Effect of Atomic Ordering on Phase Stability and Elastic Properties of Pd-Ag Alloys

Abstract

Palladium (Pd) and its alloys, renowned for their good corrosion resistance, catalytic efficiency, and hydrogen affinity, find extensive use in various industrial applications. However, the susceptibility of pure Pd to hydrogen embrittlement necessitates alloying strategies such as Pd-Ag systems. This study investigates the impact of the ordering on the phase stability and elastic properties of Pd-Ag alloys through first-principles calculations. We explore a series of ordered phase structures alongside random solid solutions using Special Quasirandom Structures (SQSs), evaluating their thermodynamic stability and elastic properties. Our findings indicate the possible existence of stable ordered L1₂ Pd₃Ag and PdAg₃ and L1₁ PdAg phases, which are thought to exist only in Cu-Pt alloys. An analysis of the elastic constants and anisotropy indices underscores some pronounced directional dependencies in the mechanical responses between the random solid-solution and ordered phases. This suggests that the ordered phases not only are thermodynamically and mechanically more stable than solid-solution phases, but also display a decrease in anisotropy indices. The results provide a deeper understanding of the atomic behavior of Pd-Ag alloys, and shed light on the design of multiphase Pd-Ag alloys to improve their mechanical properties.

1. Introduction

Palladium and its alloys display high melting points, excellent catalytic performance, and a unique affinity towards hydrogen [1,2,3,4,5], and are widely used as catalysts for various hydrogenation reactions [6,7,8,9,10,11,12,13,14], including but not limited to ethanol steam-reforming reactions, C₂H₂ semi-hydrogenation to C₂H₄, and photocatalysis of carbon dioxide. Nevertheless, pure Pd exhibits strong hydrogen embrittlement due to the α ↔ β phase transitions, regardless of its use as a hydrogen permeation material or as a catalyst for hydrogenation reactions [15]. A common solution is to utilize Pd alloys instead of pure Pd to modulate the mechanical properties when subject to a hydrogen environment [16,17,18].

In past few decades, the Pd-Ag system has been considered as one of the most promising choices to improve the hydrogen sustainability and membrane separation properties of Pd-based membranes [16,19,20,21,22,23,24]. Compared to other noble metal elements such as Au and Pt [25,26], which can similarly ameliorate the hydrogen embrittlement of Pd, Ag is much cheaper and more efficient in enhancing the diffusion coefficient of H in Pd. On the other hand, although the more affordable Cu can also play a role [17,27], the addition of Cu elements actually changes the crystal structure of Pd-based membranes. On the contrary, Ag can form substitutional solid solutions with Pd, enabling a consistent and uniform microstructure across the membrane.

Nevertheless, experimental and theoretical investigations suggest the existence of ordered phases in Pd-Ag systems [28,29,30,31,32,33,34], which may overturn the mechanical properties of solid solution. Dating back to 1960s, Savitskii et al. [28] found some abnormities in electrical resistivity measurements below 1400 K for the Pd-Ag solid solution, and suggested the existence of AgPd and Ag₂Pd₃ ordered phases. Recently, more and more disorder–order transformations in Pd-based alloys have been observed in experiments [35,36,37]. However, due to the fact that the scattering powers of Ag and Pd atoms in most characteristic methods, like X-ray, electron, etc. are almost equal, the atomic ordering and its corresponding crystal structure have yet to be well established [20,38,39,40,41,42]. On the other hand, theoretical calculations have indicated the existence of various ordered phases [29,30,31,32]. For example, Müller et al. [29] predicted the L1₁ structure for a Pd-Ag ordered phase with a 1:1 composition ratio using the mixed-space cluster expansion method. However, through the screened generalized perturbation method (SGPM), Ruban et al. [31] suggested L1₁, L1₂, D0₂₂ and D0₂₃ ordered structures hardly occur in Pd-Ag alloys due to their low order energy value and low order–disorder transition temperature. In addition, the calculation of Pd-Ag alloys in different compositions or different catalytic surfaces [43,44,45] implies that hydrogen catalysis could be strongly influenced by atomic ordering and crystal structures. It is thus of great importance to understand the effect of atomic ordering on the phase stability and elastic properties of Pd-Ag alloys.

In this work, various ordered phase structures alongside random solid solutions using Special Quasirandom Structures (SQSs) [46] of Pd-Ag alloys are systematically investigated using first-principles calculations. The phase stability of the ordered and solid-solution phases is evaluated by not only the thermodynamic heat of formation but also Born’s mechanical criteria and electronic structure [47]. This work yielded disparate conclusions regarding the stability of the L1₂ structure in comparison to previous analysis solely focused on energy-related considerations [31]. The ordering range is thus singled out in the Pd-Ag systems, and its role in the transition of mechanical properties is discussed in detail. Moreover, the elastic anisotropies are presented for providing valuable data for the design of better mechanical properties of Pd-Ag systems. In the end, concluding remarks are drawn.

2. Methods and Details

First-principles calculations were performed in the renowned Vienna ab initio simulation package (VASP, V6.4.3) [48]. The computations were carried out utilizing a plane wave basis set with the projector-augmented wave (PAW) method [49]. The exchange-correlation potentials were captured through the generalized gradient approximation (GGA), developed by Perdew and others [50]. After testing within a range from 300 eV to 700 eV, energy cutoffs of 500 eV and 750 eV were selected for the plane wave basis and augmentation, respectively, as they converge at the ~1 meV scale. We adopted the k-mesh of 15 × 15 × 15 after the k-point convergence test [51]. In terms of k-space integration, the temperature smearing technique proposed by Methfessel and Paxton for relaxation calculations is adopted [52], while employing the modified tetrahedron method introduced by Blochl, Jepsen, and Andersen for static calculations [53].

The elastic properties of the alloy were estimated using the Hill averaging method [54], which integrates the Voigt and Reuss approximation. For a cubic crystal system, the Voigt and Reuss approximation are given by

$$G_V={\displaystyle \frac{1}{5}}\left[\left(C_{11}-C_{12}\right)+3C_{44}\right]$$

(1)

$$B_V={\displaystyle \frac{1}{3}}\left(C_{11}+2C_{12}\right)$$

(2)

$$G_R={\displaystyle \frac{5C_{44}\left(C_{11}-C_{12}\right)}{3{\left(C_{11}-C_{12}\right)+4C}_{44}}}$$

(3)

$$B_R={\displaystyle \frac{1}{3}}\left(C_{11}+2C_{12}\right)$$

(4)

where the G_V (B_V) and G_R (B_R) are the shear (bulk) modulus determined by the Voigt and Reuss [55] approximation, respectively. And C₁₁, C₁₂ and C₄₄ are single-crystal elastic constants in the stiffness matrix of solid solutions or ordered phases. Then, the shear modulus G, the bulk modulus B, Young’s modulus E, and the Poisson rate υ in the Hill approximation were expressed by

$$G={\displaystyle \frac{1}{2}}\left(G_V+G_R\right)$$

(5)

$$B={\displaystyle \frac{1}{2}}\left(B_V+B_R\right)$$

(6)

$$E={\displaystyle \frac{9BG}{3B+G}}$$

(7)

$$\nu ={\displaystyle \frac{3B-2G}{2\left(3B+G\right)}}$$

(8)

3. Results and Discussion

3.1. Phase Stability

A series of ordered Pd-Ag structures were chosen in this work and depicted in Figure 1 using VESTA software (ver. 3.5.8 64-bit Edition) [56]. The choice of those structures was based on the possible stable phases reported in the literature. For example, the L1₁ structure, which is an ordered FCC phase, was found to be stable in many precious metal alloys, like Pt-Cu and Pt-Ag for an extremely narrow composition rage [57]. Additionally, it is also predicted to be stable for Pd-Ag through density function theory. Other common ordered FCC structures, like L1₂, L1₀, and L6₀, which have been demonstrated to exist in other Pd-based binary alloys in experimental works were taken into consideration [58,59,60,61,62]. In addition, some ordered hexagonal close-packed (HCP) structures, D0₁₉ and B_h, were also considered for comparison. On the other hand, Special Quasirandom Structures (SQSs), constructed by Pezold et al. [63], were used for modeling the solid-solution Pd-Ag phases.

In order to demonstrate the reliability of the SQS method, a radial distribution function between L1₂ Pd₃Ag and SQS Pd₃₂Ag₈ after relaxation was demonstrated in Figure 2. The radial distribution function represents the probability of an atom existing within a unit volume from the center, with the function being centered on a selected atom. The blue bars in Figure 2 depict the radial distribution of the L1₂ Pd₃Ag ordered phase. The radial distribution of L1₂ is similar to that of the FCC structure, which is also evident in the L1₂ supercell structure on the right side of the figure. The red bar in the figure indicates the radial distribution of the SQS Pd₁₂Ag₄ structure after relaxation. As illustrated by the comparison of two distribution functions, the distances of the atoms in the SQS structure are shifted by a minor amount in the original FCC structure. It can be concluded that the SQS method can simulate the random atomic structure of the Pd-Ag solid solution while avoiding deviating from the FCC Bravais lattice.

The thermodynamic stability of ordered and solid-solution Pd-Ag structures was determined by their heat of formation (ΔH_f). The ΔH_f is calculated by

$$\Delta H_f=\left(E_{P{d}_{x}A{g}_y}-x{E}_{Pd}-y{E}_{Ag}\right)/\left(x+y\right)$$

(9)

where the $E_{P{d}_{x}A{g}_y}$, $E_{Pd}$ and $E_{Ag}$ are the total energy of Pd_xAg_y, pure Pd and pure Ag in the ground state; meanwhile, the x and y are the atom ratios of Pd and Ag. Figure 3 depicts the calculated ΔH_f of the aforementioned phases, in addition to the available experimental values of the Pd-Ag system as in reference [64]. Several features can be discerned from this figure.

Firstly, it can be seen from Figure 3 that all the calculated ΔH_f values, both of the solid solutions and ordered phases, are negative, implying that these Pd-Ag structures are thermodynamically stable.

Secondly, the ΔH_f of the SQSs is in good agreement with that derived from experiments [65]. Even though the experimental values are obtained at 1173 K, the shape of the calculated one is similar to the experimental counterpart. For example, the lowest ΔH_f is −5.64 kJ/mol at Pd_0.4Ag_0.6 from experiments, which is consistent with −5.00 kJ/mol from Pd₇Ag₉ to Pd₆Ag₁₀. Considering the ΔH_f will normally decrease as a function of temperature, the SQS model adopted in this work can describe the solid-solution Pd-Ag phases well.

Thirdly, the three ordered phases, L1₂ Pd₃Ag, L1₁ PdAg, and L1₂ PdAg₃, based on the FCC lattice, display the most thermodynamic stability. Therefore, the ordered phases to be discussed the following sections are L1₂ Pd₃Ag, L1₁ PdAg and L1₂ PdAg₃ with Ag contents of 0.25, 0.5, and 0.75, respectively. The ΔH_f values of L1₂ Pd₃Ag, L1₁ PdAg, and L1₂ PdAg₃ are −2.49, −6.19, and −5.51 kJ/mol, respectively. They are lower than the HCP ordered counterparts at the same relative concentration. In addition, the calculated ΔH_f values of these three ordered phases are also lower than those of the Pd-Ag solid solutions at each composition, indicating that ordering could happen in Pd-Ag systems.

Fourthly, one can discern from Figure 3 that the ΔH_f of the SQSs exhibits an anomalously high value of −4.28 kJ/mol for the component Pd₈Ag₈. Conversely, the ΔH_f of the L1₁ ordered phase at the same Ag concentration exhibits a minimum value among all the compounds. This indicates the Pd₈Ag₈ solid solution is likely to be in non-equilibrium and more likely to form an L1₁ ordered phase, a phase observed in other precious metal alloys [57]. This singularity will be discussed in the following sections.

The lattice constants of solid-solution and stable ordered Pd-Ag phases, L1₂ Pd₃Ag, L1₁ PdAg, and L1₂ PdAg₃, are shown in Figure 4a. The lattice constants almost linearly increase from pure Pd to pure Ag. The lattice constants of ordered L1₂ Pd₃Ag, L1₁ PdAg, and L1₂ PdAg₃ are close to those of the solid-solution phases. For example, L1₁ and SQS PdAg have almost the same lattice constant, 4.05 Å.

In order to further examine the lattice distortion in Pd-Ag systems, the deviations of the lattice constants from Vegard’s law are investigated, and the results are shown in Figure 4b, together with available calculation and experimental values [64]. Unlike the lattice constants shown in Figure 4a, the deviations in the lattice constants in Figure 4b are more pronounced. The negative deviations indicate that the lattice shrinks after forming either solid-solution or ordered structures. It can also be seen that the deviations of the SQSs display a similar trend with ΔH_f. That is, the more negative the ΔH_f, the larger the deviation. For ordered structures, the situation is different. L1₂ Pd₃Ag presents a larger deviation than the corresponding solid solution, while L1₂ PdAg₃ is smaller. On the other hand, L1₁ PdAg has the same deviation of −7.17 × 10⁻³ Å. The lattice distortion is known to reflect the electronic interaction in the alloys, which is not related to the thermodynamic stability but also results in quite distinct mechanical properties. Additionally, we found that the SQSs show better agreement with the experimental values in comparison to the exact muffin-tin orbital (EMTO) method described in the literature [64]. This gives strong support for the accuracy of the first-principles calculation in this work.

3.2. Mechanical Stability

Based on the solid-solution structures from SQSs and the most stable intermetallic structures at Ag concentrations of 0.25, 0.5, and 0.75, a first-principles calculation is conducted to obtain the elastic properties of Pd-Ag alloys and L1₂ Pd₃Ag, L1₁ PdAg, and L1₂ PdAg₃ ordered phases. The single-crystal elastic constants are calculated through the universal linear-independent coupling strains (ULICS) method [66]. The results are shown in Figure 5, as well as the available experimental values [67,68]. It can be seen that the calculated results agree well with the experimental ones. For instance, the C₁₁, C₁₂, and C₄₄ of pure Ag in this work are 112.61, 81.57, and 39.51 GPa, respectively, which match well with 124, 94, and 46 GPa from the experiments [67]. In addition, the C₁₁, C₁₂, and C₄₄ of pure Pd are also consistent with the experimental values [68].

The single-crystal elastic constants show different features with respect to the Ag concentration. For the SQSs, the elastic constants gradually decrease from Pd to Ag, with some abnormal ups and downs at Pd₃Ag, PdAg, and PdAg₃. For example, C₁₁ abruptly drops at a Ag concentration of 0.25, while C₁₂ sharply increases at this concentration. On the contrary, the single-crystal elastic constants of ordered L1₂ Pd₃Ag, L1₁ PdAg, and L1₂ PdAg₃ decrease almost linearly and close to the reference Vegard’s law.

In order to explore the intrinsic mechanism related to the elastic properties of ordered and solid-solution phases, Born’s criterion [69] is adopted in the analysis. According to Born’s theory, a solid is mechanically stable when the elastic matrix is contingently positive definite. For cubic crystals, that means that C_ij > 0, and C₁₁ > C₁₂. From Figure 6, we can see that all C₁₁, C₁₂, and C₄₄ are positive. However, C₁₁ is less than C₁₂ for the SQS Pd₃Ag, Pd₁₂Ag₆, and PdAg₃. The ordered L1₂ Pd₃Ag and PdAg₃, on the contrary, have a C₁₁ larger than C₁₂, and meet Born’s criterion [69]. The single-crystal elastic constants of ordered and solid-solution Pd₃Ag and PdAg₃ comply with the ΔH_f of those phases, that the ordered phases have a more negative ΔH_f. For ordered and solid-solution PdAg, the C₁₁ and C₁₂ are similar with each other, with the C₄₄ of the ordered L1₁ PdAg being less than its solid-solution counterpart. This indicates a softening in the mechanical strength, as the C₄₄ reflects the shear strength.

It is known that elastic constants are intrinsic properties of crystals and a direct reflection of electronic bonding [70]. To build the relation between the elastic properties and bonding features, the Debye temperature, ${\mathsf{\Theta }}_D$, is derived. The Debye temperature is derived from the elastic constants calculated above through the following formula [71]:

$${\mathsf{\Theta }}_D={\displaystyle \frac{h}{k}}{\left[{\displaystyle \frac{3n}{4\mathsf{\pi }}}\left({\displaystyle \frac{N_A\mathsf{\rho }}{M}}\right)\right]}^{1/3}{\mathsf{\nu }}_m$$

(10)

where h and k are the Plank and Boltzmann constant, respectively; n, N_A, M, and ρ are the number of atoms, Avogadro’s number, mean molecular weight, and the density, respectively. v_m is the average velocity that can be calculated by the following equation:

$${\mathsf{\nu }}_m={\left[{\displaystyle \frac{1}{3}}\left({\displaystyle \frac{2}{v_t^3}}+{\displaystyle \frac{1}{v_t^3}}\right)\right]}^{1/3}$$

(11)

$${\mathsf{\nu }}_t=\sqrt{G/\mathsf{\rho }}$$

(12)

$${\mathsf{\nu }}_l=\sqrt{\left(B+{\displaystyle \frac{4}{3}}G\right){\displaystyle \frac{1}{\mathsf{\rho }}}}$$

(13)

where the v_t and v_t are the transverse velocity and longitudinal velocity, respectively; B and G are the isothermal bulk modulus and shear modulus, which is calculated by the Voigt–Heuss–Hill average [54] with elastic constants C₁₁, C₁₂, and C₄₄.

After a series of calculations, the ${\mathsf{\Theta }}_D$ for Pd-Ag alloys is plotted in Figure 6. The ${\mathsf{\Theta }}_D$ values of the solid solutions reach a maximum of about 312 K at Ag concentrations of 0.1875 and 0.625, and then decrease to 225 K and 238 K at Ag concentrations of 0.25 and 0.6875, respectively. The sudden drop in the ${\mathsf{\Theta }}_D$ indicates that the alloy is weakly bonded when close to Ag concentrations of 0.25 and 0.6875, which are consistent with the unstable elastic constants around these Ag concentrations. In addition, the value of ${\mathsf{\Theta }}_D$ of L1₂ intermetallic Pd₃Ag is significantly higher while that of L1₂ PdAg₃ is relatively low. Accordingly, the L1₂ structure is more likely to be formed at a Ag concentration of 0.25. This observation is also supported by experiments on Pd₃Ag and PdAg₃ nanoparticles. For instance, researchers have successfully synthesized Pd₃Ag alloys and found they are stable at the nanoscale [72,73]. On the contrary, the synthesis of PdAg₃ nanoparticles seems more difficult, as it is prone to segregation in pure Ag and Pd phases [74]. The ${\mathsf{\Theta }}_D$ of L1₁ PdAg is slightly lower than the corresponding solid solution, as a reflection of the decline in the C₄₄ value.

Moreover, the electronic structure of Pd-Ag alloys is further investigated. Figure 7 shows the calculated density of states (DOS) for the ordered and solid-solution Pd₃Ag, PdAg, and PdAg₃ phases. The ordered phases have much sharper peaks compared to the solid-solution ones, demonstrating partial covalent bonding features. Nevertheless, the DOSs of the ordered phases at the Fermi level show different features in comparison to those of the solid-solution phases. For example, L1₂ Pd₃Ag has a lower DOS at the Fermi level than that of the SQS Pd₃Ag, and are 7.3475 and 5.358 states/eV, respectively. L1₁ PdAg and L1₂ PdAg₃ have smaller DOSs at the Fermi level than their solid-solution counterparts. This is consistent with the calculated ${\mathsf{\Theta }}_D$ of those ordered and solid-solution phases. Interestingly, the peaks of the SQS PdAg₃ below the Fermi level are higher than those of the L1₂ PdAg₃, suggesting the local atomic interaction may be stronger than those in the ordered L1₂ phase. As a result, the ${\mathsf{\Theta }}_D$ of the SQS PdAg₃ is much higher than that of the ordered L1₂ phase.

3.3. Mechanical Anisotropy

As both structural and functional materials, Pd-based membranes are used in hydrogen separation and catalytic reactions. The durability and selectivity of membranes are dependent upon their mechanical properties and fabrication parameters. For polycrystalline membranes, the thickness of the membrane typically influences the deformation mechanism of the material. Hosseinian et al. [75] studied thin films with 30–100 nm thickness by in situ TEM tests and concluded the mechanism through which the nano-crack forms ahead of the main crack only in thinner films. Additionally, experiments have shown that the hydrogen permeation behavior of membranes increases with membrane thickness [20,76,77]. However, an increase in the film thickness will result in higher costs and potential assembly challenges. Prior research has demonstrated that the incorporation of Ag can effectively enhance the solubility of hydrogen, thereby increasing the permeability of hydrogen [20]. It is therefore crucial to identify Pd-Ag materials with thick conditions (as, in theory, the periodic boundary condition is used to simulate the thick condition).

For monocrystalline membranes, parameters such as geometry and direction might be important because of the mechanical anisotropy, which is expected to be investigated [78]. Generally speaking, the ordered structure will display strong mechanical anisotropy compared to solid solutions. As the ${\mathsf{\Theta }}_D$ and electronic structure show some interesting features for ordered phases, it is of importance to understand the mechanical anisotropy of those ordered and solid-solution phases. Moreover, mechanical anisotropy undermines the workability and limits the application of Pd-Ag alloys as structural materials [79].

To quantitively represent the anisotropy of those ordered and solid-solution Pd-Ag phases, the universal elastic anisotropy index (A^U), proposed by Ranganathan and Ostoja-Starzewski [80], was adopted.

$$A^U=5{\displaystyle \frac{G_V}{G_R}}+{\displaystyle \frac{B_V}{B_R}}-6\ge 0$$

(14)

In addition, the A^U has a value of 0 for the isotropic materials. Additionally, the anisotropy in compressibility is also estimated by the bulk modulus within the Voigt and Reuss approximations, as follows:

$$A_G={\displaystyle \frac{G_V-G_R}{G_V+G_R}}$$

(15)

The anisotropy results are summarized in Figure 8. It should be noted that the solid-solution phases which are mechanically unstable in the previous section have been omitted in Figure 8, namely, x_Ag = 0.25, 0.3125, and 0.75. Regardless of the method used to calculate the anisotropy index, the anisotropic behavior is the same for the Pd-Ag systems. Interestingly, the ordered phases have a lower anisotropy index than the solid-solution phases at the same composition. For example, the $A^U$ of the L1₁ PdAg is 10, which is smaller than that (15) of the SQS PdAg. Except for their lower anisotropy index, the ordered phases incur abnormally high or low anisotropy for the nearby solid-solution phases. The SQS Pd₁₀Ag₈ near L1₂ Pd₃Ag shows an anisotropy index as high as 31 for $A^U$ and 0.73 for $A_B$, while the SQS Pd₇Ag₉ and Pd₉Ag₇ around L1₁ PdAg shows a lower anisotropy index compared to the ordered phase. This may originate from the fact that the solid-solution phases near the ordered phase are prone to local short-range or long-range ordering, and thus display an abnormal anisotropy index.

In order to dig into the details of mechanical anisotropy, the spatial dependence of elastic moduli, like Young’s modulus, the shear modulus, and the Poisson ratio, is investigated. The expression for the spatial dependence of Young’s modulus in cubic crystals is given by the following expression [81]:

$$E={\displaystyle \frac{1}{S_{11}+\left(-2S_{11}+2S_{12}+S_{44}\right)\left(l_1^2{l}_2^2+l_1^2{l}_3^2+l_2^2{l}_3^2\right)}}$$

(16)

where l₁, l₂, and l₃ are in the cosine direction. Figure 9 shows the calculated average Young’s modulus (E) [54] and its spatial-dependent 3D morphology in Pd-Ag systems. In the figure, the theoretically calculated values of the Young’s moduli of Pd and Ag is 96.16 GPa and 58.92 GPa, respectively, which are a little bit lower than the experimentally measured values of 112 GPa and 76 GPa [82]. This finding is in accordance with the result that the calculated values of the elastic constants are lower than the experimental values illustrated in Figure 5. The shape of the Young’s modulus of Pd-Ag, whether for the solid-solution or ordered phase, are quite similar and look like a cubic (except Pd₇Ag₉ and Pd₁₀Ag₆). The maximal E is along the crystallographic orientation <111> and the minimal E is along the <100> orientation. For Pd₁₀Ag₆, the solid solution shows the highest anisotropy of E, presenting as an eight-pointed star. Along the pointed <111> orientation, the E shows a maximum value of nearly 200 GPa, while the minimum value is lower than 80 GPa along the <100> orientation. The results align well with those of other research on the directional effects of Pd monocrystalline (111) oriented films [78]. In contrast, for Pd₇Ag₉, the Young’s modulus of the Pd-Ag solid solution has a sphere-like morphology, i.e., all the E values are within a narrow range of 130–150 GPa, showing near isotropic features. Moreover, the average Young’s modulus shows a decreasing tendency as a function of the Ag concentration, except for the solid solution with a Ag concentration around 0.25 and 0.75. The spatial-dependent 3D Young’s modulus is consistent with the anisotropy index in Figure 8, and reveals the large fluctuation for solid-solution phases near the ordered L1₂ Pd₃Ag, L1₁ PdAg, and L1₂ PdAg₃. This suggests that from a mechanical point of view, the design of Pd-Ag alloys should involve the selection of a concentration close to the stoichiometric composition near those ordered phases. Indeed, most experiments in the literature describe the selection of Pd₃Ag, PdAg, and PdAg₃ compositions and achieve better material performance [43,83,84,85].

A high shear modulus and low anisotropy reduce the susceptibility of the Pd-Ag membrane to fracture due to lattice expansion caused by hydrogen dissolution under high hydrogen flux and complex temperature conditions [20]. The spatial-dependent and averaged shear modulus (G) of Pd-Ag systems are shown in Figure 10. Note that the left part in the dashed box represents the contours of the shear modulus on the maximal shear plane, while the right part represents those on the minimal shear plane [86]. The shear moduli of pure polycrystalline Pd and Ag, as illustrated in the figure, are 30.98 GPa and 20.36 GPa, respectively. Consistent with the case of the Young’s modulus, the calculated values of the shear modulus are found to be close to the experimental values of 47.50 GPa and 28.76 GPa [87,88]. Except for Pd₇Ag₉, the max shear planes are located on the three planetary rings of (100), (010), and (001), while the minimal shear planes are like six <100> pointed stars. Similar to the Young’s modulus, solid-solution Pd₇Ag₉ presents near-isotropic features, either the maximum or the minimum shear planes, like a sphere. Similar to the Young’s modulus and anisotropy index, abnormity appears for solid-solution phases near the ordered ones. This means that except for Pd₇Ag₉, the better choice is still the stoichiometric composition, for more uniform elastic properties.

Among the elastic properties with respect to the Ag concentration, the Poisson ratio shows a divergent trend from the Young’s modulus and shear modulus, as indicated in the spatial-dependent and averaged Poisson ratios (υ) in Figure 11. Note that a dual color bar is employed here, where the upper color bar is the scale of υ on the maximal shear plane on the right, and the lower color bar is the scale of υ on the minimal shear plane on the left in the dashed box. The calculated values of the Poisson ratio for Pd and Ag are closer, 0.455 and 0.447, respectively, slightly higher than the experimental values of 0.386 and 0.335 [87,89]. In general, the shapes of υ are like flowers on the right and like six <100> pointed stars on the left. This is a feature caused by the maximum and minimum shear moduli, shown in Figure 10. Likewise, the solid-solution Pd₇Ag₉ presents near-isotropic features, and those near the ordered phases have abnormally large or small υ values. This again signifies the importance of choosing the correct concentration to achieve a better mechanical performance.

Based on the results in this section, we propose the following compositions should be chosen for Pd-Ag membranes: Pd₇Ag₉ solid solution, Pd₁₃Ag₃ solid solution, and Pd₃Ag L1₂ ordered phase. The Pd₇Ag₉ solid solution has exhibited favorable mechanical properties for polycrystalline materials and low anisotropy for monocrystalline materials, as illustrated in Figure 9, Figure 10 and Figure 11 of this manuscript. This suggests the high durability of the membranes under high hydrogen flux and stability under complex stress and temperature conditions. In addition, the high Ag content of Pd₇Ag₉ provides a greater H solubility along with lower raw material costs [20].

However, the synthesis of the Pd₇Ag₉ solid solution may be challenging. This is due to the formation energy of Pd₇Ag₉ solid solution being comparable to that of the solid solution with a higher Ag concentration, such as Pd₆Ag₁₀, and is lower than that of the PdAg L1₁ ordered phase. The alloys with an elemental ratio of Pd/Ag = 7:9 may precipitate out of the Pd₇Ag₉ ordered phase, thereby forming a mixed phase. Therefore, the Pd₁₃Ag₃ solid solution and Pd₃Ag ordered phase are selected as substitutive phases of the Pd-Ag membrane material. As these phases with more significant mechanical anisotropy are employed as monocrystalline membranes, it is essential to consider the operational conditions and geometry. This implies that the direction of loading should be aligned with the orientation exhibiting the most robust mechanical properties. In the event that the long-term structural stability of purely Pd-Ag membranes cannot be guaranteed at high hydrogen fluxes, the use of composite membranes may be a viable solution [90,91].

4. Conclusions

In this work, the thermodynamical and mechanical stability of various ordered and random solid-solution phases were systematically studied through first-principles calculations. The ordered phases, L1₂ Pd₃Ag, L1₁ PdAg, and L1₂ PdAg₃, were found to be more stable than the solid-solution ones in terms of the heat of formation. The Born criterion of the single-crystal elastic constants confirms the improved stability of the ordered phases over their solid solution counterparts. The existence of the ordered phase was analyzed through the Debye temperature and electronic signatures. The mechanical anisotropy of all the solid solutions and the stable ordered phases were determined using the anisotropy index and spatial dependence of elastic moduli, like Young’s modulus, the shear modulus, and Poisson’s ratio. The mechanical anisotropy underlines the importance of maintaining a stoichiometric composition for better mechanical uniformity. Our findings not only align well with relevant experimental results, but also deepen our understanding of the phase structure of Pd-Ag alloys.