4.2.1 CALPHAD Method

About 35 years ago, Larry Kaufman and Himo Ansara brought together a small number of scientists working on the calculation of alloy phase diagrams. This has been the origin of CALPHAD (CALculation of PHAse Diagrams) and computer coupling of phase diagrams and thermochemistry (Kaufman and Cohen, 1956). CALPHAD has become a successful and widely applied tool in many areas of materials development (Spencer, 2008). Kaufman and Agren (2014) in a recent article bring out the road map for quantitative microstructure engineering through CALPHAD (Figure 4.5), which includes atomistic calculations using methods such as DFT. DFT is a quantum mechanical modeling technique used to calculate the electronic structure in single atoms and atom clusters/molecules and condensed phases. In this technique, the properties of a many-electron system can be determined using functions of electron density, which are referred to as functionals. The name “DFT” comes from the use of functionals of the electron density. The solutions of the electron densities represented as wave functions and their associated energies are derived from the Schrodinger equation, usually utilizing one or more simplifications to reduce complexity. DFT is one of the most widely used methods in computational physics and chemistry.

Boesch and Slaney (1964) proposed the prediction of existence of harmful sigma phase in Ni-base superalloy by calculating average electron hole number and judging whether it exceeds the critical value for sigma phase formation. This is called PHACOMP (abbreviation of PHAse COMPosition). After this, a new Phacomp was invented in 1984 using d-electron concept to define the phase boundaries in terms of M_d (metal d-level), especially define the critical M_d value of gamma phase and sigma phase boundary for predicting sigma phase formation in Ni-base superalloys (Morinaga et al., 1985). Although these two methods are simple and quick ways to see the degree to which an alloy is prone to form sigma phase, they cannot predict other deleterious phases such as mu and Laves phases. Moreover, it cannot provide the temperature range of stability and phase boundaries. CALPHAD development has overcome these drawbacks for designing and developing superalloys by providing phase equilibrium, liquidus line, solidus line, transition temperature, phase amount, phase chemistry, etc.

CALPHAD can justifiably claim to be one of the major success stories in the field of materials development over the last quarter century. CALPHAD started in the 1960s but sophisticated thermodynamic data bank systems started to appear in the 1980s. In the CALPHAD method, firstly, available thermodynamic properties and phase equilibrium data of a system’s constituents are collected. Then, a thermodynamic description of the system is obtained. This description is basically a mathematical model which can be used not only to reproduce the known thermodynamic information, but more importantly, it is hoped that one can predict unknown thermodynamic properties of the system.

The CALPHAD technique makes use of the principle that the Gibbs energy of a phase that is a function of temperature and composition is enough to obtain a complete thermodynamic description of the system since almost all thermodynamic properties can be derived from the Gibbs energy function (Muggianu et al., 1975; Palumbo and Battezzati, 2008). The entire thermodynamic description is contained in a thermodynamic data base (TDB) file, which contains the Gibbs energy functions of the respective phases, constructed out of suitable “solution” models. The starting point is to write the Gibbs energy of a phase φ as,

$G^{\phi }={}^{\mathrm{ref}}G^{\phi }+{}^{\mathrm{id}}G^{\phi }+{}^{\mathrm{ex}}G^{\phi }$

The first term on the right hand side of the above equation is contribution of pure components of the phase, the second is the ideal mixing contribution, and the third is the excess Gibbs energy of mixing. The second term is given by

${}^{\mathrm{id}}G^{\phi }=RT\sum _i{x}_i\ln x_i$

where R is the universal gas constant, T is the temperature in absolute scale, and x_i is the mole fraction of the ith component. The third term is given by

${}^{\mathrm{ex}}G^{\phi }=\sum _i\sum _{j>i}{x}_i{x}_j\sum _v{}^{v}L_{ij}^{\phi }{(x_i-x_j)}^v$

where x_i and x_j are mole fractions of ith and jth components, respectively, and ${}^{v}L_{ij}^{\phi }$ are model parameters (Reidlich–Kister parameters) that are found out as best fit for available data by using statistical techniques (Durga et al., 2012). Thus, thermodynamic description of any phase can be calculated.

This information is generally read in by an optimizing engine (e.g., Thermo-calc), which uses the thermodynamic information from the databases (TDB file) to derive the required thermodynamic properties and phase diagrams. In this technique, the stable/metastable states of the system are arrived through Gibbs energy minimization. The equations which result from the optimization procedure are a set of nonlinear equations involving the Gibbs energies of the corresponding phases and the unknown compositions, which are solved through optimization routines that are part of the software engine (e.g., Thermo-calc and Pandat). The results of the calculations are the phase-coexistence lines/surfaces/hyperplanes, representing the phase diagram. At present, there are several commercial products, for example, FactSage, MTDATA, PANDAT, MatCalc, JMatPro, and Thermo-Calc, which serve as intermediate engines for reading in thermodynamic databases, and deriving useful thermodynamic information about material properties and their response to some types of materials processing such as solidification. These resources serve as useful tools in research and industrial development for saving time and experimental work.

In case of multicomponent systems, this approach has a profound influence because thermodynamic descriptions of the constituents (binaries and ternaries) can be combined and extrapolated using geometric models to develop a thermodynamic description of a multicomponent system, which is otherwise either experimentally not possible or requires stupendous number of experiments. One must note, however, that created databases must be experimentally validated to establish confidence in the derived information.

Zhang et al. (2012a) used this approach in as-cast and homogenized alloys of Al–Co–Cr–Fe–Ni system and came up with the phase diagram for the system which confirms with experimental observations fairly well. In several multicomponent systems, the stable phases, their relative amounts, and individual compositions were predicted using the CALPHAD approach with appreciable degree of success in agreement with experimental results (Durga et al., 2012; Raghavan et al., 2012). In another research (Durga et al., 2012), among the 22 multicomponent systems (quaternary and above) studied in the Co–Cr–Cu–Fe–Mn–Ni system, 14 of them show the FCC phase (Ni–Mn rich) to be the predominant phase, while 6 of them show the BCC phase (Co–Fe rich phase) to be the predominant one.

An attempt has been made to predict phase formation using a CALPHAD approach for a large number of equiatomic and nonequi-atomic HEA compositions that are known to form FCC, BCC, and a mixture of FCC and BCC phases (Raghavan et al., 2012). The stable phase is assumed to be the first phase that is formed upon cooling from liquid state with the highest driving force. The driving force for other phases at the transition for various compositions is also calculated. The results indicate that solid solution formation in multicomponent alloys is favored when the ratio of ΔS_conf/ΔS_fusion is greater than 1 and 1.2 for equiatomic and nonequiatomic alloys, respectively. CALPHAD approach appears to predict BCC phase formation much more accurately than the FCC phase formation. It is believed that this may be because of a greater presence of kinetic effects than in BCC, which is a more open structure. The results also point out that BCC phase is favored when the atomic size difference is larger, which is reflected by a higher value of mismatch entropy (ΔS_σ/k). Formation of FCC phase appears to form only when the mismatch entropy (ΔS_σ/k) is very small and the ΔH_mix is close to zero. This indicates that close-packed structures get stabilized when the system follows Hume-Rothery rules and hence is close to ideal solution. In contrast, BCC phase gets stabilized when the mismatch entropy (ΔS_σ/k) is large and the ΔH_mix is more negative, indicating that open structures (BCC) can accommodate more strain and also nonideality.

Manzoni et al. (2013a) reported on two alloy systems of same elements with significantly different compositions and properties (brittle Al₂₃Co₁₅Cr₂₃Cu₈Fe₁₅Ni₁₆ and ductile Al₈Co₁₇Cr₁₇Cu₈Fe₁₇Ni₃₃) that the phase prediction by CALPHAD method was successful in the first alloy and only partially successful in the second alloy. Senkov et al. (2013b) have done CALPHAD analysis on high-hardness and low-density refractory multicomponent alloys of the Cr–Nb–Ti–V–Zr system. The predicted equilibrium room temperature phases with their volume fractions were found experimentally only at higher temperatures (600–750°C). This indicates that still slower cooling rate (<10°C/min) from the homogenization temperature of 1200°C may be necessary to achieve the predicted structure or there is a scope for better model.

4.2.2 Ab-Initio Calculations

Ab-initio calculations such as DFT involve the direct calculation of the electronic structure of atoms through the solution of Schrodinger equations (normally simplified with assumptions).

The determination of the electronic structure serves as a powerful tool to predict materials behavior and properties, which also involves the interaction between elements. Thus, one can determine formation energies, magnetic states, and lattice parameters for binary or multinary phases. This method augments the CALPHAD method when information regarding properties such as crystal structure are not available or the literature lacks sufficient experimental data. The downside, however, is that ab-initio calculations require a large amount of numerical computation, with the computing time rapidly increasing with the number of atoms.

Li et al. (2008b) used MD simulations and ab-initio calculations to find out the type of BCC phase formed with variation in Al content in AlCrCoNiFe HEA. In this study, FORCITE OF MATERIALS STUDIO and CASTEP computer software packages were used. These computer simulations and experiments agree that due to large electronegativity difference, compound formation tendency increases in this HEA system and results in the formation of ordered phase. For example when Al and Cr were used in higher amounts (Al+Cr≥50 at.%), the solidified solid solution would have an ordered (B2) structure because Al and Cr have larger electronegativity difference. Therefore, it is suggested that while designing the alloy, higher amounts of Al and Cr can be avoided for better ductility, because B2 is a structure that is relatively brittle in nature.

Wang and Ye (2011) used first principle calculations to find out how lattice parameter and formation enthalpy are dependent on elemental composition in FeNiCrCuCo alloy. In this study, plane-wave pseudopotentials and alchemical pseudoatom methods were combined and used. It is demonstrated to be an efficient and reliable method to imitate the elemental positions in the lattice. In alloy design perspective, this study throws light on the role of Cr. Increase in formation enthalpy and decrease in lattice parameter occur with increasing Cr content. Such influence is attributed to electronic configuration and ionic radius of the element. Copper also is found to increase the formation enthalpy, but the reason is not investigated in this study.

Egami et al. (2013) reported a computer simulation study on the effect of irradiation on HEAs. Particle irradiation gives rise to atomic displacements and thermal spikes. Amorphization occurs easily in HEAs during irradiation. It occurs by atomic displacements resulting from irradiation and inherent high distortion at the atomic level of the HEAs. Following this, local melting and recrystallization occur due to thermal spikes. It is speculated by Egami et al. (2013) that such response to irradiation reduces defects in HEAs, and thus, HEAs are excellent candidates for nuclear application. Initial results of computer simulation on modeling binary alloys and an electron microscopy study on Hf–Nb–Zr alloys, demonstrate extremely high irradiation resistance of these alloys against electron damage to support this speculation.

Tian et al. (2013b) used ab-initio calculations and DFT for CoCrFeNiTi alloy system. Exact muffin-tin orbitals (EMTO) method in combination with the coherent potential approximation (CPA) is used. Accuracy of the single-site mean-field approximation is evaluated by comparing the CPA results with those generated by the supercell technique. Elastic modulus, equilibrium volume, atomic radius, and magnetic moments were estimated. Effect of variation of properties with amount of Ti was studied. Ti was found to increase anisotropy and ductility. But, calculations in this study seem to deviate significantly with experiments in several alloys indicating the scope for more extensive calculations and experiments.

One of the basic criteria to stabilize single-phase solid solution microstructure is the high-entropy effect. It arises essentially due to equiatomic composition and large number of elements. But, restricting the design only to equiatomic composition causes several constraints. Therefore, the range within which the amount of one element can be varied without changing the single-phase solid solution structure is of profound design importance. In this regard, a theoretical study using Bozzolo–Ferrante–Smith (BFS) method of atomistic modeling is reported (Del Grosso et al., 2012). By this, the lower concentration limit of each element in the alloy system with W, Nb, Mo, Ta, and V, to sustain single-phase solid solution, is estimated (Figure 4.6). It was mentioned in Chapters 2 and 3 that the effect of configurational entropy is higher in stabilizing a single phase as the temperature increases. Thus, for some alloy systems there can be a lower temperature limit (critical temperature), above which the existence of solid solution is ensured. It is not difficult to perceive that finding such a temperature limit is of major help in alloy design. Estimation of this temperature is done as a function of amount of each element in the alloy system with W, Nb, Mo, Ta, and V.

4.2.3 MD Simulation

MD simulations are atomistic techniques, which are utilized usually for prediction of structures of initial cluster formation or the material response to stimuli right-down to the scale of atoms. An important input for MD simulation is the description of pair potentials which describe the interaction forces of the different atoms in the system as a function of separation. The equations of motion of each of these atoms is then derived from simple Newtonian mechanics where the total force on an atom is arrived through the super-position of the atomic forces from all other atoms, wherein the force between two atoms is derived using the described atomic potentials. Quite clearly, the predictive capability of this method lies in the degree of detail through which the pair potentials are constructed.

An exemplary application is in using a many-body tight-binding potential model to study the effect of number of elements and size difference on the amorphous structure of HEAs (Kao et al., 2006, 2008). This is simply because this model treats the interatomic forces existing between any two unlike atoms as the geometric average of their bonding forces in their respective pure lattices, and thus treats the systems as ideal solutions (i.e., mixing enthalpy is zero). In other words, such a MD simulation rules out the effect of actual bonding energy between unlike atoms, and only investigates the effects from the number of elements and the atomic size. The alloys simulated were from traditional binary alloys to HEAs by adding one element in sequence. For example, Figure 4.7 shows the initial radial distribution function curves of the alloys at 300 K before the system was heated. Thus, we can see that the patterns of binary to quaternary alloys have well-defined peaks which indicate a crystalline structure. However, quinary alloy and sexinary alloy containing large-sized Zr have lower and broader peaks which confirm an amorphous structure. Virtually, this trend apparently shows that the amorphization is enhanced by the number of elements and large atomic size difference.

When heated up to the molten state (2200 K) the patterns typically depict a liquid structure as shown in Figure 4.8A. Moreover, as the number of elements increases, peaks become broader and distance between peaks also become larger which indicates that the liquid structure becomes more disordered. In the melt-quenched state, quinary and sexinary alloys exhibit liquid-like solid structure as shown in Figure 4.8B. But, binary to quaternary alloys display an amorphous structure because there is a splitting in their second peaks, which indicates that the structure is more ordered than the liquid structure. This again shows an increased number of elements, and thus large atomic size difference could enhance amorphization. In fact, the shape and evolution of radial distribution function curves could be explained from the close-packed hard ball model as shown in Figure 4.9. The splitting of the second peaks indicates that the second nearest neighbor shell is not fully merged with the third shell due to the insufficiency in the degree of disorder. By the hard ball model as shown in Figure 4.9, the number of atoms and radius for each shell are shown in the second and third rows of Table 4.1, respectively. Under a random occupation of sites by different atoms, the fluctuation range caused by the atomic size difference can be used to judge the merger of peaks. If the atomic size difference makes the atomic fluctuation range in the second and third shells larger than 7.2%, the second and third shells or peaks are expected to merge into each other. It can be seen from Table 4.2 that only quirary and sexinary alloys with deviation over 10% can fit this requirement. Since the deviation between fourth and fifth shell is 6.2%, all the alloys can have the merger of fourth and fifth peaks. Therefore, the judgment of peak merger by the atomic size difference is consistent with the radial distribution function calculated by MD simulation.

LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) package for MD simulations to find out the deposition and annealing processes of AlCoCrCuFeNi HEA thin films was adopted by Xie et al. (2013). Operational conditions in this simulation are magnetron sputtering of this HEA film on Si substrate. Cluster growth and thermal stability were studied during annealing in the range of 300–1500 K. The variation in root mean square displacement (RMSD) with increasing annealing temperatures is used for analysis of coalescence process. For example, at around 850 K a jump in RMSD value is observed indicating induced cluster coalescence. In computation, different methods were used for calculating interaction between different entities. For interactions among different elements in AlCrCoCuFeNi, embedded atom method force field was used. For Si–Si interactions and for Si–(AlCoCrCuFeNi) interactions, Tersoff potential and Lennard–Jones potential were used, respectively. Wang (2013b) generated Morse pair potentials for interactions among neighboring atoms using first principle calculations within DFT. This atomic model considers maximum entropy for the system. In this, using molecular statics simulation, optimized atomic configuration of AlCoCrCuFeNi alloy is found out. It is claimed that there is no short-range or long-range order in the alloys. The experimentally observed lattice structure of the alloy is only the result of average of the local disorder of atomic positions and composition in wide range.

4.2.4 MC Simulation

Similar to MD simulations, MC techniques are utilized for describing materials phenomena at the atomistic scale. The difference, however, is that while in MD simulations, the different microstates in the ensemble are traversed through physical forces describing the motion of atoms, in MC simulations these microstates of the system are derived through probabilistic arguments, which are based on the energetics of the given microstates. A useful application of the technique is in structure determination, in which multielement cluster configurations can be predicted using relatively simplistic MC computations.

Using this method, Wang (2013a) built atomic structure models of HEAs composed of four to eight elements in both BCC and FCC lattice according to the principle of maximum entropy (MaxEnt). Under the MaxEnt principle, every atom is striving for the maximum free space in the system. The driving force for this trend is called as the entropic force. The entropic force of atom i is pointed to the direction of v_i(r) increase where v_i(r) is the free space of atom i at position r. For example, consider a binary Fe–Cr alloy with a BCC lattice having n×n×n unit cells in which the Fe atoms at some random sites are replaced by the solute atoms Cr. In the next step, this initial configuration is optimized according to the MaxEnt principle in the following way: The state of the maximum system entropy for this binary phase should be such that each of Cr atom approaches to its maximum free space. The main part of the optimization program for this is constituted by a big repeat loop. The position-replacing action is executed only when the distance of solute atom i, at the new position, to its nearest solute neighbor becomes longer than at its original one. The loop terminates when the MaxEnt condition is satisfied.

In the atomic structure models for HEAs of four to eight elements in both BCC and FCC lattices, by calculating the distance of an atom to its nearest same-element atom, Wang concludes that same elements as the first nearest neighbor can be avoided for those BCC solid solutions with five or more elements. Most of the same elements in quaternary and quinary BCC solid solutions are in the second nearest neighborhood (83.0% and 65.5%). This peak area moves to the third nearest neighborhood as the element number increases. On the other hand, the highest frequency of same-element existing as its first nearest neighbor for quaternary and quinary FCC solid solutions (74.3% and 47.3%). This peak moves to the second nearest neighbors for sexinary and septenary FCC solid solutions (64.9% and 50.4%), and the third one for octonary FCC solid solution (69.8%). Only a trace (0.2%) is left in the first nearest neighbor in the octonary phase. Wang further simulated FCC FeCoCrNi, FCC CoCrFeMnNi, BCC AlCoCrFeNi, and BCC AlCoCrCuFeNi alloys. In order to consider interatomic interaction in the simulation, Chen’s lattice inversion pair-function potentials were used to obtain relaxed atomic structure models of the four alloys. In calculating lattice constant and lattice distortion, he found that BCC alloys generally have much greater lattice distortions than FCC ones. The reason is largely because the atomic radii of the component elements vary widely in the BCC phases.

This simulation demonstrates that the atomic structure with maximum entropy could be obtained for a multicomponent solid solution. Lattice constant and distortion could also be calculated. Although the real alloys may deviate from this ideal state due to the diffusion resulting from the local unevenness of physical fields or the special affinity among some elements, the MaxEnt configuration can serve as the reference standard or the starting point for the studies of real materials.

4.2.5 Phase-Field Modeling

Over the years, one of the principle tools used for understanding the microstructure evolution during phase transformations is the phase-field modeling. The phase-field modeling is mainly used for solving interfacial problems and has mainly been applied to microstructure evolution during solidification (Boettinger et al., 2002). Phase-field models were first introduced by Fix et al. (1983) and Langer et al. (1986). Although, initially limited to solidification, it is also now widely used in areas such as solid-state diffusion, deformation behavior, heat treatment, recrystallization, grain coarsening, and so on. It has now become a handy tool for metallurgists and material scientists alike (Asta et al., 2009).

The phase-field method draws its elegance from the fact that it is able to describe microstructural evolution involving the motion of interfacial boundaries with concomitant coupling to heat, mass, and momentum transfer, without explicit tracking of the location of the phase boundaries. These problems are normally of the Stefan type and are usually quite complex to solve in the framework of classical finite element methods where the problems get more difficult with the increasing complexity of the boundaries. Traditionally, these problems were solved using the boundary-integral methods, which are numerically very efficient for 2D problems. However, these methods are unable to treat complex geometric pattern formation such as catastrophic phase termination, and additionally the equations of motion become quite involved in three dimensions. The phase-field simulations on the other hand retain their simplicity across dimensions and have the potential to be applied to problems of complex geometric pattern formation while retaining numerical simplicity and generality in the construction of the evolution equations of the phases.

Through the past two decades, the phase-field modeling techniques have been fine-tuned in order to derive quantitative information regarding measures of the microstructure such as lamellar spacings, dendritic arm spacings, and other morphological features such as dendritic tip radius as a function of different processing conditions. In particular, they have contributed in providing complete theories for the morphological evolution of dendrites and eutectics. These developments have naturally led them to being applied for the simulation of materials processing of real alloys involving multicomponent systems. In this endeavor, the coupling to thermodynamic databases is essential. While a number of examples of such coupling to databases exist, there are essentially two major pathways. One of them involves, the direct calculation of the thermodynamic driving force required in the evolution equation for the phase field, from the thermodynamic engine. This has been commercialized as the TQ interface (provided by Thermo-Calc), which hides all the complex thermodynamic calculations involving the computation of the driving force through a user-friendly interface. Since a single call to this TQ interface is generally time consuming, and given that the deviations in the driving force are normally small, the driving forces are extrapolated using the thermodynamic properties (liquidus slopes, Gibbs–Thomson coefficient, etc.) of the composition of the alloy. Timely checks are performed to ensure that the deviation of the extrapolation from the correct thermodynamic value is not significant. The commerical software (MICRESS, standing for Microstructure Evolution Simulation Softwares) combines the phase-field model to the thermodynamic database using such a coupling to the TQ interface. A second approach is the direct construction of simplistic free energy information developed out of the material properties (liquidus slopes, Gibbs–Thomson coefficient, partition coefficients) pertaining to the alloy composition one is modeling. The parametric determination of the coefficients of the constructed free energy is generally kept simple, such that they can be changed dynamically when there is significant change in the alloy composition.

There are no publications so far on the use of phase-field modeling in the area of HEAs. However, this unique modeling tool offers immense scope for understanding of microstructural evolution in these novel alloys. In the next few years, we can confidently expect more research so that the promise of multiscale modeling bridging the scales from electrons, through atoms, crystals, microstructure, and components, will be applied so that the dazzling but daunting possibilities of an enormous number of HEAs can be realized.