Advances in Rubber Elasticity and Characterization of Elastomeric Networks Jean Pierre Oueslel and James E. Mark The University of Cincinnati, Cincinnati, OH 45221
The increasing importance of elastomeric materials during the past 20 years may be explained by the remarkable ability of synthetic chemists to design new polymers for specific applications and the considerable progress achieved in understanding structure-property relationships in the area of rubberlikeelasticity ( I ) . One of us has in fact presented in a previous issue of this Journal the fundamental concepts underlying comprehension of the physical properties of elastomers (2). Indeed, research in this area is still very active, and i t is new advances in molecular rubber elasticity theory and characterization of elastomeric networks that are the main topics of interest in the present review. Fundamental Postulates and Classical Models of Rubberllke Elastlclty Rubberlike elasticity, that is, high extensihility combined with a capacity for full recovery, i s a unique property of long, flexible chains, with weak interchain interactions, joined together by cross-links to form a three-dimensional network structure. Relationships between the structure of these networks and their mechanical properties have been theoretically investigated, from the very first, by statistical mechanics (3-5). These statistical mechanical theories of rubber elasticity are based on two fundamental postulates: (1)molecular chain configurations are random in undeformed amorphous polymers, and (2) the elastic response of the network originates within the chains and not to a significant extent from interactions between them. These primary premises of molecular theories have been supported by the widely investigated thermoelastic behavior of elastomeric networks (6-9). More recently, relevant small-angle neutron-scattering experiments (SANS) were performed on both amorphous bulk polymers and dilute solutions of polymers (10-13). They have confirmed that chains in bulk polymers occur in random configurations unperturhed by the neighboring chains with which they are densely packed.
Since intermolecular interactions do not affect the conformations, changes in conformations accompanying deformation do not cause changes in the intermolecular enerev. The high deformability of an elastomer and theuklastic force generated bv deformation stem from the essentiallv unlimited numbe; of configurations accessible to long milecular chains. This versatility has been characterized in numerous theoretical investigations on typical polymeric chains. They have shown that the Gaussian approximation for the end-to-end distance distribution function should be quite satisfactory for chains consisting of 100 or more bonds (3,14, 15). Non-Gaussian effects generally exist only in the very high, limited extensihility region of deformation. Stress-strain . vrooerties of elastomeric networks in the . Gaussian region can be explained only through an understanding of the relationshio between molecular chain deformarion : ~ n dmacroscopic strain. Twu l~alicmtrdds dG;~u.;sian nerworki have bcen ~ r o ~ o s lor e d this ourvoie (16.1 :I. 111 the affine network, diiplacements of junctfions and chain vectors are simple linear functions of the macroscovic strain (18). In the phantom network, on the other hand, chains are devoid of material properties (19). and macroscopic constraints operate only-onthe periphery of the network. Other junctions fluctuate without restraints from neighhoring chains. In both cases, the "reduced stress" measured in uniaxial extension is predicted to be independent of deformation. This measure of the modulus is conventionally defined as (20) ~
V*]
-
~
~~
~
~
~
~
~~~~~~~
-
~ U ~ ' ' ~ / A , ( O 08)
(1)
where a is the extension ratio defined relative to the undeformed swollen state, f the measured equilibrium force, uz the volume fraction of . polvmer in the swollen network. and . Ad the cross-sectional area of the isotropic unswollen sample. Departures from this behavior were reported very early in
Volume 64
Number 6 June 1967
491
the study of rubberlike elasticity (7,21). These observations have shown that the reduced stress decreases markedly with elongation and with dilation by swelling and have suggested that the limiting value of the reduced stress a t high elongation or high dilation is a fundamental characteristic of a given network. It became apparent later (22) that this quantity was the phantom network modulus Real networks have strong chain interpenetration, and intermolecular steric hindrances to chain motion commonly termed entanglements have been recognized as the origin of departures from both phantom and affine predictions. Different formalisms were proposed to include these intermolecular effects in the rubberlike elasticity analysis at thermodynamic equilibrium (16, 23). An attractive self-consistent model due to Flory and Erman (24) is presented below. In it, entanglements are embodied as domains of constraints acting as restrictions on junction fluctuations.
v],b
Molecular Elastlclty Theory of Real Networks Description of Flory and Erman Model In the idealized phantom model, junctions are presumed to fluctuate around their mean positions due to Brownian motion. The instantaneous distribution of chain vectors is not affine in the strain because i t is the convolution of the distribution of the mean vectors (which is affine) with the distribution of fluctuations (which are independent of the strain) (3,25). Gaussian statistics for undeformed phantom networks (denoted by the subscript 0) lead to the relationships
between the mean squared values (r2)o of the chain end-toend length, fluctuations ((ArI2)o in the chain dimensions, and fluctuations ( ( A R ) 2 )in ~ the positions of junctions from their mean locations. where brackets denote the ensemble average (31, and m is [he junction functionality. '1 h? rl3stic fret. cnrrzv
