High Conversion Diffusion-Controlled Polymerization - ACS Publications

3 High Conversion Diffusion-Controlled Polymerization

Downloaded by UNIV OF CALIFORNIA SAN FRANCISCO on December 15, 2014 | http://pubs.acs.org Publication Date: July 31, 1979 | doi: 10.1021/bk-1979-0104.ch003

F. L . M A R T E N

and A . E. H A M I E L E C

McMaster University, Hamilton, Canada L8S 4M1

In bulk, s o l u t i o n and emulsion p o l y m e r i z a t i o n dramatic p h y s i c a l changes occur during the course o f r e a c t i o n . As polymer conc e n t r a t i o n increases a p o i n t is reached where a p p r e c i a b l e chain entanglements occur and e v e n t u a l l y a g l a s s y - s t a t e t r a n s i t i o n may r e s u l t . These p h y s i c a l changes o f t e n have a s i g n i f i c a n t e f f e c t on both r a t e o f p o l y m e r i z a t i o n and molecular weight development and any attempt a t modelling such r e a c t i o n s must p r o p e r l y account f o r these phenomena. In t h i s manuscript we review the p r i n c i p l e s o f bulk and s o l u t i o n p o l y m e r i z a t i o n with p a r t i c u l a r emphasis on high conversion (high polymer concentrations) r a t e o f p o l y m e r i z a t i o n and molecular weight development. In the l i t e r a t u r e there is only one serious attempt t o devel o p a d e t a i l e d mechanistic model o f f r e e r a d i c a l p o l y m e r i z a t i o n a t high conversions ( 1 , 2 , 3 ) . T h i s model a f t e r Cardenas and O ' D r i s c o l l i s d i s c u s s e d i n some detail p o i n t i n g out its important l i m i t a t i o n s . The present authors then d e s c r i b e the development o f a semi-empirical model based on the f r e e volume theory and show t h a t t h i s model adequately accounts f o r chain entanglements and g l a s s y - s t a t e t r a n s i t i o n i n bulk and s o l u t i o n p o l y m e r i z a t i o n o f methyl methacryl a t e over wide ranges o f temperature and solvent c o n c e n t r a t i o n . P h y s i c a l Phenomena o f High

Conversions

I t i s appropriate t o d i f f e r e n t i a t e between polymerizations occuring a t temperatures above and below the g l a s s t r a n s i t i o n point(Tg) o f the polymer being produced. For polymerizations below Tg the d i f f u s i o n c o e f f i c i e n t s o f even small monomer molecules can f a l l a p p r e c i a b l y and as a consequence even r e l a t i v e l y slow r e a c t i o n s i n v o l v i n g monomer molecules can become d i f f u s i o n c o n t r o l l e d complicating the mechanism o f p o l y m e r i z a t i o n even f u r t h e r . F o r polymerizations above Tg one can reasonably assume t h a t r e a c t i o n s i n v o l v i n g small molecules are not d i f f u s i o n c o n t r o l l e d , except perhaps f o r extremely f a s t r e a c t i o n s such as those i n v o l v i n g t e r m i n a t i o n o f small r a d i c a l s .

0-8412-0506-x/79/47-104-043$07.00/0 © 1979 American Chemical Society In Polymerization Reactors and Processes; Henderson, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1979.

POLYMERIZATION

44

REACTORS AND PROCESSES

Polymerizations Above T . Let the p o l y m e r i z a t i o n begin i n pure monomer. As the c o n c e n t r a t i o n o f polymer chains i n c r e a s e s i n i t i a l l y one observes a r e l a t i v e l y small i n c r e a s e i n the terminat i o n r a t e constant. T h i s i s r e l a t e d t o the e f f e c t o f polymer conc e n t r a t i o n on c o i l s i z e . A r e d u c t i o n i n c o i l s i z e i n c r e a s e s the p r o b a b i l i t y o f f i n d i n g a chain end near the s u r f a c e and hence causes an i n c r e a s e i n k-^. Soon t h e r e a f t e r a t conversions 15-20% polymer chains begin t o entangle causing a 'dramatic r e d u c t i o n i n r a d i c a l chain t r a n s l a t i o n a l m o b i l i t y g i v i n g a r a p i d drop i n k^. The onset o f chain entanglements depends on polymer c o n c e n t r a t i o n , molecular weight and r e a c t i o n temperature. I t i s reasonable t o assume as d i d Cardenas and O ' D r i s c o l l (i.,2.,3) t h a t l a r g e r r a d i c a l chains become entangled before smaller ones and that the i n t r i n s i c t e r m i n a t i o n r a t e o f smaller not entangled r a d i c a l s would be una f f e c t e d u n t i l l a t e r i n the p o l y m e r i z a t i o n . The concept t h a t the t e r m i n a t i o n o f some smaller r a d i c a l s never becomes d i f f u s i o n c o n t r o l l e d i s however questionable and with the r e d u c t i o n o f even some f r e e volume even these r e a c t i o n s might become d i f f u s i o n c o n t r o l l e d . . The s i g n i f i c a n t r e d u c t i o n i n t e r m i n a t i o n r a t e o f t e n causes an almost e x p l o s i v e i n c r e a s e i n r a d i c a l p o p u l a t i o n and r a t e of polymerization. The extent o f the a u t o a c c e l e r a t i o n i n Rp depends a great d e a l upon molecular weight development. For example i n the bulk p o l y m e r i z a t i o n o f MMA most o f the polymer chains are produced by t e r m i n a t i o n r e a c t i o n s . There i s as a consequence a l a r g e r i n c r e a s e i n molecular weight as k^ f a l l s and t h i s g i v e s a m u l t i p l i e r e f f e c t i n i n c r e a s i n g the number o f polymer r a d i c a l s which are entangled. For p o l y m e r i z a t i o n above T the propagation r e a c t i o n s do not become d i f f u s i o n c o n t r o l l e d and as a consequence a conversion o f 100$ i s approached i n a reasonable time s c a l e .


g

g

P o l y m e r i z a t i o n Below T . For p o l y m e r i z a t i o n below Tg the s i t u a t i o n i s more complex. To i l l u s t r a t e the phenomena we r e f e r t o F i g u r e s 1, 2, 3 and h. These F i g u r e s i n v o l v e monomers which are normally polymerized below T (MMA, AN, VC). The exception i s polystyrene which i s u s u a l l y polymerized above T . Rather than begin our d i s c u s s i o n at low conversion i t i s convenient, as w i l l be seen l a t e r , t o begin a t the l i m i t i n g conversion. When p o l y m e r i z a t i o n s are done below T the monomer a c t s as a p l a s t i c i z e r and a g l a s s y - s t a t e occurs a t a conversion l e s s than 100$. When a g l a s s i s formed one experiences s o l i d - s t a t e p o l y m e r i z a t i o n with a much greater time s c a l e . In the normal time s c a l e the r a t e o f p o l y m e r i z a t i o n may be taken as zero. The existence o f t h i s g l a s s y s t a t e t r a n s i t i o n has been confirmed f o r s e v e r a l polymer systems i n bulk and emulsion p o l y m e r i z a t i o n by F r i i s and Hamielec {k) and more r e c e n t l y by Berens who has independently measured Tg values f o r PVC p l a s t i c i z e d with i t s own monomer. This information i s shown i n F i g u r e s 1, 2 and 3. F i g u r e 1 shows a l i m i t i n g conv e r s i o n o f 92% f o r PMMA-MMA a t a p o l y m e r i z a t i o n temperature o f T0°C. In other words a s o l u t i o n o f 92% wt PMMA i n Q% MMA has a g l a s s t r a n s i t i o n p o i n t o f 70°C. F i g u r e 2 shows l i m i t i n g converg

g

g

g

In Polymerization Reactors and Processes; Henderson, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1979.


3.

MARTEN AND HAMDELEC

Figure

Diffusion-Controlled

Polymerization

1. Bulk polymerization of MMA initiated by AIBN (9): temperature 70°C; (O)[I] — 0.0258 mol/L; (X) W o = 0.01548 mol AIBN/L. 0


45


46

POLYMERIZATION REACTORS AND PROCESSES

Figure

2. Polymerization temperature vs. limiting conversion for different monomer-polymer systems (4): ( V ) PMMA; ( Q ) PAN; (X) PS; (O) PVC.



3.

MARTEN AND HAMIELEC


Polymerization

100

50 —

7>

50

WT

%

- 100 10

20

VC

I 30

Figure 3. T of PVC vs. content of VC: ( V ) data measured by means of deviation from Flory-Huggins isotherm (5); (X) data measured thermomechanically (18); (O) data obtained from limiting conversion (4). g

American Chemical Society Library 1155 16th st. N. w. Washington, 0. C. 20036


47



48



3.

MARTEN AND HAMDELEC


Polymerization

49

sions p l o t t e d versus p o l y m e r i z a t i o n temperature, f o r PMMA/MMA, PAN/AN, PS/S, PVC/VC systems which have been e x t r a p o l a t e d t o a l i m i t i n g conversion o f 100$ t o estimate the Tg o f t h e polymer produced. These Tg values are i n general agreement with values measured by DSC and mechanical spectroscopy. L a t e r i n t h e manuscript i t w i l l be shown t h a t t h e e f f e c t o f r e s i d u a l monomer on g l a s s t r a n s i t i o n p o i n t as measured v i a k i n e t i c s and l i m i t i n g conversion, agrees with t h e free-volume theory. From t h e o b s e r v a t i o n o f l i m i t i n g conversions below 100$, i t i s c l e a r t h a t even r e l a t i v e l y slow propagation r e a c t i o n s i n v o l v i n g the small monomer molecule become d i f f u s i o n c o n t r o l l e d w e l l below the l i m i t i n g conversion. This i s confirmed by measurements o f kp a f t e r Hayden and M e l v i l l e (6), i n F i g u r e k where a l i m i t i n g conversion t o about Q0% was observed and i t was found t h a t kp a l r e a d y began t o drop i n value a t a conv e r s i o n o f about 50%. These observations have s i g n i f i c a n t i m p l i c a t i o n s as f a r as t e r m i n a t i o n r e a c t i o n s a r e concerned. I t must be concluded t h a t t h e magnitude o f k^ even f o r the smallest r a d i c a l r e a c t i o n s must be d i f f u s i o n - c o n t r o l l e d probably from t h e onset o f chain entanglements. In other words k-^ f o r t e r m i n a t i o n o f small r a d i c a l s must decrease s i g n i f i c a n t l y w i t h conversion. As mentioned above f o r bulk MMA p o l y m e r i z a t i o n , polymer chains are produced mainly by t e r m i n a t i o n r e a c t i o n s and hence t h e v a r i a t i o n o f k^, Rp and molecular weight a r e s t r o n g l y coupled phenomena. T h i s i s p a r t i c u l a r l y t r u e from the onset o f chain entanglements but l a t e r i n t h e p o l y m e r i z a t i o n when k^ has f a l l e n a p p r e c i a b l e t r a n s f e r t o monomer becomes an important polymer p r o ducing r e a c t i o n , l i m i t i n g t h e u l t i m a t e molecular weights t h a t can be obtained. In c e r t a i n p o l y m e r i z a t i o n s such as VC and styrene above 100°C, t r a n s f e r r e a c t i o n s c o n t r o l molecular weight development and the a u t o a c c e l e r a t i o n i n Rp i s smaller w i t h v i r t u a l l y no e f f e c t on molecular weight developments. D e s i r a b l e Features o f a P o l y m e r i z a t i o n Model a t High

Conversion.

A u s e f u l model should account f o r a r e d u c t i o n o f k t and kp with i n c r e a s e i n polymer molecular weight and c o n c e n t r a t i o n and decrease i n solvent c o n c e n t r a t i o n a t p o l y m e r i z a t i o n temperatures both below and above the T o f t h e polymer produced. For a mechanistic model t h i s would i n v o l v e many complex steps and a l a r g e number o f a d j u s t a b l e parameters. I t appears t h a t the o n l y r e a l i s t i c s o l u t i o n i s t o develop a semi-empirical model. In t h i s context t h e free-volume theory appears t o be a good s t a r t i n g p o i n t . g

Mechanistic Model - Cardenas and O ' D r i s c o l l ( l , 2, 3 ) . The b a s i s o f t h i s model i n v o l v e s the assumption that r a d i c a l s with a chain l e n g t h > x a r e entangled with the f o l l o w i n g r e l a t i o n ship based on v i s c o s i t y measurements used t o e s t a b l i s h x . c

c

K

c

=

*p

#

x

c

(1)


50

POLYMERIZATION

where

REACTORS AND PROCESSES

p i s the polymer volume f r a c t i o n . x i s the number average chain l e n g t h at the p o i n t o f chain entanglement. 3 i s an a d j u s t a b l e parameter (3 u s u a l l y u n i t f o r v i s c o s i t y measurements). K i s the entanglement constant. c

c

In a p p l y i n g equation ( l ) Cardenas and O ' D r i s c o l l use x as the c r i t i c a l chain l e n g t h f o r chain entanglement and permit x to decrease as p i n c r e a s e s during the p o l y m e r i z a t i o n according t o equation ( l ) . Therefore, during the course o f p o l y m e r i z a t i o n they note three kinds o f t e r m i n a t i o n r e a c t i o n s : c


c

k

R

#

r

+

R« s

— •

+

(R.) s

t

where r , s
x„ c

where r > x ,

s >

c

e

(R-) r

k^.

t —>.c

+

(R») —* e

e

c

c

I t i s assumed t h a t k-t i s independent o f conversion and t h a t the t e r m i n a t i o n constant f o r entangled r a d i c a l s i s given by

P and

x

s

e

n

finally

k

t

c

(

k

k

t t

)

h

( 3 )

e

T h i s model accounts f o r the c o u p l i n g between molecular weight development and a u t o a c c e l e r a t i o n i n R . However, two o f t h e b a s i c assumptions i) i s independent o f conversion ii) kp i s independent o f conversion are c e r t a i n l y not v a l i d f o r p o l y m e r i z a t i o n s below T . T h i s model does not account f o r a g l a s s y s t a t e - t r a n s i t i o n and hence cannot p r e d i c t the observed l i m i t i n g conversion. For temperatures above Tg i t may prove t o be s u c c e s s f u l . U n f o r t u n a t e l y , i t has not yet been evaluated under these c o n d i t i o n s . g

A Model Based on Free-Volume Theory. Three main problems are i n v o l v e d i n model development.


3.

1. 2.


3.

MARTEN AND HAMIELEC


51

Polymerization

Determination o f t h e conversion a t which s i g n i f i c a n t chain entanglements f i r s t occur. Development o f a r e l a t i o n s h i p w h i c h g i v e s t h e d e c r e a s e i n t h e t e r m i n a t i o n r a t e constant as a f u n c t i o n o f temperature and polymer molecular weight and c o n c e n t r a t i o n . Development o f a r e l a t i o n s h i p w h i c h g i v e s t h e d e c r e a s e i n t h e propagation r a t e constant as a f u n c t i o n o f temperature and polymer molecular weight and c o n c e n t r a t i o n .

The r a t e o f p o l y m e r i z a t i o n isothermal bulk polymerization k

where k^ k-t f k^ [I] e dp dM x t

= = = = = = = = = =

2

\

c a n b e shown t o b e i n t h e c a s e o f

/ f - k , [I]

k «t

propagation rate constant, termination rate constant, initiator efficiency, decomposition constant o f i n i t i a t o r . i n i t i a l i n i t i a t o r concentration, (dp - dj^)/dp, volume c o n t r a c t i o n f a c t o r , d e n s i t y o f polymer. d e n s i t y o f monomer degree o f c o n v e r s i o n , time

I n o r d e r t o e s t i m a t e t h e dependence o f t h e t e r m i n a t i o n r a t e c o n s t a n t o n c o n v e r s i o n , m o l e c u l a r -weight a n d t e m p e r a t u r e , t h e f o l l o w i n g i s assumed: k-^ becomes d i f f u s i o n c o n t r o l l e d when t h e d i f f u s i o n c o e f f i c i e n t f o r a p o l y m e r r a d i c a l Dp becomes l e s s t h a n o r e q u a l t o a c r i t i c a l d i f f u s i o n c o e f f i c i e n t D^ Per K ± K (5) P P c r

I t i s f u r t h e r assumed t h a t t h e t e r m i n a t i o n r a t e c o n s t a n t b e y o n d t h i s c o n v e r s i o n c a n be e x p r e s s e d b y eq. (6a) and a t t h e c r i t i c a l point (6b). k, = fc.D (6a) k, = k. D (6b) t i p t 1 p c

r

c

r

where k^ = t e m p e r a t u r e dependent p r o p o r t i o n a l i t y constant. Dp = d i f f u s i o n c o e f f i c i e n t o f polymer r a d i c a l . D = c r i t i c a l d i f f u s i o n c o e f f i c i e n t o f polymer r a d i c a l . Pcr I f no e n t a n g l e m e n t s a r e p r e s e n t , t h e d i f f u s i o n c o e f f i c i e n t p o l y m e r m o l e c u l e i s , a c c o r d i n g t o B e u c h e (T_), g i v e n a s D

p

= ( 6 /k 'M) e x p (-A/V ) o

where M 6 k A Vp Q

2

= = = = = =

2

(7)

F

molecular weight o f polymer jump f r e q u e n c y . jump d i s t a n c e . constant constant f r e e volume.

(monodispersed).


of a

52


V

F

V

F

i n the case o f b u l k or s o l u t i o n p o l y m e r i z a t i o n i s equal t o (0.025^ (T-T ^))

=

p


^ +

(0.025+a (T-T ))

g

M

(0.025+a (T-T q

))

g

g M

(8)

^

where M, P and S denote monomer, polymer and solvent r e s p e c t i v e l y . T = p o l y m e r i z a t i o n temperature. V = volume. V = t o t a l volume. Tg = g l a s s t r a n s i t i o n p o i n t o f monomer. T

=

a a£ ag

a

= =

a

£ ~ g expansion c o e f f i c i e n t f o r the l i q u i d s t a t e , expansion c o e f f i c i e n t f o r the g l a s s y s t a t e .

It i s further established that -

T„

-

#-

(9)

where T i s the g l a s s temperature o f the i n f i n i t e molecular weight polymer and M i s the cumulative number average molecular weight. Q i s a constant independent o f temperature. g o o

n

I f equation ( 6 a ) i s i n s e r t e d i n t o k

t

=

k

-l V (

2

(7)

A/

one o b t a i n s 1 Q

S /k2M)exp(- V )

( )

F

For a polymer with a molecular weight d i s t r i b u t i o n the proper molecular weight average t o use i n equations (7) and (10) can be determined u s i n g the f o l l o w i n g c o n s i d e r a t i o n s . In the case o f a heterogeneous polymer i t has been shown that (7,19 « 2 0 ) . n

=

k« M If w

f o r not entangled polymer n

=

s o l u t i o n s , and

3 , 5

k M 4 w f o r entangled polymer s o l u t i o n s . The d i f f u s i o n c o e f f i c i e n t s o f entangled polymers i n s o l u t i o n w i l l most c e r t a i n l y depend on the v i s c o s i t y o f the medium and v i c e versa. I t i s reasonable t h e r e f o r e t o expect t h a t the d i f f u s i o n c o e f f i c i e n t would c o r r e l a t e w e l l w i t h the weight average molecular weight o f the polymer. M i s t h e r e f o r e used with equation (10) giving k ^ ^ / k ^ ) exp(- / y ) = k (10a) u

w

A

F

f o r unentangled polymer

t

solutions


3.

MARTEN AND HAMIELEC


53

Polymerization

2

~ - n ( -6 / k M ) e x p ( - A / V ) = k

for

(10b)

n

k

o

2

w

F

t

entangled polymer s o l u t i o n s . I f e q u a t i o n ( 1 0 a ) i s c o m b i n e d w i t h ( 6 b ) a n d r e a r r a n g e d , one

has


*1 K-3= ( j )' = M t cr a n d

^wcrl

V

m

(10c)

exp(+A/V ) c r l

w c r i

F

m u s

" F

" t ^e e s t i m a t e d f o r e a c h p o l y m e r i z a t i o n b y crl s a t i s f y i n g equation (10c). K 3 depends o n l y on t e m p e r a t u r e a n d A i s a c o n s t a n t and t h e r e f o r e t h e r e l a t i o n s h i p between M _ a n d V* wcrl F i depends o n t e m p e r a t u r e a l o n e t h r o u g h e q u a t i o n ( 1 0 c ) At a constant temperature, t h e magnitude o f b o t h a n d V" ^ c a n c h a n g e w i t h i n i t i a t i o n rate or concentration o f solvent or chain transfer agent. The r e l a t i o n s h i p g i v e n b y e q u a t i o n ( 1 0 c ) i s h o w e v e r t h e same. I f i t i s assumed t h a t c h a i n e n g a n g l e m e n t s o c c u r s o o n a f t e r k^ becomes d i f f u s i o n c o n t r o l l e d , t h e n one h a s a s a good a p p r o x i m a t i o n c

r

F

k

t

=

e x

M

k

t

=

k

y F

)

(

1 0 d

)

w

t

o

A

P(- /

=

cr

(

~

) e

n" M „ wcrl

x

(

P "

A

/

V

} F

c r l

C o m b i n i n g e q u a t i o n s ( l O d ) a n d ( l l ) , k^ i s o b t a i n e d a s a f u n c t i o n o f c o n v e r s i o n and t h e weight average m o l e c u l a r weight

£ - ) t o k

=

(ijEEl) M w

ex (-A(^- - ± F F P

V

V

)) , c r l

(12)

W h i l e K 3 a n d A a s e x p l a i n e d l a t e r were e s t i m a t e d u s i n g a f i t to e x p e r i m e n t a l d a t a m and n a r b i t r a r i l y s e t e q u a l t o 0.5 and 1.75 r e s p e c t i v e l y . The r e m a i n i n g p r o b l e m i n t h e m o d e l d e v e l o p m e n t i s t o e s t i m a t e t h e decrease i n k as a f u n c t i o n o f c o n v e r s i o n . As t h e r e a c t i o n proceeds beyond t h e p o i n t o f c h a i n entanglement, a c r i t i c a l conv e r s i o n i s r e a c h e d where t h e p r o p a g a t i o n r e a c t i o n becomes d i f f u sion c o n t r o l l e d and kp begins t o f a l l w i t h f u r t h e r i n c r e a s e i n p o l y m e r c o n c e n t r a t i o n . A t t h e c r i t i c a l c o n v e r s i o n , one may w r i t e p


54


*3^ •where k p

M w

=

(13)

= t h e propagation constant below t h e c r i t i c a l conversion. = the d i f f u s i o n c o e f f i c i e n t o f t h e monomer at the c r i t i c a l conversion. = a proportionality factor.

Q

i|>3


kp *o

cr

Beuche (jj gives t h e f o l l o w i n g expression f o r the d i f f u s i o n c o e f f i c i e n t o f a small molecule i n a polymer s o l u t i o n . This equation a l s o known as the D o l i t t l e equation i s D = (* &1/6) exp(- B/y ) (11*) M

2

F

Beyond t h e c r i t i c a l conversion kp i s given by k

p

(15)

B

=

*

exp(- / V )

=

exp (- B( 7 " -

3

F

and ^ Po k

V

f

According t o Beuche (j)

(16) V

Fcr

2

B = 1 . 0 and t h i s value i s used here.

The general r a t e e x p r e s s i o n f o r the complete conversion interval i s

t

(

( l

d

wcrl

Q

.

° )

V/ )

( 1 - x) exp(-

x )

Conversion I n t e r v a l 1 : Interval 2 : Interval 3 :

a = 0, a = 0.875, a = 0.875,

2

B = 0, B = 0, B = 1.0,

(IT) A = 0 A = 1.11 A = 1.11

The determination of the conversion i n t e r v a l s a r e d i s c u s s e d l a t e r , a f t e r the molecular weight development. The instantaneous number and weight average degrees o f p o l y m e r i z a t i o n a r e g i v e n by k. R T

=

f L

\

=

1_

h

=

±JL

*X

+

c

+

"

c

JSi SIM1

when t e r m i n a t i o n i s s o l e l y by d i s p r o p o r t i o n a t i o n ; and t h e cumulat i v e averages by xM


MARTEN AND HAMIELEC

3.

2M


55

Polymerization

X dx

(19)


T

I t should he understood that the weight average molecular weights appearing i n equation (17) are cumulative ones. The conversion-time h i s t o r y i s obtained by simultaneous s o l u t i o n o f equations (17) and (19). The conversion i n t e r v a l s are determined i n the f o l l o w i n g way: Values of A and

K

3

i s guessed and equations (17) and (19) are i n t e g r a t e d i n i n t e r v a l 1. The c a l c u l a t e d cum and a c a l c u l a t e d p are s u b s t i t u t e d i n t o equation ( 1 0 c ) . The end of i n t e r v a l 1 i s reached when the equation i s s a t i s f i e d . The i n t e g r a t i o n i s c a r r i e d f u r t h e r i n t o i n t e r v a l 2 w i t h the a p p r o p r i a t e parameters. The e r r o r o f f i t i n these i n t e r v a l s i s noted and v

A and

K

3

a d j u s t e d a c c o r d i n g l y . T h i s procedure thus e s t a b l i s h e s the c o r r e c t end of i n t e r v a l 1 . We next guess the c r i t i c a l f r e e volume where k-p begins t o f a l l and then i n t e g r a t e through i n t e r v a l 3 t o l i m i t i n g conversion. The e r r o r o f f i t i s used t o e s t a b l i s h F and cr 2. the c r i t i c a l conversion. v

Comparison o f Simulated and Measured Rate Data. Model Parameters used with Equations Polymerization.

x 2

k

= t

U.U8

for

MMA

v

1

: c

r

l

+ K

11

• IQ^expC '

1 k

^

/

m

Q

l

) (l/mole min)

(20)

o

and p C

(19)

p

^

W

and

Data a f t e r Balke(£), I t o ( l O ) and Hayden and M e l v i l l e ( 6 _ ) were c o r r e l a t e d w i t h an Arrhenius type plot g i v i n g the f o l l o w i n g equation which was used i n a l l the s i m u l a t i o n s .

— : kj. o

M

(17)

=

3

k

— t c

Refer t o equation ( 1 0 c ) follows. = 0.563

exp(8,900

and equation (21)

cal/mol/RT)

w

where R

=

1.986

c a l / ( g mole)(°K)


which

(21)

56


T A m

in = =

(°K) 1.11 0.5

Equations (17) and (19) are i n t e g r a t e d i n I n t e r v a l 1 u n t i l the cumulative % and V s a t i s f y equation ( 1 0 c ) . T h i s provides ^wcrl F * Refer to F i g u r e 5 f o r the a c t u a l K 3 data. F

a n ( i

V

c r l


= 0.066

(22)

*cr2

with B = 1. T h i s parameter i s independent o f temperature polymer molecular weight and c o n c e n t r a t i o n .

V"

:

F

The f r e e volume i s c a l c u l a t e d u s i n g equations (9) with the f o l l o w i n g parameters. 0.U8

T

g

a

M

T a

=

llU

=

10-

g M

T

(^C)"

Q

=

-102°C

=

)

3

^ )

2.208 • 10

=

M

d^

(°

, 9 T 3

1

" -

1.2g/cm

=

(11) (11)

C

10-

and

(11) 1

(o )

=

(8)

(11)

1

(°C)

3

SS

d

(°C)-

-106

=

g

3

x icr

and

henzene as

solvent

Ig/mole]degi3ee(ii)

5

l 6 1 +

•10- -t°C)g/cm' 3

(12)

3

I t should be noted t h a t polymer volume f r a c t i o n i s r e a d i l y t e d t o conversion. k

d

:

AIBN (13)

k

d

= 6.32

• 10

1 6

exp("

1 5

'

k 6

k c a

^/

m o l e

)mln-

... BOP

(14)=

k

= 7.5*10" min" , kd if

d

1

70 f was e

:

(23a)

= 3.6^*10~

5

min"

1

50

e the volume c o n t r a c t i o n f a c t o r i s c a l c u l a t e d u s i n g dens i t y data as P " M x c = - 3 ^ — (2U) d

/

M

1

set equal t o u n i t y i n a l l the s i m u l a t i o n s .

d

C

conver-

: C

M

=

8 . 9 3 • IG"- expC 'f 2

^J™^)

o

l

(25)



MARTEN AND HAMIELEC

Figure


5. Arrhenius

plot of

Polymerization

k

s


58



Bulk P o l y m e r i z a t i o n o f MMA B a l k e d Data. F i g u r e s 1, 6 , 7 and 8 show r a t e data a f t e r Balke i n t h e temperature range, 50 - 90°C. A l s o i n c l u d e d i s one r a t e curve a f t e r Nishimura (l£). There i s o b v i o u s l y e x c e l l e n t agreement even a t l i m i t i n g conversion. F i g u r e s 9 and 10 show a comparison o f measured and p r e d i c t e d weight average molecular weights. The agreement w i t h % i s e x c e l l e n t , but w i t h M^ o n l y f a i r a t intermediate conversions near t h e onset o f c h a i n entanglements. The r e p r o d u c i b i l i t y o f % when a h i g h molecular weight spike i s generated i s r a t h e r poor and perhaps t h i s may e x p l a i n some o f the d e v i a t i o n . I t o ' s Data. F i g u r e 11 shows I t o s ( l 0 i ) r a t e data a t 1+5°C f o r a very wide range o f i n i t i a t o r c o n c e n t r a t i o n s (AIBN: 0 . 2 - 0 . 0 0 6 2 5 gmole/£). The agreement i s e x c e l l e n t showing t h a t the l a r g e changes i n molecular weights can be accounted f o r i n our model. f

S o l u t i o n P o l y m e r i z a t i o n o f MMA Schulz's Data ( l 6 ) . F i g u r e s 12 and 13 show e x c e l l e n t agreement between simulated and measured r a t e s f o r a wide range o f solvent concentrations (benzene: 0 - 0 . 9 2 7 l i t e r benzene t o 0.103 l i t e r MMA and benzoyl peroxide: 0.0^+13 gmole/£ a t 50°C and 70°C). No doubt measured and p r e d i c t e d molecular weights would have been i n good agreement. T r a n s f e r t o benzene was n e g l e c t e d i n the simulations. I t should be mentioned t h a t the p r e d i c t e d curve a t highest benzene l e v e l i n F i g u r e 13 agrees with c l a s s i c a l k i n e t i c s (no diffusion-control). I t i s not c l e a r t h e r e f o r e why measured data at even higher benzene c o n c e n t r a t i o n s do not agree w i t h c l a s s i c a l k i n e t i c s . There may be some s u b t l e chemical i n t e r a c t i o n s a t these h i g h solvent l e v e l s . D u e r k s e n ( l 7 ) found s i m i l a r e f f e c t s w i t h styrene p o l y m e r i z a t i o n i n benzene and had t o c o r r e c t kp f o r s o l vent. Conclusions A new r a t e model f o r f r e e r a d i c a l homopolymerization which accounts f o r d i f f u s i o n - c o n t r o l l e d t e r m i n a t i o n and propagation, and which g i v e s a l i m i t i n g conversion, has been developed based on free-volume theory concepts. The model g i v e s e x c e l l e n t agreement w i t h measured r a t e data f o r bulk and s o l u t i o n p o l y m e r i z a t i o n o f MMA over wide ranges o f temperature and i n i t i a t o r and solvent concentrations .



3.

MARTEN AND HAMD2LEC

Figure

6.


59

Polymerization

Bulk polymerization of MMA at 50°C: (X) [I]o = 0.02018 mol U (O)[I] = 0.01548 mol AIBN/L (9). 0


AIBN/



60

Figure

7.

Bulk polymerization of MMA at 50°C: (x) [I] = 0.05 mol (15); (O)[I] = 0.0258 mol AIBN/L (9). 0

0


AIBN/L

MABTEN AND HAMIELEC

Diffusion-Controlled Polymerization


3.


61



62

Figure

9. Bulk polymerization of MMA at 70°C: effect of conversion on molecular weight averages. (X)& ;(0)M . [I] = 0.01548 mol AIBN/L (9). n

w

0



Polymerization


MARTEN AND HAMIELEC




64



MARTEN AND HAMIELEC


500

1000

Polymerization

1500

Figure 12. Solution polymerization of MMA with benzene as solvent: temperature 50°C; [I] = 0.0413 mol BOP. (O) zero, Benzene = B, 1.030 L MMA; (X) 0.206 L B, 0.824 L MMA; (+) 0.412 L B, 0.618 L MMA; (A) 0.618 L B, 0.412 L MMA; (U) 0.824 L B, 0.206 L MMA; ( V ) 0.927 L B, 0.103 L MMA (16). 0




66

Figure 13. Solution polymerization of MMA with benzene as solvent: temperature 70°C; [I] = 0.0413 mol BOP. (O) zero Benzene = B, 1.057 L MMA; (X) 0.211 L B, 0.846 L MMA; (+) 0.423 L B, 0.634 L MMA; (A) 0.634 L B, 0.423L MMA; ([J) 0.846 L B, 0.211 L MMA; (V) 0.915 L B, 0.142 L MMA (16). 0


3.

MARTEN AND HAMIELEC


67

Polymerization

Nomenclature A

constant

B

constant

C. .

c h a i n t r a n s f e r c o n s t a n t t o monomer

Cg

chain t r a n s f e r constant t o solvent

M

D. .

diffusion coefficient

o f t h e monomer

D..

diffusion

o f t h e monomer a t t h e c o n v e r s i o n


M M

cr D

coefficient

w h e r e t h e p r o p a g a t i o n becomes d i f f u s i o n diffusion

coefficient

o f a polymer

controlled

radical

P D ^cr

d i f f u s i o n c o e f f i c i e n t o f a polymer r a d i c a l a t t h e conversion where t h e t e r m i n a t i o n becomes d i f f u s i o n controlled

d^

d e n s i t y o f monomer

dp

d e n s i t y o f polymer

f

initiator

efficiency

entanglement

constant

K3

temperature

k^

decomposition

k

propagation r a t e constant at zero

p

dependent

constant

rate constant conversion

kp°

propagation r a t e constant

kj.

t e r m i n a t i o n r a t e c o n s t a n t i n t h e absence o f g e l

k^°

termination rate constant

kj. c

t e r m i n a t i o n r a t e c o n s t a n t between e n t a n g l e d and radical

cr

t e r m i n a t i o n r a t e c o n s t a n t o f t h e c o n v e r s i o n where t h e t e r m i n a t i o n becomes d i f f u s i o n controlled

kj_

kj.

t e r m i n a t i o n r a t e c o n s t a n t between two e n t a n g l e d

k

temperature

dependent

constant

k^

temperature

dependent

constant

k^

constant

k^

constant

k^

constant

M

molecular weight

[M]

monomer c o n c e n t r a t i o n

M

molecular weight

1

q

weight

non-entangled

radicals

o f monodispersed polymer

number a v e r a g e m o l e c u l a r

M

effect

weight

o f monomer

average molecular

weight


DO



^wcrl

w e i

r t

&* average molecular the g e l e f f e c t s t a r t s

m

constant

n

constant

Q

constant

R

gas c o n s t a n t

weight o f t h e conversion

[s]

concentration of solvent

T

polymerization

T

g l a s s t r a n s i t i o n p o i n t o f monomer

T

where

temperature

g l a s s t r a n s i t i o n p o i n t o f polymer gp

T

glass t r a n s i t i o n of solvent s

s

T

g l a s s t r a n s i t i o n p o i n t o f polymer w i t h i n f i n i t e

molecular

CPoo &

weight

t V„ F V

time f r e e volume

F

crl V

F

cr2

fraction

f r e e volume f r a c t i o n a t t h e c o n v e r s i o n where t h e g e l e f f e c t starts f r e e volume f r a c t i o n a t t h e c o n v e r s i o n where t h e p r o p a g a t i o n becomes d i f f u s i o n c o n t r o l l e d

V., M Vp

volume o f polymer

V

volume o f s o l v e n t

o

V

v o l u m e o f monomer

total

volume

number a v e r a g e d e g r e e o f p o l y m e r i z a t i o n X^

weight average degree o f p o l y m e r i z a t i o n

x

conversion

X

number a v e r a g e c h a i n l e n g t h a t t h e p o i n t o f c h a i n ment

o f monomer

Greek Symbols a

constant,

exponent

01^

expansion c o e f f i c i e n t

for the glassy state

expansion c o e f f i c i e n t

for the l i q u i d

3

constant,

e

volumetric

6

jump d i s t a n c e

state

exponent contraction

coefficient


entangle-

MARTEN AND HAMiELEC

3.


6


Polymerization

69

jump d i s t a n c e

2

Ψ]_

t e m p e r a t u r e dependent lumped

constant

Ψ2

t e m p e r a t u r e dependent lumped

constant

Ψ3

lumped

constant

η

viscosity

τ

t h e r e c i p r o c a l i n s t a n t a n e o u s number a v e r a g e d e g r e e o f p o l y m e r i ζation

φ

volume f r a c t i o n o f polymer

o

jump f r e q u e n c y

φ2

jump f r e q u e n c y

Literature Cited 1.

C a r d e n a s , J. a n d 14, 8 8 3 .

2.

C a r d e n a s , J. a n d O'Driscoll, K. F., J. P o l y m . Sci. ( 1 9 7 7 ) A-1, 15, 1883. C a r d e n a s , J. a n d O'Driscoll, K . F . , J. P o l y m . Sci. ( 1 9 7 7 ) A - 1 , 15, 2097. Friis, N. a n d H a m i e l e c , A.E., ACS Symposium Series ( 1 9 7 6 ) 24, 8 2 , " G e l Effect in E m u l s i o n Polymerization of Vinyl Monomers". B e r e n s , A.R., ACS Symposium Series ( 1 9 7 8 ) 3 9 , 2 3 6 , "The Sorption of Gases a n d V a p o r s in PVC P o w d e r s " . H a y d e n , P. a n d Melville, Sir Harry, J. P o l y m . Sci. (1960) 43, 2 0 1 . B e u c h e , F., Interscience, New Y o r k ( 1 9 6 2 ) , "Physical Proper ties of P o l y m e r s " . A b u i n E. a n d Lissi, E.A., J. M a c r o m o l . Sci. Chem. ( 1 9 7 7 ) A-11, 287.

3. 4.

5. 6. 7. 8. 9.

O'Driscoll,

K . F . , J. P o l y m . Sci. ( 1 9 7 6 ) A - 1 ,

B a l k e , S.T. a n d H a m i e l e c , A . E . , J. Appl. P o l y m . Sci. ( 1 9 7 3 )

17, 905. 10. 11.

Ito, K., J. P o l y m . Sci. ( 1 9 7 5 ) A - 1 , 13, 401. H o r i e , Κ., Mita, I. a n d Kambe, M., J. P o l y m . Sci. (1968) A-1, 6, 2663. 12. B a l k e , S.T., "The Free Radical Polymerization of Methyl Methacrylate to High C o n v e r s i o n " , Ph.D. Thesis, M c M a s t e r University, Hamilton, Ontario (1972). 1 3 . A b d e l - A l i m , A.H. a n d H a m i e l e c , A.E., J. Appl. P o l y m . Sci. (1972) 1 6 , 7 8 3 . 14. P o l y m e r Handbook ( 2 n d ed.), B r a n d r u p , J . a n d Immergut, E.H., Wiley, New Y o r k ( 1 9 7 5 ) . 15. N i s h i m u r a , N., J. M a c r o m o l . Sci. ( 1 9 6 6 ) 1 , 257. 16. S c h u l z , G.V. a n d Harborth, G., M a k r o m o l . Chem. ( 1 9 4 7 ) 1, 1 0 6 .



70 17.

18. 19. 20.

Duerksen, J.H., "Free R a d i c a l P o l y m e r i z a t i o n o f Styrene i n Continuous S t i r r e d Tank Reactors", Ph.D. T h e s i s , McMaster U n i v e r s i t y , Hamilton, Ontario ( 1 9 6 8 ) . Ibragimov, I.Y. B o r t , D.N. and Efremova, V.N., Vysokomol. Soedin. Ser.B, (1974) 16 (5), 3 7 6 . Rudd, J . F . , J. Polym. Sci. ( 1 9 6 0 ) , 44, 459. Bueche, F., J. Polym. Sci., ( 1 9 6 0 ) , 43, 5 2 7 .


RECEIVED February 9, 1979.


High Conversion Diffusion-Controlled Polymerization - ACS Publications

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