- Email: [email protected]
The diffusion of boron in nickel
scripta Metallurgica et Materialia, %I. 33, No. 8, pp. 1265-1267.1995 Ekevia Science Ltd copyright 0 1995 AcLa Metallurgica Inc. Printed in the USA. AlI rights rrservcd 0956716X195 $9.50 + .OO
0956-716X(95)00365-7
THE DIFFUSION OF BORON IN NICKEL R. B. McLellan Department of Mechanical Engineering and Materials Science William Marsh Rice University Houston, Texas 77251-1892 U.S.A. (Received May 23,1995)
Introduction The atoms B and He, dissolved in j5z.c. Fe and Ni , form solid solutions. Evidence exists that they are of the hybrid type, i.e. that the solute atoms are located at both interstitial and substitutional sites in the metal lattice. The statistical mechanics of hybrid solutions has been discussed previously [l], and a simple theory of solute diffusion in such a solution crystal has been presented [2]. In the case of B in jc.c. iron, the diffusion activation energy is 88 kJ/mol (0.91 eV) [3], even smaller than that found for C in ~c.c. Fe of 142 kJ/mol (1.47 eV) [4]. This small activation energy indicates that 1I diffuses interstitially in the 6c.c. Fe lattice, but the decrease in X-ray lattice parameters [.S]with increasing B - content indicates substitutional solution formation. However, Wang, Zhang, and He [6] have recently used particle tracking auto-radiography to study B -diffusion in the temperature range 1173- 1473 K in a series of Fe - based solutions and in Ni . The results of this work for Ni - B may be expressed in Arrhenius form D=Do exp(-Q/kT), (1) with Do = 6.6 x 10m7 m*/sec and the activation energy Q = 96.3 kJ/mol (0.99 eV). These results are in good aco3rd with those of Chu [7], who obtained Do = 1.1 x 10m6 m*/sec and Q = 96.3 kJ/mol (0.99 eV). The data of Wang et al. [6] for the fc.c. Fe - based alloys yield Q -values in the range lll115 kJ/mol (1.1.4- 1.19 ev). These Q -values are smaller than those of Busby (142 kJ/mol, 1.47 eV) [3] for B in y -Fe. The small Q -values lead Wang et al. [6] to conclude that B - atoms occupy interstitial sites in ~c.c. Fe. If the magnitude of Q is the sole criterion for site occupancy, the conclusion would also be made that B occupies interstitial sites in Ni . The purpose of the present contribution is to point out that such deductions may not be made since the kinetic behavio:r of hybrid solid solutions may be essentially determined by a small fraction of interstitial solutes, whereas the predominant location of solute atoms may be substitutional. Discussion It was shown previously that for hybrid solid solutions containing solvent (u), solute (u) atoms, and vacancies, the fraction of solute (u) atoms in interstitial sites Fi is given by [l] Fuj = [l+x(l+ZKCo)]-l. 1265
(2)
1266
DIFFUSION OF BORON
Vol. 33, No. 8
This formula refers to octahedral occupation in the ~c.c. lattice, and restricts vacancies to having zero or one nearest - neighbor u - atoms in a substitutional (s) position. Z is the coordination number, and x
=
exp(AE/kT)
co
=
exp (-EfvlkT)
K
=
exp (Ebl/kT) ,
(3)
where AC = Ei - Ei , the difference in energies of the u - atom in the interstitial (j) and substitutional (s) sites, Ef, is the formation energy of a monovacancy, and Ebr is the binding energy of the u(s) -vacancy pair. The vacancy formation entropy has been neglected. Now the u -atoms can diffuse by making hops between interstitial sites with the motion energy E?‘, or by exchanging with vacancies with the motion energy EF” . When AC > 0 (X >> 1) and s -occupancy predominates, it is easy to show that the diffusion activation energy is given by Q g E;‘i + AC _ Kf ij f2
AE,
e-AE/kT
e-AEajkT
[
_ ,-(AE+AE)kT
I
II
,
(4
where K’ and C are of order unity, f is the correlation factor for the exchange mechanism, AE, = m,s - EyGp, A E is given by [2] L AE = Eiv - Ae + ( E,m3$- E,m’j ) ,
(5)
and Erip is the solvent-vacancy motion energy for a solvent atom nearest-neighbor to a substitutional solute. It has been shown that [2] f g [ I+ 3 e-AWkT l-1 , (6) so that f2 approaches unity for AE, > 0 and f2 < 0 for A Ea < 0. Furthermore, since both AE > 0 and A E > 0, the entire third term in Eqn. (4) is much smaller than E,“” + AC , since the upper bound of AE, e-AEafkT is kT e- 1 = 3.67 k.I/mol (0.004 eV) at 1200 K. Thus, Eqn. (4) may be written as
Q=E,m’j+A&
(7)
The justification for taking A E > 0 is the large value of Ef,
for both y-Fe
(164 kJ/mol, 1.7 eV [S]),
and for Ni (156 kJ/mol, 1.6 eV [9]). Since Er3’ and Ervi will be comparable in magnitude, Eqn. (5) shows that A E > 0 for reasonable values of AE . The counterpart of Eqn. (7), for the case when AE is so large that Fi = 1 - Fi is essentially unity and j - mode diffusion is suppressed, is [2] f QsEE,,+E,,
ms .
(8)
Since EzvS > 0, this situation clearly does not apply to the known diffusivity measurements for B in Fe andNi since Q < E,,f . Due to the fact that, in the temperature range of the measurements of Wang et al. [6], the Arrhenius relation is well obeyed and Q seems to be reasonably constant, we may assume that J’-mode diffusion behavior predominates and Eqn. (7) applies. The important point is that Q is still given by this relation even when the predominantly occupied sites are substitutional. If we take, conservatively, AE = 40 k.I/mol (0.41 eV), then for lbi = 0 we have, at 12OOK, Fl = 0.985 for Fe and Fi = 0.984 for Ni . Unfortunately, there are no reliable data for Ebi for B - Ni . However, for B -Fe, Chapman and Walker [lo] have used elastic methods to calculate Ebi = 54 kJ/mol (0.57 ev), close to the value estimated by
Vol. 33, No. 8
DIFFUSION OF BORON
1267
Williams et al. [ll] (48.2 kJ/mol (0.5 ev>). If ebl has a similar value for B -Ni , then because of the low values of Co ( 1.8 x 10s7 for Ni and 8.14 x lo-* for Fe ), the value of F,” is hardly affected. It is, of course, true that in the limit when AE assumes negative values, Fi rapidly approaches unity and Q w E:‘j. This situation is, however, not in accord with the X-ray lattice parameter measurements for Fe [5]. Measurements on the effect of B - additions to the lattice parameter ofNi are currently underway in the author’s laboratory. Conclusions The low values of the activation energy for diffusion found recently by Wang et al. [6] and by Chu et al. [7] for B in ~c.c. Ni - and Fe -based solutions does not, of necessity, indicate that the B -atoms are predominantly located in interstitial sites in the solvent lattice. Such behavior is consistent with rapid interstitial jumping in a hybrid solid solution in which the solute (B) atoms are located predominantly in the substitutional sites. Acknowledgment The author is grateful for the support provided by the Robert A. Welch Foundation. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
R. B. McLellan, J. Phys. Chem Solids SO, 49 (1989). R. B. McLellan, J. C/rem. Phys. 54,465 (1993). I? E. Busby, M. E. Warga, and C. Wells, Truns. AIME 197,1463 (1953). I. I. Kcvenskl, Fir MetaL i Metalloved. 16, 107 (1963). R. M. Goldhoff and J. W. Spretnak,J. Met 9,1278 (1957). W. Wang, S. :Zhang, and X. He, Actu MerafL Mater. 43,1693 (1995). Y Y. Chu, P. Ji, and T. Kc, Acta MetalL Mater. 27,8303 (1991). S. M. Kim and W. J. L. Buyers,J. Phys. F. 8, L 103 (1978). W. Wycisk and M. Feller-Kniepmeier,J. Nut. Mater. 70,616 (1978). M. A. V. Chapman and R. G. Faulkener,Actu Metall. 31,677 (1983). ‘I. M. Williams, A. M. Stoneham. and D. R. Harris, Metals Sci. lo,14 (1976).
0956-716X(95)00365-7
THE DIFFUSION OF BORON IN NICKEL R. B. McLellan Department of Mechanical Engineering and Materials Science William Marsh Rice University Houston, Texas 77251-1892 U.S.A. (Received May 23,1995)
Introduction The atoms B and He, dissolved in j5z.c. Fe and Ni , form solid solutions. Evidence exists that they are of the hybrid type, i.e. that the solute atoms are located at both interstitial and substitutional sites in the metal lattice. The statistical mechanics of hybrid solutions has been discussed previously [l], and a simple theory of solute diffusion in such a solution crystal has been presented [2]. In the case of B in jc.c. iron, the diffusion activation energy is 88 kJ/mol (0.91 eV) [3], even smaller than that found for C in ~c.c. Fe of 142 kJ/mol (1.47 eV) [4]. This small activation energy indicates that 1I diffuses interstitially in the 6c.c. Fe lattice, but the decrease in X-ray lattice parameters [.S]with increasing B - content indicates substitutional solution formation. However, Wang, Zhang, and He [6] have recently used particle tracking auto-radiography to study B -diffusion in the temperature range 1173- 1473 K in a series of Fe - based solutions and in Ni . The results of this work for Ni - B may be expressed in Arrhenius form D=Do exp(-Q/kT), (1) with Do = 6.6 x 10m7 m*/sec and the activation energy Q = 96.3 kJ/mol (0.99 eV). These results are in good aco3rd with those of Chu [7], who obtained Do = 1.1 x 10m6 m*/sec and Q = 96.3 kJ/mol (0.99 eV). The data of Wang et al. [6] for the fc.c. Fe - based alloys yield Q -values in the range lll115 kJ/mol (1.1.4- 1.19 ev). These Q -values are smaller than those of Busby (142 kJ/mol, 1.47 eV) [3] for B in y -Fe. The small Q -values lead Wang et al. [6] to conclude that B - atoms occupy interstitial sites in ~c.c. Fe. If the magnitude of Q is the sole criterion for site occupancy, the conclusion would also be made that B occupies interstitial sites in Ni . The purpose of the present contribution is to point out that such deductions may not be made since the kinetic behavio:r of hybrid solid solutions may be essentially determined by a small fraction of interstitial solutes, whereas the predominant location of solute atoms may be substitutional. Discussion It was shown previously that for hybrid solid solutions containing solvent (u), solute (u) atoms, and vacancies, the fraction of solute (u) atoms in interstitial sites Fi is given by [l] Fuj = [l+x(l+ZKCo)]-l. 1265
(2)
1266
DIFFUSION OF BORON
Vol. 33, No. 8
This formula refers to octahedral occupation in the ~c.c. lattice, and restricts vacancies to having zero or one nearest - neighbor u - atoms in a substitutional (s) position. Z is the coordination number, and x
=
exp(AE/kT)
co
=
exp (-EfvlkT)
K
=
exp (Ebl/kT) ,
(3)
where AC = Ei - Ei , the difference in energies of the u - atom in the interstitial (j) and substitutional (s) sites, Ef, is the formation energy of a monovacancy, and Ebr is the binding energy of the u(s) -vacancy pair. The vacancy formation entropy has been neglected. Now the u -atoms can diffuse by making hops between interstitial sites with the motion energy E?‘, or by exchanging with vacancies with the motion energy EF” . When AC > 0 (X >> 1) and s -occupancy predominates, it is easy to show that the diffusion activation energy is given by Q g E;‘i + AC _ Kf ij f2
AE,
e-AE/kT
e-AEajkT
[
_ ,-(AE+AE)kT
I
II
,
(4
where K’ and C are of order unity, f is the correlation factor for the exchange mechanism, AE, = m,s - EyGp, A E is given by [2] L AE = Eiv - Ae + ( E,m3$- E,m’j ) ,
(5)
and Erip is the solvent-vacancy motion energy for a solvent atom nearest-neighbor to a substitutional solute. It has been shown that [2] f g [ I+ 3 e-AWkT l-1 , (6) so that f2 approaches unity for AE, > 0 and f2 < 0 for A Ea < 0. Furthermore, since both AE > 0 and A E > 0, the entire third term in Eqn. (4) is much smaller than E,“” + AC , since the upper bound of AE, e-AEafkT is kT e- 1 = 3.67 k.I/mol (0.004 eV) at 1200 K. Thus, Eqn. (4) may be written as
Q=E,m’j+A&
(7)
The justification for taking A E > 0 is the large value of Ef,
for both y-Fe
(164 kJ/mol, 1.7 eV [S]),
and for Ni (156 kJ/mol, 1.6 eV [9]). Since Er3’ and Ervi will be comparable in magnitude, Eqn. (5) shows that A E > 0 for reasonable values of AE . The counterpart of Eqn. (7), for the case when AE is so large that Fi = 1 - Fi is essentially unity and j - mode diffusion is suppressed, is [2] f QsEE,,+E,,
ms .
(8)
Since EzvS > 0, this situation clearly does not apply to the known diffusivity measurements for B in Fe andNi since Q < E,,f . Due to the fact that, in the temperature range of the measurements of Wang et al. [6], the Arrhenius relation is well obeyed and Q seems to be reasonably constant, we may assume that J’-mode diffusion behavior predominates and Eqn. (7) applies. The important point is that Q is still given by this relation even when the predominantly occupied sites are substitutional. If we take, conservatively, AE = 40 k.I/mol (0.41 eV), then for lbi = 0 we have, at 12OOK, Fl = 0.985 for Fe and Fi = 0.984 for Ni . Unfortunately, there are no reliable data for Ebi for B - Ni . However, for B -Fe, Chapman and Walker [lo] have used elastic methods to calculate Ebi = 54 kJ/mol (0.57 ev), close to the value estimated by
Vol. 33, No. 8
DIFFUSION OF BORON
1267
Williams et al. [ll] (48.2 kJ/mol (0.5 ev>). If ebl has a similar value for B -Ni , then because of the low values of Co ( 1.8 x 10s7 for Ni and 8.14 x lo-* for Fe ), the value of F,” is hardly affected. It is, of course, true that in the limit when AE assumes negative values, Fi rapidly approaches unity and Q w E:‘j. This situation is, however, not in accord with the X-ray lattice parameter measurements for Fe [5]. Measurements on the effect of B - additions to the lattice parameter ofNi are currently underway in the author’s laboratory. Conclusions The low values of the activation energy for diffusion found recently by Wang et al. [6] and by Chu et al. [7] for B in ~c.c. Ni - and Fe -based solutions does not, of necessity, indicate that the B -atoms are predominantly located in interstitial sites in the solvent lattice. Such behavior is consistent with rapid interstitial jumping in a hybrid solid solution in which the solute (B) atoms are located predominantly in the substitutional sites. Acknowledgment The author is grateful for the support provided by the Robert A. Welch Foundation. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
R. B. McLellan, J. Phys. Chem Solids SO, 49 (1989). R. B. McLellan, J. C/rem. Phys. 54,465 (1993). I? E. Busby, M. E. Warga, and C. Wells, Truns. AIME 197,1463 (1953). I. I. Kcvenskl, Fir MetaL i Metalloved. 16, 107 (1963). R. M. Goldhoff and J. W. Spretnak,J. Met 9,1278 (1957). W. Wang, S. :Zhang, and X. He, Actu MerafL Mater. 43,1693 (1995). Y Y. Chu, P. Ji, and T. Kc, Acta MetalL Mater. 27,8303 (1991). S. M. Kim and W. J. L. Buyers,J. Phys. F. 8, L 103 (1978). W. Wycisk and M. Feller-Kniepmeier,J. Nut. Mater. 70,616 (1978). M. A. V. Chapman and R. G. Faulkener,Actu Metall. 31,677 (1983). ‘I. M. Williams, A. M. Stoneham. and D. R. Harris, Metals Sci. lo,14 (1976).