Critically Percolated States in High-Entropy Alloys with Exact Equi-Atomicity
2019 Volume 60 Issue 2 Pages 330-337
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2019 Volume 60 Issue 2 Pages 330-337
The formation of site-percolated states of exact equiatomic high-entropy alloys (HEAs) with body-centered-cubic (bcc) and face-centered-cubic (fcc) structures was investigated where their critical concentrations (pcsite) are given as 0.245 and 0.198, respectively, from conventional percolation theory. Molecular dynamics simulations were performed for WNbMoTa and WNbMoTaV HEAs with a bcc structure and AuCuNiPt and AuCuNiPdPt HEAs with an fcc structure. The simulation conditions included a generalized embedded atom method potential under NTp ensemble where the number of elements (N), absolute temperature (T), and pressure (p) were maintained constant. N-element alloys (N = 4 and 5) with a fraction of constituent elements (x = 1/N) were initially prepared in 10 × 10 × 10 supercells randomly in terms of chemical species and were simulated under atmospheric pressure at T = 1000 K. The total pair-distribution functions of the alloys revealed that the nearest neighbor distance (dn) for fcc ranged from 0.20 to 0.33 nm, whereas dn and the second neighbor distance (dnn) for bcc ranged from 0.235 to 0.305 nm and 0.305 to 0.370 nm, respectively. A 3-dimensional topological analysis for atomic correlations revealed that the alloys were in percolated and isolated states, respectively, when x ≥ pcsite and x < pcsite and that the values of 1/pcsite correspond to the ideal values of N for exact equi-atomic HEAs. Furthermore, it was observed that exact equi-atomic quaternary alloys (N = 4) with a bcc structure and quinary alloys (N = 5) with an fcc structure are in the critically percolated states.
Recently, significant scientific interest has been paid to high-entropy alloys (HEAs),1–3) which are exact or near-equiatomic multicomponent alloys in the form of solid solutions. Specifically, HEAs form simple solid solutions of face-centered-cubic (fcc),4,5) body-centered-cubic (bcc)6) or hexagonal close-packed (hcp)7,8) structures such as pure metals and primary solid solutions containing minor solute elements. Some HEAs exhibit a mixture of fcc and bcc phases.1) Single and dual solid solutions without intermetallic compounds are formed in HEAs when the constituent elements are selected carefully, even in multicomponent alloys without primary constituent elements. For instance, single-phase HEAs include WNbMoTa and WNbMoTaV alloys6) with a bcc structure and AuCuNiPt and AuCuNiPdPt alloys9) with an fcc structure. The crystallographic structures of these HEAs are slightly affected by those of the constituent elements. This effect is simply interpreted by the fact that W, Nb, Mo, Ta and V elements exhibit bcc structure and are frequently called as bcc-forming elements (bcc-formers), whereas noble metals are considered fcc-formers. According to the above interpretation, it is assumed that HEAs may be formed owing to the extension of the complete solid solutions from binary to multicomponent systems.
Analogically, an extendable tendency also can be observed in percolation theory10) where its generalization11) has been performed from a two-species (black-and-white) random process to a multispecies (polychromatic) process. Initially, a fundamental percolation theory based on a “black-and-white” model has been constructed for simple lattices, such as simple cubic, bcc, and fcc lattices in binary systems. This basic model corresponds to binary alloys with two constituent elements in the field of metallurgy. Subsequently, the percolation theory has been extended so that it can be applied to complicated systems by introducing the concept of “color”. Here, the number of colors can be interpreted as the number of constituent elements (chemical species) in metallurgy. Thus, it is worth examining the applicability of polychromatic percolation theory to HEAs by utilizing the analogy underlying the percolation theory and metallurgical alloying.
Hereafter, only exact equi-atomic HEAs are considered to simplify and derive the essence of the percolation phenomena in multicomponent systems. As for the exact equi-atomic HEAs with N-elements, the fraction of each constituent element (x) is readily calculated to be x = 1/N, which corresponds to 100/N atomic percent (at%). Here, the WNbMoTa and WNbMoTaV alloys refer to W25Nb25Mo25Ta25 and W20Nb20Mo20Ta20V20 alloys (at%), respectively. These exact equiatomic HEAs encouraged us to perform theoretical and computational studies on percolation phenomena in HEAs. Specifically, the motivation of the present study resulted from early studies10) of percolation in which a critical concentration of site percolation (pcsite) of the nearest neighbor12) is given as 0.198 (19.8 at%) for fcc and 0.245 (24.5 at%) for bcc structures. The authors realized that these values of pcsite are considerably close to x = 1/N of exact equiatomic quaternary and quinary alloys. This led to an early report by Zallen,11) in which the percolation phenomena in multi-species were discussed systematically as polychromatic percolation for a close-packed structure (fcc) and a simple cubic structure for 3- and higher-dimensional lattices. These results obtained by Zallen11) should be ideal data of percolation for multicomponent HEAs. However, there remain uncertainties about the validity of pcsite in actual HEAs. For instance, the actual HEAs possess slightly-different atomic sizes and weak negative heat of mixing among the constituent elements.13) In addition, the actual HEAs are affected by thermal fluctuations at a given temperature. To investigate these effects of HEAs on the formation of percolated states, it is convenient to use molecular dynamics (MD) simulations for actual HEAs at a certain temperature using realistic MD potentials.
The purpose of the present study is to investigate the possibility of WNbMoTa and WNbMoTaV HEAs with a bcc structure and the AuCuNiPt and AuCuNiPdPt HEAs with an fcc structure to form percolated states through a computational method utilizing MD simulations.
Classical MD simulations were performed using the commercial software SCIGRESS ME Ver. 2.3 (Fujitsu).14) The target alloys were the WNbMoTa and WNbMoTaV alloys6) with a bcc structure and HEAs comprising noble metals, represented by the AuCuNiPt and AuCuNiPdPt alloys9) with an fcc structure.
Crystal lattices of 10 × 10 × 10 supercells composed of randomly-distributed constituent elements were constructed computationally for bcc and fcc structures by using the commercial software. The supercells contained 2000 and 4000 atoms for bcc and fcc structures, respectively, and periodic boundary conditions were imposed in all directions while maintaining the shape of the cubic structure during the MD simulations. Subsequently, the alloys were structurally relaxed at T = 100 K for a certain time while monitoring the changes in pressure (p) and volume of the supercell (V) to obtain the initial structures stabilized and optimized for MD simulations. Subsequently, these relaxed alloys were annealed at 1000 K computationally under an isothermal-isobaric (NTp) ensemble at a pressure of 0.1013 MPa for 100 ps to homogenize them using equilibrium MD simulations. The annealing temperature of 1000 K was selected owing to the expectation that effective atomic displacements sufficient for statistical analysis would occur at this temperature. A time step (Δt) of 1 fs and a cut-off distance (dcut) of 1.0 nm were maintained in all MD simulations where the optimized values recommended by the software for these alloys were estimated to be Δt = 4 fs and dcut = 0.54 nm. In part, supplemental MD simulations were carried out under volume constant ensemble (NTV).
The atomic structures were analyzed from the molecular trajectories by using a pair-distribution function (PDF), g(r), as a function of the distance (r) of atomic pairs (i-j) of the constituent elements and total values. The MD potentials were carefully selected from the “Generalized Embedded Atom Method” (GEAM) library equipped in the software.14) The GEAM potential14) based on the Embedded Atom Method (EAM) scheme15) enables us to deal with 16 metallic elements and their mixtures as alloys with bcc, fcc, and hcp structures. The actual elements available in GEAM are Cu, Ag, Au, Ni, Pd, Pt, Al, and Pb with fcc, Fe, Mo, Ta, and W with bcc, and Mg, Co, Ti, and Zr with hcp structures. Notably, the above elements and the resultant alloys could be handled with the GEAM potential of the software.14) The GEAM potential14) lacks the data for V and Nb with a bcc structure, and thus, they were acquired from the data of Ta by differentiating the inter-atomic distances of V and Nb separately.
Furthermore, supplemental MD simulations for bcc structure were also performed to compare the percolated and un-percolated for simple binary alloys with compositions of Mo20Ta80, Mo25Ta75, Mo80Ta20 and Mo75Ta25. These binary alloys with the solute contents of 25 at% and 20 at%, respectively, corresponded to percolated and un-percolated states.
2.2 Analysis of percolationFirst, the ranges of the primary peak in the PDF profiles for total pairs were measured. The atomic correlations of the MD in the supercells were analyzed by selecting an appropriate element and its connection of atoms in the nearest neighbor distance (dn). The selected elements can be exemplified by W for the bcc HEAs and Pt for the fcc HEAs, as W and Pt possess intermediate atomic radius (relem)16) among the constituent elements in each system as summarized in Table 1.
The connections of atoms in dn and in the largest cluster were highlighted in the ball-and-stick view by utilizing the graphical functions equipped in commercial software CrystalMaker® X for Windows,17) Version 10.2.1 by considering the periodic boundary conditions. Specifically, the procedure to determine the largest cluster by utilizing the software was to first select an arbitrary atom, followed by “extend selection” of “molecule” by trial and error repeatedly using the software.
In addition, the length of mean square displacements (LMSD) was calculated using the MD software as reported in a previous literature.18) Specifically, LMSD is defined by eq. (1) with the coordination at a distance (r) of an atom of interest during time interval between t and 0 (tinterval) from a set of time-series data where M and N are the numbers of time-series data and total atoms, respectively, and tk is the starting time of the k-th time-series data.
| \begin{align} L_{\text{MSD}} &= \langle|r(t_{\text{interval}}) - r(0)|^{2}\rangle\\ &= \frac{1}{\text{NM}}\sum_{\text{i=1}}^{\text{N}}\sum_{\text{k=1}}^{\text{M}}|r_{\text{i}}(t_{\text{k}} + t_{\text{interval}}) - r_{\text{i}}(t_{\text{k}})|^{2} \end{align} | (1) |
The present MD simulations under the conditions described in Section 2 were successfully performed without overshooting as shown in Fig. 1 as ball-and-stick views. Figure 1 demonstrates that the simulated atomic arrangements of (a) WNbMoTa and (b) WNbMoTaV alloys with a bcc structure and (c) AuCuNiPt and (d) AuCuNiPdPt alloys with an fcc structure appear to maintain their crystalline structures without transforming to other crystalline phases. It appears that the WNbMoTa and WNbMoTaV alloys with a bcc structure exhibited higher crystallinity close to their ideal bcc structure than AuCuNiPt and AuCuNiPdPt alloys did to an fcc structure. This difference in degree of crystallinity between the simulated bcc and fcc alloys was presumably due to the reproducibility of the GEAM potential.
Ball-and-stick views of the simulation results at t = 100 ps on (100) projections. (a) WNbMoTa and (b) WNbMoTaV HEAs with a bcc structure, and (c) AuCuNiPt and (d) AuCuNiPdPt HEAs with an fcc structure.
The dn was determined by the total PDF, gtotal(r). Figure 2 revealed that the gtotal(r) functions of the AuCuNiPt and AuCuNiPdPt alloys exhibit a sharp primary peak with a peak distance of approximately r = 0.265 nm. The primary peak profiles of gtotal(r) are distributed in the range approximately 0.20 to 0.33 nm, which correspond to the dn of the fcc structure with the coordination number (Z) of 12. The partial PDF, gi-j(r), of the constituent atomic pair, i–j, slightly differs from the gtotal(r) in terms of intensity and the range of distance of the first peak, but gi-j(r) functions of the first peaks are also almost plotted over the range approximately 0.20 to 0.33 nm. From Fig. 2, the authors determined that dn = 0.20 to 0.33 nm. The correlations of the constituent elements were analyzed in a 3-dimensional view as hypothetical bonds by selecting the atomic connections within dn. The results are shown in Fig. 3 for the AuCuNiPt alloy and Fig. 4 for the AuCuNiPdPt alloy where these figures are drawn with ball-and-stick views of Pt atoms from four different projection directions. In Figs. 3 and 4, Pt atom was representatively selected because its atomic radius is the intermediate among those of the constituent elements. In Figs. 3 and 4, the balls (atoms) within 0.20 ≤ dn/nm ≤ 0.33 are drawn with sticks (bonds) in the supercell denoted by dotted lines, whereas those forming the largest cluster are highlighted. Figure 3, representing the AuCuNiPt alloy, shows that the highlighted balls and sticks are distributed over the supercells and they penetrate the boundaries of the supercell through three sets of two parallel planes of the supercell along the x-, y- and z-directions. Thus, it was interpreted that the AuCuNiPt alloy was in a percolated state. Figure 4, representing the AuCuNiPdPt alloy, shows that the highlighted balls and sticks occupy a part of the whole supercells. For instance, the connections of the highlighted balls and sticks in Fig. 4(c) narrowly penetrate the y-z parallel planes along the x-direction, but they do not pass through the x-z planes along the y-direction. Overall, the highlighted balls and sticks barely penetrate the supercell because of the presence of the other clusters comprising the non-highlighted balls and sticks. Thus, the status of connections of the constituent elements shown in Fig. 4 can be understood to be close to the critically percolated state. Thus, it was observed that the AuCuNiPt and AuCuNiPdPt alloys were in completely percolated and near-critically percolated states, respectively. The authors have confirmed the same tendency by selecting other elements than Pt. The completely and near-critically percolated states are consistent with the relationships between x and pcsite in that x = 0.25 > pcsite = 0.198 for the AuCuNiPt alloy, whereas x = 0.20 ∼ pcsite = 0.198 for the AuCuNiPdPt alloy.
Total and partial PDFs, gtotal(r) and gi-j(r), of (a) AuCuNiPt and (b) AuCuNiPdPt alloys where dn is determined from the primary peak of gtotal(r) as dn = 0.20 to 0.33 nm.
Ball-and-stick views at t = 100 ps for the AuCuNiPt HEA for Pt atoms on (a) (100), (b) (010), (c) (001), and (d) (111) projections where the dotted lines indicate the boundary of the supercell. The view directions are along the three axial vectors (⟨100⟩, ⟨010⟩ and ⟨001⟩) and ⟨111⟩ from the upward direction symbolized as “$ \odot $”. The sticks within the nearest neighbor distance, 0.20 ≤ dn/nm ≤ 0.33 only are drawn. The balls and sticks forming the largest cluster are highlighted from the analysis utilizing the software by considering the periodic boundary conditions.
Ball-and-stick views at t = 100 ps for the AuCuNiPdPt HEA for Pt atoms on (a) (100), (b) (010), (c) (001), and (d) (111) projections. The implications of the symbols and illustrations are the same as those in Fig. 3.
The features of profiles of gtotal(r) of the WNbMoTa and WNbMoTaV alloys shown in Fig. 5 demonstrates a primary peak and a secondary peak as a shoulder to the primary one. These peaks correspond, respectively, to dn and the second nearest distance (dnn), which is a considerably short distance in a bcc structure. In ideal pure metals with a bcc structure, the ratio dn/dnn is as small as 1.15. Thus, dn and dnn in the profiles of g(r) are hardly separated completely. In other words, it is difficult to deconvolute dn and dnn because of a mixture of the profiles caused by thermal fluctuations around the ideal site positions of atoms in an actual alloy and in an alloy simulated using MD with a bcc structure. Thus, dn and dnn were determined carefully. Figure 5 reveals that the profiles exhibit a sharp primary peak at a peak distance approximately r = 0.28 nm and the peaks spread in the range 0.235 to 0.305 nm. The secondary peaks are observed at a peak distance approximately r = 0.32 nm and the peaks spread in the range 0.305 to 0.37 nm. Thus, the subsequent analysis was performed with the distances dn = 0.235 to 0.305 nm and dnn = 0.305 to 0.37 nm. The former and latter correspond to Zn = 8 and Znn = 6, respectively. Here, it should be noted that comparison of the gi-j(r) profiles between Figs. 5(a)–(b) at r ranging 0.235 to 0.370 nm revealed the presence of sharp subpeaks (shoulder peaks) in Fig. 5(b), such as Mo–Mo around r ∼ 0.29 nm and Nb–Nb around r ∼ 0.30 nm as denoted by arrows. These subpeaks were characterized by their difference in intensity from gtotal(r) profiles. These subpeaks that deviate from gtotal(r) may be a symptom of the un-percolation observed from PDF profile. Such apparent difference in the gi-j(r) from gtotal(r) hardly observed in alloys with fcc structure shown in Figs. 2(a) and (b) where both alloys are in percolated states. This symptom will be discussed in the next Sub-Section.
Total and partial PDFs, gtotal(r) and gi-j(r), of (a) WNbMoTa and (b) WNbMoTaV alloys where dn was determined from the primary peak of gtotal(r) as dn = 0.235 to 0.305 nm and dnn = 0.305 to 0.370 nm. Arrows denote the presence of sharp subpeaks (shoulder peaks) in gi-j(r) profiles, such as subpeaks in Mo–Mo and Nb–Nb pairs.
The correlations of the constituent elements were analyzed in a 3-dimensional view by selecting the atomic connections within dn = 0.235 to 0.305 nm in Fig. 6 for the WNbMoTa alloy and Fig. 7 for the WNbMoTaV alloy where W was representatively selected in the analysis because of its intermediate atomic radius among the constituent elements. The atomic arrangements of the WNbMoTa alloy shown in Fig. 6(c) indicate that the highlighted balls and sticks barely penetrate the x-z parallel planes along the y-axis from y = 0 to 1. This situation in Fig. 6 is similar to Fig. 4 for the AuCuNiPdPt HEA. The highlighted balls and sticks of the WNbMoTaV alloy shown in Fig. 7 demonstrate the presence of completely-isolated clusters. Thus, it was interpreted that the WNbMoTa and WNbMoTaV alloys formed near-critically percolated and un-percolated states, respectively. The presence of the isolated percolated cluster shown in Fig. 7(a)–(d) was interpreted from the relationships between x and pcsite in that x = 0.20 and pcsite = 0.245 for the WNbMoTaV alloy.
Ball-and-stick views at t = 100 ps for the WNbMoTa HEA for W atoms on (a) (100), (b) (010), (c) (001), and (d) (111) projections. The sticks within the nearest neighbor distance, 0.235 ≤ dn/nm ≤ 0.305, only are drawn. The implications of the symbols and illustrations are the same as those in Fig. 3.
Ball-and-stick views at t = 100 ps for the WNbMoTaV HEA for W atoms on (a) (100), (b) (010), (c) (001), and (d) (111) projections. The sticks within the nearest neighbor distance, 0.235 ≤ dn/nm ≤ 0.305, only are drawn. The implications of the symbols and illustrations are the same as those in Fig. 3.
Subsequently, the authors considered the effects of dnn on the percolations by extending the threshold length to dnn = 0.305 to 0.37 nm from the PDF analysis shown in Fig. 5. This extension was according to the early study,11) which demonstrated that increasing Z tends to decrease pcsite for a close packed (cp) structure at a dimension (d) from 3 to 8 as given in eq. (2).
| \begin{equation} p_{\text{c}}^{\text{cp}}(d)Z_{\text{cp}}(d)\approx 2.4,\quad 3\leq d\leq 8 \end{equation} | (2) |
Ball-and-stick views at t = 100 ps for the WNbMoTaV HEA for W atoms on (a) (100), (b) (010), (c) (001), and (d) (111) projections. The sticks within the nearest neighbor distance and the second nearest neighbor distance from PDF analysis, 0.235 ≤ dn∼nn/nm ≤ 0.370, are drawn. The implications of the symbols and illustrations are the same as those in Fig. 3.
First, the atomic displacements of the MD simulations were quantitatively analyzed for LMSD. Figure 9 depicts LMSD of the alloys calculated during t up to 100 ps. Figure 9 shows that (a) AuCuNiPt and (b) AuCuNiPdPt alloys with fcc structure exhibit LMSD > 0.002 nm2, whereas LMSD of (c) WNbMoTa and (d) WNbMoTaV alloys with bcc structure is in the range of 0.001 to 0.002 nm2. Thus, the present MD simulations for the alloys with fcc structure tended to provide larger LMSD than those with the bcc structure. In details, the values of LMSD of (a) AuCuNiPt and (b) AuCuNiPdPt alloys with fcc structure demonstrate that the magnitude of LMSD was approximately in the order of Cu ∼ Ni > Pd > Au > Pt. Thus, they roughly tended to decrease with increasing atomic number or atomic weight of the constituent elements that are summarized in Table 1. On the other hand, the magnitude of LMSD in (c) WNbMoTa and (d) WNbMoTaV alloys with bcc structure was approximately in the order of Nb > Ta ∼ V ∼ Mo > W, and thus, they did not vary consistently in the order of atomic number or atomic weight of the constituent elements. This inconsistency observed in the alloys with bcc structure may presumably be due to the assumption of GEAM parameters of Nb and V from Ta. Excepting for Nb and V, the magnitude of LMSD of the Mo, Ta and W did not exhibit to decrease with decreasing atomic number or atomic weight because of higher LMSD value of Ta. There remains uncertainty that this inconsistency including the assumption of GEAM parameters of Nb and V may influence the percolation or un-percolation states. However, authors think that inconsistency does not provide completely and drastically different states between percolated and un-percolated states, since the present MD results for WNbMoTa and WNbMoTaV alloys were provided as stable bcc structures as shown in Figs. 1(a) and (b). In other words, this inconsistency will not provide percolated state for the alloys that should be un-percolated state in fact, and vice versa.
The changes in LMSD calculated during t up to 100 ps of (a) AuCuNiPt and (b) AuCuNiPdPt alloys with fcc structure and (c) WNbMoTa and (d) WNbMoTaV alloys with bcc structure.
Incidentally, the decreasing tendency of LMSD with decreasing atomic number or atomic weight found in the present MD simulation for alloys with fcc structure was supported by a previous study19) in a CrMnFeCoNi HEA where the average mean-square atomic displacements (MSADs) are evaluated to be 25.2 × 10−6 nm2. Here, it should be noted that MSAD = 25.2 × 10−6 nm2 is approximately 100 times smaller than LMSD in the present results. In part, this disagreement in magnitude between LMSD and MSAD might be due to the inclusions of thermal expansion and fluctuation of the supercells at T = 1000 K in the present MD simulations under NTp ensemble. However, supplemental MD simulations for alloys with bcc structure (WNbMoTa and WNbMoTaV) and fcc structure (AuCuNiPt and AuCuNiPdPt) denied this possibility. Specifically, supplemental MD simulations with superlattice constants acquired from T = 1000 K and t = 100 ps from Fig. 1 with initial ideal atomic arrangements provided almost the same order of LMSD (LMSD > 0.002 nm2 and 0.001 to 0.002 nm2 for bcc). Furthermore, other supplemental MD simulations with the same lattice constants of the supercell under NTV ensemble also gave almost 100 times larger LMSD value than MSAD. Thus, the authors came to conclude that present MD simulation essentially give the larger LMSD than MSAD from the previous study19) in a CrMnFeCoNi HEA. At any event, the present MD simulation revealed that alloys with bcc structure exhibited the smaller deviations of the atomic displacements from their ideal sites than alloys with fcc structure.
Next, the percolation was analyzed with the characteristics of the PDF profiles. In principle, the PDF gives the probability of the constituent elements that are found at a distance around an atom, as a coordination number of the coordination polyhedral. When the solvent and solute atoms are apparent in the conventional alloys, such as medium- and low-high entropy alloys (MEA and LEA), the formation of clusters in MEAs and LEAs or dispersion of solute atoms in the matrix comprised of solvent atoms in LEA gives different shape in PDFs to the complete chemically disordered solid solutions. In contrast, the HEAs are the alloys with characteristics of the absence of the concept of solvent and solute atoms, in particular, for the HEAs with exact equiatomicity. With this concept, it is expected that the absence of percolated state in HEAs, such as the WNbMoTaV alloy in an un-percolated state among the four alloys simulated in the present study, may lead to change in intensity of the gi-j(r) profiles from gtotal(r). In reality, the authors pointed out a symptom of the un-percolation observed from PDF profile for Mo–Mo and Nb–Nb pairs in Fig. 5(b) in the previous Sub-Section.
To investigate the difference more precisely, PDFs of the Mo–Ta binary systems are shown in Fig. 10. As a result, it is shown that gtotal(r) profiles do not exhibit apparent difference between percolated and un-percolated states, as shown in Figs. 10(a) and (b) as well as in Figs. 10(c) and (d). However, the gi-j(r) profiles demonstrates the difference in the intensity at distances denoted by upward and downward arrows. As a whole, there is a weak tendency denoted by three connected arrows that alloys in percolated states shown in Figs. 10(a) and (c) give difference in the intensity of all the gi-j(r) profiles. On the other hand, alloys in un-percolated states shown in Figs. 10(b) and (d) give difference in the intensity of the gi-j(r) from gtotal(r) at distanced denoted by arrows for the solute atomic pairs of Mo–Mo in Fig. 10(b) and Ta–Ta in Fig. 10(d). Thus, Fig. 10 shows that occurrence of percolation could barely be detected by analyzing gi-j(r) profiles in Mo–Ta binary alloys. This detectability of percolation in binary alloys was presumably due to the decrease in contents of solute elements from 25 to 20 at% as a compensation of increase in the contents of solvent elements from 75 to 80 at%. In contrast, such apparent changes in gi-j(r) profiles were not clearly observed in Fig. 5(b) for all gi-j(r) in HEAs. Presumably, this is due to the fact that all the constituent elements decrease in their contents from 25 to 20 at% due to the compositional characteristic in HEAs in Fig. 5. The detection of percolated state from gi-j(r) profile will be examined in the authors future researches by performing MD simulations for other HEAs, such as AlxCoCrFeNi alloys20) that exhibit transitions from fcc, fcc+bcc and bcc with increasing Al content.
Total and partial PDFs, gtotal(r) and gi-j(r), of (a) Mo25Ta75, (b) Mo20Ta80, (c) Mo75Ta25 and (d) Mo80Ta25 alloys simulated under NTp ensemble (T = 1000 K) for 100 ps. The arrows denote the remarkable difference in gi-j(r) from gtotal(r) where some of which are accompanied by the presence of sharp subpeaks (shoulder peaks). Three-connected arrows exhibit the place where all gi-j(r) difference from gtotal(r) considerably.
MD simulations using the GEAM scheme were performed for quaternary bcc WNbMoTa and fcc AuCuNiPt and quinary bcc WNbMoTaV and fcc AuCuNiPdPt HEAs. The MD results analyzed for PDFs revealed the percolated and isolated states, respectively, when x ≥ pcsite and x < pcsite where pcsite of the nearest neighbor of the bcc and fcc structures is given as pcsite = 0.245 and 0.198, respectively, from the conventional percolation theory. Further investigation revealed that the exact equi-atomic alloys in quaternary alloys (N = 4) with a bcc structure and quinary alloys (N = 5) with an fcc structure are in a critically-percolated state in terms of site percolation.
This research was supported in part by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (JSPS): Grant Program of Scientific Research (B) with the program title of “Alloy Design of High-Entropy Alloys (HEAs) Based on Screening Hypothesis and Statistical Decision Theory and Fabrication of New HEAs” (Grant No. 17H03375).
