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J O U R N A L OF M A T E R I A L S SCIENCE L E T T E R S 6 (1987) 431-433
Self-diffusion coefficient of
-zirconium
A. J. MARZOCCA, F. POVOLO* Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Dto. de Fisica, PabellOn 1, Ciudad Universitaria, (1428) Buenos Aires, *and also ComisiSn Nacional de Energ/a At6mica, Dto. de Materials, Av. del Libertador 8250, (1429) Buenos Aires, Argentina G. H. RUBIOLO Universidad Nacional de Rosario, Facultad de Ciencias Exactas e Ingenierfa, Instituto de Ffsica Rosario, Av. Pellegrini 250. (2000) Rosario, Argentina
In recent years several investigators have studied the self-diffusion coefficient of zirconium, Dv, both in the alpha phase of the pure metal and in dilute zirconium alloys [1-9]. In fact, the first results were published by research groups of the Soviet Union [1-4] and Argentina [7] and, in both cases, the measurements were performed at temperatures near the ~ to/~ phase transition and in polycrystals. Hood and Schultz [8] measured Dv in e-Zr single crystals at 1124 K, oriented for diffusion parallel and perpendicular to the hexagonal axis. The most recent and important work in this field, however, was presented by Horvfith et al. [9] where the self-diffusion coefficient was measured between 779 and 1128K, using ion-beam-sputtering techniques. The authors found an anomalous behaviour in the metal, because Dv does not follow an Arrhenius law. No physical model has been given by the authors to explain this anomaly. Hood [10] has suggested that such anomaly might be produced by the more rapid diffusion of impurities present in e-Zr but, to date, no clear explanation has been given for this behaviour. It is the purpose of this letter to introduce new elements into the discussion of the self-diffusion coefficient of e-Zr, including new values obtained from an analysis of results obtained by mechanical testing, in the temperature region 633 to 695 K. The results found by Horv~ith et al. [9] indicate that Dv is, approximately, two orders of magnitude lower than the value reported by Dyment and Libanati [7], which was attributed to a higher dislocation density, causing a higher diffusion rate through the dislocations core. This interpretation agrees with the theoretical work of Hart [11] and, more recently, with the works of Le Claire and Rabinovitch [12-14], who proposed that the apparent diffusion coefficient, Da, is related to Dv through the relationship
Ov[l+( Dv
(1 - s) +
Dv = Do exp ( - Q / k T )
s
(2)
where s = rta2Q. On following the estimation made by 0261-8028/87 $03.00 + .12 © 1987 Chapman and Hall Ltd.
(3)
where D o is the pre-exponential factor, Q is the activation energy (strictly the enthalpy) for self-diffusion, k is the Boltzmann constant and T the absolute temperature, it is possible to fit the values obtained at high temperatures to [9] Dvl =
1.59
× 10 -I3
x exp(-lll.87kJmol
1/kT)mesec-1
(4)
and, at low temperatures to [9] Ov2
=
10.8
exp ( - 3 5 1 . 0 4 k J m o l - l / k T ) m 2 s e c
i
(5) It is interesting to point out that the anomalous variation of Dv with the temperature can be accounted for by considering an analogous electrical model of fluxes in series [15], where Dv = DlvD2v/(D~v + D2v)
where D' is the diffusion coefficient in the dislocation core of radius, a, and ~ is the dislocation density in the material. Equation 1 can be written in normalized form as Da -
Horvfith et al. a = 5 x 10-~°m, and it is possible to plot Equation 2 as Da/D v against s, for different values of D'/Dv, as shown in Fig. 1. It can be appreciated in this figure that if D'/Dv ~ 107 there will be a difference of two orders of magnitude between D a and Dv when log s = - 5, which implies • ,,~ 10~3m2. Furthermore, if D'/Dv decreases, the dislocation density must increase to maintain the difference of two orders of magnitude between Da and Dv. Moreover, it is also evident from Fig. 1 that D, ~ Dv if Qdecreases (lower s). As mentioned above, the anomalous behaviour of Dv, measured by Horvfith et al., has not been explained within the context of a physical model. Observation of Fig. 5 of Horvfith et al.'s paper [9] indicates that there might be, in a first approximation, two different mechanisms, acting at low and at high temperatures, respectively, even if the identity of each mechanism is not known. In fact, on considering an Arrhenius law
(6)
Fig. 2 shows the curve given by a plot of Dv against temperature, calculated using Equation 6 and the values of Dv~ and Dv2 given by Equations 4 and 5. It is seen that the fitting to the experimental data of Horv~ith et al. is excellent. Even if the model of fluxes in series is used mainly in the study of diffusion in a matrix with the presence of a second phase (which, in principle, is not the 431
Figure 1 Plot of Equation 2 for different values of D'/Dv.
nl
o ._1
107
0 -10
-9
-8
-7
-6
-5
_;,
_~
.~
Log s
present case), future investigations should be centred towards the identification of the mechanisms acting at low and at high temperatures. Povolo and collaborators [16-20] have estimated the self-diffusion coefficient for zirconium, in the temperature range 653 to 693 K, through analysis of creep and stress-relaxation data in Zircaloy-4, using a specific dislocation model for plastic deformation. The model used was proposed by Gittus [21] and is based on a theory proposed bvy Barrett and Nix [22] for non-conservative motion of screw dislocations controlled by jog drag. On adding a component due to cell formation, Gittus has obtained the following expressions, relating the applied stress, o-, to the plastic strain rate, 8, (~/~.),/3
+
sinh-' [(~/~*)~/~fl]
worked, stress-relieved, tensile axis forming different angles with respect to the rolling direction, etc.), showing that, in all cases, the experimental log a log ~ curves could be fitted to Equation 7, leading to Dv = 1.58 4- 0.56 x 10-21m2sec -~, calculated from Equation 13. Povolo and Capitani [19] have interpreted their stress-relaxation data in bending, obtained at 633 and 673 K in Zircaloy-4 with different thermomechanical treatments, in terms of the model described by Equations 7 and 13. The parameters obtained from the stress-relaxation curves lead to the following values for the self-diffusion coefficient: Dv = 2.0 x 10-22 m 2sec ~at 633 K and Dv = 5.3 x 10-22 m 2sec-I at 673 K, respectively.
(7)
-15
-16
where fl
=
G2ABK 6
(8)
4"
=
B(kT)3K6/b613
(9)
c~ =
b2l/kT
(10)
B
= qDvb/G~kT
(11)
A
=
(12)
b2/2Oc~Dv
b is the Burgers vector, G is the shear modulus, l is the distance between neighbouring jogs, c3 is the thermal jog concentration and K is the ratio of cell-diameter to mean dislocation spacing. On combining Equations 7 to 12 it is easy to show that fi = C2t,2~2~*/20Dv
(13)
The parameters ~, ~* and fl change with the creep strain, during an experiment, saturating their values in the steady state regime. These parameters can be obtained by fitting the experimental creep curves to the theoretical model, described by Equation 7. Once the parameters are known, Dv can be obtained from the creep data by using Equation 13. Povolo et al. [16-20] and Marzocca [23] have performed creep experiments in Zircaloy-4 at 673 K, for different initial conditions of the material (cold432
[] 013 []
I -17 ~7 Io
-18
-19
-20 o Ao
-21
4-
0
+ -22 I -23
I
8
9
10
11
12
13
14
I
15
16
! l l O 4 K-1 ) 7 Figure 2 Arrhenius plot o f D v for a-zirconium. The full curve represents the values given by Equation 4. (O) [9], (D) [7], (,7) [8], (zx) [23], ( + ) [191, (o) [241.
Recently, Povolo and Rubiolo [24] have studied the load-relaxation behaviour of polycrystalline ~-Zr, in the temperature region 645 to 695 K. The results were analysed within a dislocation model that considers the non-conservative motion of screw dislocations and the formation of cells, adding an internal stress which takes into account the segregation of impurities toward the dislocations. A value for Q of the order of 95 kJ mol ~ was obtained from the stress-relaxation curves, which is close to the value obtained by Povolo and Capitani [19], i.e. 87kJmol ~. In addition, TEM observations of the zirconium specimens lead to a cell diameter, L, of the order of 2 pm for a strain, ~, of 0.18 and K was estimated to be of the order of 10. On considering the relationships [24] 0
=
(K/L) 2
temperature has been given. Finally, values for Dv, obtained from an analysis of mechanical testing of a-zirconium and Zircaloy-4, within a model based on dislocation theory, have been presented.
Acknowledgements The authors thank the Consejo Nacional de Investigaciones Cientificas y T6cnicas (CONICET), the Comisi6n de Investigaciones Cientfficas de la Provincia de Buenos Aires and the "Programa Multinacional de Tecnologia de Materiales" OAS-CNEA, for support.
References 1.
(14) 2.
and Om =
CJ'
(fl' = 0.66)
(15)
3.
where 0 and Qmare the total and the mobile dislocation densities, respectively, and, on taking [25]
4.
Om ~- 0.30
(16)
5.
leads to C = 2.54 x 10~3m2. Furthermore, since CDo = 0.95sec ~ [24] then Do = 3.74 x 10-14m2 sec 1and Dv can easily be obtained by using Equation 3. Fig. 2 shows the different values of Dv, obtained from mechanical testings, performed at the different temperatures, together with the most important values obtained by diffusion techniques. It can be seen that the values obtained from mechanical testings are higher than the values expected according to the data of Horvfith et al. [9]. This might be due to a contribution of the applied stress to the diffusion or to the presence of a higher dislocation density in the material, aiding the more rapid diffusion along the dislocations core. It might also be possible, that some vacancy-trapping mechanism is present, at low temperatures, in the specimens used by Horvfith et al. This would decrease Dv, due to the lowering of the concentration of mobile vacancies. In fact, several hysteresis effects have been observed, below about 1000K, on the high-temperature internal friction of zirconium [26]. Such a trapping mechanism would not be important enough to affect the values for Dv obtained from mechanical testing, since the diffusion is stress-assisted and enough vacancies are always available. Furthermore, this would explain the fact, as shown in Fig. 2, that the values obtained through mechanical testings coincide well with those given by an extrapolation of Equation 4 to lower temperatures. It is clear that further work is needed in this field. In conclusion, recent results on the self-diffusion coefficient of a-zirconium have been discussed and a relationship to fit the anomalous behaviour of Dv with
6. 7. 8. 9. I0. I1. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
P. L. G R U Z I N , V. S. E M E L Y A N O V , G. R I A B O V A and G. F E D E R O V , 2rid International Conference for Peaceful Uses of Atomic Energy, Vol. 19 (1958) p. 187. E . V . BORISOV, Yu. G. G O D I N , P. L. G R U Z I N , A. I. Y E V S T Y U B I N and V. S. E M E L Y A N O V , Metall. Metallov. Isdatel'stvo Akad. Nauk. SSSR (1958) 196. V. S. L Y A S H E N K O , B. N. BIKOV and L. V. PAVLINOV, Phys. Metall. Metallov. 8 (1959) 362. G. F E D E R O V and F. I. Z H O M O V , Met. i. Metallov. Chistykh Metal. ! (1959) 162. M. C. N A I K and R. P. A R G A W A L A , Acta Metall. 15 (1967) 1521. P. F L U B A C H E R , EIR-Berieht 49 (1963). F. D Y M E N T and C. M. L I B A N A T I , J. Mater. Sei. 3 (1968) 349. G. M. HOOD and R. J. S C H U L T Z , Aeta Metall. 22 (1974) 459. J. H O R V A T H , F. D Y M E N T and H. M E H R E R , J. Nucl. Mater. 126 (1984) 206. G. M. HOOD, ibid. t35 (1985) 292. E. W. H A R T , Acta Metall. 5 (1957) 597. A. D. Le C L A I R E and A. R A B I N O V I T C H , J. Phys. C 14 (1981) 3863. Idem, ibid. 15 (1982) 3455. Idem, ibid. 16 (1983) 2087. B. S. B O K S H T E I N , "Diffusion in Metals" (MIR, Moscow, 1978). F. POVOLO and A. J. M A R Z O C C A , J. Nucl. Mater. 118 (1983) 224. Idem, ibid. 119 (1983) 78. F. POVOLO, A. J. M A R Z O C C A and J. C. C A P I T A N I , Phil. Mag. A 48 (1983) 3299. F. POVOLO and J. C. C A P I T A N I , J. Mater. Sei. 19 (1984) 2969. F. POVOLO, A. J. M A R Z O C C A and G. H. R U B I O L O , Res. Meehaniea 12 (1984) 27. J. H. G I T T U S , Phil. Mag. 34 (1976) 401. C. R. B A R R E T T and W. D. NIX, Aeta Metall. 13 (1965) 1247. A. J. M A R Z O C C A , P h D thesis, University of Buenos Aires (1986). F. POVOLO and G. H. R U B I O L O , Phil. Mag. submitted. A. ORLOV~& and J. C A D E K , Mater. Sci. Engng 77 (1986) 1. F. POVOLO and B. J. M O L I N A S , J. Mater. Sci. 20 (1985) 3649.
Received 10 September and accepted 23 September 1986
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