A new model for the temperature-dependent yield strength of polycrystalline Ni-based superalloysPolycrystalline Ni-based superalloys at high strain ratesStrain rate (10−3 to 10−4 s−1) featuring multimodal precipitatePrecipitates microstructuresMicrostructure is presented. It extends existing mean field models based on the summation of particle strengthening and other strengthening contributions, which were originally created to describe behavior at room temperature. Temperature dependenceTemperature dependence is introduced for each contribution using existing models or datasets from the literature while maintaining physical validity and avoiding unnecessary fitting parameters. Further, the expected change in the precipitatePrecipitates distributions at high temperaturesHigh temperature is modeled using CALPHADCALculation of PHAse Diagrams (CALPHAD). By considering such microstructural changes, the model can predict the yield strength from room temperature to beyond the solvus temperature of the precipitatesPrecipitates. Due to the strategy for combining different strengthening contributions, the model can also predict the active deformation modes at any temperature and can be extended easily to incorporate additional deformation mechanismsDeformation mechanisms. The model’s sensitivity to microstructural parameters and its strengths and weaknesses are also discussed. Application of the model to three commercial alloys shows good agreement over the whole temperature range.

Within recent years, considerable progress has been made regarding high-performance solvers for partial differential equations (PDEs), yielding potential gains in efficiency compared to industry standard tools. However, the latter largely remains the status quo for scientists and engineers focusing on applying simulation tools to specific problems in practice. We attribute this growing technical gap to the increasing complexity and knowledge required to pick and assemble state-of-the-art methods. Thus, with this work, we initiate an effort to build a common taxonomy for the most popular grid-based approximation schemes to draw comparisons regarding accuracy and computational efficiency. We then build upon this foundation and introduce a method to systematically guide an application expert through classifying a given PDE problem setting and identifying a suitable numerical scheme. Great care is taken to ensure that making a choice this way is unambiguous, i.e., the goal is to obtain a clear and reproducible recommendation. Our method not only helps to identify and assemble suitable schemes but enables the unique combination of multiple methods on a per-field basis. We demonstrate this process and its effectiveness using different model problems, each comparing the resulting numerical scheme from our method with the next best choice. For both the Allen–Cahn and advection equations, we show that substantial computational gains can be attained for the recommended numerical methods regarding accuracy and efficiency. Lastly, we outline how one can systematically analyze and classify a coupled multiphysics problem of considerable complexity with six different unknown quantities, yielding an efficient, mixed discretization that in configuration compares well to high-performance implementations from the literature.

This work investigates the influence of the hexamethylenetetramine (hexa) content on the rheological features, pyrolysis reactions, mass-loss kinetics and char yield for two phenolic resins, each with different novolak-hexa mixtures from 0 wt.-% to 15 wt.-%. Curing and viscosity of resins was studied using a rheometer. Pyrolysis, evolved gases and char yield were investigated with TG-FTIR-GC/MS. The obtained results show that the curing of resins shifts to lower temperatures whereas the viscosity and the residual carbon amount both increase with increasing hexa-contents. The presented data suggests a hexa-induced microgel formation for phenolic resins. Models for viscosity and char yield were developed and present a valuable tool to tailor the resin-hardener mixture for the desired application, enable a more efficient processing technology and contribute to a deeper understanding of the pyrolysis of novolak resins.

The strength of a Ni-base superalloy depends strongly on its microstructure consisting of cuboidal γ^^' precipitates surrounded by narrow channels of γ matrix. According to the theory of Orowan, a moving dislocation has to crimp through the minimal inter-precipitate spacing to admit the plastic deformation. We present a novel approach to evaluate the matrix channel width distribution of a matrix/ γ^^' microstructure in binary representation. Our method relies on precise determination of the matrix/precipitate interfaces and requires no additional user input. For each matrix channel between two neighboring precipitates, we identify the minimal interface to interface distance vector with its length being the channel width. The performance of this method is demonstrated on the example of the commercial alloy CSMX-4. We show that, in contrast to conventional line sectioning approaches, the approach consistently handles experimental 2D micrographs and 3D phase-field simulation data. The identified distance vectors correlate to the underlying crystal symmetry independent of the image orientation. The obtained channel width distributions compare well between the 2D and 3D data. This is in terms of similar median and σ of a log-normal distribution. The presented method overcomes limitations of the conventional line slicing approaches and provides a versatile tool for automated microstructure characterization.

Coherent tetragonal precipitates, such as the Ni3Nb phase γ″ found in Ni-base superalloys, appear as plate-shaped particles. These shapes are the result of anisotropic elastic misfit strains. We present 3D sharp phase-field simulations that capture this circumstance well due to the inclusion of the elastic effects from the misfit. These simulations reveal that the ripening behavior of γ″ precipitates deviates significantly from the classical LSW theory of Ostwald ripening. A ripening exponent of 2 rather than 3 describes the simulated γ″ size evolution at temperatures between 700 °C and 760 °C best. Employing a quantitative distinction argument, we show that 60 pct of this deviation is attributed to the elastically induced size dependence of the precipitate shapes. With increasing precipitate size, the minimization of elastic energy leads to steadily increasing plate aspect ratios. The precipitate ripening kinetics accelerate with increasing aspect ratio. Fitting the newly received square root time dependence to experimental data yields a physically conclusive activation energy of ripening close to the activation energy of Nb diffusion in the alloy.

The influence of the withdrawal rate and wall thickness on the primary and secondary dendrite arm spacings of single‐crystal specimens manufactured by the Bridgman process is investigated. Thin‐walled specimens with wall thicknesses of 0.4, 0.8, 1, and 2 mm are compared with conventional cast single‐crystal cylindrical specimens with a diameter of 15 mm of the nickel‐based superalloy MAR M247LC. It is found that the average primary dendrite arm spacing decreases slightly with decreasing wall thickness, whereas the secondary dendrite arm spacing is relatively independent of the cast wall thickness. The slight decrease in primary dendrite arm spacing relates to an ≈ 0.2 mm wide boundary zone next to the ceramic mold/cast metal interface that is not only present in the thin‐walled specimens, but also within cylindrical single‐crystal specimens. Within this boundary zone, the primary dendrite arm spacing is anisotropic. While the spacing of two dendritic cores sitting aside each other (y‐direction) remains at the bulk value, the spacing of two dendritic cores sitting in line with the interface normal (x‐direction) is significantly reduced.

The phase-field method provides a powerful framework for microstructure evolution modeling in complex systems, as often required within the framework of integrated computational materials engineering. However, spurious grid friction, pinning and grid anisotropy seriously limit the resolution efficiency and accuracy of these models. The energetic resolution limit is determined by the maximum dimensionless driving force at which reasonable model operation is still ensured. This limit turns out to be on the order of 1 for conventional phase-field models. In 1D, grid friction and pinning can be eliminated by a global restoration of Translational Invariance (TI) in the discretized phase-field equation. This is called the sharp phase-field method, which allows to choose substantially coarser numerical resolutions of the diffuse interface without the appearance of pinning. In 3D, global TI restricts the beneficial properties to a few specific interface orientations. We propose an accurate scheme to restore TI locally in the local interface normal direction. The new sharp phase-field model overcomes grid friction and pinning in three-dimensional simulations, and can accurately operate at dimensionless driving forces up to the order of 10 4 . At one-grid-point interface resolutions, exceptional degrees of isotropy can be achieved, if further the largely inhomogeneous latent heat release at the advancing solid-liquid interface is mitigated. Imposing a newly proposed source term regularization, the new model captures the formation of isotropic seaweed structures without spurious dendritic selection by grid anisotropy, even at one-grid-point interface resolutions.

Computational microstructure design aims to fully exploit the precipitate strengthening potential of an alloy system. The development of accurate models to describe the temporal evolution of precipitate shapes and sizes is of great technological relevance. The experimental investigation of the precipitate microstructure is mostly based on two-dimensional micrographic images. Quantitative modeling of the temporal evolution of these microstructures needs to be discussed in three-dimensional simulation setups. To consistently bridge the gap between 2D images and 3D simulation data, we employ the method of central moments. Based on this, the aspect ratio of plate-like particles is consistently defined in two and three dimensions. The accuracy and interoperability of the method is demonstrated through representative 2D and 3D pixel-based sample data containing particles with a predefined aspect ratio. The applicability of the presented approach in integrated computational materials engineering (ICME) is demonstrated by the example of γ″ microstructure coarsening in Ni-based superalloys at 730 °C. For the first time, γ″ precipitate shape information from experimental 2D images and 3D phase-field simulation data is directly compared. This coarsening data indicates deviations from the classical ripening behavior and reveals periods of increased precipitate coagulation.

Diffuse interface descriptions offer many advantages for the modeling of microstructure evolution. However, the numerical representation of moving diffuse interfaces on discrete numerical grids involves spurious grid friction, which limits the overall performance of the model in many respects. Interestingly, this intricate and detrimental effect can be overcome in finite difference (FD) and fast Fourier transformation (FFT)-based implementations by employing the so-called sharp phase-field method (SPFM). The key idea is to restore the discretization-induced broken translational invariance (TI) in the discrete phase-field equation by using analytic properties of the equilibrium interface profile. We prove that this method can indeed eliminate spurious grid friction in the three-dimensional space. Focusing on homogeneous driving forces, we quantitatively evaluate the impact of spurious grid friction on the overall operational performance of different phase-field models. We show that the SPFM provides superior degrees of interface isotropy with respect to energy and kinetics. The latter property enables the frictionless motion of arbitrarily oriented diffuse interfaces on a fixed 3D grid.

We develop a phase-field model for the simulation of chemical diffusion limited microstructure evolution. The model is applied to γ′-precipitation under the influence of realistic multi-step aging treatments in multi-component nickel-based superalloys with industrially relevant chemical complexity. The temperature-dependent thermodynamic and kinetic input parameters are obtained from CALPHAD calculations using ThermoCalc. Further, the model accounts for the lattice-misfit between the precipitate- and the matrix-phase. The required temperature-dependent elastic stiffness and lattice-misfit can be measured using resonance ultrasound spectroscopy and high temperature X-ray diffraction, respectively. This allows to account for realistic shaping of γ′-particles in the simulation. The comparison to shapes of γ′-particles in experimental microstructures serves as an important cross validation of the model. The application of the model to investigate the effect of the subsequent aging treatment on the precipitation microstructure after a brazing process is discussed.

The effectiveness of the mechanism of precipitation strengthening in metallic alloys depends on the shapes of the precipitates. Two different material systems are considered: tetragonal γ′′ precipitates in Ni-based alloys and tetragonal θ′ precipitates in Al-Cu-alloys. The shape formation and evolution of the tetragonally misfitting precipitates was investigated by means of experiments and phase-field simulations. We employed the method of invariant moments for the consistent shape quantification of precipitates obtained from the simulation as well as those obtained from the experiment. Two well-defined shape-quantities are proposed: (i) a generalized measure for the particles aspect ratio and (ii) the normalized λ2, as a measure for shape deviations from an ideal ellipse of the given aspect ratio. Considering the size dependence of the aspect ratio of γ′′ precipitates, we find good agreement between the simulation results and the experiment. Further, the precipitates’ in-plane shape is defined as the central 2D cut through the 3D particle in a plane normal to the tetragonal c-axes of the precipitate. The experimentally observed in-plane shapes of γ′′-precipitates can be quantitatively reproduced by the phase-field model.

The effects of anisotropic interfacial properties and heterogeneous elasticity on the growth and ripening of plate-like θ′-phase (Al2Cu) in Al-1.69 at.% Cu alloy are studied. Multi-phase-field simulations are conducted and discussed in comparison with aging experiments. The precipitate/matrix interface is considered to be anisotropic in terms of its energy and mobility. We find that the additional incorporation of an anisotropic interfacial mobility in conjunction with the elastic anisotropy result in substantially larger aspect ratios of the precipitates closer to the experimental observations. The anisotropy of the interfacial energy shows comparably small effect on the precipitate’s aspect ratio but changes the interface’s shape at the rim. The effect of the chemo-mechanical coupling, i.e., the composition dependence of the elastic constants, is studied as well. We show that the inverse ripening phenomenon, recently evidenced for δ’ precipitates in Al-Li alloys (Park et al. Sci. Rep. 2019, 9, 3981), does not establish for the θ′ precipitates. This is because of the anisotropic stress fields built around the θ′ precipitates, stemming from the precipitate’s shape and the interaction among different variants of the θ′ precipitate, that disturb the chemo-mechanical effects. These results show that the chemo-mechanical effects on the precipitation ripening strongly depend on the degree of sphericity and elastic isotropy of the precipitate and matrix phases.

The goal of an aging heat treatment in Ni-based superalloys is to control the volume fraction and size of the strengthening precipitates. We predict the precipitate size evolution of primary L12 and D022 precipitates in the industrially relevant alloys CMSX-4, René80, and IN718 using a one-dimensional phase-field model with an artificial Gibbs–Thomson driving force considering the shape of non-spherical precipitates. In comparison with the classical theory of precipitate ripening, the presented phase-field model considers all alloying elements, non-isothermal cooling, and heating stages and off-equilibrium volume fraction. We implicitly consider elastic effects between precipitate and fcc solid solution matrix by adjusting the mobility parameters and considering size dependent non-spherical D022 precipitate shapes. Literature data reveals the strong predictive potential of this multi-scale method for integrated computational materials engineering. We identify individual aging stages during which precipitate growth is dominated by either ripening or precipitation from the supersaturated matrix phase.

The precipitation of the metastable gamma'' phase has been studied in a Nb containing nickel-based alloy in single crystalline state, with a composition close to the commercially available superalloy IN718. Solution heat treatment and isothermal aging treatment are applied to undeformed single crystalline samples. We focus on the investigation of heterogeneous gamma'' precipitation on dislocation sites, without the interference of grain boundaries or co-precipitation with other phases. The gamma'' precipitation microstructure after aging is found to be strongly related to the dislocation microstructure before aging. The dislocations found in the undeformed specimens have been generated during the dendritic solidification or the water quenching after solution heat treatment. Two specific patterns dominate the precipitation microstructure after aging. First, some gamma'' particles precipitate along curved lines formed initially by the dislocations during solution heat treatment along the dendritic boundary. Second, gamma'' particles are found to be arranged along the dislocation in slip planes, which can be seen as largely extended straight lines parallel to the < 011 > directions within the (001) plane, similar to a basket weave structure. Such an aligned arrangement of gamma'' particles has been previously observed in polycrystalline IN718 within individual grains (Choi and Choi, 1972). In contrast, the bands of numerous gamma'' particles can span the whole sample of millimeter size in single crystals, cutting several hundred dendrite cores. All gamma'' particles are in the shape of an oblate spheroid, where the normals of the plates are pointing in < 001 > directions.

The excellent mechanical properties of the Ni-based superalloy IN718 mainly result from coherent gamma '' precipitates. Due to a strongly anisotropic lattice misfit between the matrix and the precipitate phase, the particles exhibit pronounced plate-shaped morphologies. Using a phase-field model, we investigate various influencing factors that determine the equilibrium shapes of gamma '' precipitates, minimizing the sum of the total elastic and interfacial energy. Upon increasing precipitate phase fractions, the model predicts increasingly stronger particle-particle interactions, leading to shapes with significantly increased aspect ratios. Matching the a priori unknown interfacial energy density to fit experimental gamma '' shapes is sensitive to the phase content imposed in the underlying model. Considering vanishing phase content leads to 30 % lower estimates of the interfacial energy density, as compared to estimates based on realistic phase fractions of 12 %. We consider the periodic arrangement of precipitates in different hexagonal and rectangular superstructures, which result from distinct choices of point-symmetric and periodic boundary conditions. Further, non-volume conserving boundary conditions are implemented to compensate for strains due to an anisotropic lattice mismatch between the gamma matrix and the gamma '' precipitate. As compared to conventional boundary conditions, this specifically tailored simulation configuration does not conflict with the system's periodicity and provides substantially more realistic total elastic energies at high precipitate volume fractions. The energetically most favorable superstructure is found to be a hexagonal precipitate arrangement.

We investigate the influence of artificial grid friction in different phase-field models by considering the stationary motion of an interface between two phases at different bulk free energy levels. Following the striking idea of a Sharp Phase-Field Model (SPFM) from Finel et al., we proof that restoring translational invariance indeed eliminates artificial grid friction effects during stationary interface propagation. Over a largely extended range of possible driving forces the theoretic interface velocities are reproduced by orders of magnitude more accurately, even if the diffuse interface profile is only marginally resolved by just one grid point. We propose a new variant of the SPFM, which restores translational invariance locally in the direction of interface motion. It is shown that, even for marginally resolved interface-profiles, the new SPFM variant provides frictionless motion for arbitrarily oriented planar interfaces. Finally, by considering thermal diffusion limited solidification, we demonstrate the capability of the SPFM-approach to also deal with inhomogeneous driving forces using a one-gird point interface resolution.

We develop a phase-field model for the simulation of precipitate microstructure pattern formation in nickel-base superalloys. The model accounts for the local effects from inhomogeneous and anisotropic elastic deformations, which mainly result from the lattice misfit between the precipitates and matrix phase. Further, in each time-step, we consider the chemical driving force for precipitate ripening to instantaneously equilibrate to a homogeneous value, leading to conserved phase volumes. The model is applied to study the equilibrium shape of a 2D single γ′-particle embedded in the γ-matrix with varying lattice misfit and γ/γ′ interface energies. Further, we apply the method of moment invariants to quantify the resulting equilibrium shapes of precipitates, which turns out to be a size independent characterization of the particle shape. Resulting values for the 2D moment invariants of experimental as well as simulated particle shapes are discussed and compared. Considering ideally spherical particles, we find that large values for the γ/γ′-interface width lead to systematic deviations in the resulting moment invariants.

We present analytical and numerical results for the dominant mechanisms of pattern selection in two growth regimes which are crucial in elastically influenced solid-solid transformations like the bainitic one. The first growth regime comprises the very early regime, when in a nucleation scenario the size of the nucleus is so small that the bulk crystal structure is typically not yet fully developed and the phase is elastically softened. Here we see a dominant effect of curvature effects in analogy to the theory on growth of lenticular melt inclusions. The second growth regime is of specific interest to bainitic steels. During the bainitic reaction subunits form, grow up to a point where the thermodynamic driving force is kinetically overcome by a deformation-induced growth barrier, stop growth and then nucleate new subunits. Thus, the regime prior to the new subunit nucleation corresponds to the limiting case of vanishing growth velocity. For both, analytical and numerical approach, we use sharp interface descriptions of the problem, for the numerical approach we invoke a representation of the problem in terms of boundary integral equations.

We develop a continuum phase-field model for the simulation of diffusion limited solid-solid phase transformations during lithium insertion in LiFePO4-nano-particles. The solid-solid phase boundary between the LiFePO4 (LFP)-phase and the FePO4 (FP)-phase is modeled as a diffuse interface of finite width. The model-description explicitly resolves a single LiFePO4-particle, which is embedded in an elastically soft electrolyte-phase. Furthermore, we explicitly include anisotropic (orthorhombic) and inhomogeneous elastic effects, resulting from the coherency strain, as well as anisotropic (1D) Li-diffusion inside the nano-particle. The effect of the nano-particle's size on the kinetics of FP to LFP phase transformations is investigated by means of both model. Both models predict a substantial increase in the steady state transformation velocity as the particle-size decreases down to dimensions that are comparable with the width of the interface between the FP and the LFP-phase. However, the extra kinetic parameter of the Allen-Cahn-type description may be used to reduce the strength of the velocity-increase with the decreasing particle size. Further, we consider the influence of anisotropic and inhomogeneous elasticity on the lithiation-kinetics within a rectangularly shaped LiFePO4-particle embedded in an elastically soft electrolyte. Finally, the simulation of equilibrium shapes of LiFePO4-particles is discussed. Within a respective feasibility study, we demonstrate that also the simulation of strongly anisotropic particles with aspect ratios up to 1/5 is possible.

We develop a phase-field model for the simulation of chemical diffusion limited microstructure evolution, with a special focus on precipitation growth and ripening in multicomponent alloys. Further, the model accounts for elastic effects, which result from the lattice-misfit between the precipitate particles and the parent matrix phase. To be able to simulate particle growth and ripening in one dimension, we introduce an extra optional driving-force term, which mimics the effect of curved interfaces in one dimension. As a case study, we consider the one-dimensional (1D) \(\gamma \)′-precipitation growth and ripening under the influence of a realistic multistep heat treatment in the multicomponent Ni-based superalloy CMSX-4. The required temperature-dependent thermodynamic and kinetic input parameters are obtained from CALPHAD calculations using the commercial software-package ThermoCalc. The required temperature-dependent elastic parameters are measured in-house at the chair of Metals and Alloys, using resonance ultrasound spectroscopy and high-temperature X-ray defraction. Finally, the model is applied to calculate the equilibrium shape of a single \(\gamma \)′-particle with periodic boundary conditions. Relations to the shapes of \(\gamma \)′-particles in respect of heat-treated experimental microstructures are discussed.