[Dissertation] A Computer Simulation of the Evolution of Coherent Composition Variations in Solid Solutions

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N0RTHWES1ERN UNIVERSITY

A COMPUTER SIMULATION OF THE EVOLUTION OF COHERENT COMPOSITION VARIATIONS IN SOLID SOLUTIONS

A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS for the degree

DOCTOR OF PHILOSOPHY Field of Materials Science

By

Didier de Fontaine

Evanston, Illinois June 1967

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Summary of the Dissertation,

A COMPUTER SIMULATION OF THE EVOLUTION OF COHERENT COMPOSITION VARIATIONS IN SOLID SOLUTIONS

By

Didier de Fontaine

A mathematical model governing the behavior of coherent composition variations in cubic crystals of binary and multicomponent systems has been given and computer-generated numerical solutions have been obtained in some relatively simple cases.

A single non-linear partial differential diffusion

equation, originally derived by Cahn, or systems of diffusion equations, governed all processes, from the initial to the final coarsening stages

for

any composition and temperature in the coherent phase.diagram. The case most extensively studied was that of isothermal decompo­ sition of AA-Zn alloys inside the spinodal.

The diffusion equation was

solved for one-dimensional composition variations in periodic domains.

It

was concluded that the characteristics of the composition profiles appear to vary continuously as the average alloy composition is altered from the center to the edge of the miscibility gap; in other words, there is no abrupt change in the morphology of the resulting structures at the spinodal compo­ sitions.

For alloys -towards the center of the miscibility gap, a quasi-

sinusoidal structure developed and grew in amplitude until the coherent phase boundaries were reached.

The resulting structure was quite regular and the

average spacing between particles was practically independent of the initial conditions.

As the average composition approached the spinodal, the sinusoidal

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profiles gave way to composition variations resembling discrete Guinier zones (central enriched regions surrounded by depleted zones) distributed almost randomly.

A structure consisting of regularly spaced precipitates

developed with the spacing between precipitates closely related to the radius of the Guinier zones.

For average compositions close to the spinodal,

an initial coarsening mechanism became important, and the final structures depended critically on the initial composition fluctuations. An experimental verification was provided by small angle X-ray scattering studies performed by Rundman.

Sequences of experimental and

calculated intensity spectra were similar with respect to overall shape, position of the maxima, integrated intensity and progressive sharpening and shift towards longer wavelength of the main satellite envelope.

Theo­

retical calculations for the continuous cooling process were also confirmed by the experimental results.

Some calculations were also performed on two-

dimensional and spherically symmetric composition profiles. A theoretical study of the initial stages of spinodal decompo­ sition in n-component systems was undertaken.

It was shown that the

spinodal consists of n-1 surfaces enclosing regions in which the solid solution becomes unstable for different sets of directions in compositionspace.

It was also shown that the coherent spinodal surfaces must pass

through points where tyro or more chemical spinodal surfaces intersect.

At

these points, spinodal decomposition can occur initially without coherency strains.

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Ce travail est dedicace & Danielle.

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i

.

’r

.

TABLE OF CONTENTS page INTRODUCTION

1

1. DERIVATION OF DIFFUSION EQUATIONS

k

1.1. The Diffusion Equation for Two-Component Systems

k

1.2. The Diffusion Equations for Multicomponent Systems-

9

2. SOLUTION FOR THE INITIAL STAGES OF SPINODAL DECOMPOSITION

15

2.1. Spinodal Decomposition in Binary Systems

l6

2.2. Spinodal Decomposition in Multicomponent Systems

23

3. SOLUTION OF NON-LINEAR EQUATIONS

37

3.1. Polynomial Approximation of Helmholtz Free Energy

-^0

3.2. Discussion of General Method of Solution

UU

3.3. The Aluminum-Zinc System

58

3.^+. Isothermal Aging of One-Dimensional Composition Variations

63

3.5. One-Dimensional Simulation of Continuous Cooling

92

•3.6 . Solid Solutions with Plane-Wave Imperfections

99

3-7* Two-Dimensional Composition Fluctuations

105 .

3.8. Spherically Symmetric Composition Fluctuations

llU

CONCLUSION

126

REFERENCES

132

ACKNOWLEDGMENTS

135

VITA

. 137 i

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i i

page

Appendix A : Notation

139

Appendix B : Green's Theorem

1^2

Appendix C : Interpretation of Ti Parameters

lMl-

Appendix D : Systems of Ordinary Linear DifferentialEquations

1^7

Appendix E : Relationship Between Chemical andCoherentSpinodals Appendix F : Polynomial Free Energy Approximations

152 .

155

Appendix G : Expressions for M(u) and D(u).

159

Appendix H : Description of Program THERMO

163

Appendix I : Description of Program PERI0D1D

171

Appendix J :Description of

Program FYSPIN1P

l8l

Appendix K :Description of

Program INKPLOT

191

Appendix L :Description of

Program FYFILM

197

Appendix M :Description of

Program SPICOC

2014-

Appendix N :Description of

Program FYSPIMFY

211

Appendix P :Description of

Program FYSPIN2D

217

Appendix Q :Description of

Program PLOTOODY

223

Description of Program FYSPHERE

230

Appendix R

Appendix S : Diffraction Satellites from Spherically Symmetric Composition Variations

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236

LIST OF FIGURES page

Fig. 1.

Vertical section CD through the.spinodal surfaces for

a ternary regular solution model.

3*+

The full curves represent

the chemical spinodal, the broken curves represent the coherent spinodal.

Fig. 2.

Isothermal section through the chemical spinodal

3^

surfaces for a ternary regular solution model at kT/io = l/b. Spinodal decomposition can occur for composition fluctua­ tions inside the dark areas, shown at selected points along CD.

Fig. 3. Helmholtz free energy data points (open circles) and

39

fourth-degree polynomial approximation (upper full curve). The broken curve represents a parabolic approximation . about the point c = cQ. The second derivative of the fourth-degree polynomial free energy is shown in the lower part of the figure.

Points c' and c" are the

spinodal compositions.

Fig. k. CsCl structure represented by a b.c.c. array of tetrakaidekahedra instersected by the (110) plane.

*.

symmetry about point r

L

39

Anti­

I I I 1 is indicated. 44 J

Fig. 5. Aluminum-rich side of the Al-Zn phase diagram.

The

heavy curve shows the coherent miscibility gap, the

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57

broken curve is the coherent spinodal.

The open circles

indicate the alloy systems investigated.

Fig. 6 . Helmholtz free energy data (open circles) for the ... Al-Zn system and the corresponding fourth-degree polynomial approximation (full curves) at selected temperatures below the critical point.

The open triangles represent

additional constructed points.

Fig. 7*

Coherent Helmholtz free energy curves derived from

the polynomial approximations of Fig. 6 by the addition of the term TfYc(c-0.7)• The calculated coherent equi­ librium and spinodal compositions-are indicated by open circles. Fig. 8 . Composition profile used in calculating the normal­ ization factor a0 used in the percent completion calcul­ ations .

Fig. 9*

Calculated values of the logarithm of the intensity

(amplitude squared) versus aging time for the first three Fourier components of an Ag-Au layered structure.

Fig. 10 a. Composition profiles corresponding to the spectrum of Fig. 9 at the initiation of the calculations and at time t =

ho

rain.

The broken horizontal lines indicate

the pure components Ag and Au.

Fig. 10 b. Continuation of Fig. 10 a.

Continued on Fig. 10 b.

Composition profiles

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at time t = 100 and 200 min.

The broken horizontal

lines indicate the pure components Ag and Au.

ig. 11.

Amplitude spectra (absolute magnitude) and corres­

ponding composition profiles calculated according to Eq.(7*0 for a 0.225 Al-Zn alloy aged at 100°C in a 1+00 K domain with periodic boundary conditions and random initial conditions of small amplitude.

The broken

horizontal lines indicate the coherent equilibrium compositions. (Computer-generated plots)

'ig. 12.

Amplitude spectra (absolute magnitude) and corres­

ponding composition profiles calculated according to Eq.(7^) for a 0.200 Al-Zn alloy aged at 100°C in a 1+00 A domain with periodic boundary conditions and same initial conditions as in Fig. 11.

The broken horizontal lines

indicate the coherent equilibrium compositions.

(Computer-

generated plots)

Fig. 13.

Amplitude spectra (absolute magnitude) and corres­

ponding composition profiles calculated according to Eq. (7I+) for a 0.375 Al-Zn alloy aged at 100°C in a 300 A domain with periodic boundary conditions and same initial conditions as in Fig. 11.

The broken horizontal lines

indicate the coherent equilibrium compositions.

(Computer­

generated plots)

Fig. lU.

Amplitude spectra (absolute magnitude) and corres-

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vi

page

ponding composition profiles calculated according to the linear diffusion equation for a 0.225 Al-Zn alloy aged at 100°C in a UOO A domain with periodic boundary conditions and same initial conditions as in Figs. 11 and 12.

The

broken horizontal lines indicate the coherent equilibrium compositions. (Computer-generated plots)

Fig. 15.

Amplitude spectra (absolute magnitude) and corres-

8l

ponding composition profiles calculated according to Eq.(7*+) showing the coarsening stage for a 0.225 Al-Zn alloy aged at 100°C in a 1+00 I domain with periodic boundary conditions (the initial stages were shown in Fig. 12).

The broken horizontal lines indicate the

coherent equilibrium compositions. (Computer-generated plots)

Fig. l6.

Calculated percent completion curves for 0.225 Al-Zn

alloys aged at 100°C.

83

Results for small-amplitude random

initial conditions fall within the shaded band.

The

upper full curve is that for a large quenched-in com­ position fluctuation.

The broken curves show the corres­

ponding percent completion values calculated according to the linear equation.

The open circles indicate un­

normalized percent completion values for a spherically symmetric composition variation.

Fig. 17*

Intensity spectra (amplitude squared) calculated

accordingto Eq.(8o) for a 0.225 Al-Zn alloy aged at

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85

150°C.

The initial condition spectrum (curve no. l)

simulates an experimental quenched-in spectrum (curve no. 1 of Fig. l8).

The critical (Xc) and optimum (\B)

wavelengths are indicated by arrows.

Fig. l8 . Intensity specta obtained by Rundman

23

from small-

angle X-ray scattering data on a 0.225 Al-Zn alloy aged at 150°C.

The corresponding calculated spectra are

shown in Fig. 17.

Fig. 19 a and b.

Intensity spectra (amplitude squared) cal­

culated according to Eq.(8o) for a 0.200 Al-Zn alloy aged first (a)-at 100°C (inside the spinodal) and sub­ sequently (b) at 200°C (outside the spinodal).

The

critical (xc) and optimum (xn) wavelengths are indicated by arrows in Fig. 19 a.

Fig. 19 c.

Continued on Fig. 19 c.

Continuation of Fig. 19 a and b : calculated

intensity spectra for the final aging treatment at 100°C (inside the spinodal) of a 0.200 Al-Zn alloy. The critical (Xc ) and optimum (Xn) wavelengths areindicated by arrows.

Fig. 20.

Amplitude spectra (absolute magnitude) and corres­

ponding composition profiles calculated according to Eq.(7*0 for a 0.100 Al-Zn alloy aged at 100°C (outside the spinodal) in a 600 A domain with periodic boundary conditions.

Three one-dimensional Guinier zones constitute

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the initial condition (at time zero).

The broken horizontal

lines indicate the equilibrium compositions.

(Computer­

generated plots)

Fig. 21.

hypothetical cooling curve from the solution-treatment

temperature T^ to the quenching bath temperature T^. temperature Tq is located just below the spinodal.

The Inter­

mediate temperatures indicate assumed ranges of validity of free energy curves.

Fig. 22.

Exponential cooling curves calculated according to

Eq.(85) (solid curves).

The quenching rates were :

(l) 103 °C/sec, (2) 5 x 103 °C/sec, (3) 104 °C/sec, 0 0 3 X 104 °C/sec, (5) 6 x 104 °C/sec, (6 ) 10s °C/sec. The broken lines are rough sketches of 2, 10 and 65 percent completion curves for a 0.225 Al-Zn alloy.

Fig. 23-

Coherent derivative modulation (a), and composition

profiles (b,c,d) calculated according to Eq.(89) for a 0.225 Al-Zn alloy aged at 100°C in an imperfect 1+00 A domain with insulating boundary conditions.

The initial

condition is shown in Fig. 25 a.

Fig. 2k.

Coherent derivative modulation (a), and composition

profiles (b,c,d) calculated according to Eq.(89) for a 0.225 Al-Zn alloy aged at 100°C in an imperfect ^00 A domain with insulating boundary conditions.

The initial

condition is shown in Fig. 25 a.

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ix

page

Fig. 25.

Composition profiles calculated according to Eq.(89)

103

for a 0.225 Al-Zn alloy aged at 100°C in an imperfection$

free kOO A domain with insulating boundary conditions. Figure (a) is the common initial condition for Figs. 23} 2b and 25.

Fig. 26.

Acrylic sheet and nylon model of a two-dimensional

108

Fourier spectrum (absolute magnitude) calculated according to Eq.(90) for a 0.225 Al-Zn alloy aged at 100°C for

1500 sec (corresponding composition profile is shown in lower frame of Fig. 27 c ).

Fig. 27 a.

Two-dimensional composition profiles synthesized

109

from amplitude spectra of Eq.(90) for a 0.225 Al-Zn alloy aged at 100°C for 0 and 300 sec. computer-generated plots)

(Traced-over

Continued on Fig. 27 b, c

and d.

Fig. 27 b.

Continuation

of Fig.

27 a. Two-dimensional profiles110

synthesized from amplitude spectra of Eq.(90) for a 0.225 Al-Zn alloy aged at 100°C for 600 and 900 sec. Continued on Fig. 27 c and d.

Fig. 27 c.

Continuation

of Fig.

27 h. Two-dimensional profiles111

synthesized from amplitude spectra of Eq.(90) for a 0.225 Al-Zn alloy aged at 100°C for 1200 and 1500 sec. Continued on Fig. 27 d.

Fig. 27 d.

Continuation

of Fig.

27 c- Two-dimensional profiles112

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X

page

synthesized from amplitude spectra of Eq.(90) for a 0.225 Al-Zn alloy aged at 100°C for 1800 sec.

Fig.

28. Amplitude spectra (absolute magnitude) and corres-

122

ponding spherically symmetric composition profiles cal­ culated according to Eq.(l06) for a 0.225 Al-Zn alloy aged at 100°C (inside the spinodal) in a 300 A radius spherical domain.

Initial condition is a small composition

fluctuation at the origin.

The broken horizontal lines

indicate the equilibrium compositions.

(Computer­

generated plots)

Fig.

29- Amplitude spectra (absolute magnitude) and corres-

12^

ponding spherically symmetric composition profiles cal­ culated according to Eq.(l06) for a 0.100 Al-Zn alloy aged at 100°C (outside the spinodal) in a 300 A radius spherical domain.

Initial condition (at time zero) is

a spherically symmetric Guinier zone.

The broken horizontal

lines' indicate the equilibrium compositions.

(Computer­

generated plots)

Fig. Al.

Interpretation of the 7]^ (j = l,...,n) parameters

for multicomponent systems.

A hypothetical lattice

parameter versus composition surface is shown in per­ spective for the ternary subsystem (j,k,n).

The plane

tangent to this surface at aQ (average lattice parameter) is also indicated.

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1U5

xi

page

Fig. A2.

Ideal entropy (full curve) and its fourth-degree

it 5

Legendre polynomial approximation (open circles).

Fig. A3.

Diffusion equation coefficient D(u) (solid curve)

160

and its parabolic Taylor's .expansion about u = 0 (c0 = 0.225 at. fract. Zn) in the case of an Al-Zn solid solution at 100°C.

True spinodal compositions

are indicated by open circles, incorrect ones (derived from the parabolic expansion) by open triangles.

Fig. At.

Photographic reproduction of typical Fourier

spectrum and composition profile plots generated by the subroutine GRAPH used in conjunction with programs FYSPIN1P or FYSPIMPY.

The symbols AXMIHA.,

DAXA, MAX, AXMINU, BAKU were added manually.

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182

First used page

LIST OF PRINCIPAL SYMBOLS

A

Amplitude vector

19

A

Time derivative of amplitude vector

19

A0

Initial amplitude vector

19

A

Amplitude vector in diagonalyzed system

27

Ak

Fourier coefficients, amplitudes

17

A°

Initial amplitudes

19

A

Fourier cosine coefficients

65

AK

Sums of products of amplitudes

kQ

a

Lattice parameter

a,

Coefficients of fourth-degree polynomial in u

k2

B

Matrix in system A = BA

19

B

Diagonalized matrix B

27

B*

Fourier sine coefficients

65

B.

Sums of products of amplitudes

5

118

Elements of matrix B

19

Ck

Complex Fourier coefficients

6k

C^

Sums of products of complex Fourier coefficients

65

C11,C12,C44

6

Elastic constants

c

Composition (atomic fraction) in binary system

k

cQ

Average composition in binary system

k

Cj

Composition (atomic fraction)

9

c?

Average composition in multicomponent system

in multicomponent system

9

c ,c

Coherent spinodal compositions

^3

C 5C _

Coherent equilibrium compositions

k3

cr p

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Page

Diffusion equation coefficient

D

Do, Dl5D2 ^

D

Coefficients of expansion of D

b5

*

13

Tracer diffusivities

7

E

Diagonalizing operator of matrix B

27

Ei

Eigenvectors of matrix B

30

F

Free energy of non-uniform system

F

Polynomial approximation of free energy f

h2

F

Matrix of second derivatives of free energy

31

F

Matrix of minors of matrix F

32

Ek

Amplitudes for spherically symmetric fluctuations

109

> Fourier coefficients of coherent second derivative

17

f k ,k

Helmholtz bulk free energy

f

fojfp,f'",fo

Bulk-free energy and derivatives at u = 0

5

5 16,56

Second derivatives of free energy (multicomponent systems)

2k

G

Coefficient matrix for multicomponent systems

25

G°

Coherent second derivative matrix for multicomponent systems

28

G1J

Elements of matrix G

25

Gn

Function x-1 sin x integrated from 0 to mn

J

Flux for binary systems

Jlc

Flux for multicomponent systems

k

Boltzmann's constant

k

Wave index

M

Mobility in binary systems

M

Mobility matrix in multicomponent systems

117 k 12 7 6k 5 25

M ojM15M2 , Coefficients of expansion of M

^5

M kl

12

Elements of mobility matrix

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xiv

page Ny

Number of atoms per unit volume

7

n

Number of components in multicomponent systems

9

n

Maximum number of Fourier coefficients retained

6h

p

Percent completion

67

Q

Activation energy

62

Q

Quadratic form

23

Q

Source term vector

19

Qk

Fourier coefficients of source term

q

Source term in binary

qk

Source term in multicomponent systems

lU

R

Gas constant

62

R

Radius of spherical domain

115

r

Radial polar coordinate

11*+

r_

Position vector

T

Absolute temperature

t

Time

U

Free energy of volume

element of non-uniformsystem

U0

Free energy of volume

element of uniformsystem

u

Composition variation c - c0

u

Composition variation vector in multicomponent systems

28

u

Normal composition variation vector

29

u0

Initial composition fluctuation

9

u1

Composition variation Cj - c£

9

V

Domain volume

v

Function r u

W

Coherent strain energy

systems

’ 17 5

^ •' 7 ^

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6 10 *+

17 115 5jl0

XV

page

Xt

Coefficients of fourth-degree polynomial F

*+2

x

Cartesian distance coordinate

^3

Yk

Strain energy parameter

5

Eigenvalue of matrix B, amplification factor o. k

Ok

Eigenvalue of matrix

B, amplification factor

Amplification vector

'

21,50 2o ^

J

Wave number

17

$

Critical wave number

20

j3B Optimum wave number

90

At

21

Iteration time step Linear expansion per unit composition change

1j

Vector of components Tiparameters for multicomponent systems

h

Gradient energy coefficient in binary systems

Hjj

Gradient energy coefficient in multicomponent systems

X

Wavelength

5 31 13 5 13 6U

Xa

Optimum wavelength

cp

Potential for binary systems

cpj

Potential for multicomponent systems

11

a

Integrated (c - c0)2

67

a0

Normalization factor for percent completion

67

t

Reduced temperature RT/oj

33

a'

Interaction parameter for regular solution model

32

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53,60 5

INTRODUCTION

1,2 Recently, theories have been developed by Hillert by Cahn

3-6

and

to describe the kinetic behaviour of coherent composition

variations in binary solutions.

Both of these theories are based

on diffusion equations which differ from the well-known Darken equation by the addition of a fourth-order spacial derivative which accounts for the incipient interfacial energy associated with composition gradients.

By using a finite-difference diffusion equation, Hillert

2

was able to take into account the discrete nature of the crystal lattice, thereby covering, in principle, both clustering and ordering phenomena, i.e., long and short wavelength composition fluctuations. The treatment was limited, however, to one-dimensional composition variations in binary systems whose bulk free energy was that of a regular solution model. medium, Cahn

By treating the solid solution as a continuous

derived a partial differential diffusion equation

which incorporated the effect of coherency strains introduced by small-amplitude coherent composition variations in cubic crystals. In order to obtain analytical solutions, Cahn was obliged to linearize his equation, thereby limiting the treatment to solid solutions whose free energy curve is parabolic with composition.

Cahn applied the

theory to the initial stages of the decomposition of binary solutions

1

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2

inside the spinodal (where the free energy curve is concave downwards). 1 The later stages of spinodal decomposition and the behaviour of composition fluctuations outside the spinodal cannot be treated by a linear partial 7 differential equation, however. Cahn has therefore recently extended his original theory to show the influence of the non-linear terms of the differential equation on the later stages of spinodal decomposition. The aim of the present study is to construct a general mathematical model covering all stages in the evolution of coherent composition variations in cubic crystals, for both binary and multicomponent systems.

The Helmholtz free energy is assumed to be analytic

in the composition variables.

Since the model is based on non-linear

partial differential diffusion equations, the crystal is implicitely regarded as an elastic continuum and, therefore, dispersion effects associated with very short wavelength composition fluctuations cannot be treated correctly within this framework.

An extension to systems

8

exhibiting a tendency to order is forthcoming . The diffusion equations for two and multi-component systems are described in Sect. 1.

Following Cahn, the derivation of these

equations is based on the expression for the free energy of non9 uniform solutions given by Cahn and Hilliard . The diffusion equation parameters are, in general, space and time dependent.

The space

dependence allows one to treat crystalline imperfections which give rise to long range elastic fields.

The time dependence allows one

to treat continuously varying temperatures and diffusivities.

Further­

more, the addition of an appropriate source term to the diffusion equations allows one to treat, in principle, the statistical generation

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3

of composition fluctuations, an essential feature of any nucleation theory. In Sect. 2,the diffusion equations are linearized in order to obtain analytical solutions.

These solutions are valid only for

a specific process however : the initial stages of spinodal decomposition. In that section, spinodal decomposition in multicomponent systems is studied in some detail with a ternary regular solution model given as an example. Special cases of the general diffusion equations of Sect. 1 are analysed in Sect. 3*

Iterative techniques are used to find numerical

solutions of the appropriate non-linear partial differential equations in the case of one and two-dimensional composition variations and also of three-dimensional composition variations exhibiting spherical symmetry.

Both isothermal aging of a metastable solid solution and

continuous cooling are examined.

The effect of crystalline imper­

fections is also briefly touched upon.

Although these numerical

solutions describe particular events, some general conclusions will be drawn regarding such complete processes as spinodal decomposition, growth and coarsening.

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1. DERIVATION OF DIFFUSION EQUATIONS

General diffusion equations will be derived, first for binary systems, then for multicomponent systems.

In order to preserve

maximum generality, only a minimum of restrictive assumptions will be made regarding the form of the equations and the precise nature of their parameters.

The crystal containing coherent composition

variations will be treated as a continuum and it will be assumed that its free energy is given by an expression analogous to that derived by Cahn and Hilliard^ for a two-component fluid phase and extended by Cahn^ to crystals of cubic symmetry by the addition of a coherent strain energy function.

The present treatment is thus

limited to clustering phenomena in solid solutionsof cubic symmetry.

1.1. The Diffusion Equation for Two-Component Systems

The composition variation will be defined by

u(r,t) = c(r,t) - c0

in which c(r,t), a continuous function of position r and time t (the notation is explained in Appendix A), denotes the mole fraction of component B, c0 being the mean composition. The continuity equation of continuum mechanics can be written in the present case as follows :

du/dt = - div J + q(r,t)

expressing the fact that the time variation of the

(l)

composition u in

a small volume of solution is proportional to the negative of the divergence of the flux J of B atoms into that element of volume plus

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5

whatever source q(r,t) may be operating at time t.

For small deviations

from equilibrium, the flux will be proportional to a driving force expressed as the gradient of some potential cp :

J = - M grad cp

(2)

where M, the mobility, is a positive proportionality factor.

We shall

now make the assumption that Eq. (2) holds even in the case of large departures from equilibrium. We must now seek an expression for the potential function cp. Consider first a one-dimensional coherent composition variation in a volume V of the solid solution.

Its Helmholtz free energy F according

to Cahn^ is given by the following integral

F =

[f(u) + w(u) + x(vu)2] dr

(3)

V

in which f(u) is the Helmholtz bulk free energy per unit volume and 9 n(vu)2is the gradient free energy introduced by Cahn and Hilliard . Equation (3) was derived for small gradients vu; we shall assume, however, that this expression is valid for arbitrary gradients.

For

simplicity, the gradient energy parameter x will be regarded as composition and position independent.

The strain energy W, arising

from coherency strains, is given by

W(u) = TfYKu2

with

11 = (l/a)(da/dc) c=c0 j

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(h)

6

a being the lattice parameter.

The parameter YK is a function of

the elastic constants

^)|)|

Material and of the directional

cosines of the vector K giving the direction of the composition variation (the convention regarding upper case subscripts such as K is given in Appendix A).

The explicit dependence of YK 'on. K has been derived

k by Cahn . We shall use Eq. {k) for the strain energy, but without the subscript on Y, even in cases where the composition variation is no longer one-dimensional.

This turns out to be a valid procedure

provided all computations are performed on the Fourier spectrum of the function u(r,t) since it can be shovm that the strain energy, due to a general three-dimensional variation u can be expressed as the sum of the strain energy contribution from its individual Fourier components.

Under these conditions Eq. (3) will be valid for an

arbitrary composition variation u(r,t). Let us denote the integrand of Eq. (3) by U(u,Vu).

5

Cahn , consider a small change in composition 6u.

Following

Correspondingly,

there will be a small change in free energy 6F given by

(5)

where cpis the variational derivative of U : (6 )

cp = (3U/du) - v(dU/3Vu) •

We see that

cp = 6U/ 6u .

The function cp thus appears as a generalized chemical potential and will be used as potential function in E q . (2).

The mobility M will

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7

then be determined by requiring that the resulting diffusion equation agree with Darken's equation when strain and gradient energy are omitted.

Using this method, Huston, Cahn and Hilliard‘S obtained

the following expression for M

M = [c(l-c)/%lcT][Djc + Dg(l-c) ]

where

(7)

is the number of atoms per unit volume, k is Boltzmann's ^

y

constant, T is the absolute temperature and DA and Dg are the tracer diffusivities of atomic species A and B respectively. It is now possible to combine Equations (l) and (2) to obtain the diffusion equation

du/ot = v(MVcp) + q

(8)

in which, in accordance with Eq. (6), the potential cp will be given by

cp = f '(u) + 2TfYu - 2 k V2u ,

the prime denoting differentiation with respect to u.

(9)

Equation (8)

is valid in any coordinate system since the space derivatives are expressed by means of the invariant nabla operators.

This diffusion

equation is a non-homogeneous, non-linear partial differential equation of parabolic type in three-dimensional anisotropic space.

Various

simplified versions of this equation will be solved in the following sections. To complete the mathematical formalism, one must also specify the boundary and the initial conditions.

It is not necessary to

describe the composition variation over the whole crystal; it is simpler to consider the crystal as a periodic array of identical

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8

small domains

, each domain containing the same composition variation

regarded as sufficiently representative of the crystal as a whole (essentially the same procedure was used "by Ham in his papers on pre­ cipitation12’1^).

Since only crystals of cubic symmetry will be

considered, we shall select, as periodic domains, right rectangular prisms of edge lengths 2Xx, 2X2, 2}^ in the (100) directions.

It

is then natural to impose periodic boundary conditions on the function cp and its normal derivative (see Appendix B), as was suggested by Cahn1^ : cp(-s) = cp(_s)

(10) 3cp(-_s)/5n = -9cp(_s)/3n ,

_s being the value of r on the domain walls.

These conditions insure

that the following holds : t

u (r,t)d r = 0 .

JV This equation expresses the fact that the average composition cq will remain constant within any domain since, because of the periodic conditions, any flux J entering the domain through one wall will be exactly compensated by the same flux leaving through the opposite wall.

For some calculations, the more stringent condition of vanishing

flux on the domain walls was imposed.

This insulating boundary condition

is expressed mathematically by the equation

dcp(s)/3n = 0 .

Unless otherwise specified, we shall use the periodic conditions of Eq. (10), which, in turn, will give rise to periodic boundary conditions

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'

hr.

9

on the function u(r,t) itself and on

as will he shown later.

Finally, an initial composition fluctuation

u0(r) = u(r,t0)

(at time t0) must he specified in each case.

1.2. The Diffusion Equations for Multicomponent Systems

Consider an n-component system.

The composition will

he defined hy n variables

Cj ~ Wj/W

(i

1,...,n)

where 1'^ represents the number of moles of component Cj and N the total number of moles of the solid solution.

Let us also define the

composition variations

ui = ci " ci , c° denoting the average of the variable ct over the whole crystal. Since the equation

i=i holds, we must have

Y ut = 0 l

(11)

(unless otherwise specified, the summations will extend from 1 to n). We shall now define the free energy of a small volume dr of solution by the function U(ui,...,un;vu15...,Vun)dr. This constitutes a natural extension of the Cahn and Hilliard treatment of binary

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1°

systems^ but limited, here, to the n composition variables u t and their first spacial derivatives yu.l. let us expand the function U about its value UQ at zero gradients.

By definition, we have

U0(Ui,...,un) = U(ux,...,un;0,...,0) = f(u1,...,un) + Vf(u^,...,un),

where, in analogy with the results of the previous section, f represents the Helmholtz free energy and V/ the coherent strain energy [which must be interpreted as explained in connection with Eq.. (^)) :

W(ui3...,un) = i

-

with

^

= (l/aKSa/Bu^^^o .

The Taylor’s expansion, limited to second order terms, can be written, even in the case of dependent variables (see Appendix A) :

,... ,un;vu -l ,«•«, Vun) = UQ + ^ ^

(°/3vu j)J UQ

J

+ ( 1 / 2 ) [ £ w j (9/dVUj)Ju0 .

(13)

J

By using the same argument as that used by Cahn and Hilliard'*, it can be shown that the linear terras in Eq .• (13) must vanish.

By analogy

i

with Eq. (3)j we can now express the total free energy F of a volume V of solid solution as

F = f U(u1,... ,un;V ^ , ..., vun)dr "V or as

F = J {U0 + (l/2)[ I v

U0) dr .

i

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11

Consider small variations in composition 6ut giving rise to a variation in free energy of

6F = f 6U dr . JV

The variations 6u t are not independent and Courant and Hillert^ consider such a variational problem with the constraint

G ( V * ’un) s E U J = 0 J

as required by Eq. (ll).

The variations 6ut can now be regarded as

independent by introducing a Lagrangian multiplier l(r) and by defining the auxiliary function

U* = U + XG .

We may now write

6F

jy

6Us a r =

f

) cpi6u,dr

Jv L

with

cp* =

(S U ^ /c U j) -

^ ( S U ^ '/S N T U i)

•

From the equation

cp* =

6 U */ dUj

* we see that the cpt can be regarded as potential functions.

%=

(au/aui) - tCau/avui) + x(aG/aut)

or

9* = cpi+ X

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We have

12

if we define

cp,. = ( s u / a u j - vCsu/aWi)

(i*0

ana since I 3••*n) •

dGr/aU| — 1

For simplicity, we shall now assume that the parameters of the function U are position and time independent.

Then, by performing the differen­

tiations indicated by Eq. (l^+) we obtain

CPl = af/aui + 2Y; TMIijTMI Jj‘U -j J - 2Y hu V2Uj i i

(15)

where we have defined

Mlj = (i/2)(a3uo/avu1avuj) •

(16)

16 In general, the flux of component k is given by

Jjj =

V

Mjjj Vcj>i

(k=l,... ,n) .

1 Conservation of matter requires that

k

1 k

Since this equation must hold identically, we have

J Mjjj = 0

(i=l,... ,n) .

(IT)

k

The Onsager relations further limit the number of independent mobilities M kl since

M k l = M Ilc.

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(18)

13

By making use of Eqs. (17) and (l8) we may now write n-l flux equations : n-1

I k = - ^ M klv(cpt-^) 1=1 (k.”13 *•*>n-l)

oi* n-l

j l k = - ^ Mk l v(cpt - cpn) .

(1 9 )

1=1 The second of these expressions will be used since it does not contain the multiplier \.

It is useful to express the differences cpt - cpn in

terms of n-l independent composition variations ut and gradients W j • Let us eliminate un and vun by using the equations n-l

v = “ Zj u j J= i and n-l =

-

Y,

•

1=1 We then obtain

“-1

n-i

kj= *ji by virtue of Eqs.(22) and (l6).

In general, the matrix D will not

be symmetric,however; but we shall show that its eigenvalues

are

nevertheless real, thereby ruling out unacceptable oscillatory solutions. The proof rests on a theorem given at the end of Appendix D : if a matrix, say D, can be expressed as a product of a symmetric matrix and a positive definite matrix (whose eigenvalues are all positive), then the eigenvalues of D are all real.

We shall now show that one

can reasonably expect the mobility matrix M to be positive definite. Indeed, if its off-diagonal elements were negligible (as is often assumed in practice), then the eigenvalues M kk would all be positive by definition of the mobilities.

If M were not diagonal, a diagonalizing

operator could be found which would convert the system (19) into a set of independent flux equations, with the eigenvalues of M as proportionality factors.

It is reasonable to assume that these constants

are all positive and that M is thereby positive definite.

The theorem

of Appendix D then states that the eigenvalues a'k of D are those of the symmetric matrix

S = M ^ G M^ 3

being the real positive square root matrix of M defined by

mV

*

= M . ‘

’

If the number n of components or chemical species is not

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27

too large, 3 or k typically, the linear system (*tl) can be solved exactly by evaluating the eigenvalues and eigenvectors of D (or S). Let E, a matrix composed of the eigenvectors of D, be an orthogonal operator which, by definition, diagonalizes D.

In Appendix D it is

shown how the substitutions

A = EA ,

A 0 = EA° ,

E 'BE = B

(^3)

transform the system (hi) into a diagonal one

A = BA

A 0 given

,

whose explicit solution is [Eq.(A 13)3 : A k(t) = A k exp(akt)

(k = l,...,n-l) ,

where the Gk are the eigenvalues of the matrix D. From our knowledge of the vector A for each triplet of integers K, we can obtain A by the first of Eqs.(h3) and then introduce the components A k (actually A ^ s since all of these calculations refer to the Fourier space direction defined by the triplet of integers K) into the Fourier expansion for uk(r,t) given by Eq.(37)»

The system

of diffusion equations (36) is thus formally solved, but, before leaving the subject, let us try to explain the role of the "normal" amplitudes A K in multicomponent spinodal decomposition. We must first define the limit of stability for multicomponent systems.

Let u be an (n-l)-component vector in composition space,

for example the vector u. u = s,

in the ABC composition triangle of a ternary system. in free energy

The difference

between a solution containing a given infinitesimal

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composition fluctuation u and the uniform solution (u=0) is given by

“

! [ 5 > i (3/SuJ)I

f °

i

+

[T v

J

y

•

i

This is just the n-component extension of the intercept rule, with the coherency strain energy term included.

If we rewrite £f in terras

of n-l independent composition variables wre obtain the quadratic form n-l n-l

Q = 2Af = I I (fU + 2TliV ) uiu.j 1=1j=l

.

or, in matrix notation,

Q = u 'G°u

u ' being the row vector transpose of u.

Notice that G° is just the

matrix of the elements Gt 3 of Eq.(^O) but with jS2 set equal to zero; hence the use of the above notation.

The limit of metastability is

defined as the locus of points in the phase diagram which separates t the regions of absolute stability from the metastable ones.

The

region of absolute stability, in turn, is defined as that for which an arbitrary composition fluctuation u imparts to Q (or df) a positive sign.

In this region, Q, is a positive definite quadratic form and

its matrix G°, whose eigenvalues will all be positive, is also said to be positive definite.

The solid solution will be absolutely unstable

in a region where 0, is negative for arbitrary u.

The quadratic form o and its matrix are then negative definite and all eigenvalues of G are negative.

These two regions are separated by intermediate ones

where the solid solution is unstable with respect to certain composition fluctuations only.

In these intermediate regions, Q (or G°) is

indefinite, some eigenvalues of G° being positive, others negative. t In discussing stability in this Section, we shall always assume that the fluctuations referred to are of infinitesimal amplitude and of

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29

Any tvo regions are separated by a surface (in n-dimensional space) over which one of the eigenvalues of G° vanishes, that particular eigenvalue having opposite signs on either side of.the surface.

Let

us Call these surfaces the "spinodal surfaces" of the n-component system.

The limit of metastability can then be redefined as the

uppermost spinodal surface, i.e. the one corresponding to the highest temperature.

In general, the spinodal will consist of n-l surfaces,

each surface corresponding to the vanishing of one of the eigenvalues of the (n-l)x(n-l) matrix G°. We shall now give surfaces.

an equivalent definition, of the spinodal

Consider what we shall call a "normal" sinusoidal composition

fluctuation in the reciprocal space direction J K, related to the eigenvalue c/k and defined by

uk(r,t) =

exp(cvkt + ij-r) ,

where, again, the subscript K is omitted.

(U5)

Consider the locus of

points in the phase diagram for which the fluctuation uk of very large wavelength

but of infinitesimal amplitude just becomes unstable.

This locus is then defined mathematically by the equation :

lim c/v = 0 . P-0 Now the eigenvalue ok of D can vanish only if the determinant of D, Jd|, vanishes.

Thus, by Eq.(h2), we seek the condition

lim |D| = - £.2 lim |m |* |g| = 0 |5-*0 p-»0 that is to say

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30

since

is strictly positive.

We therefore recover the previous

definition of the spinodal surfaces since the vanishing of |G°| implies the vanishing of at least one of the eigenvalues of G°.

The determinental

equation |G° j = 0 then constitutes the most compact mathematical definition of the spinodal.

In particular, at the limit of metastability (the

uppermost spinodal surface), all eigenvalues |D | is proportional to — |G ] and if the

of D are negative (since

j parameters constitute a

positive definite matrix) but one, say Qi, which just becomes positive, enabling the corresponding normal fluctuation ux (of infinite wavelength) to grow in amplitude, according to Eq.(^5).

Spinodal decomposition

can thus occur at or close to the limit of metastability only in the composition-space direction [see the first Eq.(U3)l

u = Eu = (El5...,En )•

= u1E 1 ,

\w l Ejj being the eigenvector of D corresponding to uk. The unstable sine wave (of infinitesimal amplitude and very large wavelength) can thus be represented by a vector whose components on the concentration axes are proportional to the components of the eigenvector E ^ The definition of the limit of metastability given above differs from that of Gibbs strain energy term

20

n-

only by the addition of the coherent

1

J=1 Since this expression is alwrays positive (or zero), its presence in Eq.(^t) always tends to make the quadratic form more positive, that is to say, the coherent strain energy tends to stabilize the solid solution with respect to small composition fluctuations.

In other

words, the "coherent" limit of metastability, defined by the uppermost surface Jg ° | = 0 must lie below the uppermost surface |F | = 0 defining R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.

31

the "chemical" spinodal, F being a matrix whose elements are the f t i of Eq.(35). It would appear, at first sight, that spinodal decomposition could occur without strain in a multicomponent system whenever a fluctuation u were found which would satisfy the following equation : n-1

Tfu = ^ TljUj = 0 .

0*6)

J=i Actually, since spinodal decomposition can only occur at the limit of metastability along the composition-space direction u 15 Eq.(^6 ) merely expresses a condition on T] which, in general, will not be satisfied by an arbitrary set of n-1 strain parameters T|j.

However,

at the points in the phase diagram where two uppermost chemical spinodal surfaces happen to intersect, then spinodal decomposition can indeed occur without strain regardless of the values of the Tlj parameters. This can be shown as follows : if two chemical spinodal surfaces intersect, two eigenvalues of F vanish simultaneously and two normal fluctuations, say uj and u3, become unstable. N u = • \oi

But then all fluctuations

lo\ + bp?a \0

,

lying in the composition plane defined by (ux, u2) are also unstable and one of these u can always be found which will also satisfy Eq.(^6 ) for arbitrary T)j1s. In other words, coherent spinodal surfaces must always pass through points where two or more chemical spinodal surfaces intersect.

A more rigorous proof of this property is given in Appendix E

where it is also shown that the determinants of the matrices G° and F are related by the equation :

|g° | = |f| + 2(t/ f *n)y

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0*7)

32

in which F is a matrix whose elements are the minors (or cofactors) of P and T|' is the transpose of the column vector T] of components T)j. These concepts will now he illustrated for the case of a 3-component regular solution model.

Rrigogine

21

defines the free

energy (per atom) of such a system by

£ (c i ,c 2,c3;T) = a^ciC;} + a^3c2c3 + UA-jCgCj + kT(c1log ^

+ c2log c2

+ c3log c3) .

In this equation, the constants a'* j = tojt are interaction parameters, k is Boltzmann's constant anu T the absolute temperature.

The quantities

fjj of Eq.(35) then take on the explicit expressions (with n=3) :

flX =(kT/cJ + (kT/Cg) - 2w31 f l2

= f si = (kT/cg ) + u)12 -

= (kT/c ) + (kT/c3) - 2

the

spinodal will consist of two surfaces, as expected, since n-1 = 2 in the present case.

These surfaces intersect the binary diagram (l,2),

defined by c3 = 0, along the curves

T = 0

and

2u) - (kT/Clca) = 0

the second of these equations giving the well-known parabolic spinodal of a binary regular solution.

Let us now examine the median vertical

section CD defined by

The spinodal surfaces intersect section CD along the two curves

T = c

and

T = 3c(l - 2c) .

These curves are plotted in Fig. 1.

They intersect at point c = l/3,

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+ De f .

Fig. 1. Vertical section CD through the spinodal surfaces for a ternary regular solution model. The full curves represent the chemical spinodal, the broken curves represent the coherent spinodal.

1/3 3 h ■* 1/4 I t-

I ndef .

I ndef . /

-Def.

0

1/8

1/4

1/3

1/2

+ ,Def>

Fig. 2. Isothermal section through the chemical spinodal surfaces for a ternary regular solution model at kT/w = l/U. Spinodal decomposition can occur for composition fluctuations inside the dark areas, shown at selected points along CD.

-.D e f.

I ►k)b

In d e f.

3^

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35

t

= 1/3 and define three regions : an upper (or outside) region where

the solution is stable to all fluctuations^(F is positive definite), a lower (or inside) region where the solution is unstable to all fluctuations (F is negative definite), and an intermediate region where the solution is unstable to certain fluctuations only (F is indefinite).

In this intermediate region, the quadratic form

Q(ul5u2) = flxu I + 2 £ ^ y x a + f32u2

is also indefinite and we can define two "asymptotic directions"

22

(in composition space) which will make Q, = 0 :

V u2 = [“fi2 ±

•

These two asymptotic directions are real since [F | (the determinant of the matrix F) is negative in the indefinite region.

In Fig. 2,

we have plotted some of these pairs of asymptotic directions along the section CD and for a value of

t

equal to 1/^.

cq = 1/8 we find that u^/Ug = 2 ± yields

J

v(M^cp) exp(-ijJ*r) dr =

ijJ

Mvcp exp(-i§*r) cb_ .

(53)

The surface integral vanishes because of the periodic boundary conditions, Eq.(lO).

We now seek an explicit expression for the

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function Mvcp.

Consider cp as a function of u(r,t) and ^u.

Its

gradient will then be given by

Vcf = (Bcj/ou)tu + (dcf/o^ujv^u

.

Therefore, by Eq.(9)»

MVcp = D(u)w - 2^M(u)v3u ,

(5l+)

in which the composition dependent coefficient D(u) is given by

D(u) = M(u)[f"(u) + 2T|2Y] .

(55)

Since, according to Sect. 3*1, the free energy f(u) has been approximated by a fourth degree polynomial, its second derivative f"(u) must be quadratic in u.

To keep the algebra treatable, we

shall also express D(u) as a quadratic in u, which implies either (a) that the mobility must be regarded as a constant M , or (b) that the product M(u)f"(u) must be Taylor-expanded about u=0 up to terms quadratic in u.

The computer programs to be described were set up

in such a way as to accomodate either alternative; one merely has to define differently the constants Mj and Dj appearing in the following quadratic expressions :

M(u) = M 0 + MjU + M 2u 2

(56)

D(u) = D 0 + D xu + D 2u 2 .

(57)

Explicit expressions for the coefficients under procedure (a) or (b) are given in Appendix G along with reasons for preferring one alternative over the other. If we substitute for D and M the quadratic expressions

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h6

(56) and (57) into MVcp [Eq.(5^)j and insert this explicit expression into Eq.(53)> we obtain, under the integral sign, terms of three different types, namely, linear, quadratic and cubic in the function u.

But, in order to use the Fourier transform method of Sect. 2,

which tolerates only linear terms, we must, at this stage, use a linearization process which is somewhat arbitrary.

We shall adopt

the following procedure : let us assume that at some time t0 a solution u0 = u(r,t0) is known (the initial condition for example). Let us then pick a time interval At such that the solution u(r,t) at a later time t = t0+At does not differ appreciably from u0. Then within the time interval At, and to this degree of approximation, we may write u0 for u in the expressions for D and M to obtain, for the integral of Eq.(53) :

i§| Mvcp exp(-ij-r) dr = ijf (Do + Dxu0 + Dsu^)vu exp(-if-r) dr V JV - 2ijnf (M0 + Miu0 + M u®)v*u exp(-ij*r) dr . JV

(58)

Next, in order to eliminate the space dependence of the quantities in parentheses in the above expression, we expand the known solution uQ in a Fourier series :

UQ(r) =

exp(ij'-r) K'

which then implies

uq

= X Z A °'

*

k' k "

In these expressions

and

stand for _§K/ and

respectively.

Let us insert these values of u0 and u^ in Eq.(58) and then apply

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^7

Green’s theorem the required number of times, noting the vanishing of the surface integrals because of the periodic boundary conditions on u, t3u, gu/gn and g^u/gn.

After grouping terms we obtain the

three expressions :

-p2(D0+2vM0f>2)J u(r,t) exp(-ij*r) dr ,

These three terms, which we shall label linear, quadratic and cubic, respectively, replace the first integral to the right of the equal sign in Eq.(52).

Let us define new subscripts by the (triplet

of integers) substitutions : in the quadraticterms : implying

L = K - K ', J3L= J - J ',

in the cubic terms

:

L = K - K '- K" ,

implying

J L= J -

- J"

•

We may now use Eqs.(26) and (28) to obtain the system of ordinary linear differential equations :

A = BA + Q, ,

A° given

with the vectors A, A 0, A and Q, defined as in Sect. 2.1.

(59)

The

matrix B has the elements

b KL = _§k '( . s £ A°_l

^KK = Jk "

+

^V l )

j

K,- 5^ L

+ .2* A0.)

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with the vectors a* given by

d, =

(i = 0,1,2)

(61)

and the scalars Aj given by

(62)

Since the matrix B depends on the current values of the amplitudes A°, solving system (59) can only yield an "instantaneous" solution, valid for a small time interval At.

Varga's matrix iterative

method is thus ideally suited for the present problem since, at each iteration, the matrix elements will have to be calculated anew.

By using Eq.(A l3) of Appendix D and Eq.(6o) we thus obtain

the explicit instantaneous solution for the amplitudes A K(tQ + At) as a function of the known amplitudes A° :

•^k ~ -^k + At]?-*• j>{< A° + ^ (c^ A k_l + af A k_l)A° J + AtQK . L

(63)

Equation (63) is the basic equation around which the computer programs are constructed. A new vector A (at time t0 + At) is calculated from an old A 0 (at time tQ).

The new vector A is then

labeled A° and substituted back into Eq.(63) to yield the vector A at time tQ + 2At.

This procedure is repeated m times until a

desired later time t = t0 + mAt is attained.

With non-linear

equations, it is difficult to derive a general stability criterion for At^.

Therefore, a trial and error method was used : first

a At which satisfied the linear equation criterion of Sect. 2.1 was selected, then, if the solution A did not remain bounded, a smaller value of At was tried and the computer program rerun until

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a stable solution was obtained.

Notice that Eq.(63) does not require

that the matrix B be stored in the computer.

This results in a

considerable storage saving, especially if the number of triplets of integers K is large.

The number of significant Fourier components

retained will-depend on the convergence of the series and on the size of the domain.

Although Eq.(63) appears rather innocent,

the number of algebraic operations which it implies can be staggering. Suppose we choose n Fourier components in each - t—

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OO

6l

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62 and

Y = 1.20 x 1012 dynes/cm2.

These values, inserted in Eq.(72), yielded the required fourth degree polynomial coherent free energies which are plotted as a function of composition (atomic fraction Zn) in Fig. 7*

By applying

the common tangent rule to these free energies one obtains, for the corresponding temperatures, (coherent) equilibrium compositions c and c which were used in constructing the coherent miscibility o

p

gap of Fig. 5.

Likewise, by equating the second derivatives of these

polynomials to zero and by solving the resulting quadratic equation one obtains the coherent spinodal compositions which yield the coherent spinodal of Fig. 5. The tracer diffusivities D* were calculated, according to the formula

D* = D exp(-Q/RT)

where R is the gas constant and T is the absolute temperature, The coefficient D and the activation energy Q for Zinc in an Al-Zn alloy containing 22.5 at. pet. Zn were taken from the work of 31 Hilliard, Averbach and Cohen :

D = 0.1 cm2/sec Q = 24.7 kcal/mole .

For simplicity, these values were used throughout the phase diagram. By comparison, the corresponding values of Aluminum are vanishingly small and need not be taken into account in Eq.(7). We now have enough data to compute the parameters M t

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and Dj of Eqs.(5&) and (57)*

In order to perform kinetic calculations

according to the fundamental Equation (63), we need one more parameter : the gradient energy coefficient h«

The following value was obtained

from small angle scattering data on a 22.5 at. pet. Zn by a method described by

Rundman^

h

;

- s

= 1.050 x 10

erg/cm

and it was assumed to be independent of composition and temperature. This value of h was later revised along with some free energy data. The earlier values will be used here, however, since we are seeking consistency rather than strict fidelity to a physical system.

The

calculations referred to in this Section are performed by the computer program THERMO described in Appendix H.

3A. Isothermal Aging of One-Dimensional Composition Variations Consider a one-dimensional (or plane-wave) composition variation in the Fourier space direction K in an imperfection-free cubic crystal of a binary solid solution.

We will suppose that the fluctuation is

along the x axis in direct space.

We can now reduce the problem to

a one-dimensional one by the substitution

r

-*

X

K

k

(k - 0,±1, ±2,

_§K 1 & -

1 «k

(i = 0,1,2)

provided the source q has a plane wave expansion in the same direction K only.

The function u(x,t) then denotes the average composition

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6H

variation in the plane of the fluctuation.

Equations (66) and (67)

then indicate that the composition variation will remain one-dimensional when the Fourier component interactions take place.

If 2X denotes

the length of the domain, a Fourier component of wave index k will he associated with a wave number

Pk = ku/X

and a wave length

Xk = 2X/k .

Let us denote the complex amplitudes by the letter C rather than the letter A used in previous sections.

An arbitrary composition

variation will be given by CO u(x,t) = V Ck exp(iPkx ) .

(72)

k=-co

For computer calculations, it is advantageous to use positive integers k only.

We can easily eliminate the negative indices by the relation

c-k = ck

>

C0 = 0 + iO

the star denoting the complex conjugate.

(73) Notice also that

Bk = " and = -ajj

(i = 0,1,2) .

Let us impose a high-frequency (short wavelength) cutoff n such that

C_n = 0 + iO

for

m > n .

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Under these conditions the one-dimensional version of Eq.(63) is :

Cjc =

+ Atpk^(o^ + of C 0)C° - (of

+ of C i){)Ck

n

•*.

+ S(of

C°_h +

of

C k_h)C° -

(of

C°k4h +

of c kth) c f ]}

h=l

(7*+)

(k=l,...,n)

with n

^ 3 = 1 (c° c°-i + ci* C°I) i=i

(J =

(75)

(2n quantities Cj must be considered in Eq.(7^) in order to obtain better convergence). If, in Eq.(7*+), k-h were negative, the corresponding complex amplitudes would have to be replaced by C°*k and cf_k since, by Eq.(75),

c3 = c3 . Equation (7^-) enables us to compute explicitly

the nunknown

amplitudes C k at time t = t0 + At as a function of theknown

C k at

time t0. By performing m iterations, we can obtain the components Ck for any required later time t0 + m£t.

At any stage, the composition

variation profile may be obtained from Eq.(72) or, by restricting ourselves to k>0 , from the equation n

u(x,t) = 2 £ (Ak cospkx - B k sin|3kx )

(76)

k =l

if we set

Ck = A k + iBk .

(77)

The calculations corresponding to Eqs.(7^-)> (75) and (76)

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c o Ui

Fig. 8.

0 01

Composition profile used in calculating the normalization factor ct0 of

E

o

C0

o

Eq.(78).

x

0

2\

Distance

■o —o

,-2

Fig. 9-

(A

C

40

>io

jjo

>40

400

Wlr«4*A|(ft (ll

Fig. 1^.

Amplitude spectra (absolute magnitude) and corresponding compo­ sition profiles calculated according to the linear diffusion equation for a 0.225 Al-Zn alloy aged at 100°C in a itOO A domain with periodic boundary conditions and same initial conditions as in Figs. 11 and 12. The broken horizontal lines indicate the coherent equilibrium compositions. (Computer-generated plots)

78 R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.

79

at comparable values of the percent completion [Eq.(78)3j are shown in Figs. 11, 12 and 13 within one periodic domain.

The domains are

drawn to the same scale in order to show the variation in average wavelength Xm across the coherent phase diagram (see Table l).

The

vertical scales of the amplitude plots are likewise identical.

The

computer program which generated these plots is described in Appendix K. Figures 11 through 13 clearly show the influence of the non-linear terms of the diffusion equation; the same initial fluctuation, at 100°C and with c0 = 0.225, is shown in Fig. 1^ evolving according to the exponential solution [Eqs.(6U) and (65)3 of the linear diffusion equation^.

The fourth plot, corresponding to t = 2000 sec, could

not be included since, according to the linear treatment, the amplitude of the composition profile would be of the order of several thousand percent.

As expected, non-linearities are, initially, least apparent

in the case of the nearly symmetric 0.375 alloy whose third derivative fo is relatively small (Table l).

The composition profile (Fig. 13)

is a quasi-symmetric sine wave of modulated amplitude and the associated spectrum envelope consists mainly of one broad diffuse peak (firstorder satellite) whose maximum is close to that predicted by the maximum amplification factor o?,, of the linear equation.

A large

portion of the spectrum was included beyond (ic [Eq.(33)3 in order to show the very weak third-order satellite.

The second-order

satellite, also located beyond Pc, is practically absent.

An identical

computer calculation was performed on this alloy with a spectrum which was terminated before the third-order satellite and composition profiles were obtained which differed from the ones shown in Fig.13 by only one or two percent.

In both cases, the final solutions were

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80

very stable to subsequent aging, which seems to indicate that the second harmonic (third-order satellite) does not play an essential role in stabilizing the composition variation at the coherent phase boundaries; interactions within the fundamental itself (first-order satellite) appear to perform this function. As the average composition approaches the coherent spinodal (c ' = 0.175, Fig. 5)s quadratic interactions become predominant, the amplitude spectra become very diffuse on the short wavelength side, the composition profiles become increasingly asymmetric and one-dimensional fluctuations of the same form as Guinier zones make their appearance at a very early stage (compare, for example, Figs. 11 and 1^ at time t = 600 sec).

The average distance between subsidiary

maxima of the zone profiles is consistent with the value of \n determined from the linear equation whereas the spacing between zones depends on the particular initial condition chosen and is therefore random.

The evolution of the zones depends on the average composition

of the solid solution with respect to the coherent spinodal.

In the

case of the 0.225-alloy (Fig. ll), the zones are unstable and subsidiary maxima develop into full-fledged precipitates by growing away from the center of the zones into untransformed solid solution.

Since

the average wavelength of the fluctuation thus tends to increase, one observes a corresponding shift of the first-order satellite envelope towards the longer wavelengths. A first metastable structure appears : regularly spaced precipitates of composition Cp in a matrix of composition c&. In the case of the 0.200 alloy (Fig. 12), the subsidiary Guinier zone maxima do not grow to the equilibrium phase boundaries but quickly dissolve to yield a metastable structure

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tt .M • r

M # I fc . c t m c t i o - f« .i»

1 l

A

I

J.HO • 10'

S£C

K . COflCUOH • 19.SB

A i 5

3. so . io'

sec

fc . c t m c n w • w.19

•

| .| — X, l I I I | I I I 1 > «...» , N (/) c a> c

2.0

0.5

2.5

/3 (c m "') (* I07 ) Fig. 19 c. Continuation of Fig. 19a and b : calculated intensity spectra for the final aging treatment at 100°C (inside the spinodal) of a 0.200 Al-Zn alloy. The critical (lc) and optimum (Xn) wavelengths are indicated by arrows.

89

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90

of large amplitude, we can still distinguish the three stages of spinodal decomposition, growth and coarsening although the spinodal decomposition stage is no more than a short transient due to the rapid breakdown of the solution of the linear diffusion equation (see the upper curves of Fig. l6, for example). Figures 19 a 3 b and c show the effect of aging outside the spinodal on the intensity spectrum’s harmonics.

Again an 800 A

domain was used with insulating boundary conditions.

A large "quenched-

in" initial condition was used with plus signs assigned to the amplitudes A°.

A 0.200 alloy was aged first at 100°C (inside the

spinodal) for 1000 sec, then at 200°C (outside the spinodal, but inside the miscibility gap; see Fig. 5) for 1 sec, then again at 100°C for an additional 800 sec.

Note that the final spectrum in

each figure is the initial one in the next.

The curves of Fig. 19 a

are very similar to those of Fig. 17 with, however, a less pronounced second-order satellite.

The third-order satellite is totally absent

but the fourth-order one, not visible in Fig. 19 a, can be clearly detected in the corresponding amplitude spectrum.

The spectrum maximum

again shifts to a value well inside the value of j5m corresponding to the optimum wavelength \n. When the solid solution is aged outside the spinodal (Fig. 19 b), a sharp drop in integrated intensity occurs since the amplitude of the composition variation must decrease in order to conform to the new coherent equilibrium phase boundaries. Subsequently, the short wavelength components continue to decrease in amplitude while the long wavelength components increase slowly. After one second at 200°C, the higher-order satellites have completely disappeared.

A final aging inside the spinodal (Fig. 19 c) produces

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• ll 0.00 • i o '

src

r c . c im E H O N • m . m

X £

9

■t

9 .0 0 .

10*

SCC

fC . C tm e H O N . 55.39

i ‘ “ I

9 J

i

1111111

,,

1. 0 0 .

io 1

sec

r e . c t m e n o H • s3.sb

9

9

LUJ.LU.,USO. 10*

SCC

re . c a n c n o n . s t.w

^ | i| I i| I i . i . i I I mii.ll m m m

mm

Fig. 20.

Wmli*rhttl

Amplitude spectra (absolute magnitude) and corresponding compo­ sition profiles calculated according to Eq.(7*0 for a O.lOO Al-Zn alloy aged at 100°C (outside the spinodal) in a 600 A domain with periodic boundary conditions. Three one-dimensional Guinier zones constitute the initial condition (at time zero). The broken hori­ zontal lines indicate the equilibrium compositions. (Computer­ generated plots) 91

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92

an increase in the integrated intensity without noticeable change in the peak position and the fourth-order satellite, not visible in the intensity plot, reappears. Finally, Fig. 20 shows the evolution of three Guinier zones in a 0.100 alloy aged at 100°C, outside the spinodal. was used with periodic boundary conditions.

A 600 A domain

As expected, the two small

zones tend to dissolve quickly while the larger zone reaches an equilibrium shape.

The final spectrum shows an amplitude |Ck | decreasing mono-

tonically with increasing wave index k except very near the origin of reciprocal space where a faint maximum is observed.

Analogous

35 experimental spectra have been reported by Bonfiglioli and Levelut In another computer calculation, a single supercritical precipitate was aged under the same conditions in a ^-00 A domain with insulating boundary conditions.

As expected, the composition profile evolved

towards a very stable precipitate of composition c

~ .66k in a matrix P

of composition c

= .0^3* Again the spectrum intensity decreased o: monotonically with increasing wave index k.

3.5. One-Dimensional Simulation of Continuous Cooling

In a continuous cooling experiment, a solid solution is quenched from a high temperature into the spinodal according to a specified cooling law

T = T(t) ,

T being the absolute temperature and t the time.

(81+)

In order to simulate

such an experiment a diffusion equation with temperature-dependent

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9)

k.

3 4> cl

E

v h-

h

Time

Fig. 21.

Hypothetical cooling curve from the solution-treatment temperature Tj_ to the quenching bath temperature Tf. The temperature T0 is located just below the spinodal. Inter­ mediate temperatures indicate assumed ranges of validity of free energy curves.

200

in

(60

i -n iir w 'ff

2 pet. ,65pct.

o O „ 120 fa.

3 d

+■*

I E

80

01 I40

2x10

,"4

.-3

-2

T i m e (sec.)

Fig. 22.

Exponential cooling curves calculated according to Eq.(85) (solid curves). The quenching rates were: (l) 103 °C/sec, (2) 5 X 103 °C/sec, (3) 104 °C/sec, (It-) 3 X 104 °C/sec, (5) 6 x 104 °C/sec, (6) 10s °C/sec. The broken lines are rough sketches of 2, 10 and 65 percent completion curves for a 0.225 Al-Zn alloy. 93

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9*+

parameters and with idealized initial conditions was used.

Consider,

for example, the hypothetical cooling curve of Fig. 21 : Tf is the solution-treatment temperature, TQ is a temperature just below the spinodal and Tf is the temperature of the quenching bath.

Since it

is very difficult to evaluate statistical composition fluctuations just above the spinodal where a higher-order extension of the Landau and Lifshitz^ treatment is required, the upper part of the cooling curve (Tf -» T0 : broken curve of Fig. 2l) was replaced by a step function (solid curve).

In other words, the fluctuations at Tf, calculated

according to Eq.(82), were used as initial conditions at time tQ = 0 (corresponding to the temperature TQ). The decomposition of the solid solution along the lower part (Tq -> Tf) of the cooling curve, Eq.(81+), was obtained by the repeated use of Eq.(8o) whose parameters iteration.

were altered at each

For a usual drop in temperature Tf -* Tf, the mobility M

can be expected to vary over several orders of magnitude; accordingly, a variable time step At was adopted.

The free energy parameters will

vary more slowly, however, and, as an approximation, their values f

were held constant over fairly wide temperature ranges.

The temperature

intervals over which a single free energy curve was used are indicated by horizontal lines in Fig. 21.

To complete the simulation, an

exponential decay of the high-temperature equilibrium concentration of vacancies was also incorporated. Let us now formulate the problem analytically.

The following

exponential law for the cooling curve below TQ was adopted :

T = Tf + (T0 - Tf ) exp(-bt)

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(85)

95

where

b = -[(dT/dt)t=0]/(To - Tf) .

It was assumed that the total activation energy for diffusion, Q, is the sum of two contributions : an activation energy of motion and an activation energy of formation of vacancies

Q = Qjn + Qf • We are, of course, considering a binary system where diffusion occurs primarily by the interchange of vacancies with one of the atomic species; in the present case :Z n in an Al-Zn solid solution.

It

was also assumed, as in the previous sections, that the mobility M does not vary appreciably with composition so that

M s: MQ .

Corresponding to the two activation energies Qq and % , there are two mobilities

and Mf such that

M q = MmMf .

(36)

From Eq. (7), the first of these"mobilities"is given by

^

= [c0(l-c0)/%kT] D|n (l-c0)

with

DZn = DZn expC-Q^RT) . The second mobility M£ contains the exponential vacancy decay law :

Mj = Ms + Mj. exp (-at)

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96

in which Mg is the steady-state term (at t = ■

co

1.1 X io_1

l

57-0

58

3 x 104

>. 103

0

Ik.2

57

6 x 104

> 103

0

6.7

55

105

> 103

0

5.0

5H

3 x 104

5.5 X 10'3

192

69.3

88

’

Table 2 summarizes the results obtained with six different quenching rates (dT/dt)o, but with identical initial conditions obtained from the Landau and Lifshitz formula with a random distribution of

0

positive and negative amplitudes A k. In each case the computation was continued until either 65 pet. completion had been attained (i.e. almost complete decomposition) or a decomposition time of 103 sec

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93

had elapsed (at this stage the mobility is so low that decomposition can no longer proceed at an appreciable rate).

Table 2 gives the

values of time, temperature, pet. completion and approximate average wavelength XD at which a stable structure was reached, according to the two criteria just, stated.

The corresponding continuous cooling

diagram is shown in Fig. 22; the broken lines are rough sketches of the completion curves.

It is apparent that quench rates slower than

about 104 °C/sec yield total decomposition whereas faster ones yield partial decomposition.

Quench rates faster than about 5 X 104 °C/sec

essentially retain the untransformed solid solution.

The experimental

quenching rates obtained by Rundman^ were in the range of 104 to 5 X 104 °C/sec and partial decomposition was observed with a pet. completion of about 15 to 25 and an average wavelength of hO to 50 A. For both the experimental results and the theoretical calculations, a very broad, diffuse intensity spectrum was obtained.

Since the

agreement between theory and experiment is satisfactory, we can therefore conclude that partial decomposition occurs at very low temperature by a non-equilibrium vacancy assisted diffusion process. A similar quench on a 0.375 alloy, however, produced'a sharp experimental intensity spectrum with a maximum corresponding approximately to a 100 A average wavelength.

A comparison with the

theoretical results of Table 2 again shows that the agreement is quite good.

For this composition, seven free energy curves were

used : those for 300, 250, 200, 150, 100, 65 and 0°C, the 250°C curve being constructed by interpolation.

The temperature T0 was chosen

equal to 325°C since the coherent spinodal is at approximately 335°C. The extreme temperatures Tf and Tf were, again, lf00 and 0°C respectively.

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The computer program SPICOC, by means of which continuous cooling was simulated, is described in Appendix M.

3.6. Solid Solutions with Plane-Wave Imperfections

In general, the presence of crystalline imperfections will make the solid solution lose its cubic symmetry and the diffusion equation for such a medium becomes extremely complex; for example, the gradient coefficient n becomes a tensor with position-dependent components-

In this section, we shall merely consider a hypothetical

one-dimensional case such that the second derivative of the coherent free energy fc(u) at u = 0 is given by : 00

(83)

For simplicity we shall also assume that the other derivatives of fc(u) at u = 0 are constant parameters.

By choosing an initial

fluctuation u0(x) in the same direction x and a vanishing source term q, the diffusion problem then reduces to a one-dimensional one. We shall not inquire into the -nature of the imperfections which give rise to an analytical expression such as that of Eq.(83) since we merely wish to show mathematically how the position-dependent para­ meters of Sect. 2.1 can be incorporated into the non-linear diffusion equation. By combining the one-dimensional versions of Eqs.(3l) and (60) we obtain, in the present case and with Mi = M 2 = 0, the matrix elements

^kh

^k-h +

^k-h “

^k ^*k-h)

(k T^h)

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,

100

^lck ~ Pk(°? + °k

Pk -^o) *

Therefore, if insulating boundary conditions are used on the periodic domains, we obtain the modified form of Eq.(80) :

A-k = A? + A t p ^ a ® - a]; A°k + G^(A0-A2k) - gk(B0+B 2!c)]A° n

+ Y j t^ ( A fk-hl-A-°*h) + C;h(^|k-hrA k+h) “ Pk (B|k-hl+Bk+h)^A hj h= l #k

(®9)

with

B k = MoF k

(k = l,...,n) .

Because the insulating boundary condition, expressed by the vanishing of the first and second derivatives, must also be respected by the second derivative modulation [Eq.(88)J, we find that

B-k = B k . A computer program based on Eq.(89) is described in Appendix N. Actual computations were performed for the following example : consider a second derivative modulation consisting of the truncated Fourier expansion of a 6 function at x = xQ : n

6(x - xQ) =

cos(knx0/A) cos(knx/l) ,

0 < x0 < X .

k=l If we choose a UOO A domain (\ = 200 A) and truncate the 6 function expansion at n = 10, we obtain the profiles of Fig. 23a and 2l+a respectively for xQ = 56 A and xQ = 150 A.

The Fourier coefficients

of Eq.(88) were

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H

-p

•< • » o c a •» a

i\\J90rttj

d

G o O d o

r— 1 Tf o O d o •H aj h P -p d Jh"H > •H o -a < h d o < D Tf

a \°

p co s d 0) • cj M CJ’TJ 0) H |

.d O

o

m

OJ

(lujtjjad oituoiv) ° 3 - 3

to •H

(ui3/sauAp0|oi) uo!iD|npo^ »a!| d m j 3 q puojas luajaqoo

P


w

100

50

200

150

-20,

50

D istan ce (A)

100

150

200

150

200

D ista n ce (A)

H O ro 40

40

2 0 0 sec.

6 0 sec. c

V

u 20

20

(I

CL

0

c.

1o < uIo u

- 20 ,

50

100 D ista n ce (A)

150

200

-20,

50

100 D ista n ce (A)

Fig. 2b. Coherent derivative modulation (a), and composition profiles (b,c,d) calculated according to Eq.(89) for a 0.225 Al-Zn alloy aged at 100°C in an inperfect UOO I domain with insulating boundary conditions. The initial condition is shown in Fig. 25 a.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

40

40

2 0 sec.

20

20

a.

-20

50

100

150

-20,

200

50

100

ISO

200

150

200

D isto n e e (A)

D istance (A)

H 40

40

2 0 0 sec.

6 0 sec.

S 20

20

u

a.

u

io
10) .

g The value of 2T|2Y was 5-7 x 10 dynes/cm.

In previous Sections, the

value corresponding to the parameters of Sect. 3*3 had been used : 21]2Y = 1.6 x 109 dynes/cm.

Figures 23 and 2k show a sequence of

three composition profiles for a 0.225 Al-Zn alloy aged at 100°C for 200, 600 and 2000 sec.

For comparison, the corresponding profiles

for an "imperfection-free" domain are given in Fig. 25.

Figure 25a.

shows the common initial condition for these three examples (the same initial condition was used, for the movie film).

The presence of

imperfections appears to hasten the coarsening reaction since the final structure consisting of a single broad precipitate is quickly attained in both cases (a more complete sequence of profiles shows, that the small precipitate of Fig.2^d is on the verge of complete dissolution).

Yfe can perhaps analyse these results in the following

way : Fig. 25 shows the growth of two Guinier-type zones in an imperfection-free domain. about 75 and 150 A.

The final precipitates are located at

In the example of Fig. 23, the 75 A zone is

located in a "soft" region of the crystal (corresponding to a trough in the modulation profile of Fig. 23a); this zone will therefore grow preferentially.

In the example of Fig. 2k, the 150 A zone is located

in a "hard" region and therefore quickly dissolves while one of the o

'subsidiary maxima of this zone, located at about 175 A, in a soft region, will tend to grow until, perhaps pushed aside by its mirror image at 225 A (not shown) it will move into a hard region and finally dissolve.-

This explanation is only tentative since the presence

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105

of non-linear terms in the diffusion equation make the qualitative discussion of these phenomena rather hazardous.

3.7. Two-Dimensional Composition Fluctuations

In this section we shall consider two-dimensional composition fluctuations in periodic domains of perfect cubic crystals.

The

general three-dimensional problem can then be reduced to a two-dimensional one by the substitutions :

K

-

(k l5 k a )

P -» ( T r / X ) ( k l 5 k 2 )

.

Since the two-dimensional calculations require considerable computer time, let us use a small square domain with insulating boundary conditions, thereby eliminating the necessity of using the computer's time-consuming c,implex arithmetic mode.

If we assume that the composition fluctuation

/1. /

has two mirror planes along the axes

and x , we need solve the

diffusion equation in one quarter of the domain only.

This restrictive

hypothesis was made in order to save additional computer execution time.

The insulating boundaries and the mirror planes impose the

following conditions on the Fourier cosine series coefficients of u(r,t) :

A

00

= 0

A -k,,-kj = A -k,,k1 = A kl,-ki = A k,,Vct 5 and all the sine coefficients B k k vanish.

We may thus limit the

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106

calculation of amplitudes to one quadrant of reciprocal space in which all indices are positive : A v

1

(k_ ,k„ = 0,1,...,n). 1 2

The reduction

of the general Equation (63) to the positive quadrant of two-dimensional reciprocal space is accomplished as in the one-dimensional case of Eq.(8o).

For simplicity, we shall also assume that the motility does

not vary appreciably with composition (Mx — M2 =r0). then reduces explicitly

Equation (63)

to the following : n

= A ° jkz + £torknkiA ° >k2 - At(n/\)2 £

n

^ { (kih i+k2h a )[Di(Al°rM,|VHi|

h,=0 h,=0 A k,+h|,k^-s-h* ) + ®2^jkrh||, |ki-h2| “ •^k,+h,Jk^hj )1

+ (k1h1-kah2)[D1(Aj^_h(|jki+hj - A^h,, |ki-h2|)

+ D2^Ajlcl-h1j,ki+hz

A k,+h,, |kz-h2|)^J' A ?,,hi »

(9°)

with n

= Z i=o

u

Z

A ° , h ^A|°i-ii|,|h -i:| + A|°,-i. | »Jj+i*

+ A °i+ i., |J;~h|

1=0 + A °,+i,,jj,+h )

(Ji.Ja = 0}lj***j2n)

(91)

and

“ k,,^

=

- 2 ( t t / x)2 (k2+k|)M0 [a2

+ i f Y ^

+ K(rr/l)2 (k2-f-k2 ) ]

(92)

in which a2 is the coefficient of u2 in the polynomial expression of f(u), Eq.(50).

The parameter Y, as defined by Cahn^ is given by :

- i (=11 + 2C i S)

1-

C n + 2C 12

13

- Cl, + 2(20, 4- 0 11+c ; ; j ? J

with

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'93)

107

k = kjkjj/Ocf + k|) .

(9^)

The calculations corresponding to Eqs.(90) to (9^-) were handled by the computer program FYSPIN2D described in Appendix P. Calculations were performed for conditions corresponding to a 0.225 Al-Zn alloy at 100°C.

The only new parameters needed were the three

elastic constants Clx, C12 and C44
38

If we expand the general diffusion Eq.(8) and make use of Eqs.(9)j (55) and (57)> we obtain du/dt = D(u)v2u +

d

'(u )(v u )2 - 2>;M(u)v'iu - 2 vK /(u )v u V3 u

where the primes denote differentiation with respect to u. Equation (95), which is expressed in an invariant form, can be rewritten in spherical polar coordinates with the help of the following operators :

V = S/3r V2 = (l/r2)(s/Sr)(rsa/5r) 't3 = (a/oOv2 v* = v2^

,

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(95)

115

in which r is the radial

coordinate.

The. partial derivatives with

respect to the angular coordinates 9 and cp are not used since we are assuming spherical symmetry : i.e., u(r,t) = u(r,t). In dealing with linear partial differential equations with spherical symmetry, it is usual to make the substitution

u = v/r .

Let us make this substitution for some of the terms in u in the non­ linear Eq.(95) in which the differential operators defined above have been used.

We obtain, after multiplying through by r :

dv/dt = D(u)(d2v/3r2) + D'(u)(3u/&r)(dv/ar) - 2 mM(u )(54v/dr4) - 2*m '(u)(Su/gr)(c)3v/3r3) - uD'(u)(du/cir) (96)

- 2vJ!i/(u) (terms in u) .

The "terms in u" need not be calculated explicitly

since it will

be assumed that the mobility is composition independent : M /(u) = 0. Equation (96) is a linear partial differential equation in the function v(r,t) with a non-vanishing source term

q(r,t) = -u D'(u)(3u/5r) .

Equation (96) is therefore easily linearized by the substitution u -» u q of Sect 3-2, and then solved in a spherical domain of radius R. If the crystal is regarded as a periodic array of quasi-spherical domains of volume (!+/3)ttR3, one must then impose insulating boundary conditions on the polyhedral domain walls

12 13 * . In this case, one

obtains, for the spacial expansion of v(r,t), a complete set of

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116

eigenfunctions with eigenvalues k which are the zeros of the trans­ cendental equation

tan kn = kn .

Such a formulation obviously leads to considerable computational difficulties and was therefore no-t attempted.

Instead, much simpler

boundary conditions were used :

v(R,t) = 0

(97a)

[32v(r,t)/9r2] =0 r=R

(97b)

expressing the fact that, at a distance R from the origin, the uniform solid solution is untransformed.

These boundary conditions do not

insure that the average composition within the spherical domain remains constant since the domain can exchange atoms with the outside region. Besides the conditions expressed by Eqs.(97a) and (97b), there is also the requirement that u be finite at the origin, which implies

v(0) = 0 .

The boundary conditions suggest the use of a Fourier sine series for v : . CO

v(r,t) = r u(r,t) = ^A^(t) sin(knr/R) .

(98)

k =1

Since v is odd about the origin, u must be an even function of r which can then be expressed by a cosine series

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117

w

u(r,t) = VAi(t) r"1 sin(knr/R)

(99a)

oo = Bo(t)/2 + X Bh(t) cos(hrtr/R) .

(99b)

h= l

However, it is more convenient to work with an exponential series, the Fourier coefficients of which are given by :

A k(t) = (1/2)A; = (i/2R) £ v(r,t) exp(-ikrcr/R) dr ~R

(k>0)

(lOO)

and

Bh(t) = (1/2 )B', = (1/2R) £ u(r,t) exp(-ihTir/R) dr —R

(lOl)

or

B h(t) = (l/2R)

f

r_1v(r,t) exp(-ihnr/R) dr .

4

If, in the last equation, v is replaced by its exponential series expansion, derived from Eq.(98), one obtains the following relation between the A k and Bh coefficients : £ Is Bh = (l/R) ) A k r_1 sin[ (Tr/R)(k-h)r3 dr Ic ——00

which can be written

Bh = I A kGk_h

(102)

k = -«

with Jim Gb = (l/R) j x_1 sinx dx . tl0

(103)

Equation (96) can now be solved for a short time interval At by the following procedure.

Let the initial condition u 0 be given

by its spectrum A°; then the corresponding B° spectrum can be obtained

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118

from Eq. (102) and the function u in Eq.(96) can be replaced by

00 (10k)

u0(r) = ^ B ° exp(ihrtr/R) . h-00

Each term of the linear partial differential in the variable v(r,t) is then multiplied by [i/(2R)] exp(-iknr/R) dr and integrated from -R to R.

Integrating by parts the required number of times and noting

that the integrated terms vanish because of the boundary conditions, one finally obtains, by Eq.(lOO), as'in the previous sections, the system of ordinary linear differential equations

A = BA + Q .

(105)

Since the homogeneous part of Eq.(96), linear in v, is formally identical to the linearized equation for u in the one-dimensional case of Sect. the elements of the matrix B are

b kh =

+ a* Bk_h )

b kk = P|e(o£ + 4

(k j#h)

B0)

with CD

These formulae differ from Eq.(6o) only by the addition of the term cj B° which does not vanish since, in general, B0

0 even if the

average composition in the spherical domain is exactly equal to c0. The inhomogeneous, or source, term can be evaluated from the expression [valid for M*(u) = 0] JR — [i/(2R)]

(D1u0 + 2D2u§)(Bu0/Sr) exp(-iknr/R) dr . -R

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119

By making use of Eq.(lOl), one obtains the source term components at t = t

: 00

Q? = X Ph(DiBth+ 2D2B k_h )B°h . : h=» The short-time solution of System (105) can thus beobtained by the matrix iterative technique leading to Eq.(A l8) :

A = A 0 + At(BA° + Q°) .

(A l8)

Let us impose a short wavelength cutoff n and use positive indices k and h only.

The explicit expression of Eq.(A l8) for the amplitudes

at time t = tD + At as a function of the amplitudes at to is then

Aj{ = A° + At Pk{[o° + ok(B° + B°k) + o*(B0 + B 2k)]A° + [DX(B° - B°k) n

+ 2D2(B0 - B 2k)]B°} + At £ {pk[oi(B1 o*|+ B°k+h) + 0 ® ^ +

Bkth)]A°

h=l

+

B°^) + 2D3(b [m - B k+h)]B°}

(k = l,...,n) (106)

with n

B J = B0 B° + I B?(Bi°-ll+ B J+i)

(j = l,...,2n) .

(107)

(h = 1,...,2n)

(108)

1=1

Likewise, Eq.(102) becomes n

K

= lA°k(Ok_h + Gk+h) k=l

with

G_b = -Gn

(m = 1,... ,3n)

where 3n GB quantities must be used in order to achieve proper convergence.

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120

It can be shown that

lim Bh = 0 h-*“

'

since, by Eq.(l03), and for k :£ n ,

lim Gk+h = -lim Gk_j; = n/(2R) • h-» 525 (l96l).

2.

M. Hillert, D.Sc. Thesis, Massachusetts Institute of Technology, Cambridge (1956).

3.

J. V/. Cahn, Acta Met., 9> 795 (1961).

k.

J. W. Cahn, Acta Met., 10, 179 (1962).

5.

J. W. Cahn, J. Chem. Physics, h2, 93 (1965).

6 . J. W. Cahn, Acta Met., 10, 907 (1962)7.

J. W. Cahn, Acta Met., (in press).

8.

H. E. Cook, D. de Fontaine and J. E. Hilliard, (to be published).

9-

J. W. Cahn and J. E. Hilliard, J. Chem. Physics, 28, 258 (1958).

10.

E. L. Huston, J. W. Cahn and J. E. Hilliard, Acta Met.,

1h, 1053 (1966). 11.

D. de Fontaine, Ch. 2 in Local Atomic Arrangements Studied by X-ray Diffraction, J. E. Hilliard and J. B. Cohen, Ed., Gordon and Breach, New York, (in press).

12.

F. S. Ham, J. Phys. Chem. Solids, 6, 335 (1958).

13.

F. S. Ham, J. Appl. Phys., 30, 1518 (1959).

ll+.

J. W. Cahn (private communication).

15.

R. Cowant and D. Hilbert, Methods of Mathematical Physics, Vol I, p. 219, Interscience Publishers, Inc., New York (1953).

16 . S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, Ch. XI, North-Holland Publishing Company, Amsterdam (1962). 17.

T. Mura, Proceedings Royal Soc., A, 280, 528 (196^).

132

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133

18.

R. D. Richmeyer, Difference Methods for Initial-Value Problems, Interscience Publishers* Inc., New York (1957)•

19-

R. S. Varga, Matrix Iterative Analysis, Ch. 8, Prentice-Hall, Englewood Cliffs, N.J. (1962).

20.

J. W. Gibbs, Scientific Papers, Vol I, Ch. Ill, Dover Publications, New York (1961).

21.

I. Prigogine, Bull. Soc. Chim. Bel., 8-9, 115 (19^3)•

22.

Ch.-J. de La Vallee Poussin, Cours d'Analyse Infinitesimale, Vol II, p. H81, Librairie Universitaire, Louvain (19^9)*

23.

K. B. Rundman, PH.D. Thesis, Northwestern University, Evanston,

111. (1967). 2k.

L. J. van der Toorn and T. J. Tiedema, Acta Met., 8 , 711 (i960).

25-

D. de Fontaine and J. E. Hilliard, Acta Met., 13, 1019 (1965)*

26. B.E. Sundquist, Trans. Met. Soc. AIME, 236, 1111 (1966). 27.

M.E. Rose, Quart. Appl. Math., l*t, 237 (1956).

28.

J. W. Cahn, Acta Met., 12, 1^57 (196*+).

29.

A. Guinier, Acta Met., 3_, 510 (1955).

30.

J. E. Hilliard, B. L. Averbach and M. Cohen, Acta Met., 2,

621 (195^). •31*

J*

Hilliard, B. L. Averbach and M. Cohen, Acta Met., 7,

86 (1959). 32.

K. B. Rundman and J. E. Hilliard (to be published).

33*

H. E. Cook, Ph.D. Thesis, Northwestern University, Evanston,

111. (1966). 3*+. ,L. D.. Landau and E. M. Lifshitz, Statistical Physics, translated by E. Peieris and R. F. Peierls, p. 366, Addison-Wesley, Reading, Mass. (1958).

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13*t

0

35.

A. F. Bonfiglioli and A.-M. Levelut, Communication A la VI Assemblee Generale de l'Union Internationale de Cristallographie, Rome (Italy), 9-19 septembre 19^3•

36 .

R. S. Leigh, Phil. Mag., b2 [vii], 876 (1951).

37.

R. Graf, J. de Phys. et Rad., 23, 819 (1962).

38.

R. Graf and M. Lenormand, C. R. Acad. Sci. Paris, 259, 3^9^ (19^).

39.

A. Guinier and G. Fournet, Small Angle Scattering ofX-rays, Wiley, New York (1955).

1+0.

J. W. Cahn and J. E. Hilliard, J. Chem. Physics, 31j 688 (1959).

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ACKNOWLEDGMENTS

I wish to express my sincere thanks to Professor J. E. Hilliard who gave me the opportunity to study the subject of spinodal decom­ position and who guided my work with sustained interest. I also wish to thank Professor J. W. Cahn for valuable criticism and suggestions which greatly clarified my ideas on the subject.

His

sending me a preprint of his forthcoming paper on the later stages of spinodal decomposition was particularly appreciated. I learned practically all of the basic mathematical concepts used in these pages from Professor I. Stakgold's stimulating lectures on partial differential equations and related topics.

I would also

like to thank Professor Stakgold for many helpful suggestions concerning the mathematical aspects of my work. It was a marvelous opportunity for me to study with Professor Hilliard and his students, all'of us working in closely related fields. Hardly a day went by without lively discussions ranging far and wide. I am particularly indebted to Karl Rundman for interesting conversations concerning the fascinating Al-Zn system. The use of the CDC 3*+00 of the Northwestern University Computing Center is gratefully acknowledged.

Thanks are also due

to Mrs. Lois Schneider who did the drawings and to Mr. A. Nelson who photographed them.

135

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136

I am particularly grateful to my wife who typed my thesis, a difficult manuscript, with patience and loving care. The financial support of the W. P. Murphy Fellowship during the academic year I963-6U was very much appreciated as was the support of the Materials Research Center at Northwestern University.

For

the remainder of the time, this work.was sponsored by the Army Office of Research - Durham.

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vim

Name

Didier de Fontaine

Born

Cairo, Egypt on April 11, 1931

Education

Metallurgical Engineering ' "Ingenieur Civil Me'tallurgiste", October 1955, University of Louvain, Louvain, Belgium

Positions

Military training in Belgian Navy (Lieutenant J.G.), Aug. 1956 - Jan. 1958 "Charge de Mission", Belgian Ministry of Foreign Affairs, Brussels, March 1958 - Nov. 1958 Assistant, Metallography Lab., University of Louvain, Dec. 1958 - Feb. 1962 Instructor, Lovanium University, Leopoldville, Rep. of Congo, Feb. 1962 - Feb. 1963 Research Assistant, Northwestern University, Evanston, Illinois, March 19&3 - Oct. 1986 Walter P. Murphy Fellow, Northwestern University, Evanston, Illinois, Sept. 1963 ■ June 196*+

Publications

Theoretical Determination of the Slip System with Highest

Resolved Shear Stress in a Fee Crystal for any Orientation of the Tensile Axis, Trans AIMS, 22k, 869 (19^2)

137

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138

Comment on"Precipitation in gold-platinum alloys : Thermo­ dynamics11(with J. E. Hilliard), Acta Met., 13, 1019 (19^5)

A Theoretical and Analogue Study of Diffraction from One-Dimensional Modulated Structures, Chapter II in the hook Local Atomic Arrangements Studied by X-ray Diffraction, J. E. Hilliard and J. B. Cohen, Ed., Gordon and Breach, New York, (in press)

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Appendix A : Notation

The symbols of vectors in 3-dimensional space will be underscored; thus, for example, the position vector will be denoted r.

If a Cartesian coordinate system is used, then r is the column

vector

\ X 3 /

of coordinates x t. Likewise, the symbols of 3-dimensional reciprocal space vectors will be underscored; thus the position vector in reciprocal space (or wave number) will be denoted J.

If a Cartesian coordinate

system is used, then _§ is the row vector

_§ =

tt[

(k^y

), (k2/\2)s(^3/ X3)1

of integral components kt,

(A l)

being the edge length of the unit

cell of volume V = 8 X.iX.2ta’ As an exception to this rule, we shall denote the reciprocal space vector of integral components kt by the corresponding upper case letter.

Thus

K = (k , k , k ) 1 3 3 L = (lj, 12, I3) etc... . Another exception will be the notation for the vector ’’gradient". Thus, in the expression

J = - grad cp or J = ■ Vcp 139

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lUO

the underscore will not be indicated in the right hand members. The following conventions will bemade with regard to Fourier series.

If a 3-dimensional Cartesian coordinate system is used, we

have the following expansion for an arbitrary function of position r and time t :

u(r,t) = ^ C K(t) exp(iJK-r) K

(A 2)

= A 0 + 2^T [Ak (t) cos (j3K*r) - BK(t) sin(JK-r)] . K

(A3)

In the above expressions, it is understood that the summations extend over the three inte'gers kx, k2 and k3, each k t running from -

0.8 o

0.6 Fig. A2. Ideal entropy (full curve) and its fourth-degree Legendre poly­ nomial approxi­ mation (open circles).

0 .4

+ o

=

0.2

N 0 1 0 . 2 5i

0 .5 0

0 .7 5

1.00

Composition, c 1U5

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lU6

a0 + [(aa0/acJ)-(aa0/acB)]u-j = o .

(A 10)

By considering a point of the binary system to be defined by the single independent variable

we see that the intersection given

by Eq.(A 10) can alternatively be expressed by

(A 11)

ao + (da0/dcj)u j = 0 • Then, by comparing Eqs.(A 10) and (A ll) we find

da0/dCj = (5a0/3Cj) - (da0/Bcn) .

If we divide both sides of this equation by aQ we obtain

\ which corresponds to Eq.(2l) in the text.

Thus the Tjj's are proportional

to the actual partial derivatives of the lattice parameter function with respect to the (dependent) variables c^, whereas the T]j's are the slopes (measured along the positive Cj axis) of the intersection of the tangent plane with the binary systems (j,n), n referring to the component eliminated from the independent flux equations.

Of

course, if the lattice parameter varies linearly with composition, the surface a is everywhere identical to its tangent plane and a0T]j is then just the slope of the lattice parameter versus composition plot of the binary system (j,n).

The meaning of the Tjparameters

is illustrated in Fig. A 1 in the case of a three-component system (cJ,ck,cn).

The T1 parameters cannot be illustrated graphically.

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Appendix D ; Systems of Ordinary Linear Differential Equations

The Fourier transformation of the partial differential equations studied here often lead to systems of simultaneous ordinary linear differential equations with the time t as the only independent variable.

Such systems can be written symbolically as follows :

A =

BA + Q

(A 12)

in which A, A and Q, are n-dimensional column vectors of complex components A k, dAk/dt and Qk respectively. initial conditions vector, is given.

We assume that A°, the

The symbol B stands for an

nyn matrix of complex elements h kh, say. Let us first consider the case of a time-independent non­ singular Hermitian matrix B (i.e. b kh = b^k, the star denoting complex conjugate quantities).

The n eigenvalues ak of B are then real.

Let

us further assume that they are distinct (the case of multiple characteristic values, a rare occurrence in numerical calculations, will not be discussed here).

It is then always possible to find

an orthogonal operator E which diagonalizes B :

E_1B E = E'B E = B ,

B being a diagonal matrix whose non-zero elements are the eigenvalues 0^ y 1 •6 •

^kk = G'k

b kh = 0

(k^h).

1^7

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1 U8

Since E is orthogonal, its inverse E-1 is equal to the transpose matrix E 7 (i.e. e(j = e 31).

Define the vectors A, A, A 0 and Q by the inverse

of the orthogonal transformations

A = EA,

The

A = EA,

A 0 = EA°,

Q = EQ .

system (A 12) can then be written

EA = BEA + EQ .

Nov; multiply

on the left

by E'1 (or by E 7) to

A = BA + Q , A

obtain thesystem :

given .

Since B is diagonal, the solution of this system can be written down explicitly for each component A h :

A h = A° exp(c*ht ) + exp(a?ht)

pL Qh(T) exp(-ahT) dT .

(A 13)

The quantity uh will be called the amplification factor for the corresponding component (or amplitude) A h; it alone determines the behaviour

of the amplitude A h .in the absence If the order n

of a source term Q.

of B is large, the search foreigenvalues

cyh becomes impractical and approximate methods must be used. adopt a matrix iterative method described by Varga

We shall

19 , Let us first

define the exponential of a matrix X by the expansion

exp(X) = I + X + (1/2)X2 + (1/3! )X3 + ... ,

I being the unitary matrix.

(A

The solution of system (A 12) for the

time interval At = t-t0 can then be written symbolically as

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19

1*0

A(t0+/Vt) = exp(AtB) A0 + exp(tB) j exp(-TB)

Q (

t

)

dr .

(A 15)

to

By the integral of a vector Y(t), say, we mean, of course, the vector whose components are the integrated components of Y.

If the components

of Q do not vary appreciably during the small time interval At we may use the approximation

Q(t) — Q(t0) = Q° ,

say.

(A 16)

Now from Eq. (A 1*1) it can be shown that pt J exp(Xr) dT = [exp(Xt) - exp(Xto)]X'1 to

for an arbitrary constant matrix X.

Then, for small At, we have,

from Eqs.(A 15) and (A l6)

A = exp(AtB) A0 - [I - exp(AtB) JB"1 Q° .

Let

usnow use the forward difference matrixapproximation

exp(AtB) s I + AtB ,

(A 17)

the repeated application of which converges in a stable manner to the correct solution provided that the following inequality is satisfied

At 5 2 (Max |»h |)_1 . b=l,n

If the matrix exponentials of Eq.(A 15) are approximated by I + AtB [Eq.(A 17)]} one obtains

A = A 0 + At(BA° + Q°) .

The components A k at time tQ+At can thus be calculatedexplicitly

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(A 18)

150

from the (given) components

and A£ at time t by the equation

(A 19) h= 1

the explicit form of Eq.(A l8).

By stepping ahead in At time increments

one can successively find A(to+2At), A(tQ+3At), ....

Equation (A 19)

is the central one in the numerical solution of the non-linear diffusion equation (see Sect. 3)The above methods can be used with B matrices with somewhat less restrictive conditions.

Suppose, first, that we have

B = MlT

where M(t) is a scalar function of time and B is a constant Hermitian matrix.

Define the function

(A 20)

then

dor/dt = M(t)

and, by dividing Eq.(A 12) by M(t) and by making use of Eq. (A 20) we obtain the system

dA/do = B A + Q,

(A 21)

where

Q = 0/M .

If Eq.(A 20) expresses a one-to-one relation between t and ct, the system (A 21) can be solved as a function of the independent variable

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150*.

cr, by themethods

outlined above, forexample.

Thetime dependence

is obtainedby setting a = a(t) in the finalsolution. Another useful generalization concerns a non-Hermitian nxn matrix B which can be expressed in the form

B = BP2

(A 22)

where B is Hermitian and P2 is a real positive definite nxn matrix (i.e. all eigenvalues of which are positive) whose (real) positive square root is the nxn matrix P.

Following Faddeev and Faddeeva^"

we shall now show that all the eigenvalues of B are real.

Equation

(A 22) can be written

B =I B P 2 = F 1(PBP)P =

F ’-S P

(A 23)

with

S = PBP ,

which shows that the matrix S is Hermitian, and therefore has real eigenvalues.

But B and S are related by a similarity transformation

[Eq.(A 23)], and therefore have the same eigenvalues. eigenvalues of B are real.

Thus, the

Furthermore, if E diagonalizes B, and E

diagonalizes S, the diagonal matrix B is equal to

B = E"1 B E =E'1 P'1 S P E = r 1S E .

Therefore

E = PE

t D. K. Faddeev and V. N. Faddeeva, Computational Methods of Linear Algebra, p. lfl, Freeman and Co., San Francisco (1963)

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151

and

E = P _1E . This last equation completes the reduction of the problem of diagonalizing the matrix B to the simpler one of diagonalizing the Hermitian matrix S. We could have proved the same theorem for the matrix

B = P2B . In the latter case, E, a diagonalizing operator of B, will be related to E, a diagonalizing operator of S by

E = PE

.

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Appendix E •. Relationship between Chemical and Coherent Spinodals

It was shown, in Sect. 2.2, that the coherent spinodal surfaces are defined by the equation

|g ° | = o

where |G°| is the determinant of a matrix G° whose elements are

= fu +

2\\Y .

(The quantities ftJ , Tij and Y are defined in Sect.2.2.)

The chemical

spinodal, on the other hand, is defined by

1F l= 0

»

F being the matrix of the f ' s .

We shall now show that |P J and

|G° | are related to each other by Eq.(^7) of Sect.2.2. Since each element of G° is the sum of two terms, we can expand its determinant in the following way

f11

^12

••• ("Hi

) ••• -^*1,11-1

V 1 |G°| = |F| + 2Y I

I

J=1 f n-l,l ^n-1,2

where ^

**• f n-i,n-

represents a sum of determinants having two or more columns

equal to the vectors

T|kT|, etc....

Each of these determinants

will vanish since, after division by Tlj\, say, it will contain two or more identical columns.

Therefore ^ = 0 and we obtain n-l

n-1

j= l

i= l

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153

in which f^ |F|.

is the cofactor (or minor) of f tJ in the determinant

Let us denote by F the matrix of the cofactors of

. The

preceding equation, the last term of which is a quadratic form in Tjj, can then be written in the compact notation of Eq.(*+7) :

|G0 |.= |f| + 2 ( V F Tl)Y .

(A 23)

It is easy to show that if Xt (i = l,...,n-l) are the eigenvalues of F, those Xj of F will be given by

li

=

|^l/^l

=

ll

^-1-1

X W

•••

ln-1

(A

>

2^)

since

|F | = Xi ••• Xn_i • Therefore, F is positive definite whenever F is positive definite and thus |G° ] cannot change sign in a region where F is positive definite, regardless of the values of the strain parameters

.

Sine F and G° must both be positive definite at high temperature, this shows that the coherent limit of metastability can never lie above the chemical limit of metastability, regardless of the strain energy parameters, a result already derived in Sect. 2.2. From Eq.(A 23) we see that the chemical and coherent spinodals will always coincide at points for which

7]' F H = 0

identically, i.e. for all possible vectors T|.

This can only happen

when all the eigenvalues of F are equal to zero; but, according to Eq. (A 2k)} the sufficient condition for this to occur is that two

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15^

eigenvalues of F vanish simultaneously.

We have thus proved that

the coherent spinodal must pass through points where two or more chemical spinodal surfaces intersect.

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Appendix F : Polynomial Free Energy Approximations

Let us first approximate the negative of the ideal entropy of mixing :

S(c) = c log c + (l-c)log(l-c)

by aLegendre polynomialexpansion.

Since

(A 25)

S is defined between the

values 0 and 1 of the variable c and the Legendre polynomials are defined between the values -1 and 1 of their arguments x, let us change variables in the latter by the substitution :

x = 2c - 1 .

Since the

ideal entropy is symmetric about c =

(A 26)

l/2 (x = 0)

only the

even polynomials

P0(*) = 1 P2(x) = (1/2)(3x2 - 1) P4(x) = (1/8 )(35x4 - 30x + 3)

are required.

Under the substitution of Eq.(A 26 ) these become

Q 0 (c) = 1 Q2(c) = 6c2 - 6c + 1 Q4(c) = 70c4 - lt-Oc3 + 90c2 - 20c + 1 .

We seek coefficients p n such that

S(c) s p 0 + P 2Q s (c ) + P 4Q 4(c )

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1 56

pn = (2n+l) JqS(c) Qn(c) dc ,

or, by Eq. (A 25),

pn = (2n+l) JoQn(c) c log c dc + (2n+l) JoQ„(c) (l-c)log(l-c) dc or, since Qn(c) = Qn(l-c),

pn = 2(2n+l) JQQn(c) C.1°S c dc • We shall make use of the definite integral pl

-2

1 c“log c dc = -(m+l) "O

(m > -l) .

We then find

p. = 2 f c log c dc = -1/2 u Jo p 2 = loj (6c3 - 6c2 + c) log c dc = 5/12 p, = 18f (70c5 - l^Oc4 + 90c3 - 20c + c) log c dc = 1/20 . Jo Thus

S(c)

ST-(1/2) + (5/ 12)Q2(c) + (l/20)Q4(c) ,

and, by making use of the explicit expression of Qn and by grouping terms we finally obtain :

S(c)

a (7/2)c4 - 7c3 + 7cs - (7/2)c - (1/30) .

The function -S(c) is plotted as a solid line in Fig. A2, the open circles represent its fourth-degree polynomial approximation.

This

polynomial, multiplied by kT and added to the enthalpy term ac(l-c)

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157

yields the required polynomial approximation to the regular solution model. We shall now use the least squares method to approximate the n values

= f(cj) _

(i = 1,...,n)

by the polynomial expressions

Fj = F(c J = X x + X 2c + X3c2 + X4c3 + X5c4

(A 27)’

(the subscript zero cannot be handled conveniently in computer programs, hence the slightly different notation from that used in Sect. 3»l). We must minimize the expression n ^T(fl

- Fj )

= c p ( x 1 , • • • 3 3CS ) ,

1=1

say.

This implies solving the system of 5 linear equations in 5 unknowns

Stf/j = o

(□ = la^a3>^+>5)

5

or, explicitly,

XiI ci_1+X2I ci+XaI Ci+1+X*I Ci+2+X5I ci*3=1 l

l

i

l

1

1

Let us define the following quantities :

v i ' i 4 Y k - I ^ r 1 1=1

Ach =

.

The linear system can then be written, in matrix notation,

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•

153

AX = Y

(A 28)

where A is the matrix of the A kh and Y is a column vector of components Y k. The solution vector X yields the desired polynomial coefficients Xj.

These calculations are performed in the computer program THERMO,

to be described in Appendix H.

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Appendix G : Expressions for M(u) and P(u)

Equation (7) expresses the mobility M as a polynomial of the third degree in the composition c.

The diffusion equation

coefficients, however, are limited to polynomials of the second degree in the variable u [Eqs.(56) and (57)]-

By making the sub­

stitution

c = u + c0

in Eq.(7) and by identifying the resulting expression with Eq.(56), one obtains the coefficients

M0 =(MykT)_1c0 (l-c0)[D^c0 + D*(l-c0)]

= (NvkT)"1 {(l-2c0)[D*c0 + D*(l-c0)] +c0 (l-c0)[D*- Dj]}

M2 =(ycT)"1 {-[Djc0 + D*(l-c0)] + (l-2c0)[D^ - D*]j. .

(A 29a)

(A 29b)

(A 29c)

The coefficient of u3,

M3 = -(NykTj^tDJ - D*] ,

is not used. Let us now evaluate the coefficients of the function D(u), defined by Eq.(55) ' •

D(u) = M(u)[f"(u) + TfYu2] ,

where, by Eq.(50),

159

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(55)

1.2

0.8 0.4 o o U)

-- ----

C S I

«

to

-0.4

O

0.8

-

Q

-

1.2

-

1.6

-

2.0

-2.4

0.2

0.4

0.6

Atomi c Fract i on

0.8 Zn

Fig. A3. Diffusion equation coefficient D(u) (solid curve) and its parabolic Taylor's expansion about u = 0 (c0 = 0.225 at. fract. Zn) in the case of an Al-Zn solid solution at 100°C. True spinodal compositions are indicated by open circles, incorrect ones (derived from the parabolic expansion) by open triangles.

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l6i

f"(u) = 2a2 + 6a3u

+ 12a4u2.

Since M(u) is also quadratic in u, D(u) must be of the fourth degree. A plot of such a function is shown as a solid line in Fig. A3*

This

curve intersects the horizontal axis at four points, two of which are the zeros of M(u), the other ‘ two being the zeros of the coherent second derivative, i.e., the coherent spinodal compositions (open circles).

As mentioned earlier, only quadratic expressions can be

used in the diffusion equation and the coefficients of the parabolic expression for

D [Eq.(57)]are then given by

D0 =

2(a2 + TfY)M0

Dx =

2(a2 + 7)2Y)Mx +

6aaMQ

D2

2 (a s + T12 y )m 2 +

6 a 3M i + 1 2 a 4M o

=

(A 30a)

(A 30b)

‘

(A 3 0 c )

This parabolic approximation is plotted as a broken curve in Fig. A3; it is apparent that this is just a Taylor's expansion of the full curve about the point u = 0 (c0 = 0.225, full circle in Fig. A3). Unfortunately, this parabola is unacceptable since its "spinodal compositions" (intersection with the horizontal axis, open triangles in Fig. A3) are incorrect.

It follows, of course, that the corresponding

equilibrium' compositions will also be incorrect.

The simplest way

to obtain a correct behaviour of the solution of the diffusion equation with respect to the equilibrium phase boundaries is then to assume Slt>-

a composition independent mobility.

In that case, D(u) is automatically

quadratic in u with coefficients given by

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Dq

= 2(a2 +T12Y)M0

(A 31a)

Dt

= 6a3M0

(A 31b)

D

= 12a4M0,

(A 31c)

obtained by setting

M

1

=M = 0 2

in Eqs.(A 30b) and (A 30c).

This procedure was always adopted in

calculations referring to the Al-Zn system.

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Appendix H : Description of Program THERMO

Program THERMO computes the coefficients of the fourth-degree polynomial approximation of the Helmholtz free energy and evaluates the coefficients of D(u) and M(u)-needed in the diffusion equations.

Operations Performed

The values

(Sect.3.l) are read in and a multiple regression

analysis is performed according to the method described in Appendix F. The 5 x 5 linear system AX = Y is set up [Eq.(A 28)] and is solved by calling the linear equation solving subroutine

SIMULT which returns

the X t (i = 1,...,5) coefficients of the free energy polynomial, Eq.(A 27).

If these coefficients are already known from a previous

calculation, this section of the program is bypassed, and the X t coefficients must be read in instead of the ft. This polynomial F(c) and its second derivative F"(c) are evaluated along with the coherent free energy and its second derivative, the former obtained from F(c) by the addition of the quadratic term

TfYc(c - cwJ

.

Usually cMA/would be set equal to one, as in Eq.(72). of the Al-Zn system, however, cw

In the case

was chosen equal to 0 .7 .

Next, if the coherent free energy contains a metastable portion, as in Fig. 3> the coherent equilibrium compositions are calculated (common tangent rule) by the Newton-Raphson method of

163

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16^

successive approximations^ and the coherent spinodal compositions are calculated by equating the parabolic second derivative to zero and by solving the resulting quadratic equation.

The parameter cQ

is also calculated according to the profile of Fig. 8 (see Sect. 3*^0* If the free energy curve is everywhere concave upwards, this section of the program is bypassed and an appropriate message is printed out. The three significant coefficients of f(u) : a2, a3 and a4 are calculated according to Eq.(5l) for the chemical free energy (not the coherent one) and then the diffusivities

and Dg (see Sect. 3-3).

Next, the coefficients M0, Mx and Mg are computed according to Eq.(A 29) and the coefficients Dq, D1 and D2 are computed according to either Eqs.(A 30) or (A 3l)«

Finally, 1+0 amplification factors

are computed according to Eq.(65).

Input

Data carpi No. 1_.

F0RMT(l1+,1iF^.3,6E10.3)

N = no. of fj data points used (gradient energy coefficient (erg/cm). NL = Nl, number of atomic planes per unit length (cm-1).

Ca?Ld jL0;jL: F0RMAT(E10.3,F5.3,3F5.0) SIGMA = a0 = [(c/s-c0)(c0-cJ]V'3 (see page 67).

If the free energy

curve used has no common tangent, this parameter o0 can be set equal to one. CC = Co (atomic fraction). T = isothermal aging temperature (°K). TI = Tj, initial (quenching) temperature (°K), see Fig. 21. TS = Ts, spinodal temperature at the composition c0 (°K).

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17^

Options on Initial Conditions

NT = 1 : No initial conditions are read in or generated, the computer uses the final stored values from a previous run.

This option

is used when some parameters have to be changed in the course of a calculation.

A first calculation is initiated by appro­

priate initial conditions and run for AMAX iterations.

The

calculation can then be continued with other parameters (for example : an abrupt change in aging temperature) simply by adding to the deck another set of data cards with appropriate parameters and NT = 1.

The initial condition for this second

calculation will thus be the final spectrum of the preceding one.

This procedure can be repeated as many times as desired.

NT = 2 : The initial conditions are specified by I'rcomplex quantities C° on data cards which are to be placed immediately after the thermodynamic parameter cards.

The FORMAT is (^C(E10.3 ,E10.3)).

NT = ±3 : The initial amplitudes ((Ck |) are generated by the computer according to Eq.(82).

The complex coefficients C° are then

computed according to Eq.(83).

If NT = + 3s the phases 2 CALL VE C T O R ( 0 . 0 » 0 . 0 » 1 . 0,0.0) CALL VECTOR(0.0»YA,0.0,YB) 60 CALL VECTOR(1.0»YA»1.0»YB) DO 25 J=1,A0,2 XI = •025*(J - l ) XF = .025 + XI CALL VECTOR(XI»UA»XF»UA) 25 CALL VECTOR(XI»UB»XF,UB) U1 = 0.0 DO AO K = 1»MAX AO U1 = U1 + 2.*A(K) DO 50 1=2,NX U2 = 0.0 DO 51 K=1,MAX 51 U2 = U2 + A (K )* F (K » I ) 52 CALL V E C T O R (X (I- 1 ) ,U1 50 U1 = U2 CALL ADVANCE 1A IF(M O D (A M ,A P R I N T )) 16,17,16

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17 TIME = AM*DELT PRINTOUT DO 70 K=1,N 70 I N T (K ) = A(K)**2 SUM = 0.0 DO 81 K=1»MAX 81 SUM = SUM + INT tK ) PER = 2 0 0 • * S O R T (S U M )/SIGMA DO 82 I= 1 »NX »MULT U(I ) = 0.0 DO 82 K = 1♦MAX 82 U (I ) = U 1 1 ) + A (K )* F (K » I ) PRINT 105 » AM » TIME, (K, A(K)> INT(K ), K = 1 , N ) 105 FORMAT(1H1»10X»23HF0URIER SPECTRUM AFTER ,15 ,22H TIME STEPS, 1TI ME =>E10.3,5H S E C . //9 X ,1H K ,1I X ,9H A M P L I T U D E ,1I X ,9HI N T E N S I T Y / / 2( 110 *2E20.3)) PRINT 110, P E R ,(X X (I ), U(I), 1=1,NX,MULT) 110 F O R M A T (1 H 0 / 3 0 X ,*REAL SPACE COMPOSITION VAR IAT I O N * ,1 0 X ,20HPERCENT C 10MPLETI ON =,F7.2//17X,1HX»18X,AHC-C0//(F20.2,F20.A) ) 60 TO 16 3 CALL C H A N G E (3 ) DO 71 K = 1 »15 CALL LABEL(0.1,0.0,7HTHE E N D ,1 ) 71 CALL ADVANCE CALL EXIT END

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Appendix M ; Description of Program SPICOC

Continuous cooling (Sect.3.5) of one-dimensional composition variations is simulated by means of program SPICOC which combines some of the functions of programs THERMO (Appendix H) and FYSPIN1P (Appendix j) Eq.(80) is used with temperature dependent parameters which vary at each (variable) time step it.

Program SPICOC is not as general as

the previous ones, however; it is presently assumed

that diffusion

occurs primarily by the interchange of vacancies with only one of the atomic species, and also that the mobility M does not vary appreciably with composition.

Operations Performed

The initial amplitudes |Ck | are generated internally according to Eq.(32) and the numbers R k of Eq.(S3) are read in on the initial conditions card(s).

The iteration process starts with a calculation

of the temperature T according to Eq.(85).

A first free energy data

card,assumed to be valid from T0 to T1 (Fig.2l), is read in and the mobility is calculated according to the appropriate equations given in Sect.3-5.

New amplitudes A k are then computed according to Eq.(8l)

and a modif'ication of Eq.(8o).

When the temperature drops below Tx,

a second free energy data card is read in, etc..

The mobility-, however,

is recalculated at each time step by using the current temperature and vacancy concentration.

Spectra and profile printouts are given

20^

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205

at specified temperature intervals.

Two pet. completion values are

given with each printout : one relative to the current equilibrium compositions (relative), the other relative to the equilibrium compo­ sitions at the final temperature Tp (absolute). The calculations are terminated when either a specified pet. completion has been attained or when a specified total decomposition time has elapsed.

If desired, the last spectrum obtained can be punched

outT'O'h cards which can then be used as initial conditions input for program FYSPIN1P.

Input (Cards are listed in the order of their appearance in the deck)

Instruction Card. : F0RMAT(E8.3,6l3,2E7.1,5A8)

IWL, N, MAX, MIN, NX

(> 0)

(see Appendix I).

NCARD = number of free energy data cards used. MPRINT = number of spectra and profile printouts (initial conditions excluded) requested at regular temperature intervals between T0 and Tp (temperature at t = », see Fig. 2l).

If MPRINT

is a negative integer, the last spectrum will, in addition, be punched out on cards. TMAX = tteAX, maximum decomposition time allowed (sec). CRIT = C, value used in calculating the time step At according to Eq.(87) (erg cm"5). ID

(see Appendix i).

Thermodynamic Rirame_ter_Card_ : F0RMAT(5E10.3,F2,2FU.3,*+F5)

AB = average lattice parameter aQ (cm). t

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20 6

ETA, Y

(=E),

QBAR = total Z, CO,

KAP

(see AppendixH)..

activation energyQ= + Q,f (cal/mole).

CMAX (see Appendix H).

TI, TS (°C) (see Appendix i). TO = T0, temperature (°C) at time t = 0 (T0 ^ Ts). TF = T|>, temperature at time t = » (°C).

Initial Conditions Card(s) : F0RMAT(20F^.2) Contain

the numhers R k of Eq.(83).

Contin.uou.s_Cooling__ Parame_ter_ Card_: FORMAT(2E10.3,3F^.3, F6.2)

B = quenching rate (dT/dt)^^

(°C/sec).

ALPHA = reciprocal half life of vacancies (sec-1). PART = p, fraction of Q, which is equal to Q^. :

= pQ.

CAF, CBF = coherent equilibrium compositions ca and c^ at temperature T|. (atomic fract.). PMAX '= maximum pet. completion to which computation is to be carried ($).

Free_ Ene^y__Data_Cardj3__ : FORMAT(5E10.3,2Fh.3 5F5)

(There must be NCARD of these.) A2, A3, A1! = coefficients of f(u) polynomial (see Appendix H). DD = D, preexponential term (page 62) (cm2/sec). Q = total activation energy Q, equal to QBAR if one neglects the temperature dependence (cal/mole). CA, CB = current coherent equilibrium compositions Cq,, Cp (atomic fract.). TEMPF = lowest temperature (°C) at which free energy curve is valid.

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207

Printed Output

The identification ID is printed out followed by the domain length, tjie average composition, T^, T^, tw4,, C (=M0At), the quenching rate, the reciprocal vacancy half life and p (=Qf/Q).

The spectra

and profiles are plotted as in program FYSPIN1P (Appendix j) at regular temperature intervals AT [=(TO-TF)/MPRINT]. Along with these printouts, the current values of the following parameters are also given : the number of iterations, the time, At, the temperature, the mobility M, Q, D (the preexponential terra), c^, and c.j, the absolute and relative pet. completions.

Punched Output

If MPRINT is a negative integer, the last A k spectrum calculated will be punched out according to FORMAT(8E10.3).

These cards will be

preceded by a card containing the final values of the following para­ meters : the number of iterations, the time, the relative pet. completion and the identification ID according to F0RMAT(l^r,E10.35F7.2,19X,5A 8 ).

Approximate Execution Time

No estimate can be given since the total number of iterations cannot be evaluated ahead of time.

For comparable numbers of iterations,

however, programs SPICOC and FYSPIN1P should require about the same execution time for identical N.

The CDC FORTRAN listings for program SPICOC follow on the next pages.

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PROG R A M SPICOC DIMENSION I D (5), X(100)» U(100)» BETA(IOO)* A(IOO)* AO(IOO)* 1A T (100 ) » I N T (100)* R (100)* F(55,85)» TP(50) E QUIVALENCE (R,AO,INT) REAL M* M0» Ml, KAP , N V K , NL, L W L , INT, LAMBDA 1 READ 100, LWL, N, MAX, MIN, NX, N C A R D , MPRINT, T M A X * CRIT, ID 100 FORMA T ( E 8 . 3 » 6 I 3 » 2 E 7 . 1 » 5 A 8 ) I F (E O F ,60) 3,2 2 READ 101, AB, ETA, Y, KAP* QBAR, Z, CO* CMAX, TI, TS* TO* TF 1C1 F O R M A T { 5 E 1 0 . 3 » F 2 » 2 F 4 . 3 » 4 F 5 ) READ 1 0 2 , ( R I K ). K=1>N) 102 F O R M A T (2 0 F 4 •2) READ 103, B, ALPHA, PART* C A F , C B F » PMAX 103 F O R M A T ( 2 E 1 0 . 3 , 3 F A .3,F6 .2) MP = MPRINT MPRINT = IABS(MP) HL = N*LWL $ TL = 2.0*HL PRINT 10A » ID* TL, CO* TI* TF* TMAX* CRIT, B* ALPHA* PART 104 F O R M A T (1 H 1 * 2 0 X » 5 A 8 / / 1 0 X , 2 7 H C U 5 I C DOMAIN OF LENGTH 2L = , E 1 0 . 3 » 4 H CM 1•* 2X »31HAND OF INITIAL C O M P O S I T I O N CO = * F 6 .3/ / 1 0 X »1 3 H Q U ENCHED FROM 2 »F6 *0»22H DEGREES C AND HELD AT*F6.0»14H DEGREES C FOR,E10.3» 5H S 3 E C . / / 1 0 X , 1 1 H C R I T E R I O N = , El 0.3 *3 X »22H E X P O . QUE N C H I N G RATE =,E10.3» 43X , 15HVACANCY DECAY = , E 1 0 . 3 , 3 X ,13HQ PARTITION =*F6.3/> TI = TI + 273.2 $ TF = TF + 273.2 TO = TO + 273.2 S TS = TS + 273.2 BK = 1.37E-16 $ MA = 0 S N2 = 2*N $ LT = 0 LC = 1 NL = 1.0/AB NVK = N L * * 3 * B K DO 39 K=1»N LAMBDA = HL/K X I K ) = 2.E + 08*LAMBDA B E T A (K ) = 3 . 14159/LAMBDA 39 a(K) = R ( K ) * S Q R T ( T I / ( 2 . * ( N L * H L ) * * 3 * ( 2 . * ( T I - T S ) + K A P * B E T A (K )**2/ 1NVK ) )) PRINT 130, (K, X (K ), K=1»N) 130 F O R M A T (1 H 0 / / 6 X ,1 H K »A X » 9 H A N G S T R O M S / / (17*F I 1.2)) DO 5 K = N »9 9 5 A (K + l ) = A O (K + 1) = A T (K + 1 ) = 0.0 XO = H L /(N X - 1)

A3

70 71

10

X (1 ) = 0.0 DO A3 1=2,NX X(I) = X(I-l) + XO DO 71 1 = 1,NX DO 70 K = 1 *MAX F (K , I ) = 2 0 0 . * C 0 S ( B E T A ( K ) * X ( I )) X (I ) = l.E + 08*X(I ) TT = TO - TF B = B/TT DT = I F I X (T T )/MPRI NT T P (1) = TO - DT DO 10 L =1»MPRINT T P (L + l ) = TP(L) - DT T P (MPRI N T ) = TF + 1.0 SIGMAF = S Q R T ( 2 . * ( C B F - C 0 ) * ( C 0 - C A F ) ) COO = C O * C O * (C MAX- C O )/NVK PQ = - P A R T * Q B A R / 1 . 9 8 7 Ml = E X P (P Q / T I ) EE = E T A*ETA*Y

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209

r> r»

KAP = 2**KAP Q = QBAR $ TEMPF = TI $ T = TO TIME = DELT = M = DD = CA = CB = 0.0 GO TO 17

C C

C C

C C

$

„ SIGMA = 1.0

START OF ITERATION 16 MA = MA + 1 " TIME = TIME + DELT IF (TIME.GE.TMAX .OR. PER.GE.PMAX) 11.12 11 TP(LT) = T P (L T + 1 ) = TI 12 T = TF + TT#EXP(-B*TIME) I F (T - TEMPF) 90.90,91 90 READ 120, A 2 , A3, A A , D D , Q» CA, C B , TEMPF 120 F ORMAT(5E10.3,2FA.3,F5) LC = LC + 1 TEMPF = TEMPF + 273.2 SIGMA = S Q R T ( 2 . # ( C B - C 0 ) * ( C 0 - C A ) ) QM = Q/1.987 + PQ CD = COO*DD DO = 2 • * ( A2 + EE) D1 = 6.*A3 D2 = 12•*AA 91 MO = E X P (P Q / T ) M = C D * E X P (- Q M / T )*(MO + (Ml - M O )# E X P (-ALPHA*TI M E ))/T DELT = CRIT/M DO 61 K=MIN »N 61 A (K ) = S I G N F ( M I N 1 F ( A B S F ( A ( K ) )»1•E - O A ), A ( K ) ) CALCULATION OF AT ARRAY DO 30 J = 1»N NJ = N + J AO(J) = A (J ) $ A T (J ) = AT(NJ) = 0.0 DO 30 1=1,N IF (I.EQ.J) GO TO 31 A T (J ) = A T (J ) + A ( I ) * ( A ( I A B S C J - I )) + A(J + D ) AT(NJ) = AT(NJ) + A ( I ) * A ( N J - I ) GO TO 3 0 31 A T (J ) = A T (J ) + A(I)*A(2#I) 30 CONTINUE ATO = 0.0 DO 32 1=1,N 32 ATO = ATO + 2.*A(I)**2 C ALCULATION OF NEW A M P L I T U D E S DO 20 K=1»N BB = BE T A (K )**2 DO 20 L=1»N IF(L.EQ.K) GO TO 21 KL = IABS(K-L) S LK = K+L A (K ) = A (K ) - C R I T * B E T A ( K ) * B E T A ( L ) * ( D 1 * ( A 0 ( K L ) - AO(LK)) + D2* 1{A T (K L ) - A T ( L K ) ) ) # A 0 ( L ) GO TO 20 21 A (K ) = A (K ) - CRIT*BB*( DO + KAP#BB - D1*A0(2*K) + D2*(AT0 1 A T ( 2 * K ) ) )*AO(K) 20 CONTINUE IF (T - T P ( L T ) ) 17,17,16 FOURIER SYNTHESIS 17 DO 50 K = 1»N

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INT(K) = A(K)**2 SUM = 0 . 0 DO 81 K=1»MAX 81 SUM =. SUM + I N K K) SO = 200»*SQRT(SUM) PERF = SO/SIGMAF PER = SO/SIGMA DO 82 1=1,NX U( I ) = 0.0 DO 82 K=1,MAX 82 U < 1) = U ( I ) + A(K)*F tK * I)

n n

50

PRINTOUT TEMP = T - 273.2 PRINT 105, MA, TIME, DELT, TEMP, M, Q, DD, CA, CB, (K, A (K ), 1INT(K), K= 1,N ) 105 FORMAT(1H1*10X»23HF0URIER SPECTRUM AFTER ,15 ,22H TIME STEPS, 1TIME =,E10»3,5H S E C .,3 X ,6HDELT =,E 10.3 / 9 X ,7 H T E M P . =,F5,2H C,3X, 2 lOHMOBILITY =,E 1 0 .3,3 X ,19HACTI VAT ION ENERG. = , El 0.3 ,3X ,AHDO =, 3 E 1 0 . 3 , / 9 X , 2AHEQUILIBRIUM COMPOS I T I O N S ,FI 0.3,AH , »F 8 •3 / / 9 X ♦1HK » A 1 1 X , 9 H A M P L I T U D E , 1 1 X , 9 H I N T E N S I T Y / / (I10,2E20.3) ) PRINT 110, PERF, PER, (X(I), U(I), 1=1,NX) 110 F O R M A T (1H0/30X,*REAL SPACE COMPOSITION VAR IAT I O N * ,1 0 X ,31HPERCENT C 10MPLETI ON (ABSOLUTE) = , F 7 .2/9I X ,1 2 H (R E L A T I V E ) =,F7.2// 217X,1HX,18X,AHC-C0//(2F20.2)) LT = LT + 1 IF (TO - T P ( L T ) ) A , A , 16 A 18

DO

L = LC , NCARD

18

READ

1 2 0 ,

1 AO

PUNCH

AA-.

DD,

Q,

CA,

CB ,

1 AO ,

MA ,

T IM E ,

PER,

ID ,

( A ( K ) ,

F O R M A T ( I A , E 1 0 .3 , F 7 . 2 i> 1 9 X , 5 A 8 / ( 8 E 1 0 . 3 ) GO

3

A3,

TO

TEMPF

6 ,1 ,1

IF(M P ) 6

A2 ,

K=1*N) )

1

STOP END

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Appendix TT : Description of Program FYSPIKPY

Program FYSPIMFY simulates the behaviour of one-dimensional coherent composition variations in solid solutions with plane-xave imperfections in periodic domains vith insulating boundary conditions (Sect.3.6).

The calculated Fourier spectra and composition profiles

are plotted through the use of the CALCOMP plotter.

This program thus

constitutes another modification of program FYSPIN1P.

Operations Peformed

The Fourier coefficients F k (k = 0,1,...n) of the coherent second derivative modulation defined by Eq.(38) are read in and the function n

k=0

is synthesized and printed out at selected, values of x.

The initial

conditions are either read in or generated internally according to Eq.(82). in.

In the latter case, the numbers R k of Eq.(S3) must be read

The remainder of the calculation proceeds as in program FYSPIN1P

except that the amplitudes A k are calculated according to Eq.(89) rather than Eq.(8o).

No punched output can be obtained in this version

of the program.

211

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212

Input (Cards are listed in the order of their appearance in the deck)

Instruction Card_: FORMAT(e 8.3,5I3,2I5,E7.1,5A8) IWL, N

(see Appendix j).

MAX = maximum number of Fourier coefficients used in the Fourier synthesis (ill).

.

MIN, NX, NT, APRINT, AMAX, DELT, ID

(see Appendix j).

Thermodynam_ic_Parame ter Card (see Appc-nd.ix I).

Plot_ InstrucUon_CartL (see Appendix J).

Imperfe ctions_ Card( s_) The N + 1 values Fk (k = 0,1,...,n) (ergs/cm3) defined by Eq. (88) are read in according to F0R1YAT(8e 10.3) .

In:y;ial_Cond_itions_ Cards (See Appendix J.

Note, however, that the random numbers R k

cannot be generated internally in the present version of program FYSPIMPY).

Printed Output

•The printed output is identical to that of program FY8PIN1P (Appendix j) with, in addition, the synthesized coherent second deri­ vative modulation printed out after the kinetic parameter arrays.

Plotted Output

(See Appendix J.)

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213

Subroutine Required

GRAPH, plotting subroutine, also used in conjunction with program FYSPIN1P (Appendix j).

Approximate Execution Time

Comparable to program FYSPIN1P.

The CDC FORTRAN listings for program F/SPIMFY follow on the next pages.

The listings for subroutine GRAPH are given in Appendix J.

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n n

2114-

PROGRAM FYSPIMPY KINETICS OF D E C O M P O S I T I O N FOR O N E - D I M E N S I O N A L N O N - L I N E A R EQUATION B(K) GIVE M O D U L A T I O N OF DO, IMPERFECTIONS DIMENSION M (3), D C3)* ID(5)» X(100)» ALPHA I3»100)» U(100), 1 B E T A (100)♦ A (100), A0C100), AT (100 ) ». INT (100 ) » R(100), A F (100)» 2 F (55,85), D A T A (2000), B(100) • C 0 M M 0 N / C 1 / XX, NAX, NX, LL, FR, E X » P E R EQUIVALENCE ( R , A O , I N T ) , (AF,AT) INTEGER AM, AMAX, APRINT REAL M, KAP, NVK, NL, LWL, INT, L A M B D A CALL P L O T S C D A T A C 1),2000,15) CALL P L 0 T ( 0 . 0*1.0,-3) 1 READ 100, LWL, N, MAX, M I N , NX, N T , A P R I N T , A M A X , D E L T , I D ,E7.1*5A8) 100 F O R M A T ( E 8 . 3 » 5 I 3 » 2 I 5 I F (E O F ,60) 3,2 T I ♦ TS 2 READ 101» M, D, KAP, N L , SIGMA, C O i 101 FO R M A T ( 8 E 1 0 . 3 / E 1 0 . 3 » F 5 . 3 » 3 F 5 . 0 ) READ 109, NAX, M U L T ,L L » M P L O T A ♦ A X M I N A , D A X A » A X M I N U , D A X U F O R M A T (313, 15, 2F7.4.2F7.2) 109 NP = N N = IABS(NP) READ 199, BO, {B (K )» K=1,N) F O R M A T {8 E 1 0 .3) 199 TL = 2 . 0 * H L $ HL = N*LWL $ TIME = A M AX*DELT PRINT 104, ID, TL, CO, TI, T, TIME, D E L T F O R M A T { 1 H 1 * 2 0 X , 5 A 8 / / 1 0 X » 2 7 H C U B I C DOMAIN OF LENGTH 2L = , E10.3»4H CM 104 1•» 2X »31HAND OF INITIAL C O M P O S I T I O N CO = , F 6 .3/ / 1 0 X ,13HQUENCHED FROM 2 »F 6 •0 »22H DEGREES K AND HELD A T , F 6 . 0 , 1 4 H DEGREES K F O R ,E 1 0.3,27H S 3EC. TIME INTERVAL DELTA = , E 1 0 . 3 , 5 H SEC.//) BK = 1.37E-16 $ AM = 0 $ N2 = 2*N NVK = N L * * 3 * B K BO = B O * M ( 1 ) DO 39 K = 1 »N LAMBDA = HL/K X (K ) = 2 • E + 0 8 * LAMBDA B E T A (K ) = 3. 1 4 1 5 9 / L A M B D A

B (K ) = B (K )* M (1) DO 41 L = 1» 3 A L P H A (L ,K ) = -BETA(K)*(D(L) + 2.* K A P * M (L )* B E T A (K )**2 ) 41 39 A F (K) = BETA(K)*A L P H A ( 1 , K ) IF(NP) 10,10,11 10 READ 1 0 2 , ( R (K ), K=1»N) 102 F O R M A T (2 0 F 4 . 2) DO 12 L = 1 »N 12 ACL) = R ( L ) * S Q R T I T I / ( 2 . * ( N L * H L ) * * 3 * ( 2 . * ( T I - T S ) + K A P * B E T A ( L ) * * 2 / 1 N V K ))) GO TO 13 11 IF (NT.LT.O) GO TO 13 READ 1 0 7 , ( A ( K ) , K = 1» N ) F O R M A T (8 E 1 0 . 3) 107 PRINT 103, (K, X CK ), A F (K ) * (A L P H A (L ,K ), L = l,3)» K = 1»N) 13 1 C 3 F O R M A T ( 1 H O » 3 0 X » * K I N E T I C PAR A M E T E R A R R A Y S * / / 6 X ,1H K ,4 X , 1 7 X ,5 H A M P L F ,9 X ,6HALPHA0 » 9 X * 6 H A L P H A 1 » 9 X * 6 H A L P H A 2 / / ( I 7 , F 1 1 . 2 , 4 E 1 5 . 3 ) ) DO 5 K = N ,99 ACK+l) = AOCK+l) = BCK+l) = 0.0 DO 6 K = 1»100 A T (K ) = 0.0 XO = HL/(NX-1 ) XC1) = 0.0

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DO 43 I=2,NX 43 X (I ) = X ( 1-1 ) + XO DO 71 1=1,NX DO 70 K=1»MAX 70 F (K »f) = 2 0 0 . * C O S ( B E T A ( K ) * X ( I )) 71 X {I ) = 1•E + 0 8 * X ( I ) DO 84 1=1,NX,MULT U( I ) = BO DO 84 K=1»MAX 84 U (IJ = U(I) + B(K)*F(K»IJ/100.0 PRINT 202, (X(I), U(I), 1= 1 ,NX,MULT) 202 F O R M A T (1H 1 , 1 6 X , 1 H X , 1 9 X , 2 H D 0 / / ( F 2 0 . 2 , E 2 0 . 3 )) XX = 7.0/NAX X (N X + 1 ) = 0.0 $ X (N X + 2 ) = HL/8.E-08 U (N X + 1 ) = AXMINU $ U (N X + 2 ) = -DAXU F R = DELT $ EX = 0.0 $ TM = 1.0 73 I F ( F R - 10.0) 69,72,72 72 FR = FR/10.0 TM = T M * 10.0 EX = EX + 1.0 GO TO 73 75 FR = FR*10.0 TM = T M / 10.0

on

on

on

69

215

EX = EX - 1.0 I F (FR - 1.0) 75,17,17

START OF ITERATION 16 AM = AM + 1 I F (A M . G T . A M A X ) GO TO 1 DO 61 K=MI N ,N 61 A (K ) = SIGNF(MIN1F(ABSF( A(K) ) , l.E-0'4) ,A(K) ) CALCULATION OF AT ARRAY DO 30 J = 1»N NJ = N + J AO {J ) = A {J ) $ AT (J ) = AT(NJ) = 0.0 DO 30 1=1,N IF ( I.EQ.J) GO TO 31 AT (J ) = A T (J ) + A (I )*(A ( IABS(J - I )) + A(J + I)) AT(NJ) = AT(NJ) + A (I )* A (N J - I ) GO TO 30 31 A T (J ) = A T (J ) + A(I)*A(2*I) 30 CONTINUE ATO = 0.0 DO 32 1= 1,N 32 ATO = ATO + 2.*A(I)#*2 CALCULATION OF NEW AMPLITUDES DO 20 K=1»N DO 20 L = 1»N ' IFIL.EQ.KJ GO TO 21 KL = IABS(K-L) $ LK = K+L A (K ) = A(K) +DELT*(BETA(K)*(ALPHA(2»L)*(A0(KL) - A O (L K )) + 1ALPHA(3»L)*(AT(KL) - A T (L K )))- B E T A (L )**2*(B (K L ) + B (L K )))*A O (L ) ' GO TO 2 0 21

A ( K )

=

A(K)

+

D E L T *B E T A (K )*(A L P H A (1 ,K )

1 A L P H A ( 3 » K ) * ( ATO

-

20 CONTINUE IF (ABS(A(1) ) ,GE.

A T ( 2 * K ) )

10.)

-

-

B E T A ( K ) * ( BO

ALPHA( 2 , K )* A 0 ( 2 * K ) +

B ( 2 * K ) ) ) * A 0 ( K )

GO TO 4

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+

on

no

on

IF ( M 0 D ( A M , A P R I N T ) ) 16*17*16 17 TIME = AM*DELT FOURIER SYNTHESIS DO 50 K= 1»N 50 INT(K) = A(K)**2 SUM = 0.0 DO 81 K=1»MAX 81 SUM = SUM + IN T (K ) PER = 200.*SQRT(SUM)/SIGMA DO 83 I= 1 »NX U ( I ) = 0.0 DO 83 K=1»MAX 83 U (I ) = U t I ) + A(K)*F(K»I) PRINTOUT PRINT 105* AM* TIME* (K* A tK )* I NT(K )* K = 1*N) 105 FORMAT(1H1*30X,23HF0URIER SPECTRUM AFTER *15 »22H TIME STEPS* 1TIME =*E10»3 *5H S E C .//9X »1H K ,1I X »9H A M P L I T U D E ,1I X »9HI N T E N S I T Y / / 2 ( I 1 0 . 2 E 2 0 . 3 )) PRINT 110, PER, (X (I ), U (I ) * I = 1 » N X , M U L T ) 110 F O R M A T (1H0/30X,*REAL SPACE COMPOSITION VAR IAT IO N * »1 0 X ,20HPERCENT C 10MPLETION = ,F7.2//17X,1HX , 1 8 X , 4 H C - C 0 / / ( 2 F 2 0 . 2 ) ) IF(NT.LE.O) GO TO 57 PRINT 108* ATO, (K» A T {K J, K=1»N2) 10 8 F O R M A T ( l H 0 / / 9 X , l H K » 1 4 X , 5 H A T ( K ) / / 9 X , l H 0 » E 2 0 . 3 / ( I 1 0 » E 2 0 . 3 ) ) PLOTS 57 FR = TIME/TM 58 I F (FR - 10.0) 59*60*60 60 FR = FR/10.0 TM = TM*10.0 EX = EX + 1.0 GO TO 58 59 I F (M O D (AM * M P L O T A )) 16,91,16 91

CALL GO

4

GRAPHtA, X , U , 9HAMPLITUDE , 9 , 1 5HC-C0

TO

TIME

IN

PERCENT, 1 5 , AXM INA, DAXA)

16 =

AM*DELT

PRINT 105, AM, TIME, (K, A (K ) »- INT IK ) , K = 1,N) GO TO 1 3 CALL P L O T d O . 0,-1.0,999) STOP END

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Appendix P : Description of Program FYSPIN2D

The computations relative to the isothermal aging of twodimensional composition variations in the (imperfection-free) anisotropic space of cubic crystals are handled by program FYSPIN2D.

The diffusion

equation is solved in one quarter of- a two-dimensional domain with insulating boundaries as explained in Sect.3.7-

It is assumed that

the mobility is composition independent.

Operations Performed

The pertinent parameters and the 2 x 2 array of initial amplitudes are read in.

(real Fourier coefficients of the exponential series) The elastic parameter and amplification factor matrices

are then calculated according to Eqs.(93) and (92) and printed out. The iteration process starts with the application of the convergence factor : the amplitudes of components located outside a circle of radius n in Fourier space are set equal to zero; the amplitudes of components located in the annular region between the radii m (=MII'f)'* and n (=N) are reduced to £L04 if they exceed this number in magnitude (the sign chosen being that of the amplitude before artificial conver­ gence was applied).

The 2n x 2n A,J|>d ,i array is then set up [Eq.(9l)]

and the amplitudes A k?ki are calculated according to Eq.(90).

Every

APRINT iterations starting with zero, the initial condition, the n x n amplitude spectrum is printed out and the corresponding two-dimensional profile u(x1,x2) is synthesized and printed out along with the value

217

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of the pet. completion.

In addition, the Fourier spectrum is punched

out on cards which can serve as input for the plotting program PLOTOODY (Appendix Q).

Input (Cards are listed in the order of their appearance in the deck)

In^tru^t^n_Card_ : FORMAT(e 3.3,5I3j2I5,E7.1j5A8) IWL

(see Appendix i).

N = number n of Fourier components retained in each (100) direction (*15).

MAX, MIN (> 0)

(see Appendix J).

NX = number of points on each Cartesian (100) axis, including the origin, at which the composition profiles are to be calculated (s l6). NT = an integer just larger than N*N. APRINT, AMAX, DELT, ID

(see Appendix j).

Thermodynamic Parame_ter_Cards_ Card No^l j FORMAT (8S10.3) M = M0

(see Appendix i).

Dl, D2 = D-l, D2

(see Appendix i).

A2 = ag coefficient of f(u) (erg/cm3) (see Appendix H). ETA

(see Appendix H).

Cll, C12, ClfU = CL1, Cl2, C44, elastic constants (dynes/cm2).

Card_No_.2_: F0RMAT(E10.3 ,F5•3,3F5.0,E10.3 ) SIGMA, CO, T, TI, TS KAP

(see Appendix i).

(see Appendix il).

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219

Initial Condition Card_s The consecutive elements of each row of the

} matrix

are written on W o cards according to F0RMAT(8E10.3/8E10.3)•

Printed Output

(See description under heading "Operations Performed".)

Punched Output

At each printout, the amplitude spectrum is punched out on cards according to the same FORMAT as that used for reading in the initial conditions.

Each set of spectrum cards is preceded by a card

containing the values of the parameters AM, TIME, PER (see Appendix i).

Approximate Execution Time

A calculation performed with N = 12 iterated 100 times and with 11 printouts took approximately US min.

It is estimated that

each iteration (for N = 12) takes about 0.U5 minutes.

The execution

time is roughly proportional to the fourth power of N.

The CDC FORTRAN listings for program FYSPIN2D follow on the next pages.

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220

n rs

PROGRAM

FYSPIN2D

' DECOMPOSITION

F OR

TWO-DIMENSIONAL

NON-LINEAR

EQUATION

D I M E N S I O N A ( 3 0 53 0 ) » AG(30»30)» A T(30,30), ALPHA(15 »15)» 1L 0(30 ) » ID (5 )» X ( 1 6 ) > 0 ( 16 » 1 6 ) » I A R O ( 1 6 , 1 6 ) , Ar(30,30)» R E A L M > KAP , LWL I N T E G E R AM » A M A X » A P R I N T R E A D I OC , L W L , N , M A X , M I N » NX , FORMAT(E8.3 ,5 1 3 , 2 I 3 ,E7.1,5A8)

1 100

I F ( E OF » 6 0 ) 3,2 R E A D 1 1 L , H» D l ,

2 1 101

D2,

A2,

ETA,

NT,

C 11 ,

A P R I NT »

C12,

AMAX »

CAA ,

rCAP F 0 R MA T ( 6 E 1 0 . 3 / E 1 0 . 3 , F 5 • 3 , 3 F 3 . 0 , E 1 0 • 3 ) H l = M#LWl S T I ME = A M A X * D E L T $ Tl = M2 = -11 N* i * l I N N = N + 1 S MIN = MIN + 1 6 MAX = MAX PRINT lOA, ID, T L , CO, T I , T, T I M. E, D £ L T

ULLT j

SIGMA,

CO,

£ £(1 3 ,1 5 ), Ks(30»30)

ID

1,

FI,

Ts»

2 • 0 * hL +

1

10A

F OR MA F ( 1 H 1 , 2 C X , 5 A 8 / / 1 0 X , 2 7 r i C U B I C D O h m I N u r L t n G T n 2 l= , E 1 0 . 3 , ^ h CM l • , 2 X , 3 1 H A N 0 OF I N I T I A L C O M P O S I T I O N CO = , F 6 • 3 / / 1 0 X , 1 ENChEL. ' F ROM 2 , F 6 « 0 , 2 2 H D E G R E E S R AND h t m a T»F6.0,1A H Dc. GRn. ES is F UK » £ 1 0 » 5 , 2 7r1 S 3EC. TI ME INTERVAL DELTA = , E 1 0 . 3 , 5 H S E C . / / ) AM = 0 S ' N2 = 2 N DO A 1 = 1 , 1 6 A L u ll) = 1 - 1 DU 5 I = 1 » N2 DO 5 J = 1» N 2 Ar ( I , J ) — O . u

o 6 107

A3

A ( I ,J ) - 0 • D Du 6 I = 1 , N REA D 10 7 , ( A ( I , J ) , J = 1 , N ) F o R MA T ( 6E 10 • 3 / 8 E 1 0 • 3 ) 62 = ( 3 . 1 A 1 5 9 / H L ) * * 2 DP = D E L T " o 2 FI = DEL M XO = X ( 1 ) DO A3 X II) CC = CD CE DO

D1*')P S F2 = = - 2 . d 2 * M -:‘ is2

C E* ( 3 . 0 - C C /IC ll+ EE(J1,J2) = I ASS ( J . l - J 2 )+ 1 = I A R G ( J 1 , J 2) = D E L M * S o * ( A2+ = ALPhA (J 1 , J2 )

£E(J1»J2)

PRINT 1 0 3 , ( LG ( J ) , J = 1 , N ) F ORMAT I 1 H C / 3 0 X » * E L A S T I C ENERGY DO 5 0 PRINT

1=1,N 1u 2 ,

LG( I ) , ( EE( I , J ),

CD*(X 1*K 2/S U )**2) )

+

Xa P * d 2 * S Q )

P A R A M E T E R * / / / 7X , 17 I 7 / / )

J =1,N,2)

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221 102

F OR MA T C 1 H 0 » I 3 » 8 E 1 4 - . 3 )

50

PRINT

112 113

51

16

112,

( EE( I , J ),

J=2,N,2)

FORMAT( 1 1 X » 8 E 1 A . 3 ) PRINT- 1 1 3 , ( LO(J ) , J =1 ,N ) FORMAT(1H1,20X,'^AMPLIFICATION DO 5 1 PRINT

I = 1 ,N 102,

PRINT

112,

GO

17

TO

LO ( I ) ,

( ALPr i A ( I , J ) ,

(Al PHA(I»J)»

61

-

TIMES

DELTA

T * / / 7 7 X , 17 I 7 / / )

J=1,N,2)

J = 2 , n »2>

S T A R T OF I T E R A T I O N AM - AM + 1 I F ( A M . G T . A M A X ) GO TO 1 '’ DO 6 1 X 1 = 1 » N DO 6 1 X ^ = 1 , N I F I X S I X 1 , X 2 ) . l T .M2 . OR. A I X 1» K2 )

FACTOR

KS ( X 1 , X2 ) . GT .

NT )

GO T O

61

5 1 G N F I M I N 1 F I A d S F I A I X 1 » X 2 )) » 1 • E - 0 A ) , A I X 1 , X 2 ) )

CONTI NOc CALCULATION

uF

AT

ARRAY

DO 3 1 J a - 1 , N DO 3 1 J 2 = 1 , N AO ( J 1 , J 2 ) = A ( J 1 , J 2 ) A(J 1 ,J 2 ) 31

=

A (J 1, J2 ) ■ “• F 2

AF(J 1 ,J 2 ) = A0(J1,J2)#F1 DO 3 0 J i = 1 , N N J 1 = N + J1 DO 3 C J 2 = 1 , N N J 2 = “I + J 2 A T ( J 1 , j 2) = A T ( NJ 1 , J 2) = AT(J1,NJ2) = DO 2 9 1 2 = 2 , N NN2 = N J 2 - I 2 + 1 S X2 = I A N G ( J 2 , I 2 ) AO I = A u ( 1 , 1 2 ) AT(J 1 ,J 2 ) = AT(J 1 ,J 2 ) + A0I*(A(J1»K2) AT ( J 1 , N J 2 ) AT( NJ1 , J 2 )

29

= =

AT ( J 1 , N J 2 ) + ATINJ1, J2 ) +

AO I = A O ( I 1 , 1 2 ) AT ( J 1 , J 2 ) = AT ( J 1 , J 2 ) 1 A ( L 1 , L2 ) ) ATINJ1, J 2 )

C C

.

» +

L2

=A T I N J 1 ,

) + A0 J2 ) + AO N J 2 ) + AO MJ 2 ) + A0 X2 + J2 )

0.0

J2+I2-1

A(NJ1,L2))

LI =

J l+ Il - 1

I I A I X 1 »J 2 ) + A ( L 1 , J 2 ) ) I *A I NN1 , J2 ) I * I A(X1 ,NJ2 ) + AIL1.NJ2)) I * A ( NN 1 , NJ 2)

= I A R G ( J 2 9 I 2) A0 I * ( A I X 1, X 2 ) +

=

=

A(J1,L2))

AO I * A ( J 1 , N N 2 ) A 0 I * ( A ( N J 1 , A 2 )+

A T ( N J 1 , N J 2 ) = A T ( N J 1 > N J 2 ) + AO I {N J 1 ,N N 2 ) DO 3 0 I 1 = 2 , N • NN1 = NJ 1- I 1 + 1 S X I = I ARG ( J 1 I 1 ) 6 Ao I = A O ( 1 1 , 1 ) AT I J 1» J 2 ) = AT ( J 1 , J 2 ATINJ1, J2 ) = A T ( N J 1» ATI J 1 , NJ 2) = ATI J 1 , AT(NJ1»NJ2) = AT ( NJ 1 , DO 3 0 I 2 = 2 , N NN2 = NJ2-I2 + 1 S

30

AT(NJ1,NJ2)

S +

L2 =

A(X1,L2)

AO I * I A I N N 1 , X 2 )

J2+I2-1 +

A ( L 1 * :< 2 )

+

A(NN1,L2))

A T I J 1 , NJ 2 ) AT( NJ1»N J 2 )

=ATI J 1 » NJ 2 ) + AO I * I A I X 1 , N N 2 ) + = AT I N J 1 , N J 2 ) + AO I * A I N N 1 , N N 2 )

A(L1,NN2))

CALCULATION

OF

NEW

AMPLITUDES'

N1 = 2 DO 2 1 K. 1 = 1 , N XXI = XI 1 DO 2 0 X 2 = N 1 , N IF(XS(X1»X2) -

NT)

22,22,21

R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.

+

222 22 K K 2 =

K2

A ( K1»K2 ) DO 12 19 A( 1 DO 11

“ 1 =

A O ( K 1» K 2 ) # ( 1 . 0

1 9 l 2 = 2 »N = I ARG ( K 2 * L 2 ) $ K.1 * K 2 ) = A ( K 1» K 2 ) A T { K 1»J 2) ) * A 0 ( 1 , L 2 > 2 0 L 1 = 2 »N = I ARG ( . I 2 )

J 1

=

KK 1 +

- AF(K1,J2)

+AT(K1,I2)

LI

KK1*(L1-1)

A ( K 1 , K2 ) = A ( K 1 » K 2 ) 1 A T ( J 1> K 2 ) ) # A 0 ( L 1 » 1 ) DC 2 0 L 2 = 2 » N 1 2 = I ARG ( K2 >L 2 ) S KL 2 = K K 2 * ( L 2 - 1 ) 20

+

JQL 1 * ( A F ( 1 1 • K 2 )

J2

=

K.K.2 +

-

A F ( J 1 > K2 )

+

AT(I1,K2)

L2

A(K1,K2) = A (K1*K2) ( ( K L 1 + K L 2 ) * ( A F ( I 1 , I 2) - A F ( J 1 » J 2 ) + 1 A T (I1 ,U ) - AT ( J 1 > J 2 ) ) + ( KL 1 - K L 2 ) * ( A F ( I 1 » J 2 ) - A F U 1 . I 2 ) 2 A T ( 1 1 * J 2 ) - A T ( J 1 , 1 2 ) ) ) * A0 ( L 1 , L 2 )

21

N1 = 1 I F ( MOD ( AM , A P R I. NT ) )

C C

105

1 T I M E = » E 10 . 3 »5 H SUM = 0 . 0

DO

81

20 3

206 52

SI

+

16,17,16

PRINTOUT TIME = AM*DELT PRINT 1 0 5 > AM, T I M E , (L O U ), J = 1»N) F OR MA T ( 1 H 1 , 1 0 X , 2 3 H F O U R I E R SPECTRUM AFTER

17

-

,15

,22H

ITERATIONS,

SEC . / / / 7 X , 17 I 7 / / )

K 1 = 1, M A X

DO 8 1 K 2 = 1 , MAX SUM = SUM + A ( K 1 , K 2 ) # * 2 PER = 2 0 0 . * S Q R T ( S U M ) / S I G M A PUNCH 2 0 3 , AM, T I M E , PER FORMAT( I A , t l 0 . 3 , F 7 . 2 ) DO 5 2 I = i , N PUNCH 2 0 6 , ( A ( I , J ) , J = 1 , N ) F O R M A T ( SE 1 0 . 3 / 3 E 1 0. . 3 ) PRINT 1)2, LO ( I ) , ( A ( I , J ) , J = 1 , N , 2 ) PRINT 112, M» S X » 3 Y » FORMAT(4 I 3»2 E l 1 • 4 , F 6 • 1 , F 6 • 3 , 4 A d ) IF (E O F ,60) 3,2 MM - X MAX OF ( Mi, 0 ) M = XABSF(M) NN = ( N X - 1 ) * M + 1 N = N + 1 6 MAX = AX = NX 1 i AL

MAX = -

+

Tt i ETA»

ALPHA»

ID

1

2.*(

(NX-1)/2)

ARG = 3 . 1 4 1 5 9 * T H E T A / 1 3 0 . XX = 3 . 1 4 x 3 9 / A X BX = A L PHA C 0 3 ( ARG ) oY CX CY Bd XL

= ALPHAS I n URG) = bX "XX 6 DX = S X * C X =■ BY- ;;- X X 3 DY = 3 X * C Y = d Y G X / ( 200 . * 3 Y ) S JO = SX*XX 3 SL = XL " AX

SLDX 4

5

=

SL

+

DX

3

DXDX

=

3 3 = 3

i_X = D X » A X • 3 RX = D X # A L LY = D Y * A X 4 RY = D Y * A L -C Y*SX/(200.*6 Y ) XLXL = 2.*XL 3 X L L X = LX + XL

2.*DX

3

DY DY

=

2 . * l>Y

DO 4 I = i , 2 0 LO( I ) = 1-1 Xu = X X / M

X (1)

= Y (1)

DO 3 X (J ) YIJ)

J=2,NN = X ( J —1 ) + = dX *X (J)

= 2(1)

= 0.0

XO

101

Z ( J ) = ob-"-X(J) X (NN+2) = Y ( NN+2) = 1.0/SX U(NN+2) = . 005/SY PRINT 1 0 1 , ID F 0 R H A T ( i ri u / / / 3 0 X , 4 A d )

20 103

R E A D 1 0 3 , A M , T I M E , P 'cR r uNi mmT ( l K X i u b i , " ~1 • d. )

103

CALL P L o T t 9 • 0 , 0 • 0 , - 3 ) I F ( PER. cG. C . 0 ) GO—TO 1 PRINT 1 1 3 , AM, T I M E , ( L 0 ( J ) ,J = 1» N ) F u R M A T ( 1 H 1 , 1 CX , 2 3 H F O U R I ER S PE CT RUi - i

AFTER

,13

,22ri

I T c Ra T I ui^

,

1 T I M E = , E 1 0 . 3 , 5 rI S E C . / / / 7X , 17 I 7 / / ) DO 3 0 I = 1 , N 5 0 RcAD L O S , ( A ( I , J ) , J =1 , N ) 106 f o r m a t r b e i o . 3 / b e i o . 3 )

1 Q2 52 112 1. 10

DO 5 2 I = 1 , N PRINT 1 0 2 , L 0 ( I ) , ( A ( I , J ) , J=1,N,2) F O R M A T ( 1H0 , I 3 , 8 E 1 4 . 3 ) PRINT 1 1 2 , (A( I , J ), J=2,N,2) FORMAT( i l X , 8 E 1 4 . 3) PRINT 1 1 0 , P ER, ( L O ( J ) , J=1,NX) F u R M A T ( 1 H 1 , 3 OX , # R E AL S P A C E C O M P O S I T I O N V A R I AT I O N * , 1 0 X , 2 0 H P E R C E . N T 1 0 M P L E T I ON = , F 7 . 2 / / / 7 X - , 1 7 1 7 / / / ) CALL S Y M o O L ( - 4 . 5 , 3 • 3 , . 2 0 , 1 6 H P C • C OMP L E T I ON = , 0 . 0 , 1 6 ) CALL N UMBE R( - 1 . 6 , 3 . 5 , . 2 0 , P E R , 0 . 0 , 1 ) DO 8B I = 2 , MAX A ( I , 1 ) = C . 5 *A ( I , 1 )

R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.

C

88

A ll» I ) 1 = DO U(J DO XK DO

13

11

=

0 • 5 *A (1, I )

1 11 J = 1 , N N ) = 0.0 11 K = i s M A X = ( N. - 1 ) -;