On the kinetics of stage III recovery in cold worked metals

Scripta METALLURGICA

Vol. 2, pp. 327-329, 1958 Printed in the United States

ON THE KINETICS

OF STAGE III RECOVERY

IN COLD WORKED ~ T A L S

A. van den Beukel voor Metaalkunde, Technische The Netherlands

Labora~orium

Pergamon P r e s s , Inc.

Hogeschool,

Delft,

(Received April 10, 1968) The results metals

of the study of reaction kinetics

as found in the literature

can roughly

I) The point defect concentration c The isothermal cribed

annealing

in the stage III range o£ cold worked

be divided

c decreases

in two groups:

with

- exp -( ~)P

time according (I).

o curves of cold worked AI in stage lll can be very well des-

by this relation with p - 0,49 as has been shown by Frois

for Fe (Cuddy,

(2)) with p = 0,43.

p - 0,5 describes contains

obtained

2) According

density,

to Wintenberger

diffusion

of vacancies

which can be evalueted

in this way by Frois

the kinetics

(I) with

to dislocations.

The constant

from the experimental

(for instance

of interstitials

The result

(4),

to immobile

for this second order reaction,

tributed and have equal concentrations.

(I). The same is true

(3) the relation

results.

(I) seem reasonable.

to a second group of authors

annealing is due to the migration derived

According

the stress assisted

the dislocation

The values

to the equation

Waits

(6)

when both defects are uniformly

can be written

C O - 1 = Kit½ + K2t O

(5)) stage III

vacancies.

dis-

as follows:

(2)

wi th

K 1 - 8~½r2D½c c

o

(3)

K 2 - 4W/oDc o where

c = number of migrating c

o

=

the

same

at

D = diffusion r ° = capture combines

test

spontaneously

t

=

defects

0

coefficient

radius; with

per cm 3

of the migrating

an interstitial

defects

being at a distance ~ r ° from a vacancy

re-

the vacancy.

In (2) the first term dominates for short times, the second one for longer times. c the validity o f ( 2 ) u s u a l l y ~ - I i s p l o t t e d vs t ~ f o r s h o r t t i m e s and vs t f o r

longer times.

In both cases Stals and Nihoul

Nb. They conclude

that stage III recovery

of about 10 atomic distances, Recently

which

(5) obtain a straight

obeys equation

To

line for cold worked

(2). For r ° they obtain a value

seems to be rather large.

it has been noted by Cuddy

(2),

327

that the isothermals

of Stals and Nihoul

(5)

328

S T A G E HI R E C O V E R Y

IN C O L D W O R K E D

METALS

Vol. 2, No. 6

7Co_, C 6.

Y

I

i

T

Second

order

reaction

satisfying

for cold worked Nb, fitted

to equation

F I G . "I test to a set

equation

just as those of Schultz

(with their totally different

physical

points

(7) for W and of Martin

(I) with p values between 0,40 and 0,46.

why the same set of data can be described

description

of

(I) with p = 0,~

satisfactorily

backgrounds)

(8) for Mo can be

In this note we shall examine

by both equations

(I) and (2)

and how one can possibly

decide which

is the correct one.

First we have constructed curve was submitted

a curve satisfying

the rel~tion (I) with p = 0,5. This o to the second order test by plotting ~-- ~ I vs t. iThe result is

shown in fig.

I. After a small initial part which satisfies _~o c - I,-, t~ as follows from (I), for a large range of c_.. values a straight line is obtained. Figure I is remarkably c similar to the corrsspondin~ plot of Stals and Nihoul (5) for cold worked Nb. The plot of fig.

I seems good enough

to conclude

to second order kinetics,

which is clearly not

correct. After

series expansion

of the exponent Co c

equation

-i.(9)

(I) can be written as follows:

............ (4)

For not too long times the series can be cut off after equations

(4) and (2) have

the same form, which explains

data to both of them. The difference,

however,

the second

is that for the constants

K I and K 2 in (2) and (3) the relation can be deriveds 2

El

B = ~2 whereas

in (4) K I = ~ - ~

= 16r3c

term;

then

the 6cod fit of the experimental

(5)

O O

and E 2 = (2~) -I so that in this case B = 2.

of proportionalit7

Vol. 2, No. 6

S T A G E IH R E C O V E R Y

IN C O L D W O R K E D

METALS

329

From the isothermals of Stale and Nihoul (5) we find values of B between 2 and 3. From their analysis of the isochronal recovery of cold worked Nb, S~als, Nihoul and Gevers (9) conclude that the isochronal can be described by Waite second order kinetics with 4r5c = 1 or 0,25, giving B = 4 or 1. These authors argue that the first value o o is the correct one. The quoted values of B are, however, not sufficiently different from B = 2 to exclude the possibility that the recovery should be described preferentially by equation (1). At any rate the value of B is concentration dependent when equation (2) applies. Therefore it should be useful to measure the stage III reaction kinetics for different degrees of deformation corresponding to point defect concentrations differing by an order of magnitude. References

(1) c. Frois, Acta Met. 14, 1325 (1966). (~) L.J. Cuddy, Acta Met. !~6, 23 (1968). (3) M. Wintenberger,

Acta Met. I, 549 (1959).

(4) F. Ramsteiner, W. Sch~le and A. Seeger, Phys. Stat. Sol. ~, 937 (1964). (5) L. ~tals and J. Nihoul, Phys. Stat. Sol. ~, 785 (1965). (6) T.R. Waits, Phys. Rev. 107, 463 (1957). (7) H. Schultz, Acta Met. 12, 649 (1964). (8) D.G. Martin, Acta Met. ~, 571 (1957). (9) L. Stale, J. Nihoul and R. Gevers, Phys. Stat. Sol. 15, 717 (1966).