K. D. ALLARD, T. B. FLANAGAN, AND E. WICKE
298 conduction in capacitors could be accomplished by sealing the ions in the pores as suggested earlier. Ionic conduction devices might be desirable in some cases, especially if alternating current was used and ionic recombination can be accomplished. If the molecular parameters described here are desired, most of these problems are automatically circumvented because currents can be kept low and temporary. Lattice and hydration energies were early considered. A neglect of these quantities, however, provided (for at least the diatomic molecules considered) electron affinities which were within the experimental error of values found by photodetachment. For this reason it was considered desirable for the present study to neglect the additional quantities. There is probably cancellation of surface, lattice, and hydration energies which permits a neglect of these quantities for a number of molecules. This was considered a major advantage of the experiment. Investigation is continuing with some molecules apparently showing some hydration effects a t low solvent contents. When this further study has progressed sufficiently the results will be published. Hydration could help explain the progres-
sive apparent increase in the calculated dielectric constant for the nitrates studied. It has been reasoned that the released carrier leaves behind a neutral particle and thus I should be reduced by K rather than K 2 . I n other words J
= Ce-(Ed+KI)/KzkT
If one takes into account the experimental results, this equation does not apply. Consider the case where E d is much smaller than I. This is very often the case. For example, look a t the parameters for KC1. For this situation one would need to multiply the experimental obtained resistance slopes by approximately K to fit the new equation. If K were approximately 2 as was found for most cases in this paper, one would need to approximately double the empirical resistance slope values. The value of 2 for K was found both by experimental measurement and by solving the two basic equations. Observe that the total process, dissociation plus ionization, leaves two net charges. K 2 arises, in the hydrogen model, from a reduction in energy caused by a field decrease due to the interposition of a dielectric between two charges.
The Diffusion of Hydrogen in Boron-Palladium Alloysla by K. D. Allard, Ted B. Flanagan, C'hemistr.y Department, University of Vermont, Burlington, Vermont
and E. Wicke Institut jilr physikalische Chemie Universitut Milnster, MBnster, Germany
(Received J u n e 23,1069)
The diffusion of hydrogen in a series of boron-palladium alloys has been investigated using an electrochemical relaxation technique. Results show that the values of the diffusion coefficients determined at %H + 0, where n H = H/Pd, atomic ratio, decline steadily with boron content from that of pure palladium. This behavior is in marked contrast to that of substitutional alloys of palladium. The dependence of the diffusion coefficients upon boron content supports the view that in these alloys boron occupies interstitial sites. A quantitative theory is advanced which satisfactorilyexplains the results. Introduction Before about 1960, data reported for the diffusion of hydrogen in palladium have been very divergent.Ib Recently, however, data reported from different laboratories have been in excellent agreement. For example, in the low hydrogen content cr phase, values selected from five different investigations give an energy of activation within 300 cal/mol of each other, AEa = 5730 f 150 cal/mol,2-6 and correspondingly good agreement for the absolute magnitudes of the diffusion The Journal of Physical Chemistry
constants. This close agreement extends over a very wide temperature range, i.e., from -78 to 379".2v6 In (1) (a) This research was performed a t the Institute far physikalische Chemie der Universitat Miinster, Germany; (b) e.g., R. M. Barrer, "Diffusion in Metals," Cambridge University Press, 1951. (2) 0. M. Kats and E. A. Gulbranson, Rev. Sci. Instrum., 31, 616 (1960). (3) J. Simons and T. B. Flanagan, J. Phys. Chem., 69,358 (1965).
(4) G. Bohmholdt and E. Wicke, 2. Phys. Chem. (Frankfurt am Main), 56, 133 (1967). ( 6 ) G. Holleck and E. Wicke, ibid., 56, 155 (1967). (6) H. Zilchner, Diplomarbeit, Munster, Germany (1967).
THEDIFFUSIONOF HYDROGEN IN BORON-PALLADIUM ALLOYS addition there have been several investigations at a single temperaturey-9 which also agree quite well with those cited above. The experimental techniques employed for these determinations have been quite different, ranging from conventional gas-phase permeation techniques2 to an electrochemical relaxation method.6 It is now clear that the major cause of the discrepancies in the past has been that the degree of cleanliness, and catalytic activity of the surface was insufficient for the rapid establishment of the Hz 2H equilibrium at the
*
surface; i.e., the early investigators were not measuring solely bulk diffusion but the slow surface reaction contributed to their measured rates. An exception to these remarks is the research of Jost and Widman.'O They employed palladized palladium spheres at higher temperatures and obtained diffusion constants comparable to those of the recent studies. The diffusion system of hydrogen in palladium and its alloys constitutes a most important tool for basic studies of solid-state interstitial diffusion. The hydrogen content can be varied over wide limits in pure palladium and in certain palladium alloys." The electronic and geometric properties of the metal substrate can be altered in a controlled manner by the appropriate choice of substitutional alloy partner. The thermodynamics of absorption are known for palladiumhydrogen and for many alloys of palladium with hydrogen." Experiments utilizing isotopic substitution of deuterium or tritium can be readily carried out. Neutron diffraction and scattering studies have located the positions and measured the vibrational frequencies of the hydrogen atoms in the palladium lattice.12J3 Finally, of all the possible systems involving interstitial diffusion in metals, the diffusion of hydrogen is probably the most amenable to fundamental interpretation. A new electrochemical technique for the determination of diffusion coefficients in palladium and its alloys was recently developed by Holleck and Wicke.6 This relaxation technique consists of potentiostatically charging an electrode to a uniform concentration profile. This uniform profile is then perturbed by the electrolytic addition or subtraction of an increment of hydrogen. During the perturbation a constant flux of hydrogen passes through the surface of the specimen. The relaxation of the nonuniform concentration profile thus established is followed by measurements of the electrode potential of the specimen as a function of the time. (The electrode potential can be related to the concentration at the surface if equilibrium absorption data are available.) For specimens of various geometries the diffusion equation can be solved exactly. Flat plates have been employed here. The absorption of hydrogen by boron-palladium alloys has been investigated by Sieverts and BruningI4 and more recently by Burch and Lewis15 and Husemann.16 Schaller17 and Brodowsky have also investi-
299
gated the absorption of boron itself by palladium. Indirect evidence suggests that boron atoms occupy interstitial sites within the palladium l a t t i ~ e ; ' ~ -e.g., l~ the lattice expansion which occurs upon boron addition is greater than one would expect on the basis of a substitutional alloy. Holleck and Wicke6 and Bohmholdt and Wicke4 have reported values of the diffusion coefficient of hydrogen in several palladium-rich alloys for both high and low values of n~ where n~ is the atomic hydrogen-to-metal ratio. I n the low hydrogen content region, which is the region of interest here, the value of DnH+ (25") changes upon alloying with substitutional metals from 2.6 X lo-' cm2 sec-' (Pd) to 2.3 X lo-' cm2 sec-' (10% Ag/Pd), 1.95 X cm2sec-l(25% Ag/Pd), and cmz sec-l (5% Sn/Pd). Thus the values 1.95 X of DnH+, do not fall off appreciably from that of pure palladium for additions of substitutional metals up to approximately 25 atomic per cent. This behavior has been supported by additional work in this laboratory using the relaxation method18 and by Zuchnerlg using a breakthrough time technique7 with membranes. I n addition, KussnerZ0and also Makrides and coworkersz1 have reported values of DnH+Oof 3 X om2 sec-l and 3.5 X em2 sec-l a t 25", respectively, for a 25% silver-palladium alloy. Since boron is believed to form an interstitial alloy with palladium, in contrast to those systems studied to date, it is of interest to examine the dependence of DnBdOupon the atomic ratio of boron in the alloy. Experimental Section The alloys in the form of thin plates containing 0.035, 0.075, 0.135, 0.175, and 0.208 atomic boron-topalladium ratio, n ~ were , prepared by H. Schaller (Institut fur physikalische Chemie, Munster). Their
(7) M. Devanathan and Z. Stachurski, Proc. Roy. Soc., A270, 90 (1962). (8) M. von Stackelberg and P. Ludwig, 2. Natur/orsch., 19a, 93 (1964). (9) D.N. Jewett and A. C. Makrides, Trans. Faraday Soc., 61, 932 (1965). (10) W. Jost and A. Widman, 2. Phys. Chem., B29,247 (1935). (11) F. A. Lewis, "The Palladium-Hydrogen System," Academic Press, New York, N. Y., 1967. (12) J. E. Worsham, M. K. Wilkinson, and C. G. Shull, S. Phys. Chem. Solids, 3, 303 (1957). (13) J. Bergsma and J. A. Goedkoop, Physica, 26,744 (1960). (14) A. Sieverts and K. Broning, 2. Phys. Chem., 168A,411 (1934). (15) R.Burch and F. A. Lewis, private communication. (16) H. Husemann, Ph.D. Dissertation, Monster, 1968,and H. Husemann and H. Brodowsky, 2. Naturforsch., 23a, 1693 (1968). (17) H. J. Schaller, Diplomarbeit, Monster, 1967. (18) K. Allard and T. B. Flanagan, to be published. (19) H. Ztichner, Ph.D. Dissertation, Monster, 1969. (20) A. Kassner, 2. Phys. Chem., (Frankfurt am Main), 36,383 (1963). (21) M. Lord, S. Axelrod, K. Brummer, and A. Makrides, J. Electrochem. Soc., 110, 179 (1963). Volume 74, Number 2 January 22, 1970
300 preparation and properties are described elsewhere.1 7 , 2 2 I n the alloys employed here as quenched from elevated temperature there was no phase segregation, i.e., the boron was homogeneously distributed within the palladium matrix. The thicknesses of the specimens were ( n =~ 0.175) to 9.0 X cm ( n =~ from 3.8 X 0.075) and the areas were typically 1 cm2. The thicknesses of the rolled specimens were determined by carefully measuring an edge with a measuring microscope. Ten such measurements were made for each specimen and an average value was employed. The standard deviations were from approximately 8 to 17% of the average value except for the sample with n~ = 0.208 which was larger. It is believed that these large deviations are introduced by the cutting process itself and are much greater than any irregularities present in the plates themselves. The specimens were carefully cleaned and rubbed with fine emery paper prior to their activation by deposition of platinum black on both sides of the plate. It has been shown elsewhere that the diffusion results obtained using this technique for a palladium alloy are identical whether palladium or platinum black is used for the activation.18 The samples were sealed into tubes with epoxy resin so that only the alloy was exposed to the electrolytic solution. The solution was 1 N H2S04. Measurements were made in a glass reaction vessel which was placed inside a plexiglass container through which water was circulated at the temperature of the run (*0.2”). Experimental details can be seen el~ewhere;~the only significant modification was that a recorder with a faster response and chart speed was employed (Siemens Kompensograph). Typical currents and times emA and 4 sec, ployed for the perturbation were 3 X respectively. Results were obtained using both anodic and cathodic currents for perturbation. There were no systematic differences in the results obtained with either method. The electrode potential (w. nhe) was approximately 120 mV before a run was commenced. After the perturbation the potential was changed from this value by i l mV. For the determination of a diffusion coefficient for a given alloy, anodic and cathodic runs were made consecutively until a total of ten runs were completed. Between the runs the specimen returned to a stable electrode potential indicative of a uniform concentration profile. All ten runs were analyzed and averaged for the value of the diffusion coefficient reported, the standard deviation usually amounting to about 11% of the average. The electrode potential employed here, 120 mV us. nhe, corresponds to a small hydrogen content and so the measured value of D should be the limiting, concentration-independent value, DnH-+~.In the palladium-hydrogen system an electrode potential of 120 mV (30’) corresponds to a hydrogen content of H/Pd (atomic ratio) 5 0.001.23 The Journal of Phgsical Chentistrg
K. D. ALLARD,T. B. FLANAGAN, AND E. WICKE Diffusional Relaxation The diffusion equation can be solved for the perturbation and its subsequent relaxation process. Rather simple approximations for the evaluation of the experimental data can be obtained, as will be shown, if the flux of hydrogen passing into or out of the specimen is constant with time during the course of the perturbation. I n order to approximate this condition, the cathodic or anodic currents employed for the perturbation were maintained constant. The flux of hydrogen atoms into or out of the surface may not correspond to the Faradaic current because in the case of a cathodic perturbation some of the discharged hydrogen atoms may recombine at the surface and be evolved into solution as molecular hydrogen, or alternatively in the anodic case additional hydrogen atoms may arrive a t the surface from molecules dissolved in the solution. J,= 0
-S
0
x-
S
Figure 1. Schematic drawing of concentration profiles before and after application of perturbation.
From the known solubility of hydrogen molecules in the electrolyte at p ~ * atm (the pressure corresponding to 120 mV a t 30”) it can be estimated that the magnitude of the contribution of the forward or reverse steps of the reaction H2(solution) $2H is negligible
*
compared to the measured flux during typical perturbaA for 4 sec. It can be assumed tions, e.g., 3 X therefore that constant current implies constant flux. This conclusion is supported by the close agreement of the experimental data with the exact solution of the diffusion equation (below). In addition, if the hydrogen flux into the specimen were to vary with time, it would not be expected that its time variation would be identical for the anodic and cathodic perturbations. The close agreement of the results for the two types of perturbation argues that the flux is indeed invariant with time. For the initial and boundary conditions of c = co a t t = 0 and (bc/bz)o = 0 and ( b c / d ~ ) *=~ - J k 8 / D (Figure l),Crank24gives for the concentration at t = to (22) H. Brodowsky and H. Schaller, Trans. Met. Soc. AIME, 245, 1015 (1969). (23) E. Wicke and G. Nernst, Ber. Bunsenges. Phgs. Chem., 68, 224 (1964). (24) J. Crank, “Mathematics of Diffusion,” Clarendon Press, Oxford, 1956, p 58.
THEDIFFUSION OF HYDROGEN IN BORON-PALLADIUM ALLOYS
where s is the half-thickness of the specimen and tois the duration of the perturbation. The subsequent relaxation of the perturbation under the experimental conditions of ( b c / b ~ ) ~= , ~0, is given asz4 f(x’)dx’
cos ?? [f(x’) S
cos __ dx’ S
=
- at0 + 2K ,-
K =
” 1
2J,s
n=.l
exp(-atn2)[1
- exp(-aton2)]
(3)
where a = Dn2/s2. The summation can be replaced by integrals for small values of atoso that eq 3 can be approximated by
exp(-a(t 2
+-1
to))
(4)
+
where the (1/2) exp(-at) and (1/2) exp(-a(t to)) terms are included to make the integrals better approximations t o the summation. If eq 4 is integrated by parts, eq 5 results = 2-Jss{z
Da2 2
da(t For small values of approximated by
- E(s,t)
(2)
- c(s,m) = Da2 c,x
- c(s, w )
E(s,m)
where
n?rx‘
where f(x) is given by eq 1 and x‘ is the variable of integration. At x = s it can be shown that eq 2 reduces to
c(s,t)
nhe) can be substituted for concentrations a t x = s. Following this substitution and using c = ~ H / ‘ I / I M ~ , where VMeis the molar volume of the hydrogen-containing alloy, and expanding exp(E, - E)F/RT to the first power (this is a good approximation since experimentally ] E , - El is less than 1 mV) brings eq 6 to the form
+ -2 c” exp(-:2n2t> S n=l
c(x,t)
301
exp(-at)
+
t0)n erfc
-
z/,(t
+ to)}
(5)
4 2 and m) eq 5 can be
If E(s,w ) - E(s,t) is plotted against dtfto - d-t? the value of D can be obtained from the slope S, and the intercept I , i.e. D = U2s2/S2nto2 (8) Instead of replacing the summation in eq 3 by integrals it may be computed exactly for selected values of a, lo, and t. For this purpose an IBM 360 Computer was utilized using values of a from 3 X lom4to in steps of and from to in steps of and in steps of Values of to from to 4 X were from 2 to 15 see in steps of 1 see and t in steps of 1see to 15 or 100 see. The summation was carried out until the value of a term in the sum was less than 10-5 of the sum up to that term or alternatively when 237 terms were included in the summation. The first criterion was met in most cases before 237 terms were evaluated . The availability of the nearly exact value for the summation allows the approximate treatment to be evaluated. Table I shows some typical errors introduced by the approximation used in obtaining eq 6. It can be seen from Table I that the approximation is extremely good except when a is large, 2 0 . 1 and to is large.
Table I: Comparison of “Exact” Sum,
“ 1
- e--rrtnz
n=1n2
(1 - e--rrtonz),with the Approximation to the Sum, ( 6{ - 4 for Typical Values of a, to and t
1,
4-0
+
Approximation 0
3
Because the data are obtained in regions of hydrogen content where Sieverts’ law is valid, PH2’/’ = exp. (- EF/RT) = KHnH,measured electrode potentials (vs.
n2D F - - exp(-E,F/RT) 2JasKHVllleR T
x
to, sec
10-4
1 x 10-8 1 x 10-3 1 x 10-8 1 x 10-2 1 x 10-1 1 x 10-1 1 x IO-’
3 3 10 3 3 3 15 30
t , sec
Exact sum
(0s 6)
4 4
0.01937 0.03469 0.09262 0.01988 0.09946 0.21194 0.19248 0.23524
0.01932 0.03462 0.09247 0.01990 0.09915 0.2123 0.228 1.394
4 14 4
4 I4 14
Volume 74, Number
$2
Januarg $28,1970
302
E(. D. ALLARD,T. B. FLANAGAN, AND E.
WICRE
For large values of a! a more suitable approximation is that previously employed by Holleck and Wickes in which a parabolic initial condition is obtained for the solution of eq 1when to is large and the terms of the sum may be neglected. When this parabolic distribution is inserted into eq 2 and the summation is limited to the first term, then the following equation results
E(s,t) - E ( s , m ) = const exp(-D?r2t/s2)
(9)
Alternatively, for small times and again under conditions where the parabolic initial distribution holds, to large, the plate can be treated as a semi-infinite medium26and the general solution is
e w ( - (t
+ X > ~ / ~ Dd tO )(10)
QO
0
This can be evaluated for the initial parabolic distribution, &, which extends up to a thickness T , r < s, i.e.
45)
3
6
9
t isec)
12
15
Figure 2. Typical relaxation data obtained for several runs, A, A, V; n~ = 0.175, 30”. Solid curve calculated from eq 7.
=
(5 2 7) I
It is convenient here to consider the surface to be located a t x = 0 rather than x = f s as in the derivations of eq 9 and 11. If erfcx is approximated by the
+0.3-
summation 1
> where z = s2/4Dt and s2 expression results tc(0,m)
>> 4Dt,
- c ( O , t ) I / [ c ( O , ~ ) - C(0,O)l
E
then the following
30.0is
B
w -
=
-
and
d E ( 0 , a)
- E(0,t)
= d E ( 0 , W ) - E(0,O) d E ( O , m ) - E(0,0)[3Dt/~~]”*(12)
0.0
03
06
mo - Ji- , s e c 1 I 2 where measured electrode potentials (us. nhe) have been substituted for concentrations in eq 11 in going to eq 12 using relationships given above. The diffusion coefficient is obtained from the slope of the plot of d E ( 0 ,m ) - E(0,t) us. t1I2and the intercept a t t = 0.
Results and Discussion Some typical diffusion runs are shown in Figure 2 (0.175 atomic ratio boron, 30’). A typical analysis of diffusional data via eq 7 is shown in Figure 3. The matching of the “exact” summation (eq 3) with the experimental data is shown also in Figure 2 where the value of a! and the multiplicative factor before the summation are determined from the ‘lope and intercept of eq 7. It can be seen that the experimental data and The Journal of Physical Chemietry
Figure 3.
- E(t)l vs. E , = E(s, ) and
Plots of experimental values of ] E ,
d t q -~dta t 30’ for n~ = 0.175.
E ( t ) = E(s,t).
the values from the “exact” solution agree reasonably well especially with regard to the shape of the curves. This agreement helps to justify the assumption discussed above that the flux is constant during the perturbation. Figure 4 shows an analysis of data with eq 12 using the 0.035 atomic ratio boron alloy where, among the alloys investigated here, eq 12 should be most ap(25)
w. JOB$,
6dDiffusion,V9 Academic press,New York, N. Y., 1960,
303
THEDIFFUSION OF HYDROGEN IN BORON-PALLADIUM ALLOYS plicable, i.e., DnH+0 is large. Each bar represents the spread of values obtained for ten anodic and cathodic runs. The deviation from straight line behavior observed a t large times is expected from the assumptions employed in the derivation of eq 12. Data were obtained for alloys of 0.035, 0.075, 0.137,
0.175, and 0.208 atomic ratio boron using eq 7,9, and 12. Within experimental error results generally agree except for alloys of nB 2 0.137 where the more exact analysis is needed (eq 7). The latter analysis gave smaller diffusion coefficients; these coefficients are the more correct for alloys of nB 2 0.137. Values of the diffusion coefficients are shown in Figure 5 (30'). The behavior of the diffusion coefficients for the boron-palladium alloys is in marked contrast to that of the substitutional alloys which have been investigated to date (Figure 5). This contrast, in itself, strongly supports the view that boron occupies interstitial rather than substitutional positions in the palladium matrix. Activation energies of 5.8 2.0 kcal/mol and 5.8 0.8 kcal/mol were determined for the diffusion of hydrogen at nH+O for the alloys with n B = 0.137 and 0.175, respectively, from 0 to 71". These values are essentially unchanged from that of pure palladium.2-6 It has been suggested that hydrogen diffuses through pure palladium via the tetrahedral holes from one octahedr-l interstitial site to a n ~ t h e r ; ~ this constitutes the lowest energy pathway. The decline in the values of DBnH+Owith boron content-despite the fact that the energy of activation remains unchangedsuggests that the presence of boron blocks possible diffusion paths for the hydrogen. It will be assumed that each isolated boron atom in the alloy matrix completely blocks all eight of its surrounding, neighboring tetrahedral interstitial holes as diffusion pathways for hydrogen atoms. There are N/4 unit cells in a specimen containing N palladium atoms and 2N tetrahedral holes. The fraction of blocked tetrahedral holes is 4 n so ~ that the diffusion coefficient a t nIl -+ 0 in the first, rough approximation is given by
*
Figure 4. US.
-
Typical relaxation data plotted as Z / E m E(t) E m = E(0, ) and E ( t ) = E(O,t).
4;;n~ = 0.035, 30".
DBnH-,co= DPdnH+O[l - 4nB]
*
(13)
and dDBnH40/dnB = -4DPdnH+0
0
0.I
0.2
0.3
"Bi
Figure 5. Values of D B n ~ - + 0US. n~ (30'). A, data evaluated using eq 7; A, data evaluated by eq 9 and 12 and averaged. Each point represents the average of 10 runs. Solid line, calcd by eq 13; dashed line calod by eq 17; broken line calcd by eq 18. The shaded area indicates comparable data as plotted against added metal content for substitutional alloys.1801g
This relationship is shown by the full line in Figure 5 ; it agrees quite well with the experimental data. By contrast, the assumption that boron blocks all twelve surrounding octahedral sites does not predict a slope in good agreement with the experimental data. An alternative analysis of the dependence of DBna+O upon nB can be made in analogy to Rayleigh's treatment of electrical conductivily of heterogeneous particles of insulating mates y ~ t e m s . ~ ~If*spherical ~' rial are uniformly dispersed in a conducting medium such that the total fractional volume occupied by the spherical particles is v and v
