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H owever , in a s emib a t ch or con t inu ou s -f low r ea ct or wit hin f low a n d ou t f low of polym er a n d m on om e r , a dd it ion a lba la n ce s a r e n ee ded fo r t h e m ole s of ea ch m on ome r j int h e r ea ct or t h a t a r e b oun d a s polyme r , i.e.

T h e mole s of mon om e r j boun d in t h e polym e r (N pol j )a r e a r e s u lt of polyme r iza t ion in t h e pa r t icle s (R p j pV p)a n d in t h e wa t e r ph a s e (R p j wV w) a s well a s in f low (F p j ,in )a n d ou t f low ((N pol j / V T)v ou t ). P olyme r iza t ion in t h e m on o-m e r dr op le t s is n egle ct ed for t h e r ea s ons pr e s en t edea r lie r .

A s epa r a t e ba la n ce on t h e nu m be r of m ole s of m on o-me r j in t h e wa t er ph a s e i s u s ed t o c a lcu la t e t h e volu meof mon ome r s in t h e w a t e r p h a s e. Du r in g t h e in t e r va lwh en m on ome r dr op le t s exis t , t h e nu mbe r of m ole s of m on ome r s in t h e wa t e r ph a s e i s h el d co ns t a n t f r om t h ein it ia l va lu e s ba s ed on t h e pa r t it ion in g eq u a t ion s s h ownea r li e r (s ee eq 116 ). W h en t h e dr op le t s a r e dep le t ed ,t h e m on ome r in t h e w a t e r p h a s e i s ca lcu la t ed by:

T h e polyme r iza t ion r a t e s a r e def in ed a s

wh e r e [R TOT ]w is t h e t o t a l con cen t r a t ion of r a d ica ls in

t he w

a te

r p

h a se (

see e q 29 ).F r om t h e a bove b a la n ce s , a nu m be r of va r ia ble s of

in t e r e s t ca n be ca lcu la t ed d ir ect ly. F or exa m p le, t h et o t a l m a ss con ve r s ion of mon om e r s t o polym e r ca n beca lcu la t ed by

T h e in s t a n t a n eou s polyme r com pos it ion (F j ) in t h epa r t icle s ca n n ow be ca lcu la t ed a s :

E xp r ess ions fo r t h e ins t a n t a n eous polym e r com pos it ionba s ed s olel y on t h e r ea ct ivit y r a t ios (def in ed e a r lie r ineq 16 ) a n d t h e mon om e r f ee d mole f r a ct ions we r ede r ived f r om t h e Alf r ey- Gold f in ge r eq u a t ion s for t e r -polym e r s (A lf r ey a n d Gold f in ge r , 1944 ). Th e Alf r ey-Gold f in ge r eq u a t ion s a r e s h ow n bel ow :

Af t er s om e a lgeb r a ic m a n ipu la t ion , on e m a y ob t a in

wh e r e nu m (F i / F j ) r ef er s t o t h e nu me r a t or of t h e eq u a -t ion s for F i / F j (s ee eq s 130 a n d 131 ) a n d d en r ef e r s t ot h ei r den omi n a t or . E qu a t ions 132 - 134 m a y be r edu cedt o t h e h om o- o r copolym e r iza t ion ca s e b y s e tt in g t h er ea ct ivit y r a t ios , wh ich in clu de t h e miss in g m on om e r ,eq u a l t o 1.

Th e cu m u la t ive polyme r com pos it ion (F h j ) is ca lcu la t eda s

Th e i ns t a n t a n eous polym e r co m pos it ion is ca lcu la t edm a in ly fo r it s u s e a s a weigh t in g f un ct ion fo r t h e va r iou spa r a me t e r s in t h e m odel a n d is ca lcu la t ed for t wos epa r a t e ca s e s . On e i s ba s ed on t h e polyme r bei n gfo r m ed in t h e pa r t icle s a n d t h e ot h e r on t h a t be in gfo r m ed in t h e wa t e r ph a s e. On t h e ot h e r h a n d , t h ecu m u la t ive polyme r co m pos it ion t a k e s in t o a cco un t t h epolyme r for me d in bo t h p h a s e s , i.e., t h e t o t a l polyme rfo r me d in t h e r ea ct or . Th e cu m u la t ive polyme r com -pos it ion in clu de s bot h w a t e r ph a s e a n d p a r t icle p h a s epolym e r iza t ions in or de r t o a cco un t fo r t h e co n t r ibu t iont o t h e com pos it ion du e t o t h e ca p t u r e of oligome r icr a d ica ls f r om t h e w a t e r p h a s e b y t h e p a r t icle s .

A ba la n ce i s n ow pe r fo r me d on t h e s olven t , w h ich isin t h e em u ls ion ca s e, wa t e r , com p r is in g of in f low a n dou t f low :

Si m il a r ly, a b a la n ce for t h e e m u ls if ie r s r e su lt s in

N o t e t h a t t h e em u ls if ie r ca n be l oc a t ed in fou r a r ea s :

dN pol j d t

) F p j ,in - N pol j

V Tv ou t + R p j pV p + R p j wV w (124 )

dN m j w

d t ) - R p j wV w (125 )

R p j p ) Y o[M ]pf j "i ) 1

N

k p i j i (126 )

R p j w ) [R TOT ]w[M ]wf j w"i ) 1

N

k p i j i w (127 )

x )" j ) 1

N

(N pol j M W j )

" j ) 1

N

((N m j + N pol j )M W j )

(128 )

F j )

R p j p

" j ) 1

N

R p j p

(129 )

F 1F 3

)

f 1(f 1r 23 r 32 + f 2r 23 r 31 + f 3r 21 r 32 )(f 1r 12 r 13 + f 2r 13 + f 3r 12 )f 3(f 1r 12 r 23 + f 2r 13 r 21 + f 3r 12 r 21 )(f 1r 32 + f 2r 31 + f 3r 31 r 32 )

(130 )

F 2F 3

)

f 2(f 1r 13 r 32 + f 2r 13 r 31 + f 3r 12 r 31 )(f 1r 23 + f 2r 21 r 23 + f 3r 21 )f 3(f 1r 12 r 23 + f 2r 13 r 21 + f 3r 12 r 21 )(f 1r 32 + f 2r 31 + f 3r 31 r 32 )

(131 )

F 1 ) nu m (F 1 / F 3) - 2nu m (F 2 / F 3) - 2den

nu m (F 1 / F 3) - nu m (F 2 / F 3) - den(132 )

F 2 ) nu m (F 2 / F 3)

nu m (F 1 / F 3) - nu m (F 2 / F 3) - den(133 )

F 3 ) den

nu m (F 1 / F 3) - nu m (F 2 / F 3) - den (134 )

F h j )N pol j

" j ) 1N

N pol j

(135 )

dN sd t

) F s ,in - N SV T

v ou t (136 )

dN emd t

) F em ,in - N em

V Tv ou t (137 )

9 82 In d . E n g. Ch em . Res ., Vo l. 36, N o. 4, 1997

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wh e r e y 1 is t h e m ole f r a ct ion of mon ome r 1 in t h eco polym e r a n d F 1 is t h e ove r a ll m ole f r a ct ion of m on o-me r 1 in t h e co polyme r a n d w a s given ea r lie r in eq 129a n d

Th e p s eu do- k in e t ic r a t e co n s t a n t s fo r m u lt ico m pon en tpolyme r iza t ion wer e given ea r lie r by eqs 61 (fo r t r a ns f e rt o CT A) , 63 (fo r t r a ns f e r t o m on om e r ), 83 (fo r diff us ion -co n t r olle d t e r mi n a t ion ), a n d 84 (fo r diff u s ion -co n t r olle dp r opa ga t ion ).

T h e in s t a n t a n eou s mole cu la r wei gh t dis t r ib u t ion of t h e copolyme r is g iven by

F or lon g polyme r ch a in s , on e ca n a ss u me wit h s m a llerr or t h a t a ll ch a ins p r odu ced ins t a n t a n eous ly h a ve t h es a m e com pos it ion F 1, F 2, .... Th es e ove r a ll com pos it ionsca n be foun d u s in g eq 129.

T h e bi va r ia t e dis t r ib u t ion for polyme r a cc u m u la t edin a ba t ch or s emi ba t ch r ea ct or m a y be foun d byin t eg r a t ion of eq 146 a s follow s :

In s t a n t a n eou s nu mbe r - a n d weigh t -a ver a ge ch a in len gt h sa r e g iven by

N u m be r - a n d wei gh t -a ve r a ge ch a in len g t hs of t h ea ccu m u la t ed p olyme r in ba t ch or s emib a t ch r ea ct or s a r egiven by

Tobit a a n d H a mi elec (1991 ) gen e r a lized St oc k m a ye r sbi va r ia t e dis t r ib u t ion t o a cco un t for ch a in -le n g t h de-pen den ce of polyme r r a d ica l / polyme r r a d ica l t e r mi n a -t ion . E xp r e ss ion s eq u iva le n t t o eq s 155 a n d 156 wh en

ch a in -le n g t h -depen den t t e r mi n a t ion is s ign if ica n t m a ybe foun d in Zhu a n d H a mi elec (1989 ).

2 .1 0 .2 . P o ly m e r s w i t h L o n g B r a n c h es . Th eme t h od of in s t a n t a n eou s m olecu la r weigh t dis t r ibu t ion wh ich is su ch a power f u l m e t h od fo r t h e ca lcu la t ion of m ole cu la r wei gh t d is t r ibu t ion of lin ea r polyme r ch a insm u s t be m od if ie d wh en lon g-c h a in b r a n ch in g r ea ct ion ss u ch a s t r a n s f e r t o polyme r a n d r ea ct ion w it h in t e r n a lor t e r mi n a l dou ble b on d s a r e s ign if ica n t . In t h is ca s e,dea d p olyme r ch a in s a r e n o lon ge r in e r t a n d ca n r ea ct

w it h polyme r ic r a d ica ls (a n d ot h e r r a d ica l t ype s ) a n dn ow t h e i ns t a n t a n eous m ole cu la r wei gh t dis t r ib u t ion isn o lon ger a pe r m a n en t qu a n t it y bu t ch a n ge s w it h t imea s it s ch a in s r ea ct . On e m u s t k ee p t r a ck of t h es e ch a in sin or de r t o p r ed ict t h e M W D of t h e a cc u m u la t ed lin ea rch a ins . In p r in cip le , t h is ca n be don e fo r ch a ins h a vin g1, 2, or mor e lon g br a n ch e s (Soa r e s a n d H a miele c,1996 c) . Alt e r n a t ivel y, on e ca n u s e M on t e Ca r lo s im u la -t ions t o ca lcu la t e t h e f u ll MW D for ch a ins wit h lon gb r a n ch e s a s ill u s t r a t ed by T obit a (1995 a ,b ).

Teym ou r a n d Ca m pbell (1994 ) devel oped a n ovelm e t h od co in ed nu m er ica l f r a ct ion a t ion t o c a lcu la t e t h ef u ll MW D of s ol befo r e a n d a f t e r t h e gel a t ion poin tw it h ou t t h e n or m a l d is co n t inu it y t h a t pla gu ed ea r lie r

me t h od s u s in g t h e me t h od of mome n t s . Th is me t h odis p r es en t ly be in g com p r eh ens ively eva lu a t ed by v a r iousr e s ea r ch gr ou p s , a n d we a w a it t h e r e s u lt s of t h e s es t u d ie s . Of co u r s e, wh en t h e f u ll M W D is n o t ess en t ia la n d a ve r a ge mole cu la r wei gh t s will do, it is r ecom -me n ded t h a t t h e me t h od of m ome n t s be us ed . Th e us eof t h e ps eu do-k in e t ic r a t e co n s t a n t me t h od fo r copolym -er iza t ion w it h lon g-ch a in b r a n ch in g h a s bee n ill u s t r a t edby X ie a n d H a miele c (1993b, c) .

U s in g t h e me t h od of m ome n t s , t h e le a d in g m ome n t sof t h e dis t r ib u t ion (Q 0, Q 1, Q 2) ca n be ca lcu la t ed a sfo ll ow s . In vok in g t h e s t a t ion a r y-s t a t e h ypo t h e s is forr a d ica ls , on e ca n r ea d ily de r ive t h e follow in g mome n teq u a t ion s (R a y , 1972 ; B r oa dh ea d e t a l ., 1985 ; H a miele ce t a l ., 1987 ; X ie a n d H a m iele c, 1993 a ,b ).

wh e r e

K ) (1 - 4F 1(1 - F 1)(1 - r 1r 2)) (149 )

) k t dY ok p[M ]p

+ C f m + C fc t a [CTA] p

[M ]p+

C f m s i[M S I ]p[M ]p

(150 )

) k t cY ok p[M ]p

(151 )

W (r ) ) - ##

W (r , y ) d y ) ( + )( + / 2 ( + )r )r exp (- ( + )r ) (152 )

W h (r , y ) ) 0

t

W (r , y ) R pW (t ) V (t ) d t

0t

R p W (t ) V (t ) d t (153 )

R pW (t ) ) "i ) 1N

R p i M W i (154 )

r N ) 1

+ / 2 (155 )

r W ) 2 + 3 ( + )2

(156 )

r jN ) 0

t

R pW (t ) V (t ) d t

0t

(1 / r N)R pW (t ) V (t ) d t (157 )

r j W ) 0

t r W R pW (t ) V (t ) d t

0t

R p W (t ) V (t ) d t (158 )

dV pQ 0d t

) ( + / 2 - C k Q 1[M ]p - K Q 0[M ]p)k p[M ]pY oV p (159 )

dV pQ 1d t

)

(1 + C f m + C fc t a [CTA] p[M ]p - C f m s i[MS I ]p

[M ]p )k p[M ]pY oV p(160 )

dV pQ 2d t ) (f a ct o r + 2(1 +

K Q 1 + C k Q 2[M ]p )b r a ck e tden om +

(b r a ck e tden om )2)k p[M ]pY oV p (161 )

) k t dY ok p[M ]p

+ C f m + C fc t a [CTA] p

[M ]p+

C f m s i[MS I ]p[M ]p

(162 )

) k t cY ok p[M ]p

(163 )

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ch a in -t r a n s f e r a gen t (gen e r a lly h yd r ogen ) a n d con s e-qu en t ly pr odu ce polyme r wit h ch a in -le n g t h a ve r a ge st h a t d iff e r s pa t ia lly ins ide t h e polym e r ic p a r t icle. F orco polyme r iza t ion , m on ome r s wit h diff e r en t eff ect ived iff us ivit ie s a n d r ea ct ivit ie s m a y be r e s pons ible fo rs pa t ia l com pos it ion a l h e t e r ogen ei t y in t h e polyme r icpa r t icle . In a dd it ion , if t h e r e i s a pp r ecia ble h ea t -t r a n s f e r r e s is t a n ce, h o t s po t s ca n occ u r in s ide t h epolyme r pa r t icle , a lt e r in g r ea ct ion r a t e s a n d f u r t h e rb r oa den in g MW D a n d CC D (F igu r e 3 ). Models ba s edon t h is a pp r oa ch a r e gen e r a lly ca lle d p h y s i c a l m o d e l s (Sch me a l a n d St r ee t , 1971, 1972 ; Si n gh a n d Me rr ill ,1971 ; C r a b t r ee e t a l ., 1973 ; N a gel e t a l ., 1980 ; T a ylor e t a l ., 1983 ; F loyd e t a l ., 1986 a - c; Ga lva n a n d Tirr ell,1986 a ; H u t ch ins on a n d R a y , 1988 ; S k om or okh ov e t a l .,1989 ; Sa r k a r a n d Gu p t a , 1991, S oa r e s a n d H a miele c,1995b ).

A s eco n d ca t egor y of m a t h em a t ica l m odel s n egle ct sh ea t - a n d m a ss -t r a ns f e r r e s is t a n ce s in t h e polyme rpa r t icle. In t h e s e models , m u lt ip le t ype s of ca t a ly t ics it e s a r e r e s pons ible fo r t h e p r odu ct ion of polym e r w it hb r oa d M W D s a n d CCD s ( c h e m i c a l m o d e l s ). E a ch s it et ype h a s it s own k in e t ic co n s t a n t s a n d p r odu ces polyme rw it h of t en ver y diff er en t M W D s , CCD s , a n d s t er eor egu -la r it ie s , a s sh own in F igu r e 4 for a t h r ee -s it e-t ypeca t a lys t (T a it a n d W a n g, 1988 ; de Ca r va lh o e t a l ., 1989 ;

R in co n -Ru bi o e t a l ., 1990 ; M cA u le y e t a l ., 1990 ; Lor en -zin i e t a l ., 1991 ; Vel a -Es t r a da a n d H a miele c, 1994 ; Leee t a l ., 1994 ; Soa r e s a n d H a miele c, 1995b ). Th e r e i s va s texpe r ime n t a l eviden ce su pp or t in g t h e e xis t en ce of m u l-t ip le s it e t ypes on h e t e r ogen eous Zie gle r - N a tt a ca t a -lys t s (Cozew it h a n d Ve r S t r a t e, 1971 ; B oo r , 1979 ; K eii,1982 ; Zu cc h in i a n d Cecc h in , 1983 ; Keii e t a l ., 1984 ;Ch ie n e t a l ., 1985 ; U s a mi e t a l ., 1986 ; Spit z, 1987 ; Ch en ga n d K a ku go, 1991 ; S oa r e s a n d H a miele c, 1996 a ,b ).

Th e e ff ect s of t r a n s f er r es is t a n ces a n d of m u lt ip le s it et ype s on M W D a n d CCD b r oa den in g a r e e viden t ly n o texclus ive ; t h e h e t e r ogen ei t y ca us ed by t h e p r e s en ce of m u lt ip le s it e t ype s ca n be i n cr ea s ed by t r a n s f e r r e s is -t a n ce s du r in g t h e polyme r iza t ion . Some m odel s com -bin in g t h es e t wo eff ect s h a ve bee n p r opos ed r ecen t ly a n ds eem t o be h igh ly su ccess f u l in p r edict in g t h e qu a li t a t ivebe h a vior of polym e r iza t ion wit h Zie gle r - N a tt a ca t a -lys t s . Th e s e m odel s w ill be ca lle d h y b r i d m o d e l s s in cet h ey combi n e t h e t w o pr eviou s modeli n g s t r a t egie s(G a lva n a n d T irr ell, 1986b ; F loyd e t a l ., 1988 ; Ra y, 1988 ;H u t ch ins on e t a l ., 1992 ; Sa u a n d Gu p t a , 1993 ; Soa r e sa n d H a mi elec, 1995b ).

Ph y s i c a l M o d e l s . Diff e r en t m a t h em a t ica l m odel sh a ve bee n p r opos ed t o de s cr ibe i n t e r pa r t icle a n d in t r a -pa r t icle m a ss a n d h ea t t r a n s f e r du r in g polyme r iza t ionw it h h e t e r ogen eou s Zie gle r - N a tt a ca t a lys t s . Some a r ever y s im p li f ie d p ict u r e s of t h e polyme r iza t ion pr oc e ssa n d a r e on ly us ef u l a s a r ef e r en ce poin t fo r co m pa r is onw it h m or e s oph is t ica t ed m odels . Am on g t h os e, t h e m os t

F i g u r e 2 . F r a gm en t a t ion o f h e t e r ogen eou s Zie gle r - N a tt a c a t a lys t (s eco n d a r y) p a r t icle s d u r in g p olym e r iza t ion . Th e g r ow in g p olym e rch a in s b r ea k t h e s eco n da r y ca t a lys t pa r t icle s in t o p r im a r y pa r t icle s , fo r mi n g a n expa n di n g p a r t icle m a de of c a t a lys t f r a gme n t s s u rr oun dedby de a d a n d li vin g polyme r ch a in s . Th is beh a vio r is r es pon s ib le for t h e r epli ca t ion ph en ome n on ob s e r ved i n h e t e r ogen eou s Zie gle r -N a tt a c a t a lys is .

F i g u r e 3 . Gen e r ic r ep r es en t a t ion of ph ys ica l model s . T )t em pe r a t u r e; [M ] ) mon ome r con cen t r a t ion ; % M 1 ) pe r cen t of com on ome r t ype 1 in t h e p olyme r ; M n ) nu m ber -a ve r a ge m olecu la rwei gh t . In ph ys ica l m odel s , in t r a pa r t icle m a ss - a n d h ea t -t r a n s f e rr es is t an ces a r e r es pon s ib le fo r M W D an d CCD b r oa de n in g.

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r ep r e s en t a t ive fo r t h is r evie w a r e t h e s o li d co r e m o d e l a n d t h e p o l y m e r i c co r e m o d e l (F igu r e 5 ).

T h e s olid co r e m odel doe s n o t model t h e b r ea ku p of t h e ca t a lys t pa r t icle. Th e polym er is co ns ider ed t o gr owa r oun d a s olid ca t a lys t co r e con t a in in g a ll a ct ive s it eson it s s u r f a ce. Th is m odel u s in g a s in gle t ype of c a t a lys ts it e ca nn o t p r ed ic t br oa d M W D s (Sch me a l a n d S t r ee t ,

1971, 1972; N

agel e

t a l ., 1980

). T

his r

es u

lt s h ou

ld n o t

come a s a su r p r is e s in ce t h e polyme r iza t ion t a k es p la ceon ly a t t h e su r f a ce of t h e ca t a lys t . F or a g iven polym -e r iza t ion t ime, t h e m on ome r co n cen t r a t ion is co n s t a n ta t t h e s u r f a ce of t h e ca t a lys t a n d t h e r efo r e a ll polyme rch a in s a r e p r odu ced w it h t h e s a me a ve r a ge p r ope r t ie s .On ly if t h e co n cen t r a t ion of m on ome r a t t h e su r f a ce of t h e ca t a lys t s ch a n ged s ign if ica n t ly in t im e (be ca us e of in cr ea s in g t r a ns f e r r e s is t a n ce ca us ed by t h e g r ow t h of polyme r a r oun d t h e ca t a ly t ic pa r t icle ) w ou ld t h e s olidco r e m odel pr ed ict s ign if ica n t br oa den in g of MW D .Add it ion a lly, t h e s olid co r e m odel i s cle a r ly in con t r a dic-t ion w it h expe r im en t a l d a t a on ca t a lys t b r ea ku p . De-s p it e of t h es e li m it a t ions , B r oc k m eie r a n d Roga n (1976 )a d ju s t ed t h e s olid co r e m odel t o da t a colle ct ed in a

s emi ba t ch r ea ct or a n d us ed t h is model t o pr ed ict t h ebe h a vior of a co n t inu ous r ea ct or . Th e pr ed ict ions we r ea cc u r a t e for polyme r yiel d , bu t n o mole cu la r wei gh tr e s u lt s w e r e s h own .

W it h t h e polyme r ic co r e m odel , polyme r gr ow s a r oun da n on expa n d in g polyme r ic co r e fo r me d by polyme r a n dca t a lys t pa r t icle s (F igu r e 5 ). Alt h ou gh t h is m odel i s a nim p r oveme n t ove r t h e f ir s t on e, i t s t ill i s n ot a ble t o

a cco un t for br oa d MW D s (Sch me a l a n d St r ee t , 1971,1972 ; Si n gh a n d Me rr ill, 1971 ). I t s h ow s , h oweve r , t h eim por t a n ce of a co r e of polyme r - ca t a lys t f r a gme n t s int h e e xp la n a t ion of b r oa d M W D .

Th e m odel s t h a t be s t r ep r es en t t h e polym er iza t ion int h is ca t egor y of s in gle -s i t e-t ype, t r a n s f e r -co n t r olle dm odel s a r e t h e s o-c a lle d ex p a n s i o n m o d e l s . E xpa n s ionm odel s con s ide r t h e f r a gme n t a t ion of t h e ca t a lys tpa r t icle a n d t h e for m a t ion of a n expa n d in g p a r t icle of polym e r a n d ca t a lys t f r a gm en t s .

G r ow in g polyme r ch a in s a n d ca t a lys t f r a gme n t s fo r ma co n t inuu m i n t h e p o l y m e r i c f l o w m o d e l (Sch me a l a n dS t r ee t , 1971, 1972 ; Si n gh a n d Merr ill , 1971 ; Ga lva n a n dT irr ell , 1986 a ). Diff us ion of r ea gen t s a s well a s h ea tt r a ns f er occ u r s in t h is polym er ic pa r t icle. I f t h e r ea ct ionis diff u s ion -co n t r olle d , t h e r a d ia l pr of ile s of mon ome ra n d ch a in -t r a n s f e r a gen t in t h e pa r t icle m a y ca u s eM W D a n d CCD br oa den in g. Th e e ff ect ive m a ss - a n dh ea t -t r a ns f er pa r a me t e r s in t h e polyme r ic pa r t icle h a vet o be e s t im a t ed t o us e t h is model (F igu r e 6 ). Th epolym er ic f low m odel ca n p r ed ict s ign if ica n t br oa den in gof M W D u s in g r ea s on a ble p h ys ica l p a r a me t e r s .

Th e m u lt ig r a in m odel (C r a b t r ee e t a l ., 1973 ; N a gel e t a l ., 1980 ) co ns ider s t wo level s of m a ss - a n d h ea t -t r a ns f err e s is t a n ce s . Th e polym e r ic pa r t icle ( m a c r o p a r t i c l e ors eco nd a r y p a r t i c l e ) is for m ed by a n a gg lom e r a t e of m i c r o p a r t i c l e s or p r i m a r y p a r t i c l e s . E a ch mi cr opa r t icleco n s is t s of a f r a gme n t of t h e or igin a l ca t a lys t pa r t icle,w it h a ll a ct ive s it e s on it s su r f a ce, su rr oun ded by dea d

F i g u r e 4 . Gen e r ic r ep r es en t a t ion of c h em ica l m odel s . 1, 2, an d 3 in dica t e s it es of diff e r en t t ypes ; o pen an d s ha ded cir cle s in dica t e diff e r en tt ypes of m on ome r s . In ch emi ca l m odel s , m u l t ip lici t y of a ct ive s i t e t ypes is r es pon s ib le fo r M W D a n d CCD b r oa den in g.

F i g u r e 5 . Sch em a t ic r ep r es en t a t ion of t h e s o lid co r e m o d e l an dof t h e p o l y m e r i c co r e m o d e l .

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wh e r e

a n d F h1 ) a ve r a ge m ole f r a ct ion of mon ome r t ype 1 onco polym e r , M 1, M 2 ) mole cu la r wei gh t s of mon om e rt ype s 1 a n d 2, r e s pect ivel y , r ) ch a in len g t h , r 1, r 2 )r ea ct ivit y r a t ios , a n d y ) devia t ion f r om a ve r a geco polyme r co m pos it ion .

F or s in gle -s it e ca t a lys t s , t h e p r in cipa l co n clus ions of t h e polym e r ic m u lt il a ye r model con f ir m a n d com p le-me n t t h e co n clu s ion s ob t a in ed w it h t h e polyme r ic f lowm odel a n d t h e m u lt ig r a in model : (1) Ma ss -t r a n s f e rr es is t a n ces ca n r edu ce t h e polyme r iza t ion r a t e, decr ea s em ole cu la r wei gh t a ve r a ge s , a n d a ff ect copolyme r com -pos it ion for la r ge, h igh ly a ct ive ca t a lys t s . (2) P a r t icu -la r ly im por t a n t fo r su pp or t ed ca t a lys t s , t h e co n cen t r a -t ion of h igh ly a ct ive s it e s ca n in cr ea s e m a ss -t r a n s f e rr e s is t a n ce s a n d h a ve un de s ir a ble r e s u lt s in ca t a lys tpe r fo r m a n ce a n d p r odu ct qu a li t y. (3) Ma ss -t r a ns f e rr e s is t a n ce s m a y a ls o be a s ou r ce of co polym e r co m pos i-t ion h e t e r ogen ei t y fo r h igh ly a ct ive a n d la r ge ca t a lys tpa r t icle s , if t h e com on ome r s h a ve r ea ct ivit ie s t h a t diff ers

ign

if ica n t

ly . (

4) Tem pe

r a t u re g

r ad ie

n t s i

n t he p

oly

-me r ic pa r t icle a r e n o t expect ed t o be a s ign if ica n t f a ct orfo r r ea ct ions ca rr ie d ou t in s lu rr y r ea ct or s .

Th e con clus ions ob t a in ed w it h s in gle -s it e-t ype e xpa n -s ion m odel s a r e e s pecia lly im por t a n t fo r t h e t echn ologyof s u pp or t ed me t a lloc en e ca t a lys t s , wh er e s in gle -s it e,h igh ly a ct ive ca t a ly t ic s pecie s m a y be s u b ject ed t os ign if ica n t m a ss - a n d h ea t -t r a ns f e r r e s is t a n ces (Soa r e sa n d H a mi elec, 1995 a ,b ).

C h e m i c a l M o d e l s . I t is gen e r a lly a ccep t ed t h a t ,un de r mos t polyme r iza t ion con d it ion s , t h e e ff ect of m u lt ip le -s it e t ype s on polyme r pr ope r t ie s is f a r mor eim por t a n t t h a n m a ss - a n d h ea t -t r a ns f e r r e s is t a n ce s(R a y , 1988 ; Soa r e s a n d H a mi elec, 1995b ). Un de r t h e s eco n d it ions , ea ch s it e t ype ins t a n t a n eous ly pr odu ce s

polyme r t h a t h a s F lor y s m os t p r oba ble M W D . Th e r e-fo r e, t h e ins t a n t a n eous M W D of a cc u m u la t ed p olym e rm a de w it h h e t e r ogen eou s Z ie gle r - N a tt a ca t a lys t s ca nbe con s ide r ed a n a ver a ge of t h a t pr odu ced by t h ein d ividu a l s it e t ype s , wei gh t ed by t h e wei gh t f r a ct ionof polyme r pr odu ced by ea ch s it e t ype, a s sh ow n inF igu r e 8 (Soa r e s a n d H a mi elec, 1995 c) .

T h e s a me t r ea t me n t ca n be e x t en ded t o co polyme r sby u s in g S t oc k m a yer s bi va r ia t e dis t r ib u t ion t o des cr ibet h e i n s t a n t a n eou s biva r ia t e dis t r ibu t ion of c h a in le n g t ha n d com pos it ion . Th e r efo r e, if on e a ssu me s t h a t e a chs it e t ype p r odu ces co polym e r t h a t obe ys d is t in ct bi va r i-a t e d is t r ib u t ion s of ch a in le n g t h a n d co polyme r co m po-s it ion , t h e bi va r ia t e dis t r ib u t ion of t h e a cc u m u la t edco polyme r ca n be con s ide r ed a n a ve r a ge of t h a t pr o-du ced by t h e in d ividu a l s it e t ype s , a s sh own in F igu r e9 (Soa r e s a n d H a mi elec, 1995e ).

I f t h es e h ypo t h es es a r e va lid , b r oa d M W D s a n d CCD sr e s u lt f r om t h e s u pe r pos it ion of n a rr owe r d is t r ib u t ion sp r odu ced on ea ch s it e t ype. Add it ion a lly , f r om t h ekn ow le dge of t h e globa l M W D s a n d CCD s , on e ca n us e

F i g u r e 7 . P r edi ct ion of t h e biva r ia t e di s t r ibu t ion of ch emi ca lcomp os it ion a n d ch a in len g t h w it h t h e p o l y m e r i c m u l t il a ye r m o d e l .r ) cha in le n g t h ; W (r ) d r ) cha in -le n g t h di s t r ibu t ion ; y ) de via t ionf r om copolym e r a ve r a ge comp os i t ion ; W ( y ) d y ) comp os i t iondis t r ibu t ion ; s 1, s 2, s 3 s i t e t ype s ; l1, l2, ..., l9 ) m odel l a ye r s ; l1 ist h e inn e r m os t la ye r ; l9 is t h e ou t e r m os t la ye r .

) F h1(1 - F h1)K (211 )

K ) [1 + 4F h1(1 - F h1)( r 1r 2 - 1)]0.5 (212 )

) 1 - M 2 / M 1

M 2 / M 1 + F h1(1 - M 2 / M 1)(213 )

F i g u r e 8 . In s t an t an eou s cha in -le n g t h di s t r ibu t ion (CLD ) of apolyole f in m a de w it h a m u l t ipl e-s it e-t ype ca t a lys t a s a s u pe r pos i-t ion of fo u r in dividua l F lo r y s m os t p r ob a ble wei gh t cha in -le n g t hdis t r ibu t ion s (s olid lin e in dica t es CLD of wh ole polyme r , a n ddo tt ed li n es r ep r es en t CLD of polyme r m a de on di s t in ct a ct ive s it et ype s ).

F i g u r e 9 . In s t an t an eou s ch em ica l comp os it ion dis t r ibu t ion (CCD )of a bina r y ole f in copolyme r ma de wit h a mu l t ip le -s it e -t ypeca t a lys t a s a s u pe r pos it ion of f ive in di vid u a l S t oc k m a ye r s CCD sco n s ide r in g a ll ch a in len g t h s (s olid li n e in di ca t e s CCD of wh olepolyme r , an d d o tt ed li n e s r ep r es en t CCD s of polyme r ma de ondi s t in ct a ct ive s i t es ).

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me t h od s a n d con f iden ce in t e r va ls we r e usu a lly n o tgiven . Th e r ea ct ivit y r a t ios a r e h igh ly co rr el a t ed w it hm a jor dis a gr eeme n t in va lu e s me a s u r ed in diff e r en tla bor a t or ie s (O D r is co ll a n d Reill y , 1987 ; Du be e t a l .,1991 a ; Bu r k e e t a l ., 1993 ; Ross ign oli a n d Du ever , 1995 ).

4 .1 . S t a t i s t i c a lly V a l id M e t h o d s f o r E s t i m a t i o no f R e a c t i v i t y R a t i o s . Tr a d it ion a l m e t h od s fo r es t i-m a t in g r ea ct ivit y r a t ios (M a yo a n d L ew is , 1944 ; F in e-m a n a n d Ross , 1950 ; Br a un e t a l ., 1973 ; Kele n a n dT u d os , 1975 ) a r e b a s ed on f ir s t t r a n s fo r mi n g t h e in -

s t a n t a n eous copolyme r com pos it ion equ a t ion in t o ali n ea r for m i n t h e p a r a m e t e r s r 1 a n d r 2, for exa m p le

a n d t h en e s t im a t in g t h e r ea ct ivit y r a t ios by g r a ph ica lp lo tt in g or by lin ea r lea s t s qu a r e s . Howeve r , t h e s ea pp r oa ch es , a s ide f r om r eq u ir in g t h a t t h e i n s t a n t a n eou sco polyme r com pos it ion eq u a t ion be va lid , a r e s t a t is t i-ca lly uns oun d beca us e t h e in depen den t va r ia ble

a

b

F i g u r e 1 4 . W eigh t cha in -le n g t h d is t r ibu t ion fo r polym e r popu la t ion s co n t a in in g d iff e r en t nu m be r s of lon g-cha in b r an ch es pe r p olym e rch a in : (a ) wei gh t ch a in -le n g t h dis t r ibu t ion n or m a lized wi t h r es pect t o t h e wei gh t of in dividu a l popu la t ion s ; (b ) wei gh t ch a in -le n g t hdi s t r ibu t ion n or m a lized w it h r es pe ct t o t h e t o t a l w ei gh t of t h e w h ole p olyme r . T h e g lob a l di s t r ibu t ion (ove r a ll p olyme r p opu la t ion s ) isin di ca t ed by t h e b old li n e . T h e t a ble i n t h e t op r igh t co r n e r in di ca t e s nu m be r a n d w ei gh t f r a c t ion s of e a ch p opu la t ion .

(f 1 / f 2)(F 2 / F 1 - 1) ) - r 2 + r 1(f 1 / f 2)2(F 2 / F 1) (224 )

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(f 1 / f 2)2(F 2 / F 1) h a s err or , a n d t h e depen den t va r ia ble(f 1 / f 2)(F 2 / F 1 - 1) doe s n o t h a ve co n s t a n t va r ia n ce. Bo t hof t h e l a tt e r a ss u m p t ion s a r e n ece ss a r y fo r lin ea r le a s ts qu a r e s t o be a s t a t is t ica ll y v a lid es t im a t ion me t h od .A s a r e su lt , t h ey h a ve bee n sh own of t en t o le a d t o ve r ypoo r es t im a t e s wit h mi s lea d in g con f iden ce in t e r va ls(T idwell a n d M or t ime r , 1965, 1970 ). Th e co mm on us eof t h es e s t a t is t ica lly in va lid es t im a t ion pr oc edu r e s int h e p a s t is on e of t h e r ea s on s fo r t h e w ide va r ia t ion in

r ea ct ivit y r a t ios r epor t ed by d iff e r en t r e s ea r ch e r s .I t sh ou ld be me n t ion ed h er e t h a t t h e Kele n a n d T u d os

(1975 ) me t h od , a lt h ou gh s t a t is t ica lly in va lid , ca n beu s ed t o a t le a s t ob t a in good in it ia l r 1 a n d r 2 es t im a t e s ,p r ovided t h e e xpe r im en t s h a ve bee n su it a bl y de s ign ed .M or e de t a il s a r e given in McF a r la n e e t a l . (1980 ) a n dLa u r ie r e t a l . (1985 ).

Rea ct ivit y r a t ios a r e n ow a da ys m os t usu a ll y es t i-m a t ed us in g pr oc edu r e s ba s ed on t h e s t a t is t ica lly v a liderr or -in -va r ia ble s m odel (E VM ) or on i t s m od if ica t ions(Box, 1970 ; Br itt a n d Lu eck e, 1973 ; Su tt on a n d M a cG r e-gor , 1977 ; Ga r cia -Ru bio e t a l ., 1985 ; Y a m a da e t a l ., 1978 ;Va n de r Mee r e t a l ., 1978 ; P a t in o-Lea l e t a l ., 1980 ;P a t in o-Lea l a n d Reill y, 1981 ; Du be e t a l ., 1991 a ; Bu r k e

e t a l ., 1993 ; Ross ign oli a n d Du eve r , 1995 ). E x t ens ionof t h es e E VM me t h od s t o t h e e s t im a t ion of r ea ct ivit yr a t ios in t e r polyme r iza t ion s h a s bee n dis cu ss ed inDu eve r e t a l . (1983 ). Th es e me t h od s a llow on e t o t a k ep r ope r ly in t o a cco un t a ll t h e s ou r ces of expe r im en t a le rr or . N o gr ou p s of va r ia ble s a r e cons ide r ed t o bein depen den t a n d f r ee of err or or depen den t wit hco n s t a n t e rr or . All me a s u r ed va r ia ble s (e. g., [M j ]m , j ) 1,2 com on om er s ) a r e co ns ider ed on a n eq u a l ba s is a n dr ep r e s en t ed a s comi n g f r om s ome t r u e (bu t unkn own )va lu e [M j ] t wh ich is con t a mi n a t ed wit h me a s u r eme n te rr or , t h a t is

Th e e s t im a t e s ob t a in ed by t h e s e E VM a pp r oa ch e s a r esu pe r ior t o t h os e ob t a in ed by a r bit r a r y g r a ph ica l ora r bit r a r y lea s t -s qu a r es p r oc edu r e s .

A lt h ou gh E VM p r oc ed u r e s r eq u ir e m or e in fo r m a t iont h a n a r bi t r a r y lea s t -s qu a r e s pr oc edu r e s , n a m el y, t h eme a s u r eme n t err or va r ia n ces a n d cova r ia n ces , t h e poin te s t im a t e s of t h e r ea ct ivit y r a t ios ob t a in ed by t h e s ep r oc edu r es h a ve bee n sh own t o be ve r y ins ens it ive t och a n ges in t h es e cova r ia n ce va lu es . Th e gr ea t im p r ove-

m en t p r ovided by t h e E VM es t im a t ion m e t h od s co m e sf r om s im p ly a cco un t in g co rr ect ly fo r t h e m a jor s ou r ce sof e rr or .

4 .2 . D es i g n o f E x p e r i m e n t s . F or a n y exper ime n t a lin ve s t iga t ion , a n op t im a l e xper ime n t a l des ign is of gr ea tim por t a n ce s in ce it en a bles on e t o per fo r m t h e mi n im u mnu m be r of expe r im en t s a n d ob t a in t h e mos t pr ecis epa r a me t e r es t im a t es . In copolyme r iza t ion s , s u ch ade s ign is t h a t u s in g t h e Tidwell - M or t ime r (1965 )cr it er ion . Th e cr it e r ion em ploys t h e M a yo- Lew is m odel(e. g ., s ee O D r is co ll a n d Reill y (1987 ) or H a mi elec e t a l .(1989 )) a n d is ba s ed on t h e s ens it ivit y of t h e r ea ct ivit yr a t io e s t im a t e s t o t h e e rr or s e n co un t e r ed in t h e de t e r -mi n a t ion of co polyme r com pos it ion . Tidwell a n d M or -t ime r (1965 ) r eco mme n ded r unn in g s eve r a l r ep lica t e sa t t wo diff e r en t m on ome r f ee d com pos it ions , f !1o a n d f !!1o,def in ed by t h e follow in g eq u a t ion s :

I t is eviden t f r om eq s 226 a n d 227 t h a t pr eli m in a r ye s t im a t e s of t h e t wo r ea ct ivit y r a t ios , r 1 a n d r 2, a r en ee ded . Th e s e e s t im a t e s ca n be ob t a in ed f r om t h eli t e r a t u r e (e.g., P o l y m e r H a nd b oo k , 1989 ), a s e t of

F i g u r e 1 5 . E ff ect of v a r yin g m a cr om on om e r a dd i t ion r a t e on wei gh t c ha in -le n g t h d is t r ibu t ion .

[M j ]i m ) [M j ] i

t + i (225 )

f !1o ) 2

2 + r 1(226 )

f !!1o ) r 2

2 + r 2(227 )

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P o l i cy II :

d [M 1]d t

) d [M 2]

d t ) 0 (233 )

Si n ce f 1 a n d F 1 a r e con s t a n t , 1 is a ls o co n s t a n t .G iven t h e t ime va r ia t ion of t h e t o t a l polyme r r a d ica l

co n cen t r a t ion , [P ], on e ca n r ea d ily s olve for F 1, in a n dF 2, in , t h e t ime -va r yin g mon ome r f ee d r a t e s , us in g t h ea bove eq u a t ions .

W e w ill n ow in ves t iga t e P olicie s I a n d II w it h r e s pectto c

ro

ss-li

n k in

g u s

in

g t h

e crite

rio

n of o

n se

t of gel

a tio

n

a lr ea dy d is cu ss ed . Mon ome r 1 is t h e cr oss -li n k e r (d i-vin yl m on ome r ) a n d a ch a in -t r a ns f er a gen t (CT A) isus ed t o dela y t h e ons e t of gela t ion .

6 .1 . P o l i cy I . F or co ns t a n t [P ] a n d F 2,i n ) 0, eq 229m a y be s olved a n a ly t ica lly fo r N 2 t o g ive

wh e r e

F or co n s t a n t N 2 / N 1, N 1 is g iven by

On e m a y n ow s olve eq 228 for F 1,i n t o ge t

Ass u mi n g t h a t t h e den s it ie s of m on ome r s a n d copolyme ra r e t h e s a me (Fm 1 ) F m 2 ) F p), on e ca n in t egr a t e e q 230a n a ly t ica lly t o ge t

Th e t o t a l m on ome r con cen t r a t ion in t h e r ea ct or ist h e r efo r e g iven by

wh e r e

W e will n ow in ves t iga t e gel a t ion for a bin a r y copo-lyme r iza t ion of vin yl a n d d ivin yl m on ome r s . Th e divin -yl m on ome r is co n s ide r ed t h e m or e r ea ct ive m on ome r ,a n d t h e r efo r e, wit h P olicy I , it is f ed in t o t h e r ea ct orove r t im e. Low le vel s of t h e cr oss -li nk in g m on om e r a r et o be u s ed a n d t h e r efo r e t o s im p li f y t h e a n a lys is , on eca n a ss u me t h a t t h e volu me of t h e r ea ct in g mi x t u r e, V ,is co n s t a n t a n d eq u a l t o t h e i n it ia l volu me, V 0. W e n owa pp ly eq 161 fo r co ns t a n t V , ) C p1 ) C p2 ) 0 a n d )k f CT A[CTA]/( k p[M ]) w it h P olicy I t o g ive

w it h Q h2 a t t ) 0 w h e r e

a n d

Q 2 goe s t o in f in it y a t t h e gel a t ion poin t t ) t c a n d t c isgiven a pp r oxim a t el y by

W e em p loy t h e fo llow in g s e t of k in e t ic pa r a me t er s : k f CT A) 10 2 L m ol- 1 s - 1, [CTA] ) 0.1 m ol L - 1, a co ns t a n t k *p*) 10 2 L m ol- 1 s - 1, F 1 ) 0.01, k p ) 10 3 L m ol- 1 s - 1, Y 0 )10 - 7 m ol L- 1, [M ]0 ) 10 m ol L - 1, a n d t c ) 5000 s (1.4h ).

E m p loyin g eq s 159, 160, a n d 176 fo r t h is exa m p le of P olicy I , it ca n r ea d ily be sh ow n t h a t , a t t h e gel a t ionpoin t ( t ) t c), BhN4 = 1 / 3. In o t h e r wor d s , a t t h e gel a t ionpoin t on e polyme r m ole cu le i n t h r ee on t h e a ve r a ge h a sa t e t r a f un ct ion a l lon g-ch a in b r a n ch poin t .

6 .2 . P o l i cy II . To s im p li f y t h e a n a lys is , we a ssu met h a t t h e dens it ie s of m on om e r s a n d copolym e r a r e t h es a me a n d t h a t t h e m ole cu la r wei gh t of t h e m on ome r sa r e t h e s a me, i n a dd it ion t o co n s t a n t [P ]. E qu a t ion 229m a y be s olved a n a ly t ica lly t o g ive t h e follow in g :

wh e r e

Si n ce t h e r a t io F 1, in / F 2, in is in depen den t of t ime , t h e t wom on ome r s m a y be pr emi xed a n d f ed t o t h e r ea ct or int h e s a m e s t r ea m .

W e will n ow in ve s t iga t e gela t ion fo r a bin a r y copo-lym e r iza t ion of a vin yl a n d d ivin yl m on om e r . Tos im p li f y t h e a n a lys is , it is a ss u me d t h a t t h e co n cen t r a -t ion of ch a in -t r a n s f e r a gen t , [CT A] , is m a in t a in edco ns t a n t . W e n ow a pp ly eq 161 un de r t h e foll ow in gco n d it ions : ) C p1 ) C p2 ) 0 a n d ) k f CT A[CT A]/

(k p[M ]) con s t a n t

wit h t

ime in t h

e con t

ext

of P

olicy II

.

w it h Q h2 ) 0 a n d t ) 0. Th e def in it ion s of Q h2 a n d a r et h e s a me a s t h os e u s ed w it h eq 241. E qu a t ion 251 ca nbe s olved nu me r ica lly (s e tt in g ) 4.54 10 - 4 s - 1 a n dR ) 1.515 10 - 4 s - 1) t o g ive

N 2 ) N 2oe - 2t

(234 )

2 ) (k 12 1 + k 22 2)[P ] (235 )

N 1 ) N 1oe - 2t (236 )

F 1, in ) N 1o(1 - 2)e- 2 t (237 )

V ) V o + (N 1oM W 1Fm 1 (1 - 2

2 ))(1 - e - 2t ) (238 )

[M ] ) N 1 + N 2

V )

(N 1o + N 2o)e- 2t

V o + R(1 - e - 2t )

(239 )

R ) N 1oM W 11 - 22 / Fm 1

(240 )

dQ h2 / d t ) (1 + Q h2)2e - 2t

(241 )

Q h2 ) k *p*Q 2k p[M ]

(242 )

) 2F 1k *p*k p[M ]0Y o

k f CT A[CTA] (243 )

t c ) 1 / )

k f CT A[CTA]

2F 1k *p*k p[M ]0Y o(244 )

N 1 ) N 1oeRt (245 )

N 2 ) N 2oeRt

(246 )

F 1, in ) (R + 1)N 1oeRt (247 )

F 2, in ) (R + 2)N 2oeRt (248 )

R ) f 1o1 + f 2o2

V oFm(M W (N

1o + N

2o) - 1) (249 )

V ) V oeRt

(250 )

dQ 2 / d t ) (1 + Q h2)2 - R Q h2 (251 )

t (s ) 0 100 200 300 400 500 700 900Q h2 0 0.0472 0.0983 0.154 0.215 0.282 0.438 0.636t (s ) 1100 1300 1500 1700 1900 2100 2150 2200Q h2 0.894 1 .249 1.769 2.61 4.22 8.57 11.13 15.60t (s ) 2250 2300 2310 2320 2330Q h2 25.4 64.0 91.1 157.0 556.0

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cr oss -li n k e r (s t y r en e / e t h yle n e d ime t h a cr yla t e, s t y r en e / d ivin ylben zen e) us in g s imil a r t echn iqu es .

F u r t h er a pp lica t ion s of t h e f r ee -volu me t h eor y w e r edem ons t r a t ed by X ie e t a l . (1991 ) fo r t h e sus pens ionpolym e r iza t ion of vin yl ch lor ide a n d by P en lid is e t a l .(1992 ) fo r t h e s olu t ion polym e r iza t ion of MM A . Th ela tt er s h owed h ow t o s im plif y t h e gen er a l m odel (s imil a rt o t h e on e de s cr ibe d h e r ei n ) fo r a pa r t icu la r ca s e. Of m os t s ign if ica n t n o t e, h oweve r , is t h e wor k by G a o a n dP en lid is (1996 ) wh ich dem ons t r a t ed t h e gen e r a l a p -plica bili t y of t h e m odeli n g t echn iq u es p r es en t ed h e r ei n .Th ey modele d bu lk a n d s olu t ion polyme r iza t ion s of s eve r a l m on ome r s a s well a s s ome s u s pen s ion polym -er iza t ion ca s e s (s ee Ta ble 1). Th e m os t con vin cin ga s pect s of t h a t wor k we r e t h e goo d model pr ed ict ionsof da t a f r om a la r ge nu m be r of r e s ea r ch gr ou p s a n dw idel y v a r yin g r ea ct ion con d it ion s .

7 .2 . S t y r e n e / E t h yl A c r yl a t e M o d e l P r e di c t i o nC a se S t u d y . An expe r ime n t a l s t u dy of t h e b u lk f r ee -r a d ica l co polyme r iza t ion of s t y r en e (S t y)/ e t h yl a cr yla t e(E A) w a s ca rr ie d ou t by M cM a nu s a n d P en lid is (1996 ).Th is t h or ou gh k in e t ic s t u dy in clu ded con ve r s ion , co m -pos it ion , a n d mole cu la r wei gh t da t a for r ea ct ions a t ava r ie t y of f ee d com pos it ion s , in it ia t or con cen t r a t ion s ,a n d t em pe r a t u r e s , wit h a n d wit h ou t a dd ed ch a in -t r a n s f er a gen t . Th e m odel des cr ibe d h e r ei n w a s a pp lie dt o t h e s e da t a , a n d s ome r ep r e s en t a t ive r e su lt s a r esh own be low .

In F igu r e s 16 a n d 17 a r e sh own model pr ed ict ionswh ich co rr ect ly p r ed ict t h e co n ver s ion v s t im e d a t a fo rt h r ee diff e r en t mon ome r f ee d com pos it ion s a t 50 a n d

60 C , r e s pect ivel y , fo r a g iven in it ia t or co n cen t r a t ion .Th e cu m u la t ive copolyme r com pos it ion is plott ed a ga in s tcon ver s ion in F igu r e 18 fo r t h r ee diff er en t m on ome r f ee dcom pos it ion s a t 60 C . Th es e r e s u lt s a r e r ep r es en t a t iveof a ll of t h e com pos it ion plo t s . Th e goo d a g r eeme n tbe t wee n t h e com pos it ion da t a a n d t h e m odel p r edict ions

T a b l e 1 . S om e E x a m p l es o f P r a c t i c a l A pp li c a t i o n s o f C o p o ly m e r i z a t i o n M o d e l s

B r oa dh ea d e t a l ., 1985 b a t ch , s emi b a t ch , an d co n t inu ou s em u ls ion s t y r en e (S t y)/ bu t a di en eG a r cia -Ru b io e t a l ., 1985 b u lk S t y / a cr ylon i t r ile (A N )B ha tt a cha r ya an d H a miele c, 1986 b u lk S t y / p -me t h yls t yr en eJ on es e t a l ., 1986 b u lk p -me t h yls t y r en e / MM AY a r a s k a vit ch e t a l ., 1987 b u lk p -me t h yls t y r en e /ANS t o r t i e t a l ., 1988 em u ls ion S t y /AN , A N / MM A , S t y /AN / MM AP en lidi s e t a l ., 1989 s olu t ion MM AS t o r t i e t a l ., 1989 em u ls ion A N / S t y / MM AX ie e t a l ., 1991 s u s pe n s ion V CX ie a n d H a miele c, 1993 c s olu t ion S t y / e t h yle n e dime t h a cr yla t e , S t y / di vin ylben zen eG a o a n d P en lidi s , 1996 b u lk a n d s olu t ion MM A , S t y, A N , BA , e t h yl a cr yla t e , me t h yl a cr yla t e ,

VAc , p -me t h yls t y r en e , ca r boxy lic a cid

F i g u r e 1 6 . Bu lk S t y / E A: model p r edi ct ion s of co n ve r s ion v s t imeda t a f r om M cM anu s an d P en lidi s (1996 ) fo r [A I BN ] 0.05 M a t 50C.

F i g u r e 17 . Bu lk S t y / E A: model p r edi ct ion s of co n ve r s ion v s t imed a t a f r om M cM a nu s a n d P en lidi s (1996 ) fo r [A I BN ] 0.05 M a t 60C.

F i g u r e 18 . Bu lk St y / E A: model pr edict ion s of co polyme r com -pos i t ion v s con ve r s ion da t a f r om McM a nu s a n d P en lidi s (1996 )fo r [A I BN ] 0.05 M a t 60 C.

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is n ot su r p r is in g s in ce t h e r ea ct ivit y r a t ios us ed in t h iss t u dy w e r e e s t im a t ed us in g t h e s t a t is t ica ll y v a lidme t h od s ou t li n ed e a r lie r .

F in a lly, r ep r e s en t a t ive p lo t s of nu mbe r - a n d wei gh t -a ve r a ge m ole cu la r wei gh t v s con ve r s ion a r e sh ow n inF igu r es 19 a n d 20 fo r t wo d iff e r en t in it ia t or co n cen t r a -t ions a t 60 C . Th e m odel co rr ect ly p r ed ict s t h e t r en dof in cr ea s ed m ole cu la r wei gh t a s a r e s u lt of a n in cr ea s ein t h e in it ia t or co n cen t r a t ion . Th e im por t a n t poin t t oco n s ide r in t h is ca s e s t u dy is t h a t on ly on e s e t of da t a ba s e pa r a me t e r s w a s us ed t o p r ed ict t h e da t a ove ra w ide v a r ie t y of r ea ct ion con d it ions .

7 .3 . B A / MM A / VA c M o d e l P r e di c t i o n C a se S t u d y .A co m p r eh en s ive s t u dy of t h e b u t yl a cr yla t e (BA)/ me t h ylme t h a cr yla t e (MM A)/ vin yl a ce t a t e (VAc) t e r polyme rs ys t em w a s r ecen t ly r epor t ed in Du be (1994 ) a n d Du bea n d P en lid is (1995 a - c, 1997 ). W h ile t h e e ven t u a lob ject ive of t h e w or k w a s t o devel op a n un de r s t a n d in gof t h e em u ls ion t e r polyme r iza t ion s ys t em, b u lk , s olu -

tio

n,

a nd em

uls

ion h

om o- a n

d copolym eriz

a tio

n d

a t a

we r e a ls o co lle ct ed . Th e m odel de s cr ibe d h e r ei n wa sa pp lie d t o t h e s e s ys t em s , a n d s ome r ep r e s en t a t iver es u lt s a r e d is cu ss ed bel ow . Th e pa r a me t er va lu es u s edin t h e m odel a r e fo un d in D u be, 1994.

A s a n exa m p le of a bu lk h om opolyme r iza t ion , t h ep r ed ict ion of bu lk BA d a t a f r om D u be e t a l . (1991b ), issh own in F igu r e 21. F r om t h e s e da t a , va lu e s for t h ep r opa ga t ion a n d t e r mi n a t ion r a t e co n s t a n t s (k p a n d k t ,r e s pect ively) we r e e s t im a t ed us in g on ly on e s e t of da t a ,h en ce, m or e a cc u r a t e va lu es of t h e s e r a t e co ns t a n t s a r er eq u ir ed (s ee Du be , 1994 ). U pd a t ed va lu es of t h es e r a t eco n s t a n t s a r e n ow a va il a ble f r om pu ls ed -la s e r polym -er iza t ion s t u d ie s (Bu ba ck e t a l ., 1989, 1994 ; Bu ba ck a n dDegen e r , 1993 ; Ly ons e t a l ., 1996 ). Va lu e s fo r t h e f r ee -volu m e p a r a m e t e r s A a n d V F pcr i t we r e f it t o t h e da t a a ton e s e t of r ea ct ion co n d it ion s . F or t h e BA s ys t em, r a t eco n s t a n t s fo r t r a n s f e r t o polyme r a n d t e r mi n a l dou ble -bon d r ea c t ions a r e unkn own , t hus n ece ss it a t in g t h ea cqu is it ion of molecu la r weigh t da t a . Th is is pa r t icu -la r ly d iff icu lt du e t o t h e pr e s en ce of s ign if ica n t a m oun t sof gel i n t h e s a m p le s (Du be e t a l ., 1991b ). Alt h ou gh n om ole cu la r wei gh t p lo t s a r e s h own fo r t h e s e da t a , i t w a sn o t ed t h a t M h n a n d M hw a r e ex t r emel y s ens it ive t o t h eva lu e of k f m . As men t ion ed ea r li er , a t h or ou gh in ve s -t iga t ion of bu lk a n d s olu t ion m odeli n g s h ow in g a la r ges e t of exa m p le s (in clu d in g B A , MM A , a n d VAc bu lk h om opolyme r iza t ion ) is foun d in Ga o a n d P en lid is(1996 ).

Bu

lk co

po

lymeriz

a tion

da t a r

epor t

ed in

Du

be a n

dP en lid is (1995 a ) (t h e r ea ct ion con d it ions ca n be foun dt h e r ei n ) a r e plo tt ed a lon g w it h m odel pr ed ict ions inF igu r e s 22 - 25. Con ve r s ion v s t ime da t a for b u lk B A/ MM A r un s a r e plo tt ed in F igu r e 22. Th e f r ee -volu mepa r a me t e r s A , K 3, a n d V F pcr it we r e f it t o d a t a a t on e s e tof r ea ct ion con d it ions . Th e r ea de r is r ef e rr ed t o D u bea n d P en lid is , 1995 a , fo r t h e m odel p r ed ict ion of co poly-me r co m pos it ion . Mole cu la r wei gh t d a t a a r e s h own inF igu r e 23. N o pa r a me t er f itt in g t o t h e m olecu la r wei gh tda t a wa s a tt em p t ed . As wa s t h e ca s e fo r t h e bu lk h om opolym e r iza t ion modeli n g of BA , t h e molecu la rwei gh t p r ed ict ions we r e q u it e s ens it ive t o t h e va lu e fo rt h e t r a n s f e r t o m on ome r co n s t a n t , k f m , fo r BA . Re s u lt sf r om a bu lk BA/VAc a n d a bu lk MM A/VAc r un a r e

F i g u r e 19 . Bu lk St y / E A: model pr edict ion s of nu m be r - an dwei gh t -a ve r a ge m ole cu la r wei gh t v s con ve r s ion da t a f r om Mc-M a nu s a n d P en lidi s (1996 ) fo r [A I BN ] 0.05 M a t 60 C.

F i g u r e 2 0 . Bu lk St y / E A: model pr edict ion s of nu m be r - an dwei gh t -a ve r a ge m ole cu la r wei gh t v s con ve r s ion da t a f r om Mc-M a nu s a n d P en lidi s (1996 ) fo r [A I BN ] 0.10 M a t 60 C.

F i g u r e 2 1 . Bu lk BA: model p r edi ct ion s of co n ve r s ion v s t ime d a t af r om D u b e e t a l . (1991b ) a t 60 C.

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s h own wit h model pr ed ict ion s in F igu r e s 24 a n d 25,r e s pect ivel y. As wa s t h e ca s e for t h e BA/ MM A r un s ,t h e f r ee -volu me pa r a me t e r s A, K 3, a n d V F pcr i t we r e f itt o t h

e da t a

. Th

e pred i

ction of t h

e co

m pos

ition

da t a fo r

bo t h B A/VAc a n d MM A/VAc a r e r epor t ed in D u be a n dP en lid is , 1995 a .

V a lu e s for t h e f r ee -volu me pa r a me t e r s A, K 3, a n dV F pcr it wer e f it t o t h e BA/ MM A/VAc co n ver s ion da t a f r omDu be a n d P en lid is , 1995b, for on e s e t of r ea ct ionco n d it ions . Con ve r s ion v s t im e da t a a n d m odel p r ed ic-t ion s for t h e b u lk t e r polyme r iza t ion s a r e plo tt ed inF igu r e s 26 a n d 27. Th e pr ed ict ion of t h e com pos it ionda t a is plo tt ed in F igu r e 28. Th e pr ed ict ion s of t h et e r polyme r com pos it ion da t a wer e a ch ie ved t h r ou gh t h eus e of co p o l y m e r i z a t i o n r ea ct ivit y r a t ios es t im a t ed us in gt h e op t im a l met h ods des cr ibe d ea r lie r . Molecu la r weigh tp r ed ict ion s a ls o fo llowed s imil a r t r en d s (s ee Du be ,1994 ).

In t h e p r edict ion of em u ls ion polym er iza t ions , a la r gernu m be r of ph ys ica l cons t a n t s a n d r a t e pa r a m e t er s wer er eq u ir ed . I t wa s n o t poss ible t o ob t a in a ll of t h eco n s t a n t s f r om t h e open li t e r a t u r e, n or w a s i t poss iblet o o

bt a

in

em pirica

l es t

ima t

es of t h

e pa r a

mete

r s. T

he

re

-fo r e, m a n y of t h e p a r a me t e r s us ed in t h e p r ed ict ion of t h e e m u ls ion polym e r iza t ion da t a we r e a rr ived a t byw a y of ca lcu la t ed gu e ss es . In s eve r a l ca s e s , un k n ownpa r a me t e r s we r e given va lu e s pr eviou s ly r epor t ed fo rot h er m on ome r s (s ee Du be , 1994 ). P r edict ion of co n ver -s ion fo r t h e BA em u ls ion polyme r iza t ion r un de s cr ibedin Du be a n d P en lid is (1995 c) is plo tt ed in F igu r e 29.De s p it e t h e un ce r t a in t y in t h e va r ious r a t e pa r a m e t e r sfo r BA , t h e m odel pr ed ict ion w a s qu it e goo d . E m u ls iont e r polyme r iza t ion com pos it ion v s con ve r s ion da t a a r es h own in F igu r e s 30 a n d 31. Th is is ye t a n o t h erco n f ir m a t ion of t h e a cc u r a cy of t h e r ea ct ivit y r a t ios a swell a s t h e pa r t it ion coeff icie n t s us ed in t h e model(Du be , 1994 ).

F i g u r e 22 . Bu lk B A/ MM A: model p r edi ct ion s of co n ve r s ion v s t ime d a t a f r om D u b e an d P en lidi s (1995 a ) fo r f 10 ) 0.439 a t 60C.

F i g u r e 2 3 . Bu lk BA/ MM A: model p r edi ct ion s of m olecu la r weigh tv s co n ve r s ion da t a f r om Du b e a n d P en lidi s (1995 a ) fo r f 10 ) 0.439,[A I BN ] ) 0.005 m ol / L a t 60 C.

F i g u r e 2 4 . Bu lk BA/VAc: model p r edi ct ion of co n ve r s ion v s t imed a t a f r om Du b e a n d P en lidi s (1995 a ) fo r f 10 ) 0.8, [A I BN ] )0.00054 m ol / L a t 60 C.

F i g u r e 25 . Bu lk MM A/VAc: model p r edi ct ion of co n ve r s ion v s t ime d a t a f r om Du b e a n d P en lidi s (1995 a ) fo r f 10 ) 0.3, [A I BN ] )0.01 m ol / L a t 60 C.

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http://pubs.acs.org/action/showImage?doi=10.1021/ie960481o&iName=master.img-019.png&w=234&h=223http://pubs.acs.org/action/showImage?doi=10.1021/ie960481o&iName=master.img-018.png&w=232&h=223http://pubs.acs.org/action/showImage?doi=10.1021/ie960481o&iName=master.img-017.png&w=236&h=225http://pubs.acs.org/action/showImage?doi=10.1021/ie960481o&iName=master.img-016.png&w=232&h=225

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7 .4 . S e m i b a t c h P r o d u c t i o n o f M u l t i c om p o n e n t

P o ly m e r s . Th e p r a ct ica l im p leme n t a t ion of m on ome rf ee d p olicie s r eq u ir es t h e u s e of on -li n e (o r poss ibl y off-li n e) m ea su r em en t s t o pe r mi t on e t o a d jus t for un co n -t r olle d va r ia t ion s in r ecipe im pu r it ie s s u ch a s O2 (wh ichis a r a d ica l s ca ven ge r a t t em pe r a t u r e s bel ow a bou t 100C) a n d o t h er im pu r it ies in t h e r ecipe com pon en t s wh ichca n a ff ect r a d ica l con cen t r a t ion . An ex t en s ive r evie wof o n -li n e s ens or s for polym e r iza t ion r ea ct or s wa spu bli sh ed by C h ie n a n d P en lid is (1990 ). E xa m p le s of on -li n e ca lor ime t r y, ga s ch r om a t og r a ph y , a n d d en s it o-m e t r y a r e d is cuss ed be low .

7 .4 .1 . C a lo r i m e tr i c C o n tr o l o f M o n om e r Fee d i na C o p o ly m e r i z a t i o n . Th e ins t a n t a n eous h ea t gen er a -t ion d u e t o polyme r iza t ion (V Q ) is g iven by

wh e r e V is t h e volu me of polyme r izin g mi x t u r e, Q ist h e in s t a n t a n eou s h ea t gen e r a t ion r a t e du e t o polym -e r iza t ion in ca l L - 1 s - 1, - H 1 is t h e h ea t of polym e r i-za t ion wh en m on ome r 1 a dd s t o ei t h e r polyme r r a d ica lt ype a n d s imil a r ly fo r - H 2.

Th e f ee d r a t e of M 1, F 1, in , is g iven by

fo r a n y m on ome r f ee d p olicy, w h e r e

D ividin g eq 257 b y eq 258 a n d t hu s elimi n a t in g t h e t ot a lpolyme r r a d ica l con cen t r a t ion , on e ob t a in s

F i g u r e 2 6 . Bu lk B A/ MM A/VAc ( 30 / 30 / 40 w t %): model p r edic-t ion s of co n ve r s ion v s t ime d a t a f r om D u b e an d P en lid is (1995b )a t 50 C.

F i g u r e 2 7 . Bu lk B A/ MM A/VAc ( 30 / 30 / 40 w t %): model p r edi c-t ion s of co n ve r s ion v s t im e d a t a f r om D u b e an d P en lid is (1995b )a t 70 C.

F i g u r e 28 . Bu lk B A/ MM A/VAc ( 30 / 30 / 40 w t %): model pr edic-t ion s of t e r polyme r co mp os i t ion v s co n ve r s ion d a t a f r om Du b e a n dP en lidi s (1995b ).

F i g u r e 2 9 . E m u ls ion BA: model p r edi ct ion of co n ve r s ion v s t imed a t a f r om D u b e e t a l . (1995 c) .

V Q ) [(k 11 1 + k 21 2)N 1(- H 1) + (k 12 1 +k 22 2

)N 2(- H 2)] (257 )

F 1, in ) [ 1 - 21 - R (N 1 / N 2)]N 1 (258 )

R ) F 2, in / F 1, in (259 )

http://pubs.acs.org/action/showImage?doi=10.1021/ie960481o&iName=master.img-023.png&w=229&h=221http://pubs.acs.org/action/showImage?doi=10.1021/ie960481o&iName=master.img-022.png&w=236&h=219http://pubs.acs.org/action/showImage?doi=10.1021/ie960481o&iName=master.img-021.png&w=230&h=224http://pubs.acs.org/action/showImage?doi=10.1021/ie960481o&iName=master.img-020.png&w=232&h=223

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