2024 Volume 65 Issue 9 Pages 1025-1033
In this study, thermodynamic quantities were investigated using cluster expansion and the variational method for high-entropy and medium-entropy alloys (HEAs and MEAs, respectively). In the alloy systems combined with elements that tend to form HEAs, the absolute enthalpy value was small, and the entropy term was close to the value of an ideal solution. The tendency of ordering and the accompanying change in mean square atomic displacement (MSAD) were investigated in detail for four MEAs; namely, the FeCoNi, MnFeNi, MnCoNi, and CrCoNi systems. For these alloys, the ordering tendency and the phase-separation tendency in the low-temperature region were investigated. In addition, the presence of short-range order (SRO) in the high-temperature range was confirmed from the analysis using the Warren–Cowley order parameter. Thermal equilibrium structures were used to evaluate the MSADs at various temperatures. Long-range ordering developing at low temperature tended to reduce MSAD, although in the high temperature range the effect on MSAD from the SRO was very small.
Fig. 12 Change in MSAD as a function of temperature. MSAD is calculated from the atomic arrangement from CE-MCM at each temperature. The dotted line shows the MSAD of the SQS structure.
A high-entropy alloy (HEA) is a multicomponent system in which elements are mixed in equiatomic or near-equiatomic ratios, and is defined as an alloy showing a disordered solid solution of a single phase. Typically, HEAs consist of more than five elements [1, 2]. In recent years, medium-entropy alloys (MEAs) composed of three elements have attracted attention from the viewpoint of alloy design. Unlike the conventional steels, aluminum or titanium alloys that were made by adding small amounts of alloying elements to a certain principal metal, HEA alloys have a fundamentally different concept, and a new material that significantly expands the options of the material matrix phase is expected. From a practical viewpoint, the excellent high-temperature strength, the high room-temperature strength and ductility, and the outstanding low-temperature fracture toughness of HEA materials are worth noting.
The high mechanical strength has a strong correlation with the local atomic strain. This correlation arises because HEAs contain various atomic species with different atomic diameters, leading to strong disturbances in their atomic positions. The mean square atomic displacement (MSAD) is an indicator of the turbulence of the atomic positions and is known to exhibit a linear correlation with strength [3].
In contrast, HEAs have traditionally been interpreted to be thermodynamically stabilized at high temperatures due to their higher configurational entropy. In recent years, however, the presence of local atomic ordering [4, 5] in this alloy has been under discussion. Such a locally ordered state is expected to affect the mechanical properties of the alloy. Utilizing the local ordering for structural control, such as stacking faults, may lead to further property improvement in HEAs [6]. However, the effect of the ordering itself on strength has not been found to be consistent, and the relation between MSAD and ordering is not fully understood [7]. This study aims to confirm the existence of local atomic ordering in HEAs and discuss the relation to MSAD and the effect of this ordering on the strength of alloys.
In this study, FeCoNi, MnFeNi, MnCoNi, and CrCoNi of MEA were particularly selected as our target alloys. These alloys are relatively easy to synthesize because they have fewer components than HEAs and there is extensive experimental knowledge for their property [8–11]. The MnCoNi and CrCoNi alloys present a large difference in strength despite the fact that they only differ by one constituent element, and the strength of the CrCoNi alloy is significantly higher. Namely, since the MSAD of CrCoNi, calculated from first principles, is larger than that of MnCoNi, it is concluded that CrCoNi, with a larger MSAD, has higher strength [3]. However, MSAD is evaluated using the special quasirandom structure (SQS), whereas actual alloys may possess local atomic ordering. Essentially, short-range ordering (SRO) is observed by neutron diffraction in MnCoNi and is characterized by L10-type ordering, which is known as a long-range ordering (LRO) state in the fcc lattice [10]. Therefore, it is considered that analysis considering this effect is required. In this study, the cluster expansion-cluster variation method (CE-CVM) was used to study the tendency of ordering of these alloys, and the relation between atomic ordering and MSAD was discussed.
This section describes how the free energy, MSAD, and the atomic ordering behavior of fcc, body-centered cubic (bcc), hexagonal close-packed (hcp) solid solutions were calculated. The CE-CVM was used for the free-energy calculation. The MSAD computation in solid solution with the ordering was performed by applying the cluster-expansion method. To calculate the ordering behavior of elements in solid solution, the cluster-expansion Monte Carlo (MC) simulation method (CE-MCM) was applied.
2.1 Calculation of free energy of formation by the CE-CVMThe free energies of formation of the fcc, bcc, and hcp phases were calculated using the CE-CVM. In the cluster-expansion method, it is considered that a certain ordered structure R is composed of various clusters of different sizes, and the total energies ER of the ordered structures R are expressed as eq. (1), using the effective cluster interactions (ECIs) eα of the clusters α [12].
| \begin{equation} E_{\mathbf{R}} = \sum_{\alpha}^{\alpha_{\text{max}}}e_{\alpha}\langle\phi_{\alpha}\rangle, \end{equation} | (1) |
where ⟨ϕα⟩ represents the correlation function corresponding to the density of the cluster. The correlation function is a function of the spin operator σi, representing the atomic species occupying site i in the cluster, and is obtained as the average value of the spin product in the cluster α contained in the ordered structure. In addition, ER is reproduced by summing the products of the correlation function and ECI for the cluster group (subcluster) contained in αmax, which is the largest size cluster used in the calculation. In this study, various ordered structures were created by changing the atomic arrangement in the basic structure of interest, and the ER on the left side of eq. (1) for each ordered structure was calculated by first principles calculations. By contrast, eα, which are unknown parameters, were determined by the least-squares method, establishing multiple relationships between various ordered structures and energies. Furthermore, by considering a number of cluster configuration cases, the free energy at temperature T was expressed as eq. (2).
| \begin{equation} F(T) = \sum_{\alpha}^{\alpha_{\text{max}}}e_{\alpha}\langle\phi_{\alpha}\rangle - T\sum_{\alpha}^{\alpha_{\text{max}}}\gamma_{\alpha}S_{\alpha}. \end{equation} | (2) |
Here, γαSα is the contribution of entropy from clusters using Kikuchi–Barker coefficients [13]. The free energy was calculated by applying the variational method to eq. (2) and obtaining the correlation function where F(T) is minimized.
For the energies of ordered structures, the first principles computational codes, Vienna ab initio simulation (VASP), based on the projector-augmented wave method, were used [14, 15]. The generalized gradient approximation was used for the exchange-correlation term, and the energy cutoff of the plane wave was assumed to be 400 eV.
The calculations were performed on the fcc, bcc, and hcp crystal lattice models. For these ordered structures, calculation conditions that relax the volume and the structural atomic positional coordinates were applied. In contrast, we applied fixed conditions for the crystal lattice shapes to preserve the fcc, bcc, and hcp lattice, respectively. In all structural models, the calculations were performed under the condition of a nonmagnetic state. The CE-CVM calculation was performed by the iCVM-code, which was developed by Sluter [16].
2.2 Cluster expansion for mean square atomic displacementThe cluster-expansion method was applied to MSAD by replacing the energetics given in eq. (1) with MSAD in the respective constructions.
| \begin{equation} \mathrm{MSAD}_{\mathbf{R}} = \sum_{\alpha}^{\alpha_{\text{max}}}d_{\alpha}\langle\phi_{\alpha}\rangle \end{equation} | (3) |
MSADR is the MSAD of the ordered structure R, and dα on the right-hand side corresponds to MSAD assigned to cluster α. Hereafter, we will refer to dα as the effective cluster MSAD.
MSAD was estimated by averaging the sum of squares of the displacements of the atomic positions from the fcc lattice points before and after the calculation to relax the atomic positions by the first-principles calculation.
| \begin{equation} \mathrm{MSAD} = \frac{1}{\text{N}}\sum_{\text{i}}^{\text{N}}(\boldsymbol{x}_{\mathbf{i}} - \boldsymbol{x}_{\mathbf{0}})^{2}, \end{equation} | (4) |
where xi is the atomic coordinate in the unit cell after the atomic position relaxation, and x0 is the coordinate of the fcc lattice without distortion.
By deriving the effective cluster MSAD by cluster expansion, MSAD can be promptly calculated in any large structural model from the correlation functions. By using the equilibrium atomic profile at a finite temperature obtained from CE-MCM, MSAD is evaluated at that temperature.
2.3 Assessment of ordering behavior using CE-MCMMC simulation was performed to calculate the atomic arrangements in the fcc phase appearing at finite temperatures based on the ECIs evaluated in 2.1. A supercell was constructed by extending the fcc cell by a factor of 12 along each axis. The structure in which atoms were randomly placed was used as the initial structure for the simulation for 6,912 lattice points. In addition, the canonical ensemble was generated by iteratively executing the following steps 1)–3) for a given temperature T.
The process of evaluating the exchange of all atomic sites once is defined as one MC step (MCS). By starting the calculation at T = 5000 K and decreasing the temperature by 50 K intervals, 3000 MCS were performed under fixed conditions at each temperature to obtain the thermal equilibrium distribution at each T.
To investigate the ordering effect on the strength of MEAs, the entropy change (ΔS) with ordering was calculated. In the calculation of ΔS, ECIs obtained by CEM were introduced directly into the MC simulation, and the enthalpy of the alloy was first calculated. From the temperature dependence of this enthalpy, the time evolution of the ΔS and the atomic configuration with the ordering were investigated. At this time, ΔS was evaluated based on the configuration entropy in the high temperature limit of T = 5000 K, assuming a sufficiently random configuration. Note that although this temperature range is significantly above the melting point, the fcc structure is preserved in the MC simulation because the atoms are bound to the lattice point fcc and only the atomic positions are exchanged. This ΔS is a physical property value corresponding to the amount of decrease in entropy when the ordered structure is formed from the random state, and the tendency of ordering becomes stronger as this value is negatively larger. Therefore, it is considered that the magnitude of ΔS roughly corresponds to the degree of ordering of the alloy.
First, in order to understand the thermodynamic features of HEAs and MEAs, free-energy calculations by CE-CVM were carried out in several HEA systems. Similar calculations were also carried out to bulk metallic glass systems (BMG), and differences in the trends of HEAs and MEAs were examined. In addition, in the four MEAs of FeCoNi, MnFeNi, MnCoNi, and CrCoNi, the tendency of ordering in the low-temperature range was examined, and by calculating MSAD based on the atomic configuration, the relation between the ordering and MSAD was examined.
3.1 Free energy of the solid solutionFree-energy calculations of solid solutions were performed for five alloy systems, namely, fcc: CrMnCoFeNi, bcc: VNbMoTaW, TiZrNbHfTa, and TiZrNbMoHf [17–19], and hcp: MoRuRhWIr [20], for which the experimental formation of HEAs has been reported. Specifically, three elements constituting each alloy system were selected, and the free energy of T = 1000 K in the ternary system was evaluated.
In Fig. 1, the free energy of the fcc solid solution in the Mn–Co–Ni ternary system is calculated. As illustrated in the three parts of Fig. 1, (a), (b), and (c) show the free-energy changes with respect to the composition axis across the equiatomic alloy MnCoNi, with the composition ratio of 1:1 in the relevant binary systems as the starting points. The enthalpy (H) and the entropy term (−TS) in the free energy (G) are also shown. The enthalpies in each section are roughly between 0 and 2.5 kJ/mol, and their lower values suggest weaker interatomic interactions. In addition, the composition position shown by the dotted line in Fig. 1 corresponds to the equiatomic alloy (MnCoNi), and the entropy term for the composition is approximately −8 kJ/mol. The entropy term for mixing the ideal solution of the ternary equilibrium composition at T = 1000 K is −RT ln 3 = −9.13 kJ/mol and the −8 kJ/mol is nearly equal to this value. Therefore, it can be seen that the alloy of this composition has properties close to that of the ideal solution.
Free energy of the fcc solid solution in Mn–Co–Ni ternary systems at T = 1000 K in cross sections of the composition axes of (a) MnCo–Ni, (b) MnCo–Ni, and (c) MnCo–Ni.
In the five calculated alloy systems, all the other alloys have similar features to the MnCoNi alloy shown in Fig. 1. In other words, the absolute value of the enthalpy is small, and the entropy term is close to the entropy (∑i xiR ln xi) of mixing the ideal solution.
Similar computations were carried out for the alloy systems that show the BMG for the purpose of comparing with HEA and MEA. The free energy was calculated for the fcc, and bcc lattices for Ti–Ni–Cu–Zr–Hf with higher metallic glass-forming ability [21]. The free energies of T = 1000 K in the three compositional axes of fcc alloys in the Cu–Ni–Ti ternary system are shown in Fig. 2. Because the enthalpy of formation shows a large negative value, and the interatomic interaction is fairly strong in this system, and as a result, the ordering tendency appears to be strong. Therefore, the mixing entropy follows the ordering tendency and becomes a small value near zero. Similar tendencies were confirmed in the combinations of the other elements, and all of them suggested a stronger tendency of the interaction compared with HEA and MEA.
Free energy of the fcc solid solution in Ti–Ni–Cu ternary systems at T = 1000 K in cross sections of the composition axes of (a) TiNi–Cu, (b) TiCu–Ni, and (c) NiCu–Ti.
To compare the thermodynamic properties of the alloys quantitatively, the Ω–δ diagram for the respective computations was developed. The Ω–δ diagram is a plot of the stabilization parameter (Ω) of the solid solution on the vertical axis and the misfit factor (δ) based on the atomic radii differences on the horizontal axis proposed by Yang et al. [22]
| \begin{equation} \Omega = \frac{T\Delta S_{\text{mix}}}{|\Delta H_{\text{mix}}|} \end{equation} | (5) |
| \begin{equation} \delta = \sqrt{\sum\nolimits_{\text{i} = 1}^{\text{N}}x_{\text{i}}\left(1 - \frac{r_{\text{i}}}{\displaystyle\sum\nolimits_{\text{j} = 1}^{\text{N}}x_{\text{j}}r_{\text{j}}}\right)^{2}}, \end{equation} | (6) |
where ΔHmix and ΔSmix in eq. (5) represent the enthalpy and entropy of mixing, respectively. The larger the value of Ω, the closer it is to the ideal solution and the more random mixing occurs. Using ri and rj of the atomic radius of elements i and j, δ is expressed by eq. (6). The larger the deviation from the mean of the atomic radius, the larger the value of δ. These two parameters were analyzed for all calculated systems and are presented in Fig. 3. The squares represent the fcc phase, circle bcc, and triangle hcp. The cross symbol is the calculation result of the bulk metallic glass.
Relationship between the stabilization parameter Ω for solid solution and the misfit factor δ in HEA and BMG.
According to Fig. 3, the fcc and bcc systems are both located in the upper-left area, indicating weak interaction and atomic diameter is small. These characteristics classify them as alloy systems where solid solution formation is easily achieved. In contrast, the HCP alloy is somewhat biased toward the lower right compared to the fcc and bcc ones, suggesting a weaker tendency to form a solid solution. The systems with a strong tendency to form metallic glasses deviate significantly from the HEA group. These systems occupy the lower-right area. Thus, the free-energy estimation by CE-CVM also confirms the correlations between the stability of the solid solution and the stability of the HEA as depicted in the Ω–δ diagram.
3.2 Computing MSAD using cluster expansionMSAD in solid solution with ordering was evaluated by applying the cluster-expansion method. First, the accuracy of predicting MSAD by the cluster-expansion method was examined. The analysis was performed under the condition that the subcluster group in which the 8th proximity site was at the maximum distance and the size of up to three bodies was the maximum cluster. The results of predicting MSAD by applying the cluster-expansion method compared with the calculated results by the density functional theory (DFT) are presented in Fig. 4. The dotted lines show complete agreement between CEM and DFT. The root-mean-square error (RMSE) is 1.47 ps2/atom, and some deviations exist, but the magnitude relations are reproduced.
Comparison of MSAD predicted values by CEM with values from the DFT data set used for CEM.
The MSAD value was evaluated from the first principles computation using a SQS of 256 atoms [4]. In Fig. 5, the values were plotted on the horizontal axis, and on the vertical axis were plotted the MSAD values calculated using effective cluster MSAD obtained in this study. The values calculated by the CEM method that agreed with those of DFT are shown by the dotted line, and it can be seen that the CEM value tends to be more underestimated than does the DFT value. This underestimation tendency is attributed to the frequent use of structures with small MSAD values (Fig. 4), because the MSAD values used for regression are skewed to an area of 20 pm2 or less. The reason for the MSAD bias toward the structures with small MSAD values in the data set is attributed to the use of many highly symmetric ordered structural models, which could potentially be improved by adding values into the data set using a large random structural model. However, the introduction of a large structural model requires a lot of computational time. Nevertheless, the magnitude relations of these four structural MSADs predicted from CEM were generally consistent with those of the previous work [3], and therefore, it was considered that they could be sufficiently evaluated as MSADs in solid solutions with ordering.
Comparison of MSAD predictions from CEM and DFT calculations [7] in SQS.
In Section 3.1, it was shown that the interaction between elements that compose HEA or MEA had weak features. Then, the possibility of ordering including local atomic ordering was examined for four kinds of MEA, FeCoNi, MnFeNi, MnCoNi, and CrCoNi, in which experiments on the correlations between strength and MSAD were carried out. The atomic configurations of thermal equilibrium were calculated by CE-MCM for these four types of equiatomic alloys, and the calculated enthalpy changes with temperature are shown in Fig. 6. In all these alloy systems the enthalpy decreases monotonously with decreasing temperature. In particular, the decrease rate of the enthalpy increased in all these alloys from the following regions T ∼ 1000 K. In contrast, a gradual decrease tendency of the enthalpy was confirmed even in the high-temperature range above 1000 K. These are discussed in detail in Section 3.4, together with the analysis by the order parameter of Warren–Cowley; however, it is considered that the large entropy drop in the low-temperature side is an effect accompanied by transitions to the ordered state, and that change in the high-temperature side originated from the forming of the SRO appearing in the thermal fluctuation.
Calculated enthalpy changes with temperature in MEAs by using CE-MCM.
The absolute value of enthalpy and the quantitative degree of change with temperature are slightly different in each alloy. Therefore, this behavior was analyzed using the entropy change in more detail. The change in enthalpy with temperature is expressed by eq. (7).
| \begin{equation} \Delta S = S(T)-S(T_{0}) = \int_{T_{0}}^{T}\frac{1}{T'}\frac{\partial H}{\partial T'}dT' \end{equation} | (7) |
The ΔS in eq. (7) represents the change in entropy at a temperature T relative to the entropy at T0. In this study, assuming that the atomic configuration is completely randomized at T0 = 5000 K, the reference entropy at this temperature was assumed to be S(T0) = R ln(1/3) = 1.10R. In addition, the entropy S at the desired temperature is obtained by S(T) = ΔS + S(T0). The temperature dependence of entropy evaluated based on eq. (7) is shown in Fig. 7. Only the enthalpy changes due to the atomic configuration were targeted in this study, and therefore we could consider that this entropy reduction was entirely attributed to the changes in atomic arrangement. A large entropy reduction is observed at low temperatures below T ∼ 1000 K (Fig. 5). The decrease in entropy was also clearly confirmed in the higher temperature region, indicating an increase in the fraction of stable atomic configuration with decreasing temperature. Comparing the entropy decrease in the high-temperature range, the entropy decrease in CrCoNi in the that range differed clearly from alloy to alloy, and the amount of entropy decrease is the largest in Fig. 5.
Temperature dependence of the entropy of MEAs evaluated based on eq. (7).
Incidentally, because the reduction of large entropy is confirmed in the region between 1000 K and 500 K in all these alloys, an order–disorder transformation was considered to have occurred in this temperature region. However, the CEM performed in this study does not take into account the effect of lattice vibration. Although the effect of such lattice vibration tends to stabilize the disordered state [23, 24], the ordered temperature in this study might be overestimated compared with the actual transition temperature.
3.4 Calculation of atomic distributions by the MC method and evaluating the order parameters of Warren–CowleyChanges in the atomic distribution in real space associated with changes in enthalpy and entropy were investigated. The atomic configurations obtained from CE-MCM in the two alloys of MnCoNi and CrCoNi are shown in Figs. 8 and 9, respectively. The enthalpy and entropy in Section 3.3 show a change in the temperature range between 1000 K and 5000 K, but the difference in atomic profiles in both MnCoNi and CrCoNi in this temperature range is not clear. By contrast, the atomic distribution at T = 100 K was clearly biased. MnCoNi decomposes into a Ni-rich region and a (Mn,Co)-rich region, and CrCoNi shows a tendency to separate into two phases of Cr–Ni and Cr–Co. The atomic arrangement of L10–MnCo was confirmed in the (Mn,Co)-rich region in MnCoNi with the periodic atomic arrangement in the region where two elements are concentrated. By contrast, pure Ni, L10–CrNi, and L12–CrCo3 were identified in the CrCoNi alloy.
Atomic configurations obtained from CE-MCM in MnCoNi at (a) T = 5000 K, (b) T = 3000 K, (c) T = 1000 K, and (d) T = 100 K.
Atomic configurations obtained from CE-MCM in CrCoNi at (a) T = 5000 K, (b) T = 3000 K, (c) T = 1000 K, and (d) T = 100 K.
For a more quantitative evaluation of the statistical atomic distribution deviations, the order parameters of Warren–Cowley [25] were evaluated. The Warren–Cowley order parameter at the n-th neighbor is given by eq. (8).
| \begin{equation} \alpha_{\text{n}} = 1-\frac{y_{\text{i,j}}}{c_{\text{i}}c_{\text{j}}}. \end{equation} | (8) |
Here, yi,j is the fraction of pairs in which the selected element i, j is present in the n-th neighbor site, and ci, cj are the concentrations of elements i and j. yi,j is equal to the product of cicj for perfect randomness, and thus αn = 0. In contrast, the second term on the right-hand side becomes larger as the number of pairs increases due to attractive interactions between atoms. This yields αn < 0. Conversely, repulsive interactions between atoms reduce the number of pairs, and then αn > 0.
The changes in the Warren–Cowley parameter (α1, α2) of the first (n = 1) and second (n = 2) neighbor sites of the MnCoNi and CrCoNi alloys are shown in Figs. 10 and 11, respectively. The term α1 of MnCoNi in Fig. 10 shows that the number of pairs of Ni–Ni, Mn–Co, Mn–Ni, and Co–Ni changes greatly in the low-temperature range, whereas the number of Mn–Ni and Co–Ni pairs decreases, and Ni–Ni and Mn–Co tends to increase. The term α2 shows that the number of pairs of similar atoms (Mn–Mn, Co–Co, and Ni–Ni) is increasing and the number of pairs of dissimilar atoms is decreasing. α1 of CrCoMn in Fig. 11 shows that the number of pairs of Cr–Cr, Co–Co, Ni–Ni, and Co–Ni changes significantly in the low-temperature regime, increasing the number of pairs of Co–Co and Ni–Ni and decreasing that of Cr–Cr and Co–Ni. Although the change is smaller than the other pairs, the number of pairs increases in Cr–Ni and Co–Ni at the first nearest neighbor. α2, on the other hand, has an increasing number of pairs (Cr–Cr, Co–Co, and Ni–Ni) of the same element. The increase and decrease of the first and second nearest neighbor pairs in the low-temperature region are consistent with the formation of L10 and L12 type ordered structures, which is determined from the atomic configuration. The existence of SRO in L10-CoMn in MnCoNi alloys has been confirmed from the experiments, but the existence of ordered phases has not been confirmed in CrCoNi alloys [10].
Change in the Warren–Cowley parameter (α1, α2) of the first (n = 1) and second (n = 2) neighbor sites of MnCoNi.
Change in the Warren–Cowley parameter (α1, α2) of the first (n = 1) and second (n = 2) neighbor sites of CrCoNi.
In contrast, it can be confirmed that there are several pairs deviating from α = 0 in MnCoNi and CrCoNi, even in the high-temperature region. The real-space image of the atomic arrangement shows existence of SRO that cannot be visibly distinguished, which may lead to a decrease in entropy and enthalpy (see Section 3.3). The change in α1 in the high-temperature region corresponds to the change in large α1 from below about 1000 K, and thus, it is considered that precursors of the ordered phase appearing in the low-temperature region exist as fluctuations in the high-temperature structure. The change in α2 is somewhat unclear because the interatomic interaction at the second neighbor site is weaker than that at the first neighbor site.
As described above, the ordering with different features progresses in both the high- and low-temperature sides. The ordered state on the high-temperature side has a small ordering range and a small degree of order in the real space. On the low-temperature side, the ordering has the feature that its range is broadened, and furthermore, the degree of order is also high. In this study, the ordering of the high-temperature side is discussed by using SRO, and that of the low-temperature side by using LRO. In addition, although the discussion of the existence of SRO in HEA has attracted attention, it is thought that the thermodynamic origin of SRO in the high-temperature range is the tendency of α1 to develop LRO as seen from the temperature dependency. That is to say, in the low-temperature region, the contribution of the entropy is small, and the LRO grows dominantly, but in the high-temperature region, SRO is retained by the balance with the entropy.
3.5 Evaluation of MSAD in structural models developed by SROUsing the structural modeling of the thermal equilibrium atomic distribution at every temperature obtained from CE-MCM based on the enthalpy of the configuration, the MSADs were calculated at each temperature by effective cluster MSAD. The calculated MSAD values vs. temperature are shown in Fig. 12. The dotted lines in the figure represent the MSAD calculated using SQS for each of the alloys shown in Fig. 5. When we focus on the low-temperature range (below T ∼ 1000 K), the MSAD of each alloy tends to decrease. This change appears to correspond to the steep entropy decrease below T ∼ 1000 K (Fig. 7). Simultaneously, the Warren–Cowley order parameter indicates the development of LROs for L10 and L12 from below T ∼ 1000 K. Therefore, the LROs of L10 and L12 are thought to be responsible for the decrease in MSAD. This can be understood as L10 and L12, being more symmetric than the random state, result in smaller displacements of the atomic positions. Therefore, it is considered that this reduces the total amount of strain. For all four alloys, in contrast, the temperature dependency of the entropy in Fig. 7 is clearly smaller than that of the ideal solution, even at higher temperatures (above T ∼ 1000 K). Furthermore, the Warren–Cowley parameter given in Figs. 10 and 11 is α ≠ 0, SRO has been observed even in the temperature range above T ∼ 1000 K, but the effect on MSAD was very small.
Change in MSAD as a function of temperature. MSAD is calculated from the atomic arrangement from CE-MCM at each temperature. The dotted line shows the MSAD of the SQS structure.
As described above, ordering progresses from the high to the low-temperature region, even in MEAs, but it is concluded that SRO barely affects MSAD. The strength of HEA is discussed from the viewpoints of such MSAD and SRO. That is, in HEA, the solid solution of elements with different atomic diameters inevitably causes disturbance of the atomic positions indicated by the MSAD. Because the MSAD remains relatively constant at high temperatures, it is considered to bear the strength at high temperatures by solid solution strengthening. In contrast, the entropy change ΔS is almost zero at high temperatures, and the contribution to the strength is small. At low temperatures, conversely, LRO is indicated by the abrupt decrease of the entropy change ΔS. This LRO is responsible for the strength at low temperatures through the mechanism of precipitation strengthening. In contrast, MSAD is reduced and strength is slightly reduced. It is important to predict the dominant effect on material strength from the results of these contradictory contributions. As the strength is also affected by the shape, density, and interaction with dislocations of precipitates, the construction of a prediction model based on this information is a future issue. To improve the strength of HEAs, from the MSAD viewpoint, the atomic diameter of the alloy can be further expanded with very different elements. As another method, it might also be possible to improve strength by interfering with the dislocation motion by the ordered structure by adding an element that raises the ordering temperature. For example, the CrFeCoNiPd alloy, obtained by substituting Pd for the Mn of the Cantor alloy, has been revealed to have excellent mechanical properties at room temperature [26].
In this study, the thermodynamic quantities of HEA, MEA based on the cluster expansion and variational method were investigated. Four MEA types (FeCoNi, MnFeNi, MnCoNi, and CrCoNi with synthetic cases) were investigated in relation to and ordering tendencies. The results are summarized below.
The authors thank Dr. Y. Ikeda for his helpful advice. This study was supported by the Japan Society for Promotion of Science (JSPS) KAKENHI (Grant Number 18H05454). One of the authors (ME) was supported by the GIMRT program of the Institute for Materials Research, Tohoku University (202112-CNKXX-0201, 202112-SCKXX-0207).
