A Computer Simulation Method for Sintering in Three-dimensional Powder Compacts

8/14/2019 A Computer Simulation Method for Sintering in Three-dimensional Powder Compacts

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JOUR NAL OF MATERI ALS SCI ENCE 29 (1994) 754 761

A c o m p u t e r s im u l a ti o n m e t h o d f o r s in t er in gin t h re e - d im e n s i o n a l p o w d e r c o m p a c t sW . J . S O P P E , G . J . M . J A N S S E N , B . C . B O N E K A M P , L . A . C O R R E I A ,H . J . V E R I N G ANethe r lands Energy Research Fo und at ion ECN, PO Box 1, 1755 ZG Pet ten, The Nethe r lands

A new s imu la t ion method fo r s in te r ing in th ree-d imens iona l powder pack ings is p resen ted .The mode l i s based on two-par t ic le s in te r ing in te rac t ions and is ab le to descr ibe mass t ranspor tmechan ism s based on g ra in -b oun dary d i f fus ion as we l l as on la t t i ce d i f fus ion . I t is sho wn tha t thes imu la t ion method p rov ides an adequate descr ip t ion o f s t ruc tu ra l reorgan iza t ion e f fec ts wh ichoccu r dur ing the ear ly s tage o f s in te r ing . S imu la t ion resu lts a re p resen ted fo r the s in te r ing o fa random pack ing a nd o f some crys ta l l ine - type pa ck ings in wh ich var ious in it ia l de fec ts have beencreated,

1 . In t ro d u c t io nThe modelling of sintering processes has a history thatgoes back more than 45 years [1]. One of the mainchallenges through these years, has been to predict thestructural reorganization effects, which occur duringthe sintering of irregular packings [2-5]. These reor-ganization effects lead to large fluctuations in theamo unt of shrinkage, and therefore have an enormousinfluence on the strength of the sintered material.Reorganization takes place when angles between con-necting lines of particle centres change. The change inangle is typically a many-particle result: the process atone junction influences the net results at another.Note that for this to happen, different necks them-selves need not overlap. Sintering models developedby other authors [6-81 have in common that theparticle packing is treated essentially as an entity inthe analysis of the sintering kinetics. Such a treatmenteither rules out the description of the effects of localinhomogeneities [6] or raises problems of topologicalcharacter [7, 8].The aim of the present work was the modelling ofstructural reorganization effects taking place in theearly stage of sintering. A many-particle simulationmodel is presented, in which the sintering kinetics aredescribed by a summation of local two-sphere interac-tions. This approach ensures that the simulationscheme, in principle, can be applied to any initialpacking and that topological constraints intrinsicallyare obeyed.

The two-sphere interaction model is a description ofthe local mechanism of sintering at one grain-bound-ary area. In the past, many models for the two-sphereinteraction have been proposed. In general, thesemodels are based on thermodynamic and atomisticconsiderations, combining concepts of the drivingforce, i.e. minimization of the free energy of the sur-face, with material transport mechanisms. Well-known are the interaction models suggested by Coble[9] and Kingery and Berg [10] involving mass trans-754

port through grain-boundary diffusion, lattice diffu-sion and surface diffusion. Models based on material,transport through viscous flow and through evapor-ation and recondensation have also been developed.A survey of these models can be found in the reviewpaper by Exner [-11]. In the present simulationscheme, any of these models can be implemented; theonly restriction being that the neck formation iscoupled to centre-to-centre approach:The simulation scheme and its implementation isdescribed first. Next, the simulations of sintering ofa random packing and of some typical crystallinepackings with various initial defects are presented anddiscussed.

2 . M e t h o dConsider a system of N equal spheres, each withradius R. The position of the ith particle at time t isdenoted by r ~ ( t ) . As a result of local interactions withadjoining spheres, the coordinates of a sphere willchange in time. The net displacement of a sphere i isgiven by

d r i ( t ) _ ~ f ( r l j ) ~ i j (1)d t i ~ j

Here, r i j = r i - r j , ~ i j is the unit vector in the directionof r q . The summat ion is restricted to particles j incontact with particle i, i.e. ( r l j ) = 0 for rq > 2R. Thefunetionf(r,-j) represents the local two-particle sinter-ing interaction and will be discussed below. An iter-ative numerical integration scheme with discrete timesteps At is employed to describe the configurat ionalchange during the sintering process_ The time depen-dency of r ~ ( t ) is approximated by the Taylor expansion

1r i ( t + A t ) = r i ( t ) + v i ( t ) A t + ~ a,(0(At) 2 + O(At) 3( 2 )

0 0 2 2 -2 4 6 1 9 1994 Chapman & Hall


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U s i n g t he A d a m s - B a s h f o r t h t w o - s te p m e t h o d f o r in -t eg ra t ion [12 ] , i . e . s e t t ing

a i ( t ) = [ v i ( t ) - v i ( t - A t ) ] / A t (3 )a n d s u b s t i t u t i n g E q u a t i o n 1 i n t o E q u a t i o n 2 , o n eo b t a i n s

r i ( t + At) = r~(t) + A t ~ {3i ~ j ~ f [ r i j ( t ) ]

~ f [ r i j ( t - At ) ]} ~ ,j (4)W e n o w t u r n t o t h e l o c a l t w o - s p h e r e i n t e r a c t i o n

m o d e l i n w h i c h t h e m e c h a n i s m s o f th e s i n t e ri n g a r ei n c o r p o r t e d . I n F i g . 1 a s e c ti o n a l o n g a l i n e b e t w e e nt h e c e n t r e s o f t w o s p h e r e s i s s h o w n . I n g e n e r a l , th en e c k g r o w t h c a n b e w r i t t e n a s a r a t h e r s i m p l e f u n c t i o no f t i m e

R " (5)w h e r e m a n d n a r e e x p o n e n t s . T h e c o n s t a n t C M c o n -t a i n s p h y s i c a l q u a n t i t i e s a s s u r f a c e t e n s i o n , d i f f u s i v it ya n d t e m p e r a t u r e . W i t h t h e a p p r o x i m a t i o n h = p Z / 4 Rt h i s c a n b e r e w r i t t e n a s

( ~ ) " / 2 - - 2 ~ i m t ( 6 )w h e r e 9 i s t h e n e c k r a d i u s , h i s t h e i n t e r p e n e t r a t i o nd i s t ance and t i s t he s in t e r ing t ime , c f , F ig . 1 . In h i sr e v ie w p a p e r , E x n e r [ 1 1 ] h a s s h o w n t h a t t h e a p p r o x i -m a t i o n h = p 2 / 4 R i s a p p l i c a b l e t o s i t u a t i o n s f a r b e -y o n d t h e l i m i t o f p / R < 0 .3 . Hence , fo r the in i t i a l andi n t e r m e d i a t e s t a g es o f s i n te r in g , t h e a p p r o x i m a t i o n i sva l id .I t is n o w a n e a s y m a t t e r t o d e r i v e a n e x p r e s s i o n f o rt h e l o c a l f u n c t i o n s f ( r ~ j ) . F o r t w o i n t e r p e n e t r a t i n gs p h e r e s w e c a n w r i t e

r l j = 2 R - - 2 h i j (7 )a n d h e n c e

d r i j _ 2 d h i j ( 8 )d t d t

W e h a v e a d d e d t h e s u b s c r i p t i j t o h i n o r d e r t oe m p h a s i z e t h e f a c t t h a t w e a r e d e a l i n g w i t h a p a i rp r o p e r t y . F r o m E q u a t i o n 6 w e f i n d

d h i j _ C Md t n 2 " - 1 R ( 2 m - , ) /2 h i1 ; ,/2 (9)o r

d r~ = C M h i - , /2 (10)d t n 2 " 2 R(2m -n) /2 - - ijW i t h d r o / d t = 2 f ( r i j ) , w e n o w c a n e x p r e s s f [ r i j ( t ) ] int e r m s o f h o

C Mf [ r i j ( t ) ] = - - n 2 , _ l R ( 2 , , _ , ) / 2 h r (11)W i t h t h i s r e s u l t, t h e d e s c r i p t i o n o f t h e s i m u l a t i o ns c h e m e i s c o m p l e t e . S u m m a r i z i n g : a t e a c h g i v e n t i m et t h e v a l u e h ~ j ( t ) i s c a l c u l a t e d f r o m e a c h p a i r o f t o u c h -i n g s p h e r e s . E q u a t i o n s 4 a n d 1 1 t h e n a r e u s e d t ou p d a t e t h e c o o r d i n a t e s a n d t h e w h o l e p r o c e d u r e i sr e p e a t e d f o r t i m e t + A t .

T h e m e c h a n i s m o f v a c a n c y d i f f u s i o n f r o m t h e , ,n ec kz o n e d e t e r m i n e s t h e e x p r e s s i o n u s e d f o r C ~ t , a n d t h ev a l u es f o r t h e e x p o n e n t s m a n d n . C M , n a n d m c a n b ed e r i v e d f r o m t h e o r e t i c a l c o n s i d e r a t i o n s , b u t v e r y o f t e nt h e y a r e o b t a i n e d f r o m e x p e r i m e n t s . R a n g e s f o r t h e see x p o n e n t s , a s w e l l a s p l a u s i b l e v a l u e s f o r C ~ t a r e g i v e ni n t h e r e v i e w b y E x n e r [ 1 1 ] . I n g e n e r a l , s e v e r a l d i f f u -s i o n m e c h a n i s m s w i l l o c c u r a t t h e s a m e t i m e a n da l in e a r c o m b i n a t i o n o f i n t e r a c t i o n m o d e l s m a y b eu s e d . M o d e l s d i s c u s s e d b y E x n e r i n c l u d e g r a i n -b o u n d a r y d i f f u s i o n a n d l a t t i c e d i f f u s i o n . F o r g r a i n -b o u n d a r y d i ff u s io n , w i t h p h y s i c a l c o n s t a n t C ~ , t h ev a l u e s o f t h e e x p o n e n t s a r e m = 4 a n d n = 6 , l e a d i n gto

C G h i ~ 2 ( t ) (12)f [ r i j ( t ) ] - 3.26 RF o r l a t t i c e d i f f u s i o n , w i t h p h y s i c a l c o n s t a n t C L , t h ev a l u e s a r e m = 3 a n d n = 4 , a n d h e n c e

C L h i j l ( t ) . (13)f [ r l j ( t ) ] - 3 2 R

\, /% .,F i g u r e 1 T h e t w o - b o d y s i n t e r i n g m o d e l .

B o t h t h e s e m o d e l s a r e i m p l e m e n t e d i n t h e p r e s e n ts i n te r i n g p r o g r a m . T h e e x t e n s i o n t o o t h e r i n t e r a c t i o nm o d e l s i s s t r a i g h t f o r w a r d .T o s t a r t t h e c a l c u l a t i o n , f [ r l j ( - A t ) ] is a s s u m e d t ob e z e r o . A s i n g u l a r i t y e x i s t s w h e n h ~ i s z e r o a s m a ye a s i l y o c c u r a t t h e s t a r t o f th e s i m u l a t i o n . T h e r e f o r e , i na l l c a l c u l a t i o n s w e r e p l a c e h i j b y ( h ~ + a R ) w h e r e ~ i sa n a r b i t r a r i l y s m a l l n u m b e r . T h e s i m u l a t i o n e n d se i t h e r w h e n a g i ve n m a x i m u m n u m b e r o f t i m e s te p sh a s b e e n a c h i e v e d o r w h e n a t s o m e i n s t a n c e , l o c a l l y ,f o r o n e p a i r i j , a l i m i t i n g v a l u e o f h ~ j R i s exceeded .T h e r e s u l t i n g c o n f i g u r a t i o n t h e n c a n b e a n a l y s e d b yc a l c u l a t i n g v a r i o u s c h a r a c t e r i z i n g q u a n t i t i e s . I n o r d e rt o m o n i t o r c o n f i g u r a t i o n a l c h a n g e s , s e v er a l s t r u c t u r a lc h a r a c t e r i s t i c s s u c h a s t h e o v e r a l l s h r i n k a g e , t h e c o -o r d i n a t i o n n u m b e r , G z, a n d t h e d e n s i fi c a t i o n , z~ , o fe v e r y p a r t i c l e , a r e e v a l u a t e d e v e r y n m t i m e s t ep s . I nt h i s p a p e r , t h e e x t e r n a l s iz e s o f t h e p a c k i n g , d e t e r -m i n e d b y t h e e x t r e m e s o f th e c o o r d i n a t e s o f th e p a r -

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t ic l e s, a re used fo r t he ev a lua t i on o f t he sh r ink age o ft h e p a c k i n g d u r i n g t h e s i n t e r i n g . T h e d e n s i f i c a t i o npar am eter , z l , of par t ic le i , i s def ined as 1-13]

2 n . ~ 1L (14)Z i : G i [ r i j [j = l

w h e r e t h e c o o r d i n a t i o n n u m b e r G i is t h e n u m b e r o fne igh bou rs j , w i th ] rl jl < 2R. Th e l oca l dens i t y , P i,asso cia ted w i th par t ic le i the n i s p~ = poZ~ wh ere Po ist h e i n it i a l b u l k d e n s i t y . T h e a b o v e m e n t i o n e d l i m i t i n gv a l u e o f h i j / R cor re spo nds t o Z l im . By de f in i t i on

RZlim R - h ! ira (15)~tJ

In p rac t i ce , t h i s pa rame te r wi l l be used i n t he s imu-l a t i o n s i n o r d e r t o d e t e r m i n e w h e r e a r u n s h o u l d b es t o p p e d .

3 . D e t a i l s o f t h e s i m u l a t i o n sT h e s i m u l a t i o n s a re s t a r t e d b y b u i l d i n g a t h r e e - d i m e n -s i o n a l c o n f i g u r a t i o n c o n s i s t i n g o f a c l o se p a c k i n g o fseve ra l hund red s o f sphe r i ca l pa r t i c l es . In t h i s pape r ,w e p r e s e n t s i m u l a t i o n s o f r e g u la r , c r y s t a l li n e p a c k i n g sin which de fec t s i n t he fo rm o f i n t e r face s and v acanc i e sa r e in t r o d u c e d , a n d a s i m u l a t i o n o f a r a n d o m p a c k i n g .At t h i s i n i t i a l s t age , e ach pa r t i c l e i s t ouch ing( I r l j l = 2 R ) o n e o r m o r e n e i g h b o u r s . N o w ( a t t i m et = 0) , for each par t ic le i , a l i s t i s made in which a l ln e i g h b o u r i n g p a r t ic l e s j w i t h I r ~ j l -< R N a re s t o red . Int h i s p a p e r , w e h a v e u s e d R N = 2 x ~ R . I n o r d e r t or e d u c e c o m p u t a t i o n a l e f f o rt s , w e d o n o t u p d a t e t h e s el is t s b u t w e a s s u m e t h a t d u r i n g t h e e n t i r e s i m u l a t i o n ,o n l y t h e p a r t ic l e s c o n t a i n e d i n i t s i n i ti a l n e i g h b o u r l i stc an i n t e rac t ( i . e . c an c om e in con t ac t ) w i th a ce r t a inpar t ic le .

T h e n w e s t a r t t h e s i m u l a t i o n o f t h e s i n t er i n g k i n -e t ic s . The n e t d i sp l acem ent o f each p a r t i c l e i i s c a l -c u l a t e d f o r m a n y i t e r a t i o n s t e p s , u s i n g E q u a t i o n 4 .T h e s i m u l a t i o n e x a m p l e s i n t h i s p a p e r a r e r e s t r i c t e d t os i n t e r i n g p r o c e s s e s c o n t r o l l e d b y g r a i n - b o u n d a r y d i f -fus ion ; i . e . u s ing Equa t ion 12 a s an expre ss ion fo r

f ( r i j ) . A n e x p e r i m e n t a l l y o b t a i n e d v a l u e f o r a l u m i n a ,h a s b e e n . . s u b s t i t u t e d f o r C c . A s m e n t i o n e d , w e o n l yhave t o eva lua t e h i j fo r pa r t i c l e s j w h ich a re i n t hein i t i a l ne ig hbo ur l i s t o f pa r t i c l e i , wh ich impl i e s t ha tt h e c o n s u m p t i o n o f c o m p u t e r t i m e in c r e a se s l i n e a r lyw i t h N , t h e t o t a l n u m b e r o f p ar t ic l es .

T h e r e q u i r e d c o m p u t e r t i m e c a n b e r e d u c e d f u r t h e ri f one r ea l ize s t ha t t he s i n t e r i ng ve loc i t y i s kn ow n todec rea se ex pone n t i a l l y . Th i s ve loc i t y dec rea se j us t i f ie sa l i kew i se i nc rea se o f t he t ime s t ep , At, a f t e r e achi t e ra t i on , n , i. e. At ,+1 =A to '1 0 t a" ). Ato has t o bechos en su f f i c i en t ly sma l l i n o rde r t o ob t a in a r e l i ab l es im ula t i o n o f t he c ruc i a l f i r st s t age o f the s i n t e r i ngp r o c es s . T h e v a l u e o f a h a s b e e n c h o s e n s u c h t h a tAt , = 1 s fo r n = 5000 (see Tab le I ). Howe ve r , i t ma yo c c u r t h a t d u r i n g t h e s i n t e r i n g p r o c e s s , n e w c o n t a c t sa re c rea t ed . In o rde r t o p reven t un rea l i s t i c l a rge d i s -p l a c e m e n t s o n b o t h s i d e s o f t h i s n e w g r a i n b o u t l d a r y ,t h e t ime s t ep ( fo r a ll pa r t i c le s ) i s se t back t o A to a f t e rs u c h a n e v e n t . A f t e r s u f fi c ie n t ly m a n y t i m e s t e p s ( t y p i -c a l l y o f t h e o r d e r 1 04 ) t h e s i m u l a t i o n i s t e r m i n a t e d .7 5 6

A ru n o f 20 000 t ime s t eps a nd 450 pa r t i c le s , r equ i reda b o u t 1 0 m i n C P U t i m e o n a C o n v e x C - 2 3 0 co m -pu te r .

T h e c r y s t a ll i n e m o d e l s y s t e m s , w e h a v e s i m u l a t e d i nth i s s t udy , c an be cha rac t e r i zed a s fo l l ows . Bas i c en t i -t i es a re ( a lmos t squa re ) sec t i ons o f t he {0 0 1} p l anes i na f c c c rys t a l , e ach cons i s t i ng o f 45 pa r t i c l e s and wi than ex t e rna l d im ens ion o f 9 x / / 2R x 8 x /~R ( see F ig . 2) . I fA re fe r s t o t he (00 1 ) p l ane and B to t he (002) p l ane ,t hen a s t ack ing sequ ence o f ABA BAB. . . ( and a d i s -t ance be tween the p l anes equa l t o x / / 2R g ives r i se t oa f ace cen t red cub i c ( f cc ) s t ruc tu re ; AA AA A . .. ( andwi th p l ane d i s t ance 2R) y i e lds a s imple cub i c ( sc )s t ruc tu re and . . . .A B A B A B A A A A A A . .. d e s cr ib e s afc c - s c i n t e r face . In t he se s t ruc tu re s , a dd i t i o na l de fec t sa r e c r e a t e d b y r e m o v i n g f r o m t h e c e n t r a l l a y e r o fa r o w o f p a r t ic l e s in e i t h e r t h e ( 1 0 0 ) , ( 0 1 0 ) o r t h e(1 1 0 ) d i r ec t i on .

T h e r a n d o m p a c k i n g t h a t h a s b e e n s i m u l a t e d , w a sg e n e r a t e d b y u s i n g a d e p o s i t i o n m o d e l a n d a c o n -s e c u t i v e c o m p r e s s i o n m o d e l , d e s c r i b e d i n a p r e v i o u sp a p e r [ 1 4] . I n t h e p a c k i n g s g e n e r a t e d b y t h e s e m o d e l s ,t h e p a r t i c l e s d o n o t t o u c h t h e i r n e i g h b o u r s e x a c t l y ;t h e r e fo r e , w e h a v e u s e d f o r p a c k i n g n u m b e r 8 a v a l u efo r ~ t ha t i s s l i gh t l y l a rge r t han fo r t he s imula t i ons o ft h e c r y s t al l in e p a c k in g s . B e c a u s e t h e r a n d o m p a c k i n gi s bu i ld up wi th in a f i n i t e con t a in e r , w a l l e ff ec ts c auset h e m e a n v o l u m e f r a c t i o n o f t h is s t r u c t u r e b e f o r e

T A B L E I N u m e r i c a l va l u e s f o r p a r a m e t e r s u s e d in th i s w o r kC o / R 4 4 .1 x ! 0 - S s - l ae 1 .0 x 1 0 - 3 f o r t h e c r y s t a l l i n e - t y p e p a c k i n g s

5 . 0 x 1 0 - 3 f o r t h e r a n d o m p a c k i n g .A to 1.0 1 0-13 R4/ C G sR n 2 x ~ Ra 2 . 1 0 - 4 1 o g ( 1 / A t o ) ( r e s u l t i n g i n A t . = 1 s f o r n = 5 0 0 0 )n m 4 0 0 0Znm 1.15a E x p e r i m e n t a l l y d e t e r m i n e d , n o n - i s o t h e r m a l b u l k s h r i n k a g e f o ra l u m i n a ( T = 1 5 0 0 ~ t h e c o r r e s p o n d i n g m e a n pa r ti c le s iz e , R ,m e a s u r e d w i t h a M a l v e r n M a s t e r s i z e r , w a s 0 .6 ~ tm .

t6 5

XFigure 2 B a s i c l a y e r A . T h e n u m b e r s r e f e r t o t h e s t r u c t u r e s i n w h i c ht h e c o r r e s p o n d i n g p a r t i c l e s h a v e b e e n r e m o v e d f r o m t h e c e n t r a ll a y e r .


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T A B L E I I C h a r a c t e r i s t i c s o f t h e m o d e l s t r u c t u r e s u s e d i n t h i s w o r kS t r u c t u r e S t a c k i n g s e q u e n c e D e f e c ts a N u m b e r o f E x t e r n a l s iz e ( R 3 ) G ~ . .

{0 0 1} p l a n e s p a r t i c le s X x Y x ZM e a n v o l u m e f r a c t i o n b

1 A A A A A A A A A A N o n e 4 5 0 1 4 .7 2 8 x 1 3 .3 1 4 x 2 0 . 0 00 5 . 0 00 0 0 . 4 8 0 62 A B A B A B A B A B N o n e 4 5 0 1 4 .7 2 8 x 1 3 .3 1 4 x 1 4 .7 2 8 9 . 6 4 0 0 0 . 6 52 73 A A A A A B A B A B I 4 5 0 1 4 .7 2 7 x 1 3 . 3 1 4 x 1 7 .6 5 7 7 . 0 6 22 0 . 5 4 4 44 A B A B A A A A A A B A B A B I 6 7 5 1 4 .7 2 8 x 1 3 .3 1 4 x 2 5 . 31 4 7 . 81 6 3 . 0 . 5 6 965 A B A B A A A A A A B A B A B I + 5 1 11 0 ] 6 7 0 1 4 .7 2 8 x 1 3 .3 1 4 x 2 5 . 31 4 7 . 8 0 60 0 . 5 6 5 46 A B A B A A A A A A B A B A B I + 9 1 0 10 ] 6 6 6 1 4 .7 2 8 x 1 3 . 31 4 x 2 5 . 3 1 4 7 . 8 19 8 0 . 5 6 2 07 A B A B A A A A A A B A B A B I + 5 11 00 ] 6 7 0 1 4.7 28 x 13 .314 x 2 5 . 3 1 4 7 . 8 1 7 9 0 . 5 6 5 48 R a n d o m p a c k i n g 1 0 0 0 1 7 . 0 00 x 1 7 . 0 0 0 x 2 9 . 7 9 6 3 . 4 1 2 0 0 . 4 8 6 4a T h e d e f e c t s re f e r to a n i n t e r f a c e ( I ) o r t o a m i s s i n g r o w o f p a r t ic l e s i n t h e l a y e r t h a t i s c e n t r a l in t h e Z d i r e c t io n . 5 11 1 0 ] i m p l i e s th a t i n l h a tl a y e r , s t a r t i n g f r o m ( X , Y ) = ( 0, 0 ), a r o w o f f iv e p a r t i c l e s i n t h e [ 1 1 0 ] d i r e c t i o n a r e m i s s i n g . ( S e e a l s o F i g . 2 . )b T h e m e a n v o l u m e f r a c t i o n i s d e t e r m i n e d b y t h e e x t e r n a l d i m e n s i o n s o f t h e s i m u l a t i o n b o x . O w i n g t o f i n it e si z e e f f ec t s, t h e s e v a l u e s a r es m a l l e r t h a n f o r b u l k m a t e r i a l s . ( F o r s c s t r u c t u r e s c f 1 , t h e b u l k v a l u e i s 0 . 5 2 4 a n d f o r f c c s t r u c t u r e s c f 2 , t h e b u l k v a l u e i s 0 . 7 4 0 .

s i n t e r i n g t o b e r a t h e r m o d e s t : 0 .4 9. T h e v a l u e o f t h em e a n v o l u m e f r a c t i o n i s f u r t h e r r e d u c e d b y t h e c i r -c u m s t a n c e t h a t t h e m e t h o d o f c o m p r e s s i o n i s l es se f fi c ie n t a t t h e t o p o f th e p a c k i n g t h a n a t t h e b o t t o m .I n t h e l o w e r r e g i o n s o f th e p a c k i n g , d e n s i ti e s u p t o0 . 6 0 c a n b e o b s e r v e d .

I n T a b l e I I , t h e p r e - s i n t e r i n g c h a r a c t e r i s t i c s o f t h em o d e l s t r u c t u r e s t h a t w e h a v e s i m u l a t e d , a r e b r i e f l yp r e s e n t e d . I t s h o u l d b e n o t e d t h a t t h e e x t e r n a l s i z e s o ft h e p a c k i n g s , p r e s e n t e d i n t h i s t a b l e , a r e u p p e r l i m i t sd e t e r m i n e d b y t h e e x tr e m e s o f t h e X , Y a n d Z c o o r -d i n a t e s o f t h e p a r t i cl e s a n d t a k i n g i n t o a c c o u n t t h er a d i i o f t h e p a r t i c le s . C o n s e q u e n t l y t h e v o l u m e f ra c -t i o n s g i v e n c a n b e c o n s i d e r e d a s l o w e r l i m i t s o f t h eb u l k v a l u e s .

4 . R e s u l t s a n d d i s c u s s i o nT h e s i n te r i n g s i m u l a t i o n o f e a c h p a c k i n g , d e s c r i b e d i nT a b l e I I , h a s b e e n c o n t i n u e d u n t i l s o m e w h e r e i n t h es t ru c tu re the l imi t ing v a lue o f z (Zllm = 1 .15 ) ha s bee na c h i e v e d . T h e c r y s t a l l i n e t y p e s t r u c t u r e s t y p i c a l l y n e e da f e w t h o u s a n d t i m e s t e p s t o r e a c h Z i im . D u r i n g t h e s es i n t e r i n g p r o c e s s e s , n o n e w p a r t i c l e c o n t a c t s a r e c r e -a t e d . T h e r a n d o m p a c k i n g , h o w e v e r , s h o w s a r a t h e rd i f fe r e n t b e h a v i o u r . R i g h t f r o m t h e b e g i n n i n g o f th es i n t e r i n g , n e w p a r t i c l e c o n t a c t s a r e c r e a t e d . B e c a u s eAt i s s e t back to i t s i n i t i a l va lue , A to , e ach t ime a newp a r t i c l e c o n t a c t a r i s e s , a l a r g e r n u m b e r o f t i m e s t e p s i sr e q u i r e d t o s i m u l a t e t h e s a m e r e a l s i n t e r i n g t i m e a s f o rt h e c r y s t a l l i n e ty p e p a c k i n g s . I n f a c t , t h e s i m u l a t i o n o ft h e r a n d o m p a c k i ng r e q u i r e d a n a m o u n t o f 3 7 00 0 0t ime s t eps .

T h e o t h e r s i n t e r i n g c h a r a c t e r i s t i c s a l s o s h o w a q u i t ed i f f e r e n t b e h a v i o u r f o r t h e c r y s t a l l i n e t y p e s t r u c t u r e st h a n f o r t h e r a n d o m p a c k i n g . T h e s e d i f fe r e n ce s a r ed u e t o t h e f a c t t h a t t h e c r y s t a l l i n e t y p e p a c k i n g s c o n -s i st o f o n e l a r g e c l u s t e r o f c o n n e c t i n g p a r t i c l e s , w h e r et h e r a n d o m p a c k i n g , d u e t o i t s f o r m a t i o n h i s t o r y ,c o n s i s t s o f m a n y c l u s t e r s o f v a r i o u s s iz e . C o n s e -q u e n t l y , a d i s p l a c e m e n t o f a p a r t i c l e i n a c r y s t a ll i n et y p e p a c k i n g w i ll a f f e ct t h e p o s i t i o n o f a l l o t h e r p a r -t ic le s o f t h a t p a c k i n g , b u t i n t h e r a n d o m p a c k i n g , t h ee f f ec t o f a d i s p l a c e m e n t o f a p a r t i c l e i s v e r y l o c a l : it i sr e s t r i c t e d t o t h e o t h e r p a r t i c l e s o f t h e s a m e c l u s t er .

I n T a b l e I I I , t w o t y p i c a l s i n t e r i n g c h a r a c t e r i s t i c s a r ep r e s e n t e d : t h e r e a l s i n t e r i n g t i m e o f th e s i m u l a t i o n sa n d t h e v o l u m e s h r i n k a g e . I n o r d e r t o c o m p a r e t h es i n t e r i n g t im e s o f t h e p a c k i n g s , o n e s h o u l d k e e p i nm i n d t h a t t h e s e s i n t e r i n g t i m e s , i n f ac t , a r e d e t e r m i n e db y t h e f a s te s t c o n t r a c t i n g p a i r s i n t h e p a c k i n g . T h e n e td i s p l a c e m e n t o f a p a r t i c l e ( se e E q u a t i o n 1 ) w i ll b el a r g e i f t h e p o s i t i o n o f i ts n e i g h b o u r s h a s a l o w m i r r o rs y m m e t r y w i t h r e s p ec t t o a n y p l an e t h r o u g h t h e p a r -t i c l e i t se l f . In reg u la r c ry s t a l l ine pack in gs , s uch pa r -t ic l es a r e f o u n d n e a r t h e s u r f a c e b u t i n th e n o n - r e g u l a rp a c k i n g s ( 3 - 7 ) s u c h p a r t i c l e s a r e a l s o f o u n d i n t h ev i c i n i t y o f t h e i n t e r f a c e s o r n e a r v a c a n c y d e f e c t s . I nt h e r a n d o m p a c k i n g , it is v e r y p r o b a b l e t h a t t h e p a i rsw i t h t h e f a s t e s t c o n t r a c t i o n a r e p r e s e n t : i s o l a t e d c l u s -t e r s c o n s i s ti n g o f t w o p a r t i c l e s . T h e s e p a i r s a r e r e s p o n -s i b le f o r t h e v e r y s h o r t s i n t e r i n g t i m e o f th i s s t r u c t u r e .L a t e r i n t h is p a p e r , w e w i l l p r o v i d e s o m e e v i d e n c e f o rt h e e x i s t e n c e o f t h e s e s m a l l c l u s te r s . F o r t h e s c p a c k -i n g , a p a r t i c l e a t t h e s u r f a c e e x p e r i e n c e s o n l y a : n e ti n w a r d i n t e r a c t i o n f r o m o n e p a r t ic l e , w h e r e f o r t h e f c cp a c k i n g t h i s n u m b e r i s 4 . T h i s e x p l a i n s t h e d i f f e r e n c ei n t h e s i n t e r i n g t i m e s o f s t r u c t u r e s 1 a n d 2 . T h e i n t r o -d u c t i o n o f a n i n t e r f a c e ( s t r u c t u r e s 3 a n d 4 ) c r e a t e sp a r t ic l e s w i t h e v e n m o r e n e t i n w a r d i n t e r a c t in g n e i g h -b o u r s ; t h e s e p a r t i c l e s c a n b e f o u n d a t t h e s u r f a c e o f t h ef c c s i d es o f t h e i n t e r fa c e . T h e s e i n t e r f a c es , t h e r e f o r e ,f u r t h e r d e c r e a s e t h e s i n t e r i n g t i m e . A s t h e v a c a n c yd e f e c ts , i n t r o d u c e d i n s t r u c t u r e s 5 - 7 , a r e p r e s e n t i n t h eT A B L E [ I I V o l u m e sh r i n k a g e a n d s i n t e ri n g t i m e s o f t h e m o d e ls t r u c t u r e sS t r u c t u r e L e n g t h s h r i n k a g e a ( % ) S i n t e r i n g t i m e ( s )

X Y Z1 94 .2 94 .0 94 .5 28 .12 81 .7 80 .2 81 .7 16 .43 84 .2 83 .5 84 .3 11 .54 84.3 83.7 88.1 11.55 87.1 86.3 90.3 11.56 86 .0 85 .9 90 .1 11 .57 88 .4 87 .1 91 .5 9 .68 99 .9 99 .9 99 .9 3 .0a T h e l a r g e s t d i m e n s i o n s a r e t a k e n i n t o a c c o u n t .

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s c l ay e r s, t h e y d o n o t c o n t r i b u t e v e r y m u c h t o t h ed i s p l a c e m e n t o f t h e p a r t i c l e s w i t h t h e l a r g e s t i n w a r dm o t i o n b e c a u s e t h e s e p a r t i c l e s s ti ll c a n b e f o u n d i n t h ef c c l a y e r s . T h i s i s r e f l e c t e d i n n e t s i n t e r i n g t i m e s w h i c ha r e a l m o s t t h e s a m e a s f o r s t r u c t u r e s 3 a n d 4 .

A s m e n t i o n e d a b o v e , t h e s h o r t s i n t e r in g t i m e f o r t h er a n d o m p a c k i n g i s m o s t p r o b a b l y d u e t o t h e p r e s e n c eo f s m a ll , i s o l a t e d c l u s t e r s i n t h i s s t r u c t u r e . T h e s e i s o -l a t e d c l u s t e r s , w h i c h w i l l a l s o b e p r e s e n t n e a r t h es u r f a c e o f t h e p a c k i n g , a r e a l s o r e s p o n s i b l e f o r t h es m a ll s h r i n k a g e o f t h e t o t a l v o l u m e a s w e h a v e d e f i n e di t. B e c a u s e o f t h e i r i s o l a ti o n , t h e n e t i n w a r d d i s p l a c e -m e n t o f t h e s e c l u s te r s w i l l b e v e r y s m a l l , k e e p i n g t h eo u t e r v o l u m e o f t h e w h o l e p a c k i n g a l m o s t u n c h a n g e d .T h e b e h a v i o u r o f t h e v o l u m e s h r i n k a g e o f th e r e g u l a rc r y s t a l l i n e p a c k i n g s ( 1 a n d 2 ) i s i n a c c o r d a n c e w i t ht h e ir m e a n c o o r d i n a t i o n n u m b e r . T h e v o l u m e s h r i n k -a g e o f p a c k i n g 3 is a m i x t u r e o f th e s h r i n k a g e o f 1 a n d2 , w h e r e i t s h o u l d b e t a k e n i n t o a c c o u n t t h a t t h es h r i n k a g e i n t h e s c l a y e r o f t h e p a c k i n g i s e n h a n c e d b yt h e p r e s e n c e o f t h e f c c l a y e r , w h i c h i n i t s t u r n h a sa s o m e w h a t l o w e r s h r i n k a g e t h a n a r e g u l a r f c c p a c k -i n g . F o r p a c k i n g 4 , a s i m i l a r r e a s o n i n g c a n b e m a d e ,e x c e p t f o r t h e s h r i n k a g e i n t h e Z d i r e c t i o n . H e r e , i t c a nb e n o t i c e d t h a t t h e p r e s e n c e o f t w o f c c l a y er s , o n e a tt h e t o p o f a n d o n e b e l o w t h e s c l a y e r , r e s u l ts i na n e a r l y z e r o c o n t r a c t i o n o f t h e s c l a y e r i n t h e Z d i r e c -t i o n . F i n a l l y , t h e v o l u m e s h r i n k a g e o f p a c k i n g s 5 ,6 a n d 7 s h o w t h a t t h e i n t r o d u c t i o n o f d e f ec t s i n th ef o r m o f va c a n c ie s , l o ca l l y m a y e n h a n c e t h e s h r i n k a g eb u t h a s a d e t r i m e n t a l e f f e c t o n t h e s h r i n k a g e o f th eo u t e r v o l u m e .

A n o t h e r i n t e r e s t in g p a r a m e t e r o f t h e s i n t er i n g p r o -cess is 2 . . . . . I n F i g . 3 t h i s p a r a m e t e r i s p l o t t e d a sa f u n c t i o n o f t h e s i n t e r i n g t i m e f o r a ll s i m u l a t e d s t r u c -t u r es , z . . . . b e t t e r t h a n t h e v o l u m e s h r i n k a g e a s w eh a v e d e f i n e d i t , i n d i c a t e s t h e c o n t r a c t i o n o f t h e netv o l u m e o c c u p i e d b y a p a c k i n g . I t c a n b e o b s e r v e d t h a tt h e m i n i m u m r e a l v o l u m e s h r i n k a g e i s o b t a i n e d f o rt h e s c p a c k i n g ( 1) w h e r e t h e f c c s t r u c t u r e ( 2) y ie l d s t h em a x i m u m r e a l v o l u m e s h r i n k a g e . T h e n e t v o l u m es h r i n k a g e o f th e o t h e r p a c k i n g s, i n c l u d in g t h e r a n d o m

p a c k i n g , i s s o m e w h e r e b e t w e e n t h e s e t w o e x t r e m e s .T a k i n g t h e s c p a c k i n g ( 1) as a r e f e r e n c e , w e s e e t h a tt h e r a n d o m p a c k i n g h a s a h i g h e r Z , ,e an w h e r e i ts v o l -u m e s h r i n k a g e i s s a m l l e r . T h i s i n d i c a t e s t h a t i n t h er a n d o m p a c k i n g , t h e p o r e s h a v e g r o w n b i g ge r . T h i s isa l s o t h e c a s e f o r p a c k i n g s 5 , 6 a n d 7 , f o r w h i c h t h ev o l u m e s h r i n k a g e i s s m a l le r t h a n f o r t h e v a c a n c y - f re ep a c k i n g s 3 a n d 4 , b u t w h i c h h a v e a z .. . t h a t i ss l igh t ly l a rge r .

A s m e n t i o n e d a l r e a d y , t h e m e a n c o o r d i n a t i o n n u m -b e r o f th e r a n d o m p a c k i n g i n c r e a se s d i r e c t ly f r o m t h eb e g i n n i n g o f t h e s i m u l a t i o n . I n F i g . 4 , G .. . i s p l o t t e dv e r s u s t h e s i n t e r i n g t i m e . I t c a n b e o b s e r v e d t h a tG . . . . a f t e r a f a s t i n i t ia l i n c r e a s e , g r o w s r a t h e r s l o w l ya n d e v e n s e e m s t o g o i n t o s a t u r a t i o n f o r s i n t e r i n gt i m e s b e y o n d t h e p r e s e n t s i m u l a t io n t im e . A n y w a y , itis r a t h e r u n l i k e l y t h a t t h e m e a n c o o r d i n a t i o n n u m b e rf o r t h i s r a n d o m p a c k i n g w i l l e v e r e x c e e d a v a l u e o f 4 .0 .T h i s i n d i c a t e s t h a t t h i s r a n d o m p a c k i n g , a f t e r s i n t e r -i n g , c o n s i s t s o f m a n y i s o l a t e d c l u s t e r s o f p a r t i c le s . I nF i g . 5, t h e d i s t r i b u t i o n o f t h e p a r t i c l e s i n t h e r a n d o m

3 . 8 03 . 7 53 . 7 03 . 6 53 . 6 03 . 5 53 . 5 03 . 4 53 . 4 0 0 . 0

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Figure 5 D i s t r i b u t i o n o f c o o r d i n a t i o n n u m b e r s i n th e r a n d o mp a c k i n g b e f o r e a n d a f t e r s in t e r i n g c o m p a r e d w i t h t h a t f o r t h e s cs t r u c t u r e ( 1 ) .


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Figure 6 Local dens i ty prameter , z . (a) The ver t ical diagonal plane of s tructure 5, (b) the middle XZ plane of s t ructure 6, and (c) the middle YZplane of s t ructure 7.

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Figure 6 (continued)

p ack i n g , acco r d i n g t o t h e i r co o r d i n a t i o n n u m b er i sshown. In o rder to a l low a c om par iso n wi th the s cs t ruc tu re (1 ), we a l so p lo t ted the data beg inn ing to th i ss t ruc tu re . The majo r i ty o f the par t ic les in the s c s t ruc-tu re , o f cour se , has a coord idna t ion n um ber o f 6 ;par t ic les w i th lower coord inat ion can be found a t thesur face o f the pack ing . The r and om pack ing showsa s ign i fican t amou nt o f par t ic les w i th one o r twoneares t ne ighbour s . Fur ther , i t can be observed tha t ,even af ter s in ter ing , no par t ic les w i th coord inat ionnum ber h igher than 8 are p resen t . Again , th i s ind ica testhat th i s pack ing cons is ts o f many smal l c luster s.

F inal ly , some fea tu res o f the s in tered s t ruc tu res w i l lbe discussed quali tat ively. In Fig. 6a, the density para-meter , zl , in the ver t ical dia gona l (X = Y) plan e ofs t ructu re 5 i s shown. Th is p lo t has been ob ta ined byimp osing a 100 x 200 gr id on this plane. Eac h gr idpo in t has been ass igned the z va lue o f the neares tpar t ic le i f i t is over lappe d by o ne o r more par t ic les andreceives the value 0 i f i t i s no t covered by any par t ic le .Before s in ter ing , the par t ic les in the second and the760

f o u r t h ( f cc ) l ay e r f r o m t h e b o t t o m an d f r o m t h e t o pd id no t cu t the dep ic ted p lane bu t in the s in tereds t ructu re the par t ic les have moved th rough th i s p laneand what we ac tual ly see in the second an d the four thlayer o f the s tack ing are the cross - sec tions o f the necksof these par t icles . The ro w of vacancies is s i tuated inthe dep ic ted p lane and thus appear s as a gap . I t can beobserved tha t , dur ing the s in ter ing , th i s gap haswidened and th i s w iden ing i s mos t enhanced a t thesur face o f the pack ing . T he local dens i ty , as cou ld beexpected , increases go ing f rom the cen t r e to the su r -f ace o f the pack ing bu t in the th i rd p lane f rom thebo t tom and in the th i rd p lane f rom the top , somepar t ic les w i th an ex t r aord inary dens i ty can be ob-served . These p ro v ide a n ice exam ple o f the ef f ects tha tlocal defects ( in this case: a row of f ive vacancies) canhave on the s in ter ing behav iour o f the bu lk ma ter ia l inthe ne ar vicini ty of the defects . Fig. 6b represe nts themidd le X Z plane of s truc ture 6 af ter s inter ing. Thesmall gap at the r ight s ide of the s tru cture is due toa vacancy in the midd le o f a row of n ine , perpen d icu lar


8/8

Figure 7 Cross -sect io n of the ra ndo m packing 8 , (a) before s in tering , (b) after s in tering .

to the plane of the paper and its width has increaseddue to the sintering. Again, next to an overall contrac-tion contro lled by the external surface of the packing,local inhomogeneities in the density caused by thevacancy defects can be observed. Fig. 6c shows thedensity parameter z in the middle YZ plane of struc-ture 7 after sintering. In this plane, the parts with thefc c structure appear rather massive because the planecuts two rows of particles in each layer. Again,a widening of the gap and density fluctuations in theregion surrounding the gap, can be observed. Finallyin Fig. 7, we show a vertical slice of the randompacking, before and after sintering. The two picturesprovide some examples of the sintering-inducedformat ion of isolated particles and isolated clusters ofparticles, as mentioned above.

5. Conclus ionsA new simulation model for sintering in three-dimen-sional powder compacts is presented. Based on two-sphere sintering interactions, the model representsa multi-particle approach of structural reorganizationeffects, occurring during sintering, in packings ofspherical particles. The s imulations of the powdercompacts used in this work (based on crystalline struc-

tures) show that the model adequately describes thereorganization effects that may result from local de-fects that are present before sintering.

R e f e r e n c e s1. J . FR EN K E L, J. Phys. (USSR) 9 (1945) 385.2 . G . PE TZ O W an d H . E . EX N ER , Z. Metallkde 67 (1976) 611.3. M . W . W E I S E R a n d L . C . D E J O N G H E , J. Am. Ceram. Soc.

69 (1986) 822.4. E . G . L I N I G E R a n d R . R A Y , Commun. Am. Ceram. Soc. 79

(1988) C-408.5. ldem , Sci. Sinterin9 21 (1989) 109.6 . E. ARZ T, Aeta Metall . 30 (1982) 1883.7 . J.. W. R O SS, W. A. M I LLE R an d G . C. WEA T H ER LY , ibid.

30 (1982) 203.8 . H . J . L E U , T. H A R E a n d R . O . S C A T T E R G O O D , ibid. 36

(1988) 1977.9. R . L . C O B L E , J. Appl. Phys. 32 (1961) 787.

1 0. W . D . K I N G E R Y a n d M . B E R G , ibid. 26 (1955) 1205.1 1. H .E . EX N E R , Rev. Powder Metall. Phys. Ceram. 1 (1979) 1.1 2. S . E . K O O N I N a n d D . C . M E R E D I T H , i n " C o m p u t a t i o n a l

Physics " (Addison-W esley New York , 1990) p . 29.1 3. H . J . V E R I N G A , J. Mater. Sci. 26 (1991) 5985.1 4. W. SO P PE, Powder Technol. 62 (1990) 189.

Received 17 November 1992and accepted 8 July 1993

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A Computer Simulation Method for Sintering in Three-dimensional Powder Compacts