Equilibrium vacancy concentration in Pd–H solid solutions

PII:

Acta mater. Vol. 46, No. 15, pp. 5593±5597, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(98)00213-4 1359-6454/98 $19.00 + 0.00

EQUILIBRIUM VACANCY CONCENTRATION IN Pd±H SOLID SOLUTIONS P. MAROEVIC and R. B. MCLELLAN{ Department of Mechanical Engineering and Materials Science, MS-321, William Marsh Rice University, Houston, TX 77005-1892, U.S.A. (Received 27 May 1997; accepted 17 February 1998) AbstractÐA simple framework has been developed for the estimation of the equilibrium concentration of vacancies in metal±hydrogen systems using the quantum results for the vacancy±hydrogen cluster energies. Vacancy formation obeys Fermi±Dirac statistics in assuming di€erent cluster con®gurations. The presence of hydrogen results in the equilibrium concentrations of vacancies that are many orders of magnitude higher than the thermodynamic vacancy concentration of the pure metal. This is in accordance with the observed changes in the kinetics of the host lattice and suggests possible implications to the mechanical properties. # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

The recent activity in the area of interstitial±vacancy interactions stems from the work of Fukai and Okuma [1], where lattice parameter measurements on Ni and Pd, which had been exposed to high H2-gas pressure (5 GPa), indicated that enormous vacancy concentrations had been generated. After the appearance of the works of Fukai and Okuma [1, 2] and Fukai [3] a series of studies reported observations of rapid di€usion of lattice atoms in Pd-based solid solutions containing H [4± 8]. This rapid di€usion (the term HILM, hydrogen induced lattice migration, is sometimes used) is clearly a consequence of the presence of large vacancy concentrations in the hydrogenated solution. Prior to the series of experimental observations described above, a statistical mechanical treatment of the general problem of interstitial±vacancy interactions had been given [9] which considered interstitial±vacancy clusters (U±VAC clusters) up to order z, where z is the number of interstitial sites nearestneighbor to a given vacancy. Less general treatments have been given previously [10, 11], in which clusters of order z = 2 were considered. The general solution [9] requires of course values for the U± VAC binding energy in order to obtain numerical estimates of the vacancy concentration. Such energies have been calculated by the e€ective medium technique [12] and have been used to calculate vacancy concentrations C1v in Pd [13, 14] and Fe [15]. Now the previous statistical mechanical calculations [9, 13, 14] utilized the Maxwell± {To whom all correspondence should be addressed.

Boltzmann (MB) approximation in the canonical ensemble in order to calculate the thermodynamic properties of the defect crystal. This technique is satisfactory provided that interstitial±interstitial (U±U) interactions may be ignored, and the temperature is suciently high that MB statistics represent an adequate approximation. However, it was estimated that at low temperatures the vacancy concentrations in the hydrogenated metals [13±15] can become so great that the induced mobility of the lattice atoms (HILM) could a€ect some mechanical properties of the solid [16]. Thus it is important to overcome the restrictions imposed by a MB treatment of the problem and to obtain a solution valid in the limit of low temperatures. The present report represents such a calculation in which Fermi±Dirac (FD) statistics is utilized concomitant to the degeneracy of the crystal sites pertinent to the set of cluster energy levels.

2. MODEL

If an initially perfect metal lattice is placed in contact with a hydrogen atmosphere, then due to the fast di€usion of hydrogen into the metal, chemical equilibrium with the gas phase would be established much faster than the equilibrium concentration of vacancies would be reached. We can therefore imagine a vacancy-free hydrogenated metal as a starting point for the process of vacancy di€usion into the lattice. The hydrogen atoms are assumed to be distributed randomly among the interstitial sites, which is a good approximation if we neglect the U±U interactions and ordered phases.

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MAROEVIC and MCLELLAN: Pd±H SOLID SOLUTIONS

As a model system, we have chosen Pd±H because of the large volume of data available and because its high temperature limit was solved previously [13]. In that f.c.c. system, H-atoms occupy almost exclusively octahedral interstitial sites. A f.c.c. lattice site is surrounded by six neighboring octahedral interstitial sites so that z = 6. A given metal atom together with its surrounding octahedral sites can be said to form a cluster composed of one lattice atom and n = 0, . . . , z nearest-neighbor Hatoms. Now we can associate the label n with a cluster and its energy. Let us also introduce the terms atom-cluster and vacancy-cluster to distinguish between the metal atom and its associated H-atoms, and a vacancy with its associated H-atoms (decorated vacancy). The random distribution of hydrogens in the interstitial sublattice of the vacancy-free host implies the following probability for a given lattice atom to be surrounded with n H-atoms, i.e. the atom-cluster of type n:     z! nH n nH zÿn pn ˆ

1ÿ : …1† …z ÿ n†!n! b
N b
N Here N is the number of lattice sites, and b the ratio between the number of lattice sites and the number of interstitial sites so that b
N is the number of available interstitial sites. In case of f.c.c. Pd, for octahedral occupancy b = 1. Since values of pn are simply the binomial distribution coecients, we can shorten the notation by using   nH pn ˆ B n,z; : …2† b
N Every lattice site in the vacancy-free hydrogenated metal is in fact an atomic cluster of some type n = 0, . . . , z. The most probable number of n-clusters is   nH Nn ˆ N
B n,z; : …3† b
N Quantum calculations of Nordlander et al. [12], using the e€ective medium technique have indicated, for the Pd±D system, that in the neighborhood of a vacancy D-atoms do not change their normal interstitial positions much. Thus when a metal atom is removed from an atom-cluster, it becomes e€ectively a vacancy-cluster. This transformation from an atom-cluster to a vacancy-cluster changes the energy of the system. The energy of the nth vacancy-cluster is the monovacancy creation

energy E f1v , lowered by the gain in energy involved in H±VAC interaction, Den1v <0 e…n† ˆ E f1v ‡ Den1v :

…4†

In our calculation for Pd, E f1v is taken to be 1.5 eV as in Ref. [13] and the values of Den1v are taken from Nordlander et al. [12], assuming that the electronic energies of the H±VAC clusters are approximately the same as for the D±VAC clusters. The values of energy are given in Table 1. A set of experimental values, obtained by ion implantation experiments [17] is also shown in Table 1, and used in calculations. Placing a vacancy into a given atom-cluster instead of a metal atom obviously obeys Fermi± Dirac statistics, since a cluster can be either ``occupied'' with a vacancy or not. Only one vacancy can be placed in an atom-cluster. In Fermi±Dirac logic, the vacancy-cluster here stands for the occupied state and the atom-cluster stands for the empty state, the hole. Full analogy can now be drawn with the Fermi± Dirac solution of the cell model in the grand canonical formulation, reported previously in Ref. [18] and the solution can be implemented directly. The concentration c of the particles (vacancies in this case) distributed among the cells (clusters in this case) is given by X 1 cˆ …5† pc …n†
1 ‡ exp‰…ÿm ‡ e…n††=kT Š n where pc …n† is the probability of ®nding an n-order cell in the ensemble. This relation is derived in some detail in Ref. [18]. Equation (5) basically consists of two factors: the ®rst can be considered as the density of states for this particular system or the degeneracy of the state, and the second factor is the familiar Fermi±Dirac distribution. The grand potential O used in Ref. [18] in deriving equation (5) is an explicit function of V, T and m, the chemical potential. The integral and partial thermodynamic functions are derivatives of O…V,T,m† so that the expressions for the thermodynamic functions contain m and, for solute particles (H-atoms) the partial quantities are obtained by numerical methods by inverting the analog of equation (5). The details of this procedure are given in Ref. [18]. In the case of the present calculations m is the chemical potential of vacancies and may be set to zero in equilibrium. Thus equation (5) leads directly to

Table 1. Values of Den1v n

0

1

2

3

4

5

6

Den1v (eV), EMT Den1v (eV), exp.

0 0

ÿ0.26 ÿ0.23

ÿ0.53 ÿ0.46

ÿ0.81 ÿ0.61

ÿ1.09 ÿ0.76

ÿ1.38 ÿ0.91

ÿ1.68 ÿ1.06

MAROEVIC and MCLELLAN: Pd±H SOLID SOLUTIONS

C1v

  nN ˆ B n,z; b
N nˆ0 z X




1 1 ‡ exp‰…E f1v ‡ Den1v †=kT Š

3. CALCULATIONS

…6†

where the probability of the cell type is given by equations (1) and (2) and the cell energies are de®ned according to equation (4). It is easy to show that for the high temperatures and hydrogen free lattice equation (6) has the normal thermodynamic vacancy concentration in this limit. If no H-atoms are present, all the clusters become 0-clusters, the FD function is factored out, the sum of B coecients is 1, and since the FD factor can be approximated with the MB function for high T, we are left with C1v …0† ˆ exp…ÿE f1v =kT †

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…7†

where the notation C1v(0) means the vacancy concentration of the pure, hydrogen free, solvent lattice. Solution (6) is expected to be satisfactory for much lower temperatures and higher concentrations than the MB limit solution given in Ref. [13]. Note that, as in the previous MB treatments of the problem, terms arising from vibrational e€ects, i.e. excess entropies, are not considered and C1v(0) does not contain the factor exp…S f1v =k†. However, since experimentally the values of S f1v =k for noble metals are small (1±2.5), the corresponding error in equation (6) will not be large.

Calculations were done for three di€erent temperatures, using equation (6) and the energy levels (Table 1) from Ref. [12], in the same manner as in Ref. [13] so that the previous Maxwell±Boltzmann approximation can be compared with this more complete solution. In addition, the experimental values of cluster energies (Table 1) are used in the FD framework. Due to lower values of binding energies, predicted vacancy concentrations are smaller but exhibit equivalent behavior. At T = 1000 K (Fig. 1) the MB approximation is, as expected, close to the FD solution. Equilibrium H concentrations in Pd for low pressures are not large at this temperature. However, the hydrogen induced vacancy concentration changes one or two orders of magnitude with respect to the non-hydrogenated crystal. The MB approximation was not applied in Ref. [13] for temperatures below 800 K due to its inherent limitations. At 600 K (Fig. 2) for instance, the MB approximation yields C1v-values which exceed the concentration given by the FD solution by about 10 times, and reaches extremely high values. As we clearly expect, the FD solution performs better than the MB approximation and therefore it is applicable to lower temperatures. Room temperature solutions, shown in Fig. 3, indicate that the MB approximation is inappropriate. After a certain H concentration is reached, the vacancy concentration becomes even greater than

Fig. 1. Comparison of the MB and FD calculations for 1000 K.

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MAROEVIC and MCLELLAN: Pd±H SOLID SOLUTIONS

Fig. 2. Comparison of the MB and FD calculations for 600 K.

unity. The FD solution however asymptotically reaches the values of about 10ÿ2 for the hydrogen concentrations that are high but easily obtained at 300 K and H2 pressures of about 1 atm. The total change in vacancy concentration induced by dissolved hydrogen is predicted to be very large for hydrogen concentrations at this level. Figure 3 shows that it can be enhanced by more than 20 orders of magnitude. The kinetics of the process need to be considered in order to evaluate the speed with which the equilibrium may be reached. At

least in layers of the material near the surface, which are of great importance for mechanical properties, vacancies can quickly di€use in from the surface. Such a dramatic change in vacancy concentration would change the lattice atom di€usion rate proportionally, and this would in e€ect also in¯uence the mechanical properties considerably. The anomalous fast di€usion in the light of this new insight is to be addressed in future work. Vacancy di€usion at the gas±solid interface of a crack in a hydrogenated lattice is clearly important.

Fig. 3. Comparison of the MB and FD calculations for 300 K.

MAROEVIC and MCLELLAN: Pd±H SOLID SOLUTIONS

An initial attempt [19] to evaluate the kinetics of vacancy migration in hydrogenated Pd has already been undertaken. In this model it was assumed that the surface of the crystal was essentially an in®nite source of random hoppers and that vacancy concentration pro®les could be calculated by the approporiate solution of Fick's Second Law. These calculations will be re®ned in the light of the present framework. 4. CONCLUSION

The Fermi±Dirac solution presented here is the full solution of the hydrogen±vacancy interaction model (or the cluster model) as de®ned previously [9, 13, 14] which was approximated originally by the use of Maxwell±Boltzmann statistics. The full solution was made possible utilizing the grand canonical approach, which is equivalent to that described for the cell model problem [18], and the solution obtained therein has been directly translated to the cluster model. Results clearly show that at temperatures where Pd dissolves large quantities of hydrogen, the equilibrium concentration of vacancies is increased by many orders of magnitude. It should be noted that vacancy concentrations concomitant with the present calculations have been reported by Fukai and Okuma [1±3]. Further studies have been undertaken in the authors' laboratory to observe the possible implications of the hydrogen induced lattice migration (HILM). AcknowledgementsÐThe authors are grateful for the support provided by the Robert A. Welch Foundation.

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