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KINETICS OF SHORT RANGE ORDERING AND
BEHAVIOR OF VACANCIES IN BINARY
SUBSTITUTIONAL ALLOYS
T. Eguchi, Y. Tomokiyo, C. Kinoshita
To cite this version:
JOURNAL DE PHYSIQUE Colloque C7. supplPment ou no 12. Tome 38. dkcemhre 1977. pnge C7-382
KINETICS OF SHORT RANGE ORDERING AND BEHAVIOR OF VACANCIES
IN BINARY SUBSTITUTIONAL ALLOYS
T. E G U C H I , Y. T O M O K I Y O
Department of Materials Science a n d Technology
a n d C. K I N O S H I T A
Department o f Nuclear Engineering, Faculty of Engineering, Kyushu University, F u k u o k a 812, J a p a n
R h m i . - La forme gknerale de l'kquation cinetique dkcrivant la mise en ordre B courte distance
dans des alliages binaires de substitution est discutk en supposant que la mise en ordre se produit par migration de lacunes qui sont creees ou absorbkes au niveau des dislocations et des joints de grains. L'Cquation fondamentale est appliquee ii l'analyse de la variation de rksistivitk observk au cours de recuits isothermes de aCu-A1 a p r b trempe de l'khantillon A des tempkratures variees. On montre que, grlce B un choix approprie de la valeur de 1'Cnergie de formation des lacunes, ces
d o n n k s suivent une l'oi unique en fonction d'une temperature rkduite appropriee. L'equation fanda- mentale n'est pas strictement lineaire mais peut &tre linearis& dans un certain nombre de cas. L'Cqua- tion cinetique simplifiee peut reprtsenter la variation de la rksistivite de la chaleur spkifique et du p a r a m h e cristallin de aCu-A1 au cours de divers traitements thermiques. On discute enfin la variation de 1'Cnergie de formation des lacunes en fonction de la composition de I'alliage.
Abstract. - The general structure of the kinetic equation describing the short range ordering in binary substitutional alloys is discussed, by assuming that the ordering takes place by the migration of vacancies, which are created or annihilated at dislocations and grain boundaries. The basic equa- tion is applied for an analysis of the observed variation in electrical resistivity of uCu-AI in the course of annealing at the same temperature, after the specimen is quenched from various temperatures. It is shown that by a suitable choice of the value for vacancy formation energy those data fit on a single master curve, if an appropriate transformation of the time scale is performed. The basic equation is not strictly linear, but can be linearized under certain limited circumstances. The approxi- mate rate equation can account for the change of the resistivity, specific heat and lattice constant of aCu-AI under various heat treatments. The dependence of the vacancy formation energy on the alloy composition is also discussed.
1. Introduction. - In some binary substitutional alloys, such as the cr phase of Cu-A1 system [l, 21,
constituent atoms are arranged with a certain degree of short range order. T h e purpose of the present work is t o investigate the mechanism of short range ordering in the alloys by a n analysis of the changes of their physical properties under various heat treat- ments.
2. General kinetic equation.
-
T h e state of short 'range order changes due t o the migration of vacancies. The change of any parameter X, which describes the
state of order a t a temperature T, may be approxi- mately represented by t h e following rate equation :
where CV is the vacancy concentration a n d G(x, T )
is a certain function of the degree of order a n d the temperature.
T h e parameter X may stand for any physical
quantity which depends uniquely upon t h e state of order. F o r a typical example we consider t h e relative change of electrical resistivity in aCu-AI during a n isothermal annealing a t a temperature T,,
after the specimen is quenched from a n equilibrium state a t T,. Figure 1 shows the change of the ratio of the resistance in the specimen of Cu-14.0 a t
%
A1 to that'of a dummy with a constant resistance, forT, = 121.4 O C a n d the various values of T,. T h e
measurement was carried out continuously within a n accuracy of 10-S by the use of a n alternative current method [3].
KINETICS O F SHORT RANGE ORDERING AND BEHAVIOR O F VACANCIES c7- ;S.:
0 200 LOO 600 800 1000
Annealing T ~ m e (S)
FIG. I. - Changes of electrical resistivity of Cu-14.0 at % AI as a function of time during annealing at 121.4 OC after quenching from various temperatures. R, and R, are the resistances of the specimen and dummy. R,/R, corresponds to the physical parameter
X in the basic eq. (I).
For an analysis of these experimental data we assume the time dependence of CV as given by
and the decay constant by
where A and B are certain constants, E, and E, the formation and migration energies of vacancy. Comparing the value of the integral
S
dx/G(x, T,) for each curve in figure l over a common range of X, we obtain E, = 1.031+
0.008 eV andThe reliability of these values is confirmed by trans- forming t to z by
and determining /3(Tq) such that .r = 0 for X = X,, a common definite value of X. The original rate equation (1) is then transformed into
d x
- -
- G ( x , T ) , with x = x , at r = 0 . (3) d rThus the curves for various quenching temperature in figure 1 should fit on a single curve under the transformation (2), as is demonstrated in figure 2. The curve for G(x, T,) vs. X is not strictly .linear,
but may approximately be regarded as linear when the experiment is carried out within a limited range of X.
7 ( a r b ~ l . unit)
FIG. 2. - Change of &/R, as a function of 7 after the transfor- mation (4).
3. Linear kinetic equations as applied for -various heat treatments. - In the following we shall adopt a linear approximation for the rate equation of the Warren-Cowley short range order parameter a as :
dcr
- -
d t --
A O ( ~-
cre(T)) CV e x ~ ( - E,,,lkT) , (4)where A, is a constant 141. Equation (4) should be solved simultaneously with an equation which gives the time variation in the vacancy concentration CV :
where N d , a and v are the dislocation density, the lattice constant and the atomic frequency factor. Equations (4) and (5) describe the process of short range ordering caused by the migration of vacancies. The equilibrium values for the short range order parameter (5, 61, cre(T), and the vacancy concentra-
tion, C,(T), are given by the thermodynamical relations, with the ordering energy 1 760 cal/mol for Cu-A1 system [7].
Equations (4) and (5) are used for analyses of the changes of electrical resistivity, specific heat and lattice parameter in crCu-A1 due to the change of short range order by various heat treatments. In solving these equations the time variation of the temperature must be given so as to represent the experi- mental heat treatment as faithful as possible. In the process of cooling from Ti to T,, for example, the temperature is expressed by
T = Tf
+
(Ti-
T,) exp(- tit,) , ( 6 )where 1, is a constant between 0.05 and 0.5 S for
water-quench, or 5 000 S for furnace-cooling.
C7-384 T. EGUCHI, Y. TOMOKIYO A N D C. KINOSHITA
I I I I I I I
6 Cu- 153 at % A l
/
FIG. 3. -Variations of resistivity as a function of quenching temperature. The results for Cu-Zn alloys were obtained by Lang et al. [8]. Dashed curve represents our calculated relative change of
short range order parameter.
(dashed curve) of the short range order parameter under the same heat treatment as in the experiment. The values for the parameters used for the numerical solution are given in the figure. Both the experimental result and the theoretical one are represented as the changes relative to the values at room temperature when the specimen is furnace-cooled. The close agreement between the two shows the adequateness of our model and the approximation. A similar beha- vior of the electrical resistivity in Cu-Zn system [8],
also shown in figure 3, may be explained on the same footing as described here.
The change of the electrical resistivity and specific heat in rCu-AI on heating the specimen by the cons- tant rate, as well as the change of its lattice parameter on isochronal step-annealing, can be accounted for by the numerical solutions of eqs. (4) and ( 9 , with an appropriate value for the vacancy migration energy. The results of the similar experiments on the filed or cold worked specimens are also reproduced by the solutions of eqs. (4) and (5) with a choice of a large dislocation density.
4. Conclusion.
-
From the analyses described in the foregoing sections we conclude that in binary substitutional alloys such as aCu-A1 the ordering takes place only through migration of vacancies, which are created or annihilated at dislocations and grain boundaries. By applying- the general kinetic equation for the process of ordering in isothermal annealing, we obtain the vacancy formation energy in aCu-A1 between 0.90 and 1.10 eV, depending slightly upon the alloy concentration. The linearized kinetic equation, which is valid only over a limited range of the short range order parameter, can account for the changes of some physical quantities under various heat treatments, if we assume the vacancy migration energy between 0.65 and 0.70 eV for aCu-Al.References
[ l ] HOUSKA, C . R., AVERBACH, B. L., J . Appl. Phys. 30 (1959) 1525. [5] FLINN, P. A., Phys. Rev. 104 (1956) 350.
[2] BORIE. B.. SPARKS, C. J . jr.. Acta Crystallogr. 17 (1964) 827. [6] KIDIN, I. I., SHTREMEL, M. A., Phys. Met. MetaNogr. 11 (1961) 1 . [3] KABEMOTO, S., Japan, J. Appl. Phys. 10 (1971) 1251. [7] KINOSHITA, C., TOMOKIYO, Y., MATSUDA, H., EGUCHI, T., [4] A N D R I ~ S , J., BOON, W. G., RADELAAR, S., Phys. Lett. 38A Trans. Japan. Itut. Met. I4 (1973) 91.
