Diffusion of H through Pdâ Ag Alloys (423â 523 K)

J. Phys. Chem. B 2008, 112, 1135-1148

1135

Diffusion of H through Pd-Ag Alloys (423-523 K) Da Wang,† Ted B. Flanagan,* and Kirk Shanahan‡ Chemistry Department, UniVersity of Vermont, Burlington, Vermont 05405, SaVannah RiVer National Laboratory, Building 999-2W, Aiken, South Carolina 29808 ReceiVed: August 20, 2007; In Final Form: October 26, 2007

H diffusion constants have been determined from steady-state fluxes through Pd-Ag alloy membranes. The upstream side is maintained at a nearly constant pup (and consequently at a nearly constant rup ) H/(Pd(1-x)Agx)) atom ratio, whereas the downstream side is at pH2 ≈ 0 (rdown ) 0) (423-523 K). It is shown that the permeability is a maximum for atom fraction Ag, XAg ) 0.23 (423-523 K) at both pup ) 20.3 and 50.6 kPa. DH has been determined for some Pd-Ag alloys as a function of r in the dilute region, and it decreases with r even at small H contents for alloys with XAg < 0.35. The concentration dependence of DH(cH) has been determined for the Pd0.77Ag0.23 alloy over a large concentration range. The effect of nonideality on DH(r) and ED(r) has been systematically determined as a function of XAg, where XAg is the atom fraction of Ag in the H-free alloy. (dDH/dr) increases with XAg up to XAg ) 0.35 and then changes from negative to positive at ≈0.35. The activation energies for diffusion, ED(r), have been determined as a function of H content in the dilute range for several Pd-Ag alloy membranes, and the conversion to concentration-independent E*D values has been carried out in several different ways. TABLE 1: DH(r)/10-6 cm2/s Measured with pup ) 20.3 kPa for Pd-Ag Alloys (423-523 K)a

Introduction Pd-Ag alloys are the most important Pd alloys with regard to H diffusion because the Pd0.77Ag0.23 alloy is employed industrially as a H2 purification membrane. There have been many studies of H diffusion in Pd-Ag alloys, but not much attention has been paid to the role of nonideality at moderately high temperatures. The dependence of the DH(r) and ED(r) (where r ) M/(PdxAg(1-x))) atom ratio on the H content will be examined here at various Ag contents. Here, DH(r) and ED(r) are the concentration-dependent H diffusion constant and the concentration-dependent activation energy for diffusion, respectively. H is assumed to occupy octahedral interstices in fcc Pd-rich alloys, as it does in pure Pd. Indirect evidence for this is that the inelastic vibrational frequency of H in Pd-Ag alloys is characteristic of H in octahedral sites rather than tetrahedral sites.1 Lovvik and Olson2 have shown from density functional calculations that H prefers the octahedral sites in Pd-Ag alloys. Solubility and diffusion data from 533 to 913 K are available from Holleck’s important work for a series of Pd-Ag alloys.3 Ku¨ssner4 has determined diffusion data for the Pd0.77Ag0.23 alloy using an electrochemical breakthrough method, and he determined DH(r) as a continuous function of r at 303 K up to r ) 0.42. He found a minimum in DH at r ≈ 0.19. Bohmholdt and Wicke5 investigated several Pd-Ag alloys up to XAg ) 0.40 (293-373 K) using a gas-phase permeation technique in which XAg is the atom fraction of Ag. They determined DH(r) values at both high and very low r. There are several publications by Zu¨chner et al. giving diffusion constants for a series of different Ag composition alloys at ambient temperature (296 K).6 These workers found that DH values are almost independent of Ag content to about XAg ) 0.30 (296 K) and then fall abruptly. † ‡

University of Vermont. Savannah River National Laboratory.

DH(r)/10-6 cm2/s XAg

423 K

0 0.10 0.15 0.19 0.23 0.30 0.35 0.40 0.45 0.50

5.3 4.5 (4.4) 3.4 (3.5) 2.7 (2.3) 2.6 (2.6) 2.5 (2.3) 2.0 1.4 0.77 0.33

453 K

473 K

7.8 (7.2) 6.4 (6.4) 5.5 (5.6) 4.9 (4.9) 4.0 (3.9) 3.0 2.2 1.2 0.60

12.4 10.7 (9.8) 8.7 (8.8) 8.0 (7.8) 6.6 (6.7) 5.3 (5.2) 3.9 2.7 1.6 0.84

503 K

523 K

15.0 (14.2) 13.3 (13.2) 12.3 (11.9) 9.8 (9.8) 8.1 (7.8) 5.9 4.0 2.4 1.2

21.8 18.9 (17.9) 15.9 (16.0) 14.7 (14.6) 12.5 (12.6) 9.9 (9.7) 7.5 5.0 3.0 1.6

a The values with and without parentheses represent different sets of alloy membranes and measurements.

Thermodynamic parameters have been determined for the Pd-Ag alloys elsewhere from H2 isotherms over the temperature range employed here.7 These data extend over a range of H contents and will be employed here to obtain diffusion constants from the permeation data. A greater number of alloy compositions will be examined in this research than in previous work on these alloys.3-6 Diffusion parameters referred to in this paper are defined by DH ) DoH exp-(ED/RT) where these refer to a given H content or to the concentration-independent variables. With regard to H membrane permeation, two types of nonideality should be considered. The first arises from employing KspH21/2 ) r in regions of concentration where Sievert’s law of ideal solubility does not apply. This is easily accounted for if pH21/2 vs r equilibrium isotherms are available. The other type of nonideality is due to the concentration dependence of Fick’s diffusion constant,8 DH(r). Wicke and co-workers have derived D*H from DH(r) values for Pd and some of its alloys8 using eq 1. When it is desired to determine the concentration-

10.1021/jp076690b CCC: $40.75 © 2008 American Chemical Society Published on Web 01/08/2008

1136 J. Phys. Chem. B, Vol. 112, No. 4, 2008

Wang et al.

Figure 1. H2 permeability, Psp, plotted against p1/2 at 473 K for the Pd0.77Ag0.23 alloy membrane. O, experimental data; 4, data corrected for deviations from Sievert’s law of ideal solubility.

Figure 2. H2 flux plotted against 1/d for Pd0.77Ag0.23 alloy membranes at 473 K and 50.6 kPa.

independent D, D*H, the thermodynamic factor, f(r), is needed (eq 1).

DH(r) ) D*H[∂ ln pH21/2/∂ ln r] ) DH* f(r)

(1)

Electrochemical time-lag techniques have frequently been employed when a membrane charged homogeneously with H is electrochemically perturbed on the upstream side, and the break-through time needed for the perturbation to reach the downstream side can be used to calculate DH(r).4,6,8 In these break-through experiments in which the r does not usually change very much, an average value can be employed in eq 1. Salomons et al.9,10 measured the dependence of DH(r) on r for Pd, a Pd0.80Cu0.20 alloy, and a disordered Pd0.91Y0.09 alloy at elevated temperatures using a gravimetric method to determine the relaxation for H uptake following sudden pH2 changes in times, and from these relaxation times, DH(r) values can be obtained. Details of the extent of the pH2 changes and the accompanying changes of r were not provided. They found that

DH(r) decreases with r, which they explained using the thermodynamic factor and a site blocking (1 - r) factor. They also found a minimum in DH(r) as a function of r for the Pd0.80Cu0.20 alloy, but not for the Pd0.91Y0.09 alloy.10 Subsequently, Stonadge et al.11 employed the same technique as Salomons9 with an ordered Pd0.90Y0.10 alloy; however, there was no comparison of their results with the earlier investigation. Stonadge et al. determined concentration-independent diffusion constants, D*H, for the ordered form of this alloy using isotherms to evaluate the thermodynamic factor. D*H values were not constant from 463 to 540 K, but increased with cH, which they attributed to unspecified traps. In experiments in which DH(r) is determined from steadystate fluxes, there is, most commonly, a large concentration profile within the membrane, that is, pup is large and pdown is small, and therefore, it may be inappropriate to employ eq 1 directly; the subscripts up and down refer to the high and lowpressure sides of the membrane. This is the usual situation for H2 purification. It has been shown recently for Pd membranes13

Diffusion of H through Pd-Ag Alloys

J. Phys. Chem. B, Vol. 112, No. 4, 2008 1137

Figure 3. H2 permeabilities plotted against atom % Ag for pup ) 50.6 kPa at 423, 473, and 523 K.

Figure 4. Log DH determined at pH2 ) 20.3 kPa plotted against XAg at 473 K. O, present experimental data; [, log D*H from corrected experiment data; 4, ref 3; 0, ref 15.

that even when rup . rdown, an average concentration can be employed in eq 1, that is, f(rav) ≈ f(rup)/2, provided that r is small. This approximation agrees with results from more exact methods at small r; however, this approximation may not be appropriate for all the Pd-Ag alloy membranes, as will be shown below. The earlier work with Pd gave and employed several methods to obtain D*H from DH(r) and E*D from ED(r). In the present investigation, the dependences of DH and ED on r will be determined for several Pd-Ag alloys, and in contrast

to the earlier studies on PdsH,13 the present one will not be restricted to low H concentrations. Experimental The alloys and membranes were prepared by arc-melting the pure components and then annealing at 1123 K for 72 h, followed by rolling and reannealing. The flux was determined from the decrease of pH2 on the upstream side of the membrane using an MKS gauge, and the


Wang et al.

Figure 5. Quantity proportional to J plotted against 1/T for Pd-Ag membranes at different but constant rup values. The XAg of the alloys is indicated.

TABLE 2: ED(r) for Pd-Ag Alloys (423-523 K) Determined at the Constant, Small rup Values Shown in the Table XAg

rup

ED (1)a kJ/ mol H

0 0.10 0.15 0.19 0.23 0.30 0.35 0.40 0.45 0.50

rup f 0 0.03 0.04 0.05 0.05 0.06 0.07 0.075 0.05 0.08

23.9 23.8 23.7 23.4 23.2 24.4 26.9 28.9 30.7 33.1

ED (2)a kJ/ mol H

DoH 2 (cm /s)/10-3

23.5 23.3 23.6 24.1 23.7 25.6 27.7 29.1 33.8

4.8 3.6 3.5 3.6 2.8 3.5 4.4 3.8 1.6 3.8

a (1) and (2) refer to separate sets of experiments for the determination of ED.

downstream side was kept at pH2 ≈ 0.13,14 There was a falloff in the fluxes during the measurements due to the decrease in pup, but this was generally small because the upstream volume is large, and in any case, the fluxes can be corrected for such small decreases. Generally, however, the fluxes were taken as initial ones before any appreciable decrease of pup. The steady-

state fluxes were established very quickly for these Pd-Ag alloy membranes at the temperatures employed as shown by the relatively constant flux, J, with time after correction for small decreases of pup. The areas of the membranes were all the same, 1.77 cm.2 The mounted membranes were enclosed by a tube furnace controlled to about (1 °C. The temperatures were set by an electronic controller, and they were found to be in good agreement with those measured by a temperature sensor in direct contact with the mounted membrane. The membranes were oxidized in the atmosphere for ≈30 min at 953 K before they were mounted in the diffusion apparatus, since this procedure has been shown to improve the reproducibility of the Pd membranes.14 Since Pd or Ag oxides do not penetrate into the alloy appreciably and are rapidly reduced in the presence of H2, the membrane thicknesses before and after oxidation are the same. Results and Discussion The flux of H, J, in the present experiments in which the downstream side is maintained at pH2 ≈ 0, is given by

J ) -DH(∂cH/∂x) ≈ -DHcup/d

(2)



Figure 6. ED and EP plotted against XAg determined from the data in Figure 5.

Figure 7. Diffusion data for the Pd0.77Ag0.23 alloy determined at pup ) 20.3 kPa. O, log DH(r)/cm2/s; 4, log cH/mol H/cm3; and 0, sum of the log DH(r) and log cH plots where the sum has been translated along the y axis to correspond to the intersection of the log DH(r) and log cH plots.

where d is the membrane thickness. The diffusion constants were determined from the fluxes using isotherms given elsewhere,7 and cH and r are related by cH ) rF/M, where F and M are the density and molar mass of the alloy. The first part of the paper will deal with demonstrations that, under the experimental conditions employed, bulk diffusion is the slow step for these membranes. The dependence of DH on XAg and T will next be considered. The second part of the paper concerns trends of the concentration-dependent DH(r) and ED(r) with r. The concentration-dependent diffusion parameters DH(r) are technologically important because they determine the actual permeabilities under the conditions of pH2 and temperature employed. The third part will deal with quantitative understand-

ing of the concentration dependence of DH(r) and ED(r) and the determination of the fundamental, concentration-independent parameters, D*H and E*D. Evidence for Bulk Diffusion Control in Pd-Ag Alloys. Under experimental conditions similar to those employed here, it was shown earlier for pure Pd13,14 that bulk diffusion is the rate-controlling step; however, because the H permeability is greater for some of the Pd-Ag alloys, for example, Pd0.77Ag0.23, than for Pd, it must be shown that bulk diffusion is also the controlling step for the Pd-Ag alloys. The Pd0.77Ag0.23 alloy was chosen for this purpose because it has the greatest H permeability and because it is the alloy generally employed for H2 purification.


Wang et al.

Figure 8. RT ln DH (r) and ED(r) as a function of rup at 423 K for the Pd0.90Ag0.10 alloy.

Figure 9. RT ln DH(r) and ED(r) as a function of rup at 473 K for the Pd0.77Ag0.23 alloy.

Figure 1 shows a plot of the specific permeabilities, P ) DH K′spH21/2, of Pd0.77Ag0.23 alloy membranes as a function of pH21/2 at 473 K, where K′s relates cH to pH21/2; that is, Sievert’s law of ideal solubility for a dissociating gas, cH ) K′spH21/2, is assumed to be valid. Figure 1 shows that the relation is linear only at small r, and at higher concentrations, there are deviations due to the dissolved H no longer obeying Sievert’s law. The initial linear slope indicates that bulk diffusion is the controlling step for permeation, but it would be more convincing if these data obeyed the P vs pH21/2 relation over the entire range. As shown for Pd membranes,14 the deviations from linearity are mainly due to Sievert’s law of ideal solubility not holding over the whole pH2 range. The pH2-r isotherm for this alloy7 can be employed to allow for this nonideality. The experimental pexp1/2

values have been multiplied by the ratios (pideal/pexp) 1/2 to give a pideal1/2 at each permeability. When this is done, a corrected relationship is obtained that is quite linear (Figure 1), indicating that bulk diffusion is the slow step over the entire range. The linear relation obtained (Figure 1) using the isotherm shows that pup is in rapid equilibrium with the upstream surface, which allows rup to be obtained from pup. A better test for bulk diffusion as the rate-controlling step is a linear dependence of flux, J, on 1/d where d is the membrane thickness. At constant pH21/2 and temperature, nonideality due to the thermodynamic factor, eq 1, does not depend on d and therefore does not influence the linearity of a J-1/d plot. Such experiments have been carried out here with the Pd0.77Ag0.23 alloy, and Figure 2 shows J vs 1/d at pup ) 50.6 kPa and 473



Figure 10. RT ln DH(r) and ED(r) as a function of rup at 473 K for the Pd0.50Ag0.50 alloy.

TABLE 3: DH(r) × 106 in cm2/s as a Function of rup for the Pd0.77Ag0.23 Alloy

TABLE 4: DH (r) as a Function of rup for the Pd0.050Ag0.050 Alloy

D*H (F(r))

D*H F(r) at (rup/2)

rup

DH (exptl)/10-6 cm2/s

423 K 3.89 3.89 3.89 3.95 3.99

3.85 3.87 3.86 3.86 3.82

3.87 3.89 3.86 3.96 4.10

0.29 0.31 0.34 0.37 0.40 0.43

7.13 7.02 6.92 6.78 6.64 6.43 6.27

473 K 7.47 7.53 7.60 7.64 7.68 7.63 7.64

0.040 0.045 0.050 0.055 0.060 0.065

7.51 7.62 7.68 7.68 7.67 7.61 7.57

8.04 8.25 8.44 8.15 8.03 7.86 7.74

0.030 0.035 0.40 0.045 0.050 0.060

0.72 0.75 0.78 0.82 0.85 0.94

12.20 12.14 12.04 11.88 11.68

523 K 13.27 13.48 13.67 13.80 13.89

13.15 13.33 13.49 13.55 13.55

13.20 13.48 13.68 13.42 13.61

0.025 0.030 0.035 0.040 0.045 0.050

1.54 1.62 1.69 1.75 1.83 1.88

rup

DH(r) (exptl)

0.040 0.050 0.060 0.070 0.080

3.37 3.25 3.13 3.05 2.95

0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.040 0.050 0.060 0.070 0.080

D*H (RIS)

D*H/10-6 cm2/s (from F(r))

423 K 0.22 0.22 0.22 0.25 0.24 0.23 473 K 0.68 0.73 0.76 0.77 0.80 0.80 523 K

K over a wide range of membrane thicknesses, d ) 120-305 µm. The observed linearity in Figure 2 shows that bulk diffusion is the controlling step. J should approach zero as (1/d) f 0, and it is seen that the data extrapolate linearly to the origin. Using a similar experimental setup,13 a similar adherence to the J-vs-1/d relation was found for pure Pd. From the results in Figures 1 and 2, it can be concluded that bulk diffusion is the controlling step for H permeation through the Pd0.77Ag0.23 alloy, and since its permeability is the greatest of the Pd-Ag alloys (Figure 3), it seems safe to conclude that bulk diffusion is also the rate-controlling step for the other alloys. Dependence of Permeabilities and the Diffusion Parameters, DH, ED, and DoH, on XAg. H Permeabilities in Pd-Ag Alloys as a Function of XAg. The permeabilities at 423, 473, 523 K for pup ) 50.6 kPa are shown in Figure 3, where the permeability is defined here as P ) J × d, and cup is obtained using the isotherms at given pup.7 The permeability is greatest for the Pd0.77Ag0.23 alloy. That and the fact that it does not form

1.39 1.43 1.42 1.43 1.46 1.45

a hydride phase at T g 298 K, are the reasons it is employed as a H2 purification membrane. The flux, or permeability, is proportional to DH(r) × cH. The DH(r) values are similar for the Pd-Ag alloys up to about XAg ≈ 0.303,6, and then they decline, as will be shown below. For pH2 ) 50.6 kPa (Figure 3), the solubilities are greater at 473 and 523 K for the XAg ) 0.23 alloy than for those with XAg < 0.23; however, at 423 K, the solubilities are greater for alloys with XAg < 0.23, even though the permeability is greater for XAg ) 0.23 (Figure 3). At these relatively large H contents, nonideality due to the thermodynamic factor, f(r), will be appreciable. Evidence will be presented below to indicate that the greater permeability of the XAg ) 0.23 alloy at 423 K is due to this nonideality, which causes DH(r) to be greater for this alloy at this r, as compared to alloys with XAg < 0.23. Similar arguments can be given for alloys with XAg > 0.23, for which the solubilities are greater but DH(r) is smaller. Similar data were obtained at pH2 ) 20.3 kPa with a maximum again found at XAg ) 0.23.


Wang et al.

Figure 11. Plots of RT ln DH (r) against rup at 423 K for the four alloys shown.

Fick’s Diffusion Constants at Constant pH2 and at Constant cup(rup). Fick’s concentration-dependent diffusion constants, DH(r), can be readily determined from the flux, pup, and the isotherms needed to obtain rup (or cup) from pup (eq 2). These concentration-dependent DH(r) are important for the practical use of Pd-Ag membranes because the flux is determined by DH(r) × cH for a given membrane and temperature (eq 2). DH(r) values have been determined here for the Pd-Ag alloys both at constant rup and at constant pup over the temperature range from 423 to 523 K, which is relevant for H2 purification applications. Table 1 shows DH(r) values that were determined at pup ) 20.3 kPa. Measurements were repeated with two sets of alloys from different preparations. The results in parentheses (Table 1) are the most recent. The results of the two sets of determinations are similar, and any differences can be attributed to compositional variations of the alloys and to measurements of d. It can be seen from the Table that DH(r) values do not vary much with XAg until >0.30, where there is a marked falloff. Figure 4 shows log DH(r) (pup ) 20.3 kPa, 473 K) as a function of XAg. There is a small decrease with XAg to about 0.30 and then a stronger decrease. The DH(r) values agree reasonably well at low XAg with the equation given by Holleck 3 based on his data from a higher temperature range, but the present values are larger at higher XAg than predicted by his equation. In addition, D*H values are shown that are generally slightly higher than DH at 20.3 kPa and will be discussed below. DH(r) values were also determined over this same temperature range for these alloys at constant rup values, for example, rup ) 0.03, 0.04, etc. The same trend with XAg is found for these constant H concentration values, but their magnitudes differ somewhat from those at constant pH2 due to the role of the thermodynamic factor. Activation Energies for Fick’s Diffusion Constants. Activation energies for diffusion, ED(r), were determined from the temperature dependence of the fluxes at constant rup. From eq 2, ln J ) -[ln DH + ln cup - ln d], and from the derivative of ln J with respect to 1/T, it follows that at constant rup, EJ(r) )

ED(r), where EJ(r) is the activation energy for the flux determined at constant rup. Plots of quantities proportional to the fluxes, normalized to d ) 100 µm, are plotted as ln (a × J) against 1/T in Figure 5, where a is a constant. Equilibrium pressures decrease with XAg, and at high XAg, it was difficult to measure fluxes at low rup, and therefore, larger rup values were employed for alloys with large XAg. The H contents held constant, rup, varied between r ) 0.03 (XAg ) 0.10) and 0.08 (XAg ) 0.50) (Figure 5). It can be seen that as XAg increases, the slopes of the plots and their intercepts become more negative. Table 2 shows values of ED(r) determined at these constant H contents, and it also gives pre-exponential factors, DoH(r). The diffusion parameters for Pd are also shown.13 Activation energies for permeability, EP(r), have been derived using EP(r) ) ED(r) + ∆HH(r) for a given r where ED(r) is positive and ∆HH(r) is negative for 1/2H2(g) absorption. This equation follows from eq 2 and P ) J × d. ED(r) has been determined from the constant rup plots (Figure 5), and the corresponding ∆HH(r) values have been estimated from ∆HoH + h1 × r where h1 is the H-H interaction energies estimates, which are available for these alloys.15 Figure 6 shows these as a function of XAg, where the trend in ED(r) agrees with that found by Holleck3 and Opara et al.6 in that the ED(r) are reasonably constant and nearly equal to that for Pd up to about XAg ) 0.30 and then increase. The location of the minimum in EP(r) shown in Figure 6 does not agree exactly with Figure 3 because the pre-exponential factors and nonideality also affect the permeabilities. Log cH/(mol H/cm3) and log DH/cm2/s are plotted against 1/T in Figure 7 for the Pd0.77Ag0.23 alloy (pup ) 20.3 kPa). After translation along the y axis in order for their sum to coincide with the intersection of the log cH/(mol H/cm3) and log DH/ cm2/s against 1/T lines, the sum is seen to be nearly independent of 1/T. This indicates that the permeability of a Pd0.77Ag0.23 membrane, which is used industrially for H2 purification, does not increase significantly with temperature (423-523 K). If Figure 3 is examined, it can be seen that there is not much difference in the permeabilities at the different temperatures for



Figure 12. Isotherms for the Pd0.77Ag0.23 alloy plotted as ln p1/2 as a function of ln r.

the XAg ) 0.23 alloy, but differences tend to increase as the composition deviates from this. Concentration Dependence of DH and ED for the Pd0.90Ag0.10, Pd0.77Ag0.23, Pd0.65Ag0.35, and Pd0.50Ag0.50 Alloy Membranes. These four alloy compositions have been chosen for detailed determinations of the dependence of DH(r) and ED(r) on rup because it was not feasible to examine all of the alloy compositions in detail, and the chosen ones are representative of the possible behavior. Experimental values of ln DH(r) (423 K) and ED(r) (423-523 K) as a function of rup are shown for the Pd0.90Ag0.10 alloy (423 K) in Figure 8, where it is seen that ln DH(r) decreases with r, which is similar to Pd-H; however, the slopes are not as steep as those for Pd.13 There seem to be two slopes: one at low rup, and the other, at higher rup. Extrapolation to rup ) 0 gives D*H ) 7.4 × 10-6/cm2/s, but extrapolation of ED(r) to rup ) 0 appears to give a value that is too low; however, this may be within the margin of error. Figure 9 shows the dependence of ln DH(r) and ED(r) on rup for the important Pd0.77Ag0.23 alloy membrane and detailed results are given in Table 3 for three of the five temperatures measured. It can be seen that at all three temperatures DH(r) values decrease with rup. It is difficult to extrapolate experimental ln DH(r) to rup ) 0 for this alloy (Figure 9) because the lowest values of rup are too large for meaningful extrapolations except at 473 K, where the data appear to extrapolate to 7.4 × 10-6/cm2/s, which is in reasonable agreement with the D*H determined by other methods (Table 3). The dependence of ln DH(r) and ED(r) on rup for the Pd0.65Ag0.35 alloy membrane was also determined, and it appears to be nearly independent of rup, in contrast to the lower Ag content alloys (Figures 8, 9). The Pd0.50Ag0.50 alloy is of interest because the thermodynamic factor is positive at low r, which should cause DH(r) to increase, rather than decrease, with rup, as for the lower Ag content alloys. Results are shown in Figure 10 (473 K) and tabulated in Table 4, where it can be seen that there is a steady increase in DH(r) with rup, as expected. For H2 purification, it would be desirable to operate a membrane in a range of H contents where f(r) is >1.0, and consequently, DH(r) > D*H.

Figure 11 shows plots of ln DH(r)/cm2/s against rup for Pd and for the Pd0.90Ag0.10, Pd0.77Ag0.23, and Pd0.65Ag0.35 alloys at 423 K. This temperature has been chosen for this plot because the nonideality is greater at the lower temperature. It can be seen that ln DH(r) decreases with XAg, and the slopes (negative) of the plots also decrease progressively with an increase in XAg. Concentration-Independent Diffusion Parameters. In this section, the dependence of DH(r), Fick’s diffusion constant, and ED(r) on rup will be considered. For example, the slopes of plots of ln DH(rup) against rup will be evaluated for the various alloys and compared to the predicted slopes. The concentrationindependent parameters, for example, D*H, will be determined from the concentration-dependent ones, for example, DH(r). The role of nonideality due to the thermodynamic factor has been discussed in some detail elsewhere for pure Pd using the present experimental setup where rup . rdown13 and will be briefly summarized here. Again, the Pd0.77Ag0.23 alloy will be employed for illustration because of its importance. Concentration-Independent D*H from the Experimental DH(r) Values at Small H Contents. The slopes of ln pH21/2-ln r isotherms, such as shown in Figure 12 for the Pd0.77Ag0.23 alloy, give the thermodynamic factor, (∂ ln p1/2/∂ ln x)T,XAg ) f(r) at each r. A plot of f(r) as a function of r can then be made for each alloy at each temperature. Figure 13 shows such a plot for the Pd0.77Ag0.23 alloy at 423 K, from which values of F(rup) can be obtained by integration of f(r) between 0 and rup according to eq 3, which was given in ref 12 and used for Pd-H in ref 13 to obtain D*H values. At this temperature and for this alloy, the thermodynamic factor, f(r), first decreases and then increases with r. D*H can be calculated at any value of rup with rdown ) 0 using

DH(rup) ) D*H

∫0r

up

(∂ ln pH21/2/∂ ln r) dr/rup

(3)

Metal-H systems have been described successfully at low H contents by simple statistical thermodynamic equations, which can be employed to evaluate F(r) in eq 3. The simplest one is


Wang et al.

Figure 13. The thermodynamic factor for the Pd0.77Ag0.23 alloy (423 K) plotted against r. The vertical dashed line indicates ideal behavior where f(r) ) 1.0.

the regular interstitial solution (RIS)16 or mean field, lattice gas model,

RT ln pH21/2 ) ∆µH0 + RT ln (r/(1 - r)) + g1 × r

(4)

where g1 is the first-order coefficient of a polynomial expansion in r of the excess partial free energy, µEH(r), and for a given alloy, it is a constant at a given temperature. Experimental isotherms are needed to evaluate g1, which is negative for Pd but becomes less negative with an increase in XAg until at large XAg, it becomes positive.7 Values of g1 for the alloys have been evaluated using β ) 1.0 in the configurational partial entropy term, -R ln (r/(β - r)), which is not strictly correct because of the selective occupation of interstices in Pd-Ag alloys,7 and therefore, β * 1.0. Nonetheless, β will be taken as 1.0, which will not affect the use of the thermodynamic factor, f(r), for obtaining D*H. From this model, the following equation is obtained for F(r)13

F(rup) ) -ln (1 - rup) + g1rup2/RT

(5)

% Ag

T/K

DH(r)

D*H (F(r))a

D*H (RIS)a

D*H (extrap)a

10 19 23 30 40 50 10 15 19 23 30 35 40 45 50 10 15 23 40

423 423 423 423 423 423 473 473 473 473 473 473 473 473 473 523 523 523 523

4.4 3.5 3.4 (0.04) 2.4 (0.06) 1.4 (0.06) 0.27 (0.04) 9.8 8.8 7.8 6.7 5.2 3.9 2.7 1.6 0.84 (0.03) 17.9 16.0 12.6 5.0

4.6 6.4 4.0 2.7 1.3 0.22 10.9 9.7 9.7 8.1 5.8 4.2 2.6 1.4 0.71 19.1 16.6 12.7 5.0

4.7

6.0 6.6 3.9

3.9 2.6 0.88 0.17 10.2 9.3 8.0 7.6 5.2

0.67 18.5 16.3 12.4 4.2

10.9 10.2 9.6 8.3 6.7 4.7 2.8 1.5 19.4 17.0 13.8 5.3

a The column headings F(r), RIS and extrap refer to the use of eq 4 directly, from the RIS model and extrapolation of DH(r) to rup ) 0.

and using this result, eq 6 can be derived.

DH(rup) ) D*H(g1rup/2RT - ln(1 - rup)/rup)

TABLE 5: D*H/(cm2/s)×10-6 Obtained from DH (r) at pup ) 20.3 kPa except for Some at 423 K Where the r Values Employed Are Shown in Parenthesis*

(6)

Under conditions that rup . rdown, it was found for Pd-H that an average r value, rav ≈ rup/2, can be employed in eq 1 to obtain D*H from DH(rup)13 (Table 3). When the rav ≈ rup/2 approximation can be employed, the evaluation of D*H is simpler than using the other procedures.13 Although at small rup, this is a good approximation for Pd-H,16 it may not be for some of the Pd-Ag alloy membranes, and this will be examined here. Table 3 shows values of D*H for the Pd0.77Ag0.23 alloy evaluated from DH(rup) at different rup values using three different methods: eq 6; eq 3 with F(r) determined from the integration of plots such as shown in Figure 13 for the Pd0.77Ag0.23 alloy; and last, using the approximation that f(rup/2) )

f(rav) in eq 1. It can be seen that generally, D*H values are larger and more constant than the DH(rup) values, as expected. It can also be seen (Table 3) that the approximation, f(rup/2) ≈ f(rav) gives somewhat greater values at 473 K but is otherwise good at small rup. Table 4 shows results for the Pd0.50Ag0.50 alloy, but only the RIS method has been employed to determine D*H for this alloy, and the values of D*H are quite constant at each temperature. Using the data employed for Figure 5, DH(r) values were obtained for all of the alloys. These are shown in Table 5 at 423, 473, and 523 K, where pH2 ) 20.3 kPa; however, at 423 K, only the Pd0.90Ag0.10 and Pd0.81Ag0.19 alloys were measured at 20.3 kPa, whereas the remainder were determined at the rup indicated. The D*H’s shown in Table 5 for the Pd0.77Ag0.23 alloy differ slightly from those in Table 3 because the former were



Figure 14. RT ln DH(r) against rup over a large H content (423 K). b, experimental data; O, 4, values using eq 3 from two different independently determined sets of F(r)-r relations; 2, using eq 3 with the (1 - r) factor included.

determined with a different membrane preparation, and consequently, errors enter due to the thickness measurements. The D*H’s in Table 5 show the same trend with XAg as the DH(r) values as shown in Figure 4. From eq 6, the slopes, d ln DH(r)/dr, are given by g1/2RT at small rup. These values of g1 can be compared with those obtained from thermodynamic data; that is, plots of RT ln p[1/ 2][(1 - r)/r] against r.7 The slopes of the plots in Figure 11 give g1 (Pd) ) -49.0 kJ/mol H (-35.9 kJ/ mol H), g1 (Pd0.90Ag0.10) ) -39.8 kJ/mol H (-34.7 kJ/ mol H), g1 (Pd0.77Ag0.23) ) -23.5 kJ/mol H (-22.7 kJ/ mol H), and g1 (Pd0.65Ag0.35) ) -5.4 kJ/mol H (-11.3 kJ/ mol H), where the values in parentheses are from the thermodynamic measurements. The agreement is good for the Pd0.90Ag0.10 and Pd0.77Ag0.23 alloys, demonstrating the validity of the analysis, although the agreement is not as good for Pd and the Pd0.65Ag0.35 alloy. The fact that diffusion parameters, which are kinetic data, can be employed to determine thermodynamic data (g1, h1) is noteworthy because generally, there is no relationship between the experimental results from kinetics and thermodynamics. An exception is that the rate constants in ionic solutions are affected by a thermodynamic parameter, the ionic strength of the solution. Concentration Dependence of DH(r) at Large H Contents and the DeriVation of D*H from These Values. The Pd0.77Ag0.23 alloy has been chosen to examine the concentration dependence of DH over a wide concentration range. The only method suitable for obtaining D*H from DH(rup) at large rup is directly from eq 3 with F(rup) obtained from equilibrium isotherms. Some results for large rup are shown in Figure 14 at 423 K plotted as RT ln DH(rup) against rup. There is a minimum in RT ln DH(rup) at rup ≈ 0.20, and assuming that rav ≈ rup/2, it occurs at rav ≈ 0.10 which is smaller than r ) 0.20 for the minimum found by Ku¨ssner;4 however, his measurements were at much lower temperatures, 303-348 K. The results of the conversion to D*H are shown by the open symbols where two different

evaluations of F(rup) were employed with similar results. The D*H obtained are not constant but decrease at about r ) 0.07 and then stay approximately constant from rup ) 0.16 to 0.27. Salomons et al.,9,10 following Wicke and co-workers,5,17 employ an interstice availability factor in addition to f(r), where for pure Pd, eq 1 would become

DH(r) ) D*H(1 - r)(∂ ln pH21/2/∂ ln r)T ) (1 - r) f(r) D*H (7) where (1 - r) is the interstice availability factor. Equation 3 for rup . rdown can be modified to include (1 - r).

DH ) D*H

( )

F(r) -( rup

∫0r

up

rf(r) dr)/rup

(8)

This equation has been employed here for the Pd0.77Ag0.23 data, and the deviations of D*H from a constant become especially significant at high rup (Figure 12). The (1 - r) factor may be too simple for an alloy in which some interstices may be unfavorable because of more than one Ag nearest neighbor. In any case, even for Pd, it is not certain that this factor is correct because the H atoms undergo extremely rapid jumps and may leave a given interstice as one enters in a concerted way, and therefore, (1 - r) may be an overcorrection. The conditions for a minimum can be obtained from (d ln DH(r)/dr) ) 0 using eq 3, leading to the result that at the minimum, the effective thermodynamic factor from eq 3, F(rup), is equal to f(rup) × rup. For the data in Figure 12, F(rup)/rup ) 0.65 at the minimum, where f(rup) ) 0.67, which is within experimental error. At rup values smaller than the minimum, where H in the majority of the membrane is smaller than rup, F(rup) > f(rup) × rup. When rup is greater than the minimum, F(rup) < f(rup) × rup. Concentration-Independent E*D and Do,* H Values from Concentration-Dependent Values. The activation energy for diffu-


Wang et al.

sion, ED(r), is also affected when f(r) * 1.0. This has been illustrated for Pd-H, for which ED(r) increases with rup.13 When eq 3 is differentiated with respect to 1/T, at constant rup, eq 9 is obtained, since at constant rup, the ln rup term drops out.

ED ) E*D - R

[(

) ( )] (

∂ ln F(rup) ∂(1/T)

-

∂ ln rup

) -R

∂(1/T)

)

∂ ln F(rup) ∂(1/T)

(9)

rup

The activation energy for diffusion has been determined here generally at constant rup values; for example, Figure 5, and therefore, from eq 9, the role of nonideality on ED(r) is given by the change of ln F(rup) with 1/T. If the dependence of ln F(rup) on 1/T is substituted into eq 9 at different temperatures, the evaluation of E*D is proved to be reasonably good at large rup. For example, E*D values determined at r ) 0.06 and 0.08 for the Pd0.77Ag0.23 alloy appeared to be reasonable; however, at lower values, small differences of ln F(rup) become important, and the results were not very accurate. Therefore, this method was generally not employed. Inserting the expression for F(rup) (eq 5) from the RIS model into eq 9 and taking the natural log and differentiating with respect to 1/T, eq 10 is obtained.

ED ) E*D +

h1rup [2 ln(1 - rup)/rup] - g1rup/RT

(10)

Since h1 is negative for alloys with XAg e 0.40,7 it would be expected from eq 10 that ED(r) > E*D, at least at relatively small values of rup. With an increase in XAg, |h1| decreases7 and ED approaches E*D. When h1 becomes positive, as, for example, for the XAg ) 0.50 alloy, then E*D > ED(r). Figures 8-11 confirm these predictions. At small r, eq 10 reduces to (∂ED/∂r)T ) -h1/2, and from thermodynamic data, h1 is found to be more negative than g1.7 This is supported here because the slopes with respect to rup are greater for ED(r) than for RT ln DH(r) (Figures 8, 9). Values of h1 obtained from the slopes of ED(rup) (Figures 8, 9) are -80 and -75 kJ/mol H for the Pd0.90Ag0.10 and Pd0.77Ag0.23 alloys, respectively, where the latter is based on some additional data not shown in Figure 9. The values of h1 from thermodynamics are not very accurate for these Pd-Ag alloys, so comparisons will not be made; however, h1 < g1 for Pd.15 The approximations for the slopes are good only at small r, and the complete equations, for example, eq 10, must be employed at higher r, which is also an approximation. The concentration-independent Arrhenius equation for interstitial diffusion is * D*H ) Do,* H exp -(ED/RT)

(11)

* Do,* H values given in Table 6 have been calculated from DH and * ED, and the latter appears to be smaller for the Pd0.90Ag0.10 alloy than for Pd, as found in ref 8. It is possible that the lattice expansion due to Ag lowers E*D while other factors, which enter at greater XAg, are not very important for the low Ag content, Pd0.90Ag0.10 alloy. The values of Do,* H are between 3.4 and 4.9 × 10-3 cm2/s without a clear trend with XAg. Holleck3 and Wicke and Brodowsky8 found small decreases with XAg, but the latter workers indicated that there is not much change with XAg for deuterium, which seems to be inconsistent with their results for H.

TABLE 6: Concentration-Independent E*D and Do,* H Values (423-523 K) % Ag

D*H (473 K)/cm2/s

E*D/kJ/ mol H

2 D0,* H /cm /s

0 10 15 19 23 30 35 40 45 50

13.3 × 10.7 × 10-6 9.7 × 10-6 9.1 × 10-6 8.0 × 10-6 5.9 × 10-6 4.2 × 10-6 2.7 × 10-6 1.5 × 10-6 0.69 × 10-6

23.9 23.7 23.9 24.3 24.1 26.1 27.7 29.4 30.6 34.9

5.6 × 10-3 3.4 × 10-3 4.2 × 10-3 4.0 × 10-3 3.7 × 10-3 4.5 × 10-3 4.8 × 10-3 4.8 × 10-3 3.6 × 10-3 4.9 × 10-3

10-6

The permeability of the Pd0.77Ag0.23 alloy is greater than that of the others at pup ) 50.6 kPa and, for example, 423 K (Figure 3), despite the fact that at this pup, r is greater for alloys with XAg < 0.23. An example is the Pd0.85Ag0.15 alloy, whose H2 solubility at 50.6 kPa (423 K) is r ) 0.36, as compared to 0.28 for Pd0.77Ag0.23 and D*H ) 4.7 × 10-6 cm2/s for the former and 4.0 × 10-6 cm2/s for the latter (423 K). The answer to this apparent contradiction is that f(r) is smaller at 50.6 kPa (423 K) for the Pd0.85Ag0.15 alloy, f(r) ) 0.54, than for the Pd0.77Ag0.23 alloy, f(r) ) 0.88. The permeability is given by P ) f(r) D*H × cup, and from the above values, the ratio P(Pd0.77Ag0.23)/ P(Pd0.85Ag0.15) ) 1.08. According to Figure 3, the ratio is 1.3, which is greater than this predicted value but in the right direction to explain the greater P of the Pd0.77Ag0.23 alloy. H Concentration Profiles for the Pd0.77Ag0.23 Alloy Membrane when pup . pdown. Concentration profiles can be determined using12

F(r) ) F(rup)[1 - (x/d)]

(12)

where x is the distance through the membrane with x ) 0 and d is the upstream and downstream distances, respectively. In order to determine the profile for a given rup, values of F(r) can be calculated as a function of r, and with these F(r) and F(rup), (x/d) can be calculated from eq 12. The F(r)-vs-r plot for the Pd0.77Ag0.23 alloy is shown in Figure 15. For a given alloy and temperature, the F(r)-vs-r relation has a unique shape. Also shown in Figure 15 is the relation calculated from the RIS model using eq 6 with g1 ) -22.7 kJ/ mol H. The agreement is quite good up to about r ) 0.12 but poor for r > 0.12, as expected. Values of (x/d) have been calculated for a series of r and rup values to determine the concentration profiles shown in Figure 16 for the Pd0.77Ag0.23 alloy at pup ) 20.3 and 50.6 kPa (423 K). It is seen that, except for the limiting values of (x/d) ) 0 and 1.0, the H concentrations are larger for pup ) 50.6 kPa than the ideal ones, but at 20.3 kPa, they are smaller. The value of rup is 0.15 at 20.3 kPa and 423 K (Figure 12), and at this rup, F(rup) ) 0.10 (Figure 16). For 50.6 kPa, rup ) 0.28 (Figure 12), and from Figure 15, F(rup) ) 0.235. It is of interest that the deviations of r through the membrane from the ideal value are positive for pup ) 50.6 kPa (rup ) 0.28) but negative for 20.3 kPa (rup t 0.15) (423 K) (Figure 16). For the former at rup ) 0.28, f(r) is >1.0, and for the latter at rup ) 0.15, f(r) < 1.0 (Figure 13). From J ) -DH(cH) (∂cH/∂x) ) D*H f(r) (∂cH/∂x), it follows that at rup ) 0.28, (∂cH/∂x) will be less negative than the ideal slope, -cup/d, and for rup ) 0.15 (Figures 13, 16) with f(rup) > 1.0, (∂cH/∂x) must be more negative than the ideal slope. To summarize, if f(rup) is >1.0, the deviations will be positive, and if f(rup) is < 1.0, the deviations will be negative.



Figure 15. F(r) against r for the Pd0.77Ag0.23 alloy at 423 K. O, experimental F(r); 4, F(r) from RIS model.

Figure 16. H-to-metal ratio in the membrane, r, plotted against distance, (x/d), through a Pd0.77Ag0.23 alloy membrane (423 K). O, at pup ) 50.6 kPa; 4, 20.3 kPa. The full straight lines indicate ideal behavior where f(r) ) 1.0 for the two pup values.

In the steady state, the flux is constant throughout a membrane; that is, J ) -DH(cH) (∂cH/∂x) where DH(cH) and (∂cH/∂x) change with x, but they compensate each other in the steady state, maintaining J constant at all (x/d) ratios and corresponding r. J must also be equal to the measured steadystate value, -DH(cup/d), and it must also be equal to D*H(cH/RT) (∂µH(r)/∂x). Calculated values of these various fluxes from these equations are close to equal for various x values for the Pd0.77Ag0.23 alloy. Conclusions It is shown from the linearity of plots of J against 1/d that bulk diffusion is the slow step for the Pd-Ag alloy membrane with the greatest permeability, Pd0.77Ag0.23. The permeabilities

of the alloys at pup ) 20.3 and 50.6 kPa are a maximum at about XAg ≈ 0.23 for the temperature range 423-523 K. The permeability in the Pd0.77Ag0.23 alloy is shown to be nearly independent of temperature at pH2 ) 20.3 kPa because cH and DH(r) compensate each other, that is, cH decreases with a temperature increase while DH increases, and therefore, for H2 purification, there is no advantage to increasing temperature within this range. The dependence of DH(r) on XAg has been determined at a given pup, and DH(r) is nearly constant up to XAg ) 0.30. For >0.30, DH(r) falls off more sharply. In the dilute phase of Pd-H, DH(r) and ED(r) vary significantly with r.13 The dependences of these parameters on r has been examined for the Pd-Ag alloys in which there is a greater

1148 J. Phys. Chem. B, Vol. 112, No. 4, 2008 H2 solubility in the R phase than for Pd. The dependence of DH(r) and ED(r) on r depends on XAg; that is, DH(r) decreases with H concentration for the alloys up to XAg ≈ 0.35, at which XAg there is only a negligible effect. ED(r) increases with H concentration for alloys with XAg < 0.35. These results are related to the thermodynamic factor and its dependence on XAg. It is noteworthy that experimental thermodynamic parameters g1 and h1 can be related to kinetic parameters, RT (∂ ln DH/∂r) and (∂ED/∂r) because there are not many instances in which experimental kinetic and thermodynamic quantities can be directly related. It is often assumed that the concentration-independent diffusion constants cannot be obtained by permeation measurements with large concentration gradients, for example;11 however, it is shown here that this is not the case, and D*H and E*D values have been obtained for these Pd-Ag alloys. For the Pd0.77Ag0.23 alloy, the dependence of DH(r) on rup has been examined over a large range of H concentrations, and these have been converted to concentration-independent D*H; however, they are not constant for rup > 0.15, but increase slightly. Acknowledgment. This work was supported by Washington Savannah River Company under U.S. Department of Energy Contract Number DE-AC09-96SR185000.

Wang et al. References and Notes (1) Chowdhury, M.; Ross, D. Solid State Commun. 1973, 13, 229. (2) Lovvik, O.; Olson, R. J. Alloys Compd. 2002, 330-332, 332. (3) Holleck, G. J. Phys. Chem. 1970, 74, 503. (4) Ku¨ssner, A. Z. Naturforschung 1966, 21a, 515. (5) Bohmholdt, G.; Wicke, E. Z. Phys. Chem. N.F. 1967, 56, 133. (6) Opara, L.; Klein, B.; Zu¨chner, H. J. Alloys Compd. 1997, 253254, 378. (7) Flanagan, T.; Wang, D.; Luo, S. J. Phys. Chem. B 2007, 111, 10723. (8) Wicke, E.; Brodowsky, H. In Hydrogen in Metals, II; Alefeld, G., Vo¨lkl, J., Eds.; Springer-Verlag: Berlin, 1978. (9) Salomons, E. Ph.D. Thesis, Vrije University, Amsterdam, 1989. (10) Salomons, E.; Ljungblad, U.; Griessen, R. Defect Diffus. Forum 1989, 66-69, 327. (11) Stonadge, P.; Benham, M.; Ross, D.; Manwaring, C.; Harris, I. Z. Phys. Chem. 1993, 181, 125. (12) Flanagan, T.; Wang, D.; Shanahan, K. Scripta Mater. 2007, 56, 261. (13) Flanagan, T.; Wang, D.; Shanahan, K. J. Membr. Sci. 2007, 306, 66. (14) Wang, D.; Flanagan, T.; Shanahan, K. J. Membr. Sci. 2005, 253, 165. (15) Kuji, T.; Oates, W.; Bowerman, B.; Flanagan, T. J. Phys. F: Metal Phys. 1983, 13, 1785. (16) Oates, W.; Flanagan, T. J. Mater. Sci., 1981, 16, 3235. (17) Holleck, G.; Wicke, E. Z. Phys. Chem. N.F. 1967, 56, 155.

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