Influence of crystallography upon critical nucleus shapes and kinetics of homogeneous F.C.C.-F.C.C. nucleation—II. The non-classical regime

Acta metall. Vol. 32, No. 10, pp. 1837-1843, 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 A N D KINETICS OF HOMOGENEOUS F.C.C.-F.C.C. NUCLEATION--II. THE NON-CLASSICAL REGIME F. K. LEGOUESt, Y. W. LEES and H. I. AARONSON Department of Metallurgical Engineering and Materials Science, Carnegie-Mellon University, Pittsburgh, PA 15213, U.S.A.

(Received 14 September 1983) A~traet--In a previous paper, the shape of the critical nucleus and its influence upon nucleation kinetics were studied at near zero supersaturation, i.e. in the classical nucleation regime, using a discrete lattice plane model of the coherent interphase boundary energy. In order to extend these studies to higher supersaturations, a discrete lattice point model, based on a formalism developed by Cook, de Fontaine and Hilliard, is employed in the present paper and applied to homogeneous nucleation of coherent f.c.c. precipitates in an f.c.c, matrix. Concentration profiles and free energies of formation of critical nuclei are calculated from this model as a function of temperature and supersaturation and compared with results obtained from the Cahn-Hilliard continuum non-classical model and the previously used discrete lattice plane classical model. As predicted in effect by Cahn and Hilliard, the three models converge at very low supersaturations, and the continuum and the discrete lattice point (but not the classical discrete lattice plane) models also do so near the spinodal. Thus the most important differences between the continuum and the discrete lattice point models develop at intermediate supersaturations. The main advantages of the discrete lattice point model are that it allows the influence of crystalline anisotropy to be taken into account, permits treatment of arbitrarily steep variations in composition and provides a more convenient milieu for the incorporation of volume strain energy, as is done in the next paper in this series. Rrsum~-Dans un article antrrieur, nous avons 6tudi6 la forme d'un germe critique et son influence sur la cinrtique de la germination pour une sursaturation proche de zrro, c'est fi dire dans le rrgime de germination classique,/t l'aide d'un modrle de plans r&iculaires discrets pour l'rnergie du joint interphase cohrrent. Pour 6tendre ces &udes ~i des sursaturations plus 61evres, nous utilisons dans cet article un modrle de noeuds discrets qui repose sur un formalisme drvelopp6 par Cook, de Fontaine et Hilliard et nous l'appliquons ~ la germination homogrne de prrcipitrs c.f.c, cohrrents dans une matrice c.f.c. Nous avons calcul6 ~i l'aide de ce modrle des profils de concentration er des 6nergies libres de formation de germes critiques en fonction de la temprrature et de la sursaturation et nous les comparons avec ceux qu'on obtient avec le modrle continu non classique de Cahn et Hilliard et avec le modrle de plans rrticulaires discrets utilis6 antrrieurement. Comme l'ont prrvu Cahn et Hilliard, les trois modrles convergent pour les trrs faibles sursaturations, ainsi que le modrle continue et le modrle de noeuds discrets (reals non le modrle de plans rrticulaires discrets) au voisinage de la spinodale. Ainsi, les diff6rences les plus importantes entre le modrle continu et le modrle de noeuds discrets se drveloppent aux sursaturations intermrdiaires. Les principaux aventages du modrle de noeuds discrets sont qu'il permet de rendre compte de l'influence de l'anisotropie cristalline, de traiter des variations de composition arbitrairement brutales et qu'il fournit un milieu mieux adapt6 pour incorporer l'6nergie de d6formation volumique, comme on le fait dans l'article suivants de cette s6rie.

Zusammenfassung--In einer vohergehenden Arbeit wurden die Form eines kritischen Keimes und dessen Einflul3 auf die Keimbildungskinetik ffir eine iibersfittigung in der N/ihe von Null, d.h. im Bereich der klassischen Keimbildung, untersucht. Hierzu wurde ein Modell diskreter Giiterebenen ffir die Energie der koh/irenten Grenzft~iche benutzt. Zur Erweiterung dieser Untersuchungen auf gr6Bere iibers/ittigungen wird in dieser Arbeit ein Modell diskreter Gitterpunkte, welches auf eine Formulierung von Cook, de Fontaine und Hilliard zurfickgeht, auf die homogene Keimbildung koh/irenter kfz. Ausscheidungen in einer kfz. Matrix angewendet. Konzentrationsprofile und freie Bildungsenergien der kritischen Keime werden mit diesem Modell in Abh~ingigkeit von Temperatur und (ibers/ittigung berechnet. Diese Ergebnisse werden mit denen verglichen, die mit dem nichtklassischen Kontinuumsmodell von Cahn und Hilliard und dem frfiher benutzten klassischen Modell diskreter Gitterebenen erhalten wurden. Wie tats~ichlich von Cahn und Hilliard vorausgesagt, konvergieren diese Modelle bei sehr kleinen fibersfittigungen. Das Kontinuumsmodell und das Modell diskreter Gitterpunkte konvergieren ebenso in der N/ihe der Spinodalen, nicht jedoch das klassische Modell diskreter Gitterebenen. Folglich entstehen die wichtigsten Unterschiede zwischen dem Kontinuumsmodell und dem Modell der diskreten Gitterpunkte bei mittleren fibers/ittigungen. Die wesentlichen Vorteile des Modells diskreter Gitterpunkte liegen derin, tPresent address: IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, U.S.A. ~:Present address: Department of Metallurgy and Mining Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A. 1837

1838

LEGOUES et al.: INFLUENCE ON CRYSTALLOGRAPHY ON NUCLEUS SHAPES---II dab mit ihm der EinfluB der Kristallanisotropie beriicksichtigt werden kann, beliebig steile ,~nderungen in der Zusammensetzung behandelt werden krnnen und die Volum-Verzerrungsenergie einfacher eingefiihrt werden kann. Letzteres ist die Aufgabe der folgenden Arbeit dieser Serie.

1. INTRODUCTION

measurements of nucleation kinetics in Cu-Co alloys in another paper in this series [5]; the regular solution model turns out to be a reasonable approximation for this alloy system. Furthermore, in f.c.c, crystals without long-range order a restriction to first nearestneighbor interactions is acceptable since the ratio between the distances to second and to first nearestneighbors is relatively large and there are 12 nearest neighbors "shielding" only 6 second nearest neighbors. Coherency strain energy is not considered in the present paper. This factor is incorporated in the next paper [6] in this series, where the effects on nucleation of the anisotropy of both strain and interface energies are simultaneously treated.

Cahn and Hilliard [1] have developed a continuum non-classical model of homogeneous nucleation based upon their earlier formulation of the free energy of an inhomogeneous system [2]. This treatment was carefully specified, however, as being applicable to liquids and to solids wherein composition variations have long wavelengths relative to the inter-atomic spacing. Furthermore the continuum nature of the model makes it very difficult to take into account crystalline anisotropy. Cahn and Hilliard [1] defined the classical nucleation regime as one in which the composition of the nucleus is sufficiently constant near its center so that the volume free energy change and the interfacial energy associated with the critical nucleus can be taken separately into account. 2. REVIEW OF THE CLASSICAL AND CONTINUUM THEORIES OF In the non-classical regime, they pointed out that this NUCLEATION cannot be done because composition varies with position throughout the critical nucleus; thus a non2.1. Gibbs' classical theory of nucleation classical type of formalism, which treats both toGibbs' [7] theory of nucleation is customarily gether, must be utilized. Hillert [3] treated the same problem with a one- applied under classical conditions. When the dimensional discrete lattice model. This approach nucleus:matrix interfacial energy, or, is taken to be does include anisotropy of the crystal lattice, but the independent of boundary orientation and nucleation one-dimensional condition prevents realistic model- is homogeneous, the work required to form a critical ing of nucleation under non-classical conditions. nucleus is then Cook et al. [4] have developed a very general model 16~o-3 4~ AF* = R*2tr (1) for the free energy of cubic systems having a non3AF 2 3 uniform composition. It was constructed by analogy with the Cahn-Hilliard [1] model, but used a discrete where A Fv is the volume free energy change attending lattice point approach instead of a continuum one. nucleation, tr applies to a flat interface of infinite This model was used to rewrite the diffusion equation extent and R* is the radius of the spherical critical in cubic crystals and to study the early stages of nucleus. However, equation (1) is also correct under non-classical conditions, provided that a can be ordering. In the present paper, this model will be used to defined and calculated as prescribed by Gibbs. Since study nucleation of f.c.c, crystals in an f.c.c, matrix, this proves quite difficult, the utility of Gibbsian in the situation where all directions and planes are theory is largely confined to the classical regime. parallel in the two lattices. The model seems particularly well adapted for this purpose since it is valid for 2.2. Continuum model of nucleation Cahn and Hilliard [1] showed that the change in composition variations arbitrarily small in extent and takes into account the anisotropy of the cubic lattice. free energy accompanying the formation of a critical The latter characteristic will enable deduction of the nucleus can be written as critical nucleus shape and demonstration that the A F = [ [AF' + K(Vc)E]dv (2) nucleus becomes faceted only when it is relatively dv large. This cannot be done using a continuum or the Hillert model. Because the Cook et al. model is very where general, any free energy functional can in principle be a f ' = f ( c ) - f ( c o ) - (c - - Co) (3) used and any number of interacting atoms can be (co) simultaneously considered. Nonetheless, numerical methods have to be employed in order to study where F(c) and F(co) are the free energies of one atom in homogeneous solutions of compositions c nucleation and, for the sakes of both simplicity and computational feasibility, only a regular solution and co, the initial and the average compositions of the model with nearest-neighbor interactions was used. matrix, and K is the gradient energy coefficient. By These calculations are compared with experimental considering that the critical nucleus has the lowest /

LEGOUES et al.: INFLUENCE ON CRYSTALLOGRAPHY ON NUCLEUS SHAPES--II free energy of formation and will thus be at a saddle point in free energy-composition space, Cahn and Hilliard obtained the following equation for the concentration profile 2KV2c + ( ? )

(Vc) 2 = 0 A F -~--c"

formation of the critical nucleus in an initially homogeneous solution, we have to write, as demonstrated by Cahn and Hilliard [1] for the continuum model

(4)

For a regular solution

+x:~_~F.[c(p +r)--c(P)l 2 za

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

+ 2kT~c(l - c)

(5)

and (6)

K = kTca2 6

where T~ is the critical temperature of the miscibility gap, k is Boltzmann's constant, T is the absolute temperature and a is the lattice parameter. K is thus seen to be independent of c and the equation to be solved for the concentration profile becomes, considering spherical symmetry [1] K ~2c

4K ~c

~AF

2 ~r~+ 7 ~ -

~c

AF=l E (~T[c~,lnS + (1 - cp)ln 1 - C p ] U,,~ ( 1 Co ~ A kT,,~ } - 2kTc(cp - Co)2 + ~ L [c(p + r) - c(p)] 2

(7)

with the following boundary conditions

and

atr=O

and

c = c o at

r=

DISCRETE LATTICE POINT MODEL OF NUCLEATION Cook et aL wrote the free energy of a non-uniform system composed of two types of atom, A and B, as the sum over the lattice sites of two terms, the free energy F(%) that an atom at Site p would have in homogeneous surroundings of composition %, and of a term proportional to the square of the local compositional gradient. Thus the free energy of a nonhomogeneous system can be written as

(10)

r

where Z is the bulk coordination number (12 for f.c.c.). The critical nucleus is the one with the lowest free energy barrier and will thus be defined by

r=oo.

3. A THREE-DIMENSIONAL

(9)

r

where A F ' is defined by equation (3). In the balance of this paper, a regular solution model will be used, but the same treatment is applicable to any solution model as long as K can be calculated and the system of equations to be derived for calculating the Cp'Sdoes not become so complicated that it is not solvable even by numerical methods. Thus, substituting equations (5) and (6) into equation (9) yields the following equation for the work to form an embryo in an initially homogeneous matrix of composition Co.

a L

~c --=0 ~r

1839

O(AF) _ 0 ~cp

for allp

or

kT

cp(1 -- Co)1 _ 4kTc(c p _ Co) + 2kTc lnc0(1 c?)J ~[c(p -2c(p)+c(p-r)]=OforaUp.

+r) (11)

Although this model is strictly applicable to only one nucleus in an infinitely large matrix, it can be safely used for "very dilute solutions" of nuclei where there is no interference between the diffusion fields of adjacent precipitates. Calculation of the concentration profile requires that equations (11) be solved simultaneously at all relevant points p. The symmetry of the f.c.c, crystal greatly reduces the number of c(pl] 2} (8) F = - ~ {1F ( c p ) K + 7a~ ~ [c(p + r ) points one has to consider. Furthermore, the number where p represents an atomic site, the summation of equations is limited by the fact that, as the vector over p includes all the atomic sites in the system, K p (representing the distance of a particular point from is the gradient energy coefficient defined by Cahn and the center of the nucleus) becomes large, Cp apHilliard [2], N~ is the number of atoms per unit proaches co in all directions, though often not univolume, cp is the probabilityt of having a solute atom, formly. These circumstances makes it poss.ible to B, at site p, r is the radius vector joining an atomic solve numerically a reasonable number (less than 200) site to one of its nearest neighbors, and the s u m - of non-linear equations. A Newton-type method was mation over r is done over all the nearest neighboring used for this purpose. The substitution .into equation sites. In order to calculate AF*, the free energy of (10) of the cp's thus found gives the free energy of formation of the critical nucleus. ~'For the sake of simplicity, Cp will be refered to as !'the composition at site p". However, it should be quite clear that a "point" does not have a composition but a probability of being occupied by a certain species of atom.

4. RESULTS

Figures 1 and 2 show the concentration profiles obtained using equation (11) at T / T c equal to 0.25

1840

LEGOUES et al.:

I N F L U E N C E OF C R Y S T A L L O G R A P H Y O N N U C L E U S S H A P E S - - I I

1.0

08-0.6-I

c o =0060

OA--

g

co =0.040

+

0.2~+ ~ + ' + ~"+"be'N++H+

ol

I

I

I

1

I 5.0

o

E

8

1.0

T

+ +++~

4-

Q6

i

CO=0.010

+

\

co =0002

O4 +

0.2

+

0

1.0

1

3.0

20

4.0

5.0

-

1

'1.0

L 30

1

2.0

;: v,',;::::

4.0

R/a

R/o

Fig. 1. Probability of having a solute atom at a point vs the distance of the point from the center of the nucleus at T / T c = 0.25, The crosses represent calculations performed using the discrete lattice point model; the continuous line was calculated from the continuum model.

1.0

0.8

06

co =O.210

co :0.240

"%++

04 • .

~

,

.

++++~,~+~

~

02

g l

0

l

I

I

I

I

I

I

I

I

g E

,3

40

08

\+ 06

-

\+\

~.

co=0.180

02

X I

I

I

I

I

0

1.0

2.0

3.0

40

5.0

04

I

0

1.0

I

2.0

¢,=0140

I

3,0

I

4.0

I

5.0

R/a

R/a T~ Tc

:0.75

Fig. 2. Probability of having a solute atom at a point vs the distance of the point from the center of the nucleus at T/Tc = 0.75. The crosses represent calculations performed using the discrete lattice point model; the continuous line was calculated from the continuum model.

LEGOUES

et al.:

INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES---II - --------

T/ Fc=0.25

Discrete l a t t i c e model Continuum model Classical theory

1841

T/T~ =0.TS

80

10

l

60

/ \

K. ,\ LL

40

~'\\

<3

\

20

0

'~.

Csp

I

I

0.01

J 002

~ 003

1 0.04

J

005

"~..

CM

Csp

I

006

0.12 0.14 0.16

0.I

018

0.20 0.22

0.24

Composition

Fig. 3. Free energy of formation of the critical nucleus as a function of composition (i.e. supersaturation) at constant temperature. and 0.75 respectively. The calculations were done in three dimensions, as the foregoing equations clearly indicate. But, for the sake of representation, the composition is here projected only as a function of distance. These figures also show the results obtained using the Cahn-Hilliard continuum model: equation (7). The equilibrium and spinodal compositions at 0.25 and at 0.75 T~ are, respectively,

0.00037 and 0.146, and 0.112 and 0.25. Thus one can notice that when the concentration profile is very diffuse, namely at high temperatures or near the spinodal, the two models give closely similar profiles. When T = 0.25 Tc, near the equilibrium composition (at co = 0.002) the two models differ, but this will have little influence upon AF* since the radius of the nucleus goes to infinity in the same way for both

Discrete l a t t i c e model Continuum t h e o r y C l a s s i c a l theory

T/~ :025

T I Tc : 0 . 7 5

4 --

2,0

J

1.5

1.0

05

3--

2 --

1 --

CM

Csp

Csp I

I

I

0.01

0.02

J

I

[

0.03 0.04 0.05

I

I I

J J 0.10

0.06

0.12

I

I

I

0.14

0.16

0.18

I

I

I

0.20 022 0,24

Composition

Fig. 4. Radius of the critical nucleus as a function of composition (i.e. supersaturation) at constant temperature.

1842

LEGOUES et al.:

INFLUENCE ON CRYSTALLOGRAPHY ON NUCLEUS SHAPES--II

models. Comparison of Figs 1 and 2 indicates that the two models differ most at intermediate compositions and at low temperatures. This conclusion is reinforced by Figs 3 and 4 which show the variations of the free energy of formation and of the radius of the critical nucleus with composition at the reduced temperatures of Figs 1 and 2. With Cahn and Hilliard [1], the critical radius has been taken arbitrarily as that corresponding to (Co+Co)~2 where cc is the composition at the center of the nucleus. In these figures, one can furthermore note, again with Cahn and Hilliard [I], that the classical theory gives correct results only at low suigersaturations. It is particularly important to observe that the discrete lattice point model always gives lower free energies of formation than the continuum model. This should be expected since, as shown in a previous paper [8] and discussed in the next section, the discrete lattice point model can predict anisotropy or faceting of the nucleus. Since AF* is proportional to the volume of the critical nucleus [7] and this volume is reduced by faceting, so also will be its AF*. 5. C O M P A R I S O N O F T H E T H R E E D I M E N S I O N A L CALCULATIONS

WITH THE ONE-DIMENSIONAL RESULTS OBTAINED IN REF. 181 In Figs 1 and 2, the concentration profiles appear nearly isotropic throughout the concentration ranges investigated. Thus, the concentration at a given point depends only on its distance from the origin, not on its specific position in space. For example, in Figs 1 and 2 the atoms situated at coordinates (3/2 3/2 0) and (2 1/2 1/2) are both 2.1 lattice parameters from the origin and have very nearly the same composition even though they are in different positions. The faceting illustrated in Ref. [8], which should show up in the concentration profiles, since facets represent isoconcentrates, appears only at smaller supersaturations. Thus, Fig. 5 shows the concentration profile obtained at 0.25 Tc for Co= 0.001. The profile o P o i n t s in the

(100)

direction

A Points

('1'1'1)

direction

in the

+ Others

+ 0 -P +'-P A l+

4.0

O8

co

:oool

T = 0 . 2 5 rc

o=

T~ 06

o E 3 0~

-1-

4 4

o Points

in the ( 1 0 0 )

direction

A Points

in the (111)

lirection

+ Others 1.0

....

+ . o , + + + A - ~ ', ', ', ',!. ~ - ~ . + •.

8 o8 •~

Co:OOOO7 r =0255

;+.'.

: :i~ ': _./( 5 5

"

06

:

>i

OO

R(111)

:~" RAV R(IO0),'

7 I

"

Q2 •. . .

I

I

I

1

2

3

I 4

5

,q'/a

Fig. 6, Concentration profile at

T/T,. = 0.25 and co = 0.0007.

is n o w a n i s o t r o p i c , e.g. the p o i n t s (3/2 3/2 0) a n d (2 1/2 1/2) n o l o n g e r h a v e the same c o m p o s i t i o n .

Figure 5 shows what the radius should be in the (111 ~ direction as well as an estimate of the composition of the points in the (111) direction (filled triangles), and in the (100~ direction (filled circles) in order to have the predicted faceting. It is thus clear that at co = 0.001, the faceting is far from being as strong as that predicted in Ref. [8]. Figure 6 shows the concentration profile at the same temperature but for Co= 0.0007, i.e. at a still lower supersaturation. Obviously, the profile is now much more anisotropic, and predicts about the same faceting as was displayed in Ref. [8]. This shows that the faceting is not only a function of temperature, but also of the supersaturation or the size of the nucleus. This is readily understandable since, for a large nucleus, the interphase boundary in any given area of the nucleus can be approximated as planar and the atoms in the (111) direction which are at the interface are very far away from the atoms in the (100~ direction which are also at the interface, for example. This is not the case when the nucleus is small. As an example consider, in Fig. 5, two points which are almost at the boundary (the boundary being defined by c = 0.5): (3/2 3/2 0) and (2 1/2 1/2); the distance between these two points is 1.2a. In Fig. 6, two points also lying at the boundary are (5/2 5/2 0) and (7/2 1/2 0). The distance between these two points is 2.2 a, which is about twice as much as before and explains why they can have such different compositions even though they are at the same distance from the center of the nucleus. In the first example, if two points so close to each other had very different compositions the resulting high gradient energy would produce an unacceptable increase in the total free energy at these points.

02

I 'i

I ---%++Ho2

3

I 4

R/a

Fig. 5. Concentration profile at T I T c = 0.25 and co = 0.001.

6. S U M M A R Y

A discrete lattice point non-classical model has been evolved from the formalism of Cook et al. [4] in order to calculate nucleus concentration profiles,

LEGOUES et al.: INFLUENCE ON CRYSTALLOGRAPHY ON NUCLEUS SHAPES--II

1843

critical radii and free energies of formation. The Foundation for support of this research through Grant present model and the continuum model of Cahn and DMR-80-07567. Hilliard [1] have been shown to converge at high temperatures and near the spinodal and to diverge mainly at intermediate compositions. The discrete REFERENCES lattice point model invariably yields smaller values of 1. J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 31, 539 R* and AF*. A comparison with the classical theory (1959). of nucleation shows, in agreement with Cahn and 2. J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 28, 258 Hilliard [1], that this theory gives appropriate results (1958). only at very low supersaturations. Finally a com3. M. Hillert, Acta metall. 9,525 (1961). parison of the discrete lattice point model with the 4. H. E. Cook, D. deFontaine and J. E. Hilliard, Acta metalL 17, 765 (1969). discrete lattice plane version of the classical nucle5. F. K. LeGoues and H. I. Aaronson, Acta metall. 32, ation model has been made and shows, as expected, 1855 (1984). that when the nucleus is large the two models con6. F. K. LeGoues, H. I. Aaronson and Y. W. Lee, Acta verge. Of particular interest is the finding that the metall. 32, 1845 (1984). 7. J. W. Gibbs, Collected Works, Vol. 1. Yale Univ. Press, pronounced faceting displayed on classical critical New Haven, CT (1948). nuclei in the results of Ref. [8] is reproduced by the 8. F. K. LeGoues, H. I. Aaronson, Y. W. Lee and G. J. discrete lattice point non-classical model only at very Fix, Proc. Int. Conf. on Solid-Solid Phase Transsmall supersaturations. formations (edited by H. I. Aaronson, D. E. Laughlin, R. F. Sekerka and C. M. Wayman), p. 427. Metall. Soc. Acknowledgements--Appreciation is expressed to the A.I.M.E., Warrendale, PA (1983). Division of Materials Research of the National Science