Influence of crystallography upon critical nucleus shapes and kinetics of homogeneous f.c.c.-f.c.c. nucleation—IV. Comparisons between theory and experiment in CuCo alloys

Acta metall. Vol. 32, No. 10, pp. 1855-1864, 1984 Printed in Great Britain. All rights reserved

0001-6160/84 $3.00+0.00 Copyright © 1984 Pergamon Press Ltd

INFLUENCE OF CRYSTALLOGRAPHY UPON CRITICAL NUCLEUS SHAPES AND KINETICS OF HOMOGENEOUS F.C.C.-F.C.C. NUCLEATION--IV. COMPARISONS BETWEEN THEORY AND EXPERIMENT IN Cu-Co ALLOYS F. K. LEGOUESt and H. I. A A R O N S O N Department of Metallurgical Engineering and Materials Science, Carnegie-Mellon University, Pittsburgh, PA 15213, U.S.A. (Received 14 September 1983)

Abstract--A discrete lattice point model (Cook, de Fontaine and HiUiard) which incorporates strain energy (Cook and de Fontaine), described in earlier papers, has been used to determine the ranges of temperature and composition at which homogeneous nucleation kinetics of f.c.c, precipitates in Cu-Co alloys would be neither too fast nor too slow to be measured. These predictions proved successful and it was possible to measure experimentally nucleation kinetics in Cu-Co alloys containing from 0.5 to 1.0 at.% Co within 50°C temperature ranges. Experimental results were compared with theoretical values obtained from the discrete lattice point, the Cahn-Hilliard continuum and the classical theories of homogeneous nucleation. Very good agreement was obtained between the experiments and all three theories. Although surprising at first, the good matching between classical theory and experiment was explained by showing that the calculated concentration profiles of critical nuclei at the temperatures and alloy compositions experimentally studied did show distinct "volumes" and "interfaces" i.e. the solute concentration did not vary continuously throughout the nuclei. In this case, as pointed out in effect by Cahn and Hilliard, classical nucleation theory indeed applies. These findings provide the first strong support for the essential correctness of homogeneous nucleation theory. R~sum~-Nous avons utilis~ le module de noeuds discrets (Cook, de Fontaine et Hilliard), qui tient compte de l'rnergie de drformation (Cook et de Fontaine), afin de d~terminer le domaine de temprrature et de composition darts lequel la cin&eque de germination homog~ne de prrcipitrs c.f.c, dans des alliages Cu-Co ne serait ni trop rapide, ni trop lente pour ~tre mesurre. Ces prrvisions se sont av~r~es exactes et nous avons pu mesurer exprrimentalement la cin&ique de germination dans des alliages Cu-Co contenant de 0,5 d 1,0 at.% Co dans un domaine de 50°C. Nous avons compar~ les r~sultats exp~rimentaux avec les valeurs throriques obtenues/t raide du modrle de noeuds discrets, du module continu de Cahn et Hilliard et des throries classiques de la germination homogrne. Nous avons obtenu un tr~s bon accord entre les exprriences et les trois throries. Nous avons expliqu~ le bon accord surprenant au premier aboid entre la th~orie classique et l'exp~rience en montrant que les profils de concentration calculrs pour les temprratures et les compositions d'alliage 6tudires exprrimentalement, prrsentaient effectivement des "volumes" et des "interfaces" distincts, c'est fi dire que la concentration en solut~ ne variait pas continfiment d travers le germe. Dans ce cas, comme l'ont fait remarquer Cahn et Hilliard, la th~orie de la germination classique s'applique. Ces r~sultats constituent le premier soutien solide pour la validit~ de la th~orie de la germination homogrne.

Zusammenfassung--Ein in den vorangehenden Arbeiten beschriebenes Modell diskreter Gitterpunkte (Cook, de Fontaine und Hilliard), welches die Verzerrungsenergie auch umfal3t (Cook und de Fontaine), wird benutzt, um Temperatur- und Zusammensetzungsbereiche zu ermitteln, in denen die homogene Keimbildung von kfz. Ausscheidungen in Cu-Co-Legierungen entweder zu rasch oder zu langsam abl/iuft, als dab sie experimentel erfaBt werden k6nnte. Die Voraussagen waren zutreffend: Experimentell k6nnte die Keimbildungskinetik in Cu-Co- Legierungen mit 0,5 bis 1,0 At.-% Co imim 50°C Temperaturbereich gemessen werden. Die experimentellen Ergebnisse wurden mit dem Modell diskreter Gitterpunkte, dem Kontinuumsmodell von Cahn und Hilliard und den klassischen Theorien der homogenen Keimbildung verglichen. Alle drei Modelle stimmten sehr gut mit den Experimenten/iberein. Die iiberraschend gute iibereinstimmung mit der klassischen Theorie beruht, wie gezeigt wird, darauf, dab die berechneten Konzentrationsprofile der kritischen Keime bei den im Experiment verwendeten Temperaturen und Zusammensetzungen ausgepr/igte "Volumina") und "Grenzfl/ichen" aufwiesen, d.h. die Konzentration der gel6sten Atomsorte/inderte sich innerhalb des Keimes nicht kontinuierlich. In diesem Falle kann die klassische Keimbildungstheorie angewendet werden, wie schon von Cahn und Hilliard hervorgehoben wurde. Diese Befunde sind eine erste starke Stiitze dafiir, dab die Theorie der homogenen Keimbildung im wesentlichen richtig ist.

1. INTRODUCTION A l t h o u g h G i b b s ' [1] theory of nucleation was developed m o r e t h a n a century ago a n d t h a t of C a h n a n d tPresent address: IBM T. J. Watson Research Center, Yorktown Height, NY 10598, U.S.A.

Hilliard [2] is n o w 25 years old, there have been very few serious a t t e m p t s m a d e so far to c o m p a r e the predictions of these theories with experimental d a t a o n solid-solid nucleation. T h e best k n o w n comparison p e r f o r m e d in solid metals is t h a t of Servi a n d T u r n b u l l [3], reported in 1966. They investigated 1855

1856 LEGOUES and AARONSON: INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES--IV homogeneous nucleation kinetics of f.c.c. Co-rich precipitates in f.c.c. Cu-rich Cu-Co alloys using resistivity measurements. Although their experiment was very well designed and permitted rapid quenching rates, some uncertainty attaches to the resistivity technique for measuring the number of precipitates per unit volume, particularly at very small particle sizes [4]. Further, because of the indirect means of obtaining the particle number density, a number of complicated steps and assumptions was necessary to secure this data, and thus the nucleation rates, reducing the accuracy of the results. Finally, in the light of the measurements made during the present investigation, it has become quite clear that most of their alloys contained Co concentrations high enough to produce excessively rapid transformation kinetics, with reaction during cooling to and from the isothermal reaction temperature and coarsening becoming important possibilities. Another attempt at comparing homogeneous nucleation theory and experiment in metallic solids was made in a series of three papers [5, 6, 7] published not long afterwards by Kirkwood and his colleagues. These studies dealt with the nucleation of ordered f.c.c. Ni3AI in disordered f.c.c. ~ Ni-AI solid solutions. The direct technique of transmission electron microscopy (TEM) was employed to measure particle number densities. However, over-rapid transformations kinetics were such a serious problem that nearly all of the data reported were obtained in the coarsening regime. Additionally, the thermodynamic data on these alloys are not well established [8], thereby interfering seriously with interpretation of even the most accurate nucleation rate measurements. Since comparisons between theory of nucleation and experiment have so far not been entirely successful in solid-solid transformations, it is logical to ascertain what has been achieved with parallel efforts on liquid-liquid and liquid-vapor transformations. At first glance, it would seem that these should follow nucleation theory better since the much higher diffusivities involved permit nucleation at very low supersaturations, where theories ought to apply more accurately. Suprisingly however, all of these experiments compared very poorly with theory [9]. This result at first cast serious doubt upon the fundamental validity of nucleation theory. That the source of the discrepancies lay in the experiments--which consisted of measurements of the "cloud point", i.e. the temperature at which nucleation first become profuse, rather than of the nucleation rate itself--was first recognized by Binder and Stauffer [10]. They pointed out that the cloud point is the resultant of not only nucleation but also growth and coarsening. Their combined treatment of these three processes was further developed by Langer and Schwartz [9] who demonstrated that: (a) the numerous cloud point experiments had not shown any really drastic disagreement between nucleation theory and experi-

ment, and (b) such experiments are much too insensitive to nucleation kinetics to enable them to be used as a test of nucleation theory. Since liquidqiquid and liquid-vapor transformations cannot be halted by quenching in a manner which preserves their status at the reaction temperature, and are usually too rapid to be followed successfully in situ, the prognosis for achieving improved comparisons between nucleation theory and experiment by means of such transformations is not very favorable. This brief review should make clear that nucleation measurements are indeed very difficult and the experiments upon which they are based must be designed with much care. First, the transformation studied should be as simple as possible in order to minimize the number of weakening assumptions required. On this basis, it was decided to study the homogeneous nucleation of an f.c.c.~f.c.c, transformation. Secondly, an alloy system had to be chosen in which the thermodynamics are well established and simple and in which the homogeneously nucleated precipitate is the equilibrium phase so that measurements as a function of time can be made without being complicated by the appearance of another phase. It was decided to use Cu-rich Cu Co alloys, following the lead of Servi and Turnbull [3], because these alloys provide a well established case of homogeneous f.c.c.~f.c.c, nucleation, long-range order (which would complicate analysis considerably) does not appear to be present in either phase, the regular solution approximation, as will be shown later, provides a reasonably good description of the thermodynamics of solid C ~ C o alloys, and there is no evidence for intervention of a transition phase. Additionally, it would be useful to compare the present results with those of Servi and Turnbull [31. Other important parameters in the design of the experiment are alloy composition and reaction temperature. As the experience of Kirkwood [5] and Kirkwood and West [6] has made particularly clear, homogeneous nucleation kinetics pass from immeasurably slow to equally immeasurably fast in very small ranges of temperature and composition. In order to avoid these problems, it was decided to attach the problem "backwards", i.e. to do most of the theoretical calculations first in order to identify the "window" of temperature and composition in which nucleation kinetics are measurable; only then were the experiments initiated. Finally, the decision was made at the outset of this investigation to follow the example of Kirkwood and his colleagues and make all measurements of particle number density vs reaction time and temperature (from which nucleation rates can be immediately obtained) by the direct and accurate method of room temperature TEM observations on isothermally reacted specimens. Use of the "window" would assure that all Co particle formation took place at the intended transformation temperature.

LEGOUES and AARONSON: INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES--IV 1857 2. THEORETICAL CONSIDERATIONS AND CHOICE OF ALLOY COMPOSITIONS AND REACTION TEMPERATURES

2.1. Nucleation theory In previous papers [11, 12] in this series, the Gibbs [1] classical theory and the Cahn-Hilliard [2] continuum non-classical theory of nucleation were reviewed and numerically compared with a discrete lattice point (DLP) theory directly evolved from a formalism due to Cook et al. [13]. These three theories were examined in the context of the homogeneous nucleation of f.c.c, precipitates in an identically oriented f.c.c, matrix. They were compared through their evaluation of AF*, the free energy of formation of the critical nucleus, and its variation with reaction temperature and alloy composition, since this is usually the most influential term in equations for the nucleation rate. With Cahn and Hilliard [2], the classical theory was found to be accurate only at small supersaturations. Agreement between the continuum and DLP theories was good, though noticeable divergences did occur at lower temperatures and at intermediate supersaturations. Despite these findings, all three theories were used as theoretical standards against which to compare the experimental results obtained during the present investigation. The DLP model has been further modified [14] in order to include coherency strain energy [15]. Although the misfit between the matrix and precipitate phases in Cu-rich C u C o alloys is only 1.7%, it was shown that strain energy can reduce nucleation rates in these alloys by as much as five orders of magnitude [14]. Thus, it will be with this refined version of the DLP model that most further comparisons between the DLP model and experiment will be made. Similar introduction of coherency strain energy proves more difficult to accomplish with the classical and continuum models [14]. Thus the influence of strain energy upon the predictions of these less general theories has not been investigated. It remains to write the so-called "pre-exponential" or "prefactor" terms in order to obtain complete expressions for the rate of nucleation. Russell [16] has commented upon this problem for the continuum theory and concluded that the classical prefactor is a reasonable approximation. Here this conclusion is accepted for both the continuum and the DLP theories. J*, the classical rate of nucleation, is given by -AF*

-z

where Z is the Zeldovich non-equilibrium factor, fl* is the attachment rate of single atoms to the critical nucleus, N,, is the number of atoms per unit volume (also taken as the number density of nucleation sites), k is Boltzmann's constant, T is the reaction terntit will be seen in the Results section that much smaller precipitate sizes and lower nucleation rates are in fact observable.

peramre, z is the incubation time and t is the isothermal reaction time. Rewriting Z, fl* and r, which have been previously evaluated in detail [17], in terms of AF*, the free energy of formation and R*, the radius of the critical nucleus, yields

3G FAF* ]I/2 1 Z = 4~3/~L~- j

fi* z -

R, 3

4xR*'-Dco a4

32xa4R*4kT 3 AF*v2aD co

(2)

(3) (4)

where G is the average volume per atom in the nucleus phase (designated as c0, D is the applicable diffusivity [16], c0 is the average composition of the alloy and a is the average lattice parameter of the nucleus and matrix phases.

2.2. Calculation of TTT curves Times to form 10u critical nuclei per cm 3 were estimated as a function of temperature and composition using equation (1). Employing the invariant field approximation for spherical, diffusioncontrolled growth [18, 19], the time required to reach radius R from the critical radius R*, can be approximated by t =

(R - R*)2(l - 2Cc) 2D (1 -- Co- co)

(5)

where cc is the composition at the center of the precipitate. From this relationship and calculated values of R*, the times needed for Co-rich precipitates to reach R = 50 & were added to the nucleation times. The values 1011 precipitates per cm 3 and R = 50 ]k were chosen because they are reasonable lower limits for observation with TEM t. Figure 1 shows the TTT curves obtained using these calculations at various compositions. The calculations were performed using two different diffusivities. One assumes that the number of vacancies present is the same as that at the solution annealing temperature while the other considers that the vacancy concentration is that of the temperature of isothermal reaction. The diffusivity data of Mackliet et al. [20] were used with the assumption that the enthalpies of formation and movement of vacancies are equal.

2.3. Choices of the alloy compositions and of the temperature ranges 2.3.1. Choice of the alloy compositions. The TTTcurves in Fig. 1 indicate that the composition "window" extends from 0.5 to 1% Co and possibly as far as 2% Co. Alloys both richer and leaner in Co clearly have transformation kinetics too rapid and too slow, respectively, to be suitable for measurement of nucleation kinetics. 2.3.2. Choice of the temperature range. The results of the calculations on AF* reported in preceeding

1858 LEGOUES and AARONSON:

INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES--IV

C=0.005

I-T_/T:

-

- -

0.4

I" 0.3

0.3

k~

O. 1

0.1 I'-

C - c--ool

03

b.o.

2

I

I

4

1

1

1

6 8 10 12 log ( t )

/

0

0.1

I

2

I

4

I

1

I

I

6 8 10 12 log ( t )

I

1

i

1

I

I

2

4

6

8

10

12

1o9 (t) --T~/Tc

--T~/r~ o.4(",_~ (~-0.5

-r,, / r~ Q-

0.4

~

k ~ 0.3

c=o.o,~

~

_

~ I

i

I

I

I

I

2

4

6

8

10

12

C=0.03

0.1 I

0

I

1

I

I

2

4

6

8 ' 10

log ( t )

~

--" (.,,~~

o.2

0.2 0.1

--

o,, ~, i...'-' o..

k" 0.3

0.2 Q1

c=o.o2

I

I

I

I

I

I

I

I

12

2

4

6

8

I0

12

log ( t )

log ( t )

5 C=0.04

0.4,

k~ ~_. 0.3 0.2

1

Number of vacancies defined by annealing temperature

.....

N u m b e r of vacancies defined by reaction t e m p e r a t u r e

0.1I -2

0

I

l

I

t

l

I

2

4

6

8

10

12

log ( t )

Fig. 1. TTT curves obtained using equations (1)-(5). The continuous line was calculated with the diffusivity corresponding to the number of vacancies present at equilibrium at the solution annealing temperature; the dashed line was calculated using the equilibrium number of vacancies at the reaction temperature. papers in this series [12, 14] were presented as a function of TITs. In order to determine the optimum range o f temperatures in which to work, it is thus necessary to determine T~. As previously mentioned the assumption is being made that the C u - C o system follows the regular solution model. The miscibility gap is thus determined by T

In ~

c

+ 2(1 - 2c) = 0.

Hence a plot of In c/(1 - c) vs 2(1 - 2c)/T should be a straight line passing through the origin with a slope of T~. Figure 2 shows such a plot. The experimental solubility data [3, 21, 22] do indeed fall on a straight line but the line itself does not pass through the origin. The solvus line is thus described by c 2To In 1 - c + T - (1 - 2c) + A = 0.

(7)

(6) This implies that the free energy of the system can be written as

/

/

F = kT[c In c + (1 - c)ln(1 - c)]

Reg. sol : T¢= 2768 K / ~ /

+ 2KTcc(1 - c) + kTAc

-7 -6 -5

c

/

/ ,t, ~9.~r

J SLope =2768 ~1nlercept - + 0.708

/

-4

//,/--

O LTurln~u;lon

-3 -2 -1 /I -04

I -08

k -12

l -16

L I I 1 -20 -2.4 -28 -32 2x10 3 I2C--1)

I -36

[ [ -40 -44

Fig. 2. Experimental determination of Tc.

(8)

where 2kT~ is equal to the regular solution constant. As described in another paper [11], F enters the nucleation rate equation through A F ' and K on both the continuum and the D L P theories, where A F ' is defined by F ( c ) - F ( c o ) - ( c - Co)aF/dcc=~o and K is the gradient energy coefficient. It can easily be demonstrated that a linear term does not alter A F t Cahn [23] showed that K =

0922 2

(9)

where ~,, the interaction distance, is determined by the

LEGOUES and AARONSON: INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES--IV 1859 number of nearest neighbors and their distances and is thus not affected by F. For f.c.c, crystals, when only nearest-neighbor interactions are considered: co = 12AE, with A E = E ~ , B - ~(EAA i + EBB), where EAB, EAA, EBB are respectively the energies of A-B, A - A and A - B bonds. With the nearest-neighbors approximation, co is evaluated from the enthalpy of the solution, AH, through the relationship [23]: A H = c o c o - c). If we thus assume that the linear term is entirely excess entropy, K will not be modified by the extra linear term. If we consider that the linear term is partly or entirely excess enthalpy, K then varies with temperature and composition. It does not seem possible at this time to determine if the extra linear term is an extra entropy or an extra enthalpy term or in which proportion it contributes to both. However, since this study deals only with low Co concentrations and the extra term is linear in c, its influence on the enthalpy is a maximum of 1070 of the total enthalpy; hence the effect upon K should be small enough to be neglected. Finally, since T~ is determined by OF3/cgx 3 = 0 and ~?F2/c~x2 = 0, an extra linear term in F will not affect its value. Thus Tc is taken as equal to 2768 K, as shown in Fig. 2. From the TTT curves of Fig. 1, it was decided that the most favorable temperatures at which to conduct isothermal reaction studies would be around the nose of the curve because the nucleation rate should not vary too sharply with temperature in this range. For example, at 0.5% Co, the best temperature range should in the vicinity of 0.275 T,. or 490°C. 3. E X P E R I M E N T A L

PROCEDURES

The alloys were made at the Metals Research Laboratories of the Olin Corporation, through the courtesy of Dr Eugene Shapiro, from high purity, zone refined copper and cobalt (99.99970 pure). Following an investigation for homogeneity with an electron microprobe, the alloys were homogenized for three days at 1000°C. They were then hot rolled down to 2.0 x 10 -3 m thick foil, under flowing argon and cold rolled to 2.5 x 10 -4 m. All of the heat treatments were performed on the 2.4 x 10 -4 m thick foils in order to assure swift changes in temperature. The samples were solution annealed for 20 rain at 870°C in one salt bath and directly quenched to the reaction temperature in another where they were held for varying amounts of time and then quenched into iced brine. They were then polished in a twin jet apparatus for TEM observation. Direct counting was used to determine the number of particle per unit area. The convergent beam technique was used to measure the thickness of the foil and thus determine the number of precipitates per unit volume. Four different areas were investigated for each heat treatment in order to reduce the statistical errors. Five compositions were initially investigated: 0.50, 0.81, 1.02, 1.5 and 2.0% Co. Preliminary experiments

showed, however, that transformation kinetics were too fast in the 1.5 and 2% Co alloys; at each temperature, the number of particles in these alloys decreased with time. Thus only the 0.5, 0.8 and 1.0% Co alloys were studied. 4. R E S U L T S

Figure 3 illustrates the distribution of precipitates after reacting the 1% Co alloy for 10 min at 620°C. This corresponds to a density of about 5 × 10 ~4 precipitates per cm 3 and a nucleation rate of 1.7 × 1012 nuclei crn -3 s -~. This type of microstructure is particularly convenient for measuring nucleation rates. The precipitates are very far apart, thereby minimizing both diffusional interactions between them and the volume fraction of the untransformed matrix in which supersaturation has been perceptibly reduced. Additionally, the number of precipitates is low enough so that it is easy to measure but high enough to permit ready acquisition of good counting statistics. Figure 4 shows a typical microstructure in the same alloy after 5 rain of reaction at 580°C. Although this temperature is only 40°C lower than that of Fig. 3, it approximates the upper limit of measurable nucleation kinetics as a consequence of overlapping diffusion fields and the rapidity with which the particle number density increases. Figure 5 represents the 0.5% Co alloy reacted for 5 h at 510°C. Again the precipitates are far apart and easy to count, but the nucleation rate is so much slower (10 t° cm -3 s -~) that the counting statistics are not as good and hence there is more scatter in the data. Finally, Fig. 6 shows the same alloy reacted 8 h at 470°C. Not only is the number density of precipitates much increased by the 40°C decrease in reaction temperature but also the individual precipitates are much smaller and more difficult to count. These figures explain why, in the three alloys investigated, the maximum range through which reaction temperature could be usefully varied was only about

Fig. 3. Precipitation in the 1% Co alloy reacted 10 rnin at 620°C. Imaging done under two beam conditions and g = (220).

1860 LEGOUES and AARONSON:

INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES--IV

Fig. 4. Precipitation in the 1% Co alloy reacted 5 rain at 620°C. Imaging done under two beam conditions and g = <220).

Fig. 6. Precipitation in the 0.5% Co alloy reacted 8 h at 470°C, Distortions of precipitate images are due to variations in depth within the foil. Imaging done under two beam conditions and g = (220). 50°C. Figures 3-6 indicate t h a t even this limited t e m p e r a t u r e region would be sharply curtailed at compositions appreciably a b o v e or below the r a n g e employed. Figure 7 shows example plots of the

T=620°C o

o

o

ooo o

~4

4

2F / JT==41"O2~J)=42.2xsOO3Ippc/tms/ 2

o ,~

L_J&I

I

180

Fig. 5. Precipitation in the 0.5% Co alloy reacted 5 h at 510°C. Imaging done under two beam conditions and g = (220).

•

360

:.

I

600

ppl/cm /s I

1200

18

16

16 r-

14

\ ..... \ ~\ •

14

12

,,,

~J

12--

10--

o.

10

Co=O.O1 - - 0 \ ,,

defined by n u m b e r of vacancies at the anneating temperature 0 defined

by n u m b e r

of vacancies

at the reaction temperature Experimental results

"X\ --

6 --

i

4

6-

co=0.01

.....

) 0.0

•

D defined by number of vacancies at the annealing temperature

2--

D defined by number of vacancies at the reaction temperature

0--

Experimental results 01

0.2

7-/~

[ 03

I

1800

Time ( s )

18

E

1_

?20

J=17xlO

Fig. 7, Number of precipitates as a function of time in the 1% Co alloy reacted at two different temperatures.

2°I w

I

540

o

~

o

, IO00A

o

o o

I 0.4

-2

0.1

1

I

I

I

0.2

0.3

0.4

0.5

r/T:

Fig. 8. (a) Variation of J*. (b) Variation of z. Experimental data on J* and z and variation with temperature of these quantities calculated from DLP theory incorporating strain energy at two levels of vacancy concentration for the 1.0% Co alloy.

LEGOUES and AARONSON:

INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES--IV 1861 Co=0.008

18

20--

D defined by number of vacancies at the annealing temperature

16

'18--

D defined by number of vacancies at the reaction temperature

..... 14

16

• '14--

•~2~

,,,"

•

12

",

u) 10

S.

~

Experimental results

a

10

%

/

--

,

i

6-i

6

%: o.oo8

4

D defined by number of vacanoes at the annealing temperature

2 ..... •

D defined by number of vacancies at the reaction temperature Experimental results I

0.0

0.1

0.2

0.3

4

2 0

I 0.4

-2

I 0.2

0.1

I 0.5

I 0.4

1 0,5

r/r~

Fig. 9. (a) Variation of J*. (b) Variation of z. Experimental data on J* and z and variation with temperature of these quantities calculated from DLP theory incorporating strain energy at two levels of vacancy concentration of the 0.8% Co alloy.

5. DISCUSSION

number density of precipitates as a function of isothermal reaction time at two reaction temperatures in the 1.0% Co alloy. Equivalent data plots for all temperatures investigated in each of the three alloys are included in LeGoues' Ph. D. thesis [24]. In view of the scatter in the data, a straight line was drawn through the earlier reaction time points; its slope was taken to be the steady state nucleation rate and its intercept the incubation time. These results are displayed as a function of reaction temperature in Figs 8, 9 and 10 for the three alloys investigated.

5.1. Comparisons of discrete lattice point theory and

experiment The solid and the dashed curves in Figs 8(a), 9(a) and 10(a) were calculated from discrete lattice point (DLP) nucleation theory corrected for strain energy and equations (1)-(4), assuming that the vacancy concentration at the reaction temperature was quenched-in from the solution annealing temperature or had its equilibrium value at the reaction tern-

18

2018-

16

'16-

14

'14~

12

CO=0 . 0 0 5 O defined by number of vacancies at the annealing temperature

\

~

E t) t 2

&

D defined by numberof vacancies at the reaction temperature

~\ . . . . . ~

Experimental results

•

40

/' "~,~'

lo j

/ /

8

~1

% ";

8

--

6

,,

\

\

I

i

6

CO= 0 . 0 0 5 defined by number of vacancies at The anneoling temperature ! . . . . . D defined by number of vacancies at the reaction temperature • Experimental results I ) -0.0 0.1 0,2 03

4 2

--O

0 ) 0.4

I

-2 0ll

02

-]-

0.3

I

0.4

- - J

0.5

r ~ rc

Fig. 10. (a) Variation of J*. (b) Variation of z. Experimental data on J* and z and variation with temperature of these quantities calculated from DLP theory incorporating strain energy at two levels of vacancy concentration for the 0.5% Co alloy. A.M. 32/10~S

1862 LEGOUES and AARONSON: INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES--IV perature, respectively. The counterpart curves for in Figs 8(b), 9(b) and 10(b) were obtained from equation (4). The experimental and calculated J* data are in very good agreement at all three alloy compositions. The comparisons tend to favor the diffusivity whose vacancy concentration is that of the reaction temperature. However, the validity of this conclusion is uncertain because the nucleation rate is so sensitive to other variables. In fact, the incubation time comparisons [Figs 8(b), 9(b) and 10(b)] seem to tend toward the inverse conclusion: the comparison between theory and experiment is again very good but the higher diffusivity seems now to be appropriate~. It should be noted, though, that the calculated value of ~ is probably only an order of magnitude estimate [17], Even with these limitations, Fig 8-10 show extremely good agreement between theory and experiment.

20 18 '16 ~n

14

E o

12

Tl cz

10

% v

8

6

%=am 4

Discrete lattice model . . . . . . Continuum model

2

- - - - - Classical theory Experimental results

• 0

O0

0.1

I 0 3

0.2

0.4

TIT c

5.2. Comparison with theoretical predietions from classical theory and the continuum theory In Figs 11-13, theoretical predictions from classical [ii] and continuum non-classical [2, 12] nucleation theory are superimposed upon the experimental J* data. As mentioned before, coherency strain energy has not been taken into account for the classical and the Cahn-Hilliard models. Thus, for the sake of comparison, the DLP model without coherency strain energy has also been included in these figures. As expected, the continuum non-classical theory yields nucleation rates quite similar to those computed from the DLP theory. However, as is not readily anticipated, classical theory is also seen to predict the measured nucleation rates about as well as the two more sophisticated and general theoriest. This success of classical theory was achieved despite the circumstance that the nucleation rate measurements were made at large supersaturations, where classical theory should no longer be accurate or even valid [2, 12]. Nonetheless, this is explained by Fig. 14 in which typical concentration profiles through critical nuclei are presented. Only one profile is shown for each alloy studied, since the temperature range utilized corresponds to only 0.02 Tc. These profiles indicates that, though the width of the interface is of the same order of magnitude as the critical radius, it is clearly possible to define a "volume" and tThe time to grow from critical nucleus size to that experimentally measured was not deducted from the measured incubation time because calculation showed this time to be relatively small. ++It should be noted that, for the sake of simplicity, nucleation rates at only one diffusivity (corresponding to the concentration of vacancies at equilibrium at the annealing temperatures) have been utilized. If nucleation rates corresponding to both diffusivities had been included, a range of theoretical rates of nucleation would have been defined, as in Figs 8-10, and thus both the classical and the Cahn-Hilliard model would be giving acceptable theoretical values, although somewhat too high since coherency strain energy was neglected.

Fig. 11. Comparison of expelimental data on J* in the 1% Co alloy with values calculated from classical, continuum non-classical and DLP theories, all in the absence of strain energy. Vacancy concentration assumed to be quenched-in from the solution annealing temperature. an "interface" separately because the concentration does not vary continuously throughout the nucleus. This indeed fulfills in spirit the Cahn-Hilliard criterion for the applicability of classical theory [2].

5.3. Comparison with the Servi and Turnbull data Figure t5 shows the Servi and Turnbull results for a 1~ Co alloy superimposed on the present data. The difference between the two sets of data is never larger than one order of magnitude. Considering the influence of small changes in AF* upon J*, this 2O

16

E

a -

i2

•

10

6 co =0.008 4 2

0

Discrete lattice model . . . . . . Continuum model - - - - - Classical theory • O0

Experimental results 0.1

0.2

~

_ 0,5

_

j 04

T/ rc Fig. 12. Comparison of experimental data on J* in the 0.8~% Co alloy with values calculated from classical, continuum non-classical and DLP theories, all in the absence of strain energy. Vacancy concentration assumed to be quenched-in from the solution annealing temperature.

LEGOUES and AARONSON: INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES--IV 1863 20

20 -

18 16

16-14

-

12--

\ ~.

;

•

10

/

/

g~ 6

T Co =O.O05

4

Discrete lattice model . . . . . Continuum model 2 - - - - - Classical theory • Experimental results 0 O.O 01 0.2 r/

t

I 0.4

0.3

difference is indeed almost negligible. Despite the use of the somewhat questionable [4] electrical resistivity technique to follow the progress of transformation, the use of a complex, multi-layered treatment to obtain nucleation rate data from the resistivity measurements, some internal inconsistencies in the application of this treatment (e.g. the exponent in the overall reaction kinetics law found experimentally often disagreed with that theoretically anticipated), neglect of coarsening during isothermal reaction (note the decrease in N at later reaction times in Fig.

=°

c o" o.ol

08

lo

!

co:ool

O defined by number of vacancies 4 at the annealing temperature . . . . . L) defined by number of vacancies 2 at the reaction temperature • This work • Service and Turnbull I ~ 1 0 OO 0.1 02 03

04

7-/Tc

rc

Fig. 13. Comparison of experimental data on J* in the 0.5% Co alloy with values calculated from classical, continuum non-classical and DLP theories, all in the absence of strain energy. Vacancy concentration assumed to be quenched-in from the solution annealing temperature.

1O(

6 ~

/

----o.%

Co=O.OO e

0.8

Fig. 15. Comparison of the Servi and Turnbull J* data on Cu = 1% Co with that of the present investigation and the predictions of DLP theory with strain energy. 7), and the probable occurrence of transformation at temperatures above and/or below the intended reaction temperature in their alloys containing more than 1% Co (up to a maximum of 2.69 wt%; compare with Fig. 1), Servi and Turnbull [3] have evidently obtained satisfactory data on Co nucleation kinetics in Cu-Co alloys. When certain adjustments are made in their analysis, these data were subsequently shown to be consistent with a somewhat simpler version of classical nucleation theory [25], and at least the 1% Co data are also in essential agreement with the three theoretical analyses applied here. However, now that TEM is available, data on nucleation kinetics which one wishes to insure are fully reliable should only be secured through the direct type of method employed during the present investigation and also by Kirkwood and co-workers [5, 6].

co6

C06 04

04

02

02 1

lo 08

6. SUMMARY

2

1

~

R/a

2

co=o.oo5 ~

7"=447°C

C 06 04 02 1

2

3

F?/O

Fig. 14. Typical concentration profiles for the three compositions investigated in the range of temperatures experimentally studied.

The first valid test of nucleation theory for any condensed phase system has been performed using high-purity Cu-Co alloys, containing 0.50,0.81 and 1.02 at.% Co, as a model f.c.c.~f.c.c, phase transformation. Measurements of precipitate number density vs isothermal reaction time were made over a temperature range of 50°C in each alloy by means of transmission electron microscopy on specimens quenched to room temperature. Nucleation rates were calculated prior to preparation of the alloys in order to select compositions in which nucleation and growth kinetics were neither too slow nor too rapid to be measured without interference of overlapping diffusion fields of adjacent Co precipitates, coarsening or anisothermal transformation. This selection process proved successful, and representative micrographs presented indicate that these difficulties did

1864 LEGOUES and AARONSON:

INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES--IV

n o t interfere with the experimental measurements. B o t h nucleation rate a n d i n c u b a t i o n time were measured as a function o f reaction t e m p e r a t u r e a n d alloy composition; the time-dependence of the nucleation rate was too small to be discerned t h r o u g h the scatter in the data. C o m p a r i s o n s were m a d e between these d a t a a n d classical, c o n t i n u u m non-classical [2] a n d discrete lattice p o i n t non-classical theories. Quite satisfactory agreement was o b t a i n e d between the m e a s u r e d data a n d the predictions o f all three theories. O n the basis of these results, one can n o w feel considerably m o r e confident t h a t h o m o g e n e o u s nucleation t h e o r y - - w h i c h originated with the work of G i b b s [1] m o r e t h a n a century a g o - - i s indeed soundly based.

Acknowledgements--Appreciation is expressed to the Division of Materials Research of the National Science Foundation for support of this research through Grant DMR-80-07567, and to Dr Eugene Shapiro and the Metals Research Laboratories of the Olin Corporation for preparation of the Cu~2;o alloys.

REFERENCES 1. J. W. Gibbs, Collected Works, Vol. 1. Yale Univ. Press, New Haven, CT (1948). 2. J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 31, 539 (1959). 3. I. S. Servi and D. Turnbull, Acta metall. 14, 161 (1966). 4. H. Herman, Metall. Trans. 2, 13 (1971). 5. D. H. Kirkwood, Acta metall. 18, 563 (1970). 6. D. H. Kirkwood and A. W. West, Scripta metall. 10, 681 (1976).

7. T. Hirata and D. H. Kirkwood, Acta metall. 25, 1425 (1977). 8. R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser and K. K. Kelley, Selected Values o f the Thermodynamic Properties of Binary Alloys, Am. Soc. Metals, Metals Park, OH (1973). 9. J. S. Langer, Lectures Prepared for Presentation at the SITGES International School of Statistical Mechanics (1980). 10. K. Binder and D. Stauffer, Adv. Phys. 25, 343 (1976). 11. F. K. LeGoues, H. I. Aaronson, Y. W. Lee and G. J. Fix, Proc. Int. Conf. on Solid Solid Phase Transformations (Edited by H. I. Aaronson, D. E. Laughlin, R. F. Sekerka and C. M. Wayman), p. 427. Metall. Soc. A.I.M.E., Warrendale, PA (1983). 12. F. K. LeGoues, Y. W. Lee and H. I. Aaronson, Acta metall. 32, 1837 (1984). 13. H. E. Cook, D. de Fontaine and J. E. Hilliard, Acta metall. 17, 765 (1969). 14. F. K. LeGoues, H. I. Aaronson and Y. W. Lee, Acta metall. 32, 1845 (1984). 15. H. E. Cook and D. de Fontaine, Acta metall. 17, 915 (1969). 16. K. C. Russell, Adv. Coll. Interface Sci. 13, 205 (1980). 17. K. C. Russell, Phase Transformations (Edited by Aaronson H. I., p. 219. Am. Soc. Metals, Metals Park, OH (1970). 18. W. W. Mullins and R. F. Sekerka, J. appl. Phys. 34, 323 (1963). 19. H. B. Aaron, D. Fainstein and G. R. Kotler, J. appl. Phys. 41, 4404 (1970). 20. C. A. Mackliet, Phys. Rev. 109, 1964 (1958). 21. A. Knappworst, Z. Phys. Chem. 12, 30 (1957). 22. J. D. Livingston, Trans. Am. Inst. Min. Engrs 215, 566 (1959). 23. J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 28, 258 (1958). 24. F. K. LeGoues, Ph.D thesis, Carnegie-Mellon Univ. Pittsburgh, PA (1983). 25. G. J. Shiflet, Y. W. Lee, H. I. Aaronson and K. C. Russell, Scripta metall. 15, 719 (1981).