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First-principles screening of PdCuAg ternary alloys as H2 purification membranes
Journal of Membrane Science 371 (2011) 189–196
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First-principles screening of PdCuAg ternary alloys as H2 purification membranes Chen Ling, Lymarie Semidey-Flecha, David S. Sholl ∗ School of Chemical and Biomolecular Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0100, USA
a r t i c l e
i n f o
Article history: Received 2 December 2010 Received in revised form 14 January 2011 Accepted 22 January 2011 Available online 28 January 2011 Keywords: Hydrogen Metal membranes Theory and modeling Alloys
a b s t r a c t Predictions based on Density Functional Theory (DFT) calculations have proven to be a useful complement to experimental efforts to characterize permeation of hydrogen through metal membranes at elevated temperatures. Previous applications of this approach have focused on relatively small numbers of alloys with specific binary or ternary compositions. We describe DFT-based calculations that provide predictions for the solubility, diffusivity, and permeability of hydrogen at elevated temperatures in PdCuAg alloys for a wide range of compositions of this ternary alloy that yield substitutionally disordered fcc materials. Our results show that the variation in permeability of hydrogen among these alloys is more closely tied to variations in solubility than variations in diffusivity. These results give a detailed description of a ternary alloy of practical interest for high temperature H2 purification and also demonstrate a general strategy for making predictions about complex metal alloys for this application. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Dense metal membranes are well known in applications for H2 purification and have many advantages over alternative purification methods [1–3]. Pure Pd, for example, is well known as a metal that can be used as a H2 purification material [4]. Despite the commercial success of Pd membranes, several factors limit their widespread applications. At temperatures below 573 K and moderate H2 pressures, a Pd hydride phase forms with a lattice constant substantially different from pure Pd. Membrane integrity can be compromised if this phase is formed. A more serious problem is that pure Pd membranes are extremely sensitive to non-hydrogen contaminants such as H2 S and CO, which can effectively reduce the H2 flux through the membranes to zero [5,6]. H2 S is ubiquitous in natural gas and CO is always present in syngas. Poisoning of membranes by these species is thus a critical issue for hydrogen separations from these gas mixtures. One route to improving the performance of pure metal membranes is to develop metal alloys as successful membranes [2]. Pd-based binary alloys, for example, have been widely examined experimentally and theoretically, with Pd being the core material for binary alloys in combination with elements including cerium, copper, gold, iron, nickel, silver, and yttrium [2,7–17]. Among these binary alloys, PdCu alloys are particularly interesting because these membranes may improve resistance to poisoning [6]. The permeability of hydrogen, however, has been observed to decrease with increasing Cu concentration in the alloys [6,18–22]. Other binary
∗ Corresponding author. Tel.: +1 412 268 4207; fax: +1 412 268 7139. E-mail address: [email protected] (D.S. Sholl). 0376-7388/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2011.01.030
alloys such as PdAg and PdAu have been shown to increase the permeability of hydrogen compared to pure Pd [12,15,23]. There is no evidence that the resistance of contaminants relative to pure Pd is improved for these binary alloys. Based on these results, identification of Pd-based alloys with considerable hydrogen flux and resistance of the contaminants is still required. Although experimental studies of Pd-based binary alloys have been reported, it is still challenging to experimentally screen a large series of alloys, since such efforts would require extremely large resources. In recent years, theoretical approaches based on first-principles calculations provide a complementary route to predict the performance of metal alloys in H2 purification [24,25]. Kamakoti and Sholl developed an ab initio method based on Density Functional Theory (DFT) calculations to effectively examine the solubility, diffusivity and permeability of hydrogen in the bulk phase of PdCu alloys without any experimental input [6,18,26,27]. Similar methods have since been used to examine PdAg and PdAu alloys [28,29], amorphous metals [30,31] and metal sulfides [32,33]. Kamakoti and Sholl’s calculation showed that the predictions of the permeability of H in PdCu alloys were in good agreement with experimental data [6]. Subsequently, Semidey-Flecha and Sholl extended this method to examine a series of Pd96 M4 binary and Pd70 Cu26 M4 ternary alloys with M = Ni, Pt, Rh, Ag, Au, Cu and Ni, Pt, Ag, and Au [23,34,35]. Here and throughout this paper, all compositions are given in at.%. This work indicated that the permeability of PdCuAg is improved compared to PdCu alloys by adding just 4 at% of Ag in the alloy [34]. An interesting question remaining from this study is whether adding higher level of Ag can lead to even more favorable outcomes. In this paper, we extend these calculations to a broad series of PdCuAg ternary alloys. The solubilities, diffusivities and permeabilities of hydrogen are examined and discussed. We
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show that for alloys with high Ag concentration, hydrogen permeability comparable to pure Pd can be reached. The structure of this paper is as follows. We first discuss the methods and the computational method used in examining ternary alloys in Section 2. A series of PdCuAg ternary alloys are then studied. We describe the solubility, diffusivity and permeability of hydrogen in these alloys in Section 3. Finally, a conclusion and the prospects for future work are given in Section 4.
2. Models 2.1. Model description A detailed description of the modeling framework we used to identify the properties of interstitial hydrogen in Pd-based alloys has been presented previously [34,35]. Here we highlight several important factors of this approach. Both PdCu and PdAg form a disordered fcc solid solution for the Pd-rich compositions we examine below [36]. No experimental description of the phase diagram of PdCuAg ternary alloys is available. It is reasonable to expect, however, that ternary PdCuAg also has a disordered fcc structure for the compositions we examine. All of our calculations make this assumption. This choice is not intended to imply that alloys associated with the bcc PdCu alloys that exist with ∼50 at.% Cu are uninteresting for membrane applications. Membranes made from these bcc alloys are known to have higher hydrogen permeability than fcc PdCu alloys with ∼30 at.% Cu [6]. This is largely due to the high diffusion coefficient of H in these bcc alloys [26], an observation that highlights the critical role of crystal structure in determining membrane properties. Even though there is considerable interest in these bcc alloys, insufficient information about the compositions for which these alloys could form as ternary PdCuAg alloys would make detailed DFT-based models of such alloys of limited value. As a result, we have confined our attention in this work to fcc PdCuAg alloys in the range of composition defined above. A key challenge in applying DFT calculations to describe interstitial H in alloys is how to describe the properties of interstitial hydrogen with configurationally distinct local environments, since substitutional disordered materials have a large variety of structurally distinct binding sites. In our study, we applied cluster expansion (CE) methods to describe the relationship between the binding energies of interstitial H in Pd-based alloys and the local environment of these sites. A detailed explanation of how to apply CE models in the prediction of hydrogen permeability in metal alloys is already available [34,35]. In the results presented below, DFT calculations were first performed for a large collection of distinct interstitial sites and transition states for H hopping. The data obtained from DFT calculations was then used in CE modeling. Once the CE model accurately representing the interstitial binding energies and site to site transition state energies obtained from DFT calculations was available, the net solubility of H was calculated by statistical mechanics calculations [6]. This method calculates the equilibrium between gaseous hydrogen and individual interstitial atomic hydrogen while ignoring the interaction between interstitial hydrogen. This approach is well justified at dilute interstitial hydrogen concentrations. As we discuss below, we find that the maximum concentration of hydrogen in our study is about 3.4 times higher than in pure Pd at 600 K, which indicates that the assumption of dilute interstitial H is still valid for PdCuAg ternary systems with moderate hydrogen pressures and elevated temperatures. Although DFT calculations can be used to quantify H–H interactions at non-dilute concentrations of interstitial H, we have not performed calculations of this kind in this work. To calculate the diffusivity of interstitial hydrogen in the alloys, we applied kinetic Monte Carlo (KMC) simulations. These KMC simulations
give the self diffusivity of H. At dilute concentration [6,35], the selfdiffusivity and transport diffusivity of interstitial H are equal [3], so the self diffusivity can be used in calculating the net flux of H through a membrane. 2.2. DFT calculation details We performed DFT calculations using the Vienna ab initio Simulation Package (VASP) using the generalized gradient approximation with the PW91 functional to describe electron exchange-correlation effects [37,38]. Ion-electron interactions were described by ultrasoft pseudopotentials. A plane-wave expansion with a cutoff of 233.73 eV was used in all calculations [37]. Geometry relaxations were performed with a conjugate gradient method until the forces on all unconstrained atoms were less than 0.03 eV/Å. A -centered grid with 4 × 4 × 4 k-points was used for all calculations. The supercell we used contained 27 metal atoms forming an fcc lattice and one H atom located in an interstitial site. All atoms were allowed to relax during the calculation, with the volume of the cell fixed at the optimized volume determined without the interstitial H. Vibrational frequencies of interstitial H were calculated in the harmonic approximation, assuming localized vibrations of hydrogen atom are decoupled from vibrations of metal atoms. Accurately characterizing the diffusion of interstitial H requires determining a large number of transition states (TS) with DFT calculations. The most common method to find a transition state with DFT, the nudged elastic band (NEB) method [38], is far more time consuming than the calculations needed to optimize the geometry of H in an interstitial site. To overcome this computational bottleneck, we developed an efficient method that allows transition states in the alloys of interest to be accurately computed with far smaller computational resources. This approach generated initial approximations for the position of a TS by noting that the TS for H hopping between two interstitial sites is generally located close to the center of the polyhedral face connecting the two interstitial sites [30]. These estimations made it possible to rigorously locate transition states using DFT calculations equivalent to an energy minimization. 3. Results and discussion 3.1. Optimization of lattice constants We chose to examine seven different PdCuAg ternary alloys with DFT: Pd92.6 Cu3.7 Ag3.7 , Pd85.2 Cu3.7 Ag11.1 , Pd85.2 Cu11.1 Ag3.7 , Pd70.4 Cu11.1 Ag18.5 , Pd70.4 Cu25.9 Ag3.7 and Pd81.5 Cu11.1 Ag7.4 , Pd66.7 Cu25.9 Ag7.4 . For comparison, we also examined three PdAg binary alloys, Pd96.3 Ag3.7 , Pd92.6 Ag7.4 and Pd81.5 Ag18.5 , and three PdCu binary alloys, Pd96.3 Cu3.7 , Pd88.9 Cu11.1 and Pd74.1 Cu25.9 . These compositions were chosen to give integer numbers of atoms of each species in DFT calculations using 27 atom supercells. The compositions listed above are summarized in Fig. 1. In the binary PdCu phase diagram, a bcc structure appears when the Pd concentration is lower than 55% [36]. In our calculations, we limit the concentration of Pd to be higher than 66% for this reason. Geometry optimization was performed to obtain the lattice constant of each alloy without interstitial H. Table 1 lists the DFToptimized lattice constant for each alloys we considered, including pure Pd. In previous studies, Vegard’s law was used to approximate the lattice constants of Pd-based binary alloys [39]. We find that for PdCuAg the lattice constants predicted by Vegard’s law are in good agreement with the value obtained from DFT calculations. Substituting Ag for Pd expands the lattice since Ag has a larger atomic size than Pd, and substituting Cu for Pd has the opposite effect due to the smaller size of Cu atoms. The maximum volume expansion
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Table 1 Optimized lattice constant (LC) for PdCuAg alloys. For comparison we also listed the lattice constant (LC* ) predicted by Vegard’s law. The volume expansion (V) is calculated as (Valloy − VPd )/VPd × 100 % , where Valloy and VPd is the volume of the unit cell of the alloy and pure Pd. For simplicity we only list the concentration of Cu and Ag. Cu/Ag (at.%)
LC (Å)
LC* (Å)
V (%)
Cu/Ag (at.%)
LC (Å)
LC* (Å)
V (%)
0/0 0/3.7 0/7.4 0/18.5 3.7/0 3.7/3.7 3.7/11.1
3.960 3.970 3.973 3.992 3.953 3.955 3.973
3.960 3.968 3.975 3.995 3.950 3.957 3.972
0.00 0.76 0.99 2.44 −0.53 −0.38 0.99
11.1/0 11.1/3.7 11.1/7.4 11.1/18.5 25.9/0 25.9/3.7 25.9/7.4
3.929 3.937 3.946 3.964 3.896 3.899 3.902
3.929 3.936 3.944 3.966 3.897 3.895 3.902
−2.33 −1.73 −1.06 0.30 −4.77 −4.55 −4.33
listed in Table 1 is observed for Pd81.5 Ag18.5 . These observations are consistent with the results found in binary systems. For each alloy, all calculations involving interstitial H were performed using the lattice constant listed in Table 1. 3.2. Cluster expansion of DFT data To characterize the behavior of interstitial hydrogen in PdCuAg alloys, one H atom is located at an interstitial site and the whole structure is optimized by allowing all metal atoms and the H atom to relax. Two possible interstitial sites exist in fcc structures: the octahedral site (O site) and the tetrahedral site (T site). For each supercell we considered, we calculated the classical binding energies of H in all possible interstitial sites and TS in the supercell as: Eb = EMH − EM − 0.5EH2
(1)
Here, Eb is the binding energy of hydrogen. EMH , EM and EH2 represent the total energy of the alloy with hydrogen, the alloy without hydrogen and H2 molecules, respectively. The zero point energy of interstitial hydrogen is defined as EZPE =
0.5hi
calculations sample the range of environments that are actually relevant in a substitutionally random material [34], we compared the distribution of binding energies observed in our DFT calculation with the distribution predicted by the CE model when applied to a large random volume containing >4000 metal atoms by plotting the cumulative probability of the site energies for each set of data as shown in Fig. 2(b). These comparisons indicate that there are no significant deviations between the CE model and DFT calculations, allowing us to conclude that our CE models can be used to describe the full set of interstitial sites and transition states in these alloys to calculate the macroscopic properties of interstitial H. 3.3. Solubilities of H in selected PdCuAg alloys We first look at the solubilities for PdAg binary alloys, normalized by the value in pure Pd, as shown in Fig. 3. The calculated
a
(2)
i
Here, vi is the vibrational frequency of H and h is Planck’s constant. The summation includes all three frequencies and the two real frequencies for hydrogen in the interstitial sites and transition states, respectively. Once DFT has been used to determine results for a large number of sites in an alloy, the cluster expansion method is then applied to model both the site energies and zero point energies obtained from DFT calculations. Fig. 2 shows the results from CE modeling for a typical example, Pd70.4 Cu11.1 Ag18.5 . We first compared the binding energies for those sites calculated by DFT with CE predictions as shown in Fig. 2(a). To further confirm that our DFT
Fig. 1. Compositions examined using DFT shown on a ternary Pd–Cu–Ag composition diagram.
b
Fig. 2. (a) Comparison of the energies of H in the octahedral site (O), tetrahedral site (T) and transition states (TS) predicted with CE modeling and the energies obtained from DFT calculations for Pd70.4 Cu11.1 Ag18.5 . (b) Comparison of the energy distribution predicted with CE models using a large substitutionally random sample with the distribution obtained from DFT calculations.
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Fig. 3. Predicted solubilities of H in PdAg alloys, normalized by the solubilities in pure Pd. The dotted experimental data for Pd90 Ag10 and Pd80 Ag20 came from Sakamoto et al. [15]. The dashed curve shows experimental data from Holleck [39].
solubilities of hydrogen increase with the amount of Ag in the binary phase at fixed temperature, and this increase is more significant at low temperatures. At 600 K, the predicted solubility of hydrogen in Pd81.5 Ag18.5 is 3.4 times higher than in pure Pd. The calculated results are in a good agreement with the experimental results from Sakamoto et al. and Holleck [15,39], indicating that our modeling approach is reliable. Sonwane and coworkers have previously used DFT-based methods to predict H solubilities in PdAg alloys [28,29]. They found that the solubilities of hydrogen increases with Ag concentration for alloys with less than 25 at.% Ag. Our results are consistent with these observations. Similar calculations were done for all the other PdCuAg alloys listed above at temperatures from 600 to 1200 K. Here we only present the results at 800 K, which capture the main trends among these alloys. These results are shown in Fig. 4. A similar trend is observed for PdCuAg ternary alloys as for PdAg binary alloys. That is, the solubilities increase with Ag concentration in the alloys (for a fixed concentration of Cu). The magnitude of this trend, however, becomes less significant at high Cu concentrations. Much of this trend can be understood in terms of lattice parameters of the alloys. Since Ag is a larger atom than Pd, adding Ag in the alloy expands the lattice, as seen in Table 1. The space available for the interstitial hydrogen atom is increased due to the expanded lattice, and, all other things being equal, this increase makes the energies of interstitial H atoms more favorable. This lattice expansion effect tends
Fig. 5. The cumulative probability of binding energies in O sites for different PdCuAg alloys as predicted from CE models.
to increase H solubility as Ag is added to an alloy. If a series of alloys with constant Ag content are considered, H solubility decreases as the Cu content is increased, consistent with what was already known for fcc PdCu alloys [6]. To further understand the solubility trends in Fig. 4, we compared the distribution of binding energies in O sites, which contribute the most for the solubilities, generated by CE models for alloys with same Cu content but different Ag concentrations, as shown in Fig. 5. The comparison supports the conclusion we drew in the last paragraph that adding Ag favors the interstitial hydrogen; this occurs due to decreases in binding energies of the most stable sites. However, we also notice that besides these sites being more favorable, other sites become more unfavorable as the Ag content is increased. Further examination revealed that the binding energies of H in O sites become less favorable as the number of Ag atoms in the nearest neighbor shell increases. This can be understood in terms of the atom size; having more Ag atoms in the nearest neighbor (NN) shell of the interstitial site decreases the free space available for H atom relative to a site with fewer or no Ag atom in the NN shell. This observation implies that adding Ag in the alloy causes two different effects. First, Ag atom expands the lattice, which makes H occupation more favorable. Secondly, with more Ag content in the alloy, more interstitial sites have one or more Ag atom in its nearest neighbor shell, which is unfavorable for the interstitial hydrogen. The overall solubility is the result of the interplay of these two effects. After obtaining the solubilities for all of the PdCuAg alloys listed in Table 1, we interpolated the solubilities for other PdCuAg alloys using a cubic spline function to fit the calculated data. In applying this approach, the maximum concentrations of Cu and Ag was limited to be 25.9% and 18.5%, respectively, and the Pd concentration was limited to be larger than 66.7%. Fig. 6 shows a contour map of the solubility of the predicted solubility of H in all PdCuAg alloys in this range normalized by the value in pure Pd from at 800 K. All compositions above (below) the solid line in Fig. 6 indicate lower (higher) solubilities than pure Pd. The alloys with high solubility typically contain more Ag than Cu. The highest normalized solubility at 800 K is about 2.2 times to the value in pure Pd; this result occurs for a binary PdAg alloy. 3.4. Diffusivities of hydrogen in selected PdCuAg alloys
Fig. 4. Predicted solubilities of hydrogen in PdCuAg alloys at 800 K, normalized by the solubility in pure Pd under the same conditions.
After the CE models for the site energy at the O sites, T sites and the transition states were available, kinetic Monte Carlo (KMC) simulations were performed to compute the diffusivity of H in PdCuAg
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Fig. 6. Contour map of the predicted solubility of hydrogen in PdCuAg alloys normalized by the value in pure Pd at 800 K.
alloys [6]. We first compare our calculated results for PdAg binary alloys with experimental reports. In Fig. 7, we show the calculated diffusivities for three PdAg binary alloys, normalized by the value in pure Pd, as well as experimental results for Pd90 Ag10 and Pd80 Ag20 [15,39]. An opposite trend is observed from the solubilities, namely, the diffusivities of H decrease as the content of Ag increases in the alloy. For example, at 600 K, our calculations give the normalized diffusivities of 0.98, 0.94 and 0.72 for Pd96.3 Ag3.7 , Pd92.6 Ag7.4 and Pd81.5 Ag18.5 , respectively. This trend is consistent with experimental reports. Fig. 8 shows the diffusivities of hydrogen in PdCuAg ternary alloys at 800 K, normalized by the value in pure Pd. For alloys with no or low Cu content, the diffusivity decreases with Ag concentration in the alloy. For alloys with high Cu content, however, the diffusivity first increases with Ag concentration, then decreases at high Ag concentration. Unlike the solubility, which is determined directly by H binding energies in interstitial sites, the diffusivity is affected by both the binding energies and the transition state energies for individual hops [18]. Achieving an intuitive understanding of the factors affecting diffusion is more difficult than understanding solubility. To provide some understanding of this kind, we considered a simplified model for the binding energies of H in interstitial sites and transition states. Our approach to finding a simplified model for the
Fig. 7. Predicted diffusivities of hydrogen in PdAg binary alloys, normalized by the diffusivities in pure Pd. Symbols show results from our DFT-based calculations, while curves without symbols show experimental data. The dotted experimental data for Pd90 Ag10 and Pd80 Ag20 came from Sakamoto et al. [15]. The dashed curve shows experimental data from Holleck [39].
Fig. 8. Predicted diffusivities of hydrogen in PdCuAg alloys at 800 K, normalized by the diffusivities in pure Pd.
O site binding energies is illustrated in Fig. 9. In this figure, O sites whose NN shell are made up of x Pd atoms, y Cu atoms, and z Ag atoms are denoted Pdx Cuy Agz . The symbols in Fig. 9 show averages of DFT calculated O site binding energies for each kind of site as a function of the lattice parameter of the alloy. For each class of binding site, the binding energies decrease approximately linearly as the lattice parameter increases. As expected, increasing the number of Ag atoms in the NN shell increases the binding energies, while this effect is less significant for Cu atoms. A similar approach was used for the transition states. To classify the available TS in a simplified description, we used the local environment of the T site to distinguish different transition states. There are four metal atoms in the NN shell of each T site, three of which form a triangular “window” that connects to an O site. We classified transition states by considering the three metal atoms forming the window separately from the fourth metal atom making up the T site. For example, a TS denoted Pd2 Ag–Ag has a window formed by two Pd atoms and one Ag atom, with the fourth metal atom defining the T site being Ag. The results in Fig. 10 indicate that the presence of Ag atoms in the window forming TS increases the energy of the TS significantly, while the effect of substituting Cu for Pd is almost negligible. We emphasize that the simplified model described above only gives moderately accurate descriptions of the energies of binding
Fig. 9. The average classical binding energies in O sites in PdCuAg alloys as a function of the alloy lattice constant. Symbols show averages of DFT data for the classes of O sites indicated in the legend, while lines show the linearly fitted binding energies. Lattice constants are in units of Å.
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Fig. 10. The average classical binding energies in transition states in PdCuAg alloys as a function of the alloy lattice constant. Symbols show averages of DFT data for the classes of transition states indicated in the legend, while lines show the linearly fitted binding energies. Lattice constants are in units of Å.
sites or transition states. Even for the sites in the same category as identified in Figs. 9 and 10, noticeable difference of the binding energies still exist as obtained from DFT calculations. Accurate evaluation of the diffusivities of hydrogen still requires the detailed cluster expansion model described above and used in Figs. 7 and 8. With this caveat, the simplified analysis highlights two main effects that exist when Ag is added to the alloy. First, the change in O site binding energy as a function of Ag content is more marked than the equivalent change in the TS energies. As a result, the activation energy for H atoms to hop out of O sites tends to increase as Ag content is increased. For example, when the lattice constant changes from 3.960 A˚ (pure Pd) to 3.992 A˚ (Pd81.5 Ag18.5 ), the binding energy in a Pd6 site decreases by 0.072 eV, while the energy in a Pd3 –Pd transition state decreases by 0.060 eV. The activation energy barrier, defined by the difference between the O site energy and the TS energy, therefore increases slightly by 0.012 eV. Secondly, Fig. 10 reveals that hopping through a triangle of atoms consisting of one or more Ag atoms is much less favorable than hopping through other kinds of transition states. This local effect means that large activation energy barriers exist for H hopping in the immediate vicinity of Ag atoms. The combination of these two effects means that throughout the PdCuAg phase diagram, adding Ag to the alloy tends to reduce the net diffusivity of H.
Fig. 11. Contour map of the diffusivity of hydrogen in PdCuAg alloys normalized by the value in pure Pd at 800 K.
Fig. 12. Predicted permeabilities of H in PdCuAg alloys at 800 K, normalized by the permeabilities in pure Pd.
In Fig. 11, we plot a contour map of H diffusivity at 800 K as interpolated from our calculated cluster expansion model data. The interpolation process was performed in the same way as described in the previous section. In this contour map, pure Pd has the highest diffusivity; all PdCuAg alloys have a lower diffusivity than pure Pd. The lowest diffusivity in this map is still about 60% of the value in pure Pd, which indicates that the diffusivity is unlikely to be a dominant factor in determining the permeabilities for these alloys. 3.5. Permeabilities of hydrogen in PdCuAg alloys Finally we examined the permeability of H in PdCuAg ternary alloys. The permeability of H in Pd-based alloys can be expressed as a product of solubility and diffusivity, as long as the permeation process is dominated by the diffusion of atomic hydrogen in the bulk phase of the alloy [6]. Surface processes such as recombinative desorption of H2 can be important for sufficiently thin membranes [32,40], but we neglect these effects here. In Fig. 12, we show the calculated permeability for each alloy from our cluster expansion models at 800 K, normalized by the value in Pd. The net permeability of H increases with Ag concentration for alloys with the same amount of Cu. Similar to the solubility, the strength of this trend is diminished at high Cu or Ag content. In Fig. 13, we showed the calculated contour map of the permeability at three different temperatures, 600, 800 and 1200 K. This is the first time that DFT-based models have been used to give a comprehensive description of a broad composition range of a ternary alloy material. In this figure, interpolation among data from our cluster expansion models was performed in the same way as in Figs. 6 and 11. The main features of the contour map of the permeability are more similar to the map of solubility (Fig. 6) than to the diffusivity (Fig. 11), which indicates that for permeability the increased solubility with Ag concentration is more dominant than the diffusivity. As the temperature increases, the normalized permeability generally decreases for most PdCuAg alloys. One important observation from Fig. 13 is that substituting Ag for Pd or Cu in a high Cu content alloy could increase the hydrogen permeability. This suggests that for PdCu alloys with useful levels of sulfur-tolerance but low hydrogen permeability an effective way to increase their performance is to use PdCuAg ternary alloys. We note here that only the permeation behavior of pure H2 is considered in models we have described here. Studying the resistance of these alloys to contaminants is beyond the scope of current quantitative models. It is of course of great interest to make experimental efforts to examine both the sulfur resistance and hydrogen perme-
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a
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compositions, giving the first comprehensive description of an alloy of this kind with DFT-based methods. The solubilities of hydrogen are observed to increase with Ag concentration while this variation causes diffusivities to decrease in most cases. The net permeability, a product of the solubility and the diffusivity, increases if Ag is substituted for Pd, indicating that for these alloys solubility is a more dominant contributor to the permeability than the diffusivity. In the contour map of the permeabilities, we have located compositions which have a higher permeability than pure Pd. These alloys are of particular interest as they might be good candidates as H2 purification membranes. Our results provide a strong motivation for further experimental examination of both the permeability and resistance to the contaminants of these alloys as H2 purification membranes. This is the first time that H permeability has been examined for a broad range of ternary compositions using DFT-based methods. We performed DFT calculations for a large enough number of distinct alloys that it was possible to generate contour maps of the H solubility, diffusivity and permeability as a function of the alloy composition. Although our research only examined PdCuAg ternary alloys, the same methods are also applicable for other ternary alloys. Theoretical predictions for H permeability in ternary alloys can, thus, provide a useful way in searching new materials with good properties for H2 purification. Acknowledgement This work was financially supported by the DOE-BES Hydrogen Fuels Initiative and the National Energy Technology Laboratory.
b
c Fig. 13. Contour map of the permeability of hydrogen in PdCuAg alloys normalized by the value in pure Pd at (a) 600 K, (b) 800 K and (c) 1200 K.
ance of these alloys as a necessary step in their application as H2 purification membranes. 4. Conclusion In this paper, we have screened a series of PdCuAg alloys as H2 purification membranes using DFT-based models. These models do not require any experimental input in order to make quantitative predictions about the performance of alloys as membranes operating at elevated temperatures. We have explored a range of PdCuAg
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[17] Y. Zhang, J. Lu, R. Maeda, C. Nishimura, Microfabricated Pd and Pd-25Ag alloy membranes, J. Alloys Compd. (2007) 579–582. [18] P. Kamakoti, D.S. Sholl, A comparison of hydrogen diffusivities in Pd and CuPd alloys using densitiy functional theory, J. Membr. Sci. 225 (2003) 145–154. [19] O.J. Kleppa, C. Shamsuddin, Picard, High temperature thermodynamics of dilute solutions of hydrogen in palladium–copper alloys, J. Chem. Phys. 71 (1979) 1656–1661. [20] F. Roa, M.J. Block, J.D. Way, The influence of alloy composition on the H2 flux of composite Pd–Cu membranes, Desalination 147 (2002) 411–416. [21] P.M. Theon, F. Roa, J.D. Way, High flux palladium–copper composite membranes for hydrogen separation, Desalination 193 (2006) 224–229. [22] M. Yoshihara, R.B. Mclellan, Thermodynamics of Pd–Cu–H Solid Solutions, Acta. Metall. 31 (1983) 61–68. [23] K.E. Coulter, J.D. Way, S.K. Gade, S. Chaudhari, D.S. Sholl, L. Semidey-Flecha, Predicting, fabricating and permeability testing of free-standing ternary palladium–copper–gold membranes for hydrogen separation, J Phys. Chem. C 114 (2010) 17173–17180. [24] D.S. Sholl, Using density functional theory to study hydrogen diffusion in metals: a brief overview, J. Alloys Compd. 446 (2007) 462–468. [25] D.S. Sholl, Y.H. Ma, Dense metal membranes for the production of high-purity hydrogen, MRS Bull. 31 (2006) 770–773. [26] P. Kamakoti, D.S. Sholl, Ab initio lattice-gas modeling of interstitial hydrogen diffusion in CuPd alloys, Phys. Rev. B 71 (2005) 014301. [27] P. Kamakoti, D.S. Sholl, Towards first principle-based identification of ternary alloys for hydrogen purification membranes, J. Membr. Sci. 279 (2006) 94–99. [28] C.G. Sonwane, J. Wilcox, Y.H. Ma, Achieving optimum hydrogen permeability in PdAg and PdAu alloys, J. Chem. Phys. 125 (2006) 184714. [29] C.G. Sonwane, J. Wilcox, Y.H. Ma, Solubility of hydrogen in PdAg and PdAu binary alloys using density functional theory, J. Phys. Chem. B 110 (2006) 24549–24558.
[30] S. Hao, D.S. Sholl, Using first-principles calculations to accelerate materials discovery for hydrogen purification membranes by modeling amorphous metals, Energy Environ. Sci. 1 (2008) 175–189. [31] S. Hao, D.S. Sholl, Self-diffusion and macroscopic diffusion of hydrogen in amorphous metals from first-principles calculations, J. Chem. Phys. 130 (2009) 244705. [32] C. Ling, D.S. Sholl, Using first-principle calculations to predict surface resistances to H2 transport through metal alloy membranes, J. Membr. Sci. 303 (2007) 162–172. [33] B.D. Morreale, B.H. Howard, O. Iyoha, R.M. Enick, C. Ling, D.S. Sholl, Experimental and computational prediction of H2 transport properties of Pd4 S, Ind. Eng. Chem. Res. 329 (2007) 153–159. [34] L. Semidey-Flecha, C. Ling, D.S. Sholl, Detailed first-principles models of hydrogen permeation through PdCu-based ternary alloys, J. Membr. Sci. 362 (2010) 384–392. [35] L. Semidey-Flecha, D.S. Sholl, Combining density functional theory and cluster expansion methods to predict H2 permeance through Pd-based binary alloy membranes, J. Chem. Phys. 128 (2008) 144701. [36] T.B. Massalski, H. Okamoto, P.R. Subramanian, L. Kacprzak, Binary Alloy Phase Diagrams, 2nd ed., ASM/NIST International, Gaithersburg, 1990. [37] G. Kresse, J. Hafner, Ab-initio molecular dynamics for open shell transition metals, Phys. Rev. B 48 (1993) 13115–13118. [38] D.S. Sholl, J.A. Steckel, Density Functional Theory: A Practical Introduction, Wiley-Interscience, Hoboken, 2009. [39] G.L. Holleck, Diffusion and solubility of hydrogen in palladium and palladium–silver alloys, J. Phys. Chem. B 74 (1970) 503–511. [40] T.L. Ward, T. Dao, Model of hydrogen permeation behavior in palladium membranes, J. Membr. Sci. 153 (1999) 211–231.
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First-principles screening of PdCuAg ternary alloys as H2 purification membranes Chen Ling, Lymarie Semidey-Flecha, David S. Sholl ∗ School of Chemical and Biomolecular Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0100, USA
a r t i c l e
i n f o
Article history: Received 2 December 2010 Received in revised form 14 January 2011 Accepted 22 January 2011 Available online 28 January 2011 Keywords: Hydrogen Metal membranes Theory and modeling Alloys
a b s t r a c t Predictions based on Density Functional Theory (DFT) calculations have proven to be a useful complement to experimental efforts to characterize permeation of hydrogen through metal membranes at elevated temperatures. Previous applications of this approach have focused on relatively small numbers of alloys with specific binary or ternary compositions. We describe DFT-based calculations that provide predictions for the solubility, diffusivity, and permeability of hydrogen at elevated temperatures in PdCuAg alloys for a wide range of compositions of this ternary alloy that yield substitutionally disordered fcc materials. Our results show that the variation in permeability of hydrogen among these alloys is more closely tied to variations in solubility than variations in diffusivity. These results give a detailed description of a ternary alloy of practical interest for high temperature H2 purification and also demonstrate a general strategy for making predictions about complex metal alloys for this application. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Dense metal membranes are well known in applications for H2 purification and have many advantages over alternative purification methods [1–3]. Pure Pd, for example, is well known as a metal that can be used as a H2 purification material [4]. Despite the commercial success of Pd membranes, several factors limit their widespread applications. At temperatures below 573 K and moderate H2 pressures, a Pd hydride phase forms with a lattice constant substantially different from pure Pd. Membrane integrity can be compromised if this phase is formed. A more serious problem is that pure Pd membranes are extremely sensitive to non-hydrogen contaminants such as H2 S and CO, which can effectively reduce the H2 flux through the membranes to zero [5,6]. H2 S is ubiquitous in natural gas and CO is always present in syngas. Poisoning of membranes by these species is thus a critical issue for hydrogen separations from these gas mixtures. One route to improving the performance of pure metal membranes is to develop metal alloys as successful membranes [2]. Pd-based binary alloys, for example, have been widely examined experimentally and theoretically, with Pd being the core material for binary alloys in combination with elements including cerium, copper, gold, iron, nickel, silver, and yttrium [2,7–17]. Among these binary alloys, PdCu alloys are particularly interesting because these membranes may improve resistance to poisoning [6]. The permeability of hydrogen, however, has been observed to decrease with increasing Cu concentration in the alloys [6,18–22]. Other binary
∗ Corresponding author. Tel.: +1 412 268 4207; fax: +1 412 268 7139. E-mail address: [email protected] (D.S. Sholl). 0376-7388/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2011.01.030
alloys such as PdAg and PdAu have been shown to increase the permeability of hydrogen compared to pure Pd [12,15,23]. There is no evidence that the resistance of contaminants relative to pure Pd is improved for these binary alloys. Based on these results, identification of Pd-based alloys with considerable hydrogen flux and resistance of the contaminants is still required. Although experimental studies of Pd-based binary alloys have been reported, it is still challenging to experimentally screen a large series of alloys, since such efforts would require extremely large resources. In recent years, theoretical approaches based on first-principles calculations provide a complementary route to predict the performance of metal alloys in H2 purification [24,25]. Kamakoti and Sholl developed an ab initio method based on Density Functional Theory (DFT) calculations to effectively examine the solubility, diffusivity and permeability of hydrogen in the bulk phase of PdCu alloys without any experimental input [6,18,26,27]. Similar methods have since been used to examine PdAg and PdAu alloys [28,29], amorphous metals [30,31] and metal sulfides [32,33]. Kamakoti and Sholl’s calculation showed that the predictions of the permeability of H in PdCu alloys were in good agreement with experimental data [6]. Subsequently, Semidey-Flecha and Sholl extended this method to examine a series of Pd96 M4 binary and Pd70 Cu26 M4 ternary alloys with M = Ni, Pt, Rh, Ag, Au, Cu and Ni, Pt, Ag, and Au [23,34,35]. Here and throughout this paper, all compositions are given in at.%. This work indicated that the permeability of PdCuAg is improved compared to PdCu alloys by adding just 4 at% of Ag in the alloy [34]. An interesting question remaining from this study is whether adding higher level of Ag can lead to even more favorable outcomes. In this paper, we extend these calculations to a broad series of PdCuAg ternary alloys. The solubilities, diffusivities and permeabilities of hydrogen are examined and discussed. We
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show that for alloys with high Ag concentration, hydrogen permeability comparable to pure Pd can be reached. The structure of this paper is as follows. We first discuss the methods and the computational method used in examining ternary alloys in Section 2. A series of PdCuAg ternary alloys are then studied. We describe the solubility, diffusivity and permeability of hydrogen in these alloys in Section 3. Finally, a conclusion and the prospects for future work are given in Section 4.
2. Models 2.1. Model description A detailed description of the modeling framework we used to identify the properties of interstitial hydrogen in Pd-based alloys has been presented previously [34,35]. Here we highlight several important factors of this approach. Both PdCu and PdAg form a disordered fcc solid solution for the Pd-rich compositions we examine below [36]. No experimental description of the phase diagram of PdCuAg ternary alloys is available. It is reasonable to expect, however, that ternary PdCuAg also has a disordered fcc structure for the compositions we examine. All of our calculations make this assumption. This choice is not intended to imply that alloys associated with the bcc PdCu alloys that exist with ∼50 at.% Cu are uninteresting for membrane applications. Membranes made from these bcc alloys are known to have higher hydrogen permeability than fcc PdCu alloys with ∼30 at.% Cu [6]. This is largely due to the high diffusion coefficient of H in these bcc alloys [26], an observation that highlights the critical role of crystal structure in determining membrane properties. Even though there is considerable interest in these bcc alloys, insufficient information about the compositions for which these alloys could form as ternary PdCuAg alloys would make detailed DFT-based models of such alloys of limited value. As a result, we have confined our attention in this work to fcc PdCuAg alloys in the range of composition defined above. A key challenge in applying DFT calculations to describe interstitial H in alloys is how to describe the properties of interstitial hydrogen with configurationally distinct local environments, since substitutional disordered materials have a large variety of structurally distinct binding sites. In our study, we applied cluster expansion (CE) methods to describe the relationship between the binding energies of interstitial H in Pd-based alloys and the local environment of these sites. A detailed explanation of how to apply CE models in the prediction of hydrogen permeability in metal alloys is already available [34,35]. In the results presented below, DFT calculations were first performed for a large collection of distinct interstitial sites and transition states for H hopping. The data obtained from DFT calculations was then used in CE modeling. Once the CE model accurately representing the interstitial binding energies and site to site transition state energies obtained from DFT calculations was available, the net solubility of H was calculated by statistical mechanics calculations [6]. This method calculates the equilibrium between gaseous hydrogen and individual interstitial atomic hydrogen while ignoring the interaction between interstitial hydrogen. This approach is well justified at dilute interstitial hydrogen concentrations. As we discuss below, we find that the maximum concentration of hydrogen in our study is about 3.4 times higher than in pure Pd at 600 K, which indicates that the assumption of dilute interstitial H is still valid for PdCuAg ternary systems with moderate hydrogen pressures and elevated temperatures. Although DFT calculations can be used to quantify H–H interactions at non-dilute concentrations of interstitial H, we have not performed calculations of this kind in this work. To calculate the diffusivity of interstitial hydrogen in the alloys, we applied kinetic Monte Carlo (KMC) simulations. These KMC simulations
give the self diffusivity of H. At dilute concentration [6,35], the selfdiffusivity and transport diffusivity of interstitial H are equal [3], so the self diffusivity can be used in calculating the net flux of H through a membrane. 2.2. DFT calculation details We performed DFT calculations using the Vienna ab initio Simulation Package (VASP) using the generalized gradient approximation with the PW91 functional to describe electron exchange-correlation effects [37,38]. Ion-electron interactions were described by ultrasoft pseudopotentials. A plane-wave expansion with a cutoff of 233.73 eV was used in all calculations [37]. Geometry relaxations were performed with a conjugate gradient method until the forces on all unconstrained atoms were less than 0.03 eV/Å. A -centered grid with 4 × 4 × 4 k-points was used for all calculations. The supercell we used contained 27 metal atoms forming an fcc lattice and one H atom located in an interstitial site. All atoms were allowed to relax during the calculation, with the volume of the cell fixed at the optimized volume determined without the interstitial H. Vibrational frequencies of interstitial H were calculated in the harmonic approximation, assuming localized vibrations of hydrogen atom are decoupled from vibrations of metal atoms. Accurately characterizing the diffusion of interstitial H requires determining a large number of transition states (TS) with DFT calculations. The most common method to find a transition state with DFT, the nudged elastic band (NEB) method [38], is far more time consuming than the calculations needed to optimize the geometry of H in an interstitial site. To overcome this computational bottleneck, we developed an efficient method that allows transition states in the alloys of interest to be accurately computed with far smaller computational resources. This approach generated initial approximations for the position of a TS by noting that the TS for H hopping between two interstitial sites is generally located close to the center of the polyhedral face connecting the two interstitial sites [30]. These estimations made it possible to rigorously locate transition states using DFT calculations equivalent to an energy minimization. 3. Results and discussion 3.1. Optimization of lattice constants We chose to examine seven different PdCuAg ternary alloys with DFT: Pd92.6 Cu3.7 Ag3.7 , Pd85.2 Cu3.7 Ag11.1 , Pd85.2 Cu11.1 Ag3.7 , Pd70.4 Cu11.1 Ag18.5 , Pd70.4 Cu25.9 Ag3.7 and Pd81.5 Cu11.1 Ag7.4 , Pd66.7 Cu25.9 Ag7.4 . For comparison, we also examined three PdAg binary alloys, Pd96.3 Ag3.7 , Pd92.6 Ag7.4 and Pd81.5 Ag18.5 , and three PdCu binary alloys, Pd96.3 Cu3.7 , Pd88.9 Cu11.1 and Pd74.1 Cu25.9 . These compositions were chosen to give integer numbers of atoms of each species in DFT calculations using 27 atom supercells. The compositions listed above are summarized in Fig. 1. In the binary PdCu phase diagram, a bcc structure appears when the Pd concentration is lower than 55% [36]. In our calculations, we limit the concentration of Pd to be higher than 66% for this reason. Geometry optimization was performed to obtain the lattice constant of each alloy without interstitial H. Table 1 lists the DFToptimized lattice constant for each alloys we considered, including pure Pd. In previous studies, Vegard’s law was used to approximate the lattice constants of Pd-based binary alloys [39]. We find that for PdCuAg the lattice constants predicted by Vegard’s law are in good agreement with the value obtained from DFT calculations. Substituting Ag for Pd expands the lattice since Ag has a larger atomic size than Pd, and substituting Cu for Pd has the opposite effect due to the smaller size of Cu atoms. The maximum volume expansion
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Table 1 Optimized lattice constant (LC) for PdCuAg alloys. For comparison we also listed the lattice constant (LC* ) predicted by Vegard’s law. The volume expansion (V) is calculated as (Valloy − VPd )/VPd × 100 % , where Valloy and VPd is the volume of the unit cell of the alloy and pure Pd. For simplicity we only list the concentration of Cu and Ag. Cu/Ag (at.%)
LC (Å)
LC* (Å)
V (%)
Cu/Ag (at.%)
LC (Å)
LC* (Å)
V (%)
0/0 0/3.7 0/7.4 0/18.5 3.7/0 3.7/3.7 3.7/11.1
3.960 3.970 3.973 3.992 3.953 3.955 3.973
3.960 3.968 3.975 3.995 3.950 3.957 3.972
0.00 0.76 0.99 2.44 −0.53 −0.38 0.99
11.1/0 11.1/3.7 11.1/7.4 11.1/18.5 25.9/0 25.9/3.7 25.9/7.4
3.929 3.937 3.946 3.964 3.896 3.899 3.902
3.929 3.936 3.944 3.966 3.897 3.895 3.902
−2.33 −1.73 −1.06 0.30 −4.77 −4.55 −4.33
listed in Table 1 is observed for Pd81.5 Ag18.5 . These observations are consistent with the results found in binary systems. For each alloy, all calculations involving interstitial H were performed using the lattice constant listed in Table 1. 3.2. Cluster expansion of DFT data To characterize the behavior of interstitial hydrogen in PdCuAg alloys, one H atom is located at an interstitial site and the whole structure is optimized by allowing all metal atoms and the H atom to relax. Two possible interstitial sites exist in fcc structures: the octahedral site (O site) and the tetrahedral site (T site). For each supercell we considered, we calculated the classical binding energies of H in all possible interstitial sites and TS in the supercell as: Eb = EMH − EM − 0.5EH2
(1)
Here, Eb is the binding energy of hydrogen. EMH , EM and EH2 represent the total energy of the alloy with hydrogen, the alloy without hydrogen and H2 molecules, respectively. The zero point energy of interstitial hydrogen is defined as EZPE =
0.5hi
calculations sample the range of environments that are actually relevant in a substitutionally random material [34], we compared the distribution of binding energies observed in our DFT calculation with the distribution predicted by the CE model when applied to a large random volume containing >4000 metal atoms by plotting the cumulative probability of the site energies for each set of data as shown in Fig. 2(b). These comparisons indicate that there are no significant deviations between the CE model and DFT calculations, allowing us to conclude that our CE models can be used to describe the full set of interstitial sites and transition states in these alloys to calculate the macroscopic properties of interstitial H. 3.3. Solubilities of H in selected PdCuAg alloys We first look at the solubilities for PdAg binary alloys, normalized by the value in pure Pd, as shown in Fig. 3. The calculated
a
(2)
i
Here, vi is the vibrational frequency of H and h is Planck’s constant. The summation includes all three frequencies and the two real frequencies for hydrogen in the interstitial sites and transition states, respectively. Once DFT has been used to determine results for a large number of sites in an alloy, the cluster expansion method is then applied to model both the site energies and zero point energies obtained from DFT calculations. Fig. 2 shows the results from CE modeling for a typical example, Pd70.4 Cu11.1 Ag18.5 . We first compared the binding energies for those sites calculated by DFT with CE predictions as shown in Fig. 2(a). To further confirm that our DFT
Fig. 1. Compositions examined using DFT shown on a ternary Pd–Cu–Ag composition diagram.
b
Fig. 2. (a) Comparison of the energies of H in the octahedral site (O), tetrahedral site (T) and transition states (TS) predicted with CE modeling and the energies obtained from DFT calculations for Pd70.4 Cu11.1 Ag18.5 . (b) Comparison of the energy distribution predicted with CE models using a large substitutionally random sample with the distribution obtained from DFT calculations.
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Fig. 3. Predicted solubilities of H in PdAg alloys, normalized by the solubilities in pure Pd. The dotted experimental data for Pd90 Ag10 and Pd80 Ag20 came from Sakamoto et al. [15]. The dashed curve shows experimental data from Holleck [39].
solubilities of hydrogen increase with the amount of Ag in the binary phase at fixed temperature, and this increase is more significant at low temperatures. At 600 K, the predicted solubility of hydrogen in Pd81.5 Ag18.5 is 3.4 times higher than in pure Pd. The calculated results are in a good agreement with the experimental results from Sakamoto et al. and Holleck [15,39], indicating that our modeling approach is reliable. Sonwane and coworkers have previously used DFT-based methods to predict H solubilities in PdAg alloys [28,29]. They found that the solubilities of hydrogen increases with Ag concentration for alloys with less than 25 at.% Ag. Our results are consistent with these observations. Similar calculations were done for all the other PdCuAg alloys listed above at temperatures from 600 to 1200 K. Here we only present the results at 800 K, which capture the main trends among these alloys. These results are shown in Fig. 4. A similar trend is observed for PdCuAg ternary alloys as for PdAg binary alloys. That is, the solubilities increase with Ag concentration in the alloys (for a fixed concentration of Cu). The magnitude of this trend, however, becomes less significant at high Cu concentrations. Much of this trend can be understood in terms of lattice parameters of the alloys. Since Ag is a larger atom than Pd, adding Ag in the alloy expands the lattice, as seen in Table 1. The space available for the interstitial hydrogen atom is increased due to the expanded lattice, and, all other things being equal, this increase makes the energies of interstitial H atoms more favorable. This lattice expansion effect tends
Fig. 5. The cumulative probability of binding energies in O sites for different PdCuAg alloys as predicted from CE models.
to increase H solubility as Ag is added to an alloy. If a series of alloys with constant Ag content are considered, H solubility decreases as the Cu content is increased, consistent with what was already known for fcc PdCu alloys [6]. To further understand the solubility trends in Fig. 4, we compared the distribution of binding energies in O sites, which contribute the most for the solubilities, generated by CE models for alloys with same Cu content but different Ag concentrations, as shown in Fig. 5. The comparison supports the conclusion we drew in the last paragraph that adding Ag favors the interstitial hydrogen; this occurs due to decreases in binding energies of the most stable sites. However, we also notice that besides these sites being more favorable, other sites become more unfavorable as the Ag content is increased. Further examination revealed that the binding energies of H in O sites become less favorable as the number of Ag atoms in the nearest neighbor shell increases. This can be understood in terms of the atom size; having more Ag atoms in the nearest neighbor (NN) shell of the interstitial site decreases the free space available for H atom relative to a site with fewer or no Ag atom in the NN shell. This observation implies that adding Ag in the alloy causes two different effects. First, Ag atom expands the lattice, which makes H occupation more favorable. Secondly, with more Ag content in the alloy, more interstitial sites have one or more Ag atom in its nearest neighbor shell, which is unfavorable for the interstitial hydrogen. The overall solubility is the result of the interplay of these two effects. After obtaining the solubilities for all of the PdCuAg alloys listed in Table 1, we interpolated the solubilities for other PdCuAg alloys using a cubic spline function to fit the calculated data. In applying this approach, the maximum concentrations of Cu and Ag was limited to be 25.9% and 18.5%, respectively, and the Pd concentration was limited to be larger than 66.7%. Fig. 6 shows a contour map of the solubility of the predicted solubility of H in all PdCuAg alloys in this range normalized by the value in pure Pd from at 800 K. All compositions above (below) the solid line in Fig. 6 indicate lower (higher) solubilities than pure Pd. The alloys with high solubility typically contain more Ag than Cu. The highest normalized solubility at 800 K is about 2.2 times to the value in pure Pd; this result occurs for a binary PdAg alloy. 3.4. Diffusivities of hydrogen in selected PdCuAg alloys
Fig. 4. Predicted solubilities of hydrogen in PdCuAg alloys at 800 K, normalized by the solubility in pure Pd under the same conditions.
After the CE models for the site energy at the O sites, T sites and the transition states were available, kinetic Monte Carlo (KMC) simulations were performed to compute the diffusivity of H in PdCuAg
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Fig. 6. Contour map of the predicted solubility of hydrogen in PdCuAg alloys normalized by the value in pure Pd at 800 K.
alloys [6]. We first compare our calculated results for PdAg binary alloys with experimental reports. In Fig. 7, we show the calculated diffusivities for three PdAg binary alloys, normalized by the value in pure Pd, as well as experimental results for Pd90 Ag10 and Pd80 Ag20 [15,39]. An opposite trend is observed from the solubilities, namely, the diffusivities of H decrease as the content of Ag increases in the alloy. For example, at 600 K, our calculations give the normalized diffusivities of 0.98, 0.94 and 0.72 for Pd96.3 Ag3.7 , Pd92.6 Ag7.4 and Pd81.5 Ag18.5 , respectively. This trend is consistent with experimental reports. Fig. 8 shows the diffusivities of hydrogen in PdCuAg ternary alloys at 800 K, normalized by the value in pure Pd. For alloys with no or low Cu content, the diffusivity decreases with Ag concentration in the alloy. For alloys with high Cu content, however, the diffusivity first increases with Ag concentration, then decreases at high Ag concentration. Unlike the solubility, which is determined directly by H binding energies in interstitial sites, the diffusivity is affected by both the binding energies and the transition state energies for individual hops [18]. Achieving an intuitive understanding of the factors affecting diffusion is more difficult than understanding solubility. To provide some understanding of this kind, we considered a simplified model for the binding energies of H in interstitial sites and transition states. Our approach to finding a simplified model for the
Fig. 7. Predicted diffusivities of hydrogen in PdAg binary alloys, normalized by the diffusivities in pure Pd. Symbols show results from our DFT-based calculations, while curves without symbols show experimental data. The dotted experimental data for Pd90 Ag10 and Pd80 Ag20 came from Sakamoto et al. [15]. The dashed curve shows experimental data from Holleck [39].
Fig. 8. Predicted diffusivities of hydrogen in PdCuAg alloys at 800 K, normalized by the diffusivities in pure Pd.
O site binding energies is illustrated in Fig. 9. In this figure, O sites whose NN shell are made up of x Pd atoms, y Cu atoms, and z Ag atoms are denoted Pdx Cuy Agz . The symbols in Fig. 9 show averages of DFT calculated O site binding energies for each kind of site as a function of the lattice parameter of the alloy. For each class of binding site, the binding energies decrease approximately linearly as the lattice parameter increases. As expected, increasing the number of Ag atoms in the NN shell increases the binding energies, while this effect is less significant for Cu atoms. A similar approach was used for the transition states. To classify the available TS in a simplified description, we used the local environment of the T site to distinguish different transition states. There are four metal atoms in the NN shell of each T site, three of which form a triangular “window” that connects to an O site. We classified transition states by considering the three metal atoms forming the window separately from the fourth metal atom making up the T site. For example, a TS denoted Pd2 Ag–Ag has a window formed by two Pd atoms and one Ag atom, with the fourth metal atom defining the T site being Ag. The results in Fig. 10 indicate that the presence of Ag atoms in the window forming TS increases the energy of the TS significantly, while the effect of substituting Cu for Pd is almost negligible. We emphasize that the simplified model described above only gives moderately accurate descriptions of the energies of binding
Fig. 9. The average classical binding energies in O sites in PdCuAg alloys as a function of the alloy lattice constant. Symbols show averages of DFT data for the classes of O sites indicated in the legend, while lines show the linearly fitted binding energies. Lattice constants are in units of Å.
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Fig. 10. The average classical binding energies in transition states in PdCuAg alloys as a function of the alloy lattice constant. Symbols show averages of DFT data for the classes of transition states indicated in the legend, while lines show the linearly fitted binding energies. Lattice constants are in units of Å.
sites or transition states. Even for the sites in the same category as identified in Figs. 9 and 10, noticeable difference of the binding energies still exist as obtained from DFT calculations. Accurate evaluation of the diffusivities of hydrogen still requires the detailed cluster expansion model described above and used in Figs. 7 and 8. With this caveat, the simplified analysis highlights two main effects that exist when Ag is added to the alloy. First, the change in O site binding energy as a function of Ag content is more marked than the equivalent change in the TS energies. As a result, the activation energy for H atoms to hop out of O sites tends to increase as Ag content is increased. For example, when the lattice constant changes from 3.960 A˚ (pure Pd) to 3.992 A˚ (Pd81.5 Ag18.5 ), the binding energy in a Pd6 site decreases by 0.072 eV, while the energy in a Pd3 –Pd transition state decreases by 0.060 eV. The activation energy barrier, defined by the difference between the O site energy and the TS energy, therefore increases slightly by 0.012 eV. Secondly, Fig. 10 reveals that hopping through a triangle of atoms consisting of one or more Ag atoms is much less favorable than hopping through other kinds of transition states. This local effect means that large activation energy barriers exist for H hopping in the immediate vicinity of Ag atoms. The combination of these two effects means that throughout the PdCuAg phase diagram, adding Ag to the alloy tends to reduce the net diffusivity of H.
Fig. 11. Contour map of the diffusivity of hydrogen in PdCuAg alloys normalized by the value in pure Pd at 800 K.
Fig. 12. Predicted permeabilities of H in PdCuAg alloys at 800 K, normalized by the permeabilities in pure Pd.
In Fig. 11, we plot a contour map of H diffusivity at 800 K as interpolated from our calculated cluster expansion model data. The interpolation process was performed in the same way as described in the previous section. In this contour map, pure Pd has the highest diffusivity; all PdCuAg alloys have a lower diffusivity than pure Pd. The lowest diffusivity in this map is still about 60% of the value in pure Pd, which indicates that the diffusivity is unlikely to be a dominant factor in determining the permeabilities for these alloys. 3.5. Permeabilities of hydrogen in PdCuAg alloys Finally we examined the permeability of H in PdCuAg ternary alloys. The permeability of H in Pd-based alloys can be expressed as a product of solubility and diffusivity, as long as the permeation process is dominated by the diffusion of atomic hydrogen in the bulk phase of the alloy [6]. Surface processes such as recombinative desorption of H2 can be important for sufficiently thin membranes [32,40], but we neglect these effects here. In Fig. 12, we show the calculated permeability for each alloy from our cluster expansion models at 800 K, normalized by the value in Pd. The net permeability of H increases with Ag concentration for alloys with the same amount of Cu. Similar to the solubility, the strength of this trend is diminished at high Cu or Ag content. In Fig. 13, we showed the calculated contour map of the permeability at three different temperatures, 600, 800 and 1200 K. This is the first time that DFT-based models have been used to give a comprehensive description of a broad composition range of a ternary alloy material. In this figure, interpolation among data from our cluster expansion models was performed in the same way as in Figs. 6 and 11. The main features of the contour map of the permeability are more similar to the map of solubility (Fig. 6) than to the diffusivity (Fig. 11), which indicates that for permeability the increased solubility with Ag concentration is more dominant than the diffusivity. As the temperature increases, the normalized permeability generally decreases for most PdCuAg alloys. One important observation from Fig. 13 is that substituting Ag for Pd or Cu in a high Cu content alloy could increase the hydrogen permeability. This suggests that for PdCu alloys with useful levels of sulfur-tolerance but low hydrogen permeability an effective way to increase their performance is to use PdCuAg ternary alloys. We note here that only the permeation behavior of pure H2 is considered in models we have described here. Studying the resistance of these alloys to contaminants is beyond the scope of current quantitative models. It is of course of great interest to make experimental efforts to examine both the sulfur resistance and hydrogen perme-
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a
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compositions, giving the first comprehensive description of an alloy of this kind with DFT-based methods. The solubilities of hydrogen are observed to increase with Ag concentration while this variation causes diffusivities to decrease in most cases. The net permeability, a product of the solubility and the diffusivity, increases if Ag is substituted for Pd, indicating that for these alloys solubility is a more dominant contributor to the permeability than the diffusivity. In the contour map of the permeabilities, we have located compositions which have a higher permeability than pure Pd. These alloys are of particular interest as they might be good candidates as H2 purification membranes. Our results provide a strong motivation for further experimental examination of both the permeability and resistance to the contaminants of these alloys as H2 purification membranes. This is the first time that H permeability has been examined for a broad range of ternary compositions using DFT-based methods. We performed DFT calculations for a large enough number of distinct alloys that it was possible to generate contour maps of the H solubility, diffusivity and permeability as a function of the alloy composition. Although our research only examined PdCuAg ternary alloys, the same methods are also applicable for other ternary alloys. Theoretical predictions for H permeability in ternary alloys can, thus, provide a useful way in searching new materials with good properties for H2 purification. Acknowledgement This work was financially supported by the DOE-BES Hydrogen Fuels Initiative and the National Energy Technology Laboratory.
b
c Fig. 13. Contour map of the permeability of hydrogen in PdCuAg alloys normalized by the value in pure Pd at (a) 600 K, (b) 800 K and (c) 1200 K.
ance of these alloys as a necessary step in their application as H2 purification membranes. 4. Conclusion In this paper, we have screened a series of PdCuAg alloys as H2 purification membranes using DFT-based models. These models do not require any experimental input in order to make quantitative predictions about the performance of alloys as membranes operating at elevated temperatures. We have explored a range of PdCuAg
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