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Vacancy concentrations in metals in equilibrium with H2-gas at high pressures
Journal of Physics and Chemistry of Solids 64 (2003) 527–530 www.elsevier.com/locate/jpcs
Technical note
Vacancy concentrations in metals in equilibrium with H2-gas at high pressures J. Mao, R.B. McLellan* Department of Mechanical Engineering and Materials Science, George R. Brown School of Engineering, William Marsh Rice University, P.O. Box 1892, Houston, TX 77251-1892, USA Received 20 March 2002; accepted 3 July 2002
Abstract The effect of equilibrium vacancy formation on the solubility of hydrogen in metals has been derived. The treatment is based on a Fermi – Dirac distribution function so that problems which arise at low temperatures in Maxwell –Boltzmann formulations are obviated. Using input data for Ni, the effect of vacancy formation on the validity of Sieverts’ law is calculated and the ðpTÞ-region in which the thermodynamic functions of the solute (H) atoms can be easily extracted from measured solubility has been mapped. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Hydrogen; Vacancies; C. High pressure; Nickel
1. Introduction The well-known phenomenon of abundant vacancy formation in hydrogenated metals was initially confirmed experimentally by Fukai and Okuma [1] by exposing Pd and Ni to H2-pressures up to 5 GPa. At such high pressures vacancy concentrations C1V can reach such high values (C1V ! 0.1!) that statistical mechanical calculations of the abundant vacancy effect begin to break down. However, it is clear that at pressures much below these limits sufficiently large vacancy concentrations can occur so that the validity of Sieverts’ law is questionable. The purpose of the present short paper is to discuss abundant vacancy formation in metals in equilibrium with H2-gas at such pressures where relatively simple statistical methods are still valid, but classical occlusion behavior is not expected. 2. Statistical calculations A formal treatment of the statistical mechanics of Hvacancy interactions has recently been given [2] using an * Corresponding author. Tel.: þ1-713-348-4993; fax: þ 1-713348-5423. E-mail address: [email protected] (R.B. McLellan).
approach in which the grand canonical ensemble is used to evaluate the thermodynamic behavior of the metal – vacancy – hydrogen [M– VAC –H] system in Fermi – Dirac statistics. The present calculations are based on a modification of these calculations. The solid, containing NM lattice atoms is equilibrated with a large atmosphere of constant mH (hydrogen chemical potential) such that a total of NV vacancies are created. These vacancies possess gNV nearestneighbor interstitial sites and such sites may be occupied by a subset of H-atoms with a binding free energy g1 ¼ e 1 2 T S 1
ð1Þ
where e 1 is the binding energy, and S 1 is the corresponding non-configurational entropy. Hydrogen atoms may also be contained in the NM 2 ðg 2 1ÞNV isolated interstitial sites, i.e. those not neighboring a vacancy. This approximation considers that all the vacancy-related sites are energetically equivalent. This simplification is consistent with the effective medium calculations of e 1 [3] as has been discussed previously [2]. The binding free energy in the subset of isolated interstitial sites is g1 ¼ e 0 2 T S 0
0022-3697/03/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 3 6 9 7 ( 0 2 ) 0 0 3 3 4 - 7
ð2Þ
528
J. Mao, R.B. McLellan / Journal of Physics and Chemistry of Solids 64 (2003) 527–530
where e 0 is identified with the partial enthalpy H H in the vacancy-free H– Metal system and S 0 is the partial excess entropy S XS H in such a system. Indexing the two subsystems by i, the Fermi – Dirac grand partition function may be written as [4] Y X QðV; T; mH Þ ¼ exp½ðmH 2 gi Þni =kT ð3Þ i
ni ¼0;1
It is easy to show [2] that the ensemble average number of H-atoms dissolved in the solid is kNH l ¼
NM 2 ðg 2 1ÞNV NV g þ 1 þ exp½ðg1 2 mH Þ=kT 1 þ exp½ðg0 2 mH Þ=kT ð4Þ
For the case when e 1 ! 1 and thus VAC-related occupied cells are not allowed lim kNH l ¼
e 1 !1
NM 2 NV ðg 2 1Þ H 2 T S XS 1 þ exp½ðH H 2 mH Þ=kT
ð5Þ
Writing the H-concentration in the solid as the atom ration u ¼ kNH l=NM ; Eq. (5) can be written as uH ð6Þ mH ¼ H H 2 T S XS þ kT ln H 1 2 CV ðg 2 1Þ where CV ¼ NV =NM : Eq. (6) is the correct expression for the chemical potential of interstitial atoms in a solution in which NV ðg 2 1Þ sites are excluded from occupancy. Note that in the previous [2] calculations S 1 and S 0 were set at zero. This omission does not constitute a large error in the results of the calculation but has been included in the present work since we apply the results to the metal Ni for which S XS H has been determined accurately. Now if the VAC-related cell groups containing 0 ! g Hatoms do not strongly interact, the Gibbs free energy may be
written as [2] GC ¼ mH kNH l þ NV ðH1V 2 TSF1V Þ 2 kT ln
ðNM þ NV Þ! NM !NV !
ð7Þ
F and SF1V are the enthalpy and entropy of where H1V formation of a vacancy. The term TSF1V was omitted in the previous calculations [2]. The computational procedure for evaluating kNH l and CV consists of combining Eqs. (6) and (7) and minimizing GC with respect to NV at given kNH l and T. The mathematical details have been given previously [2] and will not be repeated here. The numerical results presented here have been generated for the Ni – VAC– H-System. There are several reasons for this choice. On the one hand Ni was one of the metals used in the initial high-pressure work of Fukai and Okuma [1], and secondly reliable experimental data for S XS H ; HH ; F H1V and SF1V are available. Finally the effective-medium calculations of Nordlander et al. [3] have provided reliable F data for the H-VAC binding energy e 1 : The values of H1V F and S1V are 1.6 eV (154 kJ/mol) and 1:5k; respectively. These ‘best values’ compiled by Seeger and Mehrer [5] are in excellent accord with later experimental work on Ni [6– 8]. The values of H H and S XS H are 2 1.9 eV (2 184 kJ/ mol) and 4:25k: These data are taken from H – Ni equilibrium studies at atmospheric pressure [9]. The value of e 1 is taken as 2 2.21 eV (2213.3 kJ/mol). This is taken from the average value kEðnÞl of the g ¼ 6 cell energies calculated by Nordlander et al. [3] ðe 1 ¼ H H 2 kEðnÞlÞ: The value of S 1 is not known. Since the H-atom in the vacancy-related cell is in a relaxed site compared to the ‘normal’ interstitial sites, we expect that S 1 , S XS H : The experimental value of S 0 ¼ S XS H ¼ 4:25k:
Fig. 1. Equilibrium hydrogen solubility for Nickel in contact with H2-gas as a function of temperature (horizontal scale) and pressure (individual curves).
J. Mao, R.B. McLellan / Journal of Physics and Chemistry of Solids 64 (2003) 527–530
529
Fig. 2. Equilibrium vacancy concentrations in hydrogenated nickel as a function of temperature (horizontal scale) and pressure (individual curves).
In the calculations we take C0 ; expðS 1 =kÞ ¼ 5:0; i.e. S 1 ¼ 1:6k: The curves of CV and uH ¼ kNH l=NM as functions of temperature are not sensitive to the actual value of S 1 inside the range of 0 ! 3k:
3. Results and conclusions The variation of uH with temperature at a series of H2-pressures in the range 1 ! 104 atm (100 kPa ! 1 GPa)
is shown in Fig. 1. The points (†) shown on the curve (1) for p ¼ 1 atm were taken from the data compilation and measurements of McLellan and Sutter [9]. The corresponding curves (1 ! 7) for CV as a function of T are shown in Fig. 2. The numbers on each curve correspond to the same pressures that were indicated in Fig. 1. The points (†) shown on the dotted curve were calculated from the expression F CV ¼ expðSF1V =kÞ expð2H1V =kTÞ
Fig. 3. Hydrogen solubility in Nickel in equilibrium with H2-gas as a function of p1=2 in the temperature range 700–1000 K.
ð8Þ
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J. Mao, R.B. McLellan / Journal of Physics and Chemistry of Solids 64 (2003) 527–530
F using the values of SF1V and H1V given in Section 2. These values are, of course, close to the CV -values calculated for curve (1) since the corresponding values of u are small. However, it is clear that as T decreases the effect of dissolved H-atoms in increasing the vacancy concentration becomes greater and this effect is measurable even for H –Ni solutions produced by metal – gas equilibration at pressures exceeding , 10 atm (i.e. curve (2)). Values of uH for gaseous equilibrium in Ni in the range 700– 1000 K are shown as functions of p1=2 in Fig. 3. The dashed lines are the ‘Sieverts law’ lines calculated from the expression [10] ! ! H fp1=2 E0D 2 H S XS H exp ð9Þ uH ¼ 7=4 exp kT k T
which applies to the vacancy-free solution. In this expression f is a known [10] constant and E0D is the H– H dissociation energy per atom at 0 K. It is clear from Fig. 3 that when p 1/2 , 40 atm1/2 (,12,733 (N/m2)1/2) at all temperatures in the range 700– 1000 K the derivation from the Sieverts’ law constant due to vacancy formation is small. However, as the ‘condensation’ point is reached the deviation sets in rapidly. Thus provided the H2-gas pressure and temperature are in the range of values delineated in Fig. 3 there will be no H and S XS problem in extracting values of H H from measured ðp; uH ; TÞ-solubility data. If (p; uH ; T) lies outside this range the thermodynamic parameters of the solution may still be extracted from the
experimental solubility data, but the process involves numerical analysis of Eq. (4). These procedures have been outlined previously [2].
Acknowledgements The authors are grateful for support provided by the Robert A. Welch Foundation.
References [1] Y. Fukai, N. Okuma, Jpn. J. Appl. Phys. 32 (1993) 1256. [2] J. Mao, R.B. McLellan, J. Phys. Chem. Solids (2002) in press. [3] P. Nordlander, J.K. Norskov, F. Besenbacher, S.M. Myres, Phys. Rev. B 40 (3) (1989) 1990. [4] C. Kittel, Elementary Statistical Physics, Wiley, New York, 1961. [5] A. Seeger, H. Mehrer, Vacancies and Interstitials in Metals, North-Holland, Amsterdam, 1969. [6] J.L. Campbell, C.W. Schulte, R.R. Gingerich, J. Nucl. Mater. 70 (1978) 609. [7] W. Wycisk, M. Feller-Kniepmeier, J. Nucl. Mater. 70 (1978) 616. [8] A.A. Mamauli, T.D. Ositinskaya, T.D. Pervokov, V.I. Khomkevich, Sov. Phys. Solid State 10 (1968) 2290. [9] R.B. McLellan, P.L. Sutter, Acta Metall. 32 (1984) 2233. [10] R.B. Mclellan, in: P.S. Rudman, J. Stringer, R.I. Jaffee (Eds.), Phase Stability in Metals and Alloys, McGraw-Hill, New York, 1967, p. 393.
Technical note
Vacancy concentrations in metals in equilibrium with H2-gas at high pressures J. Mao, R.B. McLellan* Department of Mechanical Engineering and Materials Science, George R. Brown School of Engineering, William Marsh Rice University, P.O. Box 1892, Houston, TX 77251-1892, USA Received 20 March 2002; accepted 3 July 2002
Abstract The effect of equilibrium vacancy formation on the solubility of hydrogen in metals has been derived. The treatment is based on a Fermi – Dirac distribution function so that problems which arise at low temperatures in Maxwell –Boltzmann formulations are obviated. Using input data for Ni, the effect of vacancy formation on the validity of Sieverts’ law is calculated and the ðpTÞ-region in which the thermodynamic functions of the solute (H) atoms can be easily extracted from measured solubility has been mapped. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Hydrogen; Vacancies; C. High pressure; Nickel
1. Introduction The well-known phenomenon of abundant vacancy formation in hydrogenated metals was initially confirmed experimentally by Fukai and Okuma [1] by exposing Pd and Ni to H2-pressures up to 5 GPa. At such high pressures vacancy concentrations C1V can reach such high values (C1V ! 0.1!) that statistical mechanical calculations of the abundant vacancy effect begin to break down. However, it is clear that at pressures much below these limits sufficiently large vacancy concentrations can occur so that the validity of Sieverts’ law is questionable. The purpose of the present short paper is to discuss abundant vacancy formation in metals in equilibrium with H2-gas at such pressures where relatively simple statistical methods are still valid, but classical occlusion behavior is not expected. 2. Statistical calculations A formal treatment of the statistical mechanics of Hvacancy interactions has recently been given [2] using an * Corresponding author. Tel.: þ1-713-348-4993; fax: þ 1-713348-5423. E-mail address: [email protected] (R.B. McLellan).
approach in which the grand canonical ensemble is used to evaluate the thermodynamic behavior of the metal – vacancy – hydrogen [M– VAC –H] system in Fermi – Dirac statistics. The present calculations are based on a modification of these calculations. The solid, containing NM lattice atoms is equilibrated with a large atmosphere of constant mH (hydrogen chemical potential) such that a total of NV vacancies are created. These vacancies possess gNV nearestneighbor interstitial sites and such sites may be occupied by a subset of H-atoms with a binding free energy g1 ¼ e 1 2 T S 1
ð1Þ
where e 1 is the binding energy, and S 1 is the corresponding non-configurational entropy. Hydrogen atoms may also be contained in the NM 2 ðg 2 1ÞNV isolated interstitial sites, i.e. those not neighboring a vacancy. This approximation considers that all the vacancy-related sites are energetically equivalent. This simplification is consistent with the effective medium calculations of e 1 [3] as has been discussed previously [2]. The binding free energy in the subset of isolated interstitial sites is g1 ¼ e 0 2 T S 0
0022-3697/03/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 3 6 9 7 ( 0 2 ) 0 0 3 3 4 - 7
ð2Þ
528
J. Mao, R.B. McLellan / Journal of Physics and Chemistry of Solids 64 (2003) 527–530
where e 0 is identified with the partial enthalpy H H in the vacancy-free H– Metal system and S 0 is the partial excess entropy S XS H in such a system. Indexing the two subsystems by i, the Fermi – Dirac grand partition function may be written as [4] Y X QðV; T; mH Þ ¼ exp½ðmH 2 gi Þni =kT ð3Þ i
ni ¼0;1
It is easy to show [2] that the ensemble average number of H-atoms dissolved in the solid is kNH l ¼
NM 2 ðg 2 1ÞNV NV g þ 1 þ exp½ðg1 2 mH Þ=kT 1 þ exp½ðg0 2 mH Þ=kT ð4Þ
For the case when e 1 ! 1 and thus VAC-related occupied cells are not allowed lim kNH l ¼
e 1 !1
NM 2 NV ðg 2 1Þ H 2 T S XS 1 þ exp½ðH H 2 mH Þ=kT
ð5Þ
Writing the H-concentration in the solid as the atom ration u ¼ kNH l=NM ; Eq. (5) can be written as uH ð6Þ mH ¼ H H 2 T S XS þ kT ln H 1 2 CV ðg 2 1Þ where CV ¼ NV =NM : Eq. (6) is the correct expression for the chemical potential of interstitial atoms in a solution in which NV ðg 2 1Þ sites are excluded from occupancy. Note that in the previous [2] calculations S 1 and S 0 were set at zero. This omission does not constitute a large error in the results of the calculation but has been included in the present work since we apply the results to the metal Ni for which S XS H has been determined accurately. Now if the VAC-related cell groups containing 0 ! g Hatoms do not strongly interact, the Gibbs free energy may be
written as [2] GC ¼ mH kNH l þ NV ðH1V 2 TSF1V Þ 2 kT ln
ðNM þ NV Þ! NM !NV !
ð7Þ
F and SF1V are the enthalpy and entropy of where H1V formation of a vacancy. The term TSF1V was omitted in the previous calculations [2]. The computational procedure for evaluating kNH l and CV consists of combining Eqs. (6) and (7) and minimizing GC with respect to NV at given kNH l and T. The mathematical details have been given previously [2] and will not be repeated here. The numerical results presented here have been generated for the Ni – VAC– H-System. There are several reasons for this choice. On the one hand Ni was one of the metals used in the initial high-pressure work of Fukai and Okuma [1], and secondly reliable experimental data for S XS H ; HH ; F H1V and SF1V are available. Finally the effective-medium calculations of Nordlander et al. [3] have provided reliable F data for the H-VAC binding energy e 1 : The values of H1V F and S1V are 1.6 eV (154 kJ/mol) and 1:5k; respectively. These ‘best values’ compiled by Seeger and Mehrer [5] are in excellent accord with later experimental work on Ni [6– 8]. The values of H H and S XS H are 2 1.9 eV (2 184 kJ/ mol) and 4:25k: These data are taken from H – Ni equilibrium studies at atmospheric pressure [9]. The value of e 1 is taken as 2 2.21 eV (2213.3 kJ/mol). This is taken from the average value kEðnÞl of the g ¼ 6 cell energies calculated by Nordlander et al. [3] ðe 1 ¼ H H 2 kEðnÞlÞ: The value of S 1 is not known. Since the H-atom in the vacancy-related cell is in a relaxed site compared to the ‘normal’ interstitial sites, we expect that S 1 , S XS H : The experimental value of S 0 ¼ S XS H ¼ 4:25k:
Fig. 1. Equilibrium hydrogen solubility for Nickel in contact with H2-gas as a function of temperature (horizontal scale) and pressure (individual curves).
J. Mao, R.B. McLellan / Journal of Physics and Chemistry of Solids 64 (2003) 527–530
529
Fig. 2. Equilibrium vacancy concentrations in hydrogenated nickel as a function of temperature (horizontal scale) and pressure (individual curves).
In the calculations we take C0 ; expðS 1 =kÞ ¼ 5:0; i.e. S 1 ¼ 1:6k: The curves of CV and uH ¼ kNH l=NM as functions of temperature are not sensitive to the actual value of S 1 inside the range of 0 ! 3k:
3. Results and conclusions The variation of uH with temperature at a series of H2-pressures in the range 1 ! 104 atm (100 kPa ! 1 GPa)
is shown in Fig. 1. The points (†) shown on the curve (1) for p ¼ 1 atm were taken from the data compilation and measurements of McLellan and Sutter [9]. The corresponding curves (1 ! 7) for CV as a function of T are shown in Fig. 2. The numbers on each curve correspond to the same pressures that were indicated in Fig. 1. The points (†) shown on the dotted curve were calculated from the expression F CV ¼ expðSF1V =kÞ expð2H1V =kTÞ
Fig. 3. Hydrogen solubility in Nickel in equilibrium with H2-gas as a function of p1=2 in the temperature range 700–1000 K.
ð8Þ
530
J. Mao, R.B. McLellan / Journal of Physics and Chemistry of Solids 64 (2003) 527–530
F using the values of SF1V and H1V given in Section 2. These values are, of course, close to the CV -values calculated for curve (1) since the corresponding values of u are small. However, it is clear that as T decreases the effect of dissolved H-atoms in increasing the vacancy concentration becomes greater and this effect is measurable even for H –Ni solutions produced by metal – gas equilibration at pressures exceeding , 10 atm (i.e. curve (2)). Values of uH for gaseous equilibrium in Ni in the range 700– 1000 K are shown as functions of p1=2 in Fig. 3. The dashed lines are the ‘Sieverts law’ lines calculated from the expression [10] ! ! H fp1=2 E0D 2 H S XS H exp ð9Þ uH ¼ 7=4 exp kT k T
which applies to the vacancy-free solution. In this expression f is a known [10] constant and E0D is the H– H dissociation energy per atom at 0 K. It is clear from Fig. 3 that when p 1/2 , 40 atm1/2 (,12,733 (N/m2)1/2) at all temperatures in the range 700– 1000 K the derivation from the Sieverts’ law constant due to vacancy formation is small. However, as the ‘condensation’ point is reached the deviation sets in rapidly. Thus provided the H2-gas pressure and temperature are in the range of values delineated in Fig. 3 there will be no H and S XS problem in extracting values of H H from measured ðp; uH ; TÞ-solubility data. If (p; uH ; T) lies outside this range the thermodynamic parameters of the solution may still be extracted from the
experimental solubility data, but the process involves numerical analysis of Eq. (4). These procedures have been outlined previously [2].
Acknowledgements The authors are grateful for support provided by the Robert A. Welch Foundation.
References [1] Y. Fukai, N. Okuma, Jpn. J. Appl. Phys. 32 (1993) 1256. [2] J. Mao, R.B. McLellan, J. Phys. Chem. Solids (2002) in press. [3] P. Nordlander, J.K. Norskov, F. Besenbacher, S.M. Myres, Phys. Rev. B 40 (3) (1989) 1990. [4] C. Kittel, Elementary Statistical Physics, Wiley, New York, 1961. [5] A. Seeger, H. Mehrer, Vacancies and Interstitials in Metals, North-Holland, Amsterdam, 1969. [6] J.L. Campbell, C.W. Schulte, R.R. Gingerich, J. Nucl. Mater. 70 (1978) 609. [7] W. Wycisk, M. Feller-Kniepmeier, J. Nucl. Mater. 70 (1978) 616. [8] A.A. Mamauli, T.D. Ositinskaya, T.D. Pervokov, V.I. Khomkevich, Sov. Phys. Solid State 10 (1968) 2290. [9] R.B. McLellan, P.L. Sutter, Acta Metall. 32 (1984) 2233. [10] R.B. Mclellan, in: P.S. Rudman, J. Stringer, R.I. Jaffee (Eds.), Phase Stability in Metals and Alloys, McGraw-Hill, New York, 1967, p. 393.