advanced engineering & composite (reading paper)

8/3/2019 Advanced Engineering & Composite (Reading Paper)

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~ P e r g a m o nActa metal/, mater. Vo l. 43. No. 9. p p. 3385 3401, 1995Elsevier Science Ltd09 56 -71 51 (95 )00 040 -2 Copyright ~ 1995 Acta Metallurgica Inc.Printed in G reat B ritain. All rights reserved0956-7151/95 $9.50 + 0.00

SPINODA L DECOMPOS ITION IN Fe-Cr ALLOYS:EXPERIM ENTAL STUDY AT THE ATOMIC LEVEL ANDCOMPARISON WITH COMPUTER MODELS-- I .

I N T R O D U C T I O N A N D ME T H O D O L O G YM . K . M I L L E R I, J . M . H Y D E 2, M . G . H E T H E R I N G T O N t ~, 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 3~Meta ls and C eramics Div is ion , Oak Ridge Na t iona l L abora to ry , O ak Ridge , TN 37831-6376 , U.S .A. ,- 'Departmen t o f Mate r ia l s, Un ive rs i ty o f Oxfo rd , Pa rks R oad . Oxfo rd O XI 3PH, Eng land and3Schoo l o f Mathem at ica l and Physical Sc iences, Un ive rs i ty o f Sussex, Fa lm er , Br igh ton B NI 9QH , Eng land

(Receiced 30 Nocembe r 1994)

Abst rac t A th ree -pa r t se r ies o f pape rs is p resen ted conce rn ing the a tomic sca le ana lys is o f sp inoda ldecompos i t ion in Fe Cr a l loys . Th is f i r s t pa r t dea ls wi th the exper imen ta l techn iques and compute rs imula t ions , the second pa r t d i scusses the dynam ics o f ea r ly s tage phase sepa ra t ion , and the th i rd pa r tdescr ibes the morph o log ica l and s t ruc tu ra l cha rac te r iza t ion o f sp inoda l mic ros t ruc tu res . In th i s f ir s t pape r ,th ree -d imens iona l recons t ruc t ions o f the a tomic s t ruc tu re o f a se r ie s o f the rmal ly aged Fe Cr a l loys a reshown . Tw o method s fo r com pute r s im u la t ion o f the decompo s i t ion p rocess a re desc r ibed . The f i rs t i s ana tomis t ic s imu la t ion based on the Monte Car lo a lgo r i thm and the second i s a numer ica l so lu t ion to theCahn Hi l l ia rd Coo k theo ry . The th ree -d imens iona l a tomic sca le s t ruc tu res re su lt ing f rom decom pos i t ionwi th in the low tempera tu re misc ib i l i ty gap a re recons t ruc ted . I t i s shown tha t bo th mode ls gene ra temic ros t ruc tu res wh ich a re qua l i ta t ive ly s imi la r to those obse rved exper imen ta l ly .

I . IN T R O D U C T I O NM a n y a l l o y s a re u s e d u n d e r s e r vi c e c o n d i t i o n s i nw h i c h t h e y a r e m e t a s t a b l e o r u n s t a b l e a g a i n s t s o m ef o r m o f p h a s e s e p a r a t i o n o n t h e a t o m i c s c al e . I n as p i n o d a l a l l o y, t h e r e is n o t h e r m o d y n a m i c b a r r i e r t op h a s e s e p a r a ti o n . D u r i n g d e c o m p o s i t i o n , t h e a t o m su n m i x t o p r o d u c e a t w o - p h a s e m i c r o s t r u c t u r e . T h eF e C r s y s t e m [ 1 , 2 ] i s a n i d e a l m o d e l s y s t e m f o rs t u d y i n g t h e s p i n o d a l d e c o m p o s i t i o n p r o c e s s s in c et h e l a t t i c e m i s m a t c h i s s m a l l ( m i n i m i z i n g t h e e f f e c to f l at t ic e s t r a in ) a n d t h e o n l y c o m p e t i n g p h a s et r a n s f o r m a t i o n i s t h e s lu g g i sh p r o d u c t i o n o f s i g m ap h a s e . M o r e o v e r , t h e r e a c t i o n i s s u f f i ci e n t l y s l o wt o p e r m i t e x p e r i m e n t a l s t u d y o f t h e e a r ly s t a g e s o fp h a s e s e p a r a t i o n . A s d e c o m p o s i t i o n o c c u r s , a h ig h l yi n t e r c o n n e c t e d m i c r o s t r u c t u r e c o n s i s ti n g o f F e - r ic ha n d C r - e n r i c h e d ~ ' r e g i o n s is p r o d u c e d , w i t h ac h a r a c t e r i s t i c s c a l e ( o r w a v e l e n g t h ) o f le s s t h a n ~ 5n m .

T h e c o n v e n t i o n a l t h e o r ie s f o r s p i n o d a l d e c o m p o -s i t i o n p r e d i c t t h a t t h e p h a s e s e p a r a t i o n c a n b e d i v i d e di n t o t w o r e g i m e s . D u r i n g t h e f i r s t s t a g e a d o m i n a n tw a v e l e n g t h f o r m s a n d t h e d i ff e r e n c e i n c o n c e n t r a t i o nb e t w e e n t h e t w o p h a s e s i s p r e d i c t e d t o i n c r e a s ee x p o n e n t i a l l y t o w a r d s t h e l i m i ts d e fi n e d b y t h em i s c i b i l i t y g a p . O n f u r t h e r a g i n g , t h e s i z es o f t h e

t D e c e a s e d .:~To wh om all corres pond ence sh ould be addressed.

a n d ~ ' r e g i o n s a r e e x p e c t e d t o i n c r e a s e b u tt h e c o n c e n t r a t i o n s o f t h e t w o p h a s e s s h o u l d r e m a i na p p r o x i m a t e l y c o n s t a n t . H o w e v e r e x p e r i m e n t a l e vi -d e n c e s u g g e s t s t h a t b o t h t h e m i c r o s t r u c t u r a l s c a l e , a n dt h e d i f f e r e n c e i n c o n c e n t r a t i o n b e t w e e n t h e t w op h a s e s , i n c r e a s e c o n c u r r e n t l y i n t h e s p i n o d a l r e g i o n o ft h e F e - C r s y s t e m [ 3 , 4 ].

S e v e r a l d i ff e r e n t t e c h n i q u e s h a v e b e e n u s e d t oc h a r a c t e r i z e t h e d e c o m p o s i t i o n i n s p i n o d a l a l l o y s .R e c e n t e x p e r i m e n t a l s t u d i e s h a v e i n c l u d e d t h e u s e o fX - r a y a n d s m a l l a ng l e n e u t r o n s c a t t e ri n g t o m e a s u r et h e s t r u c t u r e f u n c t i o n [ 5 9 ]. A s c o a r s e n i n g o c c u r s a n dt h e s c a l e i n c r e a s e s , t h e r e i s a d e c r e a s e i n t h ew a v e n u m b e r o f th e p e a k o f t he s t r u c t u r e f u n c t i o n .T h e s e r e c i p r o c a l s p a c e t e c h n i q u e s a r e u s e f u l f o rf i n d i n g a v e r a g e q u a n t i t i e s a n a l y s i n g a n y p e r i o d i cm o d u l a t i o n s i n t e rm s o f F o u r i e r m o d e s . H o w e v e r , n oi n f o r m a t i o n i s o b t a i n e d c o n c e r n i n g t h e d e ta i l e dm o r p h o l o g y i n r e a l s p a c e a n d s o a d i r e c t c o m p a r i s o nw i t h b o t h t h e o r e t i c a l m o d e l s o r c o m p u t e r s i m u l a t i o n sa t t h e a t o m i c s c a l e is i m p o s s ib l e . T h e c o n v e n t i o n a la t o m p r o b e f i e ld i o n m i c r o s c o p e [ 1 0, 1 1] h a s s u f f i c i e n ts p a t ia l r e s o l u t i o n t o d i r e c tl y i m a g e t h e m i c r o s t r u c t u r ea n d i s a b l e t o c o l l e c t i n d i v i d u a l i o n s w h o s e c h e m i c a li d e n t it y m a y b e d e t e r m i n e d . T h u s l o c a l c o m p o s i t i o n sc a n b e c a l c u la t e d o n t h e s u b n a n o m e t e r s c al e . T h ed e v e l o p m e n t o f t h e t h r e e - d i m e n s i o n a l ( 3 D ) a t o mp r o b e s , s u c h a s t h e p o s i t i o n - s e n s i t i v e a t o m p r o b e( P o S A P ) [ 1 2 - 1 4 ] a n d o p t i c a l a t o m p r o b e ( O A P )[ 15 , 1 6 ] , e n a b l e s t r u e 3 D c h a r a c t e r i z a t i o n o f b o t h

AM 4~ J 3385


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3386 MILLER e t a l . : SPINODAL DECOMPOSITION IN Fe-Cr ALLOYS Icomposition and microstructure on the atomicscale. Other conven tional direct techniques such astransmis sion electron microscopy have proved ineffec-tive in resolving the fine scale microstructure of thebinary Fe-Cr system due to the lack of contrastbetween the phases.The high performance of mini-supercomputersnow enables sufficiently large 3D simulations ofatomic scale processes to yield realistic results.With these tools, numerical simulations and MonteCarlo methods become important techniques forstudying phase transitions. Following the emergenceof 3D at om probes, it is possible to compare directlythe microstructure resulting from fine-scale phaseseparation with that predicted by theoretical calcu-lations or numerical models. In this series of papers,the phase separation occurring within the low-tem-perature misc ibility gap in a series of Fe Cr alloys iscompared with two models of the decompositionprocess.

The microstructures have also been related to theobserved mechanical properties. The increase inhardness (and decrease of impact strength) duringspinodal decomposition is of crucial engineeringimportance, notabl y in the duplex stainless steels usedin the primary cooling circuits of many nuclearreactors. The time and temperature dependence of thespinodal reaction needs to be understood in order tosatisfactorily predict the long-term behavior of thesematerials.

2. MATERIALS AND HEAT TREATMENTSThe majority of the research reported in this series

of papers was performed on a high purity Fe -45 at. %Cr alloy. This composition was chosen to be in thecentre of the slightly asymmetrical miscibility gap.This alloy was fabricated from high purit y elementalFe (>99.99% purity) and Cr (>99.996% purity).Chemical analysis of the alloy after arc-melting in adry argon atmosphere revealed that it contained a totalof ~ 100 ppm o f interstitial impuri ties by weight. Wire,of diameter 0.25 mm, was fabricated from the masteringot by a com bina tion of swaging and wire drawingoperations. Special precautions were taken to preventcontamination of the material during these stages.Some additional experiments were performed onFe 17, 19, 24 and 32 at.% Cr alloys which werefabricated in exactly the same ma nne r from the sameinitial materials.

Prior to ex aminat ion in the atom probe, the wires ofthe Fe-17, 19, 32 and 45% Cr alloys were solutiontreated in argon (0.4 atm) for 2 h at 1273 K, waterquenched and isothermally aged at 773 K for times upto 500 h and water quenched. The Fe-24% Cr alloywas solution treated for 16h at 1223 K prior toisothermal aging at 773 K. In addition to these 773 Kisothermal treatments, a series of isothermal heattreatments at temperatures rang ing from 673 to 923 Kwas performed to estimate the position of the

500 ---o- - 2 4 % C r 5 0 0 C l ,,*450 [ ] 3 2 % C r 5 0 0 C ] J

- * - - 4 5 % C r 5 0 0 C I /Z " 4 0 0 I 1 /"I - f> 3 5 0 .- / I j ~= 3 0 0 . , ~ . / ~ o

2 5 02O O1 5 0 . . . . . . . . F ' ' ~ 7 q q0 1 0 1 O 0 1 0 0 0T i m e ( h )

Fig. 1. Hardness vs heat treatment for Fe Cr alloyscontaining 24, 32 and 45% Cr (200 g load).

miscibility gap by the experimental observa tion ofphase separation in the field ion images.

The change in microhardness as a func tion of agingtime at 773 K was determined from the ends of thewires with the use of a Shimadzu hardness tester witha 200 g load. The results are shown in Fig. 1 for the 24,32 and 45% Cr alloys. The increase in hardness wasgreatest for the alloys with the highest Cr content, asexpected due to the higher volume fraction of theCr-enriched ~' phase. The time exponents of thechanges in hardness were determined by fitting each setof data to a power law as indicated by the lines in Fig. 1and were 0.13 + 0.02, 0.12 + 0.01 and 0.10_+ 0.01,respectively for the 24, 32 and 45% Cr alloys.

These microhardness results correlate well with thework performed by Marcinkowski e t a l . [2]. Theirroom temperature hardness measurements on anFe~ 7.8 at.% Cr alloy quenched from 1273 K andaged at 773 K showed that the average hardnessdoubled with aging before reaching an apparentplateau of 470 VHN after 500 h aging.

3. EXPERIMENTAL TECHNIQUES AND DATAANALYSIS PROCEDURES3 . I . A t o m p r o b e m i c r o a n a l y s i s

The primary technique used to characterize themicrostructures of these alloys was atom probe fieldion microscopy (APFIM). A detailed description ofthis technique may be found in the monograph byMiller and Smith [17]. In order to perform a completemicrostructural characterization, both an energy-compensated atom probe and two variants of thenewer 3D atom probes were used, a PoSAP and anOAP.

The field ion microscope (FIM) provides an atomicresolution view of the surface of a specimen.Variations in composition often generate phasecontrast on the image but no compositionalinformation is available. The FIM images shown inFig. 2 show the increasing scale of decomposition asa functi on of aging in a series of Fe ~ 5% Cr alloys. Themorpholo gy of the microstructure may be observed bytaki ng a series of field ion micrographs as the specimenis field evapo rated to expose new layers of material (i.e.serially sectioned). Field desorption micrographs of


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MILLE R eI a /. : SPINOD AL DECO MPOSITIO N IN Fe Cr ALLOYS 1 3387

Fig. 2. Field ion micrographs from Fe 45% Cr samples aged for (a) 4, (b) 24. (c) 100 and (d) 500 h. Thebrightly imaging regions are Cr-enriched and the dark regions Cr-depleted.

the spec imen sur lTtce , genera ted bv indiv idualf i e ld-evapora ted ions in the opt ica l a tom probe , of tenshow bet te r phase co nt ras t than f i e ld ion micrographs .I n F i g . 3 a s e ri e s o f l i e l d d e s o r p t i o n m i c r o g r a p h s s h o wh o w t h e m o r p h o l o g y o f ph a s e s e p a r a te d Fe Cr a l lo y sv a r i es wi t h t h e Cr c o n t e n t . As t h e Cr c o n t e n t o f t h ea l loy i s increased , the volu me f rac t ion o1" the z" phaseincreases and there i s a t r ans i t ion in the mo rph olo gyo f th e ~ ' p h a s e f r o m a p p r o x i m a t e l y s p h e r ic a l p a r t i c le s[Fig . 3(a) ] to a fu l ly in te rconnected s t ruc ture[Fig . 3(d) ] . Al though the 24% Cr a l loy appearsto cons i s t s imply of i so la ted regions , f i eld eva por a t io nsequences de mon s t ra te tha t the :~ ' phase form s someinterconnected regions and some i so la ted par t i c les .

I n o r d e r t o o b t a i n c o m p o s i t i o n a l i n f o r m a t i o n , a na t o m p r o b e c o m p o s i t i o n a l a n a l y s i s m u s t b e p e r -f o r m e d . At o m p r o b e s c o l l e c t i n d i v i d u a l i o n s wh o s echemica l ident i ty can be de termined. The loca lc o m p o s i t i o n c a n b e c a l c u la t e d o n t h e n a n o m e t e r s c al ea n d u s e d t o g e n e r a t e a o n e - d i m e n s i o n a l ( I D) d e p t hprof i l e . The 3D a tom probes can be used to recons t ruc te l e m e n t a l m a p s o f th e s a m p l e i n t h re e d i m e n s i o n s wi t hn e a r a t o m i c r e s o l u t io n . M o r e o v e r t h e a n al y s e s m a y b eu s e d t o d e t e r m i n e s e v e r a l a d d i t i o n a l p a r a m e t e r s

re la ted to the micros t ruc tura l sca le and e lementa lconcent ra t ions , as descr ibed in the fo l lowing sec t ionsand in Parts 11 and I l l of this series [18, 19].3.2. Experime#ztal coHditions

Th e e x p e r i m e n t a l c o n d i t i o n s we r e c h o s e n t o s u i t t het y p e o f an a l y si s a n d t h e r e f o r e v a r i e d a m o n g e a c h o f t h et e c h n i q u e s u s e d . Th e f i e l d i o n m i c r o g r a p h s we r eo b t a i n e d wi t h t h e u se o f n e o n a s t h e i m a g e ga s a n d wi t ha s p e c i m e n t e m p e r a t u r e o f b e t we e n 6 0 a n d 8 5 K. Th ef i e l d d e s o r p t i o n m i c r o g r a p h s we r e r e c o r d e d wi t h as p e c i m e n t e m p e r a t u r e o f 60 K . Al l 1 D a t o m p r o b ea n a l y se s we r e o b t a i n e d wi t h t h e u se o f th e Oa k Ri d g eN a t i o n a l L a b o r a t o r y e n e r g y - c o m p e n s a t e d a t o mp r o b e ( E CAP ) [2 0] wit h a s p e c im e n t e m p e r a t u r e o f 5 0K a n d a p u ls e f r a c ti o n o f 2 0 % . I n t h e Po SA Pexper iments , a re la t ive ly h igh ana lys i s t empera ture of8 0 K wa s c h o s e n t o m i n i m i z e t h e l o ss o f i n f o r m a t i o nc a u s e d b y t he s i m u l t a n e o u s e v a p o r a t i o n o f m o r e t h a none C r a tom on a s ingle pul se [21] and a pul se f rac t ionof 15% w as used . An ion ra te of be twee n 50 and 100pulses per ion was used as a compromise be tweenm i n i m i z i n g t h e n u m b e r o f m u l t ip l e e v e n t s a n dm a x i m i z i n g t h e r a t e o f d a t a a c q u i s i t i o n .


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3388 MILLER et al .: SPINODAL DECOMPOSITION 1N Fe-Cr ALLOYS--I

(2 q

C~ (

Fig. 3. Field desorption micrographs front Fe-Cr alloys aged for 500 h at 773 K as a function of Crcontent--(a) Fe 19% Cr, (b) Fe 24% Cr, (c) Fe-32% Cr and (d) Fe~,5% Cr. The bright regions areCr-enriched and darker regions Cr-depleted.

3.3. Composition profileOn the atomic scale, it is impossible to trea t the local

composition of an alloy as a continuous variable. Acomposition must be defined over some local region,called a coar se-gra ining volume, which is small withrespect to the length scales of features of interest, butconsiderably larger than the size of an atom. Acomposition profile may be generated by dividing thesequence of ions from an ECAP analysis into blocksand plotting the block composition against blocknumber. The blocks can be chosen so that each hasan identical volume or so that each contains afixed number of atoms. In the former case, localmagnification effects will cause different blocks tocontain different numbers of ions making statisticalanalys is difficult. The size of each block is chosen soas not to smooth over the composition fluctuations,but large enough so that most of the statistical noiseis excluded. A qualitative examination of a compo-sition profile can be used to detect compositionfluctuations. The analysis may be applied to PoSAPdata, by selecting a sub-volume, usually in the form ofa cylinder of square cross-section, which is then

divided into blocks. However, much longer compo-sition profiles are obtaine d from conv ention al atomprobe analysis because of the greater depth analysed.For isotropic morphologies, the blocks are usuallychosen to be approximately cubic, with a size whichdoes not smooth over the composition fluctuations,but is large enough to exclude most of the statisticalnoise.

ECAP composition profiles through an Fe~ ,5% Cralloy aged at 500"C for 24 and 500 h are shown inFig. 4(a, b). Equivalent PoSAP depth compositionprofiles using a moving average smoothi ng with a fixedsample v olume (1 nm 3) are shown in Fig. 4(c). In theearly stages of decomposition small compositionfluctuations are evident. At later stages of decompo-sition, well defined sinusoidal type compositionmodulations are observed. Note that the observedmaximum composition in each vein does not appearconstant. This is the natural consequence ofsimultaneous sampling of both phases, rather thansampling pure Fe-rich regions and pure Cr-enrichedregions. An approximate estimate of the wavelengthcan be made by counting the number of wavelengthstraversing the volume. After 24 h the wavelength is


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M I L L E R 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 1N F e C r A L L O Y S - - I 3 38 9a )

0. 80. 70 . 6 -0 . 5 -0 . 4 -0 . 3 -0 . 2 -0.1

E C A P- 2 4 h

i i I i i5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0Dis tan ce (33 Ion Blocks)b) ECAPI - - 5 o o ],-~ 0,8~ 0 . 6 -~ 0 , 4 -~ 0.2 ~

5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0Dis tance (33 Ion B l o c k s )C ) 1 P o S A P

500 h I0.6 I 2 4 .~ 0 .6

o 0.4O~ 3 o . 2

: , . , ,

0 i i i r i i i0 4 8 12 16n m

F i g . 4 . C o m p o s i t i o n p r o fi l e s f o r tw o F e ~ 5 % C r a l lo y st h e r m a l l y a g e d f o r 2 4 a n d 5 0 0 h . i n ( a , b ) , r e su l t s f r o m a ne n e r g y c o m p e n s a t e d a t o m p r o b e u s i n g a c o n s t a n t b l o c k s iz eo f 3 3 io n s a r e s h o w n . I n ( c) d a t a f r o m P o S A P e x p e r i m e n t su s i n g a c o n s t a n t s a m p l e v o l u m e o f 1 n m ~ i s d i sp l a y e d .

approx. 2 nm, and increases to over 4 nm after 500 h.More accurate statistical techniques for analysing thedevelopment of microstructur al scale are presented inPart II of this series [18]. The increase in doma incomposit ion can be more clearly observed by plottin ga frequency distribution,3.4. Frequency dis tr ibut ions

A frequency distribution curve from a conve ntionalatom probe experiment is constructed by dividing thesequence of atoms into equal sized blocks of atoms andplotting the observed number of blocks with eachcomposition. A rando m solid solution will generate abinomial form, whereas a system in which phaseseparation has occurred will exhibit a broadening ofthis distribution. In the limiting case, a heavily

decomposed sample will yield a dist ributio n with twopeaks corresponding to the development of twoseparate phases. The development of the frequencydistribu tion is shown in Fig. 5 for the Fe~45% Cr alloyaged between 4 and 500 h at 773 K and analysed inan energy-compensated atom probe. At the earlieststages, before significant phase separation hasoccurred, the distribu tion follows the b inomia l form.During aging, the distribution first broadens and, atthe latest stages, begins to show two peaks. Methodsfor quantif ying the extent of decomposition based onthe form of the frequency distrib ution are discussed inPart II of this series [18].3 . 5 . 3D a tom probe da ta r epresen ta t i on

The simplest and most intuitive representation of3D atom probe data is to use a sphere to represent eachindividual atom. Different types of atom may berepresented by different colours and the use of variousforms of shading and lighting conditio ns can be usedto highlight various features. This method has beenused in Fig. 6 to show the distribution of Fe and Cratoms in an Fe-4 5% Cr specimen after aging for 500 hat 773 K. The lighter-coloured spheres represent Featoms and the darker spheres represent Cr atoms.Clearly, thermal aging has generated a two-phaseinterconnecte d web consisting of Cr-enriched regionsand Fe-rich regions. The figure reveals that each"domain" is only a few nanometers thick.

Since the atoms at the surface of the analysedvolume obstruct the line of sight from those atoms inthe bulk, it is impossible to visualize the internalstructure by the method used in Fig. 6. Forcomplicated morphologies, it is useful to define aregular 3D grid of composition cells, as shownin Fig. 7(a). Each cell is assigned a compositionaccording to the local environm ent of atoms. The sizeof an individual cell is close to that of an atomicspacing to ensure that spatial inf ormat ion is not lost.However, this means that each cell contains, onaverage, only one or two atoms. Since trajectoryaberratio ns occur during field evaporatio n the originallocation of each atom can not be precisely located. Thisinvariab ly means that many cells will conta in no

0 .0 5 ~ ~ 4 h Io 8 h! / '% . - - - * - - 2 4 ho .o4~ /~ 100h" ~ " ~ 1 , - 5 0 0 h

0-V T . . . . . I I I - 7 . . . . . - - " ~

0 0 . 2 0 . 4 0 . 6 0 . 8 1F e C o m p o s i t io n C r

F i g . 5 . F r e q u e n c y d i s t r i b u t i o n s f r o m E C A P a n a l y s e s o f as e r i e s o f F e - 4 5 % C r a l l o y s a g e d u p t o 5 0 0 h . A b l o c k s i z e o f3 3 i o n s w a s u s e d .


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3390 MILLER el a/.: SPINODA L DECOM POSITION IN Fe--Cr ALLOYS---I

|

5 nmFig. 6. Atomic reconstruction of an F e-45 % Cr alloy aged for 500 h. Th e lighter spheres represent Fe atom sand ~.be dark er spheres Cr atoms.

a tomic in f or mat ion l )-om which a com pos i t ion ma y becalculated. Moving aver age sampl ing methods mus tbe introduced to ensure that no "holes" remain in thedata. Since an average data set wil l contain at least200,000 atoms, ~he algor i thm must be extremelyefficient.

Two types of sampling are considered. In the f ir s t[Fig. 7(b) ] , a l l points within the sampling volumecont r ibu te equa l ly to the compos i t ion ca lc t l l a t ion(s imple smoothing) . This wil l tend to smooth out f ine

detai l present in the s tructure , The second, centre-we ighted , smooth ing method i s heavily b iased towar dsthose a toms nea r es t to the cen t r e of the smooth ingvolume [Fig. 7(c)] . This a lgor i th m is essentia l ly a "ho lef i l l ing" algor i th m s ince i t ass igns compos it ions to cel lsf r om wi th in which no a toms could be loca ted wi thouts ign i fican tly a ff ec ting the compos i t ions of sur r oundingcells . The result of this smoothin g is a gr id cont aininga high er level of f ine detai l . The s ize of the av eragingvolume is not cr i t ical provi ded th at a l l of the cel ls may


7/17

MILLER et al.: SPINODAL DECOMPOSITION IN Fe-Cr ALLOYS--I 3391a) Composition b) SimpleGrid Smoothing C) Centre-WeightedSmoothingI I

Weighting WeightingFig. 7. Principle of composition grid. The grid used to calculate composition is of a similar size to the atomicspacing (a). Individual cells on the grid may have no atom positions within their volume. Moving averagesmoothing methods are used to ensure all points on the grid have composition data. In simple smoothing(b), atom positions over the shaded area contribute equally to the composition at the grid point, whilecentre-weighting the data (c) produces a composition biased towards atoms close to the grid location.

be assigned a local composition. Typically a cubicregion with side 4 cells (simple smoothing) or 5 cells(centre-weighted smoothing) is used.

For a binary alloy, the compositions can befalse-colour coded. The range of colours in therainbow spectrum can be mapped to the compositionvalues, for instance from blue representing A-richregions through green to red representing B-richregions. A series of cross-sections through thestructure can be used to represent the 3D morpholo gyand to observe the degree of interconnectivity. Agrey-scale equivalent is shown in Fig. 8.

A binary image can be formed by defining acomposition threshold. Cells with compositions aboveand below this threshold are assumed to originatefrom regions of differer]t phases. The choice ofthreshold co mposition is critical since it determines theappare nt vol ume fraction of each phase. The simplestchoice for a threshold value is the mean alloycomposition. This choice would be suitable for analloy in which the two phases had concentrationssymmetrically positioned on either side of the meanalloy composition. However, if the volume fractions ofthe two phases were not equal this choice couldgenerate an interface within one of the phases as shownin Fig. 9. In this diagram, two frequency distribut ionsare shown, the first corresponding to a n unage d alloyand the second to the late stages of aging after twodistinct phases have formed. Both have the same meancomposition. Choosing the mean composition to bethe threshold level is clearly not suitable. Choosing thethreshold level as the comp osition of the med ian blockcould yield a similar result. An alternative choicewould be that of the mid-value between the maximu mand minimum compositions observed. However thisvalue would be influenced too much by statisticalvariations at the upper and lower end of thedistribution. The method selected attempts to avoidthese statistical variations by locating the positions of

the upper and lower deciles and using the mid valuebetween the two. As can be seen from Fig. 9, a muchbetter estimate of the true threshold composition isobtained. Several other choices for a threshold valueare possible. For instance, if the frequency distributi onconsists of two distinct peaks, the mi nim um valuebetween the two peaks could be selected.3.6 . I sosur face r econs t ruct ion

The sequence of binary cross-sections can beextended to generate the full 3D microstructure bygenerating an isosurface, the 3D equivalent of acontour. A composition level is chosen and a surfacedrawn over all points with this composition. Theresulting structures consist of thousands of polygonswhich can be automatically shaded and manipulatedin 3D using visualization software to give a full 3Dimpression. This technique has proved particularlyuseful for ex amining percolated or highly inter-connected structures.

An isosurface representation was used to show theatomic scale composition variations found in Fe-Crspecimens after aging at 773 K. A simple smoothingalgorithm was used to generate the composition grid.Alloys with a low Cr content phase separate to givemicrostructures with a low volume fraction of thesecond phase. The microstructure therefore consistsof isolated Cr-enriched particles within an Fe-richmatrix. An example is shown in Fig. 10(a), where anisosurface reconstruction, drawn at a conc entrat ion of20% Cr, has been used to represent the Cr-enrichedregions from an Fe-17% Cr alloy aged for 500 h. InFig. 10(b), the isosurface from an Fe 24% Cr alloyaged for 500 h d emonstrates that both isolatedparticles and veins traversing the analysed volume canco-exist. The Cr-enriched regions of the Fe-32% Cralloy aged for 500 h form a percolating structure withlittle evidence for large isolated particles [Fig. 10(c)].Analyses of the Fe~, 5% Cr alloys show that after 4 h


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5 n mF i g . 8. S e ri e s o f c ro s s - s e c t i o n s t h r o u g h t h e s t r u c t u r e o f a n F e ~ 4 5 % C r a l l o y a g e d f o r 5 0 0 h . T h e c o m p o s i t i o n sw e r e c a l c u l a te d u s i n g s i m p l e s m o o t h i n g . T h e d a r k r e g i o n s a r e C r - e n r i c h e d a n d t h e l i g h te r r e g i o n s F e - ri c h .T h e s e r i e s r e p r e s e n t s a d e p t h o f a p p r o x . 4 n m .


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MILLER et al.: SPINODAL DECOMPOSITION IN Fe Cr ALLOYS--1 3393In i t ia l Co mp osi t ion Dis tr ibut ionFinal Composi t ion Dis tr ibut ion

M e a nCornpos i t ion'~b -~ ]M/ , ~ 1 ' -~~ " ~ O ~ m p o s i t i n / I / \ 1- I ot- ~

i i i i i i0 2 0 . 4 0 . 6 0 . 8 1CompositonFig. 9. Broadening of composition frequency distributionduring spinodal phase separation. The lower (and upper)deciles mark the composition below (and above) which 10%of the composition values are observed. The thresholdcomposition, which represents the interface between the twophases, is defined as the mid-point between the compositionsof the upper and lower deciles.

aging there is little evidence of any decomposition.whereas after 24 h, Cr-enriched veins have begun toform. A highly interconnected two-phase micro-structure is fou nd after aging for 100 h. In Fig. 10(d),an isosurface has been drawn to represent theCr-enriched regions present afte r aging for 500 h.Figure 10(a-d) can be directly compared with the fielddesorption micrographs in Fig. 3. In Fig. 10(e)centre-weighted smoothing, which retains a higherlevel of detail tha n simple smoothing, has been used tocalculate the local composition in an Fe~5% Crsample after aging for 500 h. Although the phases arewell defined, it is apparent that the interface is roughon the atomic scale and that some very small isolatedCr-enriched regions exist. At this stage of decompo-sition, the domain s are approx. 2.5 nm across. In all ofthese isosurface reconstructions, no 'preferred growthorientation was detected, consistent with the smallstrain associated with the system.

4. COMPUTER SIMULATIONS4. i . Theories o t spinodal decomposi tion

Conventional theories of phase separation have sofar been based on Gibbs ian thermodynamics. Implicitin the models is the use of a c oarse-graining volumelarge enough for the composition to be treated as acontinuous parameter. The coarse-graining volumemust therefore be considerably larger than the scale ofa lattice site. However, the volume must also be smallin comparison with the length scales of the featuresof interest such as the wavelength of compositionfluctuations. A mathematical theory of spinodaldecomposition was developed by Cahn and Hilliard[22, 23]. The linear so lution explained the growthof composition waves and their crystallographicorientation but was only valid at the very early stagesof decomposition. Cook [24] later improved theCahn -Hil liar d model by including a term to modelthermal noise. Non -li near theories [25, 26], also based

on continuum models, have been developed todescribe the growth rate at intermediate and laterstages of decomposition but these have met with o nlylimited success, and yield no infor mati on on thedetailed morphology of the resulting microstructures.In the present work, two models of the decompositionprocess have been implemented: a discrete atomisticmodel and a numerical solution to the non-linearCahn-Hilliard-Cook equation.4.2. The d ) 'namic Ising model

Phase separa tion in alloys is generated by diffusionof atoms thro ugh the microstructure, a nd is driven bythe reduction of free energy. The Monte Carlotechnique can be used to model this process. AHamiltonian is defined in terms of "pair-wise" bondenergies summed over all possible interactio ns and thechange in enthalp y is calculated when a r and om pairof nearest neighbou r atoms is swapped. If only nearestneighbour interactions are considered, the change inenergy, AH, will be a multiple of the interactionpa rameter, e = EAA + EBB -- 2EAB, where EAA, EBB andE~B are the energies associated with a A-A, B-B andA-B interactions respectively. The probabilityP ( x ~ x ' ) that the new lattice configurat ion is acceptedis given by the Metropolis algorithm [27] and thechange in energy AH

P ( x ~ x ' ) = I : x p [ - A H / k T ] AHO

The algorithm is not designed to simulate exactmicroscopic behaviour. For this, a pre-factor in theform of(A/z) exp [ - Q/kT] is necessary where Q is theactivation energy, A a positive constant and 1/z theattempt frequency. This would simulate the latticevibration process but would also necessitate a largenumb er of attempts (determined by the magnit ude ofQ) before a swap actually occurs. In the case where thediffusion barrier is assumed consta nt, the algorithm isa good simplifying model. Clearly, in real systems, Qwill be a fun ction of the local atomic configuration. Ifthe pre-exponential term is the same for both species,this term simply scales the time, and so it can beignored.

Vacancy diffusion can be modelled by allowing a"hole" in the lattice to execute a rand om walk. In thissituation the energies of "bo nds " between the vacancyand su rroun ding atoms need to be defined. Thesimplest method is to define the A-vacancy "bond"and B-vacanc y "bon d" energies as being equal so thatno preferential segregation of vacancies to any singlephase occurs. I f only a single vacancy is allowed, thena vacancy vacancy interaction does not need to bedefined. Simulation s using a va cancy mechanism havegiven very similar results to the "unphy sical" nearestneighbou r swapping technique [28]. For simplicity, allsimulations reported here used the atom swaptechnique.


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3394 MILLER eta/. : S P I N O DA 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

(a )

Cb)5 nm

Fig. 10(a b). (Caption on p. 3397.)

5 nm


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MILLER c: al. : SPINODAL DECOMPOSITION IN Fe Cr ALLOYS--I 3395

Fig. 10(c). (Caption on p. 3397.)5 nm

In the case of a b.c.c, lattice (such as the Fe Crsystem), simply using nearest ne ighbour intera ctionsto calculate the enthalpy associated with a lattice siteis clearly not realistic, since the six second nearestneighbours are not much further away than the eightnearest neighbours and therefore poorly screened.Second nearest neighbour interactions must also beused, and weighted according to how the interatom icpotentials vary as a f unction of distance (s). In the 3dtransitio n metals, the bindin g potential falls approxi-mate ly as s ~ [29].

The nearest neighbours are situated at a distance(a~v/3)/2, where a0 is the lattice parameter. The secondnearest neighbours are situated at a distance a apart.The weighting is therefore given by

For a simple or f.c.c, lattice, an a lgorit hm using onlynearest neighbour interac tions is sufficient. T ime in the

Monte Carlo model is defined in terms of Monte Carlosteps (MCS). If there are N atoms in the simulationthen one MCS corresponds to N attempted swaps.

To model phase separation in the Fe-Cr system,it is necessary to match the phase diagram fromthe Monte Carlo simulations to the experimentallyobserved F~ C r miscibility gap. The miscibility gap inthe b.c.c, region of the Fe- Cr phase diagram was firstexperimentally located by Williams and Paxto n [1, 30],using hardness and electrical resistivity measurements.A critical temperature of ~ 830 K was found. Withi nthis miscibility gap the only competing phasetra nsfo rmatio n is a sluggish sigma phase reaction [31 ].The appro ximate form of the miscibility gap has beenconfirmed by other e xperimental studies, for instanceNishizawa et al. [32]. However Kuwa no [33], usingM6ssbauer spectroscopy to follow the time develop-ment of the Cr-rich paramagnetic phase, reported acritical temperature (Tc) of ~95 0 K. The miscibilitygap has also been located by thermodynamiccalculations which include the magnetic transition.


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3396 MILLER et al.: SPINODAL DECOMPOSITION IN Fe Cr ALLOYS--IAnders son and Sundma n [34] predicted T~ ~ 900 K ata composition C~ of ~0.5. A summary of theseexperimental results and thermo dynami c calculationsis shown in Fig. 11. Also included in Fig. 11 are datafrom FIM observations. The closed circles representexperiments in which phase separatio n was observedand the open circles represent experiments in which nophase separation was observed.

To introduce a temperature into the Monte Carlosimulations, the critical temperature for an eq uiatomicsimulati on was set to 900 K. Since the critical energyparameter ( - E / k T c ) is known for each lattice [35],choosing a critical temperature fixes the corresp ondingvalue of the inter action parameter (Q. A good ma tchis attained both with experimental results andthermod ynamic calculations (Fig. 12).

The miscibility gap in the Fe -Cr system is not quit esymmetrical and the Fe~5% Cr alloy produces a50% volume fraction of each phase at 773 K. Sincethe miscibility gap defined by the Mont e Ca rlo modelis symmetric, simulations containing 50% A and50% B atoms were chosen to model the realalloys. Simula tions were performed on a b.c.c, grid(80 x 80 80 lattice units) with bot h first a ndsecond nearest neighbour interactions for times upto 10,000 MCS at a temperature of 750 K.Further simulations h'~ve also been performed on arange of different lattice types, lattice sizes andtemperatures. During the "time scales" examined inthis series of papers, the development of phaseseparation was not influenced by the simulationsize.

(d)

Fig. 10(d). (Caption on p . 3397.)5 nm


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M I L L E R 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 - - Iga i t

3397

5 nm

Fig. 10. lsosurface reconstructions from PoSA P analyses showing how the morpholog y of the Cr-enrichedregions ofFe Cr alloys varies as a function of Cr content after aging for 500 h at 773 K. The composit ionsare (a) F ~ I 7% C r (isosurface at 20% Cr), (b) Fe- 24% Cr (isosurface at 30% Cr), (c) Fe-32% Cr (isosurfaceat 30% Cr) and (d) Fe- 45% Cr (isosurface at 40% Cr). In each simple smoothing has been used to generatethe comp osit ional inform ation. In (e) centre-weighted smoothing was used to sho w the f ine scale detail inan Fe-.45% C r alloy. The isosurface is drawn at a conc entratio n of 40% Cr.

A f t e r a g i n g f o r 5 0 0 0 M C S a t 7 5 0 K , t h e I s in g m o d e lh a d p r o d u c e d a t w o - p h a s e m i c r o s t r u c t u r e w i t h a sc a les i m i l a r to 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 i n t h e F e - 4 5 %C r a l l o y s a f t e r a g i n g f o r 5 0 0 h a t 7 73 K . T h e i s o s u r f a c er e c o n s t r u c t i o n s h o w n i n F ig . 1 2, g e n e r a t e d f r o m a4 0 x 4 0 x 4 0 s u b s e c t i o n o f t h e M o n t e C a r l o s i m u -l a t i o n , c o m p a r e s f a v o u r a b l y w i t h t h e P o S A P a n a l y s i s[F ig . 10(e) ] . The i sosur face leve l was se t to 50% ino r d e r t o r e p r e s e n t t h e i n t e r f a c e b e t w e e n t h e p h a s e s .

B o t h t h e F e - r i c h a n d C r - e n r i c h e d p h a s e s a r ei n t e r c o n n e c t e d a n d p e r c o l a t e t h e v o l u m e .4.3. Numerical solution to the Cahn-Hilliard-Cooktheory

C a h n a n d H i l l i a r d d e v e l o p e d a m e a n f i el d a p p r o a c ht o s p i n o d a l d e c o m p o s i t i o n o f a b i n a r y m i x t u r e [ 23 ]. Al o c al c o n c e n t r a t i o n v a r i a b l e u w a s d e f in e d i n t e r m s o ft h e d i f f e r e n c e b e t w e e n t h e m o l e f r a c t i o n s o f e a c h


14/17

3398 M I L L E R 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 - - I1000 E s ~ [ ]

9 0 0 t N ~

~ . 7 0 0 ~ / ; ' - - - - ",~.6 o o I ( I-- | ' 7 Wil~m8 '5 O 0 ! ' Montecar loi ; FIM (Pha ~ s ~ , )FIM (No Ph ase Separation)4 0 0 I

o o : ~ , o : , o : ~ o : ~ ' ,F e C o m p o s i t i o n C r

Fig . 11. The so lid s ta te m isc ib i l i ty gap in the Fe -C r phased iagram . The do t ted l ines represen t the therm odynam icca lcu la t ions fo r the m isc ib i li ty gap and the so l id sym bols a ref rom exper im enta l inves t iga t ions . The phase d iagram f romthe dynam ic Is ing m o del i s shown wi th a c r i t ica l tem pera tu reset to 900 K.

s p e ci e s, x a n d ( 1 - x ) , o v e r a c o a r s e - g r a i n i n gv o l u m e . T h e m o l e f r a c t i o n w i t h i n t h e c o a r s e - g r a i n i n gr e g i o n i s

1x ( r ) = N K ~ x ,w h e r e r i s t h e c e n t r e o f m a s s o f t h e c o a r s e g r a i n e dv o l u m e c o n t a i n i n g N a t o m s a n d t h e x i 's a r e 0 a n d 1c o r r e s p o n d i n g t o t h e tw o t y p e s o f a t o m s . T h ec o n t i n u o u s c o n c e n t r a t i o n p a r a m e t e r u i s g i v e n b y

u = x - ( I - x ) = 2 x - - 1, I . I - < 1.T h e G i n z b u r g - L a n d a u f re e e n e r gy f u n c ti o n w a s

u s e d i n w h i c h t h e c h e m i c a l p o t e n t i a l ~ t c o n s i s t s o f t w ot e r m s , r e l a t i n g t o t h e b u l k f r e e e n e r g y , ~ , a n d a r o o tm e a n s q u a r e e f f e c t i v e i n t e r a c t i o n d i s t a n c e },. B o t ht e r m s e x p l i c i t l y c o n t a i n t h e c o a r s e - g r a i n s c a l i n g l e n g t ht h r o u g h t h e d e f i n i ti o n o f u

i1 = @ ' ( u ) - ; , '2 k ~ I K 7 " uO Uw h e r e k B i s B o l t z m a n n ' s c o n s t a n t a n d T i s t h et e m p e r a t u r e .

T h e C a h n H i l l i a r d e q u a t i o n f o r s p i n o d a l d e c o m p o -s i t i o n i s

c;u (r,N ) _ V ' M V ( ~ ( u ) - r 'k B / V ' u )w h e r e u i s t h e c o m p o s i t i o n , M > 0 a m o b i l i t y , r is ap o s i t i o n v e c t o r a n d t th e t i m e v a r i a b l e .

A m a j o r d r a w b a c k o f t h e C a h n - H i l l i a r d t h e o r y ist h a t i t is c o m p l e t e l y d e t e r m i n i s ti c a n d i g n o r e s t h e r m a lf l u c t u a t i o n s . C o o k [ 24 ] c o n s i d e r e d a s i m p l e e x t e n s i o nt o t h e l i n e a r t h e o r y b y a d d i n g a r a n d o m f l u c t u a t i o n~ ( r , O

~ , . ( r , t ) V M V ( @ ( , , ) - :,'/.-.rV'.]: ( , ' . ,1N \ e . /

w h e r e( ~ ( r . 0 ) = 0

a n d( ~ ( r , t ) d ( r ' , t ' ) ) = - 2 k B T M V 2 6 ( r - r ') 6 ( t - t ' ) .T h e t w o c o n d i t i o n s e n s u r e t h a t t h e a l l o y c o m p o -

s i t i o n is c o n s e r v e d a n d t h a t t h e n o i s e i s u n c o r r e l a t e di n ti m e . A c o n v e n i e n t f o r m o f t h e C o o k t e r m i s aG a u s s i a n w i t h m e a n z e r o a n d v a r i a n c e 2 k B T M .

I n t h e p r e s e n t w o r k , a n u m e r i c a l s o l u t i o n t o t h en o n - l i n e a r C a h n - H i l l i a r d e q u a t i o n , f i r s t i m p l e m e n t e db y E l l i o t t a n d C o p e t t i [ 3 6 - 3 8 ], w a s m o d i f i e d t o i n c l u d et h e C o o k t e r m . T h e p a r a m e t e r s i n t h e s i m u l a t i o n w e r ec h o s e n i n a n a t t e m p t t o m o d e l d e c o m p o s i t i o n i n t h eF e - C r s y s t e m a t 7 7 3 K . S i n c e t h e e x p e r i m e n t a l r e s u l t sc o r r e s p o n d t o t h e v e r y e a r l y s ta g e s o f d e c o m p o s i t i o n ,w h e r e t h e i m p o r t a n t l e n g t h s ca l es a r e o n t h e o r d e r o fa f e w a t o m s , a n u n r e a l i s t ic c o a r s e - g r a i n e d v o l u m e o fa s i ng l e a t o m w a s c h o s e n . T h e m e a n f i el da p p r o x i m a t i o n f o r th e f r ee e n e r g y , 0 , a t a t e m p e r a t u r eo f 7 5 0 K w a s u s e d s i n c e t h e l i m i t s o f s o l i d s o l u b i l i t y a tt h is t e m p e r a t u r e a p p r o x i m a t e l y m a t c h t h o s e o b s e r v e di n t h e F e - C r s y s t e m a t 77 3 K . P o s i t iv e c o m p o s i t i o n s( u > 0) c o r r e s p o n d t o A - r i c h r e g i o n s a n d n e g a t i v ec o m p o s i t i o n s ( u < 0 ) c o r r e s p o n d t o B - r i ch r e gi o n s . I fo n l y 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 a r e u s ed ,7 = a / x / ; 3 w h e r e a i s t h e i n t e r m o l e c u l a r d i s t a n c e [ 23 ].T h e c o n s t a n t a 2 / M k B T w a s a b s o r b e d i n t o t h ed e f i n i t i o n o f t i m e s o t h a t o n e t i m e u n i t i s e q u a l t oa '- / M k B T s e c o n d s .

S i m u l a t i o n s w e r e p e r f o r m e d o n a c u b i c m e s h( 6 4 x 6 4 x 6 4 ) w i t h a m e a n c o m p o s i t i o n u = 0 ,c o r r e s p o n d i n g t o a n e q u i - a t o m i c al l o y. C o m p u -t a t i o n a l l i m i t a t i o n s p r e v e n t e d i n c r e a s i n g t h e si z e o f t h es i m u l a t i o n s . S i m u l a t i o n s w e r e p e r f o r m e d f o r v a l u e s o f( 7 / a ) 2 b e t w e e n 1 / 6 a n d 1 t o m o d e l a r a n g e o fi n t e r a c t i o n d i s t a n c e s . S i n c e t h e v a l u e o f M a t 7 5 0 K i sn o t k n o w n , s i m u l a t i o n s w e r e p e r f o r m e d w i t h a r a n g eo f v a lu e s f o r t h e m a g n i t u d e o f th e C o o k t e r m .S i m u l a t i o n s w e r e p e r f o r m e d u n t i l t h e m e a n d i s t a n c eb e t w e e n d o m a i n s h a d r e a c h e d a t l e a s t 5 n m .

T h e e v o l u t i o n o f t h e m i c r o s t r u c t u r e g e n e r a t e d b yt h e n u m e r i c a l d i sc r e t i z a t io n o f 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 w a s a n a l y se d . I n e a c h s i m u l a t i o nd o m a i n g r o w t h w a s o b s e r v e d a f t e r a p p r o x . 1 00 t i m eu n i t s , g e n e r a t i n g a h i g h l y i n t e r c o n n e c t e d s t r u c t u r e .A f t e r 1 0 0 0 t i m e u n i t s t h e s c a l e w a s s i m i l a r t o t h a to b s e r v e d i n t h e m o s t h e a v i ly a g e d F e M 5 % C r a l l o y( 5 0 0 h ) . I n F i g . 1 3, a n i s o s u r f a c e r e c o n s t r u c t i o n( d r a w n a t a c o m p o s i t i o n o f u = 0 ~ t h e m e a n a l lo yc o m p o s i t i o n ) i s s h o w n w h i c h c a n b e d i r e c t ly c o m p a r e dw i t h t h e e x p e r i m e n t a l r e s u l t s s h o w n i n F i g . 1 0 (e ) ( e a c hi m a g e is a p p r o x . 1 5 n m a c r o s s ) . A l t h o u g h t h e s c al e i ss i m i l a r , t h e d e t a i l e d m o r p h o l o g y i s d i f f e r e n t. 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 pr e d ic t s s m o o t h e ri n t e rf a c e s b e t w e e n t h e p h a s e s 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 . T h e d i f f e r e n c e s i n m o r p h o l o g y a r e d i s c u s s e df u r t h e r i n P a r t l l I o f t h i s s e ri e s [1 9 ].


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M I L L E R e t a l . : SPI NODAL DECOM PO SI TI ON I N Fe Cr ALLO YS- - I 3 39 9

5 nmFig. 12. Isosurface reconstruction of the B-rich regions from a M ontc C arlo simulation on a body centredcubic lattice aged at 750 K for 5000 MCS. T he compositions were calculated using centre-weightedsmoothing and the surface is drawn at a concentration of 50 at.%.

5 . M ATCHI NG OF M ODELS AND EXPERI M ENTI n o r d e r t o m a k e a c o m p r e h e n s i v e c o m p a r i s o n

b e t we e n e x p e r i m e n t a l d a t a a n d c o m p u t e r s i m u l a t e dr e s ul t s, t h e l i m i t a t i o n s o f t h e e x p e r i m e n t a l m e t h o dm u s t b e t a k e n i n t o a c c o u n t . Ex t e n s i v e a n a l y s e s h a v eb e e n p e r f o r m e d o n t h e p o s i t i o n s e n s i t i v e a t o m p r o b et o e n s u r e t h a t t h e e x p e r i m e n t s we r e c o n d u c t e d u n d e ro p t i m u m c o n d i t i o n s . Ho w e v e r , a c e r t ai n l o ss o f d a t awi l l a lways occ ur because the pos i t ion sens i t ived e t e c t o r c a n n o t p o s i t i o n m o r e t h a n o n e i o n f r o many s ingle pul se and the channel p la tes a re not100% ef f i c ient . Moreover , the t ra jec tory aber ra t ionsand f in i t e resolu t ion of the pos i t ion sens i t ive anoder e d u c e s t h e a c c u ra c y o f t h e r e c o n s t r u c t e d a t o m i cpos i t ions .

I n th e M o n t e Ca r l o m o d e l t h e e x a c t p o s i t io n o f e a c hatom i s known and so i t i s poss ib le to model thee x p e r i m e n t a l l i m i t a t io n s a n d o b s e r v e t h e e f fe c t o n a n ym e a s u r e m e n t s . Th e a d d i t i o n o f a Ga u s s i a n s c a t t e r o fwi d t h 1 a t o m i c s p a c i n g t o t h e l a t e r al d i m e n s i o n sa c c u r a t e l y m o d e l s t h e t r a j e c t o r y a b e r r a t i o n s a n ddetec tor spa t i a l r esolu t ion present dur ing a typica lPoSAP exper iment [39] . The f in i t e de tec t ion ef f i c iencyof the channe l p la tes , coup led wi th the loss of pos i t ioni n f o r m a t i o n f r o m m u l t i p l e e v e n t s , r e d u c e s t h ed e t e c t i o n e f fi c ie n c y o f Cr a t o m s t o 3 0 % , a n d t h edetec t ion ef f i c iency of Fe a tom s to 50% [21]. Thisr e d u c e s th e a p p a r e n t Cr - c o n t e n t o f th e a l lo y . Tos imula te th i s loss of da ta , a de tec t ion ef f ic iencyparam eter , Ex, was in t rodu ced for each spec ies X. Afrac t ion (1 - Ex) of the X type a tom s are rand oml y


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3400 MILLER 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

5 nmFig. 13. lsosurface reconstruction of the B-rich regions from the numerical solution to theCahn-Hill iard-Cook equation after aging for 1000 t ime units . The composit ions were calculated usingcentre-weighted smoothing and the surface is drawn at a concentration of 50 at .%.

r e m o v e d f r o m t h e M o n t e C a r l o s i m u l a t i o n r e s u l t sp r i o r t o a n a l y s i s . T h e e f f e ct o f t h e s e e x p e r i m e n t a ll i m i t a t i o n s o n t h e o b s e r v e d c o m p o s i t i o n s a n dm o r p h o l o g y i s d i s c u s s e d i n d e t a il i n P a r t s I I a n d I I I[18, 191.

6 . C O N C L U S I O N SR e c e n t a d v a n c e s i n a t o m p r o b e m i c r o a n a l y s i s n o w

e n a b l e t h e i n v e s t i g a t i o n o f c o m p l e x m i c r o s t r u c t u r e s i n3 D a t t h e a t o m i c l ev e l. T h e a v a i l a b i l i t y o f c o m p u t i n gp o w e r h a s l e d t o t h e s u c c e s s f u l i m p l e m e n t a t i o n o ft h e o r e t i c a l m o d e l s w h i c h m a y b e d i r e c t l y c o m p a r e dw i t h t h e e x p e r i m e n t a l r e s u l t s . T h e i s o s u r f a c e r e c o n -s t r u c t i o n s f r o m t h e p o s i t i o n s e n s i ti v e a t o m p r o b e ,C a h n - H i l l i a r d - C o o k m o d e l a n d M o n t e C a r l o a p -p r o a c h a l l g e n e r a t e s t r u c t u r e s t h a t a r e q u a l i t a t i v e l ys i m i l a r . A m o r e q u a n t i t a t i v e c o m p a r i s o n w i l l b ed e s c r i b e d i n P a r t s I I a n d I I I .

Acknowledgements--The authors would l ike to thankProfessor R. J . Brook for the provision of laboratoryfacilities, Dr B. Fisher for supp lying some of the alloysand Dr P. Camus who performed some of the earlyFIM experiments. J .M.H. would l ike to acknowledge theEngineering and Physical Sciences Research Cou ncil(EPSRC) and Wolfson College for f inancial support . A.C.thank s The Royal Society for f inancial support a nd Wo lfsonCollege for the provision of a Fellowship. This research wasfunded by the EPSRC under gran t numbe r GR/H/38485 andby the Division of Materials Sciences, U.S. Departmentof Energy , under cont rac t DE-AC05-84OR21400 wi thMartin Marietta Energy Systems Inc. The simulations wereperformed in the Materials Modell ing Laboratory at OxfordUniversity which is funded by the EPSRC under grantnumber GR/H/58278.

REFERENCES

1. R. O. Williams, Trans. AIME 212, 497 (1958).2. M. J . Marcinkowski, R. J . Fisher and A. Szirmae, Trans.Metall. Soc. AIME 230, 676 (1964).


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M . K . M i l l er a n d G . D . W . S m i t h . Sur f . Sc i . 2 6 6 , 3 7 0( 1 9 9 2 ) .5 . J . C . L a S a l l e a n d L . H . S c h w a r t z , De c ompos i t i on O!A l l oy s : The E ar l y S t age s ( e d i t e d b y P . H a a s e n , V . G e r o l d ,R . W a g n e r a n d M . A s h b y ) , p p . 1 04 1 09 . P e r g a m o nP r e s s , O x f o r d ( 1 9 8 4 ) .

6 . J . C . L a S a l l e a n d L . H . S c h w a r t z , A c t a me t a l l . 3 4 , 9 8 9( 1 9 8 6 ) .7 . M . F u r u s a k a , Y . I sh i k a w a , S . Y a m a g u c h i a n d Y . F u j i n o .J . P hy s . Soc . J apan 5 5 , 2 2 5 3 ( 1 9 8 6 ) .8 . F . Bley , A c t a me t a l l , mat e r . 40 , 1505 (1992) .

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l n s t r u m . 4 5 , 1 0 5 3 ( 1 9 7 4 ) .1 2 . A . C e r e z o , T . J . G o d f r e y a n d G . D . W . S m i t h , Rev . Se t .l n s t r u m . 5 9 , 8 6 2 ( 1 9 8 8 ) .1 3 . A . C e r e z o , T . J. G o d f r e y a n d G . D . W . S m i t h , J . P hy s i queC6--49 , 25 (1988) .1 4 . G . D . W . S m i t h a n d A . C e r e z o , b~iprocemw~ts m AtomP robe s . E u r o p e a n P a t e n t N o . 0 2 3 12 4 7 , g r a n t e d 1 0 th

O c t o b e r ( 1 9 9 0 ) .1 5 . M . K . M i l l e r , Sur f . Sc i . 2 4 6 , 4 2 8 ( 1 9 9 1 ) .1 6 . M . K . M i l l e r , Sur l . Sc i . 2 6 6 , 4 9 4 ( 1 9 9 2 ) .1 7 . M . K . M i l l e r a n d G . D . W . S m i t h , A t o m - P r o b eMicroanalys i s: Prine iples and Appl icat ions in Mater ial ,~Sc ience . M a t e r i a l s R e s e a r c h S o c i e t y , P i t t s b u r g h , P ~ t( 1 9 8 9 ) .1 8. J . M . H y d e , M . K . M i l l e r, M . G . H e t h c 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 e t a me t a l l .m a t e r . 4 3 , 3 4 0 3 ( 1 9 9 5 ) .

1 9 . J . M . H y d c . M . K . M i l l c 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 e t a m e t a l l.mat e r . 4 3 , 3 4 1 5 ( 1 9 9 5 ) .2 0 . M . K . M i l l e r , J . P hy s . (P ar i s ) 4 7 , 4 9 3 ( 1 9 8 6 ) .2 I . A . Ce r e z o , J . M . H y d e , M . K . M i l l e r , G . Be v e r i n i . R . P .S e l 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 . 2 6 6( 1 9 9 2 ) 4 8 [ .

2 2 . J . W . C a h n , A e l a me t a l l . 9 , 7 9 5 ( 1 9 6 1 ) .2 3 . J . W . C a h n a n d J . E . H i ll i a r d , J . c he m. P hy s . 2 8 , 2 5 8(19581.2 4 . H . E . C o o k , A e t a me t a l l . 1 8 , 2 9 7 ( 1 9 7 0 ) .2 5 . J . S . k a n g e r . M . B a r - o n a n d H . D . M i l l e r , P hy s . R e v . AI1 , 1417 (1975) .2 6 . M . G r a n t , M . S . M i g u e l , J . V i f ia l s a n d J . D . G u n t o n ,P h ) s . R e c . B 3 1 , 3 0 2 7 ( 1 9 8 5 ) .2 7 . N . M e t r o p o l i s . A . W . R o s e n b l u t h , M . N . R o s e n b l u t ha n d A . H . T e l l e r , J . c he m, P hy s . 2 1 , 1 0 8 7 ( 1 9 5 3 ) .2 8 . K . Y a l d r a m a n d K . B i n d e r , Z . P h y s . B - - C o n d e n s e dM a t t e r 8 2 , 4 0 5 ( 1 9 9 1 ) .

2 9 . V . H e i n e , P hy s . R e v . 1 5 3 , 6 7 3 ( 1 9 6 7 ) ,3 0 . R . O . W i l l i am s a n d H . W . P a x t o n , J . I r o n S t e e l l n s t . 1 8 5 ,358 11957) .3 I . W . C . L e s l i e , The Physical Metal lurgy ~?[Stee l s ( e d i t e d b yD . H i e b e r g , V . M . Z i o b r o a n d E . D u g g e r ) , p p . 3 3 7 - 3 4 0 .M c G r a w - H i l l . N e w Y o r k ( 1 9 8 2 ) .3 2. T . N i s h i z a w a , M . H a s e b e a n d M . K o , A e t a me t a l l . 2 7 ,8 1 7 ( 1 9 7 9 ) .3 3 . H . K u w a n o , Trans . J apan Ins t . M e t a l s 2 6 , 4 7 3 ( 1 9 8 5 ) .

3 4 . J . A n d e r s s o n a n d B . S u n d m a n , C A L P H A D 1 1 , 8 3 ( 1 9 8 7 ).3 5 . G . A . Ba k e r , Plow. Re t ' . 1 2 4 , 7 6 8 ( 1 9 6 1 ) .3 6 . C . M . E l l i o t t a n d D . A . F r e n c h , I M A J . a p p l . M a t h . 3 8 ,9 7 ( 1 9 8 7 ) .3 7 . C . M . E l l i o t t , I n t . S e t . N u m . M a t h . 8 8 , 3 5 ( 1 9 8 9 ) .3 8 . M . I . M . Co p e t t i a n d C . M . E l l i o t t , M at e r . Se t . Te c hno l .

6 , 273 (19901 .3 9 . J . M . H y d e , D . P h i l . t h e s i s , U n i v e r s i t y o f O x f o r d ( 1 9 9 3 ) .

AM 43/9--K

advanced engineering & composite (reading paper)

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