Structure-Property Relationships in Polymers - American Chemical

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Structure-Property Relationships in Polymers 1

TURNER ALFREY, JR.

Downloaded by UNIV OF CALIFORNIA SAN DIEGO on December 30, 2015 | http://pubs.acs.org Publication Date: September 25, 1985 | doi: 10.1021/bk-1985-0285.ch011

The Dow Chemical Company, Midland, MI 48640

T and Tg: Dependence on Molecular Structure Linear Viscoelasticity Large-Strain Rubber Elasticity Non-Newtonian Fluids Behavior of Glassy Amorphous Polymers Behavior of Crystalline Polymers Summary M

The common central structural feature of organic macromolecules is the chain of covalently bonded atoms. Bond lengths and bond angles are rather rigidly fixed, but restricted rotation about single bonds permits a polymer chain to assume a wide range of three-dimensional conformations. At elevated temperatures, bond rotation is frequent; the polymer chain wriggles rapidly from one conformation to another. This micro-Brownian motion confers flexibility upon a macroscopic specimen. At low temperatures, the chains are immobilized, and the specimen is hardened by either of two mechanisms: crystallization (packing into a crystal lattice), or vitrification (forming a glassy amorphous solid). The crystalline melting point, T , and the glass transition temperature, T , are important characteristics of a given polymer. The molecular structure of a particular polymer has two aspects, chemical composition and molecular architecture. The term "chemical composition" refers to the local molecular structure--nature of the units that make up the chains (including the stereochemical structure of these units). "Molecular architecture" refers to molecular structure in-the-large (average molecular weight, molecular-weight distribution, branching, cross-linking, etc.). For network polymers it refers to average molecular weight between cross-links, number and lengths of dangling tails, and many elusive aspects of network topology. M

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Deceased 0097 6156/85/0285 0241S06.00/0 © 1985 American Chemical Society

In Applied Polymer Science; Tess, Roy W., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1985.


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In approaching the f o r m i d a b l e problem of s t r u c t u r e - p r o p e r t y r e l a t i o n s h i p , the f i r s t and s i m p l e s t step i s to examine the r e l a t i o n s h i p s between molecular structure and the values of and Tg. Next i s the more d i f f i c u l t task of c h a r a c t e r i z i n g the q u a n t i t a t i v e mechanical behaviors (and r e l a t i n g them to structure) of polymers i n each of v a r i o u s regimes: high-temperature v i s c o e l a s t i c f l u i d s ; g l a s s y amorphous s o l i d s ; s e m i c r y s t a l l i n e s o l i d s c o n t a i n i n g f l e x i b l e amorphous r e g i o n s (between Tg and T^); s e m i c r y s t a l l i n e s o l i d s c o n t a i n i n g g l a s s y amorphous r e g i o n s ; h i g h l y c r y s t a l l i n e s o l i d s ; m e t a s t a b l e , s u p e r c o o l e d amorphous polymers; rubbery e l a s t i c networks; etc. I n some of these regimes, "structure" refers simply to molecular structure; i n others (notably g l a s s y and c r y s t a l l i n e s t a t e s ) , p r o p e r t i e s depend not o n l y on molecular structure but a l s o on supramolecular structure—molecular o r i e n t a t i o n , c r y s t a l l i n e morphology, etc. Tjwf and T : Dependence on Molecular Structure g

The c r y s t a l l i n e melting point and the g l a s s t r a n s i t i o n temperature of a polymer, i n themselves, provide a rough characterization of the polymer p r o p e r t i e s ; they a l s o p r o v i d e reference p o i n t s f o r the v a r i o u s regimes w i t h i n which the q u a n t i t a t i v e e v a l u a t i o n of p r o p e r t i e s must be made. How do T^ and Tg depend on m o l e c u l a r structure? In addressing t h i s question, we w i l l consider a s i n g l e molecular architecture—high molecular weight, l i n e a r chains. We s h a l l a r b i t r a r i l y c l a s s i f y such chains i n t o f i v e broad s t r u c t u r a l c l a s s e s : Class Class Class Class Class

I II III IV V

— P e r f e c t l y repeating "matched pearl necklace — Random copolymers — d-t and cis-trans "copolymers" — Block copolymers — Short-unit chains that assume helicalconformations

Class I chains, because of t h e i r s t r u c t u r a l r e g u l a r i t y , u s u a l l y pack e f f i c i e n t l y into a c r y s t a l l i n e l a t t i c e . They can c r y s t a l l i z e and u s u a l l y e x h i b i t a w e l l - d e f i n e d T^ and T . (The prototype—linear p o l y e t h y l e n e — c r y s t a l l i z e s so r a p i d l y that i t cannot be trapped i n the glassy amorphous state; consequently, i t s Tg has been a matter of d i s a g r e e m e n t . ) The c r y s t a l l i n e m e l t i n g p o i n t of l i n e a r p o l y e t h y l e n e i s a p p r o x i m a t e l y 140 °C. I f methylene groups of p o l y e t h y l e n e are r e p l a c e d , s p a r s e l y and r e g u l a r l y , by other m o i e t i e s , higher or lower v a l u e s of T^ are observed. Two f a c t o r s govern T^: chain f l e x i b i l i t y and interchain forces. F l e x i b l e units (such as ether, ester, or s u l f i d e ) r e s u l t i n lowered melting points. Rigid units (such as p-phenylene) r e s u l t i n higher melting points. Strong intermolecular forces (such as hydrogen bonds) y i e l d h i g h melting points (1). Within C l a s s I , the same s t r u c t u r a l features that promote high c r y s t a l l i n e m e l t i n g p o i n t s a l s o y i e l d h i g h v a l u e s of To. Consequently, there e x i s t s a rough c o r r e l a t i o n between T{^ and Ig for t h i s c l a s s of polymers (2). By c o n t r o l l i n g c h a i n s t i f f n e s s and i n t e r molecular forces, polymers with high T^ and Tg or low T^ and Tg can g


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r e a d i l y be designed; but i t i s not possible to independently c o n t r o l these two c h a r a c t e r i s t i c temperatures. C l a s s I I polymers—random c o p o l y m e r s — f i t l e s s n e a t l y i n t o crystal lattices. M e l t i n g points are depressed, and the degree of c r y s t a l l i z a t i o n i s reduced. (A few s p e c i a l e x c e p t i o n s e x i s t , i n which the two monomer u n i t s are s u f f i c i e n t l y matched i n geometry that they can i n t e r c h a n g e a b l y occupy s i t e s i n a common l a t t i c e . ) Because v i t r i f i c a t i o n does not i n v o l v e f i t t i n g i n t o a c r y s t a l l a t t i c e , the g l a s s temperatures of copolymers are not depressed by the c h a i n i r r e g u l a r i t y . C o n s e q u e n t l y , random copolymers do not f o l l o w the T^-Tg c o r r e l a t i o n c h a r a c t e r i s t i c of C l a s s I polymers (3). C l a s s I I I polymers are p r i m a r i l y made up of v i n y l and diene a d d i t i o n polymers. When a v i n y l monomer, C H 2 C H X , i s subjected to addition polymerization a stereochemical problem i s encountered at every second carbon atom of the chain. The substituent X can extend above or below the plane of the extended zigzag chain, corresponding to a d - or i - c o n f i g u r a t i o n of the chain carbon atom i n question (4). When t h e a d d i t i o n po1ymer i z a t i o n i s c a r r i e d out w i t h a s t e r e o s p e c i f i c c a t a l y s t , the polymer may be a r e g u l a r s t r u c t u r e : " i s o t a c t i c " ( r e p e a t i n g dddddd) o r " s y n d i o t a c t i c " ( p e r f e c t l y a l t e r n a t i n g dldldl) (5). On the other hand, f r e e - r a d i c a l addition p o l y m e r i z a t i o n tends to produce a r a t h e r random ( " a t a c t i c " ) copolymer of the d- and 1 - c o n f i g u r a t i o n : ddldl ldlddldl Id, e t c . Consequently, v i n y l polymers produced by f r e e - r a d i c a l polymerization tend to be permanently amorphous or at most to e x h i b i t only a sraal 1 amount of c r y s t a l l i n i t y . In the case of d i e n e s , a s i n g l e pure monomer can e n t e r the polymer chain i n s e v e r a l different manners. The simplest example i s butadiene, CH2=CH-CH=CH2, which upon polymerization can convert to a 1,2-chain u n i t ( w i t h a pendent v i n y l group), or to a c i s - 1 , 4 - o r a trans-1,4-chain unit:

H -CH ~C J 2

HC=CHo 1,2-unit

-HC 2

\

/

CH2

-HC \

c=c HH c i s - 1 , 4 unit

H

2

/

oc X H

\ Ci 2

trans-1,4 u n i t

P o l y b u t a d i e n e formed by h i g h - t e m p e r a t u r e , f r e e - r a d i c a l a d d i t i o n p o l y m e r i z a t i o n i s a copolymer of these t h r e e k i n d s of s t r u c t u r a l u n i t s . With isoprene (2-raethy1 butadiene), the number of ways the unit can enter the polymer chain i s s t i l l l a r g e r ; for example, the 1,2-unit with a pendent v i n y l group i s s t r u c t u r a l l y different from the 3,4-unit with a pendent isopropenyl group:


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CH3

H -CH -C-

-CH -C2

2

CH=CH

2

C=CH

2

CH


1,2-unit

3

3,4-unit

Synthetic polyisoprene, prepared by f r e e - r a d i c a l polymerization of isoprene monomer, i s a copolymer of s i x s t r u c t u r a l l y d i s t i n c t kinds of isoprene c h a i n u n i t s . U n l i k e n a t u r a l rubber, which i s a r e g u l a r l y r e p e a t i n g C l a s s I s t r u c t u r e ( c i s - 1 , 4 ) , such s y n t h e t i c polyisoprene does not c r y s t a l l i z e . On the other hand, by the use of the a p p r o p r i a t e s t e r e o s p e c i f i c c a t a l y s t , isoprene monomer can be c o n v e r t e d to a r e g u l a r C l a s s I polymer w i t h the same s t r u c t u r e as natural rubber (6). B l o c k copolymers ( C l a s s I V ) are made up of two (or more) d i f f e r e n t monomer u n i t s , arranged i n l o n g b l o c k s of each type of u n i t . For example, a c h a i n c o n s i s t i n g of a b l o c k of 500 A u n i t s f o l l o w i n g by a block of 500 B units and another block of 500 A units i s an ABA t r i b l o c k copolymer. I f an A block corresponds to a C l a s s I chain structure, i t can c r y s t a l l i z e i n the normal poly(A) c r y s t a l l a t t i c e and can e x h i b i t a Tjwf t h a t i s o n l y s l i g h t l y depressed compared to the poly(A) homopolymer. Even i f the i n d i v i d u a l blocks are n o n c r y s t a l l i z i n g a t a c t i c addition polymers, they are o r d i n a r i l y m u t u a l l y i m m i s c i b l e ( i f l o n g ) and undergo m i c r o s e g r e g a t i o n i n t o separate microphases, or "domains." These domains may develop i n t o r e g u l a r g e o m e t r i c a l a r r a y s , the form of which depends upon the r e l a t i v e volume fractions of the i n d i v i d u a l blocks. I f the volume f r a c t i o n s are a p p r o x i m a t e l y e q u a l , a l a m i n a r domain morphology emerges, with laminar thickness depending upon block lengths. If the B blocks constitute the major part of the copolymer, the B phase tends to be c o n t i n u o u s , w i t h c y l i n d r i c a l or s p h e r i c a l A domains dispersed within i t i n a regular fashion. The properties of such a block copolymer depend upon the composition and length of each block and the domain morphology assumed by the c h a i n s . Because of the m i c r o s e g r e g a t i o n , the i n d i v i d u a l components e x h i b i t t h e i r own c h a r a c t e r i s t i c T and T^f v a l u e s ( s l i g h t l y m o d i f i e d ) . Thus, a segregated block copolymer w i l l normally e x h i b i t two d i s t i n c t g l a s s t r a n s i t i o n s , i n contrast to the s i n g l e intermediate g l a s s t r a n s i t i o n commonly seen i n random copolymers (7). Whereas p o l y e t h y l e n e , p o l y a m i d e s , and p o l y e s t e r s assume an extended p l a n a r z i g z a g conformation i n the c r y s t a l l a t t i c e , many short-unit polymers twist i n t o some h e l i c a l conformation (Class V). In i s o t a c t i c p o l y o l e f i n s , the extended planar conformation i s s t e r i c a l l y f o r b i d d e n ; by t w i s t i n g i n t o a r e g u l a r h e l i x , the c h a i n r e l i e v e s the s t e r i c s t r a i n . I f the a n g u l a r t w i s t of each u n i t ( r e l a t i v e to i t s predecessor) i s a r a t i o n a l f r a c t i o n of one r e v o l u t i o n , then the s p a t i a l o r i e n t a t i o n of the methyl groups w i l l e x h i b i t a d e f i n i t e repeat distance. I f the i n d i v i d u a l twist angle i s 27r/n, s u c c e s s i v e methyl groups w i l l be o r i e n t e d at the a n g l e s 2ir/n, 47r/n, 67r/n, e t c . , and the o r i e n t a t i o n w i l l repeat w i t h a p e r i o d i c i t y of n groups. I f the i n d i v i d u a l twist angle i s 4"rr/n with g



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n odd, the chain w i l l go through two h e l i c a l turns before repeating. More g e n e r a l l y , i f the t w i s t a n g l e per group (measured i n r e v o l u t i o n s ) i s g i v e n by the i r r e d u c i b l e f r a c t i o n m/n, then the s u b s t i t u e n t group o r i e n t a t i o n would repeat a f t e r n u n i t s , and m complete t u r n s appearing i n the repeat sequence. The h e l i c a l conformation i s e f f e c t i v e l y a rod and packs p a r a l l e l to neighboring rods i n the c r y s t a l l a t t i c e (8). Although we have emphasized the effects of chemical composition upon the c r y s t a l l i z a t i o n and v i t r i f i c a t i o n p r o c e s s e s , m o l e c u l a r architecture a l s o influences these processes. Branching and crossl i n k i n g introduce points of i r r e g u l a r i t y that cannot e a s i l y f i t i n t o a c r y s t a l l a t t i c e . This can reduce the degree of c r y s t a l l i n i t y , the v a l u e of T^, and the r a t e of c r y s t a l l i z a t i o n . Thus, branched p o l y e t h y l e n e i s c o n s i d e r a b l y l e s s c r y s t a l l i n e than l i n e a r polyethylene, and consequently softer and l e s s dense. Vulcanized natural rubber c r y s t a l l i z e s much more s l o w l y than unvulcanized, and a h i g h d e g r e e of v u l c a n i z a t i o n can c o m p l e t e l y prevent c r y s t a l l i z a t i o n (9). I n t r o d u c t i o n of c r o s s - l i n k s i n t o a g l a s s y amorphous polymer increases the value of Tg (10). Linear V i s c o e l a s t i c i t y When we progress from the f o r e g o i n g q u a l i t a t i v e d i s c u s s i o n of structure-property r e l a t i o n s h i p s to the q u a n t i t a t i v e s p e c i f i c a t i o n of mechanical p r o p e r t i e s , we e n t e r a j u n g l e t h a t has been o n l y p a r t i a l l y e x p l o r e d . The most c o n v e n i e n t p o i n t of departure i n t o t h i s large and complex subject i s provided by the topic of " l i n e a r v i s c o e l a s t i c i t y . " Linear v i s c o e l a s t i c i t y r e p r e s e n t s a r e l a t i v e l y simple extension of c l a s s i c a l ( s m a l l - s t r a i n ) theory of e l a s t i c i t y . In s i t u a t i o n s where l i n e a r v i s c o e l a s t i c i t y a p p l i e s , the mechanical p r o p e r t i e s can be determined from a few experiments and can be specified i n any of s e v e r a l equivalent formulations (11). The accurate a p p l i c a b i l i t y of l i n e a r v i s c o e l a s t i c i t y i s l i m i t e d to c e r t a i n r e s t r i c t e d s i t u a t i o n s : amorphous polymers, temperatures near or above the g l a s s temperature, homogeneous, isotropic m a t e r i a l s , s m a l l s t r a i n s , and absence of mechanical f a i l u r e phenomena. Thus, the theory of l i n e a r v i s c o e l a s t i c i t y i s of l i m i t e d d i r e c t a p p l i c a b i l i t y to the problems encounted i n the f a b r i c a t i o n and end use of p o l y m e r i c m a t e r i a l s ( s i n c e most of these problems i n v o l v e e i t h e r l a r g e s t r a i n s , c r y s t a l l i n e polymers, amorphous polymers i n a g l a s s state f a i l u r e phenomena, or some combination of these d i s q u a l i f y i n g features). Even so, l i n e a r v i s c o e l a s t i c i t y i s a most important s u b j e c t i n polymer m a t e r i a l s s c i e n c e — d i r e c t l y a p p l i c a b l e i n a m i n o r i t y of p r a c t i c a l problems, but i n d i r e c t l y useful (as a point of reference) i n a much wider range of problems. In an u n - c r o s s - l i n k e d amorphous polymer, above i t s g l a s s temperature, the molecular chains are continuously w r i g g l i n g from one conformation to another. I f a mechanical stress i s imposed on such a system of w r i g g l i n g chains, i t can respond i n three d i s t i n c t ways: i n s t a n t a n e o u s e l a s t i c response; r e t a r d e d ( c o n f o r m a t i o n a l ) e l a s t i c response: or v i s c o u s f l o w . A c t u a l l y , i n order to f i t experimental data adequately, the retarded e l a s t i c element must be expanded i n t o a whole series of such elements, some with shorter and some w i t h l o n g e r response times. The l o c a l " k i n k i n e s s " of the c h a i n s can be s t r a i g h t e n e d out (by s t r e s s ) more r a p i d l y than the



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l a r g e r s c a l e c o n v o l u t i o n s . The time s c a l e s of these v a r i o u s retarded e l a s t i c contributions range over many orders of magnitude from the fastest to the slowest. In s p i t e of these complications, the v i s c o e l a s t i c response of an amorphous polymer to s m a l l s t r e s s e s t u r n s out to be a r e l a t i v e l y simple subject because of two h e l p f u l features: (1) the behavior i s l i n e a r i n the stress, which permits the a p p l i c a t i o n of the powerful superposition p r i n c i p l e ; and (2) the behavior often f o l l o w s a timetemperature e q u i v a l e n c e p r i n c i p l e , w h i c h p e r m i t s t h e r a p i d v i s c o e l a s t i c response at high temperatures and the slow response at low temperatures to be condensed i n a s i n g l e master curve. The superposition p r i n c i p l e makes i t possible to c a l c u l a t e the mechanical response of an amorphous polymer to a wide range of loading sequences from a l i m i t e d amount of experimental information. Thus, from a s i n g l e complete creep c u r v e i n pure shear or pure tension at a s i n g l e load, i t i s p o s s i b l e , i n p r i n c i p l e , to c a l c u l a t e the response to combined stresses and time-dependent stresses (e.g., s i n u s o i d a l ) . Going s t i l l further, problems i n v o l v i n g nonhomogeneous time-dependent s t r e s s e s i n v i s c o e l a s t i c o b j e c t s can be s o l v e d by means of the s u p e r p o s i t i o n p r i n c i p l e . The two common types of b o u n d a r y - v a l u e problems i n e l a s t i c i t y theory ( s u r f a c e f o r c e s or surface displacements specified) generalize simply to the analogous v i s c o e l a s t i c problems (surface forces or displacements specified as functions of both p o s i t i o n and time.) (12). The time-temperature equivalence p r i n c i p l e makes i t p o s s i b l e to predict the v i s c o e l a s t i c properties of an amorphous polymer at one temperature from measurements made at other temperatures. The major e f f e c t of a temperature i n c r e a s e i s to i n c r e a s e the r a t e s of the various modes of retarded conformational e l a s t i c response, that i s , to reduce the r e t a r d i n g v i s c o s i t y v a l u e s i n the s p r i n g - d a s h p o t model. T h i s appears as a s h i f t of the creep f u n c t i o n a l o n g the log t s c a l e to s h o r t e r times. A secondary e f f e c t of i n c r e a s i n g temperature i s to increase the e l a s t i c moduli s l i g h t l y because an e q u i l i b r i u m conformational modulus tends to be proportional to the absolute temperature (13). By use of the time-temperature e q u i v a l e n c e p r i n c i p l e , the v i s c o e l a s t i c response of a g i v e n p o l y m e r i c m a t e r i a l over a wide temperature range can be accommodated i n a s i n g l e master curve. By use the s u p e r p o s i t i o n p r i n c i p l e , t h i s master c u r v e can be used to estimate the time-dependent response to time-dependent stresses i n s i m p l e t e n s i l e or shear specimens or to nonhomogeneous t i m e dependent stresses a r i s i n g i n stressed objects and structures. The r e l a t i o n s h i p between molecular s t r u c t u r e and v i s c o e l a s t i c p r o p e r t i e s i n v o l v e s both c h e m i c a l c o m p o s i t i o n and m o l e c u l a r architecture. The short-time (low-temperature) b e h a v i o r i s r e l a t i v e l y i n s e n s i t i v e to m o l e c u l a r a r c h i t e c t u r e , but master creep curves for different architectures diverge strongly at long times (high temperatures). The curve for a network polymer approaches a l i m i t i n g asymptote and corresponds to e q u i l i b r i u m rubber e l a s t i c i t y ; that of a l i n e a r polymer increases to i n f i n i t y i n a l i m i t i n g steadystate viscous flow. The e q u i l i b r i u m rubber modulus i s r e l a t e d to the d e n s i t y of c r o s s - l i n k s . To a f i r s t a p p r o x i m a t i o n , G_ = K. T^V, where v designates c r o s s - l i n k density. In the v i c i n i t y of the g e l point, F l o r y showed that i t was necessary to correct for the wasted dangling t a i l s that are attached to the network because they cannot #



11. ALFREY


247

carry load at e q u i l i b r i u m (14). Likewise, the melt v i s c o s i t y of a l i n e a r polymer i s strongly dependent upon chain length. A l o g - l o g p l o t of melt v i s c o s i t y versus molecular weight commonly e x h i b i t s two s t r a i g h t - l i n e sections, with a slope of unity or somewhat higher i n the low m o l e c u l a r weight s e c t i o n and a s l o p e of about 3.4 i n the h i g h m o l e c u l a r weight s e c t i o n (15). The change i n s l o p e has been attributed to the onset of molecular entanglement (16). At a given temperature, two polymers of s i m i l a r architecture but different compositions e x h i b i t creep c u r v e s of s i m i l a r shape, but d i f f e r e n t l o c a t i o n s a l o n g the l o g t a x i s . When compared at "corresponding" temperatures, r e l a t i v e to t h e i r r e s p e c t i v e g l a s s temperatures, t h e i r behaviors are very s i m i l a r . In a crude sense, the v i s c o e l a s t i c properties of a given polymer can be correlated with two numbers—one that r e f l e c t s i t s chemical composition and one that characterizes i t s molecular architecture. The value of T« conveniently serves the f i r s t r o l e . The molecular a r c h i t e c t u r e or a l i n e a r polymer can be r o u g h l y s p e c i f i e d by the average c h a i n l e n g t h ; t h a t of a network polymer by the network d e n s i t y , or by the average m o l e c u l a r weight between c r o s s - l i n k s . Precise c o r r e l a t i o n of properties with structure must, of course, go deeper than t h i s : molecular-weight d i s t r i b u t i o n must be considered; a l s o , i n a polymer such as p o l y ( o c t y l methacrylate), the a l k y l side group not only influences Tg but a l s o occupies space and reduces the number of c h a i n s per u n i t volume. F e r r y (17) has c o n s i d e r e d such matters i n d e t a i l . O v e r a l l , the regime of l i n e a r v i s c o e l a s t i c i t y i s characterized by r e a s o n a b l e s u c c e s s i n e s t a b l i s h i n g structure-property r e l a t i o n s h i p s . The p r o p e r t i e s themselves are unambiguously and simply s p e c i f i a b l e . The r e l e v a n t s t r u c t u r a l features are l a r g e l y r e c o g n i z a b l e a s p e c t s of m o l e c u l a r structure. Molecular theories e x i s t that provide a bridge between the molecular structure and the macroscopic v i s c o e l a s t i c properties. Large-Strain Rubber E l a s t i c i t y The e q u i l i b r i u m s m a l l - s t r a i n e l a s t i c behavior of an "incompressible" rubbery network polymer can be specified by a s i n g l e number—either the shear modulus G or the Young's modulus E (which f o r an incompressible elastomer i s equal to 3G). This modulus being known, the s t r e s s - s t r a i n b e h a v i o r i n u n i a x i a l t e n s i o n , b i a x i a l t e n s i o n , shear, or compression can be c a l c u l a t e d i n a s i m p l e manner. (If c o m p r e s s i b i l i t y i s taken i n t o account, two moduli are required: G and the b u l k modulus B.) The r e l a t i o n between e l a s t i c p r o p e r t i e s and m o l e c u l a r a r c h i t e c t u r e becomes a s i m p l e r e l a t i o n between two numbers: the shear modulus and the c r o s s - l i n k d e n s i t y (or the c r o s s - l i n k density corrected for the dangling t a i l s ) . There can be some ambiguity as to how c l o s e l y the "effective" c r o s s - l i n k density ( c a l c u l a t e d from the e l a s t i c modulus) approaches the " c h e m i c a l " c r o s s - l i n k density (estimated from some chemical measure of c r o s s l i n k i n g ) ; however, i n many e l a s t o m e r s the " c h e m i c a l " c r o s s - l i n k density i s not known with s u f f i c i e n t accuracy to make t h i s a major concern. When we proceed to l a r g e e l a s t i c s t r a i n s , the problem becomes more complex. The s t r e s s - s t r a i n r e l a t i o n i n u n i a x i a l t e n s i o n becomes nonlinear. I t could be l i n e a r i z e d by a proper choice of the American Chemical Society Library

16th St., Tess, N . WRoy . W., et al.; In Applied1155 Polymer Science; Washington, DM. 20038 Washington, DC, 1985. ACS Symposium Series; American Chemical Society:

248


measure of deformation and the measure of s t r e s s , but a s a t i s f a c t o r y treatment must a l s o be consistent with the raultiaxial l a r g e - s t r a i n e l a s t i c b e h a v i o r . One g e n e r a l approach to t h i s problem has been through the use of a s t r a i n - e n e r g y f u n c t i o n , W (18). T h i s i s a s c a l a r f u n c t i o n of the t h r e e e x t e n s i o n r a t i o s A ^ , A and A 3 . I f ^2» ^ 3 ) known, the d e v i a t o r i c stresses s\ s , and S 3 can be c a l c u l a t e d (as functions of X\ A , and A 3 ) . The problem then becomes that of finding the proper form of the s c a l a r function W(Aj, ^ 2 » ^ 3 ) * V a r i o u s c h o i c e s have been suggested and t e s t e d . One of the most popular i s the Mooney-Rivlin equation, which introduces a second e l a s t i c parameter. For u n i a x i a l tension, the Mooney-Rivlin equation can be w r i t t e n : 2

i

s

9

2


9

true stress

=

2 C ( A 2 - I)

- 2C (A -

X

2

2

JL)

corresponding to a strain-energy function of the form: W = CxIIx - 3 ]

+

C [I 2

2

- 3]

where I\ and I are i n v a r i a n t s of the s t r a i n tensor. A multiparameter property equation c a l l s for a multi-parameter structure s p e c i f i c a t i o n . As one attempts to go beyond the e f f e c t i v e c r o s s - l i n k d e n s i t y ( c o r r e c t e d f o r danglng t a i l s ) , i t becomes d i f f i c u l t to i d e n t i f y the precise structure features responsible for the observed e l a s t i c properties. Some of these s t r u c t u r a l features are p r o b a b l y r e l a t e d to network t o p o l o g y . When c r o s s - l i n k s are introduced i n t o a strained polymer, or i n a s o l v e n t - s w o l l e n state, the r e s u l t i n g network has d i f f e r e n t properties from a network formed i n an unstrained, unswollen condition—even i f the average molecular weight between c r o s s - l i n k s i s t h e same ( 1 9 ) . No s t r u c t u r e s p e c i f i c a t i o n couched o n l y i n terms of the c o n n e c t i n g c h a i n s as network elements i s l i k e l y to capture the s i g n i f i c a n t differences among such networks. An adequate structure s p e c i f i c a t i o n probably must i n v o l v e the closed loops of the network and t h e i r t o p o l o g i c a l patterns; such aspects of structure are very d i f f i c u l t to e s t a b l i s h 2

(20).

Non-Newtonian F l u i d s At s u f f i c i e n t l y high temperatures, a l i n e a r polymer behaves as an e l a s t i c f l u i d . At very low s t r e s s l e v e l s , the s t e a d y - s t a t e f l o w behavior i s Newtonian; shear rate i s d i r e c t l y proportional to shear stress. At h i g h e r s t r e s s l e v e l s , the e l a s t i c component of deformation contributes large e l a s t i c s t r a i n s . The chain molecules are appreciably oriented by the flow process. Not only i s there a t r a n s i e n t e l a s t i c e f f e c t d u r i n g the approach to steady f l o w and f o l l o w i n g c e s s a t i o n of f l o w but a l s o the m o l e c u l a r o r i e n t a t i o n strongly affects the steady-state r e l a t i o n between shear stress and shear rate. In the low-shear region, t h i s steady-state behavior can be expressed by a c o n s t a n t — t h e " z e r o - s h e a r " v i s c o s i t y (or i t s r e c i p r o c a l , the f l u i d i t y ) . In the h i g h - s h e a r r e g i o n , a n o n l i n e a r f u n c t i o n i s r e q u i r e d to s p e c i f y the r e l a t i o n between shear s t r e s s and shear rate. This can be formulated i n various ways:


11.


ALFREY

or

t

-

f(T)

l

-

F(£)

T =

n

a p p

249

(e)

with r) a nonlinear function of e or x. By symmetry, f(x) must be an odd Tunction—that i s , f(-x) = -f(+x). A power law e x p r e s s i o n p r o v i d e s a u s e f u l a p p r o x i m a t i o n to the f l o w c u r v e s f o r many molten polymers o v e r a f a i r l y wide range of shear r a t e (21).


a

or

I

= kx

n

x

=

ra

Ke

As w r i t t e n , the power law does not s a t i s f y the requirement of being an odd f u n c t i o n . I f n e g a t i v e v a l u e s of £ and x are to be accommodated, the expression should be written i n terms of absolute v a l u e s : |e[ = k | x | . The steady-state flow behavior i s not only nonlinear but a l s o c h a r a c t e r i z e d by the development of normal s t r e s s e s , which are completely absent i n a simple Newtonian f l u i d . Thus, a steady shear flow i n the x-y plane, £ , not only leads to a (nonlinear) shearing stress, x but a l s o l e a d s to normal s t r e s s e s , 0 , °yy» and 0" . Associated^with these normal stresses are many d i s t i n c t i v e phenomena exhibited by polymeric f l u i d s , such as the Weissenberg effect, where a polymer being s t i r r e d by a t u r n i n g shaft tends to c l i m b up the shaft instead of being thrown outward by c e n t r i f u g a l action (22). Thus, non-Newtonian p o l y m e r i c f l u i d s d i f f e r from s i m p l e Newtonian l i q u i d s i n s e v e r a l ways: they e x h i b i t transient effects i n approaching steady-state flow, the steady-state flow i s nonlinear, and i t i s accompanied by normal s t r e s s effects. Consequently, many parameters are needed to specify the f l u i d properties. The r e l a t i o n of these parameters w i t h m o l e c u l a r s t r u c t u r e i s o n l y p a r t i a l l y understood but the form of the molecular-weight d i s t r i b u t i o n and the degree of branching of the chains, as w e l l as the average molecular weight, must be considered. The structure-property r e l a t i o n s h i p s i n t h i s m e l t - f l o w regime are most important w i t h r e s p e c t to the e f f i c i e n t melt-processing of thermoplastic polymers. This supplies a strong i n c e n t i v e to the development of more complete understanding of melt properties, molecular structures, and t h e i r i n t e r r e l a t i o n ships. n

? x

zz

Behavior of Glassy Amorphous Polymers At very low s t r a i n l e v e l s , a glassy amorphous polymer behaves as a simple l i n e a r e l a s t i c s o l i d , with a high Young's modulus (e.g., 4 x 10 " dyn/cm ). When forced beyond t h i s l i n e a r regime, a v a r i e t y of n o n l i n e a r , i r r e v e r s i b l e responses can o c c u r : m a c r o s c o p i c a l l y b r i t t l e f r a c t u r e , shear y i e l d i n g ( e i t h e r uniform or l o c a l i z e d ) , c r a z i n g , or some combination of these (23). The s t r e s s l e v e l at which onset of any of these modes of response occurs depends upon many v a r i a b l e s : m o l e c u l a r s t r u c t u r e ( b o t h c o m p o s i t i o n and a r c h i t e c t u r e ) , temperature, g e o m e t r i c a l c h a r a c t e r of the s t r e s s , rate of loading, contact with deleterious environmental agents, etc. 1

2



250


Change i n these v a r i a b l e s can r e s u l t i n a switch from one mechanism of response to another. Consider f i r s t the g e o m e t r i c a l c h a r a c t e r of the s t r e s s . A m u l t i a x i a l s t r e s s can be c h a r a c t e r i z e d by the t h r e e p r i n c i p a l stresses S\, S 2 , and S 3 , l i s t e d i n descending value. Shear y i e l d i n g depends p r i m a r i l y upon the difference between S\ and S 3 . The Tresca y i e l d c o n d i t i o n , S^ - S 3 = Y ( a p p l i c a b l e to m e t a l s ) , has been modified for polymers (for which the shear y i e l d stress Y increases w i t h h y d r o s t a t i c pressure) (24). The onset of c r a z i n g f o l l o w s a c o m p l e t e l y d i f f e r e n t s t r e s s c r i t e r i o n , as r e p o r t e d by S t e r n s t e i n (25). C r a z i n g i s f a v o r e d by h i g h t e n s i l e s t r e s s and a p o s i t i v e mean normal s t r e s s . B r i t t l e f r a c t u r e f o l l o w s s t i l l another s t r e s s c r i t e r i o n . The mode of response to a p a r t i c u l a r type of stress depends upon which c r i t i c a l threshold i s f i r s t crossed. The shear y i e l d stress depends upon temperature and s t r a i n rate. The s t r e s s l e v e l s f o r c r a z i n g and f r a c t u r e a l s o depend upon temperature and s t r a i n r a t e , but to d i f f e r e n t degrees. Thus a change i n temperature or s t r a i n rate can s h i f t the mode of response (26). The molecular mechanisms of these various responses and t h e i r r e l a t i o n s h i p s w i t h s t r u c t u r e are o n l y p a r t i a l l y understood. One thing, however, i s certain—we must go beyond molecular structure and consider supramolecular structure as w e l l . THe c r i t i c a l s t r e s s l e v e l s f o r y i e l d i n g , c r a z i n g , and f r a c t u r e depend s t r o n g l y (and d i f f e r e n t l y ) upon the molecular o r i e n t a t i o n of a specimen. In the case of p o l y s t y r e n e at room temperature, u n i a x i a l o r i e n t a t i o n can provide a marked increased i n t e n s i l e strength, toughness, and craze r e s i s t a n c e i n the d i r e c t i o n of o r i e n t a t i o n and a marked l o s s i n these p r o p e r t i e s i n the t r a n s v e r s e d i r e c t i o n (27). Biaxial o r i e n t a t i o n can confer strength and toughness i n a l l d i r e c t i o n s i n the p l a n e (28). Behavior of C r y s t a l l i n e Polymers C r y s t a l l i n e polymers, when forced beyond a l i m i t e d l i n e a r regime, a l s o can e x h i b i t a variety of i r r e v e r s i b l e , nonlinear responses to s t r e s s . Again, the mechanical behavior depends upon molecular structure but a l s o upon s u p r a m o l e c u l a r structure—morphology and o r i e n t a t i o n . A g i v e n polymer can e x h i b i t many d i f f e r e n t k i n d s of morphology, depending upon the h i s t o r y of temperature and s t r e s s encountered i n processing. Among the recognized morphologies—each with i t s own d i s t i n c t i v e pattern of properties—are the f o l l o w i n g . 1. 2.

3. 4.

S p h e r u l i t i c morphology (commonly developed when a polymer c r y s t a l l i z e s from an unstressed melt) (8) Drawn f i b r i l l a r morphology (developed when a s p h e r u l i t i c p o l y m e r i s s t r e t c h e d b e l o w i t s m e l t i n g p o i n t and t h e o r i g i n a l l a m e l l a r c r y s t a l l i t e s a r e f r a g m e n t e d and r e arranged into an oriented fibrous structure) (29) " S h i s h - k e b a b " m o r p h o l o g y (a d i f f e r e n t h i g h l y o r i e n t e d morphology which develops when an oriented melt i s c r y s t a l l i z e d ) (30) E x t e n d e d c h a i n c r y s t a l s (ECC) (formed when p o l y m e r c r y s t a l l i z e s under h i g h h y d r o s t a t i c p r e s s u r e , or a c r y s t a l l i n e polymer i s annealed under pressure) (31) In Applied Polymer Science; Tess, Roy W., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1985.

11. alfrey 5. 6.


7.


251

Oriented, extended chain c r y s t a l s (formed by the combination of hydrostatic pressure and orientation) (32) "Accordion" morphology or "hard-elastic" fibers (formed by appropriate sequences of tensile stress and temperature (33, 34) Various intermediate morphologies (35)

For a given polymer, the mechanical properties—modulus, tensile strength, yield stress, etc.—can show order-of-magnitude differences in these various morphologies. Also, molecular structure influences properties, both directly and also indirectly, as i t influences the development of a particular morphology (36). In spite of the above diversity of oriented crystalline morphologies, Samuels has shown that the structural state can sometimes be adequately characterized by the crystalline and amorphous orientation factors (37). For polypropylene samples prepared with different draw ratios, draw temperatures, shrinkage temperatures, etc., simple property correlation with these two orientation factors was observed: ". . . these results suggest that different fabrication processes are simply different paths along which the sample is moved to equivalent structural states. Thus, general structure-property correlations are achieved by concentrating on the final structural state of the sample and not on the path by which the state was reached." Where applicable, this is a most useful approach; however, when radically different fabrication processes and radically different morphologies are compared, the definition of "structural state" must include more subtle features than the crystalline and amorphous orientation factors. Summary The structure-property relationships of polymers include the dependence of T^ and Tg on molecular structure, and the quantitative s tress-strain-temperature-time behaviors exhibited in the various regimes relative to T^ and Tg. These quantitative behaviors, and their dependence on structure, are most completely developed in the regime of linear viscoelasticity (including the special cases of small-strain rubber elasticity and low shear rate viscous flow). Large-strain elasticity and high shear rate flow are somewhat more complicated, but are s t i l l correlated with molecular structure. In glassy amorphous polymers and crystalline polymers, supramolecular structure (for example, orientation) as well as molecular structure must be considered in developing structure-property relationships. Because molecular structure is primarily established during polymerization, and supramolecular structure is established during subsequent fabrication operations, the mechanical performance of such polymers depends upon the conditions of fabrication as well as of polymerization. Literature Cited 1. 2.

Hill, R.; Walker, E. E. J . Polym. Sci. 1948, 3, 609. Boyer, R. F. Rubber Chem. Tech. 1963, 36, 1303. In Applied Polymer Science; Tess, Roy W., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1985.

252 3. 4. 5. 6.


7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.

APPLIED POLYMER SCIENCE Alfrey, T.; Gurnee, E. F. "Organic Polymers"; Prentice-Hall: New York, 1967. Billmeyer, F. W. "Textbook of Polymer Science"; Wiley: New York, 1966. Natta, G.; Danusso, F. J . Polym. Sci. 1959, 34, 3. Stavely, F. W.; Home, S. E. Ind. Eng. Chem. 1956, 48, 778, 784. Burke, J. J.; Weiss, V. "Block and Graft Copolymers"; Syracuse University Press: New York, 1973. Geil, P. H. "Polymer Single Crystals"; Interscience: New York, 1963. Holt, W. L.; McPherson, A. T. J. Res. Natl. Bur. Stand. 1936, 17, 657. Boundy, R. H.; Boyer, R. F. "Styrene"; Reinhold: New York, 1952. Gross, B. "Mathematical Structure of the Theories of Viscoelasticity"; Hermann et Cie; 1953. Eirich, F. R. "Rheology", Vol. 1; Academic: New York, 1956; Chapter 11. Williams, M. L.; Landel, R. F.; Ferry, J. D. J. Am. Chem. Soc. 1955, 77, 3701. Flory, P. J. "Principles of Polymer Chemistry"; Cornell University Press: New York, 1953. Fox, T. G.; Allen, V. R. J . Chem. Phys. 1964, 41, 344. Bueche, F. J . Chem. Phys. 1956, 25, 599. Ferry, J. D. "Viscoelastic Properties of Polymers"; Wiley: New York, 1961. Treloar, L. R. G. "Physics of Rubber Elasticity"; Oxford: England, 1958. Halpin, J. C.; Mandelkern, L. Polym. Lett. 1964, 2, 139. Alfrey, T.; Lloyd, W. G. J . Polym. Sci. 1962, 62, 159. DeWael, A. J . Oil Coll. Chem. Assn. 1924, 4, 33. Weissenberg, K. Nature 1947, 159, 310. Schultz, J. M. "Polymer Materials Science"; Prentice-Hall: New York, 1974. Whitney, W.; Andrews, R. D. J . Polym. Sci. 1967, C-16, 2891. Sternstein, S. S.; Ongchin, L. Polym. Pre. 1969, September. Sultan, J . N.; McGarry, F. J . Polym. Eng. Sci. 1974, 14, 282. Cleereman, K. J . ; Karara, H. H.; Williams, J. L. Mod. Plast. 1953, 30 (5), 119. Thomas, L. S.; Cleereman, K. J. Soc. Plast. Eng. J. 1972, 28 (4), 6. Peterlin, A. J . Macromol. Sci. 1973, B8, 83. Andrews, A. H. Angew. Chem. Intern. Ed. 1974, 13, 113. Geil, P. H.; Anderson, R. F.; Wunderlich, B.; Aarakawa, T. J. Polym. Sci. A2, 1964, 3707. Southern, J.H.; Porter, R. S. J. Appl. Polym. Sci. 1970, 14, 2305. Clark, E. S.; Garber, C. A. Int. J. Polym. Mater. 1974, 1, 31. Quynn, R. G.; Brody, H. J . Macromol. Sci.-Phys. B5, 1971, 721. Clark, E. S.; Scott, L. S. Polym. Eng. Sci. 1944, 14, 682. Capaccio, G.; Ward, I. M. Polymer 1974, 15, 233. Samuels, R. J . J . Macromol. Sci. B8, 1973, 41.


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