Elasticity effects on the microstructure of alloys containing coherent precipitates

Progress m ~Uaterrals Science Vol. 40, pp.79-l 80, 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0079~6425196 $32.00

Pergamon

PII: SOO79-6425(%)00001-l

ELASTICITY EFFECTS ON THE MICROSTRUCTURE OF ALLOYS CONTAINING COHERENT PRECIPITATES Minor-u Doi Department of Materials Science and Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku. Nagoya 466, Japan

CONTENTS 1. INTRODUCTION 2. ENERGY STATE OF A COHERENT PARTICLE 2.1. Calcularion of Elastic Strain Energy 2.1 .l. Elasflc strain energy of inclusion 2.1.2. Elastic strain energy of inhomogeneit) 2.2. Calculation of Elastic Interaction Energy 2.2. I. Elastic mreraclion between particles 2.2.2. Elastic interaction due fo the interference of elastic strain fields 2.2.2.1. Calculations based on Yamauchi and de Fontaine 2.2.2.2. Calculations by Khachaturyan and his group 2.2.3. Feature of elastic interaction energy 2.2.4. Modulus effect on rhe elastic interaction 2.2.4.1. Calculations by Eshelby 2.2.4.2 Calculations bv Johnson and hu group 2.3. Calculation of Inreracrion Energy Between Precipitate Particle and External Stress 2.4. Calculation of Surface (Interfacial) Energ) 3. PHASE EQUILIBRIA IN ELASTICALLY CONSTRAINED SOLIDS 3.1. Restraint on Nucleation Induced by Elastic Strain 3.1.1. Classical theory of nucleation 3.1.2. Spinodal decomposirion 3.1.3. Effect of elastic strain on the spinodal decomposition 3.1.4. Relaxation of the restraint on spinodai decomposition 3.2. Coherent Phase Equilibria 3.2.1. Phase rule 3.2.2. Common tangent construction 3.2.3. Williams’ suggestion 3.2.4. Further investigations of the coherent two-phase state 3.2.5. Inapplicability of Gibbs phase rule and invalidity of common tangent rule 3.2.6. Experimental evidence for the prediction concerning coherent phase equilibria 4. MORPHOLOGY OF COHERENT PRECIPITATES 4.1. Shape of a Single Coherent Precipitate Particle 4.2. Distribution of Coherent Precipitate Particles 4.2.1. Alignment along (100) directions 4.2.2. Parameter describing the directional alignment 4.2.3. Inhomogeneous (non-uniform) distribution 4.2.4. Parameter describing the inhomogeneous distribution 4.3. Effect of External Stress on the Precipitate Morphology 5. THEORIES OF PRECIPITATE COARSENING 5.1. Ostwald Ripening-LSW Theory 5.2. Effect of Particle Volume Fraction on the Ostwald Ripening 5.3. Cornpurer Experiments on Ostwald Ripening 19

80 81 82 82 83 85 85 85 85 87 88 91 91 92 94 95 97 97 97 97 101 105 105 IO.5 106 106 108 112 114

115 115 119 119 120 121 123 124 124 125 131 132

80

Progress in Materials Science

5.3.1. Theoretical treatments based on statistical mechanics 53.2. Computer simulations 6. COARSENING BEHAVIOUR OF COHERENT PRECIPITATES IN ELASTICALLY CONSTRAINED SYSTEMS 6.1. Coarsening Kinetics and Particle-size Distribution 6.1.1. Gamma-prime precipitates 6.1.2. Eflects of elastic constraint 6.1.2.1. Constraint-free or weakly constrained y/y’ systems 6.1.2.2. Strongly constrained y/y’ systems 6.1.3. Coherent precipitates of b.c.c. type 6.2. Directional Alignment and Inhomogeneous Distribution of Precipitate Particles 7. SPLITTING OF PRECIPITATE PARTICLES 7.1. Transmission Electron Microscope (TEh4) Observations of Splitting 7.1.1. Splitting into a doublet of plates 7.1.2. Splitting into an octet of cubes 7.2. Calculations of Energy States Before and After the Split on the Basis of Microelasticity Theory 7.2.1. Splitting into doublet in Ni-AI alloy 7.2.2. Splitting into octet in Ni-Si alloy 7.3. Role of Elastic Interaction in Splitting 7.4. Difference in Split Modes 7.5. Kinetic Analysis of Split Phenomena by Means of Numerical Computer Simulations 8. NEW THEORIES OF PRECIPITATE COARSENING 8.1. Bifurcation Theory-Energetic Treatment 8.1.1. Model structure of two-phase state 8.1.2. The first introduction of the bifurcation concept to the structural stability 8.1.3. The development to accord with the actual alloy systems 8.1.4. Explanation of coarsening behaviour based on bifurcation diagram 8.1.5. Parameter for describing the structure btfurcation 8.2. Kinetic Theories Based on Interface Approach 8.3. Computer Simulations 8.3.1. Interface approach 8.3.2. Time-dependent Ginzburg-Landau (TDGL) approach 9. CONCLUDING REMARKS ACKNOWLEDGEMENTS REFERENCES

132 137 140 141 141 143 143 146 147 149 151 151 152 153 154 154 155 157 160 160 162 163 163 163 166 167 168 169 174 174 174 177 178 178

1. INTRODUCTION Many of the properties of metallic materials are caused by their internal microstructure. In particular, the properties of phase-separated alloys are closely related to the morphology of precipitate particles. Therefore, controlling the precipitate morphology is one of the most important methods for developing new types of precipitation-strengthened alloys. The production of some alloys that have useful properties can be regarded as being equivalent to the fabrication of a stable microstructure that has a desirable precipitate morphology, and thus we have focussed our attention on producing such a morphology. To obtain the phase-separated structure containing finely dispersed precipitate particles, the phase transformations which take place in the course of heat treatments are usually utilized. However, the desired phase-separated microstructure is, in general, obtained by interrupting the phase transformations, i.e. by interrupting the heat treatments, and hence it is almost always in a metastable (non-equilibrium) state from the thermodynamic point of view. Such an interesting and unstable microstructure should potentially change to some other microstructure which is thermodynamically more stable. The individual particles composing the microstructure gradually change their size, shape, distribution, and even composition, to decrease their energy state. At the same time, the useful properties are gradually lost during such structural change.

Elasticity Effects on the Microstructure

of Alloys

81

When the phase-separated structure containing precipitate particles exhibits useful properties, the particles are almost always coherent with respect to the matrix. In such a case, the elastic strain energy arises from the lattice mismatch between the particle and the matrix. Therefore, the elastic strain field forms around the individual particle. Furthermore, the overlap of the strain fields surrounding coherent precipitate particles also induces the elastic interaction energy. Such an alloy system is sometimes called an efastically constrained system. Metallic alloy systems are more or less elastically constrained and it is most likely that the elastic energies, i.e. the elastic strain energy and the elastic interaction energy, greatly influence the behaviour of precipitate particles during heat treatment. The effects of elastic energies on the phase transformation or phase equilibria are attracting considerable attention from both the experimental and the theoretical point of view. A typical example of elastically constrained systems is the r/r’ system of Ni-base alloys, where y’ particles having ordered f.c.c. structure are finely dispersed in the y matrix having disordered f.c.c. structure. In the r/r’ system, some seemingly strange behaviours of coherent precipitate particles, which cannot be predicted by the conventional and, in a sense, classical theories, have recently been introduced one after another by many researchers. Theoretical interpretation of the particle behaviour has also been under investigation for years by many investigators and some new theories, including the elasticity effects, have been proposed so far. Another example is the case of the A2/DO, system of Fe-base alloys where DO, particles with ordered b.c.c. structure are dispersed in the A2 matrix which has a disordered b.c.c. structure. The behaviour of DO, particles is essentially the same as that of y’ particles regardless of the difference in crystal structure. In the present article, the main subject will be limited to the behaviour of second-phase particles in the course of phase decomposition in elastically constrained coherent systems and, in particular, the r/v’ system of Ni-base alloys and A2/DO, systems of Fe-base alloys. The experimental and theoretical results on the effects of elastic energies on the phase decomposition and phase equilibria will be reviewed and the behaviour of y’ and DO, precipitate particles in the course of coarsening will be discussed in particular.

2. ENERGY

STATE OF A COHERENT

PARTICLE

The two-phase structure, i.e. the precipitate morphology, is characterized by both the shape of the individual second-phase precipitate particles and their distribution. The particles have various shapes, e.g. sphere, cube, plate, needle, etc., depending on the alloy systems. Furthermore, the particles are distributed in various ways in the matrix. It is apparent from the thermodynamic point of view that minimizing the energy state governs the precipitate morphology. Therefore, the ability to calculate the energy state is essential to the theoretical interpretation of structural changes. When a precipitate particle is coherent with a surrounding matrix, the lattice mismatch between the particle and the matrix causes elastic strain and an elastic strain field forms around the particle. Thus the two-phase system is elastically constrained. Furthermore, when the system consists of a number of coherent particles, the elastic strain field around each particle interacts elastically with the elastic strain fields around the other surrounding particles when the inter-particle distances are short enough. Therefore, in such an elastically constrained system, the individual coherent particle is associated with an elastic energy EELA which consists of two energies as shown by E ELA

=

Em

+

&NT

1

(1)

82

Progress in Materials Science

where EsTRis the elastic strain energy due to lattice mismatch and EINTis the elastic interaction energy between particles. The elastic interaction arises from the mutual interference of elastic strain fields around the individual coherent particles. It should be noted that once coherency is lost, the elastic energy vanishes. Another type of elastic interaction energy arises from the difference between the elastic moduli of the particle and the matrix, and this has been the subject of a number of theoretical calculations by Eshelby and co-workers,(‘) Johnson and Lee,(2) Johnson,(3) and others. This type of elastic interaction is, however, considered to give less influence on the particle morphology than the elastic interaction between the coherent particles. In addition to the elastic energy, we must consider the surface energy EsURFof the particle, i.e. the interfacial energy between the particle and the matrix. Therefore, when we characterize the two-phase structure, we must know the total energy E TTLof the system consisting of a number of particles, as follows:

G-i-,.= Em + EN + Esuw .

(2)

When we interpret the morphological change of two-phase structure, we should compare the energy states before and after the change. Strictly speaking, when we calculate the energy state, the chemical free energy should be added to the total energy. The chemical energy is the fundamental energy which shows that the two-phase state exists stably, irrespective of the actual two-phase structure being examined. Therefore, the chemical free energy can reasonably be assumed to remain constant during the structural change and hence the energy state (i.e. the total energy) to be considered can be expressed as eq. (2). As will be seen in various parts of this article, whether a certain phenomenon takes place or not often depends strongly on a subtle difference between the elastic energy and the surface energy. The actual alloys used for practical applications are, in general, subjected to an external (applied) stress. The interaction between the external stress and the individual precipitate particle should influence the morphology of the particles. In particular, when the temperature is high enough to permit atomic diffusion, it is most likely that the morphological change induced by the external stress cannot be observed in the absence of external stress. The external stress effect is especially important for y/y’ systems (y’ particles in a y matrix) of Ni-base superalloys which are used at high temperature, e.g. for jet turbine engines. For such precipitate particles in the external stress field, two kinds of elastic energy should be taken into consideration: one is a direct interaction energy between the individual particles and the external stress, and another is an extra energy arising because the particle disturbs the external stress field. When the system is under an external stress, the two energies should be added to the elastic energy EELAin eq. (2). 2.1. Calculation of Elastic Strain Energy Several excellent articles describing methods of calculating elastic strain energy have already been published by, e.g. Eshelby,c4) Khachaturyan,“’ Mura,@) and others. Here we shall briefly survey these methods. 2.1.1. Elastic strain energy of inclusion To make our discussion simple, first we will deal with the system (material D) which contains only one particle CI existing in a practically infinite and homogeneous matrix D-f2 which is either isotropic or anisotropic. The particle has the eigenstrain E;’ which is sometimes called the transformation strain or stress-free strain, and the elastic moduli of the particle

Elasticity Effects on the Microstructure

of Alloys

83

and those of the matrix are the same. Such a particle is usually called an inclusion. The elastic strain energy EsTRof the inclusion is given by E STR

1 =

-

0’ .s”dD

2

sR

”

’

I’

where U: is the stress inside the inclusion and is given as g:, = C&kC, -

Ek:t )2

(4)

where C,,,, are the elastic moduli of the matrix, & is the constrained strain, i.e. the total strain, and ak’,- ai’ is the elastic strain. This equation is really a form of Hooke’s law which states that the strain is proportional to the stress in every solid medium provided the strain does not exceed a certain magnitude. An interesting and very useful point is that if the inclusion is ellipsoidal and the eigenstrain is uniform in it, the internal stress is also uniform throughout the inclusion. Then the elastic strain energy is given in a very simple form as shown by Eshelbyc4,‘)

= E,,,,V, 3

(5)

where E,,,Clis the strain energy density depending on the aspect ratio p of the particle, and VPis the volume of the particle. Since the system which is being considered here contains only one particle, the elastic strain energy is equal to the self energy of the particle. When the particle exists coherently with respect to the matrix as in the r/r’ system, the stress-free transformation strain is pure expansion or pure contraction. In such a case, the eigenstrain is given only by the lattice mismatch aO,as follows:

ap- a, E;’

=

Gd,, =

a,

0

for

i=j

for

i#j

(6)

where 6, is the Kronecker’s delta, and aP and a, are the lattice constants of the particle and the matrix, respectively. The above discussion can be applied to a system consisting of a number of the same types of inclusions by considering that Sz is the sum of the individual inclusions, e.g. V,, in eq. (5) is regarded as the total volume of the particles. In other words, the elastic strain energy is the sum of the self energies of all the particles. 2.1.2. Elastic strain energy of inhomogeneitJ The above simple eq. (5) for calculating the elastic strain energy is valid only for the inclusion: in other words, eq. (5) is not applicable to the case where the elastic moduli of the particle and the matrix are different. In the actual two-phase systems, however, the particle and the matrix have their own different elastic moduli. Such a particle is usually called an inhomogeneity. In order to calculate the elastic strain energy of the inhomogeneity R’, the concept of equivalent eigenstrain (or equivalent inclusion or equivalent particle) is very convenient. The problem in the energy calculation is caused by the differences in the elastic moduli of inhomogeneity (c)k,) and elastic moduli of the matrix (C,,). Therefore, we first imagine an inclusion Q which has the same shape as the inhomogeneity a* and the same elastic moduli as the matrix. Then we give a new suitable eigenstrain to the R, so that the

84

Progress in Materials Science

____--- ,+A,

Plate(Disk) l.*,l’ 10-l

’

---@=l.O) a ‘11111’ 100

NC ’

’ ’ saaG*’ 10’

Aspect ratio , p Fig. 1. The aspect ratio dependence of elastic strain energy densities calculated by the Eshelby method for y’ particle in a Ni-AI alloy and DO, particle in an Fe-Al alloy. E,,(l) has a maximum value when the particle is a sphere: $,,(I) values for Ni-AI and Fe-AI are 0.971 x IO’ and I.3 x IO’ J/m’, respectively.

Q has the distortion, strain and stress which are identical to those of the a* despite the different elastic moduli. The inclusion Q is the equivalent inclusion and the newly given eigenstrain is the equivalent eigenstrain E;. The concept of equivalent inclusion was introduced for the first time by Eshelby.@) The above definition of equivalent inclusion can be formulated by

(7) with

where S,,, is the so-called Eshelby tensor and GUk,is given by Lin and Mura.(‘) Now we have introduced the equivalent inclusion having the same elastic moduli to the matrix, and we can easily calculate the stress inside the particle (ot) by using eq. (7). Then the elastic strain energy EsrR of the actual system can be obtained by using eq. (5). Figures 1 and 2 show examples of Ei,I, E:, and ET,calculated for a y’ particle in a y matrix (y/y’ system) of a Ni-Al alloy and a DO3 particle in an A2 matrix (A2/DO, system) of an Fe-Al alloy*. It is clear from these figures that the elastic strain energy and the equivalent eigenstrains strongly depend on the aspect ratio p of the particle. &l decreases as the p value deviates from 1, i.e. as its shape changes from cube to plate or needle. *Although a number of numerical values of elastic constants have been experimentally and theoretically obtained for the y/y’ system of Ni-base alloys, ‘lo)the following values are used: C;I = 2.201 x IO” N/m2, Cir = 1.460 x 10” N/m* and C& = 1.236 x 10” N/m2 for the y’ particle; ~~1C,! = 2.508 x IO” N/ml, C,, = 1.500 x IO” N/m* and C,, = 1.235 x IO” N/m’ for the Al matrix of Ni. For an FeAI alloy, the following values are used:“*’ C;, = 1.596 x IO” N/mr, CL = I.170 x IO” N/m* and CL = 1.533 x IO” N/m2 for a DO, particle (ordered b.c.c. phase); C,, = 1.712 x 10” N/m2, Cu = 1.204 x IO” N/m* and C, = 1.281 x IO” N/m* for an A2 matrix of Fe!Oat.%Al (disordered b.c.c. phase).

Elasticity Effects on the Microstructure 0.012

I I 111111, I I I11-11,

----,

8 a -qq,r 43

6, 0.01 ----._____________

85

of Alloys

NI-AI -_.

%_

s ~~0.008 -

_______

Fe-A] ----‘*..

.\.\ \\

_------------

__ 0.004-

0.0021o-z

,,,,L,’ 10-l

5 ,*~~~~,’ loo

Aspect

Fig. 2.

ratio

1

,,,,,*’

IO’

’

I”,“’

lo2

,p

The aspect ratio dependence of equivalent eigenstrams calculated by the Eshelby a 7’ partlcle m a Ni-AI alloy and DOI particle in an Fe-AI alloy.

method

for

2.2. Calculation of Elastic Interaction Energy 2.2.1. Elastic interaction between particles The total elastic energy of the system consisting of a pair of coherent particles c( and j?, which is expressed by E(CY + p), is given as

Eta + B) = E(a) + E(B) + A&x,

(9

where E(m) and E(p) are the elastic energies of u and b, respectively. When AEEx # 0, there is an elastic interaction energy between c( and /?. The elastic interaction energy which plays a very important role in the formation and stability of a two-phase structure arises from the interaction between the elastic strain fields around the individual particles. This type of elastic interaction is sometimes called strain-induced interaction.‘5’ 2.2.2. Elastic interaction due to the interference of elastic strain$elds 2.2.2.1. Calculations based on Yamauchi and de Fontaine. Yamauchi and de Fontaine”” have proposed a method for calculating the elastic interaction energy E$ between defect clusters CIand p which are separated at a distance d in an anisotropic matrix, as follows:

E$(d)

= $

CF ““(n)F(q).Ss( - q)exp(iq*d)

(10)

Eq

where V, is the total volume of the system containing CIand fi clusters, n is the unit vector along the Fourier wave-vector q, F Ib (n) is the elastic energy coefficient in Fourier space, and S”(q) and SB ( - q) are the shape functions of u. and /3 clusters in Fourier space, respectively. Here periodic boundary conditions are applied and the boundary effects are assumed to be negligible. F rr8(n) in eq. (10) can be obtained by F

‘f’(n) = !ZB - @y(G - ‘),k@,f

(11)

86

Progress in Materials Science

with

G,4n) = C,klmnn,, .

(12)

The particles have tetragonal strain fields: if the o! and /? clusters are parallel to each other, the tetragonality axis is along the [OOl]direction; if a and /I clusters are perpendicular to each other, one tetragonality axis is [OlO] and the other is [loo]. q,, is the distortion tensor in the diagonal form and is given for the parallel case as

( 1 1

?j”=$=tfO

0 1 0

0

or for the perpendicular

0 0

(13)

t

case as

vu=?

#=q

( 1 (t 1 1

0

0

0 0

0t

01

0

0

1

0 . 1

0 0

0

(14)

The scalar q is the amount of strain and t is the tetragonality ratio. The shape function of an a cluster in the Fourier space s*(q) is given by s”(q) = @+4X1-

4q)l *

(15)

Here 6(q) is 1 at the origin of the Fourier space and is 0 elsewhere. P(q) is the shape function of an a cluster in the real space. When the defect cluster is spherical, P(q) is given as

(16) where V, and r, are the volume and radius of a spherical a cluster, respectively. q is the magnitude of the wave-vector q and the cut-off wave-number qco is taken as the radius Qez of the effective spherical Brillouin zone, whose volume is equal to that of the real Brillouin zone, as follows:

Elasticity Effects on the Microstructure

of Alloys

87

where a, is the lattice parameter. The shape function of the fl cluster in Fourier space is, of course, given in the same manner as s”(q). In the above treatment by Yamauchi and de Fontaine, (13)the elastic constants of the clusters are assumed to be identical to those of the matrix, i.e. the clusters are regarded as inclusions. When we estimate the elastic interaction for precipitate particles in actual alloy systems, however, we should always remember that the particles usually have elastic constants that are different from those of the matrix, i.e. the particles are inhomogeneities. Miyazaki and his collaboratorC4) improved the above original procedure of Yamauchi and de Fontaine(‘3’ to agree with the actual alloy systems. They replaced q and t in eq. (13) or (14) with the equivalent eigenstrain .$, which is identical to the strain caused by the inhomogeneous ellipsoidal particle, as follows:

t =

&:3/E:,

.

(18)

Then the elastic interaction energy between a pair of inhomogeneous ellipsoidal particles can be calculated.When CIand B clusters are identical, the E& value at d = 0 gives the self energy of an c( or fi cluster. The self energy (E,,J calculated in such a way should be equal to the strain energy (ES& obtained with Eshelby’s eq. (5) which gives the accurate elastic strain energy when o! and /? are ellipsoidal inclusions. In fact, when tl and fl clusters are spherical, i.e. p = 1, the two values which are independently calculated are equal. As the clusters become flatter keeping their shapes ellipsoids, the difference between the two values becomes larger: as p decreases, the value of Eself/ESTRdecreases. However, the difference is at most 10% even when tl and j? are very thin plates, e.g. at p = 0.1 .(14) 2.2.2.2. Calculations by Khachaturyan and his group. KhachaturyarP) has maintained that some of the calculation methods for elastic interaction energy such as the method of Yamauchi and de Fontaine,(‘3’ etc., are based on his method or one quite similar to his method. According to Khachaturyan and Airapetyan, (N)the elastic interaction energy for spherical inclusions is given by

(19) where K is the bulk modulus, E,,is the lattice mismatch between the particle and the matrix, and n is the unit vector in the direction of k and is given by n = k/k = (n,, n,., n,) .

(20)

AA(n) = A(n) - (A(n)).

(21)

AA(n) is expressed as

Here 3K[l + 2[(n~nt + n$zt + n$z:) + 3~2n$z$Z] A(n) = ’ -

C,, + (C,, + C,,)t(n$zf + nfnz + n,‘nt) + (C,, + 2C,, + C.&‘nfnfnf

(22)

88

Progress in Materials Science

and (A(n))

where 5 is the elastic anisotropy

=

$2

d9

(23)

factor which is expressed by < = (C,, - C,, - 2C44)lG

(24)

and d9 is the element of the space angle. C,, are the Voigt constants constants) and the relations between C,, and C,,,, are, e.g.

(the cubic elastic

CL, = C,!,, C,, = C,,,,

(25)

C44= C,,,, *

8,(k) in eq. (19) is the shape function in Fourier space and is given when the inclusions are spheres having radius r, as follows: 8 (k) _ 47cr3 3[krcos(kr) - sin@)] 0 3 (kr)’

*

(26)

] S(k) I2 in eq. (19) is given by ] S(k) ] * = x,exp[ - ik(R, - Rb)] . R,.%

(27)

When we deal with a periodic distribution of inclusions of the same type we can regard the three-dimensional lattice of the inclusions as a kind of Bravais lattice. In such a case, the radius vector Ro, which is the position vector of the zero point for the unit cell for the periodically distributed inclusions, describes all the Bravais lattice points. 2.2.3. Feature of elastic interaction energy Figure 3 illustrates the variation in the elastic interaction energy E,,, with the inter-centre distance d between a pair of coherent spherical particles c( and B having the same radius. This calculation was made for the case of y’ in a Ni-Al alloy by using the method based on Yamauchi and de Fontaine.“‘) The E,((;k:’is the elastic interaction energy between CIand p particles which are aligned along the crystallographic direction (hkl). The figure clearly indicates that the elastic interaction energies are greatly affected by the distance between the paired particles. When a and fl particles are distant, the elastic interaction between them is negligibly small and EINTis almost equal to 0. As the distance between u and p particles becomes short, the elastic interaction between them becomes noticeable and the EINT numerical values become large enough to affect the particle morphology. It is clear from Fig. 3 that the EINTvalue also varies depending on the direction (hkl) along which CIand B are aligned: the elastic interaction energy is negative when they are aligned along (100) (i.e. E$$‘)) while the elastic interaction energy is positive when they are aligned along (110) or (111) (i.e. E$!$’ > 0, E#” > 0). The most important feature for particle morphology is that El;?’ has a negative minimum: the inter-centre distance giving the minimum interaction energy is designated by d,,,, and in this case, d,,, is about 1.09 x 2r. This means that y’ particles tend to be adjacent to each other along (100) as a result of elastic interaction. The same conclusion as above is also obtained for coherent particles having various shapes

Elasticity

Effects on the Microstructure



..........

<11rJ>

..I

___----

‘.......“““‘..............

. . . . . . . .._.................~.~~~~~.~.~.~~.~

89

of Alloys



----------____________ . . . . . . . . . . . . . . . . . . . . . . ..__.____...... . ..___.

NI-AI

@=I)

t

Frg 3 Elastrc interaction energies calculated for a pair of spherrcal ;I’ partrcles m a NI-AI alloy by the method of Yamauchi and de Fontaine. The elastic interaction strongly depends not only on the inter-centre distance of the paired particles but also on the direction of particle ahgnment. It should be noted that E,,, has negatrve values when the paired particles are ahgned along the elastrcaliy soft directron (100).

other than a sphere, e.g. a thin plate. Miyazaki et a1.(14’have calculated the E,,, values for a pair of ellipsoidal plates (p = 0.5) in various kinds of relative positions. Their calculation indicates that the E$!,o,O’ for a pair of plates also exhibits a negative minimum when they are adjacent to each other along the (100) direction in both the parallel and perpendicular cases. The above discussion was limited to the r/v’ system of Ni-base alloys where ordered f.c.c. particles of y’ (Cu,Au type L12 phase) are dispersed in a disordered f,c.c. matrix of 7 (Cu type Al phase). Another basic system is the system which consists of a b.c.c. matrix and b.c.c. particles, e.g. the A2/DO, or A2/B2 system of Fe-base alloy: A2 is a W type disordered phase,

Fe-Al @=I)

-0.1

8

’ 1

c

I

a

’ 1.5

’

a

’

h

’

_

’

25

Inter-centre distance , &!I-) Fig. 4. Elastic interaction energies calculated for a pair of DO3 particles in an Fe-AI alloy using the method of Yamauchi and de Fontaine. The difference in crystal structures does not cause any difference m general features: i.e. EINThas negative values when the particles are aligned along (100) which are elastically soft directions.

90

Progress in Materials Science

DO, is a BiF, type ordered phase and B2 is a CsCl type ordered phase. Figure 4 illustrates the variation in the elastic interaction energies E INTwith the inter-centre distances d between a pair of coherent spherical particles LXand fl which have b.c.c. structure and are dispersed in a b.c.c. matrix. This calculation is for the case of ordered DO, particles in a disordered A2 matrix of an Fe-Al alloy by using the method based on Yamauchi and de Fontaine.“” By comparing this figure with Fig. 3, it is clear that there is no practical difference between the elastic interaction in a b.c.c. system and that in a f.c.c. system. In a b.c.c. system also, the elastic interaction energies are strongly influenced by both the inter-centre distance and the particle alignment, e.g. E$,$O’is negative and exhibits a minimum, while E$,:O)> 0 and El&‘) > 0. Generally speaking, actual metallic alloy systems exhibit elastic anisotropy, which should always be taken into consideration when interpreting the precipitate morphology. To describe the elastic anisotropy of the system, the Zener anisotropy factor A which is defined by the following equation is sometimes used: A

2G

s c,, -

(28)

c,,

In the cubic system where A > 1, the elastically soft directions are (100). Since A > 1 for the y’ particle in a y matrix of Ni-base alloys, (100) directions are elastically soft. Furthermore, for DO, particles in an A2 matrix of Fe-base alloys, A > 1 and (100) directions are also elastically soft. It is clear from Figs 3 and 4 that the elastic interaction energy is a negative minimum when the particles are aligned along the elastically soft direction, whether or not the systems are based on f.c.c. or b.c.c. structure. On the contrary, in the systems where A < 1, not ( 100) but (111) directions are elastically soft. Therefore, in such systems, we can reasonably expect that E&y’ should be positive and E$,:” should exhibit a negative minimum. An example of the systems where A < 1 is a Nb-Zr alloy. Figure 5 illustrates the variation in the elastic interaction energies between a pair of spherical particles c1and /I in a Nb-Zr alloy matrix: the paired particles and the matrix are

Nb-zr @=l)

-0.05

I

1

I

I2

I

Ia

1.5 Inter-centre

I

L

I

I

2

I

t

I

I

2.5

distance , d@)

Fig. 5. Elastic interaction energies calculated for a pair of Zr-rich particles in a Nb-rich matrix of a Nb-Zr alloy. In this system, the elastically soft directions are not (100) but (111) because the Zener anisotropy factor is smaller than 1. This means that the general features are similar except that EINT has negative values when the particles are aligned along (111).

Elasticity Effects on the Microstructure

of Alloys

91

assumed to be Nb-20 at.%Zr and Zr-30 at.%Nb solid solutions both having a b.c.c. structure*. As anticipated above, a negative minimum of E ,NTappears in the elastically soft directions (111) but not in (100): E$,:l’ < 0 and E$$‘) > 0. An interesting result is that the elastic interaction exhibits negative values when the paired a and /I particles are aligned along (110): E$,:O)< 0. However, E&!$’ < E&!,:O), i.e. 1E$,:” ( > 1E$$” 1, and it is clear that the elastic interaction along (111) which are the elastically soft directions is more effective than that along (110). The same results as above have also been obtained for MO which is another system having A < 1 by Johnson et al. (19) An important feature of elastic interaction is that the elastic interaction energy takes a negative minimum, which is of course, the absolute minimum of elastic interaction energy, when the paired particles are adjacent to each other along the elastically soft direction. It is this negative minimum that has a great influence on the morphological changes undergone by coherent precipitates. 2.2.4. Modulus efect on the elastic interaction 2.2.4.1. Calculations by Eshelby. Eshelby”) has shown that the elastic interaction energy between two misfitting particles a and /I dispersed in a matrix is given by

Here V, and VBare the volumes of M:and /I particles, respectively, and Ap is the difference in the elastic stiffness constant between the particle and the matrix, i.e. AP. = ~0:- P Apa = P; - CL

(30)

where p:, p; and p are the elastic moduli of the a particle, B particle and the matrix, respectively. Ardell and coworkers (I) have divided the elastic interaction energy EINT of eq. (29) into the energy of direct interaction ,!!& and the energy of the shape-effect interaction ,!& which arises from the deviation in the particle shape from a sphere, as follows: &.,T = EL + EL

(31)

with

(32) *According to the coherent spinodal of the Nb-Zr binary system calculated theoretically by Flewitt,“” Zr concentrations for the phases which should coexist at 400°C are estimated to be 20 and 70at.%. The elastic constants used here are obtained by extrapolating the data (between 4.2 and 300 IQ reported by Walker and Peteflu to 673 K (4OO”C),as follows: C,, = 1.944 x 10” N/mi, Co = 1.131 x 10” N/m* and C, =-0.2822 x 10” N/m2 for the Nb-2Oat.%Zr matrix: C,, = 1.218 x 10” N/m2. C,, = 0.8816 x 10” N/m2 and C, = 0.3161 x 10” Njm’ for a Nb-7Oat.%Zr particle. ‘Even if we use the elastic’co&tants at 300 K which are obtained by Walker and Peter,“*’ the results obtained are essentially the same as in Fig. 5.

92

Progress in Materials Science

Here u and b particles have the same shear modulus p*, and Ap is expressed by A,u = p* - p . Eshelby has pointed out pure expansion or pure calculation of the direct Ardell and coworkers”’ in Ni-Al alloy by using

(33)

that when the particles are spherical and furthermore when E, are contraction, the shape-effect interaction vanishes and hence the interaction only (I!&) will suffice for spherical y ’ particles. have evaluated the elastic interaction between a pair of y’ particles the following equation: (34)

with

2(ET*)2Ap

(35)

where rmand rs are the radii of the paired particles separated at a distance d, v is Poisson’s ratio, and eT*is the eigenstrain which is pure expansion or pure contraction. The EINTof eq. (34) corresponds to the direct interaction energy of eq.(32). The above equations (34) and (35) indicate that the elastic interaction energy increases with decreasing inter-particle distance, with increasing particle size and with the square of the eigenstrain (lattice mismatch). Strictly speaking, both the direct interaction energy and the shape-effect interaction energy should, of course, be calculated for the case of y’ particles in the Ni-Al alloy because their shape is not spherical. However, Ardell and coworkers(‘) mentioned that a simple calculation of the shape-effect interaction energy is not easy and hence an evaluation of only the direct interaction energy is still instructive when interpreting the y’ precipitate morphology. The point of their calculation is that the difference in the elastic modulus between the particle and the matrix causes an elastic interaction which is sufficient to influence the morphological change of y’ particles. For example, when Ap < 0 and v is the same for y’ and the matrix, &NT given by eq. (34) is negative. They explained the directional distribution of y’ particles along (100) directions as a result of such a type of elastic interaction, as mentioned later. Onuki and Nishimori”‘) have also examined Eshelby’s interaction under the simplifying assumptions of isotropic elasticity without external (applied) anisotropic stresses. Their theoretical analysis is based on the Ginzburg-Landau approach proposed by Onuki himself2’,22)for solid solutions in which phase separation is progressing. They have obtained the following conclusions: Eshelby’s interaction exists only when the particles which are harder than the matrix are spherical because the interaction arises from the anisotropic elastic fields which are induced inside the spherical particles from other particles; such elastic fields disappear when the particle shape deviates from spherical - this effect is called ‘shape adjustment of harder domains’. When the elastic field disappears, Eshelby’s interaction should, of course, disappear. 2.2.4.2. Calculations by Johnson and his group. Johnson and Lee’*) have considered the modulus effect when evaluating the elastic interaction energy. They extended the above method of Eshelby, which is based on the isotropic assumption, to the anisotropic system, and evaluated the elastic interaction energy between a pair of inhomogeneous spherical particles in an anisotropic matrix, e.g. N&Al (7’) particles in a Ni matrix. Here the

Elasticity Effects on the Microstructure

of Alloys

93

lattice mismatch between particle and matrix is assumed to be pure expansion or pure contraction, i.e. E; = Eob,,.

(36)

According to their method, the interaction energy between a pair of misfitting particles can be obtained by calculating the difference in the total energies between the system which consists of a pair of homogeneous particles (inclusions) and the system which consists of a pair of inhomogeneous particles. The interaction energy between the particles in the inhomogeneous system (E$) is expressed as El& = AE ITL

-

AL,,

+

&NT

(37)

.

AETTLis the difference between the total energy of the inhomogeneous system (PH) and that of the homogeneous system (E”“‘). According to Eshelby,“’ AEmL is expressed as AETTL= EIH - EHM =- 21 where a:” and ai” and sjl” are, the volume of inhomogeneous

e,yMare, respectively, the stresses and strains of the homogeneous system, respectively, the stresses and strains of the inhomogeneous system, and VSis the system. AEEII is the difference between the self strain energy of the system (PGr) and that of the homogeneous system (Ezf;), i.e. AE,,, = E2r - E%’

(39)

and AE,,, does not depend on the relative position of the paired particles. EINTon the right-hand side of eq. (37) is the elastic interaction energy between the paired homogeneous particles, and is given by Eshelby’4) as

s

afl.sT’d !lrl V ’ “,

&NT= -

(40)

where ai are the stresses caused by a 1 particle. Johnson and Lee(*) obtained substantially the same results as in Fig. 3: that is, the elastic interaction energy along [loo] is negative and exhibits a minimum when the paired spherical y’ particles are adjacent to each other in the Ni-matrix, while the elastic interaction energies along [l lo] and [l 1l] are positive (i.e. E\$ c 0, E[$l > 0 and E&j > 0). Furthermore, they pointed out that the elastic interaction in their anisotropic system is much stronger than that in the isotropic system of Ardell et al.(‘) According to Johnson and Lee,C2)the reason for such a difference is that E& is proportional to d’ for the anisotropic case while E,‘& is proportional to db for the isotropic case. Johnson”) has also improved Eshelby’s approximation (I) to the elastic interaction energy induced by coherency strain due to lattice mismatch. When the eigenstrain 8:’ is pure expansion or pure contraction of E& type, the elastic interaction energy is given as E INT=

(d2

:

$3

+

(d2

2

(41)

$)3 I

JF’MS40/&B

94

Progress in Materials Science

with /j* =

87rAp( 1 + v)p’(l - 2v)(l + V’)EoZ1 + (1 - 2v)AK 9(1 - v)$(l

- 2v’)

2clU - v)

1 -’

(42)

where v* and v are the Poisson’s ratios of the particle and matrix, respectively, and AK is the difference in bulk moduli between the particle and the matrix. Johnson and Voorhees(23) have calculated the interaction energies between rectangular parallelepipeds due to the difference in elastic constants between particle and matrix on the basis of Eshelby’s method. Their calculation indicates that the elastic interaction energy is strongly influenced by the crystallographic direction along which the paired particles are (‘O”)for a pair of identical cubes having {100) faces aligned. A negative minimum appears in EINT (“‘) . They have also calculated the elastic while such a minimum does not appear in E$,:O’or EINT interaction energies for paired plates and paired needles. Their results show that the elastic interaction energy exhibits a negative minimum for the cases where a pair of plates with wide surfaces parallel are aligned along (100) and a pair of rods with long axes parallel are also aligned along (100). Furthermore, they have calculated the elastic interaction energy for more than two particles with different distributions, e.g. forming a stringer, a raft, etc. The results indicate that for the cases of three and seven particles, the elastic interaction energies between the particles are different from those obtained by summing the pairwise interactions between two particles. They have suggested that for a system of more than two particles, the simple summation of the individual pairwise interaction energies gives incorrect values not only of elastic interaction energy but also of the distance d,,,at which the minimum interaction energy is located. The above results clearly indicate that the elastic interaction due to the difference in elastic constants should potentially play an essential role in the formation of precipitate morphology. However, Khachaturyan et al. (Is)have commented that the modulus effect is interesting but cannot be sufficient to explain the experimental results on the morphological changes in an elastically constrained system such as the coherent y’ precipitates in a Ni-Al alloy. Furthermore, Miyazaki et a1.‘24)have also pointed out that in the case of y’ particles in a y matrix, the contribution of such a modulus effect is at most 10% of the strain-induced interaction energy. They have concluded that the morphological changes of particles are mostly influenced by the elastic interaction which arises from the overlap of the elastic strain fields.

2.3. Calculation of Interaction Energy Between Precipitate Particle and External Stress When the external stress is applied to a two-phase material containing inhomogeneous particles, there is an interaction between the particle and the external stress leading to the interaction energy EAT. Furthermore, the individual inhomogeneous particles disturb the external stress field and an extra energy E ,,.,Halso arises. Then the elastic energy state of a coherent inhomogeneous particle under such an interaction is expressed as pointed out by Eshelbyf4) E ELA

=

Em

+

@,.a

+

&w

.

(43)

Here it is assumed that the elastic interaction between particles (strain-induced interaction) is negligible. EST,, corresponds to the self energy of the particle as already explained.

Elasticity Effects on the Microstructure

of Alloys

95

Eshe1byC4,”has shown that when the particle can be regarded as an ellipsoid, EhT and EiNH are given by

where 0; is the applied (external) stress. In this case also, it should be considered that the particle is inhomogeneous and the idea of equivalent eigenstrain E: should be introduced. E; is an extra eigenstrain of the equivalent particle and is used to restore the external stress disturbance arising from the interaction between the inhomogeneous particle and the external stress. E: can be represented in a quite similar way to the case of eq. (7) for equivalent inclusion, as follows:

where .sfi is related to the external stress by Hooke’s law as shown by

a;= C,,kl&

(46)

and E: is the strain of the particle caused by the external stress. When the morphological change of precipitate particles under the external stress is interpreted theoretically, the contribution of not only the elastic energy mentioned above but also the surface energy should, of course, be taken into consideration. Then the energy state under the external stress is expressed as

=

&TR+ &q + EsURF+ I&, f

EINH .

(47)

2.4. Calculation of Surface (Interfacial) Energy The surface (interfacial) energy of a particle (ESURF)is expressed by the equation: E SURF

=

“?E~ .

(48)

Here yS is the surface energy density, i.e. the density of the interfacial energy between the particle and the matrix. Some examples of the interfacial energy density yS,which is sometimes called the specific interfacial free energy, for metal/metal systems are summarized in Table 1. The yS values range from 0.01 to about 1 J m-‘, depending on the types of interface. The yS values for coherent interfaces are smaller than those for incoherent interfaces. The value s in eq. (48) is the surface area of the particle. In almost all the cases, the particle can reasonably be regarded as an ellipsoid of revolution, and the surface is expressed by x2 + y* 2

a

+g=l.

96

Progress in Materials Science Tuble 1. The specific interfacial energy y. for various types of metal/metal systemsQ5)

Alloy system

Interface type

y. Jm-*

Reference

AILCU Al-Zn cu-co

Coherent Al/A&Cu Coherent Al,, ,,,/Zn,,,, Coherent Cu/Co

0.09-o. 11 0.07 0.18-0.23

Cu-Zn Fe-Al Fe-C FeCu

Incoherent m/j3 Coherent A2/DO, Incoherent a-Fe/Fe,C Coherent a-Fe, ,,, ,/f.c.c.-Cu, ,,,,, Incoherent a-Fe/f.c.c.-Cu Coherent y/y’ Coherent y/y’ Coherent y/y’

0.5 0.152 0.74 0.125 0.50 0.014 0.011 0.021

Whelan and HowarthlZ6’ Chadwick(z7’ Servi and TurnbulI(**J Ardell’29’ Purdy30’ Varshavsky and Donoso”‘l Smith’32’ Speich and Oriani”” Smith’32’ Ardell(‘41 Rastogi and Arde11’f5) Ardell’36)

Ni-Al Ni-Si Ni-Ti

Therefore,

the surface area of the particle is given by

for a = c

471a2

.S= {

(

2n a2 J& i

for a < c

arccos : >

When introducing the aspect ratio p ( E c/a) and r,, which is the radius of the spherical particle having the same volume as the ellipsoidal particle, i.e. 4 4 - 7rri = - 7ca2c, 3 3

(51)

eq. (50) can be rewritten in a more convenient form as

I f

(

I+$? l-p2 1 2-pz log I - J---J

2nri-p - 2’3 1 +

S=

[

forp

< 1

forp

= 1

forp

> 1

2\p4m$

2nripdz1’( 1 + d&arccosj)

(52)

In Table 1, it should be noted that the ys values for coherent y’ particles in the y matrix of Ni-base alloys are small enough to bring the competition between the surface energy and the elastic energy in governing the morphological change of y’ precipitate particles. Such elasticity effects completely disappear once the coherency of y’ is lost because the increase in ys accompanying the coherency-loss is large enough to bring the situation that the effects of elastic energy can be no longer compared with those of surface energy.

Elasticity Effects on the Microstructure 3. PHASE EQUILIBRIA

IN ELASTICALLY

97

of Alloys

CONSTRAINED

SOLIDS

3.1. Restraint on Nucleation Induced by Elastic Strain 3.1.1.

Classical theory of nucleation

ChristianO” has extensively discussed the classical theory of nucleation in his book to which the reader may refer for details. Only an outline will be described here. Solute concentration in supersaturated solid solution is macroscopically uniform and constant. Microscopically, however, local fluctuations in concentration exist here and there throughout the matrix due to thermal fluctuation; various aggregates of solute atoms, which are called embryos here, are appearing and disappearing repeatedly. When the supersaturated solid solution is metastable, the energy state of the system is not in the absolute minimum but in the local minimum. Therefore, any small fluctuation does not upset the stability of the system because such a small fluctuation disappears quickly as time goes by, as will be mentioned later. On the contrary, the larger fluctuation, which does not appear so frequently, continues to grow larger. The embryo having the fluctuation at the boundary between survival and disappearance is called the critical nucleus and the critical nucleus has the critical radius r*. The simplest model for the embryo is the microcrystallite whose structure, composition and properties are identical to those of a large precipitate. The driving force for nucleus formation is the difference in the free energy between the new phase or phase mixture and the original solid solution. The change in free energy is negative in the temperature range where the new phases are more stable than the solid solution. On the other hand, the restraints on nucleus formation originate from e.g. the interfacial energy between the new phase and the matrix phase, the strain energy due to the volume difference between the two phases, etc. The relation between the driving force and the restraint factors is schematically shown in Fig. 6. As the smaller embryo grows, the free energy increases and it cannot generally grow large enough to exceed the critical size Y*. However, if the embryo exceeds r* with the aid of thermal fluctuation, the free energy decreases progressively with further increase in particle size and the embryo becomes more and more stable as it grows. When the lattice mismatch is larger, the restraint is stronger and the r* value becomes larger. Therefore, the elastic energy opposes nucleation. 3.1.2. Spinodal decomposition Cahn’3842’has discussed an alternative mechanism for the nucleation of a new phase in a supersaturated solid solution. His theory indicates that while ageing in some area inside the two-phase field of the phase diagram, continuous fluctuation in concentration appears throughout the matrix and increases in amplitude thus giving spontaneous progress towards phase decomposition. The classical nucleation theory implies a critical size and assumes that a sharp interface exists between the new phase and the matrix phase. However, Cahn’s theory does not require any critical size for nucleation and need not assume a sharp interface between the two phases. This type of phase decomposition is called spinodal decomposition. As an example, we deal with the phase separation of an A-B binary alloy which is first homogenized to obtain a uniform a solid solution and is then aged at the temperature TL inside the two-phase field of a, + a2. The composition dependence of the free energy GS of an c1solid solution at T,_ is schematically illustrated in Fig. 7(a). We can draw a common tangent and the points of contact are e, and ez (O), and the 01phase decomposes into ~1, and a, phases in the composition range from e, to e,. When the ageing temperature changes, the free energy curve of the solid solution changes and hence the points

Progress in Materials Science

Radius , r

Fig. 6. Schematic illustration of the relation between the driving force and the restraint factors for the nucleus formation. The chemical energy is proportional to the cube of the radius and is always negative and promotes nucleus formation. The strain energy and the surface energy are proportional to the cube and the square of the radius, respectively, and are positive and restrain nucleation.

of contact e, and e, also change. A pair of contact points at different temperatures form a line indicated by the solid curve in Fig. 7(b), and this curve is called binodul, binodal line or sometimes precipitation line. Inside the binodal, the phase decomposition of CY+ a, + cl2 takes place provided atomic diffusion is permitted. When the alloy composition is near the centre cc in Fig. 8, the energy state of a lies on the concave part of the GS curve, as indicated by 0. If a small fluctuation in concentration occurs, e.g. s, and s2 in Fig. 8, the free energy decreases from 0 to @, and hence this fluctuation survives stably. As the fluctuation becomes larger, e.g. s,’ and s2’, the free energy decreases further, as indicated by the open arrow. The larger the fluctuation becomes, the lower the free energy is and hence the more stable the system is. Finally, the two-phase structure consisting of a, and a2 (indicated by 0) is obtained and the energy state decreases to @. This type of phase decomposition is called spinodal decomposition. The spinodal decomposition takes place in the composition range between the inflection points of the GS curve indicated by 0 in the figure. The curve obtained by connecting the inflection points at different temperatures is just the spinodal (spinodal line) and is shown by the dashed curve in Fig. 7(b). The condition that a2GS/ac2= 0 gives the spinodal, and spinodal decomposition, of course, takes place in the region of a2GS/ac2 < 0. In the region between binodal and spinodal, e.g. cE in Fig. 8, the classical phase decomposition of nucleation and growth (NG) takes place. This type of nucleation is very different from the spinodal type with respect to at least the appearance of the two-phase

Elasticity Effects on the Microstructure

of Alloys

99

b)

t. a

n

Concentrauon

, c

Fig. 7. Schematic illustrations of the concentration dependence of free energy GS of a solid solution at temperature TL (a) and of the spinodal and the binodal lines (b) of a imaginary A-B binary alloy system.

structure. The solid solution LXhaving the alloy composition cE lies at the convex part of the GS curve, as indicated by x . Even if a small fluctuation occurs, e.g. b, and b, in Fig. 8, the fluctuation is unstable and disappears because the energy state increases from x to #. However, this solid solution no doubt has a tendency toward phase decomposition. The first step towards the decomposition is that some areas having higher composition near e2 (e.g. b;) suddenly appear with the aid of thermal fluctuation. As far as this figure is concerned, the higher concentration areas are always stable because the free energy decreases from x due to its formation as indicated by the small arrow. However, it should be noted that the free energy GS is only the chemical free energy. Then such higher concentration areas are not always nuclei as discussed in Section 3.1.1. The o!solid solution actually starts decomposition only when the higher concentration areas having concentrations near e2 are larger than the critical size I-‘. Even when the higher concentration areas appear to cause the free energy in Fig. 8 to decrease, the system merely satisfies the condition based on the viewpoint of chemical free energy and hence there is not always any guarantee that the phase decomposition actually proceeds. The decomposition of NG type proceeds only when the decrement in chemical free energy, which is indicated by the small arrow in Fig. 8, overcomes the increment in surface and elastic energies. In contrast to the spinodal decomposition, since

100

Progress in Materials Science

Fig. 8. Schematic illustration explaining the difference in free energy change between spinodal decomposition and nucleation-and-growth (NG) type decomposition. Spinodal decomposition takes place near the centre of miscibility gap, while NG-type decomposition takes place near the edge of the miscibility gap.

the composition of the nucleus is already near the equilibrium concentration of a1 (i.e. eJ, the nucleus continues to grow with fairly little concentration change. In this case, the two-phase structure consisting of a, and u2 phases appears finally, which results in the decrease in the energy state from x to 0. The region between spinodal and binodal is a metastable region. In this metastable region, because the energy state does not take the absolute minimum but a local minimum, the small fluctuation sometimes spoils the stability of the solid solution but phase decomposition does not proceed until the solid solution overcomes a certain energy barrier with the aid of thermal fluctuation. In contrast, the spinodal region is the unstable region in which phase decomposition proceeds spontaneously to lower the energy state. According to Cahn and Hilliard’43,“) and Cahn,(38*4’)the Helmholtz free energy F of the system containing slight fluctuations of composition throughout an infinite isotropic solid solution of volume V, is expressed as F=&i(4+Wc)2]dV.

(53)

Here, f(c) is the Helmholtz free energy density of a homogeneous system of composition c without any fluctuation, K(VC)~is the additional free energy density due to composition gradient when the fluctuation exists, and K is the gradient energy coefficient. Now we can discuss the stability of such an inhomogeneous system containing fluctuation by introducing the free energy difference AF between the inhomogeneous system and the homogeneous

Elasticity Effects on the Microstructure

of Alloys

101

system without any fluctuation (i.e. the ideal solid solution). Expandingf(c) about the average composition co on the assumption that the amplitude of the concentration fluctuation (i.e. c - co) is very small and that the molar volume is independent of c, we obtain the free energy difference AF, as follows:

AF=

When the concentration

1

i ay

z ac2 (c - c,,)’ + ic(Vc)* dI’

.

(54)

fluctuation can be assumed to be sinusoidal as follows: c - cg =

Acos(Bx),

(55)

the free energy difference per unit volume before and after the occurrence of fluctuation is expressed as

(56)

If AF < 0, the solid solution becomes unstable and phase decomposition proceeds spontaneously. In this case, since JCcan reasonably be assumed to be positive, the condition of atf/ac* < 0 must be satisfied for an unstable solid solution. Inside the spinodal region where dtf/ac* < 0, the unstable solid solution starts phase decomposition only when the wavelength of concentration fluctuation exceeds the critical wavelength A, which is equal to 2 R//Z&,where jL$is given as the /I value at AF = 0. The above explanation is restricted to the case where the elastic strain need not be taken into consideration. 3.1.3. Eflect of elastic strain on the spinodal decomposition When the molar volume is a function of concentration, the coherent elastic strain appears due to concentration fluctuation. Cahn (38,39) has investigated the effect of coherent elastic strain on the spinodal decomposition in isotropic solids. When the amplitude of fluctuation is small, the molar volume V,(c) is assumed to be given as V,(c) = T/,[l + 3?(C -

Gl)l

(57)

where q is a linear expansion per unit concentration change, and I’,, is the molar volume of uniform solid solution of concentration c,,. Such a concentration fluctuation usually induces elastic strain E,, and elastic stress go, and hence an elastic energy arises. If a sinusoidal fluctuation is assumed again here, the elastic energy per unit volume is given by

A*q*E = I_v cos’(px) .

(58)

102

Progress in Materials Science

This elastic energy must be added to, e.g. eqs (53) and (56) when we consider the effect of elastic strain due to concentration fluctuation, as follows: F=

S[

v f(c) +

+$

(c - co)’+

I*

dV 1

AF -=V

2rj*E + I_v + 2N3* *

(60)

The boundary between stable (including metastable) and unstable states is given by the condition AF = 0. This condition can be expressed by the following equation provided j? is small:

etz+

212E=o l-v

.

The spinodal given by eq. (61) is called the coherent spinodul (or coherent spinodal line). On the other hand, the spinodal which is given by

ayjac* =0

(62)

(without any strain effect) is sometimes called the chemical spinodul (or chemical spinodal line). The coherent spinodal, of course, differs from the chemical spinodal if q # 0. When the system is inside the spinodal region, the following conditions hold good:

and ~

l-v

>o

.

The former promotes phase decomposition, while the latter restricts it. Therefore, spinodal decomposition accompanied by elastic strain does not always take place throughout the region where spinodal decomposition, which is not accompanied by an elastic strain, should actually take place. The chemical spinodal is shifted to a lower temperature due to elastic strain. At the maximum point of the chemical spinodal, the degree of the depression of the spinodal AT, due to elastic strain, that is, of course, the difference between the chemical spinodal and the coherent spinodal, is given as follows:

V2E ATs = 2(1 - V)k,N” where k, is the Boltzmann constant and NV is the number of atoms or molecules per unit volume. Furthermore, since actual metallic materials are, in general, elastically anisotropic, the effect of elastic energy is more complicated than the elastically isotropic case. Cahn@@also investigated the spinodal decomposition in elastically anisotropic materials having cubic structure. In his treatment, when n,, n2 and n3 are the direction cosines of the direction of decomposition [hkl], i.e. the direction of /I, to the cube axes, the effects of elastic anisotropy

Elasticity Effects on the Microstructure

of Alloys

103

are introduced by replacing E/(1 - v) in, e.g. eqs (59) and (60) with the elastic coefficient Ytkk,, which is given as C,, + 2C,, c,, + 2(2C44- c,, + c&z:n:

3-

Y,,, = 5 (C,, + 2G) [

+ n:n: + n:n:)

1. (66)

Then the free energy difference per unit volume before and after the occurrence of fluctuations is expressed as AF -=-

+ 2r12y(*k,+ w2

V

.

The direction of decomposition, i.e. the direction of concentration fluctuation, varies depending on the anisotropy and hence the magnitude of the anisotropy factor A. If A > 1 (i.e. 2C, - C,, + C,, > 0), Ycloo,which is given by Y(loo)

=

cc,,+ 2C,*)(C,,-

Cl,)

Cl,

(68)

is minimum, i.e. /I is parallel to (100) which are elastically soft directions. Then the solid solution first becomes unstable with respect to {lOO}plane waves of concentration fluctuation and phase decomposition takes place. Conversely, if A < 1 (i.e. 2C, - C,, + C,, < 0), Y,,,,, which is given by Y

6(G, + 2GYG.4

(“” = 4c44 + c,, + 2c,,

(69)

is minimum, i.e. fl is parallel to (111) which are elastically soft directions. Then the solid solution first becomes unstable with respect to { 11l} plane waves and phase decomposition proceeds. Then the limit of spinodal decomposition in (hkl) is given by (70) where (hkl) is (100) for A > 1 or (111) for A < 1. Now it is clear from the above discussion that the elastic anisotropy restricts the direction of spinodal decomposition. When the structure formed by spinodal decomposition is examined using transmission electron microscopy (TEM), many striations perpendicular to the elastically soft directions can be observed. A typical example is the case of spinodally decomposed Fe-MO alloys whose TEM images exhibit fine striations perpendicular to (100) as shown in Fig. 9(a). Such a structure is called a 100 modulated structure. The Fe-MO system has a large lattice mismatch (sO z 0.08) and hence the effect of elastic strain is also strong. In such a strongly constrained system, the appearance of the modulated structure is very sharp. However, when the elastic constraint is not so strong, the modulated structure is not sharp and the directional striations are rather diffuse, as can be seen in a Cu-Co alloy (Q, z 0.02) in Fig. 9(b). Furthermore, when alloy systems are accompanied by very small elastic strain, instead of the modulated structure, the so-called mottled structure is observed

104

Progress in Materials Science

Fig. 9. Transmission electron micrographs of 100 modulated structures observed in Fe-19 at.%Mo alloy aged at 773 K for 3.6 ks (a) and Cu-10 at.%Co alloy aged at 773 K for 100 ks (b). The larger the lattice mismatch, the sharper the striation. The contrast and the directionality of the modulated structure in Fe-MO are strong, while those in Cu-Co are diffuse.

Elasticity Effects on the Microstructure by TEM. The mottled directionality.

structure,

e.g. Fe-Cr

105

of Alloys

and Al-Zn

alloys,

no longer

exhibits

3.1.4. Relaxation of the restraint on spinodal decomposition In the region between the chemical spinodal and the coherent spinodal, the elastic strain (coherency strain) restrains the solid solution from spinodal decomposition. If such a coherency strain is relaxed, the restrained spinodal decomposition is sure to appear again. One way to relax such an elastic strain is electron irradiation. Kinoshita et af.(45)have investigated spinodal decomposition under electron irradiation in situ in a high voltage electron microscope (in situ HVEM). When foil specimens of some spinodal alloys such as Fe-MO, Cu-Ti, Cu-Ni, etc., are aged in situ under electron irradiation, a modulated structure appears. The point that they made was that under electron irradiation, the spinodal decomposition actually takes place even at the temperature above the coherent spinodal. The increments in spinodal temperature due to the irradiation of 1000 keV electrons at the flux of 5 x 10z3electrons m - *s_ ’ are about 50 K for Cu-Ni, about 170 K for Cu-Ti and even about 400 K for Fe-MO. Furthermore, they also found with similar in situ HVEM observations that spinodal decomposition actually takes place above the coherent spinodal in a Ni-Au alloy. Spinodal decomposition is never induced, even by the electron irradiation, until the irradiation eliminates the influence of coherency strain at a higher temperature than the coherent spinodal. The mechanism has been discussed by Kinoshita et al. as follows. When a periodic modulation of concentration is about to meet the demands of chemical free energy (driving force), the coherency strain which accompanies the modulation makes it unstable so that it disappears almost immediately. This behaviour is normal in the region between the chemical spinodal and the coherent spinodal without any electron irradiation. However, electron irradiation produces many excess defects such as vacancies and interstitial atoms. If the modulated structure forms under these circumstances, the compressive and tensile strain fields accompanying the modulation are effective sinks for excess defects. Vacancies and interstitial atoms will accumulate in the compressive and tensile strain fields, respectively, to relax the coherency strain so that decomposition can take place. In this case, it is, of course, essential from the standpoint of chemical free energy that the ageing and hence the irradiation conditions are in the region where the spinodal decomposition should intrinsically take place, i.e. inside the chemical spinodal line. It has also been found that the upper limit of the spinodal decomposition induced by electron irradiation generally corresponds to the chemical spinodal. 3.2. Coherent Phase Equilibria 3.2.1. Phase rule When a system consists of a number of phases which coexist in equilibrium, the equilibrium state is determined by the variables such as temperature, pressure, concentration of the individual phase, etc. The number of independent variables that can be changed without any change in the number of phases is called the degree of freedom of the system. Now we will deal with the system which consists of @ phases and contains k chemically independent components. Since the composition of each phase can be expressed by k - 1 different components, the number of variables which are necessary to express the compositions of @ phases is @ (k - 1). Besides the composition of phases, the equilibrium state of the system is determined by temperature and pressure which are common variables of all the coexisting

106

Progress in Materials Science

phases. Therefore, the total number of variables which are necessary to express the equilibrium state is @ (k - 1) + 2. The chemical potentials of the individual components should be equal. For a component, i, this condition is expressed as #’ = #’ =

/43’

=

..... =

/p

(71)

where ,u,@” is the chemical potential of the ith component in the nth phase. Since each component has @ - 1 independent conditions, there are in total k(@ - 1) relations required in equilibrium. The number of the variables which we can choose freely is the difference between the total number of variables and the number of relations between variables. This difference is just the degree of freedomfof the system in equilibrium and is expressed by the following equation: f=k-@+2.

(72)

This relation is the mathematical expression of Gibbs phase rule which is the most fundamental and universal principle in phase equilibrium. Every equilibrium phase diagram should be drawn in accordance with the phase rule. 3.2.2. Common tangent construction Another important principle to be considered is that every phenomenon proceeds to decrease the energy state of the system. In other words, the state having the lowest free energy appears as the equilibrium state. This principle already appeared when explaining the spinodal decomposition in Section 3.1.) and the fact that the solid solution a decomposes to a, and a2 phases is a result of the principle. Whether the system is coherent or not, i.e. whether or not the system is under the influence of elastic energy, the principle should always be valid. Here, we will consider the energy state of a phase-separated structure consisting of two coherent phases. When a solid solution of an A-B binary system has such a composition dependence of free energy, as is indicated by the GS curve in Fig. 10(a), the free energy G!: of the two-phase state after decomposing to the two-phase state of a, and a2 phases is expressed by the following equation: G; = G&E - co) + G&o - C,E) GE

-

(73)

CIE

where co is the alloy composition, c]E and czE are the compositions of co-existing a, and t12 phases, respectively, and G,, and Ga2are the free energies of tl, and a2 phases, respectively. This energy state Gt corresponds to the point 0 on the common tangent e,e2 in Fig. 10(a), and the points of contact are, of course, (cIE, G,,) and (czE, G,,). 3.2.3. Williams’ suggestion In the preceding section, we did not take the contribution of elastic energy into consideration at all. The above eq. (73) is applicable only to the case where a, and a, phases coexist incoherently and hence an elasticity effect need not be considered. It was the suggestion by Williams that when the phase-separated structure consists of coherent two phases, we must also consider the coherency energy. According to his suggestion, the energy state after coherent phase decomposition contains the extra energy term E,, +a2, as follows: G=G+&+.2.

(74)

Elasticity Effects on the Microstructure

107

of Alloys

e2

b)

I

I

I

L

A ClE

czc

co

CIC

Concentration

c2E 13

, c

Fig. 10. Schematic illustrations describmg the concentration dependence of the free energies in various two-phase states: constraint-free coherent or incoherent system (a); strongly constrained coherent system (b).t4’l

The addition of E,, +x2 shifts the concentration dependence of the energy state in coherent equilibrium from the common tangent to another line. Williams(46,47)for the first time introduced the above idea to explain a complex structure observed, e.g. in the two-phase field around 30 at.%Au in a Cu-Au system. His model structure is a coherent two-phase mixture having a multilayered structure in which an ordered Cu,Au layer (a) and the solid solution layer (8) are alternately stacked. In the multilayered model structure, the habit plane is (IOO), and CYand /I phases are perfectly coherent. Then the coherency energy E coE(, i.e. elastic energy, is given by the following equation:

E

4r#&&4,‘Afl’(a, - uJ2 CoH= (&A,’ + &A,‘)@, + a,)2

(75)

where r$ is the volume fraction and Q is the lattice parameter. A’ is given as follows:

A’

=

Cl,

+

Cl2

-

2c:,/c,,

.

(76)

108

Progress in Materials Science

When the period is defined by N unit cells and ys is the specific interfacial energy, the total interfacial energy density (&) is given by the following equation: ,l7.= IF

4y,

(77)

Na ’

Williams(47)discussed the coherent equilibrium in various cases, e.g. the case in which the composition-dependence of the lattice parameter is linear but elastic constants are independent of composition, the case in which the lattice parameters are different but independent of composition, etc. Now we will make sure of the points suggested by Williams, using Fig. 10(b) for the former case. The composition dependence of free energy for the solid solution a is expressed by GS. The conventional common tangent rule indicates that in the alloy having composition co, the solid solution a decomposes to the two-phase state of a, and CI*whose compositions correspond to the points of contact (i.e. cIE and cZE,respectively), and the free energy of the system decreases from 0 on GS to 0 on the common tangent line. If we consider the elastic energy which is supposed to be independent of volume fraction, the parabolic elastic energy is tangent to GS at the compositions cIc and cZc.Therefore, when the solid solution of co decomposes coherently to a, and a*, the free energy decreases from 0 to A, and the two phases that have the compositions cIc and cZc coexist coherently. Coherent phase decomposition does not take place in the composition ranges of clEclCand cZCclE,and the miscibility gap becomes narrower. The driving force for this coherent phase decomposition corresponds to 0-A and the coherency energy corresponds to A-m. Since an additional decrease in energy (from n to 0) accompanies the coherency loss, the energy state decreases from A to 0 in addition to the disappearance of coherency energy (i.e. A-m). Then the system reaches the incoherent equilibrium which is called ‘true’ equilibrium by Williams.(47) 3.2.4. Further investigations

of the coherent two-phase

state

Cahn and Larche(48) theoretically investigated, in detail, the coherent equilibrium after Williams’ proposal. When a single crystal of a solid solution in a binary alloy decomposes into two phases of coherent a and /I, which are homogeneous not only compositionally but also elastically, then the system reaches coherent equilibrium when the energy state (which can be expressed by the following equation) is minimized:

F = &wP + (1 - &,x)FB = FELA

(78)

where &.,. is the mole fraction of the a phase, and the effect of the interfacial energy is assumed to be negligibly small. If the lattice mismatch is pure expansion or pure contraction of E,,type, the elastic energy per mole (FELA)is proportional to E:, as shown by the following equation: (79)

where E is Young’s modulus and v is Poisson’s ratio. F” and FB are molar free energies of a and /I phases, respectively, and are supposed to be parabolic with respect to composition having the same parabola constant F”. Cahn and Larche(48) calculated the state having the energy minimum under various conditions of the alloys with composition c,, on the basis of eq. (78). Their results are

Elasticity Effects on the Microstructure

of Alloys

109

summarized in the diagram termed a ‘field diagram’ by using the parameters .seRand cc which are defined as: 4V,,,Es; & eff- (1 - v)F’(c,, - c,,) CN =

0

1

_

2(c~- C,E) c,lE- ('YE

where c&Eand cBEare the compositions of ~1and /? phases at incoherent (true) equilibrium. Two phases coexist when &,,. # 0, 1, and the compositions of the two phases (cf’ for CIand c; for /I) are expressed as follows: N

c,”= 1 -

Eeficn 4

_

EeK

The E,~value increases with increasing mismatch but the energy minimum no longer exists if E,~ > 4. When c$,,,~= 0 or 1, a single-phase state appears, and the composition of the single phase is as follows:

c,” = co” for &* = 1 .

(82)

The above results are summarized graphically in Fig. 11. When there is no elastic energy (i.e. E,~= 0), the two-phase field is - 1 < co” < 1, i.e. c,~ < co < cPE,and the compositions of the co-existing phases do not depend on CON and are constant: c,” = 1 and cp”= - 1, i.e. c,{~and c,~. This agrees well with the conventional idea of phase equilibrium (i.e. incoherent equilibrium). The important case is the coherent equilibrium in which the elastic energy is dominant (0 < E,~ < 4). The two-phase field coincides with the alloy composition range of - 1 + E&/4 < cp” < 1 - ~~~14and becomes narrower with increasing seffdue to increasing mismatch. For example, when sea = 2, the two-phase field is - l/2 < CC < l/2, which is half of the incoherent two-phase field. Furthermore, as cc varies, the tie-line ends which correspond to the coexisting compositions of two phases vary, although the length of the tie-line remains 2 as indicated by the dotted line in the figure. At about the same time, RoiIburd’49.50)also made a similar study on the equilibrium and the phase diagram for coherent phases in solids. His model for theoretical calculation is a multilayered structure consisting of coherent two phases having the same two-dimensional elastic moduli. Also in his model, the elastic energy is considered to be proportional to the square of the difference in the intrinsic phase strains (E’), where the intrinsic strain is assumed to be independent of composition. He has obtained equations for stable equilibrium of coherent phases as follows:

aft aft _~ -=-=

ac, ac,

j-t -f

0”- ~(c, - c,) + (1 - ~c#Q)E~= 0 .

Here f I,is the free energy density of undistorted IPMS 40/z-c

(83)

phase i (i = a or B) and is a function of

110

Progress in Materials Science 6

,

4$

a

B 2-

0

0 b=O ------- E&=2 ............ ecff4

2 _

j i “.$, ;

zu”

1-

-1

o-

I :\ : .\ ,. I : \\ /’ ! I \,’ tieIme! /’ I ,..’ , ,A f :

/

I

‘\

I

I

-2

-1 Alloy

0 composition

1

2

, co”

Fig. 11. Schematic illustration describing the inapplicability of Gibbs phase rule in coherent two-phase equilibrium which is under the strong influence of elastic energies (after Cahn and LarchPQ). In incoherent equilibrium, tie-line ends do not vary, i.e. c,N = 1 and c: = - 1, regardless of alloy composition ct. In coherent equilibrium when 0 < .Q, < 4, the tie-line ends vary with alloy composition: for example, when .scg= 2, the coherent two-phase region is from c,” = - l/2 to cz = l/2 and the tie-line ends move on the pair of dashed lines; one of the tie-lines is shown by the arrow.

composition ci (phase composition) and temperature, p is a Lagrange multiplier, & is the relative volume fraction of /I phase, and the elastic energy density E, is given by E, z GE=.

(84)

Then by solving eq. (83) together with the following condition: (1 - &&zz + &cs = c, = const.

(85)

where c,,is the alloy composition, the equilibrium compositions of two co-existing phases were found and a phase diagram drawn which is different from the classical phase diagram when the elastic energy does not exert any influence. Since the compositions of two phases at coherent equilibrium depend on the relative volume fraction, the idea of a tie-line has no meaning at all. This result is in accordance with the conclusion of Cahn ef al. Johnson and Voorhees(5i) have also discussed theoretically the phase equilibrium in two-phase coherent solid, the model structure of which is such that a sphere of CIphase contains a concentric sphere of j? phase. They considered a number of factors which determine the coherent equilibrium, e.g. the mechanical equilibrium, the coherency constraint which

Elasticity Effects on the Microstructure

111

of Alloys

requires the continuity of displacement and normal forces at the interface between c1and B, the isotropic coherency strain based on the difference in molar volumes of unstressed a and 1 phases, uniform diffusion potential and uniform temperature throughout the system, the interfacial condition which fixes the jump in grand canonical free energies across the interface between c( and /I, etc. Then they introduced four dimensionless parameters, 6, A, [ and @. The first parameter 6 is a function of the bulk moduli of tl and fl and expresses the difference in the moduli between the two phases. The second parameter n expresses the degree of elastic constraint and is proportional to the square of the coherency strain E due to the difference in the molar volumes of Q and /I in the unstressed state. The third parameter { expresses the difference in the curvature of free energy which is a function of composition, i.e. the second derivative of the free energy per unit volume with respect to composition, between CIand /? in the unstressed state. The last parameter @, which is a somewhat complicated function of external stress, bulk moduli, shear modulus, coherency strain, etc., is a quadratic function of external stress. By using these parameters, they obtained the equation of state, which is a function of the volume fraction of the /I phase (4p) as follows: ([26* - ns&#$ + [ - 6*(1 + w) + 46 - n(s + 2014% + [ - 26(1 + w) + 2 + /l(i - 2) + @&#Q+ n + @ - w - l}/(l + 6&)? = 0.

W)

This is practically a cubic equation with respect to the volume fraction of the /J phase because 1 + S$+ # 0 at any condition. Here the compositions of c1 and /I phases (i.e. Y’ and Yp, respectively) are given by YS= _ 1 + A( - w; - 24, + 1) + @ (1 + w,)* ys =

1

4 -

+

w; - 248+ 1) + @(1 +o.

(87)

(1 + w/d*

The difference in compositions 1 y_

of tl and /I, i.e. the length of the tie-line, is given by

yL71 =

_2_

4-w-w,+ (1 + Md*

I)+9 .

By using the above four parameters, various kinds of phase equilibrium states can be described, e.g. the state where elastic effects are absent when ,4 = 0 and @ = 0; the state where a and /I have the same elastic constants and the same curvatures of free energy curves when 6 = 0 and i = 0; the state of < # 0 where the curvatures of free energies of a and b are different; the state of 6 = 0 where the elastic constants of a and /I are different, etc. When the temperature, bulk alloy composition and external pressure are specified, we can calculate the volume fraction and the phase compositions by using the above parameters and eqs (86) and (88). In such a way, Johnson and Voorhees calculated various kinds of two-phase state, and they introduced the so-called phase stability diagram which illustrates the relation between the alloy composition and the phase compositions, i.e. the equilibrium condition in coherent solid systems. The above theoretical treatments had advantages and disadvantages. Cahn and LarchP8) studied the effect of elastic strain energy quantitatively but considered only the elastically homogeneous particles. Johnson and VoorheeP) considered the elastically inhomogeneous particles but did not take into consideration the surface energy. Then Miyazaki et u/.(~*) evaluated the effect of elastic strain on the coherent two-phase equilibria in A-B binary alloys

112

Progress in Materials Science

having the average composition which is given as follows:

X0 by utilizing the free energy of the microstructure

G,,, = Go + AES”RF+ AE,,, + AE,NT+ AGcHEM

Gsvs

(89)

where G, is the chemical free energy, AE SURF is the surface energy of precipitate particles, AESTR is the elastic strain energy, AE ,NT is the elastic interaction energies between precipitate is the excess free energy which is accompanied by the concentration particles, and AECHEM variation due to phase separation. The chemical free energy G, is given by adopting the regular solid solution approximation, as follows:

G(X, 7’) = G,( 1 - X) + G,X + Q,,( 1 - X)X + Rn( 1 - X)ln( 1 - X) + X In x]

(90)

where X is the concentration of B atoms, X, and X, are the phase compositions of CIand /J with respect to B atoms, G, and G, are the chemical free energy of pure A and B, and R,, is the atomic interaction parameter between A and B. The elastic strain energy AESTRis given by

Here the eigenstrain EF is proportional to the difference in compositions between two phases M and /3, i.e. X, - X,, as is shown by E:’ = 6,, &0(X/j- X,) .

(92)

When the ith element has the atomic weight M, and the density p,, the molar volume of microstructure R, is given by

Based on the above concept of the free energy of microstructure, Miyazaki et a1.‘j2’calculated the changes of phase compositions X, and X, as the two-phase microstructure coarsens in both coherent and incoherent states. Their calculation results were successful in reproducing the experimental observations on the y/y’ phase equilibria in a Ni-Al-Ti alloy, as will be explained later. 3.2.5. Inapplicability of Gibbs phase rule and invalidity of common tangent rule Now we must remember Gibbs phase rule which is most important for considering the phase equilibrium. When the phase rule is applied to the present system that two phases coexist at constant temperature and pressure, the freedom, i.e. the f value in eq. (72), is 0. Therefore, if the alloy composition varies, the compositions of co-existing two phases should remain constant. For example, in the theoretical study by Cahn and LarchC,(48)if cgNin eq. (81) varies, c,” and c: should be constant. However, Fig. 1I clearly indicates that in coherent equilibrium, the compositions of the individual phases actually vary depending on the alloy composition. Then we are forced to conclude that Gibbs phase rule no longer holds for coherent equilibrium. The reason why Cahn et al. arrived at such an important conclusion was that they took the contribution of elastic energy into consideration. Johnson followed Cahn’s conclusion and reported that Gibbs phase rule was not applicable

Elasticity Effects on the Microstructure

of Alloys

113

to the coherent systems. According to his paper, (5x)the degree of freedom f is expressed as NV - NC, where NVis the number of variables which are necessary to describe the system and NC is the number of system constraints. He studied the case where @ phases exist forming a series of concentric spheres in the hypothetical coherent system containing k independent components. Since the temperature of each phase is uniform, the number of variables with respect to temperature is @. Since the composition of each phase is also uniform, the composition of each phase is expressed by the mole fractions of k components. Then the number of variables with respect to composition is @(k - 1). As regards the mechanical state of a coherent solid, the condition that the gradient of stress tensor is zero (V . T = 0) must be satisfied for each phase. When assuming linear elasticity, system isotropy and spherical symmetry, we introduce 2@ variables by solving the equation. Moreover, @ - 1 variables for volume fraction and a variable for external pressure are added. In total, NV variables are needed to specify the state of the system: NV = @(k + 3).

(94)

Since the temperatures of the individual phases are the same, there are @ - 1 constraints. Regarding the chemical equilibrium, since the diffusion potential of a given component is the same in all the phases, (k - l)(@ - 1) constraints are added. As regards the mechanical equilibrium, considering the continuity of traction at each interface and the traction condition on the outer surface gives CDconstraints. With respect to coherency, there exists @ - 1 constraints. Furthermore, thermodynamic equilibrium requires @ - 1 constraints. The total number of system constraints NC becomes Nc = (k + 3)(@ - 1) + 2.

(95)

Fir lly, Johnson concluded that the degree of freedomffor

a coherent system is given by

f=N,-N,=k+l.

(96)

Gibbs phase rule of eq. (72) indicates that the freedom f‘should decrease with increasing number of phases. However, the above equation clearly indicates that in coherent equilibrium, the fvalue does not depend on the number of phases Qi. This means that Gibbs phase rule no longer holds in coherent equilibrium. The above theoretical discussion was limited to the coherent phase equilibrium under elastic constraint in imaginary alloy systems. Liu and Agren ‘54)have generalized the Williams approach so as to be applied to actual alloy systems. They calculated the two-phase coherent equilibrium in the Fe-Mn binary system. Firstly, they calculated the degree of freedom .f: When the number of components and phases are k and @, respectively, the number of independent variables is @(k - 1) + 2 + (@ - 1). The number of relations, i.e. the number of constraints, is k(@ - 1). Then the degree of freedom is expressed by the same equation as given by Johnson, as follows: f=k+l.

(97)

Next, Liu and Agren’54’calculated the coherent equilibrium between b.c.c. (CI)and f.c.c. (y) phases in the Fe-Mn system. When the molar Gibbs energies of unstressed u and y phases are G: and G:, respectively, the total molar Gibbs energy G, of the mixture of c( and 7 phases which exist incoherently is given by G, = &,,C% + (1 - &,,,)GL

(98)

where &. is the mole fraction of the CIphase. If CIand y phases co-exist coherently, the elastic

114

Progress in Materials Science

energy AC:, which arises due to the coherency stresses, should be added to the total Gibbs energy as G,,, = +,,,,R + (1 - &,,,)G; + AG: .

(99)

Then the phase boundaries at different temperatures can be obtained by minimizing globally the total Gibbs free energy G, including elastic energy due to coherency. The results obtained in such a way indicate that the coherent a + y region is located fairly well inside the normal (incoherent) a + y region. This is quite similar to the relation between coherent binodal and chemical binodal, but is different from the conclusion obtained by Cahn et al. that at coherent equilibrium, the tie-line ends for the minor phase are outside the two-phase field. Johnson and Miiller’s5) have theoretically re-examined the coherent phase equilibria in non-hydrostatically stressed solids. They cast doubt on both the inapplicability of Gibbs phase rule and the invalidity of the common tangent rule. According to their conclusions, even in coherent equilibrium, Gibbs phase rule is applicable to homogeneously deformed systems and the common tangent construction is effective in some appropriate thermodynamic space. They have introduced the idea of density lines on which the tie-line ends locate, but need not correspond to, the equilibrium phase boundaries. Pheifer and Voorhees(‘@ have shown a graphical method for constructing qualitatively the stable and metastable coherent phase diagrams for the systems stressed due to the lattice mismatch between the particle and the matrix. The point of their method is that the compositions of the co-existing phases can be given by drawing parallel tangents to the molar free energy curves of the unstressed phases. 3.2.6. Experimental evidence for the prediction concerning coherent phase equilibria As regards the prediction of various theoretical investigations on the coherent phase equilibria, the points which are generally understood can be summarized as follows: 1. Gibbs phase rule no longer holds; 2. tie-line ends giving the equilibrium compositions for co-existing phases do not coincide with the phase boundaries between single- and two-phase fields; and 3. the common tangent rule is invalid and cannot be used. At present, there are very few reports describing the experimental evidence for the theoretical predictions. One example is the recent work on item (2) above in a Ti-Al binary They measured the compositions of c1 (disordered alloy made by Johnson’s group. (57*58) h.c.p.) and a, (DO,, ordered h.c.p.) phases of Ti-16.64 at.%Al alloy in both coherent and incoherent equilibria by means of analytical electron microscopy. The Al-concentrations of the a matrix and tlZ precipitate in coherent equilibrium are 14.15 + 0.32 and 19.80 ) 0.61 at.%, respectively. However, the Al-concentrations of u and ~1~in incoherent equilibrium are 14.79 f 0.32 and 20.82 + 0.42 at.%, respectively. Comparing the results clearly indicates that the Al-concentrations in both the matrix and precipitate in coherent equilibrium are less than those in incoherent equilibrium. They concluded that the shifts in coexisting concentrations toward the Al-lean side are due to the coherency. The conventional theory indicates that the coherent two-phase field should be inside the incoherent two-phase field. However, the present results do not agree with the conventional theory, and the Al-concentration of the CImatrix is well inside the single-phase field. Then their results are considered to be direct experimental evidence for the idea that in coherent equilibrium, tie-line ends do not coincide with phase boundaries. Another example of experimental studies on the elasticity effect on the phase equilibria is

Elasticity Effects on the Microstructure

of Alloys

115

the case of the y/y’ system of a Ni-Al-Ti alloy reported by Miyazaki et a1.(52)They measured with energy dispersive X-ray spectroscopy the change in chemical compositions of co-existing y and y’ phases, i.e. the compositions of the tie-line ends, in Ni-8.16 at.%Al-4.96 at.%Ti alloy aged inside the y + y‘ two-phase region (1023 K). According to their experimental results, the tie-line ends vary as the two-phase state changes during ageing, and do not lie on the equilibrium phase boundary until the y’ particles become incoherent and large. When the y’ particles are coherent with the y matrix at an early stage of ageing, the solute concentrations of both y and y’ are inside the equilibrium phase boundaries. When y’ particles lose their coherency, the tie-line ends shift toward the higher solute concentration side: i.e. the solute concentration of incoherent y’ particles moves beyond the equilibrium phase boundary and into the y’ single phase region in the equilibrium phase diagram, and the concentration of the y matrix becomes higher than that before coherency loss. Further ageing produces the coarsening of incoherent y’ particles and finally the tie-line ends lie exactly on the equilibrium phase boundaries. Such movements of tie-line ends are justified by the theoretical calculations based on the idea of the free energy of microstructure(52) explained in Section 3.2.4. Miyazaki et al. cast doubt on the applicability of Gibbs phase rule to such elastically constrained systems as y/y’ of Ni-Al-Ti. Based not only on the theoretical calculations but also the experimental observations, they mentioned that conceivably the Gibbs phase rule collapses in the elastically constrained system because the phase rule does not suppose such two-phase equilibria as are under the strong influence of the elastic constraint. As regards the elasticity effect on the two-phase equilibria, experimental observations are insufficient by far as compared with the theoretical studies. A large number of experimental results are expected to appear in the near future to aid the understanding of coherent equilibria. 4. MORPHOLOGY

OF COHERENT

PRECIPITATES

Some examples of precipitate morphology observed in coherent two-phase systems such as y/y’ in Ni-base alloys and A2/DO, in Fe-base alloys are shown in Fig. 12. It is clear from this figure that different kinds of particle shapes exist: e.g. sphere, cube, plate (disc), rod (needle), etc. To characterize such precipitate morphology, not only should the shape of the individual precipitate particles be taken into account, but also the distribution of the particles. When the particle distribution is considered, the alignment and localization of particles are usually to be specified. Figure 12 also illustrates some different kinds of distributions of coherent particles in a matrix: e.g. homogeneous (uniform), random, directionally aligned, etc. Furthermore, the inhomogeneous (non-uniform) distribution of coherent particles forming a number of localized groups dispersed in a y matrix are sometimes observed as illustrated in Fig. 13. It seems that some relation exists between the shape and the distribution of precipitate particles: the spherical particles are randomly distributed in the matrix, while the cubic or plate-like particles exhibit directional alignment. That is because the cubic or plate-like shape and the directional alignment are all typical examples of elasticity effects, 4.1. Shape of a Single Coherent Precipitate Particle Nabarro’59,60’discussed a situation where the elastic strain influences the particle shape. He estimated the elastic strain energies of a group of ellipsoidal particles having the dimensions a x a x c, as a function of aspect ratio p ( = c/a). According to his results, the elastic strain energy E&p) is a maximum when the particle is spherical (p = l), i.e. E&l). When the p

116

Progress in Materials Science

Fig. 12. Transmission electron micrographs indicating various kinds of precipitate morphology of y’ particles in Ni-base alloys and DO3 particles in Fe-base alloys.

value deviates from 1, E&J) decreases. As the particle shape changes from sphere to plate (i.e. as p decreases), Es&) rapidly decreases and approaches zero for a very thin plate. As the particle shape changes from sphere to rod or needle (i.e. as p increases), _&&I) decreases and approaches (3/4)&,,(p). Needles have a strain energy which is intermediate between the sphere and the thin plate. This feature, of course, falls in line with the prediction of Fig. 1 in Section 2. Kelly and Nicholson w have discussed the effects of atomic mismatch on the particle shapes. They pointed out that the principle shapes of particles to be considered are of four different types: spheres, cubes, discs (plates) and needles. According to their discussions, spheres are formed when the mismatch is equal to or less than 3%, while discs are formed when the mismatch is equal to or greater than 5%. They also mentioned that cubes are the intermediate

Elasticity Effects on the Microstructure

of Alloys

117

Fig. 13. Scanning electron microscope (SEM) image illustrating the non-uniform (inhomogeneous) distribution of y’ particles in a Ni-8.0at.%Al-LOat.%Ti alloy aged at 1173 K and then reversion-treated at 1373 K.

state and can be considered to be basically the same as spheres slightly modified by the elastic anisotropy. Von Hornbogen and Roth (62)have reported the effect of lattice mismatch .sOon the shape of coherent 11’particles in a y matrix in some Ni-base alloys. Here the lattice mismatch is already defined as &()S

a.., - a.. a;.

-

(100)

where a,, and a, are the lattice constants of y’ and y, respectively. When the lattice mismatch is almost equal to 0, e.g. in the Ni-Cr-Al system, the y’ particles are spherical. However, when the lattice mismatch is large, e.g. in the Ni-Be system (Ed> 0.03), the shape of the y’ particles is plate-like or rod-like even at the beginning of coarsening. For alloy systems having an intermediate lattice mismatch, e.g. Ni-Al (a0 = 0.03-0.005) and Ni-Si (E,,= - 0.003), the y’ particles are cubes and are aligned into rows. Furthermore, if the lattice mismatch is large enough, the rows sometimes coalesce into rods. Similar experimental results have been reported by Loomis et al.(63)They obtained a number of Ni-Cr-Al and Ni-Cr-Ti-Al alloys having different degrees of lattice mismatch by means of alloying MO. In the Ni-Cr-Al system, the lattice mismatch increases with increasing MO content and the shape of the y’ particles changes from sphere to cube. However, in a Ni-Cr-Ti-Al system, the lattice mismatch decreases with increasing MO content and the shape of the y’ particles changes from cube to sphere. Now we can reasonably assume that the equilibrium shape of a coherent particle should be determined by minimizing the sum of the elastic strain energy and the surface (interfacial) energy at a constant total volume of particles. Consideration of only the elastic strain energy predicts that the thinner the plate becomes, the more favourable the shape becomes energetically, as indicated by, e.g. Nabarro in the above. At the same time when the strain energy is decreasing, however, it is apparent that the thinner the plate becomes while keeping its volume constant, the larger the surface area and hence the surface energy becomes.

118

Progress in Materials Science

Therefore, consideration of only the strain energy is very partial. The equilibrium shape depends on the balance between the elastic strain energy and the surface energy. Based on the above assumption, Khachaturyan (s.61)has extensively discussed the shape of a single coherent inclusion embedded in an elastically anisotropic matrix. According to his treatment, when a plate-like inclusion of volume V has the dimension a x a x c, the elastic modulus 1, the stress-free transformation strain &oand the surface energy density yI, the shape-dependent part of the free energy is expressed as

(101) The first and second terms on the right-hand side of eq. (101) are the shape-dependent strain energy and the surface energy, respectively. Since V x u*c and the aspect ratio p is defined as c/a, eq. (101) can be rewritten as

(102) When the volume V is a given constant value, the equilibrium shape can be obtained by minimizing E with respect to p, that is, the condition of

dE

-=

dp

0

(103)

gives the equilibrium shape having the most favourable aspect ratio pe, as shown by (104) It is apparent from this equation that the thin plate forms at smaller interfacial energy density, larger mismatch and larger size. According to eq. (104) we can predict the shape change of inclusion during coarsening. Considering that yr is the specific interfacial energy and JEWis the specific strain energy, the ratio lo which is defined as I,=

&

(105)

can be regarded as a kind of material constant depending on the system and has a dimension of length. Then eq. (104) is rewritten as Pe r

( > 2 10 3’5 -__ 3 VI/3 .

(106)

When VII3 x I,, pe x 1 and the shape of the inclusion is equiaxial, the energetically favourable inclusion is cubic. However, as the inclusion coarsens, its volume becomes larger and the condition of VII3>> I, is attained. Then pee 1 and the inclusion changes its shape to thin plate. The greater the effect of elastic energy in comparison with that of surface energy, the smaller is the lo value. The smaller the I, value, the more easily the inclusion changes its shape to plate. The strain energy has an essential effect on the shape of the inclusion and promotes the shape change to a thin plate.

Elasticity Effects on the Microstructure

of Alloys

119

In the above discussions, both the surface energy and the elastic strain energy are taken into consideration. When the surface energy is dominant, particles become spheres to decrease the surface energy. When the elastic strain energy is dominant, the surface of a particle is no longer spherical but consists of a number of planes which are elastically favourable, e.g. (100) for y’ particles in some Ni-base alloys, DO3 particles in some Fe-base alloys. Here it is not surprising that the elastic interaction energy between particles is often completely ignored because it has been widely believed that the elastic interaction energy intluences only the particle distribution. However, the elastic interaction energy actually influences not only the particle distribution but also the shape of the individual particles. A typical example is the splitting of y’ particles, which will be explained later. We should always remember the effect of elastic interaction even when we consider the shape of individual particles. The general principle for determining the particle shape is to minimize the energy state which consists of elastic and interfacial energies. In such a case, an important problem is how to calculate the elastic energies of particles having various shapes: the accurate elastic energies cannot easily be calculated for shapes other than ellipsoids of revolution. Therefore, we tended to limit our discussion to the cases with relatively simple shapes such as spheres, cubes, rods and plates (all of which are ellipsoids or at least can reasonably be regarded as ellipsoids). However, it is doubtful whether the energetically favourable shape always belongs to the ellipsoids of revolution. Voorhees et al. (65)tried to determine the thermodynamically stable morphology of misfitting particles in the course of coarsening in an elastically anisotropic system without the restriction of particle shapes. In the thermodynamically stable state, the conditions of elastic and interfacial equilibria and also of chemical equilibrium should be satisfied simultaneously. Their calculations for two-dimensional systems indicate that the thermodynamically stable shapes are not simple but are intermediate between the sphere and cube with four-fold symmetry. The shapes are quite similar to those observed for y’ particles shown in Fig. 12. There are several similar investigations. Khachaturyan’s group(66.b7’has studied the evolution of two-dimensional shape by utilizing the microscopic diffusion theory. Onuki’s group’68.h9)has also carried out computer simulations by utilizing the diffuse interface theory. Their results will be discussed elsewhere.

4.2. Distribution of Coherent Precipitate Particles 4.2.1. Alignment along (100) directions As has been shown previously, if the Zener anisotropy factor is greater than 1 (A > l), a negative minimum in the elastic interaction energy appears when a pair of particles are adjacent to each other along (100) which are elastically soft directions: e.g. in the y/y’ system of Ni-Al, the minimum is located at d,,,, as can be seen in Fig. 3. The negative minimum of elastic interaction energy has essential effects on the precipitate morphology. It can easily be imagined that every y’ particle should be uniformly distributed in the y matrix forming a simple cubic lattice, the edges of which are d,,, in length and (100) in direction. Such a structure has been proposed by Ardell et al. (I) In systems having smaller lattice mismatch, e.g. I .q,I < 0.001 for Fe-Ni-Cr-Ti, etc., y’ particles are spherical and no alignment can be seen. However, in systems with larger lattice mismatch, e.g. 1E,,1> 0.001 for Ni-Al, Ni-Ti, Ni-Si, etc., y’ particles are cubic and aligned along (100) directions. They showed that an elastic interaction actually arises from the difference in elastic moduli between particle and matrix, and concluded that y’ particles become aligned along (loo} directions due to

120

Progress in Materials Science

the negative minimum of elastic interaction energy. This is a typical effect of elastic interaction energy between coherent particles. 4.2.2. Parameter describing the directional alignment It is certain that a number of y/y’ systems in Ni-base alloys exhibit directional alignment along (100) directions. However, the discussion remains in a qualitative stage because it depends on the feeling that one structure exhibits directional alignment and the other does not. Doi and Miyazaki’70’ proposed a useful parameter which can describe the particle alignment quantitatively. The parameter Y,,, can express the degree of directional alignment along [hkl], and is based on the intensity analysis of the power spectrum of a two-phase structure. Fourier transformation of a two-phase structure containing precipitate particles gives a power spectrum containing some information on the precipitate morphology. Figure 14 illustrates some examples of power spectra obtained for various kinds of model structures of two-phase states. The region distant from the origin of the spectrum reflects the short-range correlation, i.e. the particle shape, while the region close to the origin reflects the long-range correlation, i.e. the particle distribution. When the particles are aligned along [hkl], the intensity of the power spectrum becomes high in [hkl] near the origin of the spectrum. Furthermore, when the particles are periodically aligned along [hkl’j, satellites appear in [hkl]. Therefore, an intensity-distribution analysis of the power spectrum near its origin gives the !ZJy,,, which expresses the degree of the particle alignment along [hkl]. According to Fig. 14, there are no practical differences between the central regions of two power spectra obtained for two model structures having the same particle distribution but different particle shapes. Therefore, to eliminate the influence of particle shapes, the area of the spectrum to be analysed is the area which is inside the circle around the origin whose radius is twice the distance between the origin and the first satellite, one of which is indicated by an arrow in Fig, 14; the radius of each encircled area is twice the distance between one of the first satellites and the origin and contains information only on the correlation larger than one half of the mean inter-particle distance. The parameter Yhk,is defined as (107) Here tihk,is given by

where ZhK,and I_,,,are the power spectra inside the circle in [hkl] and in all the directions, respectively. I$,,, is the ideal $ value for the perfectly random distribution of particles and is the ideal + value for a perfectly ordered distribution. When the particles are randomly $ tn‘ix distributed without any directional alignment, Y,,k,should be equal to 0. As the degree of alignment along [hkl] increases, the Y,,, also increases and finally it should become 1 when the particles are completely aligned along [hkl]. In many two-phase systems such as r/r’ of Ni-base superalloys A2/D03 of Fe-base alloys, Yloo is, o f course, important. Examples of power spectra obtained for some real two-phase morphologies are shown in Fig. 15. The calculated Yyloovalues are 0.0265 for structure (a) and 0.0360 for structure (b).

Elasticity

Effects on the Microstructure

of Alloys

Fig. 14. Examples of power spectra obtained with Fourier transformation of model structures. Since the three model structures are the same except for the particle shape, the features of the central region of each spectrum are practically identical to each other. Then from the region. we can get some information on the particle distribution without any influence on particle shape.

4.2.3.

Inhomogeneous (non-uniform) distribution

Ardell et al.(‘) also postulated that the elastic interaction energy also causes a uniform distribution of y’ particles. Since their conclusion, it has widely been recognized that the important effect of elastic interaction is to bring a homogeneous (uniform) distribution of precipitate particles which are aligned along (100) directions. In actual alloy systems of not only experimental but also commercial Ni-base superalloys, y’ particles are sometimes inhomogeneously (non-uniformly) distributed in the matrix although they are aligned along (100) directions. It is clear from Fig. 13 that y’ particles are non-uniformly distributed in Ni-Al-Ti alloy-forming groups. This localized distribution seems to be contrary to the prediction of Ardell et al.“’ based on the consideration of elastic interaction energy. However,

122

Progress in Materials Science

Fig. 15. Transmission electron micrographs (left) and power spectra (right) of two-phase structures in A2/DO, system: an Fe-Sat.%Si-lOat.%V alloy aged at 923 K (a, a’); an Fe-lat.%Si-8at.%V alloy aged at 923 K (b, b’). The YYIM values are 0.0265 for structure a and 0.0360 for structure b.

such a structure consisting of many aggregates of particles is also a result of elastic interaction between particles. Doi et al.(“) studied the inhomogeneous distribution of coherent particles which arises from the effects of elastic interaction energy. They calculated the variation of the average interaction energies for the individual y’ particles with the number of particles in the aggregate. In their calculation, the model structure of an inhomogeneous state is the case where n3 particles in the aggregate are distributed on simple cubic lattice sites forming a cube whose edges consist of n particles, while the model structure of an homogeneous state is the case where the aggregate of n3 particles forms only a part of an almost infinite simple cubic lattice of particles. In the latter uniform model, the particles in the aggregate are under the influence of the elastic strain fields of the particles outside the aggregate; while in the former non-uniform model, the aggregate is isolated from the outside. According to their calculation results which are shown in Fig. 16, there exists an energy minimum when the aggregate consists of several scores of particles. In other words, the calculation clearly indicates that coherent particles are in the most energetically favourable state when they are non-uniformly distributed, forming groups owing to the effect of elastic interaction. This conclusion is opposite to that of Ardell et al. (‘) that coherent particles should be uniformly distributed throughout the matrix due to elastic interaction energy. The non-uniform structure as in Fig. 13 cannot be explained until we introduce the new idea that coherent particles have a tendency to exhibit a non-uniform distribution.

Elasticity Effects on the Microstructure

I c

--o--

l

of Alloys

123

*....+ *......*... ..*.++.**+i l

IH-state

Fig. 16. Aggregate size dependence of the average interaction energies for the individual y’ particles in the homogeneous (H) state and the inhomogeneous (IH) state.

4.2.4. Parameter describing the inhomogeneous distribution The above discussion means that coherent particles tend to aggregate although the distribution seems rather uniform at a glance. Therefore, if some parameter which can describe the degree of non-uniformity in distribution, i.e. the degree of localization of particles, is introduced, a subtle difference in two-phase structures produced by ageing can be understood much better. Doi and Miyazaki”‘) introduced a new parameter symbolized by n to describe the non-uniformity in particle distribution quantitatively. Their method for the numerical expression of the non-uniformity in particle distribution is based on the examination of whether the lattice points drawn on a micrograph of two-phase structure are in the particle or in the matrix (see Fig. 17): the former points indicated by open circles are labelled P and the latter points indicated by solid circles are labelled M. Here the lattice constant of the grid (a,) is set to the mean particle size 2? where ? is the mean radius for spherical particles and the mean half length of the edge for cubic particles. The nearest neighbour pairs of points

Fig. 17. A square lattice drawn on a model structure of a two phase state to calculate the parameter &,,: the lattice points are indicated by the open circles (on particles) and the solid circles (on matrix).

124

Progress in Materials Science

can be described as P-P, P-M or M-M. The more uniform the particle distribution, the larger is the number of P-M pairs; the more non-uniform (i.e. the more local) the particle distribution, the larger is the total number of P-P and M-M pairs. Counting the number of each pair gives the parameter &,u, which can be used to describe the non-uniformity in particle distribution numerically, and is defined as

where n,, is the number of A-B pairs and na,, is the total number of all the pairs. The trouble with using the parameter LNUarises when the volume fraction of particles exceeds 50%, which is frequently observed in the actual microstructure of commercial superalloys. In such cases, the np_pand/or )2M_M inevitably increase and hence the 1,” also increases incorrectly. Then a parameter /1, which is defined as &lJ - &I,” A = A,,, - A,,, ’

(110)

is introduced to eliminate the influence of particle volume fraction and the degree of non-uniformity in particle distribution can numerically be described correctly. 1,,, is the minimum value of &,u obtained when the particles are randomly distributed. A,,,,, is the maximum value of ;l,u obtained when all the particles unite to form a large square particle having a flat interface. Some examples of calculating the /i values for DO, particles in Fe-Si-V alloys are /1 = 0.399 for morphology (a) and ,4 = 0.286 for morphology (b) in Fig. 14. 4.3. Effect of External Stress on the Precipitate Morphology According to the discussion in Section 2.3., the external (applied) stress should affect the shape of the individual precipitate particles. A good example of such external stress effects is shown for the case of y’ precipitate particles in Ni-Al alloys. Miyazaki et ~1.~‘~)have investigated the morphological changes of y’ particles in Ni-15 at.%Al alloy single crystals due to annealing at 1023 K under tensile and compressive loads in a [OOl] cube direction. The shape of the y’ particles which are initially distributed uniformly and regularly in the matrix changes from cube to plate through rod while annealing without any external load. The same morphological changes take place while stress-annealing regardless of load mode, i.e. tension or compression. The difference in the load mode during annealing, however, gives some preferred orientations in particle alignment. Both rods and plates of y’ are aligned to be parallel to the tensile axis [OOl] but perpendicular to the compressive axis [OOl]. The observations are justified by the theoretical calculations based on such an anisotropic elasticity theory, as is explained in Section 2.3., except for one point. The calculation cannot explain the appearance of rods due to annealing under compression. Miyazaki et al.‘“’ suggested that the elastic interaction energy between particles should be taken into consideration to explain the shape change of precipitates and the rod formation. 5. THEORIES

OF PRECIPITATE

COARSENING

In the early stage of phase decomposition, regions of high solute concentration appear here and there in the supersaturated solid solution, and the new phase, i.e. precipitate or zone, forms in the matrix. Hereafter, new phases are called particles whether they are precipitates or zones. For a short while after the formation of new particles, the individual particle

Elasticity

Effects on the Microstructure

of Alloys

125

continues to grow by absorbing the supersaturated solute atoms in the matrix around the particle. As the particles grow, the degree of supersaturation decreases, and the solute concentration in the matrix decreases to the equilibrium concentration, i.e. the solubility limit which corresponds to the maximum concentration soluble in the matrix. The formation of the new phase is then completed and its volume fraction becomes constant because the particles can no longer collect solute atoms from the matrix. It is evident that the driving force for such structural changes is the degree of supersaturation. Therefore, after the completion of new phase formation there is no longer a driving force for precipitation during further ageing. However, coarsening of particles takes place and the two-phase structure gradually changes. This kind of particle coarsening is a competitive growth because as the volume fraction of particles remains constant*: larger particles can continue to coarsen by absorbing smaller particles. This structural change is just the so-called Ostwald ripening, and the driving force is the decrease in the interfacial energy between the particles and the matrix. The coarsening behaviour of particles is very important in practice since losing the fine microstructure means losing the favourable properties. In this chapter. we will survey the general theories of particle coarsenin g which have widely been accepted.

5.1. Ostwtrld Ripewing-LS

W Theor?

The most widely recognized theory which deals with Ostwald ripening is the one presented by Lifshitz and Slyozov I”‘. and by Wagner 17”, independently. Their theory is sometimes called the LSW theory in short and has been basic to the analysis of coarsening behaviour of particles in solids and especially in metallic solids. However, the LSW theory originally dealt with the case where precipitate particles having a spherical equilibrium shape are dispersed with separations much larger than the average particle diameter in a liquid matrix having a constant equilibrium concentration. The particles coarsen at the condition of constant volume fraction. The LSW theory is based on the Gibbs-Thomson equation describing the concentration on the matrix side of the interface surrounding the spherical particle: this concentration is called the interface concentration hereafter and is designated by c(r) when the radius of the spherical interface (i.e. the particle radius) IS r. The equation indicates an important conclusion that at a temperature T. the concentration change AC(r) due to the change in the interface shape from flat (r = -L ) to spherlcal is inversely proportional to the radius, r, i.e. AC.(~) = c(r) - c(x)

z

21’,c&c(m) 1 RT 7

(111)

where y, is the interfacial energy density. R0 is the molar volume of the particle, R is the gas constant, and 2y,Q,c(c0)/(RT) is sometimes called the capillary length. The equation also describes in some way the interface effect on the particle stability. A typical example of the derivation of the Gibbs-Thomson equation will be given here. Consider the phase decomposed A-B binary alloy system consisting of a small particle having radius r, surface area s and volume I/ in a wide matrix at a temperature T. B atoms are supposed to have been the supersaturated solute atoms before phase decomposition and then the interface concentration c(r) is the mole fraction of B atoms on the matrix side of the spherical particle having the radius I’. The particle contains only B atoms of np moles, while *Strictly speaking, the volume fraction of precipitate particles changes very slightly but it can be safely regarded to remam practically constant durmg coarsenmg. JPMS 40/2-D

126

Progress in Materials Science

the matrix contains B atoms of nm moles. When B atoms of dn moles move from the particle to the matrix, i.e. B atoms of dn moles dissolve in the matrix, the changes in the content of B atoms are dn” moles for the matrix and dnP moles for the particle: i.e. dn” = dn and dnP = - dn. At the same time, the surface area of the particle also changes by ds. Then the free energy change due to the movement of B atoms is expressed as dG = GOdnP+ ysds + Gmdn” ds

(G”-Go)-7,~

1

dn.

(112)

Here Gm - Go is the partial molar free energy relative to the standard state with respect to B and is given by Gm - Go = RT In a(r) = RT ln[c(r)r(c(r))]

(113)

where a(r) is the activity of B in the matrix when the particle size is r, and y(c(r)) is the activity coefficient of B in the matrix at concentration c(r). Furthermore, since the particle shape remains spherical, the following equation holds: ds -=--=

dn

ds dV dV dn

(114)

Then eq. (112) can be rewritten as g

= RT ln[c(r)r(c(r))]

- F

(115)

when the particle and the matrix coexist in equilibrium, the condition dG/dn = 0 must be satisfied and hence the following equation can be obtained:

( > w&

c(rMC(r))= exp m

.

(116)

When we consider the limiting case where the activity coefficient does not depend on the particle size, the following relation is obtained:

y(c(r1)= 244~)) = Hence the dependence of the interface concentration curvature) is given by

44 = c(4ew

(

$gj.

(117)

of B on the particle radius (i.e. interface

wal -RT

1

r

> (118)

Elasticity Effects on the Microstructure

of Alloys

127

This Gibbs-Thomson equation indicates that the smaller the particle, the higher the equilibrium concentration of the matrix adjacent to the particle. From a different angle, the equation suggests that the solubility varies with particle size. The actual two-phase structure consists of a large number of particles having different sizes. In this condition, the solute concentration around the smaller particles is higher than the concentration around the larger particles, as predicted by the Gibbs-Thomson equation (118). Figure 18 illustrates the distribution of solute atoms around particles obtained from theoretical calculations. The interface concentration of the particle A4 having the mean size f is the same as the concentration in the matrix at infinity (c’) and neither rise nor fall in the interface concentration exists around M. The solute concentration gradually decreases from the smaller particle to the matrix and from the matrix to the larger particle. This figure clearly and visually shows how solute atoms behave to cause particle coarsening, as follows: first the solute atoms in the smaller particles dissolve into the matrix; then the solute atoms diffuse to the larger particles; finally, the larger particles collect the solute atoms. In this process, the smaller particles become smaller and smaller and finally vanish, while the larger particles become larger and larger. This phenomenon is accompanied by a decrease in the interfacial energy between the particle and the matrix because the volume fraction of particles remains constant throughout the process. This is the explanation of Ostwald ripening. The above discussion indicates that as long as the concentration at the interface C(T) is different from the concentration of the matrix cm, the diffusion of solute atoms takes place and solute atoms flow in or out of the particle, and the particle radius changes. This situation

Fig. 18. Concentration variation around three particles having different sizes calculated on the basis of multi-particle diffusion: W)the M particle has a mean size; the L particle is larger and the S particle is smaller than the M particle. According to the interface approach explained in Section 5.3.1., the solute concentration c”(X) at position X in the matrix is given by c”(X) = c* - (l/D) Z (ry i, / IX - X,1), eq. (a) Here c* is the average concentration at infinity and is a function of only time; D is the diffusion coefficient; r,, i, and X, are the radius, the coarsening rate and the position of the ith particle, respectively. Now we introduce a new function M, which is expressed as M, = a(1 - r,/ I,) - z, r,M,/(X, - X,, eq. (b). Then eq. (a) can be rewritten as (c”(X) - c*)/a = Z M,/lX - X,1, eq. (c). Therefore, if we know M, and r, by solving the simultaneous eqs (b) under’ the condition of Z,M, = 0, we can draw the relative concentration variation around the particles. This figure represents the case of three particles (i.e. i = 1, 2, 3) and the concept that the solute concentration around a particle is under the influence of the existence of other particles is introduced into the calculation.

128

Progress in Materials Science

can be formulated diffusion:

as eq. (119), provided the particle coarsening is controlled

=EJc”-c(m)(l + g +)I

by atomic

(119)

where D is the diffusion coefficient of solute B in the matrix, and the diffusion is assumed to be in steady state. As the particle radius changes with increasing time t, the concentration of the matrix also changes. Furthermore, in the actual two-phase systems, a large number of particles having different radii are dispersed in the matrix. In such a case, when we introduce the size-distribution functionf(r, t) of particles, the following equation ensures the conservation of solute atoms:

s m

c,=p+

fi

3

r3f(r, t) dr .

0

Heref(r, t) dr is the number density of particles whose radii range from r to r + dr. Since the equation of continuity is

a!@,0

at

+

(121)

(f(r, t)i) = 0

i

where i ( z dr/dt) is the coarsening rate, the time dependence of the particle-size distribution is expressed as -Wr, t) = -

at

f-{f(r,t)F[cm-e(m)(l

+ +$

:)]I.

(122)

The individual radii of the particles at a given time t are not the same and there actually exists a particle-size distribution which is expressed by the distribution function of particle sizes f(r,t). The LSW theory starts by assuming that a fairly large number of spherical particles exist, the radii of which exceed the critical radius r*(t) given by the condition that i = 0. By usingf(r,t), the mean particle size F(t) at time t is given by

s

mf(r, t)rdr

P(1) =

O

s0

(123)

m.f(r, t)dr

Again we should remember that the coarsening to be discussed here is diffusion-controlled.

Elasticity Effects on the Microstructure

129

of Alloys

Then the mean particle radius is equal to the critical radius, i.e. r(t) = r*(t). In such diffusion-controlled coarsening, the coarsening rate is given by

(124)

and the time dependence of the particle-size distribution

V(r, t)

at=-

2ymk(m) r’(t)RT

is given as

a @- r’(Olf(r, f)

ar[

r2

1

(125)

Equation (124) clearly indicates the important results that the coarsening behaviour particles varies with the particle size relative to the critical (mean) radius, as follows:

of

1. the particles with radii larger than r*(t) ( = f(t)) almost always coarsen, because i > 0; 2. the particles with radii smaller than r*(t) ( = r(t)) almost always shrink and finally vanish, because 3 -C0; and 3. the particles whose radii are approximately equal to r’(t) ( = Y(t)) may either coarsen or shrink, or remain unchanged, because i = 0. For diffusion-controlled coarsening, expressed by the following equation:

the time dependence

of the mean particle size is

(126)

This is known as the Lifshitz-Wagner equation. When the mean particle radius at t = 0 is f(O), eq. (126) can be rewritten as follows: F(t)’ - r(o)3 = ~~YJQ%(~) t . 9RT The particle-size distribution for P = r/r’(t) I

is finally given as follows:

3/2,

=

and for p = r/r’(t)

(127)

2

(1

+

$Dy3 P2MP)

(128)

312, f@,

4=0

(129)

130

Progress in Materials Science

where A is a constant, and zD’ is a constant having the dimension of time and is given by the following equation: ,_ To -

9RTr’(0)3 (130)

~YS&(~)

The time-invariant term p%(p) of eq. (128) is indicated by the solid line in Fig. 19. The important predictions of the LSW theory for diffusion-controlled coarsening are those shown by eqs (127X130): 1. the precipitate particles coarsen as the cube root of the ageing time, i.e. ‘ttB law’ holds; and 2. the self-similarity in structure is statistically maintained during coarsening, i.e. the particle size distribution can be scaled by mean particle size. Moreover, another suggestion of LSW theory is the following: 3. the number of particles per unit volume (i.e. number density of particles) is inversely proportional to the ageing time, i.e. ‘t-l law’ holds. Since the LSW theory has frequently been used to interpret the diffusion-controlled coarsening behaviour of precipitates, the so-called t’j3 law seems to be representative of the theory. However, there is an alternative important prediction of the LSW theory. If the coarsening is controlled not by diffusion but by interface reaction, the relation between the mean particle radius and the critical radius is given as F(t) = i r*(t) .

-

Diflhion

-.-

surface reaCh0”

(131)

‘

Fig. 19. Comparison between the particle-size distributions for the diffusion-controlled coarsening and the surface-reaction-controlled coarsening: the solid line expresses the time-invariant function of eq. (128) for the former and the dot-dash curve that of eq. (133) for the latter.

Elasticity Effects on the Microstructure The mean particle size is proportional

of Alloys

131

to the square root of time, as shown by (132)

The particle-size distribution

is given as follows: for p = r/r’(t)

=

and for p = r/r’ (t) 2

I

2,

(*+;,$)? PWP)

)

(133)

f(rJ) = 0

(134)

2,

where B is a constant, and rR’ is given by (135) For the case controlled by interface reaction, not the diffusion coefficient D but the rate constant of the interface reaction k appears. The time-invariant term @r’(p) of eq. (133) is indicated by the dot-dash line in Fig. 19.

5.2. Effect of Particle Volume Fraction on the Ostwald Ripening According to the LSW theory of Ostwald ripening, larger particles continue to coarsen by absorbing smaller particles to release the excess surface energy, with the microstructure exhibiting self-similarity. There are a large number of experimental results which are in accordance with the LSW theory: i.e. particles in a large number of alloy systems coarsen in direct proportion to the cube root of ageing time. However, there are also a large number of experimental results which are not in accordance with the LSW theory: i.e. coarsening sometimes does not obey the t ‘O law and in almost all cases, the size distributions of the particles are wider than that predicted from the LSW theory, i.e. the standard deviations of size distributions are almost always larger than 0.215 which is predicted by the LSW theory. The above discrepancy between the experimental results and the predictions of the LSW theory is likely to be due to the fact that the LSW theory is for the case where particles are dispersed in a liquid phase matrix too sparsely to interact with each other and, hence, the volume fraction of particles is practically zero (4 = 0). In actual alloy systems, however, the particle volume fraction is no longer equal to 0. Therefore, a large number of investigators have modified the LSW theory with respect to the particle volume fraction. In the present article, such theories will generally be termed ‘LSW volume fraction modified theories’, ‘LSW-VFM theories’ for short. Ardellu9) has, for the first time, modified the LSW theory with respect to the volume fraction of particles. His theory is called the ‘MLSW (modified LSW) theory’. The MLSW theory is for the case when the inter-penetration of the particles which are dispersed randomly in

132

Progress in Materials Science

the matrix need not be taken into consideration. This theory predicts the relation between the mean particle radius F(t) and the ageing time t as follows: F(t)’ - f(O)) = K(qb)t

(136)

where it should be noted that the rate constant K(4) is a function of the particle volume fraction 4. This equation clearly indicates that coarsening proceeds in direct proportion to z”~just like the case of LSW theory. The coarsening kinetics itself depends strongly on the volume fraction through K($). The coarsening accelerates with increasing #J. The particle-size distribution becomes rapidly broader and broader with increasing volume fraction even at its smaller values. When ~5= 0, the MLSW theory predicts the same coarsening behaviour as is predicted by the LSW theory. The MLSW theory of Ardell unnecessarily overestimates the influence of particle volume fraction on the coarsening behaviour. The coarsening kinetics is very sensitive to volume fraction and the predicted particle-size distribution is much broader than the distributions observed experimentally in actual alloy systems. To make the theory harmonize with the actual particle coarsening, Brailsford and Wynblatt (76) took into consideration the environment of the growing particles. They applied the chemical rate theory (spatially homogeneous rate theory) to the particle coarsening. Their theory is sometimes called ‘BW theory’. The BW theory predicts that the cube of the mean particle radius is proportional to the ageing time, i.e. F(t)3 z K($)t

(137)

where the rate constant is, of course, a function of particle volume fraction. This theory also predicts that the rate constant K (4) is less sensitive to volume fraction than that derived by the MLSW theory, which is compatible with the actual experimental results. When the volume fraction of particles is high, a growing particle has a greater opportunity to touch another particle and the particles sometimes coalesce into a big particle. This phenomenon is sometimes called encounter. Davies et al.(“) considered the influence of encounters on the Ostwald ripening. Their theory is known as ‘Lifshitz-Slyozov encounter modified theory’ and is called LSEM theory for short. In this theory also, the t’13law holds good although encounters between coarsening particles take place. As the volume fraction of particles increases, however, the rate constant K increases and hence the coarsening rate increases. Regarding the size distribution of particles, the LSEM predicts a wider and flatter distribution than that predicted by the LSW theory. Davies et al. have shown that the LSEM theory explains the size distribution obtained experimentally for y’ particles in a Ni-Co-AI alloy. 5.3. Computer Experiments on Ostwald Ripening 53.1.

Theoretical treatments based on statistical mechanics

The phenomenon of phase decomposition of solid solution is just the ordering process of a disordered state and Ostwald ripening can be regarded as the final stage of the ordering process for which the order parameter corresponds to the solute concentration. Then a formulation of such a final stage is based on the following idea: the order parameter remains constant throughout the system (material) except for the matrix side of the interface between the particle and the matrix, i.e. the only place where the order parameter changes is the interface region. During coarsening (the width of the interface is negligibly narrow as

Elasticity Effects on the Microstructure

of Alloys

133

compared with its curvature) the local shape of the interface can safely be assumed to be flat. The evolution of two-phase structure depends on the motion of the interface, i.e. in a sense, the interface instability. This is sometimes called the interface approach. In such computer experiments as are based on the interface approach, one of the basic equations is the following eq. (138), which describes the time-evolution of the radius r, of the ith spherical particle whose centre of gravity is located at X,:“*’

c,=

( >a l-5

_

rc

,f,

.A I

Ix

(138)

Here L, is the so-called Onsager parameter for the conservative system with respect to solute atoms, c is the isotropic interfacial energy density (isotropic surface tension), 6m is the difference in the order parameter (i.e. the equilibrium solute concentration in the present case) between the particle and the matrix and is defined as (139) r, is given by the condition that the total volume of particles is constant (i.e. the volume fraction 4 is constant), N is the number of particles, and 1X, - X, 1is the inter-centre distance between the ith and thejth particles. The effects of the volume fraction and the environment of particles enter through c,, that is the second term on the right-hand side of c,, i.e. r,C,,,c,/ 1X, - X, 1. The LSW theory starts with the equation in which this term is neglected; therefore, it is no wonder that the volume fraction effects are not reflected in the results obtained from the LSW theory. Equation (138) can be obtained according to the following basic idea:“” when the interface moves and its position changes, a force acts on the interface due to the surface tension. The work which is done by this force should be equal to the increment of free energy due to the change in the order parameter field (i.e. the change in the concentration field) which 1sinduced around the moving interface. Now let us express the coordinates inside the interface. the position of the interface and the unit normal to the interface by a (which is two-dimensional), r(a) and n(r(a)) which is along the z-axis, respectively. When the interface moves spontaneously by Jr(a) from the position r(a), the work done by the force F(r(a)) which is acting on the interface is given by -

d2aF-6r .

(140)

s

The increase in the free energy due to the change in the order parameter is given by

s d3r

6H am(r(a)> Sm

.&

h(a)

(141)

where the free energy H is a functional with respect to the order parameter m. According

134

Progress in Materials Science

to the above basic idea, eq. (140) is, of course, equal to eq. (141). By using the isotropic surface tension cr which is defined as B=

dz(dm/dz)*

(142)

s and the mean curvature of the surface K@(a)), the force F can be expressed as F=aKn.

(143)

Furthermore, the free energy change can be related to the interface change through the order parameter m by using the following equation, which is called the time-dependent Ginzburg-Landau (TDGL) equation:

Mr, 0 =

at

-Lb

( >b$& -v*

(14)

,

Here the constant Lb is the Gnsager parameter and b = 1 (i.e. Lb = L,) because we are now dealing with the conservative case. * Considering that no changes take place in the area except the neighbourhood of the interface, the velocity of the interface v(r(a)) satisfies the following equation:‘79’ (6m)* d2a’G(r(a) - r(a’))n(r(a’))*v(r(a’)) = L,aK(r(a)) s

(145)

Twhere the Green’s function G(r) satisfies the equation - V’G(r) = 6(r) .

(147)

Equation (138) can be obtained from the interface eq. (145). Weins and Cahn (8’)have deduced the same type of equation as eq. (138), which expresses the time-dependence of the mass mi of the ith particle, as follows: dmi = _ 4aD dt

2$oC(~) RT

- Kr, - r;$,

1XiM,X, I)

(148)

where K is a constant and M, is a set of constants each of which is associated with the jth particle. This equation was based on the idea that the steady-state diffusion of solute atoms is due to the solubility difference which depends on the particle size, as is given by the Gibbs-Thomson equation. The 2 y,Qc(m)/(RT) - Kri which is the sum of the first and second terms in brackets in eq. (148) is proportional to the particle radius r,. It gives the following prediction: the larger particles whose radii exceed 2y,Q,c(co)/(RTK) can coarsen steadily, while the smaller particles whose radii are smaller than 2y,f&c(co)/(RTK) shrink and finally disappear. The 2y&-,c(co)/(RT) - Kr, does not depend on the environment of the particles at all. It is the third term r,E,$,Mj/ 1X, - X, ( that depends on the environment. *For non-conservative cases, b = 0, i.e. L,, = L+ tIf the system is non-conservative, the velocity of interface v satisfies the equation of Allen and Cahn@“las follows: n-v = L+K where L0 is the Onsager parameter for a non-conservative

system.

(146)

Elasticity Effects on the Microstructure

135

of Alloys

As is shown by eq. (119), the difference between the solute concentrations of the matrix and the interface, i.e. cm - c(co), is important for the particle coarsening. The point of LSW theory is that cm is uniform throughout the matrix. In the actual case, however, the values of cm near smaller particles are large, while those near larger particles are small. Then the interaction between the concentration fluctuation and the particles is induced through the cm. According to the explanation of Kawasaki, when the particle radius, the variation of particle radius with respect to time (i.e. the rate of coarsening or shrinkage) and the position of the centre for each of the particles which are dispersed in the matrix are known (e.g. for the jth particle, r,, i, and X,, respectively), the solute concentration c”(X) at every position X inside the matrix is given as follows: c”(X) = c’ -

c,

;

IX*‘,,, . I

I

Here c’ is only a function of time and is given under the condition that the solute atoms are conserved, and D is the coefficient of steady-state diffusion of solute atoms in the matrix. When we pay attention to the environment of the ith particle, the solute concentration cm is equal to c”(X,) given by eq. (149). Then Kawasaki et al. obtained the coarsening rate of the ith particle by replacing cm in eq. (119) with cm (X,), as is shown by the following equation@ 831 d z

-

2YsQ”C(~) RT

4nDr,(t)

1 q)

-(c*(t)--@))-$<

,x,5x,,

1

(150)

A

or t = _ p

r,

F

21’,Qoe(oc) 1 RT

<

-(c*(t)-C(CO))]k$,,x:2x,,

(151)

These equations are, of course, on the same line as eqs (138) and (148). When c’(t) - C(‘%) is replaced by K which is constant, eq. (151) reduces to eq. (138). The term c’(t) - c(s3) expresses the degree of supersaturation and varies under the condition expressed as $-c*(i)-c(m))=

-

i!,$r,(t)3.

(152)

The size changes of the particles other than the ith cause the fluctuations. The effect of such fluctuations on the concentration around the ith particle is introduced by the final term in the square brackets of eq. (150) or (151). In this type of coarsening equation, the effect of volume fraction cannot clearly be seen at first glance. The volume fraction enters at the summation with respect to j. However, the summation such as l/r, C,.,rfi,/ 1X, - X, 1 in eq. (151) becomes 0 because ?, take both positive and negative values depending on the particles. Then we must manage to find some meaningful results by solving under various assumptions, e.g. the summation with respect to j is discontinued when the inter-particle distances 1X, - X, 1 reach the screening distance F/A of diffusion. In fact, some general results have been obtained: the P3 and the t-’ laws hold but the coefficients are different from those of the LSW theory; the scaling law holds but the particle-size distribution function is different from the distribution of the LSW theory.

136

Progress in Materials Science

We are now dealing with the time-evolution of the microstructure consisting of a large number of particles, i.e. a large number of interfaces. Since each of the particles (interfaces) has its own different and complicated shape, the computer experiments based on the equations of type (138) are not only laborious but also give results of very little meaning. Furthermore, the coarsening phenomenon is, in essence, a problem in the field of statistical mechanics of the non-equilibrium state. Therefore, the systems to which such equations can successfully be applied are limited to the cases where during particle coarsening, the centre of gravity of the individual particle does not move and furthermore the shape is spherical or at least can reasonably be assumed to be spherical. This premise can be justified when the volume fraction of particles is relatively small, e.g. less than 0.15.(@) A large number of theoretical attempts have been made to introduce the effect of particle volume fraction through ci for the cases of smaller volume fraction. Among them, the work of Kawasaki and his collaborators is conspicuous, as already seen in several parts in this article. Even their predictions, however, are limited to the systems having small volume fractions. They have extensively studied the phenomenon of Ostwald ripening with theoretical Their theory is called various different things, calculations by means of a computer. (83.8~88) such as TK, KTE, ETK, etc., for short. They pointed out that not only the collisionless drift processes but also the soft-collision processes are essential in the particle coarsening. Furthermore, Marqusee and Ross cE9)have studied the collisionless drift processes for competitive growth. They have investigated the effects of volume fraction on the basis of a systematic statistical method. The soft-collision processes arise from the interaction between the particles which are immobile but correlated. It seems probable that when the volume fraction is small its effect is not so important. However, the theoretical calculations based on the theory of Kawasaki et al. clearly indicate that its effectiveness is unexpectedly dominant even when the volume fraction is rather small, as can be seen in Fig. 20.

Fig. 20. Volume fraction dependence of particle-size distribution calculated by utilizing the theory of Kawasaki et al. The particle-size distribution becomes broader with increasing volume fraction.

Elasticity Effects on the Microstructure

of Alloys

137

5.3.2. Computer simulations The quickest way to succeed in understanding the particle coarsening theoretically may be by using computer simulations rather than struggling to find some assumption, which can sometimes be unintelligible, to solve e.g. eq. (151). Weins and Cahn’8’)tried to study the effects of particle volume fraction on the coarsening kinetics with simulations based on their original eq. (148). They also investigated the effect of particle arrangement in the matrix. Their simulation was the first one but the systems used were so small that they contained only 4-9 particles. About 10 years later, Voorhees and Glicksman started, for the first time in earnest, the computer simulation on Ostwald ripening. (w93)They studied the solution to the standard multi-particle diffusion and applied it to understanding the behaviour of particles in the course of Ostwald ripening for the systems having various values of particle volume fraction. The maximum number of particles in their simulation is 320 but even the particle number is still not enough to obtain the averaged quantities such as the size distribution function of particles. Their basic equation is of the same type as eq. (151): i, = B,lrf .

(153)

Here each particle is regarded as a source or a sink of solute atoms, and B, is the strength of the ith source or sink. The diffusional interaction between particles enters through B,. At the beginning of the simulations, we must determine the initial microstructure in which particles are randomly distributed at their fixed positions. Then the source/sink strengths of the individual particles (B,) are calculated by solving a set of linear equations which are rather complicated. Finally, the coarsening rate of each particle can be obtained by solving a set of differential equations (153). When we multiply the coarsening rate by a certain reasonable time-step, we obtain a set of new sizes for the individual particles. This new microstructure is the initial one for the second step of the simulation. The above procedure of simulating each step takes longer with an increasing number of particles because the number of equations to be solved also increases with increasing particle number. Although 320 particles are not enough to statistically analyse the coarsening behaviour, this number was very large for the numerical simulation, at least as carried out by Voorhees and Glicksman. Even when the particle number is adequate at the beginning, it apparently decreases as the particle coarsening proceeds because the simulation is made under the condition that the particle volume fraction is constant. To avoid the above weakness with respect to computation time, BeenakkeP) performed the summation in the following kinetic equation not for all the particles but only for neighbours because the diffusional interactions between particles separated at a distance greater than a few screening distances give practically no effect on the coarsening behaviour:

3, = r,Da&(co)

(co-c(co)- f a=))

I

>

r,c ,+,

; IXk

where co is a constant and d is a capillary length of the order of 10e9 m. BeenakkeP) himself pointed out that eq. (154) is based on the monopolar approximation and its validity is limited to f#J< 0.1.

138

Progress in Materials Science

Another attempt to reduce the computation time has been made by Enomoto (EKT).‘87’ Their starting equation is the standard multi-particle diffusion equation $

F r,(t)’ (

= 4rtDB,(t)

et al.

(155)

>

which is, of course, the same type as eq. (150). In order to reduce the above standard equation to an effective one, they considered three points: the ith particle interacts with the other particles which are located inside the sphere C(i), the centre of which is the centre of the ith particle and the radius of which is equal to the screening distance of diffusion I(t); the original three-dimensional model is mapped to the one-dimensional model in such a way as is used for grain growth; the source/sink strength B,(t) is expanded as a power of 4”‘. Then the following equations are obtained:

’+yr,(t)’

NV)

f(t)

C r,(t) ,=I

,=I

r,(t) =

(157) N(C)

N(r)

&W(f) - W) 1 r,(f) + 1 r,(f) ,=I

,=

I

where N(t) is the number of particles. When the initial particle number N(O), the initial particle sizes r,(O) and the initial screening length 1(O)are given, the kinetic equation of (155) can be integrated by using eqs (156) and (157). At suitable time-intervals, the calculations of N(t), f(t) and r,(t), and hence f(t)( = X,“lt),r,(t)/N(t)) are repeated. This approach makes the calculation time shorter and enables us to deal with larger systems as compared with the direct simulations such as given by Voorhees and Glicksman. In fact, Enomoto et al.cs7)were able to study statistically the coarsening behaviour in the system containing 10’ particles at the initial condition although the number of particles is still not enough as compared with the systems used in actual experimental studies. According to their discussions, the standard deviation and the skewness of particle-size distributions obtained by their simulations are in but not with MR theory,(89) as shown in agreement with those obtained by TK theory (*3.M6.95) Fig. 21. The difference is due to the fact that the simulation of EKT and the theory of TK include both the drift and the soft-collision processes, while the theory of MR includes only the former process. They concluded that not only the drift but also the soft-collision process plays an essential role in the particle coarsening. The so-called monopolar approximation which has been discussed in this section has now come to play an essential role in simulating the coarsening behaviour of precipitate particles, such as coarsening kinetics, particle-size distribution, standard deviation and skewness of particle sizes, etc. A number of computer simulations (numerical simulations) based on monopolar approximation indicate that the t ‘I3 law holds, and is independent of volume fraction. The simulations also show that with increasing volume fraction, the coarsening rate becomes higher and the particle-size distribution becomes more symmetric and wider. These results are in good agreement with the prediction of the statistical theories. It can safely be said that the monopolar approximation to the diffusion field in the system consisting of

Elasticity Effects on the Microstructure

of Alloys

139

“_ o.3~j

A

0

0

-ll,"",",'."""'.'.,,'l."'c 0 01

Ei\

0.2

03

Particle volume fraction , 4 Fig. 21. Volume fraction dependence of the standard deviation and the skewness of the particle-size distribution.‘*” The standard deviation CJ is defined by u = J((p - I)* ) and the skewness k, by k, = ((p - l)‘)/a’ where (....) denotes the average over the particle size distribution function. The results obtained by computer simulations of EKT reproduce the theory of Tokuyama and Kawasaki (TK) but are not in agreement with those obtained by Marqusee and Ross (MR).

immobile spherical particles is powerful when interpreting theoretically the morphological evolution during particle coarsening. However, there is a limit to the monopolar approximation. In the actual alloy systems, the particle shapes are not always spherical and the particles move in the course of coarsening. It is apparent that such situations should be introduced to the computer simulations. Especially when the volume fraction becomes higher than about 0.1, the monopolar approximation to the diffusion field fails. The particle shape loses its isotropic nature, i.e. the individual particle changes its shape from a sphere to other shapes. Voorhees and his group (%s9’)have observed the migration and shape evolution of non-spherical particles in a system having a high volume fraction and an isotropic interfacial energy, e.g. the Sn-rich phase in Sn-Pb eutectic liquid. In this system, the elasticity effect does not arise because of the liquid matrix. Voorhees et al. have utilized a boundary integral technique to solve the Laplace equation numerically in two-dimensional systems and explained that the observed particle migration which is accompanied by ellipsoidal deformation of particles is due to the effect of the inter-particle diffusion interaction.(98’ In order to take into consideration particle shapes other than spherical, a multipole expansion approach has been introduced by Imaeda and Kawasaki. (~9.‘O”) Their simulation is, of course, based on the interfacial approach which they have frequently utilized to derive the equations for morphological evolution in particle coarsening because of their full knowledge of the

140

Progress in Materials Science

approach. Their numerical simulation indicated that the particle shape deviates from spherical and the centre of gravity moves as the particle coarsening proceeds, although the number of particles in their system was very small. Furthermore, their results agree with the experimental observations by Voorhees and Schaeffer.u”‘) Since the actual microstructure, of course, spreads three-dimensionally in the materials, one- or two-dimensional simulations leave something to be desired. However, we have a lot of trouble in performing even two-dimensional simulations, let alone three-dimensional ones. The multipole expansion theory of Ostwald ripening in three-dimensional systems has been developed by Imaeda and Kawasaki. (‘O”) This theory is analogous to their previous theory for two-dimensional systems and the inter-particle diffusion interaction is taken into consideration. The numerical simulations based on their derived equations showed that the migration and deformation of particles, which are caused by the diffusional interaction, accelerate the particle coarsening. This new theory enabled them to perform threedimensional simulations in a fairly reasonable computation time but the number of particles was limited to 10 at the most. Quite recently, Akaiwa and Voorhees(‘02) made a threedimensional simulation by utilizing the same type of equation as eq. (148), using both monopole and monopole plus dipole approximations for describing the diffusion field. In their simulation, the particle shape was restricted to spherical but the particle migration was permitted and the initial number of particles was 104. The monopole approximation is good enough to describe particle coarsening for lower volume fractions ($J < 0.1). For the systems having a larger volume fraction 4 > 0.1, however, the particle migration has an effect on the coarsening behaviour and the inter-particle diffusional interaction should be taken into consideration. Hence the dipole approximation should be used. This migration makes the volume fraction dependency of coarsening behaviour more sensitive than the case of immobile particles. By and large, it seems that the above LSW-VFM theories or the computer simulations based on the interface approach, with the multipole approximation to the diffusion field, were successful. However, there are still a number of alloy systems which exhibit incredible coarsening behaviour which cannot be explained by the LSW-VFM theories. Such systems are elastically constrained systems in which the precipitate particles are coherent with respect to the matrix. 6. COARSENING BEHAVIOUR OF COHERENT PRECIPITATES ELASTICALLY CONSTRAINED SYSTEMS

IN

A two-phase microstructure accompanied by desirable properties is almost always obtained when the precipitate particles are coherent with respect to the matrix. This state is metastable because the phase decomposition (i.e. precipitation) is interrupted halfway. Then the coherent two-phase state potentially changes to more stable microstructures to decrease its energy state. Once coherency-loss takes place, the desirable coherent microstructure, of course, disappears. Even though the coherent state continues, the individual particles change their size, shape, distribution, etc. The most important change is the precipitate coarsening because it brings a serious deterioration of the properties of two-phase materials, which is known as over-ageing. In elastically constrained systems, coherent two-phase microstructures are, in varying degrees, under the influence of elastic energies. A typical example of elastically constrained systems is the y/y’ system of Ni-base superalloys. The superalloys exhibit the excellent high temperature strength which has already caused the material to be used for jet turbine engines.

Elasticity Effects on the Microstructure

141

of Alloys

Such excellent strength is a result of a particular two-phase microstructure containing finely dispersed y’ precipitate particles. The y’ particle in Ni-base alloys is, in general, Ni,X type L12 phase (X = Al, Si, Ti, etc.) having an ordered f.c.c. structure and is coherent with the y matrix. Another typical example of elastically constrained two-phase microstructures is the system containing ordered b.c.c. particles such as Fe&Y type DO, phase or FeX type B2 phase dispersed in a disordered A2 (b.c.c.) matrix, e.g. the systems of A2/DO,, A2/B2, etc. These systems are also coherent. In the coherent system, the individual particle is, in general, surrounded by the elastic strain field which results from the lattice mismatch between the particle and the matrix. This situation is schematically shown in Fig. 22 which is for the r/y’ system. When the lattice mismatch, i.e. the elastic constraint, is large, coherent two-phase systems have elastic energies which are large enough to influence the precipitate morphology, as already shown in the previous sections. Then the elastic strain is sure to influence the coarsening behaviour of coherent particles. In this chapter, we will survey the coarsening behaviour characteristic of coherent particles in elastically constrained systems. Although the discussion will be held with particular emphasis on y/y’ systems in Ni-base alloys and A2/DO, systems in Fe-base alloys, it will generally be applicable to other alloy systems. 6.1. Coarsening Kinetics and Particle-size Distribution 6.1.1.

Gamma-prime precipitates

Ardell studied the coarsening behaviour of y’ precipitates in a number of N&base alloys, i.e. Ni-AI, Ni-Ti, Ni-Si and Ni-Cr-Si .(29. 3s* 36)Transmission electron microscope (TEM) observations indicated that y’ (Ni,Al) particles in Ni-Al alloys containing 6.35 and 6.71 wt%Al are cubic and are aligned along (100) directions. In the case of cubic particles, half of the mean particle edge (G/2) can be regarded as the mean particle radius f. The cube of a(r)/2 is proportional to ageing time t: 4(t)’ - c?(O)~ = kt

64

(158)

(b)

Eo-0

E&O

Fig. 22. Schematicillustrationsof the atomic arrangementof interfacebetweeny’ particle and y matrix: (a) zero or negligiblysmall lattice mismatch(h z 0), e.g. Ni-Cr-Al alloy; (b) negative lattice mismatch (~0< 01, e.g. Ni-Si alloy. Coherent particles are usually accompaniedby an elasticstrain field.

142

Progress in Materials Science

where the rate constant k for cubic particles is given by k = 64y&(a)Q: 9RT

(159)

which is essentially the same as the original k given by Wagner. When a cubic y’ precipitate in Ni-Al alloy takes its equilibrium shape, its surface energy density ys is equal to yLI”)‘,and is calculated by using eq. (159): the average value of yL@“) is about 30 mJ rnm2. In Ni-8.74 wt%Ti alloy also, the metastable y’ precipitates (Ni,Ti) coarsen as the cube root of ageing time t. Furthermore, for y’ precipitates in Ni-6.5 wt%Si alloy, the plot of (G(z)/~)~versus t exhibits a straight line. The ys values calculated from the slopes k are 21.3 mJ rnw2 for the Ni-Ti alloy and 10 x 3 mJ m-* for the Ni-Si alloy. Coarsening kinetics obeys the so-called ‘t”3-law’ as is predicted by the LSW theory of Ostwald ripening but an important problem is on the distribution of particle size. The LSW theory indicates that the particle-size distributionf(r,t) is the product of the time-invariant function p*h(p) and a function of time only t. As has already been shown in Section 5.1., the function h(p) is:

0) =

(&-r(&-“3exp(*)

forp

1312

for p 2 312

0

(160)

where p is the particle size normalized by the mean particle size: p E r/J(t). The theoretical function p2h(p) is related distribution by the following equation:

to the observed

p2h(p) = i F(t)&, 0

(161)

histogram

of particle-size

(162)

When particles are cubic, r and f(t) in eqs (161) and (162) are replaced with a and c(t), respectively. Then the comparison between the histograms of (9/4)6(t)g(a,t) and p2h(p) is good enough to compare the observed distribution with the theoretical distribution. Ardell compared the two histograms for y’ precipitates in Ni-Al alloys. For the former histogram, the maximum is lower and the cut-off exists at a larger p value as compared with the latter: the observed distribution is broader than the predicted distribution. Ardell et al.“’ also obtained the same results for y’ precipitates in Ni-Ti and Ni-Si alloys. The actually observed distributions of y’ particles in Ni-base alloys are quite different from the theoretical distribution predicted by the LSW theory. Ardell et al. tried to clarify both theoretically and experimentally the effect of volume fraction $ on the precipitate coarsening. They observed with TEM the coarsening behaviour of y’ precipitates in the Ni-Al and NiCr-Al systems with different volume fractions, and found that y’ particles coarsen as t”‘, which is in accordance with the prediction of Ardell’s MLSW theory. The MLSW theory clearly indicates that the rate constant k is a function of volume fraction, as expressed by eq. (136). However, the experimental facts obtained by Ardell et al. were that the rate constant remains constant in the Ni-Cr-Al system although the volume fraction changes from 27.5 to 42.0% and the change in volume fraction from 9 to 60% does not cause any serious effects on the rate constant. The results cannot be explained by the MLSW theory. Ardell’s MLSW theory also indicates that the size

Elasticity Effects on the Microstructure

of Alloys

143

distribution becomes rapidly broad with increasing volume fraction. However, the volume fraction dependence of size distribution of y’ particles cannot be observed in Ni-Cr-Al and Ni-Al alloys despite the theoretical prediction. Then Ardell et al. concluded that the coarsening behaviour of y’ precipitates is not influenced by the volume fraction at all and is well explained by the original LSW theory of Ostwald ripening. Since the experimental Ni-base alloys such as Ni-Al, Ni-Ti, Ni-Si, etc., are the fundamental alloy systems in commercial superalloys, similar coarsening behaviour is expected to be observed in commercial superalloys. Footner and RichardP3’ have studied the coarsening behaviour of y’ precipitate particles in commercial Ni-base superalloys such as Nimonic 80A, Nimonic 90, Nimonic 105, etc., and found that the t”’ law holds good. They have also reported a strange coarsening kinetics of y’ particles in Nimonic 80A at the later stage of ageing: the coarsening stops and the particle size almost remains constant with further ageing. This is thought to be attributable to the re-solution effect of precipitates because the ageing temperatures are close to the solubility line. Furthermore, the size distributions of y’ precipitates in Nimonic 90 and Nimonic 105 were found to fit the Lifshitz-Slyozov encounter model (LSEM theory) proposed by Davies et al.“‘) 6.1.2.

Efects

of elastic constraint

As seen in the preceding section, some experimental results are explained by the LSW theory: e.g. the t”3 law sometimes holds. However, a large number of experimental results are not in accordance with the prediction of the LSW theory, especially on the particle size distribution: i.e. the size distribution observed experimentally is almost always broader than the LSW distribution. Such a disagreement was considered to be due to the fact that the LSW theory does not consider the effects of volume fraction during coarsening. Therefore, many attempts have been made so far to modify the LSW theory with respect to the volume fraction of second-phase particles. However, the modified theories leave something to be desired. Another possible cause of such disagreement is the effects of elastic constraint and hence elastic energies. Metallic alloy systems containing coherent precipitate particles are, in general, elastically constrained and the precipitate coarsening actually takes place in the metallic systems. We can easily suppose that the elastic constraint should have a definite effect on the precipitate coarsening. In the Ni-base systems consisting of y’ particles, the elastic constraint and hence the elastic energy arise from the lattice mismatch E,,.In constraint-free or weakly constrained systems where s0 z 0, the elastic effect is negligibly small and hence we can observe the normal coarsening behaviour, which can be explained by the conventional coarsening theories of Ostwald ripening such as LSW theory or the theories modified with respect to the volume fraction. In strongly constrained systems where the lattice mismatch is large, however, the elastic effect is dominant and hence some strange coarsening behaviour which is incompatible with the conventional theories should take place. Miyazaki, Doi and their group”‘. ‘04.lo5)h ave extensively investigated the elasticity effects on the coarsening behaviour of coherent precipitate particles and, in particular, y’ particles in Ni-base alloys. 6.1.2.1. Constraint-free or weakly constrained r/y’ systems. Ni-18.2 at.%Cr-6.2 at.%Al and Ni-7.0 at.%Sid.O at.%Al alloy systems have relatively small lattice mismatch between the y’ precipitate particle and the y matrix: ~0= 0.008% (eigenstrain sT’ = 0.00008) for the former and so = 0.10% (.sT’= 0.0010) for the latter. These are typical systems under practically constraint-free or weakly constrained conditions, and surface energy rather than elastic energy is dominant. Therefore, we can expect the normal coarsening behaviour which is predicted by the conventional theories of Ostwald ripening.

144

Progress in Materials Science

Miyazaki, Doi and their group investigated with TEM observations the coarsening behaviour of y’ precipitate particles in Ni-Cr-Al and Ni-Si-Al alloys. The volume fractions of particles are 11% for the Ni-Cr-Al alloy aged at 1073 K and 16% for the Ni-Si-Al alloy aged at 1073 K. The individual particles are spherical and maintain their spherical shape in the course of coarsening. Furthermore, the particles are randomly and uniformly dispersed in the y matrix. The coarsening kinetics of y’ particles in Ni-Cr-Al and Ni-Si-Al alloys aged at 1073 K are shown in Fig. 23(a). It is clear from this figure that the mean particle size (radius) r at an ageing time t is proportional to t ‘lm* . the plot of log ? versus log t for either system exhibits a straight line. The exponent l/m is equal to the slope and l/m = 0.33 for the Ni-Cr-Al and l/m = 0.32 for the Ni-Si-Al. The l/m values are consistent with the prediction of the LSW theory, i.e. m = 3. The standard deviation CJof the particle-size distribution, which represents the degree of scatter in particle sizes, is shown in Fig. 23(b). The c values remain practically constant during coarsening although they scatter a little. The constant (r values are about 0.25 for the Ni-Cr-Al and about 0.27 for the Ni-Si-Al. These values are larger than the cr value of 0.215 predicted by the LSW theory. Furthermore, the size distribution of Ni-Si-Al (4 = 0.16) is broader than that of Ni-Cr-Al ($I = 0.11). Therefore, the size distribution tends to become broader as the volume fraction of particles increases. The above results are in accordance with other experimental and theoretical results obtained by many investigators: that is, the driving

(4

(b)

0.111 10’

ld

ld

lo4

Ageing time , t I ks Fig. 23. Coarsening kinetics (a) and the changes in the standard deviation u of size. distribution (b) of coherent particles in weakly constrained y/y’ systems. y’ particles coarsen in accordance with both the I”’ law and the scaling law but the (r values are larger than 2.19 predicted by the so-called LSW theory.

Elasticity Effects on the Microstructure

of Alloys

145

force for coarsening is the excess surface energy, and the difference between the experimental and theoretical CTS is considered to be due to the fact that the LSW theory disregards the effect of particle volume fraction, i.e. the inter-particle diffusional interaction. Among the attempts to modify the LSW theory with respect to the volume fraction of particles, a new theory from Kawasaki and his collaborators, which is called TKE (Tokuyama-Kawasaki-Enomoto) here, is considered to be the most comprehensive one at present. The histogram in Fig. 24 indicates the actual size distribution of y’ particles in Ni-Si-Al alloy obtained experimentally with TEM observations. The volume fraction of y’ particles is 0.16. The particle-size distributions based on the theories of LSW and TKE are also shown in Fig. 24: the dot-clash line is for 4 = 0 calculated with the former and the solid line for d, = 0.16 (Ni-Si-Al) calculated with the latter. In the calculations based on TKE theory, not only the collisionless drift processes but also the soft-collision processes are taken into consideration, as is mentioned in the previous chapter. The TKE theory reproduces the actual size distribution much better than the LSW theory. In the systems in which the elastic constraint is negligibly small, the coarsening behaviour of coherent precipitate particles agrees with the prediction of the conventional coarsening theories such as the LSW theory and/or the theories modified with respect to the volume fraction of particles. In all such theories, only the interfacial energy is considered as the driving force for particle coarsening and no effects of elastic energies are included. Therefore, even when the interaction between particles is taken into consideration, the effects of the interaction are far different from those of elastic interactions.

($=0.16)

-

TKE (+=0.16)

Fig. 24. Comparison of the particle-size distribution observed experimentally for y/y’ system of Ni-Si-Al and the distributions predicted by LSW and TKE theories. The experimental observation is in accord with the solid curve calculated by Enomoto by utilizing TKE theory. This clearly indicates an essential role of inter-particle diffusion interaction in the morphological evolution during particle coarsening in weakly constrained systems.

146

Progress in Materials Science

6. I .2.2. Strongly constrained y/y’ systems. Ni-36.1 at.%Cu-9.8 at.%Si and Ni-47.4 at.%Cu-5.0 at.%Si alloy systems have relatively large lattice mismatch: E,,= - 1.3% (E” = - 0.013) for Ni-Cu-Si. These are the typical systems under elastically constrained conditions, and elastic energy is dominant. Therefore, we can expect some strange coarsening behaviour which cannot be predicted by the conventional theories of Ostwald ripening. Miyazaki, Doi and their collaborators investigated with TEM observations the coarsening behaviour of y’ precipitates in Ni-Cu-Si alloys. The shape of coherent precipitates under the effect of elastic energy in elastically constrained systems is usually not spherical but cube, plate or rod. The analysis of coarsening behaviour becomes very difficult if a shape change occurs during coarsening because we cannot deal with shapes other than cubic on the same basis. Therefore, the discussion should, in general, be restricted to the ageing for shorter duration when the particle shape remains cubic. In Ni-Cu-Si alloys, the shape of the y’ particles is substantially cubic. For cubic particles, half of the particle edge is considered to be equivalent to the particle radius. The coarsening kinetics of y’ precipitates in Ni-Cu-Si alloys are illustrated in Fig. 25(a). A linear relation between log f versus log t can be seen in the course of coarsening in each alloy. The slopes are 0.28 for Ni-Cu-Si with the lower volume fraction of 18% [Ni-Cu-Si(L4)], and 0.17 for Ni-Cu-Si with the higher volume fraction of 50% [Ni-Cu-Si(H4)]. The larger the lattice mismatch, the gentler is the slope. Furthermore, the larger the volume fraction, the gentler is the slope. It is noteworthy that a long-term ageing

Ageing

time , t / ks

Fig. 25. Coarsening kinetics (a) and the changes in the standard deviation u of size distribution (b) of coherent particles in strongly constrained y/y’ systems. At the later stage, coarsening practically stops and simultaneously the standard deviation o of size distribution decreases: i.e. the two-phase structure becomes uniform as ageing proceeds, which means that neither the t”’ law nor the scaling law hold good any longer in strongly constrained systems.

Elasticity Effects on the Microstructure

of Alloys

147

remarkably decelerates the coarsening of y‘ particles in Ni-Cu-Si alloys. Comparing two kinds of Ni-Cu-Si alloys indicates that the larger the volume fraction, the more smoothly the coarsening decelerates (i.e. the smaller the particle size at which the deceleration begins). The occurrence of decelerated coarsening clearly indicates that the so-called P3 law does not hold good in some elastically constrained systems. Elastically constrained systems also exhibit a characteristic change in the size distribution of precipitates. A typical example is the case of y’ particles in Ni-Cu-Si alloys. The standard deviation (T of the size distribution of y’ particles decreases in the course of ageing, as illustrated in Fig. 25(b). The particles having larger or smaller sizes away from the mean particle size gradually vanish during ageing, and the size distribution becomes more peaked. This means that the two-phase microstructure is no longer scaled with respect to the mean particle size Y. The decrease in 0 proceeds more smoothly with a high volume fraction than with a low volume fraction. The above experimental results clearly indicate that the two-phase microstructure containing coherent precipitates, in general, converge to a particular state during coarsening when the precipitates are dispersed in the matrix with large lattice mismatch and high volume fraction. This is just a result of elastic energies and especially the elastic interaction energies between particles. It is reasonable that the strange behaviour of coherent precipitates cannot be explained by the conventional Ostwald ripening theories because they considered only the decrease in surface energy as the driving force for particle coarsening. Such experimental facts urge us to construct some new coarsening theories in which the elastic energy is incorporated into the driving force. One of them is just the bifurcation theory which will be explained in the next section. 61.3. Coherent precipitates of b.c.c. type It is hard to believe that the remarkable coarsening behaviour explained above is characteristic of only the y/y’ system of Ni-base alloys. Such phenomena should be universal among various kinds of elastically constrained systems because they are results of elastic energies which originate from lattice mismatch. Therefore, it is most likely that the coherent particles in some other elastically constrained systems show similar effects. The same coarsening behaviour as in the case of y’ particles in Ni-base alloys can be seen in DO, particles in Fe-base alloys. The lattice mismatch between a DO, particle and the A2 matrix is very small for Fe-13.5 at.OhAl4.4 at.%Ge alloy and it is fairly large (co = - 0.93 to 0.97%) for Fe-8 at.%Si-8 at.%V and Fe-5 at.%Si-10 at.%V alloys. The DO, particles dispersed in an A2 matrix due to ageing at 923 K are spheres in an Fe-Al-Ge alloy and cubes in Fe-Si-V alloys, as shown in Fig. 26. The A2/DO, system of Fe-Al-Ge is a constraint-free system in which the surface energy is dominant and the particle coarsening is expected to be in accordance with LSW theory. Coarsening kinetics of DO3 particles in Fe-13.5 at.%Al-4.4 at.%Ge alloy aged at 923 K is shown in Fig. 27. It can be seen that there exists a linear relation between log f and log t and the slope is 0.334. Then the m value becomes 2.99 which is almost equal to the value of 3 which is just the prediction of the LSW theory. An example of A2/D09 systems which are elastically constrained strongly is the Fe-Si-V ternary alloy system. TEM images of DO, particles of Fe-Si-V alloys in Fig. 26 indicate that the shape of the particles is cubic. Therefore, the strange coarsening behaviour which is a result of elastic energies should be observed. The coarsening kinetics of DO, particles in Fe-Si-V alloys is also shown in Fig. 27 and as expected, the same coarsening behaviour is actually observed in those alloys. Coarsening is decelerated in the later stages. The volume

148

Progress in Materials Science

F?g. 26. Transmission electron microscope images of DO, particles in some Fe-base al loys aged at 923 K: Fe-13.5at.%A1-4.4at.%Ge alloy (a); F&at.%Si-Bat.%V alloy (b); Fe-5at.% Si-lOat. /oV alloy (c). The strange coarsening behaviour is expected to be observed in Fe-Si-V ’ alloys.

fractio #n of DO, particles in Fe-8Si-8V is higher than that in Fe-SSi-IOV. In the W ‘W system I also, the higher the volume fraction, the more remarkably and mol -e smoo thly the decelel ration of coarsening takes place.

Elasticity Effects on the Microstructure

of Alloys

149

Ageing time , t / ks Fig. 27. Coarsening kinetics of DO3particles m some Fe-base alloys. Particle coarsening practtcally stops in the strongly constrained A2/DO, system of Fe-8Si-8V alloy. while the t”’ law holds good in the constraint-free system of Fe-Al-Cc alloy.

6.2. Directional Alignment and Inhomogeneous Distribution of Precipitate Particles The particle distribution is greatly affected by the elastic interaction between particles. We can expect that the individual particles under the strong influence of elastic interaction 0.08

1

10

I

20

.

1 I Ni-Cu-Si

v

Ni-AI

30

1

.

(W

(H*)

0

Nt-CU-St (U:

V

NI-AI

l

N>-I

0

Ni-Cr-Al

(I.4

-

40

Mean particle size , f I nm Fig. 28. Development of directional alignment along (100) directions (Y ,& and of inhomogeneous distribution (~4) during coarsening of y’ particles in various kinds of N&base alloys having different degrees of elastic constraint.

150

Progress in Materials Science

should have a strong tendency to be aligned along a certain crystallographic direction and to form a non-uniform distribution. Doi and Miyazaki ‘I”) have introduced two useful parameters for describing the directional alignment and the non-uniform distribution: Y,, and A, respectively, as explained in Section 4.2. The experimental results on the changes in the values of Ytooand /i during coarsening of y’ particles in various kinds of Ni-base alloys are illustrated in Fig. 28.“” The elastic interaction is a result of the overlapping of the elastic strain fields caused by the lattice mismatch. Therefore, the first factor to be considered is whether the lattice mismatch co (i.e. elastic constraint) is large or not. For a system which is weakly constrained such as Ni-Cr-Al with a very small lattice mismatch (e. = O.OOS”/,),the values of Y,, and ,4 remain small during coarsening. This means that the particles in a weakly constrained system are uniformly and randomly distributed in the matrix without any directionality even when the particle coarsening proceeds. For the system which is strongly constrained such as Ni-Cu-Si which has a large lattice mismatch (a0 = 1.29%) the values of Y,oo and n are already large at the early stage of coarsening. Furthermore, as the coarsening proceeds, these values increase rapidly. This means that the particles in strongly constrained systems are already distributed rather non-uniformly and are aligned along (100) even at the beginning of coarsening. When the elastic constraint is intermediate such as Ni-Si (so = -0.30%), the parameters Y,ooand ,4 have small values such as those typical of Ni-Cr-Al at the early stage of coarsening. However, as the coarsening proceeds, the values increase. This means that the particles come to exhibit the (100) alignment and the inhomogeneous (localized) distribution as they coarsen. Similar behaviour is also observed for the Ni-Al system. However, since the lattice mismatch in the Ni-AI (e. = 0.56%) is larger than that in the Ni-Si system, the tendency toward both directional alignment and non-uniform distribution is more obvious in the former than the latter. As the particle volume fraction becomes large, the inter-particle distance becomes short enough to make the elastic interaction strong. Then the second factor to be considered is whether the particle volume fraction 4 is large or not. Comparing two systems having the same lattice mismatch but different volume fractions in Fig. 28 clearly indicates that the larger the volume fraction, the more promptly the values of Y,oo and /i increase in the course of particle coarsening: e.g. two Ni-Cu-Si systems (4 = 0.5 for H4, 4 = 0.2 for L#, and a0 = - 1.29% for both), and two Ni-Al systems (4 = 0.25 for H$, 4 = 0.2 for L4, and G, = 0.56% for both). Now it is obvious that the larger the magnitude of lattice mismatch 1&o1 the more smoothly are the particles aligned along (100) directions, The rate of alignment, which corresponds to the slope of each line in Fig. 28, is large if the magnitude of the lattice mismatch is large. The volume fraction also affects the particle alignment along (100). For example, at a given particle size, the Y,, value for higher volume fraction is larger than that for lower volume fraction. However, the difference in volume fractions seems to give little effect on the rate of alignment. Regarding the computer simulation of particle distribution, Wang et a!.@‘) have demonstrated the rearrangement of particle distribution during strain-induced coarsening with two-dimensional simulations*. During coarsening, the selective growth and translational motion of particles take place. Then the particles are rearranged to form regular arrays along (01) directions although the initial distribution of particles is random. If the particle performed under the condition of very small supercooling, the volume fraction of particles *The detail of their simulation method will appear in Section 7.5.

Elasticity Effects on the Microstructure

of Alloys

151

density is high, the tendency to form regular arrays is stronger and hence a more regular macrolattice of particles appears.

7. SPLITTING

OF PRECIPITATE

PARTICLES

The tendency for precipitate particles to converge at a particular size in the course of coarsening is obvious when the volume fraction is high. According to the discussions in the previous section, when the volume fraction is so low that the inter-particle distance is too long for particles to interact one another, the effect of elastic interaction energy on precipitate coarsening seems too small to cause the strange coarsening behaviour. However, the fact is that the elasticity effects do actually exist even in such cases. The phenomenon in the systems having lower volume fraction is much stranger than the deceleration of coarsening because it causes a decrease in the mean particle size during coarsening. The phenomenon is the splitting of coherent precipitates which is just a result of elastic interaction energy. Miyazaki and his coworkers found out for the first time the split from a single into a doublet experimentally. Since then a series of transmission electron microscope studies and theoretical studies based on microelasticity theory have been performed.

7.1. Transmission Electron Microscope (TEM) Observations of’ Splitting When a uniform solid solution of phase-separation type alloy is subjected to heating at a temperature just below the precipitation line, a number of precipitate particles appear sparsely and randomly in the matrix. If the particles are coherent, further heating sometimes brings an impressive precipitate morphology peculiar to elastically constrained systems. An example of such a change in precipitate morphology is illustrated in Fig. 29. Since the heating is performed under the condition of very small supercooling, the volume fraction of particles

Fig. 29. Transmission electron microscope aged at 993 K for 18 ks. The formation

image of DO, doublets in an Fe-8 at.%Si-8 at.%V alloy of each doublet of plates is a result of the splitting of a single cube.

152

Progress in Materials Science

is very small and the mean inter-particle distance is large. In such a case, a particle even in the coherent state cannot lower its energy state by interacting elastically with the other particles. The elasticity effect finds its expression in a different way which is more strange than the case described in the previous section.

7.1.1. Splitting into a doublet of plates The splitting phenomenon was first recognized in the y/y’ system of Ni-Al alloy.(24)When a Ni-12 at.%Al alloy is first homogenized at a high temperature followed by slow cooling to a temperature just below the y’ precipitation line, e.g. 1133 K, and is then aged at that temperature, cubic y’ particles appear sparsely and randomly in the y matrix. During coarsening due to ageing, a characteristic morphology of y’ precipitates appears just like the TEM image in Fig. 29. The morphological change illustrated in Fig. 29 is just a result of splitting. When an Fe-8at.%Si-8at.%V alloy is aged at 993 K after slow cooling to 993 K from a high temperature for solid solution treatment, cubic DO, particles appear sparsely and randomly in the A2 matrix. The ageing temperature of 993 K is just below the DO, precipitation line, so that the volume fraction of particles is very small and the particles are dispersed too sparsely to interact elastically with one another. Further ageing brings a characteristic morphology of DO, precipitate particles as shown in Fig. 29 which is the same precipitate morphology as Miyazaki et al. observed after the split into a doublet in a Ni-Al alloy. In this figure, a large number of coherent plates are dispersed in the matrix in pairs. The paired plates, i.e. a doublet of plates, are adjacent to each other along one of the (100) directions. The doublet is a result of splitting a single cube into a pair of parallel plates during coarsening. The sequence of the splitting of a single cube into a doublet of plates is illustrated in Fig. 30. Firstly, a rod of matrix phase forms near the centre of a cube. Then the rod widens along one (IOO}, changing its shape to a matrix plate. Finally, the split into a pair of parallel plates is completed. During further ageing, the shape of the paired plates becomes gradually flatter and flatter to decrease the energy state. This type of splitting is also observed for a number of y/y’ systems such as Ni-I 1 at.%Ti, Ni-40 at.%Cu-6 at.%Si and Ni-18 at.%Cr-5 at.%Si alloys.

Fig. 30. The sequence of the splitting of a single DO, cube into a doublet of DO, plates in an Fe-8 at.%Si-8 at.%V alloy aged at 993 K. As regards the y’ particles in some Ni-base alloys such as Ni-AI, etc., the splitting of a single cube into a doublet of plates proceeds in the same way as this.

Elasticity Effects on the Microstructure

of Alloys

153

7.1.2. Splitting into an octet of cubes Another type of splitting is observed for coherent precipitate particles in some elastically constrained systems. When Ni-12 at.%Si alloy is homogenized at a high temperature followed by slow cooling to the temperature just below the y’ precipitation line, e.g. 1103 K, and then aged at that temperature, y’ cubes appear sparsely and randomly in the y matrix. The y’ cubes coarsen without elastic interaction during ageing because the volume fraction is very small and hence the inter-particle distance is very large. Further ageing brings a distinctive microstructure as shown in Fig. 31. The octet of cubes is a type of precipitate morphology observed by Westbrook (‘06)for y’ (N&Al) particles in Ni-7.5 at.%Ti-7.5 at.%Al alloy. However, Doi, Miyazaki and their group (lo’,lo*)have for the first time explained clearly that such a distinctive morphology is a result of splitting. The split into an octet of cubes is also observed in Fe-base alloys. When Fe-17 at.%Al-10 at.%Co alloy is aged at 953 K which is just below the precipitation line, B2 cubes appear sparsely and randomly in the A2 matrix. The B2 cubes coarsen without elastic interaction during ageing because the volume fraction is very small and hence the interparticle distance is very large. Further ageing brings the distinctive microstructure equivalent to Fig. 31: this is just like the splitting of a single cube into an octet of cubes by TEM in the y/y’ system of Ni-Si alloy. In Fig. 31, it looks as if the unit assembly consists of four cubes. However, detailed TEM observations reveal that the unit assembly actually consists of eight cubes, i.e. an octet of cubes and a number of octets are dispersed in the matrix. Such an octet is formed by splitting of a single cube: the sequence is shown in Fig. 32 for the case of a DO, particle in Fe-Al-Co alloy. Firstly, the centre of each side face (100) of a cube is caved in by the matrix. Then the caves extend toward the centre of the cube and simultaneously widen along {lOO}planes. Finally, the split to eight small cubes is completed. In this case also, the first-neighbour particles in the group formed by splitting are adjacent to one another along (100) directions. This type of splitting is also observed in y/y’ system of Ni-10 at.%Al-4 at.%Si and Ni-8 at.%Al-5 at.%Ti alloys.

Fig. 3 I. Transmission electron microscope image of y’ particles in Ni-12 at.%Si alloy aged at 1103 K for 72 ks. The formation of an octet of cubes is a result of the splitting of a single cube.

Progress in Materials Science

Fig. 32. The sequence of the splitting of a single B2 cube into an octet of B2 cubes in Fe-17 at.%Al-10 at.%Co alloy aged at 953 K. As regards the y’ particles in various N&base alloys such as Ni-Si, Ni-Al-Si and Ni-Al-Ti, a single cube splits into an octet of cubes in the same way as this.

7.2. Calculations of Energy States Before and After the Split on the Basis of Microelasticity Theory It is apparent that the split causes an increase in surface area and hence an increase in surface energy. The usual concept is that the shape of the individual particle is determined by minimizing the sum of the surface energy and the elastic strain energy. As far as we follow the usual concept, the split phenomenon cannot be explained because the energy state of the system is increased due to the increase in surface energy by splitting. Nevertheless, the split actually takes place. The process which decreases the total energy of the system in spite of increase in surface energy is just the elastic interaction between particles. We can explain the splitting phenomenon by introducing the new idea that the shape of the individual particles is influenced by the elastic interaction energy also. The new idea will be examined by applying a microelasticity theory to the split in Ni-Al and Ni-Si alloy systems.(‘07~‘08) 7.2.1. Splitting into doublet in Ni-Al alloy In the Ni-Al system, the theoretical calculations are performed for the following case: a

Elasticity Effects on the Microstructure

155

of Alloys

single y’ particle (of volume V) which is an inhomogeneous ellipsoid of revolution (of aspect ratio p) exists in an infinite y matrix which is elastically anisotropic; the single y’ particle splits into a pair of small ellipsoidal y’ particles each of which has the volume V/2. The energy states before and after the split, i.e. PNG and EDBL,respectively, are expressed as follows:

and E DBL =2

I”E,“,,(p)+ 2s’cp)rs+ f&(P)

(164)

where s@) and s’(p) are the surface areas of ellipsoids before and after the split, respectively. E,,,,(p) is the elastic strain energy and @J&) is the elastic interaction energy between a pair of ellipsoids formed by the split. The two energies can be calculated as shown in Section 2. The value of &$&J) can accurately be calculated for the doublet as a two-body problem. In this calculation, the values of aspect ratio for the ellipsoids before and after the split are 1 and l/2, respectively, and the numerical values used for the Ni-Al system are also given in Section 2. The energy states before and after the split from a single into a doublet are illustrated in Fig. 33. The pNG is lower than EDBLwhen the particle is small: this means that a single particle is stable. However, as the particle size increases, the E DBLbecomes smaller than the ESNG.This clearly indicates that the stable shape changes from a single cube to a doublet of plates in the course of coarsening. 1.2.2. Splitting into octet in Ni-Si alloy In the Ni-Si system, the calculation is performed for the following case: before the split, there exists a single y’ particle (of volume v) which is an inhomogeneous ellipsoid of

105

Ni-AI .\

0.9

0

100

i?/

Parti size ,

400

500

nm

Fig. 33. Comparison between the energy states of y’ particles in Ni-Al alloy before and after the splitting of a single cube into a doublet of plates: ,I?“” and I!?“- are the energy states before and after the split, respectively. When a single cubic particle coarsens larger than D2,the particle becomes unstable and can potentially split into a doublet of plates because EDBLbecomes lower than _/TN”.

156

Progress in Materials Science

revolution (p = 1) in an infinite y matrix which is elastically anisotropic; after the split, there exists a group of eight small ellipsoidal y’ particles (p = 1) each of which has the volume V/8. The total energy before split, i.e. EN=, is expressed by eq. (163) and the energy state after split, i.e. Eocr, is given as:

where s”(p) is the surface area of the individual ellipsoid after the split. E$&) is the elastic interaction energy between the eight small cubes and its value cannot easily be calculated because of the many-body problem. Then it is assumed that practically the individual elastic interactions do not interact with one another. The elastic interaction depends not only on the aspect ratio p but also on the inter-particle distance x. In this case, since p = 1, the &#Q) in eq. (165) is replaced with E$(x). When the distance between the first-neighbour particles is x, the distances between the second- and third-neighbour particles are ,/%c and ,/?x, respectively. The elastic interaction energies can simply be summed as follows: &T(X) = 12E$!&) + 12E$(&

+ 4EI($(&)

(166)

where E&k, Ej& and E$& are the elastic interaction energies between the first-neighbour (in (lOO)), the second-neighbour (in (110)) and the third-neighbour (in (111)) particles, respectively. The numerical values for the ~7 and the ys used for the Ni-Si system are -3 and 11.1 mJ m-*, respectively. The elastic constants for the y’ and y phases at ageing temperature are not known and hence the elastic constants for the Ni-Al system are used in place of those for the Ni-Si system. The energy states before and after the split from a single into an octet are illustrated in Fig. 34. When the particle is small, the I? NGis lower than the E’cr: this means that a single particle is stable. As the particle coarsens, the EocT becomes lower than the pNG. The stable shape changes from a single cube to an octet of cubes during coarsening.

\

“‘__DL --A___

\

0 % .

1

Ni-Si

Eocr-E,iir

1.05 -

t

---.

iD

\,D, \

IF ‘\ 03

Fig. 34. Comparison between the energy states of y’ particles in Ni-Si alloy before and after the splitting of a single cube into an octet of cubes: EN6 and EocT are the energy states before and after the split, respectively. When a single cubic particle coarsens larger than D8, the particle becomes unstable and can potentially split into an octet of small cubes because p becomes lower than EsbNG.

Elasticity Effects on the Microstructure

of Alloys

157

7.3. Role of Elastic Interaction in Splitting Figures 33 and 34 clearly indicate that the splitting of a single into a doublet of plates or into an octet of cubes is justified by theoretical calculation on the basis of microelasticity theory. Since the conventional coarsening theories are based on the decrease in surface energy, they should never support any phenomena which are accompanied by an increase in surface energy. The split phenomena seem to be just opposite to the conventional theories: according to the conventional theories, the energy state increases, e.g. from D to D” in Fig. 34, as a result of increase in surface area due to splitting. However, the elastic interaction actually affects the shape of the individual particles. When the small particles formed by splitting are aligned along (loo), a negative interaction energy arises: in Fig. 34, the interaction energy corresponds to the decrease from D” to D’. The negative interaction energy overcomes the additional surface energy accompanying the split. The total energy seems to increase while splitting but the split does actually decrease the energy state. Whether or not the split takes place depends on the difference between the contribution of elastic interaction energy and that of surface energy. In fact, the y/y’ systems having small elastic interaction energy never exhibit any split. Therefore, we can describe the split phenomena by using a parameter which expresses the difference between the two energies. Doi and MiyazakPo7) introduced the parameter A [m ‘1 which is defined as:

Here, Z?$”is the elastic interaction energy between two identical spheres of diameter D/2 when they are separated by the distance of 1.090/2. The larger the value of 1A ( , the stronger the effect of elastic interaction energy. It seems natural that the A can successfully describe the effect of elastic interaction energy on split and/or shape change. Since the calculation of elastic interaction energy is not so easy, a new parameter, which can easily be calculated, is to be found especially from the practical point of view. Doi and Miyazaki have also introduced another useful parameter A’[m’ J-‘1 which is defined as the ratio of y/y’ lattice misfit to the surface energy density:

(168) When the 1A’ 1 value is large, the elastic energies and, in particular, the elastic interaction energy are dominant; in the system having smaller 1A’ ( , the surface energy is dominant. By using A’, the shape changes of y’ precipitate particles in Ni-base alloys are well classified into the following three categories: 1. when I A’ I < 0.2, a single y’ particle remains spherical during coarsening and does not split into small particles, e.g. Ni-7 at. OhSi- at.%Al, Ni-20 at.%Cr-10 at.%Al, Inconel 700; 2. when 0.2 I I A* I I 0.4, a single y’ cube splits into an octet of cubes during coarsening, e.g. Ni-10 at.OhAl-4 at.%%, Ni-12 at.%%, Ni-8 at.%Al-5 at.%Ti, Nimonic 115; 3. when 0.4 I ( A’ I , a single y’ cube splits into a doublet of plates during coarsening, e.g. Ni-12 at.%Al, Ni-11 at.%Ti, Ni-40 at.%Cu-6 at.%Si, Ni-18 at.%Cr-5 at.%Si. JPMS 40/2-F

158

Progress in Materials Science

As the elastic interaction energy becomes dominant, the energetically stable shape of y’ particles changes from sphere to doublet via octet. Furthermore, there exists a certain critical size D’ at which the single y’ particle exceeds the split. The larger the ) A’ 1, the smaller the D’.

The splitting is not a result of any kind of mechanical fracture. A single y’ particle splits into two or eight small particles by a pure diffusion process which is very directional under strong elastic constraint. Therefore, even if the energy state of the system decreases while splitting, the split phenomenon does not always take place: the split requires an adequate driving force. Moreover, even when the driving force is sufficient, whether or not a single particle actually splits depends on how smoothly the atoms diffuse to form a special particle morphology such as doublet or octet. Figure 35 indicates the total energies of a single y’ particle before and after the split into a doublet or an octet calculated on the basis of microelasticity theory. ESNG,EDeLand EocT are the energy states for a single cube, a doublet of plates and an octet of cubes, respectively, and the abscissa indicates the particle size before split (2r). When a single y’ particle grows larger than Dz, EDBLbecomes lower than ENG and the split into a doublet should potentially take place. However, since the driving force is required, the critical particle-size at which the split actually takes place (D’) shifts to a size larger than D2, e.g. 0; in the figure. Figure 35 also indicates that when a single y’ particle grows larger than D8, EOCTbecomes lower than EsNG and the split into an octet should potentially take place. However, the octet state is not stable but metastable until 2r becomes larger than 0;. Therefore, if the y’ particle grows larger than Di in a single state, it can split into an octet. Moreover, a doublet which appears by splitting from a single seems to split potentially into an octet when 2r > Di. In this case, however, such a split cannot take place because the difference between OCR and EDBLis so small that the driving force is insufficient. In fact, to date only two types of split, i.e. single + doublet and single + octet, have experimentally been found for y/y’ systems. If the difference between EocT and EDBLis large enough, the split of doublet + octet should take place.

1.05 Ni-Al :\ \

0.95 -

D2 Ds

;

\:‘\

1.4 h-

0.9

0

“T-‘-‘-. De* ---.

E3 -.-.-.-__ --------_____I% ml

ZOO ParliE

size

f%=

EoLT. 1OfXl

, ZP/ nm

Fig. 35. Comparison between the energy states of y’ particles in Ni-AI alloy before and after the split from a single cube to a doublet of plates or an octet of cubes. The split does not take place just after the particle size of single cube exceeds the critical size (e.g. D2 for the split of single + doublet) where the energy state after split is lower than that before split because some amount of driving force (e.g. AENG + DBLfor the split of single -+ doublet) is required.

Elasticity Effects on the Microstructure

of Alloys

159

Another theoretical analysis has been made by Khachaturyan et al.“” For a given morphology of particles, the energy state A~morpho’ogy) is expressed as AZPWho’o%Y) = AE + y,s .

(169)

The APmor”ho’ogy) is evaluated relative to the elastic energy of a thin plate having (100) habit. In eq. (169), y,s is the total surface energy of the particles and AZPdescribes the size-dependent and the distribution-dependent parts of elastic energy and is given by Khachaturyan et ~1.“~’ as follows:

- 25 > AEp = - 21 C,,(2C,,/C44 B’G v - 5) ( 4z’ + 3C”lCU 54512

(170)

where 5 is the elastic anisotropy factor, E,,is the lattice mismatch and V is the total volume of particles. The bulk modulus /I is expressed as B = C” + 2CU

(171)

and the geometrical functions I, and I1 are given by

Z2=

s

3

-$ky2(n)I e(k)I ’ $$

(172)

with y’(n) = n:nf + n’,nf + n$zt y2(n) = n2,n_$z~

(173)

where 8(k) is the Fourier transform of shape function. Since eq. (170) contains the distribution-dependent term, we can calculate the individual energy states for a single sphere (volume 8a3), a single cube (dimension of cube: 2a x 2a x 2a), a doublet of parallel plates (dimension of plate: 2a x a x 2a) and an octet of cubes (dimension of cube: a x a x a). In fact, Khachaturyan et ~1.“~)were able to calculate the energy states for the morphologies of y’ particles in Ni-Al alloy, as follows: AZJ+ere’= 0.709(2~)~E, + (36~)“~(2a)~y, AZ+J’W= 0.558(2~)~E, + 6(2a)‘y, @doublet)

=

0.483(2~)~E, + 8(2a)‘y,

AI”We”= 0.436(2~r)~E, + 12(2~)~y, . They estimated the conditions for the transitions between the morphologies,

(174) as follows:

2a 2 7.7r, for single sphere + single cube 2a 2 27r, for single cube --) doublet of plates 2a 2 50r,, for single cube + octet of cubes 2a 2 82r, for doublet of plates + octet of cubes

(175)

160 where r, is the characteristic

Progress in Materials Science length of the material and is defined by r,=

-. :

(176)

They concluded that the morphological changes in the course of coarsening should be sphere (single) + cube (single) --t doublet (of plates) + octet (of cubes). In addition to the calculation results obtained by Doi and Miyazaki, this calculation also indicates that from the energetic point of view, the split from a single cube into a doublet of plates or into an octet of cubes should potentially take place in the y/y’ system. 7.4. D$erence

in Split Modes

Splitting takes place only when the volume fraction of particles is too small and hence the inter-particle distances are too large for the individual particles to interact with one another. When the volume fraction is high, the inter-particle distances become shorter and the elastic interaction is always effective between the already existing particles. In such a case, therefore, the individual particles need not split to decrease the energy state. Such particles form queues along (100) and/or inhomogeneous distribution, resulting in a decrease in the energy state. Even when the particle volume fraction is high, however, the microstructure formed by splitting can actually be obtained. It is the continuous cooling that brings a large number of octets which are closely distributed in the matrix. The splitting behaviour of coherent particles in continuously cooled alloys is a little When a homogenized Ni-base solid solution different from that in isothermally aged alloys. (‘Og) is continuously cooled across the y’ precipitation line, the amount of y’ particles to precipitate increases with decreasing temperature and hence a large number of y’ particles appear and grow. When the continuous cooling rate is sufficiently slow, the y’ particles are given sufficient time to grow up to the critical size D’: if a particle size exceeds D’, the particle splits. The splitting of y’ precipitates has actually been observed during continuous cooling of some Ni-base alloys, e.g. a Nimonic 115 superalloy or a Ni-Al alloy. In the case of continuous cooling, the total amount of y’ particles which are about to split is large and the particles are closely distributed. Then the individual octets formed by splitting are closely dispersed in the matrix, as shown in Fig. 36: even though the volume fraction is high, a peculiar structure tending to split can be obtained. In the case of continuous cooling, only the split from single into octet has been observed so far. Furthermore, even when the alloy system exhibits the split into doublet during isothermal ageing, the split into octet can be observed in the system. The reason why the splits other than the splitting into octet have not yet been observed during continuous cooling can be explained as follows. 7.5. Kinetic Analysis of Split Phenomena by Means of Numerical Computer Simulations The theoretical analyses explained before are based on an energetic point of view. With such analyses, we can theoretically make sure whether the split which we are now watching actually takes place or not, as long as the energy states for the initial condition and the destination can be calculated. However, the intermediate stages of splitting are not always explained. Therefore, some kinetic approach is necessary for interpreting the split phenomenon. Wang et a1.(66,67’have proposed a general approach to the kinetics of the particle-shape

Elasticity Effects on the Microstructure

161

of Alloys

Fig. 36. Transmission electron micrograph of octets of 7’ cubes in Nimonic 115 superalloy continuously cooled at the rate of 0.1 K/s from 1523 to 1273 K. The split from a single cube to an octet of small cubes actually takes place in the course of continuous cooling even when the volume fraction of particles is high.

evolution during coarsening under the influence of elastic energies. Their starting equation is the Onsager equation, which is the same type as eq. (144) as follows:

(177) where m(r,t) is the non-equilibrium single-site occupation probability of finding a solute atom at a lattice site r at a time t, c is the concentration of solute atoms, k, is the Boltzman constant, L,,(r - r’) is a matrix of kinetic coefficient, and sH/sm(r,t) is the driving force for the morphological evolution. They utilized the Fourier representation of the Onsager equation as follows: dKz(k, t)

(178)

-a--=

where A(k, t), $(k) and {. .. }k are the Fourier transforms of the corresponding functions in real space, and k is the reciprocal lattice vector. The elastic energy effect enters, of course, through the driving force. The Fourier transform of the driving force is expressed as

= (V(k), + B(e))fi(k, t) + k,T

. k

L

Here e ( z k/k) is the unit vector along k, and V(k), is the Fourier transform of the atomic interaction energy [finite radius (chemical) interaction energy], as is expressed by

V(k), = Cw(r), exp( - ikr) .

(1W

162

Progress in Materials Science

The elastic energy effect is reflected in B(e) which is given by B(e) = - e,+(e),&e~

+ (&X&c+,),

Here 0: is related to the stress-free transformation

.

(181)

strain E:,, as follows:

0; = C&,

(182)

and n(e), is a Green function tensor reciprocal to C,,,e,+. The symbol (**.)e means averaging over all directions e. Wang et al.(&)considered a two-dimensional model with square lattice. They assumed the conditions that E!, = s&, A > 1 and the ratio of the interaction energy between the first neighbour to that between the second neighbour is fi. For their two-dimensional model, the function B(e) is given by B(e) x B e.te, - f ( >

(183)

e = (e,,e,J

(184)

with

B

(‘,, + 2c12)2Eg2 Cl,

=

(185)

where e, and e, are components of a unit vector e along the x and y axes (i.e. [IO] and [01]) in the reciprocal two-dimensional space. Now eq. (178) with the driving force eq. (179) can be solved numerically and m(r,t) is obtained. The m(r,t) gives not only the equilibrium morphology but also the metastable or non-equilibrium morphology in the course of particle coarsening. In fact, Wang et a/.@@succeeded in reproducing surprisingly the sequence of the splitting from a single cube into a doublet of plates with two-dimensional computer simulation utilizing the above method. They performed another computer simulation@” under different conditions, e.g. the parameter B which describes the elastic energy effect is different from eq. (185) as B

=

_

4(c,,

+

2c,2)2(c,,

WC,,

+

c,,

c,, +

2C44)

2&,)‘$ .

(186)

Then they obtained essentially the same results on splitting into a doublet of plates as before. The computer simulations which have been performed so far were, in general, the two-dimensional ones. When simulating the split into an octet, we need, of course, some three-dimensional model. To realize such three-dimensional simulations, a main problem may be how to reduce the computing time. 8. NEW THEORIES

OF PRECIPITATE

COARSENING

There are a number of factors which affect the coarsening behaviour of precipitate particles. Since the precipitate coarsening we are now discussing is a result of atomic diffusion, the factors which influence the diffusion process have, in general, effects on the particle coarsening. Some important factors to be considered are as follows: 1. volume fraction of particles; 2. elastic strain arising from the lattice mismatch between the particle and the matrix; 3. elastic interaction between particles;

Elasticity Effects on the Microstructure

of Alloys

163

4. coherency-loss of particles; 5. external stresses; 6. particle migration. In addition to those, there are still some important factors to be considered, as reviewed by Jayanth and Nash?‘O) 7. short-circuit diffusion at grain boundaries, dislocation lines, etc. Among the above factors, we should always remember the effects of particle volume fraction and of elastic energies. The volume fraction effect has already been discussed in Section 5.2. Another important factor is the effect of elastic energies, i.e. elastic strain energy and elastic interaction energy. The particle coarsening actually takes place in metallic alloy systems which are, in general, elastically constrained. Therefore, it is essential that the effects of elastic energies which are characteristic of elastically constrained systems should be incorporated into particle coarsening theories. Here we will take into consideration the elasticity effects on the coarsening behaviour. Strictly speaking, however, the volume fraction of particles is concerned with the elasticity effects through elastic interaction, as mentioned before. Therefore, the following discussions will also include the volume fraction effect to some degree. Furthermore, once the coherency has been lost, both elastic strain and elastic interaction vanish and hence any coarsening theories do not need to include elasticity effects to be discussed here. The effects of external stresses are very important especially for the practical use of two-phase alloys containing coherent particles, e.g. Ni-base superalloys. However, the precipitate coarsening under external stresses will not be discussed here.

8.1. Bifurcation Theory-Energetic

Treatment

8.1.1. Model structure of two-phase state Microstructures in actual phase-separated alloys usually contain a large number of precipitate particles. Furthermore, precipitate particles are sometimes classified into more than one phase. Such situations bring the theoretical examination into a many-body problem and make it difficult to interpret the structural stability such as the coarsening behaviour of precipitates. Then theoretical explanations are very much limited to a qualitative stage because the energy state of the system containing a large number of particles, and particularly the elastic interaction energies between many particles, cannot easily be estimated. Therefore, it is the most commonly used and reasonable procedure that we assume the simplest model of two-phase structure as follows: 1. the unit of two-phase structure consists of a pair of ellipsoidal and inhomogeneous particles which are coherent with respect to an elastically anisotropic matrix; 2. the microstructure satisfies the boundary condition that the units are distributed periodically; 3. the paired particles change their sizes under the condition that their total volume remains constant. 8.1.2. The first introduction of the bifurcation concept to the structural stability The singularity in the behaviour of the energy minimum is called bifurcation.““) Johnson and Cahn(3,“2) have for the first time introduced the concept of bifurcation to problems in the morphological change of precipitate particles and/or the microstructural stability. Johnson”’ dealt with the system where a pair of spherical particles a (radius r., volume V,)

164

Progress in Materials Science

and /3 (radius r,, volume V,) are separated at an inter-centre distance of d. When the relative sizes of the paired particles vary under the condition that the total volume remains constant, i.e.

=

8 -3 -7rr

3

G

the parameter

V, (const.)

(187)

R which is defined by Johnson and Cahn(“2) as R

~

trn- ‘,!I

(188)

(ra+ rh

is introduced to describe the relative sizes. Here - 1 -< R I 1.The condition of R = 0 corresponds to the state where the paired CIand j3 particles are identical and take the same size, i.e. the mean particle radius 7. As R increases from 0 to 1, a grows and /I shrinks and the condition of R = 1 corresponds to the state where only an tl particle exists taking the maximum radius 2”3J. On the contrary, as R decreases from 0 to - 1, a shrinks and /I grows and the condition of R = - 1 corresponds to the state where only a /I particle exists taking the maximum radius 2’13f. Furthermore, a dimensionless inter-centre distance dN is defined as (189) Johnson”) calculated the system energy ET which is expressed by ET = EEL+ E:NT+ Es”RF.

(190)

EEL, EkT and EsURFare the elastic self-energy, the elastic interaction energy due to coherency strains and the surface energy, respectively. Because the total volume V is constant in the present system, EELdoes not depend on the particle-size change, i.e. EELis independent of R. The EkNTis given by Johnson’3’ as

E%T = P*r%

rZ (&

_

,.;,,

ri -

(8 - r.2)’ 1

(191)

which is eq. (41) in Section 2.2.4.2. When we set the derivative dE,/dR equal to 0, the equation

dE, dR =’

= 2R(R2- 1) (1+ 3R2)s/3 M(R,LI’,) - 2

1

(192)

Elasticity Effects on the Microstructure

of Alloys

165

is obtained. Here M(R, dN) is a complicated function of the relative particle-sizes R and the inter-particle distance dN. The parameter 2 is dimensionless and is defined by

(193) The extremizing solutions to eq. (190) become R=O R= M(R,d,)

+l

= 2.

(194)

Figure 37 shows an example of the so-called bifurcation diagram obtained by JohnsonC3’by plotting R as a function of Z when dN = 3/2. The bold lines and the broken lines indicate the energy minima obtained as the stable solutions and the energy maxima obtained as the unstable solutions, respectively. The point of c indicated by the triangle A is called the bifurcation point and is given by Z” = lWd2, + d-‘N - 3) c (44 - 1)5 .

(195)

The point of zln indicated by the diamond 0 is given by ‘t” = (44 _

$2”’

_

1)

(196)

.

The arrows in the figure indicate the direction of energy decrease.

1

\\ \

‘\

t

e 0

‘\

‘\

.-._

--._t

*-._ ;

=.,

t

.*’ cc

A

________.___.___._ >

Z?

i---’ ,* ,/’ /

,’ I’

e

.*

I’

0.04

Fig. 37. Bifurcation diagram illustrating the stability against the coarsening of a pair of misfitting particles.‘3’ The point e is called the bifurcation point. The bold lines indicate the energy minima (stable) while the broken line indicates the energy maxima (unstable).

166

Progress in Materials Science

When V is small and hence 2 is large, the state of R = + 1 is only stable. When Z < Z;, two separate stable states exist, i.e. the states of R = 0 and R = & 1. Provided G < Z < c, the state of R = f 1 is still absolutely minimum and the state of R = 0 is metastable. Provided Z < c, the state of R = 0 comes to be absolutely minimum and the state of R = 11 becomes metastable. 8.1.3. The development to accord with the actual alloy systems Bifurcation theory proposed by Johnson and Cahn(“2) succeeded in introducing the elasticity effect to the explanation of the peculiar coarsening behaviour of misfitting particles. However, they could not but conclude that the peculiar coarsening is only realized when the particles are elastically softer than the matrix because their calculation was based on isotropic elasticity theory. In actual alloy systems, many precipitate particles are known to be harder than the matrix. Furthermore, such hard particles actually exhibit peculiar coarsening. Therefore, bifurcation theory in the original form is hardly based on the realities of actual alloy systems. Miyazaki, Doi and their collaborators developed the original bifurcation theory of Johnson and Cahn(“*) to accord with the realities of elastically constrained systems such as y/y’ systems of Ni-base alloys, Co particles in Cu-Co alloy, etc. (“3-“6)The model structure considered here is the simplest model mentioned above, i.e. two misfitting spherical particles which are inhomogeneous exist in an elastically anisotropic matrix. Assuming that the two-phase structure consists of a large number of paired a and B particles and the individual pairs are distributed with a period of L, the paired a and jI occupy the territory of 2L3. Then the volume fraction q5 of CIand /J is expressed by V 4=-=&.

(197)

This clearly indicates that the volume fraction does not depend on the total volume but depends only on the normalized inter-centre distance. As mentioned before, the energy state, i.e. the total energy ErrL, is the sum of the elastic strain energy, surface energy and elastic interaction energy. The variation of .ErrL of a pair of coherent particles such as y’ is schematically shown in Fig. 38 as a function of the mean

Fig. 38. Schematic illustration of the energy state of a pair of coherent particles. The bold line indicates the energy ridge. In Region II, the effect of surface energy is dominant and the total energy decreases from R = 0 to R = f 1. In Region II, the effect of elastic energies is important and the total energy decreases to R = 0.

Elasticity Effects on the Microstructure

of Alloys

167

particle radius r and the parameter R. The energy surface is symmetric with respect to the plane of R = 0. The bold line indicates the energy ridge where the energy state is unstable and ErrL takes the maximum at a given J. The dot-dash lines indicate the energy minima where the energy state is stable. When 7 increases, the energy ridge branches at the point which is called the bifurcation point and is indicated by the triangle (V). When ? is larger than r” indicated by the star (*) in the figure, the energy state at R = 0 is lower than that at R = &-1 and the energy state at R = 0 is absolutely minimum. The part of the ridge on this side of the bifurcation point forms the boundary between Regions I and II. In Region I, the energy minimum locates at R = + 1 and the En,_ decreases from the ridge at R = 0 (0) to R = ) 1 (0) as indicated by thin arrows, which is a result of surface energy itself. On the contrary, in Region II, the energy minimum locates at R = 0 and the &, decreases to R = 0 (M) as indicated by bold arrows, which is an elasticity effect-strictly speaking, the energy minimum appears at R = 0 as a result of subtle differences between the elastic energies (elastic interaction energy) and the surface energy. 8.1.4. Explanation of coarsening behaviour based on bifurcation diagram Figure 39 illustrates a bifurcation diagram which is obtained by projecting the energy surface of Fig. 38 onto the f -R plane. The bifurcation diagram is symmetric with respect to the line of R = 0. By utilizing the bifurcation diagram, we can clearly explain the coarsening behaviour of precipitate particles in elastically constrained systems, as follows. 1. When the particle size is small at the early stage of coarsening (e.g. at TJ, the state of paired a and p particles is in Region I where surface energy is dominant. In this region, the larger particle of the pair coarsens by absorbing the smaller particle and finally only one particle of the pair can stably exist. This clearly means that in Region I, a two-phase microstructure coarsens to decrease the surface energy and hence the total energy. 2. As the microstructure continues to coarsen and the particle size continues to increase, the state of paired a and fi particles enters Region II where elastic energy and, in particular,

Fig. 39. Schematic illustration of the bifurcation diagram obtained by projecting the energy surface onto the r-R plane. When the mean particle size is small at the beginning of coarsening, the effect of surface energy is dominant and normal Ostwald ripening takes place. As the coarsenmg proceeds, the system enters Region II and the elasticity effect appears: the particle coarsening IS decelerated and the splitting occurs.

168

Progress in Materials Science elastic interaction energy is dominant. When the particle size is large enough, e.g. at FL in the figure, the smaller particle of the pair can coarsen by absorbing the larger particle and finally both of the paired particles can stably exist having the same size f.

The coarsening behaviour predicted for Region I by the bifurcation diagram is just like the prediction of the theories of Ostwald ripening where the driving force is only the surface energy, e.g. the LSW theory or the LSW-MVF theories. The prediction of the bifurcation diagram for Region II surely corresponds to the incredible coarsening behaviour which is a result of elastic energies, e.g. slowing down of coarsening, formation of uniform structure, etc.; in particular, the prediction itself is the split from a single into a doublet. The above discussions clearly indicate that the bifurcation theory can successfully explain the coarsening behaviour of precipitate particles in elastically constrained systems whether the elastic constraint is large or not. The point is that the bifurcation theory introduces a very simple morphology consisting of only a pair of spherical particles as a model structure-in a sense, the system considered is assumed to be an aggregate of the simple models which are periodically distributed in a matrix, i.e. the whole system satisfies the periodic boundary condition. Such an extremely simple model structure cannot represent the actual two-phase structure containing a large number of particles and hence some reasonable three-dimensional (3-D) model structure should be introduced. That is true but it is not so easy to find a certain 3-D model which is the representative of various kinds of actual two-phase structures. Furthermore, the elastic interaction energies between more than two particles cannot easily be calculated. Even when we evaluate the energy state of a specific 3-D structure with much effort, it is open to question whether the calculation result represents the general two-phase structures reasonably or comes to nothing other than the result for only the special situation. Taking such things into consideration, the bifurcation theory based on the very simple model consisting only of a pair of particles is still appreciated at the moment. 8.1 S. Parameter for describing the structure bifurcation It is very useful to introduce a parameter by which we can comprehensively interpret the coarsening of precipitate particles regardless of the magnitude of elastic constraint. Doi and Miyazaki”“) proposed a new parameter for describing the structure bifurcation in two-phase alloys containing coherent particles. Before that, they had already proposed the parameter A’ which can describe the effects of elastic interaction energy on the morphological change as mentioned before. The parameter A’ has been of y’ precipitates in Ni-base alloys,~‘07~‘o*~ defined as the ratio of y/y’ lattice mismatch to the surface energy density (A’ = E~/YJ, and it is used to describe the degree of elastic constraint and, in particular, the effects of elastic interaction energy. The larger the 1A* 1 value, the more dominant the elasticity effect and the coarsening for large 1A’ 1 proceeds more slowly than that for small I A’ 1. For example, we can compare the coarsening kinetics in a Ni-Cu-Si alloy (A’ = - 1.0 mz JJ ‘) with that in a Ni-Si-Al alloy (A* = 0.07 m* JJ ‘): the coarsening rate in the former is slower than that in the latter. However, A* disregards a very important point that the elastic interaction depends on the inter-particle distance dN and hence on the particle volume fraction 4. A typical example of the shortcomings of A* appears when comparing the coarsening kinetics of y’ particles in two Ni-Cu-Si alloys: they have the same A’ value of - 1.0 m2 JJ ’ but the coarsening kinetics are different, i.e. the m values in the kinetic equation F(t) = kt””

(198)

Elasticity Effects on the Microstructure

of Alloys

169

are 3.6 for Ni-Cu-Si(L4) and 5.8 for Ni-Cu-Si(H4). This difference arises from the fact that the elastic interaction becomes more dominant with decreasing inter-particle distance dN and hence with increasing volume fraction 4: the two Ni-Cu-Si alloys have different volume fractions, i.e. 4 = 0.18 for Ni-Cu-Si(L4) and 4 = 0.50 for Ni-Cu-Si(H4). Bifurcation diagrams, of course, include the effect of particle volume fraction in addition to the effects of lattice mismatch and surface energy. In fact, the difference in bifurcation diagram not only between Ni-Cu-Si and Ni-Si-Al alloys but also between two Ni-Cu-Si alloys can be recognized in Fig. 40: Region II, which is the region where the elasticity effect is dominant, for Ni-Cu-Si(H$) extends more widely toward smaller mean particle sizes r than that for Ni-Cu-Si(L$). The new parameter proposed by Doi and Miyazaki”“’ expresses the extent of Region II and is defined as the mean particle radius at the intersection of the energy ridge and the line of R = 0.5 or -0.5, as indicated by the squares (0) in the figure: the parameter is symbolized by rlos and is termed structureshed. This parameter successfully describes the effects of elastic energies as well as surface energy on the microstructural changes during coarsening of precipitate particles, whether the particles are coherent or incoherent.

8.2. Kinetic Theories Based on Interface Approach It is obvious that bifurcation theory is a kind of energetic treatment. Therefore, we can only predict that a structure potentially changes to another structure because the energy state of the latter is lower than that of the former. Here it does not matter whether the destination of the structural change is in a stable (equilibrium) or a metastable (non-equilibrium) state if only each energy state can be calculated. However, it is very likely that there exists a number of structures having intermediate energy levels between the initial state and the destination state. Bifurcation theory cannot show what kinds of intermediate stages (structures) the structural change passes because such things depend on the kinetics of the structural change. When we interpret the structural changes in two-phase alloys, some kinetic theory which is concerned with the dynamic aspect of structural change is very essential. Kawasaki and Enomoto’84) performed computer calculations to explain the kinetics of the strange coarsening behaviour of particles observed in the elastically constrained systems. Their calculation was based on isotropic elasticity theory using statistical theory of Ostwald ripening which is incorporated with the elastic field interaction between particles. They calculated the elastic interaction energy between two spherical particles by using the method which has been given by Eshelby”’ and has been discussed by Johnson”) and Johnson and Cahn.“12) The equation used is given as

(199) where d,, is the inter-centre distance between the ith and jth particles. The 8, which describes the degree of mismatch between the particle and the matrix, has a relation to the eigenstrain .sT’which is pure expansion or contraction of E,,type and the difference in the shear moduli of particle and matrix A,u ( = ~1’- p), as follows: /I of (sT’)‘Ap . If the particles are softer than the matrix, i.e. Ap < 0, the attraction

cw

between particles

170

Progress in Materials Science 1,

I Ni-G-AI

Particle

size

, i / nm

0

1

Ni-Cu-Si(L#)

w

Region II

0

-

-1 0 Particle

50 size

Particle

size

0

100

, i / nm

50

100

, ? / nm

Fig. 40. Some examples of calculating bifurcation diagram for y/y’ system: elastically constrained NiCu-Si with high volume fraction (H4); elastically constrained Ni-Cu-Si with low volume fraction (Lc$); weakly constrained Ni-Cr-AI. The stronger the elastic constraints, the more widely the Region II, in which the elasticity effect can be seen, extends toward small particle sizes. The parameter structureshed (T;, I) for describing the two-phase structure containing precipitate particles is defined as the intersection of the energy ridge and the line of R = 0.5 or -0.5, as indicated by the squares (0).

appears. On the other hand, if the particles are harder than the matrix, i.e. Ap > 0, the repulsion between particles appears. Furthermore, the attractive or repulsive potential is proportional to 4”.

Elasticity Effects on the Microstructure

171

of Alloys

Kawasaki and Enomoto@4) have presented a new equation of motion of the competitive particle growth, including the elastic interaction energy in the driving force, as follows: = - 4rcDM,

(201)

(202) where D is the diffusion constant, IXis the capillary length, and E,, is given as

1

6, =

fori

= j

fori

#j

(203)

Here X, is the inter-centre distance between the ith and jth particles and B is given by B=

$-$

(204)

where CJis the surface tension. The parameter B expresses the dimensionless magnitude of elastic effect (elastic mismatch) relative to the interfacial effect (surface tension). It is apparent that the present theory of Kawasaki and Enomoto (KE theory) is based on the interface approach which frequently appeared in Section 5, and eqs (201) and (202) are essentially on the same line as e.g. eq. (138). The elastic interaction and the diffusion interaction are included through the first and second summations on the right-hand side of eq. (202) respectively. Their theoretical analyses of the coarsening of immobile spherical particles led to the following conclusions. 1. (Tl) At the early stage of ageing, particle coarsening proceeds as the cube root of ageing

time, i.e. the P3 law holds good, which is a prediction of LSW theory. 2. (T2) As the ageing proceeds, the particles coarsen to exceed the critical size fCwhich is given by (205) Then the effect of elastic energies becomes dominant. 3. (T3) At the later stage of ageing, when Ap -C 0, i.e. if the elastic interaction is attractive, particle coarsening slows down and simultaneously the distribution of particle sizes becomes sharp and symmetric (see Fig. 41). 4. (T4) At the later stage of ageing, when Ap > 0, i.e. if the elastic interaction is repulsive, particle coarsening accelerates and becomes proportional to t”‘, and simultaneously the distribution of particle sizes becomes broad (see Fig. 42). Ohta(“‘) has derived the interface equation of motion under the elastic field for conserved and non-conserved systems from the time-dependent Ginzburg-Landau (TDGL) model for a solid solution in which the phase separation is progressing. The Ginzburg-Landau free energy density is assumed to be an even function of order parameter S and the absolute minimum and two local minima locate S = 0 and S = f S,, respectively: S, is positive and

172

Progress in Materials Science

I

‘6

I1

I

3

I

a

’

IO’

1

Reduced time

10S

,t

-._

B=l.O

106

____--- 10s -.......

4-

-

5 x 103 0 (LSW)

2-

Fig. 41. Coarsening kinetics (a) and time evolution of particle-size distributions (b) calculated with the theory of Kawasaki and Enomoto (s+(KE theory), for attractive elastic interaction (B > 0) between soft particles.‘*4)Computer simulation results by Enomoto and Kawasaki (EK) are also inserted.“r9r KE theory can well describe the peculiar behaviour of particle coarsening in elastically constrained system although the actual alloy systems in which the decelerated coarsening is observed consist of hard particles (i.e. B < 0). The initial condition for calculations is identical to the LSW distribution. Computer simulation results of EK are well fitted to KE theory.

appears. On the other hand, if the particles are harder than the matrix, i.e. Ap > 0, the finite Ohta’s interface equation of motion does not require any specific shape of particles. Then he applied his interface equation to spherical particles and finally obtained the following equations for a non-conserved system

(206) and for a conserved system

=

Elasticity Effects on the Microstructure

I

I

I

I

L

I

,

b,

I

I lo8

10’

,t

Reduced time 3

I1

I

,

>

1,

173

EK Smulkon

0

1

of Alloys

1

(

II

I

I

f

~0.01

B=-10

-._____--

106,107 ,0x

. ...” . . . . .. .

5 x 10’

-

0 (ISW)

, 0

._

i

2

1 p

(w/q

Ftg. 42. Coarsening kinetics (a) and time evolution of particle-stze dtstributtons (b) calculated wtth KE theory for repulsive elastic interaction (B < 0) between hard parttcles “(I Computer simulatton 1s results by Enomoto and Kawasaki (EK) are also inserted. ““’ The initial condttion for calculations identical to the LSW distribution. This theoretical calculation cannot describe the particle coarsening in actual alloy systems because not the acceleration but the deceleratton IS observed at the later stage of coarsening for hard particles. In this case also, computer simulatton results are well fitted to KE theory.

Here r, is the critical radius, u is the surface tension, and I. is the coupling constant which is assumed to be positive. The shear moduli K and p are even functions of order parameter S as K=

K. f K,S’

P = PO+

p,s2

(208)

$.

(209)

and L is given by L= JPMS 40/2-G

K,+

174

Progress in Materials Science

The above eqs (206) and (207) are consistent with the results obtained by Kawasaki and Enomoto:(84) that is, KE theory based on eqs (201X203) is a special case of Ohta’s theory.

8.3. Computer Simulations 8.3.1. Interface approach

Enomoto and Kawasaki(“9) have performed computer simulations based on their kinetic equations (201) and (202). The initial number of particles was lo4 and the particles were spherical and immobile. The results are indicated by the circles in Figs 41 and 42, and are summarized as follows: 1. (Sl) At the early stage of ageing, whether the elastic interaction is attractive (Ap < 0) or repulsive (Ap > 0), particle coarsening proceeds steadily obeying the so-called t’13law of LSW theory. 2. (S2) At the later stage of ageing, if Ap < 0, particle coarsening slows down and simultaneously the distribution of particle sizes become sharp and symmetric. 3. (S3) At the later stage of ageing, if Ap > 0, particle coarsening accelerates and becomes proportional to t”*, and simultaneously the distribution of particle sizes becomes broad. It can be seen from Figs 41 and 42 that the above simulation results are in good accordance with the predictions of KE theory. The KS theory and the EK simulations seem to have succeeded in explaining the strange coarsening kinetics under the influence of elastic interaction because the points T3 and Sl reproduce the deceleration of coarsening very well. However, their theoretical analyses of coarsening kinetics involve a problem since they predict that only the system containing soft particles exhibits the deceleration of coarsening. In the actual metallic alloy systems, some particles are elastically hard and often exhibit decelerated coarsening, although they should exhibit the accelerated coarsening from the view point of KE theory. The discrepancy may be partly due to the fact that the calculation method for elastic interaction energy in the above kinetic analysis is based on isotropic elasticity theory. Furthermore, the accelerated coarsening has not been observed experimentally so far.

8.3.2.

Time-dependent Ginzburg-Landau

(TDGL) approach

Onuki and Nishimori simulated the changes of two-phase structures in binary alloy systems based on the time-dependent Ginzburg-Landau (TDGL) approach.@*,69,‘*O) They considered three kinds of elastic effects on the phase separation in coherent systems as follows: (1) the long-range interaction due to the cubic anisotropy; (2) the long-range interaction due to the difference in elastic moduli between particle and matrix; (3) the dipolar interaction due to external (applied) stresses. The factors to be discussed here are the first two effects: the last one already appeared in this article. Nishimori and 0nuki(68) have presented a two-dimensional diffusion-type equation for

Elasticity Effects on the Microstructure describing the effect of modulus inhomogeneity conditions, as follows:

=

v2 ( - 1 - v2 + c’)c +

[

gEC

(

v,v,w-

175

of Alloys

on the phase separation* under the coherent

coq

$,,vw>‘I+2&~V.V,C~,V,W - ; d,,(c-

121

(210) Here the order parameter is concentration c, V, = a/ax,, d is the spatial dimensionality, and co is the average order parameter, i.e. average concentration. In this simulation, the shear modulus p is given by (211)

P = PO+ PIG and the bulk modulus K. is constant. Furthermore, is assumed as

Cahn’s coherent free energy density f(c)

1 1 f(c) = - - rc2 + - c4 4 2 where r is positive for the unstable region. The parameter gE in eq. (210) which expresses the strength of modulus inhomogeneity, is given by

where tl is the coupling constant and KLois given by

KLO= Ko + 2Po

.

(214)

w in eq. (210) is given by the following equation v2w = c - co .

(215)

In their two-dimensional computer simulations (69)take into consideration not only the elastic mismatch, i.e. the difference in shear modulus between the particle and the matrix, but also the cubic anisotropy. Their diffusive equation based on the TDGL model is as follows:

=

V'(

-

1 - v2 + c’)c + v2 2

+ z,v’,v;w .

(216)

Here the effect of cubic anisotropy is introduced through the last term z,VtV:w into the diffusive equation containing only the effect of modulus inhomogeneity, as in eq. (210). By taking suitable values of z,, we can set the degree of the contribution of cubic anisotropy. *Since isotropic elasticity and no applied stress were assumed, the second and the third interactions automatically excluded for the simulation.

were

176

Progress in Materials Science

The term V2(6FE/Sc) arises from the elastic mismatch. FEcontains g, and the degree of the effect of elastic mismatch can be set by changing the g, value. The computer simulations performed by Onuki and Nishimori indicate that the effect of the cubic anisotropy is the formation of the two-phase structure in which both hard and soft phases are long narrow stripes, while the effect of elastic mismatch is the formation of the microstructure in which the particles of hard phase are dispersed in the matrix of soft phase. Furthermore, they concluded that the deceleration of particle coarsening is a result of elastic mismatch: provided the elastic mismatch is taken into consideration, the decelerated coarsening can be reproduced whether cubic anisotropy exists or not. Khachaturyan and his collaborators have performed two-dimensional computer simulations to study the kinetics of the strain-induced morphological evolution.@‘) They utilized the Fourier transform of the Onsager equation which has already been discussed in Section 7.5. Their model alloy system has the coherent equilibrium miscibility gap for plate-like particles of (10) habits. The simulations succeeded in reproducing the morphological evolution during coarsening of microstructure in both the spinodal region and the NG region: the mottled structure in the absence of elastic energy effect, the modulated structure along the elastic soft directions under the influence of elastic energy, the migration and the rearrangement of particles, the directional alignment of particles, etc. The advantage of the TDGL approach is that the TDGL equation can be applied to a wide variety of phenomena simply by adding a suitable term to the equation. For example, Nishimori and 0nuki(69) obtained eq. (216) by simply adding the term rsV:V$v to eq. (210) in order to introduce the cubic anisotropy effect in addition to the elastic mismatch effect. The numerical simulations based on TDGL approach have just started from the work by Onuki and Khachaturyan et al. As is seen in the above, however, such simulations in two dimensions have already succeeded in reproducing the time-evolution of two-phase microstructures which are observed in the course of ageing of actual alloys. The results obtained so far are as follows. 1. For elastically isotropic systems, the elastically softer phase forms a percolated network and wraps the particles of the harder phase. 2. For elastically anisotropic systems, the surface of hard particles become perpendicular to the elastically soft directions, and square or rectangular particles are aligned along the elastically soft directions in the soft matrix. 3. The particle coarsening is decelerated due to the elastic mismatch whether the system is isotropic or anisotropic, i.e. whether the cubic anisotropy exists or not. Abinandanan and Johnson”“. I**)introduced the interface approach including the effects of both the elastic interaction and the particle migration on the coarsening kinetics of spherical coherent particles: the coexisting phases are c1 and /I. The scaled (non-dimensional) equilibrium interfacial concentration c”(sn) at a point s, on the surface of the nth particle consists of three terms, c@),c(‘) and ~9, as follows: C”(S”)= CCC) + cys,) + c”‘(S,) .

(217)

The first term c@)is the capillarity contribution which does not depend on the position a,, because of spherical particles but on the interface curvature equivalent particle radius (R,,) as is given by c(C)= - 1 . R

(218)

Elasticity Effects on the Microstructure The second term c(‘)(s,,) is the elastic self-contribution and is expressed as

of Alloys

177

which is a result of lattice mismatch,

where P and ,?P are the stress and strain fields, respectively, due to the mismatch strain of the nth particle. The last term P(s,,) is the elastic interaction contribution, and is expressed as

c

C(‘)(S”) = -

7yb(S”)E;:.

(220)

l?l#”

This contribution is a result of the interaction between the mismatch of the nth particle and the stress fields P’” which are induced by the particles other than the nth. Then the coarsening rate v” and the particle migration rate R, of the nth particle (volume Vn, surface s,) are given as (221) and A,=

+

I

(r - R,)Vc%, ds

(222)

” 3”

respectively. Here 8, is the outward-pointing unit vector which is normal to the interface at s,,. Abinandanan and Johnson simulated numerically the morphological evolution in the system consisting of several spherical particles by using their derived interface equation. An interesting result they obtained is that the particle migration towards the elastically favourable orientations and hence the rearrangement of particles takes place during coarsening. Their simulations are three-dimensional. However, since the initial particle number is 40, only five particles remain after coarsening. According to the above discussions, computer simulations have succeeded, or are now succeeding in reproducing the actual kinetic behaviour of particles observed during coarsening in elastically constrained systems. However, all the results seem to be little more than a qualitative interpretation of the elasticity effect on the morphological changes. For example, the numerical simulations have offered little statistical information such as coarsening kinetics, the change in particle-size distribution, etc. We should expect further progress in these fields and, in particular, three-dimensional computer simulations for systems containing a large number of particles. 9. CONCLUDING

REMARKS

It is nothing new to say that the essential driving force for a phase transformation is the difference in the free energies of the initial and final configurations of the assembly (Christian(3n). The microstructural changes which we have been surveying here belong mainly to the late stage of phase decomposition that is, of course, a representative of phase

178

Progress in Materials Science

transformation. All the structural changes that we actually observe should be justified theoretically only when the change in free energy can be evaluated correctly. The reason why we feel some of the changes to be very strange from the conventional point of view is that there is something essential to be taken into consideration as the driving force. It is the elasticity effect to which we have not paid any special attention for a long time. It is obvious that the elasticity effect plays an essential role in microstructural changes in elastically constrained systems. However, it is a misunderstanding to imply that the surface energy has no important affects on the microstructural changes in an elastically constrained system. The effects of surface energy should be treated on equal terms with those of elastic energy. It depends on a delicate difference between the elastic energies and the surface energy whether or not the strange phenomena in the course of particle coarsening actually takes place. The field we have been discussing here is now rapidly advancing. However, there is no denying that many theoretical and experimental studies are still needed to understand fully the elasticity effect on the microstructural changes in elastically constrained systems. The time is coming in the near future when these problems will be fully understood. ACKNOWLEDGEMENTS The author would like to express his sincere thanks to Professor J. W. Christian of the University of Oxford for his continuous encouragement and critical reading of this article. He is grateful to Dr Y. Enomoto of Nagoya Institute of Technology (NIT) for his valuable discussions and his help in some calculations newly performed, such as Figs 18, 20, 24, etc. The author is indebted to Professor T. Miyazaki for his discussions, and also to Messrs T. Koyama, T. Okamura, Y. Suzuki and Y. Yamada of NIT, and Mr Y. Masuo of Nagoya Municipal Industrial Research Institute for their help in taking micrographs newly and/or redrawing figures. This article is the fruit of the author’s stay in Oxford as a visiting fellow sent by The Japan Society for the Promotion of Science (JSPS) under the recognition of The Royal Society.

REFERENCES A. J. Ardell, R. B. Nicholson and J. D. Eshelby, Acra Metall. 14, 1295 (1966). W. C. Johnson and J. K. Lee, Metall. Trans. lOA, 1141 (1979). W. C. Johnson, Acta Metall. 32, 465 (1984). J. D. Eshelby, Progress in Solid Mechanics, Vol. II (edited by I. N. Sneddon and R. Hill), Chapter III, p. 87. North Holland, Amsterdam (1961). 5. A. G. Khachaturyan, Theory of Sfructural Transformations in Solids. John Wiley & Sons, New York (1983). 6. T. Mura, Micromechanics of Defects in Solids, 2nd rev. edn, Kluwer Academic Publishers, Dordrecht (1987). 7. J. D. Eshelby, Proc. R. Sot. A241, 376 (1957). 8. J. D. Eshelby, Proc. R. Sot. A252, 561 (1959). 9. S. C. Lin and T. Mura, Physica Status Solidi (a) 15, 281 (1973). 10. J. E. Osburn, M. J. Mehl and M. Klein, Phys. Rev. B 43, 1805 (1991). 11. F. Wallow, G. Neite, W. Schriier and E. Nembach, Physica Status Solidi (a) 99, 483 (1987). 12. G. K&ter, K. Nembach, F. Wallow and E. Nembach, Mater. Sci. Engns A114, 29 (1989). 13. H. Yamauchi and D. de Fontaine, Acta Mefall. 27, 763 (1979). 14. T. Miyazaki, H. Imamura, H. Mori and T. Kozakai, J. Mater. Sci. 16, I197 (1981). 15. A. G. Khachaturyan, S. V. Semenovskaya and J. M. Morris Jr, Acta Mefall. 36, 1563 (1988). 16. A. G. Khachaturyan and V. M. Airapetyan, Physica Sratus Solidi (a) 26, 61 (1974). 17. P. E. J. Flewitt, Acta Metall. 22, 47 (1974). 18. E. Walker and M. Peter, J. Appl. Phys. 48, 2820 (1977). 19. W. C. Johnson, T. A. Abinandanan and P. W. Voorhees, Acta MetaN. Mater. 38, 1349 (1990). 20. A. Onuki and H. Nishimori, J. Phys. Sot. Jpn 60, 1 (1991). 1. 2. 3. 4.

Elasticity Effects on the Microstructure

of Alloys

179

21. A. Onuki, J. Phys. Sot. Jpn 58, 3065 (1989). 22. A. Onuki, J. Phys. Sot. Jpn 58, 3069 (1989). 23. W. C. Johnson and P. W. Voorhees, J. Appl. Phys. 61, 1610 (1987). 24. T. Miyazaki, H. Imamura and T. Kozakai, Mater. Sci. Engng 54, 9 (1982). 25. J. W. Martin and R. D. Doherty, Stability of Microstructure in MetaIIic Systems. Cambridge University Press,

Cambridge (1976). 26. E. P. Whelan and C. W. Howarth, J. Inst. Metals 94, 402 (1964). 27. G. Chadwick, J. Inst. Metals 93, 18 (1963). 28. I. S. Servi and D. Turnbull, Acta Metall. 14, 161 (1966). 29. A. J. Ardell, Acta Metall. 20, 61 (1972). 30. G. R. Purdy, Metal Sci. J. 5, 81 (1971). 31. A. Varshavskv and E. Donoso. J. Muter. Sci. Lett. 8. 430 (1989). 32. C. S. Smith, ketal Interfaces, b. 65. A.S.M., Metals Park, bH (i952). 33 G. R. Speich and R. A. Oriani, Trans. TMS-AIME 233, 623 (1965). 34. A. J. Ardell, Acta Metall. 16, 511 (1968). 35. P. K. Rastogi and A. J. Ardell, Acta Metall. 19, 321 (1971). 36. A. J. Ardell, Metal/. Truns. 1, 525 (1970). 31. J. W Christian, The Theory of Transformations in Metals and Alloys, 2nd edn, Part I Equilibrium and General Kinetic Theory. Pergamon Press, Oxford (1975). 38. J. W. Cahn. Acta Metall. 9, 795 (1961). 39. J. W. Cahn, Acta Metall. 10, 179 (1962). 40. J. W. Cahn, Acta Metall. 10, 907 (1962) 41. J. W. Cahn. J. Chem. Phys. 42, 93 (1965). 42. 3. W. Cahn, Trans. Metall. Sot. AIME 242, 166 (1968). 43. J. W. Cahn and J. E. Htlliard, J. Chem. Phys. 28, 258 (1958). 44. J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 31, 688 (1959). 45. C. Kinoshita, K. Nakai and S. Kitajima. Muter. Sci. Forum 15-18, 1403 (1987). 46. R. 0. Williams, Metall. Trans. llAi247 (1980) 47. R 0. Willlams. Calohad 8. 1 (1984). 48. J. W. Cahn and F. C. Larcht,‘Acta Metall. 32, 1915 (1984). 49. A. L. Roitburd, Soviet Phys. Solid State 26, 1229 (1984). 50. A. L. Roitburd, Soviet Phys. Solid State 27, 598 (1985). 51. W C. Johnson and P. W. Voorhees, Metall. Trans. 18A, 1213 (1987). 52. T. Miyazaki, T. Koyama and M. Doi, Acta Metall. Mater. 42, 3417 (1994). 53. W. C. Johnson, ~etall. Trans. 18A, 1093 (1987). 54 Z.-K. Liu and J. Agren, Acta Metall. Mater. 38, 561 (1990). 55. W. C. Johnson and W. H. Miiller, Acta Metall. Mater. 39, 89 (1991). 56. M. J. Pheifer and P. W. Voorhees, Acta Metall. Mater. 39, 2001 (1991). 57. J.-Y. Huh. J. M. Howe and W. C. Johnson. Scriota Metall. 24. 2007 (1990). 58. J.-Y. Huh, J. M. Howe and W. C. Johnson; Acta Metall. Mater. 41, i577 (1993). 59. F. R. N. Nabarro, Proc. Phys. Ser. 52, 90 (1940). 60. F. R. N. Nabarro. Proc. R. Sot. A175, 519 (1940). 61. A. Kelly and R. B. Nicholson, Prog. Muter. Sci. 10, 149 (1961). 62. E. von Hornbogen and M. Roth, Z. Metallkd. 58, 842 (1967). 63. W. T. Loomis, J. W. Freeman and D. L. Sponseller. Metall. Trans. 3, 989 (1972). 64 A. G. Khachaturyan. Son. Phys. Solid State 8, 2163 (1967). 65. P. W. Voorhees. G. B. McFadden and W. C. Johnson, Acta Metall. Mater. 40, 2979 (1992). 66. Y. Wang, L.-Q. Chen and A. G. Khachaturyan, Scripta Metall. Mater. 25, 1387 (1991). 61. Y. Wang, L.-Q. Chen and A. G. Khachaturyan, Acta Metall. Mater. 41, 279 (1993). 68. H. Nishimort and A. Onuki, Phys. Rev. R 42, 980 (1990). 69. H. Nishtmori and A. Onuki. J. Phys. Sot. Jpn 60, 1208 (1991). 70. M. Doi and T. Mtvazaki. Suoerallovs I992 (edited bv S. D. Antolovich. R. W. Stusrud. R. A. MacKav. D. L. Anton, T. Khan. R. D. Kiss’mger and D. L. Klarstrom), p. 537. The Metallurgical Society, Warrendale, PA (1992). 71. M. Doi, M. Fukaya and T. Miyazaki, Phil. Mag. A 57, 821 (1988). 72. T. Miyazakr, K. Nakamura and H. Mori, J. Mater. Sci. 14, 1827 (1979). 73. I. M. Lifshitz and Slyozov, J. Phys. Chem. Solids 19, 35 (196t). 74. C Wagner, Z. Electrochem. 65, 581 (1961). 75. Y. Enomoto and M. Doi, unpublished work (1995). 76. A. D. Brailsford and P. Wynblatt, Acta Metall. 27, 489 (1979). 77. C. K. L. Davies. P. Nash and R. N. Stevens. Acta Metall. 28. 179 (1980). 78. Y. Enomoto and M. Doi, Materia Japan, 34; 757 [in Japanese] (1995). 79. K. Kawasaki, Ann. Phys. (N.Y.) 154, 319 (1984). 80. S. M. Allen and J. W. Cahn, Acta Metall. 27, 1085 (1979).

180

Progress in Materials Science

81. J. J. Weins and J. W. Cahn, Sinrering and Related Phenomena, Materials Science Research, Vol. 6 (edited by

G. 82. K. 83. M. 84. K. 85. K.

C. Kuczynski), p. 151. Plenum Press, New York (1973). Kawasaki and T. Ohta, Physica A 118A, 175 (1983). Tokuyama and K. Kawasaki, Physica A 123A, 386 (1984). Kawasaki and Y. Enomoto, Physica A lSOA, 463 (1988). Kawasaki, Y. Enomoto and M. Tokuyama, Physica A 135A, 426 (1986). 86. Y. Enomoto, M. Tokuyama and K. Kawasaki, Acra Metail. 34, 2119 (1986). 87. Y. Enomoto, K. Kawasaki and M. Tokuyama, Acta Metall. 35, 907 (1987). 88. Y. Enomoto, K. Kawasaki and M. Tokuyama, Acra Metali. 35, 915 (1987). 89. J. A. Marqusee and J. Ross, J. Chem. Phys. 80, 536 (1984). 90. P. W. Voorhees and M. E. Ghcksman. Actu Metall. 32. 2001 (1984). 91. P. W. Voorhees and M. E. Glicksman; Acta Mefall. 32; 2013 (1984). 92. M. E. Glicksman and P. W. Voorhees, Phase Transformation in Solid (edited by T. Tsakalakos), p. 451. North Holland, Amsterdam (1984). 93. P. W. Voorhees, J. Stat. Phys. 38, 231 (1985). 94. C. W. J. Beenakker, Phys. Reo. A 33, 4482 (1986). 95. M. Tokuyama and Y. Enomoto, Physicu A 204A, 673 (1994). 96. P. W. Voorhees and R. J. Schaeffer. Aria MetaN. 35. 327 (1985). 97. S. C. Hardy and P. W. Voorhees, metall. Trans. 19A, 2713 (1988). 98. P. W. Voorhees, G. B. McFadden, R. F. Boisvert and D. I. Meiron, Acta Metall. 36, 207 (1988). 99. T. Imaeda and K. Kawasaki, Mod. Phys. Left. B 3, 207 (1989). 100. T. Imaeda and K. Kawasaki, Physica A 186A, 359 (1992). 101. P. W. Voorhees and R. J. Scharfer, Actu Mefull. 35, 327 (1987). 102. N. Akaiwa and P. W. Voorhees, Phvs. Rev. E. 49, 3860 (1994). 103. P. K. Footner and B. P. Richards, J. Mater. Sci. 17, 2141 (1982). 104. M. Doi, T. Miyazaki, S. Inoue and T. Kozakai, Dynamics of Ordering Processes in Condensed Mutter (edited by S. Komura and H. Furukawa, p. 287. Plenum Publishing, New York (1988). 105. M. Doi, Mater. Trans. JIM 33, 637 (1992). 106. J. H. Westbrook, Z. Krlstallog. 110, 21 (1958). 107. M. Doi, T. Miyazaki and T. Wakatuki, Mater. Sci. Engng 67, 247 (1984). 108. M. Doi and T. Miyazaki, Superalioys 1984 (edited by M. Gel], C. S. Kortovich, R. H. Bricknell, W. B. Kent and R. H. Radavich), p. 543. The Metallurgical Society, Warrendale, PA (1984). 109. M. Doi, T. Miyazaki and T. Wakatuki, Muter. Sci. Engng 74, 139 (1985). 110. C. S. Jayanth and P. Nash, J. Muter. Sci. 24, 3041 (1989). 111. G. Iooss and D. D. Joseph, Elementary Stability and Bifurcation Theory. Springer, Berlin (1980). 112. W. C. Johnson and J. W. Cahn, Acfu Metall. 32, 1925 (1984). 113. T. Miyazaki, K. Seki, M. Doi and T. Kozakai, Mater. Sci. Engng 77, 125 (1986). 114. T. Miyazaki, M. Doi and T. Kozakai, Solid Stare Phenomena 3/4, 227 (1988). 115. M. Doi, and T. Miyazaki, Superallqvs 1988 (edited by S. Reichman, D. N. Duhl, G. Maurer, S. Antolovich and C. Lund), p. 663. The Metallurgical Society, Warrendale, PA (1988). 116. T. Miyazaki and M. Doi, Murer. Sci. Engng AllO, 175 (1989). 117. M. Doi and T. Miyazaki, J. Mater. Sci. 27, 6291 (1992). 118. T. Ohta, J. Phys.: Condens. Matter 2, 9685 (1990). 119. Y. Enomoto and K. Kawasaki, Acra Metail. Mater. 37, 1399 (1989). 120. A. Onuki and H. Nishimori, Phys. Reo. B. 43, 13649 (1991). 121. T. A. Abinandanan and J. W. Johnson, Acta MetaN. Mater. 41, 17 (1993). 122. T. A. Abinandanan and J. W. Johnson, Actu Metall. Muter. 41, 27 (1993).