Characterization of Highly Cross-linked Polymers - ACS Publications

1 Formation and Properties of Polymer Networks Experimental and Theoretical Studies J. L STANFORD, R. F. T. STEPTO, and R. H. STILL

Downloaded by CITY UNIV LONDON on April 9, 2016 | http://pubs.acs.org Publication Date: February 16, 1984 | doi: 10.1021/bk-1984-0243.ch001

Department of Polymer Science and Technology, The University of Manchester Institute of Science and Technology, Manchester, M6O 1QD, England

Experimental results on reactions forming tri- and tetrafunctional polyurethane and trifunctional polyester networks are discussed with particular consideration of intramolecular reaction and its effect on shear modulus of the networks formed at complete reaction. The amount of pre-gel intramolecular reaction i s shown to be significant for non-linear polymerisations, even for reactions in bulk. Gel-points are delayed by an amount which depends on the dilution of a reaction system and the functionalities and chain structures of the reactants. Shear moduli are generally markedly lower than those expected for the perfect networks corresponding to the various reaction systems, and are shown empirically to be closely related to amounts of pre-gel intramolecular reaction. Deviations from Gaussian stress-strain behaviour are reported which relate to the low molar-mass of chains between junction points. Finally, a rate theory of random polymerisation i s described which enables the moduli of networks to be predicted from the molar mass, functionality, chain structure and initial dilution of the reactants used for network formation. This paper presents a survey o f published and more recent work on c o r r e l a t i o n s between network p r o p e r t i e s and r e a c t a n t s t r u c t u r e s and r e a c t i o n c o n d i t i o n s , and extends the work presented i n recent p u b l i c a t i o n s (1 2 2)· ^ r e a c t i o n systems used have been p o l y oxypropylene (POP) t r i o l s o r t e t r o l s and mixtures o f d i o l s and t r i o l s o f v a r i o u s molar masses r e a c t i n g w i t h d i i s o c y a n a t e s ( t o g i v e polyurethanes) o r d i a c i d c h l o r i d e s ( t o g i v e p o l y e s t e r s ) . Systems have been chosen so t h a t l i k e groups had equal r e a c t i v i t i e s and r e a c t i o n s have been c a r r i e d out i n bulk and a t v a r i o u s d i l u t i o n s i n i n e r t s o l v e n t s using equimolar amounts o f the d i f f e r e n t r e a c t i v e groups. E x p e r i m e n t a l l y , emphasis has been placed on the extent t o which p r e - g e l i n t r a m o l e c u l a r r e a c t i o n n e

9

9

0097-6156/ 84/0243-0001 $06.00/0 © 1984 American Chemical Society Labana and Dickie; Characterization of Highly Cross-linked Polymers ACS Symposium Series; American Chemical Society: Washington, DC, 1984.


2

HIGHLY CROSS-LINKED POLYMERS

and the consequent delay i n the g e l p o i n t beyond the i d e a l , Flory-Stockmayer g e l p o i n t (4,5) d e f i n e s the p h y s i c a l p r o p e r t i e s of the networks formed at complete r e a c t i o n . Intramolecular r e a c t i o n can introduce e l a s t i c a l l y i n e f f e c t i v e loops i n t o a rubbery network. In g e n e r a l , loops produce the opposite e f f e c t s on p h y s i c a l p r o p e r t i e s to those expected from entanglements. T h e o r e t i c a l approaches are o u t l i n e d which attempt to account f o r i n t r a m o l e c u l a r r e a c t i o n i n terms o f reactant s t r u c t u r e ( f u n c t i o n a l i t y , molar mass, and chain s t r u c t u r e ) and r e a c t i o n c o n d i t i o n s (concentrations o f r e a c t a n t s ) . The approaches a l l o w the p r e d i c t i o n o f g e l p o i n t s accounting f o r p r e - g e l i n t r a m o l e c u l a r r e a c t i o n . A d d i t i o n a l l y , account of p r e - g e l and p o s t - g e l i n t r a molecular r e a c t i o n allows the p r e d i c t i o n o f shear modulus at complete r e a c t i o n . Pre-Gel Intramolecular

Reaction

Previous studies(6») have shown how the number f r a c t i o n of r i n g s t r u c t u r e s formed during i r r e v e r s i b l e l i n e a r random polymerisa t i o n s l e a d i n g to polyurethanes may be measured. The work has been extended(7,8) t o n o n - l i n e a r polyurethane formation using hexamethylene diisocyanate(HDI) and POP t r i o l s . For n o n - l i n e a r p o l y m e r i s a t i o n s , i t i s found t h a t the number of r i n g s t r u c t u r e s per molecule(Np) i s always s i g n i f i c a n t , even i n bulk r e a c t i o n s . For example, Figure 1 shows N versus extent o f r e a c t i o n ( p ) , f o r l i n e a r and n o n - l i n e a r polyurethane-forming bulk r e a c t i o n s w i t h approximately equimolar concentrations o f r e a c t i v e groups(2,,6^,1). The much l a r g e r values o f N i n the n o n - l i n e a r compared with the l i n e a r p o l y m e r i s a t i o n are due t o the l a r g e r number o f o p p o r t u n i t i e s per molecule f o r i n t r a m o l e c u l a r r e a c t i o n i n the former type o f p o l y m e r i s a t i o n . However, the other f a c t o r s i n f l u e n c i n g i n t r a molecular r e a c t i o n i n the two systems, p a r t i c u l a r l y the number of bonds(v) i n the chain forming the s m a l l e s t r i n g s t r u c t u r e p r e d i c t more i n t r a m o l e c u l a r r e a c t i o n i n the l i n e a r system. A d e t a i l e d d i s c u s s i o n of these f a c t o r s has been given elsewhere(2.). It should be noted t h a t i t i s not p o s s i b l e t o reduce the number of r i n g s t r u c t u r e s formed i n such r e a c t i o n systems as the amounts o f i n t e r m o l e c u l a r r e a c t i o n r e l a t i v e to i n t r a m o l e c u l a r r e a c t i o n are at a maximum f o r r e a c t i o n s i n bulk. The g e l p o i n t o f the n o n - l i n e a r system shown i n Figure 1 was at ρ = 0.765 compared with the value of 0.707 expected i n the absence o f i n t r a m o l e c u l a r r e a c t i o n . Thus, although ρ at g e l i s only about Q% higher than expected, N = 0.3 at ρ = 0.765, showing t h a t at g e l about one molecule i n three contained a r i n g structure. Such r i n g s t r u c t u r e s or loops can have marked e f f e c t s on the p r o p e r t i e s of networks formed at complete r e a c t i o n ( 1 , 2 , 9 12). Developments i n the t h e o r e t i c a l aspects of the work, a l l o w i n g p r e d i c t i o n of N , the g e l p o i n t , and the shear moduli of networks formed at complete r e a c t i o n are presented i n the l a s t s e c t i o n of the present paper. r

r

r

r

Labana and Dickie; Characterization of Highly Cross-linked Polymers ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

1.

STANFORD ET

3

Formation and Properties of Networks

AL.

ΟΛ


0.3

-

non-linear HDI + L G 5 6

0.2

-

0.1

linear

—

0

HDI+PEG200

C\

Ο

0.2

0.4

0.6

0.8

1.0

Ρ Figure 1. Number o f r i n g s t r u c t u r e per molecule (N ) as a f u n c t i o n o f extent o f r e a c t i o n ( p ) f o r l i n e a r and n o n - l i n e a r polyurethane forming r e a c t i o n s i n bulk w i t h approximately equimolar concentrations o f r e a c t i v e groups, r = [ N C O l / [ p H l S 1) (6,7). r

0

0

0 - l i n e a r p o l y m e r i s a t i o n , HDI + p o l y ( e t h y l e n e g l y c o l ) (PEG200) at 70OC,[NCOj = 5.111 mol k g - , [ O H ] = 5.188 mol kg- ; number-average o f bonds i n chain forming s m a l l e s t r i n g s t r u c t u r e (v) = 25.2. • - n o n - l i n e a r p o l y m e r i s a t i o n , HDI + POP t r i o l (LG56) at 7QoC,[NCqIo = 0.9073 mol k g ' S J 0 H j = 0.9173 mol k g - ; v= 115. Reproduced w i t h permission, from Ref. 2. Copyright 1982, American Chemical S o c i e t y . 1

0

0

1

o


4


Intramolecular Reaction and G e l a t i o n An expression has been d e r i v e d ( 4 ) f o r the extent o f r e a c t i o n a t g e l a t i o n i n R A 2 + RBf random(13) or condensation p o l y m e r i s a t i o n which accounts more completely than e a r l i e r expressions(14-16)for i n t r a m o l e c u l a r r e a c t i o n . I t may be rearranged t o g i v e o ( f - l ) = (1 - X ' c

a b

)

(1)

2

Here, a = P Pu> where p and p^ are the extents o f r e a c t i o n o f A and Β groups at g e l , r e s p e c t i v e l y , and X'ab * ring-forming parameter. When λ ^ = 0, the c l a s s i c a l Flory-Stockmayer c o n d i t i o n f o r g e l a t i o n i s obtained. X'g^ i s p r e d i c t e d (4) to be p r o p o r t i o n a l to the d i l u t i o n o f a r e a c t i o n system, to i n c r e a s e w i t h f u n c t i o n a l i t y , and to decrease w i t h chain s t i f f n e s s and molar mass o f r e a c t a n t s . In d e t a i l , c

a

a

s

a


1

X

c

'ab

c

2

= int/ ext

s

where c i n t ( e r n a l ) * the c o n c e n t r a t i o n o f groups which can r e a c t i n t r a m o l e c u l a r l y w i t h a given group on a molecule and c t(ërnal) i s the c o n c e n t r a t i o n o f groups which can r e a c t i n t e r m o l e c u l a r l y w i t h the same group. ex

c

with

int

=

(f-2)Pab.(l,3/2) 2

Pab

(3)

3/2

= (3/2iTvb ) /N

(4)

where ν i s the number o f bonds i n the chain t h a t can form the s m a l l e s t r i n g , w i t h b i t s e f f e c t i v e bond l e n g t h , d e f i n e d such t h a t i t s mean-square end-to-end d i s t a n c e equals v b , and Ν i s the Avogadro constant. The p o s s i b i l i t y o f forming r i n g s o f a l l s i z e s i s accounted f o r by φ(1,3/2), w i t h 2

φ(1,3/2) =

Σ i=l

lV

3 / 2

=

2.612

(5)

Values o f c t have to be chosen a r b i t r a r i l y s i n c e λ' i s assumed to be constant f o r a given system. In p r a c t i v e , the two extreme experimental v a l u e s , c t = c + c and c t = c + c^ r e p r e s e n t i n g the i n i t i a l and g e l - p o i n t c o n c e n t r a t i o n s , are used i n the t h e o r e t i c a l treatment d e s c r i b e d ( 4 ) . The dependence o f X b on f u n c t i o n a l i t y i s allowed f o r by the f a c t o r (f-2) i n Equation 3. S i m i l a r l y , chain s t i f f n e s s and molar mass of r e a c t a n t s are allowed f o r by the f a c t o r ( v b ) '2 i Equation 4 and the dependence o f X on d i l u t i o n i s represented by i t s p r o p o r t i o n a l i t y to c t m Equation 2. F i g u r e 2 i l l u s t r a t e s r e s u l t s obtained from t r i - and t e t r a f u n c t i o n a l polyurethane-forming r e a c t i o n systems, w i t h X' b p l o t t e d against ( c + C ^ Q ) " , the i n i t i a l d i l u t i o n o f r e a c t i v e groups. I t i s apparent t h a t the p l o t s are curved r a t h e r than e x

b

e x

a 0

D 0

e x

C9

a c

f

a

2

n

f

a b

- 1

e x

a

1

a 0


1.

STANFORD ET AL.


0.4


0.3

0.2 1 ^

0.1

2

m

So* bo 9 -1 ._ _ .1 0.3 OA (

Ο

0.1

0.2

- -

r 1 / k

c

m

o

f

1

-

1, 0.5

Figure 2. Ring forming parameter (A'ab) versus i n i t i a l d i l u t i o n o f r e a c t i v e groups ( ( c + c b o ) - ). Experimental values o f a νι/ere used t o evaluate X b according t o Eq. 1· Systems: 1 and 2, HDI+POP t r i o l s ; 3 , 4,4 -diphenyl methane diisocyanate(MDI)+PQP t r i o l ; 4 and 5, HDI+POP t e t r o l s . Reactions c a r r i e d out a t 80°C i n bulk and i n nitrobenzene s o l u t i o n . System 1, HDI+LHT240, v=33; system 2, HDI+LHT112, v=61; system 3, MDI+LHT240, v=30; system 4, HDI+OPPE-NHI, v=29; system 5, HDI+0PPE-NH2, v=33.(LHT240 and LHT112oxypropylated 1,2,6 hexane t r i o l s ; OPPE-NHI and 0PPE-NH2 oxypropylated p e n t a e r y t h r i t o l s . ) Reproduced, w i t h permission,from Réf. 1. Copyright 1982, Plenum P u b l i s h i n g C o r p o r a t i o n . 1

ao

f

c

a

f


5

6


l i n e a r as p r e d i c t e d by Equation!?. D e t a i l e d d i s c u s s i o n s o f the r e s u l t s shown i n F i g u r e 1 and o f s i m i l a r r e s u l t s f o r p o l y e s t e r forming systems have been given else\i/here( 1,2,4,5). In g e n e r a l , p o l y e s t e r - f o r m i n g systems are found t o g i v e more l i n e a r p l o t s than polyurethane-forming systems and, w i t h regard t o the choice °f ext> the use o f c + c b g i v e s more l i n e a r p l o t s than c t = c + cbc« Thus, the f u n c t i o n a l dependence o f X,' b on d i l u t i o n appears t o be b e t t e r described by theory i f i n i t i a l d i l u t i o n ((c + C^Q)- ) i s used. From F i g u r e 2 i t i s c l e a r t h a t i n t r a m o l e c u l a r r e a c t i o n i n c r e a s e s w i t h d i l u t i o n and, as i n d i c a t e d i n Figure 1, w i t h f u n c t i o n a l i t y . I n a d d i t i o n , the p o i n t s on the curves a t the low e s t d i l u t i o n s r e f e r t o bulk r e a c t i o n mixtures, i n d i c a t i n g again ( c f . Figure 1) t h a t i n t r a m o l e c u l a r r e a c t i o n always occurs. The e f f e c t s o f chain s t i f f n e s s can be seen by comparing systems 1 and 3, which have s i m i l a r values o f ν but d i f f e r e n t chain s t r u c t u r e s ; t h a t o f system 3 c o n t a i n s a s t i f f e r , aromatic r e s i d u e . The i n i t i a l slopes o f the curves i n F i g u r e 2 and o f the corresponding p l o t s w i t h f c + c b c ^ s a b s c i s s a can be analysed according t o Equations 3 and 4, and values o f b found. The values obtained are given i n Table I . The two values o f b f o r each system g e n e r a l l y encompass the value expected from s o l u t i o n c

a 0

0

e x

a c

a

1


a o

a c

Table I . Values o f E f f e c t i v e Bond Length (b) o f Chains Forming the Smallest Ring S t r u c t u r e s ( o f ν bonds), (i) c t = c + c ; (ii) c t = c + cbc- \>DI i s the f r a c t i o n o f bonds due t o the d i i s o c y a n a t e r e s i d u e i n the chain o f ν bonds, Reproduced, w i t h permission, from R e f . l . Copyright 1982, Plenum P u b l i s h i n g Corp. e x

System 1. 2. 3. 4. 5.

HDI/LHT240 HDI/LHT112 MDI/LHT240 HDI/0PPE-NH1 HDI/0PPE-NH2

3 o

b Q

e x

a c

f

V

v /v

b/nm(i)

b/nm(ii)

3 3 3 4 4

33 61 30 29 33

0.303 0.164 0.233 0.345 0.303

0.247 0.222 0.307 0.240 0.237

0.400 0.363 0.488 0.356 0.347

D I

p r o p e r t i e s ( l , 2 , 4 , 5 ) . Thus the e f f e c t i v e average value o f C g £ l i e s somewhere bêtween~(c + cbo) and (cbc + cbc)» and probably nearer t o ( c + cbc)* The g e n e r a l l y s m a l l e r values o f b f o r the a l i p h a t i c t e t r a f u n c t i o n a l systems (4 and 5) compared w i t h the a l i p h a t i c t r i f u n c t i o n a l systems (1 and 2) probably i n d i c a t e a r e l a t i v e undercounting o f o p p o r t u n i t i e s f o r i n t r a m o l e c u l a r r e a c t i o n f o r growing s p e c i e s from t e t r a f u n c t i o n a l compared w i t h t r i f u n c t i o n a l reactants. X

ao

a c


1.

STANFORD ET AL.


7

Comparison o f systems 1 and 2 and systems 4 and 5 show t h a t s m a l l e r values o f b are obtained f o r the l a r g e r values o f ν or the s m a l l e r values o f Vnj/v, i n d i c a t i n g t h a t the chains w i t h the l a r g e r p r o p o r t i o n s of oxypropylene u n i t s are the more f l e x i b l e . Hence although system 1 g i v e s higher values o f X' b than system 2 because i t has a s m a l l e r value o f v, the d i f f e r e n c e between the curves f o r the two systems i n F i g u r e 2 i s reduced because b f o r system 2 i s s m a l l e r . S i m i l a r c o n s i d e r a t i o n s h o l d t r u e f o r the r e l a t i v e values o f X'ab f o r systems 4 and 5. Other aspects o f g e l a t i o n s t u d i e s which have been reported are the determination o f e f f e c t i v e f u n c t i o n a l i t i e s ^ ) and the use o f d i o l - t r i o l m i x t u r e s ^ ) to i n v e s t i g a t e the e f f e c t s o f v a r i a t i o n o f average f u n c t i o n a l i t y . The former work used a t r i o l which had been independently c h a r a c t e r i s e d w i t h respect to f u n c t i o n a l i t y and showed the shortcomings o f u s i n g g e l a t i o n data alone t o deduce the chemical f u n c t i o n a l i t i e s o f r e a c t a n t s . The l a t t e r work used mixtures o f a d i o l and t r i o l r e a c t i n g w i t h sebacoyl c h l o r i d e at d i f f e r e n t i n i t i a l d i l u t i o n s i n diglyme as s o l v e n t . The hydroxyl groups had equal r e a c t i v i t i e s and the r e a c t i o n mixtures were equimolar i n hydroxyl and a c i d c h l o r i d e qroups. At zero d i l u t i o n , the equation o f Stockmayer (5>17)> a = (fyj"\l), where f i s the weight-average f u n c t i o n a l i t y " o f the p o l y o l mixture, i s obeyed. The r e s u l t s are i l l u s t r a t e d i n F i g u r e 3, where a~ i s p l o t t e d versus i n i t i a l d i l u t i o n . The i n t e r c e p t s i n a at zero d i l u t i o n are equal t o the values o f ( f " l ) c a l c u l a t e d from the amounts o f d i o l and t r i o l i n the r e a c t i o n mixtures, and the decreases i n o^- w i t h i n i t i a l d i l u t i o n are due t o i n t r a m o l e cular reaction.


a

c

w

l

c

- 1

c

w

1

Network P r o p e r t i e s C o r r e l a t i o n s between Gel P o i n t and Shear Modulus. The r e a c t i o n systems i n F i g u r e 2 were used to form networks a t complete reaction(1,2,10,11). S o l f r a c t i o n s were removed and shear moduli were determined i n the dry and e q u i l i b r i u m - s w o l l e n s t a t e s at given temperatures u s i n g u n i a x i a l compression or a t o r s i o n pend ulum at 1Hz. The procedures used have been d e s c r i b e d i n d e t a i l elsewhere(11,12). The shear moduli(G) obtained were i n t e r p r e t e d according t o Gaussian theory(18-20) t o g i v e values o f M , the e f f e c t i v e molar mass between j u n c t i o n p o i n t s , c o n s i s t e n t w i t h the a f f i n e behaviour expected at the s m a l l s t r a i n s used (20). Equation 6 was used w i t h ρ c

G = ARTpφ

1 / 3 2

(V /V ) u

2 / 3

F

/M

c

(6)

the d e n s i t y o f the dry networK Φ2 the volume f r a c t i o n o f s o l v e n t present i n a swollen network, V the volume o f the dry, u n s t r a i n e d network, and Vp the volume at formation. A has the value (1-2/f) f o r networks showing phantom behaviour and 1 f o r networks showing a f f i n e behaviour (19,20). u



8


1.01 Ο

I 0.2

1 1 0.4 0.6 ao* bo

{ c

c

, 1 / k g

l-J

1 0.6 m

o

f

1.0

1

χ

F i g u r e 3. α " versus i n i t i a l d i l u t i o n o f r e a c t i v e groups ( ( c + c b ) - l ) f o r mixtures o f diol(PPG1025) and t r i o l ( L H T 112) r e a c t i n q w i t h sebacoyl c h l o r i d e a t 6Q0C i n diglyme. 0

a 0

0

rrpeOo/IMo = 1. PPG1025 - POP d i o l ; LHT112 - POP t r i o l (see c a p t i o n Figure 2). Curves 1, f = 2.99; 2, f = 2.82; 3, f = 2.65; 4, f = 2.50; 5, f = 2.35. Reproduced, w i t h permission, from Ref. 3. Copyright 1982, S o c i e t y o f Polymer Science, Japan. w

w

w

w

w


I.

9


STANFORD ET A L .

The r e s u l t s a r e shown i n Figure 4, where M / M ° i s p l o t t e d versus p p. The molar mass between j u n c t i o n p o i n t s o f the p e r f e c t nètwork(M °) i s c a l c u l a b l e from the molar mass and s t r u c t u r e o f the r e a c t a n t s (1,2.) and M was evaluated from the measured modulus using Equation 6 with A=l. p ^ ^ extent o f i n t r a m o l e c u l a r r e a c t i o n a t g e l a t i o n (1,2) given by the expression c

c

r

c

c

s

n e

r ? c

f

=

Pr,c 0 corresponds t o the ideal(Flory-Stockmayer) g e l - p o i n t , and M / M ° = 1 t o the p e r f e c t , a f f i n e network. In a l l cases i n Figure 4, M / M ° exceeds 1 and tends t o 1 as p -*· 0. Thus, only i n the l i m i t o f a p e r f e c t g e l l i n g system i s a p e r f e c t network achieved, f o r which a f f i n e behaviour i s p r e d i c t e d . I n t e r c e p t s equal t o 3 and 2 on the M / M ° a x i s would be r e q u i r e d f o r p e r f e c t , phantom networks o f f u n c t i o n a l i t i e s 3 and 4, r e s p e c t i v e l y . The p r e - g e l i n t r a m o l e c u l a r r e a c t i o n , which causes α t o exceed l / ( f - l ) i n value, a l s o produces some e l a s t i c a l l y i n e f f e c t i v e loops which have marked e f f e c t s on the moduli o f the dry networks. In f a c t , M / M ° i s equal t o the p r o p o r t i o n a l r e d u c t i o n i n modulus compared w i t h t h a t expected f o r the p e r f e c t , dry network. Thus, M °/M = 10 correpsonds t o a 1 0 - f o l d r e d u c t i o n . Any e f f e c t s due t o entanglements are i n a l l cases overshadowed by the reductions i n moduli due t o l o o p s .


c

c

c

c

ç c

c

c

c

c

c

c

The points at the lowest values of p . r

c

c

f o r the various

systems are those f o r bulk r e a c t i o n s and even f o r these s i g n i f i c a n t reductions i n moduli a r e apparent. In a d d i t i o n , such r e d u c t i o n s can be produced by r e l a t i v e l y s m a l l values o f p . Thus, system 1 shows a 5 - f o l d r e d u c t i o n i n modulus f o r an excess extent o f r e a c t i o n a t g e l a t i o n o f only 0.05, and system 5 a 3-fold reduction f o r p = 0.10. The r e l a t i v e p o s i t i o n s o f the l i n e s f o r the v a r i o u s systems can be r e l a t e d t o M o(or v ) , f , and the chain s t r u c t u r e s o f the reactants(1,2,9-12). The slopes o f the l i n e s show t h a t the r e duction i n modulus w i t h p r e - g e l i n t r a m o l e c u l a r r e a c t i o n i s l a r g e r f o r t r i f u n c t i o n a l compared w i t h t e t r a f u n c t i o n a l networks ( c f . systems 1 and 2 w i t h 4 and 5 ) , although higher values o f p« o b t a i n f o r t e t r a f u n c t i o n a l r e a c t i o n systems ( c f . Figure 2;! In a d d i t i o n , f o r a given f u n c t i o n a l i t y , the r e d u c t i o n i s l a r g e r f o r s m a l l e r values o f M ° ( c f . systems 1 with 2 and 4 w i t h 5 ) ; t h a t i s , f o r a given amount o f i n t r a m o l e c u l a r r e a c t i o n (or value of p ) systems with s m a l l e r loops have l a r g e r p r o p o r t i o n s o f those loops e l a s t i c a l l y i n e f f e c t i v e . The networks f o r system 3, based o f MDI, g i v e values o f M / M ° near u n i t y , corresponding t o r e l a t i v e l y high values o f t h e i r rubbery moduli. The reasons f o r t h i s phenomenon are not completely understood but a r e o b v i o u s l y r e l a t e d t o the s t i f f e r , aromatic chain s t r u c t u r e between j u n c t i o n p o i n t s i n these networks. r

c

r c

c

c

c

r > c

c

c



10


F i g u r e 4. Molar mass between e l a s t i c a l l y e f f e c t i v e j u n c t i o n p o i n t s (M ) r e l a t i v e t o t h a t f o r the p e r f e c t network(M °) versus extent o f i n t r a m o l e c u l a r r e a c t i o n a t gelation(p ) . Reaction systems as f o r F i g u r e 2. l i n e s through experimental p o i n t s f o r systems 1,2,4,5; and t h e o r e t i c a l curves f o r t r i - and t e t r a f u n c t i o n a l networks (see t e x t , l a s t s e c t i o n ) . System 1, HDI+LHT240, M °=0.635 kg mol"" , v=33; system 2, HDI+LHT112, M °=1.168 kg m o l " , v=61; system 3, MDI+LHT240, M °=0.705 kg m o l " , v=30; system 4, HDI+0PPE-NH1, M °=0.500 ! M ° . The presence o f s t a t e s 4 and 6 i n a network a t complete r e a c t i o n i s presented s c h e m a t i c a l l y i n Figure 7. I t can be seen t h a t over the complete r e a c t i o n system the number o f j u n c t i o n p o i n t s l o s t i s Ν .Ρβ> where N i s the number o f monomer u n i t s i n i t i a l l y and P i s the f r a c t i o n o f u n i t s i n s t a t e 6. Hence a t complete r e a c t i o n , the number o f e l a s t i c a l l y e f f e c t i v e f u n c t i o n p o i n t s i n N ( P i » - Ρε), where Ρι» + Ρβ = 1. Thus, r


15


STANFORD ET A L .

c

c

c

c

8

c

a

6

a

M /M ° c

=

c

1/(1-2P ) 6

(10)

The a n a l y t i c a l expression f o r Ρβ i s Pe =

2

2

3

2

Ρ(-Ρ λ+ (2λ - 4λ /3)ρ - (8λ /9Ηη(1 - 3ρ/(2λ + 3))) + 3λρ /2 2

2

3

+ 3λ ρ/2 + (3λ /2 + 3λ /4Ηη(1 - 2ρ/(λ + 2))

(11)

Here, λ i s a ring-forming parameter given by the equation ( c f . Equations 2 t o 4) λ

=

P

a b

/c

(12)

a 0

with P defined by Equation 4 and c the i n t i a l c o n c e n t r a t i o n o f A groups. U n l i k e λ'φ o f Equation 2, λ i s a uniquely defined parameter; i n the d e f i n i t i o n o f X , the denominator, c ^ , had to be chosen a r b i t r a r i l y . A corresponding equation t o Equation 11 has been d e r i v e d t o d e f i n e t h e extent o f r e a c t i o n a t q e l a t i o n ( 2 4 ) . I t enables p (see Equation 7) t o be evaluated as a f u n c t i o n o f λ and a l l o w ! p r e d i c t i o n o f the c o r r e l a t i o n between g e l p o i n t and r e d u c t i o n i n shear modulus ( v i z . M / M ° ) . The c o r r e l a t i o n i s shown as curve 1 i n Figure 8, which may be compared w i t h the experimental curves i n Figure 4 f o r RA2 + RBf p o l y m e r i s a t i o n s . a D

a 0

f

a D

e x

r

c

c


c


16

Table I I . Smallest Subset o f States o f Monomer U n i t s i n an RA3 Polymerisation. A- denotes a c o n t i n u i n g chain o f undefined l e n g t h .

State


1.

)-A A

A

1,2 1,5

2.

-AA. )-A A

2,3

2,6 3

-

A

\ >-A

A

>-A A

+ A-* A

A

A

- Α Α

A -AA

A

v

A >

3,4

>^AA-

A

*

A

Χ

A > ;

V A + A-

X

—AA

χ

A

V-AA—

W

v

>A A

+

-AA

-ΑΑ'

4.

Reaction

Reaction Route

-AA

>-A Al^Y

-AA )>-A + A- -> V A A -AA -AA v

-AAv )-AA-AA 7

5.

( V

6. f V

N-A

A'

A

5,6

S

A

(*> * *(*>

A A

^>-AA-


-


STANFORD ET AL.


Figure 7. Rate theory -occurrence of s t a t e s 4 and 6 i n network at complete r e a c t i o n from an RA3 p o l y m e r i s a t i o n . AA/A State 4: -AA-/ ; State 6: ( \-AAAAΚ V



18


M

C

/

M

C

Figure 8. P r e d i c t e d correlations(24)between r e d u c t i o n i n shear modulus a t complete reactionTM /M °) and extent o f i n t r a m o l e c u l a r r e a c t i o n a t g e l a t i o n ( p ) f o r an R A 3 polymerisation. Curve 1: From r a t e theory, accounting f o r p r e - g e l and postg e l i n t r a m o l e c u l a r r e a c t i o n (Equation 10). Curve 2: Accounting f o r p r e - g e l i n t r a m o l e c u l a r r e a c t i o n only (Equation 13). c

c

r > c


1.

STANFORD ET A L .

19


I f only p r e - g e l i n t r a m o l e c u l a r r e a c t i o n i s considered, then the number o f s m a l l e s t loops i n an RA3 p o l y m e r i s a t i o n i s 3Ν^ρ / 2 . The number o f j u n c t i o n p i n t s l o s t a t complete r e a c t i o n i s ' twice t h i s number (see Figure 7) and Γc

M

/ M

c c° =

^

^

Γ

,

Ο

'

·

(

3

)

This equation corresponds t o Equation 8 f o r an RA2 + RB3 polymer i s a t i o n . The r e s u l t i n g r e l a t i o n s h i p between M / M ° and p i s shown by curve 2 i n Figure 8, which may be compared w i t h t n e c a l c u l a t e d curves i n Figure 4. In a r e a l network o f f u n c t i o n a l i t y four o r l e s s , the s m a l l e s t loops apparently l e a d t o e l a s t i c a l l y i n e f f e c t i v e j u n c t i o n p o i n t s . In a d d i t i o n , l a r g e r loops can a l s o c o n t r i b u t e t o such d e f e c t s . The r e l a t i v e p o s i t i o n s o f the curves i n Figure 8 show t h a t , on the b a s i s o f the s m a l l e s t l o o p s , p o s t - g e l i n t r a m o l e c u l a r r e a c t i o n cannot be neglected, with approximately t h e same number o f loops o c c u r r i n g p o s t - g e l as p r e - g e l . The importance o f both p o s t - g e l and p r e - g e l i n t r a m o l e c u l a r r e a c t i o n i s a l s o apparent from F i g u r e 4 f o r RA2 + RB3 systems, where,apart from t h e data f o r the aro matic system 3, t h e c a l c u l a t e d curves g e n e r a l l y l i e w e l l below the experimental curves and have d i f f e r e n t shapes therefrom. To a l l o w d i r e c t comparison w i t h such experimental data, developments o f the r a t e theory t o evaluate M / M ° f o r RA2 + RB3 systems a r e p r e s e n t l y i n progress. c


1

c

c

r c

c

Literature Cited 1. Stanford, J.L.; Stepto, R.F.T.; Still, R.H., in "Reaction Injection Moulding and Fast Polymerisation Reactions"; Kresta, J.E., Ed.; Plenum Publishing Corp: New York, 1982; p.31. 2. Stanford, J.L.; Stepto, R.F.T., in "Elastomers and Rubber Elasticity"; Mark, J.E.; Lal, J., Eds.; ACS SYMPOSIUM SERIES No. 193, American Chemical Society: Washington D.C., 1982; Chap. 20. 3. Ahmed, Z.; Stepto, R.F.T. Polymer J. 1982, 14, 767. 4. Ahmad, Z.; Stepto, R.F.T. Colloid and Polymer Sci. 1980, 258, 663. 5. Stepto, R.F.T., in "Developments in Polymerisation - 3"; Haward, R.N., Ed.; Applied Science Publishers Ltd.: London, 1982; Chap. 3. 6. Stepto, R.F.T.; Waywell, D.R. Makromol. Chem. 1972, 152, 247, 263. 7. Stanford, J.L.; Stepto, R.F.T. Brit. Polymer J . 1977, 9, 124. 8. Ahmad, Z. Ph.D. Thesis, University of Manchester, England, 1978. 9. Stepto, R.F.T. Polymer 1979, 20, 1324. 10. Hunt, N.G.K.; Stepto, R.F.T.; Still, R.H. Proc. 26th IUPAC Int. Symp. on Macromolecules, Mainz, 1979, p.697.




20

11. Cawse, J.L. Ph.D. Thesis, University of Manchester, England, 1979. 12. Fasina, A.B.; Stepto, R.F.T. Makromol.Chem., 1981, 182, 2479. 13. Stanford, J.L.; Stepto, R.F.T. J. Chem. Soc. Faraday Trans.I 1975, 71, 1292. 14. Frisch, H.L. 128th Meeting Amer. Chem. Soc., Polymer Di v., Minneapolis, 1955. 15. Kilb, R.W. J. Physic. Chem. 1958, 62, 969. 16. Stepto, R.F.T. Faraday Disc. Chem. Soc. 1974, 57, 69. 17. Stockmayer, W.H. J. Polymer Sci. 1952, 9, 69; 1953, 11, 424 18. Dusek, K.; Prins, W. Adv. Polymer Sci. 1969, 6, 1. 19. Flory, P.J. Polymer 1979, 20, 1317. 20. Mark, J.E. Pure and Applied Chem. 1981, 53, 1495. 21. Stanford, J.L.; Stepto R.F.T.; Waywell, D.R. J. Chem. Soc. Faraday Trans.I. 1975, 71, 1308. 22. Askitopoulos, V. M.Sc. Thesis, University of Manchester, England, 1981. 23. Cawse, J.L.; Stanford, J.L.; Stepto, R.F.T. Proc. 26th IUPAC Int. Symp. on Macrmolecules, Mainz, 1979, p.393. 24. Lloyd, A.C. M.Sc. Dissertation, University of Manchester, England, 1981. 25. Gordon, M.; Temple, W.B. Makromol. Chem. 1972, 160, 263. RECEIVED

September 22, 1983


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