Distribution of vacancies among impurity atoms in quenched alloys

Scripta METALLURGICA

Vol. 4, pp. 617-622, 1970 Printed in the United States

Pergamon Press, Inc.

DISTRIBUTION OF VACANCIES AMONG IMPURITY ATOMS IN QUENCHED ALLOYS V. Ramachandran Department of Metallurgical and Materials Engineering University of Florida Gainesville, Florida

N. K. Srinivasan Henry Krumb School of Mines Columbia University New York, New York

(Received June 19, 1970) The distribution of quenched-in vacancies among impurity atoms has profound importance in the analysis of clustering kinetics and precipitation hardening mechanisms.

For this purpose some modified form of Lomer equation

(I) for vacancies in equilibrium condition is used. been reported in literature. Hasiguti (3).

Many such equation s have

See for instance Perry and Entwistle (2) and

In the case of quenched alloys, the non-equilibrium concentra-

tion of vacancies can be treated using formal equilibrium mass action relationships for reactions between vacancies and impurity atoms.

The important

condition, however, is that the total vacancy concentration should correspond to the quenched-in value.

This will be a function of time as the vacancies

anneal out at the aging temperature and this aspect has been treated separately elsewhere by the present authors (4).

The only justification for using equi-

librium methods at the aging temperature is that the concentration of quenchedin defects decreases very slowly.

The purpose of this paper is to derive most

general expressions for the distribution of vacancies among the impurity atoms in the quenched alloy and to deduce the approximate equations normally employed in studies of kinetics of clustering and in calculating vacancy-impurity binding energies. We shall employ the Lomer equation for total vacancy concentration in equilibrium at the quenching temperature.

We assume that the impurity concen-

trations are small and there is no interaction among the impurity atoms.

The

expressions are derived for the binary and ternary (binary with impurity) alloys as most experiments involve comparison between the two cases. The total vacancy concentration in the binary alloy with impurity (solute) concentration C I is given by cbinary Tq

=

(I

-

13C I

+

12CI egl/kTq) e -gf/kTq

617

(i)

618

DISTRIBUTION OF VACANCIES AMONG IMPURITY ATOMS

Vol. 4, No.8

where gl is the binding free energy between vacancy and solute, gf the free energy of formation of free vacancies in the alloy, k the Boltzmann constant, and Tq the quenching temperature. A coordination number of 12 is used throughout this paper. Similarly for the ternary alloy with the two impurity concentrations C 1 and C2,

cternary T q

= (1 - 13C I + 12clegl/kTq - 13C 2 + 12c2eg2/kTq) e-gf/kTq

(2)

where g2 is the binding free energy between the vacancy and second impurity. We employ mass action laws to find the concentration, after quenching, of free and bound vacancies at the aging temperature. For the binary alloy, vacancy + impurity I = vacancy~-+impurity I pair V + C 1 ~+ V+-~C1 [V~-*C1] _ 12egl/kTA

(3)

[v][cI] or

[V4-*Cl] = [V] [Cl] 12e gl/kTA

Now [el] = C 1 - [V÷÷C1] = C 1 as the pair concentrations are much smaller than C 1 and this approximation is valid in most practical cases. Therefore, [V~-*Cl] = [V] 12CI egl/kTA The total vacancy concentration is C binaryTA = [V] + [V~-~C1] = [V] (1 + 12C1 egl/kTA)

(43

We set this equal to a fraction of the total vacancy concentration at Tq, i.e. cbinary = a I c binary TA VTq where a I = 1 for perfect or ideal quench. centration at TA using (I) and (43.

(s) We can solve for free vacancy con-

Vol. 4, No.8

DISTRIBUTION OF VACANCIES AMONG IMPURITY ATOMS

gl/kT v ] b i n a r y = al(1 - 15C 1 + 12Cle TA

1 + 12Cle

-gf/kTq q) e

(6)

gl/kTA

[V÷÷Cl]binary = IV] binary • 12C 1egl/kTA L TA

and

619

(7)

Similarly for the ternary alloy, we treat the second impurity reaction independent of the first impurity because of the assumption of no interaction among impurities.

V + C2 ++- V'(-+C2 [V*+C2] - 12eg21kTA

(8)

[V][C 2 ] [V÷*C2] = [V] 12C2 eg2/kTA The total vacancy concentration is

cternary TA = [V] ÷ [V~-~C1] ÷ [V÷÷C2] = [V] (1 + 12Cle gl/kTA + 12c2eg2/kTA) .

(9)

As before we set this concentration equal to a fraction of the total concentration at Tq in the ternary alloy, i.e. C ternary = a 2 C ternary TA Tq

(10)

From equations 2, 9 and 10, ternary = a2(l

13Ci +

12clegl/kTq - 15C2 + 12c2eg2/kTq) e -gf/kT q

[V]T A (1 and

+

12Cle gl/kTA

[V.-.Cl]ternary = [V]TA

12CI egl/kTA

[V÷÷C2]ternary = [V]TA

12C2e

g2/kTA

+

(11)

12c2eg2/kTA)

(12) (13)

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DISTRIBUTION

OF

VACANCIES AMONG IMPURITY ATOMS

Vol. 4, No.8

Approximate Relations In most cases, the rates of diffusional processes proportional

to vacancy

concentration or vacancy-impurity pair concentration are compared between binary and ternary alloys

(5).

• ( Rate)bznary = R (Rate) ternary

Therefore,

[V]bAnary

[V~-+Cl]binary

rvl ternary t-, TA

[V~_+Cl]ternary

"

(14)

From equation 6 and ii,

~I R

(i + 12Cle gl/kTA + 12C2eg2/kTA)

:

(1 + 12Cl egl/kTA) (1

13C 1 + 12clegl/kTq) (15)

(I

13C 2 + 12c2eg2/kTq)

13C 1 + 12C1 egl/kTq

The above expression is the most general one for the ratio of rates in the binary and ternary alloys.

Note that it is independent of the free energy of

formation of vacancies gf.

Secondly,

it depends both on the quenching temp-

erature Tq and the aging temperature T A.

But certain approximations

are possi-

ble. (i)

If the quench is equally efficient for the binary and ternary alloy,

then el = e2"

This is not satisfied in most cases as the formation of dislo-

cation loops during the quench depends on the presence of impurities and on their binding energies

(5).

If el < e2' then setting ~I = e2 will lead to a

higher value for the binding energy. (ii) For high enough quenching temperatures,

the third bracketed term is

nearly equal to unity. Under these conditions R depends only on the aging temperature.

R

1 + 12Cle

gl/kTA

+ 12C2e

=

g2/kTA (16)

1 + 12Cle

gl/kTA

Vol.

4, No.8

DISTRIBUTION OF VACANCIES AMONG IMPURITY ATOMS

A similar expression is used by Perry and Entwistle

(2).

621

They further assume

that the major solute atom (Cu in the case of AI-Cu alloys) has zero binding energy and R = 1 + 12Cimpe gimp/kTA

If the major solute atom has very nearly

zero binding energy, Lomer equation may still be applicable

for these alloys

and addition of impurity to the alloy can be treated as if adding impurity to the pure metal.

For instance, we can compare the ratio for AI-Cu alloy and AI-

Cu-In alloy as if In is added to pure AI.

The free energy for formation of

vacancy in the binary alloy may be different from that for pure metal but this does not enter into the expression for R.

On the other hand, if even a small

binding energy exists for the major solute atom and vacancy,

the above treat-

ment using Lomer equation is invalid and the use of the approximate relations is still further objectionable. Careful considerations

of the various factors elaborated here should help

in extracting more reliable binding energy values from studies on clustering kinetics. methods

At present this method gives higher values than the more direct

(2).

It should be added that we have assumed that the equilibrium con-

ditions regarding the reactions tain amount

concerned are quickly established at T A.

Cer-

of time may be needed and transient effects may be observed soon

after quenching

in the presence of small concentrations

of impurities

(6).

References i.

W. M. Lomer, " V a c a n c i e s and P o i n t D e f e c t s M e t a l s Symposium, London ( 1 9 5 7 ) , p. 79.

in Metals

2.

A. J .

Metals,

3.

R. H a s i g u t i i n " P o i n t D e f e c t s and T h e i r Gordon and B r e a c h , New Y o r k , 1966.

4.

N. K. S r i n i v a s a n

5.

R. E. Smallman and A. Eikum i n " L a t t i c e R. M. C o t t e r i 1 1 e t a l . , Academic P r e s s ,

6.

A. J .

P e r r y and K. M. E n t w i s t l e ,

J.

Inst.

96,

Interactions,"

and V. R a m a c h a n d r a n , P h y s i c a S t a t u s

P e r r y and K. M. E n t w i s t l e ,

Phil.

and A l l o y s , "

J.

344 ( 1 9 6 8 ) . Ed. R. H a s i g u t i , Solidi,

36,

673 (1969)

D e f e c t s i n Quenched M e t a l s , " New York, 1965. Mag., 1 8 ,

Inst.

1085 ( 1 9 6 8 ) .

Ed.