The activation energy for self-diffusion in the Cu3Au alloy

J. Phys. Chem. Solids.

Pergamon

Press 1966. Vol. 27, pp. 1035-1039.

THE ACTIVATION

Printed in Great Britain.

ENERGY FOR SELF-DIFFUSION

IN

THE CusAu ALLOY S. BENCI and G. GASPARRINI Istituto di Fisica dell’UniversitB, Parma, Italy Gruppo Nazionale di Struttura della Materia de1 C.N.R. (Received

15 October

1965; in revisedform

3 February

1966)

Abstract-The

ordering process in the Cu3Au alloy below the critical temperature is shown to obey initially a simple rate theory, the activation energy for migration of vacancies being dependent on the long range order parameter S: EM = (0.71+0.36

S2) eV.

The self-diffusion energy is consequently found to increase from 1.65 to 2.01 eV while order is progressing. The consistency of the present results with the previous ones is discussed.

1. INTRODUCTION rate of CusAu alloy, subsequently to a quench at above the critical temperature, was investigated in a recent paper.(l) One of the main conclusions was that during the ordering process the long range order parameter S depends on annealing time according to the following equation : THE ordering

ds -

dt

=

r~AS&,(t)(l

- S)3

(1)

v being the atomic jump frequency at the temperature of isothermal annealing, AS, being the change in S which is associated with an atomic jump, if S = 0, and y being equal to 12 for a f.c.c. crystal. Provided the quenching temperature TQ were not too high, the vacancy concentration G_,(t) could be assumed to depend exponentially on time: CJt)

= A exp( - EF/KTQ)

exp( - t/T)

EF being the activation energy for vacancy formation. Equation (1) was integrated with the simplifying assumption that the activation energy EM for vacancy migration is not dependent on S, and consequently both v and the vacancy lifetime 7

were taken as constants throughout the ordering process. The comparison between the kinetics and the amount of ordering predicted by equation (1) and the experimental data led to the determination of both EF and EM, and consequently to the estimate of the self-diffusion energy in the considered alloy (1.67 eV). This value turned out to be in excellent agreement with the one obtained directly from diffusion measurements of Au195 in the disordered alloy (l-66 eV),(s) which is an argument in favour of the adequacy of the adopted scheme and of the related assumption. The self-diffusion energy was found by BASSANI et d.(3) to depend linearly on the long range order parameter S in the case of the CuZn alloy, and in the quoted paper the reasonable conclusion was reached that what is order-dependent is essentially the migration energy of vacancies. More recently GIRIFALCo(4) predicted that for a (50-50) AB alloy the self-diffusion energy should be a linear function of 29, and not of S, and the validity of this theoretical result was checked by making use of the diffusion data for p-brass by KUPER et al.@) The discrepancy between the functional-dependences of the self-diffusion energy on S which results from two mentioned papers is likely to be due to the fact that Bassani et al. used values of S given by Bethe’s first order approximation, while Girifalco assumed S values which were

1035

1036

S.

BENCI

and

G. GASPARRINI

experimentally determined for ,%brass by CHIPMAN and WARREN.@) Girifalco’s calculation does not essentially contradict the fact that only the migration energy of vacancies is order-dependent, as if is shown there that the formation energy of vacancies is very little dependent of S. In order to remove the mentioned hypothesis of the constancy of EM which might have represented a source of systematic error in the previous paper,(l) equation (1) was compared with the results of isothermal ordering experiments carried out at below the critical temperature. In these conditions C, is constant, and the dependence of EM on S was introduced. With this procedure the self-diffusion energy in the disordered CusAu alloy was redetermined, and the importance of the additional S-dependent term was estimated. We may anticipate that a final choice between the proposed dependences of the self-diffusion energy on S will be probably possible for the Cu& alloy with the support of self-diffusion measurements below the critical temperature; actually in the present paper both dependences were taken into consideration, but the interval of S which could be investigated was confined to small S values, so that defined conclusion could not be reached on this point. It is however felt that the dependence calculated by Girifalco should be preferred in principle. 2. EXPERIMENTAL PROCEDURE The long range order parameter was determined indirectly from resistivity measurements by making use of the curves given by ANQUETIL,(~) which were recalculated at 20°C at which temperature resistivity measurements were performed. CusAu wires,* about 20 cm long and 0.07 mm dia. were used. They underwent a preliminary 2 hr annealing at 95O”C, which treatment was also effective in outgassing them and stabilizing the grain size. Specimens were then quenched at 420°C with the purpose of disordering them fully; the consequently introduced small excess of vacancies was eliminated with a 1 hr isothermal annealing at 165”C, at which temperature ordering effects due to equilibrium vacancies were completely negligible. The consequence of the

* Supplied by Sigmund Cohn Corp.

described preliminary treatment was that the initial value of the long range order parameter was s = 0.1. All the annealings were performed within a previously evacuated vessel (10-s mm Hg), successively filled with accurately purified Argon at 850 mm Hg. A conventional potentiometric system was used for resistivity measurements, the limits of the relative error having been kept within + 2 x 10-s.

3.

EXPERIMENTAL RESULTS: DE!XRIPTION AND IN TERPRETATION Figure 1 shows some typical initial resistivity decreases which were noticed during isothermal annealings in the temperature range between 298 and 322°C. All curves were normalized, the initial resistivity values having been taken equal to 1. The corresponding values of the long range order parameter S, which were obtained from the resistivity data, are plotted as functions of time in Fig. 2 for some of the curves given in Fig. 1. Under the hypotheses that S varies during ordering according to equation (l), and that EM is

0+9-

0

1

2

3

4

5

6

7

8 t (min)

FIG. 1. Initial resistivity decrease during annealing.

THE

ACTIVATION

linearly order-dependent,

ENERGY

FOR

SELF-DIFFUSION

we may write

Ei3$ = E&Pi-;tS

(2)

or alternati~eIy EM = E&Q-X’S3

(2’1

h and A’ being constants and EMObeing the migration energy of vacancies in the disordered alloy. For instance, from equation (2’)

E&&i- A’S KT

THE

CusAu

ALLOY

1037

So being the long range order parameter at t = 0 and Qe = E& EJ@ being the self-diffusion energy in the disordered alloy. A direct comparison between equation (4) and the experimental data having considered unpractical, it was decided to proceed by successive approximations. The first order term of the series expansions of equation (4) in powers of (S-Se) gives

AS&S, - S)3 >

t

x A exp( - E&CT} where S, indicates the equihbrium the annealing temperature T.

(31 value of S at which of course predicts a linear dependence of S on t; this approximation is equivalent to take the self-diffusion energy as constant in the (S-Se) range within which equation (5) is assumed to be valid. h’ was obtained by selecting two different values of SO in one of the isotherms and comparing the slopes of the straight lines locally fit the experimental points. The procedure was repeated for all the isotherms and the found values of X were averaged. The average h’ value was then introduced into the formula corresponding to the second order approx~atio~ :

I

I

IN

(6)

FIG. 2. The long range order parameter as a function of the annealing time. The curves were calculated from equation (4).

Equation

s

(3) can be integrated

* exp(X’SsjKT)

so

5

as follows:

S yvoASpA exp( -

(Se-Sj3

and Qa was obtained by plotting the resuhing slopes as a function of I/T, as shown in Fig. 3. The calculation was then iterated by making use of the data derived from the described approximated procedure, and the results were as follows: x’ = 0.36 eV

Qo = (1.65 &O-03) eV

which will be commented in the following section. It has to be noticed that the given values of the parameters were not changed appreciably when a third order appro~~ion was used. If conversely equation (2) is assumed to be valid, the final

1038

S.

BENCI

and

results are X = 0.37 eV

Qs = (1.64 f 0.03) eV

which are identical with the previous ones. What is more critically dependent on the assumed relationship between EM and S is the constant yv,-,ASjA, which is anyway of the order of 1011 se+, but can be varied within a factor of two according to the starting hypothesis.

G. GASPARRINI both if the concentration of vacancies is the one typical of the thermal equilibrium, and if an excess of vacancies is present due to a quench. This statement is confirmed by the fact that the value of ASj (see equation (3)), representing the ordering efficiency by vacancies, is of the same order to magnitude found for quenched-in vacancies(l) (yvoASjA N 1011 see-1). Moreover, if the value of EF given in the above Ref. 1 EF = (0.945 0.02) eV is subtracted from Qs, we get for the migration energy of vacancies in the disordered situation EMO = (0*71& O-03) eV

-y 'O15.2

1 lb5

I 17

I

-I

'7*5 1o'/r

which is in excellent agreement with what found previously.(l) Actually in that case EM was measured in a partially ordered specimen, the maximum S value being about 0.15 : EM was consequently a mean value between those corresponding to S = 0 and S = O-15. If h’ is taken equal to 0.36 eV, the above mentioned mean value of EM is expected to be 0.72 eV according to the present results, and this value is the one to be compared with the previously reported result& (0.73 eV). As far as the A’ parameter is concerned very few data about the CusAu alloy can be found in the literature. The most direct comparison is possible with a paper by FEDER et aZ.(lO) in which the self-diffusion energy in the ordered alloy was estimated equal to 2.03 eV. This is in excellent agreement with the findings of the present research, as from equation (2’) we get for the migration energy of vacancies in the completely ordered phase.

FIG. 3. Deduction of the self-diffusion energy in the disordered CusAu alloy.

EM(ord’ = 1.07 eV and consequently

4. CONCLUSIONS The obtained value of the activation energy for self-diffusion in the disordered alloy, Qs = l-65 eV, is extremely well consistent with the preceding determinations by the same authors,(l.s) and also in good agreement with the results by NAGY et aZ.@) and by DUGDALE@) from measurements performed below the critical temperature. This fact suggests that, at least in the initial ordering stage which was explored in the present research, the features of the ordering process due to lattice vacancies in the Cu&_r alloy are identical

Q(ord) = (2.01+ 0.04) eV This value is also consistent with the one given by GOERING et aZ.(ll) who found 1.95 eV from anelastic measurements on partially long range ordered CusAu. A final check of the validity of the present scheme would be possible from direct measurements of the self diffusion coefficient at temperature below the critical one; a research in this direction is in progress, though remarkable difficulties arise from the fact that the self-diffusion coefficient

THE

ACTIVATION

ENERGY

FOR

SELF-DIFFUSION

is very small (z lo-18 cms/sec) in the convenient range of temperature.

Acknowledgement~The authors are grateful to Prof. E. GERMAGNOLIfor his advice and contribution.

REFERENCES 1. BENCI S., GASPARRINI G., GERMAGNOLI E. SCHIANCHI G., J. Phys. Chem. Solids 26, (1965). 2. BENCI S., GA~PARRINI G., GERMACNOLI E. SCHIANCHI G., J. Phys. Chem. Solids 26,

(1965).

and 2059 and

687

IN

THE

CusAu

ALLOY

1039

3. BASSANIC., CAMAGNIP. and PACE S., Nuovo Cim. 19, 393 (1961). 4. GIRIFALCO L. A., J. Phys. Chem. Solids 24, 323 (1964). 5. KUPER A. B., LAZARUSD., MANNING J. R. and TOMIZUKAC. T., Phys. Rev. 104, 1536 (1956). 6. CHIPMAN D. and WARREN B. E., J. Appl. Phys. 21, 696 (1950). 7. ANQUETIL M. C., J. Phys. Radium, Paris 23, 113 (1962). 8. NAGY E. and TOTH J., J. Phys. Chem. Solids 24,

1043 (1963). 9. DUGDALER. A., Phil. Mag. 1, 537 (1956). 10. FEDER R., MOONEY M. and NOWICK A. S., Acta

Met. 6, 266 (1958). 11. GOERING W.

and NOWICK A. S., Trans. Amer. Inst. Min. Metall. Engrs 2, 105 (1958).