Computer Applications in the Polymer Laboratory - American

19 Initiation Reactions and the Modeling of Polymerization Kinetics L. H . Garcia-Rubio and J . Mehta

Downloaded by FUDAN UNIV on November 22, 2016 | http://pubs.acs.org Publication Date: June 27, 1986 | doi: 10.1021/bk-1986-0313.ch019

College of Engineering, University of South Florida, Tampa, F L 33602

Research in the area of Polymerization Reaction Engineering over the last twenty years has resulted in a good phenomenologic understanding of the polymerization kinetics and a series of models capable of describing the reaction behaviour at high conversions. These models contain semiempirical correlations for the rate constants and a number of adjustable parameters. This paper reports on an approach used to obtain explicit expressions for the rate constants and i n i t i a t o r efficiencies as functions of measured conversions, molecular weight averages and the loading of initiator fragments onto the polymer molecules. The expressions obtained, together with experimental data, suggest mathematical representations and correlations for the kinetic parameters. The analysis done yields conditions for the applicability of the classical polymerization equations. The area o f homopolymerization r e a c t i o n e n g i n e e r i n g has been e x t e n s i v e l y s t u d i e d i n r e c e n t years (1,9). The main objective i n t h i s area has been the development of k i n e t i c models t h a t a r e capable o f e x p l a i n i n g the r a t e behavior and the polymer molecular properties observed d u r i n g the g e l e f f e c t . The m a j o r i t y o f the models proposed t o date are based on the stationary state hypothesis (SSH), the long chain approximation (LCA), and the assumption o f constant i n i t i a t o r e f f i c i e n c y (1-9,19). The SSH and the LCA are generally considered v a l i d even though i t i s recognized that changes i n the r e a c t i o n environment ( i e : v i s c o s i t y ) may affect the dynamic behaviour of the r a d i c a l population. The e f f e c t s due t o changes i n t h e r a d i c a l c o n c e n t r a t i o n a r e thus r e f l e c t e d on the observed behaviour of the e f f e c t i v e t e r m i n a t i o n , p r o p a g a t i o n and t r a n s f e r r a t e c o n s t a n t s . I n s p e c t i o n o f the term (2fkp/kt) i n the c l a s s i c a l polymerization equations i n d i c a t e s that changes i n the i n i t i a t o r

0097-6156/86/0313-0202S06.00/0 © 1986 American Chemical Society

Provder; Computer Applications in the Polymer Laboratory ACS Symposium Series; American Chemical Society: Washington, DC, 1986.


19.

GARC1A-RUBIO AND MEHTA

Modeling of Polymerization

Kinetics

203

e f f i c i e n c y a r e a l s o r e f l e c t e d on the estimates of both kp and k t . In addition, unaccounted changes i n the i n i t i a t o r concentration due t o r e a c t i o n s l i k e induced decomposition ( 1 0 ) a l s o a f f e c t the i n i t i a t o r e f f i c i e n c y and t h e o b s e r v e d b e h a v i o r o f t h e r a t e constants. I t i s t h e r e f o r e f e l t that i n order to investigate the nature of the gel effect i t i s necessary t o incorporate the changes i n both, the i n i t i a t o r concentration and i n i t i a t o r e f f i c i e n c y into the existing polymerization models. Quantitative determinations of the number of end groups per molecule by radio tracer techniques ( 1 1 » 1 3 ) t NMR ( 1 2 ) and U V spectroscopy ( 1 4 ) have p r o v i d e d new information regarding the fate of the i n i t i a t o r fragments during the polymerization of v i n y l monomers. The data, interpreted as o v e r a l l r a t e s of decomposition and i n i t i a t i o n e f f i c i e n c i e s suggests that at l e a s t part of the behavior attributed to the r a t e constants during the p o l y m e r i z a t i o n c o u l d be e x p l a i n e d i n terms of the i n i t i a t i o n reactions. T h i s paper r e p o r t s on t h e development o f v i n y l p o l y m e r i z a t i o n models that a r e capable o f d e s c r i b i n g both; the i n i t i a t o r behavior observed and the molecular weight development. Literature Review Experimental data on i n i t i a t o r e f f i c i e n c i e s based on the d i r e c t measurement of the products of the i n i t i a t o r reactions i s rather scant, p a r t i c u l a r l y at high conversions. The main reasons f o r t h i s have been the d i f f i c u l t i e s associated with the analysis of the large numbers of decomposition products known t o be p r e s e n t i n t h e r e a c t i n g m i x t u r e s . A t l o w c o n v e r s i o n s , where t h e i d e a l polymerization kinetics are expected t o be v a l i d , the experimental a p p r o a c h e s have been b a s e d on measurements o f some o f the decomposition by-products, r a t e s of p o l y m e r i z a t i o n and on t h e d e t e r m i n a t i o n o f the chain lengths ( 1 5 - 2 1 ) . The r e s u l t s obtained have been, i n general, contradictory ( 2 2 ) . A notable exception have been the semi-empirical models derived by Hamielec et a l ( 1 8 , 1 9 , 2 1 ) that correlate the i n i t i a t o r e f f i c i e n c y t o v i s c o s i t y of the reacting mixture and, the cumulative values of the e f f i c i e n c y obtained by Yenalyev and Melnichenko ( 2 1 ) from c o n v e r s i o n mesurements and the corresponding degree o f p o l y m e r i z a t i o n . The uncertanties i n the model structure, however, are included i n the e f f e c t i v e values of the other rate constants ( i e : k p / k t / ) . Theoretically, two types of models have been proposed: models based on cage e f f e c t s ( 1 0 , 1 8 , 2 0 ) and models without cage effects ( 1 9 ) . The cage models suggest that the i n i t i a t o r e f f i c i e n c y i s p r o p o r t i o n a l t o t h e monomer c o n c e n t r a t i o n and independent of the i n i t i a t o r concentration ( 1 5 ) . The second class of models indicates that the i n i t i a t o r e f f i c i e n c y i s proportional to both the monomer and the i n i t i a t o r concentration. The l a t e r models predict a decrease of the i n i t i a t o r e f f i c i e n c y as function of conversion ( 1 9 ) . 1

2


204

JL


f 0

COMPUTER APPLICATIONS IN THE POLYMER LABORATORY

I

88

2

2

(A) 2

fo + ( f o + (Mo/M) (C/Co)((2 - f o ) -

2

fo ))

V 2

where f i s the e f f i c i e n c y , C i s the i n i t i a t o r concentration, and M i s the monomer c o n c e n t r a t i o n . The s u b s c r i p t (o) r e f e r s to the i n i t i a l value. At high c o n v e r s i o n s , the e s t i m a t i o n o f the i n i t i a t o r e f f i c i e n c i e s have been based on measurements of the addition of i n i t i a t o r fragments onto polymer molecules (11-14). Calculation of the i n i t i a t o r e f f i c i e n c i e s u s i n g t h i s approach have resulted i n abnormal e f f i c i e n c y values (14). The c o n t r a d i c t o r y experimental evidence, and the discrepancy i n the t h e o r e t i c a l results indicates that a d d i t i o n a l experimental evidence i s r e q u i r e d i n order t o e l u c i d a t e the behaviour of the i n i t i a t o r e f f i c i e n c y as function of conversion. The required information can be obtained from the r a t e data and from the instantaneous values of the polymer properties. Knowledge of the instantaneous polymer p r o p e r t i e s a l l o w s the e s t i m a t i o n of the instantaneous e f f i c i e n c i e s and the point values of the rate constants. These parameter values are then c o n d i t i o n a l upon t h e i n i t i a t i o n and the p o l y m e r i z a t i o n models used. The instantaneous polymer p r o p e r t i e s can be estimated from e x i s t i n g data by proposing empirical models f o r the instantaneous properties and then f i t t i n g these models to the c u m m u l a t i v e d a t a . T h i s approach a l l o w s the s e l e c t i o n of low order models, avoids some of the known d i f f i c u l t i e s associated with d i f f e r e n c i a t i n g experimental data and permits s t a t i s t i c a l testing for model adequacy. Experimental Data L i t e r a t u r e data f o r the suspension polymerization of styrene was selected for the analysis^. The data, shown i n T a b l e I, i n c l u d e s c o n v e r s i o n , number and weight average molecular weights and i n i t i a t o r loadings (14). The empirical models selected to describe the r a t e and the instantaneous properties are summarized i n Table I I . In every case the models were shown to be adequate w i t h i n the l i m i t s of the r e p o r t e d experimental e r r o r . The experimental and calculated instantaneous values are summarized i n F i g u r e s (1) and ( 2 ) . The r a t e constant f o r the thermal decomposition of benzoyl peroxide was taken as In kd « 36.68 * 137.48/RT kJ/(gmol) (11). Model Development The accepted k i n e t i c scheme f o r 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 r e a c t i o n s ( e q u a t i o n s 1*11) has been used as b a s i s f o r the development of the mathematical equations f o r the e s t i m a t i o n of b o t h , t h e e f f i c i e n c i e s and the r a t e c o n s t a n t s . Induced decomposition reactions (equations 3 and 10) have been i n c l u d e d t o g e n e r a l i z e the model f o r i n i t i a t o r s such as Benzoyl Peroxide f o r


19.

GARCIA-RUBIO A N D MEHTA

Modeling of Polymerization Kinetics

205

Table I . Styrene Suspension Polymerization 90 °C with 2.5% by Weight Benzoyl Peroxide (14)

Time

%

M M w


Sample

Ph-C0

2 2

n

3

End Grp./

3

Min.

Conv.

x10

x 10

Molecule

PS01

7

7.13

29.50

15.09

1.558

1 .95

PS02

20

26.34

30.58

15.91

1.556

2.07

PS03

36

57.47

32.28

16.46

1.457

2.04

PS04

60

72.97

35.03

17.86

1.4156

2.09

PS05

90

79.04

37.76

18.88

1.317

2.17

PS06

120

85.74

41.78

19.82

1.246

2.19

PS07

155

88.84

43.53

19.45

1.293

2.22

PS08

240

91.59

46.65

21.56

1.253

2.40


206


Table

II.

E q u a t i o n s f o r the

I n s t a n t a n e o u s Polymer

Properties


Conversion and Rate of Polymerization X • 9i(1 * exp(*0 (t * 0 ) ) 2

Rp - P(1 + e) (1 + e X )

2

3

dX dt

(fraction/second) (moles/liter-sec)

m

2.799; 0 0.1737E-03; 9 -3l6.0; p- molar density of styrene t« r e a c t i o n time (seconds); e- volume contraction factor 2

3

Number average Molecular Weight Mn - 9520 + 5225 (1-X) Weight Average Molecular Weight Mw - 17810 + 10380

I n i t i a t o r Loading (grams of i n i t i a t o r fragments/gram of polymer) Y - 0.01642 exp(-0.5576)



19.

GARCl A-RUBIO AND MEHTA


Kinetics

207

Figure 1 . Measured conversions and c a l c u l a t e d p o l y m e r i z a t i o n rates. ( # ) experimental values. S o l i d l i n e s calculated values from equations i n Table I I .

2.0

r

.1 , 0.2

OA

0.6

0.8

1.0

CONVERSION

F i g u r e 2. Measured and calculated instantaneous and cummulative molecular weight averages. (•) experimental cummulative v a l u e s ; S o l i d l i n e s , calculated values (Table I I ) .


C O M P U T E R A P P L I C A T I O N S IN T H E P O L Y M E R L A B O R A T O R Y

208

which induced d e c o m p o s i t i o n r e a c t i o n s (10,11,13,14).

have

been

reported

Initiation: * 2

C

d

R 6

k2 R 6

o

R*+ C

(1)

o

Rc,j + B

(2)

P + R6

(3)

0

Rc

+

M

+

M

r

0


Rc

1

?'

R

;

(1) (5)

I

Propagation: Rn +

*>

M

Rft+1

(6)

Pn+m

(7)

Pn + Pm

(8)

Pn + RJ

(9)

Termination: k£c

Rh + Rm

k£d

Rn + Rm Rn +

k£m

M

kx

Rn + C

Pn + R6

k£c»

Rn + Rc

(10)

0

Pn

(11)

D e p e n d i n g on t h e i n i t i a t o r and monomer s y s t e m s e c o n d a r y decomposition (equation 2), induced decomposition (equations 3>9), primary r a d i c a l termination (equation 11) or transfer reactions may or may not be important and w i l l have to be considered accordingly i n the balance e q u a t i o n s . From the above r e a c t i o n scheme t h e f o l l o w i n g equations have been d e r i v e d under the SSH, the LCA, n e g l i g i b l e secondary decomposition and n e g l i g i b l e primary r a d i c a l termination (9,19,20): Rp « -dM » kp dt rn (t)

R*

(12)

1 + 3/2)

(13)

2T

+ 3B

(14)

( T

+

» ( T

rw (t)

-

M

3)

2


19.

GARCIA-RUBIO AND MEHTA


t = kfm + kxC + ktd Rp kp kpM kp M

(15)

$ - ktc Rp kp M"

(16)

2

2

209

2

2

In a d d i t i o n t o the above equation, i n the absence of branching reactions, i t i s also possible to write a balance f o r the number of i n i t i a t o r fragments attached to the polymer molecules.

dt Downloaded by FUDAN UNIV on November 22, 2016 | http://pubs.acs.org Publication Date: June 27, 1986 | doi: 10.1021/bk-1986-0313.ch019

(

dt

transfer t o polymer

dor dt term by disp }

A

dt

term, by comb

jdgrj der dt trans to solv dt A(

}

trans to mon.

I dpr/dt r«1

g£ t o t a l «

r e p l a c i n g the terms r e s u l t i n g from equations 7 to 10 (6,9,15), f o r an i n i t i a t o r decomposing into two a c t i v e fragments, the moles of i n i t i a t o r a t t a c h e d t o t h e polymer molecules can be r e a d i l y obtained: r-1

dn » I ((kx C + ktd I R3 ) I Rj + ktc I I dt 2 j-1 r« s1

S

Rs Rr-s) "

1

(17)

where the i n d i c e s j , r and s account f o r monomer units. 17 s i m p l i f i e s to: (T+B)

dn - 1^ Rp dt 2

A balance on the number of i n i t i a t o r groups per molecule (N(t)) r e s u l t s i n : N(t) -

T

( T

+

+ 6 3/2)

Equation

(18) fragments, or on the end

(19)

A l t e r n a t i v e l y , the grams of i n i t i a t o r bonded per grams of polymer formed Y ( t ) can also be r e a d i l y calculated. Y ( t ) « ( T • 8 ) (Mf/Mm)

(20)

where Mf i s the molecular weight of the i n i t i a t o r fragments and Mm i s the molecular weight f o r the monomer. I f the transfer reactions ( i e : transfer to i n i t i a t o r ) imply the addition of a fragment to the growing r a d i c a l , equation 20 becomes.


210

COMPUTER APPLICATIONS IN THE POLYMER LABORATORY Y ( t ) - (2T • 0) (Mf/Mm)

(20a)

Based on the l i t e r a t u r e data available f o r styrene polymerized with benzoyl peroxide, (10,12,14) transfer t o monomer and t e r m i n a t i o n by d i s p r o p o r t i o n a t i o n w i l l be neglected. For the i n i t i a t i o n step, only primary and induced decomposition reactions w i l l be considered.


Analysis of the data From the instantaneous values of the properties reported i n Table I, i t i s p o s s i b l e t o determine a maximum of four k i n e t i c parameters. E x p l i c i t expressions f o r the rate constants can be obtained d i r e c t l y from equations 12 t o 16 i n terms of the parameters T and a , and from these the values of the r a t e constants can be obtained for a variety of reaction schemes. a - (2/rw)((2 - rw/rn) + (4 - 2 r w / r n ) ) T

- 1/rn - a/2

l/2

(21) (22)

where rw and r n a r e the instantaneous weight and number average chain lengths. Note that, for the parameters a and T t o be p o s i t i v e o r z e r o , the instantaneous polydispersity (rw/rn) must be l e s s or equal to 2. This i s an important observation because the above condition must be met throughout the c o n v e r s i o n trajectory regardless of the actual values of the r a t e constants. Furthermore, i t can be shown t h a t i f there i s more than one termination mechanism present, the r a t i o (* T/8) i s f i n i t e and i t has a value o f : 2

/2

* - -2((fw/ni-2) ± ((rw/rn-2) - 4(rw/fn-2)(rw/rn-3/2))* )

(23)

(rw/rn-2) where o n l y the positive root has physical meaning. The discriminant i n equation 23 i s positive only i f 3/2 * rw/rn £ 2

(24)

Therefore, the c l a s s i c a l polymerization model i s a p p l i c a b l e o n l y t o those conversion t r a j e c t o r i e s that y i e l d polydispersities betwen 1.5 and 2 regardless of the mode o f t e r m i n a t i o n . Although t h i s i s an expected r e s u l t , i t has not been implemented, the high conversion polymerization models reported to date a r e based on the c l a s s i c a l e q u a t i o n s f o r w h i c h t h e c o n s t r a i n t g i v e n by e q u a t i o n 24 i s applicable. The r e s u l t has been piecewise continuous models, ( 1 - 6 )


19.



Kinetics

211

where the changes i n the parameter values and model s t r u c t u r e s compensate f o r t h i s l i m i t a t i o n i n t h e c l a s s i c a l model. The p o l y s t y r e n e data used here shows cummulative polydispersities which a r e s m a l l e r than 2, t h e r e f o r e , equations 11-20 are c o n s i d e r e d applicable. I f i t i s assumed, i n accordance to 0 D r i s c o l l and White, (10) that the r a d i c a l s produced from the induced decomposition reaction equation 3 have an e f f i c i e n c y of 1, a balance on the t o t a l r a d i c a l concentration yields ,

R* - (2fkd C ) ktc

1 / 2

(25)


R e p l a c i n g e q u a t i o n 25 i n t o obtained.

11 an expression for the e f f i c i e n c y i s

f - 8 Rp 2kdC

(26)

A balance on the i n i t i a t o r y i e l d s : -dC - kd C + xRp dt

(27)

Equation 27 can be n u m e r i c a l l y i n t e g r a t e d a l o n g the c o n v e r s i o n t r a j e c t o r y t o o b t a i n the i n i t i a t o r c o n c e n t r a t i o n as function of time. Therefore, c a l c u l a t i o n of x, 0 and C together with the v a l u e s of M, Rp, rw and r n from the equations i n T a b l e I I allows the estimation of the r a t i o s ( k t c / k p ) , (kx/kp) and the e f f i c i e n c y as f u n c t i o n s of conversion. Figure 3 shows the e f f i c i e n c y as function of conversion. Figure 4 shows the v a r i a t i o n of the r a t e c o n s t a n t s and e f f i c i e n c i e s normalized to their i n i t i a l values. The values for the r a t i o (ktc/kp )/(ktc/kp )o reported by Hui (18) a r e a l s o shown f o r comparison. From the d e f i n i t i o n of e f f i c i e n c y i t i s possible to d e r i v e an equation f o r the instantaneous l o a d i n g o f i n i t i a t o r fragments, 2

2

Y

f

2

- f(Mf/Mm)(2kd C + xRp) RP

(28)

The cummulative values of the i n i t i a t o r loading calculated with equations 20a and 28 should agree, betwen themselves and w i t h the measured values. As i t i s shown i n Figure 5 t h i s i s not the case, a correction for the i n i t i a t o r balances seems to be r e q u i r e d . Larger e f f i c i e n c y v a l u e s can be o b t a i n e d i f i t i s assumed that a l l the i n i t i a t o r r a d i c a l s have the same e f f i c i e n c y . Under t h i s c o n d i t i o n , the r a d i c a l concentration i s given by R* - -kxC(1-f) + ( k x C ( 1 - f ) + 4ktckd c ) 2ktc 2

2

2

> / 2

(29)


212

COMPUTER APPLICATIONS IN THE POLYMER LABORATORY (X7 0.6 0.5

er o.4 c 0

1

Q3

o

0.2


ai

02

04 0.6 0.8 conversion

1.0

Figure 3. C a l c u l a t e d e f f i c i e n c i e s . (1) From the cage e f f e c t model and no primary r a d i c a l termination (Case I ) ; (2) From the assumption o f an o v e r a l l e f f i c i e n c y and no p r i m a r y r a d i c a l t e r m i n a t i o n (Case I I ) ; (3) From the assumption o f an o v e r a l l e f f i c i e n c y and primary r a d i c a l t e r m i n a t i o n (Case I I I ) ; (4) Calculated from equation (A) with f o - 0.663. 1.00 0.50

0.2

OA

0.6

0.8

1.0

X

F i g u r e 4. V a r i a t i o n o f the r a t e parameters as f u n c t i o n s o f c o n v e r s i o n : f o r Case I : (1) £ « ( k t c / k p ) / ( k t c / k p ) ; (2) £ » (kx/kp)/(kx/kp) ; (3) S « f / f ; W 5 - ( k t c / k p ) / ( k t c / k p ) from reference (18). 2

2

0

2

0

2

0


0

19.

213


GARCIA-RUBIO A N D MEHTA

which upon s u b s t i t u t i o n into equation 11 gives a new value f o r the efficiency f -

2Rp(x + g) 4kd C + xRp

(30) 1

This equation also yields values f o r Y that are s m a l l e r than the experimental values (see Figures 4-5). F i n a l l y , i f primary r a d i c a l t e r m i n a t i o n i s c o n s i d e r e d ( i e : e q u a t i o n 11), and under the assumption of (kp « kp), a balance on both, macro and primary r a d i c a l s gives: t

(3D

Rc - fC(2kd + kxR') kpM - fkxC Downloaded by FUDAN UNIV on November 22, 2016 | http://pubs.acs.org Publication Date: June 27, 1986 | doi: 10.1021/bk-1986-0313.ch019

fC(2kd + kxR6) - ((1-f)kxC - ktc'RcjR'- ktcR*

2

- 0 (32)

Substitution of equation 31 into equation 32 and replacement of R from equation 12 y i e l d s , after some a l g e b r a i c m a n i p u l a t i o n , the following equations: f

x

- 1/rn - g/2

(33)

where x* now includes the primary r a d i c a l termination term, i e : x

f

T

X

2

» x + ktc'R6 kpM

(34)

- x' + g + ftp f(x'+g+1)

(35)

- A(\|> - x» + T) - fx(x

f -

x* — x \|>A • x(A+x* ) - T

f

- T) «0

(36) (37)

2

where: * - 2kd C/Rp

and

2

2

A - ktc'Rp/(kp M )

The i n i t i a t o r concentration at any time can be obtained from: -dC - kd C + xRpQ + f\l* + fx) dt 1 - fx

(38)

The non-linear equations 21 and equations 35-38 can be s o l v e d i t e r a t i v e l y to give d i r e c t l y , the instantaneous e f f i c i e n c i e s and the the r a t i o s of rate constants (kx/kp), (ktc/kp ) and ( k t c ' / k p ) . The values obtained f o r the r a t e constants have been summarized i n Table III and i n Figures 3 and 6. The r e s u l t s from the c a l c u l a t i o n s show a s m a l l d i f f e r e n c e ( i e : l e s s than 1$) betwen ktc and k t c . Therefore, for a l l p r a c t i c a l purposes they can be c o n s i d e r e d equal 2

2

f


214


r r 2.4

.

o o


x

2.0

0.4 .

02

0.4 0.6 0.8 conversion

1.0

Figure 5. I n i t i a t o r l o a d i n g s : ( O ) Experimental values; ( # ) C a l c u l a t e d from the molecular weight data and the assumption of two end g r o u p s p e r m o l e c u l e ; (1) C a l c u l a t e d from Case I e f f i c i e n c i e s ; (2) C a l c u l a t e d from Case I I e f f i c i e n c i e s ; (3) Calculated from Case I I I e f f i c i e n c i e s .

Table I I I . Summary of Kinetic Parameters

Equations

21,22,28,29 21,22,29,31 21,34-39

0.453 0.584 0.663

fo

ktc/kp

76.39 76.39 76.39

0.451 0.451 0.447

2


kx/kp

76.40


19.



215

at t h i s temperature. The conversion of i n i t i a t o r as function of time i s shown i n Figure 7. The large discrepancy observed between t h e i n i t i a t o r c o n s u m p t i o n c a l c u l a t e d from t h e f i r s t order decomposition and the one calculated from equation 37 accounts f o r the abnormal behaviour of the i n i t i a t o r e f f i c i e n c y obtained from uv measurements (14). The i n i t i a t o r l o a d i n g s c a l c u l a t e d from the e f f i c i e n c y values correspond t o those calculated from the Mn's and the assumption o f two end groups per molecule F i g u r e 5. I t i s noteworthy that the r a t i o s of rate constants vary only s l i g h t l y f o r the d i f f e r e n t equations r e s u l t i n g from t h e a s s u m p t i o n s made r e g a r d i n g the e f f i c i e n c y and the i n i t i a t o r balances (see Table I I I and Figures 4 and 6). However, equations 34^39 a r e the ones that g i v e the c l o s e s t agreement between the measured and calculated i n i t i a t o r loadings. I f i t i s assumed that a l l the chains are l i n e a r (and t h e r e i s no e v i d e n c e t o t h e c o n t r a r y f o r p o l y s t y r e n e synthesized at 90° C ) , then i t must be concluded that a f r a c t i o n of the primary r a d i c a l s react with the polymer chains to y i e l d excess ester groups (11,12,14). The c l a s s i c a l p o l y m e r i z a t i o n e q u a t i o n s , a p p l i e d t o the polystyrene data over the complete conversion t r a j e c t o r y , suggest t h a t a l l t h e r a t e parameters, i n c l u d i n g the e f f i c i e n c y , vary considerably from the onset of the reaction. The v a r i a t i o n o f the r a t e parameters i s function of the molecular weight as well as of the concentration of polymer i n the reacting mixture (1-8,21). The agreement with the accepted d e s c r i p t i o n o f the p o l y m e r i z a t i o n kinetics at high conversions i s not s u r p r i s i n g . What i s s u r p r i s i n g i s the behaviour o f the i n i t i a t o r e f f i c i e n c y , which i s t o t a l l y d i f f e r e n t from the expected b e h a v i o u r b a s e d on 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 (18). As i t i s shown i n Figure 3 there i s a large discrepancy between the theoretical values calculated with equation 1a, and those o b t a i n e d from the d a t a . Furthermore, none of the empirical equations proposed t o date, account f o r the change i n e f f i c i e n c y a r e capable o f r e p r e s e n t i n g the behaviour shown i n F i g u r e s 3 through 6 and account f o r the f a t e o f the i n i t i a t o r fragments (16,18,20,21,23). I t i s also evident that a l l of the rate parameters are affected i n a s i m i l a r way by the changes occuring i n the r e a c t i o n environment. The functional relationship that can be used t o represent the behaviour shown i n F i g u r e 6 i s o f the form (24):

5 - «pharh-)

( 3 9 )

where X i s the conversion and XL i s the l i m i t i n g conversion. Equation 39 has the structure proposed for the rate c o n s t a n t s on the basis of the free volume theory (1,5,9). From t h i s , i t would be expected that the models developed from the f r e e volume theory would be very successful i n predicting both, the rate behaviour and the molecular properties at high conversions. The reason why these m o d e l s have been o n l y p a r t i a l l y s u c c e s s f u l stems from the



216


.001 I 02

OA

0.6

0.8

1.0

X

F i g u r e 6. V a r i a t i o n o f the r a t e parameters as f u n c t i o n s of conversion: for Case I I I : (1) £ - ( k t c / k p ) / ( k t c / k p ) ; (2) £ (kx/kp)/(kx/kp) ; (3) C - f / f ; (4) £ - ( k t c / k p ) / ( k t c / k p ) from reference (18). 2

2

0

2

0

2

0

0

F i g u r e 7. I n i t i a t o r c o n v e r s i o n as f u n c t i o n o f time: (1) For f i r s t order thermal decomposition; (2) C a l c u l a t e d from equation (38).


19. GARCIA-RUBIO AND MEHTA


Kinetics

217

constraint dictated by equation 24. P r e l i m i n a r y c a l c u l a t i o n s , on the order of magnitude of the terms contributing to the molecular weight averages, i n d i c a t e s that c o n s i d e r a t i o n of the r a d i c a l p o p u l a t i o n i n the p r o p e r t y equations may relax the constraint on the c l a s s i c a l model.


Summary and Conclusions The instantaneous values f o r the i n i t i a t o r e f f i c i e n c i e s and the r a t e c o n s t a n t s a s s o c i a t e d w i t h the suspension polymerization of styrene using benzoyl peroxide have been determined from e x p l i c i t e q u a t i o n s based on the instantaneous polymer p r o p e r t i e s . The e x p l i c i t equations for the rate parameters have been d e r i v e d based on accepted r e a c t i o n schemes and the standard k i n e t i c assumptions (SSH and LCA). The instantaneous polymer p r o p e r t i e s have been o b t a i n e d from the cummulative experimental values by proposing empirical models for the instantaneous properties and then f i t t i n g them t o the cummulative experimental values. This has circumvented some of the problems associated with d i f f e r e n c i a t i n g experimental data. The r e s u l t s obtained show that: None of the models, theoretical or empirical, proposed i n the l i t e r a t u r e to account for the change i n e f f i c i e n c y as function of conversion, can describe the behaviour observed. The r a t e parameters f o l l o w s i m i l a r conversion t r a j e c t o r i e s . Therefore, the rate constants and the i n i t i a t o r e f f i c i e n c y can be modelled with the same equation. An equation of the form of equation 39 i s suggested. The t h e o r e t i c a l j u s t i f i c a t i o n f o r the form of equation 39 stems from the free volume theory. From the r e a c t i o n schemes i n v e s t i g a t e d , i t i s c l e a r t h a t i n d u c e d d e c o m p o s i t i o n and p r i m a r y r a d i c a l t e r m i n a t i o n reactions should be considered i n the i n i t i a t o r balances i n order t o account f o r the observed i n i t i a t o r loadings. This i s p a r t i c u l a r l y i m p o r t a n t when r e l a t i v e l y h i g h i n i t i a t o r concentrations are involved. The use of an o v e r a l l i n i t i a t o r e f f i c i e n c y , appears to be more e f f e c t i v e than the cage effect concept i n describing both the effect of the i n i t i a t o r concentration on the e f f i c i e n c y , and the i n i t i a t o r loadings. From the analysis of the rate equations i t can be concluded that the c l a s s i c a l p o l y m e r i z a t i o n model does not apply whenever the instantaneous polydispersity i s greater than 2 or smaller than 3/2. T h i s l i m i t a t i o n o f the c l a s s i c a l moael has resulted i n piecewise continuous models for high v i s c o s i t y polymerizations. P r e l i m i n a r y c a l c u l a t i o n s , on the order of magnitude of the terms contributing


C O M P U T E R A P P L I C A T I O N S IN THE P O L Y M E R L A B O R A T O R Y

218


to the molecular weight averages, indicates that c o n s i d e r a t i o n o f the r a d i c a l p o p u l a t i o n i n the p r o p e r t y equations may relax the constraint on the c l a s s i c a l model. We are c u r r e n t l y extending our work i n t h i s d i r e c t i o n and we w i l l report the r e s u l t s i n the near future.

Literature Cited 1. Marten, F. L., Hamielec A. E.; "ACS Symposium Series", 104, 43 (1979) 2. Cardenas, J., O'Driscoll, K. F.; J. Polymer Sci. A l , 14, 883 (1976) 3. Cardenas, J., O'Driscoll, K. F.; J. Polymer Sci. Al, 15, 1883 (1977) 4. Cardenas, J., O'Driscoll, K. F.; J, Polymer Sci. Al, 15, 2097 (1977) 5. Harris. B., Hamielec A. E., Marten, F. L.; "ACS Symposium Series", 18, 3199 (1980) 6. Soh, S. K., Sundberg, D. C. ; J. Polymer Sci. Polymer Chem., 20, 1299; 1315; 1331; 1345 (1952) 7. Tulig, T. J., Tirrell, M; Macromolecules, 14, 1501 (1981) 8. Tulig, T. J., Tirrell, M; Macromolecules, 15, 459 (1982) 9. Hamielec, A. E . , Friis, N; "Introduction to Chain Polymerization Kinetics", Short Course Notes, Mc Master University (1980) 10. O'Driscoll, K. F., White, P. J; J. Polymer Sci a3, 283 (1985) 11. Berger, K. C., Deb, P. C. and Meyerhoff, G.; Macromolecules, 10 (5), 1075 (1977) 12. Moad, G., Rizzardo, E., Solomon, D. H.; Macromolecules 15, 909 (1982) 13. Bevington, J. C.; Ebdon, J. R.; Huckerby, T. N.; Hutton, N. W. E. Polymer 23, 163 (1982) 14. Garcia-Rubio, L. H.; Ro, N.; Patel, R.; Macromolecules, Vol 17, 1998 (1984) 15. North, A. M., "The Kinetics of Free Radical Polymerization" Pergamon Press (1966) 16. Biesenberger, J. A . ; Sebastian, D.H., "Principles of Polymerization Engineering", John Wiley & Sons, New York, (1983) 17. Eastmond, C. G., "Chemical Kinetics, Vol.14a", Bamford, C. H and Tipper, C. F. H editors, Elsevier, (1976) 18. Duerksen, J.H. and Hamiche A.E. J. Polym. Sci. Part C, 155 (1968). 19. Hui, A.T. and Hamielec A . E . , J. Polym. Sci. Part C, 167 (1968). 20. Yenalyev, V.D. and Melnichenko, V.I., in "Emulsion Polymerization" eds I. Piirma and J.L. Gardon ACS symp. Series N° 24, 221 (1976). 21. Hui, A. PhD Thesis, McMster University, (1970) 22. Allen, P. E. M; Patrick, C. R., "Kinetics and Mechanisms of Polymerization Reactions", John Wiley & Sons, (1974) 23. Ross, R. T.; Lawrence, L. R., AIChE Symp. Series, 160, Vol 72, 74, (1974) 24. Fabian., E., Private Communication, (1982) RECEIVED December 2, 1985


Computer Applications in the Polymer Laboratory - American

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