Thermodynamic study of the palladium-yttrium-hydrogen system

J, P&a. Chew SoM %I. Printed in Great Britain.

49, No. 4, pp. 401-407.

$3.00 + 0.00 ~22-~7/~ Q 1988 prrgsmoa Reu pk

1988

THERMODYNAMIC

STUDY OF THE PALLADIUM-YTTRIUM -HYDROGEN SYSTEM

MICHIKO YOSHIHARA and REX B. MCLELLAN Department of Mechanical Engineering and Materials Science, William Marsh Rice University, Houston, Texas 77251, U.S.A. (Received 27 May 1987; accepted 26 August 1987)

A~~-~~lib~urn isobars for the ternary solid solution Pd-Y-H have been measured in the temperature range 625-1250 K and the composition range O-13 at. % Y. The maximum H content was 1.5 at. %. The isobars were determined at six different pressures ranging from 1.01 x IO3 to 1.01 x 10’ N/m*. The resulting partial thermodynamic functions for H atoms in the solid solution are in good accord with the predictions of the cell model for such systems. Keywords: Thermodynamics,

diffusion, hydrogen, palladium-yttrium,

thermore, the much larger values of 6, encountered make it essential to study this system at much lower H, pressures in order to generate data for Ri and the cor~sponding partial excess entropies Sp at ei values in the “infinite dilation” limit so that they can then be compared with values calculated by assuming that the H atoms interact with the matrix only. This is the motivation behind the work presented in this report.

INTRODUCTION In a recent survey on the thermodynamic and kinetic behavior of ternary Pd-U-H systems (V is a substitutional solute species) [I], it was shown that in general the behavior of the H atoms in the Pd-U matrix can be understood in terms of simple statistical mechanics in which the H atoms are adequately represented by Boltzmannian statistics even at the lowest temperatures at which the data were measured. In the cases of U = Ag, Au, Pt, Rh, Ni, Co, and Cu solubility isotherms at a H2 pressure of 10s N/m’ correspond to solid solutions with hydrogen concentrations of 19,less than lob2 [6, is the atom ratio of hydrogen (i) to the metal atoms]. Thus such systems can be understood in terms of statistical models in which H-H interactions can be ignored [ 11. This is, of course, not true in regions of (P,Z-T)-space where Si can assume large values. A second general feature of such systems is that the partial enthalpy Ri of the H atoms does not vary markedly when the U ~n~ntration is varied. As 0, (atom fraction of U in the Pd-U matrix) varies from 0 to 0.15, the maximum variation in Ri for all the systems cited above is N 6 kJ/mol ( N 0.06 ev). However, recent preliminary thermodynamic measurements [2] on the Pd-Y-H system indicate radical departures from the trends outlined above. The value of Ri at 10’ K decreases by 20 kJ/mol(O.21 eV) when 8, increases from 0 to 0.15. Furthermore, the H solubility in isobars at 10’ N/m2 is as high as Bi= 0.2 at 625 K (the lowest temperature measured). It is thus clear that the Pd-Y-H system represents a departure from the Pd-U-H solutions studies up to this juncture. This carries with it the implication that the experimental data will be the impetus for the generation of more interesting theoretical treatments. The much larger variation of !Ji with the composition of the “solvent matrix” makes this system a prime candidate for embedded atom calculations 131.Fur-

solid solutions.

EXPERIMENT

PR~UR~

Solid solutions of Pd-Y were made in the concentration range O-14 at. % Y from MARZ-grade Pd and Y by arc melting using a tungsten electrode under a protective atmosphere consisting of high-purity helium with 5% H,. The ~1 g samples produced were homogenized by remelting five times and then rolled into foils * 10e4 m thick. Several intermediate vacuum anneals were given under Hz gas at atmospheric pressure for 4-6 h at 770 K as work-hardening reduced the plasticity of the foils. The foils were subjected to a final annealing treatment at 1173 K under a vacuum of better than 10e5 N/m2 for 70 h. The compositions of the five alloy films made were determined by the electron microprobe. The results obtained were 2.11, 5.35, 8.17, 9.51, and 13.66 at. % Y with no discernible inhomogeneity. The films were deliberately produced with surface areas varying more than an order of magnitude in order to render visible any surface area effects in the measured isobars. No such effects were observed. The Pd-Y solid solutions were calibrated with H, gas (99.995 vol. %) in a system described in detail in earlier reports [I] and quenched into a silicone fluid at 298 K. The H contents of foils were determined by a hot extraction procedure [l], Every determination of the H content for a given H2 pressure and temperature was within f 8% of the mean of three rn~s~rnen~. Isobars were determined at six

401

402

MICHIKO

YOSHIHA~U and REXB.

different H, pressures. The H, pressures, measured with an MKS Baratron, were 1.01 x 10’ (@), 6.363 x 10’ (Q)), 1.616 x lo4 (@), 2.525 x lo4 (.), 5.686 x lo4 (@), and 1.01 x 10’ (0) N/m*. There is some evidence that Pd-Y solid solutions become ordered at low temperatures [4]. The order-disorder temperature is about 600 K. The equilibrations in this work were all in the range 625-1250 K so that it is reasonable to assume that the H atoms were introduced into an essentially disordered matrix.

MCLELLAN

-2 -3 -4 -5 -6

t

EXPERIMENTAL RESULTS The solubility isobars are shown in Figs 1 and 2 in the form of plots of ln[6,/1-0,)] vs 104/T. Each curve corresponds to a given H2 pressure and the symbol code is that given in the preceding section. The nominal Y concentration is indicated above each set of isobars. In addition to data for the Pd-Y solutions, Fig. 1 contains corresponding isobars measured for the Pd-H binary system (lowest set of curves). The adherence of the data to Sievert’s law is illustrated in Fig 3 which shows plots of the quantity OJ(l-0,) vs P’/~ (P is the H, gas pressure) at differing values of 0”. Three values of 6” have been chosen, and data at the lowest (625 K) and highest (1250 K) experimental temperatures plotted. The symbol code and differing scales are given in the upper part of the diagram. The partial molar enthalpies & and excess entropies sp were generated from the experimental data using a procedure outlined previously in detail [5]. The chemical potential pf of H atoms in the gas phase

6

1

I

9

I

IO

I

II

I

1

12 I3

1

I4

1

I3

I

I

12

13

I

I4

I

I3

where 4 is a known [5] constant and -Et the dissociation energy of the H, molecule per atom at

L

-0.-PdI3Y -..-- Pd-5Y --S--Pd

625K

t

1

I6

Fig. 1. Solubility isobars for the Pd-Y-H system. The different symbols refer to differing pressures (given in text). The nominal Y concentration is indicated above each set of

I_

I6

is given by [5]

PI’2 x10-a

104/T

.

1

II

Fig. 2. Solubility isobars for the W-Y-H system. The different symbols refer to differing pressures (given in text). The nominal Y concentration is indicated above each set of isobars.

25

6

b

IO

104/T

1230 K

-gc

I

9

(P in N/m21

3. Sieve& plots of 0,/( - OJ

.

.

.

is H

Thermodynamic study of the Palladium-Yttrium-Hydrogen

403

system

0 K. The energy zero for eqn (1) is the energy of an H atom at rest in a vacuum. The value of pf given by (1) is also equal to &(tIi& T), the corresponding potential of H atoms in the solid solution. The values of & are expanded in a power series in (l/T) and the R, and SF generated from the identities [1]

4 wn 1,

g, =

ah/T)

Sy = -($$),

(2)

+k ln(&j.

(3) I-

The only model-dependency in determining Hi and Sp is in the assumption that the H atoms dissolve interstitially in the octahedral sites of the f.c.c. Pd-Y lattice. The values of R,-Ef and Sp/k obtained are shown in Figs 4 and 5. Each curve in these diagrams represents the partial thermodynamic functions at each temperature at which the isobars were measured. The six temperatures span the range 625-1250 K. The values of Et, -Et and S:S are plotted at lixed values of Bi corresponding to the temperature as given in Table 1. The solid and dashed lines in Figs 5 and 6 are calculated curves which will be discussed later. The details of the the evolution of Ri and sy from the experimental data are given in Ref. [5]. Figures 6 and 7 show the variation of the quantities Ri and 37 with tIi and T. The variation of Ri with Bi at the various sets of values of 0. and T, clearly marked in Fig. 6, is small and, within experimental error, linear. For the sake of comparison, Fig. 6

626K

0

’

I

0

0.05

I

,

0.10

-I

0.15

8.

Fig. 5. The partial molar excess entropy of H in W-Y-H solutions as a function of 8.. the atom fraction of Y. The temperature is indicated on each curve. The lines are explained in the text.

contains a vertical marker of 0.01 eV length. The excess entropies are even less sensitive to Biwithin the range chosen, as is clearly seen in Fig. 7. Figures 6 and 7 provide the answer to one of the questions posed at the outset of the present study, i.e. to what extent do H-H interactions affect the partial thermodynamic functions? It is clear from Figs 6 and 7 that, within the range of (e,, fIi, T) encompassed in these representations, the effect of i concentration upon Ri and sp is small.

II t

9

....-.-

7. 1

. =Ir-:

IO

G5 E

>-2 AZ -3 00

**.:a-

1::;N

f

‘fQOI Pd.

1000

- PI-2Y.

lv

K

1000 K

Pd.;25

K

Pd-2Y.

625

K

-9 -10

i

-24

,

0

-23 0.05

0.10

0.16

v

9.

Fig. 4. The partial molar enthalpies of H in Pd-Y-H solutions as a function of O,, the atom fraction of Y. The temperature is indicated on each curve. The lines are explained in the text.

L 0

h

Fig. 6. The variation of the partial molar enthalpy of H in Pd-Y-H solutions at various selected temperatures and Y concentrations (indicated on each curve).

MICHIKOYOSHIHARAand

404

enables constant pairwise interactions to be used in the calculation of Rf despite the large dilation of the host lattice by Y atoms. The volume quantities pi, AV, and V, are, respectively, the partial molar volume of i (assumed to be independent of e,), the difference between the partial molar volumes of U and V (i.e. AV = i?,- RY), and the molar volume of the binary V-U system. The quantity B is the bulk modulus of the Pd-Y solutions. This has been measured [1] as a function of 0, and T and may be represented by the power series

3

tOOOK

Pd.

Pd-ZY,

1000 K

Pd-13Y,

1000

0 ” K

3,

Or\*%.

Y3 >_ lu)

n

0

o_..o

_

_

Pd. 623 n

n

PI-PY,

2-

K

625

REXB. MCLELLAN

K

3 =

so+ 6,@,+ 6*fq

(6)

in terms of the tem~rature-de~ndent coefficients 6,. The integral (6) can be evaluated in terms of the 6, in the form I= Fig. 7. The variation of the partial excess molar entropy of H in W-Y-H solutions at various selected temperatures and Y concentrations (indicated on each curve).

D~U~ION

OF RESULTS

Preliminary thermodynamic measurement on the Pd-Y-H system [2] indicated that the partial enthalpies !?, could be expressed by a simple relation of the form Ri =H~+R;+Ry.

(4)

The quantity a; is the value of Hi in the Pd-H binary system, Bf is the “statistical” part of &, i.e. the contribution arising from the interatomic pairwise interactions in the various configurational states of the system. The term @‘I is given by [1]

(5) and represents the “volume correction” corresponding to the calculation of & using the assumption that the specific volume of the V-U (i.e. Pd-Y) lattice is invariant to 6, (but not to 6J. This quasi-rigid lattice

Table 1. Representational Bivalues for Figs 4 and 5 Temp WI

1250 1111 1000 833 714 625

P 0) x 102

Wmol)

0.75 G

35.0 30.0 22.0

1:1 1.3 I.5

16.0

14.0 13.0

+;Ot+(s,--$)ln(l

+tfWJ],

(7)

where $ = AV/ Vz and Vz is the molar volume of V. The values of AV can be taken from the X-ray work of Harris and Norman [6]. (AV = 4.91 x 10-6m3/mol.) It should be noted that AV is large in comparison with all other Pd-U systems studied thus far [l]. Recent X-ray and densitometric studies have shown that the Pd-Y system is indeed substitutional in nature despite the large difference in ~old~hmidt radius f7] of the two metals. Now, to return to eqn (4). If Af is calculated from the cell model for quasi-rigid systems [8], it turns out that the experimental values of R, agree with eqn (4) to a remarkable degree when cp, the cell interaction energy, is zero. This simple state of affairs implies that in the constant-vol~e representation there is essentially no direct energetic difference between cells (interstitial sites) containing zero or 132,. . . Y atoms. This is illustrated in Fig. 8. In this diagram the symbols (a), taken from previous work [2] in which the variation of Ei -& with Bi was ignored, are experimental values of Bi. The dashed lines (---) are calculated values of @’ taken from eqn (7). It should be noted that at temperatures over lo3 K there is more deviation between $ -I?,” and Rye’. This may be due in part to the effect of Zener relaxations in the determination of E [9). Nevertheless, the implication is clear: in the quasi-rigid solution the interaction between H atoms and the Y species, relative to Pd, is essentially zero. The effect of the Y atoms is merely to dilate the host lattice. The present data, in the range of Bivalues depicted in Table 1, will be used to evaluate the cell model for this system in the light of a constant-pressure representation. In the cell model [I, IO,1 I] the central force assumption is used in such a manner as to

Thermodynamic study of the Pa~adj~-Ytt~~Hydrogen

where &(P,

system

T) is the cell partition

QzV, T ) =

405

function,

Tr@),

Z,,,,,= &,,L$ exp( - a;/kT),

(12) (13)

and Z!

D”=(Z__)lZl(l-8,)‘z-“‘e:. The quantity

Q;(P, T) is [l] Q;(P, T) = ‘I@‘),

-

(14)

(15)

calculatad

where &xi0

.Z,& = a,,,,$& Fig. 8. Measured values of the partial molar enthalpy of H in Pd-Y-H solutions. The (0) symbols were taken from previous measurements [2] in which Biwas not constant. The data sets (0) refer to Ri values at the constant 8, levels given in Table 1. The lines represent calculations discussed in the text.

describe the thermodynamics of the system in terms of the energy levels of i atoms in the (2 + 1) cells (interstitial sites) (Z is the number of host lattice atoms surrounding a given interstitial site in the f.c.c. structure). The (Z + 1) ~stinguishable cells may containn =O, I,..., Z&J atoms in the first co-ordination shell of Z host atoms. In the rigid lattice representation the spectrum of cell energies is given by e;=nep

+fye;,

(8)

where 6: is the energy required to insert an i atom from an appropriate ground state into a cell for which n=O and f;=2n(n-l)/(Z-1) is the ensemble average probability of finding a U-U nearest-neighbor pair in a cell of type n and ~Iis an additional energy con~mitant to the presence of such a cell. As pointed out already, the data are in accord with all my= 0 (i.e. Rf = 0). In the constantpressure representation, the cell energy spectrum is given by the enthalpy levels, 0; = nap +fla:+

pt?.,

(9)

where the term &, is not specific to a given cell type (n-value), but attempts to take into account the lattice dilation due to the U species. In contrast to eqn (4), the partial enthalpy is given by R.=RY+Ral’ f t t’ It is easy to show that [I]

exp( - o;/kT).

(16)

Equations (10)-(1(i) were used to calculate Rj - RY in terms of a!, aland p, The values of these parameters were stepped in units of 0.5 kJ/mol and the calculated enthalpies fitted to the data of Fig. 8 (0 symbols) using a least-squares regression. The data 0 refer to the constant values of t9, listed for each temperature in Table 1. The maximum value of 0, is 1.5 x 10e2. The eqns 10-16 are valid strictly in the case when 0, + 0, i.e. the limit of infinite dilution. The solid lines in Fig. 8 are lines calculated from the cell model using the values of a:, a;, and p generated by the regression. All values of a: were zero, all values of ai were -8.25 kJ/mol, and the p values were temperature dependent. They are given in Table 1. In view of the small values of ~9 it is hardly surprising that all a,: turn out to be zero. As a comparison, the energy E, + a: required to transfer an H atom from the vacuum rest state outside the crystal into an O-cell (n = 0) is N -220 U/m01 (i.e. -2.2 eV). As Fig. 8 shows, the agreement between experiment and theory is remarkably good. The partial entropies at constant t?, are shown in Fig. 5. The solid lines in Fig. 5 are calculated from the cell model using the values of the parameters c$, a’, and p extracted from the iTti data. The degree of agreement is satisfyingly good since there is no constraint that R, and S:S must be represented by the same set of fitted parameters. They are of course related by the identity

(17) The partial excess entropies are calculated from the values of as, a’, and p derived above by using the relation [l]

MICHIKO YOSHIHARA and REX B. MCLELLAN

406

Note that eqn (18) is the partial excess calculated by subtracting the ideal partial entropy, - k In Bi, from the partial entropy. Details of the derivation of eqns (1 l-18) are given in the recent review [l]. As can be seen in Fig. 5, the agreement between the entropy data (0 symbols) and the calculated partial excess entropies is satisfactory. CONCLUDING

REMARKS

One of the motivations of this work was to study the H,-(Pd-Y-H) equilibrium at low values of H, pressure and thus obtain thermodynamic data for the Pd-Y-H system at Bi values approaching the level of infinite dilation but spanning large ranges of 8, and temperature. This goal has been achieved and it has been shown that the experimental data are in excellent accord with the cell model using the constantpressure formulation. This representation is in some senses superior since it corresponds to the physical reality of measuring Hi and Sp at constant pressure and allowing the volume to expand as 0. is increased. However, the constant-volume representation and its concomitant values of C: are fundamentally important since the conclusions reached are more easily verifiable in terms of quantum mechanical calculations in which the specific volume of the lattice is held constant. The non-zero values of 0: in the P presentation are clearly a result of the local nature of the lattice distortion in the neighborhood of cells for which n > 0. This local distortion is superposed on the uniform dilation of the crystal. The V representation may be used to calculate the diffusivity DVmUm’ of H in the Pd-Y-H system. It is easy to show in this representation that for 0. < 1 the diffusivity ratio I = D “-“~‘/ D y-i (with obvious notation) is given by [I] 1 = Ldi’[Qz(V, T)-‘[rl

+ (1 - q)exp(-Q’lkT)],

(19)

where Q,( V, T) is the constant-volume cell partition function defined by (12-14) but replacing al by 6: and Q ’ is a barrier matrix energy which essentially reflects the perturbation of the saddle point energy due to the presence of the U atoms at constant specific volume [I]. The quantity 9 I/*= (1 - 0,)’ is the ensemble average probability that a random jump from an O-cell (n = 0) will be directed toward a neighboring O-cell. The factor Idi’ takes the uniform dilation of the lattice into account and is required since D is of course measured at constant pressure and the cell activation energies are functions of specific volume. The factor idI’ is easy to calculate from X-ray lattice parameter measurements [6] and the value of V”, the activation volume for H diffusion in the Pd lattice. This activation volume has been estimated from measurements of the diffusivity of H in Pd-Ag-H solutions [12]. The calculation of Adi’ has recently been carried out [13] and the results shown by the uppermost solid line in Fig. 9.

Fig. 9. Measured (0) values of the diffusivity ratio of H in Pd-Y-H solutions. The straight solid and dashed lines represent calculations explained in the text.

Now the results of the thermodynamic studies show that Q,(V, T) z 1 so that for Q’ >>kTeqn (19) assumes the simple form A = ~dii(l -

ey.

(20)

The quantity 22 ln(1 - 0,) is shown by the lowest solid line in Fig. 9 and the resultant value of In 1 by the central dashed line in Fig. 9. The calculated diffusivity is in good accord with the recent measurements of Sakamoto et al. [14]. Previous measurements due to Ishikawa and McLellan exhibited a more rapid decrease in D v-u-i as 19. increases [ 151. A final remark is in order. A primary reason for carrying out careful thermodynamic measurements on solid solutions is to produce information which may be compared with fundamental calculations. Because of the simple nature of the results found for a system involving very large volume changes, the Pd-Y-H system would seem to be an ideal system for which quantum mechanical calculations should be carried out. authors are grateful for support provided by the National Science Foundation under the Metallurgy Program (Grant No. DMR78-01306) and to the Robert A. Welch Foundation.

Acknowledgements-The

REFERENCES 1. McLellan R. B. and Yoshihara M., Acra Metall. 35, 197 (1987). 2. Yoshihara M. and McLellan R. B., Acra Metall. 36,385 (1988).

Thermodynamic study of the Palladium-Yttrium-Hydrogen 3. Daw M. S. and Baskes M. I., Phys. Rev. B. 29, 6443 (1984). 4. Hughes D. T., Evans J. and Harris I. R., J. LessCommon Metals 14, 255 (1980). 5. Labes C. and McLellan R. B., Acta Metall. 26, 893 (1978). 6. Harris I. R. and Norman M., J. Less-Common Metals 15, 285 (1968). 7. Yoshihara M., Pharr G. M. and McLellan R. B., Scripra MeraIl. 21, 393 (1987). 8. McLellan R. B. and Kirchheim R., J. Phys. Chem. Soli& 42, 157 (1981).

system

407

9. Yoshihara M., McLellan R. B. and Brotzen F. R., Acra Melall. 35, 775 (1987). 10. McClellan R. B., Acra Metall. 30, 317 (1982). 11. Yoshihara M. and McLellan R. B., J. P/w. Chem. Solidr 42, 767 (1981). 12. Ishikawa T. and McLellan R. B.. Acta MeraN. 34. 1825 (1986). 13. McLellan R. B., Scripta Metall. 21, 981 (1987). 14. Sakamoto Y.. Kaneko H.. Tsukahara T. and Hirata S.. Scripla Metail. 21, 415 (i987). 15. Ishikawa T. and McLellan R. B., Acta MeraN. 35, 787 (1987).