Analysis of Molecular Weight Distribution Using Multicomponent Models

4

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Analysis of Molecular Weight Distribution Using Multicomponent Models EPHRAIM BROYER Allied Corporation, Corporate Research & Development, Morristown, NJ 07960 RICHARD F. ABBOTT Allied Corporation, Allied Fibers & Plastics, Baton Rouge, LA 70805

The analysis of molecular weight distribution is an important tool in polymerization and is used widely in quality control, development of new grades, polymerization reaction engineering and in assessing the effects of processing on prop e r t i e s . In this work, a polymer is considered as a blend of defined components. Each component is descirbed by a molecular weight distribution model. In this study the models of Flory, Schulz-Zimm and Wesslau were used. A computer program was written, which calculates the weight fraction of each component and the parameters defining i t s distribution. Characterization of molecular weight distribution is transformed, therefore, to a non-linear regression analysis, which can be solved by any number of different techniques. The components of the blend can be related to catalyst activity or to operating con ditions of the polymerization reactors. The method was used to analyze GPC curves of HDPE samples. Good description of the distribution was obtained with 2-5 components. The Wesslau model requires the smallest number of components to define a mole cular weight distribution for HDPE. The importance of molecular weight distribu tion in studies of polymerization, polymer pro cessing and the physical and mechanical properties of polymers creates a need for mathe matical description of the distribution. Several models are commonly used (Flory [1], Schulz-Zimm 0097-6156/82/0197-0045$06.00/0 © 1982 American Chemical Society In Computer Applications in Applied Polymer Science; Provder, Theodore; ACS Symposium Series; American Chemical Society: Washington, DC, 1982.

COMPUTER APPLICATIONS IN APPLIED POLYMER SCIENCE


[ 2 ] , W e s s l a u [3] and Tung [ 4 ] ) . Often these s i m p l e m o d e l s do n o t d e s c r i b e a d e q u a t e l y the m e a s u r e d d i s t r i b u t i o n of C o m m e r i c a l p o l y m e r s . T h e o r e t i c a l d e r i v a t i o n s f o r the b l e n d i n g of two components w i t h known d i s t r i b u t i o n s have been reported [5]. T h e s e s t u d i e s c o n c e n t r a t e d on the e f f e c t o f t h e p a r a m e t e r s on t h e MWD of t h e blend. T h i s work o r i g i n a t e d from the need to r e p r o duce m a t h e m a t i c a l l y t h e MWD as m e a s u r e d by GPC. The p r o p o s e d m e t h o d d e s c r i b e s a p o l y m e r as a b l e n d of c o m p o n e n t s ; e a c h of them i s d e f i n e d by i t s p a r a m e t e r s and w e i g h t f r a c t i o n . I t provides the i n f o r m a t i o n needed i n the development and e n g i n e e r i n g of a p o l y m e r i z a t i o n p r o c e s s and can be r e l a t e d t o o p e r a t i n g c o n d i t i o n s a n d catalyst performance• MULTI-COMPONENT

MODELS

A p o l y m e r i n t h i s s t u d y i s a s s u m e d t o be a b l e n d of s e v e r a l components. Each component i s d e f i n e d b y i t s w e i g h t f r a c t i o n a n d a MWD, which i s d e s c r i b e d by a m o d e l . A g e n e r a l e q u a t i o n f o r the w e i g h t f r a c t i o n of m o l e c u l e s of s i z e P (P d e g r e e of p o l y m e r i z a t i o n ) i s g i v e n by N = E g. w.

W(P)

(1) (C.,P)

where N ^

g

i

=

1

N i s the number of components i n g± i s the w e i g h t f r a c t i o n of the

(2) the l t

blend

and

h

component• wj i s t h e e q u a t i o n d e s c r i b i n g t h e MWD of the i ^ h component c a l c u l a t e d f o r m o l e c u l e s of s i z e P, w i t h p a r a m e t e r s g i v e n by t h e v e c t o r C. Equation 2 i s a mass b a l a n c e f o r the system. Use of a l i n e a r combination of the m o d e l s l e a d s to a s i m p l e e v a l u a t i o n of the m o l e c l a r w e i g h t averages of the s y s t e m . I n t h i s s t u d y t h e MWDs o f the c o m p o n e n t s a r e d e s c r i b e d by t h e m o d e l s p r o p o s e d by F l o r y , S c h u l z and Zimm, and W e s s l a u . A brief d e s c r i p t i o n of e a c h m o d e l and i t s c h a r a c t e r i s t i c s is given. The equations of t h e a v e r a g e molecular weights of the b l e n d s are a l s o presented.

In Computer Applications in Applied Polymer Science; Provder, Theodore; ACS Symposium Series; American Chemical Society: Washington, DC, 1982.

BROYER AND ABBOTT


Flory

Models for MW Distribution

Distribution

Analysis

Model

The F l o r y d i s t r i b u t i o n i s a random distribution u s e f u l i n s e v e r a l modes o f p o l y m e r i z a t i o n . This distribution results from a d d i t i o n polymeriza t i o n r e a c t i o n s when t h e o n l y s i g n i f i c a n t pro cesses that i n t e r r u p t macromolecular growth are either or both of chain t r a n s f e r ( t o any s p e c i e s b u t t h e p o l y m e r ) o r t e r m i n a t i o n by d i s p r o p o r tionation. Likewise, this molecular weight d i s t r i b u t i o n describes linear condensation poly m e r i z a t i o n when e q u a l r e a c t i v i t y i s a s s u m e d f o r a l l ends o n l y when t h e r e a c t i o n i n v o l v e s an e q u i l i b r i u m between p o l y m e r i z a t i o n and depolymerization. The model d e s c r i b e s t h e d i s t r i b u t i o n w i t h one p a r a m e t e r w h i c h i s t h e number a v e r a g e molecular weight. The d i s t r i b u t i o n e q u a t i o n i s : W(P)

= y

2

P exp (-yP)

(3)

where y - P - p - p (4) n w z y i s the reciprocal o f t h e number a v e r a g e degree of p o l y m e r i z a t i o n . T h e i n d i c e s n , w, a n d z a r e f o r t h e number, w e i g h t s , and z m o l e c u l a r weight averages. The s i m p l i c i t y of t h e F l o r y distribution (1 p a r a m e t e r ) , the constant ratio between t h e a v e r a g e s , and i t s t h e o r e t i c a l deriva t i o n a r e t h e main reasons f o r i t s u s e . I n a b l e n d o f N c o m p o n e n t s , each d e f i n e d by a F l o r y d i s t r i b u t i o n , the weight f r a c t i o n of molecules of s i z e P i s given by: N

W(P)

2 = E g.y.^P e x p ^ y . P )

(5) K

D

}

Equation 5 results from combining Equations 3 and 1. A molecular weight d i s t r i b u t i o n i s defined, a c c o r d i n g t o t h i s scheme, by (2N-1) p a r a m e t e r s . The r e d u c t i o n of one p a r a m e t e r r e s u l t s from imposing t h e c o n s t r a i n t of Equation 2. Schulz-Zimm

Distribution

Model

This i s a s p e c i f i c case of the g e n e r a l i z e d expo nential distribution. I t was d e r i v e d t o d e s c r i b e polymerization with a constant rate of i n i t i a t i o n and t e r m i n a t i o n by s e c o n d o r d e r i n t e r a c t i o n with American Chemical Society Library 16thPolymer St., N.W.Science; Provder, Theodore; In Computer Applications in 1155 Applied Washington, DC 20036 ACS Symposium Series; American Chemical Society: Washington, DC, 1982.

48


t h e monomer. I t p r o v i d e s d e s c r i p t i o n f o r v i n y l and c o n d e n s a t i o n p o l y m e r s . I t i s a two p a r a meter model g i v e n by:

a+1 a T(aTiy Y

W ( P )

=

e x

(

Y P )

p -

(6)


where

Y

=

a P

a+1

=

=

a+2

P w

n

P

(7)

z

The r e c i p r o c a l of a measures t h e b r e a d t h of t h e d i s t r i b u t i o n a n d T i s t h e gamma f u n c t i o n . A l t h o u g h t h i s m o d e l p r o v i d e s an a d e q u a t e f i t for the d i s t r i b u t i o n curves, the calculated parameters may l e a d t o e r r o n e o u s molecular weight averages. I f a p o l y m e r i s c o n s i d e r e d as a b l e n d components, the d i s t r i b u t i o n equation i s : N

a

g.Y. i

+ 1

of

a

p i (8)

w ( p )

= j n ^ f i ) —

(3N-1) parameters the MWD. Wesslau

must

Distribution

e

x

p

(

Y

p

- i >

be d e t e r m i n e d

to describe

Model

T h i s i s a s p e c i f i c case of the g e n e r a l i z e d logarithmic distribution. I t i s a log-normal d i s t r i b u t i o n w h i c h a l l o w s one t o c o v e r a w i d e range of m o l e c u l a r w e i g h t s . The w e i g h t fraction of t h e l o g a r i t h m of t h e m o l e c u l a r s i z e i s assumed t o be n o r m a l l y d i s t r i b u t e d . The d i s t r i b u t i o n i s d e f i n e d by two p a r a m e t e r s and i s g i v e n by:

W(P) =

1 BP/5?

exp^

ln(P/Po)

(9)

where

°

exp( 3 /4) 2

r

exp(3 /4) 2

exp(36 /4) 2


(10)

BROYER AND ABBOTT


Analysis

P i s t h e median v a l u e of t h e d i s t r i b u t i o n , that i s , one h a l f o f t h e v a l u e s o f P a r e l e s s t h a n P , a n d i t i s t h e s t a n d a r d d e v i a t i o n o f I n P. A c o n s t a n t r a t i o o f e x p (6^/2) i s o b s e r v e d b e t w e e n the molecular weight averages. The peak of t h e MWD i s l o c a t e d a t P„ I n case t h e p o l y m e r i s c o n s i d e r e d t o be a b l e n d , t h e w e i g h t d i s t r i b u t i o n i s given by: 0


0

,

?! J i

-I

N

w(p)

r fln(P/Poi)1

^?7¥

e x p

{

\

- [ X j

As w i t h t h e S c h u l z - Z i m m m o d e l , one n e e d s t o e v a l u a t e (3N-1) parameters t o c h a r a c t e r i z e a blend. A graph of the blend cumulative d i s t r i b u t i o n on p r o b a b i l i t y l o g p a p e r w i l l r e s u l t i n a c u r v e composed o f s e v e r a l s t r a i g h t segments. Average

Molecular

Weights

E v a l u a t i o n o f t h e number, w e i g h t and Z mole c u l a r w e i g h t a v e r a g e s of t h e b l e n d i s done a f t e r these m o l e c u l a r averages a r e c a l c u l a t e d f o r each component. E q u a t i o n s 4 , 7 , a n d 10 p r o v i d e r e l a t i o n s between t h e parameters of t h e models and the m o l e c u l a r averages of the components. The m o l e c u l a r weight averages of t h e blend a r e c a l c u l a t e d a c c o r d i n g t o a d d i t i o n r u l e s and a r e a f u n c t i o n o n l y of t h e m o l e c u l a r a v e r a g e s and the weight f r a c t i o n of each component.

The

number

average

N

-T

M

The

•1

=IE g, g

^

weight

M

The

n

x

/ M

(12)

ni

average

molecular weight i s :

N = E g. M . w ^ 1 wi

Z average M



N = E g.M M ./M z i wi z i w &

1

(13)

. . (14)



CALCULATION

OF

THE

PARAMETERS


Molecular weight distribution is usually m e a s u r e d by GPC or o t h e r f r a c t i o n a t i o n tech niques. The a c c u r a c y of f r a c t i o n separation d e p e n d s on f a c t o r s l i k e t h e l e n g t h o f the c o l u m n , f l o w r a t e and c o n c e n t r a t i o n . A common GPC measurement d e s c r i b e s a d i s t r i b u t i o n by 20-30 fractions. E v a l u a t i o n of the p a r a m e t e r s i n e q u a t i o n s 5 , 8 a n d 11 c a n be d o n e b y : A. I n t e g r a t i o n of t h e s e e q u a t i o n s and comparison w i t h a c u m u l a t i v e d i s t r i b u t i o n or the differen t i a l d i s t r i b u t i o n a s p r o v i d e d by t h e normalized GPC chromatogram. JB. Differentiation of the c o m m u l a t i v e d i s t r i b u t i o n to weight fractions o f one u n i t o f d e g r e e o f p o l y m e r i z a t i o n , and d i r e c t l y c o m p a r e them w i t h e q u a t i o n s 5, 8 a n d 11. In both methods, a n o n - l i n e a r r e g r e s s i o n proce dure i s needed to e v a l u a t e the parameters. The use of the i n t e g r a t i o n p r o c e d u r e may require long computation times, especially i f a model cannot be integrated analytically. We, therefore, decided to use the d i f f e r e n t i a l form of the distribution and n u m e r i c a l l y d i f f e r e n t i a t e d the cumulative distributions. The cumulative d i s t r i b u t i o n was p i e c e - w i s e i n t e r p o l a t e d and f o r e a c h f r a c t i o n the d i f f e r e n t i a l was c a l c u l a t e d as t h e w e i g h t fraction of molecules o f s i z e P. The differentiating procedure i s described schematically in Figure 1. Differentiating by s u c h a n u m e r i c a l t e c h n i q u e i s d o n e o n c e f o r e a c h GPC reading. The fitting of these data p o i n t s i s t e s t e d w i t h the v a r i o u s models. Numerical d i f f e r e n t i a t i n g through this interpolation procedure provides adequate a c c u r a c y , but o t h e r i n t e r p o l a t i o n procedures may be used. T h e m a i n p u r p o s e o f t h i s w o r k was to r e p r o d u c e t h e w h o l e MWD and t h e o b j e c t i v e f u n c t i o n o f t h e n o n - l i n e a r r e g r e s s i o n was to minimize t h e sum o f r e l a t i v e e r r o r s . Determination of e a c h b a s i c m o d e l s t a r t s w i t h one c o m p o n e n t and t h e number of c o m p o n e n t s i s i n c r e a s e d u n t i l an a c c e p t a b l e f i t i s o b t a i n e d between the computed c u r v e and m e a s u r e d one. A g r e e m e n t h a s t o be reached a l s o b e t w e e n the v a l u e s of the computed and e x p e r i m e n t a l m o l e c u l a r averages.



4.

BROYER AND ABBOTT

Figure 1.


Analysis

Schematic description of the calculation of the differential weight fraction.


51


ANALYSIS

OF T H E G P C OF H D P E


The chromatograms o f three samples of high d e n s i t y p o l y e t h y l e n e were a n a l y s e d according to t h e p r o c e d u r e s o u t l i n e d i n t h ep r e v i o u s s e c t i o n s . The parameters c a l c u l a t e d f o r each model a r e l i s t e d on t h er e s p e c t i v e g r a p h s . Sample 1 i sa polymer w i t h a r e l a t i v e l y n a r r o w m o l e c u l a r w e i g h t d i s t r i b u t i o n w h i c h was c h a r a c t e r i z e d by 115 f r a c t i o n s . I t s weight a v e r a g e Mw= 6 . 2 2 X 1 0 * a n d i t s d i s p e r s i t y i s g i v e n by M /M = 2.68 a n d M /M = 3 . 0 7 . A l l t h e multicomponents models give adequate f i t t o t h e molecular weight averages, but the Wesslau m o d e l s p r o v i d e i t w i t h t h e l e a s t number o f components. F i g u r e s 2-5 a r e p l o t s o f s e v e r a l r e p r o d u c t i o n attempts w i t h t h et e s t e d models. The W e s s l a u model, which i s c o n s i d e r e d as t h e b e s t d e s c r i p t i o n f o r MWD o f Z i e g l e r t y p e p o l y m e r i z a t i o n , p r o v i d e d t h eb e s t reproduction of t h ed i s t r i b u t i o n c u r v e . A one component W e s s l a u model ( F i g u r e 2) p r o v i d e s good agreement with the molecular weight averages, but there are l a r g e d e v i a t i o n s between t h ee s t i m a t e s and the measurements. A two component Wesslau model ( F i g u r e 3 ) p r o v i d e s e x c e l l e n t f i t t o t h e MWD curve and t h ea v e r a g e s . The a d d i t i o n of t h e c o m p o n e n t o f h i g h m o l e c u l a r w e i g h t ( o n l y 1% w e i g h t ) and s m a l l changes i n t h e parameters of the main component r e s u l t e d i n t h i s e x c e l l e n t f i t . The S c h u l z - Z i m m ( F i g u r e 4) o r t h e F l o r y m o d e l s ( F i g u r e 5) do n o t p r o v i d e a g o o d f i t t o t h e MWD w i t h t h e s a m e n u m b e r o f c o m p o n e n t s . w

n

z

w

H D P E S a m p l e 2 i s a p o l y m e r w i t h a w i d e MWD, where M = 7.34 X 1 0 , M / M = 6.33 a n d M / M = 15.9. The l a s t d i s p e r s i t y r a t i o i s l a r g e be cause of t h et a i l of high molecular weight fractions. A 3 component F l o r y model ( F i g u r e 6) d e s c r i b e s w e l l t h e h i g h m o l e c u l a r weight t a i l b u t t h e r e p r o d u c t i o n o f t h e MWD i s n o t satisfactory. 2 component Schulz-Zimm model ( F i g u r e 7) does n o t a d e q u a t e l y reproduce the distribution. Figure 8 describes the results of a 2 component Wesslau model. Although a one component model p r o v i d e s good f i t t o most o f t h e MWD, t h e h i g h m o l e c u l a r w e i g h t t a i l (which 4

w

w

n

z

w


In Computer Applications in Applied Polymer Science; Provder, Theodore; ACS Symposium Series; American Chemical Society: Washington, DC, 1982. 0

4

Figure 2. Analysis of MWD of polyethylene sample 1 by 1-component Wesslau model . Key: • , experiment and O, model β = 1.42; M — 3.93 χ 10 .


5

55'

Î

«·* s*

s

δ

δ:

ετ "3*

δ·

Ο

W W Ο Η Η

>

Ό

>

w


Figure 3. Analysis of MWD of polyethylene sample 1 by 2-component Wesslau model. Key: • , experiment and O, model.


w

Ο

Q W

w

Η-

ο r

w ο

I—I

>

23

t—ι

oo

δ

Η

δ >

r

>

H W

a

8

-Ρ»


Figure 4. Analysis of MWD of polyethylene sample 1 by 3-component Schulz-Zimm model. Key: • , experiment and O, model.


5

U\

5>*

f

o'

•s.

s

δ

S!'

"3*

δ·

I

Η Η

w w ο

>

a

>

w


Figure 5.

Analysis of M WD of polyethylene sample 1 by 3-component Flory model. Key: • , experiment and O, model.


Q w o w

W

m α hd O r

>

2

O 2!

H

Π δ >

>

W

cH

π ο

as


Figure 6.

Analysis of MWD of polyethylene sample 2 by 3-component Flory model. Key: • , experiment and O, model.


2

U\

f

I"

δ «•·».

O

ετ

1 a-

0 H H

> w w

α

>

W

«

4^


Figure 7. Analysis of MWD of polyethylene sample 2 by 2-component Schulz-Zimm model. Key: experiment and O, model.


Ο W

25

οο Ο W

w

ο •β ο

3

>

00

ο

Η

δ >

>

Η W

C

*ύ

Ο Ο

LU 00




s*

«»*.

ετ

a-

H H

w o

00

>

o

2;

>

W

4^


a f f e c t s the rheology of the p o l y m e r ) i s not q u a t e l y a c c o u n t e d f o r . A d d i t i o n o f 1.3% o f high molecular weight f r a c t i o m improves the r e p r o d u c t i o n o f t h e MWD remarkably.

ade the

HDPE S a m p l e 3 i s a p o l y m e r w i t h a b i m o d a l d i s t r i b u t i o n , c h a r a c t e r i z e d by M = 3.10xl0 and d i s p e r s i t y d e f i n e d by M / M = 3 9 . 3 a n d M / M =6. The b i m o d a l s h a p e a f f e c t s c o n s i d e r a b l y t h e r h e o l o g i c a l and t h e m e c h a n i c a l p r o p e r t i e s of t h e d i s t r i b u t i o n and a good r e p r o d u c t i o n of t h e distribution i s therefore essential. Vast dif ferences are found between averages c a l c u l a t e d f r o m t h e GPC a n d t h e r e g r e s s e d parameters. E x a m i n a t i o n o f F i g u r e s 9-11 shows, however, t h a t acceptable f i t was o b t a i n e d between the m u l t i c o m p o n e n t m o d e l s and t h e e x p e r i m e n t s . The 3 com p o n e n t F l o r y m o d e l ( F i g u r e 9) a n d t h e 2 c o m p o n e n t Schulz-Zim m o d e l ( F i g u r e 10) g i v e o n l y a f a i r f i t . The 2 component W e s s l a w m o d e l ( F i g u r e 11) p r o v i d e s ; h o w e v e r , v e r y good r e p r o d u c t i o n of t h e MWD . 5

w


w

n

z

z



Figure 9.

Analysis of MWD of polyethylene sample 3 by 3-component Flory model. Key: • , experiment and O, model.


Os

î

S' s

S «•«·.

"3·

î I

Η Η

> w » ο

α

>

m

>
hi hd

25

t-H

δ

H

>

o

t—I

r

hd hd

>

H W

C

hd

ο ο

to




as

î

s

ï

s: «-».

a

"S"

1

O > W W 0 H H

>

m

w


64

Acknowledgment The authors are grateful to H. D. Oltmann and A. M. Kotliar for their valuable advice.


Literature Cited 1. P.J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca New York, (1953). 2. G.V. Schulz, Z Physik. Chem., B43, 25 (1939). 3. H. Wesslau, Makromol. Chem., 20, 111 (1956). 4. L.H. Tung, J . Polym. S c i . , 20, 495 (1956). 5. A.M. K o t l i a r , J . Polym. S c i . , A2, 1057 (1964). RECEIVED May 4, 1982.


Analysis of Molecular Weight Distribution Using Multicomponent Models

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