Molecular theory of rubber elasticity

Theories of rubber elasticity are reviewed and compared. Principal differences centre on the connection between the locations of network junctions and the macroscopic strain. Constraints on junctions imposed by surrounding chains have been postulated to be responsible for departures of experiments from the theory for idealized ‘phantom’ networks. Experimental evidence in support of theory developed using this hypothesis is discussed.     

The theory of rubber elasticity

It is argued that one can envisage two extreme regimes in rubber. The first is a dilute regime, where crosslinks are the only significant interaction between chains, and is the conventional model. The other extreme is one in which a very entangled situation exists where the chain statistics under deformation are remote from free Gaussian statistics. This latter case is studied and it is argued that the entropy per crosslink takes the form where α and β are independent of deformation but depend on temperature and density, and J² is the usual first invariant. The dilute regime is α = β = 0.     

A modification of rubber elasticity theory

Summary Based on a reconsideration of the packing conditions in amorphous polymers, a modification of the statistical theory of rubber elasticity is derived. The main assumption is, that the mean square end-to-end distance of a free chain in the bulk state depends on the dimensions of the surrounding chains. The theory describes quite well the deformation behaviour of real elastomeric networks.     

The essential role of network topology in rubber elasticity

This paper discusses various new aspects in the elasticity of rubbers and the statistics of elastomers. It is shown that the role of network topology is crucial in the statistics of rubbers. This is seen mostly on the influence of heterogeneities of crosslink density, which is also of practical interest. Strong heterogeneities lower the modulus, but do not change the stress-strain behavior. It is also shown that if in a given network the flexible chains are replaced by rigid rods which are flexibly hinged the entropy is still of order of that of the flexible network and that the modulus is of the same order as the Gaussian one. This holds for lower functionalities, i.e. less than six. The phase behavior of crosslinked blends and interpenetrating networks is shown to be similar of the microphase transition of block copolymers. Semi IPNs demix at a critical crosslink density of the crosslinked part, or alternatively at a critical length of the free chains, even though if the chains are of the same chemical nature as the crosslink component.     

The present status of the theory of rubber elasticity

In this review a modified equation for the free energy of a single chain is proposed, based on a recent discussion of the excluded volume problem. A simple model of a network element is used to study departures from affine behaviour and the form of the stress/strain curve. Comparison with the Mooney equation suggests that the C₂ term cannot be explained in terms of excluded volume, and it is suggested that the packing problem requires further study. Energy and volume changes on elongation are reviewed and the need for further work emphasized.     

Calorimetric study of rubber elasticity

The thermodynamics of simple elongation of a large number of rubberlike materials has been studied by deformation calorimetry with particular reference to the energy contributions to rubber elasticity. Expressions for mechanical work, heat and change of internal energy on deformation have been derived from the elastic equation of state. New equations for determination of the relative intramolecular energy contribution based on the analysis of elastic inversions of heat and internal energy are proposed. For the majority of polymers studied the free energy of deformation contains a significant intramolecular energy contribution. The temperature coefficients $\text{dln}\text{〈r}{}^2\text{〉}{}_{\mathrm{<mtext>o</mtext>}}\text{d}\text{T}$ of the unperturbed dimensions of chains obtained by calorimetry are compared with literature data. Calorimetric results for four networks in the region of 1.02 < λ < 2 have shown that the relative intramolecular energy contribution is independent of deformation and that strain-induced volume dilation and intermolecular energy and entropy changes resulting from this dilation follow the statistical theory only up to λ < 1.3 and at larger deformations the statistical theory underestimates the intermolecular changes. The data obtained for styrene-butadiene thermoplastic elastomers up to λ = 10 demonstrate that the intramolecular energy contribution is independent of deformation in the whole range of deformations. It is concluded that this is a consequence of the validity of the Gaussian statistics at large deformations.     

Contribution of entanglements to rubber elasticity

Uncrosslinked polymers show rubber-like properties above the glass transition temperature and at times short enough to prevent flow. This effect is thought to be due to a temporary entanglement network. Several studies indicate large entanglement contributions to the equilibrium elastic modulus of crosslinked networks. Three methods are discussed in greater detail: stoichiometric crosslinking with a suitable peroxide, the Langley method, and the two-network method. The two latter methods give equilibrium entanglement modulus contributions which are equal to the pseudo-equilibrium modulus of the uncrosslinked polymer. They also indicate that the entanglement contribution to the stress is responsible for departures from neo-Hookean behaviour observed in simpler extension.     

An expansion approach in rubber elasticity

We introduce an expansion approach to obtain the elastic free energy density in rubber elasticity. This new approach presents fast convergence for whole range of deformation. We apply this method to the freely jointed model and get an analytical form for the free elastic energy density in terms of the strain invariants. We use this expression to fit experimental data for PMDS network.     

Entanglement and excluded volume effects in rubber elasticity

The role of interchain excluded volume and entanglement in the elastic behavior of polymeric networks is theoretically examined. A three-chain network model is used, with each chain confined within an infinite, rectangular cylinder. The cylinders and the network crosslinks, are assumed to deform affinely. When the cross sections of the cylinders are small, the network elastic free energy equation has the form The λs represent the macroscopic deformation ratios. The constants A, B and C are functions of the number of each type of network chain (i.e., A contains the number of tie chains; C contains the number of loops and tie chains; B contains the number of dangling chain ends, unattached chains, loops and tie chains), their unperturbed dimensions and the sizes of the cylinders which confine them.     

A non-Gaussian network model for rubber elasticity

A simple constitutive model based on the non-Gaussian, Kuhn-Grün probability distribution function is derived. It is assumed that the actual macromolecular structure of a rubber-like material can be replaced by idealized equilateral tetrahedra cells that are not mutually exclusive so far as occupancy of the space is concerned. The three chains are assumed to meet at a junction point located at the centroid of the cell with their other ends being fixed at the vertices of the equilateral tetrahedron. The centroid junction point is free to fluctuate, subject to the constraint imposed by the equilibrium of chain forces. Stress-stretch constitutive equations are then derived to study homogeneous deformations of isotropic, incompressible hyperelastic rubber like materials. The accuracy of the derived constitutive equations is demonstrated by using uniaxial, equibiaxial, pure shear, and plane strain experimental data provided in the literature.     

The localisation model of rubber elasticity. II

Summary The effect of the spatial localization of a network chain by surrounding chains is incorporated into the chain probability distribution function and the network free energy is then calculated using the statistical mechanical formalism for constrained systems. In addition to a term having the classical Gaussian form, the resulting expression contains another term which depends on both the cross-link density of the network and the plateau modulus of the uncross-linked melt.     

Bound rubber theory and experiment

Adsorption of polymer on filler from bulk is known to result in a partial loss of polymer solubility (“bound rubber”). The existing theories of this phenomenon were critically examined, and the random adsorption model suggested by the author was found to provide the most complete explanation of available experimental data. Theory based on this model and containing one adjustable parameter (filler surface area per adsorption site) correctly predicts the change of molar mass distribution with adsorption on filler and satisfactorily describes the experimental dependence of fraction B of the filler-bound polymer on filler concentration and surface area and on M _w of the polymer, in both narrow and very broad molar mass distributions. No distinct effect on B of a moderate degree of branching of EPM, EPDM chains could be detected. It is concluded that the modification of the random adsorption model suggested by Shiga is not warranted, neither theoretically nor experimentally. © 1993 John Wiley & Sons, Inc.     

Molecular models of polymer networks and constitutive equations of rubber elasticity

Equilibrium elastic behaviour of crosslinked polymers (networks) is analysed both from the macroscopic (phenomenological) and microscopic (molecular) points of view. Molecular mechanisms possibly responsible for non-linear rubber elasticity are discussed. They include: conformational entropy of network chains at small and large extensions, hindered rotation of such chains and intra-chain point interactions, dipole-dipole interactions between asymmetric chain segments, and topological constraints leading to chain entanglement.     

Characterization of cross-linked polyampholytic gelatin hydrogels through the rubber elasticity and thermodynamic swelling theories

A new approach is proposed to characterize polyampholytic gelatin hydrogels, cross-linked covalently with glutaraldehyde. An experimental methodology involving simple mechanical extension and compression and equilibrium swelling tests is developed to estimate relevant microstructural parameters and electrokinetic properties of this type of hydrogel, for different cross-linker to gelatin mass ratios. The polyampholytic cross-linked matrices are studied here in the framework of the rubber elasticity and thermodynamic swelling theories. The main purposes of this work are the estimations of average mesh size and toughness of the swollen hydrogels, and the determination of the feasibility of polyion complexation between cross-linked gelatin chains and bioactive macromolecules to be delivered through hydrogel biodegradation.