CALPHAD as a Toolbox to Facilitate the Development of HEAs

Abstract

This chapter would be a comprehensive review of the successful application of the CALPHAD method in the research and development of high-entropy alloys (HEAs) in recent years. The structure of this chapter is obedient to the conventional framework in materials science, and several topics would be covered, including composition and structure design, heat treatment and solidification processing, and some properties and performances, such as mechanical and electrochemical corrosion behaviors and high-temperature oxidation characteristics. A conclusion and outlook of the CALPHAD method are also provided at the end of this chapter. Hope this chapter could offer a thermodynamic and kinetic perspective for facilitating the field of HEA research and inspire other researchers to contribute more valuable ideas and works.

1. Introduction

The concept of high-entropy alloys (HEAs), where the configurational entropy plays a significant role, is closely related to the content of thermodynamics. As a phenomenological methodology of computational thermodynamics, CALPHAD could be the most direct tool for the design of HEAs [1]. CALPHAD stands for the abbreviation of CALculation of PHAse Diagram, which is a computational approach proposed in the early 1970s to bridge the gap between the theoretical calculations and the experimental phase equilibrium information [2]. The Gibbs energy function occupies the core of the CALPHAD method. Thermodynamic properties of the materials could be described by the Gibbs energy of each phase. The phase diagrams are constructed by minimizing the total Gibbs energy of the system. The model parameters of Gibbs energy, which is the main constitute of the CALPHAD databases, are assessed by fitting the input data via CALPHAD-based software, such as Thermo-Calc [3], Pandat [4], and FactSage [5]. The experimental results would be the first choice for the input, and the first-principle calculations would be a good complement wherever there is a lack of experimental input [6]. The Gibbs energy of an element can be written as follows [7]:

$$G_i^{\mathrm{\theta }}-H{}_i{}^{\mathit{SER}}=a+\mathit{bT}+\mathit{cT}ln\left(T\right)+{\mathit{dT}}^2+{\mathit{dT}}^{-1}+{\mathit{\text{fT}}}^3+\dots$$E1

where $H_i^{\mathit{SER}}$ represents the enthalpy of the pure element i in its reference state standard element reference (SER). This reference state is defined as the most stable state of the pure element at 298.15 K and 1 bar. Coefficients a, b, c, d, … are the parameters to be optimized.

To deal with the case containing more elements, several models were proposed to describe the atomic interactions, such as compound energy formalism (CEF) [8], cluster variation method [9], quasi-chemical model [10], and so on. In this chapter, the compound energy formalism (CEF) as the most widely used model was introduced. In the CEF, the structure of a phase is considered to be constituted by a set of sublattices represented by the formula (A, C)x(B, D)y, with A, C mixing on the first sublattice and B, D occupying the second one [6].

$$G_m^{\alpha }={}^{\mathit{Srf}}{G}_m+T{}^{\mathit{cfg}}{S}_m+{}^{\mathit{Phy}}{G}_m+{}^{\mathit{Ex}}\begin{array}{c}{G}_m\\ \end{array}$$

$$=\sum P_{I_0}\left(Y\right){G_{\mathit{end}}}^{°}+\mathit{RT}\sum \sum n^s{y}_j^{s}ln\left(y_j^s\right)+{}^{\mathit{Phy}}{G}_m+\sum P_{I_1}\left(Y\right)L_{I_1}+\sum P_{I_2}\left(Y\right)L_{I_2}\dots$$E2

${}^{\mathit{Srf}}{G}_m$ is named “surface of reference,” which is a linear combination of the Gibbs energy of end-members. ${}^{\mathit{cfg}}{S}_m$ denotes the configurational entropy of the phase, considering the ideal mixing only. The sum of ${}^{\mathit{Srf}}{G}_m$ and $T{}^{\mathit{cfg}}{S}_m$ is the so-called ideal solution model. ${}^{\mathit{Phy}}{G}_m$ describes the physical effect (usually the magnetic effect), which was treated separately. ${}^{\mathit{Ex}}{G}_m$ represents the excess Gibbs energy describing the interactions between the constituents in different sublattices. $P_{I_0}\left(Y\right)$ refers to the product of site fraction in $I_0$, which means an array of zeroth order including only one constituent in each sublattice. $P_{I_n}\left(Y\right)$ corresponds to the nth-order array, which contains n extra constituents in sublattices 1, 2, …, n – 1. $n^s$ is the number of sites in sublattice s and $y_j^s$ is the site fraction of element j in sublattices. $L$ denotes the interaction parameter, which is a Redlich-Kister polynomial.

One of the aims of the CALPHAD method is to develop the reliable databases for the users of interest. The descriptions of stable binaries and ternaries are the building blocks for the multicomponent CALPHAD databases. The CALPHAD method uses extrapolation to extend the range from these stable constituent lower systems to the unstable region and higher order systems. The semiempirical nature and the unique structure of the database (Figure 1) make CALPHAD an accurate and self-consistent technique and gain the popularity in the research of structural materials [11]. Therefore, the CALPHAD has been considered as an important part of the Integrated Computational Materials Engineering (ICME) and Materials Genome Initiative (MGI) [12].

Figure 1.

The structure of the CALPHAD thermodynamic databases.

With the combination of thermodynamic and kinetic databases, the CALPHAD method could not only depict the characteristics of thermodynamics (phase diagrams and thermophysical properties) but also simulate the kinetic process (precipitation and diffusion) by solving the diffusion equations numerically. The following sections will show how the CALPHAD method serves as a powerful toolbox to facilitate the development of the HEAs.

2. Thermodynamic calculations in the composition and phase design of HEAs

The emergence of the HEA provides a new alloying strategy for the materials scientists. Different from the conventional way of adding a handful of alloying elements to a primary one, the HEA creates a mixture of the multiple principle elements with (near) equiatomic concentration [13, 14]. The most intuitive representation of this strategy would be the composition map of the phase diagram (Figure 2). The conventional alloys locate at the corner regions (blue region), and the HEAs occupy the central region of the phase diagram. Therefore, the name of “central alloys” is perhaps a good choice to eliminate the ambiguity caused by the confusing use of “high-entropy alloys,” “medium-entropy alloys,” “complex concentrated alloys (CCAs),” “multiprincipal element alloys (MPEAs),” etc. Its vastness of the composition space and the complex combination of elements open the possibilities of discovering novel alloys with exceptional properties [15]. Meanwhile, it is a more challenging task of the HEA design to find the desired point in this composition sea. As the sole technique of visualizing the phase diagram directly, the CALPHAD method that is expert in the composition and phase design for the multicomponent systems could serve as an equipped ship for the HEA researchers. Hundreds of published works have applied the CALPHAD method in the phase prediction of HEAs [16, 17]. In this section, the advantage and limitations of the CALPHAD method will be discussed with several examples.

Figure 2.

The contour map of entropy of mixing on an illustrated ternary alloy. The blue regions represent the conventional alloys with only one principal element and the bright area indicates the high-entropy region [13].

2.1 Application of CALPHAD databases in the HEA design

Phase diagrams have been considered as the road map for the research of materials science, which act as a guiding role to design the experiments and understand the behavior of the materials. As mentioned above, the accuracy of the calculated phase diagram depends on the reliability of the CALPHAD databases. Owing to the lack of specific HEA databases, the early works usually uses the Ni-base and, sometimes, the Fe-base databases to explore the alloy composition of the desired phase. Choi et al. [18] used the TCFE2000 database [19] to design a new face-centered cubic (FCC) phase HEA with nonequiatomic composition. Figure 3 shows that the prepared samples fall well on the FCC single-phase region, which verifies the validity of the CALPHAD calculation. Zhang et al. [20] employed the Thermo-Calc software [3] with database TCNI8 [19] to calculate the phase diagram of the CoCrFeNi-based HEAs. In Figure 4, the pseudobinary phase diagram of CrCoFeNi-Cux was calculated to show the alloying effect of Cu on the phase stability of the HEA. Miscibility gaps of FCC and liquid phase begin to form with the increasing Cu content. This calculated result has been experimentally confirmed by the work of Wu et al. [21]. Butler et al. [22] studied the phase stability of AlNiCoCrFe alloy using the CALPHAD approach. Figure 5d shows their prediction on the phase fractions of the Al15 HEA calculated by using the TCNi8 database. They found that the predicted phase stability was generally in agreement with the experimental measurements except the FCC phase. These findings indicate that the CALPHAD could be a powerful tool for the design of HEAs, but the quality of the databases needs to be improved. Therefore, the specific thermodynamic databases, TCHEA [23] and PanHEA [24], have been tailored in recent years to accelerate the development of HEAs. Feng et al. [25] extensively investigated the phase stability of the lightweight HEAs using both the experimental and computational methods. They calculated the phase fraction of each in Al1.5CrFeMnTi alloy (Figure 6a) and the isopleth of Al1.5CrFeMnTix (Figure 6b) using Pandat software with the PanHEA database. A reasonable agreement has been reported between the experimentally determined and predicted compositions. One of the disagreements is that the nanosized L21 phase was experimentally observed at higher annealing temperature than the predicted one. MacDonald et al. [26] thoroughly discussed the FCC phase decomposition of the equiatomic CoCuFeMnNi alloy with the aid of the CALPHAD method. They performed the calculation on the equilibrium step diagram of CoCuFeMnNi alloy at the temperature range from 400 to 1600°C employing the Thermo-Calc software with database TCHEA3 and compared the experimentally observed compositions of three phases with the CALPHAD-predicted ones at 500°C. It can be seen in Figure 7 that the composition predicted by CALPHAD reasonably agrees with the measured ones except the case in the Cu-rich FCC phase. Although the CALPHAD has been successfully applied to the phase and composition design of HEAs, its power has not been fully explored due to the imperfection of the databases. The difficulties of developing a high-quality HEA database could be attributed to several reasons. As we noted earlier, the traditional alloy databases were developed based on the binaries and ternaries with only one principal component, while the HEAs focus on the central regions of the alloy systems. Therefore, one of the difficulties is that there is lack of experimental data on this region as the input for the CALPHAD assessment [27]. Another thing that should be noted is that the ternary interaction could be significant at the highly concentrated region, so all the ternaries should be carefully evaluated [28]. Those barriers will burden the workload of assessment exponentially and make it almost a “mission impossible” to develop a perfect HEA database in a short term [27].

Figure 3.

Calculated FCC phase region of 10Co-15Cr-10 V-65 (Fe, Mn, Ni) [18].

Figure 4.

Pseudobinary phase diagram of CoCrFeNiCux [20].

Figure 5.

BSE image of Al15 HEA heated treating at (a) 700°C and (b) 1050°C, (c) XRD spectra of Al 15 HEA, and (d) calculated phase fractions for the Al 15 HEA [22].

Figure 6.

Calculated phase diagram of the (a) alloy Al1.5CrFeMnTi and (b) isopleth of Al1.5CrFeMnTix [25].

Figure 7.

(a) Calculated equilibrium step diagram of the CoCuFeMnNi alloy from 400–1600°C compared with experimentally measured phase composition from STEM EDS, (b) Fe-Co rich B2 phase, (c) Cu-rich FCC phase, and (d) the matrix FCC phase [26].

2.2 High-throughput CALPHAD (HT-CALPHAD) calculations for the HEA design

Alternatively, the high-throughput CALPHAD method and the machine learning (ML) models provide us a promising way of accelerating the design of advanced HEAs. Feng et al. [29] utilized the CALPHAD-based high-throughput calculation to screen the optimal composition of the lightweight HEAs. The vast composition space has been narrowed to a small range by meeting the criteria on phase fraction and temperature, with eight candidate alloys surviving out finally (Figure 8). In Figure 8b–e, region 1 meets only one criterion, region 2 meets two of criteria, and region 3 meets three. This approach significantly reduces the cost of experimental work. Zeng et al. [30] proposed several phase selection rules by combining the machine learning method and the CALPHAD calculations. The flowchart of this chapter is summarized in Figure 9. More than 300,000 of entries of phase equilibrium information were generated by the CALPHAD method. Then, the XGBoost method was employed to explore five most important features to depict the composition space spanned by the generated data. Based on the trained ML model, five-phase selection rules were established, which provides an efficient approach of designing a single-phase HEA. The CALPHAD method has accumulated tons of high-quality phase equilibria and thermochemical data in decades, which would be a valuable resource for the machine learning models. At the same time, ML could help CALPHAD extend its application and establish the structure–property connection quantitatively. A recent review of the HT-CALPHAD method could be found in [31].

Figure 8.

(a) Flowchart of the HT-CALPHAD, (b) Al-Cr projection, (c) Al-Fe projection, (d) Al-Mn projection, and (e) Al-Ti projection [29].

Figure 9.

The road map for the discovery of phase selection rules [30].

Besides, thermodynamic analysis on the phase stability of single solid solution (SSS) phases in HEA has also been extensively studied by using the CALPHAD method [14, 32]. Unexpectedly, the configurational entropy does not play the dominant role in many cases [1]. Other entropic contribution and enthalpy should also be valued in the analysis because the phase stability of the SSS depends on the Gibbs energy (G = H – TS) of its competing intermetallic phases [14, 15]. A widely accepted framework of this analysis is needed for further research.

3. CALPHAD-based prediction in processing optimization

3.1 Heat treatment processing optimization

Known as the idiom “the refined steel, softly winds fingers,” the condensed words were refined as the most significant information about heat treatment in the present perspective. For metallic materials, heat treatment represents some thermal as well as thermochemical processes to bring a series of complicated transformations in physical metallurgy, mechanical, and corrosion resistance performances. Therefore, to understand heat treatment, it is necessary to study the thermal phenomena, microstructure, phase stability/transformation, properties, and so on. As a kind of promising material, HEAs like other ferrous and nonferrous alloys, heat treatment is also an important process for improving formability, machinability, strength, and ductility.

As the name implies, the CALPHAD method is a technique of calculation for phase diagram initially. Therefore, in the field of the phase diagram, the CALPHAD method is inherently associated with a unique advantage in comparison with other techniques. One of the most important advantages of the CALPHAD method is to predict high-order thermodynamic information via extrapolation from the corresponding low-order subsystems. To realize the CALPHAD method, it is required to combine both computational platforms and some thermodynamic and mobility databases, and so on. Some commercial software is available to perform various calculation and simulation, including Thermo-Calc software (also add-on diffusion and precipitation module, Dictra & TC-Prisma) [3, 19], Pandat [33], and FactSage [34]. Some commercial databases also need to be matched in these software to provide various thermodynamic and mobility information, including TCHEA5 and MOBHEA2 [35], PanHEA_TH, PanHEA_MB, and PanHEA_MV [36]. The continuous update databases, which contained dozens of elements and hundreds of subsystems, provide a set of self-consistence thermodynamic and kinetic parameters to assist a high-quality prediction. With the development of thermodynamic and kinetic modeling, the CALPHAD method has played a more and more important role in materials design, processing optimization, phase transformation, mechanical and corrosion behavior investigations, and so on.

In this section, some CALPHAD-based research on the heat treatment of HEAs was collected to aim at highlighting the significant role of the CALPHAD method in the thermodynamic, kinetic, and precipitation predictions, which are the three main parts of this section.

3.1.1 Thermodynamic analysis of heat treatment

The phase diagram is one of the most important components in the field of materials science and technology, obviously in the field of heat treatment. Thus, the CALPHAD method has an inherent advantage due to the technique that was initially originated to couple the phase diagrams and thermochemistry, especially due to the extrapolation characteristic. For “multiprincipal element alloy (MPEA)” [37, 38] or “complex concentrated alloy (CCA)” [39] systems, the extrapolation characteristic from the known low-order subsystems to the unknown high-order HEAs system becomes an outstanding superiority of the CALPHAD method, particularly in comparison with the conventional one or two principal metallic materials. To complete a high-quality thermodynamic prediction needs an important prerequisite, i.e., a set of critical, reliable, and self-consistence thermodynamic modeling, especially in the whole composition and at a wide temperature range, due to the equiatomic or near-equiatomic composition region of the HEA system.

The application of the CALPHAD-based thermodynamic analysis in heat treatment will be introduced as follows.

The equilibrium mapping and stepping calculations of the phase diagram, viz., calculated isothermal/isoplethal sections and calculated equilibrium phase fraction at various temperatures, are two major applications in materials design and also in heat treatment optimization. The basic investigation approach is to study from the subsystems (i.e., binary and ternary) to the multicomponent HEA systems; some examples for determining the temperature of annealing heat treatment for a proposed alloy are shown in (Figures 10 and 11) [27, 40, 41].

Figure 10.

The calculated Co-Cr phase diagram (a1) and the phase fractions at various temperatures for Co-15Cr (a1–1) and Co-15Cr (a1–2) [40].

Figure 11.

Calculated phase diagram of the Co-Cr-Ni system. The isoplethal section along with the Cr-Cr0.5Ni0.5 pseudobinary section (a); and the isothermal section at 1200°C (b) [27].

For casting, a low liquidus temperature and a narrow window between liquidus temperature (T_liquidus) and solidus temperature (T_solidus) are desirable [42]. However, for high-temperature alloys (i.e., superalloys) and high-temperature HEAs (also known as high-entropy superalloys (HESA) [1]), a higher T_liquidus and T_solidus and also a narrow window between the two temperatures are beneficial. For homogenization heat treatment, a wide temperature range of the one-phase region at high temperature (i.e., high-temperature solid solution phase) is preferable (Figure 10). Meanwhile, for precipitation heat treatment, a wider window in-between T_solidus and the solvus temperature of the precipitation phase (T_solvus) is favorable. Therefore, in certain or designed compositions for an HEA system, these typical temperatures of heat treatment can be predicted by using the CALPHAD-based thermodynamic calculations, which are very important for facilitating various heat treatment processing optimization.

3.1.2 Kinetic analysis of heat treatment

The CALPHAD-based thermodynamic calculation is usually performed as a significant starting point for materials design, processing optimization, and so on. Sequentially, kinetic simulation tends to become an important link.

To perform kinetic simulation, it is necessary to combine a simulation module, which has been implanted in some specific kinetic models (e.g., 1D homogenization models [43, 44, 45]), and various thermodynamic and kinetic databases (Section 3.1). The simulation of diffusion is usually both time- and space-dependent and is particularly suitable for solving one-phase problems and moving boundary problems. Kinetic simulation for assisting to optimize processing parameters of homogenization heat treatment is an important application.

For homogenization heat treatment, the kinetic simulation is a valuable method to predict and optimize the specific processing parameter, especially for the relationship between temperature, time, and distance in some given phases. Meanwhile, various heat treatments can be optimized by CALPHAD-based simulations, for instance, to facilitate appropriate heat treatment procedures for manipulating solid solution strengthening and precipitation hardening of HEAs [46], and so on.

3.2 Solidification and precipitation simulations

The Scheil-Gulliver solidification simulation [47, 48], also called Scheil solidification simulation, is usually performed on the basis of the CALPHAD method. For analyzing solidification processes, some equations were used by assuming that the diffusivity in the solid phases is extremely slow to be treated as zero and the diffusion in the liquid phase is very fast. With the assumption, a type of nonequilibrium transformation can be treated as a local equilibrium state. An example of the Scheil solidification simulation can be found in [40] (Figure 12).

Figure 12.

Solidification curves and homogenization heat treatment of M5 alloy (thermodynamic and kinetic database: TCHEA2 and MOBNI2) [40] (This figure was not displayed in this reference).

Various modules and packages mentioned in Section 3.1, such as TC-Prisma and Pandat, have been developed especially to predict the precipitation problems. Taking TC-Prisma as an example; for predicting the concurrent nucleation, growth, and coarsening of dispersed precipitate phases, the module built by the Langer-Schwartz theory [49] also adopted the Kampmann-Wagner numerical method [50] and many models [51]. To simulate multimodal particle size distribution of precipitate phase is a typical application.

As shown in Figure 13, to investigate the precipitation kinetics and to predict the temperature–time-transformation (TTT) diagram for high-entropy superalloys with a typical structure of γ matrix phase (i.e., disordered structure A1) and γ’ coherent precipitate phase (i.e., ordered structure L1₂), various simulations can be performed by TC-Prisma with an available software development kits (SDKs) of TC-Python language combined with many thermodynamic and kinetic databases [52].

Figure 13.

Experimental temperature profile for the alloys conducted by homogenization and solid solution heat treatment at 1500 K for 20 h and then air cooling to room temperature (a); based on the experimental temperature profile, simulated particle size distribution and cubic factor of the γ’ precipitate phase for the alloys by continuous cooling from the solvus temperature (b); and the simulated TTT diagram for the γ’ precipitate phase from the γ matrix phase (c and d) (The details can be found in [52]).

4. Theoretical understanding of materials properties and performances

4.1 Mechanical properties

Until now in this chapter, the application of the CALPHAD method in composition and structure design and the processing optimization has been briefly described in the previous sections. On the basis of the classical framework and the conventional research route in materials science, only left the last section here, i.e., the CALPHAD-based calculation and simulations for facilitating the theoretical understanding of materials properties and performances. The materials science computational modeling is a key crosslink point to connect the other three points, including composition and structure, processing, and properties and performances. In this section, the CALPHAD-based prediction in understanding mechanical properties, electrochemical corrosion, and high-temperature oxidation performances is described as follows.

As described previously, CALPHAD calculations and simulations are a coupling technique for experimental and theoretical phase diagrams, thermodynamic and kinetic information, and so on. Therefore, the thermodynamic and kinetic databases are compiled by a set of expressions of Gibbs energy and atomic mobility on the basis of a number of composition−/temperature- and time−/distance-dependent functions (also contain other influence factors, e.g., pressure, volume, magnetic, etc.). In this framework, predicting mechanical properties is probably not the main direction of the CALPHAD method. However, the characteristic of the thermodynamic and kinetic database also provides some fundamental data and information to establish various mechanical databases to predict the corresponding mechanical performances. Take CALPHAD-based prediction for the Young’s moduli of the Ti-Nb-Zr-Ta/Mo system as an example [53, 54, 55, 56, 57, 58, 59]. Based on Young’s moduli data through experimental investigation and/or theoretical calculations, the composition-dependent Young’s moduli database can be built also from low-order systems to high-order systems like other thermodynamic and kinetic databases. In Figure 14, the predicted data are in good agreement with the experimental results for Young’s moduli of the body-centered cubic (BCC) Ti-Nb-Zr-Mo system. It shows that the CALPHAD-based prediction for Young’s moduli on the basis of the accurate database is a reliable method.

Figure 14.

The diagram of Young’s moduli of the BCC Ti-Nb-Zr-Mo system. The data (red line) [53] were predicted from Young’s modulus database in comparison to the experimental results [53, 54] (blue and green points).

4.2 Electrochemical corrosion performance

From the corrosion mechanism perspective, electrochemical corrosion and high-temperature oxidation are the most important two types of corrosion. Therefore, both of the main kinds of corrosion predicted by the CALPHAD-based calculations and simulations will be introduced in the last two sections.

The electrochemical corrosion is usually treated as wet corrosion, i.e., the metal undergoes some reactions in various specific aqueous solutions. Therefore, understanding the electrochemical corrosion mechanism in thermodynamics and kinetics are two key points. The CALPHAD-based prediction is a powerful technique to study the thermodynamic theory of electrochemical corrosion; a typical representative application is the calculation of the Pourbaix diagram.

The Pourbaix diagram, i.e., potential-pH diagrams, derived by M. Pourbaix, collected thermodynamic information for the relevant electrochemical and chemical reactions. With the development of the experimental and theoretical thermodynamic data for the Pourbaix diagram, the calculation of the Pourbaix diagram can be applied not only in corrosion science but also in hydrometallurgy and electrodeposition processing. In the field of corrosion science, the potential-pH domain represents two significant physical and chemical processes. That is, potential is an important characteristic parameter to reflect the metal either kept in an immunity state or reacted to various specific oxidation or ionic states; pH is a significant typical parameter to mirror the activities of H⁺ or OH⁻ in corrosive media or environment. The two parameters usually represent the anodic and cathodic reactions, respectively. Therefore, the calculation of the Pourbaix diagram is a unique method to predict electrochemical corrosion behaviors.

Like other thermodynamic calculations, the calculation of the Pourbaix diagram also needs many thermodynamic databases especially to combine aqueous database (e.g., TCAQ3 database [60]). A calculated Pourbaix diagram and a diagram of the potential and the phase amount of compounds for CoCrFeNi alloy at pH = 7 are shown in Figure 15 [20]. From the calculation results, some important information can be obtained, such as the main stable oxides and spinels in the passive layers, the specific potential and pH series for different oxidation and reduction reactions, and so on. It needs to notice that the potential in the Pourbaix diagram and in the real corrosion system represents equilibrium and nonequilibrium potential, respectively. Therefore, the equilibrium and non-equilibrium states should be distinguished when analyzing the corrosion behaviors.

Figure 15.

The calculated Pourbaix diagram (a) and the diagram of potential and the phase amount of oxides and spinels for CoCrFeNi alloy at pH = 7 (b) [20].

4.3 High-temperature oxidation performance

Compared with the metallic materials suffering various types of destruction at room temperature, the high-temperature alloys (i.e., superalloys) undergo much more heavily deterioration at elevated temperatures. To take the turbine as an example, as the most important core component, like the skeleton for a human, in fossil-energy/nuclear power plants and vehicles, especially in aircraft, the high-temperature metallic materials should conquer a series of complex chemical, physical, and metallurgical process, for instance, high corrosion, wear, fatigue, stress, and creep resistance performances, particularly for the interaction between these mechanics and corrosion factors. Therefore, the superalloys with high performances at elevated temperatures are the most significant materials in high-temperature components.

For the corrosion behaviors at room temperature and elevated temperature, the basic theories have some common points, such as thermodynamic and kinetic theories. However, the corrosion mechanisms of the high-temperature corrosion, usually called high-temperature oxidation, still have some specific theories including thermodynamics and kinetics in comparison with the theories for electrochemical corrosion at room temperature. To draw inferences from the above sections, perform the CALPHAD-based calculations, and simulations for high-temperature oxidation also focus on the thermodynamic and kinetic computation. That is, some predictions can be obtained from the above sections in this chapter, and some calculations and simulations can usually be performed especially to predict the behaviors of high-temperature oxidation. Some common examples are described in the following.

Some thermodynamic predictions for different usages based on the materials in specific environments or working conditions can be applied here. For instance, isothermal and isoplethal sections can be calculated for various HEA systems, not only for the alloying elements systems but also especially for the alloying element–oxygen systems. Figure 16 shows the calculated stable phases of the Fe-Cr-O system at 650°C at various oxygen activities [61]. A spinel miscibility gap appears in the S1 + S2 region (S1: Fe_3-xCr_xO₄, S2: FeCr_2-xFe_xO₄), S1 and S2 represent the spinel phase between Fe₃O₄ and FeCr₂O₄ phase, and the miscibility gap will be disappeared above ∼665°C.

Figure 16.

The calculated stable phases of the Fe-Cr-O system at 650°C at various oxygen activity. C: Corundum-type M₂O₃, S1: Fe_3-xCr_xO₄, S2: FeCr_2-xFe_xO₄, W: Fe_1-xO, BCC: Ferrite structure [61].

In the calculated phase diagram of the alloying element-oxygen systems, the oxygen can be exchanged for carbon, nitrogen, or sulfur, because the processes of oxidization, carbonization, nitridation, and vulcanization are treated as generalized oxidization.

In Sections 3.1.2 and 3.2, the application of the kinetic and precipitation simulations on heat treatment has been introduced. As described above, the thermodynamic calculations can be applied to study materials design, processing optimization, and various properties and performance; however, each prediction has some different specific emphases for different studies. The kinetic and precipitation simulation is the same situation.

Kinetic simulation has the capability to predict the formation of oxide layers during the high-temperature oxidation. The thickness of the oxide layers formed with time for the Fe-Cr-Al system for 24 h at 600°C was simulated by Dictra, as shown in Figure 17 [62]. The simulation sets an assumption that the bulk composition in alloys only is affected by the grain size and grain boundary diffusion in the oxide layers. Meanwhile, precipitation simulation is capable of predicting particle size distribution of precipitate phases. Until now, the studies of the precipitation simulation for the HEAs are not very common like steels. The reason probably is that the typical precipitation and inclusion in different HEA systems are not like M₇C₃ or M₂₃C₆ in steels. However, much stronger support for the precipitation simulation of the HEAs will be provided in the future resulting in more and more experimental investigation of HEAs [63, 64, 65, 66].

Figure 17.

A Dictra simulation of high-temperature oxidation. The oxide thickness time was dependent on the Fe-Cr-Al system for 24 h at 600°C. the simulation sets an assumption that the bulk composition in alloys only is affected by the grain size and grain boundary diffusion in the oxide.

5. Conclusions

In this chapter, the applications of the CALPHAD-based calculations and simulations in the recent advances and the new perspectives of HEAs were introduced briefly. The framework of the chapter is followed by the classical components, i.e., modeling, composition and structure, processing, and properties and performances. Here, the CALPHAD-based thermodynamic and kinetic modeling play the key role to connect the other three components. As can be seen, some examples were not only limited in the CALPHAD prediction in HEAs, but also contained some examples in different alloy systems, probably because not so much corresponding research was found. To draw inferences from the similar research, it is no problem to apply the similar thermodynamic and kinetic calculations and simulations in the specific HEAs systems if the specific alloy systems have the matched databases. Besides the content described in this chapter, there still exist a number of other research studies on the basis of the CALPHAD computations, and many techniques can also couple with the CALPHAD data or combined with CALPHAD software programs. These new trends extremely extend the boundaries of the CALPHAD method. For instance, the CALPHAD modeling can couple with first principle, phase field, machine learning, etc.; meanwhile, many conventional CALPHAD software can interact with Python or MATLAB through specific application programming interfaces (APIs).

HEA is a fast-growing field attractive to the scientists of different backgrounds. Many emerging applications of CALPHAD have not been included in this chapter, such as the refractory HEAs, high-entropy ceramics (HECs), additive manufacturing of HEAs, etc. These areas would offer new chances of exploring the unexpected and exciting features of HEAs. Hope CALPHAD could play a more important role in the new journey of discovering advanced HEAs.

Acknowledgments

W. Wang would like to acknowledge the funding from a Key R & D item in Science and Technology Development Project / International Collaborative in Science and Technology Development Project, Science and Technology Department of Jilin Province in China (Grant No. 20200401106GX and 20210402061GH). W. Wang would like to acknowledge the funding from the Opening Project of Shanxi Key Laboratory of Controlled Metal Solidification and Precision Manufacturing, North University of China (Grant No. MSPM202001). Z. Li would like to acknowledge the National Natural Science Foundation for Youth of China (Grant No. 52001098).

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this chapter.