hydriding kinetics of powders

7/27/2019 Hydriding Kinetics of Powders

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E L S E V I E R J o u r n a l o f A l l o y s a n d C o m p o u n d s 2 1 6 ( 1 9 9 4 ) 1 59 - 1 7 5

J o u r n a l o fAL~YSAND COMPOUNDS

R e v i e wHydriding k inetics of powders

M o s h e H . M i n t z a'b , Y e h u d a Z e i r i a~Nuclear Research Center, Negev, PC) Box 9001, Beer-Sheva, Israel

bThe Ben-Gurion University of the Negev, Department of Nuclear Engineering, PO B ox 6 53, Beer-Sheva, IsraelR e c e i v e d 1 8 M a r c h 1 9 9 4 ; i n f i n a l f o r m 9 M a y 1 9 9 4

A b s t r a c t

Kinetic measurements of gas-solid reactions performed on powders, are usually interpreted according to single-par t ic leanalysis (SPA) models. The use of SPA functions is not a pr ior i just if ied, s ince deviations f rom that functional dependencemay occur in the powder, owing to three factors: (i) particle size distributions, (ii) particle shape variations and (iii) timedistr ibutions for the beginning of the reaction on each one of the par t ic les composing the powder. The effects of these factorsare quanti ta t ively analysed and th eir interferenc e in the SPA procedure is est imated. I t is concluded that u nder cer ta incircumstances the S PA can yie ld the correct reaction m echanism and even enables a reasonable est im ate of the correspondingintr insic kinetic parameters. This analysis is re levant generally to gas-solid kinetics, with par t icular emphasis on hydridingreactions.Keywords: H y d r i d i n g k i n e t i c s ; P o w d e r s ; S i n g l e - p a r t i c l e a n a l y s i s

1 . I n t r o d u c t i o n

N u m e r o u s k i n e t i c s t u d i e s o n g a s - s o l i d r e a c t i o n s i ng e n e r a l , a n d m o r e s p e c i f ic a ll y o n h y d r i d i n g - d e h y d r i d i n gr e a c t i o n s o f p o w d e r e d m e t a l l i c s a m p l e s , h a v e b e e np r e s e n t e d i n t h e l i t e r a t u r e d u r i n g t h e l a s t t w o d e c a d e s .M o s t o f t h e s e s t u d i e s w e r e r e l a t e d t o i n t e r m e t a l l i ch y d r id e s s y s t e ms ( f o r a r e c e n t r e v i e w s e e [ 1 ] ) , w h e r e a sa f e w d e a l t w i t h b i n a r y h y d r o g e n - m e t a l r e a c t i o n s ( s e ef o r e x a m p le [ 2] ). I n a n y c a s e , w h e n d i f f e r e n t i n v e s ti -g a t io n s p e r f o r m e d o n s o m e g iv e n s ys t e m s a re c o m p a r e d ,c o n f u s i n g a n d d i v e r s e k i n e ti c r e s u l ts a r e o b t a i n e d , c a s t-i n g s e r i o u s d o u b t s o n t h e m e c h a n i s t i c i n t e r p r e t a t i o n so f th e s e s t u d i e s . F o r e x a m p l e , c o m p a r i n g d i f f e r e n tr e s u l t s o n t h e w e l l - s t u d i e d L a N i s - h y d r o g e n s y s t e m , av a r i e t y o f r e a c t i o n r a t e s , p r o p o s e d r a t e l a w s a n d r e l a t e dm e c h a n i s m s h a v e b e e n r e p o r t e d [ 3 -1 1 ]. S o m e o f t h e s ed i s c r e p a n c i e s w e r e a t t r i b u t e d t o h e a t t r a n s f e r e f f e c t sr e s u l t in g i n n o n - i s o t h e r m a l c o n d i t i o n s [1 0]. I n d e e d , s u c he f f ec t s m a y l e a d t o e r r o n o u s k i n e t i c r a t e l aw s , a sd e m o n s t r a t e d b y D a n t z e r a n d O r g a z [ 12 , 13 ]. H o w e v e r ,e v e n w h e n c o n t r o ll e d h e a t t r a n s f e r c o n d i t i o n s w e r ea p p l i e d [ 4 , 5 , 7 , 8 ] , i n c o n s i s t e n t r e s u l t s s t i l l o c c u r r e d .

I t h a s b e e n p o i n t e d o u t [ 14 ] t h a t i n o r d e r t o e v a l u a t ei n t r i n s i c k i n e t i c p a r a m e t e r s ( i . e . t h e p a r a m e t e r s w h i c ha r e n o t d e p e n d e n t e i t h e r o n g e o m e t r i c a l f a c t o r s o r o n

r e a c t i o n t im e , t h u s s p e c i f y in g t h e i n t r i n s i c k in e t i c s o ft h e g i v e n r e a c t i n g s y s t e m , u n d e r t h e g i v e n p r e s -s u r e - t e m p e r a t u r e c o n d i t i o n s ) i n a r e l i a b l e m a n n e r , t h ek i n e ti c m e a s u r e m e n t s s h o u l d b e p e r f o r m e d o n m a s si v es a mp le s , w i th w e l l d e f i n e d g e o me t r i c a l s h a p e s ( e . g .p l a t e s , c u b e s , c y l i n d e r s o r s p h e r e s ) . F o r s u c h s a mp le s ,w h e n r e a c t e d u n d e r c e r t a i n e x p e r i m e n t a l c o n d i t i o n sw h i c h l e a d t o t h e p r o g r e s s i o n o f t h e r e a c t i o n b y a" c o n t r a c t i n g e n v e l o p e " ( o r " s h r i n k i n g c o r e " ) m o r -p h o lo g y , s imp le e x a c t a n a ly t i c a l e x p r e s s io n s r e l a t e t h eme a s u r e d o v e r a l l k in e t i c s ( i . e . t h e t o t a l r e a c t e d f r a c t i o na v s . t ime c u r v e s ) t o t h e i n t r i n s i c k in e t i c p a r a me te r so f t h e s y s t e m. T h e d i f f e r e n t c h o i c e s o f t h e s e i n t r i n s i cp a r a m e t e r s a r e s u m m a r i z e d i n [ 14 ] fo r d i f f e r e n t p o s s i b let y p e s o f k in e t ic s . F o r e x a m p l e , f o r a c o n s t a n t ( t im e -i n d e p e n d e n t ) v e l o c it y U o f t h e h y d r i d e - m e t a l i n t er f a c e ,t h i s p a r a m e t e r i s a p p r o p r i a t e f o r c h a r a c t e r i z i n g t h er e a c t in g s y s t e m . F o r a t i m e - d e p e n d e n t v e l o c it y , o t h e ri n t ri n s ic k i n e ti c p a r a m e t e r s m a y b e c h o s e n . T h e p r e s-s u r e - t e m p e r a t u r e d e p e n d e n c e o f t h e s e k in e t ic p a ra m -e t e r s m a y t h e n p o i n t t o t h e m i c r o s c o p i c m e c h a n i s mc o n t r o l l i n g t h e r e a c t i o n r a t e [ 1 4 -1 9 ] .

F o r s i m p l e g e o m e t r i c a l s h a p e s o f t h e r e a c t i n g s a m p l e s ,t h e r e l a t i o n b e t w e e n t h e o v e r a ll r e a c t e d f r a c t i o n a a n dt h e r e a c t i o n d i s p l a c e m e n t X(t) ( a t a g iven t ime t ) i sgiven by [14,20- .22]

0925-8388:95 /$09 .50 1995 E l s ev ier Sc ience S . A. Al l r igh t s r es erve dS S D I 0 9 7 5 - 8 3 8 8 ( 9 4 ) 0 1 2 6 9 - N


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160 M.H . Mintz, Y. Zeiri /Journ al o f Alloys and Com pound s 216 (1994) 159-1753a ( t ) = ~ a , X i ( t ) (1 )

i = l

w h e r e a a r e c ons t a n t s r e l a t e d t o t he sha pe a nd in i t i a ld im e ns ions o f t he r e a c t ing sa m ple s , a nd X ( t ) is givenby

X ( t ) = X o + ' f U ( t) d tto

(2 )

wi th U ( t ) t h e v e l o c i t y o f t h e h y d r i d e - m e t a l i n t e r f a c e( w h ic h m a y b e e i t h e r t i m e d e p e n d e n t , e .g . f o r so m edi f f us ion- c on t r o l l e d c a se s , o r c ons t a n t [ 14], t o be ing thet i m e r e q u i r e d t o f o r m f ir st a c o n t i n u o u s h y d r i d e l a y e ron the sa m ple su r f a c e , a nd Xo the i n i t i a l t h i c kne ss o ftha t l a ye r wh e n f o r m e d a t to ( F ig . 1 ) . He nc e , f o r t het im e in t e r va l be twe e n to a nd t , E q . ( 1 ) a l l ows f o r t hee xa c t e va lua t ion o f X( t ) vs . t , f r om the m e a su r e d va lue sof t he ove r a l l r e a c t e d f r a c t i on a ( t ) . A p lo t o f X ( t ) vs.t y i e lds t he n the hydr ide f r on t ve loc i ty U ( t ) (given byt h e s l o p e o f t h a t c u r v e ) , w h i c h a s m e n t i o n e d a b o v e i st h e i n t r in s i c k in e t i c p a r a m e t e r f o r t h e c a s e o f a c o n s t a n t( t i m e - i n d e p e n d e n t ) v e l oc i ty .I t s h o u l d b e p o i n t e d o u t t h a t f o r t < t o , i .e . beforea c o n t i n u o u s p r o d u c t l a y e r h a s b e e n f o r m e d o n t h esa m ple , no s im ple e xpr e ss ion ( suc h a s E q . ( 1 ) ) c a n beu t i li z e d to r e l a t e t he ov e r a l l ( m e a su r e d) r e a c t e d f r a c t i ona ( t ) a n d a n y i n t r i n s i c k i n e t i c p a r a m e t e r . H e n c e t h equa n t i t a t i ve i n t e r p r e t a t i on o f t he k ine t i c da t a i n t ha tin i t i a l r a nge i s no t poss ib l e i n m os t c a se s , a nd on lyqua l i t a ti ve t r e nds m a y be e s t im a te d . T h e r e l i a b l e a p-

R R - %

( o ] ( b )

~ u/

F i g . 1. S c h e m a t i c i ll u s t r a t io n o f t h e d e v e l o p m e n t o f a p r o d u c t l a y e ro n a s p h e r i c a l r e a c t i n g p a r t i c l e w i t h a n i n i t i al r a d i u s R : ( a ) n u c l e a t i o no n t h e s u r f a c e ; ( b ) n u c l e i g r o w t h a n d p a r t i a l o v e r l a p ; ( c ) i n i t i a lf o r m a t i o n o f a c o n t i n u o u s p r o d u c t l a y e r ( w i th a n a v e r a g e t h i c k n e s sX o ) a t t i m e to ; ( d ) f u r t h e r t h i c k e n i n g o f t h e l a y e r a t t i m e t .

pl i c a t i on o f E q . ( 1 ) f o r t he e va lua t ion o f t he hy-d r i d e - m e t a l i n t e r fa c e v e lo c i ty U t h e r e f o r e r e q u i r e s t h a tXo be sm a l l c om pa r e d w i th t he i n i t i a l d im e ns ions o ft h e r e a c t in g s a m p l e . S u c h r e q u i r e m e n t m a y n o t a lw a y sbe ful f i lled, espec ia l ly for sm al l - s ize sam ples (e .g . thinf o i l s o r f i ne - pa r t i c l e powde r s ) . T h i s po in t w i l l f u r the rbe d i sc usse d l a t e r .

I n c e r t a in c a se s , whe n the s i z e o f t he sa m ple c a nbe spe c i f i e d by a s ing l e d im e ns iona l pa r a m e te r R , E q .( 1 ) c a n be r e p l a c e d by a r e l a t i on o f t he f o r m [ 14 ,20-22]kFr. g(a ) = ~ t (t > t0) (3)

whe r e F r .g ( a ) i s a spe c i f i e d f unc t ion o f t he r e a c t e df r a c t ion . T he subsc r ip t r i nd i c a t e s t ha t t he f o r m of t ha tf unc t ion de pe nds on the t ype o f r e a c t ion k ine t i c s ( e .g .k ine t i c s w i th t he c ons t a n t U , o r d i f f us ion- c on t r o l l e dk i n e t i c s w i t h t h e t i m e - d e p e n d e n t U ( t ) e t c . ) a nd thesubsc r ip t g t ha t i t de pe n ds on th e ge om e t r i c a l sha peof t he sa m ple ( e .g . a c ube , sphe r e o r w i r e ) , k i s ac ons t a n t ( e .g ., f o r a t im e - in de p e nd e n t i n t e r f a c e ve loc ity ,k i s p r opor t i ona l t o U) , a nd n i s a n i n t e ge r ( 1 o r 2 )r e l a t e d t o t he t ype o f k ine t i cs .As for Eq. (1) , Eq. (3) i s appl icable only a f te r thec om ple t ion o f a c on t inuo us p r odu c t l a ye r w i th a n i n i t ia lthickne ss o f Xo (Fig. 1) , i .e . for t > to. Th e cor re spo ndi nga pp l i c a b l e a r a nge thus de pe nds on the r e l a t i ve m a g-n i t u d e s o f Xo a n d R . F o r X o / R


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M.H. Mint z , Y . Ze M / J ournal o f A l loy s and C ompo unds 21 6 (1994) 159 -175 161

( e .g . s p h e r e s ) . O b v i o u s l y , s u c h a n a s s u m p t i o n c a n n o ta p ri o r i b e j u s ti f ie d , a n d t h e a b o v e a p p r o a c h h a s b e e ncr i t i c i zed in the l i t e ra tu re [23-25] , po in t ing to poss ib lee r r o n e o u s o r p e r h a p s i n d e t e r m i n a t e r e s u lt s c o n c e rn i n gr e a c t i o n me c h a n i s ms a n d a c t i v a t i o n e n e r g i e s , d e d u c e db y t h e s i mp l i f i e d a b o v e p r o c e d u r e .

I n t h i s w o r k , t h e d e v i a t i o n s in t r o d u c e d i n t o t h e s in g l e-p a r t i c l e a n a l y s i s ( SPA ) b y t h e d i f f e r e n t f a c t o r s c h a r -a c t e r i z i n g r e a l p o w d e r s a r e e s t i ma t e d . T h e s e f a c t o r sinc lude (1 ) par t i c l e s i ze d i s t r ibu t ions , (2 ) par t i c l e shapev a r i a t i o n s a n d ( 3 ) t i me d i s t r i b u t i o n s f o r t h e c o m-me n c e m e n t o f t h e r e a c t i o n o n d i f f e r e n t p a r t ic l e s , c o m-p o s in g t h e p o w d e r . T w o S P A m o d e l s a r e a p p l ie d , ac o n s t a n t - v e l o c i ty c o n t r a c t i n g - e n v e l o p e mo d e l , a n d a d i f -f u s i o n - c o n t r o l l e d t i me - d e p e n d e n t - v e l o c i t y mo d e l . T h ea b o v e c a l c u l a t i o n s i n d i c a t e u n d e r w h i c h p o w d e r c h a r -ac te r i s t i cs and reac t ion cond i t ions a re l i ab le d i s t inc t ionb e t w e e n t h e t w o m o d e l s m a y s t il l b e p o s s i b l e , u ti li z in ga s i mp l e SPA p r o c e d u r e . I t h a s b e e n s h o w n [ 2 5 ] t h a t ,u n d e r c e r t a i n c o n d i t i o n s , a p p a r e n t a c t i v a t i o n b a r r i e r sc a n b e e v a l u a t e d , e v e n w i t h o u t k n o w l e d g e o f e i t h e rt h e c o n t r o l li n g me c h a n i s m o r t h e p o w d e r p a r t i c le s i z ed i s t r i b u t i o n . I n t h e s e c a s e s i t i s s h o w n h e r e t h a t a ne v a l u a t i o n o f t h e p r o p e r me c h a n i s m i s a l s o p o s s i b l e b yt h e SPA p r o c e d u r e , w i t h o u t a n y c o n s i d e r a t i o n o f s i z eo r s h a p e d i s t r i b u t i o n s . H o w e v e r , t h i s s i mp l i f i e d SPAp r o c e d u r e i s l i m i t e d t o r e a c t i o n s w h e r e t h e t i me d i s -t r i b u t io n s f o r t h e i n i t ia t i on o f r e a c t i o n o n t h e d i f f e r e n tp o w d e r p a r t ic l e s a r e mu c h s ma l l e r t h a n t h e c o r r e -s p o n d i n g r e a c t i o n h a l f - t i me s t in , a n d w h e r e a c o n -t r a c t i n g - e n v e l o p e m o r p h o l o g y i s a t ta i n e d f o r v e r y s ma l la va lues ( i . e . when Xo i s much smal l e r t han the par t i c l ess ize) .

I n c o n t r a s t w i t h t h e a b o v e c o n t r a c t i n g - e n v e l o p e mo r -pho logy , , where a l l t he p roduct fo rms a def ined f i lm( d e v e l o p i n g f r o m t h e o u t e r s u r f a c e i n t o t h e b u l k ) , t h ee x t r e m e o p p o s i t e c a se i s t h a t o f r a n d o m n u c l e a ti o na n d g r o w t h i n t h e b u l k , w h e r e t h e p r o d u c t i s d i s p e r s e deven ly wi th in the re ac t ing par t i c l e . In th i s case , t hek i n e t ic e q u a t i o n s s h o u l d e v i d e n t l y b e i n d e p e n d e n t o ft h e s a mp l e ' s g e o me t r i c a l s h a p e o r s i z e . T h e e v a l u a t i o nof such ana ly t i ca l k ine t i c func t ions i s no t as s imple asf o r t h e c o n t r a c t i n g - e n v e l o p e c a s e s , a n d o n l y a p p r o x i ma t er e l a t i o n s c a n b e o b t a i n e d [ 2 6 - 2 8 ] . T h e mo s t u t i l i z e df u n c t i o n s o f th i s ty p e a r e t h e A v r a m i - E r o f e y e v f u n c ti o n s ,g i v e n i n Se c t i o n 2 . I n o r d e r t o c o mp l e t e t h e k i n e t i cana lys i s o f powders , t h i s case i s a l so inc luded .

I t s h o u l d b e e mp h a s i z e d t h a t s e l f - h e a t i n g p r o b l e ms ,e n c o u n t e r e d i n m a n y k i n e t i c e x p e r i m e n t s p e r f o r m e do n p o w d e r s , a r e n o t c o n s i d e r e d h e r e a n d i s o t h e r m a lc o n d i t i o n s a r e a s s u me d . I t i s e v i d e n t t h a t t h e i n t e r -f e r e n c e o f s e l f -h e a t i n g e f f e c t s w i ll mo d i f y t h e m e a s u r e dk i n e ti c s i n s u c h a c o mp l e x w a y t h a t n o m e a n i n g f u la n a l y s i s c a n t h e n b e ma d e .

The fo l lowing ana lys i s i s per t inen t in genera l t o avar i e ty o f gas -so l id reac t ions ; however , spec i f i ed ex -a mp l e s o n h y d r i d e s y s t e ms a r e d e mo n s t r a t e d .

2 . Calculat ion procedures2.1. Single-particle analysis

Be f o r e d i s c u s s i n g p o w d e r a n a l y s i s , s o me r e mi n d e r o fSP A ma y b e h e l p f u l. A s me n t i o n e d i n Se c t io n 1 f o ra c o n t r a c t i n g - e n v e l o p e p r o g r e s s io n , f u n c t i o n s o f t h ef o r m g i v en e i t h e r b y E q . ( 1 ) o r a l t e r n a ti v e l y b y E q .( 3 ) r e l a t e t h e r e a c t e d f r a c t i o n a t o t h e r e a c t i o n t i met. A s u mm a r y o f t h e s e f u n c t i o n s i s g i v e n in t h e l i t e r a tu r e[ 1 4 , 2 0 - 2 2 ] . S i n c e t h e c o mmo n a p p r o a c h t o p o w d e ra n a l y s i s a s s u me s t h e p o w d e r s t o c o n s i s t o f s p h e r i c a lpar t i c l es , we sha l l spec i fy now E qs . (1 ) and (3 ) fo r th isg e o me t r y o n l y . T h e k i n e t i c c l a s s e s a r e t h e n a n a l y s e dfo r the con t rac t ing-envelope cases , and a th i rd c l as s( t h e A v r a m i - E r o f e y e v t y p e ) fo r t h e r a n d o m n u c l e a ti o nand g rowth in the bu lk .

Fo r t h e c o n t r a c t i n g - e n v e l o p e p r o g r e s s i o n , o n e p o s -s i b le c a s e i s o f a c o n s t a n t i n t e r f a c e v e l o c i ty U ( h e n c e f o r t hd e n o t e d b y th e s u b s c r i p t CV ) w h e r e t h e k i n e t i c f u n c t io n sa r e g i v e n e i t h e r b yt rey ( t ) = 1 - 1 - ~ t 3U 3U 2 t3= T t - - t 2+ (4 )or , by rear rang ing Eq . (4 )F c v ( a ) = l - [ 1 - a ( t ) ] 1/3= -R t O < t < (5 )For smal l a va lues , i . e . ~


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16 2 M.H. M intz , Y. Zebi /Jou rnal o f Alloys and Compounds 216 (1994) 159-175r e a c t i o n s , k D i s given by [ 1 4 ]k D ( P , 7 3 - 1 Z o ( P ,y ( T )T ) D( 73 ( 7 )Ew h e r e P is t h e h y d r o g e n w o r k i n g p r e ss u r e , T ( K ) t h er e a c t i o n ( a b s o l u te ) t e m p e r a t u r e , y ( T ) t h e h y d r i d e c o m -posi t ion l imi t ( a t 7 3 , Zo ( P , 7 3 t he e qu i l i b r ium e xc e sshyd r oge n c om pos i t i on ( i .e . t ha t h ydr o ge n whic h d is so lve sin t he hyd r ide a t t he g ive n P - T ) a nd D ( T ) t he d i f f usiv it yo f h y d r o g e n i n t h e h y d r i d e .I t i s wor thw hi l e t o m e nt io n t ha t i n som e s tud i e s [ 21]the so - c a l l e d Ja nde r e qua t ion i s u t i l i z e d t o r e pr e se n tthe a bove d e c e l e r a t i ng ( d i f f us ion- c on t r o l l e d) k ine t ic s .T h i s Ja nd e r f unc t i on , howe ve r , is ba se d o n t he i nc or r e c ta s sum pt ion t ha t t he r a d ius o f t he sphe r i c a l sh r ink ingc o r e o b e y s a s q u a r e - r o o t t i m e d e p e n d e n c e , s i m i l a r t othe d i f f us ion- c on t r o l l e d p l a na r c a se . He nc e ,F o ( a ) = [ F c v ( a ) ]z = k t (8 )w i th Fc v( a ) - g ive n by E q . ( 5 ) a nd k - a d i f fus ion-r e l a t e d c ons t a n t . E v ide n t ly , t he u t i l i z a t i on o f p l a na rk ine t i c s t o t he sphe r i c a l ge o m e t r y is no t jus t i fi e d , a ndthe p r ope r d i f f us ion e qua t ion shou ld be so lve d , l e a d ingto t he a bove Ca r t e r - Va le ns i r e l a t i on ( E q . ( 6 ) ) .

I t i s poss ible , how ever , to s impl i fy Eq. (6) , for casesw h e r e e ~ 1 ( i . e . w h e r e t h e v o l u m e c h a n g e a s s o c i a t e dwi th t he p r o duc t f o r m a t ion i s no t l a r ge ) . I n t he se c a se s,( e - 1 ) a ( t ) < < 1 , a s e r i e s e xpa n s ion o f [1 + ( e - 1 ) a ( t ) ] 2/3r e du c e s E q . ( 6 ) to

2FD(O ~ 1 - - ~ a ( t ) - [1 - a(t)]2/3 2kD= R 2 t

T he l a t t e r s im pl i f i e d f o r m wi l l be u t i l i z e d he nc e f or thf o r t h e a na lysi s o f t he de c e l e r a t i ng- ve loc i t y d i f fus ion-c on t r o l l e d k ine t i c s . I t i s i n t e r e s t i ng t o po in t ou t t ha t ,f o r a < < 1 , e xpa nd ing FD ( a ) i n E q . ( 9 ) to a T a y lo r se ri e sa n d t a k i n g t e r m s u p t o q u a d r a t i c y i e l d

1 2kDF D ( a < < I ) = ~ a 2 (t ) = " - ~ t (9 ' )w h i c h d i s p l a y s t h e s q u a r e - r o o t - o f - t i m e a d e p e n d e n c et y p ic a l o f p l a n a r g e o m e t r y [ 1 4 ] b u t w i t h a d i f f e re n ts lope , a s d i sc usse d f u r the r i n Se c t i on 2 .2 .2 .T he a l t e r na t i ve f o r m of E q . ( 9 ) , c o r r e spond ing t othe aD( t ) VS. t k ine t ic curv es , c an be der ive d by sub-s t i tut ing into Eq. (9) :~ t) = [1 - a(t )] a/3 (10)E q . ( 9 ) t he n a s sum e s t he f o r m

( 3kDt~R/:3 - 1 . 5 + 0 . 5 - , = 0 (1 1 )

So lu t i on o f t he c ub i c E q . ( 11) y ie ld ing the phys i c a ll yme anin gfu l roo t ~o( t ) (0-


5/17

M.H . Mintz, Y. ZeM / Journal of Alloys and Compounds 216 (1994) 159-175T a b l e 1S u m m a r y o f t h e c h a r a c t e r i s t i c s o f t h e d i f f e r e n t f u n c t i o n a l f o r m s u t i l i z e d f o r g a s - - s o l i d k i n e t i c s ( s p h e r i c a l g e o m e t r y )

16 3

T y p e o f k i n e ti c s F u n c t i o n a l t i m e S l o p e o f( r ) d e p e n d e n c e r e d u c e d t i m e(F~(a) v s . t ) r e p r e s e n t a t i o n

( E q . ( 1 7 ) )

R e d u c e dt i m er a n g e a

C o n t r a c t i n g e n v e l o p ew i t h a c o n s t a n ti n t e r f a c e v e l o c i t y(r - -- CV )C o n t r a c t i n g e n v e l o p ew i t h a d e c e l e r a t i n gv e l o c i t y , c o n t r o l l e db y d i f f u s i o n t h r o u g ha n a d h e r e n tt h i c k e n i n g p r o d u c tl a y e r( r = - D )

E q . (5 ) 0 .21

E q . (6 ) 0 .37E 0 .6 3(e + 1 ) 2/3 0 .63+ - -( C a r t e r - V a l e n si ) - 1 - 1 - 1

o r 0 . 0 3 7s i m p l i f i e d f o r m(for E--, 1)E q . ( 9 )

R a n d o m O ul k E q . ( 1 4 )n u c l e a t i o n a n d ( A v r a m i - E r o f e y e v )g r o w t h n = 2 0 . 8 3 3( r - = N G ) n = 3 0 . 8 8 5

n = 4 0.921

0 ~ T ~ 4 . 80 ~ r ~ * ~ 3 . 8

0 ~ r D ~ 90 ~ z D * ~ 8

0~TNG ~0 ~ C * ~ 2 . 60~NG*~I.90 ~ G * ~ l . 6

"~-~ d e n o t e s t h e r a n g e c o r r e s p o n d i n g t o 0 ~< a ~< 1 w h e r e a s ~'~* d e n o t e s t h e r a n g e c o r r e s p o n d i n g t o 0 ~< c t~ < 0. 99 .

0 < a ~ < t , w h i c h a r e g i v e n b yFr(1)0 ) 7 ( 0 . 5 ) ( 1 8 )

F o r t h e b u l k n u c l e a t io n - a n d - g r o w t h k i n e ti c s , e v e nt h o u g h a ~ 1 c o r r e s p o n d s t o ~ 'N G~ ~ , p r a c t i c a l l y i t i ss u f f i c i e n t t o c a r r y o u t t h e c a l c u l a t i o n s w i t h i n t h e r a n g ew h e r e (~ ~< a ~ < 0 . 9 9 , w h i c h c o r r e s p o n d s t o t h e r e d u c e dt i m e r a n g e b e l o w a b o u t 2 - 2 . 6 ( d e p e n d i n g o n t h e v a l u eo f n ) . I n f a c t , t h e l a t t e r 0 ~< a ~< 0 . 9 9 r e d u c e d t i m e r a n g e s( d e n o t e d a s zr * i n T a b l e 1 ) r e l a t e d t o t h e n u c l e a t i o n -a n d - g r o w t h k i n e ti c s a r e s h o r t e r t h a n t h o s e o f t h e c o n -t r a c t i n g - e n v e l o p e k i n e t i c s . T h e s e t i m e r a n g e s { rr *} f o l l o wt h e o r d e r

( 1 9 )I t i s w o r t h w h i l e t o p o i n t t o a q u a l i t a t i v e f e a t u r e w h i c he n a b l e s a s i m p l e d i s ti n c t io n b e t w e e n t h e t w o g r o u p so f k in e t ic s , n a m e l y t h e c o n t r a c t i n g - e n v e l o p e k i n e ti cf u n c t i o n s a n d t h e b u l k n u c l e a t i o n g r o w t h f u n c t i o n s .W h e r e a s t h e t y p e s o f c o n t r a c t i n g - e n v e l o p e k i n e ti c s ( i. e.E q s . ( 4 ) a n d ( 1 2 ) ) d i s p l a y a v s . t c u r v e s w i t h g r a d u a l l yd e c r e a s i n g s l o p e s , t h e b u l k n u c l e a t i o n a n d g r o w t h f u n c -t i o n s ( E q . ( 1 3 ) ) d i s p l a y a n S - s h a p e w i t h a n i n f l e c ti o np o i n t ( i .e . a m a x i m u m s l o p e ) , w h i c h c o r r e s p o n d s t o th er e d u c e d t i m e ( z , , ) :

n - 1 ~ a /." I ' m (n ) = ~ ] ( n > / 2 ) ( 2 0 )F o r n = 2 , 3 a n d 4 , t h e c o r r e s p o n d i n g ~-~ a r e 0 . 8 5 , 0 . 9 9a n d 1 . 02 r e s p e c t i v e l y . H e n c e , r e g a r d l e s s o f t h e v a l u eo f th e n u c l e a t i o n - a n d - g r o w t h r a t e c o n s t a n t k N c , t h ei n f l e c t i o n p o i n t s i n t h e aNG(t) f u n c t i o n s a r e l o c a t e d a ta b o u t ti n ( i . e . d i s p l a y e d a t r e a c t i o n t i m e s c l o s e t o t h a tr e q u i r e d t o r e a c h a=~ ) . T h u s a q u a l i t a t i v e q u i c k d i s -t i n c ti o n b e t w e e n t h e a b o v e t w o g r o u p s o f k in e t i cs i sp o s s i b l e b y j u s t o b s e r v i n g t h e s h a p e o f t h e e x p e r i m e n t a la v s . t c u r v e s . I t m a y b e a r g u e d t h a t t h e o c c u r r e r i c eo f i n it i a l i n d u c t i o n p e r i o d s a s w e l l a s i n i ti a l s u r f a c en u c l e a t i o n - a n d - g r o w t h p r o c e s s e s ( p r e c e d i n g t h e f o r -m a t i o n o f a c o n t i n u o u s p r o d u c t f i lm ) m a y p r a c t i c al l yr e s u l t i n s o m e t u r n i n g p o i n t s a l s o in t h e a v s . t c u r v e sr e l a t e d t o t h e c o n t r a c t i n g - e n v e l o p e k i n e t i c s ( i .e . a sm e n t i o n e d i n s e c t i o n 1 t h e f i t o f E q s . ( 5 ) , ( 6 ) o r ( 9 )m a y s t a rt n o t a t t = 0 b u t a f t e r a c e r t a in to ). T h e s et u r n i n g p o i n t s , h o w e v e r , d i f f e r f r o m t h e i n h e r e n t t u r n in gp o i n t s i n t h e n u c l e a ti o n - a n d - g r o w t h f u n c t i o n s b y ( i)o c c u r r i n g a t l o w a ( a n d t ) v a l u e s , f a r b e l o w t h e a = 0 . 5(t=tl/z) v a l u e s t y p ic a l o f t h e n u c l e a t i o n a n d g r o w t hp r o c e s s e s , ( i i) b e in g m u c h s h a r p e r , r e s e m b l i n g a b r e a kp o i n t r a t h e r t h a n i n f l e c ti o n a n d ( i ii ) d e p e n d i n g o n t h es i z e ( a n d s h a p e ) o f t h e r e a c t i n g s a m p l e s . I n f a c t , t h el a t t e r d i f f e r e n c e i s v a l i d n o t o n l y f o r t h e t u r n i n g - p o i n t


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1 6 4 M.H. Mintz, Y. Zeir i /Journal of Alloys and Compounds 216 (1994) 159-175va lue s bu t a l so f o r t he whole a ( t ) vs. t curves , whichf o r t h e c o n t r a c t i n g - en v e l o p e ca s e s d e p e n d o n t h e s iz ea n d s h a p e o f t h e r e a c t in g s a m p l e s ( r e a c t e d u n d e r g i v e ne x p e r i m e n t a l c o n d i ti o n s ), w h e r e a s f o r t h e n u c l e a t io n -a n d - g r o w t h c a s e s d o n o t d e p e n d o n s i z e a n d s h a p efac tors .I t w i l l be po in t e d ou t l a t e r on ( Se c t ion 3 ) t ha t t he r em a y o c c u r s o m e s p e c i al c a s es w h e r e t h e a ( t ) vs. t curvesob ta ine d f o r we l l - de f ine d ( s ing l e - pa r t i c l e ) sa m ple s don o t d e p e n d o n t h e s i ze a n d s h a p e o f th e r e a c t i n gs a m p l e s a n d y e t o b e y c o n t r a c t i n g - e n v e l o p e t y p e o fk ine t ic s ( a nd no t n uc l e a t ion- a nd - gr owth k ine ti c s ).2 . 2 . P o w d e r a n a l y s i s

I n t he f o l l owing c a l c u l a t i ons t he e f f e c t s o f t h r e epa r a m e te r s whic h a s sum e a s ing le de f ine d va lue i n t heSPA c a se s , bu t so m e sp r e a d va lue s ( i .e . som e d i s t r i bu t ionf u n c t io n s ) i n p o w d e r s , a r e e v a l u a t e d . T h e s e p a r a m e t e r sare ( i ) s ize , (i i ) geo me tr ica l shap e an d ( i ii ) in i t ia l reac t iont im e . E v i d e n t l y , e a c h o n e o f t h e s e p a r a m e t e r s m a y v a r yf or t he d i f f e r e n t pa r t i c l e s c om pos ing the powde r . Ana na lys i s i s t hus m a de in o r de r t o e s t im a te t he i n f lue nc eof suc h va r i a t i ons on th e SP A f unc t ions d i sc usse d be f or e .T h e m a i n q u e s t i o n s a d d r e s s e d t o t h e s e c a s e s a r e t h efol lowing.(1) Can the overa l l kine t ics ( i .e . the a ( t ) vs. t curves)s t i l l be r e pr e se n t e d by the c or r e sponding SPA- l ikefunc t ions (e .g . Eqs . (5) and (9) )?( 2 ) I f t he se SPA f unc t ions a r e s t i l l a pp l i c a b l e , wha ta r e t he c or r e sponding s lope s o f t he se l i ne s?I n t he f o l l owing a na lys i s we sha l l u t i l i z e on ly t hec on ta c t ing - e nve lop e c a se s ( E qs . ( 5 ) a nd ( 9 ) ), s i nc e t hen u c l e a t i o n - a n d - g ro w t h k i n e t ic s a r e i n h e r e n t l y i n d e p e n -d e n t o f t h e s i z e a n d s h a p e p a r a m e t e r s .2 . 2 .1 . E f f e c t s o f s i z e d i s t r i b u t io n sA s s u m e a p o w d e r c o m p o s e d o f s p h e r ic a l p a rt i cl e shaving a given s ize dis t r ibut ion D ( R ) . A s s u m e f u r t h e rtha t a l l pa r t i c l e s s t a r t t o r e a c t a t t he sa m e t im e , t= 0 .E a c h r e a c t ing pa r t i c l e i s a t i s f i e s a g ive n c on t r a c t ing-e nve lop e type o f k ine ti c s r , e .g. a s g ive n by E q . ( 4 )( r = C V) o r by E q . ( 12) (r = D) . T hus , f o r t he i t h powd e r ' spar t ic le ,o~ ( t ) = a~ ( R~ , t ) (21)wi th a r ( R i , t ) give n by the a bove c on t r a c t ing- e nve lopee q u a t i o n s . T h e r e a c t e d v o l u m e o f t h e i th p a r ti c le i sg ive n by

4 4vi ( t ) = -~ ~rRia ai ( t ) = -~ "n 'Ri3 r (R i , t ) (22)a nd the num be r o f pa r t i c l e s w i th t he s i z e r a ng ingb e t w e e n R i a n d R i + d R a red N i = D ( R , ) d R (23)

H e n c e , t h e t o t a l r e a c t e d v o l u m e a t t i m e t , i n t e g r a te dover a l l powder par t ic les , i s given by

4 fr ( t) = ~ ~ r D ( R ) R 3 a ~ ( R , t ) d R (24)Rmin

wi th R m l n t he s iz e o f t he sm a l l e s t pa r t i c l e , D ( R ) a givens iz e d i s t r i bu t ion f un c t ion , a nd a ~ ( R, t ) g ive n e i t h e r byE q . ( 4 ) ( r - = C V ) o r E q . (1 2 ) ( r - D ) . T h e i n i ti a l v o l u m eVo of t he powde r i s g ive n byV o = -~ ~ r D ( R ) R 3 d R (25)

Rmin

or a l t e r na t ive lyWV o - ( 2 6 )P

w i t h W t h e p o w d e r ' s w e i g h t a n d p t h e t h e o r e t i c a l d e n s i tyof t he powde r ' s m a te r i a l .Th e c a lcu la ted tota l rea c te d f rac t io n a~ca lc(t ) (a t t im e

t ) o f t h e w h o l e p o w d e r i so~catc(t)= V r ( t ) (27)

VoSubs t i t u t i ng d i f f e r e n t d i s t r i bu t ion f unc t ions D ( R ) in toEq. (24) , and ut i l iz ing Eq. (27) , the ca lcula ted a r"~( t )vs . t k ine t i c c ur ve s a r e ob t a ine d f o r a ny g ive n type o fk in e ti cs ( r - C V o r D ) .Th e cor r esp on din g Fr ( a ~a1) vs . t curv es m ay the n bee v a l u a t e d a n d c o m p a r e d w i th t h e f u n c t i o n a l S P A f o r m sgiven in Eqs. (5) an d (9) respec t ive ly. Th e ca lcu la t ionsof t he i n t e gr a l i n E q . ( 24) a r e pe r f o r m e d us ing thet r a p e z o i d a l m e t h o d [ 3 1 ]. I n t h e i n t e g r a ti o n p r o c e d u r e ,c a r e m u s t b e t a k e n t o e l i m i n a t e t h o s e p a r t i c l e s t h a tha ve a l r e a d y r e a c t e d c om ple t e ly a t t im e t ( i. e. a , ( t ) = 1),f r om f ur the r be ing c oun te d a t h ighe r t va lue s . T h i s i sdone by d iv id ing the i n t e gr a l i n to

4 R r t )V r ( t ) = 5 D(R )R 3 dRRmm

+ ~ f D ( R ) R 3 a r ( R , t ) d R ) (28a)R r ' ( t )

with Rr ' ( t ) given byRc v' ( t ) = U t (28b)g o ' ( t) = (6kDt) '/2 (28C)

As m e nt ione d in Se c t ion 2 .1 , i t i s c onve n ie n t t opr e se n t t he k ine t i c c ur ve s F , ( a ) o n a r e d u c e d t i m esc a le ~ ', whic h f o r t he SPA c a se s y i e ld f o r e a c h o f t hek ine t i c c la s se s a " un ive r sa l " s lope , r e l a t e d t o t ha t c l as s( se e T a b le 1 ). Suc h a p r e se n t a t i on i s t hus u t i l iz e d a l so


7/17

D i s t r i b u t i o n f u n c t i o n s

IOOE+O0

G a u s s i a n D o ( ( R ) , t r; R )

L o g - n o ~ a a l D E N ( R L N , O '; R )

L a b e l l i n g ; o f d i s t r i b u t i o n f u n c t i o n sF u n c t i o n L a b e l l i n g i n f i gu r e sD c (3, 1; R) G1Do (3, 3; R) G2DLN (3, 0.4; R) L1DLN (3, 1.2; R) L2

0 ( ( R < R m i n )--L--1 e x p - (R- / R m l n )

{ 0 ( ( In R - l n R L N)~ / ( R < R ~ i . )~ e x p 2 t r2 ] (R > / R m ~ . )

0 5

0 4

_ ~ 0 3=

A - 0 2

0 1

0( o )

f o r t h e d i f f e r e n t Fr (a ~a ~c ) p l o t t e d v s . ~ - ( wi t h t l r c o r -r e s p o n d i n g t o a c alc = 0 . 5 ) . I n t h e s e c a l c u l a t i o n s a v a r i e tyo f p o s s i b l e d i s t r ib u t i o n f u n c t i o n s w e r e a p p l i e d i n c l u d i n gs y m m e t r i c ( g a u s s ia n ) f u n c t i o n s , a n d a l s o a s y m m e t r ic( l o g - n o r m a l ) f u n c t i o n s . T a b l e 2 s u m m a r i z e s t h e s e a p -p l i e d d i s t r i b u t i o n s , w h i c h a r e a l s o s h o w n i n F i g . 2 . I na l l c a s e s , R m i n w a s s e t t o 1 / z m .

T h e r e s u lt s o b t a i n e d f o r t h e c o n s t a n t - v e l o c i ty c a s ea r e p r e s e n t e d i n F i g . 3 ( a c v v s . z) a n d F i g . 4 ( F c v ( a )v s . -r ), w h i l e t h e c o r r e s p o n d i n g r e s u l t s o b t a i n e d f o r t h e

0 4

G1G2 (x3)

" , .

- - . . .q . . . . . .5 lo

R [ m i c r o n ]

750E-01

500E-01

L1L2 (x5)o 3 ' " , ~ - . . . . . . . . . .

--2 " .0 2o_o l

. . . . . . . . . . . . . . . . . . . . .o I I 4

10 20 30 40( b ) R [ m i c r o n ]

F i g . 2 . D i f f e r e n t p a r t i c l e s i z e d i s t r i b u t i o n s u t i l i z e d i n t h e c a l c u l a t i o n s( T a b l e 2 ) : ( a ) G a u s s i a n ; ( b ) l o g - n o r m a l .

250E-01

O . O O E + O 0

( a )I I

000E+O 0 2 50E+O0 5.00E+O0T

100E+O0

7 50E-01

500E-01

2.50E-01

O.OOE+O0

165.H . M int z , Y . Ze i r i I J ournal o f A l loy s and Compounds 16 (1994) 159-175T a b l e 2T y p e s o f d i s t r i b u t i o n f u n c t i o n s u s e d i n t h e c a l c u l a t i o n s , a n d t h e i r l a b e l l i n g i n t h e f i g u r e s

I IO.OOE+O0 2 50E+O 0 5,00E+O0

( b )F i g . 3 . E f f e c t s o f s i z e d i s t r i b u t i o n s o n c o n s t a n t - v e l o c i t y k i n e t i c s o fa s p h e r e a s s h o w n b y c a l c u l a t e d r e a c t e d f r a c t i o n t ~ v v s . r e d u c e dt i m e ,7- c u r v e s , f o r d i f f e r e n t s i z e d i s t r i b u t i o n s ( F i g . 2 a n d T a b l e 2 ) :( a ) g a u s s i a n d i s t r i b u t i o n s ; ( b ) lo g - n o r m a l d i s t r i b u t i o n s . T h e f u l l b r o a dl i n e r e p r e s e n t s t h e S P A r e s u l t s.

d e c e l e r a t i n g v e l o c i t y d i f f u s i o n - c o n t r o l l e d c a s e a r e p r e -s e n t e d i n F i g . 5 ( a D VS. ~ ') a nd F i g . 6 ( FD ( a ) S. ~-). Iti s s e e n f r o m t h e s e f i g u r e s t h a t t h e o c c u r r e n c e o f a s i z ed i s p e r s i o n l e a d s t o s o m e d e v i a t i o n s f r o m t h e l in e a rF r ( a ) v s . ~" r e l a t i o n s . H o w e v e r , t h e s e d e v i a t i o n s a r ed i s p l a y e d m a i n l y i n t h e h i g h e r a r e g i o n , w h e r e a s as i g n i f i c a n t r e g i o n i n t h e l o w e r a r a n g e ( i . e . u p t o a b o u ta = 0 . 5 - 0 . 6 ) d i sp l a ys t h e l in e a r S P A - l ik e d e p e n d e n c e .


8/17

16 6 M.H, Mintz, Y. Zebi /Journal of Alloys and Compounds 216 (1994) 159-175100E+00

7 50E-01

500E-01

2.50E-01

0.00E+000.00E+00 2.50E+00 5 00E+00

Ca) x

1.00E+O0

750E-01

E3 500E-01

250E-01

000E+O0

[ a )OOOE+O0 2.50E+00 5,00E+00 7.50E+00 1 00E+01

T

1.00E+00

7 50E-01 ( C t = 0 . 5 )

~ e.OOE-O, .. .. ................................., . - - - - - :

2 50E-01 A

0 00E+00 I I0.00E+00 2.50E+00 5.00E+00

(b) "~F i g . 4 . E f f e c t s o f s i z e d i s t r i b u t i o n s o n c o n s t a n t - v e l o c i t y k i n e t i c s o fa s p h e r e a s s h o w n b y c a l c u l at e d r e a c t e d f r a c t i o n f u n c t i o n F c v ( a )v s . r e d u c e d t i m e ~" c u r v e s f o r t h e d i f f e r e n t s i z e d i s t r i b u t i o n s ( a( ~- )a s o b t a i n e d i n F i g . 3 ) : ( a ) g a u s s i a n d i s t r i b u t i o n s ; ( b ) l o g - n o r m a ld i s t r i b u t i o n s ( t h e s a m e a s i n F i g . 3 ) . T h e f u l l l i n e r e p r e s e n t s t h eS P A r e s u l t .

A l s o , t h e v a l u e s o f t h e s l o p e s o f t h e s e l i n e a r p o r t i o n sa r e c l o s e t o t h e " u n i v e r s a l " SPA v a l u e s ( a s l i s te d i nT a b l e 1 ) .

Be s i d e s t h o s e " u n i v e r s a l " ( i . e . k i n e t i c s - t y p e - r e l a t e d )s lopes o f the F r (a ) vs . ~" cu rves , one can re fe r a l so tot h e s y s t e m - r e l a t e d s l o p e s o f t h e F r( a ) vs. t curves ( i .e .E q s . ( 5 ) a n d ( 9 ) ) w h i c h d e p e n d o n t h e i n t r i n s i c k i n e t i cp a r a me t e r s o f t h e r e a c t i n g s a mp l e ( i . e . U o r k D r e -s p e c t i v e l y ) a s w e l l a s o n t h e s i z e p a r a me t e r R .

T h e f o l lo w i n g q u e s t i o n s ma y t h e n b e a d d r e s s e d : i si t j u s t i fi e d t o r e p l a c e t h e s i n g l e - v a lu e d p a r a m e t e r R ,a p p e a r i n g i n t h e SPA e x p r e s s i o n s ( E q s . ( 5 ) a n d ( 9 ) ) ,b y s o m e a v e r a g e ( R ) o f th e p o w d e r ? F u r t h e r m o r e , i tis k n o w n t h a t t h e c o n v e n t i o n a l a v e r a g e ( R ) i s n o t a l w a y st h e p r o p e r q u a n t i t y t o r e p r e s e n t i n t e g r a l p a r a m e t e r sr e l a t e d t o t h e p o w d e r . F o r e x a m p l e , th e m a t e r ia l v o l u m ea n d r e l a t e d t h e o r e t i c a l d e n s i t y ( E q s . ( 2 5 ) a n d ( 2 6 )r e s p e c t i v e l y ) a r e e x p r e s s e d b y ( R 3 ) r a t h e r t h a n b y ( R ) 3.W e m a y t h e n u t i l i z e t h e m o r e g e n e r a l i z e d f o r m f o rt h e a v e r a g e m o m e n t o f o r d e r n :

(R . ) = D ( R ) R " d Rr a i n

(29)

1 . 0 0 E + O 0

o 7,50E-01 .........................

~ . O O E - O ~ f L 2

2.50E-01 - - SP AL

0.00E+00 q I I0.00E+00 3 0 0 E + 0 0 6 0 0 E + 0 0 9 . 0 0 E + 0 0

(b) ,~F i g . 5 . E f f e c t s o f s i z e d i s t r i b u t i o n s o n d i f f u s i o n - c o n t r o l l e d k i n e t i c sa s s h o w n b y a D VS. ~" c u r v e s f o r ( a ) G a u s s i a n a n d ( b ) l o g - n o r m a ld i s t r i b u t i o n s ( F i g . 2 a n d T a b l e 2 ) .

wi th n = 1 , 2 , 3 , . . . ( i .e . n = 1 co r re spo nds to th e con-v e n t i o n a l a v e r a g e ) a n d c h e c k t h e v a l i d i t y o f t h e s e ( R , )s u b s t i t u t i o n s i n t o t h e SPA e x p r e s s i o n s . I n t h e s e c a l -c u l a ti o n s , s o m e a r b i t r a r y v a l u e o f e i t h e r U ( i n E q . ( 5 ) )o r k D ( i n E q . ( 9 ) ) w e r e c h o s e n . T h e c o r r e s p o n d i n g( R , ) ( n = 1, 2 . . . ) v a l u e s a r e e v a l u a t e d , f o r a n y g i v ens ize d i s t r ibu t ion (u t i l i z ing Eq . (29) ) , and the co r re -s p o n d i n g SPA - l i k e s l o p e s a r e t h e n o b t a i n e d ( i . e . U~( R , ) f o r c o n s t a n t - v e l o c i t y k i n e t i c s a n d 2kD/(R~) z fo rd i f f u s i o n - c o n t r o l l e d k i n e t i c s ) . T h e s e s l o p e s a r e c o m-p a r e d w i t h t h e r e s p e c t i v e i n i t i a l s l o p e s i n t h e l i n e a rr e g i o n o f F r ( o f "~c) vs . t cur ves (a ca~c ev alu ate d by Eq .( 2 7 ) ) . T h e r a t i o s o f t h e SPA - l i k e s l o p e s ( f o r t h e g i v e n( R , ) ) t o t h e c a l c u l a t e d OFr(a~a~c)/ots l o p e s a r e s u m-ma r i z e d i n T a b l e 3 . I t s h o u l d b e p o i n t e d o u t t h a t , e v e nthough the spec i f i c cho ice o f a g iven in t r ins i c k ine t i cp a r a me t e r ( i . e . U o r k o ) a f f e c t s t h e a b s o l u t e v a l u e s o ft h e c o r r e s p o n d i n g s l o p e s , t h e a b o v e - me n t i o n e d s l o p era t ios a re insens i t ive to tha t cho ice , y i e ld ing s imi l a rresu l t s . I t t u rns ou t f rom Tab le 3 tha t , fo r a g iveni n tr i n si c k i n e t i c p a r a m e t e r , u t i li z a t io n o f t h e a v e r a g emo me n t s o f t h e o r d e r o f a b o u t 5 - 6 b e s t f i t s t h e i n i t i a ll i n e a r s l o p e s o b t a i n e d f r o m t h e F r( a ) vs . t cu rves . Fore x a mp l e , u t il iz i ng ( Rs ) , t h e m a x i mu m d e v i a t i o n o f t h eSPA - l i k e s l o p e f r o m t h e " r e a l " s l o p e i s a b o u t 3 0 % ,w i t h mo s t o f t h e d e v i a t i o n s r a n g i n g w i t h i n + 2 0 % .H e n c e , p e r f o r m i n g t h e k i n e t ic a n a l y s i s o f a g iv e n p o w d e r ,


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M.H. M intz , Y. Ze M /J our na l of Alloys and Compounds 216 (1994) 159-175 1674.00E-01

2.00E-01i j_ 3

000E+OC

( a )

(ct=0.5)-- G2- - - - G 1

f ~ I I I I0 I)OE+O0 2.50E+00 500E+ O0 7.50E+00 100E+ 01

1:

400E-01

(~--o.5)

...- .......... " ..... - - L 2 i.....-'"" - - - - L 1 1- - S P A

000E+0(' I I0. )0E+00 3 00E+00 6 DOE+00 9 00E+00

(b) "~Fig. 6. Effects of size distributions on diffusion-controlled kineticsas shown by FD(a) vs. ~" curves for (a) Gaussi an and (b) log-normaldist ribut ions (with OtD(~') as ob tai ned in Fig. 5).

Table 3Typical ratios of SPA-like slopes (i.e. U / ( R , ) and 2kD/(R,) 2 asappearing in Eqs. (5) and (9) respectively, for the different mech-anisms) to the corresponding OF,(aa))/Ot nitial linear slopes, for anygiven particles size distribution, the (R,) values in the SPA-likeslope s we.re calcula ted (acc ording t o Eq. (29)), with n = 1-8Type of Distributionkinetics function

0 (R)R a t i o o f S P A - l i k e s l o p e s t o t h e c a l c u l a t e dO F r ( a c a ~ c ) / ~ t s l o p e s , u s i n g t h e d i f f e r e n tm o m e n t s , ( R ~ ) w i t h t h e f o l l o w i n g n v a l u e s1 2 3 4 5 6 7 8

r~C V

r ~ D

G1 1.23 1.16 1. 11 1.07 1.04 1.00 0.98 0.95G2 2.02 1.56 1.35 1.22 1.12 1.05 1.00 0.95L1 1.40 1.29 1.19 1.10 1.02 0.95 0.88 0.82L2 4.10 2.33 1.54 1.15 0.93 0.79 0.70 0.64G1 1.53 1.36 1.25 1.16 1.08 1.02 0.97 0.92G2 4.12 2.45 1.84 1.50 1.28 1.12 1.00 0.90L1 1.96 1.67 1.43 1.22 1. 05 0.91 0.79 0.69L2 16.1 5.26 2.29 1.27 0.83 0.61 0.48 0.40

i . e . f i t t i ng a l inear Fr (a ) vs . t dependence , which y ie ldst h e c o r r e s p o n d i n g i n i t i a l l i n e a r s l o p e OFr(oO/~t, e n a b l e sn o t o n l y t h e i d e n t i f ic a t i o n o f t h e c o n t r o l li n g m e c h a n i s m( i . e . r - - C V o r D ) b u t a l s o t h e r e a s o n a b l e e s t i m a t i o no f t h e c o r r e s p o n d i n g i n t ri n s ic k i n e t ic p a r a m e t e r s ( i.e .U i n E q . ( 5 ) o r kD in E q . ( 9 ) ) , w h i c h a r e a p p r o x i m a t e db y

0 F c v ( ) Iu = ( R e ) - - - - i f - - l ,


10/17

1 68 M.H. M int z , Y . Ze i r i / J ournal o f A l loy s and C omp ound s 216 (1994) 15 9-175I f t h is e xpr e ss ion , howe v e r , is " m i s t a ke n ly" subs t i t u t e dinto the spher ica l case , i .e .Fc v, s(aw xp) = 1 - (1 - ~wexp)1/3 (37)the resul t ing Fcv, ~ (awexp) vs. r c urv e is i l lus trate d inF ig . 7 . I t i s s e e n tha t t he l i ne a r de pe nde nc e i s s t i l lm a i n t a i n e d o v e r m o s t o f t h e a , ~exp rang e . T he l inears lope c a n be c a l c u l a t e d f r om0Fcv. s(aw 'p) 0 F c v , s (~ wexp) 0aw xp= ( 3 8 )0r Oa ~p 0rwhich, by uti l iz ing Eq. (37) and E q. (36) , yie lds0Fcv, ,(awexp) 2 0.2 9= - - . ( 1 - 0 . 2 9 r ) - m (3 9)0r 3

For r < 1 , t he i n it i al s l ope i s t he n a ppr ox im a te ly 0 .19 ,whic h i s c lose t o t he va lue o f 0 .21 ob ta ine d f o r t he" r e a l " sphe r i c a l c a se .

A s im i l a r a na lys i s c a n be done f o r t he d i sk ( E q .( 34) ) . He r e , subs t i t u t i ng the " e xpe r im e nta l " a d e~p intothe sphe r i c a l e xpr e ss ionF c v , s(O~d xp)= 1 - (1 - 0 . 5 r ) ' / 3 (40)y i e lds a c ur ve whic h a ga in , up to a bou t ~ ' = 1 ( i. e .ad ~xp= 0.5) is l inear (Fig. 7) , wi th a s lop e of ab out 0 .17,a g a in n o t m u c h d i f f e r e n t f r o m t h e " r e a l " s p h e r ic a lcase .

I t c a n e a s i l y be shown tha t , a l so f o r d i f f us ion- c on-t r o l l e d k ine t i c s , subs t i t u t i on o f t he d i sk ( i . e . p l a na r )e xpr e ss ion%exp = 0.5r~a (41)into the sph er ica l Fd. ~(ot) vS. r fu nc t io n (Eq. (9) ) yie ldsa c ur ve whic h i s a lm os t l i ne a r up to a bou t oe= 0 .5 w i tha s lope of about 0 .028, c lose to the va lue of 0 .037obta ine d f o r r e a l sphe r i c a l sym m e t r y ( T a b le 1 ) .I t is thus c o nc lud e d tha t , f o r t he c o n t r a c t ing- e nve lo pek ine t i c func t ions , t he va r i a ti ons i n t he " un ive r sa l " f o r m s100E+O0 . ' " ' " ~ "- *'*****'""" ........... -7.50E-01 (Ct=0.5) . . .~

~(~ 5.00E-01",~ "" - - g =d

2 . 5 0 E - 0 1 / [ - - g =s/000E+O0 I I

000E+O0 400E+O0 800E+O0T

F i g . 7 . E f f e c t s o f s h a p e v a r i a t i o n s o n t h e c o n s t a n t - v e l o c i t y k i n e t ic s( p l o tt e d o n a " u n i v e r s a l " r e d u c e d t i m e s c al e ). T h e o r d i n a t e r e p r e s e n t sF c v , s ( otg e~ P) , t h e s p h e r i c a l f u n c t i o n ( T a b l e 1 ) , i n w h i c h o tg xp v a l u e so f r e a c t i n g n o n - s p h e r i c a l p a r t i c l e s ( g - = w i r e o r d i s c r e s p e c ti v e l y ) w e r es u b s t i t u t e d a n d c o m p a r e d w i t h t h e s p h e r i c a l c a s e .

( a nd " un ive r sa l " s lope s ) o f t he se f unc t ions , i nduc e d byg e o m e t r i c a l s h a p e c h a n g e s , a r e m u c h l es s p r o n o u n c e dtha n the va r i a t i ons i nduc e d by c ha nge s i n t he c on t r o l li ngm e c h a n i s m (i .e . r - C V o r D ) . H e n c e , a p pl y in g t h eFr . s (a ) vs . r ana lysis, u t i l iz ing the sph er ica l -ge om etryexpress ion, i s s t i l l va l id for non-spher ica l geometr ieswhic h , e ve n thou gh th e y r e su l t in som e de v ia t ions f r omlinear i ty a t higher a va lues , s t i l l d isplay a l inear de-p e n d e n c e ( f o r th e c o r r e c t m e c h a n i s m ) o v e r a r e l at iv e l ywide r a nge o f oe, w i th t h e " un ive r sa l " l i ne a r s lopeind ic a tive o f t he r e spe c t ive m e c ha n i sm . I n o th e r wor ds ,va r i a t i ons i n pa r t i c l e sha pe do no t a l t e r m uc h o f t hea bove a na lys i s, whic h p r ope r ly po in t s t o t he c on t r o l li ngm e c h a n i s m .

T h e s e c o n d i s s u e t h a t s h o u l d b e c o n s i d e r e d n o w i sthe a b i l i t y t o ob t a in t he r e spe c t ive i n t r i ns i c k ine t i cpa r a m e te r s ( i . e . U o r kD) f r om the l i ne a r r e g ions o fthe F r, s ( a ) vs. t c u r ve s , f o r p ow de r s w i th non - sphe r i c a lpa r t ic l e s . I n t h i s c a se , e xpr e ss ions o f t he t ype g ive n byEq. (30) ( for cons tant -ve lo c i ty kine t ics) or Eq. (31) ( ford i f f us ion- c on t r o l l e d k ine t i c s ) shou ld be use d , bu t w i thsom e " e f f e c t ive " sphe r i c a l r a d ius R e re p l a c ing the de -f ine d sphe r i c a l c a se . T h e qu e s t ion to be a ddr e sse d the nr e ga r ds t he t ype o f " e f f e c t ive " r a d ius whic h shou ld beu t i l iz e d a n d to w ha t a c c ur a c y doe s i t y i e ld t he r e spe c t ivein t r i ns i c k ine t i c pa r a m e te r s .

For c ons t a n t - ve loc i ty k ine t ic s , c om pa r ing the i n it ia lFc v , s (oe


11/17

M.H . Mintz, Y. Zeiri /Jour na l of Alloys and Compounds 216 (1994) 159-175 1 6 9

Defin ing the ax ia l ra t io2R e) t a = . . . . ( 4 4 )H ~

E q . ( 4 3 ) c a n b e r e w r i t t e n a sA, + 2 2 (a , + 2 )~ cv .~ =: - - U tR~ R~2

"~" (U t) 3 (45 ) (Ut) ~+ Ro---H o w e v e r , a s s u mi n g a " s p h e r i c a l - e q u i v a l e n t " p a r t i c l ewi th e f fec t ive rad ius R e reac t ing accord ing to Eq . (4 )( w i t h R ~ r e p l a c i n g R) , e q u a t i n g t h e t w o l i n e a r - t e r mc o e f f i c i e n t s i n t h e s e e x p r e s s i o n s y i e l d s t h e b e s t c h o i c efo r R ~ as

(R5 c) --- 1.5( tis) (49)( w i t h ( 6 5 ) d e f i n e d b y a n e q u a t i o n s i mi la r t o E q . ( 2 9 ) )w h e r e a s s y mm e t r ic p a rt i c le s a r e c h a r a c t e r i z e d b y as in g l e s p h e ri c a l - e q u iv a l e n t d i me n s i o n A ( o f a b o u t ar a d i u s o f a s p h e r e e n c l o s e d w i t h i n t h e c o r r e s p o n d i n gs h a p e ) w i t h(R5 ~) = (As) (50)H o w e v e r , f o r p o w d e r s w i th a w i d e d i s p e r s i o n o fs h a p e s ( e . g , c o n t a i n i n g b o t h a s y mme t r i c a n d s y mme t r i cp a r t i c le s ) , t h e c h o i c e o f ( R5 ~ ) is mo r e c o m p l i c a t e d a n d ,e v e n t h o u g h t h e c o r r e s p o n d i n g c o n t r o l li n g k i n e t i c s ( i. e.c o n s t a n t v e l o c i t y o r d i f f u s io n c o n t r o l l e d ) c a n b e e v a l-ua ted (by F~. ~(a) vs . r ana lys i s ) , accu ra te es t im at ion o fthe in t r ins i c k ine t i c parameters i s no t poss ib le .

3R e - Re (46)A , , + 2( N o t e t h a t t h is e x p r e s s i o n is d if f e r e n t f r o m t h e v o l u m e -e q u i v a l e n t s p h e r i c a l r a d i u s g i v en b y (3 /2&)raRe. ) S o m espec i f i c : cases der ived f rom Eq . (46) a re as fo l lows : ( i )sym me t r i c cy l inder , aa = 1 , fo r w hich R c =R e; ( i i) w i re-l ike geo me t ry , as > 1 , fo r wh ichR e ~ 1.SHp (n p i s the he igh t o f the p la t e ) , as d i scusseda b o v e . H e n c e , f o r v e r y a s y mm e t r i c p a rt i c le s , t h e e f f e c t iv es p h e r i c a l r a d i u s i s r e l a t e d t o t h e s ma l l e s t d i me n s i o nof the co r respond ing shape . S imi la r ly , fo r d i f fus ion-con t ro l l ed k ine t i cs , co mp ar ing the in i ti a l FD. s (a


12/17

170 M.H . Mintz, Y. Zeiri I Journal of Alloys and Compounds 216 (1994) 159-1751.00E-01

=5.00E-02

0 0 0 E+0 00 .0 0 E+0 0

( o )

~ l ; - - M =I (x0.1)" , ~ . . . . M = 0 1: ; I ~ " ~ t . . , , , ~ n - - M =O .O I( x5 ):,...,.. '~ ~ U ~

" " . . . . . . . . I I2.50E-01 5.00E-01

t [ s e c ]

1 0 0 E + 00 . . . . .

7 50E-01

5.00E-01

2.50E-01

0.OOE+00 l ( l0.25 0.5 0.75

( b t [ s e c ]Fig. 8. (a) Probability distribution of number of particles which startto react at time t, for different choices of the "marker" parameterM (see text). The widths ot of these distributions corresponding tothe different M values are as follows: M = I , o-t=0; M=0.1,o', =1. 31 0 -3 s; M=0. 05, o-,=0.13 s. (b) Effects of initiation timedistributions (as displayed in (a) on the acv vs. t kinetic curves.Labelling as in (a).

but ion f un c t i on ( G2 in t a b l e 3 ) , a s sum ing c ons t a n t -ve loc i ty k ine t i c s a nd som e g ive n U va lue s ( whic h de -t e r m i n e tl/2). Fig . 8 ( b ) sum m a r i z e s t he c or r e spond inga vs . t behav iour , display ed for di f fe rent tr , va lues . Tw oe f f e c t s a r e a ppa r e n t f r om the se k ine t i c c ur ve s : ( i ) ad i sp l a c e m e nt o f t l/ 2 t owa r ds h ighe r va lue s r e su l t e d byt h e d e l a y in t h e c o m m e n c e m e n t o f t h e r e a c ti o n f o rsom e of t he pa r t i c l e s ; ( i i ) i n f l e c t i on po in t s i n t r oduc e din to t he c ur ve s , whic h now a ssum e a n S - l i ke sha pe .T h i s be ha v iour qua l i t a t i ve ly r e se m ble s t he nuc l e a t i on-a nd- gr owth k ine t i c s ( i . e . E q . ( 13) ) ; howe ve r , t he i n -f l e c t i on po in t s i n t he l a t t e r c a se a r e l oc a t e d a t a bou tt=t l / z ( c f . E q . ( 20) ) whe r e a s , f o r t he p r e se n t k ine t i c s ,inf lec tion occ urs at t


13/17

M.H . Mintz, Y. ZeM / Journal of Alloys and Com pounds 216 (1994) 159-175 1 7 1d e t e r m i n a t i o n o f t h e a b s o l u t e v a l u e s o f i n tr i n si c k i n e t i cp a r a m e t e r s ( a t a g iv e n p r e s s u r e a n d t e m p e r a t u r e ) t h ea c c u r a t e e v a l u a t i o n o f a p p a r e n t " a c t i v a t i o n e n e r g i e s "E , a s s o c i a t e d w i t h t h e s e p a r a me t e r s [ 1 4 ] i s p o s s i b l ee v e n f o r u n k n o w n s i z e d i s t r i b u t i o n s o r s h a p e f a c t o r so f t h e p o w d e r s . A s l o n g a s th e f u n c t i o n a l f o r m o f th eSPA k i n e t i c s ( e . g . E q . ( 3 ) ) i s e s t a b l i s h e d ( w h i c h d o e sn o t r e q u i r e k n o w l e d g e o f th e s e d i s t r i b u t i o n s ) , a p a -r a m e t e r k ( T ) / ( R , ) m i s e v a l u a t e d , w i t h k ( T ) b e i n g p r o -p o r t i o n a l t o t h e ( t e mp e r a t u r e - d e p e n d e n t ) i n t r i n s i c k i -n e t i c p a r a me t e r , a n d ( Rn ) m a d i me n s i o n - r e l a t e d f a c t o r(e .g. ( Rs ) o r ( R s ) 2 in Eq . (30) o r (31) respec t ive ly ) .H e n c e , e v e n w h e n ( R n ) " i s n o t k n o w n , s t il l a n A r r h e n i u sp l o t o f l n [ k (T ) / (R n ) m] vs . 1 /T w i l l y i e l d t h e a p p a r e n tE , , i f t h e c o r r e s p o n d i n g i n t ri n s ic k i n e t i c p a r a m e t e r d o e so b e y s u ch a n A r r h e n i u s d e p e n d e n c e . T h i s c o n s id e r a t io nh a s b e e n p o i n t e d o u t p r e v i o u s l y b y K a p u r [ 25 ], a l th o u g hi n a s o me w h a t d i f f e r e n t w a y . I n f a c t , t h e p r o c e d u r eo u t l i n e d t h e r e [ 25 ] e n a b l e s t h e d e t e r m i n a t i o n o f E ae v e n w i t h o u t t h e e v a l u a t i o n o f t h e s p e c i f i c f u n c t i o n a lf o r m o f t h e k i n e t i c d a t a ( i . e . w i t h o u t t h e c h o i c e o f ap a r t i c u l a r F r , g ( a ) i n Eq . (3 ) ) .

T h e a b o v e c o n s i d e r a t i o n s a b o u t t h e e v a l u a t i o n o f E ~s h o u l d b e mo d i f i e d t o s o me e x t e n t w h e n a n a l y s i n gk i n et ic e x p e r i m e n t s p e r f o r m e d u n d e r c o n d i ti o n s w h ic ha r e c l o s e t o e q u i l i b r i u m, i . e . u n d e r s u c h w o r k i n g p r e s -s u r e s P a n d t e m p e r a t u r e s T , w h e r e P i s n o t v e r y mu c hd i f f e r e n t f ro m Pc ( T ) ( t h e e q u i l i b r iu m p l a t e a u p r e s s u r eo f t h e h y d r i d e ) . I n t h e s e c a s e s t h e i n t r i n si c k i n e t i cp a r a m e t e r s ( o r t h e r e l a t e d k r a t e c o n s t a n t s ) a r e mo d i f i e db y t e m p e r a t u r e v a r ia t io n s i n a c o m p l e x w a y o w i n g tot h e s i mu l t a n e o u s i n t e r p l a y o f t h e f o r ma t i o n a n d d e -c o mp o s i t i o n p r o c e s s e s . T h e t y p i c a l A r r h e n i u s d e p e n -d e n c e a t t a i n e d u n d e r f a r - e q u i l i b r i u m c o n d i t i o n s i s t h e nr e p l a c e d b y a c o m p l e x t e m p e r a t u r e b e h a v i o u r , d i sp l a y in ga max imum in the In k vs . 1 /T curves [14] . Hence , thee v a l u a t i o n o f E a r e q u i r e s k n o w l e d g e o f t h e p r e s s u r ed e p e n d e n c e o f k[P, Pc(T) , T] . O t h e r w i s e , d i f f e r e n t E ,v a l u e s c a n b e a s s i g n e d f o r d i f f e r e n t p o s s i b l e k i P , Pc ( T ) ]re l a t ions [7,9] , wi th no rea l ph ys ica l s ign i f i cance o f these" a c t i v a t i o n " p a r a m e t e r s .

I t is w o r t h w h i l e t o d e m o n s t r a t e s o m e q u a l i ta t i v et r e n d s e me r g i n g f r o m t h e p r e s e n t a n a l y s i s w h i c h ma ya c c o u n t f o r s o me c o n t r o v e r s i a l k i n e t i c r e s u l t s r e p o r t e di n t h e l i t e r a t u r e . O n e e x a mp l e c o n c e r n s t h e k i n e t i c so f f o r m a t i o n o f t h e L a N i s - H 2 s y s t e m ( a c t u a l ly t h eLaNia .gAlo .a a lloy) as rep or t ed in two pub l i ca t ions [7,9] .I n t h e s e s t u d i e s , t w o t y p e s o f k i n e t i c s a r e r e p o r t e d ,o n e [ 9 ] c o r r e s p o n d i n g o t t h e c o n t r a c t i n g - e n v e l o p e k i -n e t i c s ( E q . ( 5 ) ) , a n d t h e o t h e r [ 7 ] c o r r e s p o n d i n g t ob u l k n u c l e a t i o n a n d g r o w t h ( E q . ( 1 4 ) ) w i t h n = 2 . I nb o t h s t u d i e s t h e a p p l i e d e x p e r i me n t a l c o n d i t i o n s ( Pa n d 7 ) w e r e i n t h e s a m e r a n g e , a n d c a r e h a s b e e nt a k e n t o k e e p i s o t h e r ma l - i s o b a r i c c o n t r o l d u r i n g t h ek i n e t i c me a s u r e me n t s .

A c c o r d i n g t o t h e a b o v e d i s c u s s io n , t w o p o s s i b l e f a c t o rsc o u l d a c c o u n t f o r t h e c h a n g e i n t h e o b s e r v e d k i n e t i c s .O n e f a c t o r i s t h e o c c u r r e n c e o f a s i g n i f i c a n t s p r e a d( d i s t ri b u t io n ) i n t h e t i me s o f c o m me n c e m e n t o f t h er e a c t i o n ( o n e a c h o f t h e p o w d e r ' s p a r t ic l e s ) , w h e r e a st h e s e c o n d p o s s i b l e f a c t o r i s a n i n c r e a s e i n t h e X o / ( R )rat io .

A s f o r t h e f o r me r p o s s i b i l i t y , i t h a s b e e n p o i n t e dou t in Sec t ion 2 .2 .3 tha t such a t ime d i s t r ibu t ion maylead on the one hand to a s ign i f i can t increase in t a / a( i . e . s l o w e r k i n e t i c s ) a n d o n t h e o t h e r h a n d t o t h eapp ear anc e o f in f lec t ion po in t s in the a vs . t k ine t i ccurves , s imi la r to nuc lea t ion-and-growth k ine t i cs . I t i se v i d e n t t h a t b o t h s y m p t o ms a r e a p p a r e n t i n t h e d a t ap r e s e n t e d i n [ 7 ] . D i f f e r e n t r e a s o n s m a y b e p o s t u l a t e dfo r the ex i s t ence o f such an e f fec t . One poss ib i l i ty i st h e p r e s e n c e o f s o me g a s - p h a s e i mp u r i t ie s w h i c h i n h i b itt o s o me e x t e n t t h e h y d r i d i n g r e a c t i o n s , l e a d i n g t o t h ed i spersed in i t i a t ion t ime d i s t r ibu t ion . The samples s i zeu t i li z e d i n [7 ] a r e a b o u t t h r e e o r d e r s o f ma g n i t u d es ma l l e r t h a n t h o s e u s e d i n [ 9] ( o n l y 5 mg i n t h e f o r m e rc o m p a r e d w i t h a b o u t 5 g in t h e l a t te r ) . I t i s t h u s p o s s i b l et h a t c h e m i s o r p t i o n o f su c h mi n o r i mp u r i t i e s o n t h e( l a rge) su r face a r ea o f the l a rger s am ples [9 ] inh ib i tson ly a smal l and ins ign i fi can t f rac t ion o f the po wd er ,whereas fo r the smal l e r samples [7 ] such inh ib i t ione f f e c ts a r e p r o n o u n c e d . T h e r e i s, h o w e v e r , o n e p o i n tw h i c h ma y c a s t d o u b t o n t h e a b o v e r e a s o n i n g . T h el o c a t i o n o f t h e i n f l e c t i o n p o i n t i n t h e d a t a p r e s e n t e din [7 ] i s a t abou t a - -0 .5 ( i . e . t ~ t m ) . A s d i s c u s s e db e f o r e , i n f l e c t i o n p o i n t s i n d u c e d b y d i s p e r s e d c o m-me n c e me n t t i me d i s t r i b u t i o n s a r e u s u a l l y l o c a t e d a ta < < 0 . 5 , w h i c h i s i n c o n s i s t e n t w i t h t h e s e e x p e r i me n t a lresul ts .

A s f o r t h e s e c o n d p o s s ib i l it y me n t i o n e d a b o v e , i. e .an increase in the X o / ( R ) r a t i o , s u c h a n i n c r e a s e ma yb e t h e r e s u l t o f t w o f a c t o r s ; i t c o u l d b e d u e e i t h e r t oa d e c r e a s e i n ( R) ( i . e . a f i n e r p o w d e r w i t h s ma l l e rpar t i c l es ) o r to an incr ease in Xo ( i. e. t he com ple t iono f a c o n t i n u o u s p r o d u c t l a y e r c o v e r i n g t h e s u r f a c e ,occurr ing a t a l a rger av a lu e , fo r a g iven s i ze d i s tr ibu t ion) .N o e x p e r i me n t a l d e t a i ls o n t h e s iz e d i s tr i b u t io n s o f t h ep o w d e r s u t i l i z e d i n [ 7 , 9 ] w e r e p r e s e n t e d . H o w e v e r ,s i n c e i n b o t h s t u d i e s t h e s a mp l e s w e r e a c t i v a t e d b ysevera l hydr id ing-dehydr id ing cyc les , i t i s no t l ike lytha t th ese s i ze d i s t r ibu t ions cou ld d i f fe r by such ane x t e n t a s t o a c c o u n t f o r t h e o b s e r v e d c h a n g e i n k in e t ic s .M o r e o v e r , t h e t m v a l u e s o b t a i n e d i n [ 7 ] a r e mu c hh igher then those in [9 ] (under s imi la r cond i t ions ) . I ft h e r e a s o n f o r t h e c h a n g e i n t o n u c l e a t i o n - a n d - g r o w t hk ine t i cs was a t t r ib u ted to a f iner par t i c l e s i ze d i s t r ibu t ion( in [7 ] ) , t hen the oppos i t e t rend ( i . e . fas t e r k ine t i cs ,o r s h o r t e r t l/ z v a l u e s ) w o u l d b e a n t i c i p a te d . H e n c e , t h i sp o s s i b l e e x p l a n a t i o n c a n b e r u l e d o u t .

T h e o t h e r a l t e r n a t i v e f o r a n i n c r e a s e i n t h e X o / ( R )ra t io i s, as m en t io ned above , an incre ase in Xo , occurr ing


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172 M.H . Mintz, Y. Zeiri / Journal of Alloys and Compounds 216 (1994) 159-175un de r t he e xp e r im e n ta l c ond i t i ons a pp l i e d i n [ 7 ]. I t isi l l us t ra t e d i n F ig . 1 tha t t he p r e c e d in g s t a ge to t h ef o r m a t i o n o f a c o n t i n u o u s p r o d u c t l a y e r o n t h e s u r f a c eof a g ive n r e a c t i ng pa r t i c l e i nvo lve s a nuc l e a t i on- a nd-gr owth p r oc e ss . He nc e , f o r a g ive n nuc l e i g r owth r a t e ,i t is a n t i c ipa t e d t h a t a l owe r nu c l e a t i on de ns i t y w il lr e su l t i n a l a r ge r Xo va lue ( s inc e t he c om ple t e ove r l a pb e t w e e n t h e g r o w i n g n u c l e i i s t h e n d e l a y e d t o l a r g e rXo) . S inc e t he nuc l e a t i on de ns i t y i s ve r y se ns i t i ve t othe p r e se nc e o f ga s - pha se im p ur i t i e s [ 32], t he r e a son ingpr e se n t e d a bove ( i . e . sm a l l e r s a m ple s u t i l i z e d i n [ 7 ]w h i c h a r e m o r e p r o n e t o t h e e f fe c t s o f i m p u r i t ie s ) i sr e l e va n t a l so f o r t he p r e se n t c a se . T h i s e xp l a na t i on i sa l so c ons i s t e n t w i th t he i nc r e a se i n t he t l / z va lue so b s e r v e d i n [ 7 ] . H e n c e , t h e d i f f e re n c e s b e t w e e n t h er e su l t s r e por t e d i n t he two a bove pub l i c a t i ons a r er e a s o n a b ly a c c o u n t e d f o r b y t h e i n c r e a s e i n ) to d u e t oa l o w e r n u c l e a t i o n d e n s i t y u n d e r t h e c o n d i t i o n s a p p l i e din [7] .

Anothe r e xa m ple , i l l us t r a t i ng t he e f f e c t s a n t i c ipa t e df o r t h e c a s e o f a t im e d i s tr i b u ti o n f o r t h e c o m m e n c e m e n tof t he r e a c t i on on t he d i f f e r e n t pa r t i c l e s , i s g ive n byt h e d a t a p r e s e n t e d i n a r e c e n t p u b l i c a t i o n o n t h ehydr iding kine t ics of LaNis_~AIx a l loys [33] . Accordingt o t h i s s t u d y , i n c r e as i n g t h e a l u m i n i u m c o n t e n t o f t h ea l loy ( i.e . increas ing x) resul t s in a shi f t in t l/2 tow ardsh igh e r va lue s , c onc o m i t a n t w i th a n i n f l e c ti on po in t ( i nthe a vs . t curves) occur r ing a t low a va lues . This i ss im i l a r t o t he be ha v iour d i sp l a ye d i n F ig . 8 ( b ) . Suc hb e h a v i o u r h a s n o t b e e n a n a l y s e d b y Z h a n g e t a l . [33]a n d c a n b e a c c o u n t e d f o r b y t h e p r e s e n t c o n s i d e ra t io n s .

F i n a l l y , i t s h o u l d b e p o i n t e d o u t t h a t t h e p r e s e n ta na lys i s i s r e l e va n t no t on ly t o t he c a se o f powde r sbu t a l so t o som e topoc he m ic a l f o r m s d i sp l a ye d dur ingthe hydr id ing r e a c t i on s o f m a ss ive po lyc r ys t a l li ne sa m -pies . In cer ta in cases [34,35] , meta l lographic exami-na t i ons o f pa r t i a l l y hydr ide d sa m ple s ha ve r e ve a l e dtha t t h e r e a c t i o n p r ogr e s se s by fa s t d i f fus ion o f hydr o ge na l o n g g ra i n b o u n d a r i e s , a n d t h e f o r m a t i o n o f a p r o d u c thydr ide l a ye r c oa t i ng e a c h o f t he g r a ins i n t he sa m ple ,d e v e l o p i n g f r o m t h e o u t e r g r a i n b o u n d a r y i n s i d e t h eg r a in . H e n c e , e v e n th o u g h a b u l k ( n o n - p o w d e r e d ) s am -p le i s c onc e r ne d , t he ove r a l l k ine t i c s r e se m ble t hec on t r a c t i ng - e nve lo pe k ine t i c s o f a pow de r , w i th g r a ins ize dis t r ibut ions replac ing the par t ic les s ize dis t r ibu-t i ons i n t he powde r . A s im i l a r s i t ua t i on ha s be e nobse r ve d f o r t he h ydr id ing o f bu lk L a Ni5 [ 36], wh e r et h e r e a c t i o n c o m m e n c e s a l o n g p r e f e r r e d b u l k p a t h s( p r oba b ly g r a in bounda r i e s o r m ic r oc r a c ks ) , r e su l t i ngin t he pa r t i c u l a t i on o f t he bu lk sa m ple a t a ve r y e a r lys t a ge ( i .e . a ve r y sm a l l a ) o f t he r e a c t i on . A de t a i l e dd i s c u s s i o n o f t h e s e m e a s u r e m e n t s w i l l b e p r e s e n t e de l se w he r e [ 36] . I t i s, howe ve r , im p or t a n t t o r e c a l l t ha ti n s u c h c a s e s s o m e c o n f u s i o n m a y a r i s e w h e n a t t e m p t i n gto a na lyse t he k ine t i c da t a , w i thou t no t ing t he pa r t i c u l a rtopo logy o f p r oduc t p r ogr e s s ion . Assu m e tha t suc h a

polycry s ta l l ine bulk samp le , wi th a wel l -def in ed init ia lge om e t r i c a l sha pe ( e .g . a c ube w i th a n e d ge s i zeA A A) , y i e lds a c e r t a in e xpe r im e nta l a vs . t k ine t i cc ur ve , whic h a c tua l l y c or r e sponds t o t he a bove - m e n-t i one d t opoc he m is t r y ( i . e . c on t r a c t i ng- e nve lope c on-s t a n t -ve loc i t y k ine t i c s de ve lop ing on e a c h o f t he g r a insin t he bu lk sa m ple ) . Howe ve r , w i thou t no t i c ing t hec a se , t he a vs . t da t a a r e f i t t e d t o t he SPA f unc t ions(Eq. (4) or (5) ) assuming a rea l s ingle-par t ic le reac t ion( i . e . a s sum ing tha t t he sa m ple r e a c t s by c onve n t iona lc on t r a c t i ng- e nv e lope c ons t a n t - ve loc i t y k ine t ic s , w i th t hep r o d u c t d e v e l o p i n g o n t h e s u r f a c e o f t h e c u b e a n dpr ogr e s s ing i nwa r ds ) . E v ide n t ly , in bo th c a se s t he SPAfunc t ions (4) o r (5) wi ll y ie ld a very good f i t to the avs . t c u r ve s . I n t he f o r m e r ( r e a l ) c a se , howe ve r , ( Rs )( a ve r a ge d ove r t he g r a in s i z e d i s t r i bu t i on) shou ld besubs t i t u t e d , a c c or d ing t o E q . ( 30), l e a d ing t o t he c o r r e c tva lue o f U , whe r e a s f o r t he l a t t e r ( m i s t a ke n) c a se t hec ube d im e ns ion A i s u t i li z e d , l e a d ing t o a n e r r o ne o use s t im a t ion o f U . T h i s t y pe o f i nc or r e c t a na lys is c a n bep r e v e n t e d b y e i t h e r p e r f o r m i n g m e t a l l o g r a p h ic o b s e r -va t i ons on pa r t i a l l y hydr ide d sa m ple s ( no t ing t he n t her e a l m o r p h o l o g i c a l f o r m o f p r o d u c t d e v e l o p m e n t ) , o ra l te r n a ti v e ly b y p e r f o r m i n g t h e S P A p r o c e d u r e o n d if -f e r e n t s i ze s o f s a m ple s ( e .g . a s e t o f c ube s w i th i nc r e a s ingd im e ns ions ) . For a c onve n t iona l SPA c a se ( c on t r a c t i nge nve lop e fo l l owing the ge o m e t r i c a l sha pe o f t he sa m ple ) ,t he s l ope s o f t he Fc v( a ) vs . t c u r ve s ( E q . ( 5 ) ) shou ldd e c r e a s e w i t h i n c r e a s i n g A v a l u e, w h e r e a s f o r t h e g r a in -bou nda r y c ons t a n t - ve loc i ty c a se t he sa m e s lope s ( U /( Rs ) ) shou ld be d i sp l a ye d r e ga r d l e s s o f t he i n i t i a ld i m e n s i o n s ( o r s h a p e ) o f t h e r e a c t i n g s a m p l e . H e n c e ,such a s ize (or sh ape) ana lys is s imply ident i f ies thec or r e c t m or pho log i c a l c ha r a c t e r i s t i c s o f p r oduc t de -v e l o p m e n t .

4 . C o n c l u s i o n s

T he va l i d i t y i n u t i l i z i ng sphe r i c a l SPA m ode l s f o rthe i n t e r p r e t a t i on o f k ine t i c da t a ob t a ine d f o r ga s - so l i dr e a c t i ons ( e .g . hydr id ing r e a c t i ons ) pe r f o r m e d on pow-d e r s w a s e v a l u a te d . T h e S P A m o d e l s d i s c u ss e d i n c l u d e dtwo typ e s o f c on t r a c t i ng- e nve lope k ine t i c s ( c f. F ig . 1 ) :one w i th a c ons t a n t - ve loc i t y i n t e r f a c e m ov e m e nt a ndthe o th e r w i th a de c e l e r a t i ng d i f f us ion- c on t r o l le db o u n d -a r y p r ogr e s s ion . T he c or r e spond ing SPA k ine t i c f unc -t i ons f o r a sph e r i c a l ge om e t r y a r e g iven by E qs . ( 4 )a nd ( 5 ) o r by E qs . ( 9 ) a nd ( 12) r e spec t i ve ly . T he setype s o f f unc t i ons a r e c onve n t iona l l y a pp l i e d i n t h el i t e r a tu r e t o i n t e r p r e t t he k ine t i c s o f powde r s . T hejus t if i c a ti on f o r suc h a p r oc e du r e i s no t a p r io r i s t r a igh t -f o r w a r d b e c a u s e o f t h r e e f a c t o r s i n h e r e n t i n p o w d e rk i n e ti c s w h i c h m a y m o d i f y t h e s p h e r ic a l S P A b e h a v i o u r.These fac tors a re ( i ) par t ic le s ize dis t r ibut ions , ( i i )par t ic le shape var ia t ions and ( i i i ) t ime dis t r ibut ions for


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M.H . Mintz, Y. Zebi / Journal of Alloys and Compounds 216 (1994) 159-175 173

t h e c o m m e n c e m e n t o f th e r e ac t io n o n e a c h o f t h epa r t i c l e s c om pos ing the powde r .

I n t h e p r e s e n t w o r k , t h e e f fe c t s o f t h e s e p o w d e r -inhe r e n t f a c to r s a r e qua n t i t a t i ve ly a na lyse d a nd the i ri n t e r f e r e n c e in t h e S P A p r o c e d u r e is e s t i m a t ed . T h eou tc om e of t h i s a na lys i s r e f e r s t o two m a in que s t ions .( 1 ) Ca n the S PA a na lysi s y i e ld t he c or r e c t t ype o f

k ine t ic s c on t r o l l i ng the ga s - so l id r e a c t ion o f t he pow de r ,i .e . c a n i t i de n t i f y c or r e c t ly e i t he r t he c ons t a n t - ve loc i tyor t he d i f f us ion- c on t r o l le d k ine ti c s?( 2 ) Be y ond th e i de n t i f ic a t i on o f t he t ype o f k ine ti c s ,i s i t poss ible to o bta in b y such an ana lysis a quant i ta t ivee s t im a te o f t he a s soc i a t e d i n t r i ns i c k ine t i c pa r a m e te r[14] , i .e . to eva lua te U or ks in Eqs. (4) and (5) orEqs. (9) and (12) respec t ive ly?

T he c onc lus ions ob t a ine d , r e ga r d ing the que s t ionsaddressed above , a re as fol lows.( 1 ) As long a s a l l t he pa r t i c le s c om po s ing the p ow de rs t a r t t o r e a c t s im ul t a ne ous ly ( o r a t le a s t wh e n the t im es p r e a d f o r t h e c o m m e n c e m e n t o f t h e r e a c ti o n o n e a c hof t he pa r t i c l e s i s m uc h sm a l l e r t ha n the r e a c t ion ha l f -t im e t l / z ) , a nd a s l ong a s a c on t inu ous p r odu c t l a ye ri s f o r m e d on e a c h pa r t i c l e a t a ve r y e a r ly s t a ge ( i . e .Xo in Eq. (2) , i l lus t ra ted in Fig. 1 , i s much smal le rtha n the d im e ns ions o f t he r e a c t ing pa r t i c l e s ) , t heoc c ur r e nc e o f e i t he r d i spe r se d s i z e d i s t r i bu t ions o fpa r t i c l e s o r pa r t i c l e sha pe va r i a t i ons ( f r om sphe r i c a lsym m e t r y) doe s no t l e a d to s ign if ic a n t de v ia t i ons f r omthe sphe r i c a l SP A f un c t ions , ove r a r e la t i ve ly w ide r a n geof the reac t ion co urse ( i .e . up to a -~ 0.5-0.6) . This i sdisplayed in Figs . 3 and 4 for the constant -ve loc i tyk ine t ic s , a nd in F igs. 5 a n d 6 f o r t he d i f f us ion- c on t ro l l e dk ine t i c s . He nc e , t he c or r e c t i de n t i f i c a t i on o f t he c on-t r o l li ng type o f k ine t i c s ( i .e . e i t h e r c o ns t a n t ve loc i ty o rd i f f us ion c on t r o l l e d) i s poss ib l e i n t he se c a se s , e ve nwi thou t kno wle dge o f t he spe c if ic si z e d i s tr i bu t ion f unc -t i on o f t he powde r , o r i t s pa r t i c l e sha pe s .

( 2) T h e c o n t r o ll i n g ty p e o f k i n e t ic s m e n t i o n e d a b o v ec a n be i de n t i f i e d by c he c k ing the F r , g ( a ) t im e b e ha v io ur( E q . ( 3 ) ) w i th g c or r e spond ing to t he sphe r i c a l ge o m e t r y( o r a n y o t h e r c h o s e n g e o m e t r y w h i c h i n f a c t is n o ts ign if i ca n t f o r t ha t pur pose ) a nd r c o r r e spon ding e i t he rto c o ns t a n t - ve loc i ty k ine t i c s ( E q . ( 5 ) ) o r d i f fus ion- c on-t r o l l e d k ine t i c s ( E q . ( 9 ) ) . T he subs t i t u t i on o f t he e x-p e r i m e n t a l a ( t ) da ta i n to t he F r , g(te) VS. t c urv e sh ou ldy ie ld a n i n i t i a l l i ne a r de pe nde nc e f o r t he c or r e c t t ypeof c on t r o l l i ng k ine ti c s ( l i ne a r ly up to a bou t a = 0 .5 - 0 .6 ) .

A l t e r na t ive ly , p r e se n t ing the F r . g ( a ) vs. r e du c e d t im esca le ( r = t / t l / 2 ) shou ld y i e ld f o r t he p r ope r k ine t i c s al i ne a r de p e nd e nc e ( up to ~ '= 1) , w i th a " un ive r sa l s l ope "typ ic a l o f t he t ype o f k ine t i c s ( se e T a b le 1 ) . He nc e ,the d i f f e r e nc e be tw e e n th e se two a l t e r na t ive s o f da t aa na lys i s i s t ha t , whi l e t he f o r m e r t ype o f p r e se n t a t i on( i .e . Fr, g(o vs . t curve s) yie lds ( for the given pro pe rk ine t i c s ) l i ne a r s lope s whic h de p e nd o n the e xp e r im e n ta lp r e s s u r e - t e m p e r a t u r e c o n d i t i o n s ( b e c a u s e o f t h e d e -

p e n d e n c e o f t h e c o r r e s p o n d i n g i n tr in s i c k i n e ti c p a r a m -e ters [14] ), th e la t te r ( i .e . Fr .g(a) vs . ~ ') pre sen ta t io nyie lds a f ixed s lope ( for a l l the appl ied pres-s u r e - t e m p e r a t u r e c o n d i t i o n s ) t y p i c a l o f t h e t y p e o fkine t ics only.

(3) Besides ident i f ica t ion of the cont rol l ing kine t ics ,s inc e t he i n i t i a l l i ne a r s lope s ( ob t a ine d f o r t he p r ope rkine t ics) of the Fr , g(a) vs . t curves a re p rop or t io na l tothe c or r e sponding in t r i ns i c k ine t i c pa r a m e te r s , i t i sposs ible to eva lua te by this ana lysis the func t iona lt e m p e r a t u r e a n d p r e s s u r e d e p e n d e n c e s o f t h e s e k i n e ti cpa r a m e te r s ( a l t hough no t t he i r a bso lu t e va lue s ) . Fore xa m ple , Ar r he n ius p lo t s o f t he se l i ne a r s lope s c a ny ie ld t he a ppa r e n t a c t i va t ion e ne r g i e s . ( For hydr id ingr e a c t ions , e spe c i a ll y f o r non- s t a b l e i n t e r m e ta l l i c hy-dr ide s , spe c i a l c ons ide r a t i ons o f t he Ar r he n ius - typep l o t s s h o u l d b e m a d e w h e n a p p r o a c h i n g n e a r - e q u i l i b -r ium c ond i t i ons , a s m e nt ione d in t he d i sc uss ion pa r to f t h i s a r t ic l e ) . He n c e , qua l i ta t i ve p r e ssu r e - t e m p e r a tu r et r e nds a nd a ppa r e n t a c t i va t ion e ne r g i e s c a n a l so bee v a l u a t e d i n t h e s e c a s e s w i t h o u t k n o w l e d g e o f e i t h e rpar t ic le s ize dis t r ibut ions or par t ic le shapes .

( 4 ) I n o r de r t o e va lua t e t he a bso lu t e va lue s o f t heassoc ia ted int r ins ic kine t ic parameters ( i .e . U for con-s tant -ve loc i ty kine t ics and kd for di f fus ion-control ledk ine t ic s ) , i t i s ne c e ssa r y t o de t e r m ine the sha pe s a ndthe s i z e d i s t r i bu t ions o f t he powde r s . T he a c c ur a c i e so f s u c h a b s o lu t e v a lu e s e s t im a t i o n s d e p e n d o n t h e s h a p eva r i a t i ons o f t he pa r t i c l e s i n t he powde r s .

( i) For pow de r s c om pose d o f ne a r ly a x ia l ly sym m e t r i cpa r t i c l e s ( e .g . ne a r ly sphe r i c a l , ne a r ly c ub ic a nd sym -m e t r i c p o l y h e d r a ), a c c u r a c ie s b e t t e r t h a n + _ 3 0 % c a nbe obta ined, ut i l iz ing Eq. (30) or Eq. (31) for constant -ve loc i ty or d i f fus ion-co ntrol led kine t ics respec t ive ly.

( i i ) For powde r s c om pose d o f a x i a l l y a sym m e t r i cpar t ic les (e .g . pla te le ts and wire- l ike par t ic les) butun i f o r m ly sha pe d , a ve r a g ing i s m a de ove r t he sm a l l e s td im e ns ion o f t he c or r e sponding sha pe ( e .g . t he r a d iusof t he w i r e ) , a nd E q . ( 49) r e p l a c e s t he a x i a l ly sym m e t r i ca ve r a ge m om e nt ( i .e . ( R s ) ) i n E qs . ( 30) a nd ( 31).

( ii i) F or p owd e r s c om pose d o f pa r t ic l e s w i th va r ia b l eshap es ( i .e . inc ludin g bo th axia lly symm etr ic and axia llya sym m e t r i c pa r t i c le s ) , t he e s t im a t ion o f t he a bso lu t eva lue s o f t he i n t r ins i c k ine ti c pa r a m e te r s i s no t a c c ur a t e ,a nd on ly i de n t if i c a t ion o f t he t ype o f c on t r o l l ing k ine t i c sa n d q u a l i t a t i v e p r e s s u r e - t e m p e r a t u r e t r e n d s ( a s s u m -m a r i z e d in c onc lus ions ( 1 ) - ( 3 ) ) a r e a pp l i c a b l e .

( 5 ) For c a se s whe n th e t im e spr e a d f o r t he i n i t i a ti ono f t h e r e a c t i o n o n e a c h o f t h e p a r ti c le s c o m p o s i n g t h epowde r i s c om pa r a b le w i th t he r e a c t ion ha l f - t im e t l /2 ,t he sha pe s o f t he a vs . t k ine t i c c ur ve s a r e m odi f i e d ,and inf lec t ion points a re disp layed a t low a va lues ( i .e .a t a < < 0 .5 ). T he se m odi f i c a t ions ob l i t e r a t e t h e k ine t ic s -type " f inge r pr in t s " o f t he se c ur ve s , a nd no m e a n ingf u la na lys i s o f t he da t a i s t he n poss ib l e .


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174 M.H. M intz , Y. ZeM / Journ al o f Alloys and Compounds 216 (1994) 159-175A s i m il a r s i tu a t i o n m a y b e e n c o u n t e r e d w h e n t h e D ( T )f or m a t ion o f a c on t inuous p r oduc t l a ye r ( c f . F ig . 1 ) i s

de l a ye d , i .e . wh e n Xo in E q . ( 2 ) be c om e s c om pa r a b le E awi th t he s i ze o f t he r e a c t ing pa r t i c le s ( f o r m o r e de t a i l s F r, g ( a )se e t he d i sc uss ion se c t ion o f t h i s a r t i c l e ) .

( 6 ) T h e d i f f e r e nc e be tw e e n the i n f l ec t i on po in t sindu c e d in t he a vs . t k ine t i c c ur ve s by the oc c ur r e n c eof t he a bov e - m e nt ione d in i t i a t ion t im e d i spe r s ions , a ndi n f l e c t i o n p o i n t s i n d u c e d b y t h e o c c u r r e n c e o f b u l knuc le a t ion - a nd- gr o wth k ine ti c s (E qs . ( 13) a nd ( 14) )shou ld be no t e d . I n t he l a t t e r c a se t he i n f l e ct i on po in t sa r e l o c a t e d a t a b o u t t h e h a l f -r e a c t io n s t ag e ( a = 0 . 5 ; G 1 , G 2t = t l / z ) , a nd a k ine t i c a na lys i s o f t he da t a i s poss ib l e .

F ina l ly , i t shou ld be e m p ha s i z e d tha t t he t e r m " pa r - Ht i c l e " i s no t we l l de f ine d in pow de r - r e l a t e d f i elds a nd Hcis u s u a ll y a s s o c i at e d w i t h t h e m e t h o d o f m e a s u r e m e n t kof t he pa r t i c u l a r s i z e d i s t ri bu t ion ( e .g . a " pa r t i c l e " c a nbe asso c ia ted wi th a c rysta l l i te , an aggreg a te of c rys- kDta l l i t e s o r a n a gg lom e r a t e ) . I n our p r e se n t c on te x t t he kNGt e r m " p a r t i c l e " r e f e r s t o t he sm a l l e s t so l id e n t i t y r e a c t ingby the c on t r a c t ing - e nve lo pe m or p ho log y ( Fig. 1 ) a nd Kgi s no t ne c e ssa r i l y ide n t i c a l w i th t he pow de r s a ggr e ga t e sa s d e t e r m i n e d b y m i c r o s c o p ic ( e. g. s c a n n in g e l e c t r o n L 1 , L 2m i c r o s c o p y ) o r s e d i m e n t a t i o n m e a s u r e m e n t s . A n e x -a m p l e o f s u c h a n e x t r e m e d i f fe r e n c e h a s b e e n d e m - Mons t r a t e d i n t he d i sc uss ion se c t ion , f o r t he r e a c t ion o fpo lyc r ys t a l li ne bu lk sa m ple s o c c ur r ing by f a s t g r a inbou nda r y d i f fus ion . I n t h i s c a se t he t e r m " pa r t i c l e s" ni s i n f a c t r e l a t e d to g r a ins w i th in t he bu lk sa m ple . T h i s pi s sue , how e ve r , i s im po r t a n t on ly f o r t he de t e r m ina t ion Pe ( T )of a bso lu t e va lue s o f t he i n t r i ns i c k ine t i c pa r a m e te r s( whic h r e qu i r e s t he c a l c u l a t ion o f a ve r a ge d s i z e pa - Rr a m e te r s ) bu t i s i r r e l e va n t t o t he i de n t i f ic a t i on o f t he Rct y p e o f k in e t ic s a n d t h e d e t e r m i n a t i o n o f p r e s - R ~s u r e - t e m p e r a t u r e t r e n d s ( w h i c h , a s s t a t e d a b o v e , d ono t r e qu i r e knowle dge o f pa r t i c l e sha pe s o r s i z e d i s - Rmint r ibut ions) .

AcknowledgementsT h i s w o r k w a s s u p p o r t e d b y a g r a n t f r o m t h e I s ra e lC o u n c i l f o r H i g h e r E d u c a t i o n a n d t h e I s r a e l A t o m i c

E n e r g y C o m m i s s i o n .Appendix A: Nomenclaturea i c o n s t a n t s a p p e a r i n g i n t h e e q u a t i o n w h i c hr e l a t e s t he r e a c t e d f r a c t i on to t he r e a c t iond i s p l a c e m e n t ( E q . ( 1 ) ); t h e y d e p e n d o nt h e s h a p e a n d s i z e o f r e a c t i n g s a m p l e sA d im e ns ion o f a sym m e t r i c pa r t i c l e ( e .g . as p h e r e o r c u b e )( A , ) n t h a v e ra g e m o m e n t o f AD ( R ) s i z e d i s t r i bu t ion f unc t ion o f t he pa r t i c l e sc o m p o s

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