m.a. meyers, li-hsing yu and k.s. vecchio- shock synthesis of silicides-ii. thermodynamics and...

8/3/2019 M.A. Meyers, Li-Hsing Yu and K.S. Vecchio- Shock Synthesis of Silicides-II. Thermodynamics and Kinetics

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~ Pergamon Acta metall, mater.Vol. 42, No. 3, pp. 715 729, 1994C o p y r i g h t 1 9 94 E l s e v i e r S c i e n c e L t dP r i n t e d i n G r e a t B r i t a i n . A l l r i g h t s r e se r v e d0956-7151/94 $6.00 + 0.00

S H O C K S Y N T H E S I S O F S I L IC I D E S -- II . T H E R M O D Y N A M I C SA N D K I N E T IC S

M. A. M EYER S, LI-HSING YU and K. S. VECCHIODepartment of Applied Mechanics and Engineering Sciences, University of California, San Diego,La Jolla, CA 92093-0411, U.S.A.

(Received 16 September 1992; in revised fo rm 23 July 1993)Abstract--A hermodynamic and kinetic analysis of shock-induced reactions in the (Nb or Mo)-Si systemsprovides a framework for the extraordinarily high reaction rates and a quantitative interpretation of theexperimental results obtained in Part I. The thermodynamic analysis is conducted by adding the heat ofreaction to the shock energy; increases in shock pressure, temperature, and velocity are predicted. At theparticle level, melting at the silicon-metal interface is found to be a necessary condition for the initiationof reaction; heat conduction calculations enable the prediction of a critical molten (Si) region size for whichthe heat generated through the reaction exceeds the heat lost to the unreacted regions. The calculationof melt fraction (of Si) as a function of shock energy coupled with critical melt pool sizes enables thedetermination of a minimum shock energy for the initiation of shock-induced reaction. At the local level,the reaction kinetics can be rationalized through the production of a liquid-phase reaction product(NbSi2), the formation of spherical nodules (~ 2 #m diameter) of this product through interfacial tensionand their subsequent solidification (in times of 1-5 ns). The heat generated by the reaction is sufficientto melt n iobium along the interface which facilitates both the expulsion of the NbSi2nodules into the liquidSi, and the generation of fresh Nb interface for further reaction. In addition, the dissolved Nb enrichesthe surrounding Si liquid, promoting more NbSi: reaction and formation.

1. INTRODUCTIONShock-wave propagation through materials gener-ates significant structural changes; these effects havebeen the object of extensive studies, that wereinitiated with the Manhattan project in the early1940's and continue to this day. Analytical investi-gations coupled with experimental studies haveyielded mechanisms rationalizing the production ofdislocations and point defects due to shock-wavepassage through solid materials. The first mechanismwas proposed by Smith [1], followed by Hornbo gen[2] and later modified by Meyers [3] and Weertman[4]. This work is reviewed by Mogilevski [5], who alsoapplied molecular dynamics to explain the gener-ation of defects at the shock front [6]. Weak shock-wave effects have been modeled by Weertman andFollansbee [7, 8]. The effects of shock-wave passagethrough porous (powder) materials are considerablymore complex, because intense and non-uniformplastic deformation is coupled with the shock-waveeffects. Th us, the particle interiors experience primar-ily the effects of shock waves, while the surfacesundergo intense plastic deformation which can resultin melting. This localized melting leads to the bond-ing of the powder, and this gave rise to the researchfield of shock-wave compac tion. A theoretical frame-work predi cting a mini mum shock energy and shockpulse duration to create and solidify an interparticlemelting layer has been developed by Gou rdi n [9] andSchwarz e t a l . [10]. This frame work has recently been

extended by Potter and Ahrens [11] to ultrahardmaterials incorporating material strength effectsthrough a Carroll-Holt model. When potentiallyreactive powder mixtures are used, the shock pulsemay trigger these reactions and synthesize newmaterials. The energy of reaction has to be incorpor-ated into the computat ions, and efforts by Horieand Kipp [12], Boslough [13], and Yu and Meyers [t4]are noteworthy. Recently, Krueger et al . [15] andKrueger and Vreeland [16] proposed models thataddress, in a predictive mann er, shock-induced reac-tions, with the postulation of a threshold energy(provided by the shock wave) for the initiation ofreaction.

In this paper, a rational and quantitative analysisis provided for the results reported by Vecchio et al .[17], consistent with the threshold energy conceptproposed by Krueger and Vreeland [16].

The introduction of exothermic reactions in con-currence with the physical processes under shockcompression of porous materials requires specialanalyses, at the following levels:

(i) continuum--thermodynamic;(ii) local--particle level;

(iii) local --sub -part icle (interface) level.It will be shown that the extraordinarily high reac-tion rates encountered under shock conditions canbe rationalized and quantified. The experimentallyobtained regions where shock-induced reactions

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716 MEYERS e t a l . : S H O C K S Y N T H E S I S O F S I L I C I D E S - - I Io c c u r a r e c o m p a r e d w i t h c o m p u t a t i o n a l p r e d i c t i o n si n S e c t i o n 2 . A n a n a l y t i c a l f r a m e w o r k t h a t e n a b l e sa m e c h a n i s t i c u n d e r s t a n d i n g o f s h o c k s y n t h e si s o fs i l i c id e s wi l l b e d e v e lo p e d in Se c t i o n 3 -5 . Th ro u g hth i s f r a m e wo rk i t wi l l b e p o ss ib l e t o d e v e lo p q u a n t i -t a t i v e e x p re ss io n s t h a t p re d i c t t h e m a jo r e f f e c t s o fsh o c k sy n th e s i s .

2 . C O M P U T E D S H O C K P R E S S U R E S A N DE N E R G I E SIn o rd e r t o r a t i o n a l i z e t h e e x p e r im e n ta l r e su l t s o f

t h e c o m p a n i o n p a p e r ( P a r t I ) [1 7] , c o m p u t e d p r e d i c -t i o n s o f sh o c k p re ssu re s a n d e n e rg i e s wi th in t h ec a p s u l e s a re i m p o r t a n t . T h e c o n t o u r s o f p a r t ia l l ya n d fu l l y re a c t e d r e g io n s o f F ig . 2 o f Pa r t I [1 7 ]c a n b e c o r re l a t e d t o t h e p re ssu re a n d e n e rg y l e v e l se x p e r i e n c e d b y t h e p o wd e rs wi th in t h e c a p su l e s . As t r ic t ly o n e - d i m e n s i o n a l s t r a in c o m p u t a t i o n u s i n g t h ee q u a t i o n o f s t a t e f o r t h e N b - - S i m i x t u r e d e v e l o p e da c c o r d i n g t o t h e m e t h o d d e s c r i b e d i n S e c t i o n 2 i ssh o wn in F ig . 1 . Th e se p lo t s we re o b t a in e d i n as im p l e h y d r o d y n a m i c c o d e M Y I D L , d e v el o p ed b yY o sh id a [1 8 ] . Th e f i r st p re ssu re p u l se , w h ic h c a r r i e s

m o s t o f t h e e n e r g y a n d t r a ve l s t h r o u g h t h e p o r o u sp o w d e r , c o m p a c t i n g it , h a s p re s s u r es o f ~ 7 a n d1 5 G Pa , fo r t h e 1 .2 a n d 1 .9 k m /s im p a c t v e lo c it i e s( these a re the impac t ve loc i t ies used in [17]) . Ase c o n d , h ig h e r a m p l i t u d e p u l se i s i n i t i at e d a t t h e b a c ksu r fa c e o f t h e c a p su l e ( ,- , 15 a n d 3 0 GPa , r e sp e c ti v e ly )a n d t r a v e l s t h r o u g h t h e c o m p a c t e d / m a t e r i a l . T h et r i -d im e n s io n a l e ff ec ts i n t ro d u c e d b y t h e f i n it e l a te ra ld im e n s io n s o f t h e c a p su l e a re v e ry s ig n i f ic a n t a n dp r o d u c e l a rg e d e v i a ti o n s o f p r e s s u re a n d t e m p e r a t u r ef ro m th e v a lu e s o f F ig . 1 . Th e p re ssu re a n d e n e rg yc o n t o u r s p r e d i c t e d u s i n g a n a d v a n c e d , t w o - d i m e n -s io n a l C S Q c o d e b y N o r w o o d a n d G r a h a m [1 9] a r es h o w n i n F i g . 2 . T h e s e s i m u l a t i o n s w e r e c o n d u c t e du s in g ru t i le a t ~ 6 0 % th e o re t i c a l d e n s i t y a s a m o d e lm a te r i a l . Th e p re ssu re l ev e ls a re a t a t im e o f 1 .9 5 # sa f t e r t h e sh o c k wa v e e n t e r s t h e c a p su l e , a n d t h ee n e rg y l e ve ls a re a t a t im e o f 5 # s a f t e r t h e sh o c kwa v e e n t e r s t h e c a p su l e . A sp ik e is se e n a t t h e c e n t e r ,a lo n g th e a x i s o f t h e d i sk . Th i s c e n t r a l p re ss u re /e n e r g y s p i k e is d u e t o t h e c o n v e r g e n c e o f t h e s h o c kf ro m th e s id e s ( t h e sh o c k wa v e t r a v e l s f a s t e r i n t h ec a p s u l e t h a n i n t h e p o w d e r , c r e a t i n g a " p i n c e r "a c ti o n) . T h e e n e r g y w a s o b t a in e d f r o m N o r w o o d a n dG r a h a m ' s [ 19 ] b u l k t e m p e r a t u r e c o n t o u r s b y p r o p e r

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Fig. 1. One dim ensional computed pressure as a function o f t ime at front (1) and rear (2) surfaces of theshock capsule; pressure at the front represents amplitude o f the first wave, whereas pressure at the backrepresents am plitude o f the reflected pulse (a) computational configuration, (b) profiles for 1.2 km/simpact, an d (c) profiles for 1.9 km/s impact.


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M E Y E R S e t a l . : S H O C K S Y N T H E S I S O F S I L I C I D E S - - I IS H O C K W A V E( a )

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Fig. 2. The summaries o f energy contours a t a t ime of 5/ ts after shock wave enters capsule for impactvelocit ies: (a) 1.9 km/s a t roo m temperature, (b) 1.3 km/s at room temperature, (c) 1.3 km/s a t 500C, andpressure conto urs a t a t ime o f 1.95/~s after shock wave enters capsule for imp act velocit ies of( d) 1.9 km/s,and (e) 1.3 km/s.

717

c o n v e r s i o n , u s i n g a h e a t c a p a c i t y t h a t i s th e m a s sa v e r a g e

E = Cv dT . (2)F o r t h e 5 0 0 C e x p e r i m e n t , t h e t h e r m a l e n e r g y o f t h es p e c i m e n p r i o r t o s h o c k i n g w a s a d d e d t o t h e s h o c ke n e r g y : Eth = C v A T = 2 2 0 J /g . T h r o u g h o u t t h i s p a p e rw e w i l l u s e C o a n d C v in t e r c h a n g e a b l y s i n c e t h ev o l u m e c h a n g e s a r e n o t a p p r e c i a b l e . T h u s , t h e v a l u e so f t h e e n e r g i e s a r e i n c r e a s e d b y t h i s a m o u n t o n t h ecur ves f o r t he 500 C expe r i men t ; t h i s i s a r e l a t i ve l ym i n o r c h a n g e i n t h e l e v e l . F i g u r e 2 s h o w s t h a t t h et o p o f t h e c a p s u l e u n d e r g o e s p r e s s u re s o f ~ 7 a n d1 5 G P a , f o r t h e 1 .3 a n d 1 .9 k m / s i m p a c t v e l o c i t ie s ,r e s p e c t iv e l y . T h i s i s c o n s i s t e n t w i t h t h e o n e - d i m e n -s i o n a l c o m p u t a t i o n s . H o w e v e r , t h e p r e s s u re a n d e n -e r g y l e v e ls a t o t h e r r e g i o n s a r e c o n s i d e r a b l y h ig h e r ,d u e t o t h e t h r e e - d i m e n s i o n a l e f fe c ts . A l t h o u g h aq u a n t i t a t i v e a s s e s s m e n t i s n o t p o s s i b l e , th e f o l l o w i n go b s e r v a t i o n s c a n b e m a d e :

( a ) T h e p a r t i a l l y a n d f u l l y r e a c t e d r e g i o n s f o l -l o w m o r e c l o se l y t h e e n e r g y c o n t o u r s t h a nt h e p r e s s u r e c o n t o u r s . T h e p r e s s u r e , a t

t i m e s e a r l ie r t h a n s h o w n i n F i g . 2 ( d ,e )r e a c h e s v e r y h i g h v a lu e s a t t h e u p p e r e d g e so f t h e c a p s u le ( m a r k e d b y a r r o w s ) , w h e r en o r e a c t i o n w a s s e e n t o o c c u r . T h e f u l l y -r e a c t e d r e g i o n i s c o n c e n t r a t e d o n t h eb o t t o m / c e n t r a l p o r t i o n s o f t h e c a p s u l e ,c o r r e s p o n d i n g t o t h e m a x i m u m e n e r g i e s .

( b ) B y c o m p a r i n g F i g . 3 o f P a r t I [ 17 ] w i t hF i g . 2 , it c a n b e s e e n t h a t a t h r e s h o l de n e r g y o f 6 0 0 - 8 0 0 J / g i s n e c e s s a r y t oi n i t i a te th e r e a c t i o n f o r t h e r o o m t e m -p e r a t u r e e x p e r i m e n t s in t h e N b - S i s y s te m .F o r t h e 5 0 0 C e x p e r i m e n t , t h i s e n e r g ys e e m s t o b e s o m e w h a t l o w e r . T h i st h r e s h o ld e n e r g y c o n c e p t w a s p r o p o s e d b yK r u e g e r e t a l . [15, 16].

A p o s s i b l e c o m p l i c a t i n g f a c t o r n o t c o n s i d e r e d i nt h e a b o v e c o m p a r i s o n b e t w e e n o b s e r v e d a n d c o m -p u t e d r e a c t i o n p r o f i l e s is t h e p o s s i b i l i t y t h a t r e a c -t i o n s c a n c o n t i n u e a f t e r t h e p a s s a g e o f t h e s h o c kp u l s e . T h e r e a c t i o n , i n i t i a t e d d u r i n g t h e p a s s a g e o ft h e s h o c k p u l s e , c a n c o n t i n u e t o c o m p l e t i o n . T h i sc o m p l i c a t e s t h e i n t e r p r e t a t i o n o f r e c o v e r y s p e c i m e n s ,a n d h a s n o t b e e n a d d r e s s e d q u a n t i t a t i v e l y in t h ep r e s e n t o r p a s t s t u d i e s .


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718 MEYERS e t a l . : SHOCK SYNTHESIS OF S ILICIDES- - - I I3 . T H E R M O D Y N A M I C S O F S H O C K S Y N T H E S I SO F S I L IC I D E SIn th i s se c t i o n , i t wi l l b e se e n h o w th e t h re sh o ld

e n e rg y c a n b e c o r re l a t e d t o t h e s t a t e o f t h e sa m p le .T h e t h e r m o d y n a m i c s o f s h o c k -w a v e p a s s ag e t h r o u g hp o r o u s m a t e r i a l s h a s t r a d i t i o n a l l y b e e n t r e a t e d b yu s i n g t h e M i e - - G r u n e is e n e q u a t i o n o f s t a te ( E O S ) f o ra c o n s t a n t v o lu m e [20 , 2 1 ]. Kru e g e r e t a l . [15] deve l -o p e d a t h e o r y f o r th e p r o p a g a t i o n o f s h o c k w a v e st h r o u g h p o w d e r m i x t u r e s . T h e t r e a t m e n t u s e d h e r ew i l l b e t h e s i m p l e M i e - G r u n e i s e n a p p r o a c h , s i n c ea c c u ra c y i n t h e e q u a t io n o f s ta t e i s n o t e s se n t i a l t o t h ed e v e lo p m e n t . T h e " M i x t u r e " p r o g r a m d e v e lo p e d b yY o s h i d a [ 22 ] w a s u s e d i n t h e c o m p u t a t i o n o f t h ee q u a t io n o f s t a t e ; it is b a se d o n t h e s im p le m ix in g o ft h e c o n s t i tu e n t s i n t h e i s o t h e r m a l c o m p r e s s i o n c u r v e,a s p r o p o s e d b y M c Q u e e n a n d M a r s h [ 2 1 ] .

F i r s t, t h e e q u a t i o n s o f s t at e f o r t h e p o r o u s N b - S ia n d M o - S i p o w d e r m i x t u r e s w i ll b e c a l c u la t e d u n d e rs h o c k c o m p r e s s i o n , a s s u m i n g t h a t t h e y a r e i n e r t ( n ore a c t i o n ) . Se c o n d ly , t h e r e a c t i o n e n e rg y wi ll b e a d d e dt o t h e m . T h i r d l y , t h e m e l t i n g o f th e c o m p o n e n t s a s afu n c t io n o f sh o c k e n e rg y wi l l b e es t a b l i sh ed .3 . 1. E q u a t i o n o f s ta t e f o r p o r o u s m i x t u r e s

T w o c o m p o n e n t s , A a n d B ( N b a n d S i o r M o a n dS i ) a re c o n s id e re d , a n d t h e m a ss f r a c t i o n s a re m A a n dm a . Th e sp e c i f i c v o lu m e , V, i n t e rn a l e n e rg y , E , a n dh e a t c a p a c i t y , Cv , o f t h e m ix tu re i s ta k e n a s

V = m A V A + m B V BE = m A E A + m B E B

C v = m ACvA -k- mB Cvs. (3)S i m i l a r l y , t h e M i e - G r u n e i s e n p a r a m e t e r ( a t a m b i e n tp re ssu re ) , ~0, o f t h e m ix tu re i s o b t a in e d f ro m th e m a ssa v e r a g e

- - = VA0 VB0V m A - - + r o B - - . (4 )Yo YAO YBOT h e p r o c e d u r e f o l l o w e d in t h e c a l c u l a t io n i s t o o b t a i nt h e i s o t h e r m a l c o m p r e s s i o n c u r v e f o r t h e m i x t u r ef r o m t h e i s o th e r m a l c o m p r e s s i o n c u r v es f o r t h e c o m -p o n e n t s . F r o m t h e i s o t h e r m a l s t a t e s ( E r , P T , I x ) o n eo b t a i n s t h e H u g o n i o t s t a t e s ( E a , P H , Va ) fo r t h em i x t u r e b y a p p l y i n g t h e M i e - G r u n e i s e n e q u a t i o n ,wh ic h r e l a t e s o n e s t a t e t o t h e o th e r s t a t e b y t h ee q u a t i o n

E~ - E~ = ~ (e~ - ?O. (5)

T h e c o n v e r s i o n f r o m t h e r e f e re n c e s t a t e ( i s o t h er m a l )t o t h e s h o c k s t a t e ( H u g o n i o t ) i s m a d e a t a c o n s t a n tv o l u m e (V H = I x ) , i n t h e M i e -- G r u n e i se n t r e a t m e n t .T h e s h o c k t e m p e r a t u r e c a n b e o b t a i n e d f r o m t h er e f e r e n c e t e m p e r a t u r e b y

E H - - ET -- Cv d T . (6 )298

Th e in t e rn a l e n e rg y i s d e f in e d a s ( f ro m th ec o n s e r v a t i o n o f e n e r g y e q u a t i o n )

Er~ - Eo = (P + Po ) (Vo - V) . (7 )Fo r t h e p o ro u s m a te r i a l t h e i n i t i a l sp e c i f i c v o lu m eo f t h e so l i d m a te r i a l i s s im p ly r e p l a c e d b y t h e sp ec if icv o l u m e o f t h e p o r o u s m a t e r i a l , Voo(Po ~ O)

Ep - E00 = P(V00 - V ,). (8)I f t h e p o r o u s m i x t u r e r e a ct s d u r i n g t h e p a s s a g e o ft h e s h o c k w a v e , t h e i n t e r n a l e n e r g y e q u a t i o n c a n b ee x p re sse d s im p ly a s

E H R - - E o o = l p ( V o o - - V H R ) + m E R . ( 9 )m is t h e m a s s f r a c ti o n r e a c te d , a n d E R t h e e n e r g y o fr e ac t io n ; t h i s a p p r o a c h w a s i n t r o d u c e d b y B o s l o u g h[ 13 ] a n d Y u a n d M e y e r s [ 1 4] .

T h e s e c a l c ul a t i on s w e r e c o n d u c t e d f o r t h e N b - S ia n d M o - S i p o w d e r m i x t u re s , a t a n in i ti al p o r o s i t y o f4 0 % (V00 = 1 .4 V0 ) , wh ic h c o r re sp o n d s t o t h e i n i ti a lp a c k i n g o f th e p o w d e r s ; t h e r e s u l ts a r e s h o w n i nF ig . 3 . I f t h e r e a c t i o n t a k e s p l a c e a t t h e sh o c k f ro n to n e e x p e c t s a n i n c re a se i n p re ssu re a n d sh o c k v e l -o c i t y ; t h e se h a v e b e e n p re d i c t e d b y Bo s lo u g h [1 3] a n dY u a n d M e y e r s [ 1 4] . T h e t h r e s h o l d e n e r g y f o r r e a c -t i o n, a s s u m e d t o b e ~ 70 0 J / g f o r N b - S i a n d M o - S i

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Fig. 3. Equation of state for powder m ixtures at 60% of thetheoretical density and for m ixtures after reaction; (a)Nb-Si; (b) Mo--Si.


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1 0 N b S i 2

5 i i , , i , h , , i , t i , i , t , , i t I L , I , , , ,

1 0 2 0 3 0 4 0 5 0 6 0P o r o s i t y ( % )

Fig. 4 . Predic ted (one-d imens ional ) pressure required toin i t ia t e r eac t ion i n N b-S i and M o -S i sys tem s, a s a f unc t i onof in i t ia l poros i ty .

S H O C K S Y N T H E S I S O F S I L I C I D E S - - I I5O

4 0-

P2 0-

t.

tOOOI

2 0 0 0

Nb

3 0 0 0 4 0 0 0 5 0 0 0Temperature K)

Fig. 5 . The ef fect of shock pressure on the mel t ing points ofN b , M o a n d S i .

w i l l b e u s e d t o c a l c u l a t e a t h r e s h o l d p r e s s u r e a s af u n c t i o n o f p o r o s i t y . T h i s is s i m p l y d o n e b y s o l v i n gs i m u l t a n e o u s l y e q u a t i o n s ( 5 ) a n d ( 8) . F i g u r e 4 s h o w st h e p r e d i c t e d t h r e s h o l d p r e s s u r e s . H i g h e r p r e s s u r e sa r e r e q u i r e d a s th e p o r o s i t y o f t h e p o w d e r m i x t u r ei s d e c r e a s e d . T h i s a n a l y s i s i s i n a g r e e m e n t w i t he x p e r i m e n t a l r e s u l t s b y S o n g [ 2 3 ] .3.2. Melting as a funct ion of shock energy

T h e m e l t i n g o f t h e c o m p o n e n t s i s n o t c o n s i d e r e d i nt h e E O S g r a p h i c a l l y e x p r e s se d i n F i g . 3 . I t i s i m p o r t -a n t t o d e t e r m i n e w h a t t h e t h r e s h o l d e n e r g i e s c o r r e -s p o n d t o , i n p h y s i c a l t e r m s . T h e s h o c k e n e r g y i sd e p o s i te d i n t h e t w o c o m p o n e n t s . T h e m e l t i n g p o in t so f t h e c o m p o n e n t s a r e d e t e r m i n e d , a s a f u n c t i o n o fp r e s s u r e , b y t h e C l a p e y r o n e q u a t i o n

d P A H (10)dT- TmAVw h e r e T m i s t h e m e l t i n g p o i n t , A V i s t h e v o l u m ec h a n g e i n m e l t i n g , a n d A H i s t h e e n t h a l p y o f f u s io n .T h e e f f e ct o f p r e s s u r e i s e x p r e s s e d a s

S i: T = 1 6 85 + A P ( - 3 5 . 7 ~ p a )

N b : T = 2 7 4 0 + A P ( 4 5 . 7 ~ p a )

M o : T = 2 8 9 0 + A P ( 5 2 . 9 3 ~ p a ) . (1 1)

T h e m e l t i n g p o i n t s o f t h e e l e m e n t a l c o n s t i t u e n t sa s a f u n c t i o n o f p r e s s u re a r e p l o t t e d i n F i g . 5 .S i l i c o n h a s t h e l o w e s t m e l t i n g p o i n t ; a d d i t i o n a l l y ,i t s m e l t i n g p o i n t d e c r e a s e s w i t h p r e s s u r e b e c a u s et h e r e i s a v o l u m e d e c r e a s e a s s o c i a t e d w i t h i t sm e l t i n g . A t t h e o n e - d i m e n s i o n a l p r e s s u r e s o f 7a n d 1 5 G P a ( F i g . 1 ), t h e m e l t i n g p o i n t s f o r s i l i c o na r e

MpTsiGPa= 1460 KMoISGPa __ 1100 K .. ~ t S i

T h e d i s t r i b u t i o n o f th e e n e r g y d e p o s i t i o n b e t w e e n S ia n d ( M o o r N b ) c o m p o n e n t s w i l l e s t ab l i sh t h e e n e r g yl e v e l a t w h i c h m e l t i n g i s i n i t i a t e d

, : x ( y ? + , , x , ( f ?(12)

Tm~ i s th e m e l t i n g p o i n t o f S i a t t h e e n e r g y l e v e l E ,L f i s t h e e n t h a l p y o f f u s i o n , 2 i s t h e f r a c t i o n o f s i l i c o nt h a t i s m o l t e n , x i s t h e f r a c t i o n o f e n e r g y d e p o s i t e di n S i, a n d ( 1 - x ) i s t h e f r a c t i o n o f e n e r g y d e p o s i t e di n t h e ( N b , M o ) p o w d e r . I t is a s s u m e d , t o a f i rs ta p p r o x i m a t i o n , t h a t t h e t e m p e r a t u r e s i n t h e t w oc o m p o n e n t s a r e t h e s a m e ( a f te r K r u e g e r et al. [15])B y u s i n g t h e q u a d r a t i c e x p r e s s i o n s f o r t h e h e a t

Table 1. Physical and thermody namicdata of N b-Si, M o-Si systemsMaterialsProperty N iob ium Mo lybd enum Sil icon

Melting temperature (K) 2740 2890 1685Latent he at of fusion (kJ/mol) 26.79 27.47 39.62Therm al condu ctivity W/cm-K ) 0.54 1.38 1.49Heat capacity (Cp) a = 24.63 a = 24.18 a = 23.94a+b x 10 3T+c x 10ST 2 b=3 .39 b = 1 .17 b=2 .47(J/C-mol) c = 9.21 c = 0 c = 4.14Compressive strength (MPa) 200-400 300 90Main reaction product with Si NbSi2 MoSi2 - -Melting temperature of product (K) 2420 2300 - -Heat of reaction to form product (kJ/mol) 138 129.6 - -


6/15

72 0 M E Y E R S e t a l . : S H O C K S Y N T H E S I S O F S I L I C I D E S - - I Ic a p a c i t y f r o m T a b l e 1 , a n d e q u a t i o n s (1 I ) a n d ( 12 ),o n e a r r i v e s a t , f o r t h e N b - S i s y s t e m P(V 0o - V ) = x [ 0 . 2 95 10 - 3 ( 16 85 - 35 . 7P ) 2

+ 5. 7 2 (1 6 8 5 - 3 5 . 7 P ) + 0 . 9 9x 1 0 5 ( 1 6 8 5 - 3 5 . 7 P ) - 1 - 2 . 0 6 x 1 03+ 2 4 3 0 ] + ( 1 - x ) [ 0 . 4 0 5x 1 0 - 3 ( 1 6 8 5 - - 3 5 . 7 P ) 2+ 5 . 8 8 5 ( 1 6 8 5 - - 3 5 . 7 P ) + 2 . 2x 1 0 5 ( 1 6 8 5 - 3 5 . 7 P ) - ' - 2 . 5 3 x 10 3].

( 13 )T h e e q u a t i o n o f s t a te f r o m F i g . 4 ( a) c a n b e e x p r e s se da s

V 0 0 - V = 1 ( p + 0 .0 0 2 9" ]4 6 . 0 4 i n 0 - . 0 - 0 ~ j ( i n G P a ) . ( 1 4 )B y s u b s t i tu t i n g e q u a t i o n ( 1 4) in t o e q u a t i o n ( 1 3) ,o n e o b t a i n s t h e m e l t i n g f ra c t i o n , 2 , as a f u n c t i o n o fs h o c k p r e s s u r e , P , o r s h o c k e n e r g y , E . T h e r e s u l t sa r e p l o t t e d i n F i g . 6 ( a) f o r t w o v a l u e s o f x : 1a n d 0 . 38 . W h e n x = l , t h e e n e r g y i s e n t i re l ya b s o r b e d b y S i. F o r x = 0 . 3 8, i t is u n i f o r m l y d i s -t r ib u t e d a m o n g t h e tw o p h a s e s i n p r o p o r t i o n t ot h e i r m a s s r a t i o s . M e l t i n g o f S i s t a r t s a t a ne n e rg y o f ~ 4 0 0 J / g f o r x = l a n d , - ,6 0 0 J / g f o rx = 0 . 3 8 . M e l t i n g o f S i i s c o m p l e t e ( 2 = 1 ) fo r

( a )1200 - j + J +

- ~ 1 00 0 -- + J ' + . ~~" homogeneous ~800 - /+

~- 6004 0 0 I I I {0.2 0.4 0.6 0.8 1.0( b )7 - - - 600

6

~ 5

50 0

40 0o

4 I I I I 30 00.2 0.4 0.6 0.8 1.0M e l t i n g f r a c t i o n o f S i

Fig . 6 . ( a ) Me l t ing f r ac t ion o f Si a s a func t ion o f shocke ne r gy f o r hom oge ne ous ( e qua l l y , i n N b a nd S i ) a ndhe te rogeneous ene rgy di s t r ibut ion ( comple te ly in to Si ) , and(b) me l t ing f r ac t ion w hen ene rgy o f r eac tion i s added toshock ene rgy.

(a )

( b ) Nb(62)-Si(38)10.5 -- Hot spot size (radius) H,mI10 .0 0 . 1 0 _ ,

_ ~ ~ ' ~ 1 generationz " 9 . 0 - 5 o - ~ i"~ 10 I " l8. 5

i=~ 8. 0 I I ', I i I Io - 9 . 5 - 9 . 0 - 8 . 5 - 8 . 0 - 7 . 5 - 7 . 0 - 6 . 5 - 6 . 0" ~ ( C )10.5 Mo(63)-Si(37)0.1 Hot spot sizel(radius ) p~m, . = 1 0 . 00 . . . . . . . .9 . 5 - / ~ " ~ " Heat .~ generation9 .0 - - 5 e / J I

10 ~ {l8 . 5 - - l

l8 . 0 { l ' { l l l-9 .5 -9 .0 -8 .5 -8 .0 -7 .5 -7 .0 -6 .5 -6 .0

L og t i m e ( s )Fig . 7 . ( a ) M ol t en r egion wi th r adius r 0 a t t o ; hea t i sconduc ted to the envi ronment , i n which the t empera ture i sfunc t ion o f pos i t ion and t ime . C a lcula t ed hea t l os se s f rom"h ot sp ot s " for d i f f e rent hot spot r adi i r 0 (0 . I , 0 .5 , i , 5 and10#m ) a nd he a t ge ne r a t i on ( i n s t a n t a ne ous ) f r om s hoc ksynthes is of (b) Nb -Si and ( c ) M o-S i powd er mixture s ; hea tgen erat ion t ime is 10 -8 s .

~ 1 0 0 0 J / g ( x = l ) a nd ~ l 1 5 0 J / g ( x = 0 . 3 8 ) . T h et h r e s h o l d e n e r g y l e v e l o f 7 0 0 J / g c o r r e s p o n d s t o am e l t i n g f r a c t i o n , o f S i, 2 , b e t w e e n 0 . 2 a n d 0 . 5 . T h i sa n a l y s i s i s i n a c c o r d w i t h t h e e x p e r i m e n t a l r e s u l t s o fS e c t i o n 2 , w h i c h i n d i c a t e t h a t m e l t i n g o f S i is ap r e c o n d i t i o n f o r r ea c t io n . B y c o n s i d e r in g t h a t t h em o l t e n s i l ic o n r ea c t s in t e g r al l y , a n d b y c o m p u t i n g t h ec n c t g y o f r e a c ti o n , t h e p l o t o f F i g . 6 ( b) i s o b t a i n e d( f o r x = 0 . 3 8 ) . I t s h o w s t h e e n e r g y r e q u i r e d t o p r o p -a g a t e t h e r e a c t i o n i s l o w e r t h a n t h e e n e r g y r e q u i r e dt o i n i t i a t e i t , a n d t h e r e f o r e t h e r e a c t i o n t e n d s t o b es e l f - s u s t a i n i n g .

T h e r e i s , h o w e v e r , a r e m a i n i n g q u e s t i o n : W h y i s am i n i m u m f r a c t io n o f m o l t e n s i l ic o n n e c es s a r y toi n i t i a t e t h e r e a c t i o n ? I n o r d e r t o a d d r e s s t h i s q u e s -t i o n , a m o d e l i s p r o p o s e d b e l o w t h a t i s b a s e d o n


7/15

M E Y E R S et al . : SH O C K SY N T H E SI S O F SI L I C I D E S- - I I 721t h e f o l l o w i n g a s s u m p t i o n s , ( c o n s i s t e n t w i t h t h eo b s e r v a t i o n s o f P a r t 1 [1 7] ):

( a ) m e l t i n g o f s il i c o n i s r e q u i r e d f o r t h e i n i t i a t i o no f r e a c t i o n ; a n d

( b ) a m i n i m u m s i ze o f r e a c t e d m a t e r i a l s i s r e q u i r e df o r t h e p r o p a g a t i o n o f r ea c t io n .

C o n s i d e r a n i d e a l i z e d m o l t e n p o o l [ F i g . 7 ( a) ] o fr a d i u s 2 r 0 , a t Tm l a n d s u r r o u n d e d b y m a t e r i a l a ta t e m p e r a t u r e T0 . T h e r e a c t i o n o f t h i s m o l t e n S iw i t h t h e s u r r o u n d i n g s o l id ( N b o r M o ) w i ll g e n e r a teh e a t , t h a t i s t r a n s f e r r e d t o t h e s u r r o u n d i n g . A g e n e r a le q u a t i o n t h a t a d d r e s s e s b o t h t h e h e a t g e n e r a t i o n( a l o n g t h e s o l i d - l i q u i d i n t e r f a c e , w i t h a s u r f a c e e q u a lt o 4 n r g ) a n d h e a t e x t r a c t i o n f r o m t h i s c e n t r a l c o r e ,i n s p h e r i c a l c o o r d i n a t e s , i s

~ T k / O 2 T 2 ~ T 'X4nr 2Q + p C p ~ - = [- ~- yr Z r - ~ - r ) ( 15 )Q = H f e - A H / R r H ( T - Tin, (16)

w h e r e p i s t h e d e n s i t y , k is t h e t h e r m a l c o n d u c t i v i t y ,Q i s t h e h e a t g e n e r a t i o n r a t e , H ~ i s th e e n t h a l p y o fr e a c t i o n , A H t h e a c t i v a t i o n e n e r g y f o r r e a c t i o n ,H ( T - T m0 is a H e a v i s i d e f u n c t i o n ( r e a c t i o n r a t e = 0a t T < T m~ , w h e r e Tm~ i s t he m e l t i ng po i n t a t t he shoc kp r e s s u r e ) . E q u a t i o n ( 1 6 ) h a s a n i m p l i c i t a s s u m p t i o n :t h a t t h e r e a c t i o n s t o p s w h e n t h e t e m p e r a t u r e i s b e l o wt h e m e l t i n g p o i n t o f s i l i c o n .

R i d e a l a n d R o b e r t s o n [2 4] p r o p o s e d a s i m p l e rf o r m a l i s m f o r t h e d e t o n a t i o n o f e x p lo s i v es . T h e ya s s u m e d t h a t t h e h e a t w a s i n s t a n t a n e o u s l y g e n e r a t e da t a h o t s p o t . T h e y c a l c u l a t e d t h e h e a t e x t r a c t e d fr o mt h e r e a c t e d r e g i o n a s a f u n c t i o n o f t i m e b y u s i n ge q u a t i o n ( 1 7 ) ( t h is is th e F o u r i e r e q u a t i o n i n s p h e r i c a lc o o r d i n a t e s )

p C p ~ T ~ 2 T 2 0 Tf- (17)kOt ~r 2 r OrT h e h e a t g a i n e d b y t h e e n v i r o n m e n t i s e q u a l t o t h eh e a t l o s t b y t h e h o t s p o t . A t a t i m e t , t h e h e a t g a i n e db y a s p h e r i c a l s h e ll o f r a d i u s r ( r > r 0 ) a n d t h i c k n e s sd r i s

dQ = (4/~r 2 d r ) p C p T . (18)B y i n t e g r a t i n g f r o m r 0 t o i n f i n i t y , o n e o b t a i n s t h et o t a l h e a t g a i n e d b y t h e s u r r o u n d i n g ( o r l o st b y t h eh o t s p o t )

Q = 4nrEpCp T dr . (19)0

F i g u r e 7 s h o w s t h e h e a t g e n e r a t e d a n d h e a t l o s t t oe n v i r o n m e n t a s a f u n c t i o n o f t i m e f o r d if f e r e n t " h o ts p o t s " s iz e s; c a lc u l a t io n s w e r e m a d e f o r t h e N b - S i[ F i g . 7 ( b )] and M o- - S i [F i g . 7 (c )] sys t ems . T h e cu r vesf o r d i f f e r e n t h o t s p o t s i z e s c o n v e r g e t o t h e t o t a l v a l u eo f t h e h e a t c o n t a i n e d i n t h e h o t s p o t f o r t i m e s o n t h eo r d e r 1 m s . H o t s p o t r a d i i o f 0. 1 , 0 . 5 , 1 , 5 a n d 1 0 # m

w e r e c o n s i d e r e d . A s t h e h o t s p o t s i z e i n c r e a s e s , t h et i m e r e q u i r e d f o r t h e h e a t t o b e e x t r a c t e d i n c r e a s e s .T h e h e a t g e n e r a t e d i s p l o t t e d a s a h o r i z o n t a l ( d a s h e d )l i n e f o r t h e t h r e e c a s e s . T h e h e a t i s a c t u a l l y n o ti n s t a n t a n e o u s ly g e n e r a t e d , b u t o v e r a p e r i o d t h a t c a nb e a p p r o x i m a t e d a s t h e t r a n s it t im e o f t h e s h o c k w a v eo v e r a p a r t i c le . A n a p p r o x i m a t e s h o c k - w a v e v e l o c it yc a n b e o b t a i n e d f r o m F i g . 1 ; t h e c a p s u l e t h i c k n e s s i s1 0 m m , y i e l d i n g a s h o c k v e l o c i t y o f ~ 3 m m / # s . F o ra p a r t i c l e d i a m e t e r o f 40 # m , a s h o c k t r a n s i t t im e o f~ 10 - 8 s is ob t a i n ed . T he c i r c l e s on F i g . 7 ( b , c ) co r r e -s p o n d t o t h e c r i t ic a l " h o t s p o t " r a d i i f o r t h e t w om a t e r i a l s . T h e s e c r i t ic a l " h o t s p o t " s i z e s f o r r e a c t i o nt o b e i n i ti a t e d a r e e q u a l t o 3 a n d 4 # m f o r N b - S i a n dM o - S i s y s t e m s , r e s p e c ti v e l y . T h e s e p r e d i c t i o n s a r e i na g r e e m e n t w i t h o u r e x p e r i m e n t a l r e s u l t s . T h e c r i t i c a ls i z e s o f t h e s e " h o t s p o t s " a r e c o n s i s t e n t w i t h t h et h e r m o d y n a m i c p r e d i c t i o n s .

4 . M O D E L I N G S H O C K - I N D U C E D R E A C T I O N S A TT H E PA R T I C L E L E V E LA n a t t e m p t t o r a t i o n a l i z e th e a c c e l e r a t i o n o f r e ac -

t i o n k i n e t i c s o b s e r v e d i n s h o c k c o m p r e s s i o n c a n b em a d e b y a n a l y z i n g t h e v a r i o u s te r m s o f a n A r r h e n i u se q u a t i o n , i n w h i c h b o t h f o r w a r d a n d r e v e rs e re a c t i o nt e r m s a r e c o n s i d e r e d . F i g u r e 8 s h o w s t h e a c t i v a t i o nb a r r i e r n e c e s s a r y t o b e o v e r c o m e t o r e a c t A a n d Bt o p r o d u c e A x B y . A G i s t h e d i f f e r e n c e i n f r e e e n e r g yb e t w e e n p r o d u c t s a n d r e a c t a n t s , w h i l e A Q i s t h ea c t i v a t i o n e n e r g y ( e n t h a l p y ) f o r t h e r e a c t i o n . U n d e rs h o c k c o m p r e s s i o n , t h e e q u a t i o n t h a t e x p r e s s e s t h er a t e o f r e a c t i o n h a s t o b e m o d i f i e d t o i n c o r p o r a t ev a r i o u s e f f e c t s . F i g u r e 8 ( a ) s h o w s t h r e e p r i n c i p a lef fect s :

( a ) T h e r e a c t a n t s a r e i n a h i g h l y e x c i t e d s t a t e d u et o t h e s h o c k p a s s a g e . H i g h d e f e c t d e n s it i e s c h a r a c t e r -i ze t h i s s t a t e . Pan i n et al . [ 2 5 ] h a v e p r o p o s e d t h a tu n d e r t h e s e c o n d i t i o n s t h e m a t e r i a l is i n a m o r eene r ge t i c s t a t e .

( b ) T h e a c t i v a t i o n e n e r g y fo r r e a c t i o n i s d e c r e a s e db e c a u s e o f a " r e a c t i o n s t r a i n " t e r m . I f t h e p r o d u c t( e .g . N b S i 2 ) h a s a l o w e r s p e c i fi c v o l u m e t h a n t h er e a c t a n t s ( N b a n d S i ), t h e r e is a n e n e r g e t i c g a i n w i t ht h e p r o c e s s , d u e t o l o c a l r e d u c t i o n i n s t r a i n e n e r g y .F i g u r e 8 ( b ) s h o w s t h i s e f f e c t s c h e m a t i c a l l y . P r o d u c tA x B y o c c u p i e s a s m a l l e r v o l u m e t h a n r e a c t a n t s Aa n d B . B y a h y p o t h e t i c a l p r o c e d u r e d e v i s e d b y E s -he l by [ 26 ] , i t i s pos s i b l e t o e s t i ma t e t h i s ene r gy . Forp u r e l y d i l a t a t i o n a l s t r a i n s , E s h e l b y ' s e q u a t i o n s i m -p l i f i e s i t s e l f t o

E = PEr V (20 )w h e r e V i s t h e v o l u m e t r a n s f o r m e d , e v t h e v o l u m e t r i cs t r a i n a s s o c i a t e d w i t h t h e r e a c t i o n , a n d P i s t h ea p p l i e d p r e s s u r e .( c ) T h e e ff e ct o f p r e s s u r e o n t h e t h e r m o d y n a m i ce q u i l i b r i u m o f t h e p h a s e s . T h e f r e e e n e r g y d i f f e r e n c e


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7 2 2 M E Y E R S e t a l . : S H O C K S Y N T H E S I S O F S I L I C I D E S - - I I

t _el &

l. .

a

T r a n s i t i o n S t a t e $ 2 . e n e r g y d u e t or e a c t i o n s t r a i n

( e x ci te d s t a te ) / ~ Q - a Q s

S I - - A ~ + ~ B A G s " ~ ~ C u r v e a t P = O

A x B yP r e s s u r e . d e p e n d e n t C u r v eD i s t a n c e

b

Reaction Strain EnergyP

[ i i l A B ! i i i i i i i ; i l i; i i i i i i i i i i ii l i i i : : i i i ]- - ~ i f : ~ i i i ! ! i} i i : : ; ~ i i ~ ! i i i i ii i ! ~ i ! : : i ii ! ~ : : i i : :: : i ! ! ii i : ,- -

::iii::iiii::::::::iiiiiii::iii::::iii::i::!::i::ii~:~:i:i;i!i~ i i i ii i! : : :- - ~ i i ii i ~ ! i ii i ! i i i i i ~ i i : :i i ii i ii i ii iI : : i : : : k : ~ : i : ~ : ~ : : : : : : ! : : i ! ! ~ i i i : ! : i ! ! ~ i i ~ i : ! : ! : ! : ! : ! : i : i : l

!::i::ii::iiiiiii!ii!!".'..'!::ii!i::i::::iiiii!i!!~i~i~::ii!ii i~ i~ i ; i i i : : i ! i~ i : : ! i- -- - - - - i~ i~ i i ! ! ! i i : :i i i i : : i i! ! !~! ii i i i : : i ! i i : i i i i i : : i i i !: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : , i t . - . , . - .t t t t t t t t t t t t t

PP

-.~ ............,.............................. . ... ... .. .. ~ F i n a l:::::::::::::::::::::::::::::::::::::::::::::::::: Bound: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :~i: i :~::~:: : : : : : : : : : : : : : : : : : : : :- - " [ ! i i i ii : :i ~ i: i i i ~ i i i i i :: i l i i l ? : l i i i i ' ii ! : : :: : : ii ! ! ~ : : : :" "' -::~f:~::iii!~!!::~::iii::i::ii!!~r!!::~i~::~i~iii~iiiii!i~::i~!i!ii::i::ii!i!!. .. .. ~ - -i! ~ ~ i - i~ : i i i ! ~ : : : : ': ' : :: ~ : i~': I. .. .. ~ . .e - - 4 - i i i i i i i i ! i i i i i i i i i i i ~ ! i ~ . . , - - - e

- - " ~ i ~ i ii ;~!!::~!~::~!.~ii i! -.'~~.. .: : ; : : : : : : : ~ : : : : : :: : : : : : : : : ! : : : : : : ~ : : ~ : : : .- - , . : : i : : i i ! i i ! i i ! ! i i i i l i i i i ~ o , - i g ~ , a !- - ' . l i ~ i : : ~ i ~ i ~ i : : : : ! ~ i ~ i ~ ! Boundaryttttttttttttt

Unreacted

F i g . 8 . ( a ) A c t i v a t i o n e n e r g y c u r v e f o r r e a c t i o n x A + y B ~ A x B , a n d e f f e c t o f s h o c k w a v e o n p a r a m e t e r s .( b ) R e a c t i o n s t r a i n a s s c h e m a t i c a l l y i l l u s t r a t e d b y E s h e l ' b y ' s " c u t a n d p a s t e " m e t h o d [ 2 6 ] .


9/15

M E Y E R S e t a l . : S H O C K S Y N T H E S I S O F S I L I C I D E S - - I I 723b e t w e e n p r o d u c t s a n d c o m p o u n d s , A G , i s d e p e n d e n to n t h e s u p e r i m p o s e d p r e s su r e . 1

T h u s , t h e A r r h e n i u s r a t e e q u a t i o n , u n d e r s h o c kc o m p r e s s i o n p r o c e s s in g ( t h a t i n c o r p o r a t e s b o t h 0 . sf o r w a r d a n d r e v e r s e r e a c t i o n s ) c a n b e e x p r e s s e da s

e x p - ( A G + A G s ) ]= A e x p - ( A Q - A Q s ) r l -r R T L(21)

T h e t e r m s A Q a n d A G a r e t h e a c t iv a t i o n e n e r g y f o rt h e r e a c t i o n a n d t h e f r e e e n e r g y d i f f e r e n c e b e t w e e nr e a c t a n t s a n d p r o d u c t s , r e s p e c t iv e l y . T h e t e r m s A Q sa n d A G s r e s u l t s f r o m s h o c k m o d i f i c a t i o n , a s s c h e m a t -i c a l ly i n d i c a t e d i n F i g . 8 ( a ). A i s a p r e - e x p o n e n t i a lf a c t o r r e l a t e d t o t h e a t t e m p t f r e q u e n c y . N o r m a l i z e dr e a c t i o n r a t e s a r e c a l c u l a t e d i n t h e a b s e n c e a n dp r e s en c e o f s h o c k c o m p r e s s i o n , f o r t h e r e a c t i o n

N b + S i ~ N b S i 2 .T h e r e a c t i o n s t r a i n ( o r v o l u m e t r i c c o n t r a c t i o n )

a c c o m p a n y i n g t h e c o n v e r s i o n o f th e N b a n d S i t oN b S i 2 is 0 . 2 9; t h i s c o r r e s p o n d s t o a r e a c t i o n s t r a i n o f0 .1 1 . C o n s i d e r i n g a n a p p l i e d s tr e ss o f 1 4 G P a , o n eo b t a i n s a n e n e r g y o f 1 78 J / g . T h e d e f e c t e n e r g y c a nb e e s t i m a t e d f r o m a d i s l o c a t i o n d e n s i t y g e n e r a t e db y s h o c k . A s s u m i n g a d i s l o c a t io n d e n s i t y o f 5 10~/cm2 ( t y p i c a l o f m a t e r i a l s s h o c k e d t o h i g h p r e s s -u r e s ) , o n e o b t a i n s a n e n e r g y o f 9 J / g . T h i s y i e l d s :A Q s = 1 8 7 J / g . T h e f r ee e n e r g y o f r e a c t i o n c a n b et a k e n a s A G = 92 6 J / g . T h e c h a n g e i n f r e e e n e r g yw i t h p r e s s u r e c a n b e o b t a i n e d d i r e c t l y f r o m t h eC l a p e y r o n e q u a t i o n [ e q u a t i o n (1 0) ]. I t w a s f o u n d t ob e : A G s = 2477 J /g .

T h e a c t i v a t io n e n e r g y f o r t h e f o r m a t i o n o f M o S i 2f r o m e l e m e n t a l M o a n d S i h a s b e e n c a l c u l a te d b yK u m a r e t a l . [ 2 7 ] a n d f o u n d t o b e e q u a l t o9 9 .7 k J / m o l . S i n c e t h e a c t i v a t i o n e n e r g y i s n o t k n o w ne x a c t l y , a r a n g e o f v a l u e s f r o m 1 00 t o 4 0 0 k J / m o l w a su s e d i n t h e c a l c u l a t i o n s . T h e p r e - e x p o n e n t i a l f a c -t o r , A , is n o t k n o w n a n d o n l y t h e r a t io s r / A w e r ec a l c u l a t e d . T h e r e s u l t s a r e s h o w n i n F i g . 9 ( a ) f o r t h ef o r w a r d r e a c t i o n o n l y , a n d F i g . 9 (b ) f o r b o t h f o r w a r da n d r e v e r s e r e a c t i o n s . A l t h o u g h s h o c k c o m p r e s s i o ne n h a n c e s t h e r a t e r / A , t h e c h a n g e s a r e n o t d r a m a t i c .F o r t h e f o r w a r d r e a c t i o n o n l y , t h e i n c r e a s e i s ~ 1 0 % ,w h i l e c o n s i d e r i n g b o t h f o r w a r d a n d r e v e r s e r a t e s , t h ei n c r e a s e is f o u r f o l d . T h e s e i n c r e a s e s a r e m u c h i n f e r i o rt o t h e o n e s o b s e r v e d b e t w e e n t h e r a t e s o f r e a c t i o n i nq u a s i - s t a t i c a n d s h o c k e x p e r i m e n t s ; i n t h e l a tt e r ,t h e se r a t e s a r e c o m p a r e d i n S e c t i o n 3 o f th e c o m p a n -i o n p a p e r [ 1 7 ] . T h e s h o c k - i n d u c e d r e a c t i o n r a t e w a sf o u n d t o b e ~ 1 08 t i m e s h i g h e r t h a n t h e s o l i d s ta t e ,d i f f u s i o n g o v e r n e d r a t e . T h u s , u n l e s s th e p r e - e x p o -n e n t i a l f a c t o r A i s r a d i c a l l y d i f f e re n t i n t h e t w op r o c e s s e s , t h e A r r h e n i u s r a t e e q u a t i o n s h o w s t h a tt h e a c c e l e r a t i o n o f t h e r e a c t i o n c a n n o t b e a s c r i b e dt o t h e s h o c k e f f e c t s a b o v e , a n d h a s t o b e i n d i c a t i v e

~ o . 6

0 . 4

0 . 2

(a )

S I ~ - -U I ~

$ 3 / / i / / ; " L 1 , TU 11 1 0 0 k J / m o l esU43~ (" ~ : U ~ 3 00 kJ/molekJ/mle$ 4 , U 4 4 0 0 k J / m o l eU 4

S - S h o c k e dU - U n s h o c k e di i h I i i i t i i i I i i J i i i0 4 0 0 8 0 0 1 2 0 0 1 6 0 0 2 0 0 0( h ) T e m p e r a t u r e K

1 .0 /0 . 8 S 1N 0 . 6 8 2 U 1 1 0 0 k J / m o l e~ z S 3 t / $ 2 , U 2 2 0 0 k J / m o l e0 .4 / $ 3 , U 3 3 0 0 k J / m o l e

$ 4 $ 4 , U 4 4 0 0 k J / m o l eU I0 . 2 U 2U3. . . . . . . . . . . . . . _ . . . . .0 0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0

T emper a t u r e KFig. 9. Reaction rate/pre-exponential param eter rat io as afunction o f temp erature under unshocked and shock con-dit ions; (a) forward reaction only; (b) forward and r e v e r s ereactions.

o f a n o t h e r m e c h a n i s m o p e r a t i n g . T h e s p ec if ica s p e c t s o f t h i s n e w m e c h a n i s m w i l l b e d e s c r i b e d i nSec t i on 5 .

I t i s p o s s i b l e t o c a l c u l a t e t h e o v e r a l l f r a c t i o nr e a c t e d a s a f u n c t i o n o f t i m e f o r s h o c k - i n d u c e dr e a c t i o n s . T h e s e g l o b a l r e a c t i o n k i n e t i c s , a s o p p o s e dt o t h e l o c a l r e a c t i o n k i n e t i c s d i s c u s s e d e a r l i e r , a r ed e t e r m i n e d b y t h e d e n s i t y o f i n i t i a t i o n s i t e s , a n d b yt h e m a s s r e a c t e d p e r i n d i v i d u a l e v e n t a s a f u n c t i o n o ft i m e . A g e n e r a l f o r m u l a t i o n w i ll b e p r e s e n t e d t h a tc a p t u r e s t h e e s s e n t i a l p h e n o m e n a o f th e p r o c e s s .

I t w i l l b e a s s u m e d t h a t t h e i n i t i a t i o n s i t e s f o rr e a c t i o n w i l l b e a t p a r t i c l e - p a r t i c l e j u n c t i o n s . T h ej u s t i f i c a t io n f o r t h is i s t h a t m o l t e n p o o l s a r e o b s e r v e da t p a r t i c l e t ri p l e p o i n t s , i n s h o c k c o n s o l i d a t i o n ( p l a -n a r s e c t io n s ) . A s t h e p a r t i c l e s a re a c c e l e r a t e d a g a i n s te a c h o t h e r , t h e m o l t e n m a t e r i a l i s e j e c t e d a n d c a p -t u r e d a t t h e s e r e g i o n s t h a t a r e , i n a t r i - d i m e n s i o n a lv i e w , c e n t e r s o f t e t r a h e d r a . A t r i - d i m e n s i o n a l v ie wo f a t e t r a h e d r a l a r r a n g e m e n t o f sp h e r e s is s h o w ni n F i g . 10 ( a ) . A sma l l sphe r e i n t he cen t e r i nd i ca t e st h e p o s s i b l e i n i t i a t i o n s i t e . T h e i n i t i a t i o n s i t e s a r et h e r e g i o n s w h e r e t h e g r e a t e s t g a p s e x is t , a n d w h e r e ,a s a c o n s e q u e n c e , s h o c k e n e r g y i s p r e f e r e n t i a l l yd e p o s i t e d . T h e i n i t i a l d e n s i t y o f p o w d e r s f o r t h es i li c i d e s w a s 6 0 % o f t h e t h e o r e t i c a l d e n s i t y ( T D ) .T h i s j u s t if i e s t h e s e l e c t i o n o f a n i d e a l i z e d b o d y - c e n -t e r e d c u b i c a r r a n g e m e n t f o r t h e ( s p h e r i c a l ) p a rt i c l e s ,w i t h a d e n s i t y o f 6 8 % T D . I t is a s s u m e d t h a t t h e r ea r e n t e t r a h e d r a l s i te s p e r a t o m . T h e m e a n d i s t a n c ebe t w een t hese s i t e s w i t h r e spec t t o a b . c . c , c e ll , i f


10/15

7 2 4 M E Y E R S et al.: S H O C K S Y N T H E S I S O F S I L I C I D E S - - - I It h e s i t e s a r e e q u i d i s t a n t , i s d . A f t e r t h e c o n s o l i d a t i o no f t h e u n i t c e l l b y t h e s h o c k p r e s s u r e , t h e c e l l s i z ew i l l b e r e d u c e d f r o m 2 . 3 S t o 2 S ( S i s t h e p a r t i c l er a d i u s ) . T h u s

n d 3= a 3 = 23S 3 . (22 )T h e r e a r e 2 4 t e t r a h e d r a l s i t e s p e r u n i t c e l l , e a c h b e i n gs h a r e d b y t w o a d j a c e n t c e l ls i n a b . c . c , a r r a n g e m e n t .T h u s , o n e h a s s i x i n i t i a t i o n s i t e s p e r p a r t i c l e . F o rn = 6 , t h e m e a n d i s t a n c e b e t w e e n i n i t i a t i o n s i t e s i s1 . I S .

I f a l l s i t e s a r e s i m u l t a n e o u s l y i n i t i a t e d a t a t i m e %( e q u a l t o a n i n c u b a t i o n t i m e ) , t h e f r a c t i o n t r a n s -f o r m e d a t a t i m e ~ c a n b e e s t i m a t e d b y s e p a r a t i n g t h er e a c t i o n i n t o t w o s t a g e s [ F i g . 1 0 ( b ) ] :

( a ) i n d e p e n d e n t r e a c t i o n e v e n t s [ % < t < T 1;F i g . 1 0 ( b ) ( i , i i ) ];

( b ) i n t e r p e n e t r a t i n g r e a c t i o n f r o n t s I t > Z l ;F i g . 1 0 ( b ) ( i i i , i v ) ] .

T h e f r a c t i o n t r a n s f o r m e d , f , c a n b e e s t i m a t e d a s af u n c t i o n o f r e a c t i o n f r o n t a d v a n c e , r . I f n I i s t h en u m b e r o f i n i ti a t i o n s it e s p e r u n i t v o l u m e , w e h a v e ,f o r r 0 < r < d / 2

d f = n 1 d V = nl41rr2 d r . ( 2 3 )E q u a t i o n ( 23 ) a ss u m e s t h a t t h e r e a c t i o n i s o c c u r -r i n g a t t h e s u r f a c e o f a s p h e r i c a l r e a c t i o n r e g i o n .I n t e g r a t i n g f r o m ro t o r a y i e l d s t h e f r a c t i o nt r a n s f o r m e d a s

41tnl (r~ -- r~). ( 2 4 )f = - T -F o r r > d /2 w e h a v e t o s u b t r a c t t h e o v e r l a p v o l u m e so f s p h e ri c a l s eg m e n t s V0 s h o w n i n F i g . 1 0 ( b , i v ) . I fe a c h i n i t i a t i o n s i t e h a s n 2 n e a r e s t n e i g h b o r s , w e h a v e( e a c h o v e r l a p v o l u m e i s s h a r e d b e t w e e n t w o r e a c t i o nr e g i o n s )

F4~ 3 2 ]/2f = n I L- ~ - ( r I - - r ] ) - - 1 Io (25 )B u t t h e v o l u m e o f t w o j u x t a p o s e d s p h e r i c a l c a p s i s

V o : 2 - - - ~ ( r - d ) 2 ( 2 r + d ) . ( 2 6 ,T h u s :

F4n 3 nf = n , L ~ - ( r ~ - r g ) - n 2 - ~ ( r - d ) 2 ( 2 r + d ) ] .( 2 7 )

W e c a n u s e a H e a v i s i d e f u n c t i o n , H ( r - d / 2 ) , t og e n e r a l i z e t h e a b o v e e x p r e s s i o n

4 n n , r 3 H ( r - - d i n 2f = - - T - [ ( ~ - - r 3 )- - 2 / / 1 2x ( r - - d ) 2 ( 2 r + d ) ] . ( 28 )

a

I a

b O O %< t


11/15

MEYERS e t a l . : SHOCK SYNTHESIS OF SILICIDES--II 725The maximum value of t is

p 0.61pd/max = ~ rmax=The number of initiation sites per unit volume, n~,can be expressed as (six initiation sites per particle):nl = d -3 = (1.I S) -3. If one considers that each reac-tion region will overlap with six neighborin g regions,then

[ / 3 - - H ( t - - z I ) ( t - - z l )2(2t + zl )].(32)

Figure 11 shows plots of the fraction synthesized,f, as a function of time for different reaction rates,~, and partic le sizes, d. In Fig. l l(a) the reactio nrates are varied for a constant Particle size (50 #m),while in Fig. ll(b) the particle sizes are varied(10, 50, 500 #m) at a cons tant re action rate. Theseplots are only illust rative since the r eaction rates werevaried arbitrarily. The reaction rates were assumedconsistent with synthesis within the time frames ofshock-wave passage. This corresponds d r / d t valuesof 5 -50 m/s. The particle sizes were varied between10 and 500 #m. The reacti on rates, (, are o btain edfrom

d r= p ~ -- ~ . (33)These results show conclusively that the kineticsof synthesis are highly dependent upon both ofthese parameters. The fraction transformed increasesinitial ly with t 3, and for t > zj, withA t 3 - B ( t - z 1)2(2t + z~), reaching satur ation atf= 1. Beyond z~ the reaction rate slows downsensibly; this is the sphere overlap region0.5d < r < 0.61d. Depend ing on the reaction rate andparticle size, the reaction may or may not reachcompletion during the passage of the shock pulse.These results rationalize the results obtained in PartI [17] (partially and fully reacted regions).

5. REACTION MECHANISMThe analysis of the partially reacted regions in the

Nb -S i shock experiments revealed the detailed natureof the reaction sequence and mechanisms. There areclear indic ations that a t hreshold energy level existsabove which the reaction occurs. This is the confir-matio n of the Krueger-Vreeland [16] suggestion; forthe Nb--Si and Mo-Si systems this threshold energyis the level that results in the melting of the pre-estab-lished fraction of silicon. The pre ponder ance of smallNbSi 2 particles su rrou nded by silicon, as well as theexistence of NbSi2 particles attached to the niobiumparticles (Figs 3 and 4 of Part I [17]) are evidence fora r eaction me chanism in which the NbSi2 particles arecontinuous ly being generated at the interface and

i 0 " 8 1 ~ dr/dr 25 m/s /o.q / / A , . . , . . , . . , . .

T i m e , g s

C o n s t a n t R e a c t i o n R a t e 2 5 m / s1 . 0 Part i c l eS i z e 1 0 l a ~ /

Z ' 6 1 1 " ~~ 0.4,~r~ 0.2 ~ P a r t i c l e Size 500 g m0.0, - . . . . , - , - -o 2 4 6T i m e , ~ s

bFig. 11. Fraction reacted as a function of time for (a)different reaction rates (constant particle size of 50 #m) a n d(b) different particle sizes (constant interface advance rate of25 m/s).

ejected into the (molten) silicon. Thus, a permanentdiffusion barrier (i.e. a n interph ase layer) that w ouldslow down the reaction process is not formed, andreaction can proceed at a cons tant rate unti l the entireniobium or silicon are consumed. The simple calcu-lation below shows that the temperature rise due tothe reaction leads to a temperature (locally) higherthan the melting point of NbSi2. Figure 12 shows thesequence of events envisaged to occur. NbSi2 willform at a S i-N b interface, where silicon is molte n andniobium solid. Assuming an adiabatic reaction, thetemperature is

si AH~T = T M p + . (34)CpThe enthalpy of reaction of NbSi2 is (at P = 0)

equal to 926 J/g. The heat capacity of NbSi 2 wasestimated by interpolation between those of Nb andSi on a mass basis (m s~ and m N b a r e mass fractions ofSi and Nb)

C p = m S i c S i + m N b C N b . (35)This gives a value of 0.58 J/gK for NbS i/. Thus,assuming that the pressure exerted by the shock wave


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726 MEYERS et al.: S H O C K S Y N T H E S I S O F S I L I C I D E S - - I Ic o r r e s p o n d s t o t h e t h r e s h o l d e n e r g y , a t 4 0 % p o r o s i t y ,t h e m e l t i n g p o in t o f s i l i c o n i s 1 4 0 0 K ( se e F ig s 4 a n d5 ) . Th i s l e a d s t o : T ~ 3 0 0 0 K.

T h i s t e m p e r a t u r e i s c o n s i d e r a b l y h i g h e r t h a n t h em e l t i n g p o in t o f N b S i2 (2 4 2 0 K a t P = 0 ) . I t i st h e re fo re l i k e ly t h a t t h e se q u e n c e sh o w n in F ig . 1 2t a k e s p l a c e . T h e s u g g e s t i o n w a s m a d e b y G l a s s m a n[2 8] t h a t i n t e r f a c i a l e n e rg y p l a y s a n im p o r t a n t ro l e i nth e k in e t i c s o f sh o c k sy n th e s i s , wh ic h l e d t o t h ed e v e l o p m e n t o f t h e f o l l o w i n g m e c h a n i s ti c m o d e l .

Fo l lo win g F ig . 1 2 , t h e r e a c t i o n i s i n i t i a t e d a t ( a ) ,a l o n g t h e N b - S i i n t e r f a c e . A f t e r r e a c t i o n h a s p r o -c e e d e d to a c e r t a in e x t e n t ( c ) , su r f a c e ( i n t e r f a c i a l )f o r c e s b e c o m e d o m i n a n t , a n d t h e l i q u i d r e a c t i o np ro d u c t a g g lo m e ra t e s , fo rm in g a sp h e ru l e ( e) . At t h i sp o in t , r e a c t i o n k in e t i c s a re d ra s t i c a l l y d e c re a se d ,b e c a u s e o f t h e r e d u c t i o n o f t h e N b - S i i n te r f ac i a l a r e a ,a n d so l i d i f i ca t i o n o f t h e sp h e re s t a r t s ( e ) . As t h esp h e re so l i d i f i e s , n e w n u c l e i fo rm a lo n g th e Nb -S iin t e r f a c e ( f ) . Th e y g ro w, a g g lo m e ra t e i n to sp h e re swh e n th e y r e a c h a c r i t i c a l s i z e , a n d fo rm n e ig h b o r -in g sp h e re s (g ) . Th e se n e ig h b o r in g r e g io n s a re c o n -s t r a in e d a n d e x e r t fo rc e s o n t h e f i r s t sp h e re , e x p e l li n gi t (h ) a n d t h u s e x p o s in g f r e sh su r fa c e s . Th i s r e a c t i o np r o c e s s c a n c o n t i n u e u n t i l t h e r e a c t a n t s a r e c o n -su m e d . As su c h , n o d i f fu s io n b a r r i e r i s fo rm e d . Th ec o o l in g r a t e o f t h e Nb S i2 sp h e ru le s c a n b e c a l c u l a t e d ,a n d e q u a t io n (3 6 ) sh o ws th e e f f e ct o f t im e , t , a n dd i s t a n c e r , o n t e m p e ra tu re , T , fo r a sp h e r i c a l p a r t i c l e( sp h e r i c a l c o o rd in a t e s )

OT k [02T 20T '~O t = p C p ~ r Z + r - ~ r ) (36)

w h e r e k , p , a n d C a r e t h e t h e r m a l c o n d u c t i v i t y ,d e n s i t y , a n d h e a t c a p a c i t y , r e sp e c t i v e ly . Th e i n i t i a lc o n d i t i o n s a r e

T = 3 2 0 0 K a t t = 0 f o r r < a , a n dT = 1600 K a t t = 0 for r > a ,

wh e re a i s th e r a d iu s o f t h e n o d u le , w h ic h c a n b eo b t a i n e d f r o m m e a s u r e m e n t s i n F i g s 4, 5 , a n d 1 0 o fp a r t I [ 1 7 ] . A n a v e r a g e v a lu e o f 2 . 5 / ~ m w a s t a k e n i nth e c a l c u l a ti o n s , a n d v a lu e s fo r k , p , a n d C we rea v e r a g e d f r o m N b a n d S i f o r N b S i 2 . F i g u r e 1 3 s h o w st h e r e s u lt s o f t h e c o m p u t a t i o n s u s i n g e q u a t i o n ( 37 ).Th e so l i d i f i c a t i o n h e a t i s n o t i n c o rp o ra t e d i n to t h ec a l c u l a t i o n s . Ne v e r th e l e ss , i t i s c l e a r ly se e n t h a t so -l i d if i c a t io n c a n b e i n i t i a t e d 2 - 5 n s a f t e r t h e sp h e ru l ei s fo rm e d . Th u s , t h e so l i d i f i c a t i o n t im e i s a sm a l lf r a c t i o n ( ~ 1 /1 0 00 ) o f t h e sh o c k p u l se d u ra t i o n t im e .I t i s th e re fo re e a sy t o se e h o w su c c e ss iv e g e n e ra t i o n so f n o d u le s c a n b e fo rm e d , so l id if ie d , a n d e j e c te d f ro mth e r e a c t i o n i n t e r f a c e .

Th e fo rc e s t h a t c a n b e e x e r t e d b y t h e l i q u id Nb S i2o n th e so l i d Nb S i2 sp h e ru l e c a n b e c a l c u l a t e d i n a na p p r o x i m a t e m a n n e r . T h e s e f o r c e s a r e d u e t o t h ein t e r f a c i a l e n e rg y b e twe e n l i q u id Nb S i2 a n d l i q u id S ia s a r e su lt o f t h e li q u id Nb S i2 a t t e m p t in g t o b e c o m esp h e r i c a l . A l i q u id sp h e re c a n b e f l a t t e n e d i n to ap ro l a t e sp h e ro id b y a fo rc e F , a s sh o w n in F ig . 1 4 (a) .I t i s possib le to ca lcu la te th is force F , due exc lusive lyto in te rfac ia l energy , ~ 's

d E d SF = ~ = ~s ~yy. (37)

(a )

( c )

( e )

( g )

S i ( L )

Nb S )

~ Reaction

Forces mechanical)

( b )

NbSi (L )~ 2

N b S i

(d )NbSi NbS

( 0 .

NbSi (L )2

(L )

Fig. 12. Sequen ce of events in the synthesis of NbSi2 spherules at N b(solid)-Si(liquid) interface undershock loading; (a-c) n ucleation and grow th of thin layer; (d) interracial energy produces sph eroidization;(e) solidification and reduced reaction; ( 0 ad jacent reaction reg ions form in ne wl y exposed interface; (g)spheroidization and solidification of adjacent reaction region; (11) expulsion o f spherule.


13/15

MEYERS e t a l . : SHOCK SYNTHESIS OF SILICIDES--II 727S is the surface area of the spheroid, equal to

27 tabS = 2~zb2 + - sin - I E. (38)E

Here a and b are the major and minor axes of thespheroid and E is the eccentricity, defined as

~j x2,] . (39)The volume of the spheroid is constant and equalto (4/3)Tra2b. By substituting equations (38) and(39) into (37) and el iminat ing the vari able a, onearrives at

3~kF = Ys 4~zy - (1 - k -2y3 )1 /2y 1 /2r n k y - -1 /2 "1- 2rrk 1y5/2-] . 1 k - 2 y 3 ) l / 2 }+ / Jsm (,-

(40)where k = r 1 / 2 and r is the initial spherule radius.

The interfacial energy for the NbSi2 is not knownbut the surface energy of liquid NbSi2 can be esti-mated from the values of Nb a nd Si, by interpolation.These values are given by M urr [29]: ?Nb = 1.9 J/m Eat2473oc; ys i= 0.73 J/m 2. By inter polat ion on a massfraction basis; one obtains: yNbSi2= 1.46 j /m 2. Closeinspect ion of Table 3.6 from Mu rr [29] shows thaty ~ (0.8-1.2) x 10 -3 Tm for a large num ber of metals.Since the melting point of NbSi: (2420 K) substan-

tially exceeds that of Si (1685 K) and is close to themelting point of Nb (2740 K), the above estimate isrealistic.

Figure 14(b) shows the var iati on of F with distance,y, for a spherical particle with radius equal to 1/~m(V =4/ 3ng m3) . Note that the value of Fr betweeny = 0.9 and y = 1 #m is shown by a dashed line. Theformulation breaks down at high values of y. Theforce exerted by a prolat e spheroi d of r = 1 #m isapproximately equal to 1.5 10-sys.

From this value it is possible to estimate the totalforce exerted on a spherule; Fig. 14(c) shows a solidspherule surround ed by liquid prolate spheroids. Onecan assume that n spheroids surround one spherule.Thus, the total force is: F T _~ nF . The stress exerted bythese spherules is approximately equal to

n F 1.5 10-Sny~a = 4n r~ - 4nrl (41)

where r I is the contact area radius, determ ined by therelative int erfacial energies [see Fig. 14(c)]. It ispossible to estimate the stresses produced in solidspherules by these surface (interfacial) tensions. Anaverage value of 2 10-sa~ is assumed for F; r~ isassumed to be equal to r / 2 ; and one considers thateach spherule is surrou nded by four prola te spheroidsof liquid silicide. Thus, a = 28 MPa.

This stress is of the order of the strength of thesilicide. Thus, the stresses can conceivab ly eject

L

L 2 5 0 0 , 0

[ . . ,2 0 0 0 ,

1 6 8 5 " " " " -~1 6 z 16 " " " - > ~ " - 8 5

, s , \ ' ~~ ' ~ " 2 " 1 0I 110 . !

Fig. 13. Heat extraction from spherule with initial temperature of 3200 K while surroundings are at1600 K.


14/15

7 2 8 M EYERS et al.: S H O C K S Y N T H E S I S O F S I L I C I D E S - - II

FY

. ~ . . s p h e r es p h e r o i d

X

t~

00. $

#

- i - , - i -0 .6 0 .7 0 .8 0 .9Y ( m i c r o m e t e r )b

1. 0

Solid

~ lquldI I

r I

Fig. 14. (a) Prolate spheroid an d force F exerted b y surfaceenergy as i t is deformed, (b) force, F, as a funct ion ofdistance y, for spherical ra dius r = 1/zm, (c) forces, F,exerted by l iquid spheroids on sol id spherule.

t h e s o l i d s p h e r u le , a n d t h e m e c h a n i s m p r o p o s e d i sr e a l i s t i c .T u r b u l e n t f l o w o f l i q u id s i l ic o n u n d e r s h o c k c a na l s o c o n t r i b u t e t o t h e d e t a c h m e n t o f t h e s p h e r u l e sf r o m t h e i n t e r f a c e ; h o w e v e r t h i s m e c h a n i s m i sd i ff ic u lt t o q u a n t i fy . U n d o u b t e d l y t h e d r a g e x e r t ed b yt h e l i q u i d s i l i c o n a l s o p l a y s a r o l e , a s w e l l a s t h e r m a ls t r e s s e s a n d s t r e s s c o n c e n t r a t i o n s a t t h e s p h e r u l e -s u b s t r a t e ( N b ) i n t e r f a c e . N o n e o f t h e s e e f fe c t s h a v eb e e n c o n s i d e r e d i n t h i s a n a l y s i s ; h o w e v e r , t h e y a l lc o n t r i b u t e t o m o r e r a p i d d e t a c h m e n t o f th e N b S i 2s p h e r u le s f r o m t h e s o l i d N b i n t er f a c e, t h e r e b y m a k i n gn e w s o l i d - l i q u i d i n te r f a c e a v a i l a b l e fo r s u b s e q u e n tr e a c t i o n .

o f a t h r e s h o l d e n e r g y c o n c e p t b e l o w w h i c h n o r e a c-t i o n t a k e s p l a c e e n a b l e d t h e d e t e r m i n a t i o n o f t h ec r i t ic a l p r e s s u r e f o r r e a c t i o n a s a f u n c t i o n o f p o w d e rd e n s i t y .

2 . T h e f r a c t i o n o f s i l ic o n m e l t e d a s a f u n c t i o n o fs h o c k e n e r g y w a s c a l c u l a t e d a n d i t is f o u n d t h a t t h et h r e s h o l d e n e r g y c o r r e s p o n d s t o a m o l t e n f r a c t io n i nt h e r a n g e 0 . 2 - 0 . 5 f o r t h e N b - S i s y s t e m .

3 . H e a t t r a n s f e r c a l c u l a t i o n s w e r e c o n d u c t e d i na n e f f o r t t o e s t a b l i s h a c r i t i c a l m o l t e n r e g i o n s i z ef o r w h i c h t h e h e a t g e n e r a t e d b y t h e r e a c t i o n e x c e e d st h e e n e r g y d i s s ip a t e d to t h e e n v i r o n m e n t . A s s u m i n gs p h e r i c i t y , t h e s e c r i t i c a l r a d i i w e r e f o u n d t o b ee q u a l t o 3 a n d 4 / z m f o r t h e N b - S i a n d M o - S is y s t e m s , r e s p e c t i v e ly . T h e s e s iz e s a r e q u a l i t a t i v e l yc o n s i s t e n t w i t h o b s e r v a t i o n s ; t h e s m a l l e r c r i t i c a ls h o c k e n e r g y f o r r e a c t i o n o c c u r s i n t h e s y s t e m w i t ht h e l a r g e r h e a t o f r e a c t i o n a n d l o w e r t h e r m a lc o n d u c t i v i t y .

4 . R e a c t i o n k i n e t i c s c a l c u l a t i o n s w e r e p e r f o r m e du s i n g a n A r r h e n i u s e q u a t i o n w i t h f o r w a r d a n dr e v e r s e r a t e s a n d m o d i f i e d t o i n c o r p o r a t e t h e e f f e c t so f s h o c k c o m p r e s s io n . T h e r e a c t io n r a t e i n c r e a s e th a tc o u l d c o n c e i v a b l y b e p r o d u c e d b y s h o c k c o m p r e s s i o np r o c e s s in g ( a s s u m i n g t h e s a m e m e c h a n i s m ) i s m u c hl o w e r t h a n t h e o n e o b s e r v e d i n a c t u a l e x p e r i m e n t s ,l e a d i n g t o t h e c o n c l u s i o n t h a t s h o c k c o m p r e s s i o ni n d u c e s d i f f e r e n t r e a c t i o n m e c h a n i s m s .

5 . T h e g l o b a l r e a c t i o n k i n e t i c s w e r e c a l c u l a t e d b yc o n s i d e r i n g a n u m b e r o f i n i t i a t i o n s it e s p e r u n i tv o l u m e t h a t i s a f u n c t i o n o f p o w d e r s i z e. T h i sf o r m u l a t i o n e n a b l e s t h e p r e d i c t i o n o f t h e f r a c t i o nt r a n s f o r m e d a s a f u n c t i o n o f t im e a n d p a r t i c l e s iz e .6 . A r e a c t i o n m e c h a n i s m o c c u r r i n g a t t h e ( N bo r M o ) - S i i n t e r f a c e i s p r o p o s e d . T h i s r e a c t i o nm e c h a n i s m i n v o lv e s th e d i s s o l u t io n o f ( N b o r M o )i n t o m o l t e n S i , t h e r e a c t i o n p r o d u c i n g a m o l t e ni n t e r m e t a l l i c , i t s s p h e r o i d i z a t i o n , s o l i d i f i c a t i o n , a n ds u b s e q u e n t e x p u l s i o n i n t o t h e l i q u i d s i l ic o n m e l t. I nt h i s r e a c t i o n m e c h a n i s m a f r e s h s o l i d ( N b o rM o ) - l i q u i d ( S i ) i n t e r f a c e i s c o n t i n u o u s l y m a i n t a i n e d ,e n a b l i n g a h i g h r e a c t i o n r a t e .A c k n o w l e d g e m e n t s - - T h i s research was supported byNa t i o n a l Sc i e n c e Fo u n d a t i o n Gr a n t DM R 8 7 1 3 2 5 8(Materials Processing Ini t iat ive) , and by McDonnell-Douglas Research Labora tory . We thank the suppor tp r o v id e d b y Dr B . M c Do n a l d ( NS F) a n d b y D r s C . W h i t s e ttand P. Meschter (MDRL). Discussions with Professor I .Glas sman , Princeton Universi ty, w ere extremely helpful inarr iving at the mechanism for spherule formation andsolidif icat ion. The interact ion with Dr R. A. Graham,Sandia Nat ion a l L abora tor ies , th rough severa l d iscuss ionshas provided mot iva t ion for th i s work and cont r ibutedsignif icant ly to i ts approach.

6 . C O N C L U S I O N S1 . T h e t h e r m o d y n a m i c s o f s h o c k s y n th e s i s w a sa n a l y z e d b y c o n s i d e r i n g a n e q u a t i o n o f s t a te f o r t h ep o r o u s i n e r t a n d r e a c ti v e m i x t u r e s. T h e a p p l i c a t i o n

R E F E R E N C E S1. C. S. Sm ith, T r a n s . M e t a l l . S o c . A I M E 2 1 2 , 5 7 4

(1958).2 . E. Hornbogen, A c t a m e t a l l . 10, 978 (1962).3. M. A. Meyers, Scr ip ta me ta l l . 12, 21 (1978).


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M E Y E R S e t a l . : S H O C K S Y N T H E S I S O F S I L I C I D E S - - I I 7 294 . J . W e e r t m a n , i n Shock Waves and High-S t ra in -RatePhenomena in Metals ( e d i t e d b y M . A . M e y e r s a n dL . E . M u r r ) , p . 4 6 9 . P l e n u m P r e s s , N e w Y o r k .5 . M . A . M o g i l e v s k i , Phys . Rep . $7, 359 (1983).6 . M. A. M ogi levsk i , in Shock Waves and High-S t ra in -Rate Phenomena in Metals ( e d i t e d b y M . A . M e y e r s a n dL . E . M u r r ) , p . 5 3 1 . P l e n u m P r e s s , N e w Y o r k .7 . J . W e e r t m a n a n d P . S . F o l l a n s b e e , Mech. Mater . 2 ,

265 (1983).8 . P . S . F o l l a n s b e e a n d J . W e e r t m a n , Mech. Mater . 1 ,345 (1982).9 . W . G o u r d i n , J . appl. Phys. 55, 172 (1984).10 . R. B. Schwarz , P . Kas i ra j , T . Vree land J r and T . J .A h r e n s , Acta metall . 32, 1243 (1984).1 1 . D . K . P o t t e r a n d T . J . A h r e n s , J . appl. Phys. 63 , 910(1988).12 . Y. H or ie a nd M. J . Kipp , 3". appl. Phys. 63, 5718(1988).13 . M. B. Bos lough , J. chem. Phys. 92, 1839 (1990).1 4 . L . H . Y u a n d M . A . M e y e r s , J. M ater. Sci. 2,6, 601(1991).1 5. B . R . K r u e g e r , A . H . M u t z a n d T . V r e e l a n d J r , J. appl.Phys . 70, 5362 (1991).1 6 . B . K r u e g e r a n d T . V r e e l a n d J r , i n S h o c k - W a v e a n dHig h-Stra in-R ate Phenomena in Materials ( e d i t e d b yM . A . M e y e r s , L . E . M u r r a n d K . P . S t a u d h a m m e r ) ,p . 2 4 1 . M a r c e l D e k k e r , N e w Y o r k ( 1 9 9 2 ) .1 7. K . S . V e c c h i o , L . H . Y u a n d M . A . M e y e r s , A c t ametall, mater. 42, 701 (1994).1 8. M . Y o s h i d a , P r o g r a m M Y 1 D L O n e - D i m e n s i o n a l

L a g r a n g i a n H y d r o d y n a m i c C o d e , C E T R . N e w M e x i c oI n s t i t u t e o f M i n i n g a n d T e c h n o l o g y , S o c o r r o , N . M .(1986).1 9. F . R . N o r w o o d a n d R . A . G r a h a m , in S h o c k - W a v e a n dHigh -Strain-R ate Phenomena in Materials (ed i ted byM . A . M e y e r s , L . E . M u r r a n d K . P . S t a u d h a m m e r ) ,p . 9 8 9 . M a r c e l D e k k e r , N e w Y o r k ( 1 9 9 2 ) .20 . L . V. Al t ' s chu le r , S a y . P h y s . - - U S P E K H 1 8, 2 (1965).2 1 . R . G . M c Q u e e n a n d S . P . M a r s h , i n Behavior of DenseMed ia U nder High Dyna mic Pressures, p . 2 0 7 . G o r d o n& B e a c h , N e w Y o r k ( 1 9 6 8 ) .2 2 . M . Y o s h i d a , M i x t u r e P r o g r a m , R e p o r t . C e n t e r f o rE x p l o s iv e s T e c h n o l o g y R e s e a r c h , N e w M e x i c o I n s t i t u t eo f M i n i n g a n d T e c h n o l o g y , S o c o r r o , N . M . ( 19 8 6) .2 3 . I . S o n g , P h . D . t h e si s , N e w M e x i c o I n s t i t u t e o f M i n i n ga n d T e c h n o l o g y , S o c o r r o , N . M . ( 1 9 9 1 ) .2 4 . E . K . R i d e a l a n d A . J . B . R o b e r t s o n , Proc. R. Soe. 195,135 (1948).25 . V. E . Pan in , Yu V. Gr i nvae v , V. E . Egorush k in , I . L .B u e h i n d e r a n d S . N . K u l ' k o v , T r a n s l a t i o n f r o m Izv.vyssh. Zaved. Fiz. 1, 34 (1987).26 . J . D. Eshe lby , Proc. R. Soc. 9A, 376 (1957).2 7 . S . K u m a r , J . A . P u s z y n s k i a n d V . H l a v a c e k , i n C o m -

bustion and Plas ma Synthesis o f High TemperatureMaterials ( e d i t e d b y Z . A . M u n i r a n d J . B . H o l t ) ,C h a p . 2 7 , p . 2 7 3 . V C H P u b l i s h e r s , N e w Y o r k .2 8 . I . G l a s s m a n , P r i n c e t o n U n i v . , p r i v a t e c o m m u n i c a t i o n(1991).29 . L . E . M urr , Interracial Phenomena in Metals and Alloys,p . 131 . Add ison-W es ley (1975) .

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