Rubber Elasticity Modulus of Interpenetrating Heteropolymer Networks

Chapter

4

Rubber Elasticity Modulus of Interpenetrating Heteropolymer Networks

Downloaded by UNIV OF ROCHESTER on June 15, 2013 | http://pubs.acs.org Publication Date: April 18, 1988 | doi: 10.1021/bk-1988-0367.ch004

Christos Tsenoglou Department of Chemistry and Chemical Engineering, Stevens Institute of Technology, Castle Point, Hoboken, NJ 07030 This is a theoretical study on the structure and modulus of a composite polymeric network formed by two intermeshing co-continuous networks of different chemistry, which interact on a molecular level. The rigidity of this elastomer is assumed to increase with the number density of chemical crosslinks and trapped entanglements in the system. The latter quantity is estimated from the relative concentration of the individual components and their ability to entangle in the unmixed state. The equilibrium elasticity modulus is then calculated for both the cases of a simultaneous and sequential interpenetrating polymer network. This i s a t h e o r e t i c a l study on the entanglement architecture and mechanical properties of an i d e a l two-component interpenetrating polymer network (IPN) composed of f l e x i b l e chains (Fig. l a ) . I n t h i s system molecular i n t e r a c t i o n between d i f f e r e n t polymer species i s accomplished by the simultaneous or sequential polymerization of the polymeric precursors [l]. Chains which are thermodynamically incompatible are permanently interlocked i n a composite network due to the presence of chemical c r o s s l i n k s . The network structure i s thus reinforced by chain entanglements trapped between permanent junctions [2,3]. I t i s evident that, entanglements between i d e n t i c a l chains l i e further apart i n an IPN than i n a one-component network (Fig. lb) and entanglements associating heterogeneous polymers are formed i n between homopolymer junctions. In the present study the density of the various interchain associations i n the composite network i s evaluated as a function of the properties of the pure network components. This information i s used to estimate the equilibrium rubber e l a s t i c i t y modulus of the IPN. Structural Characteristics of the Network Let us assume that each of the two networks p a r t i c i p a t e s with a v-^ volume f r a c t i o n i n the IPN ( i = 1,2) and i s composed of élastomeric 0097-6156/88/0367-0059$06.00/0 © 1988 American Chemical Society In Cross-Linked Polymers; Dickie, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1988.


60

CROSS-LINKED POLYMERS

a

b

F i g . 1. (a) An i d e a l i z e d representation of an interpenetrating polymer network. Lines of unequal thickness s i g n i f y d i f f e r e n t polymeric species, c r o s s l i n k s are represented "by c i r c l e s and entanglements by l i n e interceptions. The molecular weight of the AE step i s equal to M i , of BD i s M i2> EF i s M l l , GK i s M 2, and DH i s M 21 · Here Nj_ = 3 and N£ = 1. (b) A s i n g l e component elastomeric network. The molecular weight of the PQ step i s equal to M Q I Q and N ^ Q 2. e

c

C

e

e

=

In Cross-Linked Polymers; Dickie, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1988.

4.

61

Rubber Elasticity Modulus

TSENOGLOU

strands of molecular weight M . ent (νχ = 1 = 1-V2), M Q i s the entanglements along a (1) strand system. The number of p r i m i t i v e two consecutive chain junctions)

In the absence of any other compon molecular weight between trapped and Ρχο density of the pure steps ( i . e . , subsegments defined by per strand i s then equal t o :

ci

G 1

1 + N

t n e

= 1 + M /M

10

cl

(1)

elQ

where N ^ Q i s number of steps caused by trapped entanglements. Similar quantities can be defined f o r the pure network (2), by r e placing the index (1) with (2).


t n e

The equivalence between steps, v a l i d i n a s i n g l e component net work, does not e x i s t i n an interpenetrating polymer network due to the d i v e r s i t y of i n t e r a c t i n g species. I f i j s i g n i f i e s a step l y i n g on an i chain and confined by two entanglements with j polymers, then % j i s the average number of such subsegments per strand and MQ±J i t s average molecular weight ( F i g . 1). The t o t a l number of steps per i strand i s then given by-: 1+N.

= 1 + Ν. + N. i l ι2

ι

Ί

(2)

9

where : N

M

/

ij " ci

M

( 3 ) e

i

j

and s i g n i f i e s the number of steps caused by trapped entangle ments. In an i d e a l l y structured IPN, without any phase separation, i t i s reasonable to assume that the frequency of entanglements be tween chains of the same species decreases l i n e a r l y with the f r a c t i o n a l p a r t i c i p a t i o n of t h i s polymer i n the system: M

M

elO

/ M

= V 10 N

ell

/M

N

=

/N

V

( 4 a

l

V

>

( 4 b )

e20 e22 = 22 20 = 2

I t i s also assumed that the f r a c t i o n of trapped entanglements along a chain of the one species caused by associations with chains of the other species i s equal to the f r a c t i o n a l p a r t i c i p a t i o n of- the steps of the second species i n the t o t a l step population: N

N

where

/ N

12 l /N

21 2

=

v

N

=

V

/ ( v

2

2

N

l l

/

(

N

l l v

N

l l

+

V

2*V

( 5 a )

+

V

2 2

N

( 5 b )

}

s i g n i f i e s the number density of the v.

ι

i

chains i n the IPN:

= p. v. N /M . i0 ι A c i n

A

(6)

and N^ i s the Avogadro number. Physical symmetry requires that the number density of the 12 steps i s equal to that of the 21 steps: Vl2

V

N

" 2 21


(7)


62

The fact that Equation 7 can e a s i l y be proven from Equation 5 shows the self-consistency of t h i s assumption. Combination of Equations 1-7 r e s u l t s i n the following expres sions f o r N. and M ..: ι eij N

N

+

N

M

M

l = l l 12 = c l ( V e l O

N

= N

2

+ N

2 1

= M (l/M

22

c 2

+

1/M

( 8 a )

el2> + v /M

e 2 1

2

e2Q

)

(8b)

where


M

v

el2 2

( p

20

/ p

10

) 1 / 2

M

p

= e21 V 1 0

/ p

) 1 / 2 = ( M

20

M

elO e20

) 1 / 2

(9

>

These r e l a t i o n s h i p s are adequate to q u a n t i t a t i v e l y define the architecture of the IPN. Equilibrium E l a s t i c i t y Modulus Simultaneous IPN. According to the s t a t i s t i c a l theory of rubber e l a s t i c i t y , the e l a s t i c i t y modulus ( E ) , a measure of the material r i g i d i t y , i s proportional to the concentration of e l a s t i c a l l y active segments ( v ) i n the network [3,4]. For n e g l i g i b l e perturbation of the strand length at rest due to c r o s s l i n k i n g (a reasonable assump t i o n f o r the case of a simultaneous IPN), the modulus i s given by: e

e

E

=

e

3 k

T v

( 1 0 )

B e

where k i s the Boltzmann constant and Τ the absolute temperature. For an elastomer with no defects, and with the assumption that steps caused by trapped entanglements are as e f f e c t i v e i n storing e l a s t i c energy as steps due to c r o s s l i n k s , ν i s evaluated by the follow ing r e l a t i o n s h i p : fi

e

V

e

=

V

l

( 1

+

N

ll

+

N

12

}

+

V

2

( 1

+

N

+

12

N

22

}

The presence of dangling segments i n the IPN decreases the s t r u c t u r a l s o l i d i t y of the system since steps associated with the tethered chains have a f i n i t e l i f e t i m e subject to the dynamics of the free end [5,6]. I f φ.^ i s the f r a c t i o n of strands i n the ( i ) net work that are crosslinked on both ends, then: v

e

- φ ν χ

χ

(1 + φ Ν 1

11

+ Φ Ν ) + φ ν 2

12

2

2

(1 + φ Ν 1

21

+ φ Ν ) 2

22

(12)

Combination of t h i s l a s t r e l a t i o n s h i p with Equations 6 and 8 r e s u l t s in:


4.


TSENOGLOU

63

where R i s the i d e a l gas constant. In the absence of the second intermeshing networks (v2 = 0) the theory reduces to an expression s i m i l a r to the r e s u l t by Mancke et a l . [5] f o r a single component elastomer. On the other hand, i n the case of one of the system components (e.g., 2) being an oligomer or a low-molecular weight solvent, the i n a b i l i t y of the small molecules to form to entangle ments i s accounted for by s e t t i n g M g . An a l t e r n a t i v e way to express the r i g i d i t y of the composite network i s i n terms of the r i g i d i t i e s of i t s i n d i v i d u a l components. S t a r t i n g from Equation 13 i t can e a s i l y be shown that: 00

e2

E

E

+ E

+2

(14)

e - e l e2 hWl^V^2

where E ^ i s the equilibrium modulus of the network ( i ) alone, stripped of i t s intermeshing heteropolymer neighbor; i . e . , a network with a (l-φ^) f r a c t i o n of defects, a polymer chain concentration equal to (P-^gV^) * o l e c u l a r weight between trapped entanglements equal to Μ . /ν. [5]: eiO ι —


e

anc

a

m

Λ

Ε . = RT φ.ρ. ν.(1/Μ . + φ.ν./Μ , ) ei ι ι0 ι ci ι ι eiO τ

and

E^

γ

Λ

Λ

(15)

i s the rubbery plateau modulus of a pure, undiluted f l u i d E . = 3RT p / M . N

i0

e

(16)

0

I t i s remarkable that the predicted modulus from Equation 14 i s higher i n value than the one predicted by a simple l i n e a r a d d i t i v i t y law due to the r e i n f o r c i n g e f f e c t of the heteropolymer couplings. Sequential IPN. The preceding analysis does not apply to the case of a sequential IPN. The formation of t h i s system originates with the synthesis of the network (1). Then, network (1) i s swollen with monomer (2) which i s subsequently polymerized i n s i t u to form a second network. Due to perturbed chain dimensions, the modulus of the f i r s t network i s higher than the corresponding modulus i n the unswollen state by a factor equal to v^~^' [ 6 j :

E

e -

3 k

B

T

( 1 7 )

where v ^ and v , the number densities of e l a s t i c a l l y active steps i n the two networks, are given by the f i r s t and second term on the r i g h t of Equation 12. Following the same sequence of thoughts which gave Equation 13 an expression f o r the equilibrium modulus of a sequential IPN i s obtained: e

E

e 2

1 e

/3/Ρ \ ιη

,Ροην

ο

/./q/P

3RT


64


Because ν, i s a f r a c t i o n a l quantity, Equation 18 always predicts modulus values larger than the corresponding expression f o r a simul taneous IPN(Equation 13). For the s p e c i a l case of a network with no defects or trapped entanglements (φ^ •> 1, + «>) , an e a r l i e r r e s u l t derived by Sperling i s recovered (Ref. [l] , Equation 4.8).


Discussion S t a r t i n g from simple s t a t i s t i c a l considerations concerning the coupling between s i m i l a r or d i s s i m i l a r chains, a model was con structed f o r the architecture of a class of IPN where i n t e r a c t i o n among phases occurs on a molecular l e v e l . I t was shown(Equation 13) that the contributions to the modulus due to crosslinks are subject to l i n e a r a d d i t i v i t y , while a square root a d d i t i v i t y rule holds f o r the contributions due to entanglements. Comparisons of the theory with experiment can not be presently made due to the lack of data on w e l l characterized molecular IPN. Indications about i t s v a l i d i t y can, however, be deduced by examining i t s consistency at extreme cases of material behavior. The agree ment at the one-component l i m i t , for example, provided that the rubber i s not very weak (φ not very small), has been successfully demonstrated by Ferry and coworkers [ 5 j . A useful r e s u l t i s ob tained at the version of the theory applicable to the f l u i d state ( i . e . , at the l i m i t of zero c r o s s l i n k i n g ) . From the l a s t two terms of Equation 13, the following r e l a t i o n s h i p can be derived for the plateau [7] and time dependent r e l a x a t i o n modulus of miscible polymer blends :

E

+

Ld •1Î V E

V

B

E

2 2

( 1 9 )

This simple mixing rule demonstrates s a t i s f a c t o r y agreement with experimental evidence from experiments with binary f l u i d blends [7]· Furthermore, i t i s s i m i l a r i n form with the r e s u l t from a continuum theory approach by Davis [8], applicable for IPN with dual phase continuity but which are not mixed on a molecular l e v e l . This l a s t model involves an exponent equal to 1/5 instead of 1/2 and i s quite successful i n predicting the experimental evidence [ l ] from permanent networks. The molecular theory presented i n t h i s paper i s not antagonistic but complementary to the macroscopic continuum models of multicomponent polymeric materials [9]. This i s due to the fact that material homogeneity i s rarely the case i n IPN, and that some phase separation does occur upon polymerization. As a r e s u l t , a two-phase material i s produced. Due to p a r t i a l m i s c i b i l i t y each of the two phases contains both polymeric species but at d i f f e r e n t compositions. For the modeling of t h i s heterogeneous system i t i s suggested here to f i r s t use the molecular approach f o r the description of each of the homogeneous phases and then combine these two r e s u l t s with a s u i t able phenomenological model (depending on the p r e v a i l i n g morphology) for the description of the multiphase IPN as a whole.



4. TSENOGLOU


65

Literature Cited 1. Sperling, L.H., Interpenetrating Polymer Networks and Related Materials, Plenum Press, New York, 1981, pp. 38,51,160. 2. Ferry, J.D., Viscoelastic Properties of Polymers, 3rd ed., Wiley, New York, 1980, p. 409. 3. Graessley, W.W., Adv. Polymer Sci., 16, 1 (1974). 4. Treloar, L.R.G., The Physics of Rubber Elasticity, Oxford, 1958, p. 75. 5. Mancke, R.G., Dickie, R.A., and Ferry, J.D., J. Polym. Sci., Part A-2, 6, 1783 (1968). 6. Flory, P.J., Principles of Polymer Chemistry, Cornell U. Press, 1953, pp. 492, 462. 7. Tsenoglou, C., Polymer Preprints, 28, (2), 198, (1987), and J. Polymer. Sci. Phys. Ed., submitted. 8. Davis, W.E.A., J. Phys. P., Appl. Phys., 4, 1176 (1972). 9. Dickie, R.A., in Polymer Blends, Vol. 1, (D.R. Paul and S. Newman Eds.), Academic Press (1978), p. 353. RECEIVED

October 7, 1987


Rubber Elasticity Modulus of Interpenetrating Heteropolymer Networks

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