The coherent solubilities of γ′ in Ni-Al, Ni-Si AND Ni-Ti alloys

THE

COHERENT

SOLUBILITIES OF y’ IN Ni-Al, ALLOYS* P.

K.

RASTOGIt

and

A.

Ni-Si

AND

Ni-Ti

J. ARDELLtS

The coherent solubilities of the y’ phase in Ni-Al, Ni-Si and Ni-Ti alloys have been determined from magnetic studies of y’ particle coarsening and by electron metallographic observations. Over the temperature intervals 620-775°C (Ni-Al), 620-845°C (Ni-Si), and 525-775°C (Ni-Ti), the solubilities obey the empirical equations cdl = 0.2545 exp (- 1418/RT) csi = 0.2828 exp (- 1768/RT) cri = 0.2566 exp (- 1850/RT) The solubilities above are expressed in terms of the atom fraction of solute in the saturated Xi-rich matrix and are estimated to be accurate to within * 1%. These data are compared with literature values of the maximum solubilities of Al, Si and Ti in nickel. The data on y’ in Ni-Al, in particular, are considered in light of a recent theoretical treatment of coherent two phase equilibrium by Oriani. It is concluded that previous investigators also determined the coherent solubility of y’ in Ni-Al alloys. SOLUBILITES

COHERENTES

DE

y’ DANS

LES

ALLIAGES

Ni-Al,

Ni-Si

ET

Ni-Ti

Les solubilites coherentes de la phase y’ dans les alliages Ni-Al, Ni-Si et Ni-Ti ont et& determinees a partir des etudes magnetiques du grossissement des particules de y’ et par des observations au microscope electronique. Pour les intervalles de temperatures 620-775°C (Ni-Al), 620~845°C (Ni-Si) et 525-775°C (Ni-Ti), les solubilites obeissent aux relations empiriques suivantes cbr = 0,2545 exp (csr = 0,2828 exp rc~i = 0,2566 exp (-

1418/RT) 1768/RT) 1850/RT)

Les solubilites ci-dessus sont exprimees en fonction de la fraction d’atome de solute contenue dans la matrice saturee riche en Ni et on estime que ces resultats sont p&is a * 1%. Ces resultats sont compares aux valeurs don&es par la litterature pour les solubilites maximum de Al, Si et Ti dans le nickel. En particulier, les auteurs examinent les resultats relatifs a y’dans Ni-Al a la lumiere d’une theorie recente de l’equilibre de deux phases coherentes Btablie par Oriani. 11s concluent que les precedents chercheurs ont aussi determine la solubilite coherente de y’ dans les alliages Ni-Al. DIE

KOHARENTEN

LOSLICHKEITEN VON Ni-Ti-LEGIERUNGEN

y’ IN

Ni-Al-,

Die koharenten Loslichkeiten der y’-Phase in Ni-Al-, Ni-Si- und magnetischen Untersuchungen der y’-Teilchenvergroberung und Beobachtungen bestimmt. In den Temperaturintervallen 620-775C 525-775*C (Xi-Ti) gehorchen die Lijslichkeiten folgenden empirischen

Ni-Si-

UND

Ni-Ti-Legierungen wurden aus aus elektronenmikroskopischen (Ni-Al), 620~845°C (Ni-Si) und Gleichungen

0,2545 exp (-1418/R!I’) csr = 0,282s exp (- 1768/RT) Cri = 0,2566 exp (- 1850/RT)

CA1 =

Die oben angegebenen Losliohkeiten sind in Atomprozenten des gelijsten Stoffes in der gesattigten Ni-reichen Matrix ausgedrtickt und schatzungsweiseauf & 1% genau. Diese Daten werden mit veroffentlichten Werten der maximalen Loslichkeit von Al, Si und Ti in Nickel verglichen. Insbesondere die Loslichkeit von y’ in Ni-Al wird in Hinblick auf eine neue theoretische Behandlung des koharenten Zweiphasen-Gleichgewichts von Oriani betrachtet. Es wird geschlossen, daB friihere Autoren such die koharente Loslichkeit von y’ in Ni-Al-Legierungen bestimmten.

1. INTRODUCTION

The dynamic

solubility

of

equilibrium

a dispersed

where K is a rate constant, phase

at thermo-

is one of the important

param-

of solute in the matrix.

c, is a true

thermodynamic

eters which can be obtained from studies of particle coarsening. According to the Lifshitz-Slyozov theory

equilibrium

of

capillarity).

diffusion

concentration asymptotically

controlled

coarsening,(l)

the

average

c, = (/&)-1’3,

17, MAY

(1)

1969

of infinite

size (i.e. no

This is the state of the system that will

exist at t = co. To obtain c, at a given temperature,

To date,

i W. M. Keck Laboratory of Engineering Materials, California Institute of Technology, Pasadena, California. of Engineering, University of $Now at: Department California, Los Angeles, California. VOL.

a particle

solubility

of the matrix in

it is necessary only to plot c vs. tp1j3 and extrapolate the curve to tk113 = 0.

* Received September 12, 1968.

ACTA METALLURGICA,

with

The parameter

equilibrium

because it defines the solute content

of solute, c, in the matrix phase varies with time, t, as

c -

and ce is the equilibrium

concentration

595

studies

of particle

been applied in the determination because the variation to measure

by

most

coarsening

have

not

of phase equilibria

of c is fairly small and difficult methods.

In

binary

Ni-base

ACTA

596

alloys, however, 0,, is a very tration.c2)

function

Therefore,

accurately

detected

c has been function study

the ferromagnetic

strong

of aging

small

changes

already

of the y’

crystal structure)

in

in Ni-Al

concen-

c can

be

curve of Bc vs.

with respect

fully coherent aging

that

coarsening

the

(N&Al,

Ll,

Ni,Si, like

coherent solubilities and incoherent

are

during the

used in our experiments. of c, extracted ought

to

from

represent

It

between

The elastic

in Ni-Ti

Ni-Ti

solubility

The coarsening

behavior

experimental The

details peculiar to each system.

alloys

used

in the

coarsening

the compositions

at. % Al, Ni-12.68

Ni-13.14

at. % Ti.

Rod

shaped

to &l”C.

argon atmosphere

Temperature

alumel thermocouples

was measured

platinum-platinum

precipitate.

More recently,

in his

papers

Oriani(‘)

on

spinodal

has considered

(sharp-interface) solubility

must

clecomposition,(5a6)

the problem

precipitate. always

Cahn

quantitatively

Since

exceed

and

for a classical the

coherent

the incoherent

solu-

thermocouple. equation

by additional

of the validity

coarsening Fig. 1.

of y’ in Ni-Si

The magnitude

in many

coarsening

the binary Ni-base important

class

true of

alloys, which form the basis of an

of commercial

materials

hardened

the degree relative

by chromel-

of

against

and an accurately - 10 % the

rhodium

application

of

can be seen

These data were supple-

measurements

Examples

of c, reported

This is particularly

Examples

in a previous publication.(3)

bility, there is a need for data on coherent solubilities alloy systems.

of aluminum

(1) to the data on 7’ in Ni-Al

mented

and aged in

in furnaces controlled

which were calibrated

calibrated

with

1 mm2, were

treated in an inert atmosphere

a Ti-gettered

at. % Si

specimens

of approximately

with respect

tion of a coherent

were

starting materials and had

region.

has treated the problem of stabilization

studies

prepared from high-purity

solid solution

of the classical theory of nuclea-

this and

account will be presented of the

the two-phase

treatments

on

of y’ in Ni-Si

alloys will be fully discussed in future publica-

tions, and a complete

point

within

the y’

after very

results

system.

coherent

the melting

to decomposition

that

does coarsen

feel conficlent in

the

solution

This fact is implicit in standard

the experiments

doubt

short aging times, and we therefore

square cross-sections

associated

alloys

with the coherent state will stabilize a supersaturated

textbook

some other kind

including

the

of a dis-

energy

precipitate

leave

and Ni-10.50

the coherent

solubilities

strain

laboratory

the

of y’ in the three systems.

(unconstrained)

phase.

7 phase.c4)

the y’ precipitates

There is a clear distinction persed

is

to those of

controversy(lO*ll’

may be undergoing little

to the hexagonal

experiments

There has beenrecent

in this

with Ni,Al(*)).

values

for y’ in Ni-Si. that y’ in Ni-Ti

used to

with their Ni-rich matrices

times normally

follows

in addition

and Roth,(g) suggest that this is also true

been

precipitate

In all three of these systems

and our studies,

Hornbogen

However,

is a stable phase, but the y’ form of Ni,Ti

metastable

Ni-Al,@**)

1969

of kinetic growth process.

is now being applied to investigate coarsening of the y’ precipitate in Ni-Si and Ni-Ti alloys (in these Ni,Al,

17,

6, as a

alloys,c3) and this method

systems y’ is isomorphous

VOL.

of

Measurements

time have

coarsening

Curie temperature,

of the solute

once a calibration

established.

METALLURGIC-~,

for this study.

of equation and Ni-Ti

(1) for

the

are shown

of the errors in the absolute values

herein is difficult

to estimate.

studies we are primarily of accuracy

with

which

concerned

In the with

we can detect

changes in c. For this purpose the magnetic

by the y’ precipitate. The objectives

of this research were to present the

values of c, obtained a unified manner,

from our coarsening

to demonstrate

of c, are indeed coherent our data with solubility magnetic

technique

studies is described calibration

curve

and to compare

data reported in the literature.

2. EXPERIMENTAL

The

studies in

that these values

solubilities,

PROCEDURE

used

in the

coarsening

in Ref. 3, which also shows the of 8, vs. c for the Ni-Al

system.

Similar procedures were employed to establish the calibration curves for Ni-Si and Ni-Ti alloys, except that splat-cooled

specimens

were not used.

In order to use equation (1) it is necessary to be absolutely certain that the precipitate in question is growing by a diffusion-controlled coarsening process. There is no doubt that this is true for y’ in

in

t-“3

x 102(sec-“3)

FIG. 1. Examples of the application of equation (1) to

the coarsening behavior of y’ in Ni-Si and Ni-Ti alloys. The values of c,, determined from the intercept of the extrapolated curve (dashed) with the ordinate, were converted to units of at. fraction (or at.%) for this study.

RASTOGI

AND

ARDELL:

COHERENT

technique is very sensitive, and changes of c on the order of 0.002 wt. % are readily detected. On the other hand, the absolute values of c depend primarily upon the accuracy of the calibration curves, and errors in the calibration curves may vary from one system to another. We therefore did some additional experiments to check the results of the magnetic measurements. Sheet samples of each of the three alloys were aged for 48 hr at 700°C (the Ni-AI alloy used for this experiment contained 12.85 ‘A Al, rather than 13.14 %). This treatment produced homogeneous dispersions of the y’ precipitates in all of the systems. Samples of the aged alloys were then re-aged for 14 hr at various temperatures near the coherent solubility limit predicted by the results of the magnetic measurements. Specimens were prepared by normal electropolishing techniques for examination by transmission electron microscopy in a Siemens Elmiskop I, operating at 100 kV. The expected results of the dissolution experiments were that the coherent y’ precipitates would grow (apart from some dissolution which must occur on re-establishment of the new and smaller volume fraction of y’) if re-aged at a temperature below the coherent solubility limit, but would dissolve completely on re-aging above the coherent solubility limit. The dissolution experiments therefore serve a dual purpose. In addition to checking the results of the magnetic measurements, they permit an unambiguous evaluation of whether or not we are dealing with a coherent solvus. This is because the coherent solvus is a true phase boundary, and at temperatures higher than the coherent solubility limit the only remaining precipitates should be those that are incoherent, or semi-coherent (which includes precipitates on dislocations), or those that have transformed into a more stable crystal structure (e.g. the transformation of y’ to 7 in Ni-Ti). This principle was used with success to establish the coherent K-X’ solvus in concentrated Al-Zn alloys.(12)

SOLUBILITIES

cAl = 0.2545 exp (-1418/RT), csi = 0.2828 exp (-1768/RT), cTi = 0.2566 exp (-1850/RT),

y’

IS

Si

59-l

ALLOYS

where the concentrations are in units of atomic fraction of solute. The coherent solubility limits for the samples used in the dissolution experiments, calculated from the empirical equations above, are : Ni-12.85 % Al, T, = 769°C; Ni-12.68% Si, T, = 836°C; Ni-10.50% Ti, T, = 768°C. Representative microstructures of the samples used in the dissolution experiments are shown in Figs. 3-5. We found, in general, that near

i ~ 0

INVESTIGATION

COARSENING

.

STUDIES

15m f

0

;

Y

0

I

0%

HORNBOGEN AND KREYE(‘3’ WILLIAMS”” TAYLOR AND FLOYDI’SJ -RAY) TAYLOR ANO FLOYdX (METALLOGRAPHY)

COARSENING STUDIES ELECTRON METALLOGRAPHY _

m 0

‘O-

x

-

t-

0 ;

* c c .

I4lm

-800 - 700

II-

1

-

CT

0 t-

-600

z-

I

t,

’ 30e

I

I,

$1

I,

I

09

L,,

IO

log

cS,

(,]J

,

I I8

1 I

(atomic

5OO

%I

(b) CT, (atomic 7

3. RESULTS

The results of the magnetic measurements are shown in Fig. 2, along with representative literature values of the equilibrium solubilities of Al,(13-i5) Si(16) and Tio7) in Ni. The experimental values of c, all obey an Arrhenius relationship over the range of temperatures used in this study. Least-squares analyses of the data yielded the following empirical equations :

OF

-7

8

08

I,

oY_

TAYLOR AND FLOYD”” (SOLUBILITY OF THE PHASE1

IO

:

II

I,,

II

12

,

13

, /,

14

/

15

1 ,-, 9w

q

.y 10 0

I,

%I

IO

3

800

J

700

0

c

600

1

I2 -

u

;, ’ 30s

0 COARSENING ELECTRON

I

,:-:.-;;, 09

STUDIES METALLOGRAPHY

,

/

10 log

CT, (atomic

, 1 ‘8

500

%I

(0)

Fra. 2. Arrhenius plots of the equilibrium solubilities of y‘ in: (a) X-Al alloys; (b) Ni-Si alloys; (c) Ni-Ti alloys.

ACTA

METALLURGICA,

(b) FIG. 3. Electron micrographs of Ni-12.85% 700°C for 48 hr snd m-aged for I& hr at: (b) 775°C.

Al, aged at (a) 767%;

VOL.

17,

1969

lb) FIG. 4. Electron micrographs of h-i-12.6S”/, 700’C for 48 hr and re-aged for I+ hr at: (h) 85rT.

Si, aged at (a) 835’C;

RASTOGI

AKD

ARDELL:

COHERENT

SOLUUILITIES

OF

y’

IN

Ni

ALLOYS

599

T, the spatial distribution of the y’ precipitate was not uniform ; some areas of a given thin foil contained more y’ particles than others. This was not the result of failure to observe the particles because the foils were always tilted into a position of strong diffraction contrast where the particles are readily observed by either strain-field contrasP) or &fringe contrast.(lg) The non-uniformities were most likely due to small compositional heterogeneities in the alloys which persisted after fabrication. Therefore, at temperatures near T, some regions of a given sample were probably within 6he two-phase field while others were outside it. This would lead to a variation of the volume fraction of y’ from one region to another which was the observed result. In spite of this problem the results of t,he dissolution experiments are in excellent agreement with the predictions of the empirical equations. In Fig. 3 fully coherent y’ precipitates are present in a Ni-Al sample re-aged at 767% (Fig. 3a), but at 775% practically all the remaining y’ is on dislocations (Fig. 3b). Figure 4 shows the results on y’ in Ni-Si. In this system y’ was never observed at 855°C (Fig. 4b), whereas at 835°C y’ was always present (Fig. 4a). Some samples of this alloy re-aged at 845°C contained y’ whereas other did not, indicating that T, is in the vicinity of 845°C (cf. T, = 836°C from the empirical equation). The results on the Ni-Ti alloy in Fig. 5 are similar to those in the other alloys. At 766°C (Fig. 5a) coherent y’ was always present, whereas at 775°C (Fig. 5b) nearly all the y’ precipitate has been replaced by the stable 7 precipitate (the 7 phase was rarely observed in samples re-aged at 766’C). From the results in Figs. 3-5, we have assigned the following temperature intervals to represent the coherent solubility limits for the three alloys: Ni12.85 % Al (765-785°C) ; Ni-12.68 % Si (855~858’C) ; Ni-10.50 % Ti (765-785°C). These temperature intervals are indicated in Fig. 2. For Ni-Al and Ni-Ti, the upper temperature was raised by 10°C to allow for possible sampling errors. Even with this allowance, the empirical representations of c are correct to better than fl%. This is a conservative error limit which includes the uncertainties in our observations and allows for possible errors in the Al, Si and Ti contents of the binary alloys. 4. DISCUSSION

b) FIG. 5. Electron micrographs of Ni-10.50qb 700°C for 48 hr and re-aged for 14 hr at: (b) 775°C.

Ti, aged at (a) 766°C;

The results of the dissolution experiments provide positive confirmation that the solubilities determined from the magnetic measurements are indeed coherent solubilities. The predictions expressed in Section 2

600

ACTA

METALLURGICA,

were all borne out by the experiments illustrated in Figs. 3-5, and require no further discussion.

VOL.

17, 1969

simplify to the following expression:

(2) The literature data on the solubility of y’ in Ni-Si are sparse and conflicting, as discussed by Hansen. We have accepted Hansen’s judgement in choosing the curve in Fig. 2b. The extent to whioh a solid solution is stabilized by coherency strains depends largely on the fractional misfit in lattice parameter between the precipitate and matrix,(7) Aala = Since Aala in the Ni-Si @matrix - @y’)lamatrix. system is fairly small (M -0.3 %)@) the coherent solubility of y’ is probably not much greater than the incoherent solubility. Assuming that Hansen’s curve represents the incoherent solubility of y’ in Ni-Si, our coherent solubility data are consistent insofar as they are displaced to slightly higher concentrations. Our data on y’ in Ni-Ti are, to the best of our knowledge, the only reliable data on the solubility of the y’ form of Ni,Ti. The data of Taylor and Floyd,o’) shown in Fig. 2c, represent the solubility of the stable hexagonal 7 phase, which is smaller than that of coherent y’, as it must be. In contrast to the other two systems, there is a good body of reliable independent data on the solubility of y’ in Ni-Al alloys. The results of Hornbogen and Kreye,d3) Williams(14) and Taylor and Floyd(ls) are shown in Fig. 2a, where it is seen that the agreement among their investigations is generally good. It is also evident from Fig. 2a that the literature data are in reasonably good agreement with our results over the temperature range relevant to this study. According to the theory of Orian& the difference between the coherent and incoherent solubilities is proportional to (Aa/at2. The value of Aaja for y’ in M-Al is between 0.5 and 0.6%.(20) If we assume that the literature data represent the incoherent solubility of y’ in Ni-Al, the difference between our results and the literature values should be three to four times larger for y’ in Ni-Al than for y’ in Ni-Si. Instead, the agreement between our data and the literature data is about the same for both systems. To investigate the cause of this apparent Iack of consistency, we decided to calculate values of the incoherent solubility of y’ in Ni-Al, using the theoretical equations of Oriani. 4.2 E~t~~~t~on of the ~~co~e~e~t ~ol~~~~~t~off

in Ni-Al

The original equations derived by Orianic’) are moderately complicated. It is shown in the appendix that with certain assumptions Oriani’s equations

where 18G, V,f (Aa/a)2 ’ =

3 + 4K;G;-

(3)

csi is the mole fraction of Al in the matrix at inooherent equilibrium, cy, is the mole fraction of Al in y’, G, is the shear modulus of the matrix, V,,,“’ is the molar volume of y’, K,. is the compressibility of y’, and ha/a is as defined in Section 4.1 (ha/a must be calculated from the lattice parameters of the elastically unconstrained phases at the compositions corresponding to the coherent equilibrium). In the derivation of equations (2) and (3) it is assumed: (a) that both y’ and the matrix are elastically isotropic; (b) that they’ precipitates are spherical (they are actually cuboidal in shape(211); (c) that eye is the same at both incoherent and coherent equilibrium; (d) that the ratio c~~/c, is equal to the corresponding ratio of the activities. Nearly all of the parameters in equations (2) and (3) are dependent on both concentration and temperature Fortunately, there are enough data on the physical properties of Ni-Al ahoys to allow reasonable estimates of all the parameters involved. Phillip@O) has measured G, for a X-12.71 % Al alloy at 303 and 77’K. We used his two values to extrapolate G, into our temperature range, using the empirical rule that shear moduli usually decrease linearly with temperature in the absence of relaxation effects.(z2) K,, was estimated from the temperature dependence of Young’s modulus of N&Al, reported by Davies and Stoloff.(23) For this estimate a value of Poisson’s ratio equal to 0.31, the value for pure Ni,t20) was assumed. The effects of concentration and thermal expansion were both taken into account in our estimates of Aa/a. Room temperature (300°K) values of %natrixas a function of c, were taken from the data of Taylor and Floyd.(15~ The thermal expansion of Ni alloys containing 5.0% and 12.4 % Al has been measured by Rovinskii et a&(24) For both alloys danlatriJdT = 5.0 X 10e5 A/‘%, and we used this figure to extrapolate the room temperature values of amatrix into our temperature range. Values of ayt were calculated using the room temperature value of 3.5600 A and the thermal expansion coefficient of 13.7 x 10-6/“C measured diIatometrically by Taylor and Floyd. (15) A volume thermal expansion coefficient of 41.1 x 10-6/oC was used to estimate the high temperature values of I/,?’ from the room temperature

RASTOGI

ASD ARDELL:

(dyne/ems)

KY* x lOis (cnP/dyne)

urn&nix (A)

6.55 6.45 6.40 6.32

7.28 7.39 7.45 7.52

3.5681 3.5720 3.5741 3.5769

Garx lo-” c&t.%)

619 676 709 748

11.44 12.01 12.28 12.67

of

value

SOLUBILITIES

27.16 cm3/mole.(21)

The

value

of

cY, =

0.231’15) was used for all the calculations. The values of all the concentration dependent

parameters

and the calculated 2a.

The

elastic

and temperature

are summarized calculations

energy

in Table

suggest

associated

state

of y’ is sufficient

Ni-Al

alloys to a significant

with

to stabilize extent.

that

for by invoking Therefore,

1,

the

the concept

the

coherent

supersaturated If the theoretical

values of cei are correct, the small difference our data and the literature

between

data cannot be accounted of coherent

equilibrium.

the results in Fig. 2a require an alternative

explanation

for their apparent

lack of consistency.

4.3 Evaluation of the literature data on Ni-Al It has been demonstrated in Ni-Al

that the y’ precipitate

alloys will remain coherent

even when the particle coherency

between

y’

temperatures

between

results of Taylor

600 and 755°C.

at least Now,

and Floyd,

2a were obtained

ments made on undeformed highly probable

to cause total loss of

and the matrix,

for the X-ray

ture data in Fig.

with the matrix

size is very large.(1g*21) Pro-

longed aging is thus insufficient

at

except

the litera-

from

measure-

aged samples,

and it is

that these samples always contained

fully coherent y’ precipitates. Let us now consider Taylor

and Floyd.

on cold-worked samples

and annealed

the y’ precipitates

coherent. Fig.

the X-ray

This is consistent

3a, where

filings,

stable,

but the coherent and Floyd analysis

were most

produced

of

out

likely

semiin

their

metallo-

from a quantitative

c,{(at.%)

5.75 5.43 5.29 5.03

27.82 27.88 27.92 27.96

8.87 9.67 10.03 10.62

with respect to semicoherent metastable

the solubilities

samples

semicoherent

containing

lie between

the coherent

This is exactly

equilibrium,

with respect

determined precipitates

and incoherent

what is observed

with the literature tions of Oriani’s librium.

we conclude

are entirely

of coherent

determined

investigators

of y’.

and

likely

most

solubilities, theoretically

they

predicequi-

with the y’

metallographically

because

coherent solubility Floyd

consistent

two-phase

Our data are in good agreement

solubilities other

solubilities.

data and the quantitative theory

from should

in Fig. 2a.

On the basis of the above arguments, that our data on y‘ in Ni-Al

to inco-

by

the

too measured

the

The X-ray results of Taylor represent “semicoherent”

and are therefore greater than calculated incoherent solubilities,

smaller than the experimentally

determined

the but

coherent

solubilities. 5. SUMMARY

AND

CONCLUSIONS

We have shown that the solubilities determined

from

coarsening

magnetic

in Ni-Al,

studies

Ni-Si

definitely

coherent

confirmed

by dissolution

and

solubilities.

of the y’ phase, of

Ni-Ti

y’

particle

alloys,

This conclusion

experiments

are was

on each alloy

system. The incoherent

solubility

solubility

of y’ in Ni-Al

from our data according

alloys was

to the theoretical

of Oriani. The calculated incoherent of y’ is estimated to be significantly smaller

than our values for the coherent solubility solubilities is concluded

determined

by previous

that the experimental

by the other investigators

and the y’

investigators. techniques

It used

could not have produced

higher

soluThis

matrix. For this reason our data are in relatively good agreement with the results of the other investi-

by

point of view, but it

allows us to qualitatively explain the lower solubilities determined by Taylor and Floyd from their Since coherent

equilibrium

is

equilibrium

between

a

(see Fig. 2a).

that there exist states of semicoherent

measurements.

601

ALLOYS

state of incoherent

consistently

analysis

are

equilibrium which represent a continuum of metastability between the extreme states of coherent and incoherent equilibrium. This hypothesis may not be

X-ray

Si

herent equilibrium,

treatment

result, and the results in Fig. 3, can be explained

productive

IN

8,;,‘(cm3/mole)

is, in turn,

and in these

have dissolved.

that

metastable

calculated

on dislocations

y’ particles

pointed

bilities than their X-ray the hypothesis

y’

Aa/a x lo3

3.5886 3.5914 3.5930 3.5949

were made

with the observation

the y’ particles

Taylor graphic

measurements

These measurements

w(A)

which

values of csi are also shown in Fig.

theoretical

strain

OF

1. Values of the parameters used in the calculation of cei from equations (2) and (3)

TABLE

T(‘C)

COHERENT

y’ and the

gators. ACKNOWLEDGMENT

This research was supported Energy Commission.

by the U.S. Atomic

REFERENCES 1. I. M. LIFSHITZand V. V. SLYOZOV,J. phys. Chem. Solids 19, 35 (1961). 2. V. MARIAN, Ann. Phys. 7, 459 (1937). 3. A. J. ARDELL, ActaMet. 16, 511 (1968).

J. Phys. Radium, Paris 23, 830 (1962). J. W. CAHN, Acta. Met. 10, 907 (1962). J. W. CAHN, Acta Met. 14, 83 (1966). R. A. ORIANI, Acta Met. 14, 84 (1966). A. J. ARDELL and R. B. NICHOLSON, J. phys. Chem. Solid8 27, 1793 (1966). E. HORNBO~EN and M. ROTH, 2. MetaUk. 58, 842 (1967). J. B. COHEN and M. E. FINE, Scrip& Met. 2, 153 (1968). A. J. ARDELL, Scripta Met. 2, 173 (1968). A. J. ARDELL, K. NUTTALL and R. B. NICHOLSON, to be published. E. HORNBOOEN and H. KREYE, Z. Metdk. 57,122 (1966). R. 0. WILLIAMS, Trans. metall. SW A.I.M.E. 215, 1026

(1959). A.

TAYLOR and

R.

W.

FLOYD, J. Inst.

Metals

(1952-3).

HANSEN and K. ANDERICO, Constitution of Binary 2nd edition, p. 1039. McGraw-Hill (1958). A. TAYLOR and R. W. FLOYD, J. Inst. Metals 80, 577 (1951-2). M. F. ASEBY and L. M. BROWN, Phil. Mag. 8,1083 (1963). A. J. ARDELL, Phil. Mag. 16, 147 (1967). V. A. PHILLIPS, Phil. Mag. 16, 103 (1967). A. J. ARDELL and R. B. NICHOLSON, Acta Met. 14, 1295 M.

R.

G. DAVIES and

A.I.M.E.

N.

of Metals,

S. STOLOFF, Trans.

ln

81, 25

Alloys,

(1966). C. ZENER, Elasticity and AneZastieity University of Chicago Press (1948).

The expression for w used by Oriani reduces to equation (3) of the text with the substitution AVIV = 3Aala. In equation (A.2) we have used the symbolf to represent the complicated function of various molar volumes that is given in equation (7) of Oriani’s paper. The application of some algebra to equation (A.l) and (A.2) yields the expressions

p. 24.

(1 -

In $ = -

7, 73 (1959).

(1 - c,i) cr,) ln ~(1 _ ce)

=

-

gY

(A.l)

In

c,(l -

cGi) =

SF [l +

1-

cei =l

1-

c,

(1 -

c,,)j].

w(1 -%f)

k4.4)

(A.5)

RT

-XC.

RT ’

ci = 1 _ w(1 2 RT

Gtz

(A.2)

$3)

--.

Yf RT

(A.6)

After subtracting (A.5) from (A.6) and some rearrangement, we have c, c,(l -

and c,)

(A.3)

and

To derive equations (2) and (3), we start with equations (4) and (7) of Oriani’s paper, and the assumptions stated in the text. In our notation Oriani’s equations are

cei(l -

-c,,f)

The arguments of the logarithmic terms in (A.3) and (A.4) are close to unity, and on expanding, these equations become

metall. Sot.

APPENDIX

Cei

-$1

e

B. M. ROVINSKII, A. I. SAMOILOV and G. M. ROVENSKII,

c, ln C, + (1 -

=

%)

and

233, 714 (1965).

Physics Metals Metallogr. N.Y.

(1 - cei)

___

c,i

Yf c,) = RT ’

(A.7)

The substitution of (A.7) into either (A.5) or (A.6), and some more algebraic manipulation yields the final result, equation (2) in the text.