Transformation kinetics during continuous cooling

TRANSFORMATION

KINETICS

DURING

JOHN

CONTINUOUS

COOLING*

W. CAHNt

It is shown that transformations which nucleate heterogeneously will quite often obey a rule of additivity and transform nonisotbermally according to simple rate laws which can be calculated from isothermal trax~format~ondata. CINfiTIQUE

DE LA TRANSFORMATION

PAR REFROIDISSEMENT

CONTINU

L’auteur montre que des transformations it germination h&&og&ne ob&raient souvent B une rbgle d’additiviti et se transformeraient anisothermiquement suivant des lois simpIes de vitesse que l’on peut calculer B partir des don&es de la transformation isotherme. UMWANDLUNGSKINETIK

BE1 KONTINUIERLICHEM

ABKtiHLEN

Es wird gezeigt, dass Umwandlungen mit heterogener Keimbildung h&fig einer AdditivitBts-Regel gehorchen und die nichtisotherme Umwandlung nach einfachen ~chw~digkeitsg~etzen abliiuft, die man aus Daten der isothermen Umwandlung berechnen kann.

INTRODUCTION The

problem

nonisothermal importance.

of

conditions It

difficulty

kinetics

is of tremendous

has not

using fundamental

Many reactions where the nucleation

transfo~ation been

kinetic

practical

extensively

quantities.

is that both the nucleation

under

Part

of the

rate and growth

temperature,

what follows

to the

If all nucleation

that such a reaction would be additive. We then have here a condition which leads to the rule of additivity, systems.

It also becomes

would not expect to encounter such a system. Especially, for the case of the pearlite reaction, where,

deviations

from additivity.

exhibit

the property

given an isothermal

for this system.(4)

function

is heterogeneous

of sites for nucleation

expect possible

to find in real to estimate

can be stated

the

of temperature,

eontinuous

as follows:

curve for t,he time 7, as a

TTT

at which the reaction

reached a certain fraction of completion,

that all nucleat,ion occurs early

number

often

The rule of additivity

in the reaction.c5y 6, This may be due to the fact that only a limited

one could

THE RULE OF ADDITIVXTY

even isothermally, the nucleation rate is a function of time,c3) this condition would not hold. Yet, has been demonstrated

of

well for

does occur at the beginning of the

and which

Many systems in which nucleation

will hold equally

reaction, it will be shown that under certain conditions the reaction is isokinetic in the sense of Avrami, and

growth rate over a range of temperatures, the reaction is additive. This is a very special condition, and one

additivity

early in the reaction.(s)

these systems.

been found that some transformations obey an additivity rule.(l) Avrami(2) has shown that for the rate is proportional

occurred

is “pparently

by the assumption

If the number of nuclei formed is also independent

product are time-dependent However, it has is time-dependent.

case where the nucleation

also are describable

that all nucleation

studied,

rate of the transformation if the temperature

homogeneous

has

x0. Then, on

cooling, at that time t and t,e~~pera,ture

T,

when the integral

exist,

such as impurity particles or grain corners, or in case of grain surface or edge nucleation, that, these-sites can be consumed very early in the reaction. This is a general property of heterogeneous nucleation, and will be called site satura.tion. It has been shown for the

quals unity, the fraction completed will be x0. The rule has often been stated for the initiation of the

case of grain-boundary surface, edge and corner nucleation, that the expected interval in under-

reaction.

cooling from the initiation of nucleation to site saturation is small.(‘) This is due to the large temperature coefficient of nucleation. * Received January 35 3 1956 .

YOT ACTA

eneral Electric hesearch Laboratory, Schenectady, New M~TALLURGI~A,

VOL.

4, NOVEMBER

1956

572

This is a quantity difficult to define, since it

depends on the method of observation. It is better to define the time for the initiation as the time for about 1% reaction. A sufficient condition of additivity has been described by Avrami. He found that for the “isokinetic” range, i.e. when the nucleation rate is proportional

to the growth

rate, that the reaction

CAHN:

FIG.

1.

Pb-6.8%

would be additive.

Sn partially

TRANSFORMATION

precipitated

at lOO”C,

Under these conditions,

then quenched to room

he showed

be additive.

temperature

two time-temperature

for the time scale, regardless of the temperature path. It has been more generally assumed that a reaction

be

amount

of

transformation

and

2 =f(x, Thenr=

s X0

the

only of the temperature

Upon substitution

into (1) and

for additivity.

The general isokinetic

further precipitation.

parameters

Nucleation

and

general involve an integrated well as an integrated

will in general not

growth

reactions

reaction will be defined as a

reaction which can be written

in

nucleatjion parameter as

growth parameter.

However,

if

the nucleation sites saturate early in a reaction, and if the growth rate is a function of the instantaneous temperature only, the reaction will be The question of to what extent growth

T).

--. a f(? T) integrating to x0, one can see that (1) becomes identically unity, showing that equation (2) is a sufficient condition

additive.

function

dx

for

It will be noted that a reaction involving

that the reaction would take the same course, except

is additive whenever the rate is a function

573

KINETICS

of thermal

examining

history

micrographs

temperature temperature.

can

be

of a reaction

and permitted

additive. rate is a

estimated started

to continue

Fig. 1* is a micrograph

by

at one

at another

of Pb-6.8

wt.

per cent Sn held at 100°C for partial precipitation, quenched further.

to room temperature and allowed to react The spacing characteristic of the room

temperature

appears

immediately,

and

one

would

expect that the growth rate would likewise be that of the new temperature. On the other hand, Hull and Mehl(s) have published where h(T) is a function growth

It will be convenient variable,

of temperature

rate, nucleation

only, such as

rate, or diffusion

to consider

constant.

x the independent

j-h(T) dt = H(x).

(4)

Then

z=

dH (x) [--I

ax

which satisfies condition

may be a rea.1 effect.

-1

* h(T)

(2), and such a reaction will

of similar experi-

If it is the latter,

from additivity could be expected growth rate G under nonisothermal

and write (3) as

dX

micrographs

ments on pearlite in which the spacing changes over a finite distance. This may be due to a slack quench or it

describedbyG=G,

deviations

to result. conditions

If the can be

(1 +eg)whereG,isthegrowth

rate under isothermal conditions and F is a constant, then for a constant cooling rate (1) leads to a value of * I am indebted micrograph.

to Mr. H. N. Treaftis

for the use of this

574

l/l

ACTA

+

dT

E -

G?t

when

the

METALLURGICA,

nonisothermal

reaction

has

VOL.

4,

1956

because there will only be a single time-temperature parameter.

However,

if I is sufficiently

reached x,, completion.

correction

The deviation from the assumption of instantaneous site saturation can be estimated. We shall confine

high carbon steels, the correction

ourselves

For medium-carbon

to point

slight modification The number

sites.

The same arguments

with

will apply to other types of sites.

of sites per unit volume

i:

time between t and t + & is

at

[S

0

4rr 3 IN exp where

1

=

tG

sites

R(T)]3 at

(5)

-

Integrating

s0

s

‘1e

probably

fortuitous. precipitations.

In this case it is possible to

l/2

and that his solution is a solution for a D dt [s ti:e-dependent diffusion coefficient D if Dt is replaced

-

R(E)]3&.

A

that the equilibrium

concen-

of temperature.

This is a

are independent

such a reaction

change

because also

in

the

the

R(1)12 G di

that

remains

(6) for

new

deviations,

same

volume

of precipitate

because

a sudden

phase,

which

change

previously

suddenly

Therefore,

are relatively

only

may

not but

equilibrium close to the

been

close

to

or supersatur-

restriction

must be

applied that the bulk of the reaction occur at tempera-

additivity [R (t)]3 1 (

had

the additional

tures where the composition

x) = ;

in the

undersaturated

in the reaction,

with

not

of completion,

will make the material

instantaneous saturation. The second term is the correction term. If saturation occurs relatively early

--In (1 -

compositions

produce

to the same fraction

equilibrium, ated.

will also be additive.

equilibrium

will

concentrations x [R(t) -

to

1

t

temperature

term

with

success is

Zener(ll) has given a solution for growth of diffusion-

the reaction,

[R(t)

by parts results in

first t,erm is the only

rate increases

Its quantitative

condition that is fairly often approached at large undercoolings, and if all nucleation occurs early in

“I& ’

- is a correction s 7 sign, since the ratio of the

undercooling.

correspond

The

curve, it

at

part of

rate to the growth

increasing

trations

Then

x) = 4GN

has the correct

nucleation

by the s D dt, provided

cit.

s 0

--In (1 -

in

steels, if the bulk of the reaction

show that the radius of a particle is proportional

dt [R(t) -

R(t)

reaction

term will be small.

will also be small, and will increase with decreasing

controlled

-f1 0

For the pearlite

occurs in the vicinity of the nose of the TTT

which

of the untransformed

[S

to G.

which requires neglecting

nucleating per unit time, and N is the number of sites per unit volume. The extended volume(2l 7) of these sites at time t is

is proportional

cooling rate.c6) The method of Grange and Kiefer,(lO)

I dt d2

Ihlexp where I is the fraction

nucleating

large, the

terms will be small regardless of whether I

constant,

of the equilibrium

in order

phases

that the rule for

hold.

3

[exp[-rIdt]d$$

THE

.

(7)

EFFECT

OF GRAIN

SIZE

It must be noted that the proof, that equation (2) is sufficient for additivity, requires that the function F be the same function

for the nonisothermal

specimen

and the isothermal specimen.

The correction factor is of the order of the ratio of R at saturation to R(t), and becomes less as the reaction proceeds. This is the correction term for

For reactions where the nucleation sites saturate, this means that the quantity of nucleation sites must be the same for both specimens.

deviations from instantaneous saturation, and applies to isothermal as well as nonisothermal reactions.

For example, for grain-boundary nucleated reactions both specimens must have the same grain size. If isothermal data are available for a specimen which

When

substituted

into (l),

the correction

term will

enter both into 7 and clxldt. If I is proportional to G, the correction term will vanish, as Avrami has shown,

has a grain size different specimen, one has to modify

from a nonisothermal the rule of additivity to

CAHN:

take differences

in quantity

TRANSFORMATION

of nucleation

sites into

KINETICS

When this is substituted into (4), it gives an expression for H(x) for nonisothermal

account. For discontinuous

or

cellular

reactions

7

For discontinuous

-_

1 _

,-&RU)12

_ -

1 _

e-4n/3A'[R(t)]a

(8)

unit volume,

H

with

respect

to time,

one

(5). Differ-

obtains

H=

upon

at

SC

respectively

?-

x?&

lated from

4 I’* Ln

Ni

If

these

are

the equations

1’3

log (1 -

n to the nonisothermal nucleated

(9) all reduce to

s

D,

ISOKINETIC

The amount calculated

if h(T)

This, however,

LJL,,

grain diameters.

RATE

LAWS

expected

is not always known.

L

(1 -

x0) [s tt

12 [s1 7

at 3 -

-2s)

%

(13)

7

nucleated reactions the quantities

and N,/N,

are easily

expressible

in

from

wishes to thank

Dr. J. H. Hollomon

and for some stimulating

suggestions.

REFERENCES

an

path can be F are known.

What is known

often is 7, and one usually has some clues of the function F. From (4) we can write an expression for h(T) in terms of T:

h(T) = ; H(x,)

for his advice

FOR

REACTIONS

and the function

2s) s “,’

ACKNOWLEDGMENTS

1.

for any temperature

2) = $log

-x)=$log(l

For grain-boundary S,/S,,

The author

of transformation

reaction

log(l

(10)

where D, and Di are the respective

(1 -

terms of the grain size.

-=q i-

NONISOTHERMAL

5) =$log

sites.

then be calcu-

2

grain-boundary

dt

of nucleation

of transformation’can

log (1 -

(9)

the amount of transformation is x,,. The subscript i refers to the isothermal specimen from which T was and the subscript

for the three types

The amount

(4 = C-1 3,

isokinetic

(rln (1 - x)j*‘*

into (1) that when

-

THE

=

and N is the number of point sites per

s= _

reactions,

which

which nucleate on surfaces, edges,

unit volume, and R is defined in equation substitution

or cellular transformations

saturate, where H is R, one obtains from (8)

or points respectively. S is the area of nucleating sites per unit volume, L is the length of nucleating sites per

specimen.

s at ~~

-

J: = 1 _ ,-2SRU)

for transformations

obtained

reactions

which

H(x) = H(x,)

saturate”)

entiating

515

(11)

J. H. HOLLOMON, L. D. JAFFE, and M. Trans. Amer. Inst.Min. Met. Ens. 167.

R. 419

NORTON

(19461. ’ ’ 2. M. AVRAMI J. Chem. Whys. 8, 212 (1$40). 3. F. C. HULL, R. A. COLTON, and R. F. MEHL Trans. Amer. Inst. Min. Met. Eng. 150, pp. 185-207 (1942). 4. H. KRAINER Archiv Eisenhuttenw. 9, 619 (1936). 5. D. TKJRNBULL~~~H.N.TREAFTIS ActaMet.3,43(1955). 6. J. W. CAHN The kinetics of the pertrlite reaction. Submitted to the Journal of Metals. I. J. W. CAHN Acta Met. 4. 449 (1956). 8. c: WERT J. _4ppZ. Phy8.120, 9i3 (1649). 9. F. C. HULL and R. F. MEHL Trans.Amer. Sot. Metals 30, 381 (1942). 10. R. A. GRANGE and J. M. KIEFER Tmns. Amer. Sac. Metals 29, 85 (1941). 11. C. ZEXER J. Appl. Phys. 20, 950 (1949).