University of Thessaly. Department of Mechanical Engineering. Laboratory of Materials. High Entropy Alloys. Ioannis S.Aristeidakis

Transcription

1 University of Thessaly Department of Mechanical Engineering Laboratory of Materials High Entropy Alloys Ioannis S.Aristeidakis Maria-Ioanna T.Tzini Physical Metallurgy M.Sc Supervisor: G. N.Haidemenopoulos January 2016, Volos

2 Abstract The High Entropy Alloys (HEAs) have triggered a new interest in materials design. The main concept of this study is based on the diffusion and the phase transformations, which occur in these alloys. The processing routes for the synthesis of HEAs and the techniques used are classified in four states and each state is described. Thermodynamics parameters and their contribution to the prediction of the HEAs properties have been considered. The kinetics of phase transformation in HEAs are slower than the kinetics of the conventional steels. This behavior is attributed to the sluggish diffusion effect. The diffusion in HEAs is characterized as sluggish due to the complex interactions between the alloying elements and the high activation energy. The competition between configurational entropy and mixing enthalpy is a key factor for the determination of phase transformations. Diffusive transformations, such as the Nucleation and Growth Transformations (NGT) and the Continuous Transformations (CT), have been observed. Nanoprecipitation and eutectic transformations in NGT, as well as order/disorder transformations and spinodal decomposition in CT are the most reported amongst the HEAs references. Simulation models and new thermodynamic parameters have been developed during the past years, in order to predict the microstructure in HEAs. Despite the difficulties regarding computational calculations and experimental procedure, the High Entropy Alloys can lead to properties beyond those of the traditional alloy systems. 1

3 Contents 1. Introduction Synthesis and Processing Thermodynamic Analysis Diffusion Phase Transformations Simulation and Prediction References

4 1. Introduction High Entropy Alloys or HEAs for short, are alloying systems composed of at least 5 elements at almost equal quantities. Rely on the maximization of Configurational Entropy, which is able to stabilize solid solution phases, against intermetallic compounds. HEAs where firstly reported in 1996 by Huang KH and Yeh JW, however the interest for the field didn t develop until 2004, when the two separate teams of Jien-Wei Yeh and Brian Cantor published some significant results. The field of High Entropy Metallurgy is still in research today as there are many phenomena not completely understood. Their multicomponent nature, raises complexity in the system and thus makes it difficult to analyze and predict their behavior. HEAs exhibit some quite exotic effects, most of which have only been observed in High Entropy systems. These phenomena are called Core Effects and they originate from complex interactions among the constituent elements. The most important of them are the High Entropy Effect, the Sluggish Diffusion Effect, the Severe Lattice Distortion Effect and the Cocktail Effect. The High Entropy Effect is the base on which HEAs are created. It implies that a disordered solid solution phase can be stabilized when the entropy is sufficiently high. HEAs have extremely increased entropy as they are near equimolar. The Sluggish Diffusion Effect proposes that kinetics transformations in HEAs are very slow in comparison to conventional alloys. This is partly because of the increased activation energy for substitutional diffusion and partly due to local atomistic phenomena that eventually stall diffusion. Furthermore, the Severe Lattice Distortion Effect suggests that the crystalline structure of HEAs is extremely deformed. Although disordered solid solution phases resemble common BCC, FCC or HCP structures, are altered significantly, due to the different size among constituent atoms. Finally the Cocktail Effect infers that the strength of a high entropy solid solution is much higher than simply the weighted average of the composing element strength. The addition of low strength and density alloying elements like Al, can actually make the system more resilient. Again this effect originate from complex atomic interactions. Core Effects endow HEAs with extraordinary properties, very useful in various applications. In general single phase FCC systems tend to be very ductile, whereas single phase BCCs exhibit increased yield strength due to a pronounced solid solution strengthening. A mixture of FCC and BCC phases takes advantage of both properties, resulting in a high strength ductile material. Precipitation of sigma phase can further increase the yield stress, however it greatly diminishes ductility. For example, AlCoCrFeNiTi 0.5 has been reported to have a yield strength of 2.26 GPa and a plasticity of 23%. Furthermore, many HEAs perform exceptionally well at elevated temperatures as a result of the decreased kinetics. They have excellent creep behavior as they are very stable and resist thermal softening, even at very high temperatures. In addition, due to their low diffusivity HEAs have found implementation as diffusion barrier coatings. Recent studies have also shown that several HEAs are able to withstand a range of highly corrosive environments even at elevated temperatures. It should be noted that although their remarkable properties, HEAs have not yet found widespread application other than as protective coatings and high temperature components. The main reason for that is the increased cost associated with the material and the large scale manufacturing. 3

5 2. Synthesis and Processing During the manufacturing of High Entropy Alloys many techniques are used, such as arc melting, Bridgman solidification, mechanical alloying, sputtering, laser cladding and electrodeposition. The processing routes for the synthesis of HEAs can be classified according to their initial states for alloy preparation. Specifically, these states are distinguished into the liquid state, the solid state, the gas state and last the electrochemical process. 2.1 Preparation from the liquid state The majority of HEAs have been produced using liquid processing methods, such as arc melting, induction melting and Bridgman solidification. During arc melting, the torch temperature in the furnace is very high, approximately 3000 o C, and it is controlled by the adjustment of the electrical power. This method is not suitable for the low melting elements such as Mg, Zn or Mn because they can evaporate. In these cases the induction heating is a more adequate method. The Bridgman solidification or the Bridgman-Stockbarger method is a technique used in order to grow single crystal ingots. During this method, the material is heated above its melting point and then it is slowly cooled down from the one end of its container, where a seed crystal is located. A single crystal of the same crystallographic orientation as the seed material is grown on the seed, and progressively formed along the length of the container. Bridgman method is used to produce certain semiconductor crystals and can be carried out in a horizontal or vertical geometry. For coating fabrication in HEAs, a thermal spray technology was applied. In this process the material is gradually heated up until melting and a thermal spraying gun strikes its surface to flatten and to form thin compatible platelets. Next, the sprayed particles are accumulated on the substrate during cooling, creating in this way a cohesive structure (coating). Figure 1: A schematic diagram of (a) the arc-melting method and (b) the Bridgman solidification from the liquid state preparation. 4

6 2.2 Preparation from the solid state The solid state processing route is performed by mechanical alloying (MA) technique. This method involves repeated cold welding, fracturing and re-welding of powder particles in a high energy ball mill. The mechanical alloying is capable of synthesizing a variety of equilibrium and non-equilibrium alloys starting from blended elemental or pre-alloyed powders. The MA process occurs in three steps. At the beginning, the alloy materials are combined in ball mill and stranded in fine powders. Then the powders are compressed and sintered during a hot isostatic pressing process. In the final step, the powders are relieved by internal stresses during heat treatment. Many high heat turbine blades, aerospace components and a variety of advanced materials have been made with the MA process. 2.3 Preparation from the gas state Magneton sputtering and plasma nitriding techniques are quite popular among the methods for preparation from the gas state. These techniques are used to produce thin films or layers of High Entropy Alloys on the surface of the substrates. The magneton sputtering is a technique for depositing a thin film into a substrate by sputtering away atoms from a target under the bombardment of charged gas ions. The magneton sputtering can be distinguished between DC sputtering and radio frequency sputtering (RF). In DC sputtering the deposition rates are controlled by the voltage and argon pressure. The voltage and the argon pressure values are higher than the values of the RF sputtering. As a result the RF sputtering technique is more adequate for insulating materials, where the argon pressure have to be low. The magnetron sputtering technique has been used in biomedical applications as well, and it was observed that the produced coatings had excellent wear resistance and good biocompatibility in simulated body fluids. Plasma nitriding technique is used in order to make surface hardened layers for protection. Although with the use of this technique the layers are thicker than the layers of magnetron sputtering, the hardness which is accomplished is much higher. Hence, the wear resistance of the nitride samples is almost 49 to 80 time higher than the unnitrided samples. Also, the molecular beam epitaxy process is used to get high entropy metallic or ceramic films. Figure 2: A schematic diagram of the MA process from the solid state preparation. Figure 3: A schematic diagram of the sputter method from the gas state preparation. 5

7 2.4 Electrochemical process The use of the electrochemical method in HEAs is limited, because the combination of many electrochemical parameters affect the predicted structure. Despite this fact, with this processing route, unique properties can be accomplished. With the potentiostatic electrodeposition method, a film can be produced at low temperatures. It was observed that in the as-deposit films, soft magnetic behavior and magnetic anisotropy take place. By adjusting the electrochemical parameters, the thickness and the morphology of the structure can be controlled and high deposition rates can be obtained. Generally, the electrochemical deposition is one of the most simple and efficient methods which allow easy control of the nucleation and growth of metal nanoparticles. 3. Thermodynamic Analysis Before the discovery of High Entropy Alloys, it was believed that combining multiple elements in near equimolar compositions would result in a mixture of brittle compounds with no significant technological application. In contrast to those beliefs HEAs commonly consist of a sole solid solution or a mixture of solid solution phases. These phases resemble the common FCC, BCC and HCP structures, although they might be severely distorted due to the different atomic volumes among the constituent atoms. Considering Gibbs Phase Rule (Eq.1) it is possible to calculate the maximum number of phases possible to form at thermodynamic equilibrium. For example in a 5 component alloy such as the CoCrFeMnNi HEA, by setting the degrees of freedom (F) equal to zero and solving for (P) we get a theoretical maximum of 6 phases. = +1 (.1) Nonetheless, experimental and computational procedures have shown that after solidification this alloy is composed of a sole FCC solid solution. Although HEAs are theoretically able to form a large number of phases, in reality only a few are stable. This phenomenon is attributed to the High Entropy Effect. As the name suggests, HEAs exhibit increased configurational entropy, which acts as a stabilizing factor for solid solutions. A phase is stable only if its formation results in reduction of the Gibbs free energy (Eq.2a). In equivalence the resulting free energy of mixing should be negative (Eq.2b) so that a sole solid solution is able to form. = (.2) = (.2) Mixing entropy is correlated with the possible atomic arrangements that the system can take and for multicomponent systems can be calculated using Eq.3. = ln (.3) Where n is the number of Moles, R is the Ideal Gas Constant and is the Atomic Fraction in a j element system. Using Eq.3 it can be proven that is maximized when the system is equimolar. Mixing entropy can also increase by the addition of more alloying elements. Subsequently, solid solutions in HEAs have extraordinarily high configurational entropy. In near equimolar compositions and when temperatures are sufficiently high the term overpowers mixing enthalpy. As a result the term can become negative even 6

8 though is positive. On the other hand, for intermetallic and other ordered compounds the configurational entropy is practically equal to zero since there is only one possible atomic arrangement. The system will select to form the phase that yields the greater reduction in Gibbs free energy. Typically in HEAs, the free energy of solid solutions is lower than that of an intermetallic ( "" < $%& ), thus solid solution phases are stabilized. Still this is not always the case. If the formation enthalpy has a very large negative value, then the formation of the ordered intermetallic structure minimizes the total free energy ( $%& < "" ). In contrast, if the has a very large positive value, then > 0 and the system tends to separate via either NGT or Spinodal Decomposition. For instance, consider the equimolar CoCrFeMnNi HEA. A construction of Cr and Mn isopleth, in Figures 4a and 4b, reveals that the system is most likely to form a single disordered FCC phase after solidification. An alloy containing 20% Cr and Mn, solidifies at approximately 1570K or 1297 o C. During solidification the stable FCC structure forms, as it results in a reduction of the Gibbs free energy. This effect is clearly illustrated in Figure 5a and 5b, which indicate the free energy curves as a function of Cr and Mn concentration at 1000K (or 727 o C). Due to the high configurational entropy, the FCC structure exhibits the lowest Gibbs energy and thus remains the only stable phase for temperatures over 790K (or 517 o C). At that point the positive becomes larger than the entropic term and a separation of phases occur. Considering only Figure 5a in near equimolar composition, the FCC phase should separate via Spinodal Decomposition into two phases, one rich and one poor in Cr. However this doesn t happen as the FCC and BCC curves cross in Figure 5b, meaning that a FCC-BCC mixture will precipitate via NGT. This transformation results in greater reduction of Gibbs energy and thus it is stable. It is evident that the transformation theory describing binary systems is not entirely applicable in multicomponent alloys. For higher Mn contents, the phase diagram Figure 4b predicts the formation of Sigma phase (σ). In reality the chances of forming Sigma are extremely low, since is stable at very low temperatures, where diffusion is very sluggish. The equilibrium diagrams featured have been produced using the CALPHAD Methodology and as will be discussed later on, the results might not be completely reliable. Yet the conclusions derived from the analysis are still valid. Figure 4: Cr (a) and Mn (b) Isopleths for the CoCrFeMnNi HEA. 7

9 (a) (b) (c) (d) Figure 5: Gibbs Free Energy Curves for the CoCrFeMnNi HEA, at 700K (a, b) and at 1000K (c, d) as a function of Cr and Mn concentration respectively. The thermodynamic analysis described above, takes into account only chemical contributions to the Gibbs free energy. However, most of the known HEAs are governed by intense internal strain field caused by a large difference in size among constituent atoms. Strain energy increases the Gibbs free energy of the system and thus is able to destabilize a disordered solid solution. To account for this effect a supplementary non chemical energy term should be added. = +) * (.2 ) As a result Eq.2b can be written as Eq.2b, where * indicates strain energy difference per unit volume and ) the total volume of the system. It should be noted that the accurate calculation of * can be a quite difficult process, however it is possible to be expressed indirectly by the misfit parameter, in Eq.12a and 12b. Furthermore it has been observed that multicomponent systems exhibit various local phenomena that can affect equilibrium behavior. One of these is the existence of rapid variations in potential energy among lattice sites. These factors raise complexity in the problem, making it difficult to predict accurately the structure of HEAs at thermodynamic equilibrium. 8

10 4. Diffusion Extensive experimental procedures have shown that generally diffusion is significantly slower in HEAs, in comparison to other common alloys, such as Carbon or Low Alloyed Steels. This fact has great implications to the manufacturing methods as well as to the high temperature properties of HEAs. For example HEAs find implementations as diffusion barrier coatings or as creep resistant structural parts, by taking advantage of the decelerated kinetics. Sluggish diffusion can be attributed mainly to the complex interactions between the alloying elements. In HEAs, this phenomenon can be expressed mathematically through a low interdiffusion coefficient as a result of the increased activation energy for diffusion. HEAs consist almost exclusively out of substitutional alloying elements. Substitutional diffusion is not possible without vacancy formation, thus bulk diffusion in these systems is controlled by vacancy diffusion. Both the concentration and the mobility of the vacancies affect the rate of diffusion, through the activation energy (-. ). 0 = 0 1 exp (Eq.4) -. = - : +- ; (Eq.5) The interdiffusion coefficient (Dij) is proportional to the exponential function of the activation energy (-. ) for diffusion therefore a small increase in -. results in a large decrease in D, effectively stalling the diffusion process. For the case of substitutional diffusion the activation energy is given by Eq.5, where the term - : corresponds to the enthalpy associated with vacancy formation and the term - ; with vacancy migration enthalpy. There is a strong relation between - ; and the mobility of an element in the system since both of them depend on the bonding energy between atoms. In the case of interstitial atoms, the term - : is equal to zero since vacancy formation is not required in order to activate diffusion. As a consequence, the energy barrier for interstitial diffusion is lower and the diffusivity significantly higher. In common carbon steels the dominant mechanism that allows atoms to migrate is interstitial diffusion, fact than can explain the huge difference in diffusivities between common carbon steels and HEAs. Although this reasoning works well for carbon steels, it cannot explain the slower diffusion rate of HEAs in comparison to other systems dominated by substitutional diffusion. Table 1 indicates some characteristic values for the Self-Diffusion coefficients and the Activation Energy of each element in the CoCrFeMnNi HEA and the Fe-FCC matrix. These values are a product of theoretical calculations using a quasi-binary diffusion model and are validated experimentally. It can be seen that the diffusivity of a specific element is significantly lower for the CoCrFeMnNi system, in comparison to the equivalent Fe binary system. The temperature dependence of the diffusivity is illustrated in Figure 6. The slope of the graph indicates the activation energy for diffusion, whereas the intersection point with the y axes, the pre-exponential factor 0 1. Specifically, the preexponential coefficient is generally lower and the activation energy, normalized by the melting temperature A is higher in the case of HEAs. 0 = BlnC D BlnE D 6 (Eq.6) Since diffusivity is exponentially related to the activation energy, HEAs exhibit a lower diffusion rate for the same homologous temperature. As shown in Figure 7, the Intrinsic-Diffusion 9

11 coefficients also depend on the concentration through the thermodynamic factor (Eq.6). For alloys near the equimolar composition diffusivity remains almost constant since the thermodynamic factor is quite stable. Element Matrix MN KL G H IJH O P QR G S TU MVW X Y M Z [\ QR G Y M G YM IJHKJ] MN O P Co CoCrFeMnNi Fe-Fcc Cr CoCrFeMnNi Fe-Fcc Fe CoCrFeMnNi Fe-Fcc Mn CoCrFeMnNi Fe-Fcc Ni CoCrFeMnNi Fe-Fcc Table 1: Comparative Diffusion Parameters between HEA and Fe-FCC Matrices. Figure 6: Temperature Dependence of the Diffusivity in Different Matrices. 10

12 Figure 7: Concentration Dependence of the Diffusivity in CoCrFeMnNi System. In order to explain the reason that activation energies for substitutional diffusion is significantly higher in High Entropy Alloys, it is necessary to consider the complex atomic interactions of the alloying elements. HEAs exhibit an absence of a true solvent element, such as Fe in Steel, since compositions range near the equimolar composition. If full homogenization is assumed, then each atom in the crystal has almost the same chances of neighboring with any other element. However, not every bond is equally preferable as some yield in a greater potential energy reduction, i.e. some bonds are stronger than others. The difference in the bonding energy between each lattice position causes the Lattice Potential Energy (LPE) to fluctuate. These energy density fluctuations are present in conventional alloys such as steel as well, but their magnitude is considerably smaller. Large variance in bonding energy contributes to the sluggish diffusion effect, as atoms get trapped in areas where the potential energy is minimized. That behavior raises the activation energy for the entire system, effectively slowing down the diffusion process. An illustration of the lattice potential energy as a function of the distance between two lattice sites, for the diffusion of Ni in three different alloys is given in Figure 8. For a pure metal the LPE exhibits symmetry which means that the potential energy before (position L) and after migration (position M) is exactly identical. In the case of a Fe-Cr-Ni Alloy or a CoCrFeMnNi HEA, that symmetry is lost, since the LPE is increased in position M with respect to position L. In other words the M site has turned into a metastable position. The fluctuations in LPE are expressed in Figure 8 through the Mean Difference (MD) of the potential energy between the two lattice sites. The activation energy required so that an atom jumps from L to M is equal to E b+md and for the opposite movement is E b-md, where E b is the activation energy for pure metal. Because the M position is metastable and the energy barrier from L to M is higher, the chances of atoms migrating to this direction diminishes. If an atom manages to jump from L to M, there is high probability that it will jump back to its original position, as it is energetically more favorable. Migrating atoms are trapped in low energy potential regions and therefore require more thermal energy in order to overcome the energy barrier for diffusion. The stronger the fluctuations in LPE, the more prominent the effect gets. Since the Mean Difference in potential energy is somewhat larger for the High Entropy Alloy, the activation energy is higher 11

13 and thus diffusivity is greatly reduced. In the case of the Fe-Cr-Ni Alloy, the MD is rather small in comparison to the activation energy for pure metal (E b) resulting in a less pronounced effect. Another factor that results in an increased activation energy in HEAs has to do with the intrinsic deformation of the crystalline lattice that these alloys possess. It has been observed that HEAs form greatly deformed lattices due to the large difference in atomic size and shear modulus among its constituent elements. This phenomenon is named Sevier Lattice Distortion Effect and it is accompanied by a strong internal stress-strain field, which interacts with dislocations, impeding their movement. These interactions are responsible for the pronounced solid solution strengthening that HEAs display. However, stress field is not uniform as it is influenced by the local atomic properties of the system. Large atoms cause compression, whereas smaller atoms cause tension in the lattice. The energy of the bond influences the field as well, since stronger bonds have a tendency to minimize their length to the equilibrium bonding distance, inducing an intense stress field in the process. Because of these interactions High Entropy Alloys exhibit sharp local stress gradients that are able to stall diffusion, by raising the activation energy. Figure 8: Variations in LPE between lattice sites for different alloying systems. The main mechanisms by which internal stress field influences the diffusion rate, are through the activation energy and the driving force. To accommodate the effect of the stress field to the diffusivity, an extra term W has been added to the activation energy according to Eq.7. The term W is associated with the mechanical work done by the application of a stress field, and is negative for tension and positive for compression. Stress fields cause atoms to shift slightly from their equilibrium positions. As atomic bonds stretch and contract, the distance between atoms changes, enabling or disabling other atoms to squeeze through. The energy required to shift the atoms is expressed via the term W and it depends on the applied stress as well as the topology and nature of the bonds. In the case of conventional alloys such as Fe-Cr-Ni systems, the term W remains almost constant with respect to the applied stress, in fixed crystallographic 12

14 indices and does not fluctuate strongly, since bonding energy is almost identical in each site. On these terms, the additional energy W simply affects the energy barrier for diffusion E b (Figure 8) and leaves the MD unaffected. If the local stress is tensile then E b decreases whereas if it is compressive the activation energy is increases. Either way the symmetry among lattice sites is preserved. In HEAs bonding energy vastly differs from lattice site to lattice site and therefore W fluctuates in a fashion similar to that of the LPE. In addition to the effect on the E b, these fluctuations contribute to a more pronounced asymmetry of the potential energy among the lattice sites. The variations in W and in LPE are superimposed resulting in grater mean difference (MD). Atoms become trapped in low potential sites with greater ease and thus the macroscopic apparent activation energy increases. 0 = 0 1 exp^ -. + w ` (Eq.7) Inhomogeneous stress fields also affect diffusion by inducing an additional driving force. The dominant driving force that causes diffusional flux in alloys is the chemical potential gradient, however as indicated by Onsager, a flux can be generated by other potential gradients as well. According to Eq.8 the gradient of the hydrostatic pressure multiplied by the local volumetric change that is caused, acts as an additional driving force. The internal stress field in High Entropy Alloys is governed by sharp changes in intensity due to large difference in size and properties of the atoms. These local gradients assist atom migration from compression to tension regions. Atoms with large atomic radii cause symmetric compressive stresses around them, while smaller atoms cause symmetric tensile stresses. As a result the region near small atoms is typically accompanied by a lower potential energy in comparison to the rest of the alloy. Drifting atoms tend to fall in these energy pits and segregate, forming nano-precipitates. In order to overcome this barrier, more thermal energy is required and thus the diffusion is once again stalled. It is worth mentioning that the low potential regions originating from stress gradient are in the order of several atomic diameters. In contrast, those caused by LPE fluctuations, exhibit shorter range interactions in the order of one atomic diameter. % b = c d f + V pi (Eq.8) kl Many transformations require the cooperative movement of atoms. One of these transformations is the grain growth. During this process chemical composition inside the grain should remain unchanged. However, in order to be feasible a transformation of that nature, atoms of each composing element must migrate to the boundary, so that the new interface is formed. Since the presence of all atoms is necessary to continue the growth, elements with low diffusivity slow down the process as they force the system to wait for their arrival. As a result these elements are the rate limiting factor for transformations such as grain growth. It has been theorized and then proven experimentally that in CoCrFeMnNi HEAs the element with the slowest diffusion rate for a given temperature is Ni (Eq.9). 0 % > 0 mn > 0 op > 0 mq > 0 r (.9) 13

15 5. Phase Transformations In order to design the High-Entropy Alloys and fully understand their outstanding properties, the phase transformations have to be investigated. In High-Entropy Alloys (HEA) the diffusion is a prerequisite for phase transformation. Martensitic transformation is a diffusionless, displacive transformation, accompanied by a large shear deformation of the crystal. Since Martensite formation does not require the diffusion of atoms in the lattice, which in HEAs is very sluggish, it could be possible that some HEAs exhibit martensitic transformation. However, at present time there are no experimentally tested High Entropy Alloys indicating that. Although it is theoretically feasible to form Martensite from a HEA with FCC structure, in reality the transformation is never initiated due to a high activation energy for nucleation. This effect can be expressed via the Ms Temperature, indicating the critical point at which Martensite can start forming. The Ms temperature can either be calculated experimentally or approximated using Andrews Equation (Eq.10). For instance consider the equimolar CoCrFeMnNi HEA, which consist of a single FCC phase at 1200 o C. If Eq.10 is applied, it can be proven that Ms is well below ambient temperature and thus a rapid cooling to room temperature will not result in martensite formation. tu = t 7.5w +30xy 12.1z 17.7{w 7.5t (.10) The phase transformation kinetics in High-Entropy Alloys are much slower than the conventional alloys and the growth takes place under diffusion control with change in composition. The key factor for the determination of phase transformation is the competition between the entropy and enthalpy. For solid solutions: uu = }w~ }w~ (.2 ) Boltzmann s relation about the mixing configurational entropy of j-elements with X i atomic percent contribution is given in Eq.3. Where R is the gas constant with value 8.31 J/Kmol. In Figure 9, the entropy of mixing as a function of elements for equimolar alloys in disordered states, is depicted. Considering the fact that the mixing entropy should be high enough to counterbalance the mixing enthalpy, the number of the selected elements must be more than four. Beyond thirteen elements, the contribution in mixing entropy is insignificant. For intermetallic compounds: w = (.2 ) A thermodynamic parameter φ will be used in order to determine the stability of random solid solution. = "" K ; (.13) Where ΔGss is the change in Gibbs free energy for the formation of a fully disorder solid solution from a mixture of its individual elements. ΔGmax is the lowest (intermetallic) or the highest (segregated) possible Gibbs free energy obtained from the formation of binary systems from the constituents of the mixture. If 1 indicates the formation of a random solid solution. For negative values of φ the formation enthalpy is highly positive and the solid solution will not be formed. More details about this thermodynamic parameter will be given to the next chapter. 14

16 Figure 9: Mixing entropy vs the number of elements of equimolar alloys. HEA s microstructure is possible to consist of random solid solution phases (e.g. FCC, BCC), ordered solid solution (e.g. B2 and L1 2) and intermetallic phases (e.g. Laves phases). It was considered that alloys consisting of several elements, will form complicated and brittle structures. On the contrary to this expectation, it was found that due to the mixing high entropy the HEA alloys were composed of few solid solution phases or even one single phase. The number of phases was smaller than the predicted maximum number. This is attributed to the enhanced mutual solubility between the elements, which prevents the separation into terminal solution phases or intermetallic compounds. Although, if the bond between some elements is strong, then some intermetallic compounds could be formed. Even in this case, these phases include many elements and with the increase of the temperature the overall degree of order in HEAs decreases. Hence, when a high entropy alloy contains ordered phases, it is possible with the increase of the temperature, these phases to transform to random phases. On the other hand, if the formation enthalpy of an intermetallic compound is high enough to overcome the effect of the mixing entropy, then this phase will remain stable at high temperature. An accurate classification is essential, in order to avoid confusion. According to the structure and ordering, a phase is identified as simple ordered phase (SOP), simple disordered phase (SDP) or complex ordered phase (COP). In a simple phase the structure is identical to FCC, BCC or HCP or derived from them. If a phase is not simple, then its structure is characterized as complex. For example, the Laves phase belongs to the COP category. The study of phase transformation in the HEAs steels can be distinguished into two parts: Nucleation and growth (NGT) and Continuous Transformations (CT). 5.1 Nucleation and Growth Transformations The NGT transformations were observed in High-Entropy Alloys. The main characteristics of NGT transformation is the downhill diffusion and the sharp interface between the phases Precipitation The precipitation of phases constitutes the main mechanism of strengthening in High-Entropy Alloys (HEA). A new generation of these alloys has been developed, exploited for high temperature applications, the high entropy superalloys (HESA). The high solvus temperature 15

17 of the precipitation phase enables its stability in high temperatures and enforce the strengthening of the alloy. The Al-Co-Cr-Fe-Ni system has been employed in order to study thermodynamically the formation of an intermetallic compound. For ordered intermetallic: = 0 w = -w Al + Co + Cr + Fe + Ni (AlCoCrFeNi)ss ΔGss = kj/mol and ΔΗss< 0 Al +Ni AlNi ΔGin = kj/mol and ΔΗin<0 Al + Co + Cr + Fe + Ni AlNi +FeCr+ Co ΔGmax = kj/mol. Equating to, φ=0.36, also predicting a single phase solid solution is not stable. A non-equimolar 8.9Al-17.2Co-9.2Cr-8.2Fe-6Ni-50.5Ti (at %) system was investigated for the case of the High Entropy Superalloys (HESA). The as-cast dendritic microstructure consists of the FCC (γ) matrix and the coherent ordered L1 2 γ (Ni 3(TiAl)) phase, as shown in Figure 10. The solvus temperature of the γ in this alloy was measured to be 1150 o C approximately. After isothermal ageing, below the solvus temperature of the spherical γ phase, the γ/γ microstructure remained stable and no other detrimental phases were formed, such as the μ, σ, δ, hexagonal Ni 3Ti (η), or cubic Ni(AlTi) β phases. On the contrary in conventional Fe-Ni alloys, during isothermal heating (650 o C) the metastable γ phase transforms to η phase, resulting to strengthening loss. Accordingly, Nb addition to the conventional steels, faces the same problem, since with thermal exposure the γ (Ni 3Nb) can be replaced by the δ phase. The δ phase forms coarse plates and the softening takes place. In HESAs the Fe content is able to accommodate sufficient amount of Al without decreasing in expense the phase stability, due to the high content of Co that stabilizes the microstructure at high temperatures. In investigated alloys, the Thermocalc software predicts the formation of BCC and μ phase. However, the sluggish phase transformation rate prevents the formation of these phases and the microstructure remains stable even in high temperatures. Figure 10: TEM analysis on HESA sample (a) the γ particles in FCC matrix (b) and superlattice diffraction in γ phase, are depicted. The high mix entropy in HESA cause higher degree of randomness in the γ and γ phase. The strength of HESA at high temperatures is attributed to the high volume fraction of the γ phase, to the high degree of solid solution strengthening due to lattice distortion and last, to the increase of the APB energy (0.25 J/m 2 ). When the growth of ordered domains take place, a new boundary will be formed, known as antiphase domain boundary, and the atoms will have wrong neighbors. Hence, the high energy boundaries are associated with the APB energy. With the increase of the APB energy, the dislocations pairs have to confront bigger energy barrier and 16

18 their motion is prevented. The precipitates γ and the FCC matrix (γ) are coherent, which leads to low interfacial energy and low coarsening rate. In addition, the lattice misfit (δ) enforce the strengthening due to its negative value. Directional coarsening of the precipitates, known as rafting, can be used in order to hinder the climbing of the dislocations and decrease the creep rate, if the load is applied perpendicular to the rafting. The high temperature strength, the oxidation resistance and the high phase stability make promising the use of these materials Allotropic Transformation In High Entropy Alloys, the formation of simple solid solutions with FCC or BCC structure was observed. In addition, similar disordered and ordered versions of the same base structure often coexist. For example the coexistence of BCC and ordered B2 phases or the coexistence of FCC and ordered L1 2 phases were frequently reported. Due to nanoprecipitates, the allotropic transformations were extremely rare. In YGdTbDyLu and GdTbDyTmLu alloy systems an atomic arrangement of a HCP structure was found. During cooling in these steels an allotropic transformation takes place and the BCC phase transforms to FCC phase. Unfortunately, the microstructure consisted of needle-shaped and particle-shaped inclusions. More research is needed Eutectic Transformation The exploitation of the eutectic transformations have been considered in HEAs, in order to enhance the strengthening and ductility of the alloys. HEAs with FCC structure have good ductility and poor strength, whereas HEAs with BCC structure have great strength but they lack in plasticity. Hence, aiming to design an alloy with combine strength and ductility, the eutectic high entropy alloys (EHEA) have been developed. The CoCrFeNiNb x system was investigated for x=0.1, 0.25, 0.5 and 0.8, corresponding to the hypo-eutectic structure (0.1, 0.25, 0.5) and the hyper-eutectic structure (0.8). Specifically, the eutectic structure is accomplished for x equal to 0.63 or 14% Nb mole fraction. The addition of the Nb element induces the formation of the eutectic structure and cause lattice distortion, due to larger size in contrast to other elements. For the case of the hypo-eutectic composition, the proeutectic FCC phase will appear, while in the hyper-eutectic composition the proeutectic Laves phase will occur. The eutectic lamellar structure is depicted in Figure

19 Figure 11: SEM images of CoCrFeNiNbx alloys for hypo-eutectic (a) x=0.1 (b) x=0.25 (c) x=0.5 and hyper-eutectic (d) x=0.8 structures. The eutectic volume fraction increases with the addition of Nb and the lamellar spacing decrease. So the hyper-eutectic composition is expected to form higher eutectic volume fraction and a denser structure. For the CoCrFeNiNb 0.1, CoCrFeNiNb 0.25 and CoCrFeNiNb 0.5 alloys, the proeutectic region lacks of Nb, while the eutectic structure is rich in Nb. On the contrary, for the CoCrFeNiNb 0.8 alloy, the proeutectic Laves phase is Nb-rich. During the solidification process, for the hypo-eutectic compositions the proeutectic FCC phase nucleate and grows and when the eutectic reaction takes place the two phases grow simultaneously, creating the lamellar structure. In the hyper-eutectic case, the process is similar. Although, a retardation of growth is observed due to the interactions of particles of eutectic FCC phase. The CoCrFeNiNb 0.8 alloy has the maximum hardness. It is apparent that with the increase of the Nb element, the volume fraction of Laves phase becomes higher. The intermetallic Laves phase is harder than the FCC phase and considering the attribution of the small lamellar spacing, the alloy is expected to have excellent strength. When Nb content decreases, the ductility of the alloys increases. Hence, understanding the eutectic transformation, the desired alloy properties can be accomplished with the control of the Nb content Massive Transformation The massive transformations have not been observed in High Entropy Alloys, since this kind of transformation require high diffusivity rate. 5.2 Continuous Transformation In continuous transformation the main characteristics are the diffusive interfaces between two phases with gradually change in composition and that the decomposition proceeds with uphilldiffusion, against the concentration gradient. In High-Entropy Alloys the spinodal decomposition and the Order-Disorder transformation have been observed. An interesting fact in HEAs is that spinodal decomposition and ordering take place simultaneously, which is thermodynamically contrasting. This phenomenon was explained due to the atomic size difference and the high elastic interactions. More details are presented below Spinodal Decomposition The Al xcocrcufeni and Al xfecocrni 2-x high entropy alloys have been investigated for 18

20 spinodal decomposition. With the increase of Al ratio the BCC structure becomes dominant, while the increase of Ni content promote the FCC structure. Studies of Al xcocrcufeni alloys with high Al content (x>=1) have identified the occurrence of spinodal decomposition and ordering in the dendritic constituent, leading to the formation of the coherent A2 and B2 phases, accompanied by precipitation of Cu-rich particles. The predicted phase diagram of Al xcocrcufeni system with variable Al content (at %), is depicted in Figure 12. However, the spinodal decomposition has been observed in the Al 0.5CoCrCuFeNi in the as-cast state for the formation of two FCC phases. The Al 0.5CoCrCuFeNi system contains two FCC phases and an ordered L1 2 phase. The L1 2 phase was rich in Ni and Al, with some solubility in Cu. The first FCC1 phase was rich in Ni, Fe, Cr and Co while the second FCC2 phase was only Cu-rich. All phases were present in both dendritic and interdendritic structure. In Figure 13, the nano-scale spinodal structure in the dendritic material with the L1 2 phase, is depicted by STEM-EDX mapping. Due to high mixing entropy, the FCC1 and FCC2 phases are stable in high temperatures. During cooling the FCC1 spinodally decomposes into FCC1 and FCC2, relieving any Cu supersaturation. In addition, in FCC2 phase the spinodal driving force is eliminated, resulting in the decrease of solubility and the precipitation of the L1 2 phase. Hence, the spinodal decomposition scale increases with the decrease of the cooling rate. This behavior leads to finer microstructure, since with the decrease of the temperature, the maximum wavelength (λ max) is shorter and the amplification factor R(β) is higher. In addition, the coherency between the phases enhances the resistance to coarsening. Figure 12: The phase diagram of Al xcocrcufeni system with variable Al content (at %). 19

21 Figure 13: STEM-EDX mapping of the dendritic material in the as-cast state for (a) Co (b) Cr (c) Cu (d) Fe (e) Ni (f) and Al elements. The AlCoCrFeNi high entropy alloy consisted of equiaxed grains with alternating Fe-Cr and Al-Ni rich plates, corresponding to the A2 and B2 phases respectively, and L1 2 precipitates. In the Fe-Cr rich region, strong anti-correlated fluctuations of Fe and Cr atoms and a highly intertwined morphology have been observed and attributed to the spinodal decomposition. The amplitude of the fluctuations is more distinct in the AlCoCrFeNi alloy than the equiatomic AlCoCrCuFeNi alloy, due to the absence of Cu. As it is claimed above, spinodal decomposition coexists with an ordered phase (L1 2). The ordered phase develops with continuous transformation, since none antiphase boundary was found. However, these transformations thermodynamically are opposite. In the case of the spinodal decomposition, the mixing enthalpy is positive (ΔΗmix>0), whereas the ordered phases exhibit with negative mixing enthalpy (ΔΗmix<0). The bond energy and the atomic interactions between two dissimilar atoms explain the short range order (SRO) interactions, but fail to explain the long range order (SLO) interactions and the spinodal decomposition (continuous transformations). It was proposed then by Ren et al. that the elastic interactions were responsible for this phenomenon and explain the long range order interactions. The high atomic difference causes extreme lattice deformations and strains, resulting in the increase of the elastic energy and promoting ordering and spinodal decomposition concurrently. The driving force of the elastic interactions is expressed in Eq.8 for the initiation of transformations. This theory comes to agreement with the observation that with the increase of Al content the spinodal decomposition becomes possible, since the aluminum atoms are bigger than other elements. The coexistence of spinodal decomposition and continuous ordering have been observed in other systems as well. For example, in a Fe-Be supersaturated alloy the A2 phase transforms to the B2 phase, then to α+b2 phase with eutectoid transformation and last the α phase decomposes spinodally to the α1 and α2 phases. In both systems, the probability of the spinodal decomposition and ordering coexistence increases at the middle of the respective phase diagram. The addition of Co, increase the critical temperature for spinodal decomposition close to 1073 K. In the annealed alloys, the cooling rate affects the formation of σ phase, which is competitive to the spinodal decomposition. Increasing the Ni content, the onset of σ phase transformation is delayed and the final microstructure is softer. Hence, the optimization of the Al/Ni ratio is the key factor for the design of HEAs with better mechanical properties Order-Disorder Transformations In order-disorder transformations the enthalpy of mixing is negative ( }w~ < 0) and the 20

22 formation of solid solutions is exhibited. In High Entropy Alloys, the CoCrFeNiAl 0.3, CoCrFeNiTi 0.3, CoCrFeNiAl 0.3Mo 0.1 and CoCrFeNiAl 0.3Mo 0.3 systems have been investigated for the formation of ordered L1 2 and disordered FCC structures. Previous researches discovered that both Al and Ti possess the dual characteristics of being a strong BCC stabilizer and a high solid solution strengthener. In HEAs, FCC nanoparticles was discovered in the FCC matrix of CoCrCuFeNiAl 0.5 and CoCrFeNiAl 0.3 alloys, and the BCC nanoparticle was observed in the BCC matrix of AlCoCrFeNiTi alloy. The CoCrFeNiAl 0.3 alloy consists of FCC crystal structure and multi-elemental nanoparticles with L1 2 ordered structure. In Figure 14a, the L1 2 ordered structure is depicted by the selected area diffraction pattern (SAD), corresponding to the (11 0) superlattice spot. The multi-elemental nanoparticles are depicted in the bright-field image in Figure 14b. Figure 14: TEM (transmission electron microscopy) microstructures of the CoCrFeNiAl0.3 alloy where (a) the dark-field image corresponds to the (11 0) superlattice spot of SAD pattern and (b) brightfield image with SAD pattern of FCC [0 0 1] zone axis. Due to the high mixing entropy and sluggish cooperative diffusion of substitutional solute atoms, it was pointed out that HEAs system could form simple solid solutions or nanoprecipitates during solidification. The L1 2 ordered structure is usually a γ -Α 3Β stoichiometric intermetallic compound or solid solution (LRO), in which A corresponds to the most electronegative elements such as Ni, Co or Fe, and B corresponds to the most electropositive elements Al or Ti. Al Co Cr Fe Ni Mo Ti Al Co Cr Fe Ni Mo Ti Table 2: The mixing enthalpies of atom-pairs between each element. Hence, it is possible to predict the composition of the L1 2 structure by computing the mixing 21

23 enthalpies of atom-pairs between the elements, as shown in Table 2. The Ni-Al and Ni-Ti atom pairs are more negative than the others, therefore Ni is likely to combine Al or Ti during solidification. In addition, due to the fact that the ΔGss is more negative than the ΔGin, an ordered solid solution will be formed instead of an intermetallic compound. In Figure 15, the schematic diagram of the L1 2 ordered lattice, is depicted. The Ni content affect the formation of the ordered phase and the minimum contents of Al and Ni are 6.25 at% and 25 at%, respectively. The CoCrFeNiTi 0.3 alloy consists of multi-elemental nanoparticles with disordered FCC structure. The disordered FCC structure can be explained by the low Ti content, which is unable to achieve the minimum 6.25 at% requirement of ordering. The microstructure of CoCrFeNiAl 0.3Mo 0.1 alloy contains a disordered FCC structure, indicating that the ordered nanoparticles in the CoCrFeNiAl 0.3 alloy are transformed into a disordered structure after adding 0.1 mole Mo. In this case as well the Al content is less than the 6.25 at% required for ordering. In the CoCrFeNiAl 0.3Mo 0.1 no nanoparticles was found and the FCC disordered structure was found. It is known that with the increase of the temperature an ordered solid solution is transformed to a disordered. In HEAs this critical temperature is higher than the conventional steels. Hence, a criterion of ordering is essential for designing concept. In this system the Ni content should be more than 25 at% and the Al or Ti contents should be at least 6.25 at%. Figure 15: Schematic diagram of L1 2 ordered lattice of the CoCrFeNiAl0.3 alloy: (a) unit cell and (b) eight unit cells. 6. Simulation and Prediction Macroscale properties, such as strength and ductility, depend vastly on the microstructure of the material. Knowing the volume fraction and the chemical composition of phases formed after solidification, it is possible to predict the mechanical properties of the material. Using computational thermodynamics and kinetics it is possible to determine the alloy s microstructural characteristics and thus to a large extent the mechanical properties. These algorithms promote the discovery of new alloys with highly optimized characteristics while reducing experiments to a minimum. 6.1 Computer aided design Most computational methods have been developed and optimized for commercial alloying systems such as steel or aluminum alloys. In these systems, calculating and predicting 22

24 microstructural characteristics is rather trivial and fairly accurate. In contrast, HEAs are not that easy to model for two major reasons, complexity and lack of available data. The multicomponent nature of HEAs, accompanied by high concentration of alloying elements, results in an increased computational complexity. As the number of constituent elements becomes larger, atomic interactions become more intricate. In addition, the effect of factors like atomic size and shear modulus difference, valance electron concentration, electronegativity and reactivity, becomes amplified as concentration increases. Local atomistic phenomena can influence macroscopic material properties in unexpected manners. In order to simulate these effects, very elaborate and computationally intensive models must be employed. Another large factor that impedes HEA modelling is the lack of experimental data. Currently only 185 High Entropy Alloys have been reported and experimentally studied, a number extremely small in comparison to the number of steel alloys in use. The topic is quite new and aspects of it are not entirely understood. Yet, existing algorithms in order to produce reliable results, require precise input data. In HEAs these data are commonly inaccurate or entirely missing. These factors raise complexity in the problem, making it difficult to predict accurately the structure of HEAs at thermodynamic equilibrium. However, some interesting approaches have been developed to determine the microstructural properties of As-Cast HEAs. In many cases this is an excellent approximation of the actual behavior, since several HEAs are used straight after solidification without further heat treatment. In this form the alloy is inhomogeneous and out of thermodynamic equilibrium. Even after long annealing treatments it is improbable to reach equilibrium conditions, due to the extremely low diffusivity that HEAs possess. One of the first approaches was to apply the CALPHAD Methodology. The method works by constructing a model of the Gibbs free energy for each phase and then tries to minimize the entire Gibbs energy of the system. The combination of phases that yield the lowest free energy is the thermodynamically stable state for a given temperature. Since the history of HEAs is rather short, there is not currently a database created suitable for this specific application. In some cases it is possible to use existing databases to reveille some general characteristics and trends of the system, although computed results might differ significantly from reality. For instance CALPHAD predicts that the equimolar AlCoCrCuFeNi consists of FCC and BCC phases, however the calculated fractions may not be very accurate. Recent studies have shown that for specific systems such as AlCoCrFeNi HEAs, it is possible to increase calculation accuracy by extrapolating and combining existing databases. Even so, for the majority of HEAs the creation of entirely new databases is required so that CALPHAD produces valid results. The most common method used today for predicting the properties of HEAs is through the use of semi-empirical indices. These factors originate from thermodynamic analysis and provide a convenient way to quantify some thermodynamic properties of the system. They are accompanied by a set of observational rules meant to forecast microstructure, based on previous experience. Rules implemented in HEAs are somewhat comparable in functionality to the Hume-Rothery rules for solid solution formation in a binary system. Hume-Rothery rule imply that a binary sole solid solution is not stable unless the two elements are not very different in size, electronegativity, crystalline structure or valance electron configuration. Similarly HEA formation rules suggest some boundary values to the thermodynamic parameters in order to 23

25 promote solid solution phase formation. The results of this technique are usually referred to the conditions after solidification and thus work quite well to predict the As-Cast microstructure. Although it cannot forecast equilibrium behavior, it provides excellent insight of the system s properties. 6.2 Predicting solid solution HEAs Indices used to describe thermodynamic behavior often tend to be dimensionless to enable comparison between different alloys. One of the first factors created was the Š parameter Eq.11. In Eq.11 ; is the melting temperature of the system and can be calculated by averaging the melting temperature of each element, weighted by the corresponding atomic fraction. Š is a measure of the solid solution stabilization due to the increased configurational entropy. Increased Š values, higher than 1.1, indicate a high probability of simple disordered phase formation during solidification. An additional parameter used to express the difference in atomic size is,, given in (Eq.12a, 12b). Where z and correspond to the atomic radius and molar fraction of the element w in a { component system respectively. The misfit parameter, provides indirect information about the system s internal strain due to atomic mismatch. When, is very large sole solid solution is is stabilized as a result of the increased strain energy. In particular, a, value greater than 6.6 is thought to destabilize completely single phase HEAs. Š = ; mq% (.11) r, = Œ 1 z z kl 100% (.12) z = z r kl (.12) Categorizing HEAs through the use of Š and, allows some conclusions to be made, however it cannot provide concrete evidence for single phase stabilization. Figure 16a illustrates a scatter plot of Š vs, for 185 experimentally examined HEAs. The horizontal and vertical dotted lines correspond to the boundary values Š = 1.1 and, = 6.6 respectively. As indicated in Figure 16a alloys that don t satisfy the conditions Š > 1.1 and, < 6.6 are not able to form a single solid solution, still those that fulfill them, are not guaranteed to form one. In other words the conditions are necessary but not sufficient. Additional experimental studies, propose some complementary conditions necessary for single phase solid solution stabilization. They suggest that the enthalpy of mixing should be higher than -15 kj/mol and lower than 5 kj/mol. A more relaxed version of that condition implies that if the mixing enthalpy is between -22 kj/mol and 7 kj/mol it is probable that either a single or mixture of solid solution phases will occur. The condition aforementioned allows for the existence of some ordered or intermetallic phases as well. These supplementary restrictions certainly improve the predicting capabilities of the model, however there are still many cases that cannot be explained with the existing model. In order to draw a more solid conclusion a new thermodynamic parameter has recently been employed (Eq.13). The term "" corresponds to the free energy variance from the formation of a random solid solution at ;. On the other hand, ; is the highest free energy difference, in absolute terms, that can occur from the combination of any two constituent alloying elements. To be more specific, ; refers to the maximum difference that is 24

26 theoretically possible from the creation of binary system containing two of the present elements. For intermetallics, since the configurational entropy is equal to zero, Gibbs free energy is equal to formation enthalpy, i.e. %& = %&. Considering the scatter plot (Figure 16b) of vs,, there is an apparent separation of alloys into two major clusters. The horizontal dotted line, again indicates the critical value of =1. This method suggests that a single phase HEA is able to form at ; only if '1 and,#6.6. Although there are suitable HEAs that do not fulfill the conditions, the model is still extremely useful since if the conditions are satisfied, then the system has a very high probability of being a single phase HEA. "" ;.13 Figure 16: Scatter Plots of (a) Š vs, and (b) vs, for 185 experimentally examined HEAs. The addition of the parameter to the existing model, greatly increases its accuracy. However, with the parameters described up to this point, it is possible to predict only if a solid solution is stable and not the structure that will form. Experimental evidence indicate a strong correlation between the Valence Electron Configuration (VEC) of the alloy and the lattice structure stable in a solid solution. The VEC of an alloy is defined as the average VEC of the constituent elements, weighted by the corresponding atomic fraction (Eq.14). Alloys with VEC below 6.87 are expected to form BCC, while those above 8 are expected to form FCC structure. A mixture of BCC and FCC phases can occur if the VEC is in between those values. This effect is clearly shown in Figure 17, where open symbols indicate a sole BCC phase, closed symbols a single FCC phase and half open symbols a mixture of BCC and FCC. ) r kl ) r r ).14 kl 25

27 Figure 17: Phase Structure as a function of VEC. Open symbols for BCC, Closed Symbols for FCC and partialy filled for BCC+FCC mixture. Due to their simplicity these guidelines (Table 3) are a very common way of predicting the existence as well as some key features of a possible High Entropy Alloy. However, the implementation of the calculations required to make the method perform well are not always straight forward. The problem usually boils down to the accurate determination of the mixing enthalpy. is correlated with the energy of the bonds formed and thus depends strongly on the atomic properties of the constituent atoms. In multicomponent alloys like HEAs, these type of calculations tent to be difficult and time consuming due to complex atomic interactions. Currently there are two dominant methods for enthalpy modeling, a multicomponent extension of Miedama s model and through the use of Density Functional Theory (DFT). The DFT simulation uses ab initio calculations to very accurately determine enthalpy, however it tends to be very computationally intensive. Miedama s model implements a macroscopic description of the formation enthalpy and is usually much easier to implement. Although DFT is more precise, in both cases the error is fairly small and thus their results are comparable. In contrast, the mixing entropy of the system is much simpler to be determined, using Eq.3. Single Solid Solution Š'1.1 BCC,#6.6 )#6.87 '1 FCC 15 5 kj/mol )'8 Mixture of Phases Ordered and Disordered,#6.6 BCC+FCC 22 7 kj/mol 6.87 ) 8 Table 3: Conditions for HEA Formation. 26