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PergamonAcra mrrd. marer. Vo l. 43, No. 9, pp. 3403-34 13, 1995Elsevier Scrence Ltd0956-7151(95)00041-0 Copyright C 1995 Acta Metal lurgica Inc.Pr inted in Great B r i tain. Al l r ights reserved0956-7151195 $9.50 + 0.00

SPINODAL DECOMPOSITION IN Fe-Cr ALLOYS:EXPERIMENTAL STUDY AT THE ATOMIC LEVEL ANDCOMPARISON WITH COMPUTER MODELS-II.

DEVELOPMENT OF DOMAIN SIZE AND COMPOSITIONAMPLITUDE

J . M. HYDE, M. K. MILLER*, M. G. HETHER INGTONT, A. CEREZO,G . D . W. SMITHS and C. M. ELLIOTT3

Department of Materials, University of Oxford, Pa rks Road, Oxford OX1 3P H, England, 2 Metals andCeramics Divis ion, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6376, U.S.A. and School ofMathem atical and Physical Sciences, University of Susse x, Falmer. Brighton BN I 9QH, England(Received 30 Norember 1994)

Abstract-The three-dimensional interconnected microstructures resulting from spinodal decomposition ina series of thermally aged Fe-Cr alloys have been analysed in terms of scale and composit ion amplitude.The development of the microstructure scale was found to f it a power law w ith a t ime exponent considerablysmaller than that predicted by the LS W theory but in agreement with Monte Carlo simulations of thedecomposition. Numerical solutions to the classical non-linear Cahn-HilliarddCook equation were foundto tit the classical LS W theory. A model, based on the non-linear theory of spinodal decomposition byLanger et al. is used to quantify the composition amplitu de at any stage of the phase separation. A detailedcomparison between the atomic scale experimental results and computer simulations of spinodaldecomposition is given,

1. INTRODUCTIONIn Part I of this series [ l] , the use of atom probetechniques for analysing the three-dimensional (3D)atomic structure of alloys was discussed . The result ingspinodal decomposit ion in a series of thermally agedFe-Cr alloys was compared qualitatively with twocomputer simulations of the decomposit ion process: aMonte Carlo model and a numerical solution to thenon-linear Cahn-H illiard-Cook equation. In thispaper, the kinetics of phase separation in the binaryFe-Cr sys tem are determined and quantitativelycompared with the two computer simulations. Twoquantitative parameters are studied: geometr icaldomain size and composit ion amplitude.

2. DEVELOPMENT OF SCALE2.1. Theory

At the late stages of spinodal decompos it ion, thedomains reach their equilibr ium compo sit ions. Thedriv ing force for further decom posit ion is thereduction of interface area between the domains, s incethese are associated with exce ss surface energy. As thedomains coarsen, both the area and curvature of theinterfaces are reduced. For materials with low volume+Deceased.j:To whom al l correspondence should be addressed.

fractions of partic les. the larger partic les will grow atthe expense of the smaller ones, a process knownas Ostwald r ipening. Independently, Lifshitz andSlyozov [2] and Wagner [3] developed a theory for thetime dependent coarsening of partic les, known as theLSW theory. It was assume d that the partic les werespherical and widely separated so that interactionsbetween them could be ignored, and that thesupersaturation was small ( f ields quasi-static). Asdecomposit ion proceeds, the rate of growth of thepartic les slows , since the distance or diffusion pathbetween the remaining partic les gradually increases.The LSW law predicts the power law growth of themean precipitate size R(t)

More recently, H use [4] generalized the LSW theory toapply to arbitrary volume fractions of the two phasesin the late stage limit.

In the LSW formulation, the only method ofinterface motion or growth is by long range diffusion.Clearly. in sys tems with interconnected micro-structures, long range diffusion is coupled withdiffusion along the interface. To date, neither a growthlaw nor scaling formula has been derived directlyfor the case when both of these mecha nisms occursimultaneously. The free energy will be minimized ifthe diffusing atoms adhere to the interface at points of

3403

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3 40 4 J . M . H Y D E e t a l . : S P I N O D A L D E C O M P O S I T I O N I N F e - C r A L L O Y ~ - I Iminima l curvature. Atoms ad hering at points of highercurvature will either diffuse along the inte rface or leavethe interface and diffuse to another domain. Althoug hthis diffusion along the interface does not contrib uteto domai n growth, it can limit the effectiveness of longrange diffusion particularly during the early stages ofdecomposition when high curvat ures exist. A t the laterstages of decomposition, there is less need for furthertranspo rt of the ato ms along the interfaces because theinterfaces are smoother. Generally, the thermodyn-amic driving force for diffusion along the interface willbe much greater than the driving force for bulkdiffusion. However, when high curvatures exist, localrearrangement s are important . This results in a slowercoarsening during the early stages of decomposition.Therefore, the growth exponent is expected to increaseIo the limiting value of 1,'3. However, the behavio urduring early stages of phase separation, before thelimit is reached, has not been extensively studied.2.2. Determination of microstructural scale from atomprobe data

For isolated particles, the scale of the microstruc-ture may be defined in terms of the mean particle sizeand the mean particle separation. The equivalentmeasures for interconnected morphologies are themean domain width and the mean distance betweendomai ns. In Part I of this series [1], it was shown tha ta simple composi tion profile could be used to estimatethe microstructu ral scale. An aut ocorrela tion analysis[5, 6] is useful for quant ifying regular modul ation s incomposition and deriving averaged spatial infor-matio n from the compo sition fluctuations present in asample. The one-d imensional (1D) autocorrela tionfunction Rk, at lag k, may be defined as

R , = ~ , ~ , ( c , - C o ) ( C , ,+ ~ , - C o )where C, is the composition of sample i, N the totalnumb er of samples, Co the mean compositio n and cr2the variance of the compositions C~ given by

~r2= L(C-Co) . 1

For each pair of samples situated at a lag k apart,the compos ition differences between the samplecompositi on and me an value are multiplied together.The sum of these pairs is divided by the variance of thecomp osit ion values to yield the cor relat ion coefficientR~. Comparison of two low conc entra tion samples ortwo high concentration samples yields a positivecon tri but ion to the correla tion coefficient, whereascompariso n between a low conce ntrat ion sample anda high concentration sample yields a negativecon tribut ion to the correlation coefficient. If theindividual contributions are randomly positive andnegative, their sum will be zero. Thus, the cor rela tioncoefficient calculated from a sinusoida l comp ositi on

distribu tion in an alloy would be large and positive fork = 2 and large and negative for k = 2/2 (where 2 isthe wavelength of the modulations). The correlationcoefficient for k = 0 is always equa l to 1 since thecorrelation of a value with itself is perfect.

The development of the autocorrelation functionfrom 1D atom probe analyses of a series of thermally"aged Fe~5% Cr alloys is shown in Fig. 1. Thecompositions were calculated using a block size of 33ions. Fro m the positions of the first minima, k0, and thefirst maxima, k., a nd their variation with aging time,it is possible to estimate the wavelength of thecomposition fluctuations and the rate of decompo-sition with aging time. The position at which theautocorr elation fu nction crosses zero may also be usedto estimate the thickness of the ~' phase. The increasein magnitude of these parameters with aging timeindicates the increasing microstructural scale duringphase separation. Although the autocorrelationfunctions are noisy, it is easier to determine the scaleof decomposition, from either the posi tion of the firstmin imum or the first maxim um, than from a directanalysis of compositi on profiles as shown in Part I ofthis series [1].

The 1D autocorrelation function may be extendedto use 3D data by using a radial function

R , ~ ~ l r m a " -- k0 "~ E ( f i r ) - - Co)(C(r -Fk ) - Co)r - - O

where C is the compo sition of a spherical shell atradius r from the chosen centre point , rm,x is themaximal radius at which there is compositionalinformation and er2 s the variance of the compositions.Note that the mean Co is the mean of the compositionsC,,~which may be slightly different from the m ean alloycomposition.

Since the volume enclosed by a spherical shell ofunifor m thickness varies as r 2, the numb er of atomscontributing to each sample volume is not constant.However, the significance of the autocorrelationfuncti on is determined only by the numb er of samplevolumes. The shells are therefore made as thin aspossible to maximize the numb er of samples. For the

ECAP124 h I0.8 500 h

0 . 6 -

0. 20 '

- 0 . 2 -- 0 . 4 [ I I i I I

0 20 40 60 80 100Lag, k

F i g . 1 . D e v e l o p m e n t o f a u t o c o r r e l a t i o n f u n c t i o n f r o m 1 Datom probe analyses of a series of Fe~J,5% Cr alloysthermally aged at 500C.

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J . M . H Y D E e t a l . : S P I N O D A L D E C O M P O S I T I O N I N F e - C r A L L O Y S I I 3 40 5a )

R k

1 ] P o S A P A n a l y s is0 . 8 ~ F e - 4 5 % C r0"61111 4 h0 "4 ~I \ \ / 2 4 h

1 / ~ 100 h 500 h

0 2 4 6 8 10n m

b )1 0 -

mY , 1

G r a d i e n t= 0 .2 8 + 0 .0 5F i r st M a x i m u m

J J . . . . Gr adi ent 0.22-+0.05 ~ / F i rs t M i n i m u m

. . . . . . . . i , i

1 0 1 0 1 0 0 0Ag ing (h )

Fig . 2 . (a ) Deve lopm ent o f the 3D au tocorre la t ion func t ionca lcu la ted f rom PoSA P ana lyses o f a se ries o f the rm al ly agedFe~45% Cr a l loys . In (b ) the cor responding m easurem en t o fscale with t ime is shown. Multiple points represent resultsfrom different experiments.

a n a l y s i s o f th e e x p e r i m e n t a l d a t a , a s h e l l t h i c k n e s s o f0 .2 n m w a s c h o s e n , a n d f o r th e c o m p u t e r s i m u l a t i o n s ,a c o m p a r a b l e s h e l l t h i c k n e s s o f 0. 5 la t t i c e u n it s w a su s e d . S a m p l e s w i t h l a r g e r a d i i h a v e c o m p o s i t i o n sa v e r a g e d o v e r a w i d e a r e a , r e s u l t i n g i n v a l u e s r o u g h l ye q u a l t o t h e m e a n c o m p o s i t i o n o f th e a l lo y . T h u s , R~t e n d s t o z e r o a t l a r g e k , a n d n o s i g n i f i c a n c e s h o u l d b ep l a c e d o n t h e a u t o c o r r e l a t i o n f u n c t i o n b e y o n d t h e f ir s tm a x i m u m .

I n o r d e r t o r e m o v e t h e b i a s a s s o c ia t e d w i t h t h es e l e c ti o n o f a c e n t r a l p o i n t , m a n y r a d i a l a u t o -c o r r e l a t i o n s m a y b e c a l c u l a te d c e n t r e d o n r a n d o mp o s i t i o n s i n t h e a n a l y s e d v o l u m e a n d t h e r e s u l t sa v e r a g e d . S a m p l i n g t h e d a t a m a n y t i m e s i s c o m p u t a -t i o n a l l y i n t e n s i v e , s i n c e f o r e a c h a t o m a d i s t a n c em u s t b e c a l c u l a te d f r o m e a c h c e n t ra l p o i n t . I n t h er e s u l t s t h a t f o l l o w 2 00 r a n d o m c e n t r a l p o i n t s w e r ec h o s e n a n d t h e a v e r a g e d a u t o c o r r e l a t i o n f u n c t i o n w a sc a l c u l a t e d .

2 . 3 . A t o m p r o b e r e s u l t sA f t e r 4 h a g i n g a t 7 7 3 K , t h e a u t o c o r r e l a t i o n

f u n c t i o n d o e s n o t s h o w a n y s i g n i f i c a n t p h a s es e p a r a t i o n i n t h e F e ~ , 5 % C r a l l o y , w h e r e a s a f t e r 24 ha sc a l e c a n b e m e a s u r e d [ F i g . 2 ( a) ]. A m e a s u r e m e n t o ft h e f i r s t m a x i m u m , a f t e r 2 4 h a g i n g , g i v e s t h ew a v e l e n g t h o f t h e c o m p o s i t i o n f l u c t u a t i o n s a s a p p r o x .2 n m . A f t e r a g i n g f o r 5 00 h , t h e w a v e l e n g t h o f t h ec o m p o s i t i o n f l u c t u a t i o n s i n c re a s e s t o a p p r o x . 5 n m .A s t h e a g i n g p r o c e e d s t h e f i r s t m i n i m u m b e c o m e sm o r e p r o n o u n c e d [ F ig . 2( a) ].

F o r e a c h s e r i e s o f t h e r m a l l y a g e d a l l o y s , th e s c a l ea s a f u n c t i o n o f t i m e [ F i g . 2 (b ) ] w a s f i t te d t o ap o w e r l a w , R ( t ) v c t , a n d t h e t im e e x p o n e n t n w a sm e a s u r e d . A s u m m a r y o f t h e t i m e e x p o n e n t s f o rt h e d e v e l o p m e n t o f sc a le d e r i v e d f r o m t h e a u t o -c o r r e l a t i o n a n a l y s e s i s s h o w n i n T a b l e 1. T h e r e s u l t sr e f e r t o t h e s c a l e m e a s u r e d u s i n g t h e f i r s t m i n i m u ma n d f ir s t m a x i m u m o f t h e a u t o c o r r e l a t i o n f u n c t i o n .D u e t o t h e lo w v o l u m e f r a c t i o n o f t h e C r - e n r i c h e d ~ 'p h a s e , i n s u f f i ci e n t d a t a w a s o b t a i n e d f r o m t h e a n a l y s i so f t h e F ~ 1 7 % C r an d F e - 1 9 % C r f or a n a c c u r at ea u t o c o r r e l a t i o n a n a ly s i s. R e s u l t s f r o m a l l o f t h e o t h e rs e r ie s o f t h e r m a l l y a g e d a l l o y s s u g g e s t t h a t t h es c al e o f d e c o m p o s i t i o n f it s a p o w e r l a w r e l a t i o n s h i pw i t h a ti m e e x p o n e n t i n d e p e n d e n t o f a l l o y c o m p o -s i ti o n . T h e a u t o c o r r e l a t i o n f u n c t i o n s f r o m t h e e n e r g yc o m p e n s a t e d a t o m p r o b e s h o w a t im e e x p o n e n t o f

0 .2 1 a n d t h e d a t a f r o m t h e P o S A P e x h i b i t s a ti m ee x p o n e n t o f ~ 0 . 2 5 . T h e r e s u l ts a r e c o n s i d e r a b l yh i g h e r t h a n t h e e x p o n e n t d e t e r m i n e d b y B r e n n e r e t a l .[7 ] ( n ~ 0 . 1 1 ) f r o m c o m p o s i t i o n p r o fi l e s a n d F I Mm i c r o g r a p h s o f F e - 3 2 % C r a l lo y s a g e d a t a l o w e rt e m p e r a t u r e o f 7 43 K . T h e r e s u lt s a r e, h o w e v e r , i ng e n e r a l a g r e e m e n t w i t h s o m e r e c e n t s m a l l a n g l en e u t r o n s c a t t e r i n g ( S A N S ) m e a s u r e m e n t s [ 8, 9 , 1 0] .B l e y [ 8 ] s t u d i e d F e C r a l l o y s c o n t a i n i n g 2 0 , 3 5 a n d5 0 a t . % C r t h e r m a l l y a g e d a t 7 7 3 K . I n e a c h s e r i e so f a l l o y s , t h e c h a r a c t e r i s t i c l e n g t h s c a l e s w e r e f o u n dt o s c a le w i t h a t i m e e x p o n e n t o f 0 .2 f o r a g i n g l o n g e rt h a n 3 0 h . H a r w i c k [ 10 ] f o u n d a t i m e e x p o n e n t o f~ 0 . 2 5 f o r a ll o y s w i t h C r c o n t e n t s b e t w e e n 2 0 a n d

4 0 % t h e r m a l l y a g e d a t t e m p e r a t u r e s b e t w e e n 7 88a n d 8 1 8 K . L a S a l l e a n d S c h w a r t z a l s o u s e d S A N S t os t u d y p h a s e s e p a r a t i o n i n F e - 3 2 % C r a l lo y s a g e d u pt o 2 0 0 h a t 7 73 K [ 9 ] . T h e y f o u n d t h a t t h e t i m ee x p o n e n t s h o w e d a s t r o n g d e p e n d e n c e o n s o l u t i o n i z -i n g tr e a t m e n t . A t i m e e x p o n e n t o f 0 .2 w a s f o u n d f o ra l l o y s s o l u t i o n t r e a t e d a t 1 2 73 K , w h e r e a s a t i m ee x p o n e n t o f 0 .1 2 w a s f o u n d f o r a l lo y s h e a t t r e a t e d a t1123 K .

Table I. Tim e scaling exponents from the autocorrelation analyses of atom probe analyses of Fe Cr alloysTime scaling exponents from autocorrelation analysis

Alloys aged at 773 K An alys is Time region (h) First minimum First maximumFe-2 4% Cr PoSAP 24-500 0.25 +_ 0.06 0.30 _+ 0.04Fe 32% Cr PoSAP 8-500 0.27 + 0.02 0.29 + 0.04Fe~ ,5%C r PoSAP 4-500 0.22 +_ 0.05 0.28 + 0.05Fe~15%C r ECA P 4-500 0.28 +_ 0.03 0.21 + 0.02

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3 40 6 J . M . H Y D E et al.: S P I N O D A L D E C O M P O S I T I O N I N F e -C r A L L O Y S - - I Ia) 1 Monte CarloSimulation (750 K)

O.6-Rk 0.4-

0 . 2 -0 -

1 MCS/100 MCS

/ 10000 MCS/-0.2 2 4 6 8 10

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First 6 5 0K | M a x i m u m ,[ ] 7 5 0 K I/', 8 5 0 K ~

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F i g . 3 . ( a ) A u t o c o r r e l a t i o n a n a l y s i s o f a M o n t e C a r l os imu la t ions f o r an A 50% B a l loy on a b . c. c , l a t t i ce wi ths e c o n d n e a r e s t n e i g h b o u r i n t e r a c t io n s . I n ( b ) t h e c o r r e s p o n d -i n g m e a s u r e m e n t s o f s ca le w i t h t im e a r e s h o w n .

2 . 4 . M o n t e C a r l o s i m u l a t i o n sT h e t i m e d e p e n d e n c e o f d o m a i n g r o w t h f r o m

M o n t e C a r l o s i m u l a t i o n s h a s b e e n e x t e n s iv e l y s tu d i e d .A l o g a r i t h m i c g r o w t h l a w h a s b e e n f o u n d a t t h e la t es t ag e s o f a g in g b y M a z e n k o e t a l . [ 1 1 , 1 2 ] s t u d y i n gs p i n o d a l d e c o m p o s i t i o n o f t h e 2 D I s i ng m o d e l a t 5 0 %c o n c e n t r a t i o n . H o w e v e r , m o s t r e s e a rc h e r s h a v er e p o r t e d p o w e r l a w b e h a v i o u r . B o t h A m a r [1 3] a n dG r a n t e t a l . [1 4] f o u n d a g r e e m e n t w i t h L S W t h e o r y .O t h e r M o n t e C a r l o s i m u l a t i o n s h a v e s h o w n t i m ee x p o n e n t s o f 0 . 1 7 - 0 . 2 5 [ 1 5 - 1 7 ]. H u s e [4 ] p r e s e n t e dM o n t e C a r l o r e s u lt s f r o m 2 D s i m u l a t i o n s w h i c hy i e l d e d a t i m e e x p o n e n t o f 0 .2 9 . N o n e o f t h e p r e v i o u ss i m u l a t i o n s h a v e b e e n d i r ec t ly c o m p a r e d w i t he x p e r i m e n t a l r e s u l t s f r o m s p i n o d a l d e c o m p o s i t i o n i nt h e F e - C r s y s t e m .

T h e a u t o c o r r e l o g ra m s f r o m M o n t e C a r l o s i m u -l a t i o n s f o r a n A - 5 0 % B a ll o y o n a b .c . c, la t t i c e w i t hs e c o n d n e a r e s t n e i g h b o u r i n t e r a c t i o n s a r e s h o w n i nF i g . 3 ( a) . A s a g i n g p r o c e e d s , t h e f i r s t m i n i m u mb e c o m e s m o r e p r o n o u n c e d a n d i t s v a l u e i s o b s e r v e d t oi n c r e a s e . A f t e r a p p r o x . 1 0 ,0 0 0 M C S , t h e s c a l e o f t h es t r u c t u r e p r o d u c e d b y t he M o n t e C a r l o s i m u l a t i o n iss i m i l a r t o t h a t o b s e r v e d e x p e r i m e n t a l l y a f t e r 5 0 0 ha g i n g a t a p p r o x i m a t e l y t h e s a m e t e m p e r a t u r e .

T h e f i rs t m i n i m a a n d m a x i m a f r o m t h e a u t o c o r r e l a -t i o n f u n c t i o n s f r o m t h e M o n t e C a r l o d a t a p r e s e n t e d a l lf it te d a p o w e r l a w w i t h a t im e e x p o n e n t o f ~ 0 . 2 1[ F i g . 3 ( b ) a n d T a b l e 2 ] . T h e r e w a s n o e v i d e n c e o fl o g a r i t h m i c b e h a v i o u r . T h e r e su l ts a r e i n d e p e n d e n t o fl a t ti c e t y p e , la t t i c e si z e a n d a g i n g t e m p e r a t u r e . N o t et h a t b e c a u s e t h e M o n t e C a r l o a l g o r i t h m i s a s t o c h a s ti cp r o c e ss , t w o s i m u l a t i o n s p e r f o r m e d u n d e r t h e s a m ec o n d i t i o n s m a y y i e ld s l i g h t ly d i f fe r e n t r e s u lt s . I n e a c hs i m u l a t i o n , t h e t im e e x p o n e n t m e a s u r e d u s i n g t h e f i rs tm i n i m u m o f th e a u t o c o r r e l a t i o n f u n c t i o n w a s w i t h ine r r o r b o u n d s o f th e e x p e r i m e n t a l r e s u lt s, a n d t h e t i m ee x p o n e n t m e a s u r e d u s i n g t h e f ir st m a x i m u m w a sf o u n d t o b e o n l y s l ig h t ly l o w e r t h a n t h e e x p e r i m e n t a lr e s u l t s .

I t h a s b e e n s u g g e s t e d t h a t t h e d i f f e r e n t s o l u t i o n i z i n gt r e a t m e n t s c o u l d a f fe c t t h e l o n g t e r m d y n a m i c s o fp h a s e s e p a r a t i o n a n d c o u l d e x p l a in t h e d e p e n d e n c e o ns o l u t i o n t re a t m e n t t e m p e r a t u r e r e p o r t e d b y L a S a l l ea n d S c h w a r t z [ 9]. A c o m p u t e r e x p e r i m e n t w a sp e r f o r m e d u s i n g th e M o n t e C a r l o a l g o r i t h m t o a s se s st h e e ff e c t o f a s o l u t i o n t r e a t m e n t j u s t a b o v e t h em i s c ib i li t y g a p . A s i m u l a t i o n w a s p e r f o r m e d o n as i m p l e c u b ic l a t ti c e a g e d a t t h e e q u i v a l e n t t e m p e r a t u r eo f 1 0 0 0 K f o r 2 0 ,0 0 0 M C S a n d t h e n q u e n c h e d t o 7 5 0K a n d a g e d f o r a f u r t h e r 1 0 , 00 0 M C S . A f t e r t h eq u e n c h i n t o t h e s p i n o d a l r e g i o n, t h e t im e e x p o n e n t f o rt h e d e v e l o p m e n t o f s c al e w a s m e a s u r e d a n d f o u n d t ob e 0 . 21 4 - 0 . 0 1 . T h i s v a l u e a g r e e s w i t h t h e t i m ee x p o n e n t s m e a s u r e d f r o m s i m u l a t i o n s q u e n c h e d f r o ma n i n f i ni t e t e m p e r a t u r e . T w o f u r t h e r s i m u l a t i o n s w e r ep e r f o r m e d , s i m u l a t i n g a c o n t r o ll e d c o o l i n g f r o m 1 0 00t o 7 5 0 K d u r i n g ( a ) 1 00 M C S a n d ( b ) 1 0 0 0 M C S . S i n c ed e c o m p o s i t i o n o c c u r r e d d u r i n g t h e q u e n c h , t h e ti m ee x p o n e n t s f o r s u b s e q u e n t d e c o m p o s i t i o n a t 7 50 Km i g h t b e e x p e c te d t o b e l o w e r . N o t e t h a t t i m e w a s r e s e tt o 0 f o l l o w i n g t h e q u e n c h . T h e r e s u l t s ( T a b l e 2 ) s h o wt h a t g e n e r a l l y t h e t i m e e x p o n e n t s a r e l o w e r , b u t t h e y

Table 2. Tim e scaling exponents from the au tocorrelation analyses of Monte C arlo simulationsLattice Solution treatment

Time scaling exponent from 3D autocorrelationAgeing T (K) T im e egion (MCS) First minimum First maximum

Simple cubic Drop quench from T = :c 650Simple cubic Drop quench from T = ~ 650Simple cubic Drop quench from T - ~ 750Simple cubic Drop quench from T = ~ 850Body centred Drop quench from T = oo 750Simple cubic 20 ,000MCS ( 1000 K) 750Simple cubic Gradual quench from 7501000 K over 100 MCSSimple cubic Gradual quench from 7501000 K over 1 000 MCS

10-10,000 0.21 + 0.02 0.20 + 0.0210-10,000 0.22 __ 0.01 0.21 4- 0.0110-10,000 0.20 4- 0.02 0.18 __ 0.0110-10,000 0.23 + 0.03 0.22 4- 0.0110-10,000 0.20 + 0.01 0.22 + 0.0310-10,000 0.21 + 0.01 0.20 4. 0.0210-10,000 0.24 + 0.03 0.18 4. 0.03I0-10,000 0.17 4- 0.03 0.18 + 0.03

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Numerical Discretisation of1 Cahn-H illiard-Cook Equation0.8 - \ lO Time Units0.6- l / / 100 Time Units0. 4 1' 1000 Time Units0 .2 -

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!~ - . . . . . . . . i - , . . . . . . T . . . . . . . . i . . . . . . . i1 10 100 1000 1 0 4Aging (t ime units)Fig. 4. (a) Auto corre |ation analysis of the num erical solutionto t he C ah ~ H i l l i a rd -C ook equa ti on o f an A 50% B a ll oy .In (b) the corresponding measurements of scale with t ime areshown.

d o n o t p r e d i c t t h e v e r y s l o w d y n a m i c s o b s e r v e d b yB r e n n e r e t a l . [ 7 ] . V a r y i n g t h e s o l u t i o n i za t i o nt r e a t m e n t d i d n o t a f f e c t t h e l o n g t e r m d y n a m i c s .2 . 5. N u m e r i c a l d i s c r e ti z a ti o n o f t he C a h n - H i l l i a r dC o o k e q u a t i o n

T h e 3 D a u t o c o r r e l o g r a m s f o r t h e n u m e r i c a ls o l u t io n t o th e C a h n H i l l i a r d - C o o k e q u a t i o n a r es h o w n i n F i g . 4 ( a ) . I n t h e e a r l y s t a g e s o f a g i n g( 1 0 - 1 0 0 t i m e u n i t s ) a d o m i n a n t w a v e l e n g t h f o r m sa n d t h e c o m p o s i t i o n a m p l i t u d e i n c r e a s e s , s h o w nb y t h e i n c r e a s i n g a m p l i t u d e o f t h e f i r s t m i n i m u m .i n a g r e e m e n t w i t h t h e l i n e a r t h e o r y o f s p i n o d a ld e c o m p o s i t i o n [ 1 8 ] . B e y o n d a p p r o x . 1 00 ti m e u n i t s ,c o a r s e n i n g b e g i n s . T h e s i m u l a t i o n s w e r e s t o p p e d a f t e r3 00 0 t i m e u n i t s s i n c e f u r t h e r c o a r s e n i n g m i g h t h a v eb e c o m e l i m i t e d b y t h e v o l u m e o f t h e s i m u l a t i o n . A s t h ei n t e r a c t io n r a n g e w a s i n c r e a s e d , b o t h t h e m a g n i t u d e o ft h e e a r l y s t a g e d o m i n a n t w a v e l e n g t h a n d t h e t i m eb e f o r e t h e o n s e t o f c o a r s e n i n g i n c r e a s e d . T h e s i ze o f

t h e C o o k t e r m h a d n o e f fe c t o n t h e m a g n i t u d e o f t h ee a r l y s ta g e d o m i n a n t w a v e l e n g th .

D u r i n g t h e c o a r s e n i n g s t a g e ( b e tw e e n 2 0 0 a n d 3 00 0t i m e u n i t s ) t h e s c a l e o f t h e m i c r o s t r u c t u r e s p r o d u c e db y t h e n u m e r i c a l s o l u t i o n t o t h e C a h n - H i l l i a r de q u a t i o n f o l l o w e d a p o w e r l a w w i th a t i m e e x p o n e n tc l o s e to t h e v a l u e 1 /3 [ F i g . 4 (b ) a n d T a b l e 3 ] p r e d i c t e db y t h e c l a s s i c a l L S W t h e o r y [ 2, 3 ] i n m a r k e dd i s a g r e e m e n t w i t h b o t h e x p e r i m e n t a n d t h e M o n t eC a r l o s i m u l a t i o n s . N e i t h e r t h e C o o k t e r m n o r t h ep r e c i s e v a l u e o f t h e i n t e r a c t i o n r a n g e s i g n i f i c a n t l ya f f e c t e d t h e v a l u e o f t h e t i m e e x p o n e n t .

S e v e r a l r e s e a r c h e r s h a v e a s s e s s e d n u m e r i c a l m o d e l sb a s ed o n t h e C a h n - H i l l i a r d - C o o k e q u a t i o n [ 1 9 -2 1 ]a n d t h e r e i s g e n e r a l a g r e e m e n t t h a t t h e l a t e s t a g ed e v e l o p m e n t o f s c a l e f i ts a p o w e r l a w . R o g e r s e t a l . [20]u s e d f i n i te d if f e r e n c e m e t h o d s t o s t u d y a 2 Dp e r c o l a t i n g s p i n o d a l s y s t e m w i t h c o n t i n u o u s o r d e rp a r a m e t e r . T h e y f o u n d t h a t a t l a t e t i m e th e d o m a i n sc o a r s e n w i t h a ti m e e x p o n e n t o f 1 / 3 w h i c h w a si n d e p e n d e n t o f t h e r m a l n o i s e . R e c e n t l y , L a c a s t a e t a l .[ 22 ] s t u d i e d d o m a i n g r o w t h , i n 2 D , a t l o w t e m p e r a t u r eu s i n g a c o n c e n t r a t i o n d e p e n d e n t d i f f u s i o n c o e f fi c ie n t .T h e s c a l i n g e x p o n e n t f o r t h e c h a r a c t e r i s t i c d o m a i n s i z ew a s f o u n d t o i n c r e a s e c o n t i n u o u s l y f r o m 1 /4 to l / 3w i t h i n c r e a s i n g t im e . T h e c h a n g e i n t i m e e x p o n e n t w a si n t e r p r e t e d a s o r i g i n a t i n g f r o m a c r o s s o v e r f r o mi n t e r f a c i a l t o b u l k d i f f u s i o n m e c h a n i s m s .

H a r w i c k [ 1 0 ] c o n s i d e r e d t h e e f fe c t o f t h e C o o k t e r mo n a 1 D n u m e r ic a l so l u t i o n t o t he C a h n - H i l l i a r d -C o o k e q u a t i o n b y p l o t t i n g a c h a r a c t e r is t i c s iz ee v o l u t i o n a g a in s t t im e w i t h a n d w i t h o u t t h e C o o k t e r mp r e s e n t . A t l o w t e m p e r a t u r e s , i t s e ff e c t w a s n e g l i g i b l e ,w h e r e a s a t h i g h e r t e m p e r a t u r e s ( ~ Tel4) t h e n o i s et e r m c l e a r l y a c c e l e r a t e d t h e i n i t i a l g r o w t h p r o c e s s ,a l t h o u g h a t l a t e r t i m e s t h e t w o c u r v e s c o n v e r g e d .T h e d e c o m p o s i t i o n p r o c e e d e d m o r e r a p i d l y a t h ig h e rt e m p e r a t u r e s ( ~ T o /2 ), b u t t h e C o o k t e r m h a d ap r o p o r t i o n a t e l y s m a l l e r e f fe c t. T h e r e i s n o s u b s t a n t i a le v i d e n c e f o r t h e e ff e c t o f t h e C o o k t e r m o n t h e l o n gt i m e c o a r s e n i n g b e h a v i o u r [ 2 3 ] .

3 . C O M P O S I T I O N A M P L I T U D E3 . 1 . I n t r o d u c t i o n

S o m e m a n i p u l a t i o n o f t h e ra w a t o m p r o b e d a t ai s r e q u i r e d t o d e t e r m i n e t h e m a g n i t u d e o f t h ec o m p o s i t i o n f l u c t u a t i o n s p r e s e n t i n a n u l t r a f i n e f i n es c a l e t w o p h a s e m i c r o s t r u c t u r e . B r e n n e r e t a l . [24] we ret h e f i rs t t o u s e c o m p o s i t i o n f r e q u e n c y d i s t r i b u t i o n s t od e t e c t s p i n o d a l d e c o m p o s i t i o n in a n F e - 3 2 % C r a l l o y.

T a b l e 3 . T i m e s c a l i n g e x p o n e n t s f r o m t h e n u m e r i c a l s o l u t io n t o t h e C a h n - H i l l i a r d ~ C o o k e q u a t i o nT i m e s c a l i n g e x p o n e n t f r o m 3 D a u t o c o r r e l a t i o n

( 7 / aF M a g n i t u d e o f C o o k t e r m T i m e r e g i o n F i r s t m i n i m u m F i r s t m a x i m u mI/6 0 200-+3000 0.36 ,+ 0.01 0.37 ,+ 0.011 /3 0 2 0 0 - 3 0 0 0 0 . 3 5 + 0 . 0 1 0 . 3 6 , + 0 . 0 21 0 1 0 0 0 - 3 0 0 0 0 . 3 1 + 0 . 0 2 0 , 3 4 , + 0 , 0 l1 / 3 0 . 01 20 0- 3 000 0 . 33 , + 0 . 01 0 , 34 _+ 0 . 011 / 3 0 . 0 0 0 1 2 0 0 - 3 0 0 0 0 . 3 6 4 - 0 . 0 2 0 . 3 5 4 - 0 . 0 2

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3408 J.M. HYDE et al.: SPINODAL DECOMPOSITION IN Fe Cr ALLOYS IISeveral models have been developed to quantify theextent of phase separation by analysing the sampledfrequency distribution [25--29]. These models onlycontain information on the composition amplitudeand not wavelength or spatial distribution. A randomsolid solution will generate a binomial frequencydistribution whereas a system in which phaseseparation has occurred will produce a broad ening ofthis distributi on. In the limiting case, a very heavilydecomposed sample will yield a distrib ution with twopeaks corresponding to the development of twoseparate phases. Thus, i n a two phase microstructure,the phase separat ion may be quantified by deconvolut-ing the two peaks. The choice of block size (N) hascaused much debate [30, 31]. If too small a block sizeis chosen, none of the noise is smoothed out, limitingthe significance of the calculated amplitude. In thelimit of a block size of 1, the frequency dist ributio n willconsist of two delta functions of compositions 0 and1 corres ponding to pure A and pure B. However. if toolarge a block size is chosen, the compositionttuctuations will be smoothed over. Estimates byHetheri ngton and Miller [31] for the Oak RidgeNational Laboratory AP, using the mean change incompositio n as a func tion of block size suggest that theoptimu m block size ranged from 20 atoms at the earlystages of decomposi tion to 100 atoms per block in thelater stages.Since the lateral extent of the region analysed by thePoSAP is much larger than that of the conventionalatom probe, a PoSAP experiment may be consideredas a series of parallel atom probe experiments, eachwith an effective square aperture equal to the cell areashown in Fig. 5. Since the position al informa tion foreach ion is known, the cell area may be chosenarbitrarily. For an isotropic microstructure, and agiven block size (in terms of a numbe r of atoms) a cellsize is chosen to give an approximately cubic block.Non-cubic blocks may smooth the microstructuraldetail in the elongated directions. Variations indetector effciency, local magnification effects andstatistics of collection will cause some variation ofdepth of each cell as shown in Fig. 5.

Ce l l a r e a

F i x e d n u m b e r o f io n s p e r b l o ckFig. 5. Method of sampling data into approximately cubicblocks each containing the same number of atoms. Localmagnification effects and variations in detector efficiencycause a small variation of the depth of each individual cell.

3.2. Development d frequency distributionsIn Fig. 6(a), the development of the frequencydistribution using a block size of 27 atoms is shownfrom a PoSAP analysis of the thermally aged Fe ~ 5 %Cr alloys. At the earliest stages, before significant

phase separation has occurred, the distributiolafollows the binomial form. During aging, -thedistribution first broadens and, at the latest stage,begins to show two peaks. A comparison with thefrequency distribution obtained from data from theenergy compensated atom probe (see Part I [1])illustrates an important difference between the twoinstruments. The frequency distributions from thePoSAP experiments are not centred around the meanalloy composition, in contrast to the ECAP analyses.The different positioning efficiencies for Fe and Cratoms in the P oSAP [32] lead to an inaccurate estimateof the alloy composition. The frequency distributionsdemonstrate the importance of taking into consider-ation all experimental limitations when comparingmodels with experiments. In Fig. 6(b), a composit ionfrequency distribution is shown from a Monte Carlosimulatio n ofsp inoda l decomposition at 750 K. Beforecalculating the distribution, both the detectionefficiency an d trajectory aberr ations were modelled asdescribed in the Part I of this series [1]. The resultsshow a remarkable similarity with the experimentaldata. During aging up to 10,000 MCS two distinctphases were produced and the scale of the structurewas similar to that observed in the Fe-45% Cr alloyafter aging for 500 h at 773 K.

An equivalent diagram for the development of thefrequency distribution for the numerical solution tothe Cahn -Hi llia rd equation is shown in Fig. 6(c). Sincethe composition is defined in terms of a continuouscomposition variable, no sampling is required. Theinitial peak is therefore very narrow, but rapidlybroadens retaining symmetry about the meancomposition. After 200 time units, phase separationhas occurred to such an extent that some of thecomposition samples have already reached the solidsolubility imits and, after 500 time units, the frequencydistri bution consists of two well defined peaks.A direct comparison can be made with theexperimental results by modelling the experimentallimitations (as for the Monte Carlo data) and thesampling process. Each vertex in the lattice must firstbe defined as either an A or B site. This is achieved byinterpreti ng the parameter (u + 1)/2 as the probab ilitythat there is an A atom at a site with composition u.The for m of the frequency distr ibution curves, shownin Fig. 6(d), compares favourably with both theexperimental results and Monte Carlo simulations,although no phase separation is detected during thefirst 100 time units.3.3. Quant~/.~'ing the composition amplitude

In the method by Langer et al. [33], for solvingthe non-linear Cahn-Hilliard Cook equation, the

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J . M . H Y D E e t al . : S P I N O D A L D E C O M P O S I T I O N I N F e C r A L L O Y S ~ I I 3 40 9a )

l / \,, - - c~ 24 h |o % +,00a - o . o 2 - / I~- 0.0-

0 0 . 2 0 . 4 0 . 6 0 . 8 1C o m p o s i t i o n

b) 0.05~i ~ [ o 1 M C S I^ , / /'~, ~ t 0 M C S I. ~ o . , . , , , ] r , ~ ', I z~ 1 0 0 M C S |g / I ~ , / I ~ l O O O M C S I

2 0 0 3 i ~ I 1 0 0 0 0 M CS I

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0 0 0.2 0.4 0.6 0.8 1Composition

d ) O.O5 I o 1 T i m e U n i t sL ~ 1 0 0 T i m e U n i t s0 . 0 4 ~ ~ ~ 2 0 0 T i m e U n i ts= ~ I f \ I 5 0 0 m i m e U n i t s~ 0.0 al ,~ ~ [ * 1000VimeUnits0.02,g o o v

0 0,2 04 0.6 0.8 1Composit ionFig . 6 . Deve lopm ent o f the com pos i t ion f requencyd is t r ibu t ion f rom (a ) PoSAP ana lyses o f a se ries o f the rm al lyaged Fe~45% Cr a l loys , (b ) Monte C ar lo s im ula t ion o f thedecom pos i t ion p rocess , (c . d ) a num er ica l so lu t ion to thenon- l inear Cahn H i l l ia rd Cook equa t ion . In (c ) the exac tf requency d is t r ibu t ion i s shown whereas the sam pleddistrib ution is shown in (d). In (a), (b) and (d) the size of eachcom pos i t ion b lock was 27 a tom s.

p r o b a b i l i t y d i s t r i b u t i o n i s a s s u m e d t o b e th c s u mo f t w o G a u s s i a n d i s t r i b u t i o n s o f e q u a l w i d t h ( s ). T h eG a u s s i a n s a r e c e n t r e d o n # , (1 > / q > x 0) a n d / *2( 0 < / ~ 2 < x 0) , w h e r e x o i s t h e m e a n c o m p o s i t i o n o f t h ea l l o y as s h o w n i n F ig . 7 . T h e p a r a m e t e r / h - # 2 i s t h ed i f fe r e n ce i n c o m p o s i t i o n o f t h e t w o p h a s e s a n dt h e r e f o r e t a k e n a s th e c o m p o s i t i o n a m p l i t u d e o f t h es p i n o d a l

( x o e x p L 202 _]2)

+ (1 '~ - Xo) ex p ~ j -p [ x ( r ) ] = ( ~ , - i i , ) ~ v , , ' ~ f ~

I n t h i s m o d e l , t h e r e a r e t h r e e t i m e - d e p e n d e n tp a r a m e t e r s , / q , # 2 a n d 0 . T h e r a t i o o f t h e h e i g h t s o f t h et w o G a u s s i a n s i s p r e - d e t e r m i n e d s i nc e th e m e a nc o m p o s i t i o n i s f i x e d. F o r l a r g e v a l u e s o f # l - # 2 , o r o , t h et a il s o f t h e d is t r i b u t i o n e x t e n d b e y o n d t h e p h y s i c a ll i m i t s o f 0 a n d 1 0 0 % . I n t h i s c a s e , t h e p r o b a b i l i t yd i s t r i b u t i o n n e e d s r e s c a li n g , t o e n s u r e t h a t t h ei n t e g r a t e d p r o b a b i l i t y b e t w e e n 0 a n d 1 0 0 % i s e q u a lt o 1 .

I n o r d e r t o m o d e l t h e s t a t i s t i c a l s a m p l i n g p r o c e s st h a t o c c u r s w h e n t h e a t o m p r o b e c o l l e c t s a t o m s , a n yu n d e r l y i n g p r o b a b i l i t y d i s t r ib u t i o n m u s t b e c o n v o l v e dw i t h a b i n o m i a l d i s tr i b u t i o n . T h e p r o b a b i l i t y o fo b t a i n i n g n s o l u t e a t o m s i n a b l o c k o f N a t o m s i s

P ( n ) = ~ x;'(1 - xj) Nj Iw h e r e x , i s a d i s c r e t i z a t i o n o f t h e c o n t i n u o u sc o m p o s i t i o n d i s t r i b u t i o n a n d f = p ( x 3 . I n c o n t r a s t ,h o w e v e r , n o s a m p l i n g i s r e q u i r e d i n t h e a n a l y si s o ft h e n u m e r i c a l s o l u t i o n t o t h e C a h n - H i l l i a r d C o o ke q u a t i o n , s i nc e i n th i s m o d e l t h e c o m p o s i t i o n i s w e l ld e f i n e d w i t h i n e a c h l a t t i c e c e l l o n a c o n t i n u o u s s c a le .

T h e b e s t v a l u e f o r e a c h p a r a m e t e r i n ea c h m o d e l i se s t i m a t e d b y m a x i m i z i n g t h e l o g o f t he p r o b a b i l i t y o fo b t a i n i n g t h e s e t o f e x p e r i m e n t a l d a t a , u s i n g t h em e t h o d o f m a x i m u m l i ke l ih o o d . T h e m a x i m u ml i k e l i h o o d t e c h n i q u e w i l l a l w a y s y i e l d t he p a r a m e t e r sw h i c h h a v e t h e m a x i m u m l i k e l i h o o d o f b e i n g c o r r e c t .H o w e v e r , i f t h e m o d e l i s i n a p p r o p r i a t e , t h e m o d e lf r e q u e n c y d i s t r i b u t i o n m a y b e a r n o r e l a t i o n s h i p t ot h a t o b s e r v e d . T h e X : s t a t i s t ic c a n b e u s e d t o t e s tw h e t h e r t h e e x p e r i m e n t a l d a t a c o u l d h a v e b e e no b t a i n e d f r o m t h e u n d e r l y i n g d i s t r i b u ti o n p r e d i c t e db y a m o d e l . A l e v e l o f s ig n i f i c a n c e :~ % i s c h o s e n t o t e s tt h e h y p o t h e s i s t h a t t h e o b s e r v e d d i s t r i b u t io n i sc o n s i s t e n t w i t h t h e m o d e l . S i n c e t h e X 2 f o l l o w s t h e Z2d i s t r i b u t i o n , t h e n u l l h y p o t h e s i s c a n b e r e j e c t e d i f t h eX ~ v a l u e e x c e e d s t h e t a b u l a t e d ~ % p e r c e n t i le w i t hn - 1 d e g r e e s o f fr e e d o m w h e r e n is th e n u m b e r o fc o m p o s i t i o n b i n s .

I n t h e t i m e r e g i m e s s t u d i e d , t h e L B M m o d e l f i t t e db o t h t h e e x p e r i m e n t a l a n d M o n t e C a r l o f r e q u e n c y

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J. M. HYDE et a l .: SPINODAL DECOMPOSITION IN Fe-Cr ALLOYS--II 34111

=~ 0.1- ~

: - 2 o ; o0.01 r . . . . . . . . i10 100 1000 104

MCSFig. 10. Effect of positional resolution on the measurementof composition amplitude for a Monte Carlo simulation ofan A 50% B alloy on a b.c.c, lattice with second nearestneighbour interactions aged at 750 K. To simulate a uniformresolution a Gaussian scatter of width s atomic spacingshas been added to the atomic co-ordinates. The detectionefficiencies of the Fe and Cr atoms were 0.5 and 0.3respectively.

comparisons can be made directly with the probabi litydistributions predicted by the models. At the earlystages of aging, a single narrow peak dominates, whichcan be fitted to a Gaussian with a very small width.However, at the late stages &agin g, when two narro wpeaks in the composition distribution have formed,poor fits are observed with the LBM model since thepeaks are na rrow with respect to the range o f observed

a )1~ [ ~ e ~ - A m p l it u d e I f o ot l Ex ponen t i a l F i t I /

: . ' t /i o : : 1 0 1O 0 1000Ag i ng T i me (T i me U n i ts )b )

1

//.t~

0.01

0 .0 01 I . . . . . . . . . . . . . . . . ~ . . . . . .. . , , , ~I I 0 100 1000 104Ag i ng (T i me Un i t s )Fig. 11. Development of composition amplitude for thenumerical solution to the non linear Cahn-Hilliard equation.In (a) the maximum variation in composition has beenplotted as a function of time and in (b) the LBM model wasused to calculated the composition amplitude.

Table 4. Time scaling of composition amplitude from experimentaldataExponen t with Expone nt with

Dat a series Tim e region (h) block size 27 block size 100Fe- 17% Cr 8-500 0.41 + 0.14 0.35 + 0.08Fe -24% Cr 24-500 0.37 __+0.09 0.41 + 0.10F~32% Cr 8-500 0.27 ___0.03 0.29 _+ 0.04Fe~45% Cr 4-500 0.35 _4- 0.05 0.38 _+ 0.06

compositions. Therefore, in Fig. l l(a), the co mpo-sition amplitude has been defined as the differencebetween the maximum and minimum compositionobserved during the simula tion as a fu nction of time.A rapid increase in compositio n amplitude is observedbetween 10 and 100 time units during which time littlecoarsening of the domains was observed [Fig. 4(b)].The growth in amplitude between 10 and 100 timeunits is consistent with the linear solution to theCahn-Hilliard-Cook equation which predicts expo-nenti al growth in the very early stages. Forcomparison, an e xponential curve fit through the datais also shown in Fig. 1 (a). In Fig. 1 (b), the results offitting the LBM model to the sampled data are shown.Since the modelling of trajectory aberrations anddetection efficiency of each element reduces themeasured amplitude, the rapid increase in amplitudeappears to occur at a slightly later time.

The results are not, however, consistent with theobserved experimental results. Not only is exponentialgrowth predicted by the Cahn-Hilliard-Cookequation, but the composition amplitude growthoccurs with only a small increase in wavelength incontrast to the experimental results and Mon te Carlosimulations.3 , 7 . T i m e s c a l i n g o f t h e c o m p o s i t i o n a m p l i t u d e

The time exponents for the development ofcomposition amplitude (experimental data), fittedusing the method of least squares, are higher than thetime exponents for the developm ent of scale and arealso subject to large errors (Table 4). A summar y of theeffect of detector efficiency and lateral resolutionapplied to the Monte Carlo simulations are given inTable 5. As both the efficiency of detector and lateralresolution decrease, the measured time exponentincreases (Table 5). This reflects the fact that themeasured composition amplitude is reduced as boththe efficiency and resolution decrease. The effect ismore pro nounc ed at the early stages of aging than at

Table 5. Time scaling of the composition amplitude for Monte Carlosimulation of an A-5 0% B alloy on b.c.c, lattice with second nearestneighbour interactions. The effects of block size, detection efficiency(E~, and Ec0 and lateral resolution (a) are shownTime exponent Time exponentE~:. Err a (block size 27) (block size 100)

I 1 0 0.19 _+ 0.00 0.29 + 0.000.7 0.5 0 0.23 0.01 0.31 + 0.020.5 0.3 0 0.24 + 0.00 0.35 + 0.020.5 0.3 0.5 0.25 _+ 0.00 0.26 _+ 0.040.5 0.3 1.0 0.29 + 0.02 0.43 + 0.06

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3 41 2 J . M . H Y D E et al.: S P I N O D A L D EC O M P O S I T I O N IN F ~ C r A L L O Y S - - I It h e l a t e r s t ag e s . T h e f a c t t h a t t h e g r a d i e n t i s a f u n c t i o no f b o t h d e t e c t i o n e f f i ci e n cy a n d d e t e c t o r r e s o l u t i o nd e m o n s t r a t e s t h a t t h e c o m p o s i t i o n g r a d i e n t m e a s u r e dd i r e c t l y f r o m e x p e r i m e n t a l r e s u l t s i s n o t n e c e s s a r i l yc o r r e c t .

W i t h a r e a s o n a b l e v a l u e s o f d e t e c t i o n e f fi c ie n c y( E F 0 = 0 . 5 , Ecr= 0 . 3 ) a n d s c a t t e r ( a = 1) , a t i m ee x p o n e n t o f 0 . 2 9 +_ 0 . 0 2 ( w i t h a b l o c k s i ze o f 2 7 ) w a sm e a s u r e d f o r th e M o n t e C a r l o s i m u l a t i o n s o f a nA - 5 0 % B a l lo y o n a b .c . c, l at t ic e . A t i m e e x p o n e n t o f0 .3 5 _+ 0 . 05 w a s m e a s u r e d f r o m c o m p a r a b l e P o S A Pd a t a . I t h a s b e e n d e m o n s t r a t e d t h a t t h e p o s i t io n i n ge f f ic i e n c ie s i n c r e a s e d w i t h i n c r e a s i n g d e c o m p o s i t i o n[ 3 2 ] . T h i s w i l l a l s o h a v e a n e f f e c t o n t h e m e a s u r e ds c a l i n g e x p o n e n t s i n c e t h e l o w e r d e t e c t i o n e f f i c i e n c y a tt h e e a r l y s t a g e s o f a g i n g w i ll t e n d t o p u s h t h e s ee s t i m a t e s o f t h e c o m p o s i t i o n a m p l i t u d e f u r t h e r d o w n .

4 . D I S C U S S I O N A N D C O N C L U S I O N ST h e a u t o c o r r e l a t i o n a n a l y s e s s u g g e s t t h a t a t i m e

s c a l in g r e g i m e f o r t h e d e v e l o p m e n t o f d o m a i n s iz ee x i s ts . A b e s t f i t t o t h e e x p e r i m e n t a l r e s u l t s y i e l d e d at i m e e x p o n e n t o f 0 .2 5 + 0 .0 3 . T h e M o n t e C a r l o r e s u lt sf i t te d a p o w e r l a w r e l a t i o n s h i p w i t h a t i m e e x p o n e n to f 0 .2 1 +_ 0 . 0 3 w h i c h , a l t h o u g h s l i g h t l y l o w e r t h a n t h eP o S A P r e su l ts , i s s t il l w i t h i n t h e e r r o r b o u n d s o f t h ee x p e r i m e n t a l r e s u l t s . D e s p i t e t h e f a c t t h a t t h en u m e r i c a l s o lu t io n t o t h e C a h n - H i l l i a r d - C o o ke q u a t i o n g e n e r a t e d a c o m p l e x in t e r c o n n e c t e d m o r -p h o l o g y , a t i m e e x p o n e n t w a s o b s e r v e d d u r i n gc o a r s e n i n g c l o s e t o t h e v a l u e o f 1 / 3 p r e d i c t e d b y t h eL S W t h e o r y f o r t h e c o a r s e n i n g o f i s o l a te d p r e c i p i t at e s .T h i s e x p o n e n t i s o u t s i d e t h e e r r o r b o u n d s o f th ee x p e r i m e n t a l d a t a .

A c o m p a r i s o n o f t h e f r e q u e n c y d i s t ri b u t i o n ss h o w e d g o o d a g r e e m e n t b e t w e e n t h e e x p e r i m e n t a ld a t a a n d s i m u l a t i o n s . T h e m o d e l f o r m e a s u r i n gc o m p o s i t i o n a m p l i t u d e b a s e d o n t h e t h e o r y o f L a n g e ret al. h a s p r o v e d s u c c e s s f u l , f i n d i n g g o o d f i ts t o b o t ha t o m p r o b e d a t a a n d M o n t e C a r l o s i m u l a t i o n s f o ra l l o ys w i t h a r a n g e o f c o m p o s i t i o n s a n d a g i n gt r e a t m e n t s .

B e c a u s e o f t r a je c t o r y a b e r r a t i o n s a n d t h e l o ss o fd a t a a s s o c i a t e d w i t h a t o m p r o b e a n a l y s i s , t h ec o m p o s i t i o n a m p l i t u d e c a l c u l a t ed i s a n u n d e r e s t i m a t eo f t h e tr u e e x t e n t o f d e c o m p o s i t i o n . A s a r e s u l t it i sd i f f i c u lt t o f i n d t r u e a m p l i t u d e s c a l i n g r e l a t i o n s h i p sw i t h a g i n g t i m e . I t h a s , h o w e v e r , b e e n s h o w n b y u s i n gt h e M o n t e C a r l o d a t a a n d a p p l y in g a p p r o x i m a t e l yk n o w n d e t e c t i o n e f f i c ie n c i e s a n d e s t i m a t e s o f t h er e s o l u t i o n , t h a t i t i s p o s s i b l e t o m a k e a g o o d e s t i m a t eo f t h e e f f ec t .

B o t h M o n t e C a r l o s i m u l a t i o n s , p e r f o r m e d f o ra p p r o x . 5 0 0 0 M C S , a n d t h e n u m e r i c a l s o l u t io n t o t h eC a h n - H i l l i a r d - C o o k e q u a t i o n , a ge d f or 1 0 00 t i m eu n i t s , g e n e r a t e d a s t r u c t u r e w i t h a s i m i l a r s c a l e a n dc o m p o s i t i o n a m p l i t u d e t o t h a t o b s e r v e d i n t h eF e ~ 5 % C r a l lo y a f t e r a g i n g fo r 50 0 h . A q u a n t i t a t i v ec o m p a r i s o n o f th e d e v e l o p m e n t o f i n t er f a c i a l s tr u c t u r e

a n d m o r p h o l o g y o v e r t h e se t i m e r e g i o n s i s e x a m i n e di n P a r t I I I i n t h i s s e r i e s [ 3 4 ] .Acknowledgements--The au thors would l ike to thankProfessor R . J . B rook fo r the p rov is ion o f labora to ryfaci li tie s. J .M.H . w ould l ike to acknowledge the Engineer ingand Phys ica l Sc iences Research Counc i l (EPSRC) andWo lfson Col lege fo r f inanc ia l suppor t . A .C . thanks TheRoyal Soc ie ty fo r f inanc ia l suppor t and W olfson Col lege fo rthe p rov is ion o f a Fe l lowsh ip . Th is research was funded byt h e E P S R C u n d e r g r a n t n u m b e r G R / H / 3 8 4 8 5 a n d b y t h eDiv is ion o f Mater ia ls Sc iences, U .S . Dep ar tm e nt o f Energy ,u n d e r c o n t r a c t D E - A C 0 5 - 8 4 O R 2 1 4 0 0 w i th M a r t i n M a r i e t t aEnergy Systems Inc.

R E F E R E N C E S1 . M. K . Mil le r , J . M. Hyde , M. G . Hether ing ton , A .Cerezo , G . D . W. Sm ith and C . M. E l l io t t , Acta metall.mater. 43, 338 5 (1995).2. 1. M. L ifshitz and V. V. Slyozo v, J. Phys. Chem. Solids

19, 35 (1961).3 . C . Wagner , Z. Electrochem. 65, 581 (1961).4. D. A. Huse, Phys. Rev. B 34, 7845 (19863.5. J . Piller and H. W endt, Proc. 291h Int. Field EmissionSyrup., Gothenburg (ed i ted by H . Andr6n and H .Nord 6ns), pp. 265-274. Al mq vist & Wiksell , Stoc kho lm(1982).6 . M. Kenda l l , A . S tuar t and J . K . Ord , The AdvancedTheory of Statistics, 4th edn . Gr i f f in & Box , London(1983).7. S. S. Brenner, M . K. Mille r and W . A. Sofia, Scriptametall. 16, 831 (1982).8. F. Bley, Aeta metall, mater. 40, 1505 (1992).9. J. C. LaSalte and L. H. Schwartz, Acta metall. 34, 98 9(1986).10. K. A. Harwick , Ph.D. thesis, Edin burg h, Sco tland(1991).11. G . F . M azenko , O . T . Vai ls and F . C . Zhang , Phys. Rev.B 32, 5807 (19853.12. G. F. Mazenko, O. T. Vails and F. C. Zhang, Phys. Rev.B 31, 4453 (1985).13 . J . G . Am ar , F . E . Su l l ivan and R . D . Mounta in , Phys.Rev. B 37, 196 (1988).14. M. Grant, M. S. Miguel, J . Vifials and J. D. Gunton,Phys. Rev. B 31, 3027 (19853.15. J . L . Lebowitz , J . Ma rro and M. H . Kalos , Aeta metall.30, 297 (1982).16. M. R ao , M. H . Kalos , J . L . Lebowitz and J . Ma rro , Phys.Rev. B 13, 4328 (1976).17. G. S. Grest and D. J . Srolovitz, Phys. Rev. B 30, 5150(1984).18. J . W. Cahn, Acta metall. 9, 795 (19613.

19 . A . Chakrabar t i , R . Tora l and J . D . Gunton , Phys. Rev.B 39, 4386 (19893.20 . T . M . Rogers , K . R . E lder and R . C . Desa i , Phys. Rev.B 37, 9638 (1988).21. Y. Oono and S. Puri, Phys. Rev. A 38, 434 (1988).22 . A . M. Lacas ta , A . Hernandez-Machado , J . M. Sanchoand R . Tora l , Phys. Rev. B 45, 5276 (1992).23 . R . Tora l , A . Chakrabar t i and J . D . Gunton , Phys. Rev.Lett. 60, 2311 (1988).24. S. S. Brenn er, M. K. M iller and W . A. Sofia, Int. Conf.Solid Solid Phase Transformations, Pittsburgh (edited byH. I. Aaro nsons ), pp. 191 195 . The M etall . Soc. ofA1ME, Warrenda le , Pa (1982) .25 . M. G . Hether ing ton , J . M. Hyde and M. K . Mil le r ,Mater. Res. Soc. Symp. Proc. 186, 209 (1991).26 . M. G . H ether ing ton , J . M. Hyde , M . K . Mil le r andG. D . W. Sm ith , Surf. Sei. 246, 304 (19913.27 . T . J . Godfrey , M. G . Hether ing ton , J . M. Sassen andG. D . W. Sm ith , J. Physique C6-49, 421 (1988).

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J . M . H Y D E et al.: S P I N O D A L D E C O M P O S I T I O N I N F e - C r A L L O Y S - - I I 34 132 8 . M . K . M i l l e r . A . C e r e z o , M . (3 . H e t h e r i n g t o n a n dJ . M . H y d e , Sur/~ Sci . 266, 446 (1992) .2 9 . F . D a n o i x , P . A u g e r , A . B o st e l a n d D . B l a v e t t e , S w f . S c i .246, 260 (1991) .3 0. A . C e r e z o a n d M . G . H e t h e r i n g t o n , J . P hy s i que C 8 - 5 0 ,523 (1989).3 1 . M . G . H e t h e r i n g t o n a n d M . K . M i l l e r , J . P hy s i que

C8-50, 535 (1989) .

3 2 . A . C e r e z o , J . M . H y d e , M . K . M i l l e r , G . B e v e r i n i , R . P .S e t n a , P. J. W a r r e n a n d G . D . W . S m i t h , Sur f . Sc i . 266 ,481 (1992).

3 3 . J . S . L a n g e r , M . B a r - o n a n d H . D . M i l l e r , P h y s . R e v . A11, 1417 (1975),3 4 . J . M . H y d e , M . K , M i l l e r , M . G . H e t h e r i n g t o n , A .C e r e z o , G . D . W . S m i t h a n d C . M . E l l i o t t, A c t a m e t a l l.ma t e r . 43, 3415 (1995).