Hydrogen Solubility in Pd₃Ag Phases from First-Principles Calculation

Abstract

First-principles calculation was used to systematically investigate hydrogen solubility in Pd₃Ag phases. It was found that the solubility of hydrogen in Pd₃Ag phases was much greater than in face-centered cubic (FCC) Pd, suggesting that Ag atoms enhanced hydrogen solubility with respect to FCC Pd. In addition, the present calculation also revealed that the anti-site defect formation enthalpies of Pd₃Ag were close to zero, and the values of vacancy were positive and large, which indicated that Pd₃Ag distributed compactly. In the process of hydrogen separation, anti-site defects decreased the hydrogen solubility in the Pd₃Ag phases, i.e., the ordered Pd₃Ag phases bestowed excellent properties of H selectivity. The results presented not only explore the fundamental properties of Pd₃Ag phases and their various potential applications, but also agree with experimental observations reported in the literature.

1. Introduction

In recent decades, Pd–Ag bimetallic systems have attracted significant research interest for their superior hydrogen (H) catalytic and separation properties [1,2,3,4], mainly due to their unique hydrogen selectivity and permeability [5,6]. It is well known that Ag would improve a pure Pd membrane lifetime and lower the operation temperature of H separation [1,7,8]. Moreover, substituting Pd with Ag can reduce the price of the membranes, and the so-called hydrogen embrittlement would be eliminated [8,9,10]. Accordingly, after studies on hydrogenation were performed [11,12], interest in the Pd–Ag system for use in hydrogen energy has increased [13,14].

As a typical hydrogen separation material, PdAg membranes with various Ag content have been the focus of attention. PdAg phases with 23–32% of Ag have an optimal performance in hydrogen permeation experiments [9,10,15]. Very recently, the mechanism of hydrogen diffusion in Pd₃Ag phases has been reported by means of first-principles calculation [13]. Nevertheless, there is a lack of further theoretical investigations on the effects of Pd₃Ag phases on H selectivity. In addition, point defects have important effects on the physicochemical properties of PdAg phases and play a deleterious role in the performance of Pd₃Ag phases. As for the fundamental effects of point defects on H selectivity, however, there are no more results so far in the literature to the best of our knowledge.

By means of a highly accurate theoretical calculation based on density function theory (DFT), this study is aimed to systematically investigate the effects of point defects on hydrogen solubility in Pd₃Ag phases, as, based on experimental observations, this composition of PdAg membranes confers high hydrogen permeability [9,10,15]. H solubility in Pd₃Ag phases with and without point defects were also computationally predicted, and the mechanism in relation to bond charge and charge density was determined. The derived results not only are in good agreement with experimental results in the literature, but also provide a deep understanding of hydrogen separation by Pd₃Ag.

2. Theoretical Methods

The present first-principles calculation was performed by means of the well-established Vienna Ab initio Simulation Package (VASP) [16,17]. To handle the wave functions near the core region, the PAW method has been widely used [18]. The exchange-correlation potential was approximated by the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) [19]. Electron wave functions were expanded in a plane-wave basis set with a kinetic energy cutoff of 400 eV, and periodic boundary conditions were fixed in three directions.

A unit cell of 2 × 2 × 2 (32 atoms) was chosen for L1₂ Pd₃Ag. To simulate the Pd₃AgH phases, one hydrogen atom was added at the octahedral (O) and tetrahedral (T) sites of the unit cell, and each phase with H addition was required to fully relax. As typical examples, Figure 1 shows the O and T sites of the L1₂ Pd₃Ag phase. Notice from Figure 1c,d that there are two kinds of O sites for Pd₃Ag: the O1 site is only surrounded by Pd atoms, and the O2 site by four Pd and two Ag atoms.

The relaxation and static calculations were conducted in a GAMMA-centered scheme with 9 × 9 × 9 and 11 × 11 × 11 respectively, together with a k-point broadening of 13 × 13 × 13 for the calculations of density of states (DOS) and bond charge. The force criterion acting on each atom during relaxation was 1.0 × 10⁻³ eV/Å, and the criteria of energy convergence were 1.0 × 10⁻² and 1.0 × 10⁻³ meV for relaxation and static calculations respectively.

The zero-point energy (ZPE) of H is considered in the present study and is derived through the following formula:

$$E_{ZPE}\approx \frac{1}{2}{\displaystyle \sum _I\hslash {\nu }_{\mathrm{i}}}$$

(1)

where $\hslash$ and ${\nu }_{\mathrm{i}}$ are Planck’s constant and the frequency associated with the harmonic mode, respectively. It should be noted that the vibration frequencies (${\nu }_{\mathrm{i}}$) of H atoms at interstitial sites of the Pd₃Ag phases are derived through harmonic approximation by relaxing only the hydrogen atom and fixing the Pd and Ag atoms, since the Pd and Ag atoms have a much larger mass than the H atoms [20].

The solubility (C) of hydrogen in the Pd₃Ag phases can be estimated using the formula C = K_sP_H^0.5, where P_H is the H pressure in the gas phase, and K_S is the Sieverts’ constant with the following forms [21,22]:

$$K_{\mathrm{S}}=\exp (\mathsf{\beta }[{\sum }_i\frac{h{\mathsf{\gamma }}_i}{4}-E_{\mathrm{b}}-\frac{1}{2}{\sum }_{i}h{\nu }_i])\frac{1}{\sqrt{\mathsf{\alpha }}}\sqrt{1-\exp (-\mathsf{\beta }{\sum }_i\frac{h{\mathsf{\gamma }}_i}{2}})\frac{1}{{\prod }_i(1-{\mathrm{e}}^{-\mathsf{\beta }h{\nu }_i})},$$

(2)

where $\mathsf{\alpha }=(\frac{2\mathsf{\pi }m{k}_{\mathrm{B}}T}{h^2}{)}^{3/2}\frac{4{\mathsf{\pi }}^{2}I{(k_{\mathrm{B}}T)}^2}{h^2}$ and $\mathsf{\beta }=\frac{1}{k_{\mathrm{B}}T}$, k_B is the Boltzmann constant, E_b and ${\nu }_i$ denote the binding energy and vibration frequency of interstitial H atoms in the Pd₃Ag phases respectively, γ_i is the vibration frequency of gaseous H₂, h is the Plank constant and m and I represent the mass and moment of inertia of H₂ respectively. In most cases, the experimental solubility of H in metal is closer to the total solubility values predicted for its octahedral (O) and tetrahedral (T) sites. Accordingly, the Sieverts’ constant of each PdAg phase is obtained through the summation of the K_S values of all its interstitial sites.

3. Results and Discussion

To simulate the H effects, the binding energy of one H atom in the Pd₃Ag phases was calculated according to the following formula:

$$E_{\mathrm{b}}=E_{{\mathrm{Pd}}_3\mathrm{AgH}}-E_{{\mathrm{Pd}}_3\mathrm{Ag}}-\frac{1}{2}{E}_{{\mathrm{H}}_2}$$

(3)

where $E_{{\mathrm{Pd}}_3\mathrm{AgH}}$ and $E_{{\mathrm{Pd}}_3\mathrm{Ag}}$ are the total energies of Pd₃Ag with and without the H atom respectively, and $E_{{\mathrm{H}}_2}$ is the total energy of the H₂ molecule. After a series of calculations,

Table 1. Lattice constants (a), zero-point energy of hydrogen (ZPE), binding energies (E_b) of pure Pd and Pd₃Ag phases. The corresponding experimental and calculated data in the literature are also listed for comparison [13,23,24,25].

Phase	Lattice (Å)		H Site	ZPE (eV/atom)	Eb (eV)
	This Work	Others			This Work	Others
Pd	3.950	Exp. 3.89 a Cal. 3.954 b	T	0.182	−0.082	Cal. −0.058 d
			O	0.056	−0.132	Cal. −0.11 d
Pd3Ag	3.993	Exp. 3.928 c Cal. 4.002 b	T	0.181	−0.051	-
			O1	0.042	−0.227	-
			O2	0.045	0.046	-

^a Ref. [23]. ^b Ref. [13]. ^c Ref. [24]. ^d Ref. [25].

was compiled listing the lattice constants (a), the zero-point energy of hydrogen (ZPE), and the binding energies (E_b) of the Pd₃Ag phases. Meanwhile, the corresponding data of H in the interstitial sites of FCC Pd were also considered for comparison. The results of pure Pd from a supercell of 2 × 2 × 2 are also present in Table 1. One can clearly see from this table that the calculated lattice constants of FCC Pd and L1₂ Pd₃Ag from the present simulation were not only in good agreement with experimental values reported in the literature [23,24], but also consistent with the other calculated results [13]. Moreover, the obtained binding energies of PdH also agreed well with the data of GGA-PW91 in the literature [25]. Such nice agreements suggest that the present PAW-PBE method would be suitable for revealing the behavior of H in Pd₃Ag phases.

Consequently, it can clearly be observed from Table 1 that the E_b value of PdH (O) was much lower than that of PdH (T), suggesting that the O site is energetically more stable than the T site for H in pure Pd. This phenomenon is in excellent agreement with similar experimental observations in the literature [26]. Next, as Ag atoms occurred in Pd, it is important to recognize from this table not only that Pd₃AgH (O1) was energetically more favorable with a much lower E_b value than Pd₃AgH (T) and PdH (O), but also that the difference of E_b values between the O1 and T sites of Pd₃AgH was much bigger than the corresponding value between PdH (O) and PdH (T). Such a phenomenon predicts that a super-high hydrogen selectivity could be obtained with both Ag–H and Pd–H bonds. In other words, substituting Pd with Ag had no effect on the preferred location of H found in pure Pd, and a lower E_b value of Pd₃AgH (O1) could enhance the solubility of H in the O site of Pd.

According to Equation (1), Figure 2 plots the temperature-dependent solubility of H atoms in the Pd₃Ag phases. The corresponding solubility of H in pure Pd is also shown in this Figure for comparison. It can be clearly observed that the determined H solubility in pure Pd was in excellent agreement with similar experimental and calculated values available in the literature [25,27]. For instance, H concentration in pure Pd at the temperature of 770 K was 1.27 × 10⁻², which matched well with the corresponding values of 1.15 × 10⁻² and 1.41 × 10⁻² from experimental and calculated data available in the literature [25,27].

Figure 2 shows that as the temperatures rose, there was a decline in H solubility in pure Pd and Pd₃Ag phases. Such a downtrend is consistent with experimental observations reported in the literature [28,29]. In addition, by directly comparing the values reported in Figure 2 we found that H solubility was slightly higher in the Pd₃Ag phase than in pure Pd, which suggests that a low amount of Ag in the alloy would increase H solubility with respect to pure Pd. This may be the primary mechanism leading to the observed excellent hydrogen permeability of the Pd₃Ag phase [30,31,32]. The calculated values also matched well with those obtained from similar experimental measurements found in the literature [28,29,30,31,32]. All the above agreements suggest that the theoretical method here presented should be relevant when it comes to reflecting the intrinsic features of the solubility of H in Pd₃Ag phases.

It is important to investigate the intrinsic mechanism of the increase of H solubility in Pd₃Ag phases and to calculate the charge transfers of PdH and Pd₃AgH. The Bader analysis [33,34] was used to handle the charge transfers between H and Pd or Ag atoms in both O and T sites of PdH and Pd₃AgH. As a typical example, the bond charge in the O (0.071 e) site, O1(0.170 e), and O2 (0.006 e) sites of PdH and Pd₃AgH were selected and discussed. Interestingly, the bond charge in the O sites of PdH and Pd₃AgH decreased in the order Pd₃AgH (O1) → PdH(O) → Pd₃AgH (O2), which is the exact opposite of that of the binding energy listed in Table 1. Such a descending trend confirmed that the chemical bond formed in Pd₃AgH (O1) was much stronger than that in PdH (O), which helps explain the higher H solubility in the Pd₃Ag phase found in the experiments.

We then focused on the point defects in the Pd₃Ag phases. All possible point defects introduced into Pd₃Ag phases are represented in Figure 3. It can be seen clearly from this Figure that there were four kinds of single point defects in Pd₃Ag: (a) Ag vacancy (V_Ag), (b) Pd vacancy (V_Pd), (c) Ag anti-site (Ag_Pd), and (d) Pd anti-site (Pd_Ag). In the following figure, according to the Wagner-Schottky model, the defect formation enthalpy (H_d) in the Pd₃Ag phases is calculated by the following Equation [35,36]:

$$H_{\mathrm{d}}=\frac{\Delta H_{\mathrm{d}}-\Delta H_{\mathrm{f}}}{X_{\mathrm{d}}}$$

(4)

where ΔH_d and ΔH_f represent the formation enthalpy of L1₂ Pd₃Ag with and without a single point defect respectively, and X_d is the defect’s concentrations. It is well known that one possible selection criterion to describe stability is formation enthalpy. A lower formation enthalpy normally means a more stable structure and vice versa.

Table 2. Structure, lattice constants (a), and formation enthalpy (ΔH_f) of Pd₃Ag with and without a single point defect. The corresponding four kinds of defect formation enthalpy (H_d) werealso derived and are listed for comparison.

Phases	Structure	Lattice Constant (Å)			ΔHf (eV)	Defect Formation Enthalpy, Hd (eV)
		a	b	c
Pd3Ag	FCC	3.993	3.993	3.993	−0.022	-
Pd24Ag7 (VAg)	FCC	3.977	3.977	3.977	0.0232	1.401
Pd23Ag8 (VPd)	FCT	3.963	3.963	4.027	0.0142	1.122
Pd25Ag7 (AgPd)	FCC	3.991	3.991	3.991	−0.0193	0.086
Pd23Ag9 (PdAg)	FCT	3.997	3.997	4.017	−0.0225	−0.016

lists structure, lattice constants (a), Ag percentage, and formation enthalpy (ΔH_f) of Pd₃Ag with and without a single point defect. The corresponding four kinds of defect formation enthalpy (H_d) were also derived and are listed for comparison. It can be clearly observed from this table that the PdAg phase could keep the original FCC structure in the presence of less than 25% of Ag, while a face-centered tetragonal (FCT) structure was obtained with more Ag atoms. It should be also noted that the calculated formation enthalpy of Pd₃Ag was small and negative, as well as those of Pd₂₅Ag₇ (Ag_Pd) and Pd₂₃Ag₉ (Pd_Ag). More importantly, the values of defect formation enthalpies (H_d) of Ag_Pd and Pd_Ag were close to zero, which implies that the dominant single point defects were anti-site defects in the Pd₃Ag phases. Additionally, the defect formation enthalpies of vacancy defects had positive and large values, suggesting that vacancy defects were difficult to form.

We now discuss a little bit more about the effects of point defects on the solubility of H in Pd₃Ag. Accordingly, Pd₃Ag with Ag_Pd and Pd_Ag were selected to investigate H behavior. Figure 4 shows various interstitial sites of Pd₃Ag with Pd and Ag anti-sites, and

Table 3. Zero-point energy of hydrogen (ZPE) and binding energies (E_b) of the Pd₂₅Ag₇ and Pd₂₃Ag₉ phases.

Phases	H Site	ZPE (eV/atom)	Eb (eV)
Pd25Ag7	T1	0.158	−0.082
	T2	0.183	−0.173
	O1	0.049	−0.232
	O2	0.046	0.02
	O3	0.072	−0.136
Pd23Ag9	T1	0.179	−0.015
	T3	0.185	0.180
	O1	0.051	−0.196
	O2	0.047	0.095
	O3	0.070	−0.146
	O4	0.107	0.137

lists the zero-point energy values of hydrogen (ZPE) and the binding energies (E_b) of the Pd₂₅Ag₇ and Pd₂₃Ag₉ phases. Several features could be deduced from Table 3 as well as from Table 1 and Figure 4.

Firstly, it can be clearly seen from Figure 4 that, for the Pd₂₅Ag₇ and Pd₂₃Ag₉ phases, there were two kinds of T sites and different kinds of O sites, indicating that Ag and Pd anti-sites would lead to O sites with different surrounding atoms of Pd and Ag. Secondly, one notices in Table 3 that the value of Pd₂₅Ag₇H (O1) was lower than that of Pd₂₅Ag₇H (T1), Pd₂₅Ag₇H (T2), Pd₂₅Ag₇H (O2), and Pd₂₅Ag₇H (O3), and a similar phenomenon could be observed for Pd₂₃Ag₉H. Such a direct comparison suggests that anti-site defects had a negligible effect on the preferred location of H in Pd₃Ag. Thirdly, for the Pd₃Ag, Pd₂₅Ag₇ (Ag_Pd), and Pd₂₃Ag₉ (Pd_Ag) phases, the calculated E_b value (−0.227 eV) of Pd₃AgH (O1) seemed lower and higher than its corresponding values at the O1 site of Pd (−0.196 eV) and at the Ag (−0.232 eV) anti-sites respectively, suggesting that anti-site defects had different effects on the thermodynamic stability of Pd₃AgH. In addition, Figure 5 plots the solubility of hydrogen in the Pd₃Ag, Pd₂₅Ag₇ (Ag_Pd), and Pd₂₃Ag₉ (Pd_Ag) phases. It can be observed from this Figure that H solubility in the Pd₃AgH phases with and without point defects decreased in the following order: Pd₃AgH → Pd₂₅Ag₇H → Pd₂₃Ag₉H, suggesting that anti-site defects would decrease hydrogen solubility in the Pd₃Ag phases. Specifically, the solubility of hydrogen in the Pd₃Ag phases would be severely decreased because of the appearance of a Pd anti-site. Considering the above effects of anti-site defects in the Pd₃Ag phases, a useful suggestion for future experiments might be that Pd anti-sites should be prevented.

To understand the difference of H solubility, the charge density of H at all interstitial sites was considered and calculated. As another example, Figure 6 shows charge density plots of (a) Pd₂₅Ag₇ and (b) Pd₂₃Ag₉ with an H atom at the O1 site. The charge densities of H–Pd interactions in the Pd₂₅Ag₇ phases were denser than those in the Pd₂₃Ag₉ phases, which implies that the H atom formed a stronger bond in the O1 site of Pd₂₅Ag₇. Moreover, it can be observed from this Figure that the charge densities from H to Ag were sparse, suggesting a repulsive nature of the Ag–H bond. The results from this section also contribute to our understanding of the higher H solubility in Pd₂₅Ag₇ shown in Figure 5.

4. Conclusions

We performed a comparative first-principles calculation to examine the effects of point defects on hydrogen solubility in Pd₃Ag. The defect formation enthalpies of Pd₃Ag were first calculated and discussed, and it was found that the membrane possessed the perfect compact structure for hydrogen separation, as the vacancy defects were difficult to form. Our simulations also showed that in the Pd₃Ag phase there was a slightly higher hydrogen solubility, and H presented the same preferred location as in pure Pd. In addition, a Pd anti-site (H_d = 0.086 eV) is more stable an Ag anti-site (H_d = −0.116 eV), thus suggesting that the favorable anti-site in the Pd₃Ag phases was a Pd anti-site. The calculated results of H solubility in the Pd₃AgH phases predicted that anti-site defects have different effects on the Pd₃Ag phases, that is, Ag and Pd anti-sites have a neglected and an extremely negative effect on H solubility respectively. The calculated results not only agree well with similar experimental observations in the literature, but also provide a deep understanding of the fundamental effects of Ag on hydrogen solubility in Pd₃Ag phases.