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Thermodynamic properties of the Pd-H system
Journal
of the Less-Common
THERMODYNAMIC
Metals,
88 (1982)
PROPERTIES
411
- 424
OF THE Pd-H
411
SYSTEM*
W. A. OATES Department of Metallurgy, 2308 (Australia) (Received
June
The
University
of Newcastle,
Shortland,
New
South
Wales
1, 1982)
Summary A satisfactory interpretation of the thermodynamic properties of Pd-H alloys must account for the configurational entropy contribution to the excess chemical potential pnE of hydrogen at ambient temperatures and for the known superlattice structures at low temperatures. The development of our understanding of the variation in pnE with hydrogen concentration is reviewed. Models based on nearest-neighbour central pairwise H-H interactions are inadequate but it is possible to interpret the essential features of the system using a near-neighbour model in which the pair interaction energies vary with hydrogen concentration. However, a lattice simulation computation demonstrates the presence of substantial many-body effects in the elastic H-H interactions. Further the composition variation of the volume energy and structure-independent electronic contributions to /.~n”are not known. It seems probable, therefore, that progress towards a thorough quantitative understanding of the thermodynamic properties will be slow.
1. Introduction Both from an experimental and a theoretical standpoint, the Pd-H system is the most extensively investigated metal-hydrogen system [ 11. Many of these investigations have been directed towards measuring and interpreting the thermodynamic properties of the alloys and we would have hoped, by now, for a substantial understanding of this prototype of other metal hydrides. Indeed, there have been periods during the last 50 years when it has seemed that a satisfactory understanding of the thermodynamic properties was imminent only, however, for these periods of optimism to be followed by an awareness of the inadequacy of earlier interpretations.
*Paper presented at the International of Metal Hydrides, Toba, Japan, May 30
Symposium
on the Properties
and Applications
- June 4,1982. 0 Elsevier
Sequoia/Printed
in The Netherlands
412
In this paper the progress present position is summarized. 2. The experimental
in our understanding
is reviewed
and the
data
A substantial amount of Pu, --r-T data (r = [H] /[Pd]) showing (see Fig. 1) the familiar miscibility gap and sharply rising isotherms were available for both Pd-H and Pd-D systems by the mid-1930s [l].In later investigations emphasis has been placed on obtaining more accurate results from which reliable thermodynamic properties could be obtained. The equation relating the external hydrogen pressure pu, and the solubility, expressed as fraction 0 of the available sites occupied, to thermodynamic quantities can be written as follows [ 31 :
(1) where j,&* is the chemical potential of hydrogen dissolved in a hypothetical infinitely dilute SOhtiOn and & E is the excess chemical potential of hydrogen which is defined as the difference between the actual and the ideal chemiCd potentials Of hydrogen. pn E is zero in the infinitely dilute solution. Both ,.&* and ,+” are functions of temperature (and hydrostatic pressure) and can be split into their enthalpic, entropic and heat capacity contributions &, ,!& and cu.
300K
-2. 0
0.1
0.2
0.3
0.4
1.5
0.6
0.7
h@dl
Fig. 1. Pressure-composition
isotherms for the Pd-H system (after ref. 2).
413
k’l/bi Fig. 2. Partial excess thermodynamic properties of hydrogen in the Pd-H system in the temperature range 0 - 300 “C (after ref. 2).
Figure 2 summarizes the recent results of Kuji et al. [2] which were based on isotherm, calorimetric and phase boundary measurements in the temperature range 0 - 300 “C. Related investigations by Picard et al. [4] and Wicke and Blaurock [5] give slightly different results but all are in agreement that, at r values up to 0.5, there are considerable negative contributions to SHE. Kuji et al. have made estimates of the magnitude of the various possible contributions to Sn”. Those arising from the optical and acoustic mode vibrations, from electronic excitations and from image interactions were estimated to increase steadily in total from zero at r = 0 to about 8 J K-’ (mol H)-’ at r = 1. It follows that the large negative contributions to Sn” at low r must be configurational in origin. This receives confirmatory support from the sudden change from a negative to a positive value at large r as shown in Fig. 2. Since the integral excess configurational entropy must be zero at r = 1, i.e. /$SnE(config) dr = 0, the negative area swept out by SnE at low r must be compensated by a positive area at large r. Neutron scattering studies have also made important contributions to our understanding of Pd-H thermodynamics. Diffraction [6], diffuse scattering [ 71 and inelastic scattering [ 8,9] results have assisted in providing information about the type of site occupied, the existence of short- and long-range order and the vibrational partition function. The neutron work, supplemented by other techniques [lo] , has been particularly successful in developing our understanding of the very low temperature region of the phase diagram at high r [ll] . Concentration-dependent superlattice configurations have been found [ 121 . The superlattices on the interstitial sublattice do not have cubic symmetry but exist as coherent domains within the
414
parent metal lattice. On heating above the transition temperature region short-range order remains and is known to persist at room temperature [ 131, in agreement with the configurational interpretation of Sn”. These new results on Sn” and the occurrence of the low temperature superlattices must be taken into account in any satisfactory interpretation of the thermodynamic properties of the Pd-H system.
3. Nearest-neighbour
pairwise models
In two classic papers, Lather [ 14, 151 applied the Fowler model [16] of localized adsorption to interpreting the pH,--r-T results for the Pd-H(D) systems. The model considers interactions between the dissolved hydrogen atoms only, and when a constant nearest-neighbour pairwise interaction energy is assumed then, in the zeroth approximation, it is found that
(2) where WHH is the interaction energy coefficient. Combination of eqns. (1) and (2) gives the p,,-8-T relation once &* and WHH are fixed. Conversion to pn -r-T requires a further relation between 6 and r. Lather assumed 0 = r/Ok9 whereas it is clear from Fig. 1 that values of r well in excess of 0.59 are attainable and from Fig. 2 that &.r” is not a linear function of r. Wagner [ 171 appears to have been the first to appreciate that pnE is not a linearly decreasing function of r at low r and also that it subsequently increases rather rapidly at r 2 0.6 in the manner shown in Fig. 2. These factors are responsible for the displacement of the critical composition r, from 0.5 to the observed lower value and for the asymmetry of the isotherms about r C’
Fowler and Smithells [ 181 had shown, using the protonic drogen atom dissolution, that pH*
=
pH+ *
+/-c
model for hy(3)
where the subscripts H, H’ and e refer to hydrogen atoms, protons and electrons respectively. By combining this with the rigid band explanation proposed by Mott and Jones [ 191 for paramagnetic susceptibility changes in Pd-Ag and Pd-H alloys, Wagner was able to explain the manner in which peE changes with hydrogen concentration (very small increases at low r when the high density of states (DOS) 4d band is being filled but rapid increases at r 2 0.6 when the low DOS 5s band is being filled). When Wagner combined this with an assumed linear decrease in &r+E the desired variation in pHE with r could be explained. Wagner [20] also showed how the same explanation could be used to interpret the solubility of hydrogen in palladium-based alloys. Later Simons and Flanagan [ 211 and Brodowsky [22] adopted the Wagner model and used it in a more quantitative manner. Brodowsky combined the constant pairwise quasi-chemical approximation for pH+E with peE
415
obtained by difference from pnE(exp) --- /~n+~, and considered the values obtained for peE to be in good agreement with those expected from the application of the rigid band model to physical property results. The protonic interaction was identified as being a short-range elastic interaction. In a series of subsequent papers Brodowsky and coworkers [ 23, 241 used the same approach for palladium-based alloys, and this work has been well summarized in a recent review by Wicke and Brodowsky [ 251. Recently, Tanaka [ 261 has represented this same model in the pairwise quasi-chemical approximation. Some weaknesses of the Wagner-Brodowsky nearest-neighbour model are listed below. (1) In the zeroth approximation the calculated phase diagram at high r is inconsistent with the third law of thermodynamics, i.e. the presence of random solid solutions at 0 K. In the quasi-chemical approximation a phase diagram of the wrong topology is predicted [ 271. An exact calculation by Monte Carlo methods [28] also predicts a phase diagram in disagreement with experiment, i.e. the wrong superlattice compositions and structures. It is only by taking into account more distant neighbours that these problems can be overcome. (2) The interpretation of the phonon dispersion curves also requires that interactions between more distant neighbours be taken into account
[291. (3) Although the interaction energy is envisaged as elastic, any changes in relaxations and hence interactions as the lattice becomes more fully occupied are ignored. (4) The influence of lattice dilatation is ignored. (5) Modern theory [ 30, 311 would regard the idea of equating /.~n+~ with the H-H interaction and pf with the Fermi energy change, calculated on the basis of the rigid band model, as being too facile. The situation is more complex than simply splitting /.~nE into a protonic part and an electronic part, as in eqn. (3), since the two are intertwined. (6) The model does not give a satisfactory explanation of certain features of hydrogen solubility in palladium-based alloys. These features can, however, be explained by a central atoms model [ 321. Some of these points are taken up in Sections 5 and 6. 4. Some other explanations
of in”
Several workers have suggested that the lattice dilatation and/or changes in electronic structure, which take place on hydrogen absorption, will influence all three types of pairwise interactions involved. In a wholly pairwise scheme we can write H,k = HPdPdE + H,,nE + H& The subscripts refer to the contributions pairwise interaction.
(4) to Hn
from the different
types of
416
Harasima et al. [33] were the first to suggest that there may be contributions to the configurational H n” other than those arising from constant pairwise H-H interactions. They modified the Lather model by assuming that the H-H interaction energy decreases slightly (becomes more attractive) whilst HPdPdE + HpdHE increases ’ markedly with increasing r owing to the lattice expansion. The effect of dilatation on pairwise interactions has also been considered more recently by Rudman [34] and Machlin [35]. Rudman considered nearest-neighbours only and assumed HHHE to be linear in r with the H-H interaction being repulsive. A force balance relates the variation in H P~IW~ By considering all potential-distance and HpdHEwith composition. curves to be parabolic and the Pd-H and Pd-Pd spring constants to be equal, Rudman obtained H
E
H
_- Art2 +r) _
l+r
+Br
where the first term is the net Pd-H, Pd-Pd contribution and the second term is the repulsive H-H contribution. Machlin [ 351 used a 4-8 Mie potential and considered interactions out to distant neighbours. From the empirically constructed potentials for the three types of pairs he concluded that the attractive (cf. Rudman [ 341) H-H interaction becomes more so as r increases and that there is a decrease in the Pd-H repulsion which is approximately offset by an increase in the Pd-Pd potential as lattice dilatation proceeds. Machlin was concerned only with explaining the initial decrease in HHE at low r, whereas Rudman aimed to explain the decrease followed by the increase at high r. Ebisuzaki and O’Keeffe [36] proposed that the initial decrease is due to the change in the proton screening energy with composition. By assuming the interchangeability of silver and hydrogen atoms, they were able to obtain the effect of increasing hydrogen concentration on HHE from the variation in the heat of solution of hydrogen in dilute solution in Pd-Ag alloys. It seems more likely that the observed variation in mH* with the silver concentration at low silver concentrations is due principally to a lattice expansion effect [ 321 , making the assumed interchangeability of silver and hydrogen atoms rather questionable. McLellan [37] proposed that the increase in pn” at high r is due to a change from octahedral to tetrahedral site occupancy. With the differences in the site energies proposed by McLellan there would be equal occupancy of the two kinds of site at r = 1, which is a rather unlikely possibility.
5. Near-neighbour
pairwise models
An important step in the understanding of the contributions to c(n” derives from Alefeld’s appreciation [38] of the relevance of Eshelby’s work
417
[39] on the theory of defects in elastic continua. The volume energy (or image infraction energy) ~nE(vol) was shown to be a major cont~butor to to pHE of the dilatathe decrease in @nE at low r. ~~~(~01) is the contribution tional effect on the Pd-Pd and Pd-H interactions. It therefore plays no part in determining hydrogen atom correlations on the interstitial sublattice. Its effect in the pairwise scheme is to modify eqn. (4) to ElnE = (/&ME
+ lUeanE + p,uE)(short
range) + ~.1us(vol)
If a dilute solution of isotropic defects in an isotropic assumed the volume energy is found to be [ 381 /Q(vol)
(6) continuum
is
y’BVH2r
= - ~-
V;. where y’ = 1 -B/Cil, B is the bulk modulus, C,, is the elastic constant, V, is the partial molar volume of hydrogen and V, is the molar volume of palladium. For the Pd-H system at low r this gives a contribution of approximately -12r kJ (mol H)-’ at room temperature. Subsequently, Homer and Wagner [40] used the lattice theory of defects [41] to give a clearer interpretation of local relaxation and dilatation effects in alloys with a free surface. In the general case the volume energy contribution to hnE becomes
where Wab is the strain-induced pairwise interaction energy between two hydrogen atoms at positions a and b. The superscripts indicate samples with and without a free surface. E.abWa$Teeis known exactly [40] and EobWabm is obtained from the short-range interaction model. On adding the effect of the direct chemical H-H interactions to the short-ranged strain-induced interactions the final equation of the Ho~e~Wa~er model for pnE is llH E =
E.cuE(short-range elastic + chemical)
+ c (Wabfree - Wabrn) r t I ab
(9)
Within the framework of the model this equation correctly separates the direct H-H interactions from the other contributions to pnE_ The limitations of the model are discussed in Section 6. Dietrich and Wagner 1421 calculated IYabrnfor the Pd-H system from the static lattice Green function for pure patladium, which was obtained from a Born-von Karman fit to the measured phonon frequencies, and from the strength of the force dipole tensor, which was obtained from I’,. Wabm was calculated out as far as the eleventh nearest neighbour; they found the nearest-neighbour interaction energy Vl to be negative and the next-nearestneighbour interactions V2 to be repulsive and stronger. As the separation dis-
418
tance increases, the strength of the interaction decreases with the tail oscillating between positive and negative values. Two calculations of the short-range chemical interactions between hydrogen atoms in palladium have been reported [42 - 441. Dietrich and Wagner [42] used a model based on linear screening in a simple metal. However, it is known that this model is inappropriate for protons in transition metals [45] so that their results are probably unreliable. They did find that V2 was greater than Vl, however, in agreement with a more sophisticated calculation by Demangeat and coworkers [43, 441. The latter found that the repulsive interaction V2 was exactly twice as large as Vl but the magnitudes of their calculated interaction energies (0.19 and 0.38 eV) seem impossibly high, It should also be mentioned that both these calculations refer to an isolated pair of defects in the infinitely dilute solution and the results would not be expected to be relevant to more concentrated solutions. Numerous theoretical methods are available which yield approximate solutions to the Ising model. In recent years the cluster variation method has been used very successfully for alloy phase diagram and structure calculations [46] . An alternative approach is to carry out a Monte Carlo simulation, a procedure which was first used for the calculation of metal-interstitial solution thermodynamic properties in 1969 [47], and which recently has been used for the Pd-H system [9,42,48]. Dietrich and Wagner attempted to carry out a fully quantitative calculation of the thermodynamic properties using the elastic and chemical interaction energies referred to above. The agreement with the experimental isotherms and phase diagram was only fair and, as they point out, one of the major weaknesses was the symmetry of the phase diagram about the r = 0.5 composition in disagreement with experiment. This matter is discussed more fully below. Bond and Ross [ll] had a different aim from that of Dietrich and Wagner in their Monte Carlo calculations for the system. Their major concern was to make appropriate choices of the interaction energies which would stabilize the desired ground state structures at high r. By choosing V2/ Vl = l/4 and ignoring more distant neighbour interactions they were able to calculate a phase diagram at high r which contains ordered phases with the structures I4,/amd at r = 0.5 and I4/mmm at r = 0.8. The Monte Carlo calculations of both Dietrich and Wagner and Bond and Ross give rise to symmetry of the phase diagram about the r = 0.5 composition. In the case of Dietrich and Wagner, where a substantial contribution from the volume energy is included, two o-o’ phase envelopes were calculated. In contrast, Bond and Ross ignored the volume energy contribution and as a result they found a symmetrical phase diagram with the transformations at low temperature. In order to have an o-o’ phase transition with rc = 0.29 and T, = 570 K together with phase transitions in the region of 55 K at high r it is necessary, if the pairwise scheme is to be maintained, to remove the assumption of composition-independent interaction energies. Hasebe and Oates [48] have recently carried out a Monte Carlo study in which this has
419
Fig. 3. Schematic reproduction tem by composition-dependent
of the short-range contributions to /.JH~in the Pd-H sysinteractions of the first three neighbours (after ref. 48).
been done. Their results are best explained with reference to Fig. 3. The broken curve is an estimate of the short-range contributions to /.I~~, and the manner in which this can be reproduced by the Monte Carlo calculations by letting Vl, V2 and V3 vary with composition is also shown schematically. The magnitude of Cizi Vi where z is the ith shell coordination number determines the location of the baseline about which pnE is symmetric, and the relative values of Vl, V2 and V3 can be adjusted to obtain the desired ground state compositions and structures at selected values of r. The ground state requirements were taken from the work of Kanamori and Kakehashi [49]. It is possible to reproduce the essential features of Pd-H thermodynamics at all r by means of this purely phenomenological method involving compositiondependent interactions between the three nearest neighbours.
6. Discussion Although it is only phenomenological, there are several problems involved in applying the Homer-Wagner model to the Pd-H system in a quantitative manner as was attempted by Dietrich and Wagner [42] . The following assumptions are made concerning the short-range strain-induced elastic interaction: (a) in the evaluation of W,,- the Pd-Pd interactions are harmonic and the Pd-H interaction is linear; (b) the values of Wabm calculated for the infinitely dilute solution can be used in calculating ~~~(~01) and (ppdpdE + ppdnE + /.~un~)(short range) at all r; (c) the elastic interaction is truly pairwise.
420 TABLE 1 H-H elastic interaction energies for different Pd-H force models Neighbour
1 2 3 4
Interaction energy (meV) Model I
Model 2
Model 3
-10.93 +21.45 -2.83 +7.03
-24.8 +2.51 +3.04 +3.35
-37.21 +9.84 +0.15 +6.74
Model 1 is a linear force model with forces on the first nearest neighbour only. Model 2 is also linear but the forces act on the first and second nearest neighbours. Model 3 uses a Born-Mayer potential. TABLE 2 Many-bodied nature of the H-H elastic interaction energy calculated using model 1 Defect
NN pair NN triad NN tetrahedra Shared edge tetrahedra Shared corner tetrahedra -
Actual interaction energy (mev)
Pairwise calculated interaction energy
-10.93 -34.84 -3.71 -114.78 -172.68
-10.93 -32.67 -65.34 --82.39 -115.17
(mev)
NN, nearest neighbour.
Recently, Oates and Stoneham [50] have examined the validity of these assumptions. The Hades code [ 511 for the computer simulation of lattices was used. This code minimizes the energy of the lattice as a function of the atom displacements and has the great advantage over the analytical method that it is unnecessary to make the assumption (a) and that extended defects are readily examined. The major results from this investigation [ 501 can be summarized as follows. sensitive to the Pd-H interatomic potential (1) w&I= is extremely chosen. The important part of the potential can be estimated from the strength of the force dipole tensor and, if necessary, from the optical mode frequency. The sensitivity of Wabm to the chosen interatomic potential is illustrated in Table 1 where it can be seen that the magnitudes of Wabm for different Pd-H potentials are substantially different and that even the sign of the interaction energy may change. (2) Many-body effects come into play in determining the interaction energies of extended defects so that the pairwise assumption is inadequate. Some examples of this effect are given in Table 2 where the large difference between the interaction energy of the actual extended defect and that calculated on the basis of pairwise interactions alone can be seen.
421
The validity of using the infinitely dilute solution results for the elastic interactions at high hydrogen concentrations should also be considered. As the interstitial sublattice is filled the average local relaxation energy around a hydrogen atom must decrease because of the likelihood of there being other hydrogen atoms adjacent to the palladium atoms neighbouring a given hydrogen atom. At the PdH composition, for example, with all the sites occupied there will be a dilatation contribution only since local relaxations are not possible. The effect of this will be to make the relative contributions to psE from Web- and ~nE(vol) change markedly with composition, with the former decreasing and the latter increasing in importance as r increases. It should also be appreciated that the lattice expansion with increasing hydrogen concentration will also change the Pd-Pd and Pd-H force constants. This effect will also influence the magnitude of Wabm and /..+E(vol). Thus whilst it is possible in a phenomenological pairwise scheme to obtain a satisfactory reproduction of the pHE versus r curve and the phase diagram at low and high r, any pairwise model is inadequate and the calculation of the significant ~ont~bution from pnn(vol) as a function of r with any degree of accuracy is difficult. The principal value of these Monte Carlo studies would seem to be in the semiquantitative interpretation of the thermodynamic and diffuse scattering results. They would, of course, be more valuable if it were possible to combine a Hades-type calculation with a Monte Carlo calculation so that all the interactions were taken into account without the necessity of having to make the pairwise assumption about the strain induced H-H interactions. First-principle calculations also point up the weakness of the phenomenological models in showing the presence of configuration-independent contributions to Hus. The energy of solution for a proton and an electron dissolving in a metal can be expressed in terms of the changes in kinetic, electrostatic and exchange correlation energies. Sholl and Smith [31, 451 have attempted to calculate these changes for hydrogen in palladium, but the calculation is exceedingly difficult to perform with the desired accuracy. In the present context, however, more interest centres on how the structure changes when hydrogen is added to palladium and, adapting an equation of theirs for the present purpose, we can write Hn* (band structure)
= 2ELBE + ndEdE + AEF
(10)
where ELBE and EdE are the changes in the average lower band and d band energies respectively, AEF is the absolute change in the Fermi energy relative to a vacuum and nd is the number of electrons in the d band. The lower band, which is associated with the formation of a Pd-H bond, decreases in energy whereas the unhybridized d state is not markedly affected as the hydrogen concentration increases. It is not known how AE, changes, as the change in the work function is inappropriate because of the contribution from the surface dipole layer. Gelatt et al. [ 301 do not provide any information on the electrostatic and exchange correlation contributions
422
to HHE and this may be an important omission since the band contribution may be approximately cancelled by the electrostatic contribution [52]. There is also no indication as to which parts of the band structure energy might be described in pairwise terms. Pseudopotential theory may ultimately provide a better link between ab initio calculations of the energy and the expression of this energy in a form suitable for statistical modelling. Although there are problems in applying the theory to tr~sition~ metals these do not appear to be insuperable [ 531. Katsnelson et at. [ 54 ] have reviewed the application of pseudopotential theory to the caiculation of short-range order parameters in substitutional alloys, including transition metal alloys, and claim some success. However, the volume changes associated with the introduction of hydrogen into palladium mean that the configurational potentials will change with composition and the question of local relaxations would also have to be allowed for. The elastic calculations on extended defects [ 501 seem to indicate that it may never be possible to express the structure-dependent energy in a pairwise form.
7. Conclusions We seem to be at a difficult stage in our progress towards a complete understanding of the thermodynamic properties of Pd-H alloys. We can give a satisfactory “explanation” of the major features at all compositions by a phenomenological pairwise model but only if we take into account more than nearest neighbours and if we allow the interaction energies to vary with composition. Going beyond this point will be very difficult. Ab initio calculations of the lattice energy which include predicted local lattice relaxations and volume changes at all hydrogen concentrations appear some way off, although a start has been made in this direction [ 551. If this stage is reached then it will still be necessary to express parts of this energy in a pairwise or cluster fashion in order that the configurational partition function can be calculated, although expressing the energy in this manner will probably involve major approximations. This same dilemma pertains to the whole of alloy thermodyn~ics. It seems that for some time we shall have to remain satisfied with semiquantitative interpretations using phenomenological models whilst continuing to learn more about the limitations of those models.
Acknowledgment The author is grateful financial support.
to the Australian
Research
Grants Scheme
for
423
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of the Less-Common
THERMODYNAMIC
Metals,
88 (1982)
PROPERTIES
411
- 424
OF THE Pd-H
411
SYSTEM*
W. A. OATES Department of Metallurgy, 2308 (Australia) (Received
June
The
University
of Newcastle,
Shortland,
New
South
Wales
1, 1982)
Summary A satisfactory interpretation of the thermodynamic properties of Pd-H alloys must account for the configurational entropy contribution to the excess chemical potential pnE of hydrogen at ambient temperatures and for the known superlattice structures at low temperatures. The development of our understanding of the variation in pnE with hydrogen concentration is reviewed. Models based on nearest-neighbour central pairwise H-H interactions are inadequate but it is possible to interpret the essential features of the system using a near-neighbour model in which the pair interaction energies vary with hydrogen concentration. However, a lattice simulation computation demonstrates the presence of substantial many-body effects in the elastic H-H interactions. Further the composition variation of the volume energy and structure-independent electronic contributions to /.~n”are not known. It seems probable, therefore, that progress towards a thorough quantitative understanding of the thermodynamic properties will be slow.
1. Introduction Both from an experimental and a theoretical standpoint, the Pd-H system is the most extensively investigated metal-hydrogen system [ 11. Many of these investigations have been directed towards measuring and interpreting the thermodynamic properties of the alloys and we would have hoped, by now, for a substantial understanding of this prototype of other metal hydrides. Indeed, there have been periods during the last 50 years when it has seemed that a satisfactory understanding of the thermodynamic properties was imminent only, however, for these periods of optimism to be followed by an awareness of the inadequacy of earlier interpretations.
*Paper presented at the International of Metal Hydrides, Toba, Japan, May 30
Symposium
on the Properties
and Applications
- June 4,1982. 0 Elsevier
Sequoia/Printed
in The Netherlands
412
In this paper the progress present position is summarized. 2. The experimental
in our understanding
is reviewed
and the
data
A substantial amount of Pu, --r-T data (r = [H] /[Pd]) showing (see Fig. 1) the familiar miscibility gap and sharply rising isotherms were available for both Pd-H and Pd-D systems by the mid-1930s [l].In later investigations emphasis has been placed on obtaining more accurate results from which reliable thermodynamic properties could be obtained. The equation relating the external hydrogen pressure pu, and the solubility, expressed as fraction 0 of the available sites occupied, to thermodynamic quantities can be written as follows [ 31 :
(1) where j,&* is the chemical potential of hydrogen dissolved in a hypothetical infinitely dilute SOhtiOn and & E is the excess chemical potential of hydrogen which is defined as the difference between the actual and the ideal chemiCd potentials Of hydrogen. pn E is zero in the infinitely dilute solution. Both ,.&* and ,+” are functions of temperature (and hydrostatic pressure) and can be split into their enthalpic, entropic and heat capacity contributions &, ,!& and cu.
300K
-2. 0
0.1
0.2
0.3
0.4
1.5
0.6
0.7
h@dl
Fig. 1. Pressure-composition
isotherms for the Pd-H system (after ref. 2).
413
k’l/bi Fig. 2. Partial excess thermodynamic properties of hydrogen in the Pd-H system in the temperature range 0 - 300 “C (after ref. 2).
Figure 2 summarizes the recent results of Kuji et al. [2] which were based on isotherm, calorimetric and phase boundary measurements in the temperature range 0 - 300 “C. Related investigations by Picard et al. [4] and Wicke and Blaurock [5] give slightly different results but all are in agreement that, at r values up to 0.5, there are considerable negative contributions to SHE. Kuji et al. have made estimates of the magnitude of the various possible contributions to Sn”. Those arising from the optical and acoustic mode vibrations, from electronic excitations and from image interactions were estimated to increase steadily in total from zero at r = 0 to about 8 J K-’ (mol H)-’ at r = 1. It follows that the large negative contributions to Sn” at low r must be configurational in origin. This receives confirmatory support from the sudden change from a negative to a positive value at large r as shown in Fig. 2. Since the integral excess configurational entropy must be zero at r = 1, i.e. /$SnE(config) dr = 0, the negative area swept out by SnE at low r must be compensated by a positive area at large r. Neutron scattering studies have also made important contributions to our understanding of Pd-H thermodynamics. Diffraction [6], diffuse scattering [ 71 and inelastic scattering [ 8,9] results have assisted in providing information about the type of site occupied, the existence of short- and long-range order and the vibrational partition function. The neutron work, supplemented by other techniques [lo] , has been particularly successful in developing our understanding of the very low temperature region of the phase diagram at high r [ll] . Concentration-dependent superlattice configurations have been found [ 121 . The superlattices on the interstitial sublattice do not have cubic symmetry but exist as coherent domains within the
414
parent metal lattice. On heating above the transition temperature region short-range order remains and is known to persist at room temperature [ 131, in agreement with the configurational interpretation of Sn”. These new results on Sn” and the occurrence of the low temperature superlattices must be taken into account in any satisfactory interpretation of the thermodynamic properties of the Pd-H system.
3. Nearest-neighbour
pairwise models
In two classic papers, Lather [ 14, 151 applied the Fowler model [16] of localized adsorption to interpreting the pH,--r-T results for the Pd-H(D) systems. The model considers interactions between the dissolved hydrogen atoms only, and when a constant nearest-neighbour pairwise interaction energy is assumed then, in the zeroth approximation, it is found that
(2) where WHH is the interaction energy coefficient. Combination of eqns. (1) and (2) gives the p,,-8-T relation once &* and WHH are fixed. Conversion to pn -r-T requires a further relation between 6 and r. Lather assumed 0 = r/Ok9 whereas it is clear from Fig. 1 that values of r well in excess of 0.59 are attainable and from Fig. 2 that &.r” is not a linear function of r. Wagner [ 171 appears to have been the first to appreciate that pnE is not a linearly decreasing function of r at low r and also that it subsequently increases rather rapidly at r 2 0.6 in the manner shown in Fig. 2. These factors are responsible for the displacement of the critical composition r, from 0.5 to the observed lower value and for the asymmetry of the isotherms about r C’
Fowler and Smithells [ 181 had shown, using the protonic drogen atom dissolution, that pH*
=
pH+ *
+/-c
model for hy(3)
where the subscripts H, H’ and e refer to hydrogen atoms, protons and electrons respectively. By combining this with the rigid band explanation proposed by Mott and Jones [ 191 for paramagnetic susceptibility changes in Pd-Ag and Pd-H alloys, Wagner was able to explain the manner in which peE changes with hydrogen concentration (very small increases at low r when the high density of states (DOS) 4d band is being filled but rapid increases at r 2 0.6 when the low DOS 5s band is being filled). When Wagner combined this with an assumed linear decrease in &r+E the desired variation in pHE with r could be explained. Wagner [20] also showed how the same explanation could be used to interpret the solubility of hydrogen in palladium-based alloys. Later Simons and Flanagan [ 211 and Brodowsky [22] adopted the Wagner model and used it in a more quantitative manner. Brodowsky combined the constant pairwise quasi-chemical approximation for pH+E with peE
415
obtained by difference from pnE(exp) --- /~n+~, and considered the values obtained for peE to be in good agreement with those expected from the application of the rigid band model to physical property results. The protonic interaction was identified as being a short-range elastic interaction. In a series of subsequent papers Brodowsky and coworkers [ 23, 241 used the same approach for palladium-based alloys, and this work has been well summarized in a recent review by Wicke and Brodowsky [ 251. Recently, Tanaka [ 261 has represented this same model in the pairwise quasi-chemical approximation. Some weaknesses of the Wagner-Brodowsky nearest-neighbour model are listed below. (1) In the zeroth approximation the calculated phase diagram at high r is inconsistent with the third law of thermodynamics, i.e. the presence of random solid solutions at 0 K. In the quasi-chemical approximation a phase diagram of the wrong topology is predicted [ 271. An exact calculation by Monte Carlo methods [28] also predicts a phase diagram in disagreement with experiment, i.e. the wrong superlattice compositions and structures. It is only by taking into account more distant neighbours that these problems can be overcome. (2) The interpretation of the phonon dispersion curves also requires that interactions between more distant neighbours be taken into account
[291. (3) Although the interaction energy is envisaged as elastic, any changes in relaxations and hence interactions as the lattice becomes more fully occupied are ignored. (4) The influence of lattice dilatation is ignored. (5) Modern theory [ 30, 311 would regard the idea of equating /.~n+~ with the H-H interaction and pf with the Fermi energy change, calculated on the basis of the rigid band model, as being too facile. The situation is more complex than simply splitting /.~nE into a protonic part and an electronic part, as in eqn. (3), since the two are intertwined. (6) The model does not give a satisfactory explanation of certain features of hydrogen solubility in palladium-based alloys. These features can, however, be explained by a central atoms model [ 321. Some of these points are taken up in Sections 5 and 6. 4. Some other explanations
of in”
Several workers have suggested that the lattice dilatation and/or changes in electronic structure, which take place on hydrogen absorption, will influence all three types of pairwise interactions involved. In a wholly pairwise scheme we can write H,k = HPdPdE + H,,nE + H& The subscripts refer to the contributions pairwise interaction.
(4) to Hn
from the different
types of
416
Harasima et al. [33] were the first to suggest that there may be contributions to the configurational H n” other than those arising from constant pairwise H-H interactions. They modified the Lather model by assuming that the H-H interaction energy decreases slightly (becomes more attractive) whilst HPdPdE + HpdHE increases ’ markedly with increasing r owing to the lattice expansion. The effect of dilatation on pairwise interactions has also been considered more recently by Rudman [34] and Machlin [35]. Rudman considered nearest-neighbours only and assumed HHHE to be linear in r with the H-H interaction being repulsive. A force balance relates the variation in H P~IW~ By considering all potential-distance and HpdHEwith composition. curves to be parabolic and the Pd-H and Pd-Pd spring constants to be equal, Rudman obtained H
E
H
_- Art2 +r) _
l+r
+Br
where the first term is the net Pd-H, Pd-Pd contribution and the second term is the repulsive H-H contribution. Machlin [ 351 used a 4-8 Mie potential and considered interactions out to distant neighbours. From the empirically constructed potentials for the three types of pairs he concluded that the attractive (cf. Rudman [ 341) H-H interaction becomes more so as r increases and that there is a decrease in the Pd-H repulsion which is approximately offset by an increase in the Pd-Pd potential as lattice dilatation proceeds. Machlin was concerned only with explaining the initial decrease in HHE at low r, whereas Rudman aimed to explain the decrease followed by the increase at high r. Ebisuzaki and O’Keeffe [36] proposed that the initial decrease is due to the change in the proton screening energy with composition. By assuming the interchangeability of silver and hydrogen atoms, they were able to obtain the effect of increasing hydrogen concentration on HHE from the variation in the heat of solution of hydrogen in dilute solution in Pd-Ag alloys. It seems more likely that the observed variation in mH* with the silver concentration at low silver concentrations is due principally to a lattice expansion effect [ 321 , making the assumed interchangeability of silver and hydrogen atoms rather questionable. McLellan [37] proposed that the increase in pn” at high r is due to a change from octahedral to tetrahedral site occupancy. With the differences in the site energies proposed by McLellan there would be equal occupancy of the two kinds of site at r = 1, which is a rather unlikely possibility.
5. Near-neighbour
pairwise models
An important step in the understanding of the contributions to c(n” derives from Alefeld’s appreciation [38] of the relevance of Eshelby’s work
417
[39] on the theory of defects in elastic continua. The volume energy (or image infraction energy) ~nE(vol) was shown to be a major cont~butor to to pHE of the dilatathe decrease in @nE at low r. ~~~(~01) is the contribution tional effect on the Pd-Pd and Pd-H interactions. It therefore plays no part in determining hydrogen atom correlations on the interstitial sublattice. Its effect in the pairwise scheme is to modify eqn. (4) to ElnE = (/&ME
+ lUeanE + p,uE)(short
range) + ~.1us(vol)
If a dilute solution of isotropic defects in an isotropic assumed the volume energy is found to be [ 381 /Q(vol)
(6) continuum
is
y’BVH2r
= - ~-
V;. where y’ = 1 -B/Cil, B is the bulk modulus, C,, is the elastic constant, V, is the partial molar volume of hydrogen and V, is the molar volume of palladium. For the Pd-H system at low r this gives a contribution of approximately -12r kJ (mol H)-’ at room temperature. Subsequently, Homer and Wagner [40] used the lattice theory of defects [41] to give a clearer interpretation of local relaxation and dilatation effects in alloys with a free surface. In the general case the volume energy contribution to hnE becomes
where Wab is the strain-induced pairwise interaction energy between two hydrogen atoms at positions a and b. The superscripts indicate samples with and without a free surface. E.abWa$Teeis known exactly [40] and EobWabm is obtained from the short-range interaction model. On adding the effect of the direct chemical H-H interactions to the short-ranged strain-induced interactions the final equation of the Ho~e~Wa~er model for pnE is llH E =
E.cuE(short-range elastic + chemical)
+ c (Wabfree - Wabrn) r t I ab
(9)
Within the framework of the model this equation correctly separates the direct H-H interactions from the other contributions to pnE_ The limitations of the model are discussed in Section 6. Dietrich and Wagner 1421 calculated IYabrnfor the Pd-H system from the static lattice Green function for pure patladium, which was obtained from a Born-von Karman fit to the measured phonon frequencies, and from the strength of the force dipole tensor, which was obtained from I’,. Wabm was calculated out as far as the eleventh nearest neighbour; they found the nearest-neighbour interaction energy Vl to be negative and the next-nearestneighbour interactions V2 to be repulsive and stronger. As the separation dis-
418
tance increases, the strength of the interaction decreases with the tail oscillating between positive and negative values. Two calculations of the short-range chemical interactions between hydrogen atoms in palladium have been reported [42 - 441. Dietrich and Wagner [42] used a model based on linear screening in a simple metal. However, it is known that this model is inappropriate for protons in transition metals [45] so that their results are probably unreliable. They did find that V2 was greater than Vl, however, in agreement with a more sophisticated calculation by Demangeat and coworkers [43, 441. The latter found that the repulsive interaction V2 was exactly twice as large as Vl but the magnitudes of their calculated interaction energies (0.19 and 0.38 eV) seem impossibly high, It should also be mentioned that both these calculations refer to an isolated pair of defects in the infinitely dilute solution and the results would not be expected to be relevant to more concentrated solutions. Numerous theoretical methods are available which yield approximate solutions to the Ising model. In recent years the cluster variation method has been used very successfully for alloy phase diagram and structure calculations [46] . An alternative approach is to carry out a Monte Carlo simulation, a procedure which was first used for the calculation of metal-interstitial solution thermodynamic properties in 1969 [47], and which recently has been used for the Pd-H system [9,42,48]. Dietrich and Wagner attempted to carry out a fully quantitative calculation of the thermodynamic properties using the elastic and chemical interaction energies referred to above. The agreement with the experimental isotherms and phase diagram was only fair and, as they point out, one of the major weaknesses was the symmetry of the phase diagram about the r = 0.5 composition in disagreement with experiment. This matter is discussed more fully below. Bond and Ross [ll] had a different aim from that of Dietrich and Wagner in their Monte Carlo calculations for the system. Their major concern was to make appropriate choices of the interaction energies which would stabilize the desired ground state structures at high r. By choosing V2/ Vl = l/4 and ignoring more distant neighbour interactions they were able to calculate a phase diagram at high r which contains ordered phases with the structures I4,/amd at r = 0.5 and I4/mmm at r = 0.8. The Monte Carlo calculations of both Dietrich and Wagner and Bond and Ross give rise to symmetry of the phase diagram about the r = 0.5 composition. In the case of Dietrich and Wagner, where a substantial contribution from the volume energy is included, two o-o’ phase envelopes were calculated. In contrast, Bond and Ross ignored the volume energy contribution and as a result they found a symmetrical phase diagram with the transformations at low temperature. In order to have an o-o’ phase transition with rc = 0.29 and T, = 570 K together with phase transitions in the region of 55 K at high r it is necessary, if the pairwise scheme is to be maintained, to remove the assumption of composition-independent interaction energies. Hasebe and Oates [48] have recently carried out a Monte Carlo study in which this has
419
Fig. 3. Schematic reproduction tem by composition-dependent
of the short-range contributions to /.JH~in the Pd-H sysinteractions of the first three neighbours (after ref. 48).
been done. Their results are best explained with reference to Fig. 3. The broken curve is an estimate of the short-range contributions to /.I~~, and the manner in which this can be reproduced by the Monte Carlo calculations by letting Vl, V2 and V3 vary with composition is also shown schematically. The magnitude of Cizi Vi where z is the ith shell coordination number determines the location of the baseline about which pnE is symmetric, and the relative values of Vl, V2 and V3 can be adjusted to obtain the desired ground state compositions and structures at selected values of r. The ground state requirements were taken from the work of Kanamori and Kakehashi [49]. It is possible to reproduce the essential features of Pd-H thermodynamics at all r by means of this purely phenomenological method involving compositiondependent interactions between the three nearest neighbours.
6. Discussion Although it is only phenomenological, there are several problems involved in applying the Homer-Wagner model to the Pd-H system in a quantitative manner as was attempted by Dietrich and Wagner [42] . The following assumptions are made concerning the short-range strain-induced elastic interaction: (a) in the evaluation of W,,- the Pd-Pd interactions are harmonic and the Pd-H interaction is linear; (b) the values of Wabm calculated for the infinitely dilute solution can be used in calculating ~~~(~01) and (ppdpdE + ppdnE + /.~un~)(short range) at all r; (c) the elastic interaction is truly pairwise.
420 TABLE 1 H-H elastic interaction energies for different Pd-H force models Neighbour
1 2 3 4
Interaction energy (meV) Model I
Model 2
Model 3
-10.93 +21.45 -2.83 +7.03
-24.8 +2.51 +3.04 +3.35
-37.21 +9.84 +0.15 +6.74
Model 1 is a linear force model with forces on the first nearest neighbour only. Model 2 is also linear but the forces act on the first and second nearest neighbours. Model 3 uses a Born-Mayer potential. TABLE 2 Many-bodied nature of the H-H elastic interaction energy calculated using model 1 Defect
NN pair NN triad NN tetrahedra Shared edge tetrahedra Shared corner tetrahedra -
Actual interaction energy (mev)
Pairwise calculated interaction energy
-10.93 -34.84 -3.71 -114.78 -172.68
-10.93 -32.67 -65.34 --82.39 -115.17
(mev)
NN, nearest neighbour.
Recently, Oates and Stoneham [50] have examined the validity of these assumptions. The Hades code [ 511 for the computer simulation of lattices was used. This code minimizes the energy of the lattice as a function of the atom displacements and has the great advantage over the analytical method that it is unnecessary to make the assumption (a) and that extended defects are readily examined. The major results from this investigation [ 501 can be summarized as follows. sensitive to the Pd-H interatomic potential (1) w&I= is extremely chosen. The important part of the potential can be estimated from the strength of the force dipole tensor and, if necessary, from the optical mode frequency. The sensitivity of Wabm to the chosen interatomic potential is illustrated in Table 1 where it can be seen that the magnitudes of Wabm for different Pd-H potentials are substantially different and that even the sign of the interaction energy may change. (2) Many-body effects come into play in determining the interaction energies of extended defects so that the pairwise assumption is inadequate. Some examples of this effect are given in Table 2 where the large difference between the interaction energy of the actual extended defect and that calculated on the basis of pairwise interactions alone can be seen.
421
The validity of using the infinitely dilute solution results for the elastic interactions at high hydrogen concentrations should also be considered. As the interstitial sublattice is filled the average local relaxation energy around a hydrogen atom must decrease because of the likelihood of there being other hydrogen atoms adjacent to the palladium atoms neighbouring a given hydrogen atom. At the PdH composition, for example, with all the sites occupied there will be a dilatation contribution only since local relaxations are not possible. The effect of this will be to make the relative contributions to psE from Web- and ~nE(vol) change markedly with composition, with the former decreasing and the latter increasing in importance as r increases. It should also be appreciated that the lattice expansion with increasing hydrogen concentration will also change the Pd-Pd and Pd-H force constants. This effect will also influence the magnitude of Wabm and /..+E(vol). Thus whilst it is possible in a phenomenological pairwise scheme to obtain a satisfactory reproduction of the pHE versus r curve and the phase diagram at low and high r, any pairwise model is inadequate and the calculation of the significant ~ont~bution from pnn(vol) as a function of r with any degree of accuracy is difficult. The principal value of these Monte Carlo studies would seem to be in the semiquantitative interpretation of the thermodynamic and diffuse scattering results. They would, of course, be more valuable if it were possible to combine a Hades-type calculation with a Monte Carlo calculation so that all the interactions were taken into account without the necessity of having to make the pairwise assumption about the strain induced H-H interactions. First-principle calculations also point up the weakness of the phenomenological models in showing the presence of configuration-independent contributions to Hus. The energy of solution for a proton and an electron dissolving in a metal can be expressed in terms of the changes in kinetic, electrostatic and exchange correlation energies. Sholl and Smith [31, 451 have attempted to calculate these changes for hydrogen in palladium, but the calculation is exceedingly difficult to perform with the desired accuracy. In the present context, however, more interest centres on how the structure changes when hydrogen is added to palladium and, adapting an equation of theirs for the present purpose, we can write Hn* (band structure)
= 2ELBE + ndEdE + AEF
(10)
where ELBE and EdE are the changes in the average lower band and d band energies respectively, AEF is the absolute change in the Fermi energy relative to a vacuum and nd is the number of electrons in the d band. The lower band, which is associated with the formation of a Pd-H bond, decreases in energy whereas the unhybridized d state is not markedly affected as the hydrogen concentration increases. It is not known how AE, changes, as the change in the work function is inappropriate because of the contribution from the surface dipole layer. Gelatt et al. [ 301 do not provide any information on the electrostatic and exchange correlation contributions
422
to HHE and this may be an important omission since the band contribution may be approximately cancelled by the electrostatic contribution [52]. There is also no indication as to which parts of the band structure energy might be described in pairwise terms. Pseudopotential theory may ultimately provide a better link between ab initio calculations of the energy and the expression of this energy in a form suitable for statistical modelling. Although there are problems in applying the theory to tr~sition~ metals these do not appear to be insuperable [ 531. Katsnelson et at. [ 54 ] have reviewed the application of pseudopotential theory to the caiculation of short-range order parameters in substitutional alloys, including transition metal alloys, and claim some success. However, the volume changes associated with the introduction of hydrogen into palladium mean that the configurational potentials will change with composition and the question of local relaxations would also have to be allowed for. The elastic calculations on extended defects [ 501 seem to indicate that it may never be possible to express the structure-dependent energy in a pairwise form.
7. Conclusions We seem to be at a difficult stage in our progress towards a complete understanding of the thermodynamic properties of Pd-H alloys. We can give a satisfactory “explanation” of the major features at all compositions by a phenomenological pairwise model but only if we take into account more than nearest neighbours and if we allow the interaction energies to vary with composition. Going beyond this point will be very difficult. Ab initio calculations of the lattice energy which include predicted local lattice relaxations and volume changes at all hydrogen concentrations appear some way off, although a start has been made in this direction [ 551. If this stage is reached then it will still be necessary to express parts of this energy in a pairwise or cluster fashion in order that the configurational partition function can be calculated, although expressing the energy in this manner will probably involve major approximations. This same dilemma pertains to the whole of alloy thermodyn~ics. It seems that for some time we shall have to remain satisfied with semiquantitative interpretations using phenomenological models whilst continuing to learn more about the limitations of those models.
Acknowledgment The author is grateful financial support.
to the Australian
Research
Grants Scheme
for
423
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