Interaction Energies Among Rh Impurities in Pd and Solvus Temperatures of Pd-Rich PdRh Alloys

Abstract

We present the ab-initio calculations for the solvus temperatures (T_solvus = 820∼860 K) of Pd_1−cRh_c (0.09 ≤ c ≤ 0.12) in which the Rh atoms are treated as impurities in Pd. The interaction energies (IEs) among the Rh impurities in Pd, being used in the real-space cluster expansion for the internal energies in the free energies, are determined by the ab-initio calculations based on the full-potential Korringa-Kohn-Rostoker Green’s function method, combined with the generalized gradient approximation in the density functional theory. The configurational entropy calculations are based on the cluster variation method within the tetrahedron approximation in which the 2∼4 body IEs are treated exactly within a tetrahedron of the 1st-nearest neighbor (nn) pairs. In order to take into account the 2-body IEs at the long-distance neighbors, we renormalize the 1st-nn 2-body IE by including the 2-body IEs up to the 10th-nn, because the 9th-nn 2-body IE is comparatively large. To realize the precise calculations for the T_solvus of Pd_1−cRh_c, we also investigate the following three effects on the IEs among the Rh impurities: (1) the electron excitation due to the Fermi-Dirac distribution, (2) the thermal lattice vibration by the Debye-Grüneisen model, and (3) the local lattice distortion for the 1st-nn 2-body IE. The calculated results for the T_solvus of Pd_1−cRh_c agree fairy well (within the error of ∼50 K) with the observed T_solvus.

1. Introduction

In a previous study,¹⁾ we successfully reproduced the observed solvus temperatures (T_solvus = 820∼860 K) of the Pd-rich Pd_1−cRh_c alloys (0.09 ≤ c ≤ 0.12),²⁾ by the free energy calculations based on the cluster variation method within the tetrahedron approximation (CVMT) for the configurational entropy. The internal energies in the free energies were determined by the real space cluster expansion (RSCE)^3–6) in which the Rh atoms were treated as impurities in Pd. The 2∼4 body interaction energies (IEs) among the Rh impurities, up to the tetrahedron formed by the 1st-nearest neighbor (nn) pairs, were calculated by the Korringa-Kohn-Rostoker Green’s function (KKR) method for the perfect and impurity systems. We found that the 2-body interaction between the Rh impurities in Pd is considerably short-range within a distance of the 8th-nn and that the 3-body and 4-body interactions in the tetrahedron formed by the 1st-nn pairs are considerably weak. We have shown that the T_solvus obtained by the free energy calculations based on the CVMT, with these 2∼4 body IEs for the internal energy, agree well with the observed T_solvus, if the dominantly attractive 1st-nn 2-body IE is renormalized by including the weakly repulsive 2-body IEs at the 2∼8th nns, the sum total of which slightly reduces the attractive 1st-nn 2-body IE.

In the previous calculations, we fixed the temperature (T_FD) of the Fermi-Dirac (FD) distribution at 800 K as in the usual KKR calculations,^7,8) and did not investigate T_FD dependence for the IEs because we believed that T_FD dependence is very low. We did not investigated the 2-body IEs beyond the 8th-nn either because the 2-body IEs from the 4th-nn to 8th-nn are almost zero, as seen in Fig. 4 of Ref. 1. However, we very recently found that the T_FD dependence of the IEs among Ru impurities in Pd is very important to reproduce the observed T_solvus of the Pd-rich PdRu alloys and that the 2-body IE at the 9th-nn is also comparably large.⁹⁾ In addition, we found that the local lattice distortion effect makes the attractive 1st-nn Ru–Ru interaction considerably stronger.

Moreover, Mohri and Chen succeeded in describing the phase stability and phase equilibria for many binary alloys using the ab-initio free energy calculations based on the CVMT for the configurational entropy and suggested the importance of the thermal vibration effect.^10,11) For example, they reproduced the experimentally determined transition temperature (1023 K) of the FePd alloys very well (1030 K) by including the thermal lattice vibration effect.

Our previous calculations for Pd-rich PdRh alloys did not included the above-mentioned effects. Nevertheless, the calculated T_solvus are in good agreement with the experimental results. We have had a question why the calculated results for the T_solvus of the Pd-rich PdRh alloys agree with the experimental results, without these effects. Thus, in the present study, we investigate the following four effects for the IEs among the Rh impurities in Pd-rich PdRh alloys, being neglected in Ref. 1): (1) the 2-body IEs beyond the 8th-nn, up to the 20th-nn, (2) the electron excitation due to the FD distribution, (3) the thermal vibration effect, and (4) the local lattice distortion effect for the dominant 1st-nn 2-body IE.

In Sect. 2, we describe the calculation procedures for the FPKKR with the electron excitation effect due to the FD distribution, the thermal lattice vibrational effect, and the free energy calculations based on the CVMT. In Sect. 3, we first discuss the calculated results for the IEs among the Rh impurities in Pd. We concentrate on the effects due to the neglected terms in the previous study and show that the main part of the local lattice distortion effect (a considerably strong attraction) may be cancelled out by the fairy strong 2-body repulsion at the 9th-nn. It is also shown that the thermal vibration effect becomes very weak, while the T_FD dependence is very high. We clarify that the selection of T_FD = 800 K in the previous calculations was an unexpected good fortune. We then show that the CVMT calculation results for the T_solvus of the Pd_1−cRh_c (0.09 ≤ c ≤ 0.12) alloys, including the above-mentioned four effects, agree fairy well with the observed T_solvus (820∼860 K) for the Pd_1−cRh_c alloys (0.09 ≤ c ≤ 0.12). It is noted that the calculated results are just a little higher (∼50 K) than the experimental results because the renormalized 2-body attractive interaction at the 1st-nn becomes a little stronger by including the neglected terms in the previous study. It is also shown that the small discrepancy (overestimation) between the observed and CVMT calculation results may be reduced from ∼50 K to ∼20 K by the free energy calculations based on the cluster variation method within the tetrahedron-octahedron approximation (CVMTO), in which the 1st- and 2nd-nn IEs can be treated exactly. Section 4 summarizes the main results of the present study.

2. Calculation Procedures

We briefly describe the calculation procedures for the FPKKR with the electron excitation effect due to the FD distribution (in Sec 2.1), the thermal vibration effect by the Debye-Grüneinsen model (in Sec 2.2), and the free energy calculations based on the CVMT for the configurational entropy (in Sec 3.3), because the details of the calculational procedures have already been described in Refs. 1, 3–10, 12–14).

2.1 Total energies of impurity systems

The calculations for the total energy of the Rh impurities in Pd are based on the density functional theory (DFT) in the generalized gradient approximation (GGA). In order to solve the Kohn-Sham equations, we used a multiple scattering theory in the form of the Korringa-Kohn-Rostoker (KKR) Green’s function method for the full-potentials (FP). We used the screened version of the FPKKR band calculations for the Pd host, which significantly simplifies the numerical calculations by introducing the short-range structural Green’s functions.^13,14) In order to simplify the total energy calculations in the GGA formalism, we used the electronic densities self-consistently obtained by the local spin density approximation in the DFT. The accuracy of the present GGA calculations has been described in Refs. 7 and 12).

In order to calculate the Rh-concentration dependence of the T_solvus of the Pd_1−cRh_c alloys, we must first calculate the total energy difference between the disordered and segregated states. The total energies (per tetrahedron basic cluster) of the disordered and segregated states within the CVMT approximation are written as follows:   

\begin{equation} E_{\textit{dis}}=2\sum_{ijkl}E_{ijkl}\omega_{ijkl} \end{equation} (1)

where i, j, k, and l distinguish Pd or Rh, E_ijkl is the total energy (per tetrahedron basic cluster), and ω_ijkl is the cluster probability. E_ijkl was calculated using the IEs up to the 4-body, obtained in the RSCE.^1,3–6,9) The characteristic feature of the RSCE is that the IEs among the Rh impurities in the Pd-rich PdRh alloys are uniquely and successively determined from the low-order to high order (for the first time from the Pd-host, and one-body, two-body, ….) and independent of the concentration.^3–6)

In the present study, we calculated the T_FD-dependence of the IEs due to the FD distribution. As shown in Ref. 8), the introduction of the FD distribution into the ab-initio Green’s function calculations is very useful to simplify the ground state calculations. Based on a contour integration in the complex energy plane, we evaluated the residues at a few Matsubara frequencies, while the remaining integral was analyzed by the Sommerfeld expansion. The imaginary part (δ) in the complex energy plane is related to the T_FD of the FD distribution (δ = πkT_FD). The total energy was calculated by the grand canonical potential for which a simple formula is given. Since the total energy change due to the electron excitation is usually expected to be negligible, the FD distribution at T_FD = 800 K has been used to efficiently perform the ground state calculations for metals.^7,8) However, in the present paper, we show that the T_FD-dependence of the IEs among the Rh impurities in Pd is not as low as we have usually been expecting it to be.^7,15)

In the present study, we also evaluated the local lattice distortion effect on the 1st-nn 2-body IE in Pd.^16–20) The local lattice distortion energy is defined as the total energy change caused by the lattice distortion around the impurities.^18–20) The positions of the displaced atoms were determined by the condition of the Hellman-Feynman force = 0. The detailed discussion for the local lattice distortion has been given in Ref. 16).

2.2 Vibrational free energies

In addition to the local lattice distortion effect due to the difference in the atomic size, even existing at T = 0 K, the effect of the lattice softening due to the thermal vibration at finite temperatures was considered in the present study. The vibrational free energy for the lattice softening was calculated using the Debye-Grüneisen model which is based on the quasi-harmonic approximation. By following Ref. 10), the vibrational free energy ($F_{\text{vib}}^{(n)}(a,T)$) of each phase (n) in the basic Pd_4−nRh_n (n = 0∼4) cluster used in the CVMT (see Fig. 2 in Ref. 1); H = Pd, I = Rh) is written as follows:   

\begin{equation} F_{\text{vib}}^{(n)}(a,T)=E_{\text{vib}}^{(n)}(a,T)-TS_{\text{vib}}^{(n)}(a,T) \end{equation} (2)

where $E_{\text{vib}}^{(n)}(a,T)$ is the vibrational internal energy and $S_{\text{vib}}^{(n)}(a,T)$ is the vibrational entropy, depending on the lattice parameter (a), temperature (T), and the phase (n) specified by the basic cluster (Pd_4−nRh_n). These terms are given by the following formulas:   

\begin{equation} E_{\text{vib}}^{(n)}(a,T)=\frac{9}{8}k_{B}\cdot\Theta_{D}^{(n)}+3k_{B}\cdot T\cdot D\left(\frac{\Theta_{D}^{(n)}}{T}\right) \end{equation} (3)

  

\begin{equation} S_{\text{vib}}^{(n)}(a,T)=3k_{B}\cdot\left[\frac{4}{3}D\left(\frac{\Theta_{D}^{(n)}}{T}\right)-\ln\left\{1-\exp\left(-\frac{\Theta_{D}^{(n)}}{T}\right)\right\}\right] \end{equation} (4)

where $\Theta _{\text{D}}^{(n)}$ is the Debye temperature of the phase (n) in the basic Pd_4−nRh_n (fcc structure for Pd₄ and Ru₄, L1₂ for Pd₃Ru and PdRu₃, and L1₀ for Pd₂Ru₂) cluster, determined by using the lattice parameter dependence of the total energy.⁹⁾ D(x) is a Debye function and the first term of the vibrational internal energy (eq. (3)) represents the zero-point energy.

2.3 Free energy differences between disordered and segregated states and solvus temperatures

In the present study, we calculated the free energy difference (eq. (5)) between the disordered and segregated states,   

\begin{equation} \Delta F(a,T)=\Delta E(a,T)-TS_{\text{con}}(a,T) \end{equation} (5)

  

\begin{equation} \Delta E(a,T)=E_{\text{dis}}(a,T)-E_{\text{seg}}(a,T) \end{equation} (6)

where $E_{\text{dis}}(a,T)$ and $E_{\text{seg}}(a,T)$ are the internal energies corresponding to the disordered and segregated states, respectively, depending on the lattice parameter (a) and temperature (T), and include the contributions from the electron excitation due to the FD distribution and the thermal vibration effect (eq. (2)) based on the Debye-Grüneisen model.⁹⁾ $S_{\text{con}}(a,T)$ is the configurational entropy calculated by CVMT. $S_{\text{con}}(a,T)$ and $\Delta E(a,T)$ are expressed by the cluster probabilities {ω_ijkl} of the tetrahedron, where i, j, k, and l distinguish Pd or Rh. Thus, by minimizing the free energy difference (eq. (5)) with respect to the cluster probabilities {ω_ijkl}, we can determine the cluster probabilities in the equilibrium state.⁹⁾ The Rh-concentration limit at a given temperature (T_solvus), was calculated using the calculated values for {ω_ijkl}, as the Rh-concentration = $\sum_{jkl = \text{Pd,Rh}}\omega _{\text{Rh},jkl} $. In the present calculations, the equilibrium lattice parameter of the Pd_1−cRh_c alloy at a given temperature (T) was approximated by the minimum of $E_{\text{Pd}_{4}}(a,T)$ with the FD distribution and the thermal vibration effects, discussed in Sects. 2.1 and 2.2, since the difference between the equilibrium lattice parameters of Pd_1−cRh_c and Pd is very small for the low Rh-concentration, as shown in Sec. 3.2.

3. Calculated Results

We first discuss the following four effects on the IEs among the Rh impurities in Pd: (1) the 2-body IEs beyond the 8th-nn, up to the 20th-nn, (2) the electron excitation due to the FD distribution, (3) the thermal lattice vibration at finite temperatures, and (4) the local lattice distortion for the 1st-nn 2-body IE. We then show that the calculated results for the T_solvus of the Pd_1−cRh_c (0.09 ≤ c ≤ 0.12) alloys, including the above-mentioned effects, agree fairy well (within error of ∼50 K) with the observed T_solvus (820∼860 K) of the Pd_1−cRh_c alloys (0.09 ≤ c ≤ 0.12). It is also shown that the small discrepancy (overestimation) between the observed and CVMT calculation results may be significantly reduced by the free energy calculations based on the cluster variation method within the tetrahedron-octahedron approximation (CVMTO), in which the 1st- and 2nd-nn 2-body IEs can be treated exactly.

3.1 Interaction energies among Rh impurities in Pd

In the CVMT, we can exactly treat the IEs only up to the 4-body in the tetrahedron formed from the 1st-nn pairs, the definitions of which are given in Ref. 1). However, as shown in Fig. 1, the 9th-a-nn 2-body IE is comparatively large, being neglected in the previous study,¹⁾ although the attractive 1st-nn 2-body IE is dominant. The positive energy means repulsion, while the negative attraction. It is noted that the 1st $(0.5,0.5,0)$- and 9th-a $(1.5,\text{1}.5,0)$-nn sites align along the ⟨110⟩ direction, although we have not understood the micro-mechanism of the comparatively large repulsion at the 9th-a-nn. Thus, as the first approximation, we renormalized the 1st-nn 2-body IE by including the 2-body IEs up to the 10th-nn as follows:¹⁾   

\begin{align} \Delta\tilde{E}_{1}^{\text{2-body}}&=\Delta E_{1}^{\text{2-body}}+\frac{6}{12}\Delta E_{2}^{\text{2-body}}+\frac{24}{12}\Delta E_{3}^{\text{2-body}} \\ &\quad+\cdots+\frac{24}{12}\Delta E_{10}^{\text{2-body}} \end{align} (7)

where the coefficient in the nth-term is the ratio of the coordination number of the nth-nn shell to that of the 1st-nn shell. It is noted that the sum total of the long-range 2-body IEs is kept in the renormalized 1st-nn IE. In the present paper, the CVMT calculations with the 1st-nn 2-body IE renormalized by the 2-body IEs up to the 10th-nn are written as CVMT10. The thermal vibration effect on the renormalized 1st-nn IE was taken into account in the present calculations, as discussed in Ref. 9).

Fig. 1

Calculated results for the distance dependence (1∼20th nn) of the Rh–Rh IEs in Pd at the lattice parameter of 7.40 a.u. and with the FD distribution at T_FD = 800 K. There are two nonequivalent sites for the 9th-, 13th-, 16th-, 17th-, and 18th-nns. The 9th-a and 9th-b nn sites are $(1.5,1.5,0)$ and $(2,0.5,0.5)$ in the unit of the lattice parameter (7.40 a.u.), respectively. See the text for details.

The 3-body and 4-body IEs are listed in Table 1 as their lattice parameter (a) dependence with the FD distribution at T_FD = 800 K. These values are considerably small, compared to the 1st-nn 2-body IEs shown in Fig. 1. It is also noted that the contributions from the 3-body and 4-body are proportional to the 3rd- and 4th-power of the Rh-concentration. Thus, in the present study for the low Rh-concentration, we did not consider the contributions from the 3-body and 4-body IEs of the larger clusters. Figure 2 shows the calculated results for the distance (1∼10th)- and T_FD (= 200, 400, 600, 800, 1000 K)-dependence of the 2-body IEs, at the lattice parameter of 7.40 a.u. It is noted that the 2nd-nn IE changes from 0.0093 eV (T_FD = 200 K) to 0.0024 eV (T_FD = 1000 K); the T_FD-dependence of the FD distribution, being usually neglected in the ab-initio calculations, is not as small as we have usually expected it to be, and the electron excitation due to the FD distribution attractively works on the 2-body IEs of the Rh impurities in Pd. However, it is also obvious that the 2-body IEs are almost unchanged at the narrow temperature region of 800 K∼900 K (for example, 0.0036 eV∼0.0030 eV for the 2nd-nn 2-body IE). Thus, in the present calculations for the observed T_solvus = 820∼860 K, the T_FD was fixed at 800 K. As a result, it may be concluded that the selection of T_FD = 800 K in the previous calculations was an unexpected good fortune.

Table 1 Calculated results (in eV) of the 3-body and 4-body IEs among the Rh impurities in Pd shown in Fig. 2 (H = Pd, I = Rh) in Ref. 1) for the lattice parameters (a = 6.8∼8.0 a.u.). The T_FD is fixed at 800 K. See the text for details.

Fig. 2

Calculated results for the distance (1∼10th nn) dependence of the Rh–Rh IEs in Pd, at the lattice parameter of 7.40 a.u. and with the Fermi-Dirac distribution at T_FD = 200, 400, 800, and 900 K. The repulsive 2nd-nn IE becomes smaller and smaller with the increasing T_FD. See the text for details.

The vibrational internal energy (eq. (3)) and vibrational entropy (eq. (4)) for the phase (n) specified by the basic Pd_4−nRh_n (n = 0∼4) clusters used in the CVMT, were calculated using the Debye temperature which are determined from the lattice parameter dependence of the total energies per $E_{\text{Pd}_{4 - n}\text{Rh}_{n}}$, as discussed in Sect. 2.2. The lattice parameter dependences (6.8∼8.0 a.u.) of the total energies ($E_{\text{Pd}_{4 - n}\text{Rh}_{n}}$, n = 0∼4) without the thermal vibration effect are shown in Fig. 3(a) and those with the thermal vibration effect at T_vib = 700, 800, and 900 K are shown in Figs. 3(b), (c) and (d), respectively. The FPKKR calculations for the lattice parameter dependence of the total energies of the basic $E_{\text{Pd}_{4 - n}\text{Rh}_{n}}$ (n = 0∼4) clusters were carried out in intervals of 0.1 a.u. Table 2 shows the calculated results for the renormalized 1st-nn IE with the thermal vibration effect. It is noted that the T_vib dependence is very low.

Fig. 3

Lattice parameter dependence (6.8∼8.0 a.u.) of the total energies per basic cluster ($E_{\text{Pd}_{4 - n}\text{Rh}_{n}}$) of the five phases consisting of Pd_4−nRh_n (n = 0∼4) shown in Fig. 2 of Ref. 1): (a) the total energies without the thermal vibration effect; (b), (c) and (d) those with the thermal vibration effect at T_vib. = 700, 800 and 900 K, respectively. All these calculations were executed with the FD distribution at T_FD = 800 K. See the text for details.

Table 2 Calculated results (in eV) for the renormalized Rh–Rh 1st-nn IEs with the thermal lattice vibration at T_vib. = 600, 700, 800, 900, and 1000 K. The T_FD is fixed at 800 K. See the text for details.

In the present study, we also took into account the local lattice distortion effect on the dominant 1st-nn 2-body IE in the Pd-rich PdRh alloys. The lattice parameters of the Pd_1−cRh_c (c ≤ 0.12) around T_vib = 700, 800, and 900 K are the values between 7.5 and 7.6 a.u., as seen in Figs. 3(b), (c), (d). The calculated lattice distortion energies for the 1st-nn 2-body IEs at a = 7.5 and 7.6 a.u. are considerably large (−0.0123 and −0.0125 eV); the local lattice distortion effect also attractively works on the 1st-nn 2-body interaction as well as the T_FD effect shown in Fig. 2. In the following calculations for the T_solvus, the lattice distortion energies at the equilibrium lattice parameters, corresponding to the given temperatures (T_vib), were estimated by interpolating or extrapolating these two values at 7.5 and 7.6 a.u. because the lattice parameter dependence of them is very low.

3.2 Solvus temperatures of Pd_1−cRh_c

We now discuss the calculated results for the Rh-concentration dependence of the T_solvus of the Pd_1−cRh_c (0.09 ≤ c ≤ 0.12) alloys. In order to clarify the thermal vibration effect, we carried out the three kinds of calculations: (a) without the thermal vibration effect (a is fixed at 7.46 a.u.), (b) with the temperature-dependent equilibrium lattice parameters of the Pd-host, being expanded by the thermal vibration effect, but neglecting the vibrational free energies (eq. (5)) for the IEs, and (c) with the overall thermal vibration effect (the thermal expansion of Pd-host and the IEs with the contributions from the vibrational free energies). It is obvious that the lattice expansion is caused by the vibrational free energies. However, we carried out the calculation for case (b) since it may be interesting to study the lattice expansion effect on the T_solvus of the Pd_1−cRh_c alloys.

Figure 4(a) shows the calculated result for the above-mentioned case (a). We found that the T_solvus is overestimated (∼50 K), although the previous calculations reproduce very well the experimental results (see Fig. 9 in Ref. 1)). The difference (∼50 K) between these calculated results may be easily understood by considering the sum total of the neglected terms in the previous calculations. Since the attraction due to the local lattice distortion slightly overcomes the 2-body repulsion at the 9th-nn, the main part of the attraction due to the local distortion is cancelled out by the 2-body repulsion at the 9th-nn and the remaining weak attraction slightly raises the T_solvus. We also found that the observed T_solvus is very well reproduced by the calculations with the thermal expansion effect (the above-mentioned case (b)), as shown in Fig. 4(b), but again overestimated (∼50 K) by the calculations with the overall thermal vibration effect (the above-mentioned case (c)), as shown in Fig. 4(c). These differences among Figs. 4(a), (b), and (c) are mainly due to the differences in the renormalized 1st-nn IEs $( - 0.0449, - 0.0429, - 0.446\,\text{eV})$ used in the three kinds of calculations ((a), (b), (c)). As a result, we can conclude that the present CVMT calculations including all the above-mentioned effects reproduce fairy well (within the error of ∼50 K) the observed T_solvus (820∼860 K) for the Pd_1−cRh_c (0.09 ≤ c ≤ 0.12) alloys, as shown in Fig. 4(c). This successful result may be mainly due to the fact that the weakly repulsive 2-body IEs at the distant neighbors may be appropriately renormalized in the dominant 1st-nn 2-body IE, by using eq. (7).

Fig. 4

Rh-concentration dependence of the T_solvus of the Pd_1−cRh_c (0.09 ≤ c ≤ 0.12) alloys obtained by the three kinds of CVMT10 calculations at T_FD = 800 K, in which the 2-body IEs are taken into account up to the 10th-nns; (a) without the thermal vibration effect, (b) with the thermally expanded lattice parameters of the Pd-host, neglecting the vibrational free energies for the IEs, and (c) with the thermal lattice vibration effect, including the thermal expansion of Pd-host and the vibrational free energies for the IEs. See the text for details.

We now discuss the theoretical limit of the CVMT10 approximation, in which the 1st-nn 2-body IE is renormalized by the 2-body IEs at the distant-neighbors, as shown in eq. (7). According to the Monte Carlo simulations by Schweika,^21,22) the overestimation due to this approximation becomes significant for the binary alloys with the competing 2-body interactions at the 1st- and 2nd-nns, in which the 2nd-nn 2-body interaction is considerably repulsive against the strongly attractive 2-body interaction at the 1st-nn. We have already shown that the overestimation of the T_solvus, due to the approximation of eq. (7), is significant for the Pd-rich PdRu alloys, being the binary alloys with the competing 2-body interactions at the 1st- and 2nd-nns.⁹⁾ In order to clarify the influence of this approximation for the present systems, we carried out the calculations based on the cluster variation method within the tetrahedron-octahedron approximation (CVMTO), in which the 1st- and 2nd-nn 2-body IEs can be taken into account. We treated the repulsive 2-body IE at the 2nd-nn, independently from the attractive 2-body IE at the 1st-nn, and renormalized the 1st-nn 2-body IE by including the 3∼10th-nn 2-body IEs (eliminating the 2nd-nn part from eq. (7)). In the present paper, the CVMTO calculations with the 1st-nn 2-body IE renormalized by the 2-body IEs up to the 10th-nn are written as CVMTO10. It is noted that the thermal vibration effect on the 2nd-nn 2-body IE was calculated by the process discussed in Ref. 9), with the basic cluster (Pd_6−nRh_n) of the octahedron. We found that the overestimation (∼50 K) due to the CVMT10 approximation is reduced to ∼20 K by the CVMTO10 approximation, as seen in Fig. 5(a). In order to study the effect due to the repulsive 2-body IEs at the further distant-neighbors, we then carried out the CVMTO10 calculations with the different renormalization of the 1st- and 2nd-nn IEs: (1) the 2nd-nn 2-body IEs including the 4th-nn 2-body IE, (2) the 2nd-nn 2-body IEs including the 9th-a-nn 2-body IE, (3) and the 2nd-nn 2-body IEs including the 3∼10th-nn 2-body IEs, in which all of the remaining 2-body IEs were included in the 1st-nn 2-body IE. It is noted that the T_solvus decreases with the increase in the repulsive 2nd-nn 2-body IE renormalized by including the repulsive 2-body IEs at the distant neighbors, as shown in Fig. 5(b). We found that the overestimation of ∼20 K changes to the underestimation of ∼50 K by the different renormalization of the 2nd-nn 2-body IE including the distant-neighbor 2-body IEs. Although the discrepancies (overestimation) between the observed and CVMT10 calculation results are not so large, if we want to more accurately reproduce the observed T_solvus, we must carry out the CVM calculations with the larger basic cluster.²³⁾

Fig. 5

Rh-concentration dependence of the T_solvus of the Pd_1−cRh_c (0.09 ≤ c ≤ 0.12) alloys obtained by the CVMTO10 calculations with the different renormalizations of the 1st- and 2nd-nn IEs, in which the 2-body IEs are taken into account up to the 10th nns; (a) the 2nd-nn IE is treated independently from the attractive 2-body IE at the 1st-nn, including the 3∼10th-nn 2-body IEs, and (b) the four kinds of different renormalizations for the 2nd-nn 2-body IEs, while the 1st-nn 2-body IE renormalized by the remaining 2-body IEs. The T_FD is fixed at 800 K. See the text for details.

Finally, we discuss the error caused by using the equilibrium lattice parameters of Pd instead of those of the Pd_1−cRh_c alloys. In order to take into account the Rh-concentration dependence of the equilibrium lattice parameters of Pd_1−cRh_c, we must prepare the concentration-dependent equilibrium lattice parameter $a(T,c)$ by fitting the five equilibrium lattice parameters of Pd_4−nRh_n (n = 0, 4) at a given temperature T, shown in Fig. 3. The Rh-concentration limit at a given temperature (T) was self-consistently determined by using the obtained $a(T,c)$. The CVMT10 calculation results with the lattice parameter $a(T,c)$ are shown in Fig. 6 together with the CVMT10 calculation results with the lattice parameter $a(T,0)$ which were already shown in Fig. 4(c). We found that the CVMT10 calculation results with the lattice parameter ($a(T,c)$) of the Pd_1−cRh_c alloys are almost the same as the calculated CVMT10 results with the lattice parameter ($a(T,0)$) of Pd. For example, the maximum difference (at c = 0.12) between the calculated results with and without the Rh-concentration dependence of the lattice parameter is less than 10 K. Thus, we can conclude that the above-mentioned results, obtained by using the equilibrium lattice parameters of Pd instead of those of the Pd_1−cRh_c alloys, are almost unchanged.

Fig. 6

Rh-concentration dependence of the T_solvus of the Pd_1−cRh_c (0.09 ≤ c ≤ 0.12) alloys, obtained by the CVMT10 calculations with and without the Rh-concentration dependence of the lattice parameters ($a(T,c)$ and $a(T,0)$). The T_FD is fixed at 800 K. See the text for details.

4. Summary

We presented the ab-initio calculations for the T_solvus of the Pd_1−cRh_c (0.09 ≤ c ≤ 0.12) alloys, in which the Rh atoms were treated as impurities in Pd. The internal energies in the free energies were calculated by the present RSCE with the IEs up to the 4-body, all of which were uniquely and successively determined from the low-order to high order by the ab-initio calculations based on the FPKKR Green’s function method for the perfect and impurity systems, combined with the generalized gradient approximation in the density functional theory. The configurational entropy calculations were based on the cluster variation method within the tetrahedron approximation (CVMT) and tetrahedron-octahedron approximation (CVMTO). We clarified the neglected effects in the previous study.¹⁾

We first showed that the 2-body interaction is repulsive at the 2nd-, 4th-, 5th- and 9th-a-nn, although it is very weak compared to the large attraction at the 1st-nn. In addition to the study of the 2-body IEs at the distant neighbors, we also investigated the following three effects on the IEs among the Rh impurities in Pd: (1) the electron excitation due to the FD distribution, being usually neglected in the ab-initio calculations, (2) the lattice softening due to the thermal vibration at high temperatures, and (3) the local lattice distortion for the 1st-nn 2-body IE, even existing at T = 0 K. We found that the effects (1) and (3) attractively work on the IEs. For the thermal vibration effect (2), the lattice parameter expansion by the thermal vibration effect repulsively works on the IEs (decrease for T_solvus), if the contributions from the vibrational free energies for the IEs are neglected. On the other hand, the contributions from the vibrational free energies attractively work on the IEs (increase in T_solvus). As a result, the overall thermal vibration effect becomes very weak for the IEs among the Rh impurities in Pd and does not contribute to the T_solvus of the Pd_1−cRh_c alloys. It is also noted that the main part of the attraction due to the lattice distortion effect (3) is cancelled out by the 2-body repulsion at the 9th-nn.

We then showed that the free energy calculations based on the CVMT10 approximation, in which the 1st-nn 2-body IE is renormalized by including the 2∼10th-nn 2-body IEs, fairy well (within the error of ∼50 K) reproduce the observed T_solvus (= 820∼860 K) of the Pd_1−cRh_c alloys. We discussed that the selection of T_FD = 800 K is very important. We also showed that the small discrepancies (overestimation) between the observed and CVMT10 results may be reduced from ∼50 K to ∼20 K by the CVMTO10 calculations, in which the repulsive IE at the 2nd-nn IE can be treated exactly, independent from the 1st-nn IE renormalized by the 3∼10 IEs.

Acknowledgement

The authors are grateful for the financial support from the Ministry of the Education, Culture, Science and Technology (JSPS KAKENHI Grant Nos. 15K06422 and 16K06710). This study was partly supported by JSPS KAKENHI Grant Number JP15K14103 and by the Structural Materials for Innovation of the Japan Science and Technology (JST). One of the authors (T. Mohri) appreciates their support.

This study was almost completed when T. Hoshino was at the Institute for Material Research, Tohoku University. C. Liu and T. Hoshino would like to express their sincere thanks to the crew of the Center for Computational Materials Science of the Institute for Materials Research, Tohoku University, for their continuous support of the SR16000 supercomputing facilities.

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