional (2D) phase-field (PF) simulations. It is revealed that the equiaxed dendritic morphology and the secondary dendritic arm spacing (SDAS) in the 2D PF simulations are in a reasonable agreement with the experimental data. Increasing the cooling rates results in a smaller SDAS, as predicted by the analytical Kattamis-Flemings model. The transformation kinetics of solid fraction can be described by the Johnson-Mehl-Avrami-Kologoromov (JMAK) theory, but quantitative differences between the experiments and 2D PF simulations are significant. The maximum solute concentration Cmax in liquid is approximately equal to the equilibrium concentration, which depends on the undercooling rather than the cooling rate. But the minimum solute concentration Cmin in solid decreases with the cooling rate, thus leading to a larger segregation ratio SR = Cmax/Cmin. Moreover, the liquid gravity-driven natural convection is considered in simulations. The liquid flow slightly increases the SDAS but has no apparent effect on solid fraction, and the segregation ratio is slightly reduced by the liquid convection, which could be attributed to the almost same Cmax and enlarged Cmin.

Two-dimensional Phase-field Simulation of Pore Evolution in High Temperature Self-lubrication Composites

The phase field model was established based on the free energy function method to simulate the evolution process of microporous matrix. High temperature self-lubricating microporous matrix was prepared by powder metallurgy method, then its porosity was measured under different pore forming agent ratio and addition amount, in order to verify the phase field model. The results show that the simulation value of porosity by phase field method is basically consistent with the experimental value, and the phase field model can be used to simulate the effect of pore forming agent on the pore evolution process of matrix.

Phase Field Study of the Microstructural Dynamic Evolution and Mechanical Response of NiTi Shape Memory Alloy under Mechanical Loading.

For the purpose of investigating the microstructural evolution and the mechanical response under applied loads, a new phase field model based on the Ginzburg-Landau theory is developed by designing a free energy function with six potential wells that represent six martensite variants. Two-dimensional phase field simulations show that, in the process of a shape memory effect induced by temperature-stress, the reduction-disappearance of cubic austenite phase and nucleation-growth of monoclinic martensite multi-variants result in a poly-twined martensitic microstructure. The microstructure of martensitic de-twinning consists of different martensite multi-variants in the tension and compression, which reveals the microstructural asymmetry of nickel-titanium (NiTi) alloy in the tension and compression. Furthermore, in the process of super-elasticity induced by tensile or compressive stress, all martensite variants nucleate and expand as the applied stress gradually increases from zero. Whereas, when the applied stress reaches critical stress, only the martensite variants of applied stress-accommodating continue to expand and others fade gradually. Moreover, the twinned martensite microstructures formed in the tension and compression contain different martensite multi-variants. The study of the microstructural dynamic evolution in the phase transformation can provide a significant reference in improving properties of shape memory alloys that researchers have been exploring in recent years.

Overgrowth behavior at converging grain boundaries during competitive grain growth: A two-dimensional phase-field study

Herein, three types of converging grain growth were investigated during directional solidification, through two-dimensional phase-field simulations. The three types were distinguished by the orientation of the favorably oriented (FO) dendrites. The growth orientation of the FO dendrites was parallel to the thermal gradient direction in type I, whereas the orientation of FO dendrites was inclined in the same and opposite lateral directions as those of the unfavorably oriented (UO) dendrites in types II and III, respectively. The largest difference between FO and UO grain inclination angles—that is, the threshold angle, for unusual overgrowth in different alloy systems were clearly different, in type I and III cases. In all three types of converging grain growth, overgrowth of UO dendrites by FO dendrites (usual overgrowth) was observed to occur more frequently than that of FO dendrites being overgrown by UO dendrites (unusual overgrowth). The asymmetry of the inclined dendrite diffusion layer caused different solute interactions in the three types of converging grain growths. Based on the different solute interactions, unusual overgrowth behavior occurred most readily in type III cases—and with most difficulty in type II cases. We also saw that the local GB orientation may alternate between FO and UO grain orientations in type III cases.

Effects of Cr content on compositional evolution and precipitation kinetics of γ′ phase in Ni–Al–Cr alloy: 2D phase-field simulation

The two-dimensional phase-field simulation coupled with the sublattice model is utilized to simulate the γ′ precipitation in Ni-12 Al-x Cr at.% (x = 8, 9, 10) alloys aged at 1073 K for 831 h. The compositional evolution of Cr and Al, and the precipitation kinetics of γ′ phase are studied. At the nucleation and growth stage, the concentration distribution of Cr and Al is characterized by an Al enrichment and Cr depletion at the γ/γ′ interface. As Cr content increases from 8 to 10 at.%, the partition ratio of Al and Ni into γ′ phase is promoted, and the equilibrium volume fraction of γ′ phase is increased, while the Al and Ni atoms at the corner site of L12-Ni3(Al, Cr) sublattice are replaced by Cr atoms partly. Besides, the coarsening rate constants of steady-state coarsening (166.2 < t < 831 h) are less than that of growth and coarsening stage (4.2 < t < 50.0 h), the increased Cr content promotes the coarsening of γ′ phase, and the average radius of γ′ phase obeys the time exponent 1/3 well for a high Cr content alloy.

Calibration of a concentration-driven nucleation mechanism for phase-field simulations of eutectic and off-eutectic compositions in AlCu-5Ag

The present work is an extension of the studies conducted by Kellner et al. [M. Kellner et al., Acta Materialia, vol. 182, pp 267–277, 2020.], to show the transferability of their concentration-driven nucleation mechanism to other systems. The implementation of this method improves the adjustment process of directionally solidifying eutectic structures in two-dimensional phase-field simulations. To quantitatively evaluate the nucleation mechanism, the velocity-spacing relationships of different eutectic and off-eutectic melt compositions are investigated for AlCu-5Ag. A good agreement between simulations with and without the nucleation mechanism is found for all studied compositions.

Phase-Field Microstructure Solidification of Al–2 wt% Si Alloys

In this work, we develop one- and two-dimensional phase-field simulations to approximate dendritic growth of a binary Al–2 wt% Si alloy. Simulations are performed for both isothermal as well as directional solidification. Anisotropic interface energies are included with fourfold symmetries, and the dilute alloy assumption is imposed. The isothermal results confirm the decrease in the maximum concentration for larger interface velocities as well as reveal the presence of parabolic, dendrite tips evolving along directions of maximum interface energy. The directional solidification results further confirm the formation of distinctive secondary dendritic arm structures that evolve at regular intervals along the unstable solid/liquid interface.

Two-dimensional phase-field simulations of competitive dendritic growth during laser welding

A phase-field model is applied to study competitive dendritic growth between different grains with different crystalline orientations during laser welding of Al-Cu alloy. An unfavorable oriented (UO) grain is set to locate between two favorable oriented (FO) grains to study the dendritic growth behaviors at not only converging grain boundary (GB) but also diverging GB. It is found that the cellular dendrites appear firstly nearby the GBs at initial instability stage. At early competitive growth stage, the UO misalignment angle influences the competitive dendritic growth at both GBs. After completely entering competitive growth stage, the leading dendrites that are much faster than other dendrites at GBs firstly grow in lateral direction. The side branches growing from leading dendrites block growth path of nearby dendrites and change the UO/FO grain widths. The converging GB deflection angle and the side branch development at diverging GB as characteristic competitive dendritic growth behavior at relatively steady growth stage also significantly affect UO and FO grain widths. The numerical results are basically consistent with microstructure obtained by experiment.

Phase field modeling of stressed grain growth: Effect of inclination and misorientation dependence of grain boundary energy

The effect of anisotropic grain boundary energy on the kinetics of stressed grain growth is investigated by two-dimensional phase field simulations. The simulations are performed for a two-dimensional representative volume element of polycrystalline copper with elastic cubic symmetry under both uniaxial elongation and shear loading with elastic strains in the order of a few thousandths. The Read-Shockley relation for the misorientation–dependence and the cubic symmetry for the inclination–dependence of grain boundary energy are assumed. The simulation results have generally shown that the misorientation–dependent anisotropy, compared to the inclination–dependent anisotropy, significantly affects the evolution of microstructure and texture during stressed grain growth. Particularly, the misorientation–dependent anisotropy of grain boundary energy results in (i) the elongated morphology of grains in the evolved microstructure; (ii) the aggregation of grains with similar orientations in the evolved microstructure; (iii) the strengthening of lower–misorientation angle grain boundary segments; and hence (iv) decelerating the overall rate of stressed grain growth and texture evolution and (v) decelerating stress relaxation under constant–strain loading. These results are based on the assumption that the isotropic grain boundary energy is equal to the high angle grain boundary energy of the misorientation–dependent grain boundary energy and the inclination–dependent grain boundary energy varies sinusoidally around the isotropic grain boundary energy.

Fine Grains Reduce Cracking Susceptibility During Solidification: Insights from Phase-Field Simulations

Hot cracking is commonly observed in welding and metal additive manufacturing processes. In this work, the effects of grain size on cracking susceptibility during solidification were investigated using two-dimensional phase-field simulations. Al-3.0 wt.% Cu alloy was chosen as an example, and the grain size was controlled by adjusting the primary dendritic arm spacing. It was found that fine grains can significantly reduce liquid channel segregation and facilitate earlier coalescence of adjacent grains to resist cracking. Moreover, the cracking susceptibility indexes were predicted from the microstructurally complex phase-field data, showing good agreement with the simulated liquid channel morphology and segregation. Both the liquid channel characteristics and cracking susceptibility index demonstrated how grain refining reduces cracking susceptibility during solidification in welding and metal additive manufacturing processes.

A Trial to Design γ/γ′ Bond Coat in Ni–Al–Cr Mode TBCs Aided by Phase-Field Simulation

Phase-field modeling coupled with calculation of phase diagram (CALPHAD) database was utilized to perform a series of two-dimensional phase-field simulations of microstructure evolution in the γ + γ′/γ + γ′ Ni–Al–Cr mode bond coat/substrate systems. With the aid of phase-field simulated microstructure evolution, the relationship between the interdiffusion microstructure and the cohesiveness/aluminum protective property with different alloy compositions and bond coat thicknesses was fully discussed. A semi-quantitative tie-line selection criteria for alloy composition of the bond coat/substrate system with the identical elements, i.e., that the equilibrium Al concentrations of γ′ and γ phases in the bond coat should be similar to those in substrate, while the phase fraction of γ′ in the bond coat tends to be higher than that in the substrate, was then proposed to reduce the formation of polycrystalline structure and thermal shock from the temperature gradient.

Primary dendrite spacing selection during directional solidification of multicomponent nickel-based superalloy: multiphase-field study

The primary dendrite spacing selection in a multicomponent Ni-based superalloy during directional solidification was systematically studied using two-dimensional phase-field simulations. The alloy thermodynamic and kinetic data were obtained from Pandat software with PanNickel database and directly coupled into the multiphase-field model. All the simulations were performed on a GPU server, and an optimized computing scheme using GPU shared memory was adopted. First, the morphology of the solidification front was studied, and the segregation pattern was investigated and compared with the experimental results. Then, the dendritic spacing distribution under a wide range of pulling velocities Vp (10–500 μm s−1) and temperature gradients G (2–200 K mm−1) was obtained and analyzed. The simulation results agree well with analytical model that the primary dendrite spacing scales as $$ \varLambda \propto V_{\text{p}}^{ - b} G^{ - c} $$ . The coefficient b is near a constant value of 0.38 and varies slightly between 0.34 and 0.42, while coefficient c increases monotonously from 0.27 to 0.56 with the increasing G. The predicted dendritic spacing agrees well with the experimental data, but exhibits a major difference when under very low cooling rate (R < 0.1 K s−1). The effect of grain inclination angle θ on the final primary dendritic spacing was also studied, and an abnormal decrease in dendritic spacing was found under low grain orientation where θ < 10°. When the grain inclination angle exceeds 20°, the dendritic spacing increases with θ as the power law.

Comparison of solidification cracking susceptibility between Al-Mg and Al-Cu alloys during welding: A phase-field study

Based on two-dimensional phase-field simulations, we demonstrated the possible reasons why Al-Mg alloys can have better resistance to solidification cracking than Al-Cu alloys despite their wide freezing temperature range. Using Al-4.0wt% Cu and Al-4.0wt%Mg alloy as examples, we found that back-diffusion is negligible due to the relatively high cooling rate, and extensive coalescence occurs at earlier solidification stage in Al-4.0wt%Mg alloy compared with Al-4.0wt% Cu alloy. With considering coalescence predicted by phase-field simulations, the calculated solidification cracking index of Al-Mg alloy is reasonably lower than Al-Cu alloy.