A Computational Framework for Material Design
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Abstract
A computational framework is proposed that enables the integration of experimental and computational data, a variety of user-selected models, and a computer algorithm to direct a design optimization. To demonstrate this framework, a sample design of a ternary Ni-Al-Cr alloy with a high work-to-necking ratio is presented. This design example illustrates how CALPHAD phase-based, composition and temperature-dependent phase equilibria calculations and precipitation models are coupled with models for elastic and plastic deformation to calculate the stress-strain curves. A genetic algorithm then directs the search within a specific set of composition and processing constraints for the ideal composition and processing profile to optimize the mechanical properties. The initial demonstration of the framework provides a potential solution to initiate the material design process in a large space of composition and processing conditions. This framework can also be used in similar material systems or adapted for other material classes.
Keywords
Ni-based superalloy Integrated computational materials engineering Genetic algorithm Material designIntroduction
With the announcement of the Materials Genome Initiative advocating for a 50% reduction in time and cost to develop and deploy new materials, the need to accelerate computational material design approaches has become essential. While there are many different approaches to computational materials design [63, 69, 71, 77], all of the approaches integrate processing, structure, property (PSP) relations to predict a set of desirable material properties for a given application. As noted by Kuehmann and Olson [50], a key challenge of computational materials design is the optimization of several conflicting requirements, which are represented by a variety of processing-structure-property models, to achieve the desired materials performance. To achieve this optimization, the models used must be integrated such that composition, structure, processing variables are tracked as needed. The biggest challenges in this design process are the integration of a variety of models needed and the available experimental and computational data, the flexibility to exchange different models depending on the design application, and the ability to re-assess model parameters for specific alloy systems. There have been a variety of attempts to develop platforms and tools to support materials design for different kinds of material classes and applications. In particular, for Ni-based superalloys, there have been a variety design optimization strategies employed, including using trade-off diagrams [21, 50, 69, 100] and search algorithms [40, 86, 87]. An essential part of all these designs is the control of the precipitation of the γ^{′} strengthening phase and the high temperature properties.
Many strategies are used to design novel alloys for various applications. Reed et al. [69] used thermodynamic equilibria and composition-based models to approximate the PSP relations of a Ni-based superalloy. The alloy composition is selected according to the trade-off diagrams to fulfill the performance requirements. Gheribi et al. [34] combined CALPHAD software, FactSage, with the mesh adaptive direct search algorithm to search the optimum alloy composition and processing conditions for different design objectives. This software provides effective search in a larger parameter space. These two works are computationally efficient but ignored the kinetics of phase transformation during the heat treatment. Saunders et al. [80] integrated the Johnson-Mehl-Avrami equation for γ^{′} precipitation with mechanistic models for yield and creep rupture properties to interpret the PSP relations of Ni-based superalloys. This chain of forward calculations is ideal to optimize the processing conditions, but an extra module should be expected to select the material. Olson used multiple tools to improve the computational material design quality and reduce the number of design iterations [63]. An extra connection between tools and data may create a smooth workflow to initiate a new design project. Another trade-off diagram approach was taken by Crudden et al. [21] which coupled the CALPHAD approach with a data-driven model using an artificial neural network (ANN) to estimate yield strength. Based on the concurrent knowledge and data, this ANN model faithfully interpreted the information in a certain composition domain, but this non-physics based tool may not be applicable in a wider design space. To summarize these design strategies, an integration of phase-based models and data with search algorithm is needed for the next generation of material design.
These design examples demonstrate some of the complexity of materials design and the need to optimize within a framework of conflicting objectives. A variety of efforts have been made to design efficient algorithms, including genetic algorithms (GA) coupled with CALPHAD-based tools [58, 86, 87], atomistic simulations [13, 31, 40], and data-driven approaches [42, 55, 78, 102]. The goal of this work is to develop a platform that integrates federated experimental and computational data repositories with CALPHAD-based tools and mechanistic property models to predict materials behavior and enable materials design using a GA. A model ternary Ni-Al-Cr alloy is chosen to demonstrate this platform.
To optimize this design problem, an efficient search algorithm is needed to obtain the solution from the large composition and processing condition spaces. In the present work, we propose a tool which integrates the simulation of γ^{′} precipitation and mechanical property models within the framework of a genetic algorithm (GA) to design a high work-to-necking (E_{WTN}) Ni_{(1−x−y)}Al_{x}Cr_{y} ternary alloy.
Computational Framework
Infrastructure Framework
After casting, hot working, solution treatment, and coating, the microstructure of a γ/γ^{′} Ni-base superalloy ideally contains a homogeneous, disordered face centered cubic (FCC, γ) phase. During the tempering treatment at processing temperature (T_{p}), the strengthening phase, ordered FCC (γ^{′}), precipitates (Fig. 1). The radius and volume fraction of the γ^{′} particles are the important microstructure features affecting the yield stress [19, 49]. The next four sections will review the models used to predict the precipitation, yield strength, elastic and plastic deformation.
γ^{′} Precipitation
The models for the precipitation kinetics, including classic nucleation, diffusion controlled growth, and coarsening, are well developed [12, 14, 66, 67, 85] and have been successfully applied to simulate the γ^{′} precipitation in Ni-based superalloys [3, 5, 64, 65, 68, 74]. Olson et al. validated a similar precipitation model implemented in PrecipiCalc for 3rd generation disk alloys [64, 65]. The phase equilibria and diffusivity data that are needed for these models are obtained from CALPHAD (calculation of phase diagrams) computations using thermodynamic and diffusion mobility databases [11, 25, 71, 79]. In this work, we implement a module to simulate the γ^{′} precipitation during the tempering treatment. The precipitate size and volume fraction are calculated as a function of the processing time. The yield stress (σ_{ys}) at service temperature (T_{s}) will be estimated from these parameters and the processing time is optimized once the yield stress is maximized.
Nucleation
The alloy composition and model parameters used in kinetic simulation: E_{int} and N_{0} are determined by data fitting; \(\protect |\Delta H^{\gamma - \gamma ^{\prime }}|\) and α_{int} are the calculated values
Experimental conditions | Model parameters | ||||||
---|---|---|---|---|---|---|---|
Sample | Composition | T_{p} K | Ref | E_{int} mJ/m^{2} | N_{0} 1/m^{2} | \(\protect |\Delta H^{\gamma - \gamma ^{\prime }}|\times 10^{4}\) J/mol | α_{int} × 10^{−6} mol/m^{2} |
Kt1 | Ni-7.5Al-8.5Cr | 873 | [6] | 15 | 1.5 × 10^{26} | 1.52 | 0.99 |
Kt2 | Ni-9.8Al-8.3Cr | 1073 | [84] | 24 | 5.0 × 10^{27} | 1.34 | 1.80 |
Kt3 | Ni-6.5Al-9.5Cr | 873 | [7] | 18 | 4.0 × 10^{26} | 1.64 | 1.10 |
Growth
Coarsening
Parameter Assessment for γ^{′} Precipitation Model in Ni-Al-Cr System
In this example design for an Ni-Al-Cr system, the interfacial energy E_{int} and initial number density N_{0} are the two remaining parameters that depend on composition and processing conditions. These two parameters are determined using a regression analysis of available experimental data for number density, mean radius of γ^{′} particles, and volume fraction [6, 7, 84]. For Ni-Al-Cr, there are three available data sets that are listed in Table 1 and the digital data is stored in the MDCS. For each alloy composition, the γ^{′} precipitation is simulated and the predicted number density, mean radius, and volume fraction of γ^{′} as a functions of time are compared to the experimental results. The precipitate radius can be correlated to the number density at each numerical time step by the size distribution function [1, 14].
For the present work, it is assumed that the composition dependence of E_{int} is represented by \(|\Delta H^{\gamma - \gamma ^{\prime }}|\). The initial number density of particles is the other adjustable parameter for the phase transformation modeling and is related to the initial microstructure. In this work, we adopt a constant value of N_{0} = 4.0 × 10^{26} according to the Kt3 sample in Table 1. Ideally a constant value should only be assumed when alloy preparations are the same to ensure similar microstructures.
The two model parameters, E_{int} and N_{0}, are composition and processing history dependent and need to be re-adjusted for new alloy compositions and processing histories. To apply these phase transformation models for material design, i.e., new composition and processing domains, the ability to predict these parameters is needed.
Yield Stress
During tempering, γ^{′} precipitation occurs as described in the “γ^{′} Precipitation” section. To optimize the strength of the alloy, the optimum tempering conditions must be determined. The strengthening contributions from dislocation shearing or bowing around the γ^{′} precipitates are modeled taking the microstructure and chemistry of the alloy into account. For these models, the volume fraction and mean radius of γ^{′} (Vfγ^{′} and \(\bar {R}^{\gamma ^{\prime }}\)) are the needed microstructure parameters [49, 54, 93] and the anti-phase boundary energy (E_{APB}) is an important chemical parameter that determines whether a weak or strong dislocation coupling effect dominates over the dislocations shearing effect [19, 20, 22]. The models for additional strengthening mechanisms, such as solid solution strengthening and Hall-Petch effect, have also been implemented based on the statistical analyses [49, 73]. Using these strengthening models, the processing conditions (processing temperature and time) are chosen to maximize the yield strength of the microstructure [3, 41, 91].
Elastic Deformation
The materials knowledge systems (MKS) package is used to determine the elastic deformation with inputs from the yield stress predictions. MKS is a statistical tool using a response variable of the local material state to estimate the local phase or stress status of the microstructure in an applied thermal or strain field [28, 29, 46]. MKS provides efficient calculations to simulate the elastic deformation of the microstructure. The local response variables are obtained from a regression process using the results from a finite element method. In this work, we use the Python version of MKS (PyMKS) to simulate the elastic deformation of the alloy [97]. Finite element calculations using SfePy software [17] are carried out to generate the reference data for PyMKS. The model training in PyMKS is conducted using simple geometries, which are so-called delta microstructures. SfePy calculates the elastic stress of the delta microstructure and PyMKS assessed the model parameters using two-point statistics. With PyMKS, the model can then be applied to more complicated geometries.
Plastic Deformation
After yield strength prediction and PyMKS modeling to find the elastic limit, the plastic deformation is simulated to complete the stress-strain curve. Different approaches are available to model the plastic deformation. The constitutive models to simulate the plastic deformation, such as the power law [36, 37, 96, 100] and hyperbolic sine law [56, 95, 99], require model parameters that capture the strain hardening behavior of the alloy. After fitting the experimental data, the effects from material chemistry, geometry, and testing conditions are no longer distinguished by the model parameters, and, therefore, the trained models can only be used under specific conditions. Other constitutive models include more physical phenomena and provide generic, high-quality results compared to the experiments [15, 26, 45, 47] and require the assessments of the model parameters. The finite element method (FEM) provides an even higher level of accuracy but is computationally more expensive and is commonly not used within an integrated computational design framework [16, 32, 48, 51, 61, 92, 101]. As shown in Fig. 3, we adopt the constitutive model with irreversible thermodynamics model to simulate the plastic deformation of the alloy.
The test samples for plastic deformation model [98]; σ_{ys} and ε_{ys} are stress and strain at the yield point
T_{p}, K | σ_{ys}, MPa | ε_{ys},% | 𝜃 | ΔG_{ρ}, eV |
---|---|---|---|---|
1123 | 129 | 0.4 | − 40 | 3.06 |
1173 | 89 | 0.3 | ||
1273 | 48 | 0.2 | − 45.3 | 3.31 |
1423 | 27 | 0.1 |
Optimization by Genetic Algorithm
To optimize the design parameters presented in Fig. 1b using the models and data described in the “γ^{′} Precipitation” section to “Plastic Deformation” section, a genetic algorithm is implemented. A GA is used as it enables random but directional iterative optimization of the design parameters [35]. It has been successfully applied in many material research related problems [13, 42, 58, 87]. Unlike the gradient-based or the other grid search algorithms, each GA process requires a significant number of iterations to converge but it is efficient for multi-objective, multi-dimensional optimizations [35, 75].
The data ranges used in GA optimization
X_{Al} | X_{Cr} | T_{p}, K | |
---|---|---|---|
min | 0.03 | 0.00 | 973 |
max | 0.24 | 0.30 | 1473 |
The GA initially randomly selects 12 samples as the 1st generation and evaluates the resulting alloy samples following the computational steps described in Fig. 3. After the evaluation, this Python-based framework outputs the data and meta-data of each sample to a XML file which is then stored in MDCS using REST API. To continue the search process, only the two samples with highest work-to-necking are kept and the others are ignored, which is the so-called elimination. The reproduction operator duplicates these two samples five times to maintain the same number of the samples for each generation. The match operator and the crossover operator exchange the label numbers in the five new pairs of samples to create the next generation. The last applied operator before the next evaluation is a mutation that is designed to create the variance of the samples. More detailed information about the genetic algorithm can be found in the literature [18, 35, 81]. The crossover and mutation rates are minor as the γ + γ^{′} composition space as determined by the phase diagram is relatively small, and this improves the efficiency of the search. In this work, we apply single crossover by random selection of the label sections and the mutation rate is adopted as 1/(total memory size) (1/ 6 × 6 × 6) [18]. The converge rate is defined as 98%: once the binary label number of the top two samples are 98% identical, the generation is claimed as converged and the search is restarted by random selection for the next generation. Repeating these steps, the GA directs the search until the prescribed number of generations is reached. To optimize the three variables in this work, 12 samples in each generation of 30 generations were used for the search.
Results
Model Testing
Predicted input parameters: E_{int}, E_{APB}, and Vfγ^{′} for the yield strength model for the two example alloys and the tempering temperature, T_{p}
Sample | X_{Al} | X_{Cr} | T_{p} | E_{int} | E_{APB} | Vfγ^{′} |
---|---|---|---|---|---|---|
Kelvin | mJ/m^{2} | J/m^{2} | percent | |||
YS1 | 0.136 | 0.175 | 1230 | 25.5 | 0.118 | 44 |
YS2 | 0.141 | 0.102 | 1328 | 27.5 | 0.157 | 26 |
Design Optimization Results
Figure 12 summarizes the predicted yield stress in the input domain while considering the other mechanistic properties (13). After the same processing history of the raw samples, the lattice friction, grain boundary, and dislocation-dislocation hardening stresses are independent from the alloy composition and processing temperature. According to Eq. 14b, the addition of 0.01 to 0.2 mole fraction Al and Cr increases solid solution stress by 22.5 to 100 MPa and 33.7 to 150 MPa, respectively.
The samples selected during the optimization: Opt-H as the highest E_{WTN} and Opt-L as the the lowest one among 214 samples
Sample | X_{Al} | X_{Cr} | T_{p}, K | t, min | E_{APB}, J/m^{2} | σ_{ys}, MPa | E_{WTN}, MPa |
---|---|---|---|---|---|---|---|
Opt-H | 0.071 | 0.184 | 1051 | 8 | 0.141 | 465 | 41.25 |
Opt-M1 | 0.134 | 0.158 | 1238 | 12 | 0.131 | 766 | 25.28 |
Opt-M2 | 0.127 | 0.045 | 1067 | 412 | 0.167 | 762 | 25.44 |
Opt-L | 0.192 | 0.005 | 977 | 13166 | 0.196 | 1117 | 9.62 |
The phase transformation behaviors of Opt-M1 Opt-M2 and Opt-L are very similar: high nucleation rate dominates the precipitation process that results in high number density, small γ^{′} particles and higher yield stress. Comparing Opt-M1 to Opt-M2 shows that different alloy compositions and processing conditions can also lead to similar work-to-necking. Figure 14 shows that after different processing times at different temperatures, the combinations of volume fraction, mean precipitate radius, and anti-phase boundary energy result in similar maximum yield stresses as the peak values of Opt-M1 and Opt-M2 in Fig. 14c. Opt-L possesses highest maximum yield stress among these four alloys because it has the highest volume fraction, highest anti-phase boundary energy and smallest mean precipitate radius.
Discussion of Model Assessment and Parameters
Other composition-dependent parameters are the energy barrier for dislocation annihilation, ΔG_{ρ}, and the temperature dependent factor, 𝜃, which are determined for the conditions given by the base element, strain rate and service temperature [38, 72]. For Ni-based alloys, undergoing the same strain rate testing, these two can be treated as service temperature dependent parameters as recommended in the previous section. The microstructure-dependent parameters are determined by the processing history (thermal and mechanical processes). For example, the initial number density of nucleation sites is highly related to the morphology, grain size and shape. This parameter needs to be adjusted for different processing histories, however, as the alloys in the present work are treated by similar pre-processing steps, it is acceptable to use the same value of N_{0} in all the calculations. The atomic bond ratio, α_{int}, is an indirect but important parameter for extropolating the interfacial energy, E_{int}, into a broader composition domain. α_{int} is determined by E_{int} and \(\Delta H^{\gamma - \gamma ^{\prime }}\) which are obtained by the best fit to the experimental number density and CALPHAD calculation. The α_{int} may need to be re-assessed if different pre-processing steps are used that result in different initial microstructures, dislocation density, etc. or a different thermodynamic databases is used. Since the same thermodynamic database and the same initial conditions are used in the present work, α_{int} needs to be assessed only once. The remaining model parameters, grain diameter, d^{γ}, and initial dislocation density, \(\rho _{0}^{\gamma }\), are assigned as physically reasonable constants, 10 μm and 10^{13} 1/m^{2}, because of the lack of data or models for predicting these quantities. To validate these models with the assessed parameters requires extensive processing-structure-property data from a consistent experimental environment. As mentioned in the previous paragraph, the model refinements are expected to be carried out in the next design iteration while this framework narrows the composition space by decreasing the γ + γ^{′} phase region.
Summary
A modular python-based framework has been developed that integrates the computational models for desired processing-structure-property correlations as an initial step of the material design process. The goal of the present work is to demonstrate that a modeling chain can be developed and implemented with a GA to identify the potential region in a composition space that satisfies the design requirements based on the selected models and data. The present framework is developed to accommodate the modularized codes that are programmed in various languages. The smooth data flow within this framework supports a plug and play feature that allows switching of individual models for different design objectives.
We selected the Ni-Al-Cr ternary system and work-to-necking as the only design target to demonstrate the design process using this framework. In the present work, the computational models are used to identify the chemical composition and processing conditions in a domain with three variables. The training of these models is conducted individually using published results to avoid error propagation through the simulation process. The simulation process is started using pre-selected compositions and processing temperature regime. A GA performs a search for better performance alloys using the results from the integrated models. The reliability of the search results depends on the generality of the models for a wider input domain and the quality of the data used for model training.
The approach presented here can be expanded to multicomponent systems by including the additional elements from databases and re-assessing the model parameters following the “Discussion of Model Assessment and Parameters” section. The objective/utility function for the GA to evaluate the samples could also be revised for different target performance. Based on similar PSP relations, this framework could also be applied to precipitation hardened alloys such as Co-based superalloys or stainless steels. Also, other design objectives such as the castability, alloy density, and grain coarsening, could be considered by adding additional models to the framework. Therefore, a full alloy design project may require a more sophisticated design optimization strategy to complete the PSP relations. As mentioned, this Python environment easily accommodates user implemented modules and the data can be transmitted straightforwardly through the MDCS. This data-enabled design framework can also be applied to other material systems by switching to other user selected PSP relations. Programming-language friendly features and the potential for automatic parameter assessment, support the future expansion of the design framework. The general concept of the present framework can be extended for the design of novel commercial alloys by employing models with predictive capabilities.
Notes
Acknowledgements
S. Li is grateful for the discussions in the material design course, MAT_SCI 390, by Professor Gregory B. Olson at Northwestern University and discussions with William J. Boettinger at NIST.
Endnote
Commercial products are identified in this paper for reference. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.
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