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Computational prediction of durable amorphous metal membranes for H2 puriﬁcation Shiqiang Hao, 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 11 May 2011 Received in revised form 7 July 2011 Accepted 17 July 2011 Available online 22 July 2011 Keywords: Hydrogen Metal membranes Amorphous metals Theory and modeling Alloys

a b s t r a c t Amorphous metals are interesting candidates as membranes for H2 puriﬁcation. Identifying materials with high permeability for H2 remains a challenge in this ﬁeld. We apply recently developed methods that combine ﬁrst principles density functional theory calculations and statistical mechanics to make predictions of the properties of interstitial H in amorphous metals. Our calculations greatly expand the number of amorphous metals which have been considered as membranes, and predict several materials with promising properties. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Hydrogen has the potential to play an important role in creating large-scale changes to the mix of energy sources used by our global society, particularly in transport applications that currently dependent on liquid hydrocarbons. If hydrogen is to be used in fuel cells for vehicular transport, stringent requirements on the purity of hydrogen must be met. Membrane-based separations using dense metal ﬁlms provide an excellent strategy for achieving separations from syngas [1]. Even though a large number of binary and multicomponent crystalline alloys have been tested experimentally as metal membranes [2], the search for durable and cost effective alloys that show high permeability for H2 continues to be an active area. A useful benchmark for considering the permeability of new membrane materials is that the permeability of pure Pd when used with pure H2 as the feed gas is ∼10−8 (mol m−1 s−1 Pa−0.5 ) at 600 K [3]. A number of experimental studies have examined amorphous metal ﬁlms as membranes for H2 puriﬁcation [4–6]. Amorphous ZrNi and ZrNiNb ﬁlms have been shown to have H2 permeabilities comparable to pure Pd [3,4,7–9]. Because the solubility and diffusion of interstitial H in amorphous metals is qualitatively different from crystalline metals [10], amorphous ﬁlms offer intriguing opportunities as H2 membranes [11,12].

∗ Corresponding author. Tel.: +1 404 894 2822; fax: +1 404 894 2866. 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.2011.07.026

A key challenge in developing new materials as membranes is the need to efﬁciently screen a variety of candidate materials to ﬁnd promising examples for more detailed testing. Quantitative theoretical models that make predictions without experimental input can play an important role in this area. Models of this kind based on density functional theory (DFT) calculations for the properties of interstitial H in metals have been used for a number of years to examine crystalline Pd-based alloys [13–16]. We have recently developed and veriﬁed a DFT-based computational strategy to predict hydrogen permeation rates through amorphous metals [17,18]. We have shown that our approach gives good agreement with experimental results for amorphous Zr36 Ni64 and Zr30 (Ni0.6 Nb0.4 )70 without requiring any experimental input [18]. In this paper, we use these methods to predict H2 permeability through nine amorphous metals with a broad range of compositions. These results greatly expand the number of amorphous metals that have been considered as H2 membranes. A potential problem with using amorphous ﬁlms as membranes is that if crystallization occurs, the favorable properties associated with the amorphous structure can be irreversibly lost. To identify materials potentially suitable for H2 puriﬁcation at 300–500 ◦ C, which are typical for H2 production from hydrocarbons, we surveyed the literature to ﬁnd amorphous metals with high crystallization temperatures. Information on 269 amorphous metals is given in Fig. 1 and Table S1. To simplify our calculations, we have focused in this work on binary alloys. Two of the binary alloys with high crystallization temperature have already been examined experimentally as membranes [3]: Ni62 Nb38 (932 K [19]) and Zr35 Ni65 (863 K [20]). We have performed calculations

No. of materials (arb. units)

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H2 permeability (mol/m/s/Pa )

S. Hao, D.S. Sholl / Journal of Membrane Science 381 (2011) 192–196

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Fig. 1. Number of amorphous metals as a function of crystallization temperature. The red sticks label binary phases among which Ta40 Ni60 (1043 K), Ni62 Nb38 (932 K), Zr35 Ni65 (863 K), Ti33 Co67 (850 K), Hf44 Cu56 (831 K), and Zr54 Cu46 (746 K) have Tx > 700 K. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of the article.)

for the nine additional binary and ternary materials in Fig. 1 with crystallization temperatures above 700 K: Ta40 Ni60 (1043 K) [21], Ta25 Ni60 Ti15 (970 K) [21], Ti33 Co67 (850 K) [22], Hf44 Cu56 (831 K) [23], Hf25 Cu60 Ti15 (805 K) [24], Zr45 Cu45 Al10 (776 K) [25], Zr30 Cu60 Ti10 (763 K) [26], Zr54 Cu46 (746 K) [27], and Nd60 Fe30 Al10 (722 K) [19]. Throughout our results, compositions are indicated in at.%. During H2 puriﬁcation, hydrogen permeates through a metal ﬁlm by dissociation of H2 , diffusion of atomic H through interstitial sites in the metal, and recombination of atomic H as H2 [1,2]. Throughout our calculations, we consider the common situation where surface dissociation process is not rate limiting and bulk diffusion is rate determining. In experiments with amorphous ﬁlms, this is often achieved by using a thin layer of Pd as a catalytic layer on the ﬁlm’s surface [12]. When surface processes can be neglected, the net permeability of a ﬁlm can be predicted if the solubility and diffusivity of H in the bulk material is known [2,13]. 2. First principles calculations The details of our DFT calculations for interstitial H in amorphous metals have been described in previous reports [17], so we summarize the main points here. Our calculations were performed with the Vienna ab initio Simulation Package (VASP) using plane wave DFT with the PW91 GGA functional [28]. Supercells approximating amorphous samples were prepared using ab initio Molecular Dynamics (MD) at 3000 K. These liquid-like samples were then quenched using conjugate gradient relaxation. The liquid state volumes are ∼7% greater than the volume of the quenched amorphous samples [29]. Test calculations for Ta40 Ni60 and Ta25 Ni60 Ti15 indicated that including spin polarization did not affect the calculated energies, so the MD and quenching calculations were performed without spin polarization. Other materials containing magnetic species (Ti33 Co67 , Zr55 Co25 Al20 , and Nd60 Fe30 Al10 ) were calculated with spin polarization for all MD relaxation, binding energy and transition state calculations. Calculations using 32 atom supercells sampled reciprocal space with 3 × 3 × 3 k-points. A limited number of materials were also examined with 108 atom supercells, for which only the -point was used in k-space. Binding sites for interstitial H were optimized in calculations using one H atom per supercell. We used the methods described

500

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Temperature (K)

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Tx (K)

108 atom cell 32 atom cells 64 atom cell

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193

Fig. 2. Permeability comparison of Zr55 Co25 Al20 with different sizes of supercell.

elsewhere to efﬁciently locate all of the interstitial sites and transition states between adjacent interstitial sites for H in the computational volume [30,31]. The procedure normally located 85–95 distinct interstitial sites and 240–290 transition states in each 32 atom supercell. Energies of interstitial sites and transition states were found from DFT calculations that allowed all atomic degrees of freedom in the supercell to relax. Below, we use DFT-based calculations to predict H solubility and diffusivity in the nine amorphous materials listed above. These calculations are computationally intensive, and the computational effort grows rapidly as the number of atoms in the calculation is increased. In test calculations, we examined Zr55 Co25 Al20 with supercells of 32, 64, and 108 atom cells. As shown in Fig. 2, the permeability predicted with the 32 atom calculations are consistent with (although not identical to) the larger calculations. Motivated by this observation, we used 32 atom supercells to examine the nine candidate materials. We subsequently reexamined the permeability of the four candidates with the highest predicted permeability, Zr54 Cu46 , Zr30 Cu60 Ti10 , Hf44 Cu56 , and Hf25 Cu60 Ti15 , using 108 atom supercells. 3. Results Our calculations of H solubility in amorphous metals begin by using DFT to compute the binding energy of a single interstitial H in each interstitial site in a sample. Because the solubility of H in these materials can be signiﬁcant under practical conditions, H–H repulsion plays an important role in determining the overall solubility. We used the Westlake criterion to characterize H–H interactions. This criterion predicts that two interstitial sites separated by less than 0.21 nm cannot be simultaneously occupied by H atoms [32]. We have shown previously that this approach gives results in close agreement with more detailed DFT characterization of H–H interactions in amorphous Fe3 B [30] and ZrNi [18]. With this deﬁnition of the binding energy for each interstitial site and description of H–H repulsion, the net solubility of H in our amorphous samples can be calculated as a function of temperature and H2 pressure using Grand Canonical Monte Carlo (GCMC) simulations [30]. These GCMC calculations equate the chemical potential of interstitial H and the gaseous phase, which was treated as an ideal gas. The solubility calculated in this way for Zr54 Cu46 at two H2 pressures is shown in Fig. 3 in terms of the ratio of H to metal atoms, H/M. There are many conditions where the concentration of H is high. Good agreement can be seen between calculations performed with 32 and 108 atom supercells. A high concentration of interstitial H can induce lattice expansion, and this expansion can affect H binding energies and therefore solubility. We accounted for this effect

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108 atom cell, 0.3 108 atom cell, 0.2 108 atom cell, 0.1 32 atom cell, 0.3 32 atom cell, 0.2 32 atom cell, 0.1

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D0 (m /s)

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3 atm, 108 atom cell, ΔE 3 atm, 108 atom cell 3 atm, 32 atom cell 0.01 atm,108 atom cell, ΔE 0.01 atm, 108 atom cell 0.01 atm, 32 atom cell

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using an iterative method introduced in earlier calculations with ZrNiNb [18]. Including these effects, as denoted by E in Fig. 3, slightly increases the predicted solubility, especially at the lower temperatures shown. The solubility of H in Zr54 Cu46 is representative of all of the amorphous materials we examined. The predicted solubility of H in each material is summarized in Table 1. All these calculations are averaged over two independent calculations using 32 atom supercells and additional calculations were performed for the top four candidates (Zr54 Cu46 , Zr30 Cu60 Ti10 , Hf44 Cu56 , and Hf25 Cu60 Ti15 ) based on their permeability with 108 atom supercells. All data in Table 1 includes lattice expansion effects. There are large variations in the H solubility among the materials we considered. At 700 K and low pressure, the predicted solubility in Nd60 Fe30 Al10 is >1300 times larger than in Ta40 Ni60 . To investigate net transport of hydrogen through a membrane, the relevant description is Fick’s law, which relates the ﬂux of H, J, to the transport diffusion coefﬁcient, Dt , and the gradient in the H concentration gradient, c, by J = − Dt (c) ∇ c. The high solubilities of H in amorphous metals mean that the concentration dependence of Dt must be considered to reliably describe H diffusion. One convenient approach to calculate the transport diffusivity is to note that Dt is related to the corrected diffusivity, D0 (c), by [33], Dt (c) = D0 (c)(∂ ln f/∂ ln c). Here, f is the fugacity of H in the gas phase in equilibrium with the solid and the derivative is known as the thermodynamic correction factor. The advantage of expressing the transport diffusivity using this equation is that the corrected diffusivity and the thermodynamic correction factor can be computed

Table 1 Calculated solubility of H in nine amorphous metals at selected temperatures and pressures, shown as a ratio of the number of hydrogen to the number of host metal atoms, H/M.

Ta40 Ni60 Ta25 Ni60 Ti15 Ti33 Co67 Hf44 Cu56 Hf25 Cu60 Ti15 Zr45 Cu45 Al10 Zr30 Cu60 Ti10 Zr54 Cu46 Nd60 Fe30 Al10

500 K

1.2

1.4

1.6

1.8

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2.2

1000/ T (K )

Fig. 3. Calculated H solubility in amorphous Zr54 Cu46 . Lines are to guide the eye. Results denoted by E include the effects of H-induced lattice expansion as described in the text. The vertical line indicates the crystallization temperature of Zr54 Cu46 from Ref. [27].

300 K

1

-1

Temperature (K)

700 K

3 atm

0.01 atm

3 atm

0.01 atm

3 atm

0.01 atm

0.11 0.16 0.31 0.63 0.47 0.56 0.67 0.71 1.05

0.07 0.10 0.27 0.59 0.41 0.51 0.64 0.66 1.01

0.04 0.06 0.22 0.52 0.29 0.46 0.58 0.55 0.99

0.008 0.02 0.18 0.46 0.18 0.39 0.48 0.41 0.86

0.007 0.01 0.16 0.41 0.17 0.39 0.41 0.35 0.81

0.0004 0.001 0.08 0.29 0.08 0.33 0.24 0.19 0.55

Fig. 4. Corrected diffusivities of H in Zr54 Cu46 from the KMC simulations described in the text. The value of H/M for each set of calculations is indicated in the legend.

independently from the model of interstitial H we have deﬁned above [34]. We used Kinetic Monte Carlo (KMC) simulations of H hopping in each amorphous material to calculate the corrected diffusivity. In these simulations, DFT-calculated energies and vibrational frequencies of H at each interstitial site and transition state were used to deﬁne site to site hopping rates by applying quantum corrected transition state theory. These KMC simulations generate dynamically correct trajectories for each H atom as a function of time. H–H interactions were deﬁned using the Westlake criterion. Corrected diffusivities were computed by averaging over 20 independent KMC simulations at each state point of interest. We did not explicitly consider the effects of H-induced expansion, because the only effect of this expansion on diffusion is that the hopping distance associated with a site to site hop changes slightly. Including these expansion effects would increase our calculated diffusivities by at most a few percent under the highest H/M concentrations we considered. Other details of the KMC methods are available in our previous reports [34]. The resulting corrected diffusion coefﬁcients at various interstitial concentrations for Zr54 Cu46 are shown in Fig. 4. The most obvious feature of Fig. 4 is that the corrected diffusivity of H in these amorphous metals is strongly concentration dependent. At 600 K, for example, D0 increases by almost an order of magnitude as the H concentration is increased from H/M = 0.1–0.3. This result is consistent with earlier experiments with other amorphous metals [35], and our earlier calculations [34]. This behavior occurs because of the broad distribution of H binding energies in amorphous materials. At dilute concentrations, most H atoms are trapped in the most favorable binding sites with large barriers to moving H away from these sites. At higher concentrations, a fraction of the H atoms occupy and hop among the less favorable sites, which dominate the net diffusion of H in the material. The qualitative trends shown in Fig. 4 for Zr54 Cu46 are also found for the other materials we considered. Diffusivity data at representative H concentrations for each material are listed in Table 2. For each material, concentrations were chosen that are representative of the feed and permeate side of a membrane in the permeability calculations we describe below. The corrected diffusivities of all the amorphous materials are lower than the calculated self-diffusivity of H in pure Pd, which is 2.6 × 10−9 (1.6 × 10−8 ) m2 /s at 500 (800) K [4,13,36]. The thermodynamic correction factor (TCF) can be computed from the same GCMC simulation we used to determine the solubility data [37]. Results for Zr54 Cu46 are shown for a variety of gas phase pressures and temperatures in Fig. 5. These results included

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Table 2 Corrected diffusivity of amorphous metals at selected temperatures and concentration. The values are in m2 /s, data in brackets are concentrations of interstitial H deﬁned as H/M. 500 K Ta40 Ni60 Ta25 Ni60 Ti15 Ti33 Co67 Hf44 Cu56 Hf25 Cu60 Ti15 Zr45 Cu45 Al10 Zr30 Cu60 Ti10 Zr54 Cu46 Nd60 Fe30 Al10

700 K

4.24E−13(0.01) 7.63E−13(0.01) 3.52E−14(0.1) 4.30E−13(0.1) 1.65E−12(0.1) 5.21E−14(0.1) 3.07E−12(0.1) 6.71E−13(0.1) 5.51E−13(0.5)

3.56E−11(0.1) 8.89E−12(0.1) 1.08E−12(0.3) 2.10E−12(0.3) 1.05E−11(0.3) 1.02E−13(0.3) 1.45E−11(0.3) 1.36E−11(0.3) 1.59E−13(0.8)

the effects of H-induced expansion as described above. The TCF approaches unity at dilute concentrations, and this limit can be observed in the data at high temperature (e.g. 700 K) and low pressure (e.g. 10−3 atm) in Fig. 5. At lower temperatures where the solubility of H is high, the thermodynamic correction factor monotonically increases with pressure. The results for a-Zr54 Cu46 are representative of the other materials we investigated. At 600 K, the TCF is in the range of 8–13 (5–10) at 1 atm (0.01 atm) H2 pressure for all the materials we examined. After ﬁnding the corrected diffusivities and thermodynamic correction factors, we can calculate the transport diffusivities for each material. The corrected diffusivities were speciﬁed by ﬁtting continuous curves to the results such as those shown in Fig. 4. The TCF for each concentration of interest were evaluated using GCMC by varying the bulk H2 pressure until the observed value of H/M matched the value used in the KMC simulation within a small tolerance. Once the solubility and transport diffusivity of H in a metal ﬁlm are known, it is straightforward to evaluate the performance of the ﬁlm when it used as a membrane. Membrane performance is typi1/2 1/2 cally reported in terms of permeability k = JL/(Pfeed − Pperm ), where, Pfeed (Pperm ) is the H2 pressure on the feed side (permeate side) of the membrane, and L is the membrane thickness [2,13]. This quantity is independent of the pressures and ﬁlm thickness for crystalline materials that obey Sieverts’ law [2,13]. For materials such as amorphous metals where Sieverts’ c law is not satisﬁed, the net hydrogen ﬂux can be written as JL = c feed Dt (c)dc ∼ = (cfeed − cperm )Dt (¯c ) when perm

transport of H through the bulk of the membrane is the domi-

4. Discussion and conclusion It is useful to consider whether the variation in permeability of H through amorphous materials is dominated by solubility, diffusion, or both. Understanding which phenomenon dominates the behavior of these membranes would assist future efforts to screen materials. Fig. 7 shows the predicted permeability, solubility and diffusivity for all amorphous materials that have been considered in our calculations, including earlier reports. If permeability was dominated by solubility, Fig. 7(a) would reveal a simple correlation between these two quantities. It is clear that this is not the case. For example, Zr54 Cu46 and Zr45 Cu45 Al10 , have quite similar solubilities (0.35 and 0.39) but the permeability of the Al-containing

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8.03E−10(0.1) 3.48E−10(0.1) 1.27E−10(0.3) 1.92E−11(0.3) 4.06E−10(0.3) 2.31E−12(0.3) 6.11E−10(0.3) 5.23E−10(0.3) 2.35E−12(0.8)

nant transport resistance. Here, cfeed (cperm ) is the concentration of interstitial H on the feed (permeate) side, and c¯ = (cfeed + cperm )/2. The predicted H2 permeabilities for all nine amorphous materials using a feed (permeate) pressure of 3 atm (0.01) atm are shown as symbols connected by lines in Fig. 6. The dotted region shows a range of permeabilities that have been reported for crystalline Pd [3,13]. The two materials with the highest predicted permeability, Zr54 Cu46 and Zr30 Cu60 Ti10 , have permeability similar to Pd for temperatures above ∼600 K. Hf25 Cu60 Ti15 , Hf44 Cu56 , Nd60 Fe30 Al10 , and Zr45 Cu45 Al10 have lower permeability than Pd. For example, Hf25 Cu60 Ti15 , the predicted permeability at 600 K is lower than Pd by a factor of 10–20. The remaining three materials, Ti33 Co67 , Ta25 Ni60 Ti15 , and Ta40 Ni60 , are predicted to have permeabilities that are 2–4 orders of magnitude lower than Pd over the temperature range we considered.

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7.19E−11(0.01) 5.67E−11(0.01) 6.94E−12(0.1) 6.14E−11(0.1) 8.93E−11(0.1) 1.22E−12(0.1) 9.06E−11(0.1) 4.10E−11(0.1) 6.84E−12(0.5)

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Pressure (atm) Fig. 5. Thermodynamic correction factor of Zr54 Cu46 at different temperature and pressure for 32 and 108 atom supercell. Lines are to guide the eye.

Zr54Cu46 Zr30Cu60Ti10 Hf25Cu60Ti15 Hf44Cu56 Nd60Fe30Al10 Zr45Cu45Al10 Ti33Co67 Ta25Ni60Ti15 Ta40Ni60

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Temperature (K) Fig. 6. Predicted H2 permeability of nine amorphous metals as a function of temperature. The dotted region indicates the permeability of crystalline Pd. Results are only shown below the experimentally reported crystallization temperature for each material.

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Fe 75 B 25 Ta 25Ni 60 Ti 15

Zr 45Cu45 Al10

Nd60Fe30 Al10 Nd60Fe30 Al10

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Ta 25Ni 60 Ti 15 Ta40Ni60

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Zr 55Co25 Al20 Zr36Ni64 Zr 45Cu45 Al10

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Zr 30 Ni 42Nb28

Hf44Cu56

Hf44Cu56

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Zr36Ni64 -10

Zr 30 Ni 42Nb28

Zr30Cu60 Ti10

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Fig. 7. Calculated permeability dependence on (a) solubility and (b) diffusivity. The permeabilities were calculated at feed and permeate pressures of 3 and 0.01 atmospheres at 600 K. In (a), the solubilities corresponding to 0.05 atm of H2 at 600 K. In (b) the diffusivities are at the average concentration of feed and permeate sides at 600 K. The data for Fe75 B25 , Zr36 Ni64 , and Zr30 Ni42 Nb28 are from our previous work [18,34].

material is ∼40 times smaller than the binary alloy. This pair of materials also suggests that adding relatively small amounts of additional elements to known amorphous materials has potential for signiﬁcant changes in membranes properties, suggesting a fruitful direction for future work. Fig. 7(b) indicates that it is the variation in diffusivity among various materials that accounts for most of the differences that exist in overall permeability. The exceptions to this trend are Ta25 Ni60 Ti15 and Ta40 Ni60 , which have very low permeability because of low H solubility in these materials. This discussion suggests that efforts to identify amorphous materials with favorable properties as membranes for H2 should focus on ﬁnding materials in which H diffusion is rapid. In summary, we have used ﬁrst-principles based methods to greatly expand the number of amorphous metals for which information about hydrogen permeability is available. Materials were selected based on the need to use alloys with high crystallization temperatures in practical membrane applications. A key feature of our calculations is that they do not require any experimental input, allowing us to use our methods to screen a broad range of materials. Among the materials we examined, Zr54 Cu46 and Zr30 Cu60 Ti10 are predicted to have permeabilities similar to pure Pd, making them excellent candidates for experimental study. Other materials are predicted to have permeability that is orders of magnitude lower than Pd, making them of limited value as membranes. In realistic applications, the robustness of membranes to contaminants in the feed stream is critical. It is not currently feasible to make predictions about this topic with computational modeling, so this important issue needs to be examined for the most promising materials experimentally. Acknowledgements Financial support for this work came from the US DOE Ofﬁce of Basic Energy Sciences and the US DOE National Energy Technology Laboratory. Discussions with K. Coulter were greatly appreciated. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.memsci.2011.07.026.

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