Computer Applications in the Polymer Laboratory - American

23 A Kinetic Study of an Anhydride-Cured Epoxy Polymerization C. C. Lai, Delmar C. Timm, B. W. Eaton, and M . D. Cloeter

Downloaded by PENNSYLVANIA STATE UNIV on July 6, 2012 | http://pubs.acs.org Publication Date: June 27, 1986 | doi: 10.1021/bk-1986-0313.ch023

Department of Chemical Engineering, University of Nebraska, Lincoln, NE 68588-0126 Dynamic data are simulated u t i l i z i n g a kinetic model descriptive of a thermoset, step-growth polymerization. Monomeric concentrations and the cumulative molar concentration of polymeric species fit experimental data, y i e l d i n g estimates of model parameters and constants. To fit population density distribution dynamics of oligomeric species contained within the sol fraction, kinetic rate constants must be dependent on chemical f u n c t i o n a l i t y , v i s c o s i t y plus s t e r i c hindrance factors. Numerical integrations utilized a Runge-Kutta-Gill algorithm coupled to a least squares objective function.

Knowledge o f the fundamentals which control a polymerization cure are paramount t o d e v e l o p i n g s t r a t e g i e s t o o p t i m i z e a m a t e r i a l s physical performance. Extensive k i n e t i c studies have been r e p o r t e d on t h e r m o p l a s t i c r e s i n s , p a r t i c u l a r l y i n the r e g i o n s of i n i t i a l rates of polymerization and i n d i l u t e solutions. Considerably l e s s work has been published a t h i g h c o n v e r s i o n s f o r b u l k p o l y m e r i z a tions. The p r e s e n t r e s e a r c h i n i t i a t e s a long-term objective t o comprehensively model bulk p o l y m e r i z a t i o n s of thermosetting r e s i n s . The degree of network development within the r e s i n w i l l u l t i m a t e l y be e x t e n s i v e , w i t h the average molecular weight between crosslinks approaching the order o f 100. Goals i n c l u d e the demonstration of procedures f o r d e t e r m i n i n g r e a c t i o n r a t e s as f u n c t i o n s o f t h e environment and the development of strategies which w i l l optimally control c r o s s l i n k average molecular weight, the d i s t r i b u t i o n of c r o s s l i n k average molecular weights, and the average molecular weight of the network megamolecules. The r e s i n system s e l e c t e d t o i n i t i a t e these s t u d i e s i s a step-growth anhydride cured epoxy. The approach t o the k i n e t i c analysis i s t h a t which i s p r e v a l e n t i n the chemical e n g i n e e r i n g l i t e r a t u r e on reactor design and a n a l y s i s . Numerical simulations of oligomeric population density d i s t r i b u t i o n s approximate experimental data d u r i n g the e a r l y stages o f the c u r e . Future research w i l l 0097-6156/86/0313-0275S06.00/0 © 1986 American Chemical Society

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

COMPUTER APPLICATIONS IN THE POLYMER LABORATORY

276

extend the simulation to higher extents of reaction. P r i o r research (1-5) provides a f o u n d a t i o n based on a s t a t i s t i c a l approach, but model c o n s t r a i n t s do n o t i n c o r p o r a t e s u c h f a c t o r s as s t e r i c h i n d r a n c e o r v i s c o s i t y c o n t r o l l e d k i n e t i c r a t e s . The c u r r e n t r e s e a r c h shows t h a t s u c h f a c t o r s a r e n e c e s s a r y even p r i o r t o gelation f o r t h i s step growth polymerization. Reagents S h e l l ' s Epon 828, a blend o f oligomers bisphenol A (DGEBA, n=0), was used.

of d i g l y c i d y l ether o f

CH CHCH (OC H^C (CH ) ^HÔCH^HCH,,) ^ C ^ C 0 OH

(CE^) C H^0CH CHCH 0


2

2

6

3

2

6

2

2

The r e s i n i s more t h a n 90% DGEBA w i t h the remaining m a t e r i a l predominately a t n=1. The c u r i n g agent, n a d i c methyl anhydride (NMA), i s manufactured by A l l i e d Chemical. The catalyst was b e n z y l dimethylamine (BDMA). A l l materials were stored i n a desiccator to minimize moisture pickup. Curing

Reactions

Tanaka and Kakiuchi (6) proposed c a t a l y s t a c t i v a t i o n v i a a hydrogen donor such as an alcohol as a refinement to the mechanism d i s c u s s e d by F i s c h e r (7) f o r anhydride cured epoxies i n the presence of a t e r t i a r y amine. The b a s i c c a t a l y s t e l i m i n a t e s e s t e r i f i c a t i o n reactions (8). Shechter and Wynstra (2) f u r t h e r observed t h a t a t r e a c t i o n c o n d i t i o n s BDMA does not produce a homopolymer i z a t ion of oxiranes. The chemistry of the cure i s as follows: K -+ ROH + NR. V R0 IfNR. +

k

+

K

1 1 ~ 2 R0 HNR + CR.CO ' R0CR.C0 HNR j f R0CR.C0H + NR 00 * 00 * 00 -+ k . -+ K R0CR.C0 HNR. + H C CHR Z * R0CR C0CHCH 0 HNR + 0 0 * 0 * 0 0 , J

Q

3

0

A

A

Q

3

A

3

l

9

0

A

F

A

o

Q

4

R0CR C0CHCH 0H + NR. 0 0 Rg * A

9

Polymeric molecules contain reactive hydrogen s i t e s i n the form of alcohols and c a r b o x y l i c a c i d s . The r e a c t i o n sequence, alcohol plus anhydride and a c i d plus oxirane, r e s u l t s i n an increase o f one i n the degree of polymerization i f the oxirane i s DGEBA. Polymeric species also supply o x i r a n e s v i a the pendant R „ . These reactions are generalized by the notation:


23.

An Anhydride-Cured

LAI ET AL.

Epoxy

Polymerization

277

K

1 A. + C J J

A.C

+

(1)

J

k

K

1 , j

A.C + A

A.AC

+

k A

fi

j + 1

C


k AC + A A k

A.A + C

(2)

K

AjAC + E l

A j

2

J A

+ C

j + 1

(3)

K ^

A

j + k

CA

A

j + k

A + C

(5)

Equation 1 expresses a state of equilibrium between an alcohol A. on a molecule whose degree of polymerization i s j , the c a t a l y s t C and the a l k o x i d e anion A.C. In Relation 2 t h i s activated intermediate reacts w i t h monomericr anhydride A, forming an a c i d adduct A.AC, which dissociates, forming an unassociable carboxylic acid A.A. Reactions 3-5 depict the union of a c a r b o x y l i c i n t e r m e d i a t e w i t h a monomeric epoxide E, or with pendant oxiranes on macromolecules containing a t l e a s t one h y d r o x y l group, A, , or one carboxyl group, AÂ. Reaction 4 y i e l d s a molecule that contains at l e a s t two hydroxyls. Reaction 5 describes a product molecule that contains at l e a s t one alkoxide anion and one c a r b o x y l group. The a c t i v a t e d i n t e r m e d i a t e i n v o l v e d i n R e a c t i o n 5 i s t h e r e f o r e d e s c r i b e d by A. ^CA. The A.C segment i s the a l k o x i d e anion; the segment A A represents the i a r b o x y l i c a c i d . The notation e x p l i c i t l y d e s c r i b e s the end group p a r t i c i p a t i n g i n the r e a c t i o n , but the molecule's functionality i s implicit. The i n t e n t i s to i d e n t i f y a macromolecule with a high f u n c t i o n a l i t y but once, either as A or as AÂ. The rate constant k. . w i l l account f o r the number of polymerization s i t e s per molecule!'' The i n i t i a l index on the r a t e constant k. . d e s c r i b e s the chemical moiety of the catalyst complex. The seddnd index describes the molecular weight of the reactive adduct. The e q u i l i b r i u m constant K. i s s i m i l a r l y d e f i n e d . I f i=1 , an alcohol group i s singled out, i f i=2, a carboxylic acid participates i n the r e a c t i o n . The t o t a l number of reactive s i t e s i s i n v a r i a n t d u r i n g a batch cure, but each c o u p l i n g of macromolecules (see Equations 4 and 5) y i e l d s a specie that accumulates these hydrogen sites. I f e a c h s i t e has an equal p r o b a b i l i t y of r e a c t i n g , a molecule t h a t c o n t a i n s m u l t i p l e s i t e s w i l l , on the average, experience reactions that are p r o p o r t i o n a l to i t s f u n c t i o n a l i t y , which i n turn i s a function of the molecular weight of the molecule. In the present work, the catalyst concentration i s nearly equal to the concentration of reactive s i t e s . J

J

+

fc

fc

K i n e t i c Reaction Modeling I f the a l c o h o l and a c i d complexes a r e a t e q u i l i b r i u m , t h e i r respective molar concentrations may be expressed by


278


(A^C) = ^

A^ C

(AjAC) = K

(6)

AjA C

2

(7)

For batch, isothermal polymerizations, the p r i n c i p l e of conservation of population y i e l d s f o r alcohol adducts


dUjO/dt-*^ A

(AjO)*^

E ( A ^ A C H ^ k ^

( A ^ O P j ^ /2 (8)

The several k i n e t i c rates are a consequence of Reactions 2-5* The f i r s t two r e p r e s e n t monomeric a d d i t i o n s ; the t h i r d describes a l l r a t e s by which two p o l y m e r i c m o l e c u l e s s m a l l e r than j form a molecule of size j . The molecule A AC contains a carboxylic anion; the second molecule P. supplies an oxirane and, i n general, i s any polymeric molecule of^s?ze j - n .

Equation 10 denotes reactions i n v o l v i n g a c i d d(A AC)/dt « k , ^ A (AjC) - k j

2

J

(AjAC) [E + P

T 0 T

adducts. ] = 0

(10)

Monomeric additions r e s u l t i n the f i r s t and second rate expressions. Polymeric molecules react with the activated intermediate a t a rate that i s proportional to t h e i r cumulative molar concentration "Pôf* P

T0T

=

2P

j

The rate of anhydride a d d i t i o n , see E q u a t i o n 10, i s also the rate l i m i t i n g step. The a d d i t i o n o f Equations 8 and 10, s u b j e c t t o e q u i l i b r i u m constraints, y i e l d s

K

=

1 " j ' * *

K

2

E

[k

k

A

2,j-1 V l

K

" 2,j 2

A

j

A P

k

A

A

- 2,j j

]+

K

k

A

2^ 2,n n n=l

A

P

j-n

/ 2

T0T

(12)

Equations 6, 7 and 10 may be solved simultaneously, y i e l d i n g k

i

= k

K

2,j 2 t

E + P

T0T

]

V

/

[

k

K

1,j1

A

]

(

1

3

)

The i n i t i a l f o r m u l a t i o n o f Epon 828 and NMA must be such t h a t oxirane equivalents equal the c o n c e n t r a t i o n of anhydride groups. A balanced stoichiometric r a t i o enables the polymerization t o develop macromolecules a t high extents of reaction (10). Therefore

k

r \ i

K

2 i k

k

/

k

u

K

i

(

u

)

The c o n c e n t r a t i o n o f carboxylic acid s i t e s i s proportional t o the concentration of hydroxyl s i t e s . Equations 9 and H may be solved, yielding


23.

An Anhydride-Cured

LAI ET AL.

[

V

k

K

+

U 1

k

2J

K

2

]

Epoxy

k

V

/ 1J

K

Polymerization

279

( 1 5 )

1

Equation 12 may now be expressed i n terms of the experimentally measurable population density d i s t r i b u t i o n P^. d

P

i

/

d

t

=

k


+

E

i,r k

(

k

1

2,M

/

k

k

2,j

P

P

p

1,i * 2 , n n j V n—l

2

k

M-V k

P

P

( 1 6 )

2J- 1J T0T j

I n i t i a l l y , oligomers of DGEBA are present i n Epon 828 a t a s p e c i f i c , but low r a t i o . I t i s also l i k e l y t h a t some r e s i d u a l a c i d i s cont a i n e d w i t h i n NMA ( l e s s than a few p e r c e n t ) . Both contribute to reactive hydrogen s i t e s . This y i e l d s the i n i t i a l conditions PÔ)

= AÔ)

+ A A(0);

P^(0) = 0

1

, j = 2,3,4...

(17)

Monomeric reactions, see Equations 2 and 3, indicate that dE/dt = -C K

2

E Z k ^

AÂ

(18)

A Z k ^

A^

(19)

2

dA/dt = -C

Since the molar c o n c e n t r a t i o n o f monomer i s s u b s t a n t i a l l y greater than the molar concentration o f polymer, Equations U 18 and 19 p r e d i c t a balanced s t o i c h i o m e t r y d u r i n g the cure. The summation terms r e p r e s e n t t h e i n v a r i a n t number o f r e a c t i v e s i t e s on a continuously decreasing number of polymeric species. The i n t e g r a t i o n of Relationship 18 y i e l d s f

E = E(0) exp (-C K

2

Zk ^

AÂ

t)

(20)

The cumulative molar concentration of polymeric s p e c i e s PmQip may be evaluated from the population density d i s t r i b u t i o n , Equation 16. The f i r s t two r a t e e x p r e s s i o n s r e p r e s e n t monomeric additions which do not change the molar concentration of polymeric molecules. A rate constant that describes f u n c t i o n a l i t y as a separable function of t h e m o l e c u l e ' s d e g r e e o f p o l y m e r i z a t i o n s a t i s f i e s t h i s constraint. The s i m p l e s t , r e a l i s t i c f u n c t i o n i s the l i n e a r expression k

i,j

=

k

i,1

( b

+

"

j )

( 2 1 )

When the number of molecules i s reduced by a f a c t o r o f two, they w i l l on the average contain twice as many reactive s i t e s . Summation for j=1,2,3... y i e l d s dP

T Q T

/dt = - P

T Q T

E k, ^

Pj / 2

(22)

The sum i s i n v a r i a n t and equals the cumulative number of reactive hydrogen s i t e s present. Equations 18 and 22 may be s o l v e d simultaneously, y i e l d i n g the power function



280

P

P

T0T / T 0 T

( 0 )

"

(

E

I

E(0) )

a

(23)

where a =

z k, . P /(2C K. j

Z k^ J A. A)

(24)

Polymerization dynamics are expected t o be d e s c r i b e d by Equations 16, 20, and 23.


Experimental

Design

The r e s i n contained 100.00 parts Epon 828, 80.00 parts NMA and 2.00 parts of BDMA. A part i s a u n i t of mass. This formulation y i e l d s a r e s i n with good mechanical performance. The formulation was cured i n small t e s t tubes t h a t were p l a c e d i n an e l e c t r i c a l l y - h e a t e d , forced-air c i r c u l a t i n g oven which was controlled within 0.1 C of the s e t temperature. Specimens were removed with i n c r e a s i n g time, thermally quenched and stored a t -25°C. I f t h e c o n t e n t o f t h e t e s t t u b e was s o l i d a t a m b i e n t temperature, an e l e c t r i c d r i l l equipped with a carbide b i t produced, upon d r i l l i n g , a t h i n ribbon that had a high surface area t o volume ratio. This was leached with tetrahydrofuran a t ambient conditions u n t i l chromatography a n a l y s i s o f the s o l f r a c t i o n i n d i c a t e d that equilibrium had been established. A gram of r e s i n was leached with 25 ml of solvent. A Waters A s s o c i a t e s g e l permeation ( s i z e exclusion) chromatograph was equipped w i t h a d i f f e r e n t i a l r e f r a c t o m e t e r and three M i c r o s t y r a g e l columns o f n o m i n a l s i z e 100, 100 and 500 A. Chromatograms were n u m e r i c a l l y i n t e r p r e t e d by the procedure d e s c r i b e d by Timm and co-workers (IJ.,12). A D i g i t a l LSI-11/23 m i c r o c o m p u t e r was i n t e r f a c e d w i t h t h e c h r o m o t o g r a p h . F o r c a l i b r a t i o n , r e a c t a n t s were used f o r monomeric standards; l i n e a r epoxy resins were used t o c h a r a c t e r i z e oligomeric species leached from the r e s i n . These polymeric standards where manufactured from phenyl g l y c i d y l e t h e r , n a d i c methyl anhydride and benzyl dimethyl amine and a r e d i s t r i b u t e d by a P o i s s o n molar d i s t r i b u t i o n . The numerical algorithm e f f e c t i v e l y corrects chromatograms f o r e f f e c t s of i m p e r f e c t r e s o l u t i o n . Analyses o f thermoplastic resins y i e l d molar d i s t r i b u t i o n s o f polymeric s p e c i e s t h a t are consistent with k i n e t i c theory (13,1^). E f f e c t s of molecular configurations on hydrodynamic volume have not been e x p l i c i t l y i n c o r p o r a t e d i n t o the a l g o r i t h m ; however, a n a l y s i s o f the s o l f r a c t i o n o f thermoset r e s i n s i n d i c a t e s that chromatography estimates o f c r o s s l i n k a r c h i t e c t u r e are consistent with independent observations v i a dynamic mechanical spectroscopy (15-18). Therefore, a reasonable l e v e l of confidence e x i s t s i n the assignment of molecular weight i n the present study. Hydrodynamic volume i s a function of branching. Nonlinear chains i n general have a smaller mean end-to-end distance. Zimm (12) calculated s t a t i s t i c a l l y that random branching with f i v e t r i f u n c t i o n a l branch points per molecule reduced the radius of g y r a t i o n o n l y by a f a c t o r o f about 0.7. The c u r r e n t research evaluates oligomeric molecules which on the average c o n t a i n l e s s t h a n f i v e b r a n c h p o i n t s due t o the


23.

LAI ET AL.

An Anhydride-Cured Epoxy Polymerization

r e l a t i v e l y h i g h r a t i o o f b i f u n c t i o n a l monomer t o f u n c t i o n a l , but d i l u t e polymer.

281 the

multi-


Results T y p i c a l d a t a i n t h e f o r m o f o b s e r v e d chromatograms a r e p r e s e n t e d by F i g u r e 1. S a m p l e s w e r e o b s e r v e d f o r up t o f o u r h o u r s . The o l i g o m e r i c m o l e c u l e s w i t h i n the l e a c h a t e e l u t e p r i o r t o 19.6 m l o f e l u e n t v o l u m e . F o r t h e f i r s t 1.5 h o u r s , t h i s m a t e r i a l i n c r e a s e s b o t h i n mass c o n c e n t r a t i o n and i n average m o l e c u l a r w e i g h t s i n c e the peak grows i n a r e a and s h i f t s t o l o w e r e l u t i o n v o l u m e s . The o l i g o m e r i c m a t e r i a l u l t i m a t e l y d i m i n i s h e d as a consequence o f p o l y m e r i z a t i o n s w i t h t h e i n s o l u b l e network f r a c t i o n . The monomers DGEBA a n d NMA e l u t e n e a r 21.6 and 23.6 m l , r e s p e c t i v e l y . I n c r e a s i n g t i m e results i n diminishing concentrations. Observed monomer c o n c e n t r a t i o n s a r e p r e s e n t e d by F i g u r e 2 as a f u n c t i o n o f c u r e t i m e and t e m p e r a t u r e ( s e e E q u a t i o n 2 0 ) . At h i g h monomer c o n v e r s i o n s , t h e d a t a appear t o approach an asymptote. As t h e e x t e n t o f n e t w o r k d e v e l o p m e n t w i t h i n t h e r e s i n a d v a n c e s , the r a t e o f r e a c t i o n d i m i n i s h e s . M o l e c u l a r d i f f u s i o n o f macromolecules, i n i t i a l l y , and o f monomeric m o l e c u l e s , u l t i m a t e l y , becomes s e v e r e l y r e s t r i c t e d , r e s u l t i n g i n d i f f u s i o n - c o n t r o l l e d reactions (20). The m a t e r i a l u l t i m a t e l y becomes a g l a s s . Monomer c o n c e n t r a t i o n dynamics a r e no l o n g e r e x p o n e n t i a l d e c a y s . The r a t e c o n s t a n t s become t i m e d e p e n d e n t . F o r t h e c u r e a t 60°C, monomer c o n c e n t r a t i o n c a n be d e s c r i b e d by an e x p o n e n t i a l f u n c t i o n . M o l e c u l a r c h a r a c t e r i z a t i o n of the s o l f r a c t i o n produced r e p r e s e n t a t i v e d a t a s h o w n i n F i g u r e 3. The a b s c i s s a i s t h e l o g a r i t h m o f t h e moles o f o l i g o m e r i c m o l e c u l e s l e a c h e d p e r gram o f r e s i n , P.. The o r d i n a t e i s d e g r e e o f p o l y m e r i z a t i o n w h i c h e q u a l s molecular* weight/328 where t h e c o n s t a n t o f c a l i b r a t i o n i s 328. The e x t e n t o f network development has a pronounced e f f e c t on o l i g o m e r i c d i s t r i b u t i o n s a s was o b s e r v e d w i t h monomeric c o n c e n t r a t i o n s . At higher cure t e m p e r a t u r e s , i . e . h i g h e r e x t e n t s of c r o s s l i n k development, 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 d a t a p r e s e n t e d a r e representative. The l o c u s o f maxima o f the p o p u l a t i o n d e n s i t i e s becomes more v e r t i c a l ; c o n c e n t r a t i o n s and average m o l e c u l a r w e i g h t s r a p i d l y d i m i n i s h . However, o l i g o m e r i c f r a c t i o n s a t t h e end o f t h e c u r e become d o m i n a t e d by e x p o n e n t i a l d i s t r i b u t i o n s ( V 7 ) , an i n d i c a t i o n t h a t the r e a c t i o n i s l i k e l y more complex t h a n t h a t w h i c h i s d e s c r i b e d by t h e p r e s e n t model. R e l a t i o n s h i p 23 p r o v i d e s a method f o r e v a l u a t i n g the p a r a m e t e r " a " t h a t i s d e f i n e d b y E q u a t i o n 24The c u m u l a t i v e m o l a r c o n c e n t r a t i o n of polymeric s p e c i e s P was n u m e r i c a l l y e v a l u a t e d v i a integration of population density d i s t r i b u t i o n s . The c o n t r i b u t i o n o f n e t w o r k m o l e c u l e s t o t h e z e r o t h moment o f t h e d i s t r i b u t i o n i s n e g l i g i b l e . R e s u l t s a r e p r e s e n t e d by F i g u r e 4 and show t h a t 1

T

0

T

(25) The v a l u e f o r P ( 0 ) may be e v a l u a t e d s i n c e E ( 0 ) = 1.65 X 10"^ moles/g. R e s u l t s i n d i c a t e t h a t P ( 0 ) i s about 10% o f E ( 0 ) . This i s c o n s i s t e n t w i t h known l e v e l s o f o l i g o m e r s o f DGEBA i n Epon 828 and t h e r e s i d u a l a c i d c o n t e n t o f NMA. T O T

T 0 T


282



0.0

1.0

2.0

3.0

TIME (hours) Figure 2. Monomer dynamics.



LAI ET AL.

An Anhydride-Cured

DEGREE

Figure 3.

Figure

Epoxy

OF

Polymerization

POLYMERIZATION

Population density d i s t r i b u t i o n dynamics.

Correlation of monomer and polymer concentrations.



284

The rate constant k. . i s a f u n c t i o n of reaction temperature, f u n c t i o n a l i t y and environmental f a c t o r s which i n c l u d e molecular diffusion. S t e r i c h i n d r a n c e may be s i g n i f i c a n t with these chemically complex monomers. These functions l i k e l y are separable. To address f u n c t i o n a l i t y , c o n s i d e r the number average molecular weight of the polymeric phase MW/328 = (E(0) - E ) / P

T 0 T

= DP

(26)

The number average degree of polymerization i s DP. A c o r r e l a t i o n of TQ.T. t ^ average degree of p o l y m e r i z a t i o n DP yfelaed a l i n e a r r e l a t i o n s h i p , which may be generalized to Downloaded by PENNSYLVANIA STATE UNIV on July 6, 2012 | http://pubs.acs.org Publication Date: June 27, 1986 | doi: 10.1021/bk-1986-0313.ch023

P

k

a

i,j

=

s

k

a

i,1

f

u

n

c

Computer Applications in the Polymer Laboratory - American

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