Reading Paper 3

8/3/2019 Reading Paper 3

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~ PergamonActa metall , mater. Vo l. 43, No. 9, pp. 3415-3426, 1995Elsevier Science Ltd0956-7151(95)00042-9 Copyright t995 Act a Metallurgica Inc.Printed in G reat Britain. All rights reserved0956-7151/95 $9.50 + 0.00

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 :E X P E R I M E N T A L S T U D Y A T T H E A T O M I C L E V E L A N DC O M P A R I S O N W I T H C O M P U T E R M O D E L S - - I I I .

D E V E L O P M E N T O F M O R P H O L O G YJ . M . H Y D E 1, M . K . M I L L E R z, M . G . H E T H E R I N G T O N t l, A . C E R E Z O 1,G, D. W. SMITH:~ and C. M. ELL IOTT

~Departmen t o f Mate r ia l s , Un ive rs i ty o f Oxfo rd , Pa rks Road , Oxfo rd O XI 3PH, Eng land , 2Meta ls andCeramics Div is ion , Oak Ridge Na t iona l Labora to ry , Oak Ridge , TN 37831-6376 , U.S .A. and 3Schoo l o fMa them at ica l and P hys ica l Sc iences , Un ive rs i ty o f Sussex , Fa lmer , Br igh ton BN1 9Q H, En g land( R e c e i v ed 3 0 N o v e m b e r 1 9 9 4 )

Abst rac t - -The f ine -sca le th ree -d imens iona l mic ros t ruc tu res fo rmed du r ing sp inoda l decompos i t ion inFe -C r a l loys a re cha rac te r ized us ing two nove l me thods . In the f i rs t, a f rac ta l ana lys is i s used to cha rac te r izethe in te r face be tween the phases and , in the second , the in te rconnec t iv i ty o f the s t ruc tu re i s de te rmined f romtopo logy . I t i s found tha t the in te r face be tween Fe - r ich e and Cr-en r iched :~' r eg ions in the exper imen ta lda ta and Mo nte Car lo s im u la t ions exh ib i t frac ta l behav iou r whereas the mic ros t ruc tu res f rom the so lu t iont o t h e C a h n - H i l l i a r d ~ C o o k m o d e l d o n o t . T o p o l o g i c a l m e t h o d s a r e u s e d t o c h a r a c t e ri z e t h e c o m p l e x ~ 'mic ros t ruc tu res in te rms o f the num ber o f cav i tie s and loops . The dec rease in the num ber o f la rge scale loopsin the mic ros t ruc tu re , du r ing the rm al age ing , i s shown to co r re la te wi th the inc reas ing mic ros t ruc tu ra l scale .The n umb er o f sma l l scale loops i s found to co r re la te w i th the complex i ty o f the in te r face be tween the :~and e ' regions.

1 . I N T R O D U C T I O NI n t hi s p a p e r , t w o m e t h o d s o f m o r p h o l o g i c a lc h a r a c t e r i z a t i o n o f p e r c o l a te d s t r u c t u r e s a r e c o n -s i d e r e d , f r a c t a l a n d t o p o l o g i c a l a n a l y s e s . I n P a r t I I [1 ],i t w a s s h o w n t h a t a f t e r a g i n g a n F e - 4 5 % C r a l l o y f o r5 0 0 h a t 7 73 K , s p i n o d a l d e c o m p o s i t i o n h a d g e n e r a t e ds t r u c t u r e s w i t h a s i m i l a r s c a l e a n d c o m p o s i t i o na m p l i t u d e t o th o s e g e n e r a te d b y a M o n t e C a r l os i m u l a t i o n a f t e r 5 0 0 0 M C S a n d a n u m e r i c a ld i s cr e t iz 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 nc o m p u t e d f o r 1 0 00 ti m e u n it s . D e t a i l e d c o m p a r i s o n sa r e m a d e b e t w e e n t h e m i c r o s t r u c t u r e s i n a s e ri e s o fs p i n od a l ly d e c o m p o s e d F e - C r a l lo y s a n d c o m p u t e rs i m u l a t i o n s o v e r t h e s e t i m e p e r i o d s.

2 . F R A C T A L A N A L Y S I S2 . 1 . I n t r o d u c t i o n

T h e r e a r e s e v e r a l p o s si b l e m e t h o d s o f c h a r a c te r i z i n gt h e m o r p h o l o g y a n d t o p o l o g y o f c o m p l e x m i c r o st r u c -t u r e s . S i n c e f r a c t a l g e o m e t r y c a n d e s c r i b e h i g h l yd i s o r d e r e d m o r p h o l o g i e s , f r a c t a l a n a l y si s [ 2 -4 ] m i g h tb e a s u i t a b l e t o o l f o r a n a l y s i n g r o u g h i n t e r f a c e s o rp o r o u s s t r u c t u re s . H o r n b o g e n [5 ] a p p l i e d f r a c ta la n a l y s i s t e c h n i q u e s t o d i s l o c a t i o n s , g r a i n b o u n d a r i e s ,p a r t i c l e d i s t r i b u t i o n s a n d s u r f a c e s u s i n g o p t i c a l a n d"['Deceased.~To w hom a l l co r respondence shou ld be addressed .

e l e c t ro n m i c r o s c o p y . S i m i l a r m o r p h o l o g i e s w e r ef o u n d f o r a w i d e r an g e o f m a g n i f i c a t i o n s i n d i c a t i n g af r a c t al c h a r a ct e r i st i c . H e a l s o f o u n d t h a t m a n yd i s o r d e r e d m i c r o s t r u c t u r e s w e r e n o t f ra c t a l. U n l i k em a t h e m a t i c a l m o d e l s , a l o w e r a n d u p p e r c u t - o f f o f s i zes c a le s n e e d e d t o b e i n t r o d u c e d , t h e l o w e r c u t - o f f b e in gd e t e r m i n e d b y t h e r e s o l u ti o n o f t h e i m a g i n g t e c h n i q u ea n d t h e u p p e r b y s p e c i m e n s i z e o r g r a i n d i a m e t e r .W r i g h t a n d K a r l s s o n [6 ] c o n s i d e r e d t h e a p p l i c~ i b i li t yo f f r a c ta l a n a l y s e s o f s t r u c t u r es , c o m p a r i n g d i f f e re n tm e a s u r e m e n t p r o c e d u r e s o n b o t h r e a l s u r f a c e s a n dm a t h e m a t i c a l m o d e l s . T h e i r p r e l i m i n a r y r es u l tsi n d i c a t e d t h a t a f r a c t a l a n a l y s i s i s u s e f u l a s a n i n d i c a t o ro f s iz e , s h a p e a n d s e l f - s i m i l a r i t y b u t t h e y d r e wa t t e n t i o n t o t h e f a c t t h a t i n s u f f i c i e n t s t a t i s ti c s l i m i t e dt h e a c c u r a c y o f t h e i r r es u lt s . C h e n g e t a l . [ 7] u s e d s m a l la n g l e X - r a y a n d n e u t r o n s c a t t e ri n g e x p e r i m e n t s t os t u d y t h i n f i l m s a d s o r b e d o n f r a c t a l s u r f a c e s b yl o o k i n g a t t h e r a n g e o f s ca l in g b e h a v i o u r a n d d e r i v i n ga s c a l i n g f o r m u l a f o r t h e s c a t t e r e d i n t e n s i t y a s af u n c t i o n o f f il m t h i c k n e s s .

C a m u s [ 8 ] s u g g e s t e d t h a t f r a c t a l a n a l y s e s c o u l d b eu s e d t o q u a n t i f y i 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 so b s e r v e d b y F I M . M o r e r e c e n t l y M i l l e r a n d R u s s e l l [9 ]h a v e u s e d t w o - d i m e n s i o n a l ( 2 D ) d i g i t i z ed f ie l de v a p o r a t i o n m i c r o g r a p h s t o e x a m i n e t h e i n t e r f a c em o r p h o l o g y o f a s e ri e s o f p h a s e s e p a r a t e d F e - C rs p e c im e n s . T h e i n t e r f a c e m o r p h o l o g y w a s f o u n d t o b er o u g h a n d f r a c t a l a n a l y s i s y i e l d e d f r a c t a l d i m e n s i o n s

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3416 HYDE et al . : SPINODAL DECOMPOSITION IN F~Cr AILOYS--IIIbetween 1.1 and 1.5. Hetherin gton and Miller havediscussed the use of fractal analyses in determining anoptimum block size at which the composition andother physical parameters may be reliably measured[lOl.2.2. Fractal structure"

The isosurface representation has already beenintroduced as a method to represent the interfacebetween two phases (Part I of this series [1 t]). A seriesof isosurfaces, generated from the PoSAP analysis ofan Fe~5% Cr specimen aged for 500 h at 773 K, isshown in Fig. 1. Each of the surfaces has beengenerated by a grid with a different resolution. Thecompositions in each grid cell were calculated usingfive-point centre-weighted smoothing and the surfacesdrawn over every fifth, third, and first points inFig. l(a)-(c), respectively. As the resolution becomesliner, more detail is seen and more "'particles" appear.This is analogous to the way that the appar ent lengthof a coastline increases with increasing magnification.Choosing the scale on which the isosurface or interfaceshould be defined is equivalent to choosing acoarse-graining volume. From an experimental pointof view, it is equivalent to trying to decide which atomsconstitute a second phase and which are simplyrandom solute atoms in the matrix phase. Unfortu-nately, there is no simple solution to this problem, soinstead we turn the problem on its head an d say thatan i mport ant characteristic of the structure is that it isnot a simple geometric shape. Instead of trying to findan "ideal" length scale, it is accepted that the structurevaries with different length scales and then the way inwhich the structure varies is characterized. Thecharacterization of scaling is the way in which fractalsare analysed, and so we can define a "fractal"dimensio n for the microstructures tbrmed by spinodaldecomposition.2.3. Methods of . f i ' actal analys is

It is important to appreciate that there is no single"fractal dimension". A fractal dimension df may bedefined by an equation such as

M(2L) = 24M(L)where M ( L ) is the mass within a l ength scale L and )~is a constant. Using this relation, both diffusion limitedaggregation (DLA) and a percolated structure mayyield the same value for df (for instance df = 2.5 for anembedding dimension [2] of 3) yet look completelydifferent. There must be some other fractal parameterthat differs. Stanley [12] lists ten different measures o fdimension necessary to determine some criticalexponents for various systems and finds that, as withthe critical point exponents, not all the fractaldimensions are independent. Some are connected bysimple relations.

A fractal dimension was defined by studying thescaling behaviour of the interface area between theCr-enriched and Fe-rich regions. A compos itional grid

was constructed from the data as described in Part Iof this series [1 1] with the use of five-pointcentre-weighted smoothing. Each grid point wasassigned to a particular phase according to the localcompositi on to create a binary array consisting of c~and e' phases (Fig. 2). The surface area may then becharacterized by the number of surface nodes orsurface links. In Fig. 2(b), the effect of decreasing theresolution of the image (increasing the size of the rulerover which the grid is defined) is shown. The way inwhich the surface area of the e' regions, defined by thenum ber of ~-~ ' interface links, varies as a funct ion of"ruler lengt h" for an Fe- 45% Cr alloy aged for 500 hat 773 K is shown in Fig. 3. The gradient of each plotgives an estimate of one fractal dimension. A nor malgeometric surface has a dimension of 2 (in anembedding dimension of 3), whereas a completelyrandom structure would have a dimension of 3. Tounde rstan d this, consider a compositi on cell belongingto the e ' phase. This cell is considered a surface elementif a neighbour ing cell belongs to the e phase. If thevolume fra ction of the c~ phase is p, and the structureis random, then the probability that a specificnei ghbour belongs to phase c~ is simply p, irrespectiveof the leng th scale chosen. The numb er of surface cellsis therefore propor tiona l to the total numb er of cells,which is proportional to the volume. Beforedecomposition occurs, the alloy is quenched from ahigh temperature above the miscibility gap an dtherefore has a random structure and each phase hasa df = 3 surface. As the alloy ages, well defineddomains form and, at the very late stages, the surfacedimensi on must tend towards 2. Therefore, as agingproceeds, the dimension df should decrease from 3towards 2. The straight line fit (shown in Fig. 3) hasa goodness of fit correlation coefficient greater t han0.9995, however the ra nge of "'ruler lengths" is limitedby the finite size of the analysed volume and themaximum resolution of the microscope. The struc-tures scale over this range, but it has not been provedthat they are fractal, since a fractal must be self similarover many decades. It is however an important toolsince it characterizes a property of microstructuresthat is not observed in simple geometric objects suchas spheres. Since the fracta l dime nsio n varies, it can beused together with other conven tional parameterssuch as wavelength or spinodal amplitude tocharacterize microstructures.2 .4 . F r a c t a l a n a ly s i s o f P o S A P d a ta

Analysis of PoSAP data from the series of Fe~ ,5%Cr alloys thermally aged at 773 K shows a mono toni cdecrease in fractal dimension of the surface of theCr-enriched phase with aging [Fig. 4(a)]. Even after500 h aging, the fractal dimension has only decreasedto approx. 2.6, demonstrat ing that the interfacebetween the domains is not smooth. In order to mak ean exact comparison between the simulations andexperimental results, the effects generated bytrajectory aberrations, anode resolution and the finite


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(a )

AM 43 ~9-~L

F i g I (a z ~ d b ) . ( C L~ p t io~ o r e r l e a l . )


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3418 HYDE el al.: SPINODAL DECOMPOSITION IN Fe- Cr ALLOYS--1II(c)

Fig. 1. PoSAP isosurface reconstructions showing the Cr-enriched regions of an Fe~45% Cr alloy aged for500 h at 773 K. In (a) every fifth composition point has been used to define the surface. In (b) every thirdpoint and in (c) every point has been used.

detec tion efficiency on the measured fractal dimensionneed to be ascertained.2.5. Monte Carlo results

In Fig. 4(b), the decrease in fractal dimensionwith aging is shown for the Monte Carlo simulationof spinodal decomposition. When the perfect MonteCarlo crystal is analysed, the fractal dimensiondecreases to a lower value than that observed in theexperimental alloys, even though, during the timeperiods examined, the experimental results andsimulations show a similar increase in both scaleand composition amplitude (see Part II [1]).However a good match is observed when realisticvalues of detection efficiency and lateral scatter areused to model the experimental limitations[Fig. 4(b)]. The addition of a Gaussian scatter witha width of 1 atomic spacing to the atomicco-ordinates increases the interface roughness whichresults in the measurement of a higher fractaldimension.

2.6. Numerical discretization of the Ca hn -H il li ar ~Cook equation

Fractal analyses of the structures generated by thenumerical solution to the Cahn-Hilliard-Cookequat ion are shown in Fig. 4(c). Three plots are dr awnto show the effect of the Cook term, and the finitedetection efficiency and lateral resolution of thePoSAP. The Cook term is significant only during thevery early stages of aging, before coarsening begins. Asdecomposition proceeds and the phases begin tocoarsen, the fracta l dimension rapidly decreases. Thefractal dime nsion measured is close to 2 [Fig. 4(c)].Even though the scale of the structure is small, thesurfaces are ordinary 2D objects in contrast to thoseobserved experimentally or as a result of the MonteCarlo simu lation. This results from the fact that thenumerical simulation is a continuum model in whichthe composition is defined on a smooth continuousscale whereas the experimental data and Monte Carlosimulations show the graininess associated withdiscrete atoms. Even when the effects of lateral scatter,


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a )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 - Cr A L L O Y S I I l

b )3419

cz' Region O Vo lum e Node Sur faceI i LinkD cz Reg ion Surface Node

Fig. 2. The are a of the interface between two phases can be charac terized b y the num ber o f surface no desor l inks. The are a is a function of the grid size used. In (a) every cell is used to define the b oun dary betwe enthe two phases whereas in (b) every other cell is used.

t i n i t e d e t e c t o r r e s o l u t i o n a n d t r a j e c t o r y a b e r r a t i o n sa r e i n c l u d e d , t h e n u m e r i c a l m o d e l s t i ll p r e d i c t s m u c hs m o o t h e r i n t e r f a c e s ( i . e . s m a l l e r f r a c t a l d i m e n s i o n )t h a n a r e o b s e r v e d e x p e r i m e n t a l l y .

3 . T O P O L O G I C A L A N A L Y S I S3.1. Introduction

I n P a r t I o f t h i s s e r i e s [ 1 l ] , 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 w e r e u s e d t o s h o w t h a t t h e p h a s e ~ " o f t e n f o r m sa s p o n g e - l ik e i n t e r c o n n e c t e d s t r u c t u r e. E x t e n d i n g t h et h e o r i e s o f s t r u c t u r e p r o p e r t y r e l a t i o n s h i p s t o t h e s em a t e r i a l s r e q u i r e s a m e t h o d o f c h a r a c t e r i z a t i o n i n 3 Dw h i c h i s e q u i v a l e n t t o c o u n t i n g t h e n u m b e r o f p a r ti c l e sf o r u n c o n n e c t e d s t r u ct u r e s.

I n t o p o l o g i c a l a n a l y s i s , t w o s t r u c t u r e s a r e c o n -s i d e r ed i d e n t i c a l if o n e m a y b e t r a n s f o r m e d i n t o t h eo t h e r w i t h o u t r e q u i r i n g a n y c u t s o r t h e f o r m a t i o n o fa n y c l o s e d l o o p s . A c o f f e e c u p a n d r i n g d o u g h n u t a r et h e r e f o r e t o p o l o g i c a l l y i d e n t i c a l . B o t h s t r u c t u r e s

1 0 6'1~ G r a d i e n t = 2 . 5 8z~ 1 5gg~

I000l0 . 1 11/(ruler length)

F ig . 3 . Logar i thm ic- logar i thm ic p lo t o f su r face a reame asured by the numb er o f surface links against the inversesepa ration of the points r ( in units o f grid spacings). T hegrad ien t i s a m easure o f f rac ta l d im ens ion . The da ta a re f roman F e4 5% Cr a l loy aged fo r 500 h at 773 K .

c o n t a i n a s i n g le h a n d l e o r l o o p . M o v i n g a d i s l o c a t i o nt h r o u g h a c o n n e c t e d m i c r o s t r u c t u r e g e n e r a l ly r e q u i re sc u t t i n g t h r o u g h t h e h a n d l e s o f t h e s t ru c t u r e , a n dt h e r e f o r e t he h a n d l e d e n s i t y ( n u m b e r o f h an d l e s o rc l o s e d l o o p s p e r u n i t v o l u m e ) o f a p e r c o l a t e d s t r u c t u rei s t h e a n a l o g u e o f t h e p a r t i c l e d e n s i t y f o r a s y s t e mc o n t a i n i n g i s o l a t e d p a r t i c l e s .

C o m p n t i n g t h e t o p o l o g i c a l p r o p e rt i e s o f a b i n a r ys t r u c t u r e i s a n a r e a o f r e s e a r c h k n o w n a s d i g i t a lt o p o l o g y [ 1 3 ] . C e r e z o et al. [ 1 4 ] u s e d t o p o l o g i c a lt e c h n i q u e s t o c h a r a c t e r i z e 3 D i n t e r c o n n e c t e d m i c r o -s t r u c tu r e s o b s e r v e d f r o m P o S A P a n a l y s es o f a r a n g eo f t w o p h a s e a l lo y s . T h e y f o u n d t h a t a c o m p l e xm o r p h o l o g y c o u l d b e r e d u c e d t o a b a s i c f r a m e w o r ka n d c h a r a c t e r i z e d i n t e r m s o f th e n u m b e r o f ca v i t ie sa n d l o o p s , i n t h i s s e c t i o n , th e t e c h n i q u e s o f d i g i t a lt o p o l o g y h a v e b e e n u s e d t o c h a r a c t e r i z e t h ei n t e r c o n n e c t e d s t r u c t u r e r e s u l t i n g f ro m s p i n o d a ld e c o m p o s i t i o n i n t h e F e C r s y s te m a n d a c o m p a r i s o nh a s b e en m a d e w i t h t h e c o m p u t e r m o d e l s .

3.2. Euler characteristicD e s i g n i n g a c o m p u t e r p r o g r a m t o d i r e c tl y c o u n t t h e

h a n d l e d e n s i t y i n a c o m p l e x s t r u c t u r e w o u l d b ee x t r e m e l y d i f f i c u l t . H o w e v e r , t o p o l o g y p r o v i d e s u sw i t h a n i n d i re c t m e t h o d o f m e a s u r i n g t h e h a n d l ed e n s i t y , m a k i n g u s e o f t h e E u l e r c h a r a c t e r i s t i c E .

T h e E u l e r c h a r a c t e r i s ti c E(S) or a se t S i s at o p o l o g i c a l i n v a r i a n t . I f S i s a p o l y h e d r a l i n 2 D t h e nE(S) i s e q u a l t o t he n u m b e r o f c o n n e c t e d c o m p o n e n t so f S m i n u s t h e n u m b e r o f h a n d l e s in S . I n 3 D E(S) ise q u a l t o th e n u m b e r o f c o m p o n e n t s o f S p lu s t h en u m b e r o f c a v i t ie s m i n u s t h e n u m b e r o f tu n n e l s. E(S)m a y a l s o s h o w n t o b e e q u i v a l e n t t o ( N o . o fp o i n t s ) - ( N o . o f e d g e s ) + ( N o . o f t r i a n g l e s ) - ( N o .o f t e t r a h e d r a ) .


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3420a )

HYDE et al.: SP1NODAL DECOMPOSITION IN Fe Cr ALLOYS--I ll

2.85 ~D2 , 8 ~ .

g 2.75 ~ ~ ._ 2.7 1 Or',, 2"65 !2 o 414.. l2 . 5 5 ~

2 . 5 i . . . . . . . .0 10

P o S A P A n a l y si sF e - 4 5 % C r

81O0 1000

A g i n g ( h )

b) 2 . 8 q !2 . 7 5 c

[ ] ." 5 2 . 7 - "E~, 2.65-

2 . 6 -la. 2 , 5 5 7

2 . 5 ~ - -1

@ Mon te Car l o S imu la t ion3 - - M o d e l l e d E x p e r i m e n t a lL i m i t a t i o n s

1~ 0 1 0 0 1 0 0 0 1 ; 4MCSc )

Cook Modelled xperimental2 . 8 ~ T e r m L i m il at io n s2 (a) NO NO2.7 ~ \ (b) Yes No.~ 2 6 L ] ~ (C) Yes Yes

2 . 3u.

2 1 1 1 0 1 0 0 1 0 0 0T i m e U n i t s

Fig. 4. Development of fractal dimension with aging from (a)PoSAP analysis of a series of thermally aged Fe-45% Cralloys, (b) Monte Carlo simulation of the decomposition ofan A 50% B alloy on a b.c.c, grid and (c) a numerical solutionto the non-linear Cahn-Hilliard Cook model for spinodaldecomposition.

Several methods o f computing the Euler character-istic in 3D have been analysed [15, 16]. Thus in 3Dwhere for instance the number density of handles isrelated to the inte rconnectivity of a structure, it iscomputationally feasible to calculate the number ofhandles from the Euler relation.

Euler and Poincare [17] proved that for eachindividual closed surface

E = n - e + f = 2where n is the numbe r of nodes on the ne t, f he numb erof faces and e the number of edges. A simple cube[Fig. 5(a)] has 8 nodes, 12 edges and 6 faces yieldinga Euler characteristic of 2. More complicated surfaces

may be formed by addin g more lines and then applyinga topologically invariant transformation. The ad-dition of a line between A and B in Fig. 5(a) createsone new face, three edges and two nodes, leaving thesum n- e +f unchanged. However, this equationdoes no t hold if the surface encloses tunnels or cavities.If the closed surface S encloses h handle s and c cavities(enclosed regions) then

E = n - e + f = 2 - 2 h + 2 c .For a cube with one handle [Fig. 5(b)], there are 16

nodes, 32 edges and 16 faces, and therefore an Eulercharacteristic of 0. A cube enclosing a cubic cavity has16 nodes, 24 edges and 12 faces which yields an Eulercharacteristic of 4. The latter may be considered ashaving two closed surfaces (E = 2 for each) since acavity is effectively enclosed by a closed surface.

In discussing the skeletonization of 3D digitalimages, Lobregt e t a l . used the Euler characteristic todefine a connectiv ity numb er N

N = E(2 - 2h)summed over all surfaces. The number of closedsurfaces minus the n umbe r of tunnels is equal to N / 2 .A modified version of the algorithm designed byLobregt e t a l . [15], which uses lookup tables to storeeach possible comb inat ion ofa 2 x 2 x 2 cubic lattice,was implemented. The cavities in the structure are firstfilled in, which allows the number of surfaces to becalculated simply by coun ting the num ber of isolatedparticles. Connectivity values are then calculatedwhich directly yields the n umbe r of handles. Analyseswere performed on the simple cubic lattice structuresgenerated by the smoothing algorithms discussedpreviously in Part II [1].3 . 3 . P o S A P r e s u l t s

For each material studied, the total number ofhandles was calculated for the structures generatedusing both simple and centre-weighted smoothing.The handle density is calculated as the number ofhandles per atom. The number of atoms fieldevaporated during the experiment was estimated fromthe number of ions detected and number of multipleevents observed during the analysis but no accountwas taken for the finite detection efficiency of thechannel plates (approx. 60%). In the unaged(quenched) condition, the handle density is very highcorresponding to a large number of small handles inthe finely percolating random structure. As themateri al ages, the nu mbe r of handles decreases (Fig. 6)as diffusion of the atomic species generates atwo-phase microstructure whose amplitude and scaleare increasing. At lower solute contents, the numb er ofhandles is smaller, since the volume fraction of thesecond phase is smaller. For specimens with a lowsolute content, isolated particles form with only a fewsmall handles around the interface region.

The results indicate that simple smoothingeffectively removes all the very small hand les, whereas


7/12

HYDE et al.: SPINODAL DECOMPOSITION IN Fe Cr ALLOYS--III 3421

a ) b )

Fig. 5 . ( a ) A simple cube (8 nodes, 12 edges and 6 faces). (b) Cube with one handle (16 nodes, 32 edges and16 faces).

the centre-weighted smoothing preserves the fine scaledetail associated with the roughness of the interface(Fig. 6). From the difference between the handledensities with the two smoothing techniques, it can beseen that most (approx. 90%) of the handles are verysmall. There is considerably more scatter on theanalysis of alloys with a low solute content. For theseries of Fe-17% Cr alloys, the scatter completelymasks the decrease in handle density with increasingaging time whereas the trend is clearly shown for theFe~,5% Cr series of thermally aged alloys.3 . 4 . Mont e Carlo simulations

Since the Monte Carlo model simulates atomicdiffusion on a perfect lattice it is a relatively simple

a ) 0 . 012 ~ ) O00.01

~ 0 0 0 8 -a 0.006 - :

0 0 o ~ -1"0.002 -

o i ~ . . . . . . . .0 10

F e - 4 5 % C rC S i m p l e s m o o t h i n g

O 'C ,

0 i1 0 0 1 0 0 0A g i n g ( h )

b ) F e - 4 5 % C ro o 6 - C e n t r e - w e i g h t e d s m o o t h i n g0.05 Zo o4 ~~" . ? ,~

a 0 . 0 3 ~ C '"~ 0.02 - O'"1-

0 O l -0 ~ f . . . . . . . . . . . . . . . . . . . . . . .

0 10 100 100oA g i n g ( h )

F i g . 6. H a n d l e d e n s i t y f o r t h e F e ~ , 5 % C r a l l o y s a s a fu n c t i o no f a g in g . S i m p l e s m o o t h i n g h a s b e e n a p p l i e d i n (a ) a n dc e n t r e - w e i g h t e d i n ( h ) .

matter to calculate the number of handles in themicrostructure. However, to enable a direct compari-son with the experimental results, the finite detectorefficiency of the position sensitive detector had to beallowed for. Atoms were rand omly removed from thesimulations (according to the detec tion efficiencies foreach ion species) before applying the smoothing, aspreviously described for the analysis of bothcomposition amplitude and measurement of fractaldimensions. The handle density, defined as the numbe rof handles divided by the number of atoms (afterremoving the "lost" atoms) in the Monte Carlo

a )0 . 0 1 2 ,! . EF~ Ecr

0 . 0 1 % ~ . . . . " " 1 . 0 1 . 0 " " 0 . 7 0 . 5~ . \ " - - -z : o oo -- o3\ " ,

~ 0 . 0 0 6 \ ""\~ 0 . 0 0 4 - ' ,"1 - " ~ " i . . .

0 , 0 0 2 - ~ ~0 - i -1 1 0 1 0 0 1 0 0 0 1 0 4

M C S

b ) 0 . 0 6 EF~ Ecr1 .0o o 5 \ _ _ o , l o :004.. 03

0 . 0 10 . . . . r , ;1 1 0 1 0 0 1 0 ' 00 1 0 4

M C SF i g . 7 . E ff e c t o f d e t e c t o r e f f ic i e n cy o n m e a s u r e d h a n d l ed e n s i t y ' . S i m p l e s m o o t h i n g h a s b e e n a p p l i e d i n ( a ) a n dc e n t r e - w e i g h t e d s m o o t h i n g i n (b ) . T h e d a t a i s fr o m a M o n t eC a r l o s i m u l a t i o n p e r f o r m e d o n a b . c . c , l a t t i c e w i t h s e c o n dn 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 t t h e e q u i v a l e n t o f 7 50 K . E xi s t h e d e t e c t i o n e f f i c i e n c y f o r e l e m e n t X .


8/12

3422 HYDE ez a l . : SPINODAL DECOMPOSITION IN Fe-Cr ALLOYS llIa )

o o 1 2 I 0 - = o .oo. o I - ! ""~- '~. o= o.5

~ \ , I . . . . . o = I . Oo . o o - I , ,= 2 . 0o.ooo-' 0 . 0 o 4 1o o 0 2 1

1 0 1 0 0 1 0 0 0 1 0 4M C S

b) r _ _ 0-=0.00 .04 ] - - - 0 "=0 .50 .03 5 - - / 0 -=1 .00.03 / 0 - = 2 . 0

0 . 0 2 5 . . . . . - .. ~ 0 .02

0 . 0 1 50 .01

0 . 0 0 5 1 0 1 0 0 1 0 0 0 1 04M C S

Fig. 8. Effect of lateral resolution on handle density. (a)Simple smoothing and (b) centre-weighted smoothing. Thesimulations were performed on a b.c.c, lattice with secondnearest neighbour interactions and the detection efficienciesfor Fe and Cr set to 0.5 and 0.3 respectively.

simulation is shown in Fig. 7. The data is for thesimul atio n on a b.c.c, lattice with both first and secondnearest neighbou r interactions aged at the equivalentof 750 K. With simple smoothing , Fig. 7(a), decreasingthe detec tion efficiency slightly increases the numbe r ofhandles measured, whereas an opposite and largereffect is seen after centre-weighted smoothing[Fig. 7(b)]. The loss of atoms from the simulationdestroys some of the fine scale detail. Remova l of thisdetail from a ran dom structure still results in a randomstructure, and since the numbe r of atoms in the sample

has decreased, so must the number of fine scalehandles. With centre-weighted smoothing, the hand ledensity decreases with decreasing detection efficiencyand the effect is more pronounced during the earlystages of aging. With the simple smoothing, the finescale c ompositio n fluctuations are smoothed over, andso the effect is not as pronounced. Another moreimportant effect produced is an increase in handledensity, since the smoothing of cells with no data(which increases as the efficiency decreases) willgenerate links between phases and increase the num berof handles.

The effect of modelling a decrease in the detectorspatial resolution by adding a Gaussian scatter ofwidth a to the x a nd y position co-ordinates is shownin Fig. 8. In these figures, the dete ction efficiency of theFe atoms and Cr atoms was set to 0.5 and 0.3respectively. After 10 MCS, the scatte r is clearly shownto increase the handle density for both simple andcentre-weighted smoothing. There are two effects.Mixing in the interface region generates some smallscale loops which are calculated with the centre-weighted smoothing. I n additio n, the nu mber of cellswith no composition data increases and the simplesmoothing technique fills in the holes and generateslinks between the phases.

The handle density measured for these structures atintermediate to long aging times is of the same orde ras that measured for the structures obtainedexperimentally (Fig. 8). In Table 1, a numericalcompariso n is made between the experimental resultsand Mo nte Carlo simulat ions on a b.c.c, lattice withsecond nearest neighbour interactions. A best fitbetween the Mont e Carlo simulati ons at 750 K an d theexperimental data occurred when a Gau ssian scatter,with width s ~ 1 atomic spacing, was added to theatomic positions (Table 1).

The curves for the Mont e Carlo simulations on asimple cubic lattice at the three temperatures collapsetowards a single line when using simple smoothing,suggesting that the overall structures are the same a ndonly the interface structure is different [Fig. 9(a)].Analysis on the detailed structure (centre-weighed

T a b l e 1 . C o m p a r i s o n b e t w e e n h a n d le d e n s i ty m e a s u r e d f r o m P o S A P d a t a a n d M o n t e C a r l o~ i m u l a t i o n s . S i m p l e s m o o t h i n g a p p l i e d (t o p ) a n d c e n t r e - w e i g h t e d s m o o t h i n g a p p l i e d ( b o t t o m )P o S A P d a t a F c - 4 5 % C r M o n t e C a r l o A - 5 0 % B o n b . c. c, l a tt ic e 7 5 0 K . Ev e = 0 . 5 , E c , = 03

A g i n g H a n d l e d e n s i ty A g i n g H a n d l e d e n s it y ( s i m p le s m o o t h i n g )( h ) ( s i m p l e s m o o t h i n g ) ( M C S ) a = 0 a = 0 . 5 a = 1 . 0 a = 2 . 0

0 0 . 0 0 7 5 1 0 . 0 1 1 6 0 . 0 1 1 2 0 . 0 1 0 8 0 . 0 0 9 5 04 0 0 0 7 3 1 0 0 . 0 0 7 8 5 0 . 0 0 7 9 1 0 . 0 0 8 2 5 0 . 0 0 8 6 4

2 4 0 . 0 0 6 0 1 0 0 0 . 0 0 3 4 1 0 . 0 0 3 1 3 0 . 0 0 4 2 3 0 . 0 0 5 9 31 0 0 0 0 0 1 5 1 0 0 0 0 . 0 0 1 0 5 0 . 0 0 0 9 2 0 . 0 0 0 9 8 0 . 0 0 1 6 35 0 0 0 . 0 0 0 3 4 1 0 0 0 0 0 . 0 0 0 3 4 0 . 0 0 0 2 5 0 . 0 0 0 2 4 0 . 0 0 0 7 0

P o S A P d a t a F e - 4 5 % C rA g i n g H a n d l e d e n s i ty( h ) ( C - W s m o o t h i n g )

M o n t e C a r l o A 5 0 % B o n b . c .c , l at t i ce 7 5 0 K . Ew = 0 . 5 , E c r = 0 . 3A g i n g H a n d l e d e n s i ty ( c e n t re - w e i g h te d s m o o t h i n g )

( M C S ) ~ = 0 a = 0 . 5 a = 1 . 0 a = 2 . 00 0 . 0 2 8 1 0 . 0 3 1 9 0 . 0 3 1 4 0 . 0 3 3 0 0 . 0 2 7 94 0 . 0 3 0 1 0 0 . 0 2 5 4 0 . 0 2 7 4 0 . 0 2 8 9 0 . 0 2 7 6

2 4 0 . 0 2 7 1 0 0 0 . 0 1 9 7 0 . 0 1 9 1 0 . 0 2 3 5 0 . 0 2 4 31 0 0 0 . 0 1 7 1 0 00 0 . 0 1 3 3 0 . 0 1 3 2 0 . 0 1 5 8 0 . 0 1 8 25 0 0 0 . 0 0 4 0 1 0 0 0 0 0 . 0 1 0 1 0 . 0 0 9 1 0 . 0 1 0 6 0 . 0 1 2 5


9/12

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 IN F e C r A L L O Y S I ll 3423a )

O.0 1q I '1C 6 5 0 K I~ , ~ ~ - 7 5 0 K I~ . 0 ' 0 0 8 1 ~ ~ 8 5 0 K I

o . o o "\0 . 0 0 4 ~ ~ ,

- r 0 . 0 0 20 - - - - ~ ' ~1 1 0 1 0 0 1 0 0 0 1 0 4

M C S

b )0 . 0 7 ~ C' 6 5 0 K0 . 0 6 ~ X { ~ 7 5 0 K0 0 5 1 ~ - - z > - 8 5 0 K

1 1 0 1 0 0 1 0 0 0 1 0 4M C S

F i g. 9 . T h e n u m b e r o f h a n d l e s p e r a t o m i n th e M o n t e C a r l os i m u l a t i o n o f a n A 5 0 % B a ll o y o n a s i m p l e c u b i c l a t t ic e a sa f u n c t i o n o f a g i n g a n d a g i n g t e m p e r a t u r e , l a ) T h e r e s u l ts f o rt b u r - p o i n t s i m p l e s m o o t h i n g , w h e r e a s l i v e - p o i n t c e n t r e -weighted smoothing was used in (b) . Simple smoothingremoves the f ine scale interface detail and all the graphscollapse to form one,

smoothing) shows a marked te mperature dependence[Fig. 9(b)] similar to that observed by fractal analysis.

The ratio of the centre-weighted smoothing tosimple smoothing handle densities, a dimensionlessmeasure of the number of small scale loops to largerscale loops, compares bulk structure with the finescale. Moreover, because the numbe r of atoms cancelsout when calculating the ratio, a direct comparison canbe made with Monte Carlo data. The experimentalresults show that the ratio increases with aging, whichimplies that the number of large scale loops decreasesmore rapidly than small scale loops. Similar beha viouris observed for the Mo nte Carlo simulations both onb.c.c, and simple cubic lattices.

3 .5 . N on- l i r a ,a t C ahn H i / l i a rd C ook m ode /The handle density' l~r the structures generated b y

the solution to the non-lin ear Cahn Hilliard Cookequation is shown in Fig. 10. At early aging times(small amplitude) no order exists and the structure isfinely percolated with a high handle density (as is thecase for the initial structure in the Monte Carlosimu lation see Fig. 9). An analys is of the autocorre-lation function shows that no coarseni ng occurs before100 time units. However. during this time period bo ththe numb er of small scale handles [Fig. 10(b)] and the

numbe r of larger handles [Fig. 10(a) ] rapidlydecreases. Thereaft er, the structure coarsens and thenumber of handles decreases to lower values thanobserved either from the experimental data or Mon teCarlo s imulations. This is due to the smoothness of theboundary between the two phases in this model, adirect result of using a continuous compositionvariable. Even the addition of the Cook term, and themodelling of the detection efficiency and trajectoryaberrations, does n ot generate values as high as thoseobserved from the experimental alloys. A comparis onwith the Monte Carlo results using four-point simplesmoothing, which lacks the fine scale detail of theinterface, still shows a different evo lution of the handledensity. Although the results generated may have asimilar scale and amplitude to the Monte Carlosimulations, the details of the interface topology arequite different and so a topological analysis yieldsdifferent results. The ratio of handle densitiesdecreases with aging, tending towards the limit 1, atwhich point no small loops exist.

4 . D I S C U S S I O N A N D C O N C L U S I O N SDifferent measures of fractal dime nsion can be used

to parameterize complex microstructures. The resultsobtained are sensitive to both detection efficiency andspatial resolution. Data from the experiments andboth the simulations showed that the fractaldimension of the interface between the two phases

a ) o . o o 6 ~ Simp le smoo th ingCookTerm Mode l l ed xper imenta l0.005~ Limi tat ions~ [ ( a ) I . o . o~ 0 . 0 0 4 \ , ~ I I b ) I v ~ NOo .oo3~ \0 . 0 0 2o . o o i - "~ ~

( b ) ~O- , , - " ~ 40 . 1 1 1 0 1 0 0 1 0 0 0 1 0A g i n g (T i m e U n i t s )

b ) Cen t re -we igh ted smoo th ing0 . 0 5 ~ M o d e l l e dExperimentalCookT e r m L i m i t a t i o n sI : ; : :~" 0 .03 J ' {& ~ (c ) Yes Y=

0 0 2 J0 . 0 1 - ; ~ (c )

0 .1 1 1 0 1 0 0 1 0 0 0 1 0A g i n g ( T i m e U n i t s )F i g . 1 0 . D e v e l o p m e n t o f h a n d l e d e n s i t y w i t h a g i n g f o r t h enumerical solution to the Cahn-Hilliard equation. In (a)simple smoothing has been applied and in (b) centre-weightedsmoothing.


10/12

3424 HYDE et a l . : SPINODAL DECOMPOSITION IN Fe Cr ALLOYS IIldecreased with increasing decomposition from ~3towards 2. However, even at the later stages of aging.both the microstructures observed experimentally andfrom Mon te Carlo simulations exhibited fractal form.In contrast, the smooth interfaces generated duringcoarsening by the numerical solution to theCahn-Hilliard-Cook equation rapidly approachsimple 2D surfaces.

Digital topology provides a new technique formeasuring and characterizing the microstructure inalloys. The Euler relation, and the well developedskeletonization algorithms, linked with the computi ngpower now available, enable a complex morphology tobe reduced to a basic framework which may beanalysed in terms of the number of cavities andhandles. This in turn can be used as a structuralparameter for correlation with the mechanical,magnetic or electronic properties of the material. It istherefore important to extend the experimental workto achieve correlation with properties. A high degreeof correlation has been found between the experimen-tally observed Fe-C r microstructures and those froma Monte Carlo simulat ion of spinodal decomposition.The smoother interfaces obtaine d from the numericalsolution to the Ca hn-Hi lliar d-Cook equation resultedin the measuremen t of a lower handle density at laterstages of evolu tion and a decrease in the ratio of handledensities with aging. The information obtaineddepends on the type of smoothin g employed, and, bymeasuring a handle density, it is possible tocharacterize the structure in terms of both large scaleand small scale loops. The larger loops have beencorrelated with the scale of the structure and the smallloops with interface measurements.

a )

gg.E_

3

Lk

3

2 . 9 5

2 9

2 . 8 5

2 8 -

2 7 50 . 0 0 1

P o S AP Dat a

oo

Oo o oo

O OO0 . 0 1

Handle Dens i ty (G -W smoothing)i

0. 1

b )3

O " 6 5 0 K I N 12 9 - ~ 7 5 0 K

_~ ~ 85 0 K2 8 -gE 2 7 -

25- / " ~

2 4 ~ 10. 1. 0 0 1 0 . 0 1

Handle Densi ty (C-W sm ooth ing)Fig. 11. (a) Relationship between handle density ofcentre-weighted structure and fracta l dimension of thesurface area (measured by counting surface nodes). The datais for the series of thermally aged Fe-45% Cr specimens. (b)Relationship between handle density of centre-weightedstructure and fractal dimension of the surface area (measuredby counting surface nodes) for Monte Carlo simulations ofan A 50'" ; B alloy on a simple cubic lattice performed at arange of temperatures.

5 . C O R R E L A T I O N O F P A R A M E T E R SAs has been previously stated, the centre-weighted

smoothin g preserves the fine scale detail, and, with thissmoothing, most of the handles measured are smallscale loops around the interface region. Since thetYactal dimension gives a measure of interfaceroughness, there should be a relationship betweenfractal dimension and ha ndle density. To confirm thissupposition, a plot of fractal dimensio n against handledensity is shown in Fig. 11. Alth ough the experimenta lresults show some scatter, the trend of decreasingtYactal dimension with handle density is clear[Fig. ll(a)]. The Monte Carlo results for a range oftemperatures collapse ont o a single line [Fig. l l(b)].The results demonstrate a direct relationship.indepe ndent of aging temperature, between the fractaldimensio n of the interface and the n umber of fine scaleloops.

A similar relationship exists between the scale of astructure and handle density after simple smoothing(Fig. 12). Agai n there is some scatter from theexper imenta l data [Fig. 12(a)]. The Monte Car lo datashows a small temperature dependence; at highertemperatures the coarsening process appears to

proceed slightly more rapidly, whereas the handledensity decreases slightly less rapidly. The combi-nation is clearly shown in Fig. 12(b).

It is possible to characterize microstructures interms of bulk properties (scale of decomposition,number of large scale loops) and interface properties(fractal dimension of the interface area and fine scaleloops) as shown in Fig. 12. However, an importantquestion remains unanswered--what principallydetermines the hardness and embrittlement of t h e s espinodally decomposed alloys? Analysis of bothexperimental data and the simulations showed that asdecomposition proceeds, the scale of the domainsincreases, the compositio n amplitude increases and theinterfaces between domains become smoother. Therelationship between these parameters and the alloyhardness is shown in Fig. 13. Once decomp ositi on h a dproceeded to such an extent that a scale could bedetermined, the hardness of the alloys was found t oincrease linearly with increasing dom ain size as s h o w nin Fig. 13(a). Figure 13(b) shows the increase incomposition amplitude with alloy hardness. Figure13(c) exhibits a mono toni c increase in hardness with


11/12

HYDE e t a l . : SPINODAL DECOMPOSITION IN Fe-Cr ALLOYS--III 3425decreasing fractal dimension. Without analysing theeffect of heat treatment temperature on thedevelopment of scale, composition amplitude andmorphology, no firm conclusion can be reached onwhich parameter gives the best guide to the observedphysical properties.

6. CONCLUSIONSThe advances in atom probe microanalysis now

enable the 3D reconstruction of compositionvariations on a sub-nan ometre scale. The complexmorphology formed during spinodal decompositionof thermally aged Fe-Cr alloys has been analysed.Quantitative parameters have been developed whichcan fully characterize the domain size, the domainconcentrations, the interface morphology and thetopology of the microstructure.

Two models of the dec omposition process have beenimplemented and directly compared with theexperimental data. Results from the Monte Carlomodel accurately matched the experimental data andpredicted the observed kinetics of phase separation.The numerical discretization of the Cahn-Hilliard

(a)

v03

1 01 0 Sca le ( F i rs t M ax im um ) It ,~ Scale (Firs t Minimum ) I

i

3 0 0 i i i i3 5 0 4 0 0 4 5 0 5 0 0Hardness (VHN)

( b )

go .o(.5

0 . 4 q

0 3 10 2 q !0 1 q

0 i 2 0 0 2 5 0

OO @

Oi i i i3 0 0 3 5 0 4 0 0 4 5 0 5 0 0Hardness (VHN)

a )

~oO0

3

2 . 5 -

2 -

1. 5

PoSAP dataOO .

O

1 T0 . 0001 0 . 001 001Handle Density (Simple smoothing)

0. 1

b) 1 2. ~ 1 0

8

6 -

2

q ~O

[ ]o 6 5 0 ! 1[ ] 7 5 0O 850

C~0.0001 0.001 0.01

Handle Density (Simple smoothing)Fig. 12. (a) Relationship between scale and handle density ofPoSAP microstructures after simple smoothing. The scalewas determined by using the fi rst minimum of theautocorrelation function. (b). Relationship between scale andhandle density of Monte Carlo microstructures on a simplecubic grid after simple smoothing for a range of temperatures.The scale was determined by using the first minimum of theautocorrelation function.

( c )

f,gEc5

u_

2 , 8 !iI2 . 7 5 ~iI2 7 iii2 . 6 5 I!

2 . 6 ! iI2.55 j2 5 0

8 O

OOo

i i [ [ i3 0 0 3 5 0 4 0 0 4 5 0 5 0 0Hardness (VHN)Fig. 13. (a) Relationship between microhardness and scale ofdecomposition, (b) relationship between microhardness andcomposition amplitude and (c) relationship betweenmicrohardness and fractal dimension.

Cook equation generated qualitatively similar struc-tures, but the domain interfaces were smoother thanexperimentally observed. Moreove r the time expon entfor coarsening was found to be higher than theexperimental results.

With only a single alloy and single heat treatmenttemperature it was not possible to determine whetherthe physical properties exhibited by these alloys areprimarily determined by the scale of decomposition,the compositi on of the phases, the complexity of theinterface between the doma ins or a com bina tion ofthese parameters. It is therefore important to extendthe experimental work to achieve correlation withproperties. It will then be possible to use the predictivecapability of the models to determine the long termperformance and stability of alloys.


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3 4 2 6 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 IAcknowledgements--The a u t h o r s w o u l d l i k e t o t h a n kP r o f e s s o r R . J . B r o o k l b r t h e p r o v i s i o n o f l a b o r a t o r yf a c i li t ie s . J . M . H . w o u l d l i k e t o a c k n o w l e d g e t h e E n g i n e e r i n g~ , n d P h y s i c a l S c i e n c e s R e s e a r c h C o u n c i l ( E P S R C ) a n dW o l f s o n C o l l e g e f o r f i n a n c i a l s u p p o r t . A . C . t h a n k s T h eR o y a l S o c i e ty f o r f in a n c i a l s u p p o r t a n d W o l f s o n C o l l e g e f o rt h e p r o v i s i o n o f a F e l l o w s h i p . T h i s r e s e a r c h w a s f u n d e d b yt 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 eD i v i s io n o f M a t e r i a ls S c i e n c es . U . S. D e p a r t m e n t o f E n e r g y ,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 t h M a r t i n M a r i e t t aE n e r g y S y s t e m s I n c . T h e s i m u l a t i o n s w e r e p e r f o r m e d i n t h eM a t e r i a l s M o d e l l i n g L a b o r a t o r y a t O x f o r d U n i v e r s i t y w h i chi s f u n d e d b y t 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 / 5 8 2 7 8 .

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