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The diffusivity of hydrogen in deformed palladium
Scripta
METALLURGICA
V o l . 15, p p . 5 0 1 - 5 0 2 , 1981 Printed in the U.S.A.
Pergamon P r e s s Ltd. All rights reserved
THE DIFFUSIVITY OF HYDROGEN IN DEFORMED PALLADIUM
Rex B. McLellan William Marsh Rice University Department of Mechanical Engineering and Materials Science Houston, Texas 77001, U.S.A. (Received February 17, 1 9 8 1 ) ( R e v i s e d M a r c h 5, 1 9 8 1 )
The general problem of the affect upon diffusivity of the interaction between interstitial atoms and trapping sites associated with lattice defects is complicated. Hirth and Carnahan [i] have recently used the isotropic elastic approximation to calculate the interaction between H-atoms and the stress field of a dislocation line. A statistical approach to the problem is only tractable when there is a finite distribution of available sites to which energy levels may be assigned. Such an approach has recently been taken [2] in which the interstitial atoms have available either "normal" lattice sites or "defect" sites associated with lattice imperfections. Since in a crystal of finite size the numbers of each available type of site is also finite, the distribution function giving the numbers of interstitial atoms occupying a given site with a maximum of one atom per site must be a Fermi-Dirac distribution. This distribution function has the form [2],
NB=(NK/2){ l+(@i/K) - t
l+(@i/K)2 + 2 @i(1-2~/K)/K t1~}
(1)
Where N i s the t o t a l number of s i t e s , NB i s the number of H-atoms in t r a p p i n g s i t e s , Gi is the H-concentration (@i=number of H-atoms per m e t a l atom), ¢ i s the t r a p p i n g s i t e d e n s i t y (number of t r a p p i n g s i t e s per s i t e ) , and K i s g i v e n by K = ~ +
(i - , ) x
(2)
where x=e &E/kT and AE is the relative trapping depth (ie. difference between the depths of "trapping" and "normal" sites). This formulation ignores the temperature-independent AS-term in equ. (2) which arises due to the difference in non-configurational entropy between an H-atom in the two kinds of site. Using this distribution function and the absolute rate theory it is possible to calculate the diffusivity of the H-atoms through the defect lattice. This calculation in the limit when ~ > 6 i and ~i<< 1 yields the simple result [2i X = DdC/D pc = (x/x+ ~)2
(3)
where D dc is the H-diffusivity through the defect lattice and D pc in the "perfect" lattice. This simple expression is in good agreement [3] with the sharp drop in the diffusivity of H in bec iron observed near 320K using the recent estimates of AE (-59.9~ 4.6 kJ/mol) and given by Kumnick and Johnson [4] for well-annealed iron (~ = 10 -8 ) and 30% cold-worked iron (+ = 6xi0-7). Now if the trapping site density is increased such that O i ~ ~, then at low temperatures the effect of the Fermi-Dirac statistics will be observed. At T ÷ 0 the mobility will tend toward zero if ~ > @i since virtually all the H-atoms will be in trapping sites and have a low jump frequency. If ~ < @ i some H-atoms will remain untrapped at all temperatures and can diffuse normally. The generalized form of equ. (3) is given by [2]
501 0036-9748/81/050501-02502.00/0 Copyright ( c ) 1981 P e r g a m o n P r e s s
Ltd.
502
DIFFUSIVITY
OF HYDROGEN
IN Pd
Vol.
% = x~(l-8i) { (Kx/20i)(l+(@i/K)-~)(x-l) + 1 }/(i-~)6 i where
and
No.
5
(4)
= (K/2~) [l+(@i/K ) -o] O ={l+(@i/K)2 + 2 @ i ~ / K } ½ ~
~ = i - (2~/K)
15,
(5) (6)
(7)
The experimental conditions described above have been realized by the recent measurements by Kirchheim [5] of the diffusivity of H through 50% cold-deformed 3.2 Pd at 295K. An electrolytic technique was used which o I0-" enables D to be measured G) ¢/) at low temperatures as a function of @i at low concentrations [6]. The E ,1" i .... ~,m" J results are depicted in Fig. (i). The upper dashed / / . . . . . ~ • 10" s , A E • - 4 1 . 6 8 kJ/mol ¢:3 10-12 line represents measure/ 2xlO'S, A E ' - 2 1 kJ/mol ments on well-annealed Pd and the solid line represents data for the deformed L'I"" I I I materials (mean of two measuring techniques [5]). i0 -s 0 .4 i0 -z There was in fact an increase found at very low H-concentrations FIG i. Measured and calculated variation of the diffusivity of (<5xl0-5ppm) which hydrogen in deformed palladium. Kirchheim ascribes to the effects of diffusion along dislocation pipes [5]. The sharp decrease between 8.=10 -4 and 10 -5 is, however, clear, i
tO-m]
r/,J / _ _; :::;.l..._.,. ...o, ,"
........~=
I0 "l~ "/~
8i
dc The set of equations (4-7) has been used to calculate D uslng the trapping depths -21, -31.4, and -41.7 kJ/mol and trapping concentrations inthe range ~=2xlO -3 - 10 -5 . This range of #-values was chosen since there is some indication [7] that the dislocation conconcentration of 50% deformed Pd is ~1015 m -2. Assuming one trapping site per intersection with a close packed plane shows that reasonable values of ~ are in the range 10 -5 - 10 -4 . The calculated diffusivities are clearly compatible with the experimental findings. Provided the traps are deep enough at the temperature considered, the shape of the curve does not much depend o n t h e actual value of AE since essentially all the H-atoms will be condensed when 0 i < ¢. Acknowledgements The author is grateful for the support provided by the U.S. Army Research Office. References [i] [2] [3] [4] [5] [6] [7]
J.P. Hirth and B. Carnahan, Acta Metall. 26, 1759 (1978). R.B. McLellan, Acta Metall. 27, 1655 (1979). R.B. McLellan, Scripta Metall., 14, 513 (1980). A.J. Kumnick and H.H. Johnson, Acta Metall. 28, 33 (1980). R. Kirchheim, Scripta Metall. 1-4, 905 (1980). R. Kirchheim and R.B. McLellan, J. Electrochem. Soc. 127, 2439 (1980). J. Friedel, "Dislocations", Pergamon Press, Oxford (1967).
METALLURGICA
V o l . 15, p p . 5 0 1 - 5 0 2 , 1981 Printed in the U.S.A.
Pergamon P r e s s Ltd. All rights reserved
THE DIFFUSIVITY OF HYDROGEN IN DEFORMED PALLADIUM
Rex B. McLellan William Marsh Rice University Department of Mechanical Engineering and Materials Science Houston, Texas 77001, U.S.A. (Received February 17, 1 9 8 1 ) ( R e v i s e d M a r c h 5, 1 9 8 1 )
The general problem of the affect upon diffusivity of the interaction between interstitial atoms and trapping sites associated with lattice defects is complicated. Hirth and Carnahan [i] have recently used the isotropic elastic approximation to calculate the interaction between H-atoms and the stress field of a dislocation line. A statistical approach to the problem is only tractable when there is a finite distribution of available sites to which energy levels may be assigned. Such an approach has recently been taken [2] in which the interstitial atoms have available either "normal" lattice sites or "defect" sites associated with lattice imperfections. Since in a crystal of finite size the numbers of each available type of site is also finite, the distribution function giving the numbers of interstitial atoms occupying a given site with a maximum of one atom per site must be a Fermi-Dirac distribution. This distribution function has the form [2],
NB=(NK/2){ l+(@i/K) - t
l+(@i/K)2 + 2 @i(1-2~/K)/K t1~}
(1)
Where N i s the t o t a l number of s i t e s , NB i s the number of H-atoms in t r a p p i n g s i t e s , Gi is the H-concentration (@i=number of H-atoms per m e t a l atom), ¢ i s the t r a p p i n g s i t e d e n s i t y (number of t r a p p i n g s i t e s per s i t e ) , and K i s g i v e n by K = ~ +
(i - , ) x
(2)
where x=e &E/kT and AE is the relative trapping depth (ie. difference between the depths of "trapping" and "normal" sites). This formulation ignores the temperature-independent AS-term in equ. (2) which arises due to the difference in non-configurational entropy between an H-atom in the two kinds of site. Using this distribution function and the absolute rate theory it is possible to calculate the diffusivity of the H-atoms through the defect lattice. This calculation in the limit when ~ > 6 i and ~i<< 1 yields the simple result [2i X = DdC/D pc = (x/x+ ~)2
(3)
where D dc is the H-diffusivity through the defect lattice and D pc in the "perfect" lattice. This simple expression is in good agreement [3] with the sharp drop in the diffusivity of H in bec iron observed near 320K using the recent estimates of AE (-59.9~ 4.6 kJ/mol) and given by Kumnick and Johnson [4] for well-annealed iron (~ = 10 -8 ) and 30% cold-worked iron (+ = 6xi0-7). Now if the trapping site density is increased such that O i ~ ~, then at low temperatures the effect of the Fermi-Dirac statistics will be observed. At T ÷ 0 the mobility will tend toward zero if ~ > @i since virtually all the H-atoms will be in trapping sites and have a low jump frequency. If ~ < @ i some H-atoms will remain untrapped at all temperatures and can diffuse normally. The generalized form of equ. (3) is given by [2]
501 0036-9748/81/050501-02502.00/0 Copyright ( c ) 1981 P e r g a m o n P r e s s
Ltd.
502
DIFFUSIVITY
OF HYDROGEN
IN Pd
Vol.
% = x~(l-8i) { (Kx/20i)(l+(@i/K)-~)(x-l) + 1 }/(i-~)6 i where
and
No.
5
(4)
= (K/2~) [l+(@i/K ) -o] O ={l+(@i/K)2 + 2 @ i ~ / K } ½ ~
~ = i - (2~/K)
15,
(5) (6)
(7)
The experimental conditions described above have been realized by the recent measurements by Kirchheim [5] of the diffusivity of H through 50% cold-deformed 3.2 Pd at 295K. An electrolytic technique was used which o I0-" enables D to be measured G) ¢/) at low temperatures as a function of @i at low concentrations [6]. The E ,1" i .... ~,m" J results are depicted in Fig. (i). The upper dashed / / . . . . . ~ • 10" s , A E • - 4 1 . 6 8 kJ/mol ¢:3 10-12 line represents measure/ 2xlO'S, A E ' - 2 1 kJ/mol ments on well-annealed Pd and the solid line represents data for the deformed L'I"" I I I materials (mean of two measuring techniques [5]). i0 -s 0 .4 i0 -z There was in fact an increase found at very low H-concentrations FIG i. Measured and calculated variation of the diffusivity of (<5xl0-5ppm) which hydrogen in deformed palladium. Kirchheim ascribes to the effects of diffusion along dislocation pipes [5]. The sharp decrease between 8.=10 -4 and 10 -5 is, however, clear, i
tO-m]
r/,J / _ _; :::;.l..._.,. ...o, ,"
........~=
I0 "l~ "/~
8i
dc The set of equations (4-7) has been used to calculate D uslng the trapping depths -21, -31.4, and -41.7 kJ/mol and trapping concentrations inthe range ~=2xlO -3 - 10 -5 . This range of #-values was chosen since there is some indication [7] that the dislocation conconcentration of 50% deformed Pd is ~1015 m -2. Assuming one trapping site per intersection with a close packed plane shows that reasonable values of ~ are in the range 10 -5 - 10 -4 . The calculated diffusivities are clearly compatible with the experimental findings. Provided the traps are deep enough at the temperature considered, the shape of the curve does not much depend o n t h e actual value of AE since essentially all the H-atoms will be condensed when 0 i < ¢. Acknowledgements The author is grateful for the support provided by the U.S. Army Research Office. References [i] [2] [3] [4] [5] [6] [7]
J.P. Hirth and B. Carnahan, Acta Metall. 26, 1759 (1978). R.B. McLellan, Acta Metall. 27, 1655 (1979). R.B. McLellan, Scripta Metall., 14, 513 (1980). A.J. Kumnick and H.H. Johnson, Acta Metall. 28, 33 (1980). R. Kirchheim, Scripta Metall. 1-4, 905 (1980). R. Kirchheim and R.B. McLellan, J. Electrochem. Soc. 127, 2439 (1980). J. Friedel, "Dislocations", Pergamon Press, Oxford (1967).