Log-normal distribution of relaxation times or two processes in the kinetics of SRO-relaxation?

Scripta

METALLURGICA

V o l . 19, pp. 1 9 9 - 2 0 3 , 1985 Printed in t h e U . S . A .

Pergamon P r e s s Ltd. All rights reserved

LOG-NORMAL DISTRIBUTION OF RELAXATION TIMES OR TWO PROCESSES IN THE KINETICS OF SRO-RELAXATION? W. Pfeiler, R. Reihsner and D. Trattner Institut f~r Festk6rperphysik der Universit~t Wien, A-I090 Vienna, Austria (Received November 6, (Revised November 26,

1984) 1984)

Introduction The deviation of alloy atoms from a random distribution known as short-range order (SRO) is a frequent property of solid solutions. A number of such alloys show a relaxation behaviour, which is not in accordance with a single exponential process (1-3). Two different explanations for this behaviour are given by various authors: (i) The existence of two simultaneously acting processes with different relaxation times is responsible for the complex relaxation process; this is in good agreement with the SRO-model of disperse order (4), ascribing an adjustment of the volume fraction of ordered zones in a disordered matrix to the faster process, and the regulation of the number of these zones to the slower one. (ii) Starting from a Gaussian distribution of activation energies of processes involved, a log-normal distribution of the relaxation times gives deviations from a singleexponential behaviour; this model is usually applied to stress-induced relaxation (5) and was first applied to SRO resistivity-relaxation by Balanzat and Hillairet (6). Until recently we were convinced that two experiments give evidence for the existence of just two processes in SRO-relaxation kinetics in contrast to a log-normal distribution of relaxation times: isochronal annealing and the cross-over experiment. It is the purpose of the present paper to show that in the case of CuZn an appropriately chosen continuous distribution of relaxation times (e.g. a log-normal distribution) can account for all experimental findings with similar accuracy as for the previous use of only two relaxation times (3). Isothermal Annealing

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(b) FIG.

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Resistivity change during isothermal annealing for a temperature increase from 200 ° to 210°C. (a) Experimental data; (b) semi-logarithmic plot of (a). The curves are calculated using a log-normal distribution of relaxation times with T and B of table 1. (O) 20at% Zn, (A) 25at% Zn, (D) 30at% Zn.

199 0036 9748/85 $ 3 . 0 0 + .00 Copyright (c) 1 9 8 5 P e r g a m o n Press

Ltd.

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KINETICS OF SR0-RELAXATION

Vol. 19, No. 2

Fig. 1 shows the isothermal kinetics of electrical resistivity measured on Cu-20,25,30at% Zn (a) and the logarithm of this resistivity change against linear annealing time (b). For more details of measurement see (3). The points in plot ib deviate significantly from linearity; this indicates that the kinetics cannot be described by a single exponential process. The solid lines of figs. la and Ib are calculated by a least mean square fit to experimental points of a lognormal distribution of relaxation times (equ. I). Ap/~p O = la(x)exp(-t/T(x))dx a(x) = (I/(~/~))exp(-x2/S 2)

(i)

T(x) = {exp(x) Good accordance with experimental points is obtained. Table I gives the fit-parameters determined in this way. TABLE 1 Fit Parameters ~ and B of Log-normal Distribution to Experimental Results on CuZn (7 in min.) T(°C)

170

180

190

200

210

220

230

240

Czn(at%) 20 25 30

182 1.6

66

1.5

128 0.5

71

1.1

39

1.2

73

30

1.4

14

1.3

1.0

20

1.2

7.4 1.4

32 1.4 18 1.3 8.3 1.3 2.7 1.6 Accuracy: AT = ±10%, AB = ±0.15

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100 150 200 250

300

ternperoture (°C) FIG. 2 Relative change of resistivity during isothermal annealing (t=15min.,~T=10°C). For more clarity the curves are shifted parallel to the resistivity axis. The points are labelled as in fig. i.

O. 65100

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i

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200 T (°C)

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L 900

FIG. 3 Normalized change of resistivity calculated using a log-normal distribution of relaxation times. T =2xl0-1~s, B=1.4, Q=1.45eV. o

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201

Isochronal annealing after a n a d e q u a t e t e m p e r a t u r e q u e n c h is a m e t h o d o f t e n u s e d in studying the e s t a b l i s h m e n t of SRO b y m e a s u r i n g e l e c t r i c a l resistivity. This technique g i v e s inform a t i o n a b o u t the atomic m o b i l i t y and the sign and m a g n i t u d e of the c o n t r i b u t i o n of SRO to e l e c t r i c a l resistivity. Further one can decide w h e t h e r single e x p o n e n t i a l k i n e t i c s take p l a c e or not b y looking at the m i n i m u m or m a x i m u m b e f o r e the r e s i s t i v i t y reaches the " e q u i l i b r i u m line". In the case of d e v i a t i o n s f r o m single e x p o n e n t i a l k i n e t i c s as o b s e r v e d for CuZn, small r e s i s t i v i t y m a x i m a are s u p e r i m p o s e d o n the rather b r o a d r e s i s t i v i t y m i n i m u m of isochronal a n n e a l i n g (fig. 2). The e x t e n s i o n of the f o r m a l i s m p r e v i o u s l y u s e d (7) to a log-normal distrib u t i o n of r e l a x a t i o n times fits the e x p e r i m e n t a l data very well (fig. 3). Cross-over Kinetics V e i t h et al. p r o p o s e d a m e t h o d of thermal t r e a t m e n t to test w h e t h e r there is o n l y a single e x p o n e n t i a l process, or w h e t h e r there are two s i m u l t a n e o u s p r o c e s s e s w i t h d i f f e r e n t r e l a x a t i o n times r e s p o n s i b l e for S R O - k i n e t i c s (8). Fig. 4 shows such an e x p e r i m e n t on C u - 2 5 a t % Zn as an example for all c o n c e n t r a t i o n s i n v e s t i g a t e d (3). The t r a n s i e n t v a r i a t i o n of r e s i s t i v i t y after the t e m p e r a t u r e change at t o d o c u m e n t s that the system runs through states w h i c h do not represent e q u i l i b r i u m states of SRO. Veith et al. were able to fit their e x p e r i m e n t a l curves of C u - 1 5 a t % AI, w h i c h showed a similar c r o s s - o v e r effect as CuZn, u s i n g a sum of two e x p o n e n t i a l p r o c e s s e s with r e l a x a t i o n times, w h i c h c o r r e s p o n d to the different a n n e a l i n g temperatures. This m e t h o d was successful w h e n u s e d for CuZn, too (3). But a s p e c t r u m of r e l a x a t i o n times can cause the same c r o s s - o v e r effect if the s p e c t r u m is asymmetric enough; as d e m o n s t r a t e d b y the c a l c u l a ted solid line of fig. 4 c r o s s - o v e r curves can be fitted to similar a c c u r a c y u s i n g a log-normal s p e c t r u m w i t h m e a n r e l a x a t i o n times c o r r e s p o n d i n g to the t e m p e r a t u r e s just b e f o r e a n d after t o .

normalized reeietivity ohan~e 1.0

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FIG. 4 C r o s s - o v e r e x p e r i m e n t on Cu-25at% Zn as an e x a m p l e of the n o n - e q u i l i b r i u m b e h a v i o u r of resistivity k i n e t i c s for all c o n c e n t r a t i o n s investigated. At t=t o a n n e a l i n g t e m p e r a t u r e was c h a n g e d f r o m 230 ° to 221.5°C. The c u r v e is c a l c u l a t e d u s i n g a log-normal d i s t r i b u t i o n of r e l a x a t i o n times w i t h Tl=5.2min., ~2=12min., BI=B2=I.6. Discussion In the i n t e r p r e t a t i o n of a n e l a s t i c relaxation, a c o n t r o v e r s y similar to that in SRO resist i v i t y - k i n e t i c s arose: in m a n y s u b s t i t u t i o n a l alloys a log-normal d i s t r i b u t i o n of r e l a x a t i o n times has b e e n found; in the case of interstitial alloys, d i s c r e t e r e l a x a t i o n p r o c e s s e s h a v e b e e n u s e d for d a t a analysis. L a t e r on it was d e m o n s t r a t e d (9) that the r e l a x a t i o n s p e c t r u m in the case o f i n t e r s t i t i a l alloys also can be e x p l a i n e d in terms of a log-normal distribution. In this c o n n e c t i o n a p o s s i b i l i t y to d i s t i n g u i s h b e t w e e n a c o n t i n u o u s and a d i s c r e t e s p e c t r u m h a s

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been pointed out (9): a plot of ~Ap/~log(t) versus log(t) would give only one maximum for a continuous distribution, whereas multiple relaxation would give more than one maximum, the exact number depending on the number of components and the spread in relaxation times. Such a plot has been tried with the present data, but the relative uncertainty of the individual data points prevented us from discerning between the two interpretations. Because the two analyzing methods stand for different interpretations of microstructure, the heterogeneous and homogeneous interpretation of SRO-microstructure seem to be equally justified. However, the result of the cross-over experiment (fig. 4) and the special behaviour of isochronal annealing (fig. 2) give evidence for the transient existence of SRO-states which do not correspond to equilibrium microstructures. This obviously is what happens when SROrelaxation times are log-normally distributed. Such a distribution can be considered as a consequence of heterogeneity: fluctuations of the local alloy concentration alter the distribution of atoms in the vicinity of an atom-vacancy pair. This in turn affects the energy barrier, which has to be overcome by the moving atom. Hence if one assumes the heterogeneous microstructure to be represented by a Gaussian distribution of activation energies of local diffusion processes, this leads to a log-normal distribution of SRO-relaxation times (5). Thus the interpretation of SRO-microstructure as being heterogeneous is still justified, though in a wider sense: besides highly ordered particles in a highly disordered matrix, concentration fluctuations in addition to the simplest conception of statistical SRO will account for experimental observations in the same way. This means it is not necessary to assume deviations from the mean alloy concentration exceeding the statistical deviations, a case which needs rather complicated considerations in order to explain a thermodynamically stable structure (i0). Instead, statistical deviations from the mean alloy concentration, which are always present in any alloy at temperatures where diffusion is still possible, explain the experimental findings in a sufficient way. However, when evaluating activation energies, essential differences between both methods of analysis show up: On the one hand the assumption of two processes yields, for both processes, equal activation energies, which show a strong dependence on alloy concentration and are not in accordance with the literature (3). On the other hand the values obtained using a log-normal spectrum (fig. 5) show no essential dependence on concentration and are closer to values from the literature (table 2). Nevertheless it seems questionable whether this is a decisive criterion in favour of one of the two methods.

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2.2

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IOOOIT (IIK) FIG. 5 Arrhenius plot of characteristic ordering times of table i. The points are labelled as in fig. i.

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KINETICS

OF SRO-RELAXATION

203

TABLE 2 Activation Energy of SRO-adjustment Using a Log-normal Distribution of Relaxation Times Czn(at%)

Q (ev) fig.5, present work

tracer,(ll)

20

1.42 ±0.20

1.83

25

1.56 ±0.15

30

1.59 ±0.15

1.70

anelastic relaxation,(6)

1.59

Conclusions Experimental deviations from single exponential SRO-relaxation which have earlier been described by two exponential processes can be fitted equally well with a continuous log-normal distribution of relaxation times. Though the activation energies derived in these two ways differ considerably we are at present not able to distinguish clearly between the two methods of data analysis. The consideration of concentration fluctuations in addition to the statistical concept of SRO is equivalent to the model of disperse order. Acknowledgements The continued interest of Professor Dr. K. Lintner is gratefully acknowledged. The authors are indebted to Dr. K. Siebinger and Dr. M. Zehetbauer for valuable discussion and to Dr. W. P~schl for a critical reading of the manuscript. The work was financially supported by the Austrian "Fonds zur F0rderung der wissenschaftlichen Forschung" grant number 4134. References i. 2. 3. 4. 5.

6.

7. 8. 9. I0. ii.

L. Trieb and G. Veith, Acta Met. 26, 185 (1978). F. Adunka, L. Trieb and M. Zehetbauer, phys. stat. sol. a62, 213 (1980). D. Trattner and W. Pfeiler, J. Phys. F: Met. Phys. 13, 739 (1983). H. Warlimont and H.P. Aubauer, Z. Metallkde. 64, 484 (1973) and H.P. Aubauer and H. Warlimont, Z. Metallkde. 65, 297 (1974). A.S. Nowick and B.S. Berry, IBM J. Res. Dev. 5, 297, 312 (1961) and A.S. Nowick and B.S. Berry, Anelastic RelaxatTon in Crystalline Solids, Academic Press, N.Y. 1972. E. Balanzat and J. Hillairet, J. Phys. F: Met. Phys. II, 1977 (1981) and J. Hillairet et al., Internal Friction and Ultrasonics Attenuation in Solids, Ed. C.C. Smith, p. 143, Pergamon Press, Oxford 1980. D. Trattner and W. Pfeiler, Scripta Met. 17, 909 (1983). G. Veith, L. Trieb and H.P. Aubauer, Scripta Met. 9, 737 (1975). S.N. Tewari, Scripta Met. 8, 371 (1974). H.P. Aubauer, Acta Met. 20, 165, 173 (1972). K.J. Anusavice and R.T. DeHoff, Met. Trans. ~, 1279 (1972).