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## Creator

[Toshio Osada](https://orcid.org/0000-0003-1539-9264), Toshiyuki Koyama, [Dmitry S. Bulgarevich](https://orcid.org/0000-0002-7086-8396), [Satoshi Minamoto](https://orcid.org/0000-0003-4023-5800), Makoto Osawa, [Makoto Watanabe](https://orcid.org/0000-0002-5064-9583), [Kyoko Kawagishi](https://orcid.org/0000-0001-7652-9232), [Masahiko Demura](https://orcid.org/0000-0002-7308-3041)

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[Virtual heat treatment for γ-γ′ two-phase Ni-Al alloy on the materials Integration system, MInt](https://mdr.nims.go.jp/datasets/25a16612-575a-42d1-a3fb-17baac17fd1b)

## Fulltext

Virtual heat treatment for γ-γ′ two-phase Ni-Al alloy on the materials Integration system, MIntMaterials & Design 226 (2023) 111631Contents lists available at ScienceDirectMaterials & Designjournal homepage: www.elsevier .com/locate /matdesVirtual heat treatment for c-c0 two-phase Ni-Al alloy on the materialsIntegration system, MInthttps://doi.org/10.1016/j.matdes.2023.1116310264-1275/� 2023 The Authors. Published by Elsevier Ltd.This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).⇑ Corresponding author.E-mail address: OSADA.Toshio@nims.go.jp (T. Osada).Toshio Osada a,⇑, Toshiyuki Koyama b, Dmitry S. Bulgarevich a, Satoshi Minamoto a, Makoto Osawa a,Makoto Watanabe a, Kyoko Kawagishi a, Masahiko Demura aaNational Institute for Materials Science, Tsukuba, JapanbDepartment of Materials Design Innovation Engineering, Nagoya University, Nagoya, Japanh i g h l i g h t s� A novel computational workflowimplemented on the MaterialsIntegration system (MInt) for virtualheat treatment for Ni-Al alloy wasdeveloped.� The workflow implemented in MIntwas constructed by integrating the PFsimulation, image analysis, andmechanical property predictionmodules.� The workflow calculates the effects ofaging process conditions, precipitatessize, volume fraction, and Alconcentration on 0.2 % proof stress.� Virtual heat treatment on the systemenables efficient design of optimalaging treatment conditions.g r a p h i c a l a b s t r a c ta r t i c l e i n f oArticle history:Received 4 November 2022Revised 9 January 2023Accepted 13 January 2023Available online 16 January 2023Keywords:Ni-Al alloyc-c0 two-phasePhase field simulationImage analysisProof stressa b s t r a c tAiming to designing the aging heat treatment conditions to maximize the 0.2 % proof stress of c-c0 two-phase Ni-based superalloys, we develop the automated computational workflow for c-c0 two-phase Ni-Albinary alloy that serves at the system foundation. This consists of phase-field (PF) simulation, image anal-ysis, and mechanical property prediction with the design of input and output data ports. The workflow isimplemented on the Materials Integration system (MInt), which computationally links process, structure,property, and performance. Users may calculate any patterns in heat treatment scheduling for Ni-Alalloys, with various Al contents, by allowing MInt to conduct the workflow. First, MInt conducts multipleparallel runs of the PF simulation to generate statistically sound datasets. Subsequently, MInt extractsstatistics of various microstructure/phase-geometrical/composition attributes by image analysis.Finally, it predicts the proof stress according to the reported superposition of multiple strengtheningmodels. The established computational workflow provides an in-depth understanding of the effect ofaging conditions on alloy strength, which is favorable for optimizing process.� 2023 The Authors. Published by Elsevier Ltd. This is an open access article under the CCBY license (http://creativecommons.org/licenses/by/4.0/).1. IntroductionNi-based superalloys with c-c0 two-phase structure are used ascore parts of high-temperature turbine components in aircrafthttp://crossmark.crossref.org/dialog/?doi=10.1016/j.matdes.2023.111631&domain=pdfhttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/https://doi.org/10.1016/j.matdes.2023.111631http://creativecommons.org/licenses/by/4.0/mailto:OSADA.Toshio@nims.go.jphttps://doi.org/10.1016/j.matdes.2023.111631http://www.sciencedirect.com/science/journal/02641275http://www.elsevier.com/locate/matdesT. Osada, T. Koyama, D.S. Bulgarevich et al. Materials & Design 226 (2023) 111631engines and land-based gas turbines because of their excellenthigh-temperature mechanical properties and oxidation resistance.In the latest Ni-based superalloys, the required performance forhigh-temperature strength properties is satisfied by multiplestrengthening mechanisms (activated by more than ten alloyingelements) [1] and a complex hierarchical microstructure [2,3]. Inaddition, their microstructural features are adjusted by casting, bil-leting, forging, and heat treatment at high temperatures, and themicrostructure is fixed using final aging heat treatment. Optimiz-ing the conditions in these multiple processes is typically a trial-and-error process requiring considerable workforce, a high budget,and years of development. Therefore, an accurate and systematicunderstanding of the process-structure–property-performance(PSPP) linkage is vital for component design, particularly in aircraftengine components requiring a high degree of reliability.Numerous reports have presented the efforts carried out todevelop PSPP prediction systems. Pioneering works include thoseof iron and steel materials reported by Bhadeshia et al. [4] andthose of Ni-based superalloys for high-temperature applicationsreported by Harada et al. [1,5] and Reed. [2,6] In particular, theTMS alloy designed with the program [1] developed by Haradaet al. has been used as the latest aircraft engine turbine compo-nent; this demonstrates that a computational prediction systemcan help design new alloys. More recently, several computationalmaterials design techniques, in integrated computational materialsengineering, have been proposed to support the improvement ofcomponent performance by predicting the distribution of mechan-ical properties and microstructural features in engine components.For example, the prediction system proposed by Olson et al. [7] inMaterials Genome Initiative in the U.S. and the method proposedby C. Rae et al. [8] in the U.K. indicate the possibility of perfor-mance prediction from process and microstructure information.Furthermore, the commercial software, JMatPro, [9,10] can predictphysical andmechanical properties from the microstructure of var-ious alloys with a certain degree of accuracy. As a result, it iswidely used in manufacturing, particularly in casting and forgingcompanies.Materials Integration (MI) [11–15] is a concept used to acceler-ate the research and development of materials by computationallylinking the PSPP using any prediction model, such as numericalsimulation, theoretical or empirical theory, and data-driven regres-sion. Based on this concept, a versatile system for materials design[16] was investigated by our group. This system was named MInt,denoted by Materials Integration with Network Technology [11].Computational materials design tools, particularly the efficientPSPP prediction system, have attracted considerable attention inrecent years in developing advanced components. In addition tohigh-temperature strength properties, predicting multiple otherproperties, such as fatigue and creep properties [14], by optimizingthe process conditions and microstructural features is important.This study aims to demonstrate the usefulness of MI for devel-oping the c-c0 two-phase Ni-Al binary alloy. We developed a com-putational workflow that can predict the high-temperaturestrength from the microstructure evolution via the heat treatmentprocess. The workflow consisted of three modules, including thephase-field (PF) simulation module, image analysis module, andmechanical property prediction module. These modules wereimplemented and consistently connected to form a workflow inMInt, allowing automatic computation of 0.2 % proof stress atany high temperature for any heat treatment scheduling. Thevalidity and basic performance of the constructed workflow inNi-Al binary alloys were examined, which is a step prior to apply-ing the workflow for practical superalloys. Based on the analysis ofthe computed virtual data set for several isothermal aging condi-tions, a process map with a wide range of process conditions was2constructed, demonstrating the practicality and usefulness of theworkflow to effectively design heat treatment scheduling.2. Workflow to high-temperature strength from heat treatmentscheduling2.1. Workflow designIn this study, we focused on the aging heat treatment thatdetermines the final performance in Ni-Al alloys with a c-c0 two-phase structure and developed a workflow to predict 0.2 % proofstress at high temperatures from the heat treatment scheduling.Fig. 1 presents a screenshot of the constructed workflow takenfrom the graphical user interface (GUI) for the workflow designin MInt.The initial field generation module was prepared to generate aninitial field for the PF simulation. Each of the modules receives nec-essary information through input ports and outputs the computedresults through output ports. In Fig. 1, input ports, modules, andoutput ports are shown in blue, yellow, and gray, respectively.First, the alloy compositions and aging conditions were fed fromthe input ports to the PF simulation module, which uses the initialfield provided by the initial field generation module. Next, thecomputed microstructure output from the PF simulation modulewas fed through the input port to the image analysis module.The image analysis module extracts the statistics of the composi-tion and microstructure, such as the mean Al concentrations inthe c and c0 phases, the average size, and mole fraction of the c0phase. The output and input ports are concealed in the linebetween the two modules (Fig. 1). To obtain reliable statistics,duplicate computations were conducted in parallel, starting fromdifferent initial fields. Here, 30 individual computations were con-ducted, the number of which can be specified through the inputports of the initial field generation module. Finally, the mechanicalproperty prediction module receives the composition andmicrostructure information from the image analysis module andoutputs 0.2 % proof stress at an arbitrary test temperature, whichcan also be specified through the input port of this module. Asmentioned in Section 2.4, 0.2 % proof stress was computed accord-ing to the theoretical and empirical equations [3,9,17,18] based onthe following inputs: test conditions (strength test temperature),composition and microstructure information (alloy composition,c composition, size, and volume fraction of c0 precipitations), andmaterial properties (Poisson’s ratio, Burgers vector, Taylor factor,anti-phase boundary energy, and melting point).Another feature of this workflow is its flexibility to adjust to thepurpose of the user. Making changes in the phase diagram, proper-ties used in the PF simulation module, and physical properties usedin the mechanical property prediction module can be carried outwith relative ease. This demonstrates that the workflow can widenthe prediction applicable range from the binary system to a morecomplicated one for practical Ni-based superalloys.The obtained composition-process-structure–property data setsare systematically stored in MInt for analyzing inverse problems,where one may optimizes the composition and/or process fromthe desired property. This study focuses on the forward problemanalysis; the inverse analysis using this workflow will be reportedin the future.2.2. Phase-field simulation moduleThe c0 precipitation and the microstructure developments in Ni-based alloys were simulated based on the PF model [19–24]. The PFmodel proposed by Vaithyanathan and Chen [23] was employed asFig. 1. Screenshot of the constructed workflow design for high-temperature mechanical properties prediction taken from graphical user interface in Materials Integration bynetwork technology (MInt) system.T. Osada, T. Koyama, D.S. Bulgarevich et al. Materials & Design 226 (2023) 111631a base model in this study, and the chemical free energy part wasslightly modified to introduce the Kim–Kim–Suzuki (KKS) model[25], as mentioned below. In this section, the outline of the simu-lation method is explained using the phase decomposition in Ni-Albinary alloy system as an example.The order parameters, which describe the microstructurechanges, are the Al solute composition field cðr; tÞ and the long-range order parameter fields si r; tð Þ; i ¼ 1;2;3ð Þ represent the L12ordering of c0 phase, where the subscript i is utilized to distinguishthe four variants in the L12 ordered structure [23]. These orderparameters are the functions of local position r ¼ ðx; y; zÞ and timet. The governing equations to calculate the microstructure changesare given as equations (1) and (2):@cðr; tÞ@t¼ Mcr2 dGsysdcðr; tÞ� �ð1Þ@siðr; tÞ@t¼ �MsdGsysdsi r; tð Þ ; i ¼ 1;2;3ð Þ; ð2Þwhere Mc and Ms are the mobility of atom diffusion and the relax-ation coefficient of the long-range order parameter field, respec-tively. Moreover, the total free energy of the microstructure Gsys isdefined by Gsys � Gc þ Egrad þ Estr in terms of the chemical freeenergy Gc, gradient energy Egard, and elastic strain energy Estr. Theseenergies are evaluated using the following equations.Gc ¼ZrfGccðf m; TÞf1� hð/Þg þ Gc0c ðf p; TÞhð/Þ þW12gð/Þgdr; ð3ÞEgrad ¼Zrf12jsðjrs1j2 þ jrs2j2 þ jrs3j2Þgdr; ð4ÞEstr ¼ 12Rr Cijklfecijðr; tÞ � e0ijðr; tÞgfecklðr; tÞ � e0klðr; tÞgdr¼ 12Rk BðnÞQðk; tÞQð�k; tÞ dk2pð Þ3 ;ð5ÞGccðf Þ ¼12W1f2; Gc0c ðf Þ ¼12W2 1� fð Þ2 ð6Þf ðr; tÞ � cðr; tÞ � c0mðTÞc0pðTÞ � c0mðTÞð7Þhð/Þ � /3ð10� 15/þ 6/2Þ; gð/Þ � /2ð1� /Þ2 ð8Þ/3ðr; tÞ � s1ðr; tÞs2ðr; tÞs3ðr; tÞ ð9Þ3e0ij r; tð Þ � dijg0 c r; tð Þ � c0f g ¼ dije0c0p Tð Þ � c0m Tð Þ c r; tð Þ � c0f g; ð10ÞB nð Þ¼ C11þ2C12ð Þe20� 3�ðC11þ2C12Þ 1þ2nðn21n22þn22n23þn23n21Þþ3n2n21n22n23h iC11þnðC11þC12Þðn21n22þn22n23þn23n21Þþn2ðC11þ2C12þC44Þn21n22n23h i2435;ð11Þwhere f ðr; tÞ is a local volume fraction field of the c0 phase; c0mðTÞ andc0pðTÞ are the equilibriumcompositions of the c and c0 phases, respec-tively, given as functions of temperature T; and Gccðf ; TÞ and Gc0c ðf ; TÞare the Gibbs energies of the c and c0 phases, respectively. Herein,the Gibbs energy functions were approximated to quadratic formwith respect to f ðr; tÞ, where the coefficients W1, W2, and W12 weredetermined using the Gibbs energy curve calculated from the ther-modynamic Gibbs energy parameters in the calculation of the phasediagram (CALPHAD) method [26]. The interpolation function hð/Þand energy barrier function gð/Þ are the standard functions, whichare typically employed in the multi-phase field method [21]. Thephase-field order parameter /ðr; tÞ, which calculates the probabilityof finding a c0 phase at position r and time t in the microstructure, isdefined as a function of siðr; tÞ. Note that the above definition/3 � s1s2s3 provides the four variants of the L12 ordered structurebecause the condition/3 � s1s2s3 ¼ 1 corresponds to four cases [23]:ðs1; s2; s3Þ ¼ ð1;1;1Þðs1; s2; s3Þ ¼ ð1;�1;�1Þðs1; s2; s3Þ ¼ ð�1;1;�1Þðs1; s2; s3Þ ¼ ð�1;�1;1Þ ð12ÞIn addition, any two si; ði ¼ 1;2;3Þ are guaranteed to be zero atthe antiphase boundary. Here, js is the gradient energy coefficientwith respect to the long-range order parameter field siðr; tÞ. Thecomposition gradient energy term was not considered becausethe Kim–Kim–Suzuki (KKS) model [25] was used to calculate thelocal chemical potential. In the evaluation of elastic strain energy,the phase-field micro-elasticity theory was used [22–24], whereecij and e0ij are the total strain and the eigen strain, respectively.Herein, Qðk; tÞ is the Fourier transform of cðr; tÞ � c0, where c0, k,and n are the average solute composition, Fourier vector, and theunit vector along the k direction, respectively. Cijkl is the elastic stiff-ness which was assumed to be constant in this calculation, that is,T. Osada, T. Koyama, D.S. Bulgarevich et al. Materials & Design 226 (2023) 111631the elastically homogeneous case was considered for simplicity.According to the Vegard’s law [22], the eigen strain e0ij is propor-tional to the local composition; g0 and dij are the lattice mismatch;and the Kronecker delta: dij ¼ 1 for i ¼ j and dij ¼ 0 for i–j. Here, g0is defined as g0 � e0=fc0pðTÞ � c0mðTÞg, and e0 is the eigen strainbetween the cmatrix and c0 precipitate, each of which has the equi-librium composition [27].The temporal evolution of the microstructure change was sim-ulated based on the conventional deference method, and dimen-sionless formalism was applied. Following the assumption of thedimensionless mobility terms, Mc ¼ 1 and Ms ¼ 1, the diffusionconstant DðTÞ of the solute Al atoms [28] was utilized to convertthe dimensionless aging time to real-time. The details of thenumerical calculation are available in reported works [19,24],and the materials parameter values utilized in this study are sum-marized in Table 1. Note that we assumed the gradient energycoefficient to be constant over the calculation temperature range,since the influence of the temperature dependence part in gradientenergy coefficient on the c0 evolution is neglectable. The initialmicrostructure was prepared by using the initial field generationmodule (Fig. 1), which is the short-term phase-field simulationfrom a supersaturated solid solution with a small composition fluc-tuation, generated by the random number computationally. Theoutput simulation results are the order parameter fields, cðr; tÞand siðr; tÞ; ði ¼ 1;2;3Þ, and the field data was saved in the visual-ization toolkit (VTK) file format [29].In the PF simulation module, because the basic unit of the heattreatment was set as isothermal aging followed by continuouscooling (or heating), any heat treatment process can be repre-sented by combining units. To improve the simulation speed, thetemperature dependence of the equilibrium compositions, in thec0 and c0 phases, was calculated in advance and approximated byusing the expansion formula (Table 1). As isothermal aging is con-sidered in the current work, the detailed explanation about han-dling the complex heat treatment was omitted in this paper.2.3. Image analysis moduleThe outputted VTK files from the PF simulation module wereused for extracting microstructure geometrical and phase composi-tion attributes with the image analysis module. The module con-verts the VTK datasets with siðr; tÞ and Al at. % to thecorresponding image stacks. The VTK data were collected and ana-lyzed at the end of the heat treatment schedule. If needed, the timedomain data could be pulled from PF simulations. siðr; tÞ, the c0-phase volume fraction f V, as well as various geometrical attributesfor each c0-object (including area, perimeter, Feret diameters, circu-Table 1Numerical values used for the phase-field simulation.Coefficients in chemical free energy* (J �mol�1)Gradient energy coefficient [28], js ðJ �m2 �mol�1)Elastic constants [29] (GPa)Eigen strain between c matrix and c0 precipitate [27], e0Diffusion coefficient of Al atoms [29], DðTÞ ðm2 � s�1ÞEquilibrium compositions (atomic fraction of Al) of c and c0 phases*:c0m Tð Þ ¼ 1:077535� 10�2 þ 1:554659� 10�4T � 7:521288� 10�8T2 þ 3:366849� 1c0p Tð Þ ¼ 2:118598� 10�1 þ 1:548849� 10�4T� 1:836197� 10�7T2 þ 4:357137� 10�11T3 þ 1:052733� 10�14T4* Numerically approximated by using the available Gibbs energy [25].4larity, solidity, aspect ratio, and their corresponding statisticalmeans)were extracted from the image stack. Consequently, the usercan judge based on their standard deviations and, if necessary,increase the number of simulated areas in repeated PF simulations.From the image stackwith Al at. % distribution, themean Al concen-trations of cmAl in c, cpAl in c0, and cAl in all phases were extracted. Sub-sequently, they were passed, together with the mean diameter of acircle with an equal projection area for c0-objects (d ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiArea�=pq),to the mechanical properties prediction module. Here, we shouldnote that area fraction f A obtained from 2D PF simulations is equalto volume fraction f V, since 2D PF simulation result was calculatedby using the phase diagram information (c0mðTÞ and c0pðTÞ) as shownin Table 1.2.4. Mechanical properties prediction moduleThe mechanical property prediction module is based onreported theoretical or empirical models for Ni-based superalloyswith c-c0 two-phase structure [3,8,17,18]. The outline of the calcu-lation method for 0.2 % tensile proof stress, r0.2, as well as thesolid-solution and precipitation strengthening in the Ni-Al binarysingle crystal alloy system is explained as follows.Herein, a strengthening model described by the superpositionof all the strengthening factors [17] was utilized. Additionally, solidsolution Drss and precipitation Drc0 strengthening were consid-ered as follows.r0:2 ¼ rNi þ Drss þ Drc0 ¼ M sNi þ Dsss þ Dsc0� �; ð13Þwhere sNi is the strength of pure nickel single crystal and M is thereciprocal number of the Schmid factor of alloy,M = 2.449 (for poly-crystalline FCC alloy, M is the Taylor factor; M = 3.1). In this study,the reported value for the strength of pure Ni was used [2]. Thedetails of the superposition of each strengthening model used canbe found in the references [3,17,18].The conventional theory for solid-solution strengthening (SSS)in the alloy has been proposed by Fleisher [30] and Labusch [31].A model of SSS for multi-component [32], such as superalloy [33]and high-entropy alloy [34], has been proposed. The criticalthreshold stress of SSS for binary alloy, known as Labusch limit,is found to be [31,35,36]:Dsss ¼ F4mwe4Gb9 !13c23; ð14Þwhere b is the magnitude of Burgers vector of the edge dislocationin the c matrix; G is the shear modulus; c is the atomic concentra-W1 ¼ 2:236� 103;W2 ¼ 3:106� 103;W12 ¼ 1:379� 1031:826� 10�15C11 ¼ 250:8;C12 ¼ 150:0;C44 ¼ 123:50.006DðTÞ ¼ 5:0� 10�4 exp � 2:60�105RT� �,R: Gas constant =J � K�1 �mol�1,T: Temperature =K0�11T3;Fig. 2. Schematic of precipitation/dislocation interaction models: (a) weakly-coupled dislocations shear model, (b) strongly-coupled dislocations shear model,(c) Orowan bypassing model.T. Osada, T. Koyama, D.S. Bulgarevich et al. Materials & Design 226 (2023) 111631tion of solute atom; we ( 2b to 3b) [37] is the range of interactionbetween obstacles and dislocation; and Fm is the maximum interac-tion force between a dislocation and an obstacle. Fm is approxi-mately equal to Gb2sinhc, where hc is a critical angle of dislocationinteracted with the obstacle. Thus, DsSS strongly depends on theshear modulus G of the alloy system. For the Ni-Al binary systemin this study, the value of F4mwe4Gb9� �13= 5.745 (MPa/atm2/3) was usedfrom the experimental values [36].Precipitation strengthening Dsc0 can be described as follows.Dsc0 ¼ DsROMc0 þ DsPc0 ; ð15Þwhere DsROMc0 and DsPc0 are contributions by the rule of mixture(ROM) and precipitation/dislocation interaction, respectively [19].Based on the typical mixture theory, the strength incrementscaused by the embedded secondary precipitates within the matrixcan be described by the isoworks criteria [18,38]. The value liesbetween the Voigt and Reuss models described by isostress criteria,which show the upper and lower bounds, respectively, as follows.DsVoigtc0 ¼ ðsc0 � sNi � DsssÞf V ;andDsReussc0 ¼ sc0sc0 1� f Vð Þ þ sNi þ Dsssð Þf V� 1  sNi þ Dsssð Þ;where sc0 is the critical resolved shear stress (CRSS) of c0 singlephase. In this study, we assumed that the strength incrementscaused by ROM could be estimated as the average of the upperand lower bounds in order to simplify the discussion. Thus, the ruleof a mixture can be simply described using the following equation:DsROMc0 ¼ DsVoigtc0 þ DsReussc0� �=2 ð17ÞMeanwhile, the CRSS of Ni3Al suggests an inverse temperaturedependence because the Keare-Wilsdorf (KW) locking controlsthe plastic deformation. In this study, we used an average CRSSon the (111)[�101] slip system in stoichiometric Ni3Al (Ni-25at. % Al) that was reported in reference [39].As illustrated in Fig. 2, the precipitation strengthening DsPc0 iscategorized primarily into three models, namely the weakly cou-pled pair-dislocation shear model DsPw (Fig. 2a), strongly coupledpair-dislocation shear model DsPs (Fig. 2b), and Orowan bypassingmodel DsPor (Fig. 2c). The favor precipitation model in the systemis selected depending on the volume fraction and size of c0. Numer-ous researchers have attempted to model the strength incrementby the precipitation/dislocation interaction. In this study, the pro-posed precipitation models [8,18] were used, and the one exhibit-ing the minimum value was selected as the favor model:Dspc0 ¼ Min½Dspw;s;DsPor�Dspw;s ¼2cAPBbrL2cAPBrð Þ12Gb2ð Þ12þ 2cAPBrð Þ12( ); if r< xmðWeaklyÞ2cAPBb1L2cAPBrð Þ12Gb2ð Þ12þ 2cAPBrð Þ12( )r2� r� 2cAPBGb22p� �� �2  12; if r� xmðStronglyÞ8>>>>>><>>>>>>:DsPor ¼3Gb2ðL� 2rÞ ðOrowan bypassingÞ ð18Þwhere cAPB is the anti-phase boundary (APB) energy of c0; xm is thecritical distance between leading and trailing dislocations (as illus-trated in Fig. 2b) regarding to the transition size defined as:xm ¼ 2cAPBGb22p� �; and L is the mean particle spacing, defined as:5L ¼ 2p3f v� �12 d2ð19ÞBy substituting equations (14)–(19) into equation (13), the 0.2 %proof stress can be estimated. In this study, high temperature 0.2 %proof stress was predicted by only considering the temperaturedependence of shear modulus G and assuming b and hc to be con-stant at elevated temperatures. Moreover, cAPB was assumed as atemperature-independent fitting parameter. Thus, cAPB = 0.10 J/m2 was used at all temperatures in this study. The reported exper-imental value of the APB energy of Ni3Al (cAPB) was approximately0.17 J/m2 for stoichiometric Ni3Al (cpAl = 0.25) [40]. The value of cAPBtends to decrease as the Al concentration decreases [41]. Therefore,in this study, cAPB was selected to be slightly lower than thereported value. In addition, incorporating the APB prediction intothe mechanical property prediction module could increase theaccuracy of prediction [42]; illustrates the importance of the APBenergy prediction and the future perspective.3. Experimental conditions3.1. Virtual experimental conditionsThe Ni-Al binary alloy phase diagrams including to c0mðTÞ andc0pðTÞ used in the calculations are shown in Fig. 3a. As shown thisfigure, PF simulation module can be applicable the Al compositionbelow 0.25. Further, the module can be applicable the temperaturebelow the solidus line or eutectic temperature also shown inFig. 3a. Ni-19.11 at. % Al was used in this study. Here we note thatall the temperature and Al composition ranges described in thisstudy are within the stable formation ranges of c-c0 two-phase sys-tems in the binary Al-Ni system. The two-dimensional (2D) calcu-Fig. 3. (a) Phase-diagram of Ni-Al binary alloy used for phase-field calculation. (b)Input test conditions for aging heat treatment. (c) SEM image of microstructure ofNi-19.11 at. % Al aged at 870 �C for 20 h [44].T. Osada, T. Koyama, D.S. Bulgarevich et al. Materials & Design 226 (2023) 111631lation area of PF was set to 2 lm � 2 lm, and N = 30 calculationswere performed for each aging heat treatment condition, changingthe initial microstructure. The image analysis module was appliedto the individual microstructure, and the average of the parametersobtained from the module was adopted as the output results. Theparameters used in this process are listed in Table 1. The initialTable 2Initial microstructural features in phase-field simulation of Ni-19.11 at. % Al.Alloy composition Precipitate sizedoAl concentrationin c0 , cpAlnm at. frac.Ni-19.11Al 20.15 0.22006microstructure was set using the initial field generation module.To represent the rapidly cooled and supersaturated microstructure,the initial state was defined as the microstructure isothermallyheat-treated at 1000 �C for 32 s. The characteristics of the obtainedinitial microstructures are summarized in Table 2. For these initialfields, as shown in Fig. 3b, calculations were performed using thePF simulation module (see Section 2.1) under 210 isothermal agingheat treatment conditions in the temperature range of 600–1000 �C and aging time ranging from 3.98107 � 10-3 to3.98107 � 105 s. Here, as the calculation area of the PF model usedwas 2 lm � 2 lm, the aging time was adjusted such that the pre-cipitate size would be approximately 200 nm, corresponding to1/10 of the model size under each temperature condition in con-sideration of calculation accuracy. For example, the maximumaging time was set to be approximately 10.5 min at 870 �C. Notethat the time is shorter than the experimental conditions(870 �C/20 h) [43], as mentioned in the later section (Fig. 5a).Using the image analysis module (see Section 2.2), the precipi-tate size d, volume fraction fV, Al concentration cmAl in c, and Al con-centration cpAl in c0 (which are important for the discussion onmicrostructure and property variations in Ni-Al binary alloys) wereoutputted for subsequent property prediction and discussion.3.2. Physical experimental conditionsTo confirm the accuracy of strength prediction, single-crystalmodel alloys [18] capable of evaluating precipitation strengthand SSS were fabricated. The aging and high-temperature mechan-ical properties of these alloys were evaluated. The model alloy wasNi-19.11 at. % Al (9.8 wt%) with c0 precipitation equivalent to thatof a general Ni-based single crystal superalloy for blade applica-tions, which was cast using a directional solidification furnace.Casted samples were subject to fast cooling (approximately300 �C/min) with Ar gas after homogenization heat treatment at1300 �C/5h to sufficiently homogenize microsegregation. Thehomogenized samples were subject to isothermal aging heat treat-ment at 870 �C for 20 h [43]. Moreover, c single-phase single crys-tal model alloy, Ni-10.27 at. % Al (5.0 wt% Al), was prepared toevaluate the SSS. Here, the amount of Al in this alloy was designedto be less than 14.05 at. %, which is the equilibrium c compositionof Ni-19.11 at. % Al at 870 �C to suppress the fine cooling c0 precip-itations after the homogenization heat treatment.The aged sample was machined into cylindrical specimens of9 mm � 15 mm (U � L) and subject to compression tests at tem-peratures ranging from room temperature to 1100 �C. The strainrate for the compression test was 10-5 s�1. The precipitatedmicrostructure of the aged specimen (870 �C/20 h) was observedusing field emission-scanning electron microscopy (FE-SEM; ZEISS,Gemini SEM 300). As shown in Fig. 3c, the c0 precipitates werecuboidal in shape, with a volume fraction of 56.3 ± 6.1 % and pre-cipitate size of 875 nm [44]. Here, c0 area fraction fA analyzed fromthe SEM images was 68.2 ± 4.7 %. However, in case of cuboidal pre-cipitates, fA is not equal to fV. Thus, we calculated fV from fA byusing the relation fV = fA3/2, assuming that the precipitates shapeis perfect cube shape as reported in [44].Al concentration in c cmAl Volume fraction of c0 , fVoat. frac. –0.1727 0.3916T. Osada, T. Koyama, D.S. Bulgarevich et al. Materials & Design 226 (2023) 1116314. Results4.1. Microstructural evolutionsAl concentration contour plots (Fig. 4) show the time evolutionof the microstructure at three temperatures (1000, 870, and600 �C) obtained by the PF simulation module. The coarsening pro-cess of the c0 phase was present at all temperatures. For the shortaging time, the influence of the initial field and the small size of thec0 phase resulted in the formation of irregularly shaped precipi-tates with a random Al concentration field. Subsequently, withincreasing time, the c0 precipitates changed to a cuboidal shapelike the experimental microstructure in Fig. 3c. This is becausethe lattice mismatch g0 � e0=fc0pðTÞ � c0mðTÞg used in this studyhas a relatively large positive value [27]. Thus, we concluded thatthe simulation results were in good agreement with the experi-mental data of actual Ni-Al alloys [27]. Furthermore, for long termaging, a series of cuboidal c0 shapes were observed in the h100i di-rection (top, bottom, left, and right directions in the figure). Thiswas because of the influence of elastic interactions between adja-cent c0 precipitates [45] and the replacement of the anti-phaseboundary (APB) of the c0 phase by a thin c phase [22], whichdemonstrated that the PF module reproduces these effects withgood accuracy.Fig. 5a � 5d show the aging temperature and time dependenceof precipitate size d, volume fraction fV, Al concentration cmAl in c,and Al concentration cpAl in c0, respectively. The data of the linesin Fig. 5 was estimated from equations (23) and (28), (29), (30),which are discussed in Section 5.2 and obtained via the image anal-ysis module.As shown in Fig. 5a, the precipitate size d increased withincreasing aging temperature and aging time. On the log–log plot,Fig. 4. Phase-field simulation results of microstructural7all the results for long-term aging were linear, confirming that Ost-wald ripening was exhibited according to the model presented inSection 3.2. The experimental results were located on the fittingline shown at 870 �C (which are discussed in Section 5.2); thisimplied that the model and the diffusion coefficient values of Alused in the calculations are highly valid.By contrast, the volume fraction rapidly increased with increas-ing time in the early aging stage, reaching a value close to the equi-librium volume fraction at each temperature in the late agingstage. The equilibrium volume fraction at 870 �C was f eqV = 0.553,which is in good agreement with the experimental value. There-fore, the selected equilibrium phase diagram for the Ni-Al binarysystem [25] was considered a suitable candidate.Note that the important feature of computer experiments usingthe PF method is the ability to easily output time dependence on Alconcentration cmAl and cpAl in the c and c0 phases, respectively (Fig. 5cand 5d). Three-dimensional atom probe field ion microscopy(3DAP) is a common method for measuring elemental concentra-tions in nm-order sized precipitates and matrix. However, researchon changes in element concentrations over time during aging islimited because an enormous amount of time and labor is requiredto analyze elemental concentrations under 210 heat treatmentconditions similar to the situation in this study.As shown in Fig. 5c, the Al concentration in the matrix phasesignificantly decreased with time in the early aging stage fromthe initial concentration of cmAl = 0.1727 and slowly decreasedtoward the equilibrium Al concentration at each temperature inthe late aging stage. Furthermore, the Al concentration in the c0phase rapidly increased with time in the early aging stage andslowly increased toward the equilibrium Al concentration at eachtemperature in the late aging stage, compared with the initial Alconcentration of cpAl = 0.2200.evolution in Ni-Al binary alloy (Ni-19.11 at. % Al).Fig. 5. Output results of phase-field simulation and image analysis module: (a) size of c0 , (b) volume fraction of c0 , (c) concentration of Al in c, and (d) concentration of Al in c0during aging heat-treatment in Ni-Al binary alloy (Ni-19.11 at. % Al).T. Osada, T. Koyama, D.S. Bulgarevich et al. Materials & Design 226 (2023) 1116314.2. High-temperature strengthThe output values from the PF simulation and image analysismodules can be passed to the strength prediction module to pre-dict the 0.2 % proof stress of the alloy at any given test tempera-ture. Fig. 6a shows a comparison of the 0.2 % proof stressexperimentally obtained for various model alloys and the pre-dicted 0.2 % proof stress via the mechanical property predictionmodule. The 0.2 % proof stress of the Ni-19.11 at. % Al model alloywith c-c0 two-phase structure (Fig. 3c) exhibited a clear inversetemperature dependence owing to the Ni3Al with L12 structureexhibiting a significant inverse temperature dependence [39].As mentioned in Section 2.4, the 0.2 % proof stress of the Ni-19.11 at. % Al alloy can be explained by a superposition of SSS,ROM strengthening, and precipitation strengthening. The 0.2 %proof stress predicted by inputting fV = 56.3 %, d = 875 nm, andequilibrium Al concentration in c at 870 �C, c0m= 0.1405 was in goodagreement with experimental values. Moreover, the 0.2 % proofstress of the Ni-10.27 at. % Al model alloy with c single-phasestructure decreased slightly with increasing test temperature.Additionally, the value was higher than that of pure Ni, thus imply-ing clear SSS by 10.27 at. % Al addition. Furthermore, the predictedvalues via the SSS model, as mentioned in Section 2.4, were in goodagreement with the experimental results. Therefore, the mechani-cal property prediction module was found to be useful for predict-ing the high-temperature strength of Ni-Al binary alloys.8Fig. 6b shows the time dependence of the predicted 0.2 %proof stress tested at 650 �C for the Ni-19.11 at. % Al alloy agedat 870 �C. Furthermore, the figure shows the contributions ofsolid-solution, ROM, and precipitation/dislocation interaction.The predicted 0.2 % proof stress gradually increased to the peakvalue tp = 6.309 � 10-1 s and then gradually decreased withtime. Evidently, this strength model adequately represents theover aging phenomenon of typical precipitation strengthenedalloys. Here, the appearance of the peak value can be primarilyexplained by the strongly-coupled pair dislocation model(Fig. 2b). Note that the peak value appeared slightly later thanthe transition time t* = 5.792 � 10-2 s for the weekly-coupledand strongly-coupled pair dislocation models (equation (18)).Moreover, the contribution of SSS (equation (14)) slightlydecreased during this period owing to a gradual decrease in cmAl(Fig. 5c). In addition, the module represents that the contributionof ROM strengthening (equation (17)) gradually increases as fVincreases (Fig. 5b).Furthermore, as shown in Fig. 6c, the peak value increased asthe aging temperature decreased owing to the significant increasein fV with decreasing temperature. The time tp of the peak valuebecame longer as the aging temperature decreased owing to thesignificant decrease in Al diffusion rate and the accompanying c0growth rate with a decrease in aging temperature. Note that thelines in Fig. 6b and 6c denote 0.2 % proof stress estimated by input-ting microstructural features, such as fV, d, cmAl, and cpAl, which areFig. 6. Output results of mechanical property module: (a) modelling of temperaturedependence of 0.2 % proof stress, (b) influence of aging time on 0.2 % proof stress at650 �C, and (c) 0.2 % proof stress at 650 �C for several aging temperatures.Fig. 7. (a) Temperature dependence on n value and (b) n value normalized using D(T)�t.T. Osada, T. Koyama, D.S. Bulgarevich et al. Materials & Design 226 (2023) 1116319calculated from equations (23), (28), (29), and (30), respectively(discussed in Section 5.2).5. DiscussionThe present workflow, which integrates the PF simulation,image analysis, and mechanical property prediction modules, pro-vides an in-depth understanding of the effects of aging tempera-ture, aging time, precipitate size d, volume fraction fV and Alconcentration cmAl in c on the 0.2 % proof stress at elevated temper-atures. In addition, the results obtained from the computer exper-iments are useful as virtual test results because obtaining a largeamount of composition-structure–property data sets, including cand c0 compositions, would require a large amount of time andFig. 8. Output results of phase-field simulation and image analysis module normalized using D(T)�t; (a) size of c0 , (b) volume fraction of c0 , (c) concentration of Al in c, and (d)concentration of Al in c0 during aging heat-treatment in Ni-Al binary alloy (Ni-19.11 at. % Al).T. Osada, T. Koyama, D.S. Bulgarevich et al. Materials & Design 226 (2023) 111631effort in actual experiments. Here, the estimated composition-structure–property data sets with classical theories of microstructureevolution were compared and discussed.5.1. Analysis of n-values for Ostwald ripeningThe Lifshitz–Slyozov–Wagner (LSW) theory [46,47] is com-monly known as a precipitation coarsening theory and is expressedin general form as:dn � dno ¼ Kt; ð20Þwhere K is the coarsening rate constant and is expressed in the LSWtheory as:K ¼ 64rV2mCe9RTD Tð Þ ¼ 64rV2mCe9RTDoexp�EART� �; ð21Þwhere r is the interfacial energy density; Vm is the molar volume ofsecond phase particles; Ce is the solvus limit of solute atoms, equiv-alent to cm0; R is the gas constant; and T is the aging temperature. Inthis study, Do = 5.0 � 10-4 m2s�1 and EA = 2.60 � 105 Jmol�1�K�110were used (see Table 1). The n value is known to be 2 in the earlystage of growth controlled by the interfacial reaction [48] and 3 inthe later stage controlled by diffusion [49].Furthermore, we assumed that K is constant at a constant tem-perature and only the time variation of the exponent n was consid-ered. To discuss the time dependence of n(t) at each agingtemperature T from the d estimated in this study, the coarseningrate equation was defined as dnðtiÞi � dnðtiÞo ¼ Kti at discrete time ti(i � 1) in the numerical calculation, and the following relationshipwas established at the small-time interval between ti and ti+1.dnðtiþ1Þi � dnðtiþ1ÞodnðtiÞi � dnðtiÞo¼ Ktiþ1Kti¼ tiþ1ti; ð22Þwhere K was assumed constant (constant temperature). The valueof nðtiÞ ffi nðtiþ1Þ in the small-interval (ti to ti+1) and values of nðtiÞin each interval can be calculated from equation (22).Fig. 7a shows the relationship between the n-values obtained atT and t. Although the n value contains some errors owing to thefinite discretization of time, the n value generally increased from3.5 to 4.5. In the Ostwald ripening theory, the n value is knownFig. 9. Contour map showing dependence of aging temperature and time on (a) concentration of Al in g, (b) volume fraction of c0 , (c) size of c0 , strength increment by (d) solidsolution strengthening, (e) precipitation strengthening, and (f) 0.2 % proof stress at 650 �C in Ni-Al binary alloy (Ni-19.11 at. % Al).T. Osada, T. Koyama, D.S. Bulgarevich et al. Materials & Design 226 (2023) 111631as 3. However, previous studies have experimentally demonstratedthat the n value becomes higher when elastic interactions actbetween precipitates, resulting in a delay in coarsening [50,51].As clearly shown in the microstructure in Fig. 4, the c0microstructure in this calculation results from coarsening behaviorunder strongly elastically constrained conditions; thus, the n valueis approximately 4. Further, Fig. 7 indicates that the present PF11module accurately considers the coherent strain field of the c0precipitates.5.2. Numerical analysis by using normalized timeIn this study, r and Vm were treated as temperature-independent constants in the coarsening rate constant K in equa-T. Osada, T. Koyama, D.S. Bulgarevich et al. Materials & Design 226 (2023) 111631tion (21). In the Ni-Al binary phase diagram (Fig. 3a), CeT could beapproximated to a constant in the temperature range covered inthis study. Therefore, the coefficient of D(T) in equation (21) couldbe approximated to a constant. This result indicates that the timedependence of various parameters on temperature can be scaledby summarizing the time t in equation (20) as D Tð Þ � t. As shownin Fig. 7b, variations in the n-value with temperature can be nor-malized by replacing the time on the horizontal axis of Fig. 7a withD Tð Þ � t.Furthermore, the normalized results using D Tð Þ � t are shown inFig. 8a � 8d for all output results shown in Fig. 5a � 5d, respec-tively. Here, these figures show the relationship between D Tð Þ � tand precipitate size d; the ratio of c0 volume fraction fV to the equi-librium volume fraction f eqV ; the ratio of the Al supersaturation in c,SAl, to the c0 equilibrium volume fraction f eqV ; and the ratio of the Nisupersaturation in c0, SNi, to the c equilibrium volume fraction f eqV;c.Here, fV, feqV , f eqV;c, SAl, and SNi are expressed as follows from the phasediagram in Fig. 3a.f V t; Tð Þ ¼ cm � cAlm t; Tð ÞcAlp t; Tð Þ � cAlm t; Tð Þ ; ð23Þf eqV ¼ cm � c0m Tð Þc0p Tð Þ � c0m Tð Þ ; ð24Þf eqV;c ¼c0p Tð Þ � cmc0p Tð Þ � c0m Tð Þ ; ð25ÞSAl t; Tð Þ ¼ cAlm t; Tð Þ � c0m Tð Þc0p Tð Þ � c0m Tð Þ ; ð26ÞSNiðt; TÞ ¼c0p Tð Þ � cAlp t; Tð Þc0p Tð Þ � c0m Tð Þ ð27ÞNote that all plots can be organized as a single curve inFig. 8a � 8d.Furthermore, as shown in Fig. 8a, the time dependence on d foreach aging condition can be approximated using the followingequation by setting n = 4, which is the average value in Fig. 7b. Evi-dently, the results are almost consistent with the calculationresults.d4 � d4o ¼ 7:59247� 10�15DðTÞ � t ð28ÞAdditionally, the values estimated at each aging temperaturefrom equation (28) are shown again in Fig. 5a. This illustrates thatequation (28) can be confirmed to reproduce the precipitate size inFig. 5a with high accuracy.In Fig. 8c and 8d, the curve regressions of the obtained calcula-tion results exhibit the following relationships.SAlf eqV¼ cAlm t; Tð Þ � c0m Tð Þcm � c0m Tð Þ ¼ 9:710368� 10�4 D Tð Þ � tf g�0:1585386; ð29ÞSNif eqV ;c¼ c0p Tð Þ � cAlp t; Tð Þc0p Tð Þ � cm¼ 2:677226� 10�4 D Tð Þ � tf g�0:1792772; ð30Þwhere c0m Tð Þ and c0p Tð Þ were determined from the phase diagramsummarized in Table 1; cm= 0.1911 for Ni-19.11 at. % Al; and cmAland cpAl at arbitrary aging temperature and time for this alloy canbe estimated using equations (29) and (30), respectively. Here, theR2-values for fitted lines, as shown in equations (29) and (30), were0.9868 and 0.9816, respectively. Furthermore, as shown in Fig. 8b,f V t; Tð Þ at arbitrary aging temperature and time could be estimated12by substituting the estimated cmAl and cpAl into equation (23). Again,fV, cmAl, and cpAl values estimated at each aging temperature fromequations (23), (29), and (30) are shown in Fig. 5b – 5d, respectively.Evidently, the data analysis could reproduce the fV, cmAl, and cpAl inFig. 5b � 5d with high accuracy.5.3. Composition-process-microstructure-strength mappingAs described above, d, fV, cmAl, and cpAl at arbitrary aging temper-ature and time can be easily calculated by the numerical analysisformula of the PF simulation results using classical coarsening the-ory. Furthermore, by inputting these results into the model equa-tion in Section 2.4, Drss, Drpc0 , and r0:2 can be estimated at anarbitrary aging temperature and time.Fig. 9a � 9f show the contour plots of cmAl, fV, d, Drss, Drpc0 , andr0:2 at 15,851 calculation points in the range of aging temperature(200–1400 �C) and aging time [ranging from 10-3 s to 1010 s(3.17 years)], respectively. The numerical analysis of PF simulationresults conducted over a limited range and number of conditionsenabled the creation of a wide range of high-resolution contourmaps.As shown in Fig. 9a, cmAl decreased with decreasing temperatureand time and exponentially increased when it reached the equilib-rium value. As shown in Fig. 9b, fv increased with decreasing tem-perature. Evidently, aging for more than t = 1010 s was required. Forexample, to reach the equilibrium values of c0m = 0.1115andf eqV = 0.609 at 600 �C. By contrast, as shown in Fig. 9c, the pre-cipitate size increased with increasing aging temperature and time.For example, at 400 �C, the precipitates grew only up tod = 126.1 nm even after heat treatment for t = 1010 s.Further, as shown in Fig. 9d, the SSS Drss was estimated by thecmAl value; thus, its contour map had a similar shape as Fig. 9a. Giventhat only Al solid solution strengthening was acting on the alloy,the maximum value of Drss was 116.6 MPa, which is smaller thanDrpc0 . Moreover, the minimum value was 67.90 MPa, whichappeared at 430 �C for 1010 s.By contrast, precipitation enhancement Drpc0 was estimated byfV and d, resulting in a contour map with distinct peak values.The maximum value of Drpc0 was 486.9 MPa, which appeared underthe conditions of T = 340 �C and t = 1010 s.The contour map of r0:2, estimated by the superposition of thesestrengthening mechanisms, was similar to that of Drpc0 , implyingthat the r0:2 of the c-c0 two-phase alloy had a more significant con-tribution from precipitation strengthening than SSS. The maximumvalue of r0:2 was 764.7 MPa, which appeared at T = 340 �C andt = 1010 s, is similar to Drpc0 . However, such a long aging time isimpractical. For example, if 105 s (27.77 h) is set as a realistic agingtime, the workflow can estimate that isothermal aging at 520 �Ccan realize the optimum combination of microstructural features(fV = 58.42 %, d = 41.69 nm, and cmAl = 12.70 at. %), resulting in themaximum proof stress value of 769.0 MPa at 650 �C.6. ConclusionsIn this study, we focused on aging heat treatment, an importantprocess that ultimately determines the properties of Ni-basedalloys with a c-c0 two-phase structure. Additionally, we estab-lished the computational workflow to predict mechanical proper-ties from the heat treatment scheduling through themicrostructure evaluation in Ni-Al binary alloys, which are a modelexample of Ni-based alloys. The workflow implemented in MIntwas constructed by integrating the PF simulation, image analysis,and mechanical property prediction modules. The analysis of theT. Osada, T. Koyama, D.S. Bulgarevich et al. Materials & Design 226 (2023) 111631data computed with the workflow provides an in-depth under-standing of the effects of aging temperature, aging time, precipitatesize d, volume fraction fV, and Al concentration cmAl in c on the 0.2 %proof stress at elevated temperatures. Furthermore, the implemen-tation of numerical analysis formulae on computer experimentresults using classical coarsening theory easily realized the estima-tion of d, fv, cmAl, and cpAl over a wide range of aging temperaturesand times. Therefore, the optimal aging process conditions andcombination of multiple microstructural features that maximizemechanical properties can be discussed.The most important feature of MInt is that it enables flexibleand seamless integration with various modules. The present work-flow provides a virtual test method for the aging process, which is acritical step in determining the high-temperature strength of thisclass of materials. Thus, placing the workflow after the modulesmay be helpful for forming processes modules, such as casting,forging, and 3D additive manufacturing [52]. In addition, with thisworkflow, a wide range of alloy systems can be accommodated byimplementing commercially available or user-created thermody-namic equilibrium phase diagram information. In the future, thisworkflow can be expected to connect with various types ofmachine learning algorithms, such as Monte Carlo tree search, toautomatically search for the optimal cost/performance solutionin terms of heat treatment scheduling and composition [53,54].Data availabilityData will be made available on request.Declaration of Competing InterestThe authors declare that they have no known competing finan-cial interests or personal relationships that could have appearedto influence the work reported in this paper.AcknowledgementsThis work was supported by the Council for Science, Technol-ogy, and Innovation (CSTI), Cross-ministerial Strategic InnovationPromotion Program (SIP), ‘‘Materials Integration for revolutionarydesign system of structural materials” (Funding agency: JST). Wewould also like to thank Mr. Takuma Kohata and Mr. Yuji Takatafor extending their support in sample preparation and microstruc-tural observation, and Mr. Yusuke Manaka for assisting with theimplementation of the workflow on MInt. We would like to thankEditage (www.editage.com) for English language editing.References[1] H. Harada, H. Murakami, Design of Ni-base superalloys, in: T. Saito (Ed.),Computational Materials Design, Springer-Verlag, Berlin, 1999, pp. 39–70,https://doi.org/10.1007/978-3-662-03923-6_2.[2] R.C. Reed, The Superalloys: Fundamentals and Applications, CambridgeUniversity Press, New York, 2006. https://doi.org/10.1017/CBO9780511541285.[3] T. Osada, Y. Gu, N. Nagashima, Y. Yuan, T. Yokokawa, H. Harada, Optimummicrostructure combination for maximizing tensile strength in apolycrystalline superalloy with a two-phase structure, Acta Mater. 61 (2013)1820–1829, https://doi.org/10.1016/j.actamat.2012.12.004.[4] H.K.D.H. 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