Polymerization Reactors and Processes - American Chemical Society

18 Reduction of Molecular Mobility Caused by Increasing

Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on June 12, 2016 | http://pubs.acs.org Publication Date: July 31, 1979 | doi: 10.1021/bk-1979-0104.ch018

Solution Viscosity D . C . T I M M , C . H U A N G , V . K. P A L S E T I A , and T. S. Y U Department of Chemical Engineering, University of Nebraska, Lincoln, N B 68588

The e f f e c t o f media v i s c o s i t y on p o l y m e r i z a t i o n r a t e s and polymer p r o p e r t i e s i s w e l l known. Analysis of k i n e t i c rate data g e n e r a l l y i s constrained t o prop a g a t i o n r a t e c o n s t a n t i n v a r i e n t o f media v i s c o s i t y . The c u r r e n t r e s e a r c h d e v e l o p e s an e x p e r i m e n t a l d e s i g n t h a t a l l o w s f o r the e v a l u a t i o n o f v i s c o s i t y dependence on u n c o u p l e d r a t e c o n s t a n t s , i n c l u d i n g i n i t i a t i o n , p r o p a g a t i o n and m a c r o m o l e c u l a r a s s o c i a t i o n . The s y s tem s t y r e n e , t o l u e n e n - b u t y l l i t h i u m i s u t i l i z e d . Steady s t a t e a n a l y s i s e x p l i c i t l y e v a l u a t e s model p a r a meters. Dynamic s i m u l a t i o n s p r e d i c t r e a c t o r s t a r t - u p transients. H i s t o r i c a l Review K i n e t i c Mechanism. The f o l l o w i n g i o n i c mechanism d e s c r i b e s sytrene p o l y m e r i z a t i o n i n a hydrocarbon s o l u t i o n w i t h n - b u t y l l i t h i u m as t h e i n i t i a t o r (1-6). I n i t i a t o r A s s o c i a t i o n : I -> y I y «- •*

K a

Initiation:

I + M •> A

Propagation:

A_. + M + A _ j

1

+ 1

K

p

j = 1,2,3

P o l y s t y r y l Anion * Association j k

1

( 1 )

G

A boundary c o n d i t i o n i s

TU,t)

=

K

T

( t )

I ( t )

i tot p tot >

K

A

( t

... 1

J

The molar r a t e o f change o f p o l y m e r i c s p e c i e s o f degree of polymerization n i n a well-mixed, continuous flow tank r e a c t o r i s r e l a t e d t o the k i n e t i c r a t e o f p r o pagation of unassociated p o l y s t y r y l anions plus t h e i r w i t h d r a w a l r a t e i n the r e a c t o r ' s e f f l u e n t . Feed streams a r e v o i d o f p o l y m e r i c s u b s t a n c e s , but c o n t a i n monomer i n i t i a t o r and s o l v e n t .

Henderson and Bouton; Polymerization Reactors and Processes ACS Symposium Series; American Chemical Society: Washington, DC, 1979.

18.

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Mobility

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Styrene monomer concentration i s described by dM(t) „ M _ _

l n

- M(t) _ K A. . (t) tot g

M(t)

(3)

p


Experimental i n i t i a t o r concentration i s the sum of molar concentrations of associated as w e l l as unassociated molecules of n-BuLi

f g t l l

-

;

*

-

K M(t,n 1

e

x

p

(« ^ J

(

4

i

i f i n i t i a t o r association i s assumed to be at e q u i l i brium. The t o t a l , cumulative molar concentration of macromolecules i s described by d T

to ^t

( t )

+ T.tot (t)/6

= K'M(t) [I exp( t ) ] x v n

1 / y

(5)

The unassociated and associated p o l y s t y r y l anion concentrations contribute to population density of the sample. Their molar concentration i s T(n,t)

= [1 + K

e q

A

t Q t

( t ) ] A(n,t)

(6)

The i n t e g r a l molar concentration of unassociated macroanions i s A. . ( t ) . The t o t a l cummulative molar concentrations or a l l polymeric species at time t i s described by T

t Q t

(t)

-

U+K

e q

A

t o t

(t)] A

t o t

(t)

(

?

)

The polymerization system f o r which experiments were performed i s represented by the mathematical model consisting of Equations 1 and 7. Their steady state solutions are u t i l i z e d f o r k i n e t i c evaluation of rate constants. Dynamic simulations incorporate v i s c o s i t y dependency. K i n e t i c Evaluation Experiments were performed i n an isothermal, w e l l mixed, continuous tank reactor. Uncoupled k i n e t i c parameters were evaluated as follows from steady state observations.


POLYMERIZATION REACTORS AND PROCESSES


378

Population Density D i s t r i b u t i o n . Integration Equation 1 y i e l d s a semilogarithmic r e l a t i o n s h i p .

of

In T ( n , s s ) - l n T ( l , s s )

(8)

= - T (ss) 6 K A. .(ss) p tot

(n-l)

t o t

M(ss)

Values of population density T(n,ss) at various degrees of polymerization n are obtained through gel permeation chromatography (GPC) a n a l y s i s . Experimental analysis y i e l d s numerical values for the slope and intercept. (-slope) =

tot 6K A (ss)M(ss)

T

( s s )

p

tot

intercept = T ( l , s s )

(10)

Rearrangement of Equation (9) results from which the product p t o t ^ ^ K

p tot

-

A

s s

c

a

n

i n Equation b

e

e

v

a

l

u

a

t

e

tot (-slope)6M(ss)

(11)

d

,

n

v

K ± ± )

The quantity K A. ( s s ) can independently be obtained from c o n s e r v a t ? o n ° o f styrene. t

Vtot

(

s

s

)

"

M

in 9M(ss) M

(

s

s

)

(12)

Equations (11) and (12) provide two experimental methods f o r evaluation K A. (ss), Experimental agreement confirms the accuracy 85 the population density d i s t r i b u t i o n obtained by GPC. Simulataneous s o l u t i o n of Relationships ( 7 and 9) r e s u l t s i n the working r e l a t i o n s h i p 6M(ss) (-slope) = 1

T

+

K~ P

t Q t

(ss)

2 0M(ss) (-slope) K

(13)

p

I f a set of isothermal data i s obtained at various l e v e l s of v i s c o s i t y , a regression analysis w i l l allow for the evaluation of the two rate constants as functions of v i s c o s i t y . I n i t i a t o r A s s o c i a t i o n . Experimental i n i t i a t o r concentration i s the concentration of associated and unassociated n - B u L i . I f i n i t i a t o r i s predominantly


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associated and at e q u i l i b r i u m , Equation (2) expressed as T(l,ss)K A —

(ss)

,

J ! ( s s T

T

-

where K ! = K . [K / y ]


may be

i

K

i

w

-

n

l

/

y

(

1

4

)

1 / / y

The k i n e t i c mechanism subject to quenched samples requires that each polymer molecule formed w i l l contain one b u t y l residue W

8

'

8

=

T

tot

( s s )

- W ^ n

(15)

The value of i n i t i a t o r association and the rate constant may be evaluated. V i s c o s i t y i s not expected to have a s i g n i f i c a n t cage e f f e c t as i n free r a d i c a l systems, but the extent of association may be dependent on v i s c o s i t y , or other properties of the f l u i d media. Polymerization Dynamics N u r a e r i c a l simulations of reactor start-up were programmed, p r e d i c t i n g monomer and i n i t i a t o r concent r a t i o n s , t o t a l polymer concentration, weight and number average molecular weights, v i s c o s i t y and popul a t i o n density d i s t r i b u t i o n dynamics. The following two relationships obtained from steady state observations were u t i l i z e d i n the simulation. y(t) K

- P„ - 2 . 0 5 9 x l 0 " ^ W ( t ) ) (T (t))" -}^ -19 - 2025 (t) = 1.16x10 * exp(+26 340/RT) ji ° 1

t o t

3

8 7 4

A

t o t

,

z

u

z

1

(

1

7

)

)

V i s c o s i t y of monomer feed solution i s Moment A n a l y s i s . The zeroth moment i s the molar concentration of polymer and i s expressed by Equation 5. The f i r s t moment i s proportioned to the mass of polymer formed and i s related to monomer concentration Equation 3 The second moment WA(t) i s expressed by 9

dWA(t) , WA(t) —

+

- e —

The second moment molecular weight.

2 K

-

p

ill*

M l t ) W

tQt

„

( t )

a J . (t))

(

1

8 8

eq tot i s used to evaluate weight average


) )

380


Reactor start-up simulations require i n i t i a l values of A , I T(n,t) and T . be zero. Monomer The boundary c o n d i t i o n , concentratii o n i s non-zero. Equation 2, i s expressed as T(l,t)

=

K

i

I : C

e x:p p

1 + K

e q

( t ) A

tot

( t ) J

(

1

9

)

P

K


I

( t ) ]

Population Density Response Surface. The algorithm method of c h a r a c t e r i s t i c i s used to reduce the p a r t i a l d i f f e r e n t i a l Equation (1) into a set pf coupled ordinary d i f f e r e n t i a l equations. Since T(n,t) i s an exact d i f f e r e n t i a l , then 3T(n,t) dt 3t ds

3T(n,t) dn 3n ds

dT(n,t) ds

=

(20)

Comparison of Equations (1) and (20) y i e l d s the following ordinary d i f f e r e n t i a l equation s e t . ds |ta

=

Kg_M(t)A (t) tot

dT(n,t) ds Integrating time y i e l d s

the three equations with respect to

t - t* = s n(t*,n*,t)-n* =

T(n,t) 0

t

V

( t ) A

o

tot

*tot™

( t )

dt -

* p T ° K

tot (t)

M ( t ) A

t

0

t

T(n,t) = T ( n * , t * ) e x p [ ( * - t ) / 0 ] t

( t

> (21) (22)

Constants of integration are t* and n * . For p r a c t i c a l a p p l i c a t i o n s , i n i t i a l conditions specify that n * > 0, t* = 0 and boundary conditions require n* = 1, t* > 0. If t* = 0, n * = 1, a ground curve passing through the o r i g i n can be generated. This function n ( 0 , l , t ) was evaluated through Runge-Kutta-Gill i n t e g r a t i o n . Values of population density along t h i s ground curve are evaluated using Equation (22) and the boundary condition T(1,0). To evaluate a s p e c i f i c molar concentration T ( n i , t i ) the point [ n i , t ] J i s i n i t i a l l y located. If i t l i e s above the p r i n c i p a l ground curve, i . e . , n^ > n ( 0 , l , t i ) , i t i s necessary that the ground curve passing


18.

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381

through the point [ n i , t ] J originates from the i n i t i a l condition plane and t* = 0, n* > 1. Equation (21) may be arranged such that n* = n^ - n ( 0 , l , t i ) . Equation (22) coupled with the n u l l i n i t i a l condition T(n*,0) = 0 y i e l d s a zero population densitv T ( m . t i ) = 0. Sufficient time has not elapsed f o r the formation of t h i s size macromolecule. If the point [n^t^] l i e s below the p r i n c i p a l ground curve n ( 0 l t ) the ground curve passing through [ n i , ^ ] must originate from the boundary cond i t i o n plane, t* > 0, n* = 1. To i m p l i c i t l y evaluate the constant t*, t h i s ground curve i s generated by the t r a n s l a t i o n /


Mobility

/

1

/

n t C l , ^ ) - ! ^ = n(0,l,t*)-l The i m p l i c i t constant t* i s evaluated from the p r i n c i p a l ground curve. An i n t e r p o l a t i o n of the function n ( 0 , l , t ) y i e l d s the constant of i n t e g r a t i o n t* when n ( 0 , l , t ) = n ( 0 , l , t * ) . The boundary condition T ( l , t * ) coupled with Relationship (22) y i e l d s the population density of polymeric species of size n^ at time t i « I f the p r i n c i p a l ground curve passes through the point [ n i , t i ] , then t* = 0, n* = 1. A p r i n c i p a l advantage f o r the above formulation i s the reduction of integrations r e q u i r e d . Along a ground curve, population densities are uncoupled from nearest neighbors. Thus a combination of two i n t e grations (Equations 2 3 and 24) plus variable c o e f f i cients and l i n e a r t r a n s l a t i o n s allows for the e x p l i c i t evaluation of the molar concentration of any polymeric specie. The c l a s s i c s o l u t i o n requires an ordinary d i f f e r e n t i a l equation at each degree of polymerization plus v a r i a b l e c o e f f i c i e n t s . Nearest neighbors are coupled. Arguments of i n t e g r a t i o n are simpler functions when the method of c h a r a c t e r i s t i c s i s a p p l i e d . Experimental Equipment, The reactor was 1.523 l i t e r , 316 s t a i n l e s s s t e e l c y l i n d r i c a l , jacketed vessel equipped with two multiblade, paddle-type a g i t a t o r s . Tracer studies showed the reactor was well-mixed. A thermocouple measured temperature and was recorded continuously. Feed tanks, tubing, pumps and valves were made of s t a i n l e s s s t e e l and had t e f l o n s e a l s . Procedure. Concentration of n-BuLi i n the feed was measured by t i t r a t i o n (15) . The reactor was f i l l ed completely with styrene monomer s o l u t i o n i n toluene initially. Time was measured from the moment i n i t i a t o r and monomer feed was i n i t i a t e d . The reaction was


382



allowed to continue for six to seven residence times. Reactor samples were quenched and analyzed for styrene concentration; polymeric weight was obtained g r a v i m e t r i c a l l y from dried samples. Gel Permeation Chromatography. The instrument used for GPC analysis was a Waters Associates Model ALC-201 g e l permeation chromatograph equipped with a R401 d i f f e r e n t i a l refractometer. For population density determination, polystyrene powder was dissolved i n tetrahydrofuran (THF), 75 mg of polystyrene to 50 ml THF. Three y - s t y r a g e l columns of 10 ,10 ,10 * A were used. E f f l u e n t flow rate was set at 2.2 ml/min. Total cumulative molar concentration and population density d i s t r i b u t i o n of polymeric species were obtained from the observed chromatogram using the computer program developed by Timm and Rachow (16). Q

2

3

t

Steady State Population Density D i s t r i b u t i o n s . Representative experimental population density d i s t r i butions are presented by Figure 1 for two d i f f e r e n t l e v e l s of media v i s c o s i t y . An excellent degree of t h e o r e t i c a l (Equation 8) / experimental c o r r e l a t i o n i s observed. Inasmuch as the slope of population density d i s t r i b u t i o n at a s p e c i f i c degree of polymerization i s proportional to the rate of propagation for that size macroanion, propagation rates are also observed to be independent of molecular weight. Uncoupled Rate Constants. An i n i t i a l evaluation of polymerization k i n e t i c s i s presented i n Figure (2), constrained by v i s c o s i t y i n v a r i a n t rate constants K . The slopes of these s t r a i g h t l i n e s give i n i t i a l estimates of K g / K according to Equation (14). Figure 3 presents g r a p h i c a l l y a power law r e l a t i o n s h i p between K / K and v i s c o s i t y at 21°C and at 1 6 . 6 ° C . More scatter i n Yu's data may be a t t r i b u t e d to the use of an older GPC instrument of r e l a t i v e l y low resolution. The r a t i o K / K p i s temperature-sensitive; a change of the order of f i v e times i s observed i f the temperature i s reduced by 4 . 4 ° C and v i s c o s i t y i s kept constant. Using t h i s preliminary observation a comprehensive analysis of data w i l l allow for the e l u c i d a t i o n of the v i s c o s i t y dependency. I f Kp and K are assumed to be power functions of v i s c o s i t y with an Arrhenius temperature c o e f f i c i e n t P

e

e q

p

p

2

2

e g

2

e g

K hr

K

= a exp (-E / R T ) y

b

Jr

= c exp (E

/RT) y

d


TIMM ET AL.

Molecular

Mobility


18.


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384



18.

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Molecular

Mobility

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Viscosity >(cp) Figure

3.

Propagation

and polystyryl (

anion association )16.6°C.

kinetics:

(


) 21° C;


386


The a c t i v a t i o n energy o f the p r o p a g a t i o n r e a c t i o n (Ep) and t h a t o f a s s o c i a t i o n e q u i l i b r i u m r e a c t i o n (Eeq) are r e p o r t e d t o be 6.13 Kcal/gmole and 38.6 Kcal/gmole r e s p e c t i v e l y (0/7) . A n o n - l i n e a r s e a r c h o f the d a t a ( E q u a t i o n 14) w i l l d e f i n e the c o n s t a n t s a,b,c, and d. Data a t 16.6°C and 21°C were i n c o r p o r a t e d w i t h a l e a s t square o b j e c t i v e f u n c t i o n u s i n g Luus and J a a k o l a ' s (18) method. The a n a l y s i s r e s u l t e d i n the f o l l o w i n g r e lationships : K K

p

= 4.44 = 6.77

x 10

5

exp 38

(-6130/RT)

0

y

"" 0

0 0 0 2

x l ( f exp(+50860/RT) y " -

2

0

2

5

2

( 3 2 )

K

T h i s shows t h a t K i s independent o f v i s c o s i t y . Equil i b r i u m a s s o c i a t i o n o f p o l y s t y r y l a n i o n s , i s dependent on s o l u t i o n v i s c o s i t y . I n i t i a t i o n a n a l y s i s i s p r e s e n t e d by F i g u r e 4. A power curve f i t o f the d a t a y i e l d s v a l u e s o f y and K | t o be 3.571 and 0.002137 r e s p e c t i v e l y . The d a t a s c a t t e r may be a t t r i b u t e d t o the f a c t t h a t concent r a t i o n of primary ions T ( l , s s ) i s very s e n s i t i v e to chromatogram h e i g h t s . C o n t r i b u t i o n s of m o l e c u l e s i n the low m o l e c u l a r w e i g h t t a i l o f a chromatogram are s i g n i f i c a n t t o the t o t a l molar c o n c e n t r a t i o n , which i s s u b j e c t t o a h i g h degree o f e x p e r i m e n t a l u n c e r t a i n t y . T h i s e r r o r i s f u r t h e r m a g n i f i e d i n r e a d i n g a semilogarithmic population density d i s t r i b u t i o n . Timm and K u b i c e k {19) r e p o r t a v a l u e o f y t o be 3. Thus, the c u r r e n t v a l u e i s o f s i m i l a r magnitude. Current r e s u l t s were o b t a i n e d u s i n g GPC columns w i t h p l a t e counts i n e x c e s s o f 1,000 p l a t e s / f t . The c i t e d r e s e a r c h u t i l i z e d equipment o f the o r d e r o f 100 p l a t e s / f t . p

P o l y m e r i z a t i o n Dynamics. R e l a t i o n s h i p s p r e s e n t e d were u t i l i z e d f o r the s i m u l a t i o n o f monomer concent r a t i o n , number and w e i g h t average m o l e c u l a r w e i g h t s , and p o p u l a t i o n d e n s i t y d i s t r i b u t i o n s f o r two e x p e r i mental observations. E x p e r i m e n t a l v a l u e s o f these v a r i a b l e s are i n r e a s o n a b l e p r o x i m i t y o f c a l c u l a t e d values. Monomer c o n c e n t r a t i o n dynamics are p r e s e n t e d i n F i g u r e 5. A d d i t i o n a l o b s e r v a t i o n s f o r Run 5 are a c c u r a t e l y c o r r e l a t e d d u r i n g the r e a c t o r s t a r t u p and a t f i n a l s t e a d y s t a t e . The o b s e r v a t i o n a t one r e s i dence time, Run 4, may be i n e r r o r . The t o t a l cummul a t i v e , molar c o n c e n t r a t i o n s o f macromolecules as a f u n c t i o n o f time are p r e s e n t e d i n F i g u r e 6. The e r r o r s a s s o c i a t e d w i t h t h i s dependent v a r i a b l e are a l s o e v i d e n t d u r i n g the s t e a d y s t a t e a n a l y s i s o f i n i t i a t i o n



TIMM ET AL.

Molecular

Mobility

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§ c

Polymerization Reactors and Processes - American Chemical Society

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